[    {        "title": "1406.6964v2.Spinmotive_force_due_to_motion_of_magnetic_bubble_arrays_driven_by_magnetic_field_gradient.pdf",        "content": "arXiv:1406.6964v2  [cond-mat.mes-hall]  17 Oct 2014Spinmotive force due to motion of magnetic bubble arrays\ndriven by magnetic field gradient\nY. Yamane1, S. Hemmatiyan2, J. Ieda3, S. Maekawa3, and J. Sinova1,2,4\n1 Institut f¨ ur Physik, Johannes Gutenberg Universit¨ at Ma inz, 55128 Mainz, Germany\n2 Department of Physics, Texas A&M University, College Stat ion, Texas 77843-4242, USA\n3 Advanced Science Research Center, Japan Atomic Energy Age ncy, Tokai, Ibaraki 319-1195, Japan and\n4 Spin Phenomena Interdisciplinary Center (SPICE),\nJohannes Gutenberg Universit¨ at Mainz, 55128 Mainz, Germa ny\nInteraction between local magnetization and\nconduction electrons is responsible for a variety\nof phenomena in magnetic materials. It has been\nrecently shown that spin current and associated\nelectric voltage can be induced by magnetization\nthat depends on both time and space. This ef-\nfect, called spinmotive force, provides for a pow-\nerful tool for exploring the dynamics and the na-\nture of magnetic textures, as well as a new source\nfor electromotive force. Here we theoretically\ndemonstrate the generation of electric voltages\nin magnetic bubble array systems subjected to\na magnetic field gradient. It is shown by deriv-\ning expressions for the electric voltages that the\npresent system offers a direct measure of phe-\nnomenological parameter βthat describes non-\nadiabaticity in the current induced magnetization\ndynamics. This spinmotive force opens a door for\nnew types of spintronic devices that exploit the\nfield-gradient.\nSpinmotiveforce(SMF) referstothegenerationofspin\ncurrent, which is accompanied by an electric voltage, as\na result of dynamical magnetic textures in conducting\nferromagnets[1–3]. This is due to the exchange coupling\nbetween conduction electrons and the local magnetiza-\ntion. SMF reflects the temporal- and spatial-dependence\nof the local magnetization[4–9], and thus it offers a pow-\nerful method to probe and explore the dynamics and the\nnature of magnetic textures. In addition, SMF can be\na new source of electromotive force, directly converting\nthe magnetic energy into the electric energy of conduc-\ntion electrons. While the classical electromagnetism tells\nus that the conventionalinductive electromotive force re-\nquires a time-varying magnetic flux, it has been reported\nthat an electromotive force can be generated by a static\nand uniform magnetic field via the SMF mechanism[10–\n12].\nIn the last few years, more attention has been focusing\non topologically nontrivial magnetic structures such as\nmagnetic vortices in soft ferromagnetic nanodiscs[13, 14]\nand skyrmion lattices in chiral magnetic thin films[15,\n16]. The SMF offers some insights into and gains ben-\nefit from such magnetic systems; the polarity of a mov-\ning magnetic vortex core can be electrically detected[17,\n18], and arbitrarily-large ac SMF was predicted due toskyrmion lattice motion[19]. To the best of our knowl-\nedge, however, there has been no work on SMF in-\nduced in systems that contain magnetic bubble domains.\nMagnetic bubbles are observed in ferromagnetic films\nwith out-of-plane anisotropy as spot-like closed domains,\nwhere the magnetization is oriented to the opposite di-\nrection to the one outside the bubbles. The structures\nof magnetic bubbles are similar to those of skyrmions\nand vortices in the sense that bubbles carry a topological\nnumber called skyrmion number. Since their first obser-\nvation in the 1960s, magnetic bubbles have been showing\ndistinctive and interesting behaviors[20–23].\nIn this paper, SMF due to the motion of magnetic\nbubble arrays is theoretically investigated based on\nthe steady-motion model. As the bubble motion may\nbe induced by a spatially-varying magnetic field, our\nwork reveals that a magnetic field gradient can be\nexploited to generate a spin current and an associated\nFIG. 1: Schematic of magnetic bubble structures in a thin\nfilm. Red and Blue arrows in the upper figure and black ones\nin the bottom indicate the magnetization.2\nelectric voltage. By deriving expressions for the electric\nvoltages, we demonstrate their cumulative nature, i.e.,\nthey can be proportional to the number of involved\nbubbles. An important fact to be stressed is that the\npresent system can confirm the SMF originating from\nthe non-adiabatic dynamics of the electron spin, leading\nto a direct measurement of the controversial, so-called,\nβparameter.\nSteady-motion model for magnetic bubble. Let us\nbegin by reviewing the collective-coordinate model for\nmagnetic bubble dynamics. We consider a cylindrical\nbubble domain with radius Rin a thin film. This type\nof magnetic structure can be stabilized by applying bias\nmagnetic field in the out-of-plane with appropriate mag-\nnitude. The distribution of the magnetization direction\nm= (sinθcosψ,sinθsinψ,cosθ) is assumed to be two\ndimensional and described as[20] [see Fig. 1]\nθ(r,χ,z) = 2tan−1/bracketleftbigg\nexp/parenleftbiggQ(r−R)\n∆/parenrightbigg/bracketrightbigg\n(1)\nψ(r,χ,z) =Sχ+ψ0 (2)\nwhere (r,χ,z) is the cylindrical coordinate measured\nfrom the bubble center, ∆ is the wall width parameter,\nψ0is a constant, Qis the topological parameter defined\nas\nQ=1\nπ/integraldisplay∞\n0∂θ\n∂rdr=±1 (3)\nandSis the winding number:\nS=1\n2π/integraldisplay2π\nχ=0dψ=1\n2π/contintegraldisplaydψ\ndsds, (4)\nwhere/contintegraltext\ndsis the contour integral taken counterclock-\nwise around the circumference. The domain wall sepa-\nratingthe bubble and the outside, in general, can contain\nvertical Bloch lines, i.e., there are many possibilities in\nthe way of distributing the azimuthal angle ψalong the\nperimeter. In Eq. (2) we assumed the linear dependence\nofψonχ, as we focus on the following two cases. First,\nwhen the magnetization rotates one full turn around the\nwall of the bubble with no Bloch line, ψ=χ±π/2, that\nis,S= 1 andψ0=±π/2, with the sign + ( −) corre-\nsponding to left (right) handed chirality. Eq. (2) is also a\ngood approximation when the Bloch lines are packed so\ncloselythatthe distancebetweenthe adjacentBlochlines\nis comparable to the wall width, i.e., |S| ≃R/∆. If one\nconsiders a small number of Bloch lines, the distribution\nofψwould be no longer as simple as Eq. (2).\nWhen a magnetic field is applied in the zdirection\nwith its magnitude varying in the x-yplane, a bubble is\ndriven to move in the film seeking positions with lower\nZeeman energy. In the following the steady motion of\nthe bubble under the constant gradient ∇Hzis assumed,i.e., during its motion with constant velocity vthe bub-\nble stays rigidly cylindrical with constant radius Rand\ntheψ-distribution does not change with respect to the\ncoordinate frame moving with the bubble:\nθ(x,t) =θ(x−vt), ψ(x,t) =ψ(x−vt).(5)\nAssuming Eq. (5) and that the magnetization dynamics\nobeys Landau-Lifshitz-Gilbert equation, the equation of\nmotion for the bubble is given by[20] (the derivation is\nshown in Appendices)\n∇Hz=2S\nR2γ/bracketleftbiggQα\n2R\nS∆/parenleftbigg\n1+S2∆2\nR2/parenrightbigg\nv+ˆz×v/bracketrightbigg\n,(6)\nwhereγis the gyromagnetic ratio and αis the di-\nmensionless Gilbert damping parameter. ˆzrepresents\nthe unit vector along the z-direction. An assump-\ntion that was made when deriving Eq. (6) is that\nthe spin transfer with the conduction electrons are\nnegligible. Remarkably, the bubble is deflected away\nfrom the direction of the field gradient at an angle\nthat is determined by the material parameters (see\nAppendices for the detail). The extension of the above\ndiscussionto multiple-bubble problem is straightforward.\nSpinmotive force due to bubble motion. Let us ex-\namine the SMF induced by the steady motion of bubble\nindicated by Eq. (5). We assume that a conduction elec-\ntron in the ferromagnetic film is described by a one-body\nHamiltonian\nH=p2\n2me+Jexσ·m(r,t), (7)\nwheremeis the electron’s mass. The second term repre-\nsents the exchange interaction between the electron spin\nand the magnetization, with Jexbeing the exchange cou-\npling energy. According to theory of SMF[4–6], dynami-\ncal magnetization exerts an effective electric field ±Eon\nthe electrons via the exchange coupling, which is called\nspin electric field since its sign depends on the electron\nspin (see Appendices):\n±E=±/parenleftBig\nEA+ENA/parenrightBig\n, (8)\nwith\nEA=/planckover2pi1\n2esinθ/parenleftbigg∂θ\n∂t∇ψ−∂ψ\n∂t∇θ/parenrightbigg\n,(9)\nENA=β/planckover2pi1\n2e/parenleftbigg∂θ\n∂t∇θ+sin2θ∂ψ\n∂t∇ψ/parenrightbigg\n.(10)\nThe upper (lower)signs in Eq. (8) correspondto the elec-\ntron with majority (minority) spin. EAandENAare\nreferred to as adiabatic and non-adiabatic spin electric\nfields, respectively, as β=/planckover2pi1/2Jexτsfis the dimensionless\nparameterdescribing the non-adiabaticity in the electron3\nspin dynamics[5, 6, 8, 9], with τsfthe relaxation time for\nthe electron spin flip. The spin electric fields (9) and\n(10) require both time and spatial dependences of the\nmagnetization, and this condition is satisfied around the\nperimeter of moving bubbles.\nThe spin electric field (8) induces a spin current js=\n−(σ↑\nF+σ↓\nF)Eand a charge current jc= (σ↑\nF−σ↓\nF)E\nin the sample, with σ↑(↓)\nFthe electric conductivity for\nthe majority (minority) electrons. These currents gener-\nate by-products such as the charge redistribution, the\nspin accumulation, and the charge/spin diffusion cur-\nrent. In an open circuit system, the electric field Eind=\n−∇φ−∂A/∂tappears to keep the total charge cur-\nrent zero, i.e., jc+ (σ↑\nF+σ↓\nF)Eind= 0, where φand\nAare the electromagnetic scalar and vector potentials.\nAn electric voltage between two given points xaandxb,\nwhich is given by the difference in the electric potential\nφ(xb)−φ(xa), enables one to detect the spinmotive force\nelectrically. To determine the gaugepotentials one has to\nfix the gauge, and here let us adopt the Coulomb gauge,\n∇·A= 0. From the above equation for the open circuit\ncondition, one obtains the Poisson equation\n−∇2φ=∇·F. (11)\nHereF=−∇φ=−PEis the conservative electric\nfield induced by the electric potential distribution, where\nP= (σ↑\nF−σ↓\nF)/(σ↑\nF+σ↓\nF) is the spin polarization of the\nconductionelectrons. Intheaboveargument,wehavene-\nglected the contribution from the diffusive current to the\ntotal charge current as we have metals in our mind as the\nsamples; once the effective U(1) electric field Eis given,\nthe problem of computing the electric voltage induced\nby the electric field falls within the classical electromag-\nnetism and the established transport theory, and it is\nknown that in metals the induced diffusion potential is\nmostly negligible compared to the electric potential, un-\nlike in semiconductors. Technically, the diffusive current\ncan be takeninto account by replacingthe electricpoten-\ntial−eφby the electrochemical potential µ=−eφ+ǫF,\nwhereǫFis the Fermi energy. ( ǫFmay be dependent on\nthe space and the spin electric field in complex ways.)\nEq. (11) can be applied to systems with arbitrarysam-\nplegeometryandmagnetictexture. InFig.2, weshowan\nexample of electric potential distribution by numerically\nsolving the Poisson equation (11) with spin electric fileds\n(9) and (10), where the steady motion of eight identical\nbubbles in a square-shape thin film is assumed [see Ap-\npendices for the numerics]. Here the coordinate system\nis set that the bubble array flows along the xdirection.\nIt is seen that the potential drop occurs at the position\nof the bubbles, as is expected. Notice that the adia-\nbatic field gives rise to the net potential drops only in\ntheydirection (perpendicular to the bubble flow), while\nthe non-adiabatic one only to the xdirection (along the\nbubble flow), indicating that in this setup the two contri-\nFIG. 2: The distributions of electric potential φinduced in\na thin film due to (a) the adiabatic field (9) and (b) the non-\nadiabaticfield(10), calculatedbysolvingthePoisson equa tion\n(11) numerically, where the steady motion of eight identica l\nbubbles along the xdirection is assumed. The profile of each\nbubble is given by Eqs. (1) and (2), with R= 50 nm, ∆ = 2\nnm,Q= 1,S= 1 andψ0=π/2. The other parameters\nassumed here are γ= 1.76×1011T−1s−1,α= 0.02,β≃\n0.0033,P= 0.5, and the side lengths are 900 nm. The field\ngradient makes an angle 75 .9◦to thexaxis with its strength\n|R∇Hz|= 10 Oe.\nbutions can be separately identified by longitudinal and\nperpendicular voltage measurements. While S= 1 is as-\nsumed in Fig. 2, qualitatively much the same profiles are\nalso obtained in the case of |S|=R/∆, but with height\nofeachpotential dropbeing different (not shown). The S\ndependence of the electric voltage will be discussed later.\nNext, let us adopt a “quasi-one-dimensional” approx-\nimation for the electric voltage and derive its analytic\nexpression. We limit ourselves to a rectangular thin film\nwhere bubbles move along either of two sides of sample,\nas is the case in Fig. 2. Assuming a bubble moving in\nthe positive xdirection, we estimate the electric voltage4\nVxalong the motion of the bubble by\nVx≃1\nLy/integraldisplayLy\n0dy/integraldisplayLx\n0∂φ\n∂xdx=1\nLy/integraldisplay /integraldisplay\nrdrdχPEx\n≃ −R\nLyP/planckover2pi1\n2e1+(|S|∆/R)2\n|S|∆/R\n×\n1+α2/parenleftBigg\n1+(S∆/R)2\n2S∆/R/parenrightBigg2\n−1/2\nπγβ|R∇Hz|,\n(12)\nwhereLx(y)isthesidelengthofthesamplealongthe x(y)\ndirection. The y-integral operation and the appearance\nof the factor 1 /Lyare for spatial-averaging of the elec-\ntric potential along the y-axis. Eqs. (1), (2), (5), (6) and\n(8) have been used to carry out the integrals. Eq. (12)\ngives the exact solution for Vxin one-dimensional sys-\ntems:Vx=/integraltext\n(∂φ/∂x)dx. In two-dimensional systems\nas is the present case, Eq. (12) is still a good approxi-\nmation when effects on Vxfrom∂φ/∂yand the detailed\nprofile of∂φ/∂xcan be ignored. This is often the case\nif there are some appropriate symmetries in the system,\nand the divergence of the electric field, i.e., the moving\nbubble, keeps away from the electrodes. Similar concept\nfor the electric voltage Vymeasured in the perpendicular\ndirection to the motion of the bubble leads to\nVy≃1\nLx/integraldisplayLx\n0dx/integraldisplayLy\n0∂φ\n∂ydy\n≃ −R\nLxP/planckover2pi1\n2eS\n|S|\n×\n1+α2/parenleftBigg\n1+(S∆/R)2\n2S∆/R/parenrightBigg2\n−1/2\n2πγQ|R∇Hz|.\n(13)\nThe dc electric voltages appear both in the xandydi-\nrections, being proportional to the field gradient for both\nmagnetic configurations S= 1 and |S| ≃R/∆. It is seen\nfrom Eq. (12) that VxforS= 1 is larger than that for\n|S|=R/∆ under the same applied field, because R>∆.\nWith the parameters shown in Fig. 2, VxforS= 1 is\nabout an order of magnitude greater compared to that\nfor|S|=R/∆. On the other hand, Vyhas little depen-\ndence on |S|.\nWhen there are multiple bubbles, the net electric po-\ntential distribution is given by the superposition of all\nindividual electric potentials produced by each bubble.\nIfNidentical bubbles move in the xdirection, the sim-\nplest extensions, VN\nxandVN\ny, of the above expressions\nfor the electric voltages, VxandVy, respectively, may be\nVN\nx=NVx, VN\ny=NVy. (14)\nFigure 3 compares Eqs. (14) with the electric voltages\nobtained by numerically solving the Poisson equation(11), showing good agreement between them.\nDiscussion Eqs. (12), (13) and (14) indicate key fea-\nturesofthis SMF. VN\nxis proportionalto N/Ly, and thus,\nroughly speaking, depends on the “density” of bubbles\nalongtheyaxis and the “number” of bubbles alongthe x\naxis, which contribute to, respectively, the height of each\npotential drop and the number of occurrence of potential\ndrop along the xaxis. The dependence of VN\nxon the\nnumber of bubbles and the sample geometry is demon-\nstrated in Fig. 3 (a). Comparing the configurations i and\nii, which share the same sample shape, the slope of Vx\nis twice larger for ii because the configuration ii contains\ntwice as many bubbles, i.e., the sites where the potential\ndrop occurs. It is not necessarily that the same value of\nNleads to the same magnitude of electric voltage; the\nconfiguration iii provides twice larger VN\nxthan ii does\nunder the same applied field because of the difference in\nthe factor 1 /Ly. Similar discussion is applied for VN\ny, see\nFig. 3 (b). Eq. (14) indicates that one may control the dc\nelectric voltages by adjusting the sample geometry and\nthe number of bubbles.\nRemarkably, only the non-adiabatic field ENAcon-\ntributes to VN\nx, while only the adiabatic one EAtoVN\ny.\nSince in most magnetic materials βis believed to be\nsmaller than unity and thus |ENA| ≪ |EA|, it is dif-\nficult to identify the contribution of the non-adiabatic\nfield in the conventional systems. In fact, there has been\nno experimental confirmation of its effects. Now, by the\nmeasurement of VN\nxone can detect ENAfree from the\nlargeradiabatic contribution. This can lead to direct and\nunambiguous measurement of the phenomenological pa-\nrameterβ; as bothVxandVycontain in their expressions\nP, which may be another uncertain material parameter,\none can be free from Pby measuring the ratio of Vxand\nVy.\nIn the present study, we assumed the specific and sim-\nple profile of magnetization, Eqs. (1), (2) and (5), to de-\nrive the analytic expressions for the electric voltages. Al-\nthoughtheinvestigationoftheeffectsofthebubbleshape\ndistortion, local disorder, and etc. must be interesting,\nit will require systematic study based on numerical ap-\nproaches to the magnetization dynamics, which is out of\nreach of this paper. The assumption made in the present\nstudy is reasonable when the gradient of applied field is\nsufficiently moderate,[22] and our analytical model will\nwork well in this field range.\nIn conclusion, We have shown for the first time\nthat a magnetic filed gradient can generate a SMF,\ni.e., spin currents and associated electric voltages, by\ndriving the motion of magnetic bubbles. Based on the\nsteady-motion model, expressions for the dc electric\nvoltages in longitudinal and perpendicular to the bubble\nmotion are derived, which turned out to be controllable\nby tuning the sample geometry and the number of\ninvolved bubbles. An important implication of our5\nresult is that the present system offers an experimental\ndetermination of the phenomenological parameter βthat\ndescribes non-adiabaticity in the electron spin dynamics.\nThis SMF can lead to a new route for basic study of\nthe electron-magnetization interaction as well as a new\nconcept in spintronic devices, exploiting the gradient of\nmagnetic fields.\nAcknowledgements The authors would like to thank\nCristian Cernov for valuable discussion and making the\nimage for Fig. 1. This research was supported by Re-\nFIG. 3: The field dependence of dc electric voltages induced\nby the steady motion of arrays of identical bubbles. Three\ndifferent configurations are examined, which are indicated i n\nthe top panel. The dynamics of each bubble is described by\nEqs. (1), (2) and (5) with S= 1 [see the caption of Fig. 2\nfor the other parameters assumed here]. (a) VN\nxobtained by\nEq. (14) and by numerically solving the Poisson equation (11 )\ncorrespond to solid lines and open symbols, respectively, a nd\n(b) similarly for VN\ny. The electric voltages are proportional\nto the field gradient, and depends on the number of bubbles\nand the sample geometry. See also Discussion in the main\ntext.search Fellowship for Young Scientists from Japan Soci-\nety for the Promotion of Science, Grant-in-Aid for Sci-\nentific Research (No. 24740247 and No. 26247063) from\nMEXT, Japan, and Alexander von Humboldt Founda-\ntion.\n[1] A. Stern, Phys. Rev. Lett. 68, 1022 (1992).\n[2] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98,\n246601 (2007).\n[3] J. Ieda, Y. Yamane, and S. Maekawa, SPIN 3, 133004\n(2013).\n[4] G. E. Volovik, J. Phys. C 20, L83 (1987).\n[5] R. A. Duine, Phys. Rev. B 77, 014409 (2008).\n[6] Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77,\n134407 (2008).\n[7] K. W. Kim, J. H. Moon, K. J. Lee, and H. W. Lee, Phys.\nRev. Lett. 108, 217202 (2012).\n[8] G. Tatara, N. Nakabayashi, and K. J. Lee, Phys. Rev. B\n87, 054403 (2013).\n[9] Y. Yamane, J. Ieda, and S. Maekawa, Phys. Rev. B 88,\n14430 (2013).\n[10] S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao, Q.\nNiu, M. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 102,\n067201 (2009).\n[11] P. N. Hai, S. Ohya, M. Tanaka, S. E. Barnes, and S.\nMaekawa, Science 458, 489 (2009).\n[12] M. Hayashi, J.Ieda, Y.Yamane, J.Ohe, Y.K.Takahashi,\nS. Mitani, and S. Maekawa, Phys. Rev. Lett. 108, 147202\n(2012).\n[13] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T.\nOno, Science 289, 930 (2000).\n[14] A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Mor-\ngenstern, and R. Wiesendanger, Science 298, 577 (2002).\n[15] S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A.\nRosch, A. Neubauer, R. Georgii, and P. B¨ oni, Science\n323, 915 (2009).\n[16] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,\nY.Matsui, N.Nagaosa, andY.Tokura, Nature(London),\n465, 901 (2010).\n[17] J. Ohe, S. E. Barnes, H. W. Lee, and S. Maekawa, Appl.\nPhys. Lett. 95, 123110 (2009).\n[18] K. Tanabe, D. Chiba, J. Ohe, S. Kasai, H. Kohno, S. E.\nBarnes, S. Maekawa, K. Kobayashi, and T. Ono, Nat.\nCommun. 3, 845 (2012).\n[19] J. Ohe and Y. Shimada, Appl. Phys. Lett. 103, 242403\n(2013).\n[20] A. P. Malozemoff and J. C. Slonczewski, Magnetic Do-\nmain Walls in Bubble Materials , Academic Press, New\nYork, 1979.\n[21] Pietro Tierno, Tom H. Johansen, and Thomas M. Fis-\ncher, Physical Review Letters 99, 038303 (2007).\n[22] C. Moutafis, S. Komineas, and J. A. C. Bland, Physical\nReview B 79, 224429 (2009).\n[23] S.R.Bakaul, W.Lin, andT.Wu, AppliedPhysicsLetters\n99, 042503 (2011).6\nAppendices\nSteady-motion of bubble. In orderto get to the equa-\ntion of motion for the bubble, consider the increment of\nthe stored magnetic energy U=/integraltext\nw dV, with the mag-\nnetic energy density w, due to the variations δθandδψ:\nδU=/integraldisplay/integraldisplay/integraldisplay/parenleftbiggδw\nδθδθ+δw\nδψδψ/parenrightbigg\ndxdydz\n=−2µ0MS|v|πRh\nγ/bracketleftbiggα\n∆/parenleftbigg\n1+S2∆2\nR2/parenrightbigg\ndX+2QS\nRdY/bracketrightbigg\n,\n(15)\nwherehis the film thickness and MSis the saturation\nmagnetization. δw/δθandδw/δψhave been expressed\nin terms of ∂θ/∂tand∂ψ/∂tby using Landau-Lifshitz-\nGilbert equation of motion without spin-transfer-torque\neffect, and the time derivatives have been further ex-\npressed in terms of randχassuming Eqs. (1), (2) and\n(5). The change δUin the internal energy is supposed to\nbe balanced by the external pressure on the bubble due\nto∇Hz:\nQ(2µ0MS∇Hz)πR2h=−dU\ndX, (16)\nwhereXdenotes the position ofthe center of the bubble.\nRearrangingthe aboveequation, Eq.(6) isobtained. The\nnet mobility of the bubble is found by solving Eq. (6) for\n|v|in terms of |R∇Hz|as\n|v|=Rγ\n2|S|\n1+α2/parenleftBigg\n1+(S∆/R)2\n2S∆/R/parenrightBigg2\n−1/2\n|R∇Hz|.\n(17)\nThe angleρof deflection of the bubble away from the\nfield gradient may be defined as\nρ= tan−1∂Hz/∂y\n∂Hz/∂x= cot−1Qα/braceleftbig\n1+(S∆/R)2/bracerightbig\n2S∆/R.(18)\nWithS= 1,Q= 1,α= 0.02, ∆ = 2 nm and R= 50\nnm, corresponding to the calculation in Fig. 2, one\nobtainsρ≃75.9◦.\nDerivation of spin electric fields. Under the Hamil-\ntonian (7), the Heisenberg equation of motion for the\nelectron is given by\nF= [[r,H],H]/(i/planckover2pi1)2=−Jexσ·∇m,(19)\nwithrandFdenoting the operators for the electron’s\npositionandtheforceactingontheelectron,respectively.\nThe actual motion of the electron is obtained by deter-\nmining the expectation value /an}b∇acketle{tσ/an}b∇acket∇i}ht↑↓of the electron spin\nwith majority ( ↑) and minority ( ↓) states. Notice that\nm· ∇m= 0 and thus the component of /an}b∇acketle{tσ/an}b∇acket∇i}ht↑↓that is(anti-)parallel to mdoes not contribute to the force. Let\nus decompose the electron spin as /an}b∇acketle{tσ/an}b∇acket∇i}ht↑↓=∓m+δm↑↓,\nwhere the upper (lower) sign corresponds to the major-\nity(minority)spinand δm↑↓representsaslightdeviation\nfrom∓m. The expectation value of the force is written\nas/an}b∇acketle{tF/an}b∇acket∇i}ht↑↓=−Jexδm↑↓·∇m; what causes non-zero force\ndue to the exchange coupling is a misalignment between\nthe electron spin and the magnetization. Assume that\nthe electron spin dynamics is described by\n∂/an}b∇acketle{tσ/an}b∇acket∇i}ht↑↓\n∂t=−2Jex\n/planckover2pi1/an}b∇acketle{tσ/an}b∇acket∇i}ht↑↓×m−δm↑↓\nτsf.(20)\nThe first term on the right-hand side represents the Lar-\nmor precession about the magnetization, and the damp-\ning motion toward the magnetization is phenomenologi-\ncally introduced by the second term, which describes the\nnon-adiabaticity in the electron spin dynamics, with τsf\nthe relaxation time for the electron spin flip. By sub-\nstituting the above expression for /an}b∇acketle{tσ/an}b∇acket∇i}ht↑↓into Eq. (20),\nδm↑↓is expressed in terms of m, by which one obtains\n/an}b∇acketle{tF/an}b∇acket∇i}ht↑↓=±(−eE) withEgiven by Eq. (8).\nWe have considered an open circuit condition. More\ngenerally Eq. (20) should include the divergence of the\nspin current carried by conduction electrons, leading\nto appearance of the spin magnetic field. While we\nhave followed Ref. [[9]] here, essentially the same result\nwas obtained by a different approach where Onsager’s\nreciprocal relation is taken into account.[5, 6]\nNumerical approach to the Poisson equation. In\nthe numerical calculations, the sample is divided into\nnumber of meshes, and a single magnetization vector mi\nis assigned to each mesh, where iis the index of the\nmeshes. The time evolution of {mi(t)}(i= 1,2,...,Nm),\nwhereNmis the number of the meshes, is given based on\nEqs. (1), (2), and (5). As the spin electric field {Ei}is\nupdated at eachmesh by Eq. (8), the induced electric po-\ntential distribution {φi(t)}satisfies the Poisson equation\n[see Eq. (11)]\n∇2φi(t) =P∇·Ei(t). (21)\nIn two-dimensional discrete systems, the above equation\nis equivalent to\nφi,j=1\n4(∇·Ei,j∆x∆y+φi+1,j+φi−1,j+φi,j+1+φi,j−1),\n(22)\nwhere (i,j) are the indices for the meshes in two dimen-\nsion, and ∆ x∆yis the area of the mesh. The poten-\ntial distribution is obtained by solving the above equa-\ntion self-consistently with Neumann boundary condition,\nwhere the spatial derivative of the electric potential is\nzero at the sample edge."    },    {        "title": "1805.03823v1.Spatial_dynamics_analysis_of_polarized_atom_vapor.pdf",        "content": "arXiv:1805.03823v1  [physics.atom-ph]  10 May 2018Spatial dynamics analysis of polarized atom vapor\nX.Y.Hu,1H.F.Dong,1H.C.Huang,1L.Chen,1and Y.Gao1\nSchool of Instrumentation Science and Opto-Electronics En gineering, Beihang University, Beijing 100191,\nChina.\n(Dated: 5 September 2018)\nWe analyze the spatial dynamics of polarized atom vapor and presen t a mathematical method to eliminate\nthe diffusion effect partially. It is found that the diffusion effect of po larized atoms can be regarded as a\nlow pass filter in spatial frequency domain and fits well with a Butterw orth filter. The fitted spatial filter\ncan be used to restore the original magnetic image before being blur red by the diffusion, thus improving the\nmagnetic spatial resolution. The results of spatial dynamics simulat ion and magnetic image restoration show\nthe potential usage of this method in magnetic gradiometer and ato mic magnetic microscopy.\nPACS numbers: 78.20.Ls, 87.57.nf\nKeywords: Bloch equation; spatial dynamics; magnetometer; diffu sion; polarized atom\nI. INTRODUCTION\nThe temporal and spatial dynamics of polarized\natom vapor is described by Bloch equation1, which\nis the basic model in fields such as nuclear mag-\nnetic resonance(NMR)2,3, atomic magnetometer4–6and\natomic gyroscopes7,8. In these applications the temporal\ndynamics of polarized atom vapor is analyzed theoret-\nically and verified experimentally. As most of the ap-\nplications measure the average polarization of atoms in\nthe vapor cell, the spatial dynamics is usually ignored in\nthe modeling and analysis. In the case of gradiometer\nor array measurements, the diffusion is thought as the\nessential limit for the spatial resolution9–11.\nInspired by the work of D. Giel et al, who pointed out\nthat the space-time evolution of polarization can be ex-\npanded in terms of spatial periodic functions10, we intro-\nducethe spatialfrequencyresponseoftheinput magnetic\nfield, which can describe the spatial dynamics of polar-\nized atomvaporin nonuniform magneticfield. MATLAB\nSimulink is used to simulate the polarized atom vapor\nsystem. By setting the magnetic field distribution as one\ndimensional (1D) sinusoidal waves with different spatial\nfrequency, we get the evolution of the atom polarization\nin both time and space domain. The result shows that\nthe response decreases when spatial frequency of mag-\nnetic field increases, just like a low pass spatial filter,\nwhich can be fitted well to a Butterworth filter. Assum-\ning the diffusion effect is isotropy, it can be expanded\ndirectly to a two dimensional (2D) spatial filter. By re-\nversing the 2D filter, we obtain a 2D high pass filter that\ncan be used to eliminate diffusion effect and restore the\noriginal image partially.\nThe paper is organized as follows: After a brief in-\ntroduction of background and motivation in section I,\nSection II describes the model and parameters calcula-\ntion used in the simulation. Section III illustrates and\ndiscusses the results of spatial dynamics simulation. Fi-\nnally, conclusions are summarized in section IV.II. MODELING AND PARAMETERS\nThe Bloch equation can be written with diffusion term\nas following12,13:\n∂\n∂t− →P=D∇2− →P+γ− →B×− →P+Rp(s−− →P)−− →P\nT1,T2(1)\nwhere− →Pis the polarization of alkali atoms, D is the\ndiffusion coefficient, γis the gyromagneticratio,− →Bis the\nmagnetic field, Rpis the pumping rate and T1andT2are\nthe relaxation times for polarization components parallel\nand transverse to− →B, respectively. The four terms on the\nright-hand side describe diffusion, precession, pumping\nand relaxation, respectively.\nTosimplify Eq1and obtainthe numericalresult ofspa-\ntial dynamics, we assume the pumping beam and the\nprobing beam are along z axis and x axis, respectively,\nand the direction of the magnetic field is along y axis.\nBesides, we also suppose that atoms are fully polarized\nusing a high power short pulse beam. The polarization\nvector precesses freely in x −z plane after the pumping\npulse. In this condition, Py,T1andRpcan be ignored\nand Eq.(1) can be simplified as below:\n∂\n∂t/bracketleftbigg\nPx\nPz/bracketrightbigg\n=D∇2/bracketleftbigg\nPx\nPz/bracketrightbigg\n+γBy/bracketleftbigg\n−Pz\nPx/bracketrightbigg\n−Rr/bracketleftbigg\nPx\nPz/bracketrightbigg\n(2)\nwhere relaxation rate Rr= 1/T2and diffusion coefficient\nD can be calculated according to the experimental setup.\nConsidering that the atom vapor cell can be antirelax-\nation coated, or buffered with high pressure gas and can\nworkunderspin-exchangerelaxationfree(SERF)regime,\nwallcollisionrelaxationandspin-exchangerelaxationcan\nbe neglected. Moreover, as we measure the local field\ninstead of the average field in the vapor cell, gradient\nbroadening can also be ignored. So the relaxation rate\nRris mainly decided by the spin destruction14,\nRr≈Rsd= ¯υασsd\nαnα+ ¯υqσsd\nqnq+ ¯υbσsd\nbnb(3)\nwhere the first term denotes the collision between alkali\natoms themselves, the second term denotes the quench2\ncollision and the third term denotes the collision between\nalkali atom and buffer gas atom. ¯ υ,σsdand n are the rel-\native velocity, collision cross-section of collision pair and\ndensity of atoms, respectively. The subscripts α,qand b\nare for alkali atoms, quenching gas atoms and buffer gas\natoms, respectively. The diffusion coefficient depends on\nthe temperature and the pressure of the gas10,\nD=D0/parenleftbiggP0\nP/parenrightbigg/parenleftbiggT\nT0/parenrightbigg3/2\n(4)\nwhereT0= 273.15Kis the standard temperature,\nP0= 760Torris the standard pressure and D0is the\nstandard diffusion coefficient at T0andP0. Standard\ndiffusion coefficients of Cesium atom in He and N2are\n0.39cm2/s15and 0.087(15)cm2/s16, respectively, which\nare used in our choice of D in section III.\nIII. NUMERICAL SIMULATION AND RESULTS\nANALYSIS\nA. Spatial frequency response simulation\nInthesimulation, Rr= 300s−1andD=0 ∼1cm2/sare\nchoosedaccordingtosectionIIwithtypicalparametersin\nthe atom vapor polarization experiments. And to obtain\nthe spatial frequency response we set Byas a magnetic\nfield with spatial sinusoidal distribution on z axis and\nBx=Bz= 0. Thus the input magnetic field can be\nwritten as\nBy(z) =B0sin(ωz) (5)\nwhereB0is the magnetic amplitude, ωis the spatial\nangular frequency and z is the spatial position along z\naxis.\nBy simulating the model in Eq.(2) with diffusion term,\nwecangetthetemporalandspatialpolarization ˆPx(z,td)\n. Then the output magnetic field can be calculated by,\nˆBy(z) =arcsin(−ˆPx(z,td)eRrtd)\n2πγtd(6)\npulse pumping magnetometer for different D and ω.\nTime delay td= 10µsand relaxation rate Rr= 300s−1.\nAccording to our simulation, ˆBy(z,td) also follows the\nsinusoidal distribution with the same spatial frequency,\ni.e.,ˆBy(z)≈ˆB0sin(ωz).We get the corresponding mag-\nnetic amplitude ˆB0of different ω,tdand D. With a cer-\ntain time delay tdthe amplitude magnification ˆB0/B0\ndecreases when spatial angular frequency ωincreases, as\nshownin Fig.1. The figure alsoshowsthat the cutoffspa-\ntial frequency of the system decreases with the increase\nof diffusion coefficient.10 0\n ω /m -1 10 2\n10 4 1D /cm 2s-1 0.5 \n-60 -40 -20 0\n-80 \n0Gain /dB \n-60 -50 -40 -30 -20 -10 0\nFIG. 1. (Color Online) Amplitude response of the short pulse\npumping magnetometer for different D and ω. Time delay\ntd=10s and relaxation rate Rr= 300s−1.\nSpatial frequency D(u,v) /m -1 10 010 110 210 310 4Gain H(u,v) \n00.2 0.4 0.6 0.8 1\nSimulation \nFit \n(a)\n (b)\nFIG. 2. (Color Online) (a) The spatial frequency response\nwhentd=10s and D = 0 .6cm2/s. (b) The plot of the 2D filter\ngenerated by the fit curve in (a).\nB. Reverse approximation of diffusion process\nFig.2(a) is the spatial response when td= 10µs and\nD= 0.6cm2/s, which can be fitted well with a Butter-\nworth filter 1 //radicalbig\n1+(ω/ω0)2n. The phase response of\nButterworth filter is ignored due to the isotropy of dif-\nfusion. The corresponding 2D Butterworth filter can be\nexpressed as\nH(u,v) =1\n1+(D(u,v)/D0)2n(7)\nwhereD(u,v) is the distance between a point ( u,v)\nin the frequency domain and the center of the frequency\nrectangle, and D0is the cutoff frequency17. Fig. 2(b)\nillustrates the corresponding 2D spatial filter, which is\nexpanded from the fitted curve of Fig. 2(a).\nAs the 2D filter in Fig. 2(b) is generated by the spa-\ntial frequency response of Bloch equation simulation, it\napproximately represents the diffusion effect. Further-\nmore, we can restore the original magnetic image with a\nreversed 2D filter 1 −H(u,v) in Fig. 2(b).\nTo provide a better view of the restoration effect, we\nsimulate the atom vapor system in magnetic field of two\nclose magnetic dipoles, and assume the dipoles is small3\nenough to ignore the magnetic field inside the dipoles.\nThe magnetic field Byvaires inversely with the third\npower of distance in y−zplane, as show in Fig.3(a)\n. The measured magnetic image simulated using Bloch\n(a)\n (b)\n(c)\n (d)\nFIG. 3. (Color Online) (a) The original magnetic field of two\ndipoles in y−zplane. The distance between the two dipoles is\n5mm. (b)The measured magnetic field, where the dipoles can\nhardly be distinguished due to diffusion. This image is calcu -\nlated using the simulation result of Pxand Eq.(6). (c) The\nrestored magnetic field using reversed 2D filter 1 −H(u,v).\n(d) Filtered image from (a), with filter H(u,v) applied. To\nprovide a better view, all of these images are normalized.\nmodel, i.e. Eq.(2), and 2D filter, i.e. Eq.(7), are dis-\nplayed in Fig. 3(b) and Fig. 3(d), respectively, where the\ntwo dipoles can hardly be distinguished. With a reversed\n2Dfilter1 −H(u,v),wegettherestoredimagefromFig.4\nand the result is shown in Fig. 3(c). In the restored im-\nage, the two dipoles can be distinguished again and thus\nthe spatial resolution is improved. The slightly diffrence\nbetween Fig. 3(a) and Fig. 3(c) may be due to the fit\nerror in the high frequency part.\nIV. CONCLUSION\nIn summary, the spatial dynamics of polarized atom\nvapor is analyzed based on the Bloch equation and short\npulse pumping and probe scheme. The simulated spa-\ntial frequency response fits well with a low pass Butter-\nworth filter. By passing the magnetic image through the\nreversed spatial fitter, we eliminate partially the diffu-\nsion effect and increase the spatial resolution of the im-\nage. This analysis and restoration method can be used\nin spatial magnetometry, such as magnetic gradiometerand atomic magnetic microscopy.\nACKNOWLEDGEMENT\nThe authors thank the support by National Natural\nScience Foundation of China under Grant No. 61074171\nand 61273067 and National Program on Key Basic Re-\nsearch Project of China (2012CB934104). The authors\nwould like to thank Dr. Iannis K. Kominis for the ben-\neficial discussion on spatial resolution, which is another\ngreat inspiration to the idea of this paper.\nREFERENCES\n1F. Bloch, W. W. Hansen, and M. Packard, “Nuclear induction,”\nAPS Journals ,69(1946).\n2W. S. Hinshaw and A. H.Lent, “An introduction to nmrimaging:\nFrom the bloch equation to the imaging equation,” Proceedin gs\nof the IEEE 71, 338–350 (1983).\n3R. H. Lehmberg, “Modification of bloch’s equations at optica l\nfrequencies,” Optics Communications 5, 152–156 (1972).\n4I. Savukov, “Gradient-echo 3d imaging of rb polarization in fiber-\ncoupled atomic magnetometer,” Journal of Magnetic Resonan ce\n256, 9–13 (2015).\n5J. C. Allred, R. N. Lyman, T. W. Kornack, and M. V. Roma-\nlis, “High-sensitivity atomic magnetometer unaffected by s pin-\nexchange relaxation.” PhysicalReview Letters 89,130801 (2002).\n6J. Fang, T. Wang, H. Zhang, Y. Li, and S. Zou, “Optimizations\nof spin-exchange relaxation-free magnetometer based on po tas-\nsium and rubidium hybrid optical pumping,” Review of Scient ific\nInstruments 85, 123104 (2014).\n7H. Dong, J. Fang, J. Qin, and Y. Chen, “Analysis of the\nelectrons-nuclei coupled atomic gyroscope,” Optics Commu nica-\ntions284, 2886–2889 (2011).\n8T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear\nspin gyroscope based on an atomic comagnetometer,” Physica l\nReview Letters 95, 230801 (2005).\n9I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis,\n“A subfemtotesla multichannel atomic magnetometer,” Natu re\n422, 596–599 (2003).\n10D. Giel, G. Hinz, D. Nettels, and A. Weis, “Diffusion of cs\natoms in ne buffer gas measured by optical magnetic resonance\ntomography,” Optics Express 6, 251–6 (2000).\n11K. Kim, S. Begus, H. Xia, S. K. Lee, V. Jazbinsek, Z. Trontelj,\nand M. V. Romalis, “Multi-channel atomic magnetometer for\nmagnetoencephalography: A configuration study,” Neuroima ge\n89, 143–151 (2014).\n12M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker,\nand M. V. Romalis, “Spin-exchange-relaxation-free magnet om-\netry with cs vapor,” Phys. Rev. A 77, 033408 (2008).\n13H. C. Torrey, “Bloch equations with diffusion terms,”\nPhys. Rev. 104, 563–565 (1956).\n14T. W. Kornack, “A test of cpt and lorentz symmetry using a\npotassium-helium-3 co-magnetometer,” (2006).\n15K. Ishikawa, Y. Anraku, Y. Takahashi, and T. Yabuzaki, “Op-\ntical magnetic-resonance imaging of laser-polarized cs at oms,” J.\nOpt. Soc. Am. B 16, 31–37 (1999).\n16M. S. Manalis, “Nonthermal saha equation and the physics of a\ncool dense helium plasma,” Phys. Rev. A 5, 993–994 (1972).\n17R. C. Gonzalez and R. E. Woods, Digital Image Processing\n(Pearson Education Asia ltd, 2009)."    },    {        "title": "1706.03039v1.Ultra_low_energy_Electric_field_induced_Magnetization_Switching_in_Multiferroic_Heterostructures.pdf",        "content": "arXiv:1706.03039v1  [cond-mat.mes-hall]  6 Mar 2017SPIN\nVol. xx, No. xx (2016) 1–35\nc/circlecopyrtWorld Scientific Publishing Company\nUltra-low-energy Electric field-induced Magnetization Sw itching in\nMultiferroic Heterostructures\nKuntal Roy\nSchool of Electrical and Computer Engineering\nPurdue University, West Lafayette, Indiana 47907, USA\nroyk@purdue.edu\nReceived 05 Jun 2016\nAccepted Day Month Year\nElectricfield-induced magnetizationswitching in multiferroicsisintrigu ingforboth fundamental\nstudiesandpotentialtechnologicalapplications.Here,wereviewt herecentdevelopmentsonelec-\ntricfield-inducedmagnetizationswitchinginmultiferroicheterostru ctures.Particularly,westudy\nthe dynamics of magnetization switching between the two stable sta tes in a shape-anisotropic\nsingle-domainnanomagnet using stochastic Landau-Lifshitz-Gilber t (LLG) equation in the pres-\nence of thermal fluctuations. For magnetostrictive nanomagnet s in strain-coupled multiferroic\ncomposites, such study of magnetization dynamics, contrary to s teady-state scenario, revealed\nintriguing new phenomena on binary switching mechanism. While the tra ditional method of\nbinary switching requires to tilt the potential profile to the desired s tate of switching, we show\nthat no such tilting is necessary to switch successfully since the mag netization’s excursion out of\nmagnet’s plane can generatea built-in asymmetry during switching. W e also study the switching\ndynamics in multiferroic heterostructures having magnetoelectric coupling at the interface and\nmagnetic exchange coupling that can facilitate to maintain the direct ion of switching with the\npolarity of the applied electric field. We calculate the performance me trics like switching de-\nlay and energy dissipation during switching while simulating LLG dynamics . The performance\nmetrics turn out to be very encouraging for potential technologic al applications.\nKeywords : Nanoelectronics; energy-efficient design; spintronics; straintr onics; electric field-\ninduced magnetization switching; multiferroics.\n1. Introduction\nMultiferroics are intriguing materials in which\nthere is intrinsic coupling between one or more\nferroic properties, e.g., ferroelectric, ferromag-\nnetic, ferroelastic ordering.1;2;3;4;5;6;7;8In\nmultiferroic magnetoelectrics , application of an\nelectric field can rotate the magnetization via\nconverse magnetoelectric effect, however, such\nmaterials in single-phase were thought to be\nrare.9Usually such single-phase multiferroics\nhave issues of weak coupling between polariza-tion and magnetization, and also they usually op-\nerate only at low temperatures.10;11Although,\nnew concepts may come along on switching in\nsingle-phase materials11;12;13;14;15;16;17;18\npossibly utilizing Dzyaloshinsky-Moriya (DM)\ninteraction,19;20;21;22magnetoelectric coupling\ninstrain-mediated multiferroic composites consist-\ning of a piezoelectric layer coupled to a magne-\ntostrictive nanomagnet (see Fig. 1) is shown to be\nvery effective.10;23;3;24;25;26;27Electric field-\ninduced magnetization switching in multiferroics\n12Kuntal Roy\nuses a voltage directly thereby eliminating the need\nto switch magnetization by a cumbersomemagnetic\nfield or by a spin-polarized current,28;29;30;31al-\nthough new concepts are being investigated e.g.,\nutilizing giant spin-Hall effect.32\nThe calculation of performance metrics using\nstochastic Landau-Lifshitz-Gilbert (LLG) equation\nof magnetization dynamics in strain-mediated mul-\ntiferroic composites is shown to hold profound\npromise for computing7;33;34;35;36in beyond\nMoore’s law era.37;38;39;40;41With a suitable\nchoice of materials and dimensions, when a volt-\nage of few millivolts is applied across such het-\nerostructures, the piezoelectric layer gets strained\nand the strain is elastically transferred to the mag-\nnetostrictive nanomagnet, which can rotate the\nmagnetization. Such switching mechanism dissi-\npates a minuscule amount of energy of ∼1 atto-\njoule (aJ) in sub-nanosecond switching delay at\nroom-temperature.7;42This study has opened up\na new field called straintronics .7;33;34;35Exper-\nimental efforts have been undergoing to investigate\nsuch device functionality and the induced stress\nanisotropy in magnetostrictive nanomagnets is\ndemonstrated.43;44;45;46;47;48;49;50The di-\nrect experimental demonstration of switching speed\n(rather than ferromagnetic resonance experiments\nto get the time-scale of switching) and using low-\nthickness piezoelectric layers avoiding considerable\ndegradation of the piezoelectric constants [e.g., <\n100 nm of lead magnesium niobate-lead titanate\n(PMN-PT)]51;52are still under investigation.\nFig. 1. Schematic diagram of a strain-mediated multifer-\nroic composite (piezoelectric-magnetostrictive heteros truc-\nture), and axis assignment. The magnetostrictive nanomag-\nnet is shaped like an elliptical cylinder and it has a single\nferromagnetic domain. The mutually anti-parallel magneti -\nzation states along the z-axis can store a binary bit of infor-\nmation (0 or 1). In standard spherical coordinate system, θ\nis the polar angle and φis the azimuthal angle. According\nto the dimensions of the elliptical cylinder in the respecti ve\ndirections, we term the z-axis the easy axis, the y-axis the\nin-plane hard axis, and the x-axis the out-of-plane hard axis\nfor the nanomagnet.The binary switching of magnetization be-\ntween two stable states of a shape-anisotropic\nsingle-domain magnetostrictive nanomagnet in a\nstrain-mediated multiferroic composite is very\nintriguing.53;54The understanding behind binary\nswitching from one stable state to another enables\nus to design a better switch to address our ever-\nincreasing demand to store, process, and commu-\nnicate information. From around mid-nineties, the\nmethodology of binary switching for information\nprocessing as conceived by famous scientist Lan-\ndauer and others says that an externally intro-\nduced tilt or asymmetry in potential landscape of a\nbistableelement inthe desired direction of switching\nisnecessary .55;56;57;58Thisasymmetry inpoten-\ntial landscape can be achieved by utilizing an exter-\nnalmagnetic field in a single-domain nanomagnet\nwith two stable states acting as a binary switch.58\nThe tilt generates a motion along the direction of\nswitching and the degree of tilt should be sufficient\nenough to dissuade thermal fluctuations with a tol-\nerable error probability. Such tilt or asymmetry in\npotential landscape is deemed to be necessary for\nswitching to take place successfully. Fig. 2(a) de-\npictssuchtraditionalmethodologyofbinaryswitch-\ning.\nFig. 2. Traditional methodology versus the proposed\nmethodology of binary switching. (a) In traditional method -\nology of binary switching, the potential landscape is tilte d to-\nwards the direction of switching alongwith the lowering of e n-\nergy barrier separating the two stable states. Note that the re\naretwoexternalagentsinvolvedhere,onemakesthepotenti al\nlandscape monostable and the other one that tilts the poten-\ntial landscape. At the end, the potential profile is restored\nback to that of the initial stage to complete the switching\nprocess. (b) In the proposed methodology of binary switch-\ning, the potential profile remains always ‘symmetric’, i.e. , en-\nergy barrier is lowered but the potential landscape is not\ntilted to favor the final state. Switching occurs due to inter-\nnaldynamics considering complete three-dimensional poten-\ntial landscape and full three-dimensional motion. Note tha t\nin this case only one external agent is involved since it does\nnot require tilting the potential landscape. For a nanomagn etUltra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 3\nacting as a binary switch, the bold line in its potential land -\nscape corresponds towhenmagnetization resides onmagnet’ s\nplane. Deflection of magnetization from magnet’s plane cor-\nresponds to out-of-plane excursion of magnetization, whic h is\nthe physical mechanism of switching in this case. (Reprinte d\nfrom Ref. 53.)\nNote that the magnetization switching mecha-\nnism using spin-transfer-torque,28;29;30in which\na spin-polarized current is passed through a nano-\nmagnet to switch its magnetization, is analogous\nto thetraditional methodology of binary switch-\ning. It is well known that the Slonczewski-like spin-\ntransfer-torque that acts in-plane of the nanomag-\nnet cannot be treated as an effective potential,28\nhowever, the direction ofexternally applied spin-\npolarizedcurrentinducesan equivalent tiltorasym-\nmetry and causes the magnetization switching in\nthe desired direction.\nIn Ref. 53, it is shown that it is not neces-\nsary to tilt the potential landscape by external\nmeans to switch successfully, even in the presence\nof thermal fluctuations. Fig. 2(b) illustrates the ba-\nsic concept underlying such methodology of binary\nswitching. To understand such switching, the com-\nplete three-dimensional potential landscape needs\nto be considered and thereby it is demonstrated\nin Ref. 53 that the intrinsic dynamics can provide\nan equivalent asymmetry without any requirement\nof making the potential landscape asymmetric. It\nneeds mention here that for both traditional and\nproposed methodologies, it is necessary to lower\nthe energy barrier and make the monostable well\ndeep enough to dissuade thermal fluctuations. For\na magnetostrictive nanomagnet, stress is the ex-\nternal agent modulating and eventually inverting\nthe potential landscape of the nanomagnet. The\nvoltage-induced stress can be generated on a mag-\nnetostrictive nanomagnet by elastically coupling\na piezoelectric layer, i.e., using 2-phase multifer-\nroic composites.10;3;24;59;60;61;33Such mag-\nnetization switching in multiferroic composites\ncan potentially be the basis of ultra-low-energy\ncomputing in our future information processing\nsystems.7;33;53;42\nIfwe consider only the steady-state scenario, we\nwill come to a conclusion that in a strain-mediated\nmultiferroic composite, the strain transferred by\nthepiezoelectric layer tothemagnetostrictive nano-\nmagnet, can only rotate the magnetization 90◦and\nnot the complete 180◦, which is the usual percep-\ntion. However, a complete 180◦switching facili-tates ustoachieve ahighermagnetoresistance while\nreading the magnetization state using magnetic\ntunnel junctions (MTJs).62;63;64;65;66;67;68\nAlthough there are proposals of 90◦switching\nmechanism,69;70;43as shown in Ref. 53 and de-\nscribed above, a complete 180◦switching is possi-\nble if we consider the dynamics of magnetization\ninto account rather than assuming steady-state sce-\nnario. Basically magnetization’s excursion out of\nmagnet’s plane provides an equivalent asymmetry\nto cause a complete 180◦switching.53\nFig. 3. Schematics of the interface and exchange cou-\npled multiferroic heterostructures. The unique coupling b e-\ntween the polarization in the P-layer and the trilayer\nM1/spacer/ M2allows the polarization direction to dictate\nthe magnetic ground state in the trilayer. If the polarizati on\npointsdownward( P↓),P-alignmentinthetrilayerispreferred\nwhile an upward polarization ( P↑) prefers the AP-alignment.\nApplication of a voltage with correct polarity can switch th e\npolarization and hence the magnetization M1gets switched\ntoo due to interface and exchange coupling. At the bottom\nof the figure, the axis assignment for the dynamical motion\nof magnetization M1in standard spherical coordinate system\nis shown. ( c/circlecopyrtIOP Publishing. Reproduced by permission of\nIOP Publishing from Ref. 87. All rights reserved.)\nAlthough the aforesaid switching mechanism\ninstrain-mediated multiferroic composites is in-\ntriguing and promising for technological appli-\ncations, it would be of substantial interest if\nthere exists a strong coupling between polariza-\ntion and magnetization at the heterostructure in-\nterface to harness additional asymmetry during4Kuntal Roy\nmagnetization switching in a specified direction.\nIn Ref. 71, interface and exchange coupled multi-\nferroics are proposed based on density functional\ntheory (DFT) of first-principles calculations. Al-\nthough this specific case needs to be experimentally\ndemonstrated, theproposed concept is tenable. The\nfirst-principles calculations have been proved to be\nvery useful in such respect9and with the exper-\nimental progress on similar front72;73;74;75;76\n(also using ferromagnetic oxides77;78rather than\nferromagnetic metals), there is a considerable\npromise on such polarization-magnetization cou-\npling mechanism.71;79;80;81;82;83In Figure 3,\nsuch interface and exchange coupling between po-\nlarization and magnetization in a multiferroic het-\nerostructure is depicted.71The polarization di-\nrection in the P-layer uniquely determines the\nground state of the trilayer M1/spacer/M2, i.e.,\nfor downward polarization P ↓, parallel alignment\n(P-alignment) in the trilayer is preferred, while\nfor the upward polarization P ↑, antiparallel align-\nment (AP-alignment) in the trilayer is achieved.\nThis polarization can be switched electrically and\nsuch polarization-magnetization coupling mecha-\nnism makes the switching of magnetization in the\nM1-layernon-volatile . If a voltage with certain po-\nlarity is applied and maintained, the state of the\nsystem remains unaltered too. This is advantageous\nover the strain-mediated switching, which just tog-\ngles the magnetization states and therefore requires\na read-before-write mechanism. Note that there are\nother exchanged coupled systems withan insulating\nspacer layer, however, the interlayer exchange cou-\npling energy is small.84;85There are device struc-\ntures with non-magnetic spacer layer to preserve\nlarge interlayer exchange coupling too, but a high\nelectric field is required and the switching becomes\nvolatile.86\nIn Ref. 87, magnetization dynamics in the\ninterface and exchange coupled multiferroic het-\nerostructures is studied by solving stochastic\nLandau-Lifshitz-Gilbert equation in the presence of\nroom-temperature thermal fluctuations. Such phe-\nnomenological study of switching has been very\nuseful to understand the performance metrics of\nmagnetic devices.7;33;34;35;42;53First, an in-\nterfacial anisotropy in the interface and exchange\ncoupled multiferroic heterostructures is modeled\nand subsequently the analysis is performed on\nthedynamics of magnetization. The results show\nthat switching in sub-nanosecond delay is possiblewhile expending only ∼1 aJ of energy at room-\ntemperature. The key point is that the stronginter-\nface anisotropy makes the switching error-resilient\nand fast, allowing to use nanomagnets with very\nsmall dimensions (magnetization is stable with ∼10\nnm lateral dimensions even in the presence of\nroom-temperature thermal fluctuations). Such per-\nformance metrics of area, delay, and energy are par-\nticularly promising for computing in our future in-\nformation processing systems.37;38;39;40;41\nThe rest of the paper is organized as follows.\nIn Section 2, we describe in subsequent subsections\ntwo models for electric field-induced magnetization\nswitching: (1) strain-mediated multiferroic compos-\nites, and (2) interface and exchange coupled mul-\ntiferroic heterostructures. For both the models, the\nstochastic Landau-Lifshitz-Gilbert (LLG) equation\nof magnetization dynamics in the presence of ther-\nmal fluctuations is analytically solved to get a cou-\npled set of equations, which need to be solved nu-\nmerically onwards. Section 3 presents the simula-\ntion results by solving the coupled sets of equa-\ntions numerically. Then we calculate the perfor-\nmance metrics e.g., switching delay, energy dissipa-\ntion, which show particularly promising results for\ntechnological applications. Finally, Section 4 sum-\nmarizes this review and provides the outlook on\nthe electric field-induced magnetization switching\nin multiferroic heterostructures.\n2. Model\nHere, we will review the models developed for\nstrain-mediated multiferroic composites, and inter-\nface and exchange coupled multiferroic heterostruc-\ntures in subsequent subsections. The emphasis\nwould be on the dynamical motion by solving the\nLandau-Lifshitz-Gilbert (LLG) equation of magne-\ntization dynamics.88;89We will also consider ther-\nmal fluctuations incorporated in the LLG equation\nmaking it of stochastic nature.90;53;42;87\n2.1.Strain-mediated multiferroic\ncomposites\nConsider a nanomagnet shaped like an elliptical\ncylinder with its elliptical cross section lying on the\ny-zplane; the major axis is aligned along the z-\ndirection and the minor axis along the y-direction.\n(See Fig. 1.) The dimension of the major axis, the\nminor axis, and the thickness are a,b, andl, re-\nspectively. The volume of the nanomagnet is Ω =Ultra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 5\n(π/4)abl. In standard spherical coordinate system,\nas shown in the Fig. 1, θis the polar angle and φis\nthe azimuthal angle. Note that when φ=±90◦, the\nmagnetization vector lies on the plane of the nano-\nmagnet( y-zplane).Anydeviationfrom φ=±90◦is\ntermed as out-of-plane excursion of magnetization.\nThe total energy of the single-domain, mag-\nnetostrictive, polycrystalline (i.e., no net magne-\ntocrystalline anisotropy) nanomagnet, while sub-\njected to uniaxial stress along the easy axis ( z-axis,\nthemajor axis of the ellipse) is the sumof theshape\nanisotropyenergyandtheuniaxialstressanisotropy\nenergy:91\nE(θ,φ,t) =Eshape(θ,φ)+Estress(θ,t),(1)\nwhereEshape(θ,φ) is the shape anisotropy energy\nandEstress(θ,t) is the stress anisotropy energy at\ntimet. The former is given by91\nEshape(θ,φ) =µ0\n2M2\nsΩNd(θ,φ),(2)\nwhereMsis the saturation magnetization and\nNd(θ,φ) is the demagnetization factor expressed\nas91\nNd(θ,φ) =Nd−zzcos2θ+Nd−yysin2θ sin2φ\n+Nd−xxsin2θcos2φ(3)\nwithNd−zz,Nd−yy, andNd−xxbeing the compo-\nnents of the demagnetization factor along the z-\naxis,y-axis, and x-axis, respectively. The parame-\ntersNd−zz,Nd−yy, andNd−xxdepend on the shape\nand dimensions of the nanomagnet91and are de-\ntermined from the prescription in Ref. 92.\nThe in-plane ( φ=±90◦) shape-anisotropic\nenergy-barrierbetween thetwostablestates ( θ= 0◦\nand 180◦) can be expressed as\nEbarrier=µ0\n2M2\nsΩ(Nd−yy−Nd−zz).(4)\nWitha= 100nm,a= 90nm, andl= 6nm,\nEbarrier≃44kTatroom-temperature( T= 300K),\nwhich means that the static error probability due\nto spontaneous fluctuation of magnetization is\ne−Ebarrier/kT=e−44.\nThe uniaxial shape anisotropy favors lining up\nthe magnetization along the major axis ( z-axis) by\nminimizing Eshape, which is why we call the ma-\njor axis the “easy axis” and the minor axis ( y-axis)\nthe “in-plane hard axis” of the magnet. The x-axis\nwill therefore be the “out-of-plane hard axis” of the\nmagnet and it is “harder” than the in-plane onesince the thickness is much smaller than the mag-\nnet’s lateral dimensions (i.e., l << a,b ).\nWe assume that an uniaxial stress is gener-\nated along the z-axis (easy axis) upon applica-\ntion of an electric field along the x-axis. It is pos-\nsible to constrain the expansion along y-axis to\ngenerate the uniaxial stress along the z-direction\nforpiezoelectrics likelead-zirconate-titanate (PZT).\nWe can also use piezoelectrics like lead magnesium\nniobate-lead titanate (PMN-PT), which generates\nanisotropic strain in the lateral plane (i.e., the signs\nofd31andd32coefficients are different) and there-\nfore it can produce more strain for a given voltage\nor otherwise a lower voltage is required to generate\na specified strain. The stress anisotropy energy is\ngiven by91\nEstress(θ,t) =−(3/2)λsσ(t)Ωcos2θ,(5)\nwhere (3 /2)λsis the magnetostriction coefficient\nof the single-domain nanomagnet and σ(t) is the\nstress at an instant of time t. Note that a positive\nλsσproduct will favor alignment of the magnetiza-\ntion along the major axis ( z-axis), while a negative\nλsσproduct will favor alignment on the x-yplane\n(θ= 90◦), becausethat will minimize Estress. Inour\nconvention, acompressivestressisnegativeandten-\nsile stress is positive. Therefore, in a material like\nTerfenol-Dthathaspositive λs,acompressivestress\nwill favor alignment on the x-yplane (θ= 90◦), and\ntensile along the major axis. The situation will be\nopposite with common magnetic materials like iron,\nnickel, and cobalt, which have negative λs.\nAt any instant of time t, the total energy of the\nnanomagnet can be expressed as\nE(θ,φ,t) =B(φ,t)sin2θ+C(t),(6)\nwhere\nB(φ,t) =Bshape(φ)+Bstress(t), (7a)\nBshape(φ) =µ0\n2MsΩ[Hk+Hdcos2φ],(7b)\nHk= (Nd−yy−Nd−zz)Ms, (7c)\nHd= (Nd−xx−Nd−yy)Ms, (7d)\nBstress(t) = (3/2)λsσ(t)Ω, (7e)\nC(t) =µ0\n2M2\nsΩNd−zz−(3/2)λsσ(t)Ω,(7f)\nHkis the Stoner-Wohlfarth switching field,58and\nHdis the out-of-plane demagnetization field.91\nNote that Bstresshas the same sign as the λsσ\nproduct. It will be negative if we use stress to\nrotate the magnetization from the easy axis ( z-\ndirection) to the plane defined by the in-plane hard6Kuntal Roy\naxis (y-direction) and the out-of-plane hard axis ( x-\ndirection), i.e., the x-yplane (θ= 90◦).\nIn the macrospin approximation, the magne-\ntization M(t) of the nanomagnet has a constant\nmagnitude but a variable direction, so that we can\nrepresent it by the vector of unit norm nm(t) =\nM(t)/|M|=ˆ erwhereˆ eris the unit vector in the\nradialdirection inthestandardsphericalcoordinate\nsystem represented by ( r,θ,φ). The unit vectors for\ntheθ- andφ-rotations are denoted by ˆ eθandˆ eφ,\nrespectively.\nThe torque due to shape and stress anisotropy\nis derived from the gradient of potential profile as\nTE(θ,φ,t) =−nm×∇E(θ,φ,t)\n=−ˆ er×/parenleftbigg∂E\n∂θˆ eθ+1\nsinθ∂E\n∂φˆ eφ/parenrightbigg\n=−2B(φ,t)sinθcosθ ˆ eφ\n−Bshape,φ(φ)sinθˆ eθ,(8)\nwhere\nBshape,φ(φ) =µ0\n2M2\nsΩ(Nd−xx−Nd−yy)sin(2φ).\n(9)\nThe effect of random thermal fluctuations is\nincorporated via a random magnetic field h(t) =\nhx(t)ˆ ex+hy(t)ˆ ey+hz(t)ˆ ez, wherehi(t) (i=x,y,z)\nare the three components of the random thermal\nfield in Cartesian coordinates. We assume the prop-\nerties of the random field h(t) as described in\nRef. 90. The random thermal field can be written\nas90\nhi(t) =/radicalBigg\n2αkT\n|γ|MV∆tG(0,1)(t) (i∈x,y,z),(10)\nwhereαis the dimensionless phenomenological\nGilbert damping parameter, γis the gyromagnetic\nratio for electrons, 1 /∆tis the attempt frequency\nof thermal fluctuations, MV=µ0MsΩ, Ω is the\nvolume,kis the Boltzmann constant, Tis temper-\nature, and the quantity G(0,1)(t) is a Gaussian dis-\ntribution with zero mean and unit variance.\nThethermal field and the correspondingtorque\nacting on the magnetization can be written as\nHTH(θ,φ,t) =Pθ(θ,φ,t)ˆ eθ+Pφ(θ,φ,t)ˆ eφ,(11)\nand\nTTH(θ,φ,t) =nm×HTH(θ,φ,t)\n=Pθ(θ,φ,t)ˆ eφ−Pφ(θ,φ,t)ˆ eθ,(12)respectively, where\nPθ(θ,φ,t) =MV[hx(t)cosθcosφ +hy(t)cosθsinφ\n−hz(t)sinθ], (13)\nPφ(θ,φ,t) =MV[hy(t)cosφ−hx(t)sinφ].(14)\nAdditionally, there is motion due to Gilbert\ndamping88;89(perpendicular to the precessional\nmotion) through which magnetization relaxes to-\nwards the minimum energy position on magnet’s\npotential landscape. The magnetization dynamics\nunder the action of these two torques TEand\nTTHisdescribedbythestochasticLandau-Lifshitz-\nGilbert (LLG) equation as follows.\ndnm\ndt−α/parenleftbigg\nnm×dnm\ndt/parenrightbigg\n=−|γ|\nMV[TE+TTH],\n(15)\nwhereαis the phenomenological Gilbert damping\nparameter and γis the gyromagnetic ratio of elec-\ntrons. Solving the above equations, we get the fol-\nlowing coupled equations for the dynamics of θand\nφ:\n/parenleftbig\n1+α2/parenrightbigdθ\ndt=|γ|\nMV[Bshape,φ(φ)sinθ\n−2αB(φ,t)sinθcosθ\n+(αPθ(θ,φ,t)+Pφ(θ,φ,t))],(16)\n/parenleftbig\n1+α2/parenrightbigdφ\ndt=|γ|\nMV[αBshape,φ(φ)+2B(φ,t)cosθ\n−{sinθ}−1(Pθ(θ,φ,t)−αPφ(θ,φ,t))] (sinθ∝ne}ationslash= 0).\n(17)\nWe will ignore the random thermal torque due\nto room-temperature thermal fluctuations while ex-\nplaining the magnetization dynamics first, follow-\ning Ref. 53, however, we will discuss the key conse-\nquences of incorporating thermal fluctuations with\nsimulation results later. We assume that the mag-\nnetization starts from θ≃180◦(−z-axis) and the\napplied stress attempts to switch it to θ≃0◦(+z-\naxis).\nOut-of-plane excursion of magnetization. We\nwill first intuitively describe how magnetization is\ndeflected from the magnet’s plane ( φ=±90◦, i.e.,\ny-zplane), and is stabilized out-of-plane as de-\npicted in Fig. 4(b) due to different torques act-\ning on it. The shape anisotropy energy is in gen-\neral dependent on azimuthal angle φ(rather than\nassuming φ=±90◦) and it generates additional\nmotions of magnetization in ˆeθandˆeφdirectionsUltra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 7\nFig. 4. Illustration of magnetization’s motion in three-di mensional space. (a) Cross section of the nanomagnet and axe s\nassignment. (b) The applied stress tries to lift the magneti zationMout of the magnet’s plane while the ˆeφ-component of the\nshape anisotropy torque due to Gilbert damping tries to brin g it back to the plane ( φ=±90◦). This stabilizes the value of φ,\nbut it happens only in the “good” quadrants φ∈(90◦, 180◦) andφ∈(270◦, 360◦). For the other two quadrants φ∈(0◦, 90◦)\nandφ∈(180◦, 270◦), such counteraction does not happen and we term them as “bad ” quadrants. (c) Illustration of the motion\nof magnetization Min three-dimensional space under various torques generate d due to shape and stress anisotropy alongwith\nconsidering the damping of magnetization ( αis the phenomenological damping parameter). Note that the d ependence of shape\nanisotropy energy on φhas generated two additional motions −|Bshape,φ(φ)|sinθˆeθand−α|Bshape,φ(φ)|ˆeφ. (See text for\ndetails.) The quadrant φ∈(90◦, 180◦) is chosen for illustration; choice of the other “good” quad rantφ∈(270◦, 360◦) is\nanalogous. (Reprinted from Ref. 53. B(φ) andBstressare replaced by B(φ,t) andBstress(t), respectively.)\n[see the motions containing the term Bshape,φ(φ)\nin Fig. 4(c) and equations (16) and (17)]. Both\nof these torques are proportional to sin(2φ) and\nvanish when φ=±90◦. Also, note that the φ-\ncomponent is proportional to the damping parame-\nterα. As shown in the Fig. 4(b), the applied stress\ngenerates a torque that attempts to rotate the mag-\nnetization anticlockwise and forces the magnetiza-\ntion to deflect from magnet’s plane and stay out\nof magnet’s plane. As magnetization is deflected\nfrom the plane of the magnet ( φ=±90◦), theφ-\ncomponent of the torque due to shape anisotropy\nenergy as mentioned earlier [ ∝αsin(2φ)] would at-\ntempt to bring the magnetization back to magnet’s\nplane. Because of this counteraction , the magneti-\nzation is only stable in the second or the fourth\nquadrant [i.e., (90◦, 180◦) or (270◦, 360◦)], among\nthe four possible quadrants of φ. Note that sin(2φ)\nis anegative quantity in these two quadrants and\nhence the motion due to shape anisotropy energy\ncounteracts the precessional motion due to stress.We would term these two quadrants (second or the\nfourth)as“good”quadrantsandtheothertwo(first\nand third) quadrants as “bad” quadrants, the rea-\nsoning behind which would be more prominent on-\nwards, i.e., consideration of the torques due to φ-\ndependence of shape anisotropy energy is crucial to\nthe magnetization dynamics. The key lesson is that\nφbecomes stable only in the “good” quadrants and\nit facilitates switching of magnetization in the de-\nsired direction.\nMagnetization’s motion in three-dimensional\nspace.We will now describe the motion of magneti-\nzation inthefullthree-dimensional spaceintuitively\nin the presence of various torques originating from\nthe shape and stress anisotropy energy as depicted\nin the Fig. 4(c). Note that the motion of magne-\ntization needs to be along the −ˆeθdirection since\nmagnetization is being switched from θ≃180◦to-\nwardsθ≃0◦. The applied stress generates a pre-\ncessional motion of magnetization in the + ˆeφdirec-\ntion, and the damping of magnetization generates8Kuntal Roy\nFig. 5. Field and torque acting on the magnetization Mwhen it comes on the x-yplane (θ= 90◦). (a)φ∈(0◦,90◦). The\nfield always tries to keep the magnetization on magnet’s plan e (φ=±90◦). For this “bad” quadrant φ∈(0◦,90◦), magne-\ntization backtracks towards θ≃180◦causing a switching failure. Choice of the other “bad” quadr antφ∈(180◦,270◦) is\nanalogous. (b) φ∈(90◦,180◦). The field again tries to keep the magnetization on magnet’s plane (φ=±90◦). For this “good”\nquadrant φ∈(90◦,180◦), magnetization can traverse towards its destination θ≃0◦. Choice of the other “good” quadrant\nφ∈(270◦,360◦) is analogous. Note that the motion of magnetization is oppo site to the direction of torque exerted on it since\nthe Land´ e g-factor for electrons is negative. If magnetization starts from the other easy axis θ≃0◦and we switch it towards\nθ≃180◦, the roles of the four quadrants of φwould have been exactly opposite. (Reprinted from Ref. 53.)\na motion additionally that is perpendicular to both\nthe direction of magnetization ( ˆer) and +ˆeφ, i.e.,\nin−ˆeθdirection. These two motions are shown as\n2B(φ,t)cosθˆeφand−2αB(φ,t)sinθcosθ ˆeθ, respec-\ntively in Fig. 4(c), where αis the damping param-\neter and the quantity B(φ,t) includes terms both\ndue to the shape anisotropy energy Bshape(φ) and\nthe stress anisotropy energy Bstress(t). The quan-\ntityBstress(t) is negative and it must overcome\nthe shape anisotropy term Bshape(φ) for switching\nto get started (mathematically, note that both the\nquantities B(φ,t) andcosθare negative in the in-\nterval 180◦≥θ≥90◦). Therefore, magnetization\nstarts switching towards its desired direction due to\nthe applied stress. Note that this damped motion in\n−ˆeθdirection is considerably weak because of the\nmultiplicative damping parameter α, which is usu-\nally muchless thanone(e.g., α=0.1 forTerfenol-D).\nAs described earlier [see Fig. 4(b)], due to ap-\nplied stress, magnetization rotates out-of-plane and\nstays in a “good” quadrant for φ[i.e., (90◦, 180◦) or\n(270◦, 360◦)], which generates a motion of magne-\ntization in the −ˆeθdirection due to φ-dependence\nof shape anisotropy energy. Thereby a damped mo-\ntion is generated in the −ˆeφdirection. These two\nmotions are shown as −|Bshape,φ(φ)|sinθˆeθand\n−α|Bshape,φ(φ)|ˆeφ, respectively in Fig. 4(c), whereBshape,φ(φ)∝sin(2φ). In the “good” quadrants for\nφ,Bshape,φ(φ) is negative, therefore, if the magneti-\nzationstaysoutofmagnet’splaneina“good”quad-\nrant, magnetization rotates in its desired direction.\nSincethismotiondoesnotpossessanydampingfac-\ntor [note the other damped motion in the −ˆeθdi-\nrection in Fig. 4(c)], it can eventually increase the\nmagnetization switching speed to a couple of or-\nders of magnitude higher. On the other hand, if the\nmagnetization resides out-of-plane but in a “bad”\nquadrant, the motion of magnetization in its de-\nsireddirectionofswitchingishindered.Ifweapplya\nhigher magnitude of stress, the magnetization is de-\nflected out of magnet’s plane moreinside a “good”\nquadrant (counteracting the random thermal kicks\npossibly acting in the opposite direction, which will\nbe described later). Note that the damped motion\n−α|Bshape,φ(φ)|ˆeφattempts to bringmagnetization\nback towards the magnet’s plane. As these two mo-\ntions counteract each other [see Fig. 4(c)], mag-\nnetization continues moving in the −ˆeθdirection\nand eventually reaches the x-yplane (θ= 90◦).\nNote that without damping, such counteraction\ndoes not happen and the magnetization just pre-\ncesses through “good” and “bad” quadrants con-\nsecutively.\nUpon reaching the x-yplane (θ= 90◦), if mag-Ultra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 9\nFig. 6. Stress cycle, magnetization directions, and potent ial profiles at different time instants during switching of ma gneti-\nzation. (a) Stress-cycle on the magnetostrictive nanomagn et. (b) Magnetization directions at different instants of ti me. (c)\nPotential landscapes of the magnetostrictive nanomagnet i n relaxed, compressively stressed, and expansively stress ed condi-\ntions. Note that the three-dimensional potential landscap e has never been made asymmetric to favor the final state durin g\nswitching. (Reprinted from Ref. 53.)\nnetization stays in a “good” quadrant for φ[i.e.,\n(90◦,180◦) or (270◦,360◦)], then the torque on the\nmagnetization will be in the correct direction to fa-\ncilitate magnetization’s traversal towards θ≃0◦\n[see Figs. 5(a) and 5(b)]. Once again this signi-\nfies the merit of terminology (“good” or “bad”)\nused for the four quadrants of φ. Atθ= 90◦(i.e.,\ncosθ= 0), the effect of stress on the magnetization\nrotation has diminished completely [see Fig. 4(c)].\nThe only two motions that are active at θ= 90◦\nare−|Bshape,φ(φ)|sinθˆeθand−α|Bshape,φ(φ)|ˆeφ[Fig. 4(c)]. Since α≪1, magnetization quickly gets\nout from θ= 90◦and as the magnetization vector\nis deflected from θ= 90◦towardsθ= 0◦, the effect\nof stress again comes into play.\nStress cycle, magnetization directions, and po-\ntential profiles. Fig. 6 shows the stress-cycle along-\nwith the energy profiles and magnetization direc-\ntions at different instants of time during switching\nof magnetization. At time t0, the magnetization di-\nrection is along the easy axis θ≃180◦with itse po-\ntential landscape unperturbed by stress. Note that10Kuntal Roy\nthe potential profile of the magnet is ‘symmetric’\nin bothθ- andφ-space with two degenerate minima\natθ= 0◦, 180◦and a maximum at θ= 90◦inθ-\nspace, signifying that a binary information can be\nstored in the nanomagnet corresponding to θ= 0◦,\n180◦. The anisotropy in the barrier is due to shape\nanisotropy energy of the nanomagnet only, which\nis∼44 kT at room-temperature using the nano-\nmagnet’s dimensions and the material parameters\nused for the magnetostrictive nanomagnet made of\nTerfenol-D (see Section 3 later). Note that the bar-\nrier height separating the two stable states ( θ= 0◦\nand 180◦) is meant when the magnetization stays\nin-plane (i.e., φ=±90◦) of the magnet. The barrier\nbecomes higher when the magnetization is deflected\nfromφ= 90◦as shown in the Fig. 6(c) at time t0.\nThe barrier is the highest when the magnetization\npoints along the out-of-plane direction ( φ= 0◦or\n180◦), which is due to the small thickness of the\nnanomagnet compared to the lateral dimensions.\nNote that the magnetization can start from any an-\ngleφinitial∈(0◦,360◦) in the presence of thermal\nfluctuations [see Fig. 9(b) later].\nFig. 6(c) depicts that as a compressive stress\nis ramped up on the nanomagnet between time in-\nstantst0andt1and a sufficient stress is applied, the\npotential landscape in θ-space becomes monostable\nnearφ=±90◦. Since the barrier height is high near\nφ= 0◦or 180◦, the potential landscape may not\nbecome monostable in θ-space therein. However,\nthat is not necessary for switching since application\nof stress rotates the magnetization in φ-direction\nand theretofore the magnetization can eventually\ncome near φ=±90◦, which facilitates switching\nfromθ≃180◦towards θ= 90◦(see Fig. 7). From\nFig. 6(c), we can see that the minimum energy posi-\ntion between time instants t1andt2is at (θ= 90◦,\nφ=±90◦) and the potential profile at time instant\nt1is still ‘symmetric’.\nFig. 6 shows that stress is held constant be-\ntween time instants t1andt2and the magnetiza-\ntion eventually reaches at x-yplane (θ= 90◦).\nFor a sufficiently fast ramp rate and high stress,\nmagnetization will reside in “good” quadrants (see\nFig. 13 later). This will ensure that magnetization\ntraversesinthecorrectdirectiontowards θ= 0◦and\nswitches successfully. A sufficiently fast ramp rate\nensures that magnetization would not backtrack to-\nwardsθ= 180◦even after crossing θ= 90◦towards\nθ= 0◦. If stress is held constant for longer time,\nmagnetization will have higher probability to col-lapseonmagnet’s plane( φ=±90◦) following which\nthermal fluctuations will scuttle the magnetization\neither in a “good” quadrant or in a “bad” quadrant\nwith equal probability, so the success rate would be\n50%.\nIt should be noted from Fig. 6 that the stress is\nreversed (compression to tensile) between time in-\nstantst2andt4rather than just withdrawn, which\nmakes the potential landscape of the nanomagnet\nmore steep in θ-space. However, it does not neces-\nsarily mean that the switching will be completed\nalways faster. The reversal of stress can cause mag-\nnetization to traverse into “bad” quadrants in φ-\nspace causing magnetization to precess and there-\nfore the switching delay may eventually increase.\nParticularly for higher stress levels, such increase\nin switching delay may happen. However, it is no-\nticed that reversing the stress makes the success\nrate of switching a bit ( <5%) higher in the pres-\nence of thermal fluctuations. The tensile stress is\nheld constant and when θbecomes ≤5◦, switching\nis deemed to have completed. The stress is with-\ndrawn at the end between time instants t5andt6\nto complete the switching cycle.\nSwitching failure. As magnetization leaves from\nθ= 90◦towardsθ≃0◦and stress is ramped down,\nthetorqueduetostresstriestorotatetheazimuthal\nangleφof magnetization clockwise rather than an-\nticlockwise. Mathematically, note that cosθis pos-\nitive for 90◦≥θ≥0◦andB(φ,t) is still negative\nwhen stress has not been brought down sufficiently,\ni.e., still |Bstress(t)|>|Bshape(φ)|(see Fig. 4(c) and\ncorrespondingdiscussionsin text). For a slow ramp-\nratethisclockwise rotation may be considerable\nand magnetization can stray into a “bad” quad-\nrant. Moreover, thermal fluctuations can aggravate\nthe scenario by possibly deflecting the magnetiza-\ntion into the “bad” quadrant further. Switching\nmay impede and magnetization vector may back-\ntrack towards θ≃180◦rather than traversing to-\nwardsθ≃0◦causing a switching failure. Therefore,\nswitching failure may happen even after the magne-\ntization has crossed the hard axis ( θ= 90◦) towards\nits destination θ≃0◦(see Fig. 8). This is why it\ndoes require a sufficiently fast ramp rate during the\nramp-down phase of stress. Such switching failure\nis an intriguing phenomenon, which cannot be con-\nceived if magnetization is always assumed to be on\nmagnet’s plane ( φ=±90◦) and only can be under-\nstood if we analyze the magnetization dynamics in\na complete three dimensional potential landscape.Ultra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 11\nFig. 7. Illustration of magnetization’s motion when magnet ization starts switching out of magnet’s plane ( φ/ne}ationslash= 90◦), which\ncan happen dueto thermal fluctuations [see Fig. 9(b)] and the high shape-anisotropy energy barrier therein cannot be ove rcome\nby stress anisotropy. (a) Magnetization starts in-plane of the magnet ( φ= 90◦), where the potential landscape is inverted by\nthe stress anisotropy and thus magnetization does not face a potential hill while starting to switch. (b) Magnetization starts\nfrom out-of-plane of the magnet ( φ/ne}ationslash= 90◦), where the potential landscape cannot be inverted by the st ress anisotropy and\nthus magnetization does face a potential hill at start. Howe ver, due to φ-motion of magnetization, it eventually surpasses the\npotential hill and comes near to magnet’s plane, where from i t can start switching in θ-space. (Reprinted from Ref. 53.)\nFluctuation of magnetization around the easy\naxis (stable orientation) due to thermal noise. From\nEquation (8), it can benoted that the torque on the\nmagnetization due to shape and stress anisotropy\nvanishes when sinθ= 0, i.e. when the magnetiza-\ntion vector is exactlyaligned along the easy axis\n(θ= 0◦, 180◦).42;53However, thermal fluctuations\ncan deflect the magnetization vector from the easy\naxis. Considering the case when θ= 180◦, we get\nthe following:\nφ(t) =tan−1/parenleftbiggαhy(t)+hx(t)\nhy(t)−αhx(t)/parenrightbigg\n,(18)\ndθ\ndt=−|γ|h2\nx(t)+h2\ny(t)/radicalbig\n(hy(t)−αhx(t))2+(αhy(t)+hx(t))2.\n(19)\nFrom the above equation we can clearly follow that\nthermal torque can deflect the magnetization from\nthe easy axis since the time rate of change of θ(t) is\nnon-zero in the presence of thermal agitations. The\ninitial deflection from the easy axis due to the ther-\nmal torque does not depend on the component of\ntherandomthermalfieldalongthe z-axis,i.e., hz(t).\nThis is a consequence of having ±z-axis as the easy\naxes of the nanomagnet. However, once the magne-\ntization direction is even slightly deflected from the\neasy axis, all the three components of the random\nthermal field come into play.\nThermal distribution of the initial orientationof the magnetization. Wecan determinethethermal\ndistributions of θandφwhen no stress is applied\non the magnetostrictive nanomagnet by solving the\nEquations (16) and (17) with Bstress= 0 (Refs. 42,\n53). This will yield the distribution of the magne-\ntization’s initial orientation when stress is turned\non. The θinitial-distribution is Boltzmann peaked\natθ= 0◦or 180◦, while the φinitial-distribution\nis Gaussian peaked at φ=±90◦(see Fig. 9). Ac-\ncordingtotheBoltzmann distributionof θinitial,the\nmost probable value of θis either 0◦or 180◦, where\nstress is ineffective. This will lead to a long tail in\nthe switching delay distribution, which is due to\nthe fact that when magnetization starts out from\nθ= 0◦,180◦, it needs to wait a while before random\nthermal fluctuations can set the switching in mo-\ntion. Thus, switching trajectories starting from an\neasy axis are very slow and it causes the long tail\nin the distribution of switching delay.42;53\nApplication of a bias field to shift the peak of the\ninitial distribution of magnetization from an easy\naxis.We can eliminate the long tail in the switch-\ningdelaydistributionbyapplyingasmallstaticbias\nmagnetic field that shifts the peak of θinitialdistri-\nbution away from the easy axis, so that the most\nprobable starting orientation will no longer be an\neasy axis.42This field is applied along the out-of-\nplane hard axis (+ x-direction) of the nanomagnet\nand thus the potential energy due to the applied12Kuntal Roy\nFig. 8. Magnetization can backtrack even after it has crosse d the hard axis towards its destination. (a) Magnetization h as\nstarted from θ≃180◦and crossed the hard axis θ= 90◦, but it is well possible that magnetization backtracks towa rds\nθ≃180◦even without considering the presence of thermal fluctuatio ns. Looking at the two-dimensional magnet’s plane and\nconsidering two-dimensional motion of magnetization on ma gnet’s plane, this seems unreasonable in the absence of ther mal\nfluctuations. (b) Full three-dimensional potential landsc ape of magnetization. Considering the complete three-dime nsional\nmotion of magnetization in this potential landscape of the n anomagnet, switching failure may be plausible even in the ab sence\nof thermal fluctuations since there is consequence of out-of -plane motion of magnetization. (c) Explanation behind mag netiza-\ntion’s backtracking even after it has crossed the hard axis t owards its destination. Magnetization may switch to the inc orrect\ndirection because it is in a “bad” quadrant for φand there is a motion of magnetization Bshape,φ(φ)sinθˆeθin the unintended\ndirection. The other magnetization’s motion −2αB(φ,t)sinθcosθ ˆeθdue to damping is in the intended direction but it may be\nquite small compared to the other motion and thus magnetizat ion may well backtrack. The quadrant φ∈(0◦, 90◦) is chosen\nfor illustration; choice of the other “bad” quadrant φ∈(180◦, 270◦) is analogous. (Parts (a) and (c) are reprinted from Ref. 53.\nIn part (c), B(φ) andBstressare replaced by B(φ,t) andBstress(t), respectively.)\nmagnetic field becomes\nEmag(t) =−MVHsinθ(t)cosφ(t),(20)\nwhereHis the magnitude of magnetic field. A\ntorque is generated due to this field, which is\nTM(t) =−nm(t)×∇Emag(θ,φ).(21)\nThe presence of this torque will modify Equa-\ntions (16) and (17) to\n/parenleftbig\n1+α2/parenrightbigdθ\ndt=|γ|\nMV[Bshape,φ(t)sinθ\n−2αB(φ,t)sinθcosθ\n+αMVHcosθcosφ −MVHsinφ\n+(αPθ(θ,φ,t)+Pφ(θ,φ,t))],(22)\n/parenleftbig\n1+α2/parenrightbigdφ\ndt=|γ|\nMV[αBshape,φ(t)+2B(φ,t)cosθ\n−[sinθ]−1(MVHcosθcosφ +αMVHsinφ)\n−[sinθ]−1(Pθ(θ,φ,t)−αPφ(θ,φ,t))] (sinθ∝ne}ationslash= 0).\n(23)Note that the bias field also makes the poten-\ntial energy profile of the magnet asymmetric in\nφ-space and the energy minimum is shifted from\nφmin=±90◦(the plane of the magnet) to\nφmin=cos−1/bracketleftbiggH\nMs(Nd−xx−Nd−yy)/bracketrightbigg\n.(24)\nHowever, the potential profile will remain symmet-\nricinθ-space, with θ= 0◦andθ= 180◦remain-\ning as the minimum energy positions. With the ma-\nterial parameters and the dimensions of the nano-\nmagnet (see Section 3 later), a bias magnetic field\nof flux density 40 mT would make φmin≃ ±87◦,\ni.e. deflect the magnetization ∼3◦from the mag-\nnet’s plane. Application of the bias magnetic field\nalso reduces the in-plane shape anisotropy energy\nbarrier from 44 kTto 36kTat room temperature.\nWe assume that a permanent magnet will be em-\nployed to produce the bias magnetic field and not\nby a current-carrying coil on-chip. Therefore no ad-\nditional energy dissipation needs to be considered\nfor this reason.Ultra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 13\nEnergy Dissipation. The energy dissipated dur-\ning magnetization switching has two components:\n(1) the energy dissipated in the switching circuit for\ngenerating stress in the magnetostrictive nanomag-\nnet with the application of a voltage, and (2) the\nenergy dissipated internally in the nanomagnet due\nto Gilbert damping. We will term the first compo-\nnent as ‘ CV2’ dissipation, where CandVdenote\nthe capacitance of the piezoelectric layer and the\napplied voltage, respectively. If the ramp rate is in-\nfinite, i.e., the voltage is turned on or off abruptly,\nthe energy dissipated during either turn on or turn\noff is (1/2)CV2. However, if the ramp rate is fi-\nnite,thisenergydissipationisreducedandtheexact\nvalue will depend on the ramp rate. We calculate it\nfollowing the procedure described in Ref. 93.\nThe second component, which is the internal\nenergy dissipation Eddue to Gilbert damping, is\ngiven by the expression/integraltextτ\n0Pd(t)dt, whereτis the\nswitching delay and Pd(t) is the power dissipated\nduring switching\nPd(t) =α|γ|\n(1+α2)MV|TE(t)+TM(t)|2.(25)\nWe sum up the two energy dissipations ‘ CV2’ and\nEdto get the total dissipation Etotal. The average\npower dissipation is Etotal/τ. There is no net dis-\nsipation due to random thermal torque, however,\nthermal fluctuations do affect both Edand ‘CV2’\ndissipations. It affects Edsince it raises the criti-\ncal stress needed to switch with ∼100% probability\nand‘CV2’dissipationisalsoraisedsincetheapplied\nvoltage is proportional to stress.\nSo far the model presented deals with the\nswitching of magnetization (i.e., writing a bit\nof information), however, the magnetization state\nneeds to be read too, which is usually per-\nformed by using a magnetic tunnel junction\n(MTJ).62;63;64;65;66;67;68While reading the\nbit of information, a material selection issue crops\nup since magnetostrictive materials in general can-\nnot be utilized as the free layer of the MTJ. This\nis an important issue since we need to achieve a\nhigh magnetoresistance for successful read opera-\ntion as required by technological applications. In\nRef. 94, it is shown that magnetically coupling the\nmagnetostrictive nanomagnet and the freelayer can\ncircumvent this issue. Stochastic Landau-Lifshitz-\nGilbert equation of magnetization dynamics in the\npresence of room-temperature thermal fluctuations\nis solved and it is shown that such design can even-tually lead to a superior energy-delay product.94\n2.2.Interface and exchange coupled\nmultiferroic heterostructures\nWe have earlier shownin the Figure 3the schematic\ndiagram of the interface and exchange coupled mul-\ntiferroicheterostructuredevices andtheaxisassign-\nment for the orientation of magnetization M1. The\nstandard spherical coordinate system with θas po-\nlar angle and φas azimuthal angle is chosen. The\nmagnetization M1orients along θ= 180◦if polar-\nization points downward ( P↓), while M1is along\nθ= 0◦if polarization points upward ( P↑). The lat-\neral elliptical cross-section of M1lies on the y-z\nplane (φ=±90◦) with its major axis pointing to z-\ndirection and minor axis in y-direction. The dimen-\nsions of the nanomagnet M1along the z-,y-, and\nx-axis are a,b, andl, respectively. So the magnetic\neasy axis becomes along the ±z-direction and the\nnanomagnet’s volume Ω = ( π/4)abl. When mag-\nnetization switches between its two stable states\n(θ= 0◦,180◦), magnetization being a rotational\nbody deflects out of magnet’s plane and any deflec-\ntion from the magnet’s plane ( φ=±90◦) is termed\nas out-of-plane excursion.\nThe interface anisotropy energy in the nano-\nmagnetM1is modeled as87\nEI(θ,t) =−MVHI(t)cosθ, (26)\nwhereMV=µ0MsΩ,Msis the saturation magne-\ntization, and HIis the interfacial anisotropy field.\nIfHI=−HI,max, the magnetization M1points\nalongθ= 180◦and if we vary HIfrom−HI,max\ntoHI,max, the magnetization orients along θ= 0◦.\nThe total anisotropy of the magnet is the sum\nof the interface anisotropy alongwith the other\nanisotropies like magnetocrystalline anisotropy and\nshape anisotropy,33;42. However, since the in-\nterfacial anisotropy is strong compared to the\nother anisotropies, we consider only the interfacial\nanisotropy (i.e., Etotal≃EI) for brevity.\nThe magnetization Mof thesingle-domain\nnanomagnet M1has a constant magnitude of mag-\nnetization but a variable direction. Therefore, the\nmagnetization vector can berepresentedby theunit\nvector in the radial direction ˆ erin spherical coordi-\nnate system ( r,θ,φ), i.e.,nm=M/|M|=ˆ er. The\nother two unit vectors in the spherical coordinate\nsystem are ˆ eθandˆ eφforθ- andφ-rotations, respec-\ntively. The torque TIacting on the magnetization\ndue to interface anisotropy can be derived from the14Kuntal Roy\ngradient of the energy [see Equation (26)] and is\ngiven by\nTI(θ,t) =−nm×∇EI(θ,t) =−MVHI(t)sinθˆ eφ.\n(27)\nNote that the torque TIacts along the out-of-plane\ndirection, so that the magnetization can deflect out\nof magnet’s plane ( φ=±90◦).\nThe effect of random thermal fluctuations is in-\ncorporated via a random magnetic field and the\ncorresponding torque TTHis given by the Equa-\ntion (12) (also see the Equations (13) and (14) for\nthe expressions of PθandPφ).\nThe magnetization dynamics of of the nano-\nmagnetM1under the action of the torques TIand\nTTHisdescribedbythestochasticLandau-Lifshitz-\nGilbert (LLG) equation as follows.\ndnm\ndt−α/parenleftbigg\nnm×dnm\ndt/parenrightbigg\n=−|γ|\nMV[TI+TTH].\n(28)\nSolving the above equation analytically, we get\nthe following coupled equations of magnetization\ndynamics for θandφ:\n/parenleftbig\n1+α2/parenrightbigdθ\ndt=|γ|\nMV[−αMVHI(t)sinθ\n+(αPθ(θ,φ,t)+Pφ(θ,φ,t))],(29)\n/parenleftbig\n1+α2/parenrightbigdφ\ndt=|γ|\nMV[MVHI(t)−[sinθ]−1\n×(Pθ(θ,φ,t)−αPφ(θ,φ,t))] (sinθ∝ne}ationslash= 0).(30)\nWe solve the above two coupled equations numeri-\ncally to track the trajectory of magnetization over\ntime, in the presence of room-temperature thermal\nfluctuations.87\nFrom Equations (29) and (30), we see that\nthe torque acting in the φ-direction is much more\nstronger than the torque exerted in the θ-direction\nsince the damping parameter α≪1. Although, the\nnanomagnet has a small thickness (i.e., l≪b < a\nandNd−xx≫Nd−yy> Nd−zz, whereNd−zz,Nd−yy,\nandNd−xxare the components of the demagneti-\nzation factor along the z-axis,y-axis, and x-axis,\nrespectively), magnetization cannot remain on the\nmagnet’s plane ( y-zplane,φ=±90◦) due to the\nfact that the interface coupling energy is a few or-\nders of magnitude higher than the shape anisotropy\nenergy. Thus, the magnetization keeps rotating in\ntheφ-direction, but it also traverses towards the\nanti-parallel directionin θ-space(θ≃180◦toθ≃0◦\nor vice-versa) due to damping [see Equation (29)].Note that exactly at θ= 180◦or 0◦, the\ntorque acting on the magnetization due to inter-\nface anisotropy [Equation (27)] is exactlyzero, how-\never, asdescribedearlier intheSubsection2.1,ther-\nmal fluctuations can scuttle the magnetization from\nthese points to initiate switching. At the very start\nofswitching,theinitial orientation ofmagnetization\nisnot afixedvalueratheradistributionduetother-\nmal agitations (as described in the Subsection 2.1\ntoo). Such distribution is considered during simula-\ntions.Thermalfluctuationsaffectthemagnetization\nswitching during the course of switching too.\nEnergy Dissipation. Due to the application of\nvoltage, we have ‘ CV2’ energy dissipation, where\nCandVdenote the capacitance of the ferroelec-\ntric layer and the applied voltage, respectively. We\ncalculate it following the procedure described in\nRef. 93. The energy dissipated in the nanomag-\nnet due to Gilbert damping can be expressed as\nEd=/integraltextτ\n0Pd(t)dt, whereτis the switching delay and\nPd(t) is the power dissipated at time tgiven by\nPd(t) =α|γ|\n(1+α2)MV|TI(θ(t),t)|2.(31)\nThermal field with mean zero does not cause any\nnet energy dissipation but it causes variability in\nthe energy dissipation Edby scuttling the trajec-\ntory of magnetization.\n3. Results and Discussions\nHere, we will review the simulation results\nfor strain-mediated multiferroic composites,42;53\nand interface and exchange coupled multiferroic\nheterostructures87in subsequent subsections. The\ncoupled sets of equations derived in the previous\nsection are numerically solved and the performance\nmetricse.g., switchingdelay, energydissipationdur-\ning switching are reported.\n3.1.Strain-mediated multiferroic\ncomposites\nHerewepresentthesimulation resultsforstrainme-\ndiated multiferroic composites.53;42The magne-\ntostrictive layer is made of polycrystalline Terfenol-\nD and it has the following material properties –\nYoung’s modulus (Y): 8 ×1010Pa, magnetostrictive\ncoefficient ((3 /2)λs): +90×10−5, saturation magne-\ntization ( Ms): 8×105A/m, and Gilbert’s damping\nconstant ( α): 0.1 (Refs. 95, 96, 97, 98). We choose\nthe dimension of the magnetostrictive layer as 100Ultra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 15\nnm×90 nm×6 nm, which ensures that the mag-\nnet has a single ferromagnetic domain.42;92The\ntradeoffs between area, switching delay, and energy\ndissipation have been comprehensively studied for\ndifferent dimensions of the nanomagnet in Ref. 99.\nFor the piezoelectric layer, we consider lead-\nzirconate-titanate (PZT), which has a dielectric\nconstant of 1000.42;33Since we want any strain\ngenerated in the PZT layer is transferred almost\ncompletely to the magnetostrictive layer, we as-\nsume that the PZT layer is four times thicker than\nthe magnetostrictive layer.42;33We consider that\nthe maximum strain of 500 ppm can be generated\nin the PZT layer,100;101which would require a\nvoltage of 66.7 mV because d31=1.8×10−10m/V\nfor PZT.102Assuming this strain is transferred\ncompletely to the magnetostrictive layer, the cor-\nresponding stress in Terfenol-D is the product of\nthe generated strain (500 ×10−6) and the Young’s\nmodulus(8 ×1010Pa). Therefore, a maximum stress\nof 40 MPa can begenerated in the Terfenol-D nano-\nmagnet. While avoiding considerabledegradation of\nthe piezoelectric constants at low-thickness (24 nm\nas assumed in Ref. 42) piezoelectric layers is un-\nder research,51;52we can use a higher thickness\nfor the piezoelectric layer (e.g., 100 nm) and a con-\ncomitant amount of higher voltage, since the energy\ndissipation due to applyingvoltage is miniscule. For\nthe piezoelectric layer, if we use materials with high\npiezoelectric coefficients e.g., PMN-PT, which has\nd31=–3000 pm/V, and d32=1000 pm/V (note the\nanisotropic strain, i.e., the signs of d31andd32are\ndifferent),48we can lower theoperating voltage fur-\nther to a few millivolts.\nItshouldbenoted that forthemagnetostrictive\nlayer, we need to choose a material that maximizes\ntheproduct(3 /2)λsY.Terfenol-D(TbDyFe), which\nhas 30 times higher magnetostriction coefficient in\nmagnitude than the common ferromagnetic mate-\nrials (e.g., Fe, Co, Ni), has the highest (3 /2)λsY\n(Ref. 93). If it needs to avoid the rare-earth materi-\nals (e.g., Tb and Dy in Terfenol-D), we can also uti-\nlize Galfenol (FeGa),61;103which has 6 times less\n(3/2)λs, but twice high Ythan that of Terfenol-D.\nWe consider that magnetization initially sit-\nuates around θ≃180◦and it fluctuates due to\nrandom thermal agitations. When a compressive\nstressis appliedto initiate switching, magnetization\nstarts with a certain ( θinitial,φinitial) picked from\nthe initial angle distributions. The voltage gener-\nated stress is assumed to be ramped up linearly andthe stress is kept constant until the magnetization\nreaches the plane defined by the in-plane and the\nout-of-plane hard axis (i.e., the x-yplane,θ= 90◦).\nNote that when magnetization reaches at θ= 90◦,\nthe azimuthal angle φmay not correspond to the\nmagnet’s plane ( y-zplane,φ=±90◦) and this has\nimportant consequences in switching the magneti-\nzation in the correct direction.53In any case, the\nx-yplane (θ= 90◦) is always reached sooner or\nlater with thermal fluctuations generating a distri-\nbution of time that magnetization takes to reach at\nθ= 90◦.\nAfter the magnetization reaches the x-yplane,\nthestress is rampeddown at the same rate at which\nit was ramped up, and reversed in magnitude to\naid switching. The magnetization dynamics ensures\nthatθcontinues to rotate from θ= 90◦towards\n0◦.53Whenθbecomes ≤5◦, switching is deemed\nto have completed. We perform a moderately large\nnumber (10,000) of simulations, with their corre-\nsponding( θinitial,φinitial) picked from the initial an-\ngle distributions, for each value of stress and ramp\nduration to generate the simulation results. Note\nthat thermal fluctuations affect the magnetization\ndynamics during the course of switching as well.\nFigure 9 shows the distributions of initial an-\nglesθinitialandφinitialin the presence of thermal\nfluctuations. No bias magnetic field is applied along\nthe out-of-plane direction (+ x-axis) here. Note that\nthe peak of the distribution of θinitialis exactly at\nθ= 180◦. This means that magnetization is most\nlikely to start from the easy axis θ= 180◦. If mag-\nnetization starts near from the easy axis, the torque\nacting on the magnetization would be vanishingly\nsmall. Therefore, only random thermal fluctuations\ncan help magnetization going away from the easy\naxis and then magnetization can start switching.\nWe have analyzed earlier that a bias field can shift\nthe peak of the distribution θinitialaway from the\neasy axis and facilitate switching, which we will\npresent next.\nFigure 10 shows the distributions of initial an-\nglesθinitialandφinitialin the presence of ther-\nmal fluctuations when a bias magnetic field ap-\nplied along the out-of-plane direction (+ x-axis). In\nFig. 10(a), note that the bias field has shifted the\npeak ofθinitialfrom the easy axis ( θ= 180◦) as we\ndesire. The φinitialdistribution in Fig. 10(b) spans\nmostlywithintheinterval [–90◦,+90◦]sincethebias\nmagnetic field is applied in the + x-direction. How-\never, the φinitialdistribution is asymmetric in the16Kuntal Roy\nFig. 9. Distribution of polar angle θinitialand azimuthal angle φinitialdue to thermal fluctuations at room temperature (300\nK). (a) Distribution of the polar angle θinitial. The mean of the distribution is ∼175◦, while the most likely value is 180◦. This\nis a nearly exponential distribution (Boltzmann-like). (b ) Distribution of the azimuthal angle φinitial. These are two Gaussian\ndistributions with peaks centered at 90◦and 270◦(or –90◦), which means that the most likely location of the magnetiza tion\nvector is in the plane of the nanomagnet. (Reprinted from Ref . 53.)\nFig. 10. Distribution of polar angle θinitialand azimuthal angle φinitialdue to thermal fluctuations at room temperature\n(300 K) when a magnetic field of flux density 40 mT is applied alo ng the out-of-plane hard axis (+ x-direction). (a) Distribu-\ntion of polar angle θinitialat room temperature (300 K). The mean of the distribution is 1 73.7◦, and the most likely value\nis 175.8◦. (b) Distribution of the azimuthal angle φinitialdue to thermal fluctuations at room temperature (300 K). Ther e\nare two distributions with peaks centered at ∼65◦and∼295◦. (Reprinted with permission from Ref. 42. Copyright 2012, A IP\nPublishing LLC.)\nquadrants (0◦,90◦) and (270◦,360◦), which can be\nexplained as follows. The magnetization is fluctu-\nating around θ≃180◦and the precessional mo-\ntion of the magnetization [ −|γ|//parenleftbig\n1+α2/parenrightbig\nM×H,\nwhereMis the magnetization and His the effec-\ntive field] due to the + x-directed magnetic field is\nsuch that the magnetization prefers the φ-quadrant\n(0◦,90◦) over the φ-quadrant (270◦,360◦). Hence,\nwhen the magnetization starts from θ≃180◦, the\ninitial azimuthal angle φinitialis more likely to bein the quadrant (0◦,90◦) than the other quadrant\n(270◦,360◦) in the + x-direction.\nFigure 11 plots the magnetization dynamics for\ndifferentvalues of φinitial(whilekeepingfixedvalues\nofθinitial, applied stress, and ramp rate) to signify\nthe role of out-of-plane excursion of magnetization.\nFor Figs. 11(a) and 11(b), the magnetization ini-\ntially lies on the plane of the magnet ( φinitial=\n±90◦) and the precessional motion of magnetiza-\ntion due to applied stress is in the + ˆeφdirection,Ultra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 17\nFig. 11. Temporal evolution of polar angle θand azimuthal angle φfor a fixed θinitial= 175◦and four different values of\nφinitial={90◦,270◦,0◦,180◦}. The applied stress is 15 MPa and the ramp duration is 60 ps. Th ermal fluctuations have been\nignored. (a) φinitial= 90◦. (b)φinitial= 270◦. (c)φinitial= 0◦. (d)φinitial= 180◦. Note that when θreaches 90◦or even\nearlier,φalways resides in a “good” quadrant [(90◦,180◦) or (270◦,360◦)], which makes the switching successful. No bias field\nin the out-of-plane direction is applied to generate these s imulation results. (Reprinted from Ref. 53.)\nwhich increases φwith time. So the magnetiza-\ntion starts out in the “good” quadrants (90◦,180◦)\nand (270◦,360◦) for Figs. 11(a) and 11(b), respec-\ntively. Therefore, both the motions of magnetiza-\ntion [the damped motion due to applied stress and\nthe motion due to out-of-plane excursion shown as\n−2αB(φ,t)sinθcosθ ˆeθand−|Bshape,φ(φ)|sinθˆeθ,\nrespectively in the Fig. 4(c)] are in the −ˆeθdirec-\ntion so that θdecreases with time and the mag-\nnetization rotates in the correct direction towards\nθ= 90◦. The increasing out-of-plane excursion of\nthe magnetization due to φwith time is eventu-\nally opposed by the damped motion due to out-of-\nplane excursion [depicted as −α|Bshape,φ(φ)|ˆeφin\nFig. 4(c)], which attempts to bring the magnetiza-\ntion back to the magnet’s plane ( φinitial=±90◦).\nThese two effects balance each other and φassumes\na stable value in the “good” quadrant which canbe clearly observed in the plots (the flat regions in\ntheφ-plots). When θreaches 90◦, the torque due\nto stress and shape anisotropy vanishes, however,\nφremains in the respective “good” quadrant for\nthe cases in Figs. 11(a) and 11(b). At this point,\nstress is reversed with the same ramp rate and the\ndamped motion due to stress and shape anisotropy\neventually becomes again in the −ˆeθdirection. In\nthis way, magnetization continues to rotate in the\nright direction towards θ= 0◦. Slightly past 0.4\nns, the precessional motion due to stress and shape\nanisotropy continues to rotate φand pushes it into\na neighboring “bad” quadrant, but eventually it es-\ncapes into the other “good” quadrant. This brief\nexcursion into a “bad” quadrant causes the ripple\ninθ-plots. However, magnetization ends up switch-\ning successfully.\nIn Figs. 11(c) and 11(d), the magnetization18Kuntal Roy\nFig. 12. Temporal evolution of the polar angle θand azimuthal angle φwhen magnetization fails to switch and backtracks\nto the initial state. Simulations are carried out at room-te mperature (300 K). (a) The applied stress is 10 MPa and the ram p\nduration is 60 ps. (b) The applied stress is 30 MPa and the ramp duration is 120 ps. The ringing in the φ-plots at the end is\njust due to thermal fluctuations that causes magnetization t o roam around θ= 180◦. No bias field in the out-of-plane direction\nis applied to generate these simulation results. (Reprinte d from Ref. 53.)\nFig. 13. Percentage of successful switching events at room- temperature (300 K) when the magnetostrictive nanomagnet i s\nsubjected to stress between 10 MPa and 30 MPa with ramp durati on (60 ps, 90 ps, and 120 ps) as a parameter. The critical\nstress at which switching becomes ∼100% successful increases with ramp duration. However, at h igh ramp duration (e.g., 120\nps), we may not achieve ∼100% switching probability for any values of stress and stre ss-dependence of the success probability\nbecomes non-monotonic. (Reprinted with permission from Re f. 42. Copyright 2012, AIP Publishing LLC.)\nis initially lifted far out of the magnet’s plane\n(φinitial= 0◦,180◦, respectively), where the huge\nout-of-planeshapeanisotropy energybarriercannot\nbe overcome by the stress anisotropy and B(φ,t)\nremains positive, i.e., |Bstress(t)|<|Bshape(φ)|.\nThis forces magnetization to precess in the clock-\nwise direction ( −ˆeφ) rather than in the anticlock-\nwise direction (+ ˆeφ). Therefore φdecreases with\ntime, which takes magnetization inside the neigh-\nboring “good” quadrant and eventually |Bstress(t)|\nbecomes greater than |Bshape(φ)|. Thenφassumes\na stable value due to the counteraction between\nthe damped motion and the motion due to out-of-\nplane excursion [depicted as −α|Bshape,φ(φ)|ˆeφand\n2B(φ,t)cosθˆeφin Fig. 4(c), respectively]. There-after, switching occurs similarly for the cases as in\nFigs. 11(a) and 11(b). Slightly past 0.3 ns, contin-\nuingφrotation pushes φinto a neighboring “bad”\nquadrant, but eventually it escapes into the other\n“good” quadrant. Once again, this brief excursion\ninto the “bad” quadrant causes the ripple in θ-plots\nand eventually successful switching takes place in\nthe end.\nSostrayingintoa“bad”quadrantforazimuthal\nangleφduringthe ramp-down phase does not mean\nat all that magnetization would fail to switch. Mag-\nnetization can rotate in the other “good” quadrant\nforφand complete the switching. Thus, there may\nbe ripples appearing in the magnetization dynam-\nics at the end of switching increasing the switchingUltra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 19\ntime, whichisbecauseof thetransition ofazimuthal\nangleφbetween two “good” quadrants through one\n“bad”quadrant. This happensparticularly near the\nend of switching, when the precessional motion is\nstrong.\nAn interesting comparison between the switch-\ning delays in the Figs. 11(a) or 11(b), and the\nswitching delays in the Figs. 11(c) or 11(d) reveals\nthat the switching delay decreases by 0.1 ns when\nmagnetization starts from out-of-plane ( φinitial=\n0◦,180◦) compared to when magnetization starts\nfrom in-plane ( φinitial= 90◦,270◦). This is a very\nconsequence of the reasoning that out-of-plane ex-\ncursion of magnetization in the “good” quadrants\naids magnetization to move faster in θ-space. When\nmagnetization starts out-of-plane, magnetization\nspends more time deep inside a “good” quadrant;\nhence, switching gets faster than that of the case\nwhen magnetization starts in-plane of the magnet.\nFigure 12 demonstrates a couple of examples\nwhen switching fails to occur, i.e., magnetization\nbacktracks to its original position. In Fig. 12(a),\nwhen the polar angle θreaches 90◦, the azimuthal\nangleφhas ventured into the “bad” quadrant\n(0◦,90◦) due to thermal fluctuations. Thus, switch-\ning eventually fails. In Fig. 12(b), when the polar\nangleθreaches 90◦, the azimuthal angle φis greater\nthan 90◦and in the “good” quadrant (90◦,180◦).\nHowever, after reaching around θ≃50◦,themagne-\ntization backtracks to the initial state and therefore\nswitchingfailstooccur.Thishappensbecauseofthe\nlong ramp duration which forces φto decrease over\ntime and eventually φenters into the neighboring\n“bad” quadrant (0◦,90◦). Also during the passage\nof long duration of ramp, thermal fluctuations have\nample opportunity to scuttle the switching. Such\nswitching failure is depicted intuitively in the Fig. 8\nearlier.\nFigure 13 plots the percentage successful\nswitching rates at room-temperature (300 K) when\nthe magnetostrictive nanomagnet is subjected to\nstress between 10 MPa and 30 MPa with ramp du-\nration (60 ps, 90 ps, and 120 ps) as a parameter.\nFor each value of stress and ramp duration, a mod-\nerately large number (10,000) of simulations were\nperformed to generate these results. Initial angle\ndistributions at 300 K for both θandφare taken\ninto account during simulations (see Fig. 10). The\nminimum stress needed to switch the magnetiza-\ntionwithoutconsidering thermal fluctuations is ∼5\nMPa, but at room temperature (300 K) this mini-mum stress is increased to ∼14 MPa for 60 ps ramp\nduration and to ∼17 MPa for 90 ps ramp duration.\nThe minimum stress of 5 MPa withoutconsidering\nthermal fluctuations is attributed to the removal\nof in-plane shape anisotropy energy barrier by the\nstress anisotropy, while at 300 K an increased mag-\nnitude of stress is required since magnetization is\nscuttled in “bad” quadrants due to thermal fluctu-\nations. When longer ramp duration, a higher stress\nis required to prevent magnetization traversing into\n“bad” quadrants. Therefore, it is beneficial to re-\nduce the ramp duration (i.e., having a faster ramp\nrate) to increase the success rate of switching at\na lower stress level. Simulation results show that\nwith 1 ps ramp duration, the critical stress can be\nreduced by ∼2 MPa compared to the case of 60 ps\nramp duration.\nFor 120 psrampduration, ∼100% success prob-\nability is unattainable for any value of stress since\nthermal agitations have higher latitude to scuttle\nthemagnetization into“bad”quadrantswhilestress\nis ramped down. At higher stresses accompanied\nby a long ramp duration, there occurs higher out-\nof-plane excursion pushing the magnetization into\n“bad”quadrants,whichfurtheraggravates theerror\nprobability. At very long ramp duration, the suc-\ncess (and error) probability becomes 50%, since the\nmagnetization would stay in-plane of the magnet\nand during the ramp-down phase, random thermal\nfluctuations may equally scuttle the magnetization\neitherinthe“good”quadrantsorinthe‘bad”quad-\nrants.\nFigure 14(a) shows the distribution of time\ntaken for magnetization polar angle θto reach 90◦\n(x-yplane). This wide distribution is caused by:\n(1) the initial angle distributions in Fig. 10, and\n(2) thermal fluctuations during the course of tran-\nsition from some θ=θinitialto 90◦. We do need to\ntackle such distribution by keeping the magnetiza-\ntion out-of-plane far enough so that magnetization\ndoes not collapse on magnet’s plane. We can use\na sensing element to detect when θreaches around\n90◦, so that we can ramp downthestress thereafter.\nThe sensing element can be implemented by mea-\nsuring the magnetoresistance in a magnetic tunnel\njunction (MTJ)62;63;64;65;66;67;68;53;34;35.\nWe need to get calibrated on the magnetoresis-\ntance of the MTJ when magnetization resides at\nθ= 90◦(x-yplane). Comparing this known sig-\nnal with the sensed signal of the MTJ, the stress\ncan be ramped down. Such comparator can be im-20Kuntal Roy\nFig. 14. Statistical distributions of different quantities when 15 MPa stress with 60 ps ramp duration is applied on the\nnanomagnet at room temperature (300 K). (a) Distribution of time taken for θto reach 90◦starting from ( θinitial,φinitial)\nwhere the latter are picked from the distributions in the pre sence of thermal fluctuations (see Fig. 10). (b) Distributio n of\nazimuthal angle φwhenθreaches 90◦. Note that this figure is similar to the Fig. S4 in Ref. 53, but a n out-of-plane bias field\nin the + x-direction is applied here following Ref. 42 to generate the se simulation results.\nplemented with these energy-efficient multiferroic\ndevices, i.e., energy-inefficient charge-based transis-\ntors do not need to be utilized.104;105;106The\nenergydissipationinthissensingelementisnotcon-\nsidered here. It should be mentioned that usually it\nrequires several peripheral circuitry in conjunction\nwiththe basicdevice itself, however, energy dissipa-\ntion considering in the other required circuitry does\nnot change the order of dissipation.107;108\nSometoleranceisnonetheless requiredsincethe\nsensing element cannot be perfect. Simulation re-\nsults show that the internal dynamics works cor-\nrectly as long as the stress is ramped down when\nθis in the interval [85◦, 110◦], i.e. it does not have\nto be exactly 90◦. This tolerance is due to the mo-\ntion arising from the out-of-plane excursion of mag-\nnetization in a “good” quadrant. If magnetization\nreaches at θ= 90◦(even past it towards θ= 0◦) and\nstress is not withdrawn soon enough, then magne-\ntization can fall on the magnet’s plane ( φ=±90◦,\npotential energy minima), upon which the success\nprobabilitywouldbe50%sincethermalfluctuations\ncan put magnetization in either direction of the po-\ntential landscape.\nFigure 14(b) shows the distribution of az-\nimuthal angle φwhenθreaches 90◦. Note that φ\nalways resides in the “good” quadrant [(90◦,180◦)\nor (270◦,360◦)] and has a fairly narrow distribu-\ntion. As depicted in the Fig. 13, a high stress and\nfastramprate arerequiredtoensurethat φis inthe\n“good” quadrants, which is conducive to successful\nswitching.Figure 15 shows the delay and energy distribu-\ntions for 15 MPa stress and 60 ps ramp duration\nin the presence of room-temperature (300 K) ther-\nmal fluctuations. The high-delay tail in Fig. 15(a)\nis particularly associated with the switching trajec-\ntories that start very close to θ= 180◦. In such tra-\njectories, the torque acting on the magnetization is\nvanishingly small, which makes the switching slug-\ngish at the beginning. During this time, switching\nmay also becomes susceptible to backtracking due\nto random thermal fluctuations, which may increas\nthe delay further. Nonetheless, out of 10,000 sim-\nulations of switching trajectories peformed, there\nwas not a single one in Fig. 15(a) where the de-\nlay exceeded 1 ns meaning that the probability of\nsuch happening is less than 0.01%. Since the en-\nergy dissipation is the product of the power dissi-\npation and the switching delay, similar behavior is\nfound in Fig. 15(b). Note that the variation in the\ndistribution of energy dissipation happens due to\nthe internal energy dissipation (caused by Gilbert\ndamping) in the presence of random thermal fluc-\ntuations since all the trajectories correspond to the\nsame 15 MPa stress and 60 ps ramp duration.\nDiscussion on magnetization switching. We will\nnow analyze the magnetization switching between\nthe 180◦symmetry equivalent states based on the\nsimulation results as depicted in the Fig. 16. The\nusualperceptionisthatstresscanrotatemagnetiza-\ntion of a magnetostrictive nanomagnet only by 90◦\nfrom±z-axis to±y-axis (see Fig. 16). However, ifUltra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 21\nFig. 15. Delay and energy distributions for 15 MPa applied st ress and 60 ps ramp duration at room temperature (300 K). (a)\nDistribution of the switching delay. The mean and standard d eviation of the distribution are 0.44 ns and 83 ps, respectiv ely.\n(b) Distribution of energy dissipation. The mean and standa rd deviation of the distribution are 184 kTand 15.5 kTat room\ntemperature, respectively. (Reprinted with permission fr om Ref. 42. Copyright 2012, AIP Publishing LLC.)\nwe determine the expression of torque due to stress\nfrom the stress anisotropy energy [see Eq. (5)] as\nTE,stress=−ˆ er×∇Estress=−(3/2)λsσsin(2θ)ˆ eφ,\n(32)\nwe see that the torque due to stress acts along the\nout of magnet’s plane ( ˆ eφdirection) and therefore\nmagnetization lifts out-of-plane (although the out-\nof-plane demagnetization field due to small thick-\nness of the nanomagnet attempts to keep the mag-\nnetization in-plane). This out-of-plane excursion of\nmagnetization generates an intrinsic asymmetry,\nwhich can completely switch the magnetization by\n180◦(Ref. 53). Full 180◦switching is desirable since\nit facilitates having the full tunneling magnetore-\nsistance (TMR) while electrically reading the mag-\nnetization state using a magnetic tunnel junction\n(MTJ).62;63;64;65;66;67;68\nSuch full 180◦switching of magnetiza-\ntion (i.e., memory operation) in piezoelectric-\nmagnetostrictive heterostructures was first ad-\ndressed by Roy in Ref. 33. Note that Ref. 109\nshows the Bennett clocking operation57forlogic\ndesign, which is not memory operation since us-\ning a neighboring nanomagnet to switch another\nnanomagnet is tantamount to using a magnetic\nfield,110the direction of which has to be reversed\nfor switching the memory bit in either direction.\nAlso, Ref. 109 performed an incorrect analysis by\nassuming that magnetization always resides in-\nplane, which leads to a very high switching delay\nand high energy-delay product compared to the\ntraditional transistors.111This was addressed andexplained by Roy.7;112;113\nFigure 16 illustrates the magnetization switch-\ning alongwith the simulation results of different\ndistributions in the presence of room-temperature\nthermal fluctuations. When magnetization starts\nswitching from θ= 180◦, the initial orientation\nof magnetization is a distribution due to random\nthermal fluctuations. When magnetization reaches\nθ= 90◦, the time taken to reach there is also a dis-\ntribution due to thermal fluctuations. After mag-\nnetization reaches aroundθ= 90◦, stress needs to\nbe released to switch the magnetization. However,\nthere can be two states magnetization can end up:\n(1)θ= 180◦causing a switching failure, and (2)\nθ= 0◦leading to successful switching. As shown in\nthe Fig. 16, magnetization’s excursion out of mag-\nnet’s plane in a particular direction would lead to\nsuccessful switching. As presented in the Fig. 13, a\nsufficiently fast ramp rate of stress accompanied by\na sufficiently high stress is conducive to successful\nswitching, i.e., the out-of-plane excursion of magne-\ntization in the specified directions as shown in the\nFig. 16 would be maintained. Therefore, the out-\nof-plane excursion of magnetization generated an\nequivalent intrinsic asymmetry for magnetization\nswitching in the correct direction. If magnetization\nstarts switching from θ= 0◦instead of θ= 180◦,\nthe direction of out-of-plane excursions of magne-\ntization as shown in the Fig. 16 would be exactly\nopposite.\nNote that the aforesaid magnetization switch-\ning as presented in the Fig. 16 requires to read\nthe magnetization state using spin valve/MTJ to22Kuntal Roy\nFig. 16. Illustration of magnetization switching between t wo anti-parallel states ( ±z-axis). Stress rotates magnetization out\nof magnet’s plane and when magnetization reaches the hard pl ane (θ= 90◦), intrinsic dynamics dictates that magnetization\nlifts out-of-plane in a certain direction so that a complete 180◦switching of magnetization is possible.53,42While switch-\ning along the – y-axis rather than + y-axis (shown by arrows), the directions of the out-of-plane excursions would be exactly\nopposite.53Three distributions are shown for θ= 180◦to 0◦switching: the one at θ= 180◦depicts the initial distribution\nof magnetization when no stress is active, the other two dist ributions around θ= 90◦andθ= 0◦correspond to 60 ps ramp\nperiod and 15 MPa stress,42which are Figs. 10(a), 14(a), and 15(a), respectively.\nsense when magnetization reaches aroundθ= 90◦\n(since room temperature thermal fluctuations make\nthe traversal time a wide distribution as shown) so\nthat stress can be brought down around that time.\nThis has been discussed in the context of explain-\ning the results in the Fig. 14. The sensing element\nreads the magnetization state and a comparator\ncan compare the read signal with a pre-calibrated\nvalue for the MTJ resistance when magnetization’s\norientation is around θ= 90◦(as explained in\nRef. 53). It is important to note that there is tol-\nerance around θ= 90◦, i.e., stress does not need\nto be withdrawn exactlyatθ= 90◦since the sens-\ning procedure cannot be perfect.53It is shown that\nthe internal magnetization dynamics provides such\ntolerance.53Any additional element for compari-\nson can be built using these energy-efficient multi-\nferroic devices in general.104;105;106Researchers\nare trying to replace the traditional switchbased on\ncharge-based transistors by a new possible “ultra-\nlow-energy” switch(e.g., usingmultiferroiccompos-\nites). Therefore, any circuitry can be built with the\nenergy-efficient switch itself ratherthan theconven-\ntional transistors.104;105;106Usually, it requires\nseveral peripheral circuitry in conjunction with the\nbasic switch in a system.107;108While researchers\nreport on the performance metrics of the basic\nswitch itself, the total energy dissipation consid-ering the other required circuitry does not change\nthe order of energy dissipation, utilizing the respec-\ntivedevices.107;108Thereforeenergyefficienttech-\nnologies can be envisaged using such magnetization\nswitching mechanism7;33;93;53;42and comput-\ning methodologies7;34;35based on such switching\nmechanism.\nAlso such aforesaid switching is confirmed\nby others,114;115;116;117;118;119but they do\nnot perform a detailed analysis using stochastic\nLandau-Lifshitz-Gilbert equation of magnetization\ndynamics and therefore do not conceive or mention\nthe requirement of sensing circuitry due to the wide\ndistribution of traversal times in the presence of\nroomtemperaturethermalfluctuationsasexplained\nearlier. While it is desirable that an additional sens-\ning procedure can be avoided, the methodology53\ntackles the variation in the traversal time of magne-\ntization particularly the crucial initial distribution\nof magnetization effectively for successful switching\nand it also aids in decreasing the switching delay\ngiven a certain error probability. An out-of-plane\nfield (see Ref. 42) can help breaking the symmetry\nwhile switching the magnetization but the varia-\ntion in traversal times towards the hard axis due\nto the initial Boltzmann distribution of magneti-\nzation is large enough to cause switching failures\n(note that it requires very low switching error prob-Ultra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 23\nability e.g., <10−4to be technologically viable).\nWhile further research may lower the error proba-\nbility with a symmetry breaking field or possibly by\nother means, the critical understandings behindthe\nswitching dynamics of magnetization as explained\nby Roy in Ref. 53 is an important step forward to\nthe technological applications for building memory\nand logic using multiferroics.\nIn Ref. 35, information propagation in a\nchain of nanomagnets using Bennett clocking\nmechanism57in the presence of room-temperature\nthermal fluctuations is analyzed. It is found that\nthe inherent magnetization dynamics caused by the\nout-of-plane excursion of magnetization can lead to\nswitching failure even if there is dipole coupling be-\ntween neighboring nanomagnets that can introduce\nasymmetryinswitching. Butsuchasymmetryisnot\nsufficient and therefore the asymmetry caused by\nthe out-of-plane excursion53can beutilized for suc-\ncessfulswitchingof magnetization. Onceagain, here\na sensing element is required to detect when mag-\nnetization reaches aroundθ= 90◦and multiferroic\ndevices can be utilized to build such functionality,\nwhich would not change the order of energy dissi-\npation.\nBut, one contradictory viewpoint exists that\nany other circuitry other than the switching de-\nvice itself needs to be built with traditional\ntransistors,120which however undermines the re-\nsearch behind finding a switch replacing the tran-\nsistor itself41;111and therefore incorrect lacking\nthe understandings over the development of tran-\nsistor based circuitry.107;108Such viewpoint does\nnot exist anywhere else in literature. A few con-\ntradicting facts vis-a-vis the comment made by\nBandyopadhyay and Atulasimha (referred as BA\nonwards) in Refs. 120, 121, 122 are as follows:111\n(1) BA are coauthors of Roy in Refs. 33, 93, 53,\n54, 42, where energy-efficiency is claimed, requir-\ning the sensing procedure too. In particular, en-\nergy efficiency is claimed in the presence of ther-\nmal fluctuations in Ref. 42, which requires the\nsensing element, explained in details by Roy in\nRef. 53. Also, there is a patent123following the\nresearch conceived by Roy33;93;54;42;53;7;112\nthat claims energy-efficiency, requiring the sensing\nelement therein as well. Therefore, the comment\nmade by BA in Ref. 120 has no technical basis. (2)\nNote that Ref. 122, coauthored by BA, uses pre-\ncisely shaped pulses. Such pulses need to be gen-\nerated too using some circuitry. According to BA,transistors need to be utilized to build such cir-\ncuitry and the system would dissipate too much\nenergy, invalidating the claim of energy efficiency\nin Ref. 122. Note that one additional hardware can-\nnot be shared between many devices distributed on\na chip due to interconnect delay and loading ef-\nfect. Also, note that pulse shaping is an ineffective\ncountermeasure since it is not helping much in re-\nducing the error probability, therefore building and\nusing such circuitry do not make sense either. (3)\nRef. 124, in which BA are coauthors, proposed a\n“toggle” switch (as stated that “a write cycle must\nbe preceded by a read cycle to determine the stored\nbit”), which would require a similar use of spin-\nvalve or MTJ for reading the known bit, storing it,\nand then using it for comparison . According to BA,\nsuchadditional circuitry needs to be constructed\nwith energy-inefficient transistors, invalidating the\nclaim of energy efficiency in Ref. 124.\nAccording to the unsubstantiated and\nincorrect viewpoint as stated by BA,120\nthe switching methodology confirmed by\nothers114;115;116;117;118;119will require the\nsensing circuitry too (as already explained by Roy\nin Ref. 53) and hence will be energy-inefficient.\nHowever, as explained earlier, any circuitry can be\nbuilt with the energy-efficient switch itself rather\nthan the conventional transistors104;105;106and\nfurthermore BA contradict themselves as explained\nabove. Therefore, there is no technical reason-\ning behind such point raised by BA in Ref. 120.\nThe other unsubstantiated points raised by BA in\nRef. 120 on Ref. 34 are also technically incorrect\nas explained.111Another paper125also incorrectly\nstatedthat stresshasto“todrivethemagnetization\nswitch out-of-plane at first”, however, as pointed\nout by the Eq. (32) that the out-of-plane excur-\nsion is inherent to the magnetization dynamics.53\nSuch out-of-plane motion also increases the switch-\ning speed to the order of GHz.54;53Furthermore,\nRef. 125 did not study the consequence of room-\ntemperature thermal fluctuations, which is critical\nto the magnetization dynamics and error probabil-\nity of switching.\nOne of the critical points to conceive while\nproposing energy-efficient systems is that it must\nbearea-efficient as well to compete with the tra-\nditional transistors e.g., our laptops cannot be 10\ntimes bigger. However, such area-inefficient devices\nhave been proposed in Ref. 126, which claims a\nsuperior design of magnetoelastic memory, com-24Kuntal Roy\npared to an earlier idea.127;43;128;129Compar-\natively, the lateral area chosen by Ref. 126 is an\norder higher than that of chosen by Ref. 127. Also,\nRef. 126 chooses to use two pairs of lateralelec-\ntrodes, used to apply stress at angles, understand-\ning behind which is otherwise known, e.g., Ref. 130,\nwhich has however issues with room-temperature\nthermal fluctuations. The additional lateralpads\ncannot be dispensed and particularly they need to\nbe large for application of stress, thereby consum-\ningadditional large area foreachnanomagnet apart\nfrom the area consumption by the nanomagnet it-\nself. The area consumed by the proposal in Ref. 126\n(also Refs. 124, 121) is so high that the devices be-\ncome of micro-scale size, therefore, the proposals\nare untenable for building nanoelectronics.131;41\nThere is an ongoing drive to reduce the area-\nconsumption,132;87but the proposals in Refs. 126,\n124, 121 do not bode at all on such crucial front.\nThere is also similar issue regarding area ineffi-\nciencyintheproposalbyRef.125duetoconsuming\nlarge area, which is a crucial issue and therefore un-\ntenable for meeting practical standard requirement\nof area density 1 Tb/in2.\nFurthermore, Ref. 126 performs an incorrect\nanalysis while comparing error probability and\nswitching delay with that of the Refs. 127, 43, 128,\n129, and underestimates the MTJ resistance ratio\nwith an incorrect statement “The maximum value\nof this ratio (assuming η1=η2= 1) is 2:1” (for\ndetails see Ref. 111). Also, there is a very basic is-\nsue behind the multi-step switching methodology\nproposed in Refs. 126, 124, which increases the\nswitching delay exponentially while meeting a low-\nered error probability.111It should be noted too\nthat Refs. 126, 124 have assumed instantaneous\nramp for stress, which is unreasonable and crucial,\nsince ramp that is not fast enough causes switching\nfailures.53;42\nAlso, switching delay is a major perfor-\nmance metric while proposing energy-efficient sys-\ntems, since devices with lower switching speed\nwould take more time given a computation task\nto be performed, therefore making the energy-\ndelay product a tenable performance metric. If\nwe calculate the switching delay of magnetiza-\ntion according to Ref. 109, it will come out\n∼1000 ns, which is clearly untenable for build-\ning nanoelectronics,131;41While comparing with\nspin-transfer-torque (STT) switching mechanism,\nif one compares an usual performance metric i.e.,switching delay-energy, clearly Ref. 109 performs\ninferior to STT switching. Also consider the issue\nthat 1 hour of execution would take 100 hours or\nmore using the operation presented in Ref. 109,\nwhich however performed an incorrect analysis.111\nIf charge based transistors were to operate slow,\nthe energy dissipation would not have been an\nissue.107;131;41\nIn Ref. 122, the authors proposed to reduce\nerror probability of switching in a system of two\ndipole-coupled magnetostrictive nanomagnets in\nstrain-mediated multiferroic heterostructures using\nvoltage(stress)pulseshaping.Theauthorsconclude\nthat high success probability of switching cannot\nbe achieved at high switching speed ( ∼1 ns), and\ntherefore their proposed system is only applicable\nfor niche applications (but therein too it is infe-\nrior to the traditional transistors). However, such\nconclusion lacked the critical understandings be-\nhindsuch high error probability for general-purpose\nlogic applications. Fortunately, such analysis and a\npossible solution along the line of the analysis is\npresent in literature35using Bennett clocking57for\nlogic design, on which Ref. 122 (and arXiv version\nRef. 133) has made some incorrect statements. Ear-\nlier, a subset of the authors of Ref. 122 also pub-\nlished a paper134on afourmagnet system using\ntheBennett clocking mechanism, wherethe authors\nalso came up with a similar conclusion of high error\nprobabilityandthedemiseofmultiferroicnanomag-\nnetic logic, however, without relevant analysis sim-\nilar to the case as in the Ref. 122. There are several\ntechnical issues in Ref. 122 as well.111Due to the\nincorrect analysis performed in the Refs. 134, 122,\nin Ref. 135, the authors cast doubt on that front,\nwhich is incorrect too.\nRather than using Bennett clocking\nmechanism57for logic design, Ref. 34 proposed\nuniversal logic gates (NAND/NOR) utilizing a sin-\ngledevice with multiple contacts on the device to\nadd up the strains generated in the piezoelectric\nlayer,136and aSetinput to preset the non-volatile\nmagnetization state and facilitate concatenation.\nRef. 34 uses the switching methodology presented\nin Ref. 53 and it is clearly mentioned while refer-\nring the Ref. 53 that “Computing methodologies\nutilizing such 180◦switching mechanism between\nthe two stable states of a shape-anisotropic mag-\nnetostrictive nanomagnet have not been proposed\nso far”. However, BA in Ref. 120 casts doubt on\nthat, which has no technical basis as explainedUltra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 25\nFig. 17. (a) Potential landscapes of the nanomagnet with no s tress and 6.1 MPa stress, as in Ref. 138. (b) Three-dimension al\npotential landscape showing the deflection of magnetizatio n out of magnet’s plane ( y-zplane,φ=±90◦). (c) Room-\ntemperature (300 K) thermal distribution of θwhen no stress is active. (d) Room-temperature (300 K) therm al distribution\nofθwhen 6.1 MPa stress is applied.\nearlier. Moreover, BA in Ref. 121 follow the same\nidea presented in Ref. 34 by Roy and uses a ∼90◦\n(precisely 86 .4◦) switching mechanism following an\nearlier idea by others.127;43;128;129Therefore,\nmagnetization does not switch a complete 180◦in\nRef. 121 and this leads to a low tunneling magne-\ntoresistance (TMR) while reading the magnetiza-\ntion state and it has serious consequence on read\nerror probability. Ref. 121 states that it requires a\nsensing circuitry for operation of the proposal in\nthe Ref. 34 and indicates as if it is an issue with\nthe computing proposal in Ref. 34. First, Ref. 121\ndoes not point out at all that the sensing element\nis required for complete 180◦switching only and it\ndoes not require so if 90◦switching mechanism is\nused.\nRef. 121 just uses resistors and potential di-\nvider(seeFig. 1in Ref.121)toaccommodate multi-\ninputs (rather than using intrinsic strain-addition\nproperty of piezoelectrics, while following the cen-\ntral idea in Ref. 34) and the authors want to use\ntraditional transistors for any other circuit ele-\nment, which makes the proposal in Ref. 121 energy-\ninefficient.111(Note that a circuit element cannot\nbe shared between many devices distributed on a\nchip due to interconnect delay and loading effect.)\nIt is stated in Ref. 121 (in the supplementary mate-\nrial) that “The dissipation in the resistance Rcan\nbe negligible as we can make this resistance arbi-\ntrarilyhigh.”Thisisincorrect(anystandardelectri-\ncal engineeringundergraduatetextbook can becon-\nsulted) since RCdelay will be too high. Therefore,\nthe design proposed in Ref. 121 is untenable. Thereare several technical issues in Ref. 121 as well.111\nIt should be noted that the switching as de-\npicted in the Fig. 16 just toggles the magnetization\nstate uponapplication andremoval of stress. There-\nfore, it requires to read the magnetization state be-\nfore writing a bit. If we want to switch the mag-\nnetization along a specified direction, it is possible\nto use spin-transfer-torque with spin-polarized cur-\nrent which was first proposed in Ref. 137 by Roy,\nhowever, it incurs much higher energy dissipation\ndue to utilizing current-induced spin-torque mech-\nanism. Unless new strategies to lower the energy\ndissipation in spin-transfer-torque switching can be\nachieved, a target of 1 aJ energy dissipation per\nswitching a bit of information cannot be reached.\nRef. 138 follows the idea presented in the\nRef. 137 with the addition of surface acoustic\nwave (SAW) and also follows the formulation from\nRefs. 42, 31. The basic idea was to switch the mag-\nnetization by STT along the desired direction when\nmagnetization comes at the hard-plane ( θ= 90◦)\nupon application of stress.137However, thermal\nfluctuations cause a wide distribution for the time\ntaken by magnetization to reach at θ= 90◦as ex-\nplained in Ref. 53. This is why STT current needs\nbe kept active for almost half of the duration of\nswitching.138Figure 3 in Ref. 138 shows that the\nenergy dissipationwith astressof 6.1 MPa is5 ×109\nkT at room-temperature (300 K), which is ∼20 pJ.\nThis is 4-5 orders of magnitude higher than that\nof traditional transistors and therefore untenable\nfor building nanoelectronics.131;41Also, Fig. 3 in26Kuntal Roy\nRef.138showstheenergydissipationwhennostress\nis present (only switching in STTRAM), which is\n∼40×109kT = 160 pJ. Note that Ref. 138 has con-\nsidered the material Terfenol-D as free layer in STT\nswitching to calculate the energy dissipation. How-\never,thematerial thatiscommonlyusedforthefree\nlayer is CoFeB,139;132;67;140which has Gilbert\ndamping parameter αan order lower139;132than\nTerfenol-D( αof Terfenol-D is0.1 usedby Ref. 138).\nThe critical current of switching is proportional to\nthe damping parameter139;132and the energy dis-\nsipation is proportional to the square of the switch-\ning current. Therefore, Ref. 138 calculated incor-\nrectlythe switching current an order higher (23\nmA)andenergydissipationabouttwoordershigher\n(∼160 pJ) for STT switching (Ref. 139 correctly de-\nterminedswitchingcurrent ∼1mAandenergydissi-\npation∼1pJexperimentally). Clearly, thecompari-\nson with STTswitching performed in Ref. 138 is in-\ncorrectand actually the energy dissipation in STT\nswitching is 12.5 times lower (in stead of 8 times\nhigher as incorrectly claimed by Ref. 138) than the\nhybrid scheme proposed in Ref. 138.\nAlso, the energy dissipation calculated in\nRef. 138 due to SAW is missing a crucial point that\nwill make the energy dissipation exceedingly high.\nRef. 138 says that “SAW is globaland affects ev-\nery memory cell.” This is represented in Fig. 17(a)\nthat with the application of 6.1 MPa stress the bar-\nrier height decreases but it is not sufficient enough\nto make the potential landscape monostable and\ncause switching. However, when stress is applied\nthe magnetization doesrotate. The distribution in\nFig. 17(d) upon application of stress is wider than\nthe distribution in Fig. 17(c) when no stress is ac-\ntive. Therefore, the energy dissipation upon appli-\ncation of stress and removal of stress must be cal-\nculated, which is missedby Ref. 138. This energy\ndissipation turns out to be ∼40 kT for one cell.\nConsidering just 1 MB memory, clearly such strat-\negy would dissipate an energy which is quite worse\nthan that of transistors.131;41\nRef. 141 embraces the same concept of using\nSTT (as proposed in Ref. 137) and does not con-\nsider the sensing procedure (explained in Ref. 53),\nwhich is required for successful switching in strain-\nmediated multiferroic composites in the presence\nof thermal fluctuations without STT, as explained\nearlier. This is why the Ref. 141 incorrectly terms\nRefs. 93, 53 as stochastically unstable. Since, using\nSTT leads to energy-inefficiency137;138;141, sym-metry breaking by other means rather than using\nSTT is desirable.\nIn Ref. 87, it is shown that interface and ex-\nchangecouplinginmultiferroicheterostructurescan\nfacilitate switching of magnetization in a particu-\nlar direction without using spin-torque mechanism,\nwhile incurring miniscule energy dissipation and\nthereis nonecessity ofasensingelement asrequired\nfor strain-mediated multiferroic composites.53The\nswitching methodology presented in the Ref. 87 can\nbe harnessed for logic design and computing pur-\nposes too.34;35In the next subsection, we will\npresent the simulation results for magnetization\nswitching in such interface and exchange coupled\nmultiferroic heterostructures.\nFigure 18(a) shows the thermally averaged\nswitching delay versus stress with different ramp\nduration (60 ps, 90 ps, 120 ps) as a parameter.\nOnly the successful switching events are consid-\nered here since the switching delay metric does not\nmake sense for an unsuccessful event. For a certain\nstress, if the ramp duration is decreased (i.e., the\nramp rate is increased), the stress reaches its maxi-\nmum value quicker and switches the magnetization\nfaster, thereby decreasing the switching delay. For\nramp durations of 60 ps and 90 ps, the switching\ndelay decreases with increasing stress since an in-\ncreasingstressanisotropy rotates themagnetization\nfaster. However, for 120 ps ramp duration, the de-\npendence of switching delay decreases with stress is\nnon-monotonic, due to the exactly the same reason-\ning that caused the non-monotonicity in Fig. 13. A\nhigh stress accompanied by a long ramp duration\nis harmful since during the ramp-down phase it ro-\ntates the magnetization in “bad” quadrants leading\ntoincreasedswitchingdelayandevenswitchingfail-\nures.\nFigure 18(b) shows the standard deviation of\nswitching delay distribution with stress for 60 ps\nramp duration. At higher values of stress, the in-\ncreased stress anisotropy energy makes the poten-\ntiallandscapemoredeepandresistthethermalfluc-\ntuations more effectively. Therefore, the spread in\nswitching delay diminishes with a higher stress.\nFigure 19(a) shows the thermal mean of the to-\ntal energy dissipated to switch the magnetization\nas a function of stress for different ramp durations.\nFor a certain ramp duration, the average power dis-\nsipation ( Etotal/τ) increases with stress and for a\ncertain stress it decreases with increasing ramp du-Ultra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 27\nFig. 18. (a) The thermal mean of the switching delay (at 300 K) versus stress (10–30 MPa) for different ramp durations (60\nps, 90 ps, 120 ps). Switching may fail at low stress levels and also at high stress levels for long ramp durations. Failed at tempts\nare excluded when computing the mean. (b) The standard devia tions in switching delay versus stress (10–30 MPa) for 60\nps ramp duration at 300 K. We consider only the successful swi tching events in determining the standard deviations. The\nstandard deviations in switching delay for other ramp durat ions are of similar magnitudes and show similar trends. (Rep rinted\nwith permission from Ref. 42. Copyright 2012, AIP Publishin g LLC.)\nFig. 19. (a) Thermal mean of the total energy dissipation ver sus stress (10–30 MPa) for different ramp durations (60 ps, 90\nps, 120 ps). Once again, failed switching attempts are exclu ded when computing the mean. (b) The ‘ CV2’ energy dissipation\nin the external circuit as a function of stress for different r amp durations. The dependence on voltage is not exactly quad ratic\nsince the voltage is not applied abruptly, but instead rampe d up gradually and linearly in time. (Reprinted with permiss ion\nfrom Ref. 42. Copyright 2012, AIP Publishing LLC.)\nration. A higher stress leads to more ‘ CV2’ dissipa-\ntion (see Fig. 19(b)) and also more internal dissipa-\ntion because an increasing stress anisotropy results\nin a higher torque. A slower switching (i.e. more\nadiabatic) decreases the power dissipation. But the\nswitching delay curves in the Fig. 18(a) show the\nopposite trend. At a slower ramp rate (higher ramp\nduration), the average power dissipation Etotal/τ\nis always smaller than that of a higher ramp rate.\nHowever, the switching delay does not decrease asfast as with higher values of stress (in fact switch-\ning delay may increase for higher ramp duration,\nsee Fig. 18(a)), which is why the energy dissipation\ncurves in Fig. 19(a) exhibit the cross-overs.\nFigure 19(b) plots the ‘ CV2’ energy dissipation\ndue to the application of voltage-induced stress for\ndifferent ramp durations. Stress is proportional to\nthe applied voltage V, and therefore the ‘ CV2’ en-\nergy dissipation increases with stress for a certain\nramp duration. This ‘ CV2’ energy dissipation how-28Kuntal Roy\nFig. 20. Thermal mean of the switching delay versus thermal m ean of the total energy dissipation for different stress leve ls\n(10–30 MPa) and different ramp durations (60 ps, 90 ps, 120 ps) . Once again, failed switching attempts are excluded when\ncomputing the mean.\never is a small fraction of the total energy dissi-\npation (<15%) particularly because a miniscule\nvoltage-generated stress is required to switch the\nmagnetization of a magnetostrictive nanomagnet\ninapiezoelectric-magnetostrictive multiferroic. The\n‘CV2’dissipationdecreaseswithanincreasingramp\nduration (i.e., slower ramp rate) for a certain stress\nsince the switching becomes more adiabatic . This\ncomponent of the energy dissipation would come as\nseveral orders of magnitude higher if we switch the\nmagnetization with an external magnetic field142\nor spin-transfer-torque.30\nNote that we would require an adiabatic circuit\nto take advantage of the energy saving due to using\nan adiabatic pulse. In any case, usinga piezoelectric\nlayer of higher piezoelectric coefficients e.g., utiliz-\ning lead magnesium niobate-lead titanate (PMN-\nPT) instead of lead-zirconate-titanate (PZT), the\nenergy dissipation can be reduced further. Also,\nPMN-PT can generate anisotropic strain, which\nallows us to work with lower voltage for a re-\nquired strain, thereby reducing the energy dissipa-\ntion even further. PMN-PT layer has a dielectric\nconstant of 1000, d31=–3000 pm/V, and d32=1000\npm/V(Ref.48).Withthepiezoelectriclayer’sthick-\nnesstpiezo=24 nm (Ref. 42), V= 1.9 mVs (2.9\nmVs) of voltages would generate 20 MPa (30 MPa)\ncompressive stress [ σ=Y deff(V/tpiezo), where\ndeff= (d31−d32)/(1 +ν)] in the magnetostric-\ntive Terfenol-D layer, which has Y= 80 GPa,42\nand Poisson’s ratio ν= 0.3 (Ref. 143). Modeling\nthe piezoelectric layer as a parallel plate capaci-\ntor (∼100 nm lateral dimensions), the capacitance\nC=2.6 fF and thus CV2energy dissipation turnsout to be <0.1 aJ. This is without considering\nany energy saving with the help of an adiabatic cir-\ncuit. Such miniscule energy dissipation is the basis\nof ultra-low-energy computing using these multifer-\nroic devices.33;7;34;35\nFigure 20 plots the switching delay versus en-\nergy dissipation for different ramp durations. This\nplot can be extracted from the Fig. 18(a) (stress\nversus switching delay) and Fig. 19(a) (stress ver-\nsus energy dissipation). This plot signifies that as\nthe switching delay increases, energy dissipation\ndeceases. This points out the usual delay-energy\ntrade-off. For 120 ps ramp duration, there is a op-\nposite trend at higher stress values, the reasoning\nbehind which has been already described while ex-\nplaining the results in the Fig. 13.\n3.2.Interface and exchange coupled\nmultiferroic heterostructures\nHere we present the simulation results for\ninterface and exchange coupled multiferroic\nheterostructures.87Fig. 3 shows that the nano-\nmagnets ( M1andM2layers) and the ferroelec-\ntricP-layer are made of Fe and PbTiO 3, respec-\ntively, while the spacer layer is made of Auand\nthe thicknesses of the trilayer M1/spacer/M2are\n1/4/1 monolayers.71;79;144;145The Fe layer has\na unit cell length of 0.287 nm and it possess the\nfollowing material parameters: saturation magneti-\nzation (Ms) = 1e6 A/m, and damping parameter\n(α) = 0.01.146;147;148The elliptical lateral cross-\nsection (y-zplane, Fig. 3) of the vertical stack has\na dimension of 15 nm×7nm. TheP-layer has a\nunit cell length of 0.388 nm and it has 5 layersUltra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 29\nin the vertical direction ( x-direction, Fig. 3)71.\nThe energy difference between the P-alignment and\nAP-alignment is 10 meV/atom,71and the absolute\nvalue of energy is calculated to be about 10 eV or\n385 kT at room-temperature. This huge interface\ncoupling makes potential landscape of M1monos-\ntableatθ= 180◦or atθ= 0◦depending on the\nP-alignment or the AP-alignment in the trilayer,\nrespectively. Due to the monostable energy land-\nscape and huge energy barrier, spontaneous switch-\ning of magnetization between θ= 180◦andθ= 0◦\ncannot take place. The interface coupling energy is\na few orders of magnitude higher than the shape\nanisotropicenergy and henceconsideration of shape\nanisotropy does not make any significant difference,\nhowever, it is included during the simulations.\nIf we consider the P-layer as a parallel plate\ncapacitor and use a relative dielectric constant of\n1000,149the capacitance Cof the layer becomes\n∼0.4fF.Ifthe P-layer isaccessed witha10 µmlong\nsilver wire with resistivity ∼2.6µΩ-cm,150the re-\nsistance Rcan be calculated as ∼3kΩ. Therefore,\ntheRCtime constant turns out to be the order\nof 1 ps. We assume that the ferroelectric PbTiO 3\nhas a coercive voltage of 20 MV/m151and hence a\nvoltageV≃40 mv is required to switch its polar-\nization. Note that polarization switching is possible\nin less than 100 ps152and a voltage ramp with pe-\nriodT= 100 ps or more is considered to enforce\nthe quasistatic (adiabatic) assumption ( T≫RC).\nWithout any adiabatic assumption, the metric CV2\ncan be calculated as 0 .5 aJ and hence the energy\ndissipation due to the application of the voltage is\nminiscule. With 100 ps ramp period, the “ CV2”\ndissipation is determined as a negligible value of\n0.01 aJ.93;42Note that we do not calculate any\nstandbyleakage current through the thin ferroelec-\ntric since the device operation is non-volatile, i.e.,\nit is possible to turn off the voltage without loosing\nthe information in long term. However, during the\nactive mode of operation, the leakage needs to be\nconsidered, however, the tunneling current is small\n(<1 nA, Ref. 77) leading to negligible energy dissi-\npation. Another issue that needs to be considered is\nferroelectric fatigue, which may make the coercive\nfield higher over time,153;154i.e., it would require\na higher voltage to switch the polarization. How-\never, the energy dissipation due to applied voltage\nis miniscule, and therefore it does not appear to be\na bottleneck provided the polarization switches and\nthe interface coupling between the P-layer and thetrilayer persists. In any case, further progress in the\nexperimental front can handle such issues better.\nFigure 21 presents a sample magnetization dy-\nnamics in the presence of room-temperature (300\nK) thermal fluctuations. The ramp period is con-\nsidered to be 100 ps and the it turns out from the\nsimulationthatswitchinghascompletedinlessthan\n175 ps. Note that during the course of switching,\nrandom thermal kicks have forced magnetization to\nbacktracktemporarily,however,thestronginterface\nanisotropyhaseventuallyenforcedmagnetization to\nswitch from θ≃180◦toθ≃0◦.\nFigure 22 plots the average switching delay ver-\nsus average energy dissipation for a range of ramp\nperiods (0.1 ns – 1 ns). A moderately large num-\nber of simulations (10,000) in the presence of room-\ntemperature (300 K) thermal fluctuations are per-\nformedandtheyareaveraged togenerateeachpoint\nin the curve. For each trajectory of the 10,000 sim-\nulations, when the magnetization reaches θ≤5◦,\nthe switching is deemed to have completed. Note\nthat as we increase the ramp period of applied volt-\nage across the heterostructure, the switching delay\nalso increases and less energy is dissipated in the\nswitching process, elucidating the well-established\ndelay-energy trade-off for a device in general. The\nresults clearly demonstrate that that switching in\nsub-nanosecond delay is possible while dissipating\na miniscule amount of energy of ∼1 aJ. Note that\nthe“CV2”energydissipationisacoupleofordersof\nmagnitude lower than the energy dissipation due to\nGilbert damping and it decreases with the increase\nof ramp period since the switching becomes more\nadiabatic.93;42While the Figure 22 provides the\nmean of the switching delay distribution, the stan-\ndard deviation in switching delay is also an impor-\ntant performancemetric. the standard deviation for\nramp period of 0.1 ns is about 22 ps and it increases\nabout twice when the ramp period is increased to\n1 ns. At higher ramp period, thermal fluctuations\nhave more time to scuttle the magnetization and\ncause variability in switching time, increasing the\nstandard deviation.\nTo understand the effect of temperature on the\nperformance metrics, simulations have been per-\nformed at an elevated temperature (400 K) and\nthe metrics switching delay and energy dissipation\nturnout tobesimilar (within 5%) compared tothat\nof room-temperature (300 K) case.87Interestingly,\nthe mean switching delay at a higher temperature\nT = 400 K decreases compared to the case at T =30Kuntal Roy\nFig. 21. A sample dynamics of magnetization while switching fromθ≃180◦toθ≃0◦in the presence of room-temperature\n(300 K) thermal fluctuations. The ramp period is 100 ps and the switching delay is 168.5 ps. The energy dissipation due to\nGilbert damping is 1.42 aJ. ( c/circlecopyrtIOP Publishing. Reproduced by permission of IOP Publishing from Ref. 87. All rights reserved.)\nFig. 22. Switching delay-energy trade-off as a function of ra mp period (upper axis). For a faster ramp, the switching beco mes\nfaster but the energy dissipation goes higher. Each point is generated from 10000 simulations in the presence of room-\ntemperature (300 K) thermal fluctuations and the average val ues of switching delays and energy dissipations are plotted . For\n0.1 ns ramp period, the average (max) switching delay is 175. 3 ps (330.8 ps), while the mean energy dissipation is 1.56 aJ. For\na slower ramp with period 1 ns, the average (max) switching de lay is 775.2 ps (1003.5 ps), while the mean energy dissipatio n\nis 0.58 aJ. ( c/circlecopyrtIOP Publishing. Reproduced by permission of IOP Publishing from Ref. 87. All rights reserved.)\n300 K, which can be traced out from the reason-\ning that the initial deflection of magnetization due\nto thermal fluctuations increases at a higher tem-\nperature. Hence, magnetization is likely to start far\naway from the easy axis at a higher temperature\nfor different trajectories, leading to the decrease in\nthe mean switching delay. It turns out that this de-\ncrease in mean switching delay at T = 400 K is very\nsmall(lessthan2%)comparedtothatofT=300K.\nHowever, the trend of standard deviation in switch-\ningdelay with the increase in temperatureshows an\nopposite trend to that of the mean. It can be un-\nderstoodbyconsideringthatthestandarddeviation\nof random thermal field at higher temperature in-\ncreases with temperature [see Eq. (10)]. The mean\nenergy dissipation decreases with increasing tem-perature and this decrease at T = 400 K is quite\nsmall (less than 3%) compared to the case of T =\n300 K.Thisonce again signifies theswitchingdelay-\nenergy trade-off for this device.\nSo far we have considered the writing of a bit\nof information by switching the magnetization from\none state to another, however, the magnetization\nstate needs to be read too. In this interface and\nexchange coupled structure, the giant magnetore-\nsistance (GMR)155;156of the trilayer is calcu-\nlated to be of the order of 30%,71;157which pro-\nvides a way to read the magnetization states (P-\nalignment or AP-alignment). Although this GMR is\nnot that high compared to tunneling magnetoresis-\ntance (TMR),62;63;64;65;66;67;68suitable de-Ultra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 31\nsign strategies can be possibly be devised to work\nwith this moderate value of GMR and also it may\nbe possible to increase the magnetoresistance by\nsuitable material choice and design. It is also ar-\ngued in Ref. 71 that even with the variance in the\nsmaller thicknesses of the layers, it is still possible\nto interface-couple the polarization and magnetiza-\ntion in the proposed structures. It should be noted\nthat the modeling of interface anisotropy is not lim-\nited to the way that is performed here, however,\nanystronginterface-coupled system wouldfacilitate\nswitchingofmagnetization fromonestatetothean-\nother.\nThe important consequence of having such a\nstrong interface anisotropy is that we can achieve\ndevices of very small lateral area, and it is therefore\npossible to cram an enormous amount of devices\non a single chip. If we use an area density of 10−12\ncm−2, the dissipated power would be 10 mW/cm2\nconsidering 1 aJ energy dissipation in a single nano-\nmagnet with 1 ns switching delay and 10% switch-\ning activity (i.e., 10% of the magnets switch at a\ngiven time). Such extremely dense and ultra-low-\nenergy non-volatile computing systems can be pow-\nered by energy harvesting systems without the need\nof an external battery.158;159;160;161\n4. Summary and outlook\nWe have reviewed the dynamical systems study\nfor electrical field-induced magnetization switch-\ning in strain-mediated multiferroic composites, and\ninterface and exchange coupled multiferroic het-\nerostructures. Themagnetization switching dynam-\nics in strain-mediated multiferroic composites using\nLandau-Lifshitz-Gilbert equation revealed intrigu-\ningphenomenainbinary switchingmechanism. Itis\nshownthatbinaryswitchingina‘symmetric’poten-\ntial landscape can be successful in the presence of\nroom-temperature thermal fluctuations. To achieve\nsuch symmetry-breaking, we require the following\ntwo criteria: (1) a sufficiently high stress that keeps\nthe magnetization more out of magnet’s plane in-\nside the so-called “good” quadrants; and (2) a suf-\nficiently fast ramp rate that reduces the possibility\nof backtracking causing switching failure. A high\nstress and a fast ramp rate also increase the switch-\ning speed and counters the detrimental effects of\nthermal fluctuations. This can potentially open up\na new methodology of binary switching since tilt-\ning the potential landscape would not be necessary\nand such findings would stimulate experimental re-search to establish the proposed methodology of bi-\nnary switching. As stated, it requires to sense when\nmagnetization reaches around hard-plane so that\nstress can be brought down thereafter to achieve\nsuccessful switching by breaking the symmetry of\nhaving equal probability of successful switching and\nswitching failure in a‘symmetric’ potential land-\nscape. Other ways to break this symmetry may be\nharnessed and is subject to further research. Note\nthattheaforesaid switchinginstrain-mediated mul-\ntiferroic composites just toggles the magnetization\nstatewithoutbeingabletomaintainthedirectionof\nswitching. It is shown that magnetization switching\nin interface and exchange coupled multiferroic het-\nerostructures can maintain the direction of switch-\ning.\nThe calculated performance metrics from LLG\nsimulations e.g., switching delay and energy dissi-\npation show a profound promise for technological\napplications. The results show that switching can\ntake place in sub-nanosecond delay while expending\na miniscule amount of energy of ∼1 attojoule. This\nenergy dissipation is at least 2-3 orders of magni-\ntude lower than that of the other emerging devices.\nSo multiferroic magnetoelectrics are intriguing in\nrespect to both basic physics of binary switching\nand applied physics. Also, a strong interface and\nexchange coupling in multiferroic heterostructures\nenforces error-resiliency during the switching pro-\ncess and facilitates to scale down the lateral dimen-\nsions to unprecedented dimensions of ∼10 nm even\nin the presence of room-temperature thermal fluc-\ntuations. This is very crucial since it will help com-\npeting with the traditional charge-based electronics\nand consequently for building future nanoelectron-\nics. Due to these superior performance character-\nistics of multiferroic magnetoelectrics as described\nhere, currently it is of immense interest to analyze\ndifferent possible theoretical designs followed by ex-\nperimental demonstrations. Successful experimen-\ntal implementations must tackle theissueof process\nvariation at low dimensions, which traditional tran-\nsistorsfacetoo. Processorsbuiltonsuchdevices can\nlead to unprecedented applications that can work\nby harvesting energy from the environment with-\nout the need of an external battery e.g., medically\nimplanted device to warn an impending epileptic\nseizure by monitoring the brain signals, wearable\ncomputers powered by body movements etc. More-\nover, the basic building blocks are non-volatile per-\nmitting instant turn-on computer, facilitating com-32Kuntal Roy\nputational designs, and avoiding energy dissipation\nforbitstorageelements inasystem.Thiscanpoten-\ntially perpetuate Moore’s law beyond the year 2020\nand turn out to be an unprecedented opportunity\nfor ultra-low-energy computing in our future infor-\nmationprocessingsystems.Experimentaleffortsare\nemerging and successful experimental implementa-\ntion can potentially pave the way for our future na-\nnoelectronics.\nReferences\n1. L. D. Landau and E. M. Lifshitz, Electrodynam-\nics of continuous media (Pergamon Press, Oxford,\nUK, 1960).\n2. H. Schmid, Ferroelectrics 162, 317 (1994).\n3. W. Eerenstein, N. D. Mathur and J. F. Scott, Na-\nture442, 759 (2006).\n4. G. Lawes and G. Srinivasan, J. Phys. D: Appl.\nPhys.44, p. 243001 (2011).\n5. L. W. Martin, S. P. Crane, Y. H. Chu, M. B. Hol-\ncomb, M. Gajek, M. Huijben, C. H. Yang, N. Balke\nand R. Ramesh, J. Phys.: Condens. Matter 20, p.\n434220 (2008).\n6. C. Binek and B. Doudin, J. Phys.: Condens. Mat-\nter17, p. L39 (2004).\n7. K. Roy, SPIN3, p. 1330003 (2013).\n8. M. Trassin, J. Phys.: Condens. Matter 28, p.\n033001 (2015).\n9. N. A. Hill, Annu. Rev. Mater. Res. 32, 1 (2002).\n10. N. A. Spaldin and M. Fiebig, Science309, 391\n(2005).\n11. D. I. Khomskii, J. Magn. Magn. Mater. 306, 1\n(2006).\n12. Y. Tokura, J. Magn. Magn. Mater. 310, 1145\n(2007).\n13. J. M. Rondinelli, S. J. May and J. W. Freeland,\nMRS Bulletin 37, 261 (2012).\n14. T. Birol, N. A. Benedek, H. Das, A. L. Wysocki,\nA. T. Mulder,B. M. Abbett, E.H. Smith, S. Ghosh\nand C. J. Fennie, Curr. Opin. Sol. St. Mater. Sci.\n165, 227 (2012).\n15. D. L. Fox and J. F. Scott, J. Phys. C: Solid St.\nPhys.10, p. L329 (1977).\n16. C. Ederer and N. A. Spaldin, Phys. Rev. B 74, p.\n020401(R) (2006).\n17. C. Ederer and N. A. Spaldin, Phys. Rev. B 71, p.\n060401(R) (2005).\n18. A. Roy, R. Gupta and A. Garg, Adv. Condensed\nMatter Phys. 2012, p. 926290 (2012).\n19. I.Dzyaloshinsky, J.Phys. Chem. Sol. 4,241(1958).\n20. T. Moriya, Phys. Rev. 120, p. 91 (1960).\n21. P. W. Anderson, Phys. Rev. 115, p. 2 (1959).\n22. K. Roy, Europhys. Lett. 108, p. 67002 (2014).\n23. M. Fiebig, J. Phys. D: Appl. Phys. 38, p. R123(2005).\n24. C. W. Nan, M. I. Bichurin, S. Dong, D. Viehland\nand G. Srinivasan, J. Appl. Phys. 103, p. 031101\n(2008).\n25. G. Srinivasan, Annu. Rev. Mater. Res. 40, 153\n(2010).\n26. M. Bichurin and D. Viehland, Magnetoelectricity\nin composites (CRC Press, 2011).\n27. N. A. Pertsev, Phys. Rev. B 78, p. 212102 (2008).\n28. J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n29. L. Berger, Phys. Rev. B 54, 9353 (1996).\n30. J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n31. K. Roy, S. Bandyopadhyay and J. Atulasimha,\nAppl. Phys. Lett. 100, p. 162405 (2012).\n32. K. Roy, J. Phys. D: Appl. Phys. 47, p. 422001\n(2014).\n33. K. Roy, S. Bandyopadhyay and J. Atulasimha,\nAppl. Phys. Lett. 99, p. 063108 (2011),\nNews: “Switching up spin,” Nature476, 375 (Aug.\n25, 2011), doi:10.1038/476375c.\n34. K. Roy, Appl. Phys. Lett. 103, p. 173110 (2013).\n35. K. Roy, Appl. Phys. Lett. 104, p. 013103 (2014).\n36. K. Roy, J. Phys.: Condens. Matter 26, p. 492203\n(2014).\n37. J. S. Kilby, Nobel lecture in physics (The Nobel\nFoundation, Sweden) (2000).\n38. G. E. Moore, Proc. IEEE 86, 82 (January 1998).\n39. V. V. Zhirnov, R. K. Cavin, J. A. Hutchby and\nG. I. Bourianoff, Proc. IEEE 9, 1934 (2003).\n40. S. Borkar, IEEE Micro 19, 23 (1999).\n41.http://www.src.org/program/nri/ .\n42. K. Roy, S. Bandyopadhyay and J. Atulasimha, J.\nAppl. Phys. 112, p. 023914 (2012).\n43. N. Tiercelin, Y. Dusch, A. Klimov, S. Giordano,\nV. Preobrazhensky and P. Pernod, Appl. Phys.\nLett.99, p. 192507 (2011).\n44. M. Liu, S. Li, Z. Zhou, S. Beguhn, J. Lou, F. Xu,\nT. J. Lu and N. X. Sun, J. Appl. Phys. 112, p.\n063917 (2012).\n45. M. Buzzi, R. V. Chopdekar, J. L. Hockel, A. Bur,\nT. Wu, N. Pilet, P. Warnicke, G. P. Carman, L. J.\nHeyderman and F. Nolting, Phys. Rev. Lett. 111,\np. 027204 (2013).\n46. N. Lei, T. Devolder, G. Agnus, P. Aubert,\nL. Daniel, J. Kim, W. Zhao, T. Trypiniotis, R. P.\nCowburn,L.Daniel,D.RavelosonaandP.Lecoeur,\nNature Commun. 4, p. 1378 (2013).\n47. Z. Wang, Y. Wang, W. Ge, J. Li and D. Viehland,\nAppl. Phys. Lett. 103, p. 132909 (2013).\n48. T. Jin, L. Hao, J. Cao, M. Liu, H. Dang, Y. Wang,\nD. Wu, J. Bai and F. Wei, Appl. Phys. Express 7,\np. 043002 (2014).\n49. P. Li, A. Chen, D. Li, Y. Zhao, S. Zhang, L. Yang,\nY. Liu, M. Zhu, H. Zhang and X. Han, Adv. Mater.\n26, 4320 (2014).Ultra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 33\n50. D. H. Kim, N. M. Aimon, X. Sun and C. A. Ross,\nAdv. Funct. Mater. 24, 2334 (2014).\n51. J. P´ erez de la Cruz, E. Joanni, P. M. Vilarinho\nand A. L. Kholkin, J. Appl. Phys. 108, p. 114106\n(2010).\n52. A. Chopra, E. Panda, Y. Kim, M. Arredondo and\nD. Hesse, J. Electroceram. 32, 404 (2014).\n53. K. Roy, S. Bandyopadhyayand J. Atulasimha, Sci.\nRep.3, p. 3038 (2013).\n54. K. Roy, S. Bandyopadhyay and J. Atulasimha,\narXiv:1111.5390 (2011).\n55. R. Landauer, IBM J. Res. Dev. 5, 183 (1961).\n56. R. W. Keyes and R. Landauer, IBM J. Res. Dev.\n14, 152 (1970).\n57. C. H. Bennett, Int. J. Theor. Phys. 21, 905 (1982).\n58. E.C.StonerandE.P.Wohlfarth, Phil. Trans. Roy.\nSoc. A (London) 240, 599 (1948).\n59. A. Khitun, D. E.NikonovandK.L. Wang, J. Appl.\nPhys.106, p. 123909 (2009).\n60. S. A. Wolf, J. Lu, M. R. Stan, E. Chen and D. M.\nTreger,Proc. IEEE 98, 2155 (2010).\n61. T. Brintlinger, S. H. Lim, K. H. Baloch, P. Alexan-\nder, Y. Qi, J. Barry, J. Melngailis, L. Salamanca-\nRiba, I. Takeuchi and J. Cumings, Nano Lett. 10,\n1219 (2010).\n62. M. Julliere, Phys. Lett. A 54, 225 (1975).\n63. J. S. Moodera, L. R. Kinder, T. M. Wong and\nR. Meservey, Phys. Rev. Lett. 74, 3273 (1995).\n64. J. Mathon and A. Umerski, Phys. Rev. B 63, p.\n220403 (2001).\n65. W. H. Butler, X. G. Zhang, T. C. Schulthess\nand J. M. MacLaren, Phys. Rev. B 63, p. 054416\n(2001).\n66. S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki\nand K. Ando, Nature Mater. 3, 868 (2004).\n67. S. S. P. Parkin,C. Kaiser,A. Panchula,P. M. Rice,\nB. Hughes, M. G. Samant and S. H. Yang, Nature\nMater.3, 862 (2004).\n68. W. J. Gallagher and S. S. P. Parkin, IBM J. Res.\nDev.50, 5 (2006).\n69. N. A. Pertsev and H. Kohlstedt, Nanotechnology\n21, p. 475202 (2010).\n70. T. Wu, A. Bur, K. Wong, P. Zhao, C. S. Lynch,\nP. K. Amiri, K. L. Wang and G. P. Carman, Appl.\nPhys. Lett. 98, p. 262504 (2011).\n71. M. Fechner, P. Zahn, S. Ostanin, M. Bibes and\nI. Mertig, Phys. Rev. Lett. 108, p. 197206 (2012).\n72. M. Gajek, M. Bibes, S. Fusil, K. Bouzehouane,\nJ. Fontcuberta, A. Barth´ el´ emyand A. Fert, Nature\nMater.6, 296 (2007).\n73. V. Garcia, M. Bibes, L. Bocher, S. Valencia,\nF. Kronast, A. Crassous, X. Moya, S. Enouz-\nVedrenne, A. Gloter and D. Imhoff, Science327,\np. 1106 (2010).\n74. Y. Shiota, T. Nozaki, F. Bonell, S. Murakami,\nT. Shinjo and Y. Suzuki, Nature Mater. 11, 39(2012).\n75. W. G. Wang, M. Li, S. Hageman and C. L. Chien,\nNature Mater. 11, 64 (2012).\n76. D. Pantel, S. Goetze, D. Hesse and M. Alexe, Na-\nture Mater. 11, 289 (2012).\n77. L. Jiang, W. S. Choi, H. Jeen, S. Dong, Y. Kim,\nM. G. Han, Y. Zhu, S. V. Kalinin, E. Dagotto and\nT. Egami, Nano Lett. 13, 5837 (2013).\n78. S. Dong and E. Dagotto, Phys. Rev. B 88, p.\n140404 (2013).\n79. M. Fechner, S. Ostanin and I. Mertig, Phys. Rev.\nB77, p. 094112 (2008).\n80. M. Fechner, I. V. Maznichenko, S. Ostanin,\nA. Ernst, J. Henk, P. Bruno and I. Mertig, Phys.\nRev. B78, p. 212406 (2008).\n81. C. G. Duan, S. S. Jaswal and E. Y. Tsymbal, Phys.\nRev. Lett. 97, p. 047201 (2006).\n82. H. L. Meyerheim, F. Klimenta, A. Ernst,\nK. Mohseni, S. Ostanin, M. Fechner, S. Parihar,\nI. V. Maznichenko, I. Mertig and J. Kirschner,\nPhys. Rev. Lett. 106, p. 087203 (2011).\n83. J. Opitz, P. Zahn, J. Binder and I. Mertig, J. Appl.\nPhys.87, 6588 (2000).\n84. C.-Y. Youand S. D. Bader, J. Magn. Magn. Mater.\n195, 488 (1999).\n85. M. Y. Zhuravlev, A. V. Vedyayev and E. Y. Tsym-\nbal,J. Phys.: Condens. Matter 22, p. 352203\n(2010).\n86. C.-Y. You and Y. Suzuki, J. Magn. Magn. Mater.\n293, 774 (2005).\n87. K. Roy, J. Phys. D: Appl. Phys. 47, p. 252002\n(2014).\n88. L. Landau and E. Lifshitz, Phys. Z. Sowjet. 8, 153\n(1935).\n89. T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n90. W. F. Brown, Phys. Rev. 130, 1677 (1963).\n91. S. Chikazumi, Physics of Magnetism (Wiley New\nYork, 1964).\n92. M. Beleggia, M. D. Graef, Y. T. Millev, D. A.\nGoode and G. E. Rowlands, J. Phys. D: Appl.\nPhys.38, 3333 (2005).\n93. K. Roy, S. Bandyopadhyay and J. Atulasimha,\nPhys. Rev. B 83, p. 224412 (2011).\n94. K. Roy, Sci. Rep. 5, p. 10822 (2015).\n95. R. Abbundi and A. E. Clark, IEEE Trans. Magn.\n13, 1519 (1977).\n96. K. Ried, M. Schnell, F. Schatz, M. Hirscher,\nB. Ludescher, W. Sigle and H. Kronm¨ uller, Phys.\nStat. Sol. (a) 167, 195 (1998).\n97. R. Kellogg and A. Flatau, J. Intell. Mater. Sys.\nStruc.19, 583 (2007).\n98.http://www.allmeasures.com/Formulae/static/\nmaterials/ .\n99. K. Roy, IEEE Trans. Magn. 51, p. 2500808(2015).\n100. M. Lisca, L. Pintilie, M. Alexe and C. M. Teodor-\nescu,Appl. Surf. Sci. 252, 4549 (2006).34Kuntal Roy\n101. A. J. Masys, W. Ren, G. Yang and B. K. Mukher-\njee,J. Appl. Phys. 94, 1155 (2003).\n102.http://www.memsnet.org/material\n/leadzirconatetitanatepzt/ .\n103. J. Lou, D. Reed, C. Pettiford, M. Liu, P. Han,\nS. Dong and N. X. Sun, Appl. Phys. Lett. 92, p.\n262502 (2008).\n104. K. Roy, American Physical Society (APS) March\n2014 Meeting, Denver, Colorado, Mar 3, Session\nA8.6(2014).\n105. K. Roy, Materials Research Society (MRS) Spring\n2014 Meeting, Mater. Res. Soc. Symp. Proc. 1691\n(2014), http://dx.doi.org/10.1557/opl.2014.730.\n106. K. Roy, Proc. SPIE Nanoscience (Spintronics VII)\n9167, 91670U (2014).\n107. J. M. Rabaey, A. P. Chandrakasan and B. Nikoli¸ c,\nDigital Integrated Circuits (Pearson Education,\n2003).\n108. M. Pedram and J. M. Rabaey (eds.), Power aware\ndesign methodologies (Kluwer Academic Publish-\ners, 2002).\n109. J. Atulasimha and S. Bandyopadhyay, Appl. Phys.\nLett.97, p. 173105 (2010).\n110. M. S. Fashami, S. Bandyopadhyay and J. Atu-\nlasimha, Sci. Rep. 3, p. 3204 (2013).\n111. K. Roy, arXiv:1501.05941v2 (2015).\n112. M. S. Fashami, K. Roy, J. Atulasimha and\nS. Bandyopadhyay, Nanotechnology 22, p. 155201\n(2011).\n113. A.Demming, Nanotechnology 23,p.390201(2012).\n114. M. Yi, B.-X. Xu and D. Gross, Mech. Mater. 87,\n40 (2015).\n115. M. Yi, B.-X. Xu and Z. Shen, Ext. Mech. Lett. 3,\n66 (2015).\n116. M. Yi, B.-X. Xu and Z. Shen, J. Appl. Phys. 117,\np. 103905 (2015).\n117. J. M. Hu, T. Yang, J. Wang, H. Huang, J. Zhang,\nL. Q. Chen and C. W. Nan, Nano Lett. 15, 616\n(2015).\n118. X. Li, D. Carka, C. Liang, A. E. Sepulveda, S. M.\nKeller,P. K. Amiri, G. P. Carmanand C. S. Lynch,\nJ. Appl. Phys. 118, p. 014101 (2015).\n119. Y. Gao, J. M. Hu, C. T. Nelson, T. N. Yang,\nY. Shen, L. Q. Chen, R. Ramesh and C. W. Nan,\nSci. Rep. 6, p. 23696 (Mar 31 2016).\n120. S. Bandyopadhyay and J. Atulasimha, Appl. Phys.\nLett.105, p. 176101 (2014).\n121. A. K. Biswas, S. Bandyopadhyay and J. Atu-\nlasimha, Sci. Rep. 4, p. 7553 (2014).\n122. K. Munira, Y. Xie, S. Nadri, M. B. Forgues,\nM. S. Fashami, J. Atulasimha, S. Bandyopadhyay\nand A. W. Ghosh, Nanotechnology 26, p. 245202\n(2015).\n123. J. Atulasimha and S. Bandyopadhyay, US Patent\n20120267735 (2012).\n124. A. K. Biswas, S. Bandyopadhyay and J. Atu-lasimha, Appl. Phys. Lett. 105, p. 072408 (2014).\n125. J. J. Wang, J. M. Hu, J. Ma, J. X. Zhang, L. Q.\nChen and C. W. Nan, Sci. Rep. 4, p. 7507 (2014).\n126. A. K. Biswas, S. Bandyopadhyay and J. Atu-\nlasimha, Appl. Phys. Lett. 104, p. 232403 (2014).\n127. N. Tiercelin, Y. Dusch, V. Preobrazhensky and\nP. Pernod, J. Appl. Phys. 109, p. 07D726 (2011).\n128. S. Giordano,Y. Dusch, N. Tiercelin, P. Pernod and\nV. Preobrazhensky, Phys. Rev. B 85, p. 155321\n(2012).\n129. S. Giordano,Y. Dusch, N. Tiercelin, P. Pernod and\nV. Preobrazhensky, J. Phys. D: Appl. Phys. 46, p.\n325002 (2013).\n130. N. D’Souza, J. Atulasimha and S. Bandyopadhyay,\nJ. Phys. D: Appl. Phys. 44, p. 265001 (2011).\n131.http://www.itrs.net .\n132. S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma,\nH. D. Gan, M. Endo, S. Kanai, J. Hayakawa,\nF. Matsukura and H. Ohno, Nature Mater. 9, 721\n(2010).\n133. K. Munira, Y. Xie, S. Nadri, M. B. Forgues, M. S.\nFashami, J. Atulasimha, S. Bandyopadhyay and\nA. W. Ghosh, arXiv1405.4000v1 (2014).\n134. M. S. Fashami, K. Munira, S. Bandyopadhyay,\nA. W. Ghosh and J. Atulasimha, IEEE Trans.\nNanotechnol. 12, 1206 (2013).\n135. N. Kani, S. C. Chang, S. Dutta and A. Naeemi,\nIEEE Trans. Magn. 52, p. 1300112 (2016).\n136.http://www.comsol.com .\n137. K. Roy, S. Bandyopadhyay and J. Atulasimha,\narXiv:1012.0819 (2010).\n138. A. K. Biswas, S. Bandyopadhyay and J. Atu-\nlasimha, Appl. Phys. Lett. 103, p. 232401 (2013).\n139. H. Zhao, A. Lyle, Y. Zhang, P. K. Amiri, G. Row-\nlands, Z. Zeng, J. Katine, H. Jiang, K. Galatsis,\nK. L. Wang, I. N. Krivorotov and J. P. Wang, J.\nAppl. Phys. 109, p. 07C720 (2011).\n140. W. H. Butler, Sci. Technol. Adv. Mater. 9, p.\n014106 (2008).\n141. A. Khan, D. E. Nikonov, S. Manipatruni, T. Ghani\nand I. A. Young, Appl. Phys. Lett. 104, p. 262407\n(2014).\n142. M. T. Alam, M. J. Siddiq, G. H. Bernstein, M. T.\nNiemier, W. Porod and X. S. Hu, IEEE Trans.\nNanotech. 9, 348 (2010).\n143. S. M. M. Quintero, C. Martelli, A. Braga, L. C. G.\nValente and C. C. Kato, Sensors11, 11103 (2011).\n144. L. Despont, C. Koitzsch, F. Clerc, M. G. Garnier,\nP. Aebi, C. Lichtensteiger, J. M. Triscone, F. J. G.\nde Abajo, E. Bousquet and P. Ghosez, Phys. Rev.\nB73, p. 094110 (2006).\n145. D. Damjanovic, Rep. Prog. Phys. 61, p. 1267\n(1998).\n146. K. Gilmore, Y. U. Idzerda and M. D. Stiles, Phys.\nRev. Lett. 99, p. 027204 (2007).\n147. D. Steiauf and M. F¨ ahnle, Phys. Rev. B 72, p.Ultra-low-energy Electric field-induced Magnetization Sw itching in Multiferroic Heterostructures 35\n064450 (2005).\n148. S. M. Bhagat and P. Lubitz, Phys. Rev. B 10, p.\n179 (1974).\n149. K. Iijima, R. Takayama, Y. Tomita and I. Ueda, J.\nAppl. Phys. 60, 2914 (1986).\n150.http://www.itrs.net/Links/2009ITRS/2009\nChapters 2009Tables/2009 Interconnect.pdf.\n151. T. Morita and Y. Cho, Appl. Phys. Lett. 85, 2331\n(2004).\n152. J. Li, B. Nagaraj, H. Liang, W. Cao, C. Lee and\nR. Ramesh, Appl. Phys. Lett. 84, 1174 (2004).\n153. W. L. Warren, B. A. Tuttle and D. Dimos, Appl.\nPhys. Lett. 67, 1426 (1995).\n154. M. Bhattacharyya and A. Arockiarajan, Smart\nMater. Struct. 22, p. 085032 (2013).\n155. G. Binasch, P. Gr¨ unberg, F. Saurenbach and\nW. Zinn, Phys. Rev. B 39, 4828 (1989).156. M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen-\nVanDau, F. Petroff, P. Etienne, G. Creuzet,\nA. Friederich and J. Chazelas, Phys. Rev. Lett. 61,\n2472 (1988).\n157. P. Zahn, I. Mertig, M. Richter and H. Eschrig,\nPhys. Rev. Lett. 75, p. 2996 (1995).\n158. S. Roundy, Energy scavenging for wireless sen-\nsor nodes with a focus on vibration to electricity\nconversion, PhD thesis, Mech. Engr., UC-Berkeley\n(2003).\n159. S. R. Anton and H. A. Sodano, Smart Mater.\nStruct.16, p. R1 (2007).\n160. F. Lu, H. P. Lee and S. P. Lim, Smart Mater.\nStruct.13, p. 57 (2004).\n161. Y. B. Jeon, R. Sood, J. Jeong and S. G. Kim, Sens.\nAct. A: Phys. 122, 16 (2005)."    },    {        "title": "0907.5068v1.Interactions_between_magnetohydrodynamic_shear_instabilities_and_convective_flows_in_the_solar_interior.pdf",        "content": "arXiv:0907.5068v1  [astro-ph.SR]  29 Jul 2009Mon. Not. R. Astron. Soc. 000, 1–11 (2009) Printed 19 November 2018 (MN L ATEX style file v2.2)\nInteractions between magnetohydrodynamic shear\ninstabilities and convective flows in the solar interior\nL. J. Silvers1⋆, P. J. Bushby2& M. R. E. Proctor1\n1Department of Applied Mathematics and Theoretical Physics , University of Cambridge, Cambridge, CB3 OWA, United Kingd om\n2School of Mathematics and Statistics, Newcastle Universit y, Newcastle Upon Tyne, NE1 7RU, United Kingdom\nSubmitted ????\nABSTRACT\nMotivatedbytheinterfacemodelforthesolardynamo,thispaper exploresthecomplex\nmagnetohydrodynamic interactions between convective flows and shear-driven insta-\nbilities. Initially, we consider the dynamics of a forced shear flow acro ss a convectively-\nstable polytropic layer, in the presence of a vertical magnetic field. When the imposed\nmagnetic field is weak, the dynamics are dominated by a shear flow (Ke lvin-Helmholtz\ntype) instability. For stronger fields, a magnetic buoyancy instabilit y is preferred. If\nthis stably stratified shear layer lies below a convectively unstable re gion, these two\nregions can interact. Once again, when the imposed field is very weak , the dynami-\ncal effects of the magnetic field are negligible and the interactions be tween the shear\nlayer and the convective layer are relatively minor. However, if the m agnetic field is\nstrong enough to favour magnetic buoyancy instabilities in the shea r layer, extended\nmagnetic flux concentrations form and rise into the convective laye r. These magnetic\nstructures have a highly disruptive effect upon the convective mot ions in the upper\nlayer.\nKey words: convection – instabilities – (magnetohydrodynamics) MHD – Sun: in-\nterior – Sun: magnetic fields\n1 INTRODUCTION\nThe 11 year solar magnetic cycle is driven by a hydro-\nmagnetic dynamo. However, the exact nature of this dy-\nnamo mechanism is still not fully understood, and there\nare several scenarios that seek to explain the observed be-\nhaviour. The well-known “interface” dynamo model (Parker\n1993), is based on the idea that the dynamo operates\nin a region that straddles the base of the solar convec-\ntion zone and the stably stratified region that lies be-\nneath (for some recent reviews see Ossendrijver 2003;\nProctor 2006; Dormy & Soward 2007; Silvers 2008). Al-\nthough this is a conceptually appealing model for the so-\nlar dynamo, the only numerical investigations of the in-\nterface dynamo have been based upon mean-field dynamo\ntheory (see, for example, Charbonneau & MacGregor 1997;\nChan, Liao, Zhang & Jones 2004; Zhang, Liao, Schubert\n2004; Bushby 2006). In mean-field theory, several aspects\nof the dynamo model (particularly the effects of turbulent\nconvection) are parametrised. However, the resulting coef -\nficients are poorly determined by both theory and observa-\ntions. Duetothecomputational costs involved,it hasnotye t\n⋆E-mail: ljs53@damtp.cam.ac.uk (LJS)been possible to demonstrate the operation of the interface\ndynamo by carrying out three-dimensional simulations of\ncompressible magnetohydrodynamics. Given these compu-\ntational constraints, it makes sense to investigate differe nt\ncomponents of the interface dynamo in isolation.\nAn important feature of the region below the solar\nconvection zone is the solar tachocline (see Spiegel & Zahn\n1992; Christensen-Dalsgaard & Thompson 2007, and refer-\nences therein), which takes the form of an intense radial\ngradient of the solar differential rotation. At the heart of\nthe interface dynamo scenario is the idea that weak poloidal\nmagnetic fields can be amplified by the intense shears in\nthe tachocline, leading to the production of strong toroida l\n(azmimuthal) magnetic fields. In the standard interface dy-\nnamo model, these poloidal magnetic fields are produced\nin the convection zone, and are pumped down into the\ntachocline by the fluid motions (Tobias et al.1998, 2001).\nIn flux transport dynamo models, these poloidal fields are\ntransported from the surface (to the tachocline region) by\na meridional circulation (see, e.g., Dikpati & Gilman 2009) .\nWherever the poloidal field is generated, a mechanism is\nneeded to produce flux structures that rise through the con-\nvectionzonetothesurface, wheretheyemerge toform active\nregions. Themost natural mechanism for inducingthis verti -\nc∝circleco√yrt2009 RAS2L. J. Silvers, P. J. Bushby &M. R. E. Proctor\ncal transport is magnetic buoyancy (Parker 1955; Newcomb\n1961). Strong coherent fields exert a magnetic pressure that\nleads to these magnetic regions becoming less dense than\ntheir surroundings. Provided that the ambient medium is\nnot too stably stratified, instabilities can occur that woul d\nappear to allow strong fields to rise into the convection zone\nabove. A full discussion of magnetic buoyancy and its im-\nportance in relation to tachocline dynamics can be found in\nHughes (2007), while the role of the tachocline in the solar\ncycle is described by Tobias & Weiss (2007).\nUntil recently, most studies have addressed the evolu-\ntion of magnetic buoyancy instabilities in a prescribed lay er\nof magnetised fluid (see, for example, Cattaneo & Hughes\n1988; Matthews, Hughes & Proctor 1995; Wissink et al.\n2000; Fan 2001; Kersal´ e, Hughes & Tobias 2007). However,\nit is not immediately obvious that a realistic velocity shea r\ncan produce strong enough magnetic fields to become un-\nstable to buoyancy modes, particularly in the very stably-\nstratified tachocline. To become buoyant the fields must\nexert strong Lorentz forces, which will also retard the\nflow and resist the field amplification. The linear evolu-\ntion of magnetic buoyancy instabilities in a compressible\nmagnetic layer, with an aligned velocity shear, was con-\nsidered by Tobias & Hughes (2004). They found that mag-\nnetic buoyancy instabilities tended to be stabilised by a\nstrong velocity shear. Recent numerical calculations have\nstarted to address the more complex problem of the non-\nlinear evolution of shear-driven magnetic buoyancy instab il-\nities (see, for example, Brummell, Cline & Cattaneo 2002;\nCattaneo, Brummell & Cline 2006; Vasil & Brummell 2008,\n2009).\nUsing a combination of high resolution numerical simu-\nlations and analytical calculations, Vasil & Brummell (200 8,\n2009) investigated the stability of a magnetic layer that\nis generated by the action of a strong vertical velocity\nshear upon an imposed uniform magnetic field. They ar-\ngued that no magnetic buoyancy instability would be pos-\nsible, in the stably-stratified tachocline, unless the mag-\nnitude of the velocity shear were many orders of mag-\nnitude larger than the inferred radial shear. This would\nhave profound consequences for the hydrodynamic stabil-\nity of the shear. Defining the Richardson number, Ri, to be\nthe square of the Brunt-V¨ ais¨ al¨ a frequency divided by the\nsquare of the velocity gradient, a necessary condition for\nhydrodynamic stability is that Ri >1/4 (see, for exam-\nple, Chandrasekhar 1961). In the tachocline, the Richard-\nson number is estimated to be many orders of magnitude\nlarger than that given by this stability bound, which im-\nplies that the shear is stable. However, if the velocity shea r\nis strong enough that the stability condition is not sat-\nisfied, as in the calculations of Vasil & Brummell (2008,\n2009), then this system will be subject to shear instabil-\nities (of “Kelvin-Helmholtz” type). Clearly the situation\nbecomes more complicated in the presence of an imposed\nmagnetic field, and the subsequent evolution depends cru-\ncially (and highly non-trivially) upon the strength of this\nmagnetic field. This is an interesting problem in its own\nright. Hughes & Tobias (2001) considered the linear evolu-\ntion of magnetised shear instabilities, whilst the nonline ar\nproblem has also been studied in unstratified compressible\nfluids (Frank, Jones, Ryu, Gaalaas 1996; Ryu, Jones, Frank\n2000; Palotti, Heitsch, Zweibel, Huang 2008) as well as inisothermal stratified layers (Br¨ uggen & Hillebrandt 2001) .\nThe recent review article by Gilman & Cally (2007) de-\nscribes global magnetohydrodynamic shear instabilities i n\nthe tachocline.\nAlthough the hydrodynamic stability of the velocity\nshear was discussed by Vasil & Brummell (2008, 2009), the\nmost important idea in their work was the suggestion that\nshear-driven magnetic buoyancy instabilities can only oc-\ncur at very small values of the Richardson number. Since\nthis would appear to be incompatible with the tachocline,\nthis would have dire consequences for solar dynamo mod-\nels. However, these results are not conclusive. Firstly, th eir\ncalculations were all performed using a fixed value for the\nimposed magnetic field strength. This is clearly an im-\nportant parameter, since the Lorentz force plays a cru-\ncial dynamical role. More importantly, recent calculation s\nby Silvers et al.(2009) have confirmed, as already known\nfor the onset of magnetic buoyancy without shear (Hughes\n2007), that the onset of magnetic buoyancy instabilities de -\npends upon the ratio of the magnetic to thermal diffusiv-\nities. At high Reynolds numbers, Silvers et al.(2009) have\nshown that magnetic buoyancy instabilities can be excited\nwith a weaker (hydrodynamically-stable) shear if the ther-\nmal diffusivity is much greater than the magnetic diffusivity\n(something that was not the case in the original calculation s\nof Vasil & Brummell (2008)). This is a more encouraging re-\nsult from the point of view of the solar dynamo, although\nmore work remains to be done.\nClearly, the parametric dependence of the instabilities\nof a forced shear flow, in the presence of a magnetic field,\nis still not fully understood. One of the aims of this pa-\nper is to enhance our understanding of these instabilities\nvia a partial exploration of parameter space. In particular ,\nwe focus attention upon the effects of varying the strength\nof the imposed magnetic field (although some variations in\nother parameters are also considered). Once the evolution o f\nthis system has been studied in a single convectively-stabl e\nlayer of compressible fluid, we move on to consider a more\ncomplicated “composite” model, which combines the stably-\nstratifiedshear layerwith an overlyingconvectively-unst able\nregion. This composite model enables us to address the in-\nteresting question, so far largely unexplored, of how buoy-\nant magnetic flux might interact with the fluid in the lower\nconvection zone. In order to limit the computational ex-\npense of this parametric survey, we choose a stronger veloc-\nity shear than that considered by Silvers et al.(2009). This\nenables us to drive buoyancy instabilities at lower Reynold s\nnumbers, which means that fully resolved numerical simula-\ntions can be carried out with a coarser numerical grid. Al-\nthough our chosen flow is hydrodynamically unstable, it is\nstill much weaker than the target shear that was considered\nby Vasil & Brummell (2008, 2009), being mildly subsonic as\nopposed to highly supersonic (though still much stronger\nthan that found within the tachocline). Although we are\nnot exploring the rather extreme parameter regime that is\ndirectly relevant for the tachocline, our choice of paramet ers\nallows us to enhance our basic understanding of the interac-\ntions between magnetic buoyancy and convective instabili-\nties. Future research (which will rely upon this work) will\nfocus upon these phenomena at higher Richardson numbers.\nThe plan of the paper is as follows: In the next section\nwe describe the set up of the model problem. Insection 3, we\nc∝circleco√yrt2009 RAS, MNRAS 000, 1–11Shear-driven instabilities in the solar interior 3\npresent numerical results from this model, describing the i n-\nteractions between magnetic and hydrodynamic instabiliti es\nin a single stably-stratified polytropic layer. In the follo w-\ning section, we describe the (more complicated) problem of\nshear-driven instabilities in the composite model. Finall y, in\nsection 5, we conclude with a discussion of the astrophysica l\nsignificance of these results.\n2 THE MODEL\nThe model we use is similar to that of several pre-\nvious studies of magnetoconvection (see, for exam-\nple, Matthews, Proctor & Weiss 1995; Bushby & Houghton\n2005; Lin, Silvers & Proctor 2008). We consider the evolu-\ntion of a plane layer of electrically-conducting fluid, whic h\nis heated from below, in the presence of a magnetic field\nthat is initially uniform and vertical. Accordingly, we ado pt\na Cartesian frame of reference such that the z-axis points\nvertically downwards, parallel to the constant gravitatio nal\nacceleration, g. Defining dto be some characteristic length-\nscale (e.g. the depth of the convection zone in the com-\nposite model), this fluid occupies the region 0 /lessorequalslantx,y/lessorequalslant\nλdand 0/lessorequalslantz/lessorequalslantdz0. We set z0= 1 in the single\nlayer calculations that are described in the next section\n(in which case dcorresponds to the layer depth), whilst\nz0= 3 in the composite model. Varying the parameter λ\nenables us to change the width of the domain without al-\ntering the value of d. In contrast to most previous stud-\nies of magnetoconvection (although see the recent paper\nby K¨ apyl¨ a, Korpi & Brandenburg 2008), we investigate the\nevolution of this system under the influence of a forced hor-\nizontal shear flow in the x-direction.\nThroughout this paper, we assume that the shear vis-\ncosity,µ, the magnetic diffusivity, η, the permeability of free\nspace,µ0, and the specific heat capacities at constant pres-\nsure and density ( cPandcVrespectively) are all constant\nproperties of the fluid. The thermal conductivity, K(z), is\nassumed to be a function of z. Defining ρto be the fluid\ndensity,Tto be the temperature, uto be the fluid velocity\nandBto be the magnetic field, the governing equations for\nthe evolution of this compressible fluid are given by:\n∂ρ\n∂t+∇·(ρu) = 0 (1)\nρ/bracketleftBig∂u\n∂t+(u·∇)u/bracketrightBig\n=−∇P+ρgˆ z−µ∇2[U0(z)ˆ x] (2)\n+1\nµ0(∇×B)×B+∇·(µτ)\nρcV/bracketleftBig∂T\n∂t+(u·∇)T/bracketrightBig\n=−P∇·u+∇·[K(z)∇T] (3)\n+η|∇×B|2\nµ0+µτ2\n2\n∂B\n∂t=∇×[u×B−η∇×B] (4)\n∇·B= 0, (5)\nwhere the pressure Psatisfies the perfect gas law\nP=R∗ρT, (6)\n(defining R∗to be the gas constant) and the components of\nthe viscous stress tensor τsatisfyτij=∂ui\n∂xj+∂uj\n∂xi−2\n3∂uk\n∂xkδij. (7)\nFinally the scalar quantity, U0(z), represents the horizontal\nshear flow. The corresponding forcing term in Equation 2\nensures that any imposed shear of this form is a solution\nof the horizontal component of the momentum equation (in\nthe absence of any other motions).\nThe boundary conditions for these variables are consis-\ntent with those of an idealised model. All variables are as-\nsumedtosatisfy periodic boundaryconditions in the xandy\ndirections. The upper and lower bounding surfaces (at z= 0\nandz=dz0respectively), are assumed to be impermeable\nand stress-free, and it also assumed that the magnetic field\nis vertical at these boundaries. The upper boundary is held\nat fixed temperature, whilst the heat flux passing through\nthe lower surface is assumed to be constant. This implies\nthat:\nuz=∂ux\n∂z=∂uy\n∂z=Bx=By= 0, T=T0atz= 0,(8)\nuz=∂ux\n∂z=∂uy\n∂z=Bx=By= 0,∂T\n∂z=Catz=dz0,(9)\nwhereCis a constant that will depend upon the initial con-\nditions of the model. Note that the choice of a stress-free\nboundary condition for uximplies that the imposed shear,\n∂U0(z)/∂z, should also be zero at these surfaces.\nThese equations can be expressed in non-\ndimensional form, using the scalings described by\nMatthews, Proctor & Weiss (1995). Lengths are scaled\nbyd, whilst the density and temperature are scaled by their\ninitial values at the top of the layer ( ρ0andT0respectively).\nVelocities are scaled in terms of the isothermal sound speed\nat the upper surface,√R∗T0, which suggests a natural\nscaling for time of d/√R∗T0. Magnetic fields are scaled in\nterms of the strength of the initial vertical magnetic field\nB0. Finally, we define K0to be the value of K(z) at the\nupper surface. When these scalings are substituted into\nthe governing equations, we obtain several non-dimensiona l\nparameters that are essentially identical to those describ ed\nby Matthews, Proctor & Weiss (1995). These include the\ndimensionless thermal diffusivity, κ=K0/dρ0cP/radicalbig\n(R∗T0),\nthe ratio of specific heats, γ=cP/cV, the Prandtl number,\nσ=µcP/K0, and the ratio of the magnetic to the thermal\ndiffusivity at the top of the layer, ζ0=ηcPρ0/K0. Finally,\nF=B2\n0/R∗T0ρ0µ0is the ratio of the squared Alfv´ en speed\nto the square of the isothermal sound speed at the top\nof the layer. This parameter determines the dynamical\ninfluence of any imposed magnetic field.\nWith appropriate choices for these non-dimensional pa-\nrameters, we solve the equations numerically using a parall el\nhybrid finite-difference/pseudo-spectral code. In this cod e,\ntime-stepping is carried out with an explicit third-order\nAdams-Bashforth scheme. Horizontal derivatives are eval-\nuated in Fourier space, whilst vertical derivatives are cal cu-\nlated using fourth-order finite differences (upwinded deriv a-\ntives being used for the advective terms). In order to carry\nout these simulations, grid resolutions of 128 ×128×200\nmesh points were used for the single layer cases, whilst\n256×256×300 mesh points were used for the composite\nmodel. Some calculations were also carried out at lower spa-\ntial resolution and comparisons of the different resolution s\nc∝circleco√yrt2009 RAS, MNRAS 000, 1–114L. J. Silvers, P. J. Bushby &M. R. E. Proctor\nParam. Description Value\nσ Prandtl Number 0 .05\nm Polytropic Index 1 .6\nθ Thermal Stratification 0 .5\nγ Ratio of Specific Heats 5 /3\nκ Thermal Diffusivity 0 .01\nFMagnetic Field Strength Variable\nζ0 Magnetic Diffusivity 0 .2\nTable 1. Parameter values for the single layer calculations.\nshow that the instabilities and structures that emerge are\nphysical and not artefacts of the discretization.\n3 THE SINGLE LAYER\nIn this section, we consider the evolution of this system wit h\nthe simplest possible initial configuration. An understand -\ning of this system will help us to interpret the results from\nthe next section, which deals with a much more complicated\nmodel problem. Throughout this section, we define the com-\nputational domain by setting λ= 4 and z0= 1. Recalling\nthat all lengths are scaled in terms of d, this implies that\n0/lessorequalslantx,y/lessorequalslant4 and 0/lessorequalslantz/lessorequalslant1.\n3.1 Parameters\nThe behaviour of this model depends crucially upon the ini-\ntial conditions that are imposed. The simplest non-trivial\ncase to choose is that of a polytropic layer with a con-\nstant thermal conductivity, K(z) =K0. In this case, the\ninitial conditions are completely determined by two non-\ndimensional parameters, namely the (dimensionless) tem-\nperature difference between the upperand lower boundaries,\nθ, and the polytropic index, m=gd/R∗T0θ. Neglecting the\neffects of viscous heating, it is straightforward to show tha t\nthe governing equations have the following (dimensionless )\nequilibrium solution: T= (1+θz),ρ= (1+θz)m,Bz= 1,\nux=U0(z),uy=uz=Bx=By= 0. Of course the effects\nof viscous heating will lead to a departure from this equilib -\nrium, but we have verified (by direct calculation) that the\ndeparture is negligible over the time-scales that are consi d-\nered in this paper. Therefore the above “equilibrium” solu-\ntion (together with a small, random, thermal perturbation)\nis used as an initial condition for all the simulations that a re\ndescribed in this section. Note that these initial conditio ns\nimply that the lower boundary condition for temperature\n(see Equation 9) becomes ∂T/∂z=θatz= 1.\nThis system of equations has a large number of di-\nmensionless parameters, making it impractical to conduct\na complete survey of parameter space. Therefore we focus\nprimarily upon varying the strength of the magnetic field,\nholding all other parameters fixed (although a small num-\nber of runs with different parameter values were also car-\nried out). The parameter choices are summarised in Ta-\nble 1. Note that this choice of θimplies that the layer\nis weakly stratified. Setting m= 1.6 andγ= 5/3 im-\nplies that stratification in the layer is mildly subadia-\nbatic. This choice of parameters is appropriate for the00.2 0.4 0.6 0.8 100.20.40.60.811.21.4\nzux\n00.2 0.4 0.6 0.8 1024681012\nz∂ ux/∂ z\nFigure 1. Top:uxas a function of depth for the single layer\nmodel. Bottom: ∂ux/∂zas a function of depth for the single layer\nmodel.\nstably stratified solar tachocline (Vasil & Brummell 2008;\nChristensen-Dalsgaard & Thompson 2007). Note also that\nthe parameter values that are given in Table 1 imply that\nthe dynamical effects of the magnetic diffusivity, the viscos -\nity and the thermal diffusivity are much more significant in\nthis model than in the solar interior. This is because the\ndissipative length scales associated with the solar interi or\ncan not be resolved using any current computer. However,\nby setting 1 > ζ0> σ, we ensure that these are ordered in\nthe same way as in the solar interior, i.e. the thermal dis-\nsipative cutoff scale is larger than the magnetic dissipativ e\ncutoff scale, which is in turn taken to be larger than the\nviscous scale.\nFinally, we must specify a suitable initial shear flow for\nthis system. We set\nU0(z) = 0.577(1+tanh[20( z−0.5)]) (10)\nas shown in Figure 1(top). The hyperbolic tangent gives a\nsmooth velocity field, varying from ux= 0 atz= 0 up to\nux= 1.154 atz= 1. The width of the shear region is suffi-\nciently small that the departure from a stress-free conditi on\nat the boundaries is comparable with the numerical error of\nthe scheme. The results of Vasil & Brummell (2008) suggest\nthat a stronger shear promotes magnetic buoyancy instabil-\nities. We have maximised the shear velocity subject to the\nconstraint that the horizontal flow speed never exceeds the\nadiabatic sound speed, whilst also ensuring that the peak\nmach number of the flow is identical to the peak mach num-\nber of the shear in the composite model (see the next sec-\ntion). Note that the fluid Reynolds number of this shear\nc∝circleco√yrt2009 RAS, MNRAS 000, 1–11Shear-driven instabilities in the solar interior 5\n(based upon the peak velocity and the width of the shear\nlayer) is approximately 300, which is much smaller than in\nother studies (Vasil & Brummell 2008, 2009; Silvers et al.\n2009). As discussed in the Introduction, this enables us\nto carry out fully resolved simulations with comparatively\nmodest numerical grids.\n3.2 Results\nHavingset upthemodel problem for astably-stratifiedpoly-\ntropic layer, we now investigate the effects of varyingthe im -\nposed magnetic field. In order to achieve this, we carry out\na series of numerical simulations for different values of the\nparameter, F(keeping all other parameters fixed, as shown\nin Table 1). In the absence of a magnetic field ( F= 0),\nwe find that the system is unstable to a shear flow (Kelvin-\nHelmholtz type) instability. Initially, we see rolls formi ng in\nthex−zplane, as shown in Figure 2. This then rapidly\nevolves into a three-dimensional time-dependent flow. The\nimplications of imposing a magnetic field across this compu-\ntational domain depend upon the strength of the imposed\nmagnetic field. If the initial field is weak (say F= 1/90000),\nthen the effect of the field on the evolution of the instability\nis negligible, as the solutions are virtually indistinguis hable\nfrom the hydrodynamic case. The magnetic field is simply\nadvected with the resultant flow as a passive vector field.\nHowever, if we increase the strength of the magnetic field,\nwe find that it starts to have a dynamical influence. Fig-\nure 3 shows the evolution of a simulation with a magnetic\nfield strength determined by F= 1/9000. In this case, the\nmagnetic field does reduce the vigour of the instability, lea d-\ning to more ordered motions (particularly at early times).\nThis behaviour is easy to explain. The shear flow instability\nacts so as to bend the magnetic field lines parallel to the\nvelocity shear. A strong field tends to resist this process,\nthus inhibiting the instability. However even in this simul a-\ntion, as in the hydrodynamic case, the instability eventual ly\ndevelops three-dimensional structure.\nThe character of the instability changes further as we\nincrease the strength of the imposed magnetic field. Results\nforF= 1/900 are shown in Figure 4. The resulting hor-\nizontal magnetic field is now strong enough to completely\nsuppress the shear flow instability. Rather than generating\nfluctuations parallel to the shear, the initial instability is\na short wavelength interchange instability, with almost al l\nvariation (at least initially) in the ydirection. These inter-\nchange modes are typical of a magnetic buoyancy instability\n(Newcomb 1961; Hughes 1985). During these early stages,\nthe developing structures are similar to those found in two-\ndimensional calculations of the break up of a magnetic layer\nin the absence of a shear (Cattaneo & Hughes 1988). At\nlater times some longer wavelength variation in the xdirec-\ntiondoesappear –thisthree-dimensionalevolutionis simi lar\nto that found by Wissink et al.(2000).\nWhile the focus of this section is toexplore the effects of\nvarying the magnetic field strength, we note that there are\nother parameters that can be varied (subject to computa-\ntional constraints). A reduction in the Prandtl number lead s\nto a reduction in the viscous dissipation relative to the oth er\ndiffusivities. This also increases the fluid Reynolds number .\nWe carried out runs with lower values of the Prandtl num-\nber, and found that reducing this parameter by up to a fac-\nFigure 2. Density perturbation snapshot for the hydrodynamic\ncase (F= 0) at times t= 2.59 (top), t= 5.20 (middle) and\nt= 7.69 (bottom)\nc∝circleco√yrt2009 RAS, MNRAS 000, 1–116L. J. Silvers, P. J. Bushby &M. R. E. Proctor\nFigure 3. Density perturbation snapshot for F=1/9000, at\nt=5.21 and 7.78.\ntor of ten (going down to σ∼0.005) appears to have little\neffect upon the vigour of the instability at F= 1/900. This\nsuggests that there is very little dependence upon the fluid\nReynolds number in this parameter regime. The effects of\nvarying both κandσhave not been systematically studied\nhere; Silvers et al.(2009) have shown that κin particular\ncan play an important role for larger values of the Richard-\nson number, and investigation of a full range of diffusivity\nratios will be the subject of future work.\n4 THE COMPOSITE MODEL\nIn this section, we consider a more complicated model prob-\nlem, consisting of a piecewise polytropic atmosphere. This is\nintendedto be a highly idealised representation of the regi on\nstraddling the base of the solar convection zone. In order to\nachieve this, we consider a deeper computational domain,\ncorresponding to z0= 3. We also choose a wider computa-\ntional domain, by setting λ= 8. Recalling the definitions of\nFigure 4. Density perturbation snapshot for F=1/900, at\nt=10.93 and 13.88.\nthese parameters, this implies that the computational do-\nmain is defined by 0 /lessorequalslantx,y/lessorequalslant8 and 0/lessorequalslantz/lessorequalslant3.\n4.1 Parameters\nOther than the dimensions of the computational domain,\nthe main difference between these calculations and those of\nthe preceding section is that the polytropic index of the do-\nmain is now afunction of depth. We split up the domain into\nthree layers of unit depth. In the top layer (0 /lessorequalslantz/lessorequalslant1), we\nchoose a polytropic index of m0= 1, which implies that this\nregion is convectively unstable. Like the single layer from\nthe previous section, the middle region (1 /lessorequalslantz/lessorequalslant2) is con-\nvectively stable with m1= 1.6. The lower layer (2 /lessorequalslantz/lessorequalslant3)\nis also convectively stable but with a much larger polytropi c\nindex,m2= 4. The primary purpose of the lower layer is to\nlessen the impact of the rigid lower boundary. Any descend-\ning convective plumes that reach z= 2 can simply pass into\nthe lower region without “splashing” back and interfering\nwith the other dynamics in the system. In order to achieve\nc∝circleco√yrt2009 RAS, MNRAS 000, 1–11Shear-driven instabilities in the solar interior 7\n00.511.522.5300.20.40.60.811.21.41.61.82\nzux\n00.511.522.5302468101214161820\nz∂ ux/∂ z\nFigure 5. Top:uxas a function of depth for the composite layer\nmodel. Bottom: ∂ux/∂zas a function of depth for the composite\nlayer model.\nthe required piecewise-polytropic structure, we choose th e\nfollowing depth-dependent thermal conductivity profile:\nK(z) =K0\n2/bracketleftBig\n1−tanh/parenleftBigz−1\n0.02/parenrightBig/bracketrightBig\n(11)\n+K0(m2+1)\n2(m0+1)/bracketleftBig\n1+tanh/parenleftBigz−2\n0.02/parenrightBig/bracketrightBig\n+K0(m1+1)\n2(m0+1)/bracketleftBig\n1−tanh/parenleftBigz−2\n0.02/parenrightBig\ntanh/parenleftBigz−1\n0.02/parenrightBig/bracketrightBig\n,\nwhereK0=K(0) (as before). The tanh profiles ensure that\nthe conductivity varies smoothly between each region.\nIn thiscomposite model, our aim is toinvestigate theef-\nfects that any shear instabilities in the mid-layer have upo n\nan established pattern of convection. Therefore, we only in -\ntroduce the shear once the convection in the upper layer\nhas become fully developed. This is achieved by integrating\nthe equations without any horizontal forcing until t≈40.\nThe shear is then introduced at this point, along with the\ncorresponding forcing term in Equation 2. Once it has been\nintroduced, the shear has the same structure as that in the\nsingle layer model but is now centred at the mid-plane of\nthe middle region (at z= 1.5). Thus the imposed shear now\nhas the form:\nU0(z) = 1+tanh[20( z−1.5)], (12)\nas shown in figure 5. Note that the amplitude of the shear\nis chosen so that the local Mach number of the flow is the\nsame as for the single-layer case.\nIn the absence of any imposed shear at t= 0, the ini-00.511.522.53051015\nzρ00.511.522.530246\nzT\nFigure 6. The initial temperature and density profiles for the\ncomposite model.\nParam. Description Value\nσ Prandtl Number 0.05\nm0,m1,m2Polytropic Indices 1.0, 1.6, 4.0\nγ Ratio of Specific Heats 5 /3\nκ Thermal Diffusivity 0.0385\nF Magnetic Field Strength variable\nζ0 Magnetic Diffusivity 0.1\nTable 2. Fixed Parameter Values\ntial conditions for this model differ slightly from those in\nthe single layer case. Here we choose a magnetohydrostatic\ninitial condition, setting Bz= 1 and ux=uy=uz=\nBx=By= 0. The equilibrium profiles for ρ(z) andT(z)\nare found numerically, and are shown in Figure 6. Note that\nthe choice of the thermal boundary condition at the lower\nsurface determines the extent of the thermal stratification .\nSetting∂T/∂z= 0.8 atz= 3 ensures that the temperature\nincreases by 50% across the middle layer, as was the case for\nthe single layer.\nThe parameters for this composite model are chosen\nso that the conditions in the middle layer are as similar as\npossible to those for the single layer calculation. Note tha t\nthis requires some rescaling of κandζ0. These parameters\nare shown in Table 2.\n4.2 Results\nAs for the single layer calculations in the previous section ,\nwe carry out a series of numerical simulations for different\nvalues of the imposed magnetic field (as measured by F). In\naddition to any effects upon the hydrodynamic instabilities\nof the shear, increasing Falso reduces the vigour of any\nconvective motions in the upper region of the computational\ndomain.1The range of Fis carefully chosen so that we cover\n1Note that estimates from linear theory suggest that a value o f\nF of approximately 0.5 is needed in order to completely suppr ess\nconvective flows in the upper layer. Therefore, we are not nea r\nthe convective stability boundary. If we were to increase F f ur-\nc∝circleco√yrt2009 RAS, MNRAS 000, 1–118L. J. Silvers, P. J. Bushby &M. R. E. Proctor\nFigure 7. Three-dimensional plots of the F= 0.0001 case in the\nabsence of any velocity shear. Top: The temperature perturb ation\non the side of the computational domain and close to the upper\nsurface at t≈40. Bottom: The same plot at t≈51.\nthe same values for the mid-layer plasma beta (the ratio of\nthe gas pressure to the magnetic pressure) that were covered\nin the single-layer case.\nInitially, we set F= 0.0001, which corresponds to a\nther, we would expect to see a transition to an oscillatory mo de\nof convection before we reach the regime in which convection is\ncompletely inhibited.\nFigure 8. Three-dimensional plots of the F= 0.0001 case at\nt≈51, after the shear was introduced at t≈40. Top: The tem-\nperature perturbation on the side of the computational doma in\nand close to the upper surface. Bottom: The horizontal magne tic\nfield on the same surfaces. Note that in this figure, Bxis nor-\nmalised with respect to the imposed field, ∝√\nF.\nweak imposed magnetic field. Given the relative complexity\nof this system, we first explore the dynamics that occur in\nthe absence of any velocity shear. This case is illustrated i n\nFigure 7, which shows the resulting pattern of convection\natt≈40 (top) and t≈51 (bottom). In the convectively\nunstable upper layer there is a time-dependent cellular con -\nvective pattern consisting of warm, broad upflows (which\nc∝circleco√yrt2009 RAS, MNRAS 000, 1–11Shear-driven instabilities in the solar interior 9\ncorrespond to the brighter regions in figure 7) surrounded\nby a network of narrower (darker) downflows. There is some\nmodest convective overshooting into the middle layer. Fig-\nure 8 also shows a snapshot of this system at ( t≈51)\nbut, in this case, the shear is introduced at t≈40. As\nwas found in the single-layer case, the shear is subject to\na Kelvin-Helmholtz type shear instability. This rapidly de -\nvelops three-dimensional structure, producingperturbat ions\nthat spread throughout the stably-stratified domain, pene-\ntrating into the convective layer. Comparing Figure 7 and 8\nwe see that there is little evidence of anisotropy in the con-\nvecting region. However we note that there is a significant\ninfluence on the convective transport of heat in the top part\nof the box, which is apparently due to the influence of the\nshear instability on heat transport in the middle layer.\nAs the field strength is increased from this (effectively)\nkinematic level, the solutions follow a similar trend to the\nsingle-layer case. When F= 0.001, the dynamics in the mid-\nlayeraresimilar informtothoseshowninFigure3(although\nthe overshooting convection from the upper layer adds some\nadditional complexity to the resulting flows). Therefore, t he\ndominant instability is still of Kelvin-Helmholtz type rat her\nthan a magnetic buoyancy instability, although magnetic ef -\nfects are starting to play a dynamical role. Interestingly, as\nthe shear-driven motions from the stable layer interact wit h\nthe convective layer, there appears to be a slight tendency\nfor an elongation of the convective cells in the direction of\nthe shear. This is a phenomenon that becomes more pro-\nnounced as Fis increased.\nIncreasing the field strength still further, so that F=\n0.01, we find that the dynamics change dramatically. This\ncase is illustrated in Figures 9 and 10. The imposed vertical\nmagnetic field is now strong enough to reduce the vigour\nof the convection, though it has little effect on the hori-\nzontal scales of motion. Once the shear is introduced, the\nevolution is dominated by the shear-driven instabilities a t\nthe mid-layer. As in the single layer case, a transition has\noccurred so that the dominant instability is now magnetic\nbuoyancy. Initially, this buoyancy instability takes the f orm\nof a two-dimensional (interchange) mode, although it soon\ndevelops three-dimensional structure, forming arching re -\ngions of magnetic flux that rise up through the convective\nupper layer of the domain. As these magnetic regions reach\nthe upper layers, we see some concentration of the verti-\ncal magnetic flux, which forms localised concentrations nea r\nthehorizontal boundaries oftheserising features. The sub se-\nquent motion is now strongly anisotropic, producing convec -\ntive cells that are predominantly aligned with the directio n\nof the shear and the buoyant horizontal magnetic flux con-\ncentrations. We note that the introduction of a shear flow at\nt≈40 again leads to larger temperature deviations in the\nconvectively-unstable region at later times (compared wit h\nthe unsheared case). However this effect seems to become\nless pronounced as the field strength is increased. We at-\ntribute this phenomenon to the fact that there is less mixing\nin thestronger field regime, where thereare larger structur es\npresent than in the weaker field cases.\nThe phenomena discussed above can be related to pre-\nvious work on isolated buoyant flux tubes rising through\nthe convection zone (see, for example, Jouve & Brun 2007).\nEven though the tubes are isolated they have been shown to\ninteract strongly with the convective flow. The present prob -\nFigure 9. Three-dimensional plots of the F= 0.01 case in the\nabsence of any velocity shear. Top: The temperature perturb ation\non the side of the computational domain and close to the upper\nsurface at t≈40. Bottom: The same plot at t≈51.\nlem is different in that the magnetic field is initially vertic al.\nHowever,theshear creates astronghorizontal magnetic fiel d\nand so the ultimate configuration is not dissimilar.\n5 CONCLUSIONS\nIn this paper, we have presented some novel calculations to\ninvestigate the ways in which an imposed magnetic field in-\nc∝circleco√yrt2009 RAS, MNRAS 000, 1–1110L. J. Silvers, P. J. Bushby &M. R. E. Proctor\nFigure 10. Three-dimensional plots of the F= 0.01 case at\nt≈51, after the shear was introduced at t≈40. Top: The tem-\nperature perturbation on the side of the computational doma in\nand close to the upper surface. Bottom: The horizontal magne tic\nfield on the same surfaces. Note that in this figure, Bxis nor-\nmalised with respect to the imposed field, ∝√\nF.\nteracts with a shear flow in a convectively-stable layer (bot h\nwith and without an overlying convective region). This in-\nvestigation was motivated by conditions at the base of the\nconvection zone, and is relevant to the interface scenario\nfor the solar dynamo. The most important interactions be-\ntween the shear layer and the convective region are due to\nthe rising plumes that are induced by magnetic buoyancy.Inthe solar context the tachocline region, where the shear is\nexpected to reside, is very stably stratified and there are\nquestions regarding the efficacy of magnetic buoyancy in\nthis situation (see, for example, Vasil & Brummell 2009).\nNonetheless we know that buoyancy instabilities do occur\nin the Sun, and so it seems worthwhile trying to understand\naspects of their evolution, even though the correct parame-\nter range might not yet have been reached. In this context,\nit is also worth noting that a diffusive instability, which is\neffective only when the thermal diffusivity is much higher\nthan in the present paper, appears to allow for buoyancy-\ninduced motion even when the shear is hydrodynamically\nstable, according to the Richardson number criterion. This\nmechanism been discussed for several decades (see, for ex-\nample, the discussions in Gilman 1970; Hughes 2007) but\nhas only recently been demonstrated numerically in our ge-\nometry (Silvers et al.2009).\nIn our “strong-shear” parameter regime, there is an in-\nstability of the shear (of Kelvin-Helmholtz type) even when\nthere is no magnetic field. This instability leads initially to\nperturbations that are primarily in the x−zplane. This in-\nstability subsequently develops structure in the ydirection.\nWe find that, for a sufficiently strong magnetic field, the hy-\ndrodynamic instability is suppressed, leading to a magneti c\nbuoyancy instability with strong variations in the ydirec-\ntion, i.e. the direction perpendicular to both gravity and t he\nshear flow. For calculations of the “composite model”, the\nmost important interactions between the shear layer and the\nconvective region are due to the rising plumes that are in-\nduced by this magnetic buoyancy instability. This instabil -\nity generates strong horizontal concentrations of magneti c\nflux that then rise through the convective layer. For the\nstrongest fields surveyed, the dynamical effects of these flux\nconcentrations are so significant that they destroy the con-\nvective pattern that would normally exist in the absence of\na magnetic field. As with isolated flux tubes (see, for exam-\nple,Jouve & Brun2007)theyeffectivelypushtheconvective\nmotion aside. It is important to note that even our strongest\nimposedfieldhas arelatively weak effectuponthehorizontal\nscales of convection that occur in the absence of shear.\nOur results are encouraging in that they show that\nthere exists the possibility of inducing strong buoyant mag -\nnetic flux structures through the action of horizontal shear .\nHowever they are preliminary calculations in the sense that\nthey do not allow the buoyant flux structures to rise very\nfar. We intend to perform further calculations, with a much\ndeeper convective zone, so as to understand the later evo-\nlution of the rising flux structures. We also note that there\nis a swift transition as we vary our parameters between es-\nsentially passive structures and ones that strongly disrup t\nthe convection layer. We intend to carry out a more exten-\nsive investigation of intermediate parameter ranges in whi ch\nthe role of downward pumpingis important in counteracting\nthe buoyancy effects (Tobias et al.2001). Finally, we intend\nto explore the interactions between shear-driven buoyancy\ninstabilities and convective flows at higher Richardson num -\nbers (with a hydrodynamically stable shear). This will be a\nchallenging problem to tackle numerically (requiring high\nnumerical resolution), but results from these preliminary\ncalculations constitute a firm foundation for future work.\nc∝circleco√yrt2009 RAS, MNRAS 000, 1–11Shear-driven instabilities in the solar interior 11\nACKNOWLEDGEMENTS\nThe authors thank Nic Brummell, Nigel Weiss and Geoff\nVasil for stimulating discussions.\nThe numerical calculations were carried out on the\nUKMHD cluster based in St Andrews, which is partially\nfunded by STFC. This research is supported via a rolling\ngrant from STFC that is held at DAMTP, University of\nCambridge. PJB and LJS also wish to acknowledge support\nfrom the KITP, Santa Barbara and travel grants from the\nRAS to facilitate attendance at a workshop where some of\nthe work was done.\nREFERENCES\nBushby P. J., Houghton S. M., 2005, MNRAS, 362, 313.\nBushby P. J., 2006, MNRAS, 371, 772.\nBr¨ uggen, M., Hillebrandt, W., 2001, MNRAS, 323, 56.\nBrummell, N., Cline, K., Cattaneo, F., 2002, MNRAS, 329,\nL73.\nCattaneo F., Hughes D. W., 1988, JFM, 196, 323.\nCattaneo F., Hughes D. W., 2006, JFM, 553, 401.\nCattaneo F., Brummell N.,Cline K.S., 2006, MNRAS,365,\n727.\nChan, K. H., Liao, X., Zhang, K., Jones, C. A., 2004, A&A,\n423, 37.\nChandrasekhar, S., 1961, Hydrodynamic and Hydromag-\nnetic Stability, Pub. Dover.\nCharbonneau, P. & MacGregor, K. B., 1997, ApJ, 486, 502.\nChristensen-Dalsgaard, J. & Thompson, M. J., 2007, Ch.\n3 in The Solar Tachocline, CUP.\nDikpati, M. & Gilman, P.A., 2009, Space Science Reviews,\nDOI 10.1007/s11214-008-9484-3.\nDormy E. Soward A. (eds), 2007, Mathematical Aspects of\nNatural Dynamos, CRC Press.\nFrank, A., Jones, T. W., Ryu, D., Gaalaas, J. B., 1996,\nApJ, 460, 777.\nFan Y., 2001, ApJ, 546, 509.\nHughes, D. W., 1985, GAFD, 32, 273.\nHughes, D. W., Tobias, S. M., 2001, Proc. Roy. Soc. A.,\n457, 1365.\nHughes, D. W., 2007, The Solar Tachocline, pp 275–298\n(CUP).\nK¨ apyl¨ a P. J., Korpi M. J., Brandenburg, A., 2008, A&A,\naccepted.\nGilman, P.A., 1970, ApJ, 162, 1019.\nGilman, P.A. & Cally, P.S., 2007, The Solar Tachocline, pp\n243–274 (CUP).\nKersal´ e E., Hughes D. W., Tobias S. M., 2007, ApJ, 663,\nL113\nJouve, L., & Brun, A. S., AN, 2007, 328, 10, 1104.\nLin, M.-K. Silvers, L. J. & Proctor, M. R. E., 2008, Phys.\nLett. A, 373, 1, 69.\nMatthews P. C., Hughes D. W., Proctor M. R. E., 1995,\nApJ, 448, 938.\nMatthews P. C., Proctor M. R. E. Weiss N. O., JFM, 305,\n281.\nMoffatt H. K., 1978, Magnetic field generation in\nelectrically-conducting fluids, CUP: Cambridge.\nNewcomb, W. A., 1961, Phys. Fluids, 4 391.\nOssendrijver M. 2003, Astron. Astrophys. Rev., 11, 287.Palotti, M. L., Heitsch, F., Zweibel, E. G., Huang, Y.-M.,\n2008, ApJ, 678, 234.\nParker E. N., 1955, ApJ, 121, 491.\nParker E. N. 1993, ApJ, 408, 707.\nProctor, M.R.E. 2006, EAS Pubs Series, 21, 241-273.\nRyu, D., Jones, T. W., Frank, A. 2000, ApJ, 545, 475.\nSilvers L. J, 2008, Phil. Roy. Trans. Soc. A., 366, 4453.\nSilvers, L.J., Vasil, G.M., Brummell, N.C. & Proctor,\nM.R.E., 2009, ApJL, submitted .\nSpiegel E. A., Zahn J.-P., 1992, Astron. Astrophys., 265,\n106.\nTobias S. M., Brummell N. H., Clune T. L. Toomre, J.,\n1998, ApJL, 502, 177.\nTobias S. M., Brummell N. H., Clune T. L., Toomre J.,\n2001, ApJL, 549, 2, 1183.\nTobias S. M., Hughes, D. W., 2004, ApJ, 603, 785.\nTobias, S.M. & Weiss, N.O., 2007, The Solar Tachocline,\npp 319–350 (CUP).\nVasil G. M., Brummell N. H., 2008, ApJ, 686, 709.\nVasil G. M., Brummell N. H., ApJ, 690, 783.\nWissink, J. G., Matthews, P. C., Hughes, D. W., Proctor,\nM. R. E., 2000, ApJ , 536, 2, 982-997.\nZhang, K., Liao, X. Schubert, G., 2004, ApJ, 602, 468.\nc∝circleco√yrt2009 RAS, MNRAS 000, 1–11"    },    {        "title": "2007.12462v1.Can_the_dynamics_of_test_particles_around_charged_stringy_black_holes_mimic_the_spin_of_Kerr_black_holes_.pdf",        "content": "Can the dynamics of test particles around charged stringy black holes\nmimic the spin of Kerr black holes?\nBakhtiyor Narzilloev,1,\u0003Javlon Rayimbaev,2, 3,ySanjar Shaymatov,2, 3, 4, 5,z\nAhmadjon Abdujabbarov,2, 3, 5, 6,xBobomurat Ahmedov,2, 3, 5,{and Cosimo Bambi1,\u0003\u0003\n1Center for Field Theory and Particle Physics and Department of Physics, Fudan University, 200438 Shanghai, China\n2Ulugh Beg Astronomical Institute, Astronomicheskaya 33, Tashkent 100052, Uzbekistan\n3National University of Uzbekistan, Tashkent 100174, Uzbekistan\n4Institute for Theoretical Physics and Cosmology,\nZheijiang University of Technology, Hangzhou 310023, China\n5Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, Kori Niyoziy, 39, Tashkent 100000, Uzbekistan\n6Shanghai Astronomical Observatory, 80 Nandan Road, Shanghai 200030, P. R. China\n(Dated: July 27, 2020)\nWe study the motion of electrically charged particles, magnetic monopoles, and magnetic dipoles\naround electrically and magnetically charged stringy black holes. From the analysis of the radius\nof the innermost stable circular orbit (ISCO) of electrically charged particles, we show that the\nelectric charge Qof stringy black holes can mimic well the spin of Kerr black holes; the black hole\nmagnetic charge Qmcan mimic spins up to a\u0003'0:85 for magnetic dipoles; the magnetic charge\nparametergof a magnetic monopole can mimic spins up to a\u0003'0:8. This is due to the destructive\ncharacter of the magnetic \feld and such a result excludes the existence of an appreciable black hole\nmagnetic charge in astrophysical black holes from the observation of rapidly rotating objects with\ndimensionless spin up to a\u0003'0:99. We then consider the magnetar SGR (PSR) J1745-2900 as a\nmagnetic dipole orbiting the supermassive black hole Sagittarius A* (Sgr A*). We show that Sgr A*\nmay be interpreted as a stringy black hole with magnetic charge Qm=M\u00140:4118.\nPACS numbers: 04.50.-h, 04.40.Dg, 97.60.Gb\nI. INTRODUCTION\nGeneral Relativity has been extensively tested in weak\ngravitational \felds [1], and the interest is now shifting\nto test the theory in the strong \feld regime [2{6]. This\nis also possible thanks to new observational facilities ca-\npable of providing unprecedented high quality data. In\nsome speci\fc cases, we have completely new observations,\nwhich were impossible up to a few years ago. The Event\nHorizon Telescope (EHT) collaboration has recently re-\nleased the direct image of the supermassive black hole in\nthe elliptic galaxy Messier 87 (M87) [7, 8]. The LIGO-\nVirgo experiment can now detect gravitational waves\nfrom the coalescence of black holes and neutron stars\nin compact binaries [9, 10]. On the other hand, we know\nthat General Relativity has a number of theoretical is-\nsues demanding new physics, such as the presence of\nspacetime singularities in almost all physically relevant\nsolutions, the di\u000eculties to \fnd a UV-complete theory of\nquantum gravity, and some fundamental problems aris-\ning in the black hole evaporation process. Observational\ntests of General Relativity in the strong \feld regime have\nthus the potentiality to show us deviations from the pre-\n\u0003Electronic address: nbakhtiyor18@fudan.edu.cn\nyElectronic address: javlon@astrin.uz\nzElectronic address: sanjar@astrin.uz\nxElectronic address: ahmadjon@astrin.uz\n{Electronic address: ahmedov@astrin.uz\n\u0003\u0003Electronic address: bambi@fudan.edu.cndictions of Einstein's gravity and the path towards a UV-\ncomplete theory of quantum gravity.\nThe well-known Kerr-Sen solution proposed in Ref. [11]\ndescribes black holes in a low-energy limit of heterotic\nstring theory. The solution is speci\fed by the black\nhole massM, the black hole rotation parameter a(or,\nequivalently, the dimensionless black hole spin param-\netera\u0003=a=M), and an additional electric charge Q\nassociated to a U(1) gauge \feld. Optical properties\nof Kerr-Sen black holes were studied in [12{17]. Ra-\ndial geodesics around Kerr-Sen black holes were studied\nin [18]. In Ref. [19], the Parikh-Wilczek method was used\nto study their Hawking radiation. Other thermodynam-\nical properties of Kerr-Sen black holes were extensively\ninvestigated in [20{22]. Instabilities of charged massive\nscalar \felds around Kerr-Sen black holes were analyzed\nin [23]. Particle collisions near Kerr-Sen Dilaton-Axion\nblack holes were studied in [24]. Motion of magneti-\ncally charged particles were discussed in [25]. Stringy\ne\u000bects on the relative time delay in the Kerr-Sen and\nKerr-Newmann spacetimes were studied in [26, 27]. The\noptical properties of a luminous object orbiting near the\nhorizon of a Kerr-Sen black hole were calculated in [28].\nThe approximate \fnal spin and quasinormal modes of\na Kerr-Sen black hole resulting from the merger process\nwere analyzed in [29].\nThe properties of the electromagnetic radiation emit-\nted by material orbiting an astrophysical black hole are\ndetermined by the motion of massive and massless par-\nticles in the strong gravity region of the compact object\nand the study of these properties can thus be a useful toolarXiv:2007.12462v1  [gr-qc]  24 Jul 20202\nto test General Relativity in the strong \feld regime [3, 6].\nIn General Relativity, the no-hair theorem guarantees\nthat black holes are only characterized by three parame-\nters (mass, spin, and electric charge) and therefore can-\nnot have their own intrinsic magnetic \feld [30]. However,\na black hole can be embedded into an external magnetic\n\feld, and the simplest scenario is that of an external,\nasymptotically uniform, magnetic \feld [31]. In the case\nof a rotating black hole, the e\u000bect of frame dragging al-\nters the structure of the asymptotically uniform mag-\nnetic \feld and an additional electric \feld appears. The\ntrajectories of charged particles may have a chaotic be-\nhavior [32{37]. In Refs. [38{47], the authors studied\nthe motion of particles in black hole spacetimes in the\npresence of external magnetic \felds.\nThe study of the motion of magnetic dipoles in black\nhole spacetimes in the presence of magnetic \felds can be\nseen as a tool to study the spacetime structure of the\nstrong gravity region around the compact object. The\n\frst pioneering attempts to analyze the motion of mag-\nnetic dipoles around Schwarzschild and Kerr black holes\nembedded in external magnetic \felds were presented in\nRefs. [48, 49]. The motion of magnetic dipoles around\ndeformed Schwarzschild black holes in the presence of\nmagnetic \felds was studied in [50]. Magnetic dipole\ncollisions near rotating black holes in the presence of\nquintessence were investigated in [51]. Acceleration of\nmagnetic dipoles near compact objects in the presence\nof external magnetic \feld were explored, for instance, in\nRefs. [52{55] in di\u000berent modi\fed gravity models. In our\nrecent papers, we studied the magnetic dipole motion in\nconformal gravity and in modi\fed gravity models [56, 57].\nStudies of the electromagnetic \feld around black holes in\nthe presence of an external, asymptotically uniform, and\ndipolar magnetic \feld are reported in Refs. [58{85].\nIn the present paper, we study the motion of electri-\ncally charged particle, magnetic monopoles, and mag-\nnetic dipoles in the strong gravity region of an electri-\ncally and magnetically charged Kerr-Sen black hole. The\nmanuscript is organized as follows. In Section II, we\nstudy the motion of electrically charged particles around\nelectrically charged stringy black holes. Section III is\ndevoted to the dynamics of magnetic monopoles around\nmagnetically charged stringy black holes. We explore the\nmotion of magnetic dipoles around magnetically charged\nstringy black holes in Section IV. Sections V and VI are\ndevoted, respectively, to possible astrophysical applica-\ntions and to our concluding remarks. Throughout the\npaper, we employ geometrized units in which G=c= 1\nand the spacetime signature ( \u0000;+;+;+). Greek (Latin)\nindices run from 0 to 3 (from 1 to 3).II. CHARGED PARTICLE MOTION AROUND\nELECTRICALLY CHARGED STRINGY BLACK\nHOLES\nIn this section, we study the motion of charged par-\nticles around electrically charged, static, stringy black\nholes, whose spacetime is described by the line ele-\nment [11]\nds2=\u0000N(r)dt2+1\nN(r)dr2+r2\u0012\n1 +2b\nr\u0013\nd\u00122\n+r2\u0012\n1 +2b\nr\u0013\nsin\u0012d\u001e2; (1)\nN(r) =\u0014\n1\u00002(M\u0000b)\nr\u0015\u0012\n1 +2b\nr\u0013\u00001\n;\nwhereb=Q2=2M,Qis the black hole electric charge,\nandMis the black hole mass. One can easily \fnd the\nradius of the event horizon from N(r) = 0 and the result\nis\nrh= 2(M\u0000b): (2)\nThe event horizon disappears when bext=M, or, equiva-\nlently, when Qext=p\n2M. The 4-potential of the electric\n\feld around the black hole is\nA\u0016=(\n\u0000Q\nr\u0012\n1 +2b\nr\u0013\u00001\n;0;0;0)\n: (3)\nThe components of the electric \feld measured by an\nobserver with 4-velocity u\u000bare given by\nE\u000b=F\u000b\fu\f; (4)\nwhereF\u000b\f=A\f;\u000b\u0000A\u000b;\fis the Faraday tensor and a\nsemicolon denotes a covariant derivate. The 4-velocity of\nthe proper observer is\nu\u000b=0\n@s\n1 +2M2\nM(r\u00002M) +Q2;0;0;01\nA; (5)\nand the orthonormal components of the electric \feld are\nE^r=M2Q\n(Mr+Q2)2; E^\u0012=E^\u001e= 0: (6)\nEq. (6) shows that the only non-vanishing component of\nthe electric \feld is indeed the radial one, as assumed. One\ncan easily check that in the Newtonian limit ( M=r!0)\nthe above expression reduces to\nE^r=Q\nr2; (7)\nand we recover the Reissner-Nordstr om case in the weak\n\feld limit.3\nLet us now use the Hamilton-Jacobi equation to derive\ncharged particle trajectories. The basic equation is\ng\u000b\f\u0012@S\ndx\u000b+qA\u000b\u0013\u0012@S\ndx\f+qA\f\u0013\n=\u0000m2; (8)\nwhereqandmare the electric charge and the mass of the\ntest particle, respectively. Considering the symmetries of\nthe spacetime, the action of the test particle Sin (8) can\nbe written as\nS=\u0000Et+L\u001e+S\u0012+Sr; (9)\nwhereEandLare the energy and angular momentum\nof the charged particle, respectively. For what follows, it\nis convenient to introduce the speci\fc energy E=E=m\nand speci\fc angular momentum L=L=m. The equation\nof motion (8) becomes\n\u0000(2bE+Er+qQ)2\nN(r)(2b+r)2+N(r)\u0012@Sr\n@r\u00132\n(10)\n+L2\n(2br+r2) sin2\u0012+1\n2br+r2\u0012@S\u0012\n@\u0012\u00132\n=\u00001:\nFig. 1 shows some examples of trajectories for a charged\nparticle. If we increase the value of the electric charge Q,\nwe increase the average radius of the trajectory. This fact\ncan be easily interpreted as the result of the interaction\nbetween the electric charge of the test particle and the\nelectric charge of the black hole: if both electric charges\nare positive, the force is repulsive and the average radius\nincreases while, for charges of opposite sign, the force is\nattractive and the radius decreases.\nNow we consider the equatorial plane ( \u0012=\u0019=2) as the\nplane of motion of the charged particle where the e\u000bective\npotential can be de\fned from the relation\n_r2=h\nE\u0000V\u0000\ne\u000b(r)ih\nE\u0000V+\ne\u000b(r)i\n; (11)\nwhere\nV\u0006\ne\u000b=qQ\nr \n1 +Q2\nM!\u00001\n(12)\n\u0006vuuut1\u00002M\nr\u0010\n1\u0000Q2\n2M2\u0011\n1 +Q2\nMr\"\n1 +L2\nr2\u0012\n1 +Q2\nMr\u0013\u00001#\n:\nIn what follows, we will only consider V+\ne\u000b, and we will\nsimply write it as Ve\u000b, because it is the e\u000bective potential\nassociated to positive energy orbits.\nThe radial pro\fle of the e\u000bective potential Ve\u000bis shown\nin Fig. 2. The electric charges of the stringy black hole\nand of the test particle can increase or decrease the value\nofVe\u000bwith respect to the Schwarzschild case. Generally\nspeaking, the e\u000bective potential decreases if the electric\ncharges of the stringy black hole and of the test particle\nhave the same sign and increases if they have opposite\nsign.Circular orbits are of particular interest because of\ntheir astrophysical applications. The Novikov-Thorne\nmodel [86, 87] is the standard framework for the de-\nscription of thin accretion disks around black holes. The\nmodel assumes that the particles of the disk follow nearly-\ngeodesic, equatorial, circular orbits, slowly falling into\nthe gravitational well of the black hole. The inner edge\nof the accretion disk is set at the innermost stable circu-\nlar orbit (ISCO) and when a particle reaches the ISCO\nit quickly plunges onto the black hole. The ISCO can\nthus have important observational implications when it\ncan be associated to the inner edge of the accretion disk\nof a source. The ISCO radius can be found by solving\nthe following set of equations\nVe\u000b(r) =E; V0\ne\u000b(r) = 0; V00\ne\u000b(r) = 0: (13)\nFig. 3 shows the radial coordinate of the ISCO radius as\na function of the black hole electric charge Qand the par-\nticle electric charge q. From the left panel in Fig. 3, we\ncan see quite an intersting phenomenon. If the electric\ncharge of a particle with unit mass is q=\u0000Qext=2, then\nfor a maximally charged black hole ( Qext=p\n2M) the\nISCO radius tends to in\fnity, namely no stable circular\norbits can exist, no matter how far the charged particle\nis from the black hole. The right panel in Fig. 3 simply\nshows that, when the sign of the electric charges of the\nblack hole and the test particle is the same (opposite), the\nelectrostatic interaction decreases (increases) the gravita-\ntional force, and therefore the ISCO radius has a smaller\n(larger) value.\nThe energy of an electrically charge particle in circu-\nlar orbits is shown in Fig. 4. If the attractive force gets\nstronger, the particle energy decreases. We can also see\nthat the energy of a test particle with opposite charge\nwith respect to the black hole is smaller than the case\nin which the test particle and the black hole have elec-\ntric charges of the same sign. The radial pro\fle of the\nangular momentum of an electrically charged particle in\ncircular orbits is shown in Fig. 5. The minimum corre-\nsponds to the ISCO radius for that particular parameter\ncon\fguration.\nIt is also remarkable that the electric charge of a static\nnon-rotating black hole can produce similar e\u000bects, and\nthus mimic, the spin parameter of an uncharged Kerr\nblack hole. This is shown in Fig. 6. The value of the\nISCO radius has some important observational e\u000bects.\nHowever, the same radial coordinate of the ISCO radius\ncan be associated either to a non-rotating but electri-\ncally charged stringy black hole and to an electrically\nuncharged Kerr black hole. It is also remarkable that\na maximally charged stringy black hole shares the same\nISCO radius with a maximally rotating Kerr black hole;\nthat is, there is a one-to-one correspondence between\nnon-rotating stringy black holes and Kerr black holes.\nSuch a degeneracy suggests that from the observational\npoint of view it may be challenging to distinguish non-\nrotating stringy black holes from Kerr black holes in the\nUniverse.4\nFIG. 1: Trajectories of test particles with positive electric charge for di\u000berent values of the black hole electric charge Q. The\ntop row is for a black hole with electric charge Q=M =\u00000:5, the central row is for an electrically neutral Schwarzschild black\nhole, and the bottom row is for a black hole with electric charge Q=M = 0:5.\nWe note, however, that astronomical macroscopic ob-\njects tends to have a very small electric charge. Be-\ncause of the highly ionized environment around black\nholes, we can expect that any possible initial large electric\ncharge can be almost neutralized very quickly to a non-\nvanishing, but very small, equilibrium electric charge. In\nsuch a case, a small electric charge of a stringy black hole\ncannot mimic a fast-rotating Kerr black hole.III. MAGNETIC MONOPOLE MOTION\nAROUND MAGNETICALLY CHARGED\nSTRINGY BLACK HOLES\nIn the previous section, we have discussed the electri-\ncally charged stringy black hole metric (also known as\nSen black hole metric), which is a solution of heterotic\nstring theory in the four dimensional low-energy \feld the-\nory limit. However, if the black hole metric is magneti-\ncally charged, the spacetime metric is di\u000berent [88]. One5\nSchwarzschild\nQ=0.5\nQ=1\nQ=-1\n1 2 5 10 200.00.51.01.5\nr/MVeffL=6, q=1\nq=0\nq=1\nq=2\nq=-1\n2 5 10 20-0.50.00.51.01.5\nr/MVeffL=6, Q=0.5\nFIG. 2: Radial pro\fle of the e\u000bective potential of charged particles around static and electrically charged stringy black holes\nin the equatorial plane \u0012=\u0019=2. The left panel is for di\u000berent values of the black hole charge. The right panel is for di\u000berent\nvalues of the particle charge.\nq=0\nq=3\nq=-0.65\nq=-2\n2\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.423456\nQISCO radius/M\nQ=0.5\nQ= -0.5\n0.0 0.2 0.4 0.6 0.8 1.05.455.505.555.605.655.705.75\nqISCO radius/M\nFIG. 3: Left panel: ISCO radius as a function of the black hole electric charge Qfor di\u000berent values of the particle electric\nchargeq. Right panel: ISCO radius as a function of the particle electric charge qfor di\u000berent values of the black hole electric\nchargeQ.\ncan write the action for heterotic string theory in the four\ndimensional low-energy regime as [25, 89]\nS=Z\nd4xp\u0000g e\u00002'h\nR+ 4(r')2\u0000F\u000b\fF\u000b\f\n\u00001\n12H\u000b\f\rH\u000b\f\ri\n; (14)\nwhere'is the dilaton \feld and e'is regarded as a cou-\npling constant reinforcing the stringy e\u000bects. Here the 3-\nformH\u000b\f\rconsists of the potential B\u000b\fand the Maxwellgauge \feld A\u000b, which are related by\nH\u000b\f\r =@\u000bB\f\r+@\rB\u000b\f+@\fB\r\u000b\n\u00001\n4\u0010\nA\u000bF\f\r+A\rF\u000b\f+A\fF\r\u000b\u0011\n:(15)\nFrom the action in (14), we can recover the standard\nEinstein-Hilbert-Maxwell action by setting H\u000b\f\r= 0. In\nfact, by rescaling the metric tensor through the coupling\nconstantg\u000b\f!e\u00002'g\u000b\f, one can rewrite the action in6\nSchwarzschild\nQ=0.5\nQ= -0.5\n3 4 5 6 70.81.01.21.41.61.82.0\nr/Mℰq=1\nq=0\nq=2\nq=3\n3 4 5 6 70.81.01.21.41.61.82.0\nr/MℰQ=0.5\nFIG. 4: Energy of an electrically charged particle in a circular orbit as a function of the orbital radius for di\u000berent values of\nthe black hole electric charge Q(left panel) and of the particle electric charge q(right panel).\nSchwarzschild\nQ=0.5\nQ=1\n2 5 10 201020304050607080\nr/Mℒ2\nM2q=1\nq=0\nq=1\nq=2\n5 10 201020304050607080\nr/Mℒ2\nM2Q=0.5\nFIG. 5: Angular momentum of an electrically charged particle in a circular orbit as a function of the orbital radius for di\u000berent\nvalues of the black hole electric charge Q(left panel) and of the particle electric charge q(right panel).\nthe following form\nS=Z\nd4xp\u0000g\u0010\nR\u00002(r')2\u0000e\u00002'F2\u0011\n:(16)\nThe above action satis\fes to the following equation of\nmotion of the Maxwell \feld\nr\u000b\u0010\ne\u00002'F\u000b\f\u0011\n= 0: (17)\nNote that this equation is invariant under the transfor-\nmationF!F?,'! \u0000'. From Eq. (17), F?\n\u000b\f=\ne\u00002'1\n2\u000f\u000b\f\r\u001aF\r\u001ais satis\fed as a curl-free [25, 90]. With\nsuch a symmetry, the electromagnetic duality transfor-\nmation, i.e. '! \u0000', can transform an electrically\ncharged black hole solution into a magnetically charged\none. Consequently, the spacetime metric describing a\nmagnetically charged black hole in Schwarzschild coordi-nates (t;r;\u0012;' ) is written as\nds2=\u0000f(r)\nh(r)dt2+dr2\nf(r)h(r)+r2d\u00122+r2sin2\u0012d\u001e2;(18)\nwhere\nf(r) = 1\u00002M\nr; h(r) = 1\u0000Q2\nm\nMr; (19)\nMis the black hole mass and Qmis related to the black\nhole magnetic charge. It is worth noting that the event\nhorizon of the above black hole spacetime is given by\n\u0012\n1\u00002M\nr\u0013 \n1\u0000Q2\nm\nMr!\n= 0: (20)\nFrom the above equation, it is immediately clear that\nthe event horizon is located at rh= 2M, which is the\nsame radial coordinate of the event horizon as in the7\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00.20.40.60.81.0\nQa\nFIG. 6: Relation between the electric charge Qof a non-\nrotating stringy black hole and the rotation parameter aof\nan electrically uncharged Kerr black hole when we require the\nsame value of the ISCO radius (in units of M). This plot is\nobtained in the case of electrically neutral test particles and\nsuggests that a non-rotating stringy black hole can mimic an\nelectrically uncharged Kerr black hole.\nSchwarzschild spacetime. We also note that the black\nhole magnetic charge cannot exceed the value Qm=p\n2M.\nLet us start with the study of the motion of a magnetic\nmonopole in the background geometry of a magnetically\ncharged stringy black hole. The test particle has rest\nmassm, vanishing electric charge q= 0, and magnetic\nchargeqm. In general, the Hamiltonian of the system is\ngiven by [30]\nH\u00111\n2g\u000b\f\u0012@S\n@x\u000b\u0000qA\u000b+iqmA?\n\u000b\u0013\n\u0002\u0012@S\n@x\f\u0000qA\f+iqmA?\n\f\u0013\n; (21)\nwhereSis the action, x\u000bare the spacetime coordinates,\nA\u000bis the 4-vector potential, and A?\n\u000bis the dual 4-vector\npotential. The non-vanishing components of A\u000bandA?\n\u000b\nare\nA?\nt=\u0000iQm\nrandA\u001e=\u0000Qmcos\u0012: (22)\nHere we only consider the motion of a magnetic\nmonopole,q= 0, and thus we are interested in the non-\nvanishingA?\ntof the electromagnetic \feld as the test par-\nticle has only a non-vanishing magnetic charge qm[25].\nThe case of an electrically charged particle was discussed\nin the previous section.The Hamiltonian is a constant and we can set H=k=2,\nwherekandmare related by k=\u0000m2. For the above\nHamilton-Jacobi equation, the action Sfor the motion\nof a magnetic monopoles in the gravitational \feld of a\nmagnetically charged black hole has the following form\nS=\u00001\n2k\u0015\u0000Et+L'+Sr(r) +S\u0012(\u0012); (23)\nwhereSrandS\u0012are functions of only rand\u0012, respec-\ntively. We can then rewrite the Hamilton-Jacobi equation\nin the following expanded form:\n\u0000h(r)\nf(r)\u0014\n\u0000E+qmQm\nr\u00152\n+f(r)h(r)\u0012@Sr\n@r\u00132\n+1\nr2\u0012@S\u0012\n@\u0012\u00132\n+L2\nr2sin2\u0012\u0000k= 0: (24)\nThere are four independent constants of motion in the\nabove equation: E,L,k, and a fourth constant related to\nthe latitudinal motion and arising from the separability\nof the action. However, in this study we ignore the fourth\nconstant of motion because we study the motion of the\nmagnetic monopole on the equatorial plane \u0012=\u0019=2 [30].\nFrom Eq. (24) it is straightforward to derive the radial\nTABLE I: Radial coordinates of the ISCO radius for mag-\nnetic monopoles orbiting a magnetically charged black hole\nfor di\u000berent values of the magnetic charge parameter gand\nblack hole magnetic charge Qm. Note that the radial coordi-\nnate of the ISCO radius always reduces to the Schwarzschild\ncase, i.e.risco= 6, wheng= 0 for any value of Qm.\ng\nQm 0:01 0 :05 0 :10 0 :50\n\u00000:01\u00000:05\u00000:1\u00000:50\n0.1 6.00001 6.00006 6.00015 6.00181\n5.99999 5.99997 5.99996 -\n0.2 6.00008 6.00044 6.00099 6.00979\n5.99992 5.99967 5.99944 -\n0.5 6.00136 6.00719 6.01545 6.13338\n5.99868 5.99379 5.98850 5.97221\n0.8 6.00708 6.03764 6.08148 6.90408\n5.99313 5.96765 5.93985 5.82584\n1.0 6.01869 6.10156 6.22713 7.82272\n5.98203 5.91674 5.84776 5.57606\nequation of motion for a magnetic monopole and we \fnd\n_r2=h\nE\u0000E\u0000(r)ih\nE\u0000E +(r)i\n; (25)\nwhere the radial function E\u0006(r;L;Qm;g) related to the\nradial motion is given by\nE\u0006(r;L;Qm;g) =gQm\nr\u0006s\nf(r)\nh(r)\u0012\n1 +L2\nr2\u0013\n;(26)8\ng= -0.01\ng= -0.1\ng= -0.5\n2 4 6 8 100.80.91.01.11.2\nr/MVeff\ng=0.01\ng=0.1\ng=0.5\n2 4 6 8 100.80.91.01.11.21.31.4\nr/MVeff\nQ=0.1\nQ=0.3\nQ=0.5\n2 4 6 8 100.940.960.981.001.021.04\nr/MVeff\nFIG. 7: Radial pro\fle of the e\u000bective potential for massive magnetic monopoles orbiting a magnetically charged black hole.\nLeft and central panels: Veffvsr=M for di\u000berent values of gin the case of Qm= 1. Right panel: Veffvsr=M for di\u000berent\nvalues ofQmin the case of g= 0:1.\ng=0.00\ng=0.05\ng=0.1\n4 5 6 7 8 9678910\nr/ML2\nQ=0.1\nQ=0.3\nQ=0.5\n4 5 6 7 8 91112131415161718\nr/ML2\nFIG. 8: Radial pro\fle of the speci\fc angular momentum for magnetic monopoles orbiting a magnetically charged black hole.\nLeft panel: L2vsr=M for di\u000berent values of magnetic charge parameter gforQm= 1. Right panel: L2vsr=M for di\u000berent\nvalues of the black hole magnetic charge Qmforg= 0:1.\nand we have de\fned g=qm=mandk=m2=\u00001. In\nEq. (25), _r2\u00150 must always be satis\fed, and this im-\nplies eitherE>E+(r;L;Qm;g) orE<E\u0000(r;L;Qm;g).\nHowever, we shall restrict ourselves to the positive en-\nergy case, which is physically associated to the e\u000bective\npotential, i.e. Ve\u000b(r;L;Q;g ) =E+(r;L;Qm;g). Note\nthat the e\u000bective potential reduces to the Schwarzschild\none in the case of vanishing magnetic charge Qm. We\nfurther focus on the e\u000bective potential Ve\u000b(r;L;Qm;g)\nto determine the motion of the magnetic monopole.\nThe radial pro\fle of the e\u000bective potential is shown in\nFig. 7 for di\u000berent values of gandQm. As can be seen\nfrom the radial pro\fle of Ve\u000b(r;L;Qm;g), the particle\nmagnetic charge parameter, g, and the black hole mag-\nnetic charge, Qm, have a similar e\u000bects if gis positive.\nIn the case of negative g, their e\u000bect is opposite. We also\nnote that the strength of the potential for positive gis\nslightly stronger than the one of the black hole magnetic\nchargeQm; see Fig. 7.\nLet us now consider the circular orbits of a magnetic\nmonopole orbiting a magnetically charged stringy black\nhole. For circular orbits, we need to consider the follow-\ning conditions for the e\u000bective potential\nVe\u000b(r;L;Qm;g) =E; V0\ne\u000b(r;L;Qm;g) = 0;(27)where here a prime0denotes a derivative with respect to\nr. It is then straightforward to calculate the speci\fc en-\nergyEand angular momentum Lof a magnetic monopole\nin a circular orbit. We \fnd\nE=gQm\nr+s\nf(r)\nh(r)\u0012\n1 +L2\nr2\u0013\n; (28)\nL2=r3\u0002\nf(r)h0(r)\u0000h(r)f0(r)\u0003\n\u00002g2Q2\nmf(r)h(r)3\nrh(r)f0(r)\u0000f(r)\u0002\nrh0(r) + 2h(r)\u0003\n+2gQmf(r)h(r)2\n\u0010\nrh(r)f0(r)\u0000f(r)\u0000\nrh0(r) + 2h(r)\u0001\u00112\n\u0002h\nh(r)\u0010\ng2Q2\nmh(r)\u00002r3f0(r)\u0011\n+ 2r2f(r)\u0010\nrh0(r) + 2h(r)\u0011i1=2\n: (29)\nThe radial pro\fle of the speci\fc angular momentum of a\nmagnetic monopole in a circular orbit around a magnet-\nically charged black hole is shown in Fig. 8.\nNow we determine the ISCO radius of a magnetic\nmonopole orbiting a magnetically charged stringy black\nhole. We calculate the ISCO radius from the condition\nV00\ne\u000b(r;L;Qm;g) = 0, as done in the previous section. We\nassumeg= 0. From V0\ne\u000b(r;L;Qm;g) = 0, we determine9\nthe minimum value of the angular momentum in a circu-\nlar orbit\nL2=\u0000\nQ2\nm\u00002M2\u0001\nr3\n6M2r\u00002M(2Q2m+r2) +Q2mr; (30)\nBy substituting Eq. (30) into V00\ne\u000b(r;L;Qm;g) = 0, we\n\fnd the following condition for the ISCO radius\n\u0010\nQ2\nm\u00002M2\u0011\n(r\u00006M)r3\u0010\nQ2\nm\u0000Mr\u00112\n= 0:(31)\nFrom this equation it is immediately clear that, in the\ncaseg= 0, we have risco= 6M, which is the same value\nas in the Schwarzschild spacetime. The observational\nimplications are straightforward: a faraway observer can-\nnot distinguish a magnetically charged stringy black hole\nfrom a Schwarzschild black hole. Table I shows the nu-\nmerical results of the ISCO radius for di\u000berent values of\nthe magnetic charge parameter gand of the black hole\nmagnetic charge Qm. Ifgis positive, the ISCO radius\nincreases as gincreases. If gis negative, the ISCO radius\ndecreases asjgjincreases. Note that for jgj\u001c1 we can\nwriterisco= 6\u0006\u000erisco, namely for small values of the\nmagnetic charge parameter there are only small changes\nin the value of the ISCO radius.\nWe now calculate the value of the magnetic charge g\nof a test particle orbiting around a magnetically charged\nstringy black hole and the value of the rotation parameter\naof a Kerr black hole leading to the same ISCO radius.\nOur results are shown in Fig. 9. However, unlike the\nprevious case of an electrically charged particle around an\nelectrically charged stringy black hole, it is not possible\nto reproduce the ISCO value of any Kerr black hole. It\nis only possible to recover the ISCO radius of Kerr black\nholes with spin parameters up to a=M\u00190:8, but it is\nimpossible to get smaller values of the ISCO radius as in\nthe case of faster rotating Kerr black holes.\nIV. MAGNETIC DIPOLES AROUND\nMAGNETICALLY CHARGED STRINGY BLACK\nHOLES\nIn this section, we study the dynamics of magnetic\ndipoles orbiting magnetically charged stringy black holes\nwith the spacetime metric given in (18) and the electro-\nmagnetic potential in (22). The non-vanishing compo-\nnents of the Faraday tensor F\u0016\u0017are\nF\u0012\u001e=\u0000F\u001e\u0012=\u0000Qmsin\u0012 : (32)\nThe magnetic \feld of the magnetically charged stringy\nblack hole can be derived from\nB\u000b=1\n2\u0011\u000b\f\u001b\u0016F\f\u001bw\u0016; (33)\nwherew\u0016is the 4-velocity of observer, \u0011\u000b\f\u001b\r is the\npseudo-tensorial form of the Levi-Civita symbol \u000f\u000b\f\u001b\r\nQm=1\nQm=1.2\nQm=1.4\n-1.0 -0.8 -0.6 -0.4 -0.2 0.00.00.20.40.60.81.0\nga/MFIG. 9: Rotation parameter aof a Kerr black hole vs magnetic\nchargegof a test particle orbiting a stringy black hole with\nmagnetic charge Qmleading to the same ISCO radius for\ndi\u000berent values of Qm. However, the stringy scenario can\nmimic the standard scenario of a Kerr black hole only for a\nKerr black hole spin parameter up to 0.8.\nde\fned as\n\u0011\u000b\f\u001b\r =p\u0000g\u000f\u000b\f\u001b\r; \u0011\u000b\f\u001b\r=\u00001p\u0000g\u000f\u000b\f\u001b\r;(34)\nwhereg= detjg\u0016\u0017j=\u0000r4sin2\u0012for the spacetime metric\nin (18),\u000f0123= 1, and\n\u000f\u000b\f\u001b\r =8\n><\n>:+1;for even permutations ;\n\u00001;for odd permutations ;\n0;for the other combinations ;:(35)\nThe orthonormal radial component of the magnetic\n\feld of the magnetically charged stringy black hole is\nB^r=Qm\nr2: (36)\nEq. (36) implies that the radial component of the mag-\nnetic \feld around a magnetically charged black hole is not\ne\u000bected by the spacetime geometry of the stringy black\nhole and formally coincides with the standard Newtonian\nexpression.\nWe can study the dynamics of magnetic dipoles around\nmagnetically charged black holes using the Hamilton-\nJacobi equation [48]\ng\u0016\u0017@S\n@x\u0016@S\n@x\u0017=\u0000 \nm\u00001\n2D\u0016\u0017F\u0016\u0017!2\n; (37)10\nwhere the termD\u0016\u0017F\u0016\u0017is responsible for the interac-\ntion between the magnetic dipole and the magnetic \feld\ngenerated by the magnetic charge of the stringy black\nhole. Here we assume that the magnetic dipole has the\ncorresponding polarization tensor D\u000b\fthat satis\fes the\nfollowing conditions\nD\u000b\f=\u0011\u000b\f\u001b\u0017u\u001b\u0016\u0017;D\u000b\fu\f= 0; (38)\nwhere\u0016\u0017is the dipole moment of the magnetic dipole.\nHere we determine the interaction term D\u0016\u0017F\u0016\u0017using the\nrelation between the Faraday tensor F\u000b\fand the compo-\nnents of the electric \feld, E\u000b, and of the magnetic \feld,\nB\u000b,\nF\u000b\f=w\u000bE\f\u0000w\fE\u000b\u0000\u0011\u000b\f\u001b\rw\u001bB\r: (39)\nEmploying the condition given in Eq. (38) and the non-\nzero components of the Faraday tensor, we have\nD\u000b\fF\u000b\f= 2\u0016\u000bB\u000b= 2\u0016^\u000bB^\u000b: (40)\nWe assume that the motion of the test particle is on\nthe equatorial plane and that its magnetic dipole mo-\nment is aligned along the direction of magnetic \feld lines\nof the stringy black hole. In such a case, the compo-\nnents of the dipole magnetic moment of the particle are\n\u0016i= (\u0016r;0;0). This con\fguration allows for an equi-\nlibrium state for the interaction between the magnetic\n\feld and the magnetic dipole, while other con\fgurations\ncannot provide any stable equilibrium state. This con\fg-\nuration also allows to study the particle motion and we\nmay avoid the relative motion problem by choosing the\nappropriate observer's frame. Due to the constant value\nof the magnetic moment of the particle, the second con-\ndition in (38) is automatically satis\fed. The interaction\npart can be calculated using Eqs. (40) and (32)\nD\u000b\fF\u000b\f=2\u0016Qm\nr2; (41)\nwhere\u0016=q\n\u0016^i\u0016^iis the norm of the magnetic dipole\nmoment of the particle.\nDue to the symmetries of the magnetic \feld and of\nthe spacetime, we can write the action of the magnetic\ndipole in the Hamilton-Jacobi equation (37) in the fol-\nlowing form\nS=\u0000Et+L\u001e+Sr\u0012(r;\u0012): (42)\nSince we consider the motion on the equatorial plane\n(\u0012=\u0019=2), Eqs. (40), (37) and (42) provide the following\nequation for the radial component\n_r2=E2\u0000Ve\u000b(r;L;B); (43)\nwhere the e\u000bective potential has the form\nVe\u000b(r;L;B) =f(r)\nh(r)\"\u0012\n1\u0000B\nr2\u00132\n+L2\nr2#\n(44)where\nB=\u0016Qm\nm;\nis the magnetic interaction parameter responsible for the\ninteraction between the magnetic dipole of the test par-\nticle and the proper magnetic \feld of the magnetically\ncharged stringy black hole. \f=\u0016=(mM) is a dimension-\nless parameter characterizing the magnetic dipole and the\nspacetime parameters. \fis always positive. Modeling a\nmagnetized neutron star as a test magnetic dipole with\nmoment\u0016= (1=2)BNSR3\nNSorbiting around a supermas-\nsive black hole (SMBH), we \fnd\n\f=BNSR3\nNS\n2mNSMSMBH= 0:0341\u0012BNS\n1012G\u0013\u0012RNS\n106cm\u0013\n\u0002\u0012mNS\n1:4M\f\u0013\u00001\u0012MSMBH\n3:8\u0001106M\f\u0013\u00001\n: (45)\nThe circular stable orbits of the magnetic dipole\naround the central object can be derived by the condi-\ntions\nV0\ne\u000b= 0;V00\ne\u000b\u00150; (46)\nwhich can be used to \fnd the speci\fc angular momentum\nand the speci\fc energy in circular orbits of the magnetic\ndipole:\nL2=\u0000\nr2\u0000B\u0001\nr2F(r)n\n2M2\u0010\nr3\u00005rB\u0011\n+ 4MB\u0010\n2Q2\nm+r2\u0011\n\u0000Q2\nmr\u0010\nr2+ 3B\u0011o\n; (47)\nE2=2M(r\u00002M)2\nF(r) \n1\u0000B2\nr4!\n; (48)\nwhereF(r) = 2M(Q2\nm+r2)\u0000r\u0000\n6M2+Q2\nm\u0001\n.\nFig. 10 shows the radial pro\fle of the speci\fc angular\nmomentum of a magnetic dipole around a magnetically\ncharged stringy black hole. One can see from the \fgure\nthat if we increase the magnetic charge of stringy black\nhole (the parameter \ffor the magnetic dipole), the spe-\nci\fc angular momentum of the magnetic dipole decreases\nand the inner circular orbit comes closer to the central\nobject, while the parameter \fdoes not change the dis-\ntance of the last circular orbit.\nThe radial pro\fle of the speci\fc energy of a magnetic\ndipole for di\u000berent values of the parameter \fand of the\nblack hole magnetic charge Qmis shown in Fig. 11. One\ncan see from Fig. 11 that the increase of both the black\nhole magnetic charge Qmand of the parameter \fmake\nthe speci\fc energy of the magnetic dipole in circular or-\nbits increase. However, the e\u000bect of the magnetic charge\nis stronger than the e\u000bect of the parameter \f.\nWe can get the equation for the ISCO radius taking\ninto account the conditions (46) for the e\u000bective potential11\n5 10 20 504681012\nr/Mℒ\nMSchw.BH\nQm=0.5;β=0.5\nQm=1.0;β=0.5\n5 10 20 5046810\nr/Mℒ\nMSchw.BH\nQm=1;β=0.5\nQm=1;β=1.0\nFIG. 10: Radial pro\fle of the speci\fc angular momentum in\ncircular orbits of a magnetic dipole for di\u000berent values of the\nmagnetic charge Qmand of the parameter \f.\n(44) in the following form\nr5(r\u00006M)\u0010\n2M2\u0000Q2\nm\u0011\n+B2h\n3r2\u0010\nQ2\nm+ 14M2\u0011\n\u00006Mr\u0010\n3Q2\nm+ 10M2\u0011\n+ 32M2Q2\nm\u00008Mr3i\n\u00150:(49)\nThe numerical solution of Eq. (49) with respect to the\nradial coordinate is presented in Fig. 12, where we show\nthe ISCO radius as a function of the black hole mag-\nnetic charge and of the parameter \f. We can see that\nthe ISCO radius decreases if the magnetic charge pa-\nrameterQmincreases, and the decreasing rate increases\nwhen we increase the parameter \f. Moreover, an upper\nvalue for the parameter \fexists. For \fexceeding such\nan upper value, there are no circular stable orbits for the\nmagnetic dipole. This upper value for the parameter \f\ndecreases with the increase of the magnetic charge pa-\nrameter. However, the value of the ISCO radius is the\nsame for all values of the magnetic charge parameter at\nthe upper value of the parameter \f.\nV. ASTROPHYSICAL APPLICATIONS\nIn this section, we would like to answer to the fol-\nlowing question: can the magnetic charge of a stringy\n5 10 20 500.900.951.001.051.10\nr/MℰSchw.BH\nQm=0.5;β=0.5\nQm=1;β=0.5\n5 10 20 500.900.951.001.05\nr/MℰSchw.BH\nQm=1;β=0.5\nQm=1;β=1FIG. 11: Radial pro\fle of the speci\fc energy in circular orbits\nof a magnetic dipole for di\u000berent values of the magnetic charge\nQmand of the parameter \f.\nblack hole mimic the spin of a rapidly rotating Kerr\nblack hole through its magnetic interactions with orbit-\ning material? To address this question, we study the\nISCO radius of magnetic dipoles around ( i) a magneti-\ncally charged stringy black hole, ( ii) a rapidly rotating\nKerr black hole, and ( iii) a Schwarzschild black hole im-\nmersed in an external, asymptotically uniform, magnetic\n\feld. We are going to show the cases when the magnetic\ncharge mimics the spin and magnetic interaction param-\neters providing the same ISCO radius.\nWe note that we focus on the location of the ISCO be-\ncause the latter is often the key-quantity in the interpre-\ntation of the electromagnetic spectra of black holes [3, 6].\nThe radiation emitted by material at larger radii is not\nvery informative about the spacetime metric: it is nor-\nmally di\u000ecult to determine the orbital radius because\nrelativistic e\u000bects are quite similar and at larger radii\nrelativistic e\u000bects are weaker. The radiation emitted by\nmaterial inside the ISCO is normally negligible and dif-\n\fcult to model: there are no stable circular orbits inside\nthe ISCO, so when a particle reaches the ISCO it quickly\nplunges onto the black hole. In the end, the ISCO is the\nmost sensitive quantity of the spacetime metric and it\nis relatively easy to measure with electromagnetic obser-\nvations. When the electromagnetic spectrum of a black\nhole is dominated by the thermal spectrum of its accre-12\nβ=0.5\nβ=0.8\nβ=1.0\n0.0 0.2 0.4 0.6 0.8 1.05.945.965.986.00\nQm/Mr\nM\nQm=0.5\nQm=0.8\nQm=1.0\n0 5 10 153.03.54.04.55.05.56.0\nβr\nM\nFIG. 12: ISCO radius of a magnetic dipole around magnet-\nically charged stringy black hole for di\u000berent values of the\nparameter\fand the black hole magnetic charge. The top\npanel shows the impact of the parameter \fand the bottom\npanel shows the impact of the black hole magnetic charge.\ntion disk and the Eddington-scaled disk luminosity is be-\ntween\u00185% to\u001830%, the inner edge of the disk is at\nthe ISCO of the spacetime with a good approximation;\nsee [91, 92] and reference therein.\nIn General Relativity, the radial coordinate of the\nISCO radius has not a direct physical meaning, as it de-\npends on the coordinate system, but still the value of\nthe ISCO radius is normally a good proxy to compare\nblack hole spacetimes with similar observational prop-\nerties in the electromagnetic spectrum [3, 6]. Thermal\nspectra of thin accretion disks around black holes are\nmulti-temperature blackbody spectra with a high energy\ncuto\u000b determined by the inner edge of the disk, so by\nthe ISCO radius [93]: if we \ft the data with a metric in\nwhich the ISCO radius is determined by two parameters,\nwe \fnd the typical banana shape in the plot of those\nparameters and we cannot measure simultaneously the\ntwo parameters [94]. A similar problem is found in the\nanalysis of the re\rection spectrum of the disk [95, 96],\neven if in the presence of high quality data and for an\nISCO radius very close to the black hole event horizon it\nis possible to break such a parameter degeneracy [97, 98].\nPolarimetric measurements are also a\u000bected by the sameissue [99].\nThe ISCO radius for prograde and retrograde orbits of\na test particle around a rotating Kerr black hole can be\nexpressed with the following compact formula [100]\nrisco= 3 +Z2\u0006p\n(3\u0000Z1)(3 +Z1+ 2Z2);(50)\nwhere\nZ1= 1 +\u0010\n3p\n1 +a+3p\n1\u0000a\u0011\n3p\n1\u0000a2;\nZ2=q\n3a2+Z2\n1:\nWe plan to perform the above-mentioned study of the\nISCO analyzing the motion of the magnetar SGR (PSR)\nJ1745{2900 orbiting Sgr A*. We model the magnetar as\na test particle with magnetic dipole.\nThe magnetar called SGR (PSR) J1745{2900 was dis-\ncovered in 2013 in the radio band [101]. It is orbiting\nthe supermassive black hole Sgr A*, whose mass is M\u0019\n3:8\u0002106M\f. From the analysis reported in [101], we\ncan estimate the value of the parameter \f. The magnetic\ndipole moment of the magnetar is \u0016\u00191:6\u00021032G\u0001cm3\nand its mass is m\u00191:5M\f. We thus \fnd\n\f=\u0016PSR J1745\u00002900\nmPSR J1745\u00002900MSgrA\u0003\u001910:2: (51)\nSchw BH+MF\nβ=5\nβ=10\nKerr BH0.0 0.2 0.4 0.6 0.8 1.03.03.54.04.55.05.56.0\nQm/M,,ar\nM\nFIG. 13: ISCO radius of a magnetic dipole orbiting mag-\nnetically charged stringy black holes, Kerr black holes, and\nSchwarzschild black holes immersed in external magnetic\n\felds for di\u000berent values of the parameter \f.\nFig. 13 presents the ISCO radius of a magnetic dipole\naround rotating Kerr black holes, Schwarzschild black\nholes immersed in external magnetic \felds, and mag-\nnetically charged stringy black holes as a function of\ntheir proper parameters a=M2(0;1),Qm2(0;1), and\nB 2 (0;1), respectively. We can see that the e\u000bect of\nthe magnetic charge Qmis stronger than the e\u000bect of\nthe external magnetic \feld and it become stronger if we\nincrease the parameter \f.13\nA. Magnetically charged stringy black holes versus\nrotating Kerr black holes\nFirst, we consider the motion of magnetic dipoles and\nnon-magnetized particles around magnetically charged\nstringy black holes and rotating Kerr black holes, respec-\ntively. We will show how the magnetic charge of a stringy\nblack hole can mimic the spin of a Kerr black hole pro-\nviding the same ISCO radius.\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.8\nQm/Ma\nMβ=5\nβ=10\nβ=15\nFIG. 14: Relations between the spin of a rotating Kerr black\nhole and the magnetic charge of a stringy black hole providing\nthe same ISCO radius for di\u000berent values of the parameter \f.\nFig. 14 illustrates the relation between the rotation\nparameter of a Kerr black hole and the magnetic charge\nparameter of a stringy black hole that provide the same\nvalue of the ISCO radius. We can see that the magnetic\ncharge of the string black hole can mimic the spin of a\nKerr black hole up to about 0.5 Mfor the magnetic dipole\nwith the coupling parameter \f= 5. Such an upper value\nof the spin parameter increases if we increase the value of\nthe parameter \f, but cannot exceed the value a\u0003\u00190:85.\nB. Magnetically charged stringy black holes versus\nSchwarzschild black holes in magnetic \felds\nLet us now analyze the role of external magnetic \felds\non the motion of magnetic dipoles and how the magnetic\ncharge of a stringy black hole can mimic the magnetic in-\nteraction between an external magnetic \feld and a mag-\nnetic dipole. The topic of magnetic dipole motion around\nSchwarzschild black holes immersed in external, asymp-\ntotically uniform, magnetic \felds was \frst studied by de\nFelice in [48]. In Ref. [57], we extended that study to the\nmotion of magnetic dipoles and we showed how the mag-\nnetic interaction can mimic a non-rotating black hole in\nmodi\fed gravity (MOG), which is a scalar-tensor-vector\ntheory proposed in [102]. The magnetic dipoles dynam-\nics around black holes in conformal gravity [56] and 4-D\nEinstein Gauss-Bonnet gravity [103] has also shown the\ndegeneracy of the magnetic interaction parameters with\nthe spin of Kerr black holes, where a magnetized neutronstar was treated as a magnetic dipole with the magnetic\ncoupling parameter\nB=2\u0016B0\nm=BNSR3\nNSBext\nmNS\norbiting around a supermassive black hole. Plugging in\ntypical values for a neutron star and an external magnetic\n\feld, we \fnd\nB= 0:0044\u0012BNS\n1012G\u0013\u0012RNS\n106cm\u00133\u0012Bext\n10G\u0013\u0012mNS\nM\f\u0013\u00001\n:(52)\nWe can estimate the value of the interaction parame-\nter for the case of the magnetar SGR (PSR) J1745{2900\norbiting around Sgr A* and we obtain\nBPSRJ1745\u00002900'0:716\u0012Bext\n10G\u0013\n: (53)\nIn previous studies [56, 57, 103], it was shown that a\nmagnetic dipole with a magnetic interaction parameter\nB\u00151 cannot be in a stable circular orbits due to the\ndestructive nature of magnetic \felds. This implies that\none can estimate an upper limit for the value of external\nmagnetic \felds through such a condition and then predict\nthat the orbit of the magnetar SGR (PSR) J1745{2900\naround Sgr A* is stable or not by using B<1. Simple\ncalculations show that one can expect that circular orbits\nof the magnetar are stable only if the external magnetic\n\feld in the environment of SrgA* is Bext.14G. This\nindicates that a magnetar with a surface magnetic \feld\n(Bsurf) of the order of Bsurf>1014G cannot be in a\nstable orbit in the environment of a supermassive black\nhole when the external magnetic \feld is more than about\n10G. Since the expected magnetic \feld near Sgr A* is\naround 100G, the magnetic coupling parameter for the\nmagnetar (SGR) PSR J1745-2900 is B'7:16 and one\ncan expect to observe pulsars with a surface magnetic\n\feld less than 1012G. Non-observability of radio pulsars\nand magnetars in the central part of our Galaxy in vicin-\nity of SgrA* can be caused by either their nonexistence\nin the region close to ISCO or scattering of radio signals\nbroadening them, which leads to pulsar's signal disap-\npearance. The detailed analysis performed here shows\nthat the interaction of an ambient magnetic \feld with\nthe magnetar's (pulsar's) magnetic moment is so strong\nthat its orbit in close vicinity of SgrA* would become\nvery unstable and would be unlikely to \fnd a magnetar\nthere. The only opportunity is to look for radio pulsars\nwith low surface magnetic \feld in that area.\nNow we are back to the question if the magnetic charge\nof the stringy black hole can mimic magnetic \feld ef-\nfects providing the same ISCO radius. One can compare\nthe ISCO radius of the two cases of the magnetic dipole\nmotion around a Schwarzschild black hole immersed in\nan external magnetic \feld and a magnetically charged\nstringy black hole following the results of Ref. [48] and\nEq.(49).14\nAs the next step, we will focus on the possible degen-\neracy due to e\u000bect of the magnetic coupling parameter\nand the magnetic charge of the stringy black hole.\nβ=5\nβ=10\n0.0 0.2 0.4 0.60.00.20.40.60.81.0\nQm/M\nFIG. 15: Relation between the parameters of magnetic cou-\npling and magnetic charge for the same ISCO radius for dif-\nferent values of the parameter \f.\nFig. 15 demonstrates the relation between the mag-\nnetic coupling parameter and magnetic charge parameter\nof a stringy black hole. One may see that the magnetic\ncoupling parameter may mimic the magnetic charge of\na black hole up to Qm=M= 0:7532 for the magnetic\ndipole with the parameter \f= 5 while it mimics up\ntoQm=M= 0:4118 for the particle with the parameter\n\f= 10. This implies that when we apply such a result to\nthe case of the magnetar SGR (PSR) J1745{2900 orbiting\naround Sgr A* it is impossible to distinguish the e\u000bects\nof an external magnetic \feld with B\u001414G and a mag-\nnetic charge of a stringy supermassive black hole with\nthe magnetic charge Qm=M\u00140:4118. We hope that the\nestimation for a realistic case may help to perform stud-\nies of magnetic dipoles such as radio pulsars which can\nbe observed as recycled pulsars and magnetars motion\naround the supermassive black hole SrgA* in the near\nfuture when such observations will be possible.\nVI. CONCLUSIONS\n\u000fFirst, we have studied the motion of an electrically\ncharged particle around an electrically charged\nstringy black hole. If the black hole electric charge\nincreases, the ISCO radius of the charged parti-\ncle decreases. For the maximal value of the black\nhole electric charge Qext=p\n2M, we have found\nthat there is a critical value for the particle elec-\ntric chargejqextj=Qext=2 such that if the particle\nelectric charge exceeds this value the ISCO radius\nbecomes in\fnitely large; that is, particles with an\nelectric charge exceeding qexthave no stable cir-\ncular orbits around the stringy black hole. We\nhave also shown that an electrically charged stringyblack hole can assume the same ISCO radius as a\nKerr black hole, suggesting that a similar object\nmay mimic well Kerr black holes of any spin.\n\u000fSecond, we have studied the motion of mag-\nnetic monopoles in the spacetime of a magneti-\ncally charged stringy black hole. The magnet-\nically charged black hole solution recovers the\nSchwarzschild one in the case of vanishing magnetic\nchargeQm. We have found that the event horizon\nand the ISCO radius are not a\u000bected by the value\nofQmand they have thus the same value as in the\nSchwarzschild spacetime. This leads to important\nimplications from the observational point of view\nand suggests that it would challenging for a faraway\nobserver to distinguish a static and spherically sym-\nmetric Schwarzschild black hole from a magneti-\ncally charged stringy black hole. The ISCO ra-\ndius changes only for particles with a non-vanishing\nmagnetic charge. The ISCO radius increases if the\nparticle electric charge gis positive and increases.\nThe ISCO radius decreases if the particle electric\nchargegis negative and its absolute value increases.\nThe ISCO radius cannot get arbitrarily close to the\nevent horizon in the case of negative values of the\nparticle magnetic charge g.\n\u000fLast, we have studied the dynamics of magnetic\ndipoles around magnetically charged stringy black\nholes in the weak interaction limit introducing a\nnew parameter \f, describing the interaction be-\ntween the particle magnetic dipole and the cen-\ntral object. From the study of the ISCO radius of\nmagnetic dipoles, we found that the existence of an\nISCO is determined by the values of the parameter\n\fand of the black hole magnetic charge. If the pa-\nrameter\fexceeds a critical value, set by the black\nhole magnetic charge, there are no stable circular\norbits for the magnetic dipole due to the increase\nof destructive Lorentz forces. Finally, we have in-\nvestigated how the magnetic charge of a stringy\nblack hole can mimic the spin of a Kerr black hole\nand the interaction between magnetic dipoles and\nexternal magnetic \felds providing the same ISCO\nradius. Our results show that the magnetic charge\nof a stringy black hole can mimic the spin param-\neter of a Kerr black hole up to a\u0003\u00180:85, while\nthe magnetic interaction parameter can mimic the\nmagnetic charge e\u000bects up to Qm=M= 0:7532 for\na magnetic dipole when \f= 5. We applied these\n\fndings to the magnetar SGR (PSR) J1745 orbit-\ning around the SMBH Srg A* and we argued that\nthe latter may be a stringy black hole with mag-\nnetic charge up to Qm=M= 0:4118.15\nAcknowledgement\nThis research is supported by Grants No. VA-FA-F-\n2-008, No.MRB-AN-2019-29 of the Uzbekistan Ministry\nfor Innovative Development. JR, AA and BA thank Sile-\nsian University in Opava for the hospitality during their\nvisit. B.N. acknowledges support from the China Schol-arship Council (CSC), grant No. 2018DFH009013. The\nwork of B.N. and C.B. was supported by the Innova-\ntion Program of the Shanghai Municipal Education Com-\nmission, Grant No. 2019-01-07-00-07-E00035, and the\nNational Natural Science Foundation of China (NSFC),\nGrant No. 11973019.\n[1] C. M. Will, Living Reviews in Relativity 9, 3 (2006),\ngr-qc/0510072 .\n[2] E. Berti, E. Barausse, V. Cardoso, L. Gualtieri, P. Pani,\nU. Sperhake, L. C. Stein, N. Wex, K. Yagi, T. Baker,\nC. P. Burgess, F. S. Coelho, D. Doneva, A. De Felice,\nP. G. Ferreira, P. C. C. Freire, J. Healy, C. Herdeiro,\nM. Horbatsch, B. Kleihaus, A. Klein, K. Kokkotas,\nJ. Kunz, P. Laguna, R. N. Lang, T. G. F. Li, T. Lit-\ntenberg, A. Matas, S. Mirshekari, H. Okawa, E. Radu,\nR. O'Shaughnessy, B. S. Sathyaprakash, C. Van Den\nBroeck, H. A. Winther, H. Witek, M. Emad Aghili,\nJ. Alsing, B. Bolen, L. Bombelli, S. Caudill, L. Chen,\nJ. C. Degollado, R. Fujita, C. Gao, D. Gerosa, S. Ka-\nmali, H. O. Silva, J. G. Rosa, L. Sadeghian, M. Sam-\npaio, H. Sotani, and M. Zilhao, Classical and Quantum\nGravity 32, 243001 (2015), arXiv:1501.07274 [gr-qc] .\n[3] C. Bambi, Reviews of Modern Physics 89, 025001\n(2017).\n[4] V. Cardoso and L. Gualtieri, Classical and Quantum\nGravity 33, 174001 (2016), arXiv:1607.03133 [gr-qc] .\n[5] K. Yagi and L. C. Stein, Classical and Quantum Gravity\n33, 054001 (2016), arXiv:1602.02413 [gr-qc] .\n[6] H. Krawczynski, General Relativity and Gravitation 50,\n100 (2018), arXiv:1806.10347 [astro-ph.HE] .\n[7] K. Akiyama et al. (Event Horizon Telescope), Astro-\nphys. J. 875, L1 (2019).\n[8] K. Akiyama et al. (Event Horizon Telescope), Astro-\nphys. J. 875, L6 (2019).\n[9] The LIGO Scienti\fc Collaboration and the Virgo Col-\nlaboration, ArXiv e-prints (2016), arXiv:1602.03841\n[gr-qc] .\n[10] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber-\nnathy, F. Acernese, K. Ackley, C. Adams, T. Adams,\nP. Addesso, R. X. Adhikari, and et al., Physical Review\nLetters 116, 061102 (2016), arXiv:1602.03837 [gr-qc] .\n[11] A. Sen, Physical Review Letters 69, 1006 (1992), hep-\nth/9204046 .\n[12] J. An, J. Peng, Y. Liu, and X.-H. Feng, Phys. Rev. D\n97, 024003 (2018), arXiv:1710.08630 [gr-qc] .\n[13] G. N. Gyulchev and S. S. Yazadjiev, Phys. Rev. D 75,\n023006 (2007), arXiv:gr-qc/0611110 [gr-qc] .\n[14] Z. Younsi, A. Zhidenko, L. Rezzolla, R. Konoplya,\nand Y. Mizuno, Phys. Rev. D 94, 084025 (2016),\narXiv:1607.05767 [gr-qc] .\n[15] K. Hioki and U. Miyamoto, Phys. Rev. D 78, 044007\n(2008).\n[16] S. Dastan, R. Sa\u000bari, and S. Soroushfar, arXiv e-prints\n, arXiv:1610.09477 (2016), arXiv:1610.09477 [gr-qc] .\n[17] R. Uniyal, H. Nandan, and K. D. Purohit, Classical and\nQuantum Gravity 35, 025003 (2018), arXiv:1703.07510\n[gr-qc] .[18] P. A. Blaga and C. Blaga, Classical and Quantum Grav-\nity18, 3893 (2001).\n[19] G.-Q. Li, Theoretical and Mathematical Physics 153,\n1652 (2007).\n[20] D.-Y. Chen and X.-T. Zu, Modern Physics Letters A\n24, 1159 (2009).\n[21] A. Larra~ naga, Pramana 76, 553 (2011), arXiv:1003.2973\n[gr-qc] .\n[22] F. Khani, M. T. Darvishi, and R. Baghbani, Astrophys.\nSapce Sci. 348, 189 (2013).\n[23] H. M. Siahaan, International Journal of Modern Physics\nD24, 1550102 (2015), arXiv:1506.03957 [hep-th] .\n[24] U. Debnath, arXiv e-prints , arXiv:1508.02385 (2015),\narXiv:1508.02385 [gr-qc] .\n[25] P. A. Gonz\u0013 alez, M. Olivares, E. Papantonopoulos,\nJ. Saavedra, and Y. V\u0013 asquez, Phys. Rev. D 95, 104052\n(2017), arXiv:1703.04840 [gr-qc] .\n[26] R. N. Izmailov, R. K. Karimov, A. A. Potapov, and\nK. K. Nand i, Annals of Physics 413, 168069 (2020),\narXiv:2002.01149 [gr-qc] .\n[27] A. Nathanail, E. R. Most, and L. Rezzolla, Monthly\nNotices of the Royal Astronomical Society: Letters 469,\nL31 (2017), arXiv:1703.03223 [astro-ph.HE] .\n[28] M. Guo, S. Song, and H. Yan, Phys. Rev. D 101, 024055\n(2020), arXiv:1911.04796 [gr-qc] .\n[29] H. M. Siahaan, Phys. Rev. D 101, 064036 (2020),\narXiv:1907.02158 [gr-qc] .\n[30] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravi-\ntation , Misner73 (W. H. Freeman, San Francisco, 1973).\n[31] R. M. Wald, Phys. Rev. D. 10, 1680 (1974).\n[32] S. Chen, M. Wang, and J. Jing, Journal of High Energy\nPhysics 2016 , 82 (2016), arXiv:1604.02785 [gr-qc] .\n[33] K. Hashimoto and N. Tanahashi, Phys. rev. D 95,\n024007 (2017), arXiv:1610.06070 [hep-th] .\n[34] S. Dalui, B. R. Majhi, and P. Mishra, Physics Letters\nB788, 486 (2019), arXiv:1803.06527 [gr-qc] .\n[35] W. Han, General Relativity and Gravitation 40, 1831\n(2008), arXiv:1006.2229 [gr-qc] .\n[36] A. P. S. de Moura and P. S. Letelier, Phys. Rev. E 61,\n6506 (2000), arXiv:chao-dyn/9910035 [nlin.CD] .\n[37] V. S. Morozova, L. Rezzolla, and B. J. Ahmedov,\nPhys.Rev.D 89, 104030 (2014), arXiv:1310.3575 [gr-qc]\n.\n[38] A. Jawad, F. Ali, M. Jamil, and U. Debnath, Com-\nmunications in Theoretical Physics 66, 509 (2016),\narXiv:1610.07411 [gr-qc] .\n[39] S. Hussain and M. Jamil, Phys. Rev. D 92, 043008\n(2015), arXiv:1508.02123 [gr-qc] .\n[40] M. Jamil, S. Hussain, and B. Majeed, European Phys-\nical Journal C 75, 24 (2015), arXiv:1404.7123 [gr-qc] .\n[41] Hussain, S, Hussain, I, and Jamil, M., Eur. Phys. J. C16\n74, 210 (2014).\n[42] G. Z. Babar, M. Jamil, and Y.-K. Lim, Interna-\ntional Journal of Modern Physics D 25, 1650024 (2016),\narXiv:1504.00072 [gr-qc] .\n[43] M. Ba~ nados, J. Silk, and S. M. West, Physical Review\nLetters 103, 111102 (2009).\n[44] B. Majeed and M. Jamil, International Journal of Mod-\nern Physics D 26, 1741017 (2017), arXiv:1705.04167 [gr-\nqc] .\n[45] A. Zakria and M. Jamil, Journal of High Energy Physics\n2015 , 147 (2015), arXiv:1501.06306 [gr-qc] .\n[46] I. Brevik and M. Jamil, International Journal of Geo-\nmetric Methods in Modern Physics 16, 1950030 (2019),\narXiv:1901.00002 [gr-qc] .\n[47] M. De Laurentis, Z. Younsi, O. Porth, Y. Mizuno,\nand L. Rezzolla, Phys.Rev.D 97, 104024 (2018),\narXiv:1712.00265 [gr-qc] .\n[48] F. de Felice and F. Sorge, Classical and Quantum Grav-\nity20, 469 (2003).\n[49] F. de Felice, F. Sorge, and S. Zilio, Classical and Quan-\ntum Gravity 21, 961 (2004).\n[50] J. R. Rayimbaev, Astrophys Space Sc 361, 288 (2016).\n[51] T. Oteev, A. Abdujabbarov, Z. Stuchl\u0013 \u0010k, and B. Ahme-\ndov, Astrophys. Space Sci. 361, 269 (2016).\n[52] B. Toshmatov, A. Abdujabbarov, B. Ahmedov, and\nZ. Stuchl\u0013 \u0010k, Astrophys Space Sci 360, 19 (2015).\n[53] A. Abdujabbarov, B. Ahmedov, O. Rahimov, and\nU. Salikhbaev, Physica Scripta 89, 084008 (2014).\n[54] O. G. Rahimov, A. A. Abdujabbarov, and B. J. Ahme-\ndov, Astrophysics and Space Science 335, 499 (2011),\narXiv:1105.4543 [astro-ph.SR] .\n[55] O. G. Rahimov, Modern Physics Letters A 26, 399\n(2011), arXiv:1012.1481 [gr-qc] .\n[56] K. Haydarov, A. Abdujabbarov, J. Rayimbaev,\nand B. Ahmedov, Universe 6(2020), 10.3390/uni-\nverse6030044.\n[57] K. Haydarov, J. Rayimbaev, A. Abdujabbarov,\nS. Palvanov, and D. Begmatova, arXiv e-prints ,\narXiv:2004.14868 (2020), arXiv:arXiv:2004.14868 [gr-\nqc] .\n[58] M. Kolo\u0014 s, A. Tursunov, and Z. Stuchl\u0013 \u0010k, Eur. Phys. J.\nC.77, 860 (2017), arXiv:1707.02224 [astro-ph.HE] .\n[59] J. Kov\u0013 a\u0014 r, O. Kop\u0013 a\u0014 cek, V. Karas, and Z. Stuchl\u0013 \u0010k,\nClassical and Quantum Gravity 27, 135006 (2010),\narXiv:1005.3270 [astro-ph.HE] .\n[60] J. Kov\u0013 a\u0014 r, P. Slan\u0013 y, C. Cremaschini, Z. Stuchl\u0013 \u0010k,\nV. Karas, and A. Trova, Phys. Rev. D 90, 044029\n(2014), arXiv:1409.0418 [gr-qc] .\n[61] A. N. Aliev and D. V. Gal'tsov, Soviet Physics Uspekhi\n32, 75 (1989).\n[62] A. N. Aliev and N. Ozdemir, Mon. Not. R. Astron. Soc.\n336, 241 (2002), gr-qc/0208025 .\n[63] A. N. Aliev, D. V. Galtsov, and V. I. Petukhov, Astro-\nphys. Space Sci. 124, 137 (1986).\n[64] V. P. Frolov and P. Krtou\u0014 s, Phys. Rev. D 83, 024016\n(2011), arXiv:1010.2266 [hep-th] .\n[65] V. P. Frolov, Phys. Rev. D. 85, 024020 (2012),\narXiv:1110.6274 [gr-qc] .\n[66] Z. Stuchl\u0013 \u0010k, J. Schee, and A. Abdujabbarov, Phys. Rev.\nD89, 104048 (2014).\n[67] S. Shaymatov, F. Atamurotov, and B. Ahmedov, As-\ntrophys Space Sci 350, 413 (2014).\n[68] A. Abdujabbarov and B. Ahmedov, Phys. Rev. D 81,044022 (2010), arXiv:0905.2730 [gr-qc] .\n[69] A. Abdujabbarov, B. Ahmedov, and A. Hakimov,\nPhys.Rev. D 83, 044053 (2011), arXiv:1101.4741 [gr-qc]\n.\n[70] A. A. Abdujabbarov, B. J. Ahmedov, S. R. Shaymatov,\nand A. S. Rakhmatov, Astrophys Space Sci 334, 237\n(2011), arXiv:1105.1910 [astro-ph.SR] .\n[71] A. A. Abdujabbarov, B. J. Ahmedov, and V. G. Kagra-\nmanova, General Relativity and Gravitation 40, 2515\n(2008), arXiv:0802.4349 [gr-qc] .\n[72] V. Karas, J. Kovar, O. Kopacek, Y. Kojima, P. Slany,\nand Z. Stuchlik, in American Astronomical Society\nMeeting Abstracts #220 , American Astronomical Soci-\nety Meeting Abstracts, Vol. 220 (2012) p. 430.07.\n[73] S. Shaymatov, M. Patil, B. Ahmedov, and P. S. Joshi,\nPhys. Rev. D 91, 064025 (2015), arXiv:1409.3018 [gr-qc]\n.\n[74] Z. Stuchl\u0013 \u0010k and M. Kolo\u0014 s, European Physical Journal C\n76, 32 (2016), arXiv:1511.02936 [gr-qc] .\n[75] J. Rayimbaev, B. Turimov, F. Marcos, S. Palvanov, and\nA. Rakhmatov, Modern Physics Letters A 35, 2050056\n(2020).\n[76] B. Turimov, B. Ahmedov, A. Abdujabbarov, and\nC. Bambi, Phys. Rev. D 97, 124005 (2018),\narXiv:1805.00005 [gr-qc] .\n[77] S. Shaymatov, B. Ahmedov, Z. Stuchl\u0013 \u0010k, and A. Ab-\ndujabbarov, International Journal of Modern Physics D\n27, 1850088 (2018).\n[78] B. V. Turimov, B. J. Ahmedov, and A. A. Hakimov,\nPhys. Rev. D 96, 104001 (2017).\n[79] S. Shaymatov, J. Vrba, D. Malafarina, B. Ahme-\ndov, and Z. Stuchl\u0013 \u0010k, arXiv e-prints , arXiv:2005.12410\n(2020), arXiv:2005.12410 [gr-qc] .\n[80] J. R. Rayimbaev, B. J. Ahmedov, N. B. Juraeva, and\nA. S. Rakhmatov, Astrophys Space Sc 356, 301 (2015).\n[81] S. Shaymatov, Int. J. Mod. Phys. Conf. Ser. 49, 1960020\n(2019).\n[82] J. Rayimbaev, B. Turimov, and B. Ahmedov, Inter-\nnational Journal of Modern Physics D 28, 1950128-209\n(2019).\n[83] S. Shaymatov, D. Malafarina, and B. Ahmedov, arXiv\ne-prints , arXiv:2004.06811 (2020), arXiv:2004.06811\n[gr-qc] .\n[84] B. Narzilloev, J. Rayimbaev, A. Abdujabbarov, and\nC. Bambi, arXiv e-prints , arXiv:2005.04752 (2020),\narXiv:2005.04752 [gr-qc] .\n[85] B. Narzilloev, A. Abdujabbarov, C. Bambi, and\nB. Ahmedov, Phys. Rev. D 99, 104009 (2019),\narXiv:1902.03414 [gr-qc] .\n[86] I. D. Novikov and K. S. Thorne, in Black Holes (Les\nAstres Occlus) , edited by C. Dewitt and B. S. Dewitt\n(1973) pp. 343{450.\n[87] D. N. Page and K. S. Thorne, Astrophys. J. 191, 499\n(1974).\n[88] Y. C. Ong and P. Chen, Journal of High Energy Physics\n2012 , 79 (2012), arXiv:1205.4398 [hep-th] .\n[89] G. T. Horowitz and D. L. Welch, Pys.Rev.Letters 71,\n328 (1993), arXiv:hep-th/9302126 [hep-th] .\n[90] D. Gar\fnkle, G. T. Horowitz, and A. Strominger,\nPhys.Rev.D 43, 3140 (1991).\n[91] J. F. Steiner, J. E. McClintock, R. A. Remillard, L. Gou,\nS. Yamada, and R. Narayan, Astrophys. J. Lett. 718,\nL117 (2010), arXiv:1006.5729 [astro-ph.HE] .\n[92] J. E. McClintock, R. Narayan, and J. F. Steiner, Space17\nScience Reviews 183, 295 (2014), arXiv:1303.1583\n[astro-ph.HE] .\n[93] L. Kong, Z. Li, and C. Bambi, Astrophys. J. 797, 78\n(2014), arXiv:1405.1508 [gr-qc] .\n[94] A. Tripathi, M. Zhou, A. B. Abdikamalov, D. Ayzen-\nberg, C. Bambi, L. Gou, V. Grinberg, H. Liu, and J. F.\nSteiner, Astrophys. J. 897, 84 (2020), arXiv:2001.08391\n[gr-qc] .\n[95] C. Bambi, A. C\u0013 ardenas-Avenda~ no, T. Dauser, J. A.\nGarc\u0013 \u0010a, and S. Nampalliwar, Astrophys J. 842, 76\n(2017), arXiv:1607.00596 [gr-qc] .\n[96] Z. Cao, S. Nampalliwar, C. Bambi, T. Dauser, and\nJ. A. Garcia, Phys. Rev. Lett. 120, 051101 (2017),\narXiv:1709.00219 [gr-qc] .\n[97] A. Tripathi, S. Nampalliwar, A. B. Abdikamalov,\nD. Ayzenberg, C. Bambi, T. Dauser, J. A. Gar-\ncia, and A. Marinucci, Astrophys. J. 875, 56 (2019),\narXiv:1811.08148 [gr-qc] .\n[98] Y. Zhang, A. B. Abdikamalov, D. Ayzenberg, C. Bambi,\nand S. Nampalliwar, Astrophys. J. 884, 147 (2019),arXiv:1907.03084 [gr-qc] .\n[99] H. Krawczynski, Astrophys J. 754, 133 (2012),\narXiv:1205.7063 [gr-qc] .\n[100] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, As-\ntrophys. J. 178, 347 (1972).\n[101] K. Mori, E. V. Gotthelf, S. Zhang, H. An, F. K.\nBagano\u000b, N. M. Barri\u0012 ere, A. M. Beloborodov, S. E.\nBoggs, F. E. Christensen, W. W. Craig, F. Dufour,\nB. W. Grefenstette, C. J. Hailey, F. A. Harrison,\nJ. Hong, V. M. Kaspi, J. A. Kennea, K. K. Madsen,\nC. B. Markwardt, M. Nynka, D. Stern, J. A. Tomsick,\nand W. W. Zhang, The Astronomical Journal Letters\n770, L23 (2013), arXiv:1305.1945 [astro-ph.HE] .\n[102] J. W. Mo\u000bat, JCAP 2006 , 004 (2006), arXiv:gr-\nqc/0506021 [gr-qc] .\n[103] J. Rayimbaev, A. Abdujabbarov, B. Turimov, and\nF. Atamurotov, arXiv e-prints , arXiv:2004.10031\n(2020), arXiv:2004.10031 [gr-qc] ."    },    {        "title": "1711.03135v1.Charge_pumping_induced_by_magnetic_texture_dynamics_in_Weyl_semimetals.pdf",        "content": "Charge pumping induced by magnetic texture dynamics in Weyl semimetals\nYasufumi Araki1, 2and Kentaro Nomura1\n1Institute for Materials Research, Tohoku University, Sendai 980{8577, Japan\n2Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980{8578, Japan\nSpin{momentum locking in Weyl semimetals correlates the orbital motion of electrons with back-\nground magnetic textures. We show here that the dynamics of a magnetic texture in a magnetic\nWeyl semimetal induces a pumped electric current that is free from Joule heating. This pumped\ncurrent can be regarded as a Hall current induced by axial electromagnetic \felds equivalent to the\nmagnetic texture. Taking a magnetic domain wall as a test case, we demonstrate that a moving\ndomain wall generates a pumping current corresponding to the localized charge.\nIntroduction |Magnetic textures, such as domain walls\n(DWs), skyrmions, spin spirals, etc., are currently at-\ntracting a great deal of interest in condensed matter\nphysics. In the context of spintronics, these magnetic\ntextures are expected to assume an integral role as infor-\nmation carriers in next-generation devices and in switch-\ning devices driven by electric and spin currents [1{3]. In\nparticular, the dynamical properties of such magnetic\ntextures, which are coupled to the spins of conduction\nelectrons, are the focus of intense e\u000borts to control and\ndetect them e\u000eciently, and promise a wide range of fu-\nture applications [4]. Depending on the particular con-\ntext, di\u000berent dynamical perspectives can be used to de-\nscribe the coupling between magnetic textures and con-\nduction electrons. One may view the spin-transfer torque\narising from the spins of conduction electrons as primar-\nily responsible for driving the dynamics of magnetic tex-\ntures [5], or conversely, see the dynamics of a magnetic\ntexture as inducing an external force on conduction elec-\ntrons through changes in the electron Berry phase, which\nis known as the spin motive force [6].\nIn this work, we propose that the dynamics of magnetic\ntextures in Weyl semimetals (WSMs) can invoke electric\ncharge pumping free from backscattering in a manner\nthat is distinct from that induced by the spin motive\nforce. WSMs form a class of topological materials char-\nacterized by a conical band structure and pair(s) of band-\ntouching points (Weyl points) isolated from each other in\nthe bulk Brillouin zone [7{10]. This \\Weyl cone\" struc-\nture arises from band inversion due to strong spin{orbit\ncoupling (SOC), and is associated with signi\fcant elec-\ntron spin{momentum locking around the nodal points.\nFor such spin{momentum-locked electrons, the exchange\ncoupling to the background magnetic texture is analo-\ngous to a \fctitious vector potential, which is referred to\nas an \\axial vector potential\" [11]. In the context of this\nanalogy, we can then observe a \\Hall current\" free from\nJoule heating that is induced by the axial magnetic and\nelectric \felds corresponding to the dynamics of the back-\nground magnetic texture. This \\Hall e\u000bect\" accounts for\nthe charge pumping mechanism proposed here.\nThe signi\fcance of magnetic textures in WSMs has\nbeen discussed in several recent studies, and based onthe features of the electron spin{momentum locking, it\nwas proposed that the correlation between the mag-\nnetic moments (mediated by the Weyl electrons) exhibits\nlongitudinal anisotropy, distinguishing it from that due\nto ordinary isotropic Ruderman{Kittel{Kasuya{Yosida\n(RKKY) interactions [12{14]. Such anisotropic correla-\ntions then give rise to the formation of nontrivial topo-\nlogical magnetic textures in WSMs. Moreover, once a\nmagnetic DW is formed in a WSM, it is accompanied by\na certain amount of electric charge and an equilibrium\ncurrent localized to the DW, no matter how the DW\nwas introduced into the WSM [15, 16]. While the theo-\nries proposed in these studies principally account for the\ncharacteristics of static magnetic textures in WSMs, the\ndynamical properties of magnetic textures in WSMs are\nmuch less understood and a better understanding of such\nproperties is required for the ability to read and write in-\nformation in future devices. The charge pumping e\u000bect\ndiscussed in this work is one such dynamical property at-\ntributed to magnetic textures in WSMs. Moreover, since\nit arises as a Hall current due to axial electromagnetic\n\felds, it is free from Joule heating and could lead to\nreduced power consumption in future spintronic applica-\ntions.\nAxial vector potentials |The formation of the WSM\nphase requires the breaking of either time-reversal sym-\nmetry (TRS) or spatial inversion symmetry so that the\ndegeneracy of the Weyl cones is lifted. TRS-breaking in\nWSMs is typically realized by the introduction of mag-\nnetic order in the system, such as the hypothetical case of\nferromagnetic order in Co-based Heusler alloys [17, 18],\nor chiral antiferromagnetic order in Mn-based materials\n(e.g. Mn 3Sn) [19{21]. The breaking of TRS shifts the po-\nsitions of the Weyl nodes in momentum space, and this ef-\nfect can be regarded as an emergent vector potential\" for\neach Weyl node [11]. As far as the low-energy phenom-\nena are concerned, one can then rely on this idea of an\ne\u000bective vector potential to treat the temporal and spa-\ntial variations of the magnetization in the system more\ne\u000eciently.\nHere, we consider a minimal model of a WSM exhibit-\ning ferromagnetic order, with a pair of Weyl cones dis-\npersed isotropically around each Weyl node. The low-arXiv:1711.03135v1  [cond-mat.mes-hall]  8 Nov 20172\nTABLE I: Classi\fcation of currents induced by normal and\naxial electromagnetic \felds (EMFs). The current induced by\naxial EMFs is evaluated with \u00165= 0.\nNormal EMFs: ( E;B)Axial EMFs: ( E5;B5)\nDrift j(D)=\u001bDE j(D)= 0\nAHE j(A)=\u001bA^M\u0002E j(A)= 0\nCME j(C)= (e2=2\u00192)\u00165Bj(C)= (e2=2\u00192)\u0016B5\nRHE j(H)=\u001bH^B\u0002E j(H)=\u001bH^B5\u0002E5\nenergy electrons (around the Weyl nodes) can then be\ndescribed by the continuum Hamiltonian\nH=svF(p\u0001\u001b)\u0000JM(r;t)\u0001\u001b; (1)\nwheres=\u00061 denotes the chirality of each Weyl node,\n\u001b= (\u001bx;\u001by;\u001bz) are the Pauli matrices corresponding\nto the electron spin degrees of freedom, vFis the Fermi\nvelocity, and p=\u0000iris the electron momentum oper-\nator. For convenience, we have taken ~= 1 here. The\nsecond term in this Hamiltonian describes the exchange\ncoupling between the electron spin \u001band the local mag-\nnetic texture M(r;t), with coupling constant J. Then,\nprovided that our Weyl-cone approximation is valid, the\nlocal magnetic texture M(r;t) can be viewed as a U(1)\naxial gauge potential A5(r;t) = (J=vFe)M(r;t) coupled\nto the electrons, and in terms of which our Hamiltonian\ncan be written as\nH=svF[p\u0000seA5(r;t)]\u0001\u001b: (2)\nIn contrast to a normal vector potential A, the axial vec-\ntor potential A5couples to di\u000berent chirality modes ( s=\n\u0006) with opposite sign and is not subject to Maxwell's\nequations. As we shall see in the following sections (and\nby analogy to normal vector potentials), it is this ax-\nial vector potential that is responsible for the proposed\nelectron transport.\nField-induced current |Before considering the pro-\nposed charge pumping mechanism, let us \frst review\nthe di\u000berent kinds of electric current induced by real\nEMFs. Currents induced by EMFs in a WSM can be\nclassi\fed based on their linear response to an electric\n\feldEand/or magnetic \feld B(see Table I). If only\nanE-\feld is applied to the WSM, a longitudinal drift\ncurrent,j(D)=\u001bDE, is induced, where \u001bDis the lon-\ngitudinal conductivity of a pair of Weyl cones. If TRS\nis broken in the WSM by the presence of magnetization\nM, an anomalous Hall e\u000bect (AHE) is also present [22{\n25], and drives the transverse current j(A)=\u001bA^M\u0002E,\nwhere the anomalous Hall conductivity is given by \u001bA=\n(e=2\u00192)(JjMj=vF) [26].\nOn the other hand, if only a magnetic \feld Bis ap-\nplied to the system, it induces Landau quantization with\na cyclotron frequency !c=vFp\n2eB. Nonzero Landau\nlevels (LLs) then appear symmetrically about the zeroenergy due to particle{hole symmetry, and the zeroth\nLL is linearly dispersed along the magnetic \feld. As the\ndispersion direction of the zeroth LL for each Weyl node\ndepends on the chirality s[27], it only contributes to the\nnet currentj(C)= (e2=2\u00192)\u00165Bif there is a chemical po-\ntential imbalance \u00165between the two Weyl nodes. This\ne\u000bect is known as the chiral magnetic e\u000bect (CME) and\naccounts for the negative magnetoresistances observed in\nWSMs [22, 28{31].\nFinally, a combination of EandBinduces a Hall cur-\nrent perpendicular to both, j(H)=\u001bH^B\u0002E, which\nwe have called the regular Hall e\u000bect (RHE) to distin-\nguish it from the AHE. The regular Hall conductivity\n\u001bHdepends on both the \feld strength and the amount\nof disorder present in the system. If the level broaden-\ning arising from disorder obscures the LL spacing, i.e., if\nthe cyclotron frequency !cis smaller than the relaxation\nrate 1=\u001c, the transport coe\u000ecients can be estimated in\nthe classical limit using semiclassical (Boltzmann) trans-\nport theory [32]. The zero temperature Hall conductivity\nis then given by \u001bH(c)=\u0000(\u001c2e3\u0016=3\u00192)jBjat the lowest\norder inB, where\u0016is the electron chemical potential\nmeasured from the Weyl nodes. On the other hand, in\nthe quantum limit where the disorder is dilute and the\nLLs can be regarded as well separated ( !c\u001c\u001d1), the\nHall current can be e\u000bectively described by the \\quan-\ntum Hall e\u000bect\". If the Fermi level \u0016lies just slightly\nbeyond the charge neutrality point so that it does not\ncross the higher LLs, only the zeroth LL contributes to\nthe Hall current. The Hall conductivity then reduces to\nthe universal value\n\u001bH(q)=e2\n2\u00192\u0016\nvF; (3)\nwhich can be derived from the quantum Hall conductivity\nin 2D Dirac systems such as graphene.\nCharge pumping induced by magnetic texture dynam-\nics|As we have outlined above, in order to consider the\ne\u000bect of magnetic texture dynamics on the electron trans-\nport, we can rely on the idea of axial EMFs. Speci\f-\ncally, the dynamics of the magnetic texture, i.e., r- and\nt-dependences in the axial vector potential A5, are equiv-\nalent to axial electric and magnetic \felds, E5andB5,\ngiven by\nE5(r;t) =\u0000@tA5(r;t) =\u0000J\nvFe@tM(r;t) (4)\nB5(r;t) =r\u0002A5(r;t) =J\nvFer\u0002M(r;t);(5)\nrespectively. The electron transport induced by the mag-\nnetic texture dynamics can then be treated in terms of\nthese axial EMFs, thus enabling its evaluation in similar\nfashion to normal EMFs, making the overall discussion\nquite simple. As we shall see in the following, the axial\nelectric \feld E5drives an \\axial current\" comprising a3\npair of currents \rowing oppositely to each other at the\ntwo Weyl nodes and thus yielding no net current, while\na net current is induced if it is accompanied by an ax-\nial magnetic \feld B5. Here, we note that we have ne-\nglected intervalley scattering processes, so that the elec-\ntron transport for each Weyl node could be treated sep-\narately.\nAs long as the magnetic texture dynamics are su\u000e-\nciently slow and \\adiabatic\", the axial electric \feld E5\nis so weak that its nonlinear e\u000bect can be safely dis-\ncarded. With this assumption, it then simply induces\na drift current and an anomalous Hall current for each\nWeyl node \rowing in opposite directions to one another,\ni.e., the axial current [33]. As such, it contributes no\nnet current unless there is an imbalance in the carrier\ndensities (\u001656= 0). On the other hand, the RHE con-\ntribution is same as that induced by normal EMFs, i.e.,\nj(H)=\u001bH^B5\u0002E5, since bothE5andB5couple to each\nchiral mode with opposite signs, driving the Hall current\nfor each Weyl node in the same direction. As long as the\ndisorder is weak enough compared with the level spacing,\nthe induced Hall current can be estimated in the quan-\ntum limit in similar fashion to the case for real Band\nE, yielding\nj(H)=e2\n2\u00192\u0016\nvF^B5\u0002E5; (6)\nwhich is independent of the \feld strength jB5j. More-\nover, since the zeroth LLs of the two Weyl nodes are\ndispersed in the same direction along B5[11, 15, 16, 37],\na \fnite chemical potential leads to the net current\nj(C)=e2\n2\u00192\u0016B5; (7)\nwhich we identify as the chiral axial magnetic e\u000bect\n(CAME), i.e., the axial counterpart of the CME. There-\nfore, the total current jindinduced byE5andB5in\nWSMs is given (up to the linear response in E5) by the\nsum ofj(H)andj(C).\nWe should note that typical axial EMFs arising from\nmagnetic textures are spatially inhomogeneous. How-\never, if the magnetic texture is su\u000eciently smooth over\nthe relevant length scales (e.g., the electron's mean free\npath), the axial EMFs can be regarded as \\locally\" uni-\nform and we can consider the properties of the electron\ntransport in the ballistic limit. In such cases, we can use\nEqs. (6) and (7) to estimate the local current distribu-\ntion. In the absence of normal EMFs EandB, the axial\nanomaly between the chiral modes (see [34{36]) does not\nviolate the conservation of charge [11], and we can use\nthe charge conservation relation\n@t\u001apump(r;t) =\u0000r\u0001jind(r;t); (8)\nto estimate the electric charge \u001apump(r;t) pumped by the\nmagnetic texture dynamics M(r;t) via the axial \feld-\ninduced current jind. Since the CAME part, j(C)/\u0016(r\u0002M), is divergence-free whenever the chemical po-\ntential is uniform, only the regular Hall current given by\nEq. (6) is responsible for the ensuing charge dynamics:\n@t\u001apump =e2\n2\u00192\u0016\nvFh\n^B5\u0001(r\u0002E5)\u0000E5\u0001(r\u0002^B5)i\n:\n(9)\nThis equation is the key result of this work, and directly\nrelates the magnetic texture dynamics (via E5andB5)\nto the pumped charge \u001apump.\nWe note that if the spatial variation of Mis copla-\nnar, the axial magnetic \feld B5is uniform, i.e., ^B5is\nhomogeneous over the whole system. In this case, the\nsecond term in Eq. (9) vanishes and we obtain @t\u001apump =\n\u0000(e2=2\u00192)(\u0016=vF)(^B5\u0001@tB5) =\u0000(e2=2\u00192)(\u0016=vF)@tjB5j,\nwhere we have used the relation r\u0002E5=r\u0002\n(\u0000@tA5) =\u0000@tB5. Thus, we obtain a further simpli-\n\fed relation for this restricted case\n\u001apump(r;t) =\u0000e2\n2\u00192\u0016\nvFjB5(r;t)j+ const: (10)\nwhich implies that an axial magnetic \rux (i.e., the curl\nof the magnetization) induces localized electric charge in\na WSM, irrespective of its orientation.\nExample: Magnetic domain wall |In order to estab-\nlish the validity of the relations presented above, let us\nconsider a moving magnetic DW in a WSM as a typical\nexample. We construct a DW of width 2 win theyz-\nplane, separating two regions of an in\fnite system with\nmagnetizations M(x!\u00061 ) =\u0006M0ez, and then set\nthe DW in motion with velocity VDWin thex-direction\nby hand. The resulting magnetic texture is then given\nby\nM(r;t) =M(x\u0000VDWt) =M00\nB@\u0015xsech\u0018(x;t)\n\u0015ysech\u0018(x;t)\ntanh\u0018(x;t)1\nCA;(11)\nwhere\u0018(x;t)\u0011(x\u0000VDWt)=wdenotes the relative po-\nsition from the center of the DW, rescaled by the DW\nwidth. The set of parameters ( \u0015x;\u0015y) characterizes the\ntexture of the DW, where a DW with a coplanar mag-\nnetic texture within the xz-plane (i.e., a N\u0013 eel DW) cor-\nresponds to ( \u0015x;\u0015y) = (\u00061;0), while a DW with a helical\nmagnetic texture twisting in the yz-plane (i.e., a Bloch\nDW) is given by ( \u0015x;\u0015y) = (0;\u00061). In reality, the partic-\nular type of DW assumed in such a system is microscop-\nically chosen by the mechanism stabilizing the DW (e.g.,\nthe Dzyaloshinskii{Moriya interaction), but we do not go\ninto such details here and simply consider an arbitrary\ntype of DW.\nIn order to estimate the current jindinduced by the\nDW's motion, we consider the axial gauge \feld A5=\n(J=vFe)M. The axial EMFs are then given by\nE5=JVDW\nevFwM0(\u0018);B5=J\nevFwex\u0002M0(\u0018);(12)4\nE5\nB5\nj(H)\nM(x,t)xyz\nVDW\nFIG. 1: Schematic picture showing the axial EMFs ( E5;B5)\nand the induced Hall current j(H), along with a N\u0013 eel domain\nwall moving with velocity VDW.\nwhereM0(\u0018)\u0011dM(\u0018)=d\u0018. The magnitude and ori-\nentation of the axial EMFs are shown schematically in\nFig. 1. In the case of a DW with w= 100 nm and\nVDW= 100 m=s in a magnetic WSM with vF= 106m=s\nandJM0= 100 meV, the strength of the axial \felds at\nthe center of the DW are given by jE5j= 1 V=cm and\njB5j= 1 T.\nThen, using the axial EMFs presented above and as-\nsuming that the chemical potential \u0016is slightly above\nthe point of charge neutrality so that the quantum limit\nis valid, the regular Hall current corresponding to the\ncharge pumping e\u000bect can be estimated from Eq. (6) to\n\fnd\nj(H)=\u0000e\n2\u00192JVDW\u0016\nv2\nFwh\njM0\n?jex+M0\nx(ex\u0002^B5)i\n;(13)\nwhereM?= (0;My;Mz). The \frst term in the\nabove equation represents the longitudinal current \row-\ning along the moving DW, while the second term is the\ntransverse current \rowing parallel to the DW, which is in-\ndependent of ( y;z) and does not a\u000bect the charge conser-\nvation relation [Eq. (8)] since it is divergence-free. From\nthis, we see that the charge pumping predominantly oc-\ncurs close to the DW center, where the most drastic vari-\nation inM?(x;t) occurs.\nThe amount of electric charge pumped along with the\nDW can be derived from the induced current using the\ncharge conservation relation. Since both of the DW's x-\nandt-dependences are characterized by a single variable\n\u0018= (x\u0000VDWt)=w, the di\u000berential operators on both\nsides of Eq. (8) are easy to treat, and lead to the charge\ndistribution\n\u001apump(x;t) =1\nVDWj(H)\nx(x;t) =\u0000e\n2\u00192J\u0016\nv2\nFwjM0\n?j:(14)\nThe net amounts of charge per unit area pumped by N\u0013 eel\nand Bloch DWs are then given by\nq(Neel)\npump =\u0000e\n\u00192JM0\nv2\nF\u0016; q(Bloch)\npump =\u0000e\n2\u0019JM0\nv2\nF\u0016; (15)respectively. These net charges are independent of the\nDW width w, which implies that the charge pumping\nis indeed a topological e\u000bect. Moreover, in the case of\nthe N\u0013 eel wall, q(Neel)\npump successfully accounts for the same\namount of localized charge obtained by exactly counting\nthe number of bound states that was presented in previ-\nous work [15], which provides a guarantee of the validity\nof the quantum limit employed in this work. Further-\nmore, the charge pumping picture presented here can\nbe used for any other type of DW, as long as the DW\ntexture is su\u000eciently sharp so that the quantum limit\napproximation can be applied. As the pumping current\ndiscussed here can be e\u000bectively described as a quantum\nHall e\u000bect, it is free from energy loss by Joule heating\nand thus distinct from the drift current arising from the\nspin motive force.\nConclusions and Outlook |We have discussed the re-\nlation between the dynamics of magnetic textures and\ncharge pumping in magnetic WSMs. Since the coupling\nbetween the magnetization and Weyl electrons may be\nviewed in terms of an axial gauge potential, the curl and\ntime derivative of the magnetic texture correspond to ax-\nial magnetic and electric \felds, respectively. The main\nmessage of this work [Eqs. (6) and (9)] is that these axial\nEMFs give rise to a regular Hall current, which can be\nregarded as a pumping current induced by the dynam-\nics of the background magnetic texture. If the spatial\nvariation of the magnetic texture is su\u000eciently slow, the\ninduced current can be described by semiclassical trans-\nport theory, whereas sharp variations yield a pumping\ncurrent described by the quantum Hall e\u000bect. The charge\npumping e\u000bect implies that a certain amount of local-\nized charge [Eq. (10)] is induced by the axial magnetic\n\rux, i.e., the curl of the magnetic texture. Conversely,\nit also implies that a local electrostatic potential that\nalters the local charge distribution would induce a mag-\nnetic texture in a magnetic WSM. However, verifying\nthe existence of such an e\u000bect remains an open question\nand further microscopic calculations will be required to\ncon\frm this proposal. Nevertheless, from a topological\npoint of view, the proposed pumping current and local-\nized charge are simple manifestations of the interplay be-\ntween the real-space topology and its momentum-space\ncounterpart, which can generally be traced back to Berry\ncurvatures de\fned in the global phase space [38].\nSecondly, by considering the coherent motion of a mag-\nnetic domain wall (DW) in a WSM, we were able to\ncompare the pumped charge with the localized charge\ncalculated in previous work by more direct methods [15],\nand show their equivalence. The idea of charge pump-\ning obtained here is also applicable to all kinds of mag-\nnetic textures: magnetic skyrmions and monopoles, for\ninstance, carry pointlike charge, whereas magnetic he-\nlices can be accompanied by arrays of localized charge,\ni.e., charge density waves. This concept may help us to\ndesign e\u000ecient spintronic devices that make use of mag-5\nnetic textures in magnetic WSMs, such as in a magnetic\nracetrack [39], where the motion of a magnetic texture\ncan be electrically detected as a current pulse and can\nthus be used to read out information from an array of\nmagnetic textures.\nWhile we have only considered a minimal ferromag-\nnetic toy model in this work, the concepts developed\nhere could be extended to other examples of TRS-broken\nWSMs. One particular case of interest would be that of\nantiferromagnetic order in WSMs, as exhibited in Mn 3Sn\n[19{21], which may exhibit similar TRS-breaking and ax-\nial vector potential e\u000bects to those presented here for fer-\nromagnetic order. However, since the ordering is not nec-\nessarily characterized by a single order parameter, this\ncase would require more detailed microscopic investiga-\ntions to clarify the relationship between the antiferro-\nmagnetic order and the charge degree of freedom.\nY. A. is supported by JSPS KAKENHI Grant Num-\nber JP17K14316. K. N. is supported by JSPS KAKENHI\nGrant Numbers JP15H05854 and JP17K05485. The au-\nthors would like to thank Editage (www.editage.jp) for\nEnglish language editing.\n[1] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma,\nRev. Mod. Phys. 76, 323 (2004).\n[2] A. Brataas, G. E. W. Bauer, and P. J. Kelly,\nPhys. Rep. 427, 157 (2006).\n[3] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468,\n213 (2008).\n[4] S. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura,\nSpin Current (Oxford University Press, 2012).\n[5] D. C. Ralph and M. D. Stiles,\nJ. Magn. Magn. Mater. 320, 1190 (2008).\n[6] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98,\n246601 (2007).\n[7] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane,\nE. J. Mele, and A. M. Rappe, Phys. Rev. Lett. 108,\n140405 (2012).\n[8] X. Wan, A. M. Turner, A. Vishwanath, and\nS. Y. Savrasov, Phys. Rev. B 83, 205101 (2011).\n[9] A. A. Burkov and L. Balents, Phys. Rev. Lett. 107,\n127205 (2011).\n[10] A. A. Zyuzin, S. Wu, and A. A. Burkov, Phys. Rev. B\n85, 165110 (2012).\n[11] C.-X. Liu, P. Ye, and X.-L. Qi, Phys. Rev. B 87, 235306\n(2013).\n[12] M. V. Hosseini and M. Askari, Phys. Rev. B 92, 224435\n(2015).[13] H.-R. Chang, J. Zhou, S.-X. Wang, W.-Y. Shan, and\nD. Xiao, Phys. Rev. B 92, 241103 (2015).\n[14] Y. Araki and K. Nomura, Phys. Rev. B 93, 094438\n(2016).\n[15] Y. Araki, A. Yoshida, and K. Nomura, Phys. Rev. B 94,\n115312 (2016).\n[16] A. G. Grushin, J. W. F. Venderbos, A. Vishwanath, and\nR. Ilan, Phys. Rev. X 6, 041046 (2016).\n[17] Z. Wang, M. G. Vergniory, S. Kushwaha,\nM. Hirschberger, E. V. Chulkov, A. Ernst, N. P. Ong,\nR. J. Cava, and B. A. Bernevig, Phys. Rev. Lett. 117,\n236401 (2016).\n[18] G. Chang, S.-Y. Xu, H. Zheng, B. Singh, C.-H. Hsu,\nI. Belopolski, D. S. Sanchez, G. Bian, N. Alidoust, H. Lin,\nand M. Z. Hasan, Sci. Rep. 6, 38839 (2016).\n[19] J. K ubler and C. Felser, Europhys. Lett. 114, 47005\n(2016).\n[20] H. Yang, Y. Sun, Y. Zhang, W.-J. Shi, S. S. P. Parkin,\nand B. Yan, New J. Phys. 19, 015008 (2017).\n[21] N. Ito and K. Nomura, J. Phys. Soc. Jpn. 86, 063703\n(2017).\n[22] A. A. Zyuzin and A. A. Burkov, Phys. Rev. B 86, 115133\n(2012).\n[23] P. Goswami and S. Tewari, Phys. Rev. B 88, 245107\n(2013).\n[24] A. A. Burkov, Phys. Rev. B 89, 155104 (2014).\n[25] A. A. Burkov, Phys. Rev. Lett. 113, 187202 (2014).\n[26] A vector with a hatdenotes a unit vector, i.e. ^X=\nX=jXj.\n[27] K.-Y. Yang, Y.-M. Lu, and Y. Ran, Phys. Rev. B 84,\n075129 (2011).\n[28] A. Vilenkin, Phys. Rev. D 22, 3080 (1980).\n[29] K. Fukushima, D. E. Kharzeev, and H. J. Warringa,\nPhys. Rev. D 78, 074033 (2008).\n[30] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa,\nNucl. Phys. A 803, 227 (2008).\n[31] D. E. Kharzeev, Prog. Part. Nucl. Phys. 75, 133 (2014).\n[32] S. Nandy, G. Sharma, A. Taraphder, and S. Tewari,\nPhys. Rev. Lett. 119, 176804 (2017).\n[33] K. Taguchi and Y. Tanaka, Phys. Rev. B 91, 054422\n(2015).\n[34] S. L. Adler, Phys. Rev. 177, 2426 (1969).\n[35] J. S. Bell and R. Jackiw, Nuovo Cimento A 60, 47 (1969).\n[36] H. B. Nielsen and M. Ninomiya, Phys. Lett. 130B , 390\n(1983).\n[37] D. I. Pikulin, A. Chen, and M. Franz, Phys. Rev. X 6,\n041021 (2016).\n[38] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82,\n1959 (2010).\n[39] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008)."    },    {        "title": "2302.13978v3.Rotating_hematite_cube_chains.pdf",        "content": "Rotating hematite cube chains\nM. Brics, V . Šints, G. Kitenbergs, and A. C ¯ebers\nMMML lab, Department of Physics, University of Latvia, Jelgavas 3, R ¯ıga, LV-1004, Latvia\n(Dated: June 7, 2023)\nRecently a two-dimensional chiral fluid was experimentally demonstrated. It was obtained from cubic-shaped\nhematite colloidal particles placed in a rotating magnetic field. Here we look at building blocks of that fluid, by\nanalyzing short hematite chain behavior in a rotating magnetic field. We find equilibrium structures of chains in\nstatic magnetic fields and observe chain dynamics in rotating magnetic fields. We find and experimentally verify\nthat there are three planar motion regimes and one where the cube chain goes out of the plane of the rotating\nmagnetic field. In this regime we observe interesting dynamics — the chain rotates slower than the rotating\nmagnetic field. In order to catch up with the magnetic field, it rolls on an edge and through rotation in the third\ndimension catches up with the magnetic field. The same dynamics is also observable for a single cube when\ngravitational effects are explicitly taken into account.\nPACS numbers: 47.65.-d, 61.46.Bc, 82.70.Dd\nKeywords: rotating magnetic field; weak ferromagnetism; hematite; short chains\nI. INTRODUCTION\nAt room temperature hematite is a weak ferromagnetic\nmaterial with an unorthodox magnetization orientation: for\ncubic-shaped hematite particles the magnetic moment with a\ncube’s diagonal makes an angle 12◦(see Fig. 1) in the plane\ndefined by two diagonals [1, 2]. Thus, leading to interest-\ning physical effects. In static magnetic field and low con-\ncentrations cubic shaped hematite particles arrange in kinked\nchains [1, 2]. If concentration is increased [3, 4], swarms are\nformed. If we let a lot of swarms interact, we can observe a\ntwo-dimensional chiral fluid in a rotating magnetic field [5].\nThe chiral fluid consists of individual cubes and short hematite\nchains (usually two and three cube chains) which are interact-\ning in a rotating magnetic field [4].\nIn the scientific literature there are also other interesting ex-\nperiments with hematite colloids formed by cubic particles. In\nthe article [6] authors investigated application of cube-shaped\nhematite microrobots for microblocks and impurities sweep in\nblood vessels. There approximately 2µmlarge cube-shaped\nhematite particles were guided by the rotating magnetic field\nthrough introduced rolling motion. They showed that hematite\ncubes can overcome obstacles and push small objects. Motile\nstructures formed by microrollers which were created by mi-\ncron sized polymer colloids with embedded hematite cubes\nwere demonstrated in [7]. In [8] authors demonstrated tar-\ngeted assembly and synchronization of self-spinning micro-\ngears or rotors made of hematite cubes and chemically inert\npolymer beads. In [9] was investigated a potential application\nof hematite colloidal cubes for the enhanced degradation of\norganic dyes. In [10] were examined the formations of light\nactivated two-dimensional “living crystals”.\nIn this article we investigate the building blocks of chi-\nral fluid demonstrated in [5], i.e. individual cube and short\nhematite chain behavior in a rotating magnetic field. We per-\nform analytical calculations and simulations which we later\nconfirm with experiments. To determine how important grav-\nity effects are two models were developed, one with explicit\ngravity treatment and one without.\nThe content of the paper is divided into five sections. The\nFIG. 1. Orientation of magnetic moment in a hematite cube. The\nangle ϕ= 12◦is in the plane defined by two diagonals and the\nmagnetic moment µµµpoints to the face.\nSec. I is introduction followed by Sec. II where theoretical\nmethods are described. The theoretical and experimental re-\nsults are given in Sec.III and Sec.IV respectively and conclu-\nsions in Sec. V. This article also contains supplementary ma-\nterial — videos to better illustrate different motion modes.\nII. THEORETICAL METHODS\nTo theoretically describe the behaviour of hematite chains\nin a rotating magnetic field we use a microscopic model where\nmotion of each hematite particle is described. The equations\nof motion (EOMs) are derived using the Newton mechanics\napproach. Only the essential forces are introduced to keep\nthe number of parameters minimal and ease interpretation.\nFor each hematite cube in an external homogeneous magnetic\nfield ⃗Bwith magnetic moment ⃗ m, mass m, moment of iner-\ntia tensor III, the force and torque balance is considered. From\nthe force balance (two-particle forces are shown in Fig. 2) we\nobtain that\nmjd\ndt⃗ vj=⃗FHD\nj+X\ni\u0010\n⃗Fmag\nij+⃗Fsteric\nij\u0011\n, (1)arXiv:2302.13978v3  [cond-mat.soft]  6 Jun 20232\nwhere ⃗FHD\nj is the hydrodynamics force acting on the parti-\nclej,⃗Fmag\nij is the magnetic force produced by particle i, and\n⃗Fsteric\nij is the reaction force that ensure that particles do not\noverlap. The corresponding equation obtained from torque T\nbalance then reads:\nIIIjd\ndt⃗Ωj=⃗ mj×⃗B+⃗THD\nj+X\ni\u0010\n⃗Tmag\nij+⃗Tsteric\nij\u0011\n,(2)\nwhere ⃗ mj×⃗Bis magnetic torque produced by the exter-\nnal field and ⃗THD,⃗Tmag,⃗Tstericare corresponding torques\nof forces in Eq. 1. Later, in order to explicitly incorpo-\nrate gravity effects in the model, we add the buoyancy force\n⃗Fb\nj= (ρh−ρs)ga3\n0, the reaction force ⃗Fwall\nj and torque ⃗Twall\nj\nwith the bottom of capillary acting on each particle. Here\ng= 9.81 m/s2is the gravitational acceleration, a0≈1.5µm\nis the edge length of a hematite cube and ρh= 5.25 g/cm3\nandρs= 1.00 g/cm3are densities of hematite and solvent,\nwhich in our case is water. However, in this work we do not\nconsider any other forces like friction between cubes and be-\ntween a cube and capillary or thermal fluctuations.\nNote that, in the general case, the knowledge of ΩΩΩat a spe-\ncific time point is not sufficient to determine the orientation\nof the particle. Thus, for non-spherical objects, we lack in-\nformation to calculate reaction forces and torques that ensure\nthat particles do not overlap. To overcome this issue rotational\nmatrices or quaternions at each time step have to be calculated\n[11, 12]. Here we choose to use quaternions qqqi[13] as this ap-\nproach requires to calculate time evolution of less dimensional\nquantity and equation of motion\nd\ndtqqqi=1\n2\n0−Ωi\nzΩi\nyΩi\nx\nΩi\nz 0−Ωi\nxΩi\ny\n−Ωi\nyΩi\nx 0 Ωi\nz\n−Ωi\nx−Ωi\ny−Ωi\nz0\nqqqi=QQQ(ΩΩΩi)qqqi(3)\nis always stable [11, 12], unlike EOM for Euler angles.\nThe quaternion is a four dimensional quantity, which satis-\nfies the normalization condition and provides a convenient\nrepresentation of spatial orientations and rotations of ele-\nments in three dimensional space (corresponds to a rotation\nmatrix). E.g., rotation around the axis uuu= (ux, uy, uz)\nby an angle θcan be expressed with quaternion qqq=\n(uxsinθ\n2, uysinθ\n2, uzsinθ\n2,cosθ\n2). Note that the angular ve-\nlocities in the laboratory frame are used. Quaternions are im-\nplemented using scalar last notation as internally stored in a\nC++ template library for linear algebra Eigen [14].\nFor the experimental conditions [4, 5] we are interested in,\nit turns out that corresponding Reynolds numbers Re << 1\nand inertial terms are negligible ( md\ndt⃗ v << ⃗FHD). Thus, for\nour calculation we neglect inertial terms and use the Stokes\napproximation. For hydrodynamics forces and torques, to\nkeep equations analytically analyzable, we use linear velocity\ndrag approximation which for a cubic shaped particle reads\n[15]:\n⃗FHD\nj=−ξvvvj;ξ≈3πηa0·1.384; (4)\n⃗THD\nj=−ζΩΩΩj;ζ=πηa3\n0·2.552, (5)\nFIG. 2. Forces acting on a two-cube chain.\nwhere η= 1.0 mPa ·sis the viscosity of the solvent, which\nin our case is water, and ξandζare drag and rotational drag\ncoefficients respectively.\nTo calculate particle magnetic interactions as in [2] we use\nthe dipole approximation colorblue since qualitatively the re-\nsults are the same [2], despite the fact that quantitative differ-\nences in specific arrangements are up to 18%. The magnetic-\nmagnetic particle interaction expressions for the force and\ntorque reads:\n⃗Fmag\nij=3µ0m2\n4πr4\nij˜F˜F˜Fmag\nij;⃗Tmag\nij=3µ0m2\n4πr3\nij˜T˜T˜Tmag\nij; (6)\n˜F˜F˜Fmag\nij=ˆrˆrˆrij( ˆmˆmˆmi·ˆmˆmˆmj) + ˆmˆmˆmi(ˆrˆrˆrij·ˆmˆmˆmj)\n+ ˆmˆmˆmj(rrrij·ˆmˆmˆmi)−5ˆrˆrˆrij(ˆrˆrˆrij·ˆmˆmˆmi)(ˆrˆrˆrij·ˆmˆmˆmj),(7)\n˜T˜T˜Tmag\nij= (ˆrˆrˆrij·ˆmˆmˆmi)( ˆmˆmˆmj׈rˆrˆrij) +1\n3( ˆmˆmˆmi׈mˆmˆmj), (8)\nwhere rijis the vector between i-th and j-th particle centers\n(shown in Fig. 2), µ0= 4π·10−7H/mis the magnetic per-\nmeability of vacuum and quantities with ˜denote dimension-\nless variables apart from unit vectors, which are denoted with\nˆ, e.g. rrrij=rijˆrrrij.\nAlready mentioned reaction forces ⃗Fsteric\nij and correspond-\ning torques ⃗Tsteric\nij are added to avoid cube overlap. The re-\nsults do not depend on the choice of exact expression for re-\naction forces whenever the model for reaction forces is rea-\nsonably chosen. Therefore, here we use some power function3\nab\na0\nFIG. 3. Reaction forces between two cubes. The schematic view of\nthe two-particle chain from above. As a cube lies on an edge, the\nprojection to the base is a rectangle with sides a0and√\n2a, if the tilt\nangle (the angle between the base and cube’s face) is 45◦.\nwhich has only a repulsive part:\n⃗Fsteric\nij =3µ0m2\n4πa4\n0˜F˜F˜Fsteric\nij;⃗Tsteric\nij =3µ0m2\n4πa3\n0˜T˜T˜Tsteric\nij;(9)\n˜F˜F˜Fsteric\nij =A\na2\n0\n \n1\n1−4θH(a\na0−1)!13\n−1\nˆFFFsteric\nij;(10)\n˜T˜T˜Tmag\nij=⃗d\na0טF˜F˜Fsteric\nij, (11)\nwhere θHis Heaviside step function, Ais the area with which\ncubes touch and quantities aand⃗dare defined in Fig. 3. This\napproach to calculate reaction forces and torques becomes,\nhowever, computationally very demanding for non-planar mo-\ntion. Especially, if one takes into account that particles used\nin experiments are with rounded corners, i.e. superballs [1–3].\nThus, to ease the computational task we reconstruct superballs\nout of spheres and calculate reaction forces as in [2, 16]. Each\ncube we replace with 93 spheres as in [2] and calculate repul-\nsion forces for every sphere with every sphere of other cube’s.\nFor steric repulsion we are using Weeks-Chandler-Anderson\npotential [17]. Also in a similar way we calculate reaction\nforces with the bottom of capillary for the model with gravity.\nCombining all expressions the EOM for dimensionless\nvariables reads:\n˜v˜v˜vj=ksX\ni \n1\n˜r4\nij˜F˜F˜Fmag\nij+˜F˜F˜Fsteric\nij!\n,\n˜Ω˜Ω˜Ωj= ˆmˆmˆmj׈BˆBˆB+sX\ni \n1\n˜r3\nij˜T˜T˜Tmag\nij+˜T˜T˜Tsteric\nij!\n,\nd\nd˜tqqqj=QQQ(˜Ω˜Ω˜Ωi)qqqj,(12)\nwhere nondimensionalization for time ˜t=ζt\nmBand dis-\ntance ˜rrrij=rrrij\na0is used leading to dimensionless variables\n˜vvv=vvvζ\nmBa 0and˜ΩΩΩ =ΩΩΩζ\nmB. The EOM has two controlparame-\nterssandk, from which only\ns=3µ0m\n4πa3\n0B≈6.6Bc\nB;Bc≈0.1 mT (13)is adjustable in experiments by changing the magnitude of the\nexternal magnetic field. The parameter\nk=ζ\na2\n0ξ≈0.614 (14)\nis drag coefficient ratio and thus is fixed in experiments .\nIn the case of gravity we have one additional parameter\nGm=4π(ρh−ρs)ga7\n0\n3µ0m2, (15)\nwhich is the ratio of buoyant forces to magnetic forces. The\nEOMs in this case reads:\n˜v˜v˜vj=ksX\ni \n1\n˜r4\nij˜F˜F˜Fmag\nij+˜F˜F˜Fsteric\nij!\n+ks\u0010\nGmˆggg+˜F˜F˜Fwall\nj\u0011\n,\n˜Ω˜Ω˜Ωj= ˆmˆmˆmj׈BˆBˆB+sX\ni \n1\n˜r3\nij˜T˜T˜Tmag\nij+˜T˜T˜Tsteric\nij!\n+s˜T˜T˜Twall\nj,\nd\nd˜tqqqj=QQQ(˜Ω˜Ω˜Ωi)qqqj.(16)\nIII. THEORETICAL RESULTS\nA. Single particle\nIn the case of a single cube (all other cubes are sufficiently\nfar away) and no gravity effects, equations read:\n˜v˜v˜v= 0, (17)\n˜Ω˜Ω˜Ω = ˆmˆmˆm׈BˆBˆB. (18)\nThere is no dependence on control parameters sandkas well\nas on the shape of the particle.\nIf we we assume that magnetic field rotates in xy-plane, i.e.\nˆBˆBˆB= cos(˜ ω˜t)eeex+ sin(˜ ω˜t)eeeywith ˜ω=ωζ\nmB, it is beneficial\nto introduce two angles θandα(see Fig. 4) to describe the\nmotion of the cube . αis the angle the magnetic moment\nmakes with the plane of external magnetic field and θis the\nangle which the magnetic moment’s projection in the plane\nof the rotating magnetic field makes with x-axis. In this case\nˆmˆmˆm= cos( θ) cos( α)eeex+ sin( θ) cos( α)eeey+ sin( α)eeezand the\nangular velocity can be expressed as ˜Ω˜Ω˜Ω = ˆmˆmˆm×˙ˆmmm. The EOMs,\nwhich previously were derived in [18], for a single cube in this\ncase reads\n˙α= cos(˜ ω˜t−θ) sin( α), (19)\n˙θ= sin(˜ ω˜t−θ)/cos(α). (20)\nTo analyze this equation it is beneficial to introduce the lag\nangle β= ˜ω˜t−θas for variable βunlike for θthere are fixed\npoints. The OEM for the lag angle reads\n˙α=−sin(α) cos( β), (21)\n˙β= ˜ω−sin(β)/cos(α). (22)4\nxz\nFIG. 4. The magnetic moment in rotating magnetic field.\nThe Eqs. 21 - 22 has four stationary points P(α, β):P1=\n{0,asin(˜ω)},P2={0,π−asin(˜ω)}andP3={acos(1 /˜ω),\nπ\n2},P4={−acos(1 /˜ω),π\n2}. The first two fixed points exist\nif|˜ω| ≤˜ωcand the last two when |˜ω| ≥˜ωc, where the critical\nfrequency ˜ωc= 1.\nIf|˜ω|<1then there are two stationary points P1andP2.\nAs one can see from the Fig. 5, the point P1is stable and P2\nunstable, thus acting as a think and source. Independent of\ninitial conditions after some transition time stationary point\nP1is reached. Any perturbation is suppressed. The cube ro-\ntates synchronously with the frequency of the external mag-\nnetic field and the magnetic moment is in the plane of the\nrotating magnetic field. It lags the direction of the magnetic\nfield by an angle βj= asin(˜ ω). By increasing frequency the\npoints P1andP2move closer to each other and at |˜ω|= 1\ncoincide and for |˜ω|>1disappear.\nSituation gets more interesting when |˜ω|>1. In this case\nthere are also two stationary points P3andP4. However, both\npoints P3andP4are neutrally stable and act as centers for ro-\ntation, as one can see from the Fig. 6. No stationary solution is\npossible except points P3andP4, and the system can reach P3\nandP4only if it initially was there. Depending on the initial\ncondition, two scenarios are possible. One option is that both\nangles periodically oscillate. The other option is that there\nis rotation around one of the stationary points. This means\nthat in the first case the motion is not anymore synchronous\nwith the external magnetic field and we observe back-and-\nforth motion where the magnetic moment can be out of the\nplane of rotation. In the second case we observe precession\nof the magnetic moment. The motion is asynchronous except\nfor stationary points P3andP4. The motion modes: syn-\nchronous rotation, precession of the magnetic moment, and\nback-and forth rotation arranged from left to right are shown\nin Video1 [19] in rotating frame, which is rotating with the\nmagnetic field (top row) and laboratory frame (bottom row).\nSynchronous rotation and precession is easier identified in the\nrotating frame but back-and-forth rotation in the laboratory\nframe. Note that cube can be rotated by an arbitrary angle\naround an axis parallel to the magnetic field.\nFrom Fig. 6 follows that, if the magnetic moment is ini-\ntially in the plane of the rotating magnetic field, it will always\nremain there. As this situation is analytically solvable we ex-\namine it in more detail.\nP1P2\n-1.5 -1.0 -0.5 00.5 1.0 1.5-3-2-10123-1.5 -1.0 -0.5 00.5 1.0 1.5\n-3-2-10123\nαβ\n0 5 10\nt\n0.00.10.20.3\n0 5 10\nt\n2\n02\nFIG. 5. A phase portrait of single magnetic cube in rotating magnetic\nfield calculated using Eqs. 21 - 22 with ˜ω= 0.5. The time evolution\nof two particular trajectories (red and green) is plotted below. Gravity\neffects are not taken into account.\nWhen one limits βj∈[−π, π]then\nβj= 2atan\"\n1 +√\n˜ω2−1 tan(√\n˜ω2−1˜t−˜t0\n2)\n˜ω#\n,(23)\nwith period ˜T=2π√\n˜ω2−1. The cube oscillates forth and back\nand during one period makes\nWn=1\n2πZ˜T\n0dθj=1\n2πZ2π\n0˙θj\n˙βjdβj=|˜ω|√\n˜ω2−1−1\n(24)\nwinds around its axis of rotation which is perpendicular to the\nplane of the rotating magnetic field. The particular trajectory\nfor the rotational field frequency ˜ω= 2/√\n3, which corre-\nsponds to Wn= 1, can be seen in Fig. 7. When Wn>1,\nthen lag angle βincreases faster than orientation angle θ. The\nopposite is observed if Wn<1.\n1. Gravitation\nDue to an unorthodox orientation of magnetic moment in\na hematite cube, if the magnetic moment is in the plane of\nthe rotating magnetic field, cube lies on an edge [2]. Thus, it\nis not the minimum of potential gravitational energy. There-\nfore, if maximal gravitational torque Tmax\ng =(ρh−ρs)ga4\n2is5\nP3 P4\n-1.5 -1.0 -0.5 00.5 1.0 1.5-3-2-10123-1.5 -1.0 -0.5 00.5 1.0 1.5\n-3-2-10123\nαβ\n0 10 20\nt\n0.00.51.01.5\n0 10 20\nt\n01020\nFIG. 6. A phase portrait of single magnetic cube in rotating magnetic\nfield calculated using Eqs. 21 - 22 with ˜ω= 1.5. The time evolution\nof two particular trajectories (red and green) is plotted below. Gravity\neffects are not taken into account.\n450 460 470 480 490\nt\n260270280, [rad.]\nFIG. 7. The time evolution after transition period of orientation angle\nθand lag angle βcalculated using Eqs. 21 - 22 with ˜ω= 2/√\n3,\nwhich correspond to Wn= 1.\nlarger than maximal magnetic torque Tmax\nm =MBa3, the\ncube lies on a face and the magnetic moment is always out\nof the plane of the rotating magnetic field. For hematite cube\nwitha≈1.5µm, density ρh= 5.25 g/cm3, solvent den-\nsityρs= 1.00 g/cm3, and permanent magnetization M=\n2.2×103A/m(m=Ma3) one finds that Tmax\ng > Tmax\nm\nifB < 15µT. This is much smaller magnetic field than\nthose used both here and in earlier experiments [4, 5]. We use\nBexp∈[0.3; 3] mT . However, as gravitational effects may\nchange dynamics, especially of neutrally stable points P3and\nP4, we examine them in detail.\nGmedgecornerface\n0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.5=0.1\n=0.9FIG. 8. Stationary solutions for angle αat˜ω= 0.1and˜ω= 0.9for\nincreasing gravity parameter Gm, obtained from long-time solutions\nof Eq. 16 with k= 0.614ands= 0.94and for fixed gravity pa-\nrameter Gm. In synchronous rotation cube with rounded corners can\nrotate either on a corner, face, or an edge.\nGm0.0 0.2 0.4 0.6 0.8 1.01.1251.1501.1751.2001.2251.250=0.9\nedgecornerface\nFIG. 9. Stationary solutions for lag βat˜ω= 0.9for increasing\ngravity parameter Gmfor the same parameter values as Fig. 8.\nTo do so we add to our model, the buoyancy force FFFband\nthe reaction force FFFwalland torques TTTwallwith the bottom\nof capillary and solve Eq. 16 with fixed values k= 0.614\nands= 0.94. Note, that even for a single cube to calculate\nreaction forces and torques we solve the equation for quater-\nnion Eq. 12, thus we are evolving in time a seven dimensional\nobject (3 coordinates and 4 components of quaternion). Al-\nthough, only five of them change in time as ( xandy) remain\nfixed as we do not have friction nor vertical walls.\nFor the shape of hematite cubes used in experiments (super-\nballs — cubes with rounded corners) one finds that dynamics\ngets more complicated. From our simulations we see that an\nindividual cube with rounded corners rotates on an edge, face,\ncorner or undergoes a complicated 3D motion (will be dis-\ncussed later in this section). The critical value of ˜ω, which\nseparates the synchronous motion of cube from asynchronous,\nchanges to ˜ωc=|cos(α)|in the case when a cube rotates on\nthe face and magnetic torque can not overcome gravitational\none. Note that here we neglect stationary points P3andP4\nas starting from random initial conditions the probability to\nreach them is zero.\nFor˜ω < ˜ωcthere are three possibilities: either a cube ro-\ntates on the edge, face or corner (see Video2 [19]). The mag-\nnetic moment is in the plane of the rotating field only if the6\ncube rotates on the edge. The stationary solutions for angles\nαandβwith increasing gravity parameter Gmare shown in\nFigs. 8, 9. From Fig. 8 one finds that for weak gravity there\nare two stationary solutions of synchronous rotation. Cube\nas before can rotate on an edge with magnetic moment in the\nplane of rotations or on a corner with magnetic moment point-\ning out of the plane of rotating magnetic field. Whether a cube\nrotates one a face or edge depends on the initial conditions and\n˜ω. At some point the rotation on the edge becomes unstable\nand the cube starts to rotate on the corner. This happens faster\nthe larger is the rotational frequency. The momentum goes\nout of the plane of the rotating magnetic field. The angle α\nfor fixed gravity parameter depends on the shape of the cube\n(how much corners are rounded) and rotation frequency ˜ω. If\nthe gravity parameter is increased the cube starts to rotate on\nthe face. For ˜ω >0.81there is again a region where rotation\non the edge becomes the only stable solution. The reason is\nthat in this interval, the magnetic moment can not anymore\nbalance the gravitational moment when the cube rotates on a\nface, but can when the cube rotates on an edge. This is the\ncase because the magnetic torque is smaller when the cube ro-\ntates on a face. For ˜ω= 0.9andGm>0.83synchronous\nmotion is not anymore observed. There the motion, which is\na blend of back-and-forth motion and precession is observed.\nThis motion is described in the next paragraph.\nIn the case when ˜ω >˜ωcthere are no new stationary points\nexcept P3andP4, which as follows form Fig. 6 still remain\nneutrally stable. This is a bit unexpected as a small parasitic\nstatic magnetic field which is perpendicular to the rotating\nmagnetic field leads to a situation where neutrally stationary\npoints become stable [18]. As one can see in Fig. 10, back-\nand-forth motion and precession are still observable (top row)\nas in the case without gravity. However, the oscillation am-\nplitude for angle αis now larger as figures on the top row are\nobtained from the same initial conditions as the red and green\ntrajectory in Fig. 6. There only values of gravity parameter\nGmfrom Gm= 0 toGm= 0.01were changed. Increas-\ning the parameter Gmeven more leads to a larger amplitude\nforα. At some point precession becomes unstable and we\nobserve more complicated 3D motion instead (bottom row in\nFig. 10). Initially a cube rotates on the face. The lag increases,\nbut instead of back motion to catch up with the magnetic field,\nthe cube rolls and trough rotation in third dimension catches\nup with the magnetic field. In fact this motion is a blend of\nback-and-forth motion and precession (for better understand-\ning different modes reader is advised to examine Video3 [19]).\nThere are two different modes of this complicated dynamics\nwhere the magnetic moment rotates around both fixed points\nor only around one (left and right figure in the bottom row of\nFig, 10 respectively). The left mode is observed for smaller\nvalues of Gm.\nB. Two particles\nFor the two-particle case the full system of equations has\nto be solved Eq. 16 and the solution depends on control pa-\nrameters s,k, and Gm. A 14 dimensional quantity has to be\n0.2 0.4 0.6 0.8051015\n0.8 1.01.41.61.8\n1\n 0 1\n1.01.52.0\n0.6 0.8 1.0\n1.01.52.0FIG. 10. Four different long time periodic solutions of Eq. 16 for\n˜ω > ˜ωc,k= 0.614,s= 0.94,Gm= 0.01of the asynchronous\nmotion projected in the α-βplane. The top row correspond to back-\nand-forth motion and precession obtained by solving Eq. 12 with\nGm= 0.01and˜ω= 1.5with the same initial conditions as in\nFig. 6. In the bottom row the long-time solution’s of Eq. 12 with\n˜ω= 0.9,Gm= 0.86andGm= 0.94projection to α-βplane\nare shown. Note as we present projections, trajectories can cross\nthemselves. The motion of the cube for these modes can be seen in\nVideo3 [19].\nevolved in time. There are four different long-time-limit sce-\nnarios possible. If the hematite particles are sufficiently far\nthey rotate independently as described in previous subsection.\nThe other possibilities are that cubes form a stable chain, or\nan asymmetric-chain where particles have different types of\nmotion, e.g. one particle rotates on an edge, but other on a\nface (see Video4 [19]). The fourth possibility is that parti-\ncles undergo motion, where at times a chain is formed which\nthen breaks. The long-time trajectories, in the last two cases,\nstrongly depend on initial conditions.\nIn a static external magnetic field two hematite cubes form\na straight chain [2]. If the external magnetic field is larger\nthan≈0.1mT (magnetic field is perpendicular to the grav-\nity) then the magnetic moments of cubes are parallel to the\nexternal magnetic field. In this case cubes lie on an edge and\nthe tilt angle (the angle between the base and cube’s face) is\nclose to 45◦[1, 2]. In a very slowly rotating magnetic field\nsuch a chain should rotate with almost no lag. By increasing\nthe rotational frequency the configuration changes and the lag\nincreases. Unlike a single cube, a two-cube chain can not ro-\ntate on a corner, due to geometric restrictions. Note that for a\ntwo-cube chain, if it rotates on an edge, the magnetic moment\ndoes not have to be in the plane of the magnetic field, as the\nreaction torque between particles can balance a component of\nthe magnetic torque. However, for values of gravity parame-\nterGmused in our experiment the magnetic moment is in the\nplane of the rotating magnetic field. For a two-cube chain also\nasynchronous motion is observed. But there may exist a fre-\nquency interval where the chain undergoes disassembly and\nreassembly motion. This is the case when chain breaks with7\nincrease of rotation frequency before back-and-forth motion\ncan happen.\nFor two-cube chains also in asynchronous regime not all\nmodes of a single cube motion are present due to geometric re-\nstrictions. We do not observe precession of the magnetic mo-\nment. Due to the same reason for back-and-forth motion the\noscillations of the angle αare suppressed whenever magnetic\nmoments of individual cubes are synchronised. Surprisingly,\nwe still observe complicated 3D dynamics where initially the\nchain rotates and lag increases, but instead of back motion to\ncatch up with the magnetic field, the chain rolls and catches\nup with the magnetic field through rotation in the third dimen-\nsion. But unlike for a single cube this mode is only stable for\nsmall values of Gm<0.15. For this mode not only the mag-\nnetic moment goes out of plane, but cubes are periodically on\ntop of each other (see Video5 [19]). For referring later we call\nit asynchronous out of plane rotation. This mode also should\nbe experimentally easily distinguishable as the chain’s length\nvisible in a microscope changes significantly (ratio of maxi-\nmal to minimal length is almost two). For Gm>0.15this\nmode is not anymore observable. Then one observes back-\nand-forth motion of asymmetric-chain where one cube rotates\nmore on an edge, but other more on a face (see Video4 [19]).\nIf we consider only experimental conditions ( Bexp∈\n[0.3; 3] mT anda0≈1.5µm) then for a two-cube chain there\nare four regimes of motion. Cube motion for those modes\nis shown in Video6 [19]. Three of them are planar motion\nregimes: solid-body motion, back-and-forth motion, and peri-\nodic chain disassembly and reassembly. For the fourth regime\nthe cube-chain goes out of the plane of magnetic field. Dur-\ning motion the chain rotates slower than the rotating magnetic\nfield and, in order to catch up with the magnetic field, it rolls\non an edge and through rotation in the third dimension catches\nup with the magnetic field. This mode is a blend of back-\nand-forth motion and precession. To give insight into how\nthis motion should look from an experimental perspective the\nreader can examine Video7 [19], where cubes have been lifted\nto make shadows better visible.\n1. Planar motion with α= 0\nFor planar motion, as long as two cubes form a chain, mo-\nments of cubes are synchronized θ1=θ2=θ. Therefore, the\nchain can be effectively described with two parameters: an-\ngleθand horizontal displacement of cube’s centers — shift b\nwhich are defined in Fig. 11.\nThe EOM in this case is more complicated than for a sin-\ngle particle without gravity, but can easily be obtained us-\ning Lagrange mechanics with Rayleigh dissipation function\nG(vvv,ΩΩΩ) =1\n2P\ni(ξvvvi·vvvi+ζΩΩΩi·ΩΩΩi)approach. However, the\nactual expressions in the general case are quite lengthy, thus\nthey are not provided explicitly. For the case when there ex-\nists a stable stationary solution for b, one obtains that there is\na stationary solution for the lag β= ˜ω˜t−θ\n˙β= ˜ω \n1 +1 +˜b2\n4k!\n−sin(β), (25)\nabFIG. 11. Schematic view of two-particle chain from above.\nwhere ˜b=b/a0.\nThe critical frequency ˜ωcabove which no stationary solu-\ntion for the lag angle may exist is\n˜ωc=4k\n4k+ 1 + ˜b2<1. (26)\nNote that below critical frequency there still may exist a re-\ngion where a chain disassembly and reassembly motion is ob-\nserved. Thus, two additional frequencies ˜ωsand˜ωaare intro-\nduced. The ˜ωsis the maximal frequency until which the chain\nrotates synchronously with the external magnetic field and ˜ωa\nis the minimal frequency when asynchronous back-and-forth\nmotion is observed. When ˜ωs̸= ˜ωathis means that there is\na region where a chain disassembly and reassembly motion is\nobserved. If ˜ωs= ˜ωa⇒˜ωs= ˜ωc= ˜ωa.\nFor a particular cube chain the critical frequency differs for\na clockwise and counterclockwise rotational direction of the\nmagnetic field (depends on the sign of ˜ω). If, for configura-\ntion visible in Fig. 11, the rotation speed is increased when the\ntwo-particle chain rotates clockwise, then the shift ˜breduces.\nThe opposite, however, happens when the two-particle chain\nrotates anti-clockwise as one can see Fig. 12. This means that\nfor clockwise rotation shift ˜bis smaller than for anti-clockwise\nrotation. Therefore, as follows from Eq. 26, the critical fre-\nquency is larger for the clockwise case. This breaks the sym-\nmetry. However, there exists the other two-particle configura-\ntion (see fig. 13), which is observable with the same probabil-\nity [2]. For this configuration the critical frequency is larger\nfor the counterclockwise case. This guarantees that on aver-\nage the observations are the same for both directions of the\nrotations. Therefore, without loss of of generality, only the\nconfiguration shown in fig. 11 is analyzed. For the second\nconfiguration, which is shown in fig. 13, one obtains the same\ndynamics as for the first if rotation direction (the sign of ˜ω) is\nreversed.\nOverall, if two particles are not too far apart, there are three\nregimes for long-time dynamics. For value of k= 1.84they\nare shown in Fig. 14. There in the region I ( |˜ω|<|˜ωs|), chain\nrotates as a solid object with the frequency of rotating field.\nIn the region II ( |˜ω|>|˜ωa|) the back-and-forth motion is ob-\nserved. In this case the shift ˜bis not anymore constant, but pe-\nriodically oscillates. The trajectories in the case of s= 0.94,\nk= 1.84, and|˜ω|= 0.8are shown if Fig. 15. The oscillation\namplitude for ˜bis always larger for counterclockwise rotation8\n0.75\n 0.50\n 0.25\n 0.00 0.25 0.50 0.75\n0.20.30.40.50.60.7b\ns=0.94, k=2.45\ns=0.94, k=0.61\ns=1.88, k=0.61\ns=3.76, k=0.61\nFIG. 12. The horizontal cube center displacement ˜bvs. rotation fre-\nquency ˜ωfor a two-cube chain, calculated using Eq. 12. The shape\nof the curve is determined by the product of sandk, however, the ˜ωc\naccording to Eq. 26 depends on both parameters sandk. Thus, red\nand blue curves overlap, but the blue curve extends further as |˜ωc|\nis larger for the blue curve. The positive value of ˜ωmeans counter-\nclockwise rotation while negative corresponds to clockwise rotation.\nab\nFIG. 13. Schematic view from above of the second two-particle chain\nchain configuration. There is a 50% chance that cubes arrange in this\nconfiguration.\nof magnetic field and, in general, reduces by increasing rota-\ntional frequency. In the region III ( |˜ωs|<|˜ω|<|˜ωa|of pa-\nrameters the chain break. After some time chain can reassem-\nble, but it will break again. We observe a periodic disassembly\nand reassembly of a chain, thus the long time trajectory is pe-\nriodic, however, it strongly depends on initial conditions. In\nthe case of positive ˜ωas thermal fluctuations are present in the\nexperiment, it can happen that during this motion cubes rear-\nrange in the second configuration (Fig. 13) as in this case the\nsecond configuration is energetically favorable.\nC. More than two particles\nIn this case, depending on how particles are distributed,\nseveral chains may be formed which undergo more or less in-\ndependent motion. When a single chain is formed there are\nthe same three planar motion regimes. The regime where the\ncube chain goes out of the plane of rotation magnetic field is\nobservable for chains consisting of up to four particles. For\nplanar motion qualitatively we obtain the same diagram as in\nFig. 14 only the region III increases and both region I and\n1 2\n0.51.01.5 s\nIIII II\ns\na\n2\n 1\n0.51.01.5 s\nIIII II\ns\na\n20 40 60\nf [Hz]12 B [mT]\nIII\nI\nIIs\na\n60\n 40\n 20\nf [Hz]12 B [mT]III\nI IIs\na\nFIG. 14. Borders of long-time dynamics regimes of Eq. 12 for\nk= 1.84using dimensionless variables (top row) and dimensional\nvariables for comparison with experiment (bottom row). The re-\ngions I,II, III correspond to solid body rotation, back-and-forth mo-\ntion, periodic chain disassembly and reassembly motion regimes re-\nspectively. The positive value of ˜ωmeans counterclockwise rotation\nwhile negative corresponds to clockwise rotation.\n78400 78410 78420 78430 78440 78450 78460\nt\n0.20.40.60.81.01.2b\n=-0.8\n =0.8\nFIG. 15. The time evolution after transition time of horizontal\ncube center displacement ˜bcalculated using Eq. 12 with s= 0.94,\nk= 1.84, and |˜ω|= 0.8. The positive value of ˜ωmeans counter-\nclockwise rotation while negative corresponds to clockwise rotation.\nregion II become smaller with increasing number of particles.\nFor the fixed values of sandkthe|˜ωs|decreases with increas-\ning number of particles Nin the chain, as shown in Fig. 16.\nFor maximal chain length which is observed we obtain that\nN∝1√\n|˜ω|. This is the same relation as for paramagnetic\nspherical particles with and without anisotropy [20–22]. Con-\ncerning chain shape a similar effect to bending of chains of\nspherical paramagnetic particles [20–25] is observed, but the\nshape is quite different and depends on rotation direction (see\nFig. 17).\nThe opposite behavior is observed for |˜ωa|, thus also re-\ngion II becomes smaller with increasing number of particles\nin a chain N. The value of |˜ωa|for larger Nbecomes very\nlarge. Therefore it is expected that for larger chains the asyn-\nchronous regime is not observed experimentally as we are far\naway from the validity region of our model.\nAs for two-particle chains, also here we observe that for9\n2 4 6 810 12 14 16 18 20\nNumber of cubes N102\n101\n|s|\n<0\n>0\n2 4 6 8 10 12\n|s|0.5\n51015Number of cubes N\n<0\n>0\nFIG. 16. Dependence of maximal rotation frequency |˜ωs|when chain\nrotates as solid object vs number of hematite particles in a chain. The\npoints are calculated from Eq.12 with s= 0.94andk= 1.84. The\npositive value of ˜ωmeans counterclockwise rotation while negative\ncorresponds to clockwise rotation.\na particular chain the behavior depends on rotation direction.\nFor a ten-particle chain shown in Fig. 17 we observe that crit-\nical frequency |˜ωa|is more than two times larger in the case\nof clockwise rotation. Visually the configuration also looks\ndifferent. However, when a chain breaks, it always breaks in\nthe middle. If there are an even number of particles in the\nchain then the chain breaks into two chains withN\n2particles\nin each. If there is an odd number of particles in the chain, it\nbreaks into 3 parts withN−1\n2, 1, andN−1\n2particles in each.\nThe long-time solution, as for two-particle chains, is a peri-\nodic disassembly and reassembly of chain and trajectory is\nstrongly initial condition dependent. However, we do not ob-\nserve formation of clusters after a chaotic transition period as\nfor paramagnetic particle chains [26].\nIV . EXPERIMENTAL RESULTS\nBehavior of two hematite cube chains in a rotating magnetic\nfield is measured experimentally. For this, hematite cubes\nare synthesised and characterized following the methods de-\nscribed in [4]. The cubes were found to have edge length\na≈1.5µmand shape factor q≈2.0. The same sample\nof hematite cubes was used in all experiments. Mixing of the\nsample and restoration of chemical composition needed for\npH level and suspension stability was done before each exper-\niment.\nThe hematite samples were contained in glass capillaries\nwith100µmthickness filled with liquid. The observation was\nFIG. 17. Configuration of ten particle chains (from left to right),\nwhich is in an anti-clockwise rotating magnetic field, a static mag-\nnetic field, and a clockwise rotating magnetic field. The configura-\ntions are obtained from Eq.12 with s= 0.94andk= 1.84. In the\ncase of the rotating magnetic field, the frequency |˜ω|is just below\n|˜ωs|. If|˜ω|is increased the chains break.\ndone with a microscope (Leica DMI3000B) equipped with\na camera (Basler ac1920-155um, up to 250frames per sec-\nond), using an oil immersion objective with 100×magnifica-\ntion. Image acquisition was done with the proprietary camera\nsoftware while image processing and analysis was performed\nwith MATLAB. In general terms, image analysis relied upon\ncross correlating an image of a single hematite cube to the\nexperimental image, with the two correlation maximums cor-\nresponding to the two cubes of the dimer. After thus identi-\nfying the individual cubes, information about their distancing\nand angle between the axis of the cube chain and the magnetic\nfield could be obtained. These parameters were then used to\nclassify the chain configuration (such as planar motion, peri-\nodic chain breakup and reassembly, or out of plane motion)\nand motion characteristics (correspondence between rotation\nfrequencies of the chain and magnetic field) as belonging to\none of the rotation regimes as described in Sec. III.\nThe magnetic field was generated by three pairs of coils,\npowered by DC current sources (KEPCO) that are controlled\nby a NI DAQ card using LabView code. The glass capillary\nwas placed near the point of crossing of the three coil pair axis\n(see scheme of layout in Fig. 18). To compensate for parasitic\nmagnetic field sources, such as Earth or those associated with\nlab equipment, the field was measured at the location of the\ncapillary within the microscope. This was done prior to each\nexperiment, using a magnetic sensor (HMC5883 GY-271 3V-\n5V Triple Axis Compass Magnetometer Sensor Module for\nArduino), obtaining background field values via Arduino and\nthen accounting for them in the LabView code. Such a method\nallows us to define the magnetic field with a precision ∆B∈\n(0.01; 0.03) mT .\nHematite cube chain rotation was captured as a sequence of\nimages and accompanying coil current measurements, provid-\ning information about the magnetic field. A sequence of mea-\nsurements for one hematite cube chain would involve either an\nincrease or decrease of field rotation frequency at a constant\nfield magnitude, or a change of magnitude at a constant rota-\ntion frequency, or both. Several such sequences are measured\nfor each pair of particles. Magnetic field was increased and10\nFIG. 18. Coil system and microscope setup used in experimental\nwork. A – coil system; B – placement of field sensor or sample; C –\nobjective; D – light source\ndecreased continuously, to avoid a stepwise supply of energy\nto the system. Experimental data presented here was gathered\nfrom 49 dimers, and a total of 845 measurements.\nSeveral aspects connected to cube chain rotation analysis\nhave to be noted. First, experimentally it is impossible to mea-\nsure the angle between the external field and magnetic mo-\nment. Instead, we use the lag angle between the external field\nand the axis of the cube chain. Second, initial configuration\nof the cube chain cannot be set, therefore it has to be deter-\nmined. We find it by applying a stationary magnetic field and\nobserving the angle between the external field and the axis\nof the cube chain (see [2] for more details on the cube chain\norientation in stationary magnetic fields). Due to the limited\nframe rate and optical resolution, this configuration control\nis repeated not only at the start and end of the measurement\nseries, but also during it. We only use measurements, where\nthe chain configuration has not changed between two control\nmeasurements. Third, in experiments we use rotating mag-\nnetic fields rotating both in clockwise and counterclockwise\ndirections, regardless of initial configuration of cube chains.\nThis is simpler for the experimental sequence control and al-\nlows to check if there are differences in results depending on\nthe initial configuration and rotation direction.\nAn insight into the different regimes of rotation as seen\nfrom experimental measurements is provided in Fig. 19.\nThere the relation for angle of rotation for the magnetic field\nand for the hematite cube chain is provided. Based on experi-\nmental videos and angle measurements we find the same four\ndifferent regimes as in theory (see Sec.III) of which three are\nplanar rotation and one which is associated with asynchronous\nrotation where chain goes out of the plane of the rotating mag-\nnetic field. From them only the solid-body regime follows the\nmagnetic field, while rotation frequency and therefore the ro-\ntation angle is lower for all the other regimes. The solid-body\nregime is the only one where the chain rotates synchronously\nwith the magnetic field. Example pictures of the chain, along\nwith illustrations of chain axis and field direction, in each of\nthe regimes are given in Fig. 20.\nTo have a better comparison with theoretical results, the\nmeasurements can be split in two groups, corresponding to\nFIG. 19. Relation between angle of rotation for a hematite cube chain\nto angle of rotation for magnetic field, in the four regimes of rotation.\nFor a motion out of the field rotation plane, the curve is not continu-\nous as when particles are exactly on top of each other chain rotation\nangle is not defined.\nFIG. 20. Example images from the four rotation regimes. Cube mo-\ntion measured experimentally can be seen in Video8 [19].\nthe two directions of rotation for a cube chain configuration\nshown in Fig. 11, as was done in Sec. III. In practice it means\nthat the measurements of the first initial cube chain config-\nuration with a subsequently applied clockwise field are com-\nbined with the measurements of the second initial position and\na rotating field rotating counterclockwise (the combined set\nof data points further referred to as \"clockwise equivalent\"),\nand the opposite (the combination called \"counterclockwise\nequivalent\").11\n0 10 20 30 40\n f, Hz051015 fchain, Hzchain 1\nchain 2\nchain 3\nchain 4\n010 20 30 40\nf, Hz0510152025fchain, Hz\ncounterclockwise \n equivalent\n010 20 30 40\nf, Hz\nclockwise \n equivalent\nsolid-body rot.\nback-and-forth\nout of plane rot.\nFIG. 21. Experimentally measured average chain (N=2) rotation\nfrequencies for four different two cube chains at various field rota-\ntion frequencies, B= 1 mT (top figure) compared to the simula-\ntion results (bottom figures).Experimental data points for each chain\nare marked with one marker and connected with tracer line. Pluses\nand crosses are clockwise equivalent; triangles — counterclockwise\nequivalent. For theoretical calculations in the counterclokwise equiv-\nalent case in the frequency range, where there is no solid body and\nback-and-forth rotation, periodic dis- and reassembly of chain is ob-\nserved. The black line is drawn for comparison purposes and shows\nback-and-forth motion in the clockwise equivalent case. Colors rep-\nresent different rotation regimes as in Fig.20.\nUsing experimental data from angle measurements (as\nshown in Fig.19), we can calculate the average rotation fre-\nquency for a chain and show its dependence on magnetic field\nfrequency. In Fig.21 we present experimental results for four\ndifferent two cube chains at a fixed B= 1 mT , indicated by\ndifferent markers and tracer lines, along with theoretical pre-\ndictions. Although the measurement series for each of the\nchains look similar to classical rotating rod [27], behavior is\nmore diverse. In particular, more rotation regimes are present\n- out of plane motion (red symbols) and periodic dis- and re-\nassembly (not observed for these four chains) complement\nsolid body (green) and back-and-forth (blue) rotation (same\ncolors as in Fig.20).\nThe data represented in Fig.21 provides some insight into\ndiscrepancies between our theoretical understanding and ex-\nperimental observations. Data points clearly show that, al-\nFIG. 22. A phase diagram of the rotation regimes for the counter-\nclockwise equivalent (top figure) and clockwise equivalent (bottom\nfigure) orientation with N=2 cubes per chain. The areas correspond-\ning to each rotation regime are defined by the extreme points of that\nregime.\nthough cube chains were selected as similar as visually possi-\nble, there are notable differences in both characteristic values\nand rotation regimes. And it is not possible to explain them\nwith clockwise equivalent ( +and×) and counterclockwise\nequivalent (triangles) division. Moreover, chains not only\nhave different critical frequencies at which they stop follow-\ning the solid body rotation regime, but also follow different\nregimes at the same field and frequency values or change the\nregimes several times.\nThese observations hint that particle size differences within\nthe known limits for our sample, surface effects and thermal\nfluctuations can cause significant differences in quantitative\nmeasurements among the different chains, which also goes to\nexplain the discrepancies from theoretical predictions. Calcu-\nlations indicate that a difference in size of 10% between cubes\nof the same chain would lead to considerable differences in the\nchain’s behavior and critical frequency, compared to a chain\nof equally sized cubes. We know the distribution of cube sizes\nto exceed that and, while the cube pairs are selected on a ba-\nsis of visual inspection to be as similar as possible, it is very\nlikely that differences in size between cubes of the same dimer\nare in many cases significant enough to contribute to differ-\nences in dimer behavior, both comparing several dimers and12\nexperimental results to theoretical. Note that measurements of\nout of plane regime rotation frequencies should be considered\nless reliable than for the other regimes, due to difficulties of\ndefining an angle when the chain is in near vertical position.\nSeveral points belonging to out of plane regime in Fig. 21 co-\ninciding with the curve of back-and-forth regime data points is\na result of one dimer displaying behavior characteristic of both\nregimes at the same field magnitude and frequency. In such a\ncase, the dimer would rotate as in back-and-forth regime but\nat a point transition into out of plane rotation. This transition\ncould be driven by thermal fluctuations.\nAnother reason for discrepancies between theory and ex-\nperiment in Fig. 21 is that mechanical friction between cubes\nis neglected in the theoretic model. When mechanical friction\nis taken into account, it is harder for cubes to slide along other\ncubes’ faces. This most probably is the reason why in theoret-\nical calculations the out of plane motion can be observed for\nhigher frequencies than in the experiment.\nAs a result, the information from particle pair rotation ex-\nperiments, which can be summarized in two phase diagrams,\nshown in Fig. 22, involve overlapping areas. For better read-\nability, as a frequency - field pair can correspond to several\nmeasurements, the data points have been offset. The areas\ncorresponding to each rotation regime are defined by the ex-\ntreme points of that regime.\nThe phase diagrams reveal that chain breakup in examined\nmagnetic fields occurs predominantly when rotation happens\nin counterclockwise equivalent conditions, as would be ex-\npected from theoretical considerations (Fig. 14). In exper-\nimental diagrams one observes that back-and-forth motion,\nparticle disassembly and reassembly and asynchronous out\nof plane rotation slightly overlap with the solid-body rota-\ntion regime. This overlap makes direct quantitative compar-\nison between experimental phase diagrams and those seen in\nFig. 14 impossible. While there is an overlap between all ro-\ntation regimes, the border between solid body and back-and-\nforth regimes are in the vicinity of those for entry into disas-\nsembly and reassembly and asynchronous out of plane rota-\ntion motion regimes. In both orientations, a particle chain can\nonly be reliably expected to remain in planar rotation and not\nenter 3D motion at high field - low frequency or, conversely,\nlow field - high frequency conditions.\nV . CONCLUSIONS\nIn current work we examine a single cube and a short\nhematite chain dynamics in rotating magnetic fields. The\ninvestigation is mainly theoretical, but results for two-cube\nchains are verified also experimentally. To determine how im-\nportant gravity effects are two models were developed, one\nwith gravity and one without.\nFor a single cube one finds that at low frequencies a cube\nrotates synchronously with the magnetic field and for higher\nfrequencies asynchronous motion is observed. During syn-\nchronous motion one finds that a cube with rounded corners\ncan rotate on the edge, corner or face. The magnetic moment\nis in the plane of the rotating magnetic field only if the cuberotates on an edge. Whether a cube rotates on an edge, corner\nand face depends on the magnetic field strength, frequency\nand initial conditions. In an asynchronous regime without ex-\nplicit gravity effects there are two neutrally stable fixed points.\nTwo modes of motion are observed: precession of magnetic\nmoment and back-and forth motion. Including explicit grav-\nity effects one finds that a new mode of complicated 3D mo-\ntion appears. In this motion cube rotates slower than magnetic\nfield, the lag increases, but instead of back motion to catch up\nwith the magnetic field, the cube rolls and trough rotation in\nthe third dimension catches up with the magnetic field. This\nmode has two sub-types where a cube rotates around one fixed\npoint or around both of them.\nFor a two-cube system some of the motion modes which\nwere observable for single cube disappear. This happens due\nto geometric restrictions. No rotation on an edge and preces-\nsion is possible. There appear, however, new scenarios: two-\ncube chains can break or asymmetric-chain is formed where\ncubes undergo different motion types, e.g., one cube rotates on\na face while other on an edge. If the chain breaks then periodic\nchain disassembly and reassembly is observed. For an indi-\nvidual chain dynamics depends on the clockwise or counter-\nclockwise rotation direction of the magnetic field. However,\nwhen averaged over many chains, there is no dependence. The\nreason for this is that in a large sample there are with the same\nprobability two chain types. They behave differently at a given\nclockwise and counterclockwise rotation direction of the mag-\nnetic field. But the first chain’s type dynamics in a clockwise\nrotating magnetic field is equal to the second chain’s type dy-\nnamics in an counterclockwise rotating magnetic field. Thus,\nthey balance out this effect and on average there are no differ-\nences.\nIn the case of small frequencies, the two-cube chain rotates\nsynchronously with the magnetic field. Chain’s configuration\ndoes not change for a fixed rotation frequency, thus the chain\nrotates as a solid body. With increase of the frequency of the\nrotating magnetic field two scenarios can happen. Either chain\nbreaks and enters to the mode where we observe periodic dis-\nassembly and reassembly of chain or asynchronous motion of\nchain is observed.\nThe transition to the asynchronous regime for a two-cube\nsystem happens at lower rotational frequencies of the rotat-\ning magnetic field compared to a single cube. For asyn-\nchronous motion depending on initial conditions three modes\ncan be observed. One is the back-and-forth motion of the\nchain where the magnetic moment remains in the plane of\nthe rotating magnetic field. The second is back-and-forth mo-\ntion of the chain with periodical disassembly and reassembly.\nThe third one is one of two modes: back-and-forth motion of\nasymmetric-chain or motion where the cube chain goes out of\nthe plane of the rotating magnetic field. There the chain, to\ncatch up with the magnetic field, rolls on an edge and through\nrotation in the third dimension catches up with the magnetic\nfield. The last mode to our knowledge has not been described\nbefore in scientific literature.\nFor experimental conditions ( Bexp∈[0.3; 3] mT anda0≈\n1.5µm) for a two-cube chain four regimes of motion are pos-\nsible. Three of them are planar motion regimes: solid-body13\nmotion, back-and-forth motion, and periodic chain disassem-\nbly and reassembly, and motion where the cube chain goes\nout of the plane of the rotating magnetic field. All four modes\nwere observed, identified in our experiments. Also there was\nno indication that there should exist another mode. In exper-\niments, as predicted in the theory part of this paper, one ob-\nserves that there are two chain types for which motion differs\nfor a given clockwise or counterclockwise rotation of a mag-\nnetic field. But the motion dynamics for the first chain type\nin case of a clockwise rotation of the magnetic field is equal\nwith the second chain type’s dynamics in counterclockwise\nrotation. Thus, in a large sample there is no global depen-dence on rotation direction as in equilibrium there is an equal\nnumber of particles in each configuration. Theoretical consid-\neration predicts that an applied field rotating in one direction\ncould change the balance between configurations, however, no\nexperimental indications of that are observed.\nACKNOWLEDGMENT\nM.B. acknowledges financial support from PostDocLatvia\ngrant No. 1.1.1.2/VIAA/3/19/562.\n[1] L. Rossi, J. G. Donaldson, J.-M. Meijer, A. V . Petukhov,\nD. Kleckner, S. S. Kantorovich, W. T. M. Irvine, A. P.\nPhilipse, and S. Sacanna, Self-organization in dipolar cube flu-\nids constrained by competing anisotropies, Soft Matter 14, 1080\n(2018).\n[2] M. Brics, V . Šints, G. Kitenbergs, and A. C ¯ebers, Energetically\nfavorable configurations of hematite cube chains, Phys. Rev. E\n105, 024605 (2022).\n[3] L. Rossi, Colloidal Superballs , Ph.D. thesis, Utrecht University,\nUtrecht, Nederlands (2012).\n[4] O. Petrichenko, G. Kitenbergs, M. Brics, E. Dubois, R. Perzyn-\nski, and A. C ¯ebers, Swarming of micron-sized hematite cubes\nin a rotating magnetic field – experiments, Journal of Mag-\nnetism and Magnetic Materials 500, 166404 (2020).\n[5] V . Soni, E. S. Bililign, S. Magkiriadou, S. Sacanna, D. Bartolo,\nM. J. Shelley, and W. T. M. Irvine, The odd free surface flows\nof a colloidal chiral fluid, Nature Physics 15, 1188 (2019).\n[6] W. Chen, X. Fan, M. Sun, and H. Xie, The cube-shaped\nhematite microrobot for biomedical application, Mechatronics\n74, 102498 (2021).\n[7] M. Driscoll, B. Delmotte, M. Youssef, S. Sacanna, A. Donev,\nand P. Chaikin, Unstable fronts and motile structures formed\nby microrollers, Nature Physics 13, 375 (2016).\n[8] A. Aubret, M. Youssef, S. Sacanna, and J. Palacci, Targeted\nassembly and synchronization of self-spinning microgears, Na-\nture Physics 14, 1114 (2018).\n[9] S. I. R. Castillo, C. E. Pompe, J. van Mourik, D. M. A. Ver-\nbart, D. M. E. Thies-Weesie, P. E. de Jongh, and A. P. Philipse,\nColloidal cubes for the enhanced degradation of organic dyes,\nJournal of Materials Chemistry A 2, 10193 (2014).\n[10] J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and P. M.\nChaikin, Living crystals of light-activated colloidal surfers, Sci-\nence339, 936 (2013).\n[11] I. P. Omelyan, Algorithm for numerical integration of the rigid-\nbody equations of motion, Phys. Rev. E 58, 1169 (1998).\n[12] K. I. Kou and Y .-H. Xia, Linear quaternion differential equa-\ntions: Basic theory and fundamental results, Studies in Applied\nMathematics 141, 3 (2018).\n[13] W. Hamilton, On quaternions; or on a new system of imaginar-\nies in algebra, The London, Edinburgh, and Dublin Philosophi-\ncal Magazine and Journal of Science 25, 489 (1844).[14] G. Guennebaud, B. Jacob, et al. , Eigen v3,\nhttp://eigen.tuxfamily.org (2010).\n[15] K. OKADA and A. SATOH, Diffusion coefficients of cube-\nlike particles for application to brownian dynamics simulations,\nTransactions of the JSME (in Japanese) 85, 18 (2019).\n[16] J. G. Donaldson and S. S. Kantorovich, Directional self-\nassembly of permanently magnetised nanocubes in quasi two\ndimensional layers, Nanoscale 7, 3217 (2015).\n[17] J. D. Weeks, D. Chandler, and H. C. Andersen, Role of repul-\nsive forces in determining the equilibrium structure of simple\nliquids, The Journal of Chemical Physics 54, 5237 (1971).\n[18] V . Palkar, P. Aprelev, A. Salamatin, A. Brasovs, O. Kuksenok,\nand K. Kornev, Rotating magnetic nanorods detect minute fluc-\ntuations of magnetic field, Physical Review E 100(2019).\n[19] See supplemental material at ancillary files section for further\nsimulation and experimental videos.\n[20] S. Melle and J. E. Martin, Chain model of a magnetorheological\nsuspension in a rotating field, The Journal of Chemical Physics\n118, 9875 (2003).\n[21] I. Drik ,is and A. Cebers, Molecular dynamics simulation of the\nchain of magnetic particles and flexible filament model, Mag-\nnetohydrodynamics 40, 351 – 357 (2004), cited by: 6.\n[22] J. Užulis and J. C ¯ımurs, The chain length of anisotropic param-\nagnetic particles in a rotating field, Journal of Magnetism and\nMagnetic Materials 542, 168555 (2022).\n[23] S. L. Biswal and A. P. Gast, Rotational dynamics of semi-\nflexible paramagnetic particle chains, Phys. Rev. E 69, 041406\n(2004).\n[24] A. C ¯ebers and I. Javaitis, Dynamics of a flexible magnetic chain\nin a rotating magnetic field, Phys. Rev. E 69, 021404 (2004).\n[25] A. Zaben, G. Kitenbergs, and A. C ¯ebers, 3d motion of flexible\nferromagnetic filaments under a rotating magnetic field, Soft\nMatter 16, 4477 (2020).\n[26] H. Abdi, R. Soheilian, R. M. Erb, and C. E. Maloney, Paramag-\nnetic colloids: Chaotic routes to clusters and molecules, Phys.\nRev. E 97, 032601 (2018).\n[27] B. Frka-Petesic, K. Erglis, J. Berret, A. Cebers, V . Dupuis,\nJ. Fresnais, O. Sandre, and R. Perzynski, Dynamics of para-\nmagnetic nanostructured rods under rotating field, Journal of\nMagnetism and Magnetic Materials 323, 1309 (2011), proceed-\nings of 12th International Conference on Magnetic Fluid."    },    {        "title": "2106.05676v1.Linear_Stability_of_Periodic_Trajectories_in_Inverse_Magnetic_Billiards.pdf",        "content": "LINEAR STABILITY OF PERIODIC TRAJECTORIES IN\nINVERSE MAGNETIC BILLIARDS\nSEAN GASIOREK\nAbstract. We study the stability of periodic trajectories of planar\ninverse magnetic billiards, a dynamical system whose trajectories are\nstraight lines inside a connected planar domain \n and circular arcs out-\nside \n. Explicit examples are calculated in circles, ellipses, and the one\nparameter family of curves x2k+y2k= 1. Comparisons are made to the\nlinear stability of periodic billiard and magnetic billiard trajectories.\n1.Introduction\nMathematical billiards serves as a fundamental example of a dynamical\nsystem and has been studied extensively over the last century. Connecting\ngeometry and dynamics, mathematical billiards concerns the motion of a free\nparticle (the \\billiard ball\") under inertia in a domain (the \\billiard table\")\nwhich moves at constant speed and undergoes perfectly elastic collisions\nwith the boundary of the table. The collisions with the boundary follow\nthe billiard re\rection law \\angle of incidence equals angle of re\rection\",\nwhere the tangential component of the velocity is conserved while the normal\ncomponent instantly changes sign. See e.g. [4, 15, 20] for a survey.\nWhile mathematical billiards serves as a model of certain physical phe-\nnomena, such as wave fronts and geometric optics, magnetic variants of\nbilliards, where the billiard ball is interpreted as a charged particle moving\nunder the in\ruence of a magnetic \feld Bwhich satis\fes the re\rection law at\nthe boundary of the billiard table, provides an extension of these same ideas\nto various geometric settings (e.g. [3, 12, 21]) and to problems in dynamics\nand mathematical physics (e.g. [2, 9, 17, 18, 19]).\nThe de\fnition of the magnetic \feld Bgreatly a\u000bects the dynamics but are\nalso informed by the problem which is to be solved. The study of charged-\nparticle dynamics in piecewise-constant magnetic \felds appears in a variety\nof settings, such as nano- and condensed-matter physics, semiconductor de-\nsign, and quantum mechanics [6, 13, 14, 22]. Classical, semiclassical, and\nquantum approaches to this system are each addressed to a degree in com-\npact or unbounded domains depending upon the applications of interest.\nThe magnetic billiard of interest is that of inverse magnetic billiards ,\nfollowing the naming by [22], which has only been studied in detail recently\n[10, 11]. Given a connected domain \n \u001aR2, de\fne a constant magnetic \feld\nBorthogonal to the plane which has strength 0 on \n and strength B6= 0\non its complement. The classical motion of a charged particle of charge e\n2020 Mathematics Subject Classi\fcation. 37J25, 37J46, 70H12, 70H14, 78A35.\nKey words and phrases. Inverse magnetic billiards, stability, linear stability, periodic\norbits, periodic trajectories.\n1arXiv:2106.05676v1  [math.DS]  10 Jun 20212 S. GASIOREK\nand massmwith constant speed jvjthroughout \n and its complement are\ncontinuous curves which are circular arcs outside \n and straight chords inside\n\n. The charged particle is subject to the Lorentz force outside \n, and the\nresulting motion is circular arcs of \fxed Larmor radius \u0016=mjvj=jeBj. We\ntakeeB < 0 so the Larmor arcs are traversed in the anticlockwise direction.\nThis paper is organized as follows. Section 2 constructs the inverse mag-\nnetic billiard map, its derivative, and relevant properties. Section 3 estab-\nlishes a linear stability criteria for 2-periodic trajectories in inverse magnetic\nbilliards, provides explicit examples, and contrasts the stability criteria with\nthe existing linear stability criteria for standard and magnetic billiards. In\nSection 4, we give examples of the linear stability of 3- and 4-periodic tra-\njectories with symmetries in various domains.\n2.Properties of the Inverse Magnetic Billiard Map\nWe give a brief review of inverse magnetic billiards in a convex set \n \u001aR2\nand note that additional details can be found in [10, 11].\nSuppose \n\u001aR2is strictly convex and parametrize the boundary @\n =\n\u0000(s) by arc length, sin the anticlockwise direction and let L=j@\nj. Pro-\nvided \u0000(s) is su\u000eciently smooth, the convexity of \n implies the radius of\ncurvature\u001a(s) = 1=\u0014(s) of \u0000(s) satis\fes 0 < \u001amin\u0014\u001a(s)\u0014\u001amax<1.\nIn the study of magnetic billiards, Robnik and Berry [18, 19] classi\fed the\ndynamics based upon three curvature regimes , depending upon the relative\nsizes of\u0016,\u001amin, and\u001amax:\n\u0016<\u001amin; \u001amin<\u0016<\u001amax; \u0016>\u001a max:\nThe motion of the charged particle (the \\billiard\") in this setting can be\ndescribed in terms of its two geometric components: its straight-line stan-\ndard billiard component and its magnetic Larmor arc. Suppose a trajectory\nstarts at a point P0:= \u0000(s0) with an initial velocity vector v0pointing to\nthe interior of \n. This initial velocity vector makes an angle \u001202(0;\u0019)\nwith the positively-oriented tangent vector to \u0000( s) atP0, and as the billiard\ntravels in a straight line inside \n, it will eventually meet \u0000( s) at a new point\nP1:= \u0000(s1). Let`1=jP0P1jbe the chord length and \u00121be the angle be-\ntween the tangent vector to \u0000( s) atP1and the velocity vector\u0000\u0000\u0000!P0P1. This\ncompletes the billiard-like component of motion.\nNext, because @\n acts as a permeable boundary between magnetic and\nnon-magnetic regions, the motion starting at P1in the direction v0moves\nalong a circular Larmor arc \rof radius\u0016until intersecting \u0000( s) at a point\nP2:= \u0000(s2) where the billiard re-enters \n. The velocity vector v2of the\nbilliard atP2is the tangent vector to the Larmor circle at P2and makes\nan angle\u00122with the tangent vector to \u0000( s) atP2. The re-entry point P2\nis well-de\fned, as the Larmor circle is tangent to the ray\u0000\u0000\u0000!P0P1atP1. Let\n`2=jP1P2jbe the chord length connecting the exit and re-entry points P1\nandP2of \n; let\u001f0;2be the angle between v0and the ray\u0000\u0000\u0000!P1P2; letS0;2be\nthe area inside the Larmor arc \r0;2but outside \n; let \u0014i:=\u0014(si) be the\ncurvature of \u0000 at si; and letA0;2denote the area of the \\cap\" between the\nchord`2and \u0000(s) that is inside the Larmor circle. See \fgure 1.LINEAR STABILITY OF PERIODIC TRAJECTORIES 3\nON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS (FINAL DRAFT) 434 SEAN GASIOREK\u0000(s)\nP0P1`1AP2`2S\n✓0✓2\u0000\u0000⌦v0v2\nFigure 1.The standard picture of the return map,T.\n(a)4 SEAN GASIOREK\nP1\u0000(s)P2`2✓2\u0000\u0000µ\u0000\u0000\u0000Figure 2.A magnetic arc.the particle moving fromP1along the circular arc\u0000of Larmor radiusµuntilintersecting\u0000(s) again atP2.T h u sT=T2\u0000T1.Proposition 1.Given the mapsT1andT2, the JacobiansDT1= @s1@s0@s1@u0@u1@s0@u1@u0!andDT2= @s2@s1@s2@u1@u2@s1@u2@u1!have components:@s1@s0=0`1\u0000sin(✓0)sin(✓1)@s1@u0=`1sin(✓0)s i n (✓1)@u1@s0=`101\u00001sin(✓0)\u00000sin(✓1)@u1@u0=1`1\u0000sin(✓1)sin(✓0)@s2@s1=\u0000sin(✓1)+`21cos(\u0000)sin(✓1)@s2@u1=`2cos(\u0000)sin(✓1)s i n (✓2)@u2@s1=\u0000sin(✓1)s i n (✓2)+( s i n ( 2\u0000\u0000✓2)\u00002`2cos(\u0000))(sin(✓1)+1`2cos(\u0000))`2cos(\u0000)@u2@u1=sin(2\u0000\u0000✓2)\u00002`2cos(\u0000)sin(✓1).Furthermore,det(DT1)=1anddet(DT2)=1.The details of this proof are given in Appendix A.(b)BILLIARDS INSIDE, CIRCLES OUTSIDE 5Corollary 1.LetT=T2\u0000T1. ThenDT= @s2@s0@s2@u0@u2@s0@u2@u0!with@s2@s0=sin(✓0)\u00000(`1+`2cos(\u0000))sin(✓2)@s2@u0=\u0000`1+`2cos(\u0000)sin(✓0)s i n (✓2)@u2@s0=0[2(`1+`2cos(\u0000))\u0000sin(2\u0000\u0000✓2)]\u00002sin(✓0)+2 cos(\u0000\u0000✓2) tan(\u0000)(sin(✓0)\u00000`1)`2@u2@u0=`2[2(`1+`2cos(\u0000))\u0000sin(2\u0000\u0000✓2)]\u00002`1cos(\u0000\u0000✓2) tan(\u0000)`2sin(✓0)Furthermore,det(DT)=1.From this we conclude thatTis an area-preserving map and that the Birkho↵coordinates are conjugate.\nP0`1`2P2P1P\u00000`\u00001\u0000\u0000\u0000\u0000\u0000\u0000\u0000(s)\nFigure 3.An example of two trajectories with the same value for`2where\u0000and\u00000are supplementary.Figure 2.(a) A magnetic arc. (b) An example of two trajectorieswith the same value for`2where\u0000and\u00000are supplementary.From the definition of these angles we find the following to be true:2\u0000+\"=⇡ +\"=2⇡\u0000(s)\nP0P1`1A0,2P2\n`2S0,2\n✓0✓2\u00000,2\n\u00000,2\n⌦\nv0v2\nFigure 1. The standard picture of the return map, T.\nFigure 1. A labeled diagram of both components of motion\nof the inverse magnetic billiard trajectory.\nRemark 2.1.In general, we will keep the same numbering convention de-\nscribed above, so that the quantities \u00122i,\u00122i+1,\u00122i+2,\u001f2i;2i+2,`2i+1,`2i+2,\netc. are all associated with the trajectory from P2i!P2i+2.\nFrom the above de\fnitions, we can encode the dynamics through the\ninverse magnetic billiard map Ton the open annulus R=LZ\u0002(0;\u0019)\u0019[0;L)\u0002\n(0;\u0019), which sends each successive reentry point and angle of the inward\npointing velocity vector to that of the next reentry point and angle:\nT: [0;L)\u0002(0;\u0019)![0;L)\u0002(0;\u0019); (si;\u0012i)7!(si+2;\u0012i+2):\nBy assuming that motion stops when \u0012= 0 or\u0019, we can extend the domain\nof this map to the closed annulus A:= [0;L)\u0002[0;\u0019]; that is, by continuity,\nT(s;0) = (s;0) andT(s;\u0019) = (s;\u0019).\nThe mapTpreserves the symplectic area form sin( \u0012)ds^d\u0012on the an-\nnulus A. This informs the de\fnition of area-preserving coordinates, ui:=\n\u0000cos(\u0012i).\nProposition 2.2 ([10, 11]) .Suppose@\nis of classCk. Then the inverse\nmagnetic billiard map TisCk\u00001andDT= \n@s2\n@s0@s2\n@u0@u2\n@s0@u2\n@u0!\nwith\n@s2\n@s0=\u00140`1sin(2\u001f0;2\u0000\u00121)\u0000sin(\u00120) sin(2\u001f0;2\u0000\u00121)\u0000\u00140`2cos(\u001f0;2) sin(\u00121)\nsin(\u00121) sin(\u00122)\n@s2\n@u0=`1sin(2\u001f0;2\u0000\u00121)\u0000`2cos(\u001f0;2) sin(\u00121)\nsin(\u00120) sin(\u00121) sin(\u00122)\n@u2\n@s0=\u00142sin(\u00120) sin(2\u001f0;2\u0000\u00121)\nsin(\u00121)+2 sin(\u001f0;2) sin(2\u001f0;2\u0000\u00121\u0000\u00122)(\u00140`1\u0000sin(\u00120))\n`2sin(\u00121)\n\u0000\u00140\u0012\nsin(2\u001f0;2\u0000\u00122) +\u00142`1sin(2\u001f0;2\u0000\u00121)\nsin(\u00121)\u0000\u00142`2cos(\u001f0;2)\u00134 S. GASIOREK\n@u2\n@u0=2`1sin(\u001f0;2) sin(2\u001f0;2\u0000\u00121\u0000\u00122)\u0000\u00142`1`2sin(2\u001f0;2\u0000\u00121)\n`2sin(\u00120) sin(\u00121)\n+\u00142`2cos(\u001f0;2)\u0000sin(2\u001f0;2\u0000\u00122)\nsin(\u00120):\nFurthermore, det(DT) = 1 .\nRemark 2.3.The preceding proposition shows that the entries of the de-\nrivative matrix DT do not directly depend upon the magnetic \feld or\nLarmor radius \u0016. However, simple geometry yields the equation `2i+2=\n2\u0016sin(\u001f2i;2i+2), so the quantities above can be stated using \u0016. Our analysis\nin the next sections makes use of this equation.\nThe derivative matrix DTfrom Prop. 2.2, interpreted as a linearization\nofT, can be used to determine the stability of periodic trajectories. If we\nletz= (s;u), the stability matrix is\nSn(z) =DT(Tn\u00001(z))\u0001\u0001\u0001DT(T(z))DT(z): (2.1)\nIfzis part of an orbit of period n, the orbit is hyperbolic ifjTrSn(z)j>\n2,parabolic ifjTrSn(z)j= 2, and elliptic ifjTrSn(z)j<2. To simplify\nnotation, we will drop the dependence upon zand largely refer to Tr Sn.\n3.Linear Stability of 2-Periodic Trajectories\nThe linear stability of standard billiards has been studied recently [15, 16],\nthough the formulation was known earlier in the 20th century [1, 23]. In\nparticular, the linear stability of 2-periodic billiard trajectories can be simply\nstated in terms of the length of the trajectory and the radii of curvature of\nthe boundary at the two points of impact, \u001a1and\u001a2.\nProposition 3.1 ([15]).Suppose\u001a1< \u001a 2are the radii of curvature of the\nbilliard table at the impact points of a 2-periodic billiard trajectory and let l\ndenote the distance between the two impact points. If either of the inequali-\nties\u001a1<l<\u001a 2orl>\u001a 1+\u001a2are satis\fed, the 2-periodic billiard trajectory\nwill be hyperbolic, while in the case 0<l <\u001a 1or\u001a2<l <\u001a 1+\u001a2the tra-\njectory is elliptic. If l=\u001a1,\u001a2, or\u001a1+\u001a2, then the trajectory is parabolic.\nThe proof of this proposition follows from a generating function argument\n[15] or by a calculation and analysis of Tr S2. The argument does not depend\nupon the convexity of \n and still holds if one or both of \u001a1,\u001a2are negative.\nWith this stability criteria for 2-periodic billiard trajectories in mind, we\ninvestigate the linear stability of 2-periodic trajectories in the setting of\ninverse-magnetic billiards.\n3.1.2-Periodic Trajectories in a Strictly Convex Set. Suppose \n is\nstrictly convex and consider a 2-periodic inverse magnetic billiard trajectory.\nIn particular, the periodicity means \u00144=\u00140and\u00124=\u00120. Geometrically,\n2-periodic trajectories are in the shape of stadia { a rectangle capped by two\nsemicircles on opposite sides { and was stated as an o\u000bhand remark in [11]\n(and is clear if one draws a picture). The following proposition adds more\nstructure to this statement.LINEAR STABILITY OF PERIODIC TRAJECTORIES 5\nProposition 3.2. An inverse magnetic billiard trajectory in a strictly con-\nvex set is 2-periodic if and only if the angles \u001f0;2,\u001f2;4in successive iterations\nofTare both equal to \u0019=2.\nProof. Suppose a trajectory is 2-periodic and \u001f0;2;\u001f2;46=\u0019=2. Without loss\nof generality, suppose \u001f0;2>\u0019= 2. Then the tangent lines to the Larmor arc\n\r0;2atP1,P2must intersect at some point Q. Then the second Larmor arc\n\r2;4must also be tangent to these two lines at the points P3andP4=P0\nand lie within the compact set bounded by Q, the tangent lines, and \r0;2.\nHowever, since the two Larmor arcs \r0;2,\r2;4have equal radii, the only way\nthis can happen is if \r0;2[\r2;4is the complete Larmor circle of radius \u0016,\nmaking the trajectory not 2-periodic.\nSuppose\u001f0;2=\u001f2;4=\u0019=2. Then`1and`3are parallel and the Larmor\narcs\r0;2,\r2;4are semicircles. If `1>`3, then@\n cannot be convex, as P0,\nP1, andP4are collinear. A similar argument holds for `1<`3. Thus`1=`3\nand the trajectory is a stadium. \u0003\nRemark 3.3.As 2-periodic trajectories are stadia, we also mention the strict\nconvexity of \n necessarily implies that each \u00122i;\u00122i+1must have angle mea-\nsure strictly less than \u0019=2. Further, it is also the case that `2=`4= 2\u0016.\nWith the preceding proposition, we can simplify DTfrom Prop. 2.2 into\nDT= \u00140`1\u0000sin(\u00120)\nsin(\u00122)`1\nsin(\u00120) sin(\u00122)\n(\u00140`1\u0000sin(\u00120))(sin(\u00121+\u00122)\u0000\u00142\u0016sin(\u00121))\n\u0016sin(\u00121)\u0000\u00140sin(\u00122)`1(sin(\u00121+\u00122)\u0000\u00142\u0016sin(\u00121))\n\u0016sin(\u00120) sin(\u00121)\u0000sin(\u00122)\nsin(\u00120)!\n(3.1)\nwhich in turn allows us to compute the stability matrix S2(s0;u0) and its\ntrace\nTrS2= 2\u00002\u000b(\f+\u000e) +\u000b2\f\u000e (3.2)\nwith\n\u000b=`1=\u0016; \f = cot(\u00120) + cot(\u00123); \u000e= cot(\u00121) + cot(\u00122): (3.3)\nBy Remark 3.3, \fand\u000eare positive, and it follows that \u000bis positive as it is\nthe ratio of two positive quantities. A direct calculation proves the following\ntheorem.\nTheorem 3.4. Let\u000b,\f,\u000ebe de\fned as above and suppose \nis strictly\nconvex. Let m= minf2=\f;2=\u000egandM= maxf2=\f;2=\u000eg. If\f6=\u000e, then\nthe 2-periodic trajectory is\n\u000fparabolic if and only if \u000b=m,Morm+M;\n\u000felliptic if and only if \u000b2(0;m)[(M;m +M);\n\u000fhyperbolic if and only if \u000b2(m;M )[(m+M;1).\nIf\f=\u000e, thenm=Mand the 2-periodic trajectory is\n\u000fparabolic if and only if \u000b=mor2m;\n\u000felliptic if and only if \u000b2(0;m)[(m;2m);\n\u000fhyperbolic if and only if \u000b2(2m;1).\nFigure 2 presents a graphical representation of the above statement.6 S. GASIOREK\nAs stated in Proposition 3.1, the linear stability of a 2-periodic billiard\ntrajectory is dependent upon the relative sizes of the length of the chord\nand the curvature of the boundary at the points of impact. In magnetic\nbilliards, the linear stability is dependent upon the curvature at the points\nof impact and the associated diametric chord length ([17, 18]). The linear\nstability criteria is precisely the same criteria as standard billiards, implying\nthe linear stability of magnetic billiards is independent of the magnetic \feld.\nHowever, the linear stability of periodic trajectories in the inverse magnetic\nbilliard setting is indirectly dependent upon the magnetic \feld through the\nterm\u000b. And in contrast to the standard billiard or magnetic billiard settings,\nthe linear stability of inverse magnetic billiards is not dependent upon the\ncurvature of the boundary at the exit and re-entry points.\n0E\nmPH\nMPE\nm+MPH\u000b\n0E\nmPE\n2mPH\u000b\nFigure 2. A stability diagram illustrating Theorem 3.4\nwhen\f6=\u000e(above) and \f=\u000e(below).\nExample 3.5. Consider the case where @\n is the circle of radius R. Then\n\u00120=\u00121=\u00122=\u00123and hence\f=\u000e. It is then necessarily true that \u0016<R ,\nas\u0016\u0015Rmakes the stadia larger than \n itself. A quick calculation shows\nthat\u000b= 4=\f= 2mand hence all 2-periodic trajectories in the circle are\nparabolic.\nExample 3.6. Consider the case where @\n is the ellipseE:x2=a2+y2=b2=\n1 witha > b > 0. In standard billiards, the only 2-periodic trajectories in\nthe ellipse are the major and minor axes whose stability are hyperbolic and\nelliptic, respectively. In inverse magnetic billiards, the 2-periodic trajectories\nare aligned along and symmetric about the major and minor axes of the\nellipse; that is, the centers of the Larmor circles lie on the coordinate axes\nof the ellipse. See \fgure 3. By symmetry, the angles \u00120=\u00121=\u00123=\u00124and\nwe are again in the case \f=\u000e.\nConsider the 2-periodic trajectory aligned along the major axis of the\nellipse. This constrains the Larmor radius to 0 < \u0016 < b . Two direct\ncalculations show that\n\u000b=2ap\nb2\u0000\u00162\nb\u0016and cos( \u00120) =a\u0016p\nb4+\u00162(a2\u0000b2): (3.4)\nAs\f= 2 cot(\u00120), this gives\n\f=2a\u0016\nbp\nb2\u0000\u00162: (3.5)LINEAR STABILITY OF PERIODIC TRAJECTORIES 7\nCombining these two equations gives us \u000bin terms of \f,\n\u000b=a2\nb24\n\f: (3.6)\nIn terms of Theorem 3.4 and \fgure 2, this 2-periodic trajectory has \u000b=\na2\nb2\u00012m, and hence the trajectory is hyperbolic.\nIn the case of the 2-periodic trajectory aligned along the minor axis of\nthe ellipse, we now have 0 <\u0016<a . Repeating the above calculations yields\n\u000b=b2\na24\n\f: (3.7)\nIn terms of Theorem 3.4 and \fgure 2, this 2-periodic trajectory has \u000b=\nb2\na2\u00012m, and hence the trajectory is elliptic except for when 2 b2=a2, in\nwhich case the trajectory is parabolic. It is noteworthy that this same\nobstruction to ellipticity occurs for standard billiards in the ellipse [16]. We\ndiscuss this further in section 3.3.\nFigure 3. 2-periodic trajectories in the ellipse along the ma-\njor axis (dashed) and minor axis (dotted).\nAn interesting feature of the two families of 2-periodic trajectories is that\nthe stability is only dependent upon the lengths of the semi-major and semi-\nminor axes of the ellipse. Moreover, the imposed restrictions on \u0016in both\ncases apply to the regime \u0016 < \u001aminand to a subinterval of the \u001amin<\n\u0016 < \u001amaxregime. This adds to the limited knowledge of inverse magnetic\nbilliards in the intermediate curvature regime.\nFor comparison, consider standard billiards in the ellipse \n. The only\n2-periodic billiard trajectories are the major and minor axes. A quick cal-\nculation shows that the major axis of the ellipse is hyperbolic while the\nminor axis is elliptic except when a2= 2b2, in which case the trajectory is\nparabolic.8 S. GASIOREK\nExample 3.7. Consider the case when @\n is the one-parameter family of\ncurvesx2k+y2k= 1,k2N. As the case k= 1 corresponds to the unit\ncircle, we also assume k>1. By symmetry, we again have \u00120=\u00121=\u00122=\u00123\nand are again in the case \f=\u000e.\nThe axes of symmetry of @\n gives rise to two families of 2-periodic tra-\njectories: the stadia are oriented along the horizontal or vertical axes of\nsymmetry, and the stadia which are oriented along the diagonal axes of\nsymmetry,y=\u0006x.\nConsider \frst the trajectories that are oriented about the horizontal (and\nequivalently, vertical) axis. See \fgure 4a. Geometrically, 2-periodic trajec-\ntories have Larmor radius \u0016limited to 0 < \u0016 < 1. Two calculations show\nthat\n\u000b=2(1\u0000\u00162k)1\n2k\n\u0016and cos( \u00120) =\u0010\n(\u0016\u00002k\u00001)2k\u00001\nk+ 1\u0011\u00001=2\n:(3.8)\nAgain,\f= 2 cot(\u00120), and hence\n\f= 2(\u0016\u00002k\u00001)1\u00002k\n2k: (3.9)\nCombining these to get \u000bin terms of \f, we get\n\u000b=\f(\u0016\u00002k\u00001): (3.10)\nIn terms of Theorem 3.4 and \fgure 2, we now must \fnd the relative size of\n\u000bwithm= 2=\fand 2m. Through some algebra, we see that \u000b=mand\n\u000b= 2mat the values\n\u0016\u0003=\u0010\n2k\nk\u00001+ 1\u0011\u00001\n2kand\u0016\u0003\u0003= 2\u00001\n2k; (3.11)\nrespectively, and the stability can be summarized in the following way:\n\u000fParabolic if and only if \u0016=\u0016\u0003or\u0016\u0003\u0003;\n\u000fElliptic if and only if \u00162(0;\u0016\u0003)[(\u0016\u0003;\u0016\u0003\u0003);\n\u000fHyperbolic if and only if \u00162(\u0016\u0003\u0003;1).\n(a)\n (b)\nFigure 4. 2-periodic trajectories in the curve x2k+y2k= 1\nalong the horizontal axis (a) and diagonal axis (b).\nNext, consider the case where the trajectory is oriented along the diagonal\ny=x, as in \fgure 4b. The diagonal of @\n is of length 2\u00012k\u00001\n2k, so any Larmor\ncircle must have diameter smaller than this diameter, meaning 0 <\u0016< 2k\u00001\n2k.LINEAR STABILITY OF PERIODIC TRAJECTORIES 9\nWe take a slightly di\u000berent approach than previously. Suppose P0=\n(x0;y0) = (x0;(1\u0000x2k)1\n2k) is the point in \fgure 4b with the largest value\nofy0(i.e. the point that is \\on top\"). This means x02(\u00002\u00001\n2k;2\u00001\n2k) and\ny0>x 0. In this construction,\n`1= (x0+y0)p\n2; \u0016 =y0\u0000x0p\n2=)\u000b= 2y0+x0\ny0\u0000x0:(3.12)\nUsing implicit di\u000berentiation, we \fnd the angle \u00120can be written simply in\nterms ofx0, andy0:\ntan(\u00120) =y2k\u00001\n0 +x2k\u00001\n0\ny2k\u00001\n0\u0000x2k\u00001\n0: (3.13)\nJust as in the previous examples, \f= 2 cot(\u00120). Becausem= 2=\f= tan(\u00120),\nthe stability depends upon the relative values of \u000bandm= tan(\u00120) and\n2m= 2 tan(\u00120). Through factoring the above equation, we get\n\u000b= 2mfk(x0;y0) (3.14)\nwith\nfk(x0;y0) =y2k\u00002\n0 +y2k\u00003\n0x0+\u0001\u0001\u0001+y0x2k\u00003\n0+x2k\u00002\n0\ny2k\u00002\n0\u0000y2k\u00003\n0x0+\u0001\u0001\u0001\u0000y0x2k\u00003\n0+x2k\u00002\n0: (3.15)\nThus determining the stability of the diagonal 2-periodic orbit reduces to de-\ntermining the values of fk(x0;y0). Luckilyfk(x0;y0) is a continuous, mono-\ntone increasing function on ( \u00002\u00001\n2k;2\u00001\n2k), has limiting values 1 =(2k\u00001) and\n2k\u00001 asx0!\u00002\u00001\n2kandx0!2\u00001\n2k, respectively, and satis\fes fk(0;1) = 1.\nBy the intermediate value theorem, there exists an ~ x02(\u00002\u00001\n2k;0) with\ncorresponding ~ y0such thatfk( ~x0;~y0) = 1=2, which in turn means \u000b=m.\nMoreover,fk(0;1) = 1 corresponds to \u0016= 2\u00001=2andfk( ~x0;~y0) = 1=2 corre-\nsponds to a value ~ \u0016= ( ~y0\u0000~x0)=p\n2 given by the equations above.\nUltimately, the stability can be summarized as follows:\n\u000fParabolic if and only if \u0016= 2\u00001=2or ~\u0016if and only if x0= 0 or ~x0;\n\u000fElliptic if and only if \u00162(0;2\u00001=2) if and only if x02(0;2\u00001\n2k);\n\u000fHyperbolic if and only if \u00162(2\u00001=2;2k\u00001\n2k) if and only if x02\n(\u00002\u00001\n2k;0)nf~x0g.\nFor comparison, we can apply Proposition 3.1 to this family of curves to\nstudy the linear stability of 2-periodic billiard trajectories. Such 2-periodic\nbilliard trajectories are the horizontal and vertical axes of symmetry, and\nthe diagonals, which are the widths and diameters of \n, respectively.\nConsider \frst a width of \n as the horizontal segment connecting the points\n(\u00001;0) and (0;1). As the curvature of @\n vanishes at these points, the radii\nof curvature are in\fnite. An analysis of the billiard stability matrix yields\nTrS2= 2\u00004l\u00121\n\u001a1+1\n\u001a2\u0013\n+4l2\n\u001a1\u001a2; (3.16)\nand hence this trajectory is parabolic (that is, Proposition 3.1 does not\ndirectly apply at points of in\fnite radii of curvature).\nConsider next a diameter of \n as the segment connecting the points\n(\u00002\u00001\n2k;\u00002\u00001\n2k) and (2\u00001\n2k;2\u00001\n2k). At these points the radii of curvature of10 S. GASIOREK\n@\n are at their minimum, \u001a1=\u001a2= 21\u0000k\n2k=(2k\u00001):Becausel= 2\u00012k\u00001\n2k=\n23k\u00001\n2k, it is quick to show that l>2\u001a1for allk >1 and hence the diagonal\n2-periodic billiard is hyperbolic.\n3.2.2-Periodic Trajectories in a Non-Strictly Convex or Concave\nSet. We again consider the case when n= 2 and we suppose \n is either not\nstrictly convex (i.e. contains points of vanishing curvature) or concave (i.e.\ncontains points of negative curvature). We address the same propositions\nas the convex case.\nProposition 3.8. Let\nbe convex or concave. If an inverse magnetic\nbilliard trajectory is 2-periodic, then the angles \u001f0;2=\u001f2;4=\u0019=2.\nThe proof of this is identical to the \frst part of the proof of Proposition\n3.2. With the assumption that \n can be convex or concave, this now allows\n\u0012ito be\u0015\u0019=2. The equations for DT, TrS2,\u000b,\f, and\u000eare unchanged\nbut now we have the possibility that \fand\u000ecould be nonpositive. Again,\na direct calculation proves the following.\nTheorem 3.9. Let\u000b,\f,\u000ebe de\fned as in equation 3.3 and suppose \nis\nnot strictly convex or is concave.\n(i) If\f=\u000e= 0, then the trajectory is parabolic for all \u000b2(0;1);\n(ii) If\f\u00140and\u000e\u00140and\fand\u000eare not both zero, then the trajectory\nis hyperbolic for all \u000b2(0;1);\n(iii) If either\na)\f >0and\u000e= 0; or\nb)\f >0and\u000e<0, and2\n\f+2\n\u000e\u00140,\nthen the trajectory is\n\u000fparabolic if and only if \u000b=2\n\f;\n\u000felliptic if and only if \u000b2\u0012\n0;2\n\f\u0013\n;\n\u000fhyperbolic if and only if \u000b2\u00122\n\f;1\u0013\n;\n(iv) If\f >0and\u000e<0and2\n\f+2\n\u000e>0, then the trajectory is\n\u000fparabolic if and only if \u000b=2\n\for\u000b=2\n\f+2\n\u000e;\n\u000felliptic if and only if \u000b2\u00122\n\f+2\n\u000e;2\n\f\u0013\n;\n\u000fhyperbolic if and only if \u000b2\u0012\n0;2\n\f+2\n\u000e\u0013\n[\u00122\n\f;1\u0013\n;\n(v) If\f >0and\u000e>0, then the stability is determined by the statement\nof Theorem 3.4.\nIdentical statements to (iii) and (iv) hold when \fand\u000eare switched.\nAs noted in the literature on the stability of 2-periodic billiard trajectories\n(e.g. [15, 16, 23]), the stability criteria therein allow for the cases where\nthe boundary is convex or concave at the point of re\rection. As in the\nstrictly convex case, the stability criterion for inverse magnetic billiards isLINEAR STABILITY OF PERIODIC TRAJECTORIES 11\nonly dependent upon the angles \u0012i, and the ratio `1=\u0016. These criteria are\nagain not dependent upon the curvature of the boundary at the exit or\nreentry points nor is it directly dependent on the magnetic \feld.\nExample 3.10. Suppose \n is the stadium: a rectangle with side lengths L\nand 2Rcapped by semicircles of radius Ron opposite sides (see \fgure 5a).\nIf the points P0;P1;P2;P3of a 2-periodic trajectory are all on the straight\nsides of@\n withP0;P3on one side and P1;P2on the opposite side, then the\nangles\u0012i=\u0019=2 fori= 0;1;2;3 and necessarily 2 \u0016 < L . Then\f=\u000e= 0\nand the trajectory is parabolic.\nIf instead the trajectory has the points P0; P3on one semicircular cap\nand the points P1;P2on the other semicircular cap of the stadium, then\nnecessarily \u0016 < R and\f=\u000e > 0 as all angles \u0012iare equal. We use the\nsecond part of Theorem 3.4 to analyze the stability. A direct calculation\nyields\n`1=L+ 2p\nR2\u0000\u00162; cos(\u0012i) =\u0016\nR; m =p\nR2\u0000\u00162\n\u0016: (3.17)\nIt follows that \u000b=L=\u0016+ 2m> 2m, and hence the trajectory is hyperbolic\nfor all\u0016<R .\nExample 3.11. Suppose \n is the \\telephone curve\" with the 2-periodic\ntrajectory pictured in \fgure 5b. Suppose P0is the lower right point on @\n.\nThen\u00121=\u00122> \u0019= 2 while\u00120=\u00123< \u0019= 2, and so\u000e <0 and\f > 0. Thus\nwe are in either case (iii) or (iv) of Theorem 3.9, depending upon the sign\nof2\n\f+2\n\u000e, which in turn equals tan( \u00120) + tan(\u00121) in this setting. Knowing or\ncalculating the values of `1,\u0016,\u00120;and\u00121will then determine the stability\nof the pictured trajectory.\n(a)\n (b)\nFigure 5. (a) A parabolic (dotted) and hyperbolic (dashed)\n2-periodic trajectory in the stadium ; (b) A 2-periodic tra-\njectory in the concave \\telephone curve\".\n3.3.A Digression on the Ellipse and Rotation Numbers. Example\n3.6 illustrates that the linear stability criterion for inverse magnetic bil-\nliards in an ellipse is the same as standard billiards in an ellipse: the major\naxis-aligned 2-periodic inverse magnetic billiard is hyperbolic and so is the\nstandard billiard trajectory along the major axis; the minor axis-aligned\n2-periodic inverse magnetic billiard trajectory is elliptic, as is the standard\nbilliard trajectory along the minor axis, and both fail to be elliptic when\na2= 2b2. We investigate this exception further in the context of standard\nbilliards.12 S. GASIOREK\nThe standard billiard in an ellipse is a well-known example of an inte-\ngrable system and is conjectured to be the only integrable planar billiard\n[4]. Billiard trajectories in the ellipse have the following caustic property: if\none segment of a trajectory is tangent to an ellipse confocal to the billiard\ntable, then after every re\rection the trajectory will remain tangent to the\nsame confocal ellipse; if one segment (or its extension) of a trajectory is\ntangent to a hyperbola which is confocal to the billiard table, then every\nsegment or its extension will be tangent to the same confocal hyperbola; and\nif a segment of the billiard trajectory passes through one focus, then after re-\n\rection the trajectory will pass through the other focus. The confocal ellipse\nor hyperbola with the aforementioned tangency property are called caustics\nof the billiard trajectory. If we write the ellipse E:x2=a2+y2=b2= 1 for\na>b> 0, then the confocal family of Eis given by\nE\u0015:x2\na2\u0000\u0015+y2\nb2\u0000\u0015= 1 (3.18)\nfor\u00152R. The caustics are members of the confocal family which are\nellipses for \u00152(0;b2) and hyperbolas for \u00152(b2;a2). The caustics are\ndegenerate and are contained in the major and minor axes for \u0015=b2and\na2, respectively. If \u0015 < 0 or\u0015 > a2, then the curves E\u0015are outsideEor\nimaginary, respectively.\nGiven a periodic billiard trajectory in the ellipse E, de\fne the following\nrotation function as the ratio of elliptic integrals\nrot :\u0000\n0;b2\u0001\n[\u0000\nb2;a2\u0001\n!R; rot(\u0015) =\u0002min(b2;\u0015)\n0dtp\n(\u0015\u0000t)(b2\u0000\u0015)(a2\u0000\u0015)\n2\u0002a2\nmax(b2;\u0015)dtp\n(\u0015\u0000t)(b2\u0000\u0015)(a2\u0000\u0015):\n(3.19)\nSee e.g. [5, 7] for a modern treatment, though these elliptic integrals were\nknown to Jacobi and other mathematicians of the 19th century.\nA periodic trajectory will have rot( \u0015) =m=n2Qfor coprime integers\nm\u0015n. WhenE\u0015is an ellipse, mis the winding number and nis the minimal\nperiod of the trajectory. In particular, rot( \u0015) increases monotonically from\n0 to 1 as\u0015increases monotonically from 0 to b2. IfE\u0015is a hyperbola, then m\nis the number of times the trajectory crosses the y-axis andnis the minimal\nperiod. This necessarily implies the trajectory must have even period when\nthe caustic is a hyperbola. As \u0015increases monotonically from b2toa2,\nrot(\u0015) decreases monotonically from 1 to r >0, wherercan be calculated\nexplicitly, as we will see below. Proving the (decreasing) monotonicity of\nrot(\u0015) when the caustic is a hyperbola is nontrivial and was proved in [8].\nA simpler proof was given recently [7] using a di\u000berent technique.\nTo determine the limiting value of rot( \u0015) for a hyperbolic caustic, r>0,\nconsider the approach of [8], where the confocal family has \fxed foci at\n(\u00061;0) and can be written in the form:\nC\u0017:x2\n\u0017+y2\n\u0017\u00001= 1: (3.20)LINEAR STABILITY OF PERIODIC TRAJECTORIES 13\nThe confocal curves are hyperbolas for \u00172(0;1) and ellipses for \u00172(1;1).\nAny ellipse of the form E:x2=a2+y2=b2= 1 can be homothetically scaled\nto a member of the confocal family C\u00170for some\u00170>1. Suppose the\nbilliard boundary is the member of this confocal family C\u0017corresponding to\n\u0017=\u00170>1. Then the limiting rotation number for a hyperbolic caustic is\nonly dependent upon \u00170and is given by\nr= lim\n\u0015%a2rot(\u0015) =1\n\u0019arccos\u0012\n1\u00002\n\u00170\u0013\n: (3.21)\nSee Section 11.2 and speci\fcally subsection 11.2.3.5 of [8] for details.\nIn the case of the isolated parabolic billiard trajectory along the minor\naxis of the ellipse, the ellipse Ehas parameters aandbsatisfyinga2= 2b2.\nThe confocal family C\u0017hasa2=\u00170andb2=\u00170\u00001, so that together these\nthree equations imply that the billiard table C\u00170has\u00170= 2. Moreover,\nthis implies that r= 1=2. The conclusion we draw from this is the isolated\nparabolic billiard trajectory in the minor axis of the ellipse has rotation\nnumber 1=2, and that as \u0015increases from b2toa2, the rotation function\nrot(\u0015) decreases monotonically from 1 to 1 =2. That is, the isolated parabolic\nbilliard trajectory along the minor axis of the ellipse is characterized by\nbeing the limiting trajectory of the unique elliptical billiard table whose\ntrajectories with hyperbolic caustics have limiting rotation number 1 =2.\n4.Linear Stability of 3- and 4-Periodic Trajectories\n4.1.3-Periodic Trajectories. The study of 3-periodic trajectories can be\nperformed in a similar fashion. However, we now have two distinct types\nof 3-periodic trajectories: those with rotation number 1 =3 and those with\nrotation number 2 =3. For an arbitrary 3-periodic trajectory, Tr S3is cubic\nin 1=\u0016with coe\u000ecients that are expressions in terms of `1,`3,`5, and sines\nand cosines of combinations of \u001f0;2,\u001f2;4,\u001f4;6, and\u0012ifori= 0;:::; 5. As in\nthe 2-periodic case, no curvature terms \u0014iappear.\nFigure 6. The shape of 3-periodic trajectories with dihedral\nsymmetry with rotation number 1 =3 (left) and 2 =3 (right).\nTo make the 3-periodic trajectories simpler to study, consider 3-periodic\ntrajectories with dihedral symmetry. See \fgure 6. This assumption means\n`1=`3=`5=:`and\u001f0;2=\u001f2;4=\u001f4;6=:\u001fare both constant with the\nlatter equal to \u0019=3 or 2\u0019=3 when the trajectory has rotation number 1 =3 or14 S. GASIOREK\n2=3, respectively. Then Tr S3is cubic in\u000b:=`=\u0016and has coe\u000ecients which\ncan be written in terms of sums, products, and di\u000berences of cotangents of\nthe\u0012i. For example, the constant term of Tr S3can be written as\n2\u00003\n4C23C45\u00003\n8C01(2C2345\u0007p\n3C23C45) (4.1)\nwhereCA=P\ni2Acot(\u0012i), the minus sign corresponds to the trajectory with\nrotation number 1 =3, and the plus sign corresponds to the trajectory with\nrotation number 2 =3. The cubic coe\u000ecient can be written as\n\u0000\u0007cos(\u0019\n6\u0007(\u00121+\u00122)) cos(\u0019\n6\u0007(\u00123+\u00124)) cos(\u0019\n6\u0007(\u00120+\u00125))\nsin(\u00120) sin(\u00121) sin(\u00122) sin(\u00123) sin(\u00124) sin(\u00125)(4.2)\nwith the same sign convention stated above applying to all \u0007terms.\nUltimately, the linear stability of 3-periodic trajectories is di\u000ecult to an-\nalyze in general.\nExample 4.1. Consider the case when all \u0012i=:\u0012are equal. Then\nTrS3(s;\u0012) = 2\u00009 cot2(\u0012)\u00003p\n3 cot3(\u0012)\n+\u000b\u00143 cos(\u0012)\n4 sin4(\u0012)\u0010\n5p\n3 cos(\u0012) +p\n3 cos(3\u0012)\u00003 sin(\u0012) + 9 sin(3\u0012)\u0011\u0015\n+\u000b2\u0014\n\u00003 sin(\u0019\n3+ 2\u0012)(cos(\u0012) + sin(\u0019\n6+ 3\u0012))\nsin5(\u0012)\u0015\n+\u000b3\u0014cos3(\u0019\n6\u00002\u0012)\nsin6(\u0012)\u0015\nand\nTrS3(s;\u0012) = 2\u00009 cot2(\u0012) + 3p\n3 cot3(\u0012)\n+\u000b\u00143 cos(\u0012)\n4 sin4(\u0012)\u0010\n\u00005p\n3 cos(\u0012)\u0000p\n3 cos(3\u0012)\u00003 sin(\u0012) + 9 sin(3\u0012)\u0011\u0015\n+\u000b2\u00143 sin(\u0019\n3\u00002\u0012)(cos(\u0012) + sin(\u0019\n6\u00003\u0012))\nsin5(\u0012)\u0015\n+\u000b3\u0014\n\u0000cos3(\u0019\n6+ 2\u0012)\nsin6(\u0012)\u0015\nwhen the trajectory has rotation number 1 =3 and 2=3, respectively. Once\n\u0012and`are known, the value of jTrS3jcan be computed to determine the\nstability.\nExample 4.2. Consider the case when @\n is the circle of radius R. By\nsymmetry, \u0012iis a constant of motion and we are in the same situation as\nthe previous example. First, suppose the trajectory has a rotation number\n1=3 and hence \u001f=\u0019=3. Then for each \u0016<R , there is a unique \u00122(0;\u0019=3)\nsuch that (s;\u0012) is the initial condition for the 3-periodic trajectory. In fact,\nwe can say that this \u0012satis\fes the equation\nsin\u0010\u0019\n3\u0000\u0012\u0011\n=\u0016sin(\u0012)p\nR2+\u00162\u00002R\u0016cos(\u0012)(4.3)LINEAR STABILITY OF PERIODIC TRAJECTORIES 15\n(see [11]) which has functional solution\ncos(\u0012) =3\u0016+p\n4R2\u00003\u00162\n4R: (4.4)\nIn the case of the circle, `= 2Rsin(\u0012), which turns equation for Tr S3into\nan equation in terms of Rand\u0016. In particular, the equation jTrS3j= 2 is\nsatis\fed for all \u0016<R , so the trajectory is parabolic.\nNext, consider the case when the rotation number is 2 =3 and so\u001f= 2\u0019=3.\nAgain, for each \u0016<R there exists a unique \u00122(\u0019=3;2\u0019=3) such that ( s;\u0012)\nis the initial condition for a 3-periodic trajectory. Just as before, this value\nof\u0012satis\fes\nsin\u0010\u0019\n3\u0000\u0012\u0011\n=\u0016sin(\u0012)p\nR2+\u00162\u00002R\u0016cos(\u0012)(4.5)\nand has functional solution\ncos(\u0012) =3\u0016\u0000p\n4R2\u00003\u00162\n4R: (4.6)\nEvaluating the expression for Tr S3for this value of \u0012yieldsjTrS3j= 2 for\nall values of \u0016 < R . Therefore both types of 3-periodic trajectories in the\ncircle are parabolic.\n4.2.4-Periodic Trajectories. The study of 4-periodic trajectories can be\nperformed in a similar fashion. Again, we have two distinct types of 4-\nperiodic trajectories: those with rotation number 1 =4 and those with rota-\ntion number 3 =4. For an arbitrary 4-periodic trajectory, Tr S4is quartic in\n1=\u0016with coe\u000ecients that are expressions in terms of `ifori2f1;3;5;7g\nand sines and cosines of combinations of \u001f0;2,\u001f2;4,\u001f4;6,\u001f6;8, and\u0012ifor\ni= 0;:::; 7. As in the 2-periodic case, no curvature terms \u0014iappear.\nAgain, we examine examples which incorporate symmetry to simplify the\nquartic expression for Tr S4.\nExample 4.3. Consider the case when @\n is the circle of radius R. By\nsymmetry,\u0012iis a constant of motion and `1=`3=`5=`7=:`and\u001f0;2=\n\u001f2;4=\u001f4;6=\u001f6;8=:\u001f. First, suppose the trajectory has a rotation number\n1=4 and hence \u001f=\u0019=4. Then for each \u0016<R , there is a unique \u00122(0;\u0019=4)\nsuch that (s;\u0012) is the initial condition for the 4-periodic trajectory. In fact,\nwe can say that this \u0012satis\fes the equation\nsin\u0010\u0019\n4\u0000\u0012\u0011\n=\u0016sin(\u0012)p\nR2+\u00162\u00002R\u0016cos(\u0012)(4.7)\n(see [11]) which has functional solution\ncos(\u0012) =R2+\u00162\u0000\u00162csc2(\u0019\n4\u0000\u0012) sin2(\u0012)\n2R\u0016: (4.8)16 S. GASIOREK\nIn the case of the circle, `= 2Rsin(\u0012), which turns equation for Tr S4into\nan equation in terms of Rand\u0016. Letting\u000b=R=\u0016, the equation becomes\nTrS4=2(1\u000010 cos2(\u0012) + 17 cos4(\u0012))\nsin4(\u0012)+\u000b\u001432 cos(2\u0012)(3 cos2(\u0012)\u00001)\nsin4(\u0012)\u0015\n+\u000b2\u00148 cos2(2\u0012)(5 + 7 cos(2 \u0012))\nsin4(\u0012)\u0015\n\u0000\u000b3\u001464 cos3(2\u0012) cos(\u0012)\nsin4(\u0012)\u0015\n+\u000b4\u001416 cos4(2\u0012)\nsin4(\u0012)\u0015\n:\n(4.9)\nFor the speci\fc value of \u0012which produces the 4-periodic trajectory with\nrotation number 1 =4, this expression for Tr S4simpli\fes to 2, and hence the\ntrajectory is parabolic.\nAn analogous calculation can be done to show the 4-periodic trajectory\nwith rotation number 3 =4 is parabolic.\nExample 4.4. Consider the case when @\n is the ellipse x2=a2+y2=b2= 1\nwitha > b > 0. Further, suppose the 4-periodic trajectories share the\nsame horizontal and vertical axes of symmetry as the ellipse (see \fgure 7).\nIn particular, this symmetry assumption means that \u00120=\u00121=\u00124=\u00125,\n\u00122=\u00123=\u00126=\u00127,`1=`5,`3=`7, and\u001f0;2=\u001f2;4=\u001f4;6=\u001f6;8=:\u001f,\nwhich in turn implies `2=`4=`6=`8. Suppose the Cartesian coordinates\n(x0;\u0000y0) are the initial point P0in the fourth quadrant of a 4-periodic\ntrajectory.\nFigure 7. Symmetric 4-periodic trajectories in the ellipse\nwith rotation number 1 =4 (dashed) and 3 =4 (dotted), each\nwith di\u000berent Larmor radii.\nConsider \frst the case when the trajectory has rotation number 1 =4, and\nhence\u001f=\u0019=4. We aim to express the stability of the trajectory in terms\nof the Cartesian coordinates of P0. ClearlyP1= (x0;y0), and the point P2\nwill be the intersection of the ellipse with the line y=\u0000x+x0+y0other\nthan the point P1, which is the same as the second intersection point of theLINEAR STABILITY OF PERIODIC TRAJECTORIES 17\nLarmor circle. Further, this means the only possible initial x-coordinates are\nx02(a(a2\u0000b2)=(a2+b2);a) { otherwise the quarter Larmor circle cannot\nintersect the ellipse to create the symmetric 4-periodic trajectory. Since\n\u001f=\u0019=4, we can write P2= (x2;y2) = (x0\u0000\u0016;y0+\u0016). A few calculations\nyield the following:\n`1= 2y0\nx2=a2(x0+y0)\u0000abp\na2+b2\u0000(x0+y0)2\na2+b2\ny2=b2(x0+y0) +abp\na2+b2\u0000(x0+y0)2\na2+b2\n\u0016=2abp\na2+b2\u0000(x0+y0)2\na2+b2=\u00062(b2x0\u0000a2y0)\na2+b2\n`3= 2x2\ncos(\u00120) =b2x0p\na4y2\n0+b4x2\n0\ncos(\u00122) =a2y2p\na4y2\n2+b4x2\n2=a2(y0+\u0016)p\na4(y0+\u0016)2+b4(x0\u0000\u0016)2(4.10)\nwhere the second equation for \u0016uses plus if \u001f=\u0019=4 and minus if \u001f= 3\u0019=4.\nPlugging in each of the above equations into Tr S4yields a rational func-\ntion inx0andy0with coe\u000ecients in aandb:\nTrS4= [ 16b4\u0000\na2\u0000b2\u00014x4\n0\u000016b2\u0000\na2\u0000b2\u00013\u0000\n2a4\u00003a2b2+ 2b4\u0001\nx3\n0y0\n2\u0000\na2\u0000b2\u00012\u0000\n8a8\u000040a6b2+ 49a4b4\u000040a2b6+ 8b8\u0001\nx2\n0y2\n0\n8a2\u0000\na2\u0000b2\u0001\u0000\n4a8\u000010a6b2+ 13a4b4\u000010a2b6+ 4b8\u0001\nx0y3\n0\n8a4\u0000\n2a8\u00004a6b2+ 5a4b4\u00004a2b6+ 2b8\u0001\ny4\n0]=h\na4b4y2\n0\u0000\nb2x0\u0000a2(x0+ 2y0)\u00012i\n:\n(4.11)\nAsy0can be written as a function of x0, we can then express the stability\nof the 4-periodic trajectory in terms of the coordinate x0.\nRepeating the above calculations in the case with rotation number 3 =4\nand\u001f= 3\u0019=4 yields nearly the same equations as in (4.10) and ultimately\nyields the same equation for Tr S4given above in (4.11). See Remark 4.5\nbelow for additional commentary on this case.\nTo further illustrate the stability analysis, consider the speci\fc case when\na= 3 andb= 2. ThenjTrS4j= 2 at the values\nx\u0003\n0=291\n9p\n13; x\u0003\u0003\n0=s\n88731 + 1575p\n217\n14534; x\u0003\u0003\u0003\n0=291\n13p\n61:(4.12)\nAnalyzingjTrS4jyields that the 4-periodic trajectory is\n\u000felliptic forx02(x\u0003\n0;x\u0003\u0003\n0)[(x\u0003\u0003\n0;x\u0003\u0003\u0003\n0);\n\u000fparabolic for x02fx\u0003\n0;x\u0003\u0003\n0;x\u0003\u0003\u0003\n0g; and\n\u000fhyperbolic for x02(15=13;x\u0003\n0)[(x\u0003\u0003\u0003\n0;3):18 S. GASIOREK\nRemark 4.5.The symmetric 4-periodic trajectories in the ellipse with rota-\ntion number 1 =4 and 3=4 are related to one another in the following way.\nConsider the eight ordered points P0;P1;:::;P 7on@\n which determine\na trajectory with rotation number 1 =4. Then a trajectory in the order\nP0;P5;P6;P3;P4;P1;P2;P7is a symmetric 4-periodic trajectory with rota-\ntion number 3 =4. See \fgure 8. This is partly due to the fact that for each\nsegment`2j, there are two supplementary values of \u001fwhich realize this\nsegment. This in turn creates two Larmor arcs whose union (after a suit-\nable re\rection across `2j) is a complete Larmor circle. As such, these two\n4-periodic trajectories are complementary to one another.\nFigure 8. Symmetric 4-periodic trajectories in the ellipse\nwith rotation number 1 =4 (dashed) and 3 =4 (dotted), each\nwith the same Larmor radii and same points Pi.\nRemark 4.6.There is a second family of symmetric 4-periodic trajectories\nin the ellipse: those whose Larmor circles are symmetric about the axes of\nsymmetry of the ellipse. The stability of these trajectories can be analyzed\nin a similar fashion to those above, though we do not address this example\nhere.\nExample 4.7. Consider the case when @\n is the one-parameter family of\ncurvesx2k+y2k= 1,k2N. As the case k= 1 corresponds to the unit\ncircle, we also assume k>1. By the dihedral symmetry of @\n, the 4-periodic\ntrajectories have constant \u0012i:=\u0012,`2j,`2j+1, and\u001f2l;2l+2=:\u001f. In addition\nto the two families of trajectories determined by the rotation number, we\ncan also consider two further families based upon whether the centers of\nthe Larmor circles are on the diagonal axes of symmetry of @\n or on the\nhorizontal and vertical axes of symmetry of @\n.\nFirst, consider the case when the rotation number is 1 =4 and the centers\nof the Larmor circles are on the diagonal axes of symmetry of @\n. Suppose\nthe pointP0= (x0;\u0000y0)2@\n is in the fourth quadrant and the trajectory\nhas initial velocity in the direction (0 ;1) so thatP1= (x0;y0)2@\n is in\nthe \frst quadrant. To produce the 4-periodic trajectory, it must be the caseLINEAR STABILITY OF PERIODIC TRAJECTORIES 19\nthatx02(2\u00001\n2k;1) andP2= (y0;x0). Further, we know \u001f=\u0019=4, which\nimplies the center of the \frst Larmor arc is at ( y0;y0) and hence \u0016=x0\u0000y0.\nDue to the curvature of @\n, it is possible for the Larmor arc of this \frst\nsegment of the trajectory to intersect @\n in up to four di\u000berent locations.\nTo ensure the point P2is the proper point of intersection of the Larmor\ncircle with @\n, there must be exactly two intersection points between the\nLarmor circle and @\n so that the billiard can \\round the corner\" of @\n.\nThat is, there exists ^ x2(2\u00001\n2k;1) such that\n(x\u0000y0)2+ (y\u0000y0)2=\u00162andx2k+y2k= 1 (4.13)\nhas exactly three real roots in x:x= ^x,x= 2\u00001\n2k, andx=y0, with\ny0<2\u00001\n2k<^x. Alternately, we can say that ^ xis the value of x0such that\nthe distance from the center of the Larmor circle, ( y0;y0) and (2\u00001\n2k;2\u00001\n2k)\nis exactly\u0016. For this value ^ x, these two equations (4.13) have exactly two\nreal roots for x02(2\u00001\n2k;^x), three real roots when x0= ^x, and four real\nroots when x02(^x;1). This behavior is analogous to a circle whose center\nlies on the diagonal of a square and can have multiple intersections with the\nsides and corners of the square. We now assume x02(2\u00001\n2k;^x).\nWe can now compute several of the relevant quantities that appear in\nTrS4:\n`1= 2y0\n\u0016=x0\u0000y0\n`2= 2\u0016\ntan(\u0012) =y2k\u00001\n0\nx2k\u00001\n0:(4.14)\nThese quantities produce the following expression for Tr S4in terms of x0\nandy0:\nTrS4= 2 +h\n16x2\n0\u0010\nx2k\u00002\n0\u0000y2k\u00002\n0\u0011\u0010\nx4k\u00002\n0\u0000y4k\u00002\n0\u0011\u0010\nx2k\u00002\n0\u0000y2k\u00002\n0 + 2x\u00002\n0y2k\u00001\n0\u0011\n\u0001\u0010\ny4k\u00002\n0\u0000x4k\u00002\n0+ (x0\u0000y0)x2k\u00001\n0y2k\u00002\n0\u00112\u0015\n=h\ny16k\u000012\n0 (x0\u0000y0)4i\n(4.15)\nBy assumption, 0 < y 0< x 0<1, and each term in the above rational\nfunction will be strictly positive for x02(2\u00001\n2k;^x). Therefore this 4-periodic\ntrajectory will be hyperbolic for all x02(2\u00001\n2k;^x) and integer k\u00152.\nRepeating the above calculations in the case when the rotation number is\n3=4 andx02(\u00001;2\u00001\n2k) yields the same equation for Tr S4as (4.15). Similar\nto before, this expression will be larger than 2 for x02(\u00001;2\u00001\n2k)nf\u00002\u00001\n2kg\nand be equal to 2 for x0=\u00002\u00001\n2k. Therefore the 4-periodic trajectory with\nrotation number 3 =4 is hyperbolic for all x02(\u00001;2\u00001\n2k)nf\u00002\u00001\n2kgand\nparabolic for x0=\u00002\u00001\n2k. There is also the same duality between this\ntrajectory and the trajectory with rotation number 1 =4 as is discussed in\nRemark 4.5.20 S. GASIOREK\n(a)\n (b)\nFigure 9. Symmetric 4-periodic trajectories in the curve\nx2k+y2k= 1 with rotation number 1 =4 (dashed) and 3 =4\n(dotted), with di\u000berent Larmor radii, and with Larmor cen-\nters on (a) the lines y=\u0006x, and (b) on the coordinate axes.\nNext, consider the case when the rotation number is 1 =4 and the Larmor\ncenters are on the coordinate axes. Then, as before, all \u0012i,`2i,`2i+1,\u001f2i;2i+2\nare constant. If the point P0= (x0;y0) is in the \frst quadrant, then x02\n(2\u00001\n2k;1),P1= (y0;x0), andP2= (\u0000y0;x0). Simple geometry yields\n`1=p\n2(x0\u0000y0)\n\u0016=p\n2y0\n`2= 2y0\ntan(\u0012) =x2k\n0y0\u0000x0y2k\n0\nx2k\n0y0+x0y2k\n0(4.16)\nand the Larmor centers are at the points (0 ;\u0006(x0\u0000y0)) and (\u0006(x0\u0000y0);0).\nThese quantities produce the following expression for Tr S4in terms of x0\nandy0:\nTrS4= 2\u0000h\n64x2k+1\n0y2k\n0\u0010\nx2k\n0y2\n0\u0000x2\n0y2k\n0\u0011\u0010\nx2k\n0(x0\u00002y0) + 2x0y2k\n0\u0011\n\u0001\u0010\nx4k\n0y2\n0\u0000x2\n0y4k\n0\u00002(x0\u0000y0)x2k+1\n0y2k\n0\u00112\u0015\n=h\nx2k\n0y0\u0000x0y2k\n0i8\n(4.17)\nThe above rational function equals zero at exactly one point, x0= \u0014x\nin the interval (2\u00001\n2k;1). A quick analysis of this rational function and the\nabove expression tells us the trajectory will be hyperbolic for x02(2\u00001\n2k;\u0014x),\nparabolic for x0= \u0014x, and elliptic for x02(\u0014x;1).\nWe can repeat the above calculations in the case when the rotation num-\nber is 3=4. In particular, this case is valid for x02(\u00002\u00001\n2k;1). Suppose\nP0= (x0;y0) is in the \frst quadrant and P1= (\u0000y0;x0) is in the thirdLINEAR STABILITY OF PERIODIC TRAJECTORIES 21\nquadrant. Then P2= (y0;\u0000x0) is in the fourth quadrant and the point P0\nandP1determine the start of a 4-periodic trajectory. In this case we have\n`1=p\n2(x0+y0)\n\u0016=p\n2y0\n`2= 2y0\ntan(\u0012) =x2k\n0y0+x0y2k\n0\nx0y2k\n0\u0000x2k\n0y0(4.18)\nand the Larmor centers are at the points (0 ;\u0006(x0+y0)) and (\u0006(x0+y0);0).\nThese quantities produce a similar expression for Tr S4in terms of x0and\ny0as the previous case:\nTrS4= 2\u0000h\n64x2k\n0y2k\u00002\n0\u0010\nx2k\u00002\n0\u0000y2k\u00002\n0\u0011\u0010\nx2k\u00001\n0(x0+ 2y0) +y2k\n0\u0011\n\u0001\u0010\nx4k\u00002\n0\u0000y4k\u00002\n0\u00002(x0+y0)x2k\u00001\n0y2k\u00002\n0\u00112\u0015\n=h\nx2k\u00001\n0+y2k\u00001\n0i8\n:\n(4.19)\nAn analysis of this equation yields \fve values of x02(\u00002\u00001\n2k;1) for which\nTrS4is parabolic, call them ~ xj,j2f1;:::; 5gwithxi< xjfori < j . In\nparticular ~x1= 0. Further analysis yields that the trajectory is\n\u000fhyperbolic for x02(\u00002\u00001\n2k;~x1)[(~x1;~x2)\n\u000fparabolic for x02f~x1;~x2;~x3;~x4;~x5g\n\u000felliptic forx02(~x2;~x3)[(~x3;~x4)[(~x4;~x5)[(~x5;1).\n5.Conclusions\nWe have established a linear stability criterion for 2-periodic inverse mag-\nnetic billiard trajectories in convex and concave domains and have applied\nthis to examples such as the ellipse and one-parameter family of curves given\nbyx2k+y2k= 1 fork2N. Further, comparisons have been made to the\nlinear stability criteria for standard and magnetic billiards, noting the sim-\nilarities (e.g. analogous stability results for billiards and inverse magnetic\nbilliards in the ellipse) and di\u000berences (e.g. the linear stability of inverse\nmagnetic billiards does not depend upon the curvature of the boundary at\nthe exit and reentry points compared to the linear stability of standard\nbilliards which does depend upon the curvature of the boundary at the re-\n\rection points). We also established the linear stability of symmetric 3-\nand 4-periodic inverse magnetic billiards trajectories in the same domains.\nFurther, we made a geometric characterization of an isolated parabolic 2-\nperiodic billiard trajectory in an ellipse in terms of rotation numbers and\ncaustics.\nA further direction of research in this area is reconciling the following:\nthe approach of [15] establishes linear stability criterion for billiards using\nthe generating function Lof the billiard map and its Hessian determinant\nevaluated at nondegenerate critical points. This criterion is identical to the\ncriterion constructed via the stability matrix Sn. However, this generat-\ning function approach to linear stability in the setting of inverse magnetic22 S. GASIOREK\nbilliards does not produce viable results in the 2-periodic case: the generat-\ning function Gfor inverse magnetic billiards is not su\u000eciently di\u000berentiable\nwhen\u001f=\u0019=2. An equivalent statement in the magnetic billiards setting is\nmade in [2] in the context of asymptotic expansions of the magnetic billiard\nmap. The technique of [15] establishes a method to transform a degenerate\ncritical point to a nondegenerate critical point, and \fnding magnetic and in-\nverse magnetic analogues to this method constitute one path to overcoming\nthis obstruction.\nAcknowledgements\nThis research is supported by the Discovery Project No. DP190101838\nBilliards within confocal quadrics and beyond from the Australian Research\nCouncil. A portion of this research is based upon work supported by the\nNational Science Foundation under Grant No. DMS-1440140 while the au-\nthor was in residence at the Mathematical Sciences Research Institute in\nBerkeley, California, during the Fall 2018 semester.\nReferences\n1. V. M. Babich and V. S. Buldyrev, Asymptotic methods in short wave di\u000braction prob-\nlems. vol. 1 , Izdat. \\Nauka\", Moscow, 1972, With the collaboration of M. M. Popov\nand I. A. Molotkov. (Russian). MR 0426630\n2. N. Berglund and H. Kunz, Integrability and ergodicity of classical billiards in a mag-\nnetic \feld , J. Statist. Phys. 83(1996), no. 1-2, 81{126. MR 1382763\n3. Misha Bialy, Andrey E. Mironov, and Lior Shalom, Magnetic billiards: Non-\nintegrability for strong magnetic \feld; gutkin type examples , Journal of Geometry and\nPhysics 154(2020), 103716.\n4. G. Birkho\u000b, Dynamical Systems , American Mathematical Society / Providence, RI,\nAmerican Mathematical Society, 1927.\n5. Pablo S. Casas and Rafael Ram\u0013 \u0010rez-Ros, Classi\fcation of symmetric periodic trajecto-\nries in ellipsoidal billiards , Chaos: An Interdisciplinary Journal of Nonlinear Science\n22(2012), no. 2, 026110.\n6. Giulio Casati and Toma\u0014 z Prosen, Time Irreversible Billiards with Piecewise-Straight\nTrajectories , Phys. Rev. Lett. 109(2012), no. 17, 174101.\n7. Vladimir Dragovi\u0013 c and Milena Radnovi\u0013 c, Periodic ellipsoidal billiard trajectories and\nextremal polynomials , Communications in Mathematical Physics 372 (2019), no. 1,\n183{211.\n8. J.J. Duistermaat, Discrete integrable systems , Springer Monographs in Mathematics,\nvol. 304, Springer New York, 2010.\n9. Holger R. Dullin, Linear stability in billiards with potential , Nonlinearity 11(1998),\nno. 1, 151{173. MR 1492955\n10. Sean Gasiorek, On the dynamics of inverse magnetic billiards , Ph.D. thesis, University\nof California Santa Cruz, 2019, p. 90. MR 4035388\n11. ,On the dynamics of inverse magnetic billiards , Nonlinearity 34(2021), no. 3,\n1503{1524.\n12. Boris Gutkin, Hyperbolic magnetic billiards on surfaces of constant curvature , Com-\nmunications in Mathematical Physics 217(2001), no. 1, 33{53.\n13. Bence Kocsis, Gergely Palla, and J\u0013 ozsef Cserti, Quantum and semiclassical study of\nmagnetic quantum dots , Physical Review B 71(2005), no. 7, 075331.\n14. A. Korm\u0013 anyos, P. Rakyta, L. Oroszl\u0013 any, and J. Cserti, Bound states in inhomogeneous\nmagnetic \feld in graphene: Semiclassical approach , Phys. Rev. B 78(2008), no. 4,\n045430.\n15. V. V. Kozlov and D. V. Treshch ev, Billiards , Translations of Mathematical Mono-\ngraphs, vol. 89, American Mathematical Society, Providence, RI, 1991, A geneticLINEAR STABILITY OF PERIODIC TRAJECTORIES 23\nintroduction to the dynamics of systems with impacts, Translated from the Russian\nby J. R. Schulenberger. MR 1118378\n16. V.V. Kozlov, Two-link billiard trajectories: extremal properties and stability , Journal\nof Applied Mathematics and Mechanics 64(2000), no. 6, 903{907.\n17. V.V. Kozlov and S.A. Polikarpov, Periodic billiard trajectories in a magnetic \feld ,\nJournal of Applied Mathematics and Mechanics 69(2005), no. 6, 844{851.\n18. M Robnik and M V Berry, Classical billiards in magnetic \felds , Journal of Physics A:\nMathematical and General 18(1985), no. 9, 1361{1378.\n19. Marko Robnik, Regular and chaotic billiard dynamics in magnetic \felds , Nonlinear\nPhenomena and Chaos, Bristol, 1986.\n20. S. Tabachnikov, Geometry and Billiards , Student mathematical library, American\nMathematical Society, 2005.\n21. Serge Tabachnikov, Remarks on magnetic \rows and magnetic billiards, Finsler metrics\nand a magnetic analog of Hilbert's fourth problem , Modern dynamical systems and\napplications (2004), 233{250.\n22. Z. V or os, T. Tasn\u0013 adi, J. Cserti, and P. Pollner, Tunable Lyapunov exponent in inverse\nmagnetic billiards , Physical Review E 67(2003), no. 6, 065202.\n23. Maciej Wojtkowski, Principles for the design of billiards with nonvanishing lyapunov\nexponents , Communications in Mathematical Physics 105(1986), no. 3, 391{414.\nEmail address :sean.gasiorek@sydney.edu.au\nSchool of Mathematics and Statistics, Carslaw Building F07, University\nof Sydney, NSW 2006, Australia"    },    {        "title": "1806.02517v1.Terahertz_Emission_from_Compensated_Magnetic_Heterostructures.pdf",        "content": "  \n1 \n Terahertz Emission from  Compensated Magnetic Heterostructures  \n \nMengji Chen , Rahul Mishra , Yang Wu , Kyusup Lee  and Hyunsoo Yang * \n \nDepartment of Electrical and Computer Engineering  and NUSNNI , National University of \nSingapore, 117576 Singapore  \nE-mail: eleyang@nus.edu.sg  \n \nKeywords : THz emission, ferrimagnet, antiferromagnet,  spintronics , magnetic heterostructure   \n \nTerahertz emission spectroscopy (TES)  has recently played an important role in unveiling \nthe spin dynamics at a terahertz (THz) frequency range. So far, ferromagnetic \n(FM)/ nonmagnetic  (NM) heterostr uctures  have been intensively studied  as THz  source s. \nCompensated magnets such as a ferrimagnet (FIM) and antiferromagnet (AFM) are other \ntypes of magnetic materials with interesting  spin dynamic s. In this work , we study  TES \nfrom compensated magnet ic heterostructures including  CoGd  FIM alloy  or IrMn AFM \nlayer s. Systematic measurement s on composition and temperature dependences of THz \nemission from CoGd/Pt bilayer structures are conducted . It is found  that t he emitted THz \nfield is determined by the  net spin  polarization of the laser induced spin current rather \nthan the net magnetization. The temperature robustness of  the FIM based THz emitter  is \nalso demonstrated . On the other hand, an AFM play s a different role in THz emission.  \nThe IrMn/Pt bilayer shows negligible THz signals, whereas Co/IrMn induces siz able THz \noutputs , indicating  that IrMn is not a good spin current generator, but a good detector . \nOur results  not only suggest that a compensated magnet can be utilized for robust  THz \nemission , but also provide a new approach to study  the magnetization dyn amic s especially \nnear the magnetization compensation point .  \n \n \n \n   \n2 \n 1. Introduction   \nDuring the past decades , THz technologie s have been studied intensively ,[1-4] for a wide \nrange of promising applications such as the chemical compositio n analysis ,[5, 6] integrated \ncircuits failure analysis ,[7] and spintronics characterization .[8] In order to ensure a high  \nperformance of the THz module in these applications , it is highly desired to develop efficient \nTHz components such as  the THz emitter,[9-13] detector,[14] phase shif ter,[15] and modulator .[16] \nIn particular , the development of an efficient THz emitter is  considered a s a key challenge in \nthe field of THz technology .[1] At present , in a fem tosecond laser driven system, \nphotoconductive a ntenna ,[10] electro -optic  crystal,[12, 13] and plasma[11, 17] have been widely \napplied as THz emitter s for different kinds of THz research  and applications . However, there \nare various  shortcomings in the exsiting THz emitters such as a low signal -noise ratio, high cost \nand narrow bandwidth. Therefore, an efficient and robust THz source with a low cost is still in \nhigh de mand.  \nSpintronics  materials have been utilized  to construct efficient and low cost  THz emitter s. \nIn 2004, the first THz emission from a laser induced ultrafast demagnetization of Ni was \nreported .[18] It was  after a decade in 2013  that spin-orbit interaction based terahertz emission \nwas demonstrated from the  FM/NM heterostuctures .[19] A fs laser  was used to excite an ultrafast \nspin current  𝒋𝐬 in the  FM layer . The spin current diffuses into the NM layer, where it is \nconverted into an charge current 𝒋𝐜 due to the inverse spin -Hall effect (ISHE) . Finally , this \nultrafast charge current leads to a radiation of the electromagnetic THz signal .[19] This new \napproach  has ignited interest in  development of efficient THz emitter s based on FM /NM  \nheterostructures.[20-23]  \nAntiferromagnetically coupled magnetic materials  such as  a FIM and AFM  have  the \nultrafast dynamics in the range of THz, minimal stray field s due to  the reduced  magnetization , \na resistance against external magnetic field perturbation , and a high thermal stability due to \ntheir ultrahigh anisotropy .[24, 25] In this work, we study  THz emission  from different   \n3 \n heterostructure s comprising of FI M and AFM  materials . A strong THz output is observed from \na compensated FIM/NM bilayer , Co74Gd26 (7 nm)/Pt (6 nm ) with nearly zero magnetization,  \nwhile there is no measureable THz signal from a n AFM/NM bilaye r, Ir25Mn 75 (7 nm)/Pt (6 nm). \nHowever, when the AFM is used a spin detector in an AFM/F M bilayer , Ir25Mn 75 (6 nm)/Co (3 \nnm), a sizable THz signal is obtained . In addition,  composition and  temperature dependent \nstudies of THz emissiom are performed on  the FIM/NM bilayer. It is found that the net spin \npolarization , rather than the net magnetization,  plays a  dominant role in the THz emission.  A \nFIM/NM bilayer is able to  sustain a temperature  cycle of 473 K . Our works not only provide a \nbetter understanding of compensated magnet based THz emission but also open up a possibility \nof ultilizing them  as a practical THz emitter.  \n \n2. Sample Preparation and Measurement Results  \nFor this work, thin films are deposited on the glass substrate s by magnetron sputtering \nwith a base pressure less  than 5 × 10−9 Torr. The FIM layer is deposited by co -sputtering of Co \nand Gd  and the AFM layer is deposited by sputtering of Ir 25Mn 75 target . All the films are \nprotected  by a 3  nm thick SiO 2 layer to prevent oxidization. The magnetic property of the \nsamples was measured by a vibrating sample magnetometer  (VSM) . Figure 1a shows the \nschematic of the TES system , which is based on a stroboscopic  setup. We employ a laser source \nwith a full width a t the half maximum of 120 fs, a center wavelength of 800 nm and a repetition \nrate of 1 kHz. A linearly polarized pump laser pulse  with an energy  of 220 μJ  is normally \nincident from the sample  layer side, with an external magnetic field ~1000 Oe appli ed along \nthe -y direction  in Figure 1b . With a  1 mm thick ZnTe (110) crystal  acting as a THz detector,  \nthe ellipticity of the probe laser pulse  with an energy  of 2 μJ is modulated by the THz electric \nfield due  to the electro -optical effect. Based on the el ectro -optic sampling method, the THz \nsignal is recorded in the time domain. \nWe first study the THz emission from the FIM/NM bilayer . A typical  sample structure   \n4 \n used for the experiments is shown in  Figure 1 b. A 7 nm FIM Co1-xGdx layer is deposited on the \nglass substrate with  a 6 nm NM Pt layer on top.  Various Co 1-xGdx samples with different \ncompositions varying from x = 0 to 55 are prepared. Figure 2a shows the saturation \nmagnetization of the samples measured by a VSM. The compensation point is found to be  x = \n~26. The samples for x < 26 are Co rich in which the magnetic moments of Co sublattice align \nparallel to the external magnetic field, and those  of Gd align antiparallel to the external magnetic \nfield. The samples with x > 26 are  Gd rich with the magne tic moment s of Co sublattice aligning \nantiparallel to the external magnetic field  and those  of Gd are parallel to the field . TES \nmeasurement results are shown in  Figure 2b. The intensity of THz signal keeps decreasing as \nthe Gd fraction increases , while the saturation magnetization Ms firstly decreases then increases \nafter compensation point ( x = 26 ). The Ms reaches almost zero at x = 26 , however the THz \nsignal has an abrupt sign change  around x = 26  without showing  a fully quenched THz signal . \nTherefore, it seems that the net magnetization  is not strongly correlated to the emitted THz \nsignal.  \nIn order to better understand  the THz emission from nearly  compensated  structure s, we \ncompare THz emission from a nearly compensated FIM/NM bilayer  (x = 26) , Co74Gd26 (7 \nnm)/Pt (6 nm) , with the AFM/NM bilayer structure, Ir25Mn 75 (7 nm)/Pt  (6 nm) , as shown in \nFigure 3, in which both samp les have zero net magnetization . Based on the theoretical model  \nof spin charge conversion in the THz emission,[19] 𝒋𝐜=𝛾𝒋𝐬×𝐌/|𝐌|, where  𝛾 is the effective \nspin Hall angl e and 𝐌 is the sample magnetization, it is expected that both the nearly \ncompensated FIM/NM and AFM/NM bilayer s should  produce negligible THz signal s. As \nexpected, no measurable THz signal is detected  from the IrMn/Pt bilayer  in Figure 3. However, \na nearly compensated FIM/NM bilayer structure , Co74Gd26/Pt acts as an efficient THz source. \nIt was observed  that the sign of THz signal  from the Co74Gd26/Pt bilayer  reverses  upon reversing \neither t he external magnetic field direction (+H or –H) or the sample around the x-axis (not \nshown) . The above results pinpoint that the observed THz signal from Co 74Gd26/Pt has a spin-  \n5 \n current -based origin  from the heterostructure itse lf, rather than  the demagnetization dynamics \ncontribution .[18, 26] The very different results from the nearly compensated FIM/N M and \nAFM/NM suggest that the generat ed spin current is negligible in AFM/NM, whereas the spin \ncurrent from a FIM/NM is finite even though the net magnetization  is close to zero .  \nThe THz emission from the nearly compensated FI M/NM bilayer can be understood  \nfrom the localized and delocalized nature of the electrons carrying magnetic moment s in Gd \nand Co, respectively. The spin -split bands of Gd sublattice originate  from the 4 f electrons which \nare spatially localized below the Fermi level about 8 eV  and are difficult to be excited by the \nlaser pulse . On the other hand, the spin -split bands of Co sublatt ice are much closer to the Fermi \nlevel (~1 eV) and are easily excited when the laser pulse thermalize s the CoGd  layer . Thus, the \nsuper diffus ive spin current generated in the FI M layer is dominated  by the electrons from the \nCo sublattic e, and accounts for the THz emission from the nearly  compensated CoGd/Pt \nbilayer.[27-31] It can also be con cluded that the net spin polarization rather than net magnetization \nis mainly responsible for  the THz emission from a magnetic/heavy metal het erostructure. The \ndecreasing THz emission with increasing the Gd composition as wel l as the sign change of THz \nsignal in Figure 2b  matches well with the spin polarization values of CoGd.[31] \nWhile there is no THz generation due to the lack of a spin current in the IrMn/Pt bilayer \nwith the laser excitation, an AFM is reported as an effective spin detector .[32, 33] A typical THz \nsignal from an AFM/FM bilayer, Ir 25Mn 75 (6 nm)/Co (3 nm) is shown in Figure 4a. With a spin \ncurrent in Co excited by a fs laser pump , the IrMn  layer converts the spin current to the charge \ncurrent, leading to the THz emission. On the other hand, Figure 4a also shows that there is no \ndetectable THz signal generated from an AFM/NM bilayer, Ir 25Mn 75 (6 nm)/Pt (3 nm) because \nof a lack of the spin current source. We further perform the thickness dependent study with a \nsample structure of glass substrate/Ir 25Mn 75 (0  10 nm)/Co (3 nm)/SiO 2 (3 nm), as shown in \nFigure 4b. With 1 nm IrMn  with the Co layer, the THz signal shows a peak intensity, and then \nthe THz signal becomes gradually weaker as we increase the thickness of IrMn. The above   \n6 \n thickness dependen t results can be attributed to the combined effect of  the s pin diffusion in \nIrMn , the THz absorption in the IrMn/Co structure  and the Fabry -Perot effect .[20, 22] When the \nthickness of the IrMn  layer is below the spin diffusion length , the ultrafast spin current from the \nCo layer cannot be fully converted into the charge current, resulting in a smaller THz signal. In \nthe previous study,[20] the Fabry -Perot interference effect is considered in the thickness \ndependence of THz emission, which is believed to  enhance  both the laser pump and t he emitted \nTHz waves resonantly.  However, in our thickness dependent measurement, the THz signal \ndecreases with increas ing the IrMn thickness and no critical thickness , in which the maximum \nTHz signal appears[20, 22], is observed.  Therefore, we believe a significant THz absorption in the \nIrMn layer  suppresses  the enhancement  from  the Fabry -Perot effect and the THz absorption in \nthe IrMn layer contributes more significantly  in our thickness dependent study.  As a result , the \npeak THz signal should be observed with a thickness of IrMn  near the spin relaxation length, \ndue to the maximum spin charge conversion  and less THz absorption from the thin film. Our \nobservation is consistent  to the reported spin relaxation length of Ir 25Mn 75 (< 1 nm).[33, 34] \nIn a FIM, the magnetization compensation temperatur e (Tcomp) also plays an important \nrole in the magnetization configuration and thus spin polarization . A Gd rich sample \n(Co 63Gd37/Pt) is chosen for a temperature dependen t study of THz emission , as shown in Figure \n5a. A sign reversal of the THz signal near 433 K is observed and the intensity  of THz emission  \nbecomes smaller when the temperatu re is closer to 433 K. Compared  with the temperature \ndependence data from VSM shown in Figure  5b, a sign reversal of the THz signal is observed \nacross  Tcomp. The magnetic alignment  of the Co sublattice as well as the spin polarization change \nits direction across Tcomp,[31] leading  to the inverted THz emission  sign in line  with the previous \nwork s.[35, 36] Different from  the composition dependent results, the THz signal decrease \nsubstant ially around the compensation temperature. This phenomenon can be attributed to thin \nfilm inhomogeneity in the deposition process or sputter target, which induces the nonuniform \nconcentration profiles of thin film with the mix of Co and Gd rich regions .[31, 37, 38] Near the   \n7 \n compensation temperature, although the sample has minimum magnetization, some regions of \nthe samples are Co ri ch and some are Gd rich due to inhomogeneity. As the net spin polarization \nis decided by Co, the sample near compensation ends up having a small polarization as the Co \npolarization in different regions cancels each other . A small temperature difference , where the \nTHz signal reverses and VSM  data show the minimum  value , can be attributed to the laser \ninduced heating of the sample .[27, 35]   \nThe temperature robutness is critical for real THz  applications , and a  finite THz signal \nabove  400 K  in Fig ure 5a indicate s a potential  for applications.  Thus , we investigate the  \nperformance of the FIM/NM based emitter s after heating to a high temperature . For the case of \nCo63Gd37/Pt bilayer structure (Gd rich sample)  shown in Figure 6, the THz signal reverses at \n~473 K , which is above Tcomp. Subsequently, when the sample is cooled down back to the room \ntemperature,  the THz signal can be fully recovered . Therefore, a FIM/NM system demonstrates \na non -hysteretic temperature cycle with respect to the THz signal aft er heating up and cooling \ndown to the same temperature.  \n \n3. Conclusion  \nWe study  THz emisison from compensated magnet based  bilayer  structures  with a fs \nlaser excitation . We show that the net spin polarization, rather than the  net magnetization, \naccounts for the THz generation. With a FIM acting as a spin source, a  sizable  THz output is \nobserved  dominated by Co sublattice s. It is found that AFM IrMn  plays a role as  an effective \nspin sink rather than a spin source in THz emission.  In addition, an non-hysteretic temperature \ncycle property make s this the FIM/NM THz emitter  an alternative  THz source.  \n \nAcknowledgements  \nThis research is supported by the National  Research Foundation (NRF), Prime Minister’s Office,  \nSingapore, under its Competitive Research Programme (CRP Award No. NRF CRP12 -2013 -\n01). \n \n    \n8 \n [1] B. Fergus on, X. -C. Zhang, Nat. Mater.  2002 , 1, 26. \n[2] P. H. Siegel, IEEE Trans. Microwave Theory Tech.  2002 , 50, 910.  \n[3] M. Tonouchi, Nat. Photonics  2007 , 1, 97. \n[4] R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, M. Bonn, Rev. Mod. Phys.  2011 , 83, 543.  \n[5] L. Ho, M. Pepper, P. Taday, Nat. Photonics  2008 , 2, 541.  \n[6] A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, J. E. Cunningham, Mater. Today  \n2008 , 11, 18. \n[7] M. Yamashita, K. Kawase, C. Otani, T. Kiwa, M. Tonouchi, Opt. Express  2005 , 13, 115.  \n[8] Z. Jin, A. Tkach, F. Casper, V. Spetter, H. Grimm, A. Thomas, T. Kampfrath, M. Bonn, \nM. Kläui, D. Turchinovich, Nat. Phys.  2015 , 11, 761.  \n[9] B. B. Hu, X. C. Zhang, D. H. Auston, P. R. Smith, Appl. Phys. Lett.  1990 , 56, 506.  \n[10] G. Mourou, C. V. Stancampiano,  A. Antonetti, A. Orszag, Appl. Phys. Lett.  1981 , 39, \n295. \n[11] X. Xie, J. Dai, X. C. Zhang, Phys. Rev. Lett.  2006 , 96, 075005.  \n[12] J. Hebling, G. Almási, I. Z. Kozma, J. Kuhl, Opt. Express  2002 , 10, 1161.  \n[13] X. Mu, I. B. Zotova, Y. J. Ding, IEEE J. Sel . Top. Quantum Electron.  2008 , 14, 315.  \n[14] Q. Wu, M. Litz, X. C. Zhang, Appl. Phys. Lett.  1996 , 68, 2924.  \n[15] C.-Y. Chen, C. -F. Hsieh, Y. -F. Lin, R. -P. Pan, C. -L. Pan, Opt. Express  2004 , 12, 2625.  \n[16] B. Sensale -Rodriguez, R. Yan, M. M. Kelly, T. Fang,  K. Tahy, W. S. Hwang, D. Jena, \nL. Liu, H. G. Xing, Nat. Commun.  2012 , 3, 780.  \n[17] K. Y. Kim, J. H. Glownia, A. J. Taylor, G. Rodriguez, Opt. Express  2007 , 15, 4577.  \n[18] E. Beaurepaire, G. M. Turner, S. M. Harrel, M. C. Beard, J. Y. Bigot, C. A. \nSchmuttenmaer, Appl. Phys. Lett.  2004 , 84, 3465.  \n[19] T. Kampfrath, M. Battiato, P. Maldonado, G. Eilers, J. Nötzold, S. Mährlein, V. Zbarsky, \nF. Freimuth, Y. Mokrousov, S. Blügel,  M. Wolf, I. Radu, P. M. Oppeneer, M. Münzenberg, \nNat. Nanotechnol.  2013 , 8, 256.    \n9 \n [20] T. Seifert, S. Jaiswal, U. Martens, J. Hannegan, L. Braun, P. Maldonado, F. Freimuth, \nA. Kronenberg, J. Henrizi, I. Radu, E. Beaurepaire, Y. Mokrousov, P. M. Oppeneer, M . Jourdan, \nG. Jakob, D. Turchinovich, L. M. Hayden, M. Wolf, M. Münzenberg, M. Kläui, T. Kampfrath, \nNat. Photonics  2016 , 10, 483.  \n[21] T. J. Huisman, R. V. Mikhaylovskiy, J. D. Costa, F. Freimuth, E. Paz, J. Ventura, P. P. \nFreitas, S. Blügel, Y. Mokrousov,  T. Rasing, A. V. Kimel, Nat. Nanotechnol.  2016 , 11, 455.  \n[22] Y. Wu, M. Elyasi, X. Qiu, M. Chen, Y. Liu, L. Ke, H. Yang, Adv. Mater.  2017 , 29, \n1603031.  \n[23] D. Yang, J. Liang, C. Zhou, L. Sun, R. Zheng, S. Luo, Y. Wu, J. Qi, Adv. Opt. Mater.  \n2016 , 4, 1944 . \n[24] T. Jungwirth, X. Marti, P. Wadley, J. Wunderlich, Nat. Nanotechnol.  2016 , 11, 231.  \n[25] J. Nishitani, K. Kozuki, T. Nagashima, M. Hangyo, Appl. Phys. Lett.  2010 , 96, 221906.  \n[26] T. J. Huisman, R. V. Mikhaylovskiy, A. Tsukamoto, T. Rasing, A. V. Kim el, Phys. Rev. \nB 2015 , 92, 104419.  \n[27] M. Binder, A. Weber, O. Mosendz, G. Woltersdorf, M. Izquierdo, I. Neudecker, J. R. \nDahn, T. D. Hatchard, J. U. Thiele, C. H. Back, M. R. Scheinfein, Phys. Rev. B  2006 , 74, 134404.  \n[28] S. S. Jaswal, D. J. Sellmyer, M . Engelhardt, Z. Zhao, A. J. Arko, K. Xie, Phys. Rev. B  \n1987 , 35, 996.  \n[29] R. Mishra, J. Yu, X. Qiu, M. Motapothula, T. Venkatesan, H. Yang, Phys. Rev. Lett.  \n2017 , 118, 167201.  \n[30] A. Kowalczyk, G. Chełkowska, A. Szajek, Solid State Commun.  2001 , 120, 407. \n[31] C. Kaiser, A. F. Panchula, S. S. P. Parkin, Phys. Rev. Lett.  2005 , 95, 047202.  \n[32] J. B. S. Mendes, R. O. Cunha, O. Alves Santos, P. R. T. Ribeiro, F. L. A. Machado, R. \nL. Rodríguez -Suárez, A. Azevedo, S. M. Rezende, Phys. Rev. B  2014 , 89, 140406.  \n[33] W. Zhang, M. B. Jungfleisch, W. Jiang, J. E. Pearson, A. Hoffmann, F. Freimuth, Y. \nMokrousov, Phys. Rev. Lett.  2014 , 113, 196602.    \n10 \n [34] W. Zhang, W. Han, S. -H. Yang, Y. Sun, Y. Zhang, B. Yan, S. S. P. Parkin, Sci. Adv.  \n2016 , 2, : e1600759.  \n[35] T. J. Huisman, C. Ciccarelli, A. Tsukamoto, R. V. Mikhaylovskiy, T. Rasing, A. V. \nKimel, Appl. Phys. Lett.  2017 , 110, 072402.  \n[36] T. Seifert, U. Martens, S. Günther, M. A. W. Schoen, F. Radu, X. Z. Chen, I. Lucas, R. \nRamos, M. H. Aguirre, P. A. Algarabel, A. An adón, H. S. Körner, J. Walowski, C. Back, M. R. \nIbarra, L. Morellón, E. Saitoh, M. Wolf, C. Song, K. Uchida, M. Münzenberg, I. Radu, T. \nKampfrath, SPIN  2017 , 07, 1740010.  \n[37] C. E. Graves, A. H. Reid, T. Wang, B. Wu, S. de Jong, K. Vahaplar, I. Radu, D. P . \nBernstein, M. Messerschmidt, L. Müller, R. Coffee, M. Bionta, S. W. Epp, R. Hartmann, N. \nKimmel, G. Hauser, A. Hartmann, P. Holl, H. Gorke, J. H. Mentink, A. Tsukamoto, A. Fognini, \nJ. J. Turner, W. F. Schlotter, D. Rolles, H. Soltau, L. Strüder, Y. Acrem ann, A. V. Kimel, A. \nKirilyuk, T. Rasing, J. Stöhr, A. O. Scherz, H. A. Dürr, Nat. Mater.  2013 , 12, 293.  \n[38] R. Chimata, L. Isaeva, K. Kádas, A. Bergman, B. Sanyal, J. H. Mentink, M. I. \nKatsnelson, T. Rasing, A. Kirilyuk, A. Kimel, O. Eriksson, M. Pereiro , Phys. Rev. B  2015 , 92, \n094411.  \n \n \n    \n11 \n  \n \n \nFigure 1. a) Schematics of the THz emission spectroscopy setup. b) Schematic s of film stack \nand THz emission geometry . An 800 nm polarized laser pump is normally incident from the \nthin film side and emitted THz signal is measure d along  the x axis. An external magnet ic field  \n(Hex = 1000 Oe) is applied along  the –y axis.  \n  \n  \n12 \n  \n \n-10 0 10 20 30 40 50 60 700200400600800\n-6000-30000300060009000Gd rich\nGd fraction (%)Ms (emu/cm3)\n(a)\n(b)Co richTHz Signal (a. u.)  \nFigure 2. a) The saturation magnetization measured by a VSM . b) Composition dependence of \nthe THz signal  of CoGd/Pt sample s as a function of Gd fraction . The shaded area  indicates  the \nCo rich region and the white background is for Gd rich sample s. All the THz signals  have  the \nsame time delay, but are shift ed horizontal ly according to the corresponding Gd fraction.  The \nmaximum peak or dip location of each THz curve indicates the Gd fraction.  \n    \n13 \n \n2 4 6 8-10000-50000500010000  Co74Gd26 (7 nm)/Pt (6 nm)\n Ir25Mn75 (7 nm)/Pt (6 nm)\n Co74Gd26 (7 nm)/Pt (6 nm)THz signal (a. u.)\nTime delay (ps)+H\n-H \n \nFigure 3. Typical THz signal from the nearly  compensated FIM/NM  and AFM/NM bilayer s. \nFor the  FIM/NM bilayer opposite external magnetic field s (+H and –H) are applied.   \n    \n14 \n  \n-2 0 2 4 6 8 10-50005001000\n0 1 2 3 4 5 6-300-200-1000100200300400500600\nTHz signal (a. u.)\nTime delay (ps)0 5 10Ir25Mn75 thickess (x nm)\n(a) (b)THz Signal (a. u.)\nTime Delay (ps) Ir25Mn75 (6 nm)/Pt (3 nm)\n Ir25Mn75 (6 nm)/Co (3 nm)\n \n \nFigure 4. a) Typical THz signal from AFM/NM and AFM/FM bilayer s. An external magnetic \nfield (~1000 Oe) is applied. b) IrMn thickness dependence of THz emission from glass \nsubstrate/Ir 25Mn 75 (0 - 10 nm)/Co (3 nm)/SiO 2 (3 nm) . All the THz signals have the same time \ndelay, but are shifted horizontally according to the IrMn thickness . The peak location of each \nTHz curve indic ates the IrMn thickness . \n \n    \n15 \n  \n \n-500-2500250500\n390 400 410 420 430 440 450 460 470010203040506070THz Signal (a. u.)\n(b)(a)\nCo GdGdMs (emu/cm3)\nTemp (K)CoHex  \nFigure 5. a) Temperature dependence of the emitted THz signal  from  the Co63Gd37/Pt bilayer.  \nAll the THz pulse s have the same time delay , but are shift ed horizontal ly with respect to the  \ncorresponding temperature . The maximum peak or dip location of each THz curve indicates the \nmeasured temperature . b) The saturation magnetization measured with a VSM as a function of \ntemperature.  The magnetization direction of Co and Gd is indicated with r espect to the external \nmagnetic field direction ( Hex). \n    \n16 \n  \n \n-5 0 5 10-600-400-2000200400600\nTime delay (ps) 300 K \n 473 K\n back to 300 KTHz Signal (a. u.)  \n \nFigure 6. a) Emitted THz signal from  the Co63Gd37/Pt bilayer  at different temperature s. The \nblack line shows the data  at room temperature  (300 K) , followed by a measurement at  473 K  \nshown by the red line.  The blue line  shows the measurement data at room temperature after \ncooling down to room temperature from 473 K . THz pulses are shifted horizontally for clarity.  \n \n \n \n \n   \n17 \n An efficient THz emission from compensated ferrimagnet/nonmagnetic metallic \nheterostructures is demonstrated. The THz emission is determined by the net spin \npolarization rather than the net magnetization. Antiferromagnet  plays a role as an effective spin \nsink rather than a spin source in THz emission. Ferrimagnet based THz emitters are promising \nnot only for THz applications, but also offering an opportunity  to study  the magnetization \ndynamic s. \n \nKeyword   \nTHz emission, f errimagnet, antiferromagnet, spintroni cs, magnetic heterostructure  \n \nMengji Chen, Rahul Mishra , Yang Wu , Kyusup Lee  and Hyunsoo Yang * \n \nTerahertz Emission from Compensated Magnetic Heterostructures  \n \n \n"    },    {        "title": "0911.1994v1.Wavelets_spectra_of_magnetization_dynamics_in_geometry_driven_magnetic_thin_layers.pdf",        "content": " 1 Wavelets spectra of magnetization dynamics in geometry  driven magnetic thin \nlayers \n \n \nPawel Steblinski and Tomasz Blachowicz \nInstitute of Physics, Silesian University of Techno logy \nKrzywoustego 2 str., 44-100 Gliwice, Poland \n \n \nSquared cobalt thin layers of different thickness a nd width were investigated by numerical \nsimulations. Using zero-valued externally applied m agnetic field (geometry driven regime) \nand different initial conditions the magnetization dynamics were examined. The wavelet-\nbased spectral analysis was applied. Transient stat es of different types were identified.  \n \n \n \nI. Introduction \nNowadays nanotechnology techniques enable creation of low-dimensional magnetic  single \nobjects, or objects organized in structures, which can find challengi ng applications in \nmagnetoelectronics. One of the main research tools, supporting above effo rts, is the computer \nsimulation with the use of realistic dimensions and material pa rameters. If the method bases on \nthe Finite Element (FE) method, this give an opportunity to explore prec isely influence of  shape \nand edges onto the nano-devices performance. The widely-known software of thi s type is the \nParallel Finite Element Micromagnetics Package (MAGPAR).1 \nIn the current paper we describe results of simulations of squared, f ree, magnetic thin Co \nfilms. By employing continuous wavelets spectral analysis to describe magnetization dynamics, \nwe identified different types of transient states, going on both in time and frequency domain s. \n \nII. Wavelets for peak-positions detection \nWavelet analysis is an approach which enables simultaneous analysis  of signals in frequency and \ntime domains, thus deals with transients events which can be interpre ted from both the frequency \nand time scales. Usually, the analysis is represented as a two-dimensional map which is \nconstructed from an input signal. In Fig. 1. the idea of wavelet analysis  is shown. The two time-\nevents, which amplitudes are well located at moments 1t and 2t, respectively, have the two \nfrequencies 1f and 2f, and finally were localized on the time-frequency map as the 1W, 2W, and  2 3W events. In other words, at the same moment 1t, there are two physical events 1W and 2W. On \nthe other hand, the 1f frequency is represented by the two events 1W and 3W separated in time. \nIn the current paper, in order to analyze positions of amplitudes (peak  positions), the \nfollowing, symmetrical Mexican-hat wavelet real-valued function is applied \n \n()()2 / 221zezzg−−= ,      (1) \n \nwhich can be then convoluted with the physical sign al ()tf. The result is represented by the \nwavelet spectral decomposition map ()TW,ω (Fig. 1.) calculated as an integral for a given \nfrequency ω, taken from a range of frequencies, at the given t ime-moment T.2 Thus, we have \n \n( ) ( ) ( )[ ]∫+∞ \n∞−− = dt Ttgtf TW ω ωω, .     (2)  \n \n \n \nFig. 1. The idea of 2-dimensional wavelet map ()TW,ω. The two physical events 1W and 2W \ncan be distinguished in the frequency scale, while the 2ω spectral component is composed from \nthe two events, 1W and 3W, separated in time at the moments 1t and 2t. \n \n \nIn order to calculate the inverse wavelet transform  (IWT) we can use the dual function \n()[]Tt−Ψω which satisfies the following condition of ortogon ality \n \n( )[ ] ( )[ ] ( )' '\n03ttdT TtTtg d −=−Ψ− ∫ ∫+∞ +∞ \n∞−δ ω ωωω          (3) \n  3 with the Dirac's delta function ()'tt−δ. Thus, we can express the original signal as \n \n( ) ( ) ( ) [ ] ∫ ∫+∞ +∞ \n∞−−Ψ =\n0, dT Tt TWdtf ωωωω .    (4) \n \nFrom that we can conclude, that wavelet spectral decomposi tion ()TW,ω, seen qualitatively in \nFig. 1., represents the amplitudes of dual basic functi ons of which the original signal is \ncomposed.  \n \nIII. Wavelets spectra of magnetization dynamics in thin magnetic layers \nWe perform simulations of magnetization evolution, and o btained spatio-temporal information, \nhaving applied assumed initial conditions for magnetization  distribution. The simulated \nmagnetization was geometry driven, thus, by demagnetizin g fields depending on geometrical \ndimensions. The external field intensity was set to zero  for all simulations performed. \nThe  simulator employed the Landau-Lifshitz-Gilbert (LL G) equation in the following \nform \n \n( )( )[ ]eff \nseff HMMMHMdt Md rrr rrr\n××+−×+−=2 21 1 αα\nαγ ,         (13) \n \nwhere Mr is the magnetization, γ is the gyromagnetic factor, α is the dumping coefficient, sM \nis the magnetization at saturation, and where the e ffective magnetic field eff Hr consists of the \nfollowing several contributions \n \nex dem ani eff HHHHrrrr++= .      (14) \nThus, during simulations, we observed tendency of o ur system to evolve into the equilibrium via \ncompetition between the anisotropy field ani Hr, which is dependent on the crystalline structure o f \nmaterial, next, the demagnetization field dem Hr, which is dependent on a shape, and finally, the \nexchange field ex Hr which includes contributions from quantum exchange  interactions. Within \nthe every simulation, we obtained the equilibrium s tate which satisfied the following condition \n \n0rrrr=++ex dem ani HHH ,     (15) \n  4 while, for an arbitrary, transient state of evoluti on, we had \n \n0rrrr≠++ex dem ani HHH .     (16) \n \nUsing simulations we tested squared Co ultrathin la yers of thickness ranging from 1.72nm up to \n2.2nm, possessing edges dimensions between 25nm and  250nm. This range of edge-to-thickness \nratios resulted in the in-plane states for magnetiz ation at equilibrium.3 \nThe simulations were realized for the following ini tial condition \n \n0, 0, 1 ===z y x MMM ,     (17) \n \nFor some simulated cases we obtained transient-osci llating states. However, for some cases we \nobtained simple evolutions with oscillating states excluded. The time-evolutions were obtained \nseparately for the every zyxMMM,, magnetization component.  \nIn order to perform time-resolved location of diffe rent frequency modes we used the \nwavelet spectral decomposition. For the sample of t he 150nm width and the thickness equal to \n2.2nm the interesting transient state for the zM component was obtained. The state disappeared \nafter about 1.6 ns (Fig. 2). The different example of evolution of magnetization states was found \nfor the xM component for the sample with width of 150 nm and thickness equal to 2.2 nm. The \nsystem evolved through the two transient wide state s (Fig. 3). The another evolution for the xM \ncomponent was obtained for the sample with edge len gth equal to 250nm and the thickness of \n1.72nm (Fig. 4), where the magnetization evolved th rough several, subsequent oscillating states. \nThere were obtained four types of transient regions : several increasing in frequency events (A), \nbelow 0.6ns, the wide doubled-in-time transient sta te at around 1.5ns (B), the single narrow case \nat 2.3ns (C), and finally, the frequency-doubled st ate visible at  t=4.9ns (D). \n  5 \n \nFig. 2. The z-component magnetization dynamics for the w=150nm and h=2.2nm sample. The \nwavelet map (a), the time-evolution (b). \n \n \nFig. 3. Two transient events of the magnetization x -component for the w=150 nm and h=2.2 nm \nsample. \n \nFig. 4. The wavelet map of the magnetization x-comp onent for the w=250nm and h=1.72nm \nsample. \n \n  6 V. Conclusions \nWithin the current work we obtained different frequ ency-time behavior of magnetization in the \nlow-dimensional squared cobalt nanodots. Thus, we o btained many different shapes of wavelet \ndistributions of magnetization. From the signal the ory point of view the wavelet approach is \nespecially suitable for transient events as they ta ke place in the cobalt nanoobjects we tested. \nThis type of spectral analysis can be additionally treated as a tool for vortex state \ninvestigations. This results from that fact, that a n evolution leading to vortexes is associated with \ndoubled-frequencies and higher order frequencies si gnals. 4-5 However, since the vortexes in \nsome situations are not stable in time and space, t hen their behavior can be effectively described \nusing wavelet analysis.  \n \nReferences \n1W. Scholz, J. Fidler, T. Schrefl, D. Suess, R. Ditt rich, H. Forster, V. Tsiantos,  \"Scalable Parallel \nMicromagnetic Solvers for Magnetic Nanostructures\",  Comp. Mat. Sci. 28 , 366 (2003).  \n2J. Lewalle,Wavelets without Lemmas -  Syracuse Univ ersity -VKI Lecture Series, April 6-8, \n1998. \n3A. Maziewski, V. Zablotskii, M.Kisielewski, Phys. R ev. B 73 , 134415 (2006). \n4V. Novosad, M. Grimsditch, K. Yu. Guslienko, P. Vav assori, Y. Otani, S. D. Bader, Phys. Rev. \nB 66 , 052407 (2002). \n5P. Vavassori, N. Zaluzec, V. Metlushko, V. Novosad,  B. Ilic, M. Grimsditch,  Phys. Rev. B 69 , \n214404 (2004).  "    },    {        "title": "1508.02921v1.Simultaneous_electronic_and_the_magnetic_excitation_of_a_ferromagnet_by_intense_THz_pulses.pdf",        "content": "Simultaneous electronic and the magnetic excitation of a \nferromagnet by intense THz pulses  \n \nMostafa Shalaby1, Carlo Vicario1, and Christoph P. Hauri1,2 \n1Paul Scherrer Institute, SwissFEL, 5232 Villigen PSI, Switzerland. \n2École Polytechnique Fédérale  de Lausanne, 1015 Lausanne, Switzerland.  \n*Correspondence to:  most.shalaby@gmail.com, carlo.vicario@psi.ch, and christoph.hauri@psi.ch  \n \nThe speed of magnetization reversal is a key feature in magnetic data storage. Magnetic fields from \nintense THz pulses have been recently shown to induce small  magnetization dynamics in Cobalt thin \nfilm on the sub-picosecond time scale. Here, we show that at high er field intensities, the THz electric \nfield starts playing a role, strongly changing  the dielectric properties  of the cobalt thin film . Both the \nelectronic and magnetic responses are found to occur simultaneously , with the ele ctric field response \npersistent on a time scale orders of magnitude longer than the THz stimulus .  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n The speed of magnetization reversal is a key feature for ultrafast  magnetic storage  technology [1-2]. \nIn the state of the art data recording technology, the magnetization reversal, i.e., the complete \ninversion of the direction of the magnetization vector  (𝐌) occurs on a relatively long time scale on the \norder of the nanosecond. In the past, femtosecond optical pulses have been used to induce faster \nmagnetic phase transition , but the impact of the optical pulse leads to thermal  effect s and therefore to \na long recovery time (nanoseconds), which significantly limits the magnetic writing rate  [3-6]. Using \nintense magnetic field pulses fro m a relativistic electron beam , magnetic switching on a picosecond \ntimescale was observed post mortem [1-2]. However,  time-resolved exploration of the interplay of the \nelectric and the magnetic field  component with  the ferromagnetic sample  could not be performed  at \nthis large scale facility .  \nAn a lternative  approach to wards  precessional magnetic switching on the picosecond timescale  relies  \non the  magnetic field component  of a strongly asymmetric single cycle terahertz (THz) pulse  phase-\nlocked to the spins [7-15]. As the THz pulse carrier frequency matches well the natural timescale of \nthe spin motions  direct  control  of the spin dynamics with the THz magnetic field component becomes \nfeasible by inducing a torque [16-17]. Recently , this concept has been prove n experimentally by \nVicario et al  where a small excursion of the spins  was introduced non-resonantly by an intense 0.4 \nTesla THz field [17]. The observed coherent spin dynamics  were  more than an order of magnitude \nfaster than  in previous studies . Effects associated to the  THz electric field w ere not observed. While  \nthis proof  of principle experiment  showed the  great  potential of intense  THz radiation for precessional \nspin motion  control , full magnetic switching requires much higher field intensities  than what was \navailable in ref. [17]. Here , we report on dynamics induced by a n up-scaled THz  pulse  intensit y \ntargeting large amplitude magnetization dynamics . We observed that at high field intensities, the THz \nelectric field starts playing a significant role in chang ing the dielectric properties of  the magnetic \nmaterial.  \nThe THz source used for the present studies provide s intense THz pulses with maximum \nelectric and magnetic field strengths of 6.7 MV/cm and 2.23 T, respectively . The corresponding peak \nintensity is 59 GW/cm2. The THz was generated using optical rectification of a short wavelength \ninfrared pulses ( c=1.5 µm) in an organic crystal DSTMS  [18-19].  Figure 1a and 1b show the \ntemporal field profile recorded by electro -optical sampling and the corresponding spectral amplitude. \nThe THz spot in the focus , recorded using an uncooled micro -bolometer array ( NEC Inc.)  is shown in \nFig. 1c . \n \n \n \n \n \n  \n \n \n \nFIG. 1. (a) The temporal trace of the exciting THz pulse retrieved using  electro-optical  sampling  \ntechnique  (peak of 6.7 MV/cm) . (b) The corresponding amplitude spectrum. (c) The THz spot size at \nthe sample position measured using a micro -bolometer array -based camera.  (d) Schematic diagram \nof the beams configurations on the sample.  \n \nTo measure the THz -induced nonlinear dynamics, we used a collinear  THz-pump /optical Kerr effect \n(KE) probe  scheme . The investigated  sample is a 20 nm -thick Co film  deposited on a Si substrate  and \ncapped with 2 nm -thick layer of Pt.  The latter is a protection layer which  plays no role in the \nexperiment presented  here. We verified that by comparing the measurement against uncoated fresh \nsample.  The beam configuration at the sample position i s shown in Fig. 1d: the THz pump and a \ncollinear 800 -nm centered probe beam impinge on the sample at an angle of 15°  measured from the \nnormal . The reflected probe beam was then collected and the THz -induced KE rotation was analyzed \nusing a balanc ed detection scheme  (a quarter wave plate followed by a Wollaston  prism) . An external \nmagnetic field 𝐁 bias (parallel to the sample plane and the plane of incidence) is used to prepare a \nwell-defined  in-plane magnetization state 𝐌 prior to pumping . During the measurements the THz \npolarization direction could be altered by rotating the the THz generation crystal along with the near \ninfra red pump beam  polarization . \n \nFIG.  2 The e xperimental layout in (a) the precession  ±𝑀⊥𝐻𝑇𝐻𝑧 and (b) non -precession ±𝑀 || 𝐻𝑇𝐻𝑧 \nconfigurations . (c) and (d) show the corresponding THz -induced electronic and magnetic dynamics  \n(optical probe rotation) . The transient curves  are zoomed in and plotted in (e) and (f ). The red and \nblue curves refer to the positive and negative directions of the applied static field ( 𝑩) and thus \nmagnetization ( 𝑴).  \n \nNonthermal temporal magnetic evolution under the application of an external (THz) magne tic field is \nmacroscopically governed by the Landau -Lifschitz-Gilbert (LLG) equation : \n   𝜕𝐌\n𝜕t=−|γ|𝐌×𝐇eff+α\nMs(𝐌×𝜕𝐌\n𝜕t)  (1) \nwhere γ, α and Ms are the gyromagnetic ratio , Gilbert damping coefficient,  and material -dependent \nsaturation magnetization, respectively . Heff is the effective field summing the contributions from the \ninternal and external ( here THz, 𝐇THz) fields . An intense  enough  𝐇THz (with a nonzero angle to the \ndirection of 𝐌) perturbs this alignment driving time -dependent  dynamics in  𝐌. The realignment \nprocess consists of two dynamic s, described by the two corresponding  terms in the LLG . This \nincludes ultrafast precessional dynamics taking place on the THz ( stimulus ) time scale and the \ndamping mechanism  which tries to orient  𝐌 towards the new direction of equilibrium 𝐇eff on a much \nlonger time scale (governed by the material parameters ). The damping intensity is generally much \nweaker than the ultrafast precessions.  At relatively weak excitation, the amplitude of the induced \nultrafast precessions is nearly  a linear function of 𝐇THz.   \nThe simplified  LLG model  (1) does not take into account the effect of the THz electric field  \nand thus may be inappropriate for the physics observed at high fields  to describe . High field \nintensities  (on the order of several Tesla in the sub -THz range and much more as  the excitation \nfrequency increases  [20-21]), however, are required for Terahertz -induced magnetization reversal . In \nthis article we were aiming to increase the excitation intensity  in order to study the role of  the THz \nelectric field component (𝐄THz) in the observed dynamics . Two major 𝐄THz-induced effects can be \nconsidered. The f irst mechanism is the change in the dielectric tensor due to Kerr-like and thermal \nnonlinearities . The s econd  effect is the thermal demagnetization of the  sample  induced by  heating the \nelectron s above the Curie temperature, similar to  the majority of laser -induced magnetization \ndynamics in the optical regimes . The second mechanism  depends on the excitation fluence.  In our \nexperiment al setup , the sample is exposed to two external excitations, namely the THz (electric and \nmagnetic ) fields  and the bias magnetic field. In phase -sensitive detection as we use d here, one of the \nexternal excitations is modulated at a frequency which is used as a reference for our acquisition \nsystem. This allows us to distinguish between the magnetic and non -magnetic (electronic)  dynamics . \nWhile modulating the THz pump reveals bot h dynamics, modulating the external bias shows on the \nmagnetic ones.  \nThe nonlinear dynamics induced by 𝐄THz and 𝐇THz are very different.  For simplicity, we \nassum e no coupling  between 𝐄THz and 𝐌. This assumption is manifested by the fact that changes \nin 𝐌 require a nonzero torque 𝐌×𝐇eff (i.e. it vanishes for 𝐌 |||𝐇THz| and is a maximum for \n𝐌⊥|𝐇THz|) and that  the excitation of the electronic system does not depend on the direction of 𝐌. \nSecond, while the  𝐇THz-induced changes in  the experimentally -dominant out-of-plane  𝐌 component \nare generally linear with the exciting 𝐇THz (at relatively low excitation as we show here), the 𝐄THz-\ninduced changes of the material refractive index are proportional to  (𝐄THz)2.  \nFigure 2c & 2d show the measured probe Kerr rotation under two conditions ±𝐌⊥𝐇THz and \n±𝐌 || 𝐇THz. A giant rotation is observed with an instantaneous rise time and duration  over a long \ndelay > 20 ps. A zoom -in on the initial dynamics is shown in Fig. 2e & 2f. In the case of ±𝐌⊥𝐇THz, \nclear oscillations following the THz excitation field can be observed on the sub -ps scale superimposed \non a large incoherent signal. To isolate the coherent precessions, we subtracted the trace pairs with \n±𝐁. The measured rotation has superimposed contributions from the excitation of both the electronic \nand magnetic systems.  The sign of probe rotation does not depend on the sign of the THz electric field. On the contrary, the vectorial nature of  the cohere nt precessional dynamics suggest s that the \napplication of two stimuli  with opposite signs (±𝐇THz) leads to opposite torques 𝐌×±𝐇eff and thus \nperfectly reversed out-of-plane temporal  magnetization  dynamics.  The probe polarization rotation \ndirection (sign) follows the direction of this torque.   \nIn order to distinguish between the two dynamics, we performed all the measurements in two distinct \nconditions  with +B and –B (that is, + M and –M). In this way, by subtracting the two traces the \ncoherent precessions  is obtained while  the incoherent dynamics were extracted by adding the two \nmeasurements . The results are shown in Fig. 3a and 3b. In the case of 𝐌×±𝐇eff, coherent magnetic \nprecessions are excited. The Fourier analysis  (Fig. 3c)  unravels that they have similar spectral \ncontents as the driving THz stimulus  (Fig. 1c) . A much stronger contribution to the rotation arises \nfrom the electronic excitation  (nearly a factor of 9) . On the contrary, in the case of 𝐌 |||𝐇THz|, only \nthe incoherent dynamics are observed (Fig. 3b). The measurement was performed by modulating the \nTHz pump  only. However, when we modulated the external bias instead, this incoherent  part \ndisappeared . This shows that the signal is not related to demagnetization and it corroborates  its origin \nin an electronic response . \n \nFIG. 3 Analysis of the measurements shown in Fig.2. The coherent spin precessions  (green), \ndepending  on the sign of 𝑴, and the electronic responses  (black) are isolated from the incoherent \ndynamics by subtracting and adding the two traces in Fig. 2e and Fig. 2f.  The results are shown in \n(a) and  (b). The green and black curves depicts the magnetic and electronic parts, respectively. In the \ncase of 𝑴⊥𝑯𝑇𝐻𝑧, (a) show s clear temporal oscillations representing magnetic precessions and \ncorresponding to t he THz temporal oscillat ions. (c ) reports the corresponding amplitude spectra to \ngreen curves in  (a, blue) and (b , red). \n \nFinally, we would like to mention that the magnitude of the Kerr polarization rotation depends \nstrongly on the probe initial polarization. We verified that by measuring the rotation in the case of \n𝐌 |||𝐇THz| different polarization of the probe (Fig. 4). Similar to ref. 22, we obtained the maximum \nsensitivity for an input polarization of 45°. Nevertheless, the angle sensi tivity depends on the \ndielectric properties of the film and the substrate.  \n \n \nFIG.  4. Sensitivity of the probe rotation to the polarization. 0° corresponds to P -polarization  and 90° \ncorresponds to S -polarization.  \n \nIn conclusion, we used ultra -intense THz pulses  centered at 4 THz with an electric and magnetic \nfields of 6.7 MV/cm and 2.3 T, respectively to trigger both the electronic and magnetic response  in a \nthin Cobalt film. In addition to the previously observed THz magnetic field -induced spin precession s, \nwe observed electric field -driven changes in the sample dielectric tensor and thus the optical \nproperties . The two effects are found to occur simultaneously on the ultrafast time scale of the \ntriggering THz pulse.  While the magnetic response takes plac e only during the THz stimulus, the \nelectric field -induced dynamics last up to a much longer time scale (> 20 ps). We conclude that the \nrole of the electric field can not  be excluded in the studies of THz -magnetism, particularly at higher \nfield intensities.  \n \nWe gratefully thank prof. Jan Luning ( Université Pierre et Marie Curie) for the sample and fruitful \ndiscussions. We are grateful to Marta Divall, Alexandre Trisorio, and And reas Dax for supporting the \noperation of the Ti:sapphire laser system  and the OPA . We acknowledge the support from Martin \nParaliev, Rasmus Ischebeck,  and Edwin Divall  in DAQ . We acknowledge financial support from the \nSwiss National Science Foundation (SNSF) (grant no 200021_146769). MS acknowledges partial \nfunding from the European Community's Seventh Framework Programme (FP7/2007 -2013) under \ngrant agreement n.°290605 (PSI -FELLOW/COFUND) . CPH acknowledges association to NCCR -\nMUST and financial support from SNSF (grant no. PP00P2_150732).  \n \n  \n[1] C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Weller, E. L. Garwin, and H. C. \nSiegmann, Science 285, 864 -867 (1999).  \n[2] I. Tudosa, C. Stamm, A. B. Kashuba , F. King, H. C. Siegmann, J. Stöhr, G. Ju, B. Lu,, and D. \nWeller, Nature 428, 831 -833 (2004).  \n[3] J. Hohlfeld, Th. Gerrits, M. Bilderbeek, Th. Rasing, H. Awano, and N. Ohta, Phys. Rev. B 65, \n012413 (2001).  \n[4] A. V. Kimel, A. Kirilyuk, P. A. Usachev , R. V. Pisarev, A. M. Balbashov, and Th. Rasing, Nature \n435, 655 - 657 (2005).  \n[5] K. Vahaplar, A. M. Kalashnikova, A. V. Kimel, D. Hinzke, U. Nowak, R. Chantrell, A. \nTsukamoto, A. Itoh, A. Kirilyuk, and Th. Rasing, Phys. Rev. Lett . 103, 117201 (2009).  \n[6] E. Beaurepaire, J.C. Merle, A. Daunois, and J.Y. Bigot, Phys. Rev. Lett.  76, 4250 (1996).  \n[7] K. Tanaka, H. , Hirori, and M. Nagai,  IEEE Trans. Terahertz Sci. Technol  1, 301–312 (2011 ).  \n[8] H. Y. Hwang, S. Fleischer, N. C. Brandt, B. G. Perkins Jr., M. Liu, K. Fan, A. Sternbach, X. \nZhang, R. D. Averitt, and K. A. Nelson , J. Mod. Optic.  doi:doi:10.1080/09500340.2014.918200 \n(2014 ).  \n[9] M. Lie, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G . R. Keiser, A. J. Sternbach, K. G. \nWest, S. Kittiwatanakul, J. Lu, et al. , Nature  487, 345–348 (2012 ).  \n[10] R. Ulbricht, E. Hendry, J. Shan, T.F. Heinz, and M. Bonn, Rev. Mod. Phys. 83, 543 (2011).  \n[11] T. Kampfrath, K. Tanaka, and K.A. Nelson, Nat. Photonics  7, 680 (2013).  \n[12] O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, \nM. Kira, S.W. Koch, and R. Huber, Nat. Photonics  8, 119 (2014) . \n[13] B. Zaks, R. B. Liu, and M. S. Sherwin, Nature (London) 483, 580 (2012).  \n[14] S. Fleischer, Y. Zhou, R.W. Field, and K.A. Nelson, Phys. Rev. Lett.  107, 163603 (2011).  \n[15] W. R. Huang, E. A. Nanni, K. Ravi, K. –H. H ong, L. J. Wong, P. D. Keathley, A. Fallahi, L. \nZapata, and F. X. Kartner, “Toward a terahertz -drive n electron gun ,” arXiv:1409.8668  [16] T. Kampfrath, A. Sell, G. Klatt, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitensorfer, \nand R. Huber, Nat. Photonics  5, 31–34 (2011 ).  \n[17] C. Vicario, C. Ruchert, F. Ardana -Lamas, P. M. Derlet, B. Tudu , J. Luning, and C. P. Hauri, Nat. \nPhotonics 7, 720 –723 (2013).  \n[18] C. Vicario, B. Monoszlai, & C. P. Hauri, Phys. Rev. Lett.  112, 213901 (2014).  \n[19] M. Shalaby and C. P. Hauri, Nat. Commun.  6, 5976 (2015).  \n[20] M. Shalaby, M. Peccianti, D. G. Cooke, C. P. Hauri, and R. Morandotti, Appl. Phys. Lett.  106, \n051110 (2015).  \n[21] M. Shalaby, F. Vidal, M. Peccianti, R. Morandotti, F. Enderli, T. Feurer, and B. D. Patterson, \nPhys. Rev. B  88, 140301(R) (2013).  \n[22] D. Allwood, P. Seem, S Basu, P. Fry, U.  Gibson, and R. Cowburn , Appl. Phys. Lett.  92, 2503  \n(2008). \n \n "    },    {        "title": "1908.06718v1.Steady_states_of_non_axial_dipolar_rods_driven_by_rotating_fields.pdf",        "content": "arXiv:1908.06718v1  [cond-mat.soft]  19 Aug 2019Steady states of non-axial dipolar rods driven by rotating fi elds\nJorge L. C. Domingos,1Everton A. de Freitas,1and W.P. Ferreira1\n1Departamento de F´ ısica, Universidade Federal do Cear´ a,\nCaixa Postal 6030, 60455-760 Fortaleza, Cear´ a, Brazil\n(Dated: August 20, 2019)\nWe investigate a two-dimensional system of magnetic colloi ds with anisotropic geometry (rods)\nsubjected to an oscillating external magnetic field. The str uctural and dynamical properties of\nthe steady states are analyzed, by means of Langevin Dynamic s simulations, as a function of the\nmisalignment of the intrinsic magnetic dipole moment of the rods with respect to their axial direc-\ntion, and also in terms of the strength and rotation frequenc y of an external magnetic field. The\nmisalignment of the dipole relative to their axial directio n is inspired by recent studies, and this is\nextremely relevant in the microscopic aggregation states o f the system. The dynamical response of\nthe magnetic rods to the external magnetic field is strongly a ffected by such a misalignment. Con-\ncerning the synchronization between the magnetic rods and t he direction of the external magnetic\nfield, we define three distinct regimes of synchronization. A set of steady states diagrams are pre-\nsented, showing the magnitude and rotation frequency inter vals in which the distinct self-organized\nstructures are observed.\nPACS numbers: 74.78.Na, 74.25.Ha, 74.25.Dw, 74.20.De\nI. INTRODUCTION\nAnisotropic colloids are an exciting and relevant sys-\ntem due to their primary role in soft matter systems.\nThe possibility of manipulation of their shapes and in-\nteraction brings a wide range of applications in the self-\nassembly of colloidal matter [1–3], in microfluidics [4–6]\nand in the design of functional devices such as probes\nand sensors [7–11]. As a particular class of colloids with\nanisotropic interaction, we mention magnetic colloids,\nwhich are micro-sized building blocks with an embedded\nmagnetic moment with dipolar interaction, i.e., particles\ncomposed of a magnetic mono-domain having a typical\nsize ranging from 1 to 150 nm[12].\nMagnetic nanoparticles (MN) with anisotropic shape\nare subject to higher interest in comparison to their\nspherical counterparts due to their more complex prop-\nerties and collective behavior, such as magnetic birefrin-\ngence [13], thermal conductivity [14] and orientation or-\ndering transition [15, 16]. Attention was also addressed\nto cases where the anisotropy is in the location of the\ndipole concerning the center of symmetry of the parti-\ncle. In recent theoretical works, the structure of fluids\ncontainingsphericalparticleswithembeddedoff-centered\nmagnetic dipoles [17, 18] was investigated.\nConcerning the interaction between rod-shaped MN,\nthe magnetic moment is usually parallel to their long\naxis. However, anisotropic shaped particles with non-\naxial dipole moment is being subject of many studies. It\nis known that ellipsoidal particles with permanent dipole\nmoment perpendicular to their long axis showed to be\nuseful in cell entrapment by magnetic manipulation [19].\nOther studies analyzed rods with non-axial dipole mo-\nment e.g. experimentally through superparamagnetic\nmagnetoresponsiverods [20] and peanut-shaped particles\n[21], also numerically [22] and analytically [23]. The lat-\nter studied the torque of magnetic rods with randomlyorienteddipolesforoptimizedhyperthermiaapplications.\nTherefore, in this study, we explore the direction of the\ndipole moment as a controlling parameter, allowing the\nfunctionalization of the particles.\nIn this work, we use a peapod-like model to simulate\nthe rigid magnetic rod. Such a model was already con-\nsidered experimentally [24] and numerically [15, 16, 22].\nWe aim to explore the interplay between the structure\nformation and the translational and rotational dynamics\nunder the influence of an external field. Several experi-\nmental and theoretical studies focused on single-particle\ndynamics, which has particularly important applications\nin actuators[25], microfluidics [26] and optical traps [27].\nHowever, an increasing interest has been devoted to the\ncollective behavior ofmagnetic particles subject to exter-\nnal fields, since time-dependent fields are an important\ntool to control self-organization processes.\nPrevious experiments and computer simulations reveal\nthat for sufficient strong rotating fields, the spatial sym-\nmetry can be broken by the formation of layers in the\nfield plane [28–30]. As a consequence of the rotating\nfields, an inverted dipolar pair interaction with an in-\nplane attraction and repulsion along the rotation axis\n[30, 31] replaces, averaged over time, the dipolar inter-\naction. Further interesting phenomena occur when the\ncolloidal magnetic particles are exposed to rotating fields\nin two-dimensional geometry. In this situation, the time-\naveraged dipolar potential is purely attractive and long-\nranged [32, 33]. The fact that particles follow the field\nsynchronously makes possible to explain the resulting\nself-organizedstructures from an equilibrium perspective\ninvolving the phase behavior of a many-particle system\ninteracting via a time-averaged inverted dipolar interac-\ntion [28, 32, 34].\nOuraim, inthisstudy, istoanalyzethephase-behavior\nof a two-dimensional system consisting of ferromagnetic\npeapod-like rods subject to rotating fields. According2\nto the extent of the synchronization of the rod particles\nwith the external field, different phases arise from the\ncompetition between of the rod-rod interaction, by tun-\nning the dipole misalignment, and the rod-external field\ninteraction, by tuning the field intensity and the rotation\n(oscillatory) frequency.\nThe paper is organized as follows: our model system is\nintroducedinSec. II.Thenumericalresultsarepresented\nand discussed in Sec. III, and our conclusions are given\nin Sec. IV.\nII. MODEL\nWe perform extensive Langevin dynamics simulations\nto study a two-dimensional (2 D) system consisting of\ntypically N= 840identicalstiffrodsofaspectratio l= 3.\nThe phase behaviour of a mono-dispersed system with\nthe same aspect ratio was recently studied [35], being\nconsidered a standard reference system [36]. For suspen-\nsions studied experimentally, the aspect ratio l= 3 is in\nthelowestaccessiblelimit [37]. We simulatethe magnetic\nnature of the rod by attaching a point dipole of perma-\nnent magnetic moment µat the center of each bead (see\nFig. 1).\nrjm\nabR\u0001aΨrod a rod b\n+x+xσ\nlσ\nhijdmm\nl\nk\u0000b\nFIG. 1: Schematic illustration of the interaction between t wo\nmagnetic rods with indication of the important parameters o f\nthe pair interaction potential.\nThe orientation of the dipoles concerning the axial di-\nrection of the rod is given by the angle Ψ, as illustrated\nin Fig 1. To model the dipolar particles, we use a dipolar\nsoft sphere (DSS) potential [15], consisting of the repul-\nsive part of the Lennard-Jones (LJ) potential urepand\na point-like dipole-dipole interaction part uD. The total\ninteraction energy between rods aandbis the sum of\nthe pair interaction terms between their respective dipo-\nlar spheres (DS):Ua,b(Ra,b,θa,θb) =/summationdisplay\nj/negationslash=muj,m, (1)\nuj,m=urep(ra,b\njm)+uD(ra,b\njm,µa\nj,µb\nm), (2)\nwhere:\nurep= 4ǫ/parenleftbiggσ\nrjm/parenrightbigg12\n, (3)\nuD=µj·µm\nr3\njm−3(µj·rjm)(µm·rjm)\nr5\njm, (4)\nwithσthe diameter of each bead, and ǫis the LJ soft-\nrepulsion constant, Ra,b=Rb−Rais the vector which\nconnects the center of rod bwith the center of rod a.\nThe orientation of rods aandbare given by θaandθb,\nrespectively. The vector ra,b\njmconnects the center of bead\nmof rodbwith the center of bead jof roda(see Fig 1).\nThe force on bead mdue to bead jis given by:\nfjm=−∇ujm. (5)\nThe torque on bead mis [15]:\nτm=µm×/summationdisplay\nm/negationslash=j{Bjm+B(t)}+dm×/summationdisplay\nm/negationslash=jfjm, (6)\nwheredmis the vector connecting the center of bead m\n(rodb) with the center of rod bas illustrated in Fig 1,\nandBjmis the magnetic field generated by the dipole\nmoment µjat the position of the dipole µm, andB(t) is\nthe external magnetic field. They are given by:\nBjm=3(µm·rjm)rjm\nr5\njm−µm\nr3\njm, (7)\nB(t) =B[cos(ωt)ˆ x+sin(ωt)ˆ y], (8)\nwhereB(t) is the external magnetic field, with intensity\nBand rotation frequency ω. The external magnetic field\nrotates in the same plane of the magnetic rods.\nThe summations in Eq. 6 are considered only for\ndipoles belonging to distinct rods. The orientation of\nthe rods is given by the unitary vector sgiven by s=\ndm/|dm|. The translational and rotational Langevin\nequations of motion of rod bwith mass Mband moment\nof inertia Ib, are given by:\nMbdvb\ndt=Fb−ΓT·vb+ξT\nb(t), (9)\nIbdωb\ndt=Nb−ΓRωb+ξR\nb(t), (10)\nwherevb=dRb/dt,ωbis the angular velocity, Fband\nNbare the total force and torque acting on rod b, respec-\ntively, while ΓTand ΓRare the translational tensor and\nrotational friction parameters. For rod-like particles the3\ntranslational tensor is composed by the parallel( ζ/bardbl) and\nperpendicular ( ζ⊥) components with respect to the rod\naxis, which are given by:\nζ/bardbl=2πη0lσ\nln(l)+δ/bardbl,ζ⊥=4πη0lσ\nln(l)+δ⊥, (11)\nand for rotation:\nζr=πη0(lσ)3\n3ln(l)+δr, (12)\nwhereη0is the solvent viscosity, δ/bardbl,δ⊥andδrare correc-\ntion factors for small rods extracted from Refs. [38, 39].\nAs a result, the total translational diffusion coefficient is\nDT=1\n3(D/bardbl+2D⊥) forD⊥=1\n2D/bardbl[40].\nξT\nbandξR\nbare the Gaussian random force and\ntorque, respectively, which obey the following white\nnoise conditions: /an}bracketle{tξα\nb(t)/an}bracketri}ht= 0, /an}bracketle{tξα\nb(t)·ξα\nb′(t′)/an}bracketri}ht=\n2ΓαkBTδbb′δ(t−t′),α=T,R.\nWe define the reduced unit of time as t∗=\nt/√\nǫ−1Mσ2, whereMis the mass of the rod. There-\nfore, the frequency of the rotation of external magnetic\nfield isω∗=t∗−1=ω/σ2/√\nǫ−1M. The energy is given\nin reduced units as U∗=U/ǫ, the dipole moment in di-\nmensionlessunits as µ∗=µ/√\nǫσ3, andthe dimensionless\ndistances as of r∗=r/σ. The ratio of the thermal en-\nergy to the soft-sphere repulsion constant is chosen to be\nkBT/ǫ= 1, where ǫ/kBis the temperatureunit and kBis\nthe Boltzmann constant. Periodic boundary conditions\nare taken in both spatial directions. Since the dipolar\npair interaction falls off as ( r−3), we take the simulation\nboxsufficientlylargesuchthatnospeciallong-rangesum-\nmationtechniques[41]areneeded. We define the packing\nfraction as η=Nbeadsπ(σ/2)2/L2, whereNbeads= 2520\nis the total number of dipolar beads of the system and\nL2is the simulation box area. Since Nbeads=lN, we\ncan rewrite the packing fraction as η=ρ∗lπ/4, where ρ∗\nis the dimensionless density ρ∗=ρσ2, andρ=N/L2, in\nall simulations, we set η= 0.1. The reduced time step is\ntypically in the range δt∗= 10−4−10−3.\nThe quantities of interest are then averaged over more\nthan 106time steps. All the beads from all rods have\nthe same dipole moment whose magnitude we set as\nµ∗= 4.4 which was estimated based on experiments\nat room temperature ( T≈293K) using iron nanopar-\nticles [24] with saturation magnetization Ms(Fe) = 1700\nkA/mand the radius of the particles r≈5nm. For\nexternal magnetic fields, we use B∗(t) values within the\nrange 10 ≤B∗≤50, which is related to the experimen-\ntal range 33 mT≤B≤165mTat room temperature.\nExperimental values for the magnetic fields are of the or-\nder of 0.1T[16], but ferrofluids have been found to be\nsusceptible already to B <10mT[24]. For the sake of\nsimplification, weareomittingthe*superscripthereafter\nin all dimensionless parameters.III. NUMERICAL RESULTS\nIn this section, we present our numerical results. We\nbegin by discussing the conditions for which the different\nphases are observed for fixed packing fraction, η= 0.1,\nand temperature kBT/ǫ= 1. We study the formation\nof the clusters according to the parameters Psi,B, and\nω, which are associated to the interaction between rods,\nand the interaction between the rods and the external\nmagnetic field. We base our analysis on structural and\ndynamical parameters.\nA. Non-Equilibrium Phase Diagram\nIn this section we examine the formation of differ-\nent structures for different values of the magnetic field\ne rotation frequency. As mentioned previously in this\nmanuscript, there is a relationship between the cluster-\ning process and the synchronousrotational motion which\nis related to the time-averaged dipolar potential in 2 D,\nuD(rij) =1\nτ/integraldisplayτ\n0u(rij,µi(t),µj(t))dt=−µ2\n2r3\nij. (13)\nThe previous equation is obtained when the magnetic\ndipoles are in phase, i.e., synchronized with the ex-\nternal rotating magnetic field, resulting in an effective\nisotropic and attractive pair-interaction potential be-\ntween the magnetic dipoles.\n0 1 2 3 4 5 6 78 9 10\nr’/σ-3-2-10U(r’) \nΨ = 15°\nΨ = 30°\nΨ = 45°\nΨ = 60°\nΨ = 75°\nΨ = 90°Ψθαr'a)\nb) c)\nFIG. 2: (a) The pair interaction energy as a function of\ninter-rod separation ( r′) minimized with respect to αandθ.\nSketches of the (b) ribbon-like and (c) head-to-tail arrang e-\nments.\nWe start by analyzing the dependence of the pair in-\nteraction for different values of the new feature added to4\nthe rods, the misalignment Ψ. We obtain rather differ-\nent potential profiles by changing Ψ. For low values of\nΨ (≤30◦), the minima are located at r′/σ≈3, which\ncorrespondsto the aspect ratio of the rods. The values of\nαandθ[Fig. 2(a)], which minimize the pair-interaction\nenergy indicate that rods are favorably in the head-to-\ntail bond. By increasing Ψ, the position of the global\nminimum is displaced to smaller values of r′/σ, suggest-\ning that the head-to-tail bond disappears, giving rise to\nthe ribbon-like bond configuration (see Fig. 2(b)). The\nlatter arrangement was obtained experimentally in Refs.\n[19, 21].\nIn order to determine the separation ( δc) for which we\ndefine a bond between two rods, we analyze the inter-rod\nseparation related to the minimum energy value. From\nFig. 2(a), the largest inter-rod separation related to the\nglobal minimum is located at r′≈3.4σfor Ψ = 15◦and\natr′≈1.4σfor Ψ = 90◦. In the former, the rods are\nin the head-to-tail arrangement, while in the latter they\nare in the ribbon-like configuration. In both cases, the\nshortest separation between beads of different rods is ≈\n1.4σ(bead-to-beadcenterdistance). Therefore,wedefine\nthat the two rods are bonded as the shortest separation\nbetween them is ≤1.4σ.\nSince the attraction between rods becomes stronger\nfor larger Ψ (Fig. 2), we expect that the ribbon-like\nconfigurations (Ψ >45◦) become more stable, implying\nthat the formation of clusters is facilitated in the many-\nbody case.\nThe synchronization with the external magnetic field\nplays, in addition to the dipole’s misalignment Ψ, an es-\nsential role to define the structures fo the system. We\nobserve three distinct typical structures by manipulating\nBandω, namely, Dynamic aggregates, Isotropic fluid,\nand Clustered fluid. We present in Fig. 3 the resulting\nB−ωphase diagrams of such configurations for different\nvalues of Ψ .\nDynamic aggregates result from a strong interaction\nbetween rods and the external magnetic field. If rods fol-\nlowtherotationoftheexternalfieldsynchronously,anat-\ntractive regime appears as a result of the aforementioned\ntime-averaged potential (Eq. (13)). In this regime, the\nrods are not necessarily connected as our definition of\na bond suggests (head-to-tail or ribbon-like), and they\nare rotating with respect to their centers, breaking the\nspatial symmetry that usually characterizes an isotropic\nfluid. We show in Fig. 3 that the dynamic aggregatesap-\npear in the limit of a high magnetic field as ωincreases.\nThese limits allow the majority of the rods to follow the\nfield, fulfilling the behavior of polarizable magnetic par-\nticles known to form 2 Dclusters.\nThe isotropic fluid is a disordered phase where the ki-\nnetic energy is more relevant than the interaction energy\nbetween rods. There is no formation of clusters. In-\nstead, we observe a more isotropic particle distribution\nin the system. For a given magnetic field, the interac-\ntion between the rods and the external magnetic field\ndecreases with increasing frequency because of the lackof synchronization, yielding an equivalence of the compe-\ntition between the rod-rod and rod-external field interac-\ntions. The rod-external field interaction is strong enough\nto avoid the formation clusters, but not strong enough to\nform dynamic aggregates. When the rod-external field\ninteraction becomes irrelevant, we observe a new phase,\nthe clustered fluid phase, which is a consequence of the\nrod-rod interaction, and it consists of chains of clustered\ncolloids. In Fig. 4, we illustrate examples of the typical\nphases observed.\nConcerning the synchronization of the particles with\nthe external magnetic field, the dynamical aggregate is\nin a high synchronization regime (for high Band lowω),\nwhile the isotropic fluid phase is in intermediate regime,\nand the clustered fluid phase is in a low synchronization\nregime (low Band high ω). We will discuss such regimes\nin more details in the next section.\nB. Structure and Synchronization\nIn this section, we study the structure of the phases\nand their synchronization with the external magnetic\nfield. Some configurations, even though classified as be-\nlonging to the same phase, have distinctive features in\ntheirmicrostructurethatmakethem distinguishable. We\nbaseouranalysisonthestrengthoftheexternalmagnetic\nfield. One essential tool to analyze the structure of the\nsystem is the pair correlation function [42], defined in 2 D\nas:\ng(r) =/angbracketleftBig/summationtext\na/summationtextN\nb/negationslash=aδ(r−Rab)/angbracketrightBig\n2Nπrρ∗, (14)\nwhereRabis the separation between centers of rods a\nandb(see Fig. 1). To quantify the extent of aggregation\nbetweendipolarrods, weanalyzethe polymerization[43],\nwhich is a measure of how many rods are bonded to at\nleast one other rod. We define the polymerization as\nthe ensemble average of the ratio between the number of\nclustered rods, Nc, and the total number of rods, N:\nΦ =/angbracketleftbiggNc\nN/angbracketrightbigg\n. (15)\nThe synchronization process is based on the analysis\nof the rotation dynamics of the dipoles by using the sin-\ngle particle time autocorrelation function for the dipole\nmoment (dipole-dipole autocorrelation function), given\nby:\nCµ(t) =1\nN/angbracketleftBiggN/summationdisplay\ni=1ˆµi(t)·ˆµi(0)/angbracketrightBigg\n, (16)\nwhere ˆµis the unitary vector of the magnetic moment of\nthei-th rod.5\n5 10 15 20 25 30ω1020304050\n5 10 15 20 25 30ω1020304050a) b)\n5 10 15 20 25 30ω1020304050c)\n5 10 15 20 25 301020304050d)\n5 10 15 20 25 301020304050e) f)BBB B\nω ωB\n1020304050 B\n5 10 15 20 25 30ω\nFIG. 3: Steady state phase diagrams presenting the self-org anized structures as a function of the intensity, B, and the rotation\nfrequency, ω, of the external magnetic field for different values of misali gnment: (a) Ψ = 15◦; (b) Ψ = 30◦; (c) Ψ = 45◦; d)\nΨ = 60◦; (e) Ψ = 75◦; (f) Ψ = 90◦. Symbols represent different phases: ✷Dynamic aggregate, △Isotropic fluid, /circlecopyrtClustered\nfluid. The solid lines are guides for the eyes to separate regi ons of different phases.\na) b) c)\nFIG. 4: Examples of typical steady state configurations: (a)\nDynamical aggregate;(b) Isotropic fluid;(c) Clustered flui d.\nIn Fig. 5 we show the pair correlation function and\ntheir respective steady-state structures for B= 10,\nΨ = 15◦, and different rotation frequencies of the exter-\nnal magnetic field. In the absence of the external mag-\nnetic field, the rods tend to cluster together in a head-\nto-tail arrangement, forming a chain-like structure. As\ndiscussed previously, such an arrangement is less stable\nagainst fluctuations compared with the ribbon-like con-\nfiguration. For B= 10, Ψ = 15◦, the coupling of the\ndipole moments with the external field is already strong\nenough to break the head-to-tail arrangement, especially\nforlowfrequency( ω= 5), wheresomerodstendtofollow\nthe external rotating field. We find that ≈20% of the\nrods are synchronized with the external magnetic field.\nThose particles avoid the clustering and lead the system\nto the gas-like configuration shown in Fig. 5(a). For0 3 6 9 12r/σ0.02.04.06.08.010.012.0g(r)ω = 5\nω = 10\nω = 20\nω = 30\nb) c) d) e)a)\nFIG. 5: (a) Pair correlation function for B= 10 and Ψ = 15◦\nfor different values of ω. (b)-(e) Steady state configurations\nfor Ψ = 15◦and different rotation frequencies of the external\nmagnetic field: (b) ω= 5; (c) ω= 10; (d) ω= 20; (e) ω= 30.\nω= 10, the single-dipole time autocorrelation function,\nCµ(t), presents no sine-like oscillation, but a slow decay6\nas a function time, indicating a lack of synchronization\nbetween individual rods and the external field, favouring\nagain the formation of the head-to-tail arrangement. As\nωincreases,the time-decayof Cµ(t)isevenslower,theef-\nfective interaction between rods is mostly attractive and\nstrong enough to cluster them together. For ω≥10,\nthe steady states are composed of curled up structures,\ndriven by the external rotating field. Similar structures\nwere observed in a recent study of 3 Dsystems of flexi-\nble paramagnetic filaments in precessing fields [44]. For\nω≥20 theg(r), in addition to the head-to-tail peaks\n(r/σ≈3), we also notice some correlation at r/σ <3,\nwhich is a consequence of the curled up structures.\n0 2 4 6 8 10 12r/σ0.010.020.030.040.0g(r)ω = 5\nω = 10\nω = 20\nω = 30\nb) c) d) e)a)\nFIG. 6: (a) Pair correlation function for B= 10 and Ψ = 45◦\nfor differentvalues of ω. (b)-(e)Steadystates for Ψ = 45◦and\nfor different values of ω: (b)ω= 5; (c) ω= 10; (d) ω= 20;\n(e)ω= 30.\nForB= 10, Ψ = 45◦there is no synchronization be-\ntween the dipoles and the external field. The larger de-\nviation of the dipoles from the axial direction of the rods\nfavors the formation of ribbon-like structures, as indi-\ncated in the g(r) function presented in Fig. 6. Notice\nthat the most intense peaks of g(r) are located at sep-\narationr/σ <3. The resultant configurations depend\nstrongly on the rotation frequency of the external mag-\nnetic field, and we observe an interesting reentrant effect\nconcerning the formation of clusters. I.e., for sufficiently\nlow (ω= 5) and high ( ω= 30) rotation frequency of the\nexternalmagneticfield, cluster-likeconfigurationsareob-\nserved,whilealinearconfigurationisfoundforintermedi-\nate values of the rotationfrequency ( ω= 10and ω= 20).\nThe case B= 10, Ψ = 90◦[Fig. 7] corresponds to the\nstrongest rod-rod interaction [22], where the ribbon-like\narrangement is dominant, and the clusters are linear and\nvery stable against thermal fluctuations. As indicated in\nFig.7, the separation between peaks is r/σ≈1, whichconfirms the ribbon-like arrangement. There is no im-\nportant qualitative differences in the configurations as a\nfunction of ω.\n0 2 4 6 8 10 12r/σ0.010.020.030.040.050.0g(r)ω = 5\nω= 10\nω = 20\nω = 30\nb) c) d) e)a)\nFIG. 7: (a) Pair correlation function for B= 10 and Ψ = 90◦\nfor differentvalues of ω. (b)-(e)Steadystates for Ψ = 90◦and\nfor different values of ω: (b)ω= 5; (c) ω= 10; (d) ω= 20;\n(e)ω= 30.\nWe now compare some of the previous results with the\nones obtained when the intensity of the external mag-\nnetic field is twice larger, B= 20. In Fig. 8 we show the\npaircorrelationfunctionandtheirrespectivesteady-state\nstructures for Ψ = 15◦and Ψ = 90◦, and different rota-\ntion frequencies of the external magnetic field ( B= 20).\nFor Ψ = 15◦, the spatial correlation between rods is\nlarger for ω= 5 due to the formation of a large cluster,\nwhich is favoured by the effective attractive interaction\nbetween rods, due to the synchronization of individual\nrodswiththeexternalfield. Forhigherfrequencies,wedo\nnotfindimportantdifferencesconcerningthe rangeofthe\nspatial correlation, but the micro-structure of the config-\nurations depends on ω[Fig. 8(a)]. We observe an oppo-\nsiteω-dependence of the spatial correlation for Ψ = 90◦,\ni.e., the range of the spatial correlation increases dra-\nmatically for ω >5, with the rods ordered according to a\nribbon-like arrangement [Figs. 8(b),(h),(i),(j)]. Our re-\nsults of the mean square displacement (not shown in the\nmanuscript) for the cases Ψ = 15◦and Ψ = 90◦indicate\nthat the system is always in the liquid phase.\nTheresultspresentedinFig. 8canbe understoodfrom\nthe coupling between the rods and the external magnetic\nfield. Note that in the limit where the system is fully\nsynchronized (all rods in phase with the external mag-\nnetic field), the time-dependence of the orientation of the\ndipole moment of the rods obeys:\nˆµ(t) =cos(ωt)ˆx+sin(ωt)ˆy, (17)7\nso that Eq. (16) results in:\nCµ(t) =1\nN/angbracketleftBiggN/summationdisplay\ni=1ˆµi(t)·ˆµi(0)/angbracketrightBigg\n=ns\nNcos(ωt), (18)\nwherensis the number of synchronized rods , Nis the\ntotal number of rods, and ωis the rotation frequency of\nthe external magnetic field. Therefore, if Cµ(t) presents\na sinusoidal oscillatory behaviour, its amplitude, ns/N,\nrepresents the fraction of rods in phase with the external\nmagnetic field.\nFIG. 8: Pair correlation function for B= 20 for different\nω, for: (a) Ψ = 15◦(b) Ψ = 90◦. Steady states for ω= 5,\nω= 10,ω= 20 and ω= 30, for Ψ = 15◦(c)-(f), and for\nΨ = 90◦(g)-(j).\nFor Ψ = 15◦andω= 5, more than 80% of the rods\nare in phase with the external field [Fig. 9(a)]. As a con-\nsequence, the average interaction between rods is mostly\nattractive[seeEq. 13], leadingto the formationofa large\npolarizedcluster(non-zerototalmagneticmoment)anda\nlarger range of the spatial correlation function, as shown\nin Fig. 8(a). For ω= 10, less than 40% of the rods are\nin phase with the external field [Fig. 8(a)]. The results\nindicate that those few synchronized rods avoid the clus-\ntering and lead the system to a gas-like configuration.\nThe number of rods in phase with the external field de-\ncreases with increasing rotation frequency. For ω= 30\nsuch a number is <∼5% [Fig. 8(a)].\nFor Ψ = 90◦, we find a partial synchronization only in\nthe case ω= 5, where ∼70% of the rods are in phase\nwith the externalfield. Aconfigurationsimilarto the one\nof the case (Ψ = 15◦;ω= 5) is observed. For ω >10, no\nsynchronization is observed within the simulation time,\nand due to the perpendicular orientation of the dipoles\nwith respect to the axial axis, the stronger interactionbetween rods favors the formation of the ribbon-like con-\nfiguration, which presents a longer spatial order with re-\nspect to the previous ones.\n0510152025t-1.0-0.50.00.51.0Cµ(t)\n0510152025tΨ = 15°\nΨ = 45°\nΨ = 90°c)                     d)0 25 50 75 100tCµ(t)\nω = 5\nω = 10\nω = 20\nω = 30\n0 10 20 30 40t-1.0-0.50.00.51.0Cµ(t)a) b)\nFIG. 9: Dipole-dipole autocorrelation function for differe nt\ncases: (a) B= 20 and Ψ = 15◦;(b)B= 30 and Ψ = 90◦; (c)\nB= 30 and ω= 25; (d) B= 50 and ω= 10. The legend\nin (a) is the same presented in (b). The legend in (c) is the\nsame presented in (d).\nIn Fig. 9 we also illustrate the behaviour of Cµ(t) for\nother values of the parameters B,ωand Ψ. By con-\nsidering the synchronized cases [Figs. 9(a),(b) and (d)]\nand from Fig. 3, we notice a characteristic amplitude\nofCµ(t), typically associated to the dynamical aggregate\nphase. In Fig. 10 we show the amplitude of Cµ[see\nEq. 18] as a function of Bfor the dynamical aggregate\nphases considering the ω-interval in which such a phase\nis found. We notice that the dynamical aggregates are\nusually observed for ns/N>∼0.65.\nIn Fig. 9(a), the curves for ω≥10 are related to\nthe isotropic fluid phase [see Figs. 8 (d)-(f)], where\nns/N<∼0.65 or the autocorrelation function assumes\na damped decay with a non-monotonic behaviour with\nat least a minimum at early times - underdamped be-\nhaviour [Fig. 9(a),(b)]. The latter indicates that the\nreorientation resulting from the rod-rod interaction ac-\ntuates somehow as a restoring torque, suppressing the8\nrotation of the rods driven by the external field. As a\nresult, it is observed a damped behavior of Cµ. Also, the\nrotational friction originated from the rod-rod and rod-\nsolvent interactions becomes more important with the\ndecrease of the effect of the external magnetic field. As\na result, we observe a damped oscillation behaviour for\nCµ(t). These results are in agreement for dipole corre-\nlation functions using a rotational diffusion model for a\nlarge angle reorientation of the particles [45]. We also\nobserved the damping decay in Fig. 9(c). This picture\nchanges either when there is an additional increase of the\nrod-rod interaction (increase of Ψ), or a decrease of the\nextent of synchronization (increase of ωor decrease of\nB). The increase of relevance of the rod-rod interaction\nresults in a monotonic average decay of Cµ(t), as shown\ne.g. forω= 30 in Fig. 9(b). We observe that such a time\ndependence of Cµ(t) is related to the clustered phase. In\nFig. 11, we show an example where is possible to observe\nall phases discussed so far, with their respective related\ncorrelations functions. Notice that the pair correlation\nfunction (Fig. 11(b)) presents for ω= 5 a liquid-like be-\nhavior, and a gas-like profile for ω= 10, resultant of the\ndispersed phase, suggesting that we obtain different ag-\ngregation states just by tunning the rotation frequency\nof the external magnetic field.\n202530354045 50B0.60.70.80.91.0ns/NΨ = 15°\nΨ = 30°\nΨ = 45°\nΨ = 60°\nΨ = 75°\nΨ = 90°\nFIG. 10: Critical amplitude of oscillation for dynamical ag -\ngregate phases for different Ψ as a function of B.\nThe clustered fluid phase, typically observed for large\nω, presents a polymer-like shape [Fig. 4] and a slower\nrelaxation with increasing degree of polymerization Φ, as\nshown in Table I. Notice that, for the clustered phase, a\nhigherrelaxationtime isassociatedwith amorepolymer-\nized systems. The decay of the autocorrelation function\nCµ(t) may be fitted with the Kohlrausch-Williams-Watts\nstretched exponential function, typical of glassy systems\n[46]:\nCµ(t) =exp/bracketleftBigg\n−/parenleftbiggt\nτ/parenrightbiggβ/bracketrightBigg\n, (19)\nwhereτis the characteristic relaxation time and βis the0 2 4 6\nr/σ0.02.04.06.08.0g(r)ω = 5\nω = 10\nω = 20\nω = 30\nc) d) e) f)0 20 40 60 80 100t-1.0-0.50.00.51.0Cµ(t)\nω = 5\nω = 10\nω = 20\nω = 30a) b)\nFIG. 11: Characterization of Ψ = 60◦phases for B= 20 for\ndifferent ω. (a) Dipole-dipole autocorrelation function. (b)\nPair correlation function. (c)-(f) Steady states: (c) ω= 5;\n(d)ω= 10; (e) ω= 20; (f) ω= 30.\nTABLE I: The extent of polymerization Φ for B= 20 as a\nfunction of ωfor different Ψ. High values at low ωrepresent\nthe dynamical aggregate phase\nω Ψ = 60° Ψ = 75° Ψ = 90°\n  5\n10\n15\n20\n25\n300.909\n0.245\n0.351\n0.968\n0.995\n0.9980.931\n0.492\n0.934\n0.997\n0.999\n0.9990.935\n0.851\n0.971\n0.997\n0.999\n0.999(Φ) Polymerization\nstretched exponential.\nIn the Fig. 12, we present Cµ(t) for some clustered\nphases and their respective stretched exponential fits. As\ndiscussed previously, the increase of importance of the\nrod-rod interaction results in an additional friction to ro-\ntation, and a slowing-down of the relaxation. Therefore,\nthe characteristic relaxation time increases for stronger\nrod-rod interaction, and this is due either to the increase\nof the net interaction itself by increasing Ψ (Fig. 12(a))\nor, for a fixed B, to the decrease of the effect of the ex-\nternal magnetic field by increasing ω(Fig. 12(b)). For\nβ= 1, the exponential decay is characteristic of Debye-\nlike relaxation, however, the behavior of the autocorre-\nlation deviates from the Debye-decay, since we obtained,\nfor all cases , β <1. Similar non-Debye behaviour was\nalso obtained for a 3D systems of magnetic spherocylin-\nders [47].\nIV. CONCLUSIONS\nWe investigated a self-organization of a two-\ndimensional system consisting of magnetic peapod rods9\nFIG. 12: Dipole autocorrelation function for some clustere d phases for B= 20: (a) for different Ψ and ω= 30; (b) for Ψ = 60◦\nand for different ω. The dashed lines represent the stretched exponential fit fo llowed by their respective fit parameters at the\nlegend boxes.\nin the presence of rotating fields using Langevin dynam-\nics simulation.This model was motivated by experimen-\ntal [24] and theoretical [15, 16, 22] studies. Each rod\nwas composed of 3 soft beads having a central pointlike\ndipole whose orientation is misaligned with respect the\naxial axis of the rods. The application of rotating fields\nin dipolar systems is already known to produce untypi-\ncal structures, which are a consequence of the resultant\ntime-averaged dipolar interaction [32, 33]. We investi-\ngated the configurations as a function of the strength\nandrotationfrequencyoftheexternalmagneticfield, and\nthe misalignment of the dipoles with respect to the ax-\nial direction of the rods. Particular attention was also\naddressed to the synchronization resultant of the com-\npetition between the rod-rod and rod-external magnetic\nfield interactions.\nWe found three different typical steady-state configu-\nrations, consequent of the three different synchronization\nregimes observed. In the high synchronization regime,\nobserved for high strength and low rotation frequency\nof the external field, we observed the Dynamical aggre-\ngates, which are mainly a consequence of the attractive\ntime-averaged dipolar interaction. In the intermediate\nregime, when the rod-rod and rod-external field inter-\nactions are equivalent, the resultant competition pro-\nduced a dispersed and spatially isotropic fluid. Here, thesynchronization with the external magnetic field is not\nstrong enough to produce a dynamical aggregate, but\nit is sufficient to avoid the rods to form clusters. The\nthird phase is a consequence of the further decrease in\nsynchronization. In this case, the rod-rod interaction is\nmore relevant; as a result, the rods cluster to each other\noriginating the Clustered phase. We also characterized\nthe system by studying the time-dependence of the sin-\ngle dipolar autocorrelation function. We showed that a\nconstant and high amplitude of the autocorrelation oscil-\nlation was associated with the high synchronized phase\n(Dynamical aggregates). A sufficient decrease of that\namplitude or a damping behavior of the autocorrelation\nfunction was related to the Isotropic fluid phase. The\nClustered phase was characterized by a monotonic de-\ncay of the autocorrelation function, with a characteristic\nrelaxation time, which increases with increasing of the\npolymerization of the system. Such a monotonic decay\ndeviated from the Debye relaxation ones.\nAcknowledgments\nThis work was supported by the Brazilian agencies\nFUNCAP, CAPES and CNPq.\n[1] C. S. Dias, N. A. M. Araujo, M. M. Telo da Gama, Phys-\nical Review E 87, 3 (2013).\n[2] D. M. Rutkowski, O. D. Velev, S. H. Klapp, and C. K.\nHall,Soft matter ,13, 17 (2017).\n[3] T. Heinemann, M. Antlanger, M. Mazars, S. H. Klapp,\nand G. Kahl, The Journal of chemical physics ,144, 7\n(2016).\n[4] Nisisako, T., Current Opinion in Colloid & Interface Sci-\nence25, 1 (2016).[5] Daddi-Moussa-Ider, Abdallah and Menzel, Andreas M.,\nPhysical Review Fluids 3, 9 (2018).\n[6] T. Debnath, Y. Li, P. K. Ghosh, and F. Marchesoni,\nPhysical Review E ,97, 4 (2018).\n[7] M. Adrian, and L. E. Helseth, Physical Review E ,77, 2\n(2008).\n[8] Mehdi Molaei, Ehsan Atefi, and John C. Crocker, Phys-\nical review letters 120, 11 (2018).\n[9] Savka I. Stoeva,Fengwei Huo,Jae-Seung Lee, and, Chad10\nA. Mirkin, Journal of the American Chemical Society\n127, 44 (2005).\n[10] Philip D. Howes, Rona Chandrawati, and Molly M.\nStevens, Science346, 6205 (2014).\n[11] Fabrice Cousin, Valrie Cabuil, and Pierre Levitz, Lang-\nmuir18, 5 (2002).\n[12] S. Gubin, Magnetic Nanoparticles , Wiley (2009).\n[13] B. J. Lemaire, P. Davidson, J. Ferr, J. P. Jamet, P. Pa-\nnine, I. Dozov and J. P. Jolivet, Physical review letters\n88, 12 (2002).\n[14] J. Philip, P. Shima and B. Raj, Applied Physics Letters\n91, 203108 (2007).\n[15] Jorge L.C. Domingos, F.M. Peeters, W.P.Ferreira, Phys-\nical Review E 96, 1 (2017).\n[16] C. E. Alvarez and S. H. Klapp, Soft Matter 8, 7480\n(2012).\n[17] A.I. Abrikosov, S. Sacanna, A.P. Philipse, P. Linse, Soft\nMatter9, 37 (2013).\n[18] A.B. Yener, S.H. Klapp, Soft Matter 12, 7 (2016).\n[19] F. Martinez-Pedrero, A. Cebers, and P. Tierno, Phys.\nRev. Applied 6, 034002 (2016).\n[20] Singh H, Laibinis PE, Hatton TA., Langmuir 21, 24\n(2005).\n[21] S.H. Lee, and C.M. Liddell, Small5, 17 (2009).\n[22] Jorge L. C. Domingos, F. M. Peeters, W. P. Ferreira,\nPloS One 13, e0195552 (2018).\n[23] J.Carrey, andN.Hallali, Phys. Rev. B 94, 184420 (2016).\n[24] R. Birringer, H. Wolf, C. Lang, A. Tsch¨ ope and\nA.Michels, Zeitschrift f¨ ur Physikalische Chemie 222, 229\n(2008).\n[25] N. Coq, S. Ngo, O. du Roure, M. Fermigier and D. Bar-\ntolo,Physical Review E 82, 041503 (2010).\n[26] P. Dhar, C. D. Swayne, T. M. Fischer, T. Kline and S.\nAyusman, Nano Lett. ,7, 1010 (2007).\n[27] W. A. Shelton, K. D. Bonin and T. G. Walker, Physical\nReview E 71, 036204 (2005).\n[28] J. E. Martin, R. A. Anderson and C. P. Tigges, J. Chem.\nPhys.108, 7887 (1998).\n[29] J. E. Martin, R. A. Anderson and C. P. Tigges, J. Chem.Phys., ,110, 4854 (1999).\n[30] N. Elsner, C. P.Royall, B. VincentandD. R.E. Snoswell,\nJ. Chem. Phys. 130, 154901 (2009).\n[31] T. C. Halsey, R. A. Anderson and J. E. Martin, Int. J.\nMod. Phys. B 10, 3019 (1996).\n[32] S. Jaeger, H. Schmidle, and S. H. L. Klapp, Physical\nReview E 86, 1 (2012).\n[33] S. H. L. Klapp, Current Opinion in Colloid & Interface\nScience21,76 (2016).\n[34] F. Smallenburg and M. Dijkstra, J. Chem. Phys. 132,\n204508 (2010).\n[35] K. May, A. Eremin, R. Stannarius, S.D. Peroukidis, S.H.\nKlapp, S. Klein, Langmuir 20,32 (2016).\n[36] E. De Miguel, L.F. Rull, M.K. Chalam, K.E.Gubbins,\nMolecular Physics 74,2 (1991).\n[37] K. May, R. Stannarius, S. Klein, A. Eremin, Langmuir\n30, 24 (2014).\n[38] Tirado and J. G. de La Torre, The Journal of Chemical\nPhysics71, 2581 (1979).\n[39] Tirado and J. G. de La Torre, The Journal of Chemical\nPhysics73, 1986 (1980).\n[40] J.K.G. Dhont, W.J. Briels, Rod-Like Brownian Particles\nin Shear Flow Wiley-VCH Verlag GmbH & Co. KGaA\n(2007)\n[41] M.P. Allen, D.J. Tildesley, Computer simulation of liq -\nuids. Oxford university press (1989).\n[42] D. C. Rapaport, The Art of Molecular Dynamics Simu-\nlation, Cambridge University press (2004).\n[43] S. S. Das, A. P. Andrews and S. Greer, The Journal of\nChemical Physics 102, 2951 (1995).\n[44] J. M. Dempster, and P. V´ azquez-Montejo, and M. O. de\nla Cruz, Phys. Rev. E 95, 052606 (2017).\n[45] J. Hansen, D. Levesque and J. Zinn-Justin, Liquids,\nFreezing and Glass Transition , Les Houches, Session LI\n(1991).\n[46] S. H. Chung and J. R. Stevens American Journal of\nPhysics95, 11 (1991).\n[47] C. E. Alvarez and S. H. Klapp, Soft Matter 9, 36 (2013)."    },    {        "title": "1803.01225v1.Dynamics_in_a_one_dimensional_ferrogel_model__relaxation__pairing__shock_wave_propagation.pdf",        "content": "Dynamics in a one-dimensional ferrogel model: relaxation, pairing, shock-wave\npropagation\nSegun Goh,1,\u0003Andreas M. Menzel,1,yand Hartmut L owen1,z\n1Institut f ur Theoretische Physik II: Weiche Materie,\nHeinrich-Heine-Universit at D usseldorf, D-40225 D usseldorf, Germany\n(Dated: March 6, 2018)\nFerrogels are smart soft materials, consisting of a polymeric network and embedded magnetic\nparticles. Novel phenomena, such as the variation of the overall mechanical properties by external\nmagnetic \felds, emerge consequently. However, the dynamic behavior of ferrogels remains largely\nunveiled. In this paper, we consider a one-dimensional chain consisting of magnetic dipoles and elas-\ntic springs between them as a simple model for ferrogels. The model is evaluated by corresponding\nsimulations. To probe the dynamics theoretically, we investigate a continuum limit of the energy\ngoverning the system and the corresponding equation of motion. We provide general classi\fcation\nscenarios for the dynamics, elucidating the touching/detachment dynamics of the magnetic particles\nalong the chain. In particular, it is veri\fed in certain cases that the long-time relaxation corresponds\nto solutions of shock-wave propagation, while formations of particle pairs underlie the initial stage\nof the dynamics. We expect that these results will provide insight into the understanding of the\ndynamics of more realistic models with randomness in parameters and time-dependent magnetic\n\felds.\nI. INTRODUCTION\nFerrogels, magnetic elastomers, or magnetic gels are\nsmart composite materials [1] the elastic properties of\nwhich are tunable by magnetic \felds from outside [2{5].\nNovel characteristics originate from the composite na-\nture of ferrogels which gives rise to a magneto-mechanical\ncoupling between the embedded magnetic particles and\nthe gel network [2, 6, 7]. Such a magneto-mechanical\ncoupling can be achieved by constraining the motion of\nmagnetic particles inside pockets of the matrix [6, 8, 9]\nor by directly anchoring the polymers to the surfaces of\nmagnetic particles [2, 6, 10, 11]. Utilizing this charac-\nteristic, a variety of applications such as sensors [12, 13],\nactuators [14], tunable devices [15, 16], medical sca\u000bolds\nfor tissue engineering [17, 18], and biocomposites for con-\ntrolled release [19] have been suggested.\nMuch e\u000bort has also been devoted to the theoretical\nunderstanding of the ferrogels. Several routes are sug-\ngested and investigated to model these non-trivial ma-\nterials. At the microscopic scale, bead-spring models to\nresolve the individual polymer chains connecting the em-\nbedded magnetic particles have been studied by means\nof computer simulations [20{23]. On the macroscale,\nhydrodynamic theories for ferrogels have been devel-\noped [24, 25]. Moreover, mesoscopic dipole-spring mod-\nels [26{31] represent the polymeric matrix by spring-like\ninteractions, while the magnetic particles are resolved\nand interact with each other via magnetic dipole-dipole\ninteractions. Alternatively, the elastic contributions can\nbe described by matrix-mediated interactions [32{36] in\nterms of continuum elasticity theory.\n\u0003segun.goh@hhu.de\nymenzel@thphy.uni-duesseldorf.de\nzhlowen@thphy.uni-duesseldorf.deRecently, more attention begins to be paid to dynamic\nproperties. Analogously to the dynamics of magnetic col-\nloidal systems [37{41], new con\fgurations or generally\nnovel phenomena observable only in the dynamics are\nexpected to emerge for ferrogels. As an important exam-\nple, the dynamic moduli/responses of ferrogels have been\nstudied extensively [42{48]. To fully describe the dynam-\nics far from equilibrium and the consequent transitions\nbetween qualitatively di\u000berent con\fgurations, it is nec-\nessary to address the approach and separation dynamics\nof magnetic particles under changing mutual magnetic\nattraction and repulsion. Indeed, the changes in particle\ndistances are well known to a\u000bect the material properties\nof ferrogels. One of the most widely studied phenomena\nin this regard is the formation of chain-like aggregates\nwhich can cause drastic changes in the elastic properties\nof systems [49{56]. It has been predicted theoretically\nthat the detachment of magnetic particles in chain-like\naggregates can give rise to the pronouncedly nonlinear,\nso-called superelastic stress-strain behavior [57, 58]. The\nformation of chain-like aggregates has been studied for\nvarious dipolar systems, for instance in combination with\nthe Van der Waals interaction [59, 60].\nIn a theoretical perspective, the formation of compact\nchains under magnetic attraction can be viewed as a\nhardening transition [26], if the particles can come into\nclose contact. Steep changes in elastic properties can be\nattributed to the hardening due to virtual touching. It is\nworthwhile to note that the hardening transition implies\na double-well structure in the energy. In other words,\nthere exist two di\u000berent equilibrium con\fgurations, one\nof which corresponds to the contracted and the other\nto the elongated systems. Such a con\fgurational bista-\nbility, involving the rearrangement of the magnetic par-\nticles and the deformation of the gel network, has been\nwidely discussed with di\u000berent settings [26, 32, 33, 50, 61]\nand therefore seems to be a relatively universal feature.arXiv:1803.01225v1  [cond-mat.soft]  3 Mar 20182\nMoreover, one may expect that there exists a regime in\nbetween the equilibrium points where con\fgurations be-\ncome unstable. In short, a type of dynamic mechanism\nformally similar to spinodal decomposition may play a\nsigni\fcant role if attention is extended to dynamics [62].\nSpinodal decomposition occurs in various systems, rep-\nresentatively to the phase separation of binary systems\ndescribed with the aid of the Cahn-Hilliard equation [63{\n66]. Recently spinodal lines were identi\fed for systems\nof active Brownian particles [67{72]. The wetting phe-\nnomenon [73, 74] also provides an example with a special\nboundary condition due to the presence of a reservoir.\nHowever, there are technical di\u000eculties related to the\nregularization of the problem [75, 76] which corresponds\nto the unique characteristics of each system under con-\nsideration. In the case of the Cahn-Hilliard equation, for\ninstance, the regularization term contains the free energy\ncost due to the interface and therefore governs the coars-\nening dynamics in the long-time scale. Also see, e.g.,\nRefs. 70 and 72 for further examples.\nIn this paper, we study the dynamics of a one-\ndimensional ferrogel model far from equilibrium. We\naddress questions on the touching and detachment dy-\nnamics of magnetic particles. The dipole-spring model\nis adopted as such a mesoscopic model deals with the\ncon\fgurations of magnetic particles in a direct manner:\nthe magnetic particles are explicitly resolved and the dis-\ntances between them are simply related to the lengths of\nsprings attached between them. Then a quasi-continuum\nequation governing the behavior of the system is derived\nbased on a term equivalent to the particle density. Our\nmain results show that the large-scale chain formation\ndynamics in the long-time regime are governed by shock-\nwave solutions in the continuum description. With the\naid of singular perturbation theory [77, 78] in connection\nwith the Stephan problem [66, 79], we can successfully\nquantify the propagation speed. The origin of our reg-\nularization and the relation to general phase separation\ndynamics are also discussed.\nThis paper is organized as follows. In Sec. II, a one-\ndimensional version of the dipole-spring model is intro-\nduced. Then we derive a quasi-continuum description\nof the system and discuss its theoretical properties in\nSec. III. Section IV is devoted to illustrate the various\nobserved dynamical scenarios and to develop a \\behav-\nioral diagram\" for the di\u000berent types of dynamics, which\nrepresents the main result of our study. Lastly, a sum-\nmary and an outlook are given in Sec. V.\nII. THE MODEL\nOur one-dimensional dipole-spring model for a ferrogel\nsystem consists of magnetic particles and springs [26, 45,\n80]:N+ 1 magnetic particles are connected by Nhar-\nmonic springs, forming a linear straight chain (see Fig. 1\nfor a graphical illustration). The number of particles is\n\fnite so that the chain has de\fnite boundaries at both\n  \n... ...\n... ...\nFIG. 1. A graphical illustration of our model. Blue and\ngreen spheres represent the magnetic particles inside and at\nthe boundaries of the system with the virtual diameter rc, re-\nspectively, while red arrows correspond to the magnetic dipole\nmoment~ mof the particles. Springs of spring constant kand\nundeformed length aare depicted as well. In addition, the\nconvention for quantifying the locations of the particles is\nalso illustrated in this \fgure.\nends. In this way, we can perturb the system by apply-\ning forces at the boundaries as in the laboratory. The\nlocation of the ith particle is then represented by rifor\ni= 1;:::;N +1 and the length of the ith spring between\ntheith and the ( i+ 1)th particles by ri;i+1\u0011ri+1\u0000ri\nfori= 1;:::;N . The magnetic dipole moment ~ miis\nassigned to the ith particle, which can be any vector in\nthe three-dimensional space. Below, after switching to a\nnon-zero value, it will be considered as constant in time.\nFollowing previous studies [44, 45, 47], we consider the\noverdamped dynamics of the dipole-spring model as a\nfunction of time t, governed by equations of motion of\nthe form\n\u0000dri\ndt=\u0000@Etot\n@ri: (1)\nObviously, the form of the total energy Etotdetermines\nthe dynamical properties of the magnetic chain. In this\nstudy, we adopt a simple version of the dipole-spring\nmodel in which Etotis given by the sum of elastic inter-\nactions, magnetic dipole-dipole interactions, and steric\nrepulsion [45, 47, 80]. First, the elastic energy of the\nharmonic springs takes the form of\nEel(fri;i+1(t)g) =k\n2NX\ni=1(ri;i+1(t)\u0000a)2; (2)\nwherekis the spring constant and athe length of the\nsprings in the undeformed state. Second, the magnetic\ndipole-dipole interaction energy is given as\nEm(frij(t)g) =\u00160\n4\u0019X\nj>i\"\n(~ mi\u0001~ mj)\nj~ rij(t)j3\n\u00003(~ mi\u0001~ rij(t))(~ mj\u0001~ rij(t))\nj~ rij(t)j5#\n;(3)\nwhere\u00160is the vacuum permeability, rij=rj\u0000ri, and\n~ rij\u0011rij^xwith the unit vector ^ xalong the chain axis.\nFor simplicity, we limit ourselves to the case in which the3\nmagnetic moments are identical across the whole system,\n~ mi\u0011m(cos\u0012^x+sin\u0012cos\u001e^y+sin\u0012sin\u001e^z). Then, intro-\nducingb\u00111\u00003 cos2\u0012, one can rewrite the magnetic\nenergy in a simpler form\nEm(frij(t)g) =\u00160bm2\n4\u0019X\nj>i1\njrij(t)j3: (4)\nFinally, the steric repulsion preventing a collapse of the\nsystem under strong magnetic \felds, reads\nEst(fri;i+1(t)g) =NX\ni=1UWCA (ri;i+1(t)); (5)\nwhereUWCA is a modi\fed Weeks-Chandler-Andersen\n(WCA) type potential in the form [45, 81]\nUWCA (r) = \u0002(rc\u0000r)\u000fs\u0014\u0010r\n\u001bs\u0011\u000012\n\u0000\u0010r\n\u001bs\u0011\u00006\n\u0000\u0010rc\n\u001bs\u0011\u000012\n+\u0010rc\n\u001bs\u0011\u00006\n\u0000cs(r\u0000rc)2\n2\u0015\n;(6)\nwith the Heaviside step function \u0002 and a cuto\u000b distance\nrc. Here,\u001bsandcsare chosen such that U0\nWCA (rc) = 0\nandU00\nWCA (rc) = 0 [45]. The parameter \u000fscharacter-\nizes the strength of the steric repulsion. Now, one can\n\fnd a set of dynamic equations, substituting the above\nde\fnitions directly into Eq. (1). Those equations for par-\nticles inside as well as at the boundaries of the system\nare described in detail in Appendix A.\nIII. FORMULATION OF A\nQUASI-CONTINUUM THEORY\nTo be able to develop a continuum description of the\nsystem, we as a major simpli\fcation cut the long-range\nmagnetic interaction beyond the nearest-neighbor inter-\naction. We have con\frmed from particle-resolved simula-\ntions that the overall dynamics with the nearest-neighbor\nand long-range magnetic interactions are qualitatively\nequivalent to each other, as far as uniform con\fgurations\nare adopted as initial conditions (see Sec. III C).\nA. Continuum equation\nNow the energy is only a function of the distances be-\ntween adjacent particles as follows:\nEtot(fri;i+1(t)g) =NX\ni=1e(ri;i+1(t)); (7)\nwhereeis the pairwise energy given by\ne(r) =k\n2(r\u0000a)2+\u00160bm2\n4\u00191\nr3+UWCA (r): (8)The direct analysis of this pairwise energy landscape will\nhelp in understanding the equilibrium states as well as\nthe dynamics of the systems and therefore constitutes\none of the essential parts of the continuum theory. We\naddress this issue in detail in Sec. III D.\nWe then seek a continuum description of the system,\nintroducing a continuous variable x, a positional \feld\nr(x;t), and its derivatives with respect to x, i.e.,rx,\nrxx, and so on. Following a standard transformation rule\nN+1X\ni=1!ZN\n0dx,ri+1\u0000ri!@r=@x , and@=@ri!\u000e=\u000er\n(see, e.g., Ref. 82), we directly obtain from Eq. (1) a fully\ncontinuum equation\n\u0000rt(x;t) =e(2)(rx(x;t))rxx(x;t); (9)\nwheree(i)(r)\u0011\u0000@\n@r\u0001ie(r) for a general natural number\nj. If the explicit form of the energy is inserted, the con-\ntinuum equation reads\n\u0000rt=krxx+3\u00160bm2\n\u0019rxx\n(rx)5+\u000fsrxx\u0002(rc\u0000rx)\n\u0002\u0014156\n(\u001bs)2\u0010rx\n\u001bs\u0011\u000014\n\u000042\n(\u001bs)2\u0010rx\n\u001bs\u0011\u00008\n\u0000cs\u0015\n:(10)\nTwo important characteristics of the continuum equation\nare summarized as follows: First Eq. (9) takes the form\nof a di\u000busion equation. However, the di\u000busion coe\u000ecient\ne(2)(rx(x;t)) may have a negative value depending on the\nvalue ofm. Second, the variable rx, which determines the\nsign ofe(2), is closely connected to the particle density via\nthe relation \u001a(x;t) = 1=rx(x;t). Therefore, the particle\ndensity controls the dynamics.\nB. Regularization\nWe note that, if there exists a range with e(2)(rx)<0,\nthe continuum equation becomes a type of the forward-\nbackward heat equation which does not necessarily have\na unique solution [75]. It is then mandatory to include\nan additional term as a regularization, which should be\nspeci\fc for each given physical problem [76]. In our case,\nthe regularization stems from the discrete nature of the\nsystem, similarly to the lattice regularization in critical\nphenomena.\nIndeed, the transformation rule ri+1\u0000ri!@r=@x in-\nvolves a truncation of higher order terms rxx,rxxx, and\nso on, neglecting corrections from the discreteness of the\nsystem. Here, we explicitly take such corrections into\naccount. We consider the di\u000berences \u0001 ir\u0011ri+1\u0000riin-\nstead of the di\u000berential @r=@x and utilize the functional-\nderivative technique for discrete variables. As expected,\nthis approach leads to the equations of motion for i=\n2;:::;N , described in Appendix A, which formally read\n\u0000dri\ndt= \u0001i\u00001e(1)(ri;i+1(t)): (11)4\nNow we probe a continuum description via a transfor-\nmation from the discrete variable i= 1;2;:::;N + 1 to a\ncontinuous variable xde\fned in a domain 0 <x<x max.\nWe choose the midpoint rule to weight equivalently the\nforces from the right and left sides of the particle under\nconsideration. In other words, a point iin the discrete\ndescription corresponds to a range [ x\u0000\u0001x=2;x+ \u0001x=2]\nin the continuum description, where \u0001 xcontrols the dis-\ncreteness of the system. If an asymmetric rule is consid-\nered, particles may prefer a motion in a certain direction.\nThen, with a transformation of the form x\u0011(i\u00001\n2)\u0001x,\nthe domain on which the newly introduced continuous\nvariablexis de\fned is determined as 0 \u0014x\u0014xmaxwith\nxmax\u0011(N+ 1)\u0001x. Here, we take xmax= 1 by further\nsetting \u0001x\u00111=(N+ 1). In this way, a continuum limit\nis achieved via \u0001 x!0 orN!1 . In contrast to that,\nrxxand higher order derivatives were neglected in the\ncase of the previous transformation rule to the domain0\u0014x\u0014Nwith \u0001x= 1 in Sec. III A, which was used to\nderive Eq. (9).\nWe set up a transformation rule in the form\nri(t)!r(x;t)=(\u0001x), relating the length scale of\nsprings to the corresponding continuous variable rxas\nri;i+1(t) =ri+1(t)\u0000ri(t)!r(x+\u0001x;t)\u0000r(x;t)\n\u0001x=rx(x;t) +\n\u0001x\n2rxx(x;t) +\u0001\u0001\u0001. Then we obtain a continuum descrip-\ntion from the discrete form in Eq. (9) as follows:\n\u0000rt= (\u0001x)\u001a\ne(1)\u0012r(x+ \u0001x;t)\u0000r(x;t)\n\u0001x\u0013\n\u0000e(1)\u0012r(x;t)\u0000r(x\u0000\u0001x;t)\n\u0001x\u0013\u001b\n:(12)\nRescalingx!x=(\u0001x) from the domain 0 \u0014x\u00141 to\n0\u0014x\u0014N+ 1, one indeed recovers to leading order\nthe continuum description of the dynamics described by\nEq. (9). This can be easily checked by expanding Eq. (12)\nin terms of \u0001 xas\n\u0000rt=\u0000(\u0001x)1X\ni=1(\u00001)i\ni!\u001ar(x+ \u0001x;t)\u00002r(x;t) +r(x\u0000\u0001x;t)\n\u0001x\u001bi\ne(i+1)\u0012r(x+ \u0001x;t)\u0000r(x;t)\n\u0001x\u0013\n=\u0000(\u0001x)1X\ni=1(\u00001)i\ni!8\n<\n:21X\nj=1(\u0001x)2j\u00001\n(2j)!r(2j)(x;t)9\n=\n;i8\n<\n:1X\nk=0e(i+1+k)(rx(x;t))\nk! 1X\nl=2(\u0001x)(l\u00001)\nl!r(l)(x;t)!k9\n=\n;;(13)\nwherer(i)\u0011\u0000@\n@x\u0001ir. In this way, we maintain as-\npects of the discrete nature of the system in a quasi-\ncontinuum description by letting \u0001 xbecome small but\n\fnite. Accordingly, we obtain regularization terms to\nEq. (9) from the second- and even higher-order contribu-\ntions in Eq. (13).\nNow one could seek for a precise solution, including\nall terms in Eq. (13). Instead, one may truncate them\nat a certain order, searching for approximate solutions.\nAt this point, we encounter the mathematical issue that\ndi\u000berent regularization forms may di\u000berently alter the\ndynamics of the forward-backward heat equation [76, 83].\nRather than rigorously investigating this issue, we here\ntake a pragmatic way using the next-order correction as a\nfeasible form of regularization. This leads to a regularized\nequation\n\u0000rt\u0019(\u0001x)2e(2)(rx)rxx+ (\u0001x)4\u00141\n12e(2)(rx)rxxxx\n+1\n6e(3)(rx)rxxrxxx+1\n24e(4)(rx) (rxx)3\u0015\n: (14)\nIn the end, a certain type of regularization is necessary\nto evaluate the equations. Our approach has the strong\nbene\ft of being fully systematic.C. Initial and boundary conditions\nFor simplicity, we only consider uniform initial con-\n\fgurations, i.e., rx(x;t= 0) =vinitwith constant vinit.\nBoundary conditions in the quasi-continuum theory are\ncarefully determined from the model as follows. First, it\nis clear that the quasi-continuum equation, e.g., Eq. (13),\napplies to the interior particles. For the boundary par-\nticles, additional rules are necessary. Speci\fcally, the\nboundary particles ( i= 1;N+ 1) at the left/right ends\nare governed by\n\u0000dr1\ndt=@e(r1;2(t))\n@r2;\u0000drN+1\ndt=\u0000@e(rN;N+1(t))\n@rN+1;\n(15)\nwhile we have\n\u0000dri\ndt=\u0000@e(ri\u00001;i(t))\n@ri\u0000@e(ri;i+1(t))\n@ri; (16)\nfor the particles inside the chain. To \fll this gap and\nmake the quasi-continuum equations of motion applica-\nble to the boundary particles as well, we introduce hypo-\nthetical particles i= 0;N+ 2, following the procedure in\nRef. 82. Indeed, Eq. (15) takes the same form as Eq. (16),\nif the location of the hypothetical particles, r0andrN+2,5\nare given by the positions satisfying\n@e(r0;1(t))\n@r1= 0;@e(rN+1;N+2(t))\n@rN+1= 0; (17)\nso that the forces from the hypothetical to the boundary\nparticles are zero. In the continuum limit, the above\nrelations to lowest order take the form\ne(1)(rx)\f\f\f\nx=\u000e\n= 0; (18)\nwhere\u000e\n =f0;1g.\nIn practice, the dynamics together with the initial\nand boundary conditions described above could be in-\nterpreted in two di\u000berent ways as follows: First, one\nmay imagine an in\fnitely long chain at an unstable \fxed\npoint. In this case, the dynamics are initiated by cut-\nting o\u000b the outer parts of the chain at the boundaries.\nSecondly, a stable \fnite chain with de\fnite boundaries\nmay be considered from the beginning. With m= 0, for\ninstance, we attain a homogeneous chain as the equilib-\nrium con\fguration, in which the distances between ad-\njacent particles are equivalent to the equilibrium spring\nlengthsa. The dynamics of the system is then initiated\nby turning on an external magnetic \feld. This accords\nwith a quenching procedure. In both cases, the inte-\nrior particles are still in the state of the unstable \fxed\npoint directly after the initiation procedures, because the\nforces from the left and the right particles are well bal-\nanced due to the initial homogeneity. This is true as\nlong as only nearest-neighbor magnetic interactions are\ntaken into account. The long-range magnetic interactions\nslightly a\u000bect this picture. However, also in our test sim-\nulations including long-range magnetic interactions, we\nhave not observed qualitative di\u000berences. As far as the\nuniform initial con\fgurations are considered, it is always\nthe boundaries from which the dynamics are initiated:\nfor the particles at the left/right boundaries, forces are\nonly acted from the particle on the right/left side at the\nmoment of cutting or quenching.\nD. Pairwise energy landscape\nMathematically, the pairwise energy eplays a simi-\nlar role as the free energy does in thermodynamics as\nthermal \ructuations of the particles [80] are neglected in\nthe present study. Above all, (mechanical) equilibrium\ncon\fgurations correspond to the minimum points in the\nlandscape of the pairwise energy, which are determined\nto lowest order from the condition e(1)(rx) = 0 in the\ncontinuum limit. We note that uniform equilibrium con-\n\fgurations automatically satisfy the static equilibrium\nconditionrt= 0 of our regularized equation, Eq. (14).\nIn this study, we are considering a double-well potential,\nwith at least one and at most two locally stable \fxed\n(equilibrium) points: one corresponds to the con\fgura-\ntion in which the particles touch each other (high den-\nsity) while the particles stay away from each other (low\n(a) (b) (c)\n000\n00\n0\nI I I I II III IV\nvinite(2)\ne(1)\neFIG. 2. Schematic illustration of the qualitatively di\u000berent\nlandscapes of the pairwise energies, leading to qualitatively\ndi\u000berent initial and long-time dynamical behaviors. While\nregions ofe(2)(vinit)<0 in the \frst row represent spinodal-like\nintervals, points at e(1)(vinit) = 0 in the second row indicate\nlocal extrema in the energy density landscapes (see ein the\nthird row). Based on the number of minimum points and\nthe existence of the spinodal-like interval, the landscapes are\nclassi\fed into three qualitative categories, depicted in (a), (b),\nand (c). Further analyzing the sign of e(2)and the location of\nthe minimum points, we classify the behavior according to the\nvalue ofvinitinto di\u000berent regimes I, II, III, and IV shaded\nby di\u000berent colors. See the text for the dynamic scenario\ncorresponding to each regime.\ndensity) in the other con\fguration. Tuning the magnetic\nmomentm, one can modulate the number of stable equi-\nlibrium points [26]. Furthermore, the di\u000busion coe\u000ecient\ne(2)(rx), the sign of which plays a signi\fcant role as al-\nready discussed in Sec. III B, is also modulated by the\nvalues ofm. Together with the trivial one for the initial\ncondition, we take into account two independent control\nparameters, namely, the magnetic moment mand the\ninitial uniform distance between adjacent particle pairs\nvinit. Keeping these in mind, we classify the landscapes\nof the pairwise energy into three categories as shown in\nFig. 2.\nFirst, we con\frm that e(2)(vinit)>0 for all the vinitval-\nues, if the magnetic moment mis very small [Fig. 2(a)].\nIn this case, there exists only one minimum in the pair-\nwise energy landscape and the whole range of vinit(col-\nored in yellow) belongs to the basin of attraction of the\nminimum point. From now on, the term Scenario I is\nused to indicate relaxation dynamics to the stable equi-\nlibrium corresponding to this case.\nIf the magnetic moment is very strong [Fig. 2(c)], once\nagain there is only one stable \fxed point which corre-\nsponds to a hardened touching con\fguration of the par-6\nticles [26]. In this case, however, there exists a range\nwithe(2)(vinit)<0 (shaded in green) analogous to the\nspinodal interval, which divides the range of vinitwith\ne(2)(vinit)>0 into two regions: a high-density region (in\nyellow) forming a basin of the only minimum point and a\nlow-density one (in cyan) separated from the \fxed point.\nAmong these three regions, the dynamics around the\nequilibrium (yellow) is equivalent to Scenario I, while the\ndynamics starting from the spinodal-like interval (green)\nand the low-density regime (cyan) are qualitatively dif-\nferent from Scenario I and, respectively, referred to as\nScenario III and IV in this paper.\nIf we consider magnetic moments lower than for the\nstrong-mregime, bistable landscapes appear [Fig. 2(b)].\nOnce again, separated regions with positive di\u000busion co-\ne\u000ecients (in yellow as before) correspond to Scenario\nI. In contrast to that, the dynamics starting from the\nspinodal-like interval in between (shaded in magenta) ex-\nhibits a new behavior which is called Scenario II hence-\nforth. Between the bistable [Fig. 2(b)] and very-weak-\nmregime [Fig. 2(a)], there is an interval with an energy\nlandscape similar to an inversion (e.g., by vinit!1=vinit)\nof the abscissa in Fig. 2(c). As one may expect, no further\ndynamics qualitatively di\u000berent from the ones of Scenario\nI, III, and IV are observed in this case.\nIV. SCENARIOS\nUsing Eq. (10), its regularized quasi-continuum ver-\nsion Eq. (14), and the analysis of the pairwise energy\nlandscapes in Sec. III D, we now describe the dynami-\ncal scenarios in detail. Even though the quasi-continuum\ntheory is developed using the variable x, simulation re-\nsults are presented in terms of the density as a func-\ntion of the location of the particles in real space, namely,\n\u001ai\u00111=ri;i+1versusri, if not speci\fed otherwise. Simi-\nlarly, we also use the terms \u001a(x)\u00111=rx,\u001ainit\u00111=vinit\nto lowest order, and so on. Henceforth, time, length, and\nenergies are rendered dimensionless setting \u0000 =k,rc, and\nkr2\ncas units of measurement, respectively. In this unit\nsystem, a density of \u001a=\u001ac= 1 with\u001ac\u00111=rcindicates\nthe set of touching adjacent magnetic particles. More-\nover, magnetic moments are then measured in a unit of\nm0\u0011p\n4\u0019kr5c=\u00160. In plotting the \fgures for additional\nparticle-resolved simulation results, values of a=rc= 2:5,\n\u000fs=(kr2\nc) = 1, andb=\u00002 (assuming that the dipole mo-\nments are parallel to the chain axis and all pointing into\nthe same direction) have been used and red cross sym-\nbols in the \fgures represent initial density distributions.\nEven though only results for N= 100 are shown, we\nhave observed equivalent dynamics simulating systems\nwithN=200, 400, and 800.\n 0.18 0.24 0.3 0.36 0.42\n-200 -100  0  100  200ρi / ρc\nri / rct (Γ/k)−1 = 0\n100\n500\n2500\n12500FIG. 3. Time evolution of density pro\fle for Scenario\nI. Density distributions extracted from the particle-resolved\nsimulations are presented by symbols while black lines show\nthe numerical solution of Eq. (10). Here, m= 0:1m0and\n\u001ainit\ni= 0:2\u001ac. Agreement between particle simulation and\nnumerical solution of the theory is manifested clearly. These\nresults are also depicted in more detail in MOVIE I of ESI [84].\nA. Scenario I: simple relaxation\nWe \frst describe the scenario in the yellow regimes in\nFig. 2, in which the uniform initial con\fguration belongs\nto the basin of attraction of the equilibrium point given\nbye(1)(\u001aeq) = 0. Moreover, e(2)(\u001a) is always positive dur-\ning the whole time evolution and the regularization is not\nnecessary: direct integration of Eq. (10) yields very good\nagreement with particle-resolved simulations as shown in\nFig. 3. The density pro\fles evolving in time can be either\nconcave (\u001ainit>\u001aeq) or convex ( \u001ainit<\u001aeq).\nB. Scenario II: pair formation\nNow, we consider the dynamics in the bistable regime\nwith initial con\fgurations in the spinodal-like interval,\ni.e., the range satisfying e(2)(\u001ainit)<0. As depicted by\nblack lines in Fig. 4, there are two di\u000berent states of ener-\ngetic minima, in both of which the corresponding con\fgu-\nrations are uniform. In the particle-resolved simulations,\nwe observe formations of particle pairs. As represented\nby high-density points in Fig. 4, the pairs consist of two\ntouching particles, appearing in a row along the chain.\nBoth of the two densities computed from the pairs as well\nas from the stretched springs between the pairs coincide\nwith the density values of the minimum points in the\npairwise energy. This indicates that the heterogeneous\ncon\fgurations are stable in the discrete systems in the ab-\nsence of \ructuations. Therefore, in the particle-resolved\nsimulations, the relaxation to the global minimum state\nwith a uniform con\fguration is not observed. In addition,\nwe note that, near the spinodal lines of e(2)= 0, clusters\nwith a number of touching particles larger than 2 (high-7\nρi / ρc\nri / rct (Γ/k)−1 = 0\n7500\n 0.4 0.7 1 1.3\n-80 -40  0  40  80\nFIG. 4. Density distributions in Scenario II. Symbols dis-\nplay the simulation results and thick black lines represent the\nuniform con\fgurations corresponding to the local minimum\npoints of the pairwise energy. The region shaded in gray repre-\nsents the spinodal-like interval. In this \fgure, m= 0:9m0and\n\u001ainit\ni= 0:625\u001ac. For more details, see MOVIE II of ESI [84].\ndensity spinodal line) or stretched springs with only one\nmagnetic particle in between (low-density spinodal line)\nare observed as stable con\fgurations in the simulations.\nWe then turn to the continuum theory. For this sce-\nnario, a regularization is mandatory. There are two dif-\nferent candidates for the boundary condition, both of\nwhich satisfy Eq. (18), as we consider the bistable regime.\nHere, let us take the global minimum state as a bound-\nary condition. Then it is observed that numerical so-\nlutions of the continuum equation (see Appendix C for\nfurther details) converge to the global minimum state\nwith a uniform con\fguration in contrast to the particle-\nresolved simulations. Such a disagreement may imply a\nfailure of the continuum theory in providing a full de-\nscription in this regime. Indeed, it is well known from\nthe \u0000-convergence theory that the solutions to the Cahn-\nHilliard equation asymptotically approach to the global\nminimum point [66, 85, 86]. Similarly, we conjecture that\nthe asymptotic solutions to our quasi-continuum theory\nare given as the uniform con\fguration at the global min-\nimum point. This may, for instance, be due to our non-\nexact regularization terms in the quasi-continuum de-\nscription together with the numerical scheme adopted in\nthe integration of the continuum equation that includes\nadditional di\u000busion. Thus, the bistability is at present\nonly visible in our discrete particle simulations.\nFor further insight, we inspect the individual particle\ndynamics. Let us consider a particle and its two near-\nest neighbors as well as the two springs connecting them.\nWith the two distances between the two particle pairs,\nr1andr2, the corresponding energy can be written as\nEin(r1;r2) =e(r1) +e(r2). Then introducing new vari-\nablesL\u0011r1+r2andl=r1\u0000r2, we \frst con\frm that\nthe state of l= 0 with a homogeneous con\fguration cor-responds to a \fxed point of the dynamics because\n@Ein\n@l\f\f\f\f\nl=0=\u00141\n2e(1)\u0012L+l\n2\u0013\n\u00001\n2e(1)\u0012L\u0000l\n2\u0013\u0015\nl=0= 0:\n(19)\nMeanwhile, the \fxed point of l= 0 is unstable if\ne(2)(L=2) =e(2)(r1) =e(2)(r2)<0 as one can easily\nverify from the corresponding Hessian matrix\n\u00121\n2e(2)(L=2) 0\n01\n2e(2)(L=2)\u0013\n: (20)\nFore(2)(L=2)<0, as in this case, it is straightforward\nto describe the onset of the scenario: dynamics initiated\nfrom the boundary (as already discussed in Sec. III C)\npenetrates into the inner part of the chain, perturbing\nparticles in the local maximum state. Then one may\nexpect a heterogeneity in the con\fguration (i.e., l6= 0) as\na consequence of the above spinodal-like decomposition\nmechanism, which underlies the formation of touching\nparticle pairs. As shown in Fig. 4, densities for touching\npairs and for the stretched springs between pairs agree\nwell with the values of the two local minimum points.\nConsequently, the resulting con\fguration remains stable\nonce the localized spinodal-like decomposition dynamics\nare accomplished.\nC. Scenario III: shock-wave propagation\nIn this scenario, the most important feature observed\nin the particle-resolved simulations is the generation of\nsharp interfaces which divide the chain into macroscopic\nhigh-density clusters and stretched low-density con\fgu-\nrations, as shown in Fig. 5. Speci\fcally, we observe move-\nments of the interfaces between these regions, which are\ninitially formed at the ends of the chain. Such move-\nments or propagations of the interfaces, for instance, in\nthe regime of strong magnetic \felds [cyan in Fig. 2(c)],\nmake the high-density clusters of touching particles grow\ntowards the center of the chain. As one can see, the\nwidths of interfaces are of the order of the distance be-\ntween adjacent particles. Before we proceed, we note\nthat, in this section as well as in Sec. IV D devoted to\nScenario IV, only the touching dynamics are analyzed.\nThe extension of the discussion to the detaching dynam-\nics corresponding to the case between very weak or van-\nishingmand the intermediate bistable regime would be\nstraightforward.\nAccording to the analysis on the level of individual par-\nticles, the dynamics are initiated from the boundary as\nbefore. In contrast to Scenario II, however, the pertur-\nbation from the boundary does not a\u000bect the particles\ninside immediately as they roughly remain in a locally\nstable state. If the e\u000bects from the boundaries are not too\nstrong (this corresponds to Scenario I, in which the ini-\ntial state already belongs to the basin of the equilibrium8\nρi / ρc\nri / rct (Γ/k)−1 = 0\n100\n500\n1500\n 0.3 0.6 0.9 1.2\n-100 -50  0  50  100(a)\nxs (Γ/k)1/2\nm / m0(b)\nρinit/ρc = 0.4\n0.33\n0.25\n 0 0.3 0.6 0.9 1.2\n 0.6  0.8  1  1.2  1.4  1.6  1.8xs (Γ/k)1/2\nm / m0(b)\n,\n,\n,\n 0 0.3 0.6 0.9 1.2\n 0.6  0.8  1  1.2  1.4  1.6  1.8\nFIG. 5. Results for Scenario III of shock-wave propagation. (a) Density distributions for m= 1:7m0and\u001ainit\ni= 0:4\u001acare\nshown, indicating high-density clusters at the ends and a low-density region on the inside. Symbols represent the particle-\nresolved simulation results while the equilibrium density computed from the theory is represented by a black line. Sharp\ninterfaces between high- and low-density clusters are manifested clearly. A movie (MOVIE III) describing the time evolution\ncan be found in ESI [84]. (b) Values of the coe\u000ecient xs, obtained from the particle-resolved simulations (symbols) and the\nsingular perturbation theory (lines), are compared to each other.\npoint), then the relation e(2)(L=2)>0 can still be sat-\nis\fed, keeping the con\fguration somewhat uniform. As\na consequence, in Scenario I particles persistently move\ntowards the center during the whole dynamics.\nAs a new feature, in Scenario III, the distortion at the\nboundaries is strong enough due to such a large di\u000berence\nbetween the initial and equilibrium densities that the sta-\nbility of the uniform con\fguration can be disturbed. In\nthis case, the l= 0 (homogeneous) con\fguration becomes\nunstable for particles at interfaces. With this mechanism,\nthe particles at interfaces can move into the direction op-\nposite to the motion of most other particles in this half\nof the chain as well as of the interface, resulting in touch-\ning to the high-density cluster at the corresponding end\nof the chain. Subsequently, a sharp undershoot is devel-\noped in the density distribution at the interfaces.\nIn terms of the continuum theory, this scenario corre-\nsponds to shock-wave propagations [77]. With a speci\fc\nregularization, we are able to describe the shock with the\naid of singular perturbation theory. Here, we brie\ry sum-\nmarize the procedure (see Appendix B for the details).\nAccording to singular perturbation theory, the structure\nof the shock is quanti\fed by the values of rxbehind and\nin front of the discontinuity or v\u0000andv+as de\fned in\nAppendix B, which should satisfy e(1)(v\u0000) =e(1)(v+).\nAmong the candidates satisfying the condition, certain\nvalues ofe(1)(v\u0006) are selected, depending on the spe-\nci\fc form of regularization. For the regularization in\nEq. (14), we \fnd that the shock wave satis\fes the equal\narea rule [66] or equivalently the common tangent con-\nstruction (see, e.g., Refs. 78 and 72 for other types of\nsolution). From the determined values of v\u0006, we can\nthen compute the similarity coe\u000ecient xswhich means\nthe factor in a similarity relation of the type NT=xsp\nt,\nwhereNTis the number of the particles in the high-density cluster. As this coe\u000ecient determines how fast\nthe shock-wave propagates, it is of interest to probe quan-\ntitatively its values which are presented in Fig. 5 (b). As\none can see, the overall behavior is described qualitatively\nby the theory, but with non-negligible errors. Regarding\nthe fact that here we consider the dynamics near a sin-\ngularity, this type of error seems to be acceptable.\nIn addition to that, we can further classify the den-\nsity pro\fles of this scenario into two cases: The \frst one\ncorresponds to the case of \u001ainit< \u001a +\u00111=v+to low-\nest order. As the outer layer solution should connect\nthe initial condition vinitandv+, the existence of an un-\ndershoot in the density pro\fle at the shock is expected.\nConsidering the conservation of particles involved in de-\ntermining the shock structure [66, 77, 79], we speculate\nthat a mechanism similar to the generation of depletion\nregions in solidi\fcation processes ahead of the solidi\fca-\ntion front [87] seems to play a role in this undershoot\ngeneration. If the initial density is high enough, such an\nundershoot disappears and the solution becomes mono-\ntonic. In particle-resolved simulations, one may take the\nconcavity/convexity of the interior part of chains as an\nindex to identify the existence of the undershoot in den-\nsity pro\fles.\nD. Scenario IV: shock wave of pairs\nLastly, we describe Scenario IV. In this scenario, the\ninitial con\fgurations reside in the spinodal-like interval\nase(2)(vinit)<0. Therefore, as in Scenario II, compli-\ncated con\fgurations consisting of touching particle pairs\ndevelop from the beginning of the dynamics. In contrast\nto Scenario II, however, the density extracted from the\nstretched springs does neither correspond to the stable9\nρi / ρc\nri / rct (Γ/k)−1 = 0\n15500\n 0.3 0.6 0.9 1.2 1.5\n-100 -50  0  50  100\nFIG. 6. Density pro\fles for Scenario IV. As before, symbols,\nblack line, and gray region indicate the particle-resolved simu-\nlation results, the equilibrium density predicted by the theory,\nand the spinodal-like interval, respectively. Here, \u001ainit\ni= 0:5\u001ac\nandm= 1:7m0. The pair formation as well as the shock-\nwave propagation are observed clearly. For more details, see\nMOVIE IV of ESI [84].\nsolution nor does it belong to the basin of the stable\n\fxed point. Moreover, the stretched con\fgurations are\nno longer in the spinodal-like interval, once the spinodal-\nlike dynamics are settled. Therefore, one may expect a\nshock-wave dynamics as in Scenario III. Indeed, we ob-\nserve once again a shock-wave propagation, see Fig. 6. In\nthis scenario, it is the touching of the touching pairs in-\nstead of the single particles which constitutes the dynam-\nics of the shock wave. We also con\frm that the numeri-\ncal integration of the theory exhibits similar time evolu-\ntion in the density distributions as shown in Appendix C.\nHowever, a quantitative description of the shock struc-\nture/position in terms of the theory is still in progress.\nE. Dynamical state diagram\nPutting together the four di\u000berent scenarios, we\npresent in Fig. 7 a dynamical state diagram of the one-\ndimensional dipole-spring model. No other qualitatively\ndi\u000berent scenario was found for the present energy with\nat most two equilibrium points. Schematic \fgures repre-\nsent the density pro\fles at intermediate time scales after\nthe settlement of the initial pair-formation dynamics but\nbefore full equilibration. Here, let us elucidate the ob-\nserved phenomena.\nWhen the magnetic moment is very small ( m=m 0.\n0:21), the e\u000bects of the magnetic interactions are neg-\nligible and the touching/detachment dynamics does not\nplay a signi\fcant role. If we consider the regime of strong\nmagnetic moments ( m=m 0&1:15), the magnetic inter-\nactions signi\fcantly a\u000bect the overall dynamics. As the\nmagnetic interactions are strong, it is the touching of par-\nticles separated in the low initial density regimes (cyan)that triggers the shock-wave dynamics. Here, we fur-\nther note that, phenomenologically, the contraction of\nthe chain is mainly governed by this shock-wave dynam-\nics.\nIn the case of the spinodal-like mechanism (green and\nmagenta), the pair formation rather contributes to the\nredistribution of particles and sometimes even causes an\nincrease of the chain length. Here, the dissipation of\nenergy is faster during the initial stage of the pair for-\nmation than during the shock-wave propagation. This\nseems plausible as the instabilities are localized only in\nthe vicinity of the interfaces in the case of the shock-\nwave propagation dynamics, while they are distributed\nacross the whole system in the spinodal-like case, simul-\ntaneously contributing to the energy dissipation during\nthe pair formation.\nOpposite phenomena are observed in the range of\n0:21.m=m 0.0:40. Even though the magnetic inter-\nactions play a signi\fcant role in this regime, what we ob-\nserve is mostly the separation of particles as the magnetic\ninteractions are still weak in this case. Speci\fcally, we ob-\nserve separation of magnetic particles starting from the\nboundaries and propagations of sharp interfaces extend-\ning expanded regions of the chain from the left/right ends\nto the center in the high-density regime (cyan). Similarly,\nin the intermediate-density regime (green), the separa-\ntion of particles that form pairs due to the spinodal-like\nmechanism in the initial stage of the dynamics underlies\nthe shock-wave propagation.\nIn the intermediate m-regime, at 0 :48.m=m 0.1:15\n(magenta), we observe heterogeneous con\fgurations as\nresulting equilibrium states due to the bistability of the\nenergy. However, if the dynamical theory based on the\nquasi-continuum equation of motion Eq. (14) is evalu-\nated, we are not able to describe the emergence of this\nScenario II. Only the relaxation dynamics to the global\nminimum states are found.\nF. General discussion\nLastly, we qualitatively discuss our results in compar-\nison to general aspects of the dynamics of phase sepa-\nration. First, in the present case, it is found that the\nboundaries initiate the dynamics of the systems, instead\nof thermal \ructuations as for general scenarios of phase\nseparation. Secondly, the growth mechanism of touch-\ning clusters (or their separation dynamics) following the\nspinodal-like initial dynamics is di\u000berent from the phase\nseparation due to di\u000berent conservation laws. In sharp\ncontrast to scenarios of typical phase separation, in our\ncase the overall size of the system may change over time.\nThus the particle number is conserved in the dipole-\nspring system, instead of the global density as in typical\nscenarios of phase separation. This counteracts the co-\nexistence of two phases of di\u000berent densities but rather\npromotes the transition to only one phase. Consequently,\nthe shock-wave propagation dominates the long-time re-10\nm / m0\nρinit / ρc 0 0.4 0.8 1.2 1.6\n 0  0.5  1  1.5I IIIII IV I\nIV III\nFIG. 7. Dynamical state diagram. Black dotted lines in-\ndicate boundaries between dynamical scenarios, while black\nsolid lines discriminate between the states with di\u000berent equi-\nlibrium con\fgurations, see Fig. 2. Red dashed lines repre-\nsent the equilibrium states of global energetic minima while\nthe blue dotted-dashed line corresponds to the unstable lo-\ncal maximum points in the pairwise energy. We note that\nthe black solid lines between yellow and magenta areas and\nthe black dotted lines between green and cyan regimes and\nbetween green and yellow areas constitute the spinodal line\nwithe(2)(\u001a) = 0. Additional black dotted lines inside cyan\nregions are of \u001a+=\u001acform=m 0&1:15 and of \u001a\u0000=\u001acfor\n0:21.m=m 0.0:48. We also note that green and ma-\ngenta regions are the spinodal-like intervals while yellow re-\ngions correspond to the basins of attraction of the equilibrium\npoints. The same colors as in Fig. 2 are used to identify the\ndi\u000berent dynamical scenarios. Schematic density pro\fles of\ncorresponding dynamical scenarios are indicated.\nlaxation dynamics of the system, driving the change in\nextension of the chain and promoting the overall trans-\nformation of the whole system.\nApart from that, as a technical detail, the underlying\nbackground of the regularization is also di\u000berent in our\nquasi-continuum description. While, for instance, the in-\nterface itself contributes to the free energy in the Cahn-\nHilliard equation in the form of gradient terms [63, 66], it\nis only the discreteness of the system that gives rise to the\nregularization in our case. We stress, however, that the\nspinodal-type mechanism based on the structure of the\nunderlying energy is formally rather analogous, leading\nto the emergence of pair/cluster formation.\nAltogether, the touching/detachment dynamics can be\nrelated to a spinodal-type mechanism, while the interfa-\ncial shock-wave propagation governing the long-time dy-\nnamics in certain cases may rather be comparable to a\nscenario of domain growth. Di\u000berent scenarios of touch-\ning/detachment dynamics are summarized in Table I.V. SUMMARY AND OUTLOOK\nUntil now, we have investigated the relaxation dynam-\nics of a one-dimensional dipole-spring model. We have\nrevealed that a type of spinodal decomposition mecha-\nnism plays a central role in the touching or detachment\ndynamics of magnetic particles and that shock-wave-type\npropagations can dominate the long-time relaxation dy-\nnamics to the equilibrium states. The boundary e\u000bects\nare shown to be an essential ingredient for the initia-\ntion and the subsequent qualitative appearance of the\ndynamics, while the discreteness of the system regular-\nizes the continuum equation of motion. It is remarkable\nthat even these simple one-dimensional systems exhibit\nheterogeneous scenarios in spite of the homogeneity in\ninitial and, mostly, equilibrium con\fgurations. A variety\nof rich dynamics involves the interplay between the for-\nmation of particle pairs and the shock-wave propagation.\nThere still remains plenty of space for further exten-\nsions of the present study. First of all, the response of the\nsystem to time-dependent magnetic \felds is of interest.\nSpeci\fcally, e\u000bects of the touching/detaching dynamics\non the dynamic moduli of the system [45, 47] may deepen\nthe understanding of the magneto-mechanical couplings\nin ferrogels. Extensions of the model to two- and three-\ndimensional systems are also an important step. In\npart, we anticipate similar dynamics for strong directed\nmagnetic interactions, as then, likewise, one-dimensional\nchain-like aggregations will form aligned along the direc-\ntion of an applied external magnetic \feld [45, 47, 55, 56].\nAlready, our one-dimensional simulation results suggest\nthat the global minimum states in the intermediate\nregime could be non-uniform. Even more possibilities\narise in two or three dimensions and, therefore, even\nricher dynamics are expected to be observed. In addition\nto that, e\u000bects of thermal \ructuations should be clari\fed\nas well [80]. For example, if heterogeneous initial con\fg-\nurations are taken into account, we observe the onset of\nthe shock-wave propagation in the particle-resolved sim-\nulation for long-range magnetic interactions even from\nthe interior of the chain. One may expect similar phe-\nnomena in the system induced by thermal \ructuations,\nwhich may correspond to the nucleation of dense clusters\nor soft components.\nWe expect that the results discussed in this study can\nbe con\frmed from experiments. Indeed, the experimen-\ntal technology these days enables researchers to capture\nthe con\fguration at a certain time point [55] or to pro-\nvide a temporal resolution of the dynamics of correspond-\ning systems [34, 88]. Therefore, supported by quantita-\ntive analysis of the data, the formation of particle pairs\nand the propagation of sharp interfaces might be veri-\n\fed. Still, there is a possibility that the imperfections\ninherent in experimental samples may obscure such veri-\n\fcation. However, there are e\u000borts to construct uniform\nnanocomposite samples [89]. With the aid of such an\napproach, the rigorous comparison between theory and\nexperiments could be achieved.11\nScenario \u001ai(ri) Intermediate con\fguration Equilibrium state\nII\n Heterogeneous\nIII\n Uniform,\u001a>\u001a c\nUniform,\u001a<\u001a c\nIV\n Uniform,\u001a>\u001a c\nUniform,\u001a<\u001a c\nTABLE I. Schematic graphical representations of the scenarios involving the touching/detachment dynamics. While schematic\n\fgures for the intermediate density pro\fles are displayed in the second column, corresponding intermediate con\fgurations during\nthe relaxation to stationary equilibrium states are brie\ry portrayed in the third column. Values of mand\u001ainitcan be identi\fed\nfrom the dynamical state diagram in Fig. 7 by comparing the corresponding schematic plots in the second column. In the last\ncolumn, the characteristics of their \fnal stationary equilibrium con\fgurations are summarized. In all of the three scenarios,\na spinodal-type mechanism underlies the initial touching and detachment dynamics of the magnetic particles. However, the\nlong-time dynamics are always dominated by the shock-wave propagation, except for Scenario II, in which there is no further\nlong-time relaxation dynamics.\nMeanwhile, especially in interpreting possible experi-\nmental results, randomness in the network connectivity\nas well as in the arrangement and size of the magnetic\nparticles should be taken into account. Still, one may ex-\npect a similar dynamics, consisting of pair formation and\nshock-wave propagation. For example, touching pairs\nand compact chain formation are observed even in three-\ndimensional inhomogeneous dipole-spring systems based\non experimentally observed particle con\fgurations [47].\nHowever, details such as the size of the touching clusters\nor the initiation mechanism of exit the dynamics may dif-\nfer. If heterogeneity is introduced in the spring constant,\nsofter parts of the chain may form a touching cluster\nmore easily than other parts of the system and, there-\nfore, the chain formation dynamics could be initiated in\nvarious parts of the system. In this case, we speculate\nthat a kind of coupling between the interfaces may play\na certain role. Veri\fcation of such couplings could be a\nchallenging task in theoretical as well as in experimental\nstudies.In short, we expect that our results may serve as an\nessential building block in understanding the dynamics of\nmore realistic models for ferrogels. However, we also note\nthat a further adjusted continuum theory with \fne-tuned\nregularization terms should be devised to fully describe\nthe whole dynamics, especially in the bistable regime.\nThis is left for future works.\nACKNOWLEDGMENTS\nWe thank Giorgio Pessot for providing codes which\nwere useful for the initiation of this study. We also\nthank Giorgio Pessot, Peet Cremer, J urgen Horbach,\nand Benno Liebchen for helpful discussions and com-\nments. This work was supported by funding from the\nAlexander von Humboldt Foundation (S.G.) and from\nthe Deutsche Forschungsgemeinschaft through the prior-\nity program SPP 1681, grant nos. ME 3571/3 (A.M.M)\nand LO 418/16 (H.L.).\nAppendix A: Equations of motion for the particles\nWe describe the equations of motion for the particles in the magnetic chain in detail. The equations for the boundary\nparticles are shown explicitly as well.\nObviously, the term on the right-hand side of Eq. (1) consists of three parts. The \frst one of them, resulting from\nthe elastic energy, reads\n\u0000@Eel\n@ri=k(ri+1\u0000ri)\u0000k(ri\u0000ri\u00001) (A1)\nfori= 2;:::;N and\n\u0000@Eel\n@r1=k(r2\u0000r1\u0000a);\u0000@Eel\n@rN+1=\u0000k(rN+1\u0000rN\u0000a): (A2)12\nSecond, the contributions from the magnetic dipole-dipole interaction take the form\n\u0000@Em\n@ri=3\u00160bm2\n4\u00192\n4\u0000X\nj>i1\n(rj\u0000ri)4+X\nj<i1\n(rj\u0000ri)43\n5 (A3)\nfori= 2;:::;N , and\n\u0000@Em\n@r1=\u00003\u00160bm2\n4\u0019N+1X\nj=21\n(rj\u0000r1)4;\u0000@Em\n@rN+1=3\u00160bm2\n4\u0019NX\nj=11\n(rj\u0000rN+1)4: (A4)\nWe note that the nearest-neighbor magnetic dipole-dipole interaction is obtained from the above equations by con-\nstraining the summations in jto nearest neighbors.\nLastly, we have\n\u0000@Est\n@ri=fst(ri;i+1)\u0000fst(ri\u00001;i) (A5)\nfori= 2;:::;N and\n\u0000@Est\n@r1=fst(r1;2);\u0000@Est\n@rN+1=\u0000fst(rN;N+1) (A6)\nfrom the steric repulsion energy, where\nfst(r)\u0011\u000fs\u0002(r\u0000rc)\u0014\n\u000012\n\u001bs\u0010r\n\u001bs\u0011\u000013\n+6\n\u001bs\u0010r\n\u001bs\u0011\u00007\n\u0000cs(r\u0000rc)\u0015\n: (A7)\nAppendix B: Singular perturbation theory\nWe de\fnev\u0011rxand rescale the time variable by introducing \u001c\u0011(\u0001x)2tfor convenience. Then Eq. (14) reads\n\u0000v\u001c=\u0014\ne(1)(v) + (\u0001x)2\u00121\n12e(2)(v)vxx+1\n24e(3)(v) (vx)2\u0013\u0015\nxx: (B1)\nIntroducing the extended variable ~ x=x\u0000s(\u001c)\n(\u0001x), we probe an interlayer solution which describes the behavior of\nthe system in the vicinity of the shock front at s(\u001c). Under a change of variables v= ~v(~x;\u001c) withvx=1\n\u0001x~v~xand\nv\u001c= ~v\u001c\u0000_s(\u001c)\n\u0001x~v~x, Eq. (B1), to the leading order of \u0001 x, becomes\n\u0012\ne(1)(~v) +1\n12e(2)(~v) ~v~x~x+1\n24e(3)(~v) (~v~x)2\u0013\n~x~x= 0; (B2)\nwhich has a solution of the form\ne(1)(~v) +1\n12e(2)(~v) ~v~x~x+1\n24e(3)(~v) (~v~x)2=A~x+B: (B3)\nFor the interlayer solutions ~ vand ~v~xmust approach constant values v\u0006and 0 as ~x!\u00061 and, therefore,\nA= 0; B =e(1)(v\u0000) =e(1)(v+): (B4)\nFurther multiplying by ~ v~x, we also \fnd\n(B~v)~x=\u0012\ne(~v) +1\n24e(2)(~v) (~v~x)2\u0013\n~x; (B5)13\n 0 2 4 6 8 10\n 1  2  3  4  5(a)\nv+\nv−r+\nr−\nrx/rce(1)(rx)/(kr2\nc)\n 0.69 0.71 0.73 0.75\n 1.2  1.4  1.6  1.8 0 2 4 6(b)\nv+\nv−r+\nr−\nm/m 0r−/rc, v−/rc\nr+/rc, v+/rc\nri,i+1 / rc\nit (Γ/k)−1 = 0\n100\n500\n1500\n 1 2 3\n 0  20  40  60  80  100(c)\nNT\nNT\nt (Γ/k)−1(d)\nm / m0 = 1.3\n1.5\n1.7\n 0 10 20 30 40 50\n 0  2000  4000\nFIG. 8. (a) How the equal area rule applies in practice is described: the structures of the shocks are given by the \flled blue\ncircle (v\u0000) and the \flled green square ( v+), corresponding to the intersection points between the red line representing e(1)and\nthe black dotted line, where v\u0000andv+are the values of rxbehind and in front of the shock. The gray areas above and below\nthe black dotted line need to be equal to each other. Plus and cross symbols representing the particle-resolved simulation results\nare displayed as well. Speci\fcally, r\u0000andr+are given as the lengths of the springs in front of and behind the abrupt jump in\nthe density, respectively. (b) The values of v+andv\u0000(magenta and cyan solid lines) computed from Eqs. (B4) and (B6) are\ncompared with the particle-resolved simulation results, r+andr\u0000(blue circles and red squares). We note that the values on\nthe left axis correspond to those of r\u0000(red squares) and v\u0000(magenta solid line), while r+(blue circles) and v+(cyan dashed\nline) are represented by the axis on the right. (c) Fig. 5(a) for Scenario III is replotted in terms of ri;i+1as a function of i. The\nvalues ofNTare explicitly depicted by arrows. (d) The values of NTas functions of time tare plotted for m= 1:3m0, 1:5m0,\nand 1:7m0with\u001ainit\ni= 0:4\u001ac. We extract the values of xsby numerically \ftting the results to the relation NT=xsp\nt(black\nsolid lines), which here are found to be xs(\u0000=k)1=2= 0.380 (m=m 0= 1:3), 0.600 (m=m 0= 1:5), and 0.732 ( m=m 0= 1:7).\nwhich leads to the equal-area rule [66, 78] for the interlayer solution of the form\nB(v+\u0000v\u0000) =e(v+)\u0000e(v\u0000): (B6)\nNumerically solving Eqs. (B4) and (B6), one can compute the values of v+andv\u0000, which specify the structure of the\nshocks. The construction of the equal area rule and the resultant shock structures are presented in Figs. 8(a) and (b).\nWe turn to the propagation speed of the shock-wave front s(\u001c). As our equation of motion in the leading order\ntakes the form of a di\u000busion equation, we consider a similarity solution U(z) =v(x;\u001c) with the similarity variable\nz\u0011x=p\u001c. The equation of motion in the leading order follows as\n\u0000v\u001c= (e(2)(v)vx)x (B7)\nin terms of xand\u001c. It can be rewritten as\n\u0000\n2zdU(z)\ndz=\u0000d\ndz\u0012\ne(2)(U)dU(z)\ndz\u0013\n(B8)14\nin terms of the similarity variable z. In particular, the quantity of interest is the coe\u000ecient xswhich is de\fned by a\nsimilarity relation s(\u001c) =xsp\u001c. Quantifying the values of xswith the Whitham's derivation [90], one can describe\nthe dynamics of the shock-wave propagation. For self-containedness, we brie\ry summarize the procedure, following\nRefs. 77 and 90. We also note that an equivalent result was obtained [66] with the aid of mathematical consideration\nof Stephan problems [79].\nFirst, we consider the di\u000busion \rux qin a region x1> x > x 2where a balance between the net in\row across x1\nandx2in a region is described by\n\u0000d\nd\u001cZx2\nx1v(x;\u001c) dx+q(x1;\u001c)\u0000q(x2;\u001c) = 0; (B9)\nleading to the conservation form\n\u0000@v\n@\u001c+@q\n@x= 0: (B10)\nTherefore, we de\fne the di\u000busion \rux as q\u0011\u0000e(2)(v)vx[see Eq. (B7)]. We then extend the above consideration to\na case with a discontinuity at x=s(\u001c). In this case, Eq. (B9) can be rewritten as follows [90]:\nq(x1;\u001c)\u0000q(x2;\u001c) =\u0000d\nd\u001cZs(\u001c)\nx2v(x;\u001c)dx+ \u0000d\nd\u001cZx1\ns(\u001c)v(x;\u001c)dx (B11)\n=\u0000v(s\u0000;\u001c)ds\nd\u001c+Zs(\u001c)\nx2\u0000v\u001c(x;\u001c)dx\u0000\u0000v(s+;\u001c)ds\nd\u001c+Zx1\ns(\u001c)\u0000v\u001c(x;\u001c)dx: (B12)\nWith the limits x1!s+from above and x2!s\u0000from below, we obtain\n\u0000ds\nd\u001c=q(s\u0000;\u001c)\u0000q(s+;\u001c)\nv(s\u0000;\u001c)\u0000v(s+;\u001c)\u0011[q]\n[v]; (B13)\nwhere square brackets denote the jump of the contained value across the interface. For the similarity solution,\nq(x;\u001c) =\u0000\u001c\u00001=2e(2)(U)Uz, and therefore the above equation is cast into the form [77]\n1\n2xs=\u0000[e(2)(U)Uz]\n[\u0000U]; (B14)\nwhich \fnally determines the propagation speed of the shock front. Numerically solving Eqs. (B8) and (B14), we obtain\nthe values of xswhich are presented in Fig. 5(b). Using these values, we can compute, for instance, the number of\ntouching particles NT(t) in one end. Speci\fcally, scaling back to the time t, we have\nNT(t) =s(\u001c)\n\u0001x=xsp\n(\u0001x)2t\n\u0001x=xsp\nt: (B15)\nAs expected, the number of touching particles is independent of the value of \u0001 x. Predicted values of xsare shown in\nFig. 5(b), together with those extracted from the simulations results by the procedure described in Figs. 8(c) and (d).\nAppendix C: Numerical integration of the continuum equation of motion\nIn this appendix, we describe the algorithm used in integrating the quasi-continuum equation of motion. The\nalgorithm is a modi\fed version of the upwind scheme [91, 92], which is widely used to \fnd propagating solutions to\nwave equations. However, if it is directly applied to the magnetic chain under contraction, for instance, the shrinkage\nof the chain is rather exaggerated as the particles behind an interface receive biased information towards the particles\nin front of the interface [91], which may impose a resistance to contraction. To compensate such an artifact, we\nintroduce an additional downwind-biased step and write the discretized equation for each time step \u0001 tas follows:\n\u0000r(x;t+ \u0001t) = \u0000r(x;t) +\u0001t\n2\u0014\ne(2)\u0012r(x;t)\u0000r(x\u0000\u0001x;t)\n\u0001x\u0013r(x+ \u0001x;t) +r(x\u0000\u0001x;t)\u00002r(x;t)\n(\u0001x)2\u0015\n+\u0001t\n2\"\ne(2) \nr\u0000\nx+ \u0001x;t+\u0001t\n2\u0001\n\u0000r\u0000\nx;t+\u0001t\n2\u0001\n\u0001x!\nr\u0000\nx+ \u0001x;t+\u0001t\n2\u0001\n+r\u0000\nx\u0000\u0001x;t+\u0001t\n2\u0001\n\u00002r\u0000\nx;t+\u0001t\n2\u0001\n(\u0001x)2#\n:\n(C1)15\nρ(x) / ρc\nr(x) (∆x)-1 / rct (Γ/k)−1 = 0\n2\n10\n30\n 0.3 0.6 0.9 1.2\n-100 -50  0  50  100(a)\nρ(x) / ρc\nr(x) (∆x)-1 / rc(b)\nt (Γ/k)−1 = 0\n610\n30\n 0.3 0.6 0.9 1.2 1.5\n-90 -60 -30  0  30  60  90\nFIG. 9. Density pro\fles extracted from the numerical solutions with the scheme Eq. (C1) for (a) Scenario III and (b) Scenario\nIV. In (a), \u001ainit= 0:4\u001acandm= 1:7m0have been used, while \u001ainit= 0:5\u001acandm= 1:7m0in (b). Even though the shock\nwaves obtained in this way propagate much faster than in the particle-resolved simulations, the essential features, including\nthe spinodal-like mechanisms for the touching, pair formation, sharp-interface generation, and relaxation to the equilibrium\nstate, are well veri\fed by the numerical solutions. Notably, compared to Fig. 5(a) and Fig. 6, solutions with high density, e.g.,\n\u001a(x)&1:12\u001ac, exhibit much smoother density variations in space, implying an excess numerical di\u000busion [92] as described in\nthe text.\nAs already pointed out in Ref. 76, a certain form of regularization is always involved in the numerical integrations,\nwhich are indeed discrete. In the case of the numerical scheme discussed here, the dominant correction to the fully\ncontinuum equation of motion [Eq. (10)] is given as\n(\u0001x)2\u00141\n12e(2)(rx)rxxxx +1\n6e(3)(rx)rxxrxxx+1\n8e(4)(rx) (rxx)3\u0015\n: (C2)\nInterestingly, the terms are almost equivalent to the leading order regularization in Eq. (14). Therefore, we conclude\nthat the algorithm discussed above provides solution to the continuum equation of motion but with a slightly di\u000berent\ntype of regularization.\nIt is well known that the upwind scheme introduces numerical di\u000busion of the interface [92]. The numerical\nintegration scheme described above also seems to su\u000ber from such an issue, as the numerical solutions are not consistent\nwith Eq. (B4), which should be satis\fed regardless of regularization. Speci\fcally, it has been tested by plotting a\n\fgure like Fig. 8(a) from the numerical integration results. Moreover, the propagation speed of the interface sensitively\ndepends on the structure of the shock as manifested in Eq. (B14). We \fnd that the coe\u000ecient xsextracted from a\nnumerical solution can be, roughly, 100 times larger than the one obtained by the particle-resolved simulation and the\nsingular perturbation theory. Still, the essential shapes of the solutions agree quite well with the simulation results,\nas shown in Fig. 9.\n[1] G. Filipcsei, I. Csetneki, A. Szil\u0013 agyi, and M. Zr\u0013 \u0010nyi, Adv.\nPolym. Sci. 206, 137 (2007).\n[2] P. Ilg, Soft Matter 9, 3465 (2013).\n[3] A. M. Menzel, Phys. Rep. 554, 1 (2015).\n[4] S. Odenbach, Arch. Appl. Mech. 86, 269 (2016).\n[5] M. Lopez-Lopez, J. D. Dur\u0013 an, L. Y. Iskakova, and A. Y.\nZubarev, J. Nano\ruids 5, 479 (2016).\n[6] N. Frickel, R. Messing, and A. M. Schmidt, J. Mater.\nChem. 21, 8466 (2011).\n[7] E. Allahyarov, A. M. Menzel, L. Zhu, and H. L owen,\nSmart Mater. Struct. 23, 115004 (2014).\n[8] T. Gundermann and S. Odenbach, Smart Mater. Struct.\n23, 105013 (2014).\n[9] J. Landers, L. Roeder, S. Salamon, A. M. Schmidt, andH. Wende, J. Phys. Chem. C 119, 20642 (2015).\n[10] R. Messing, N. Frickel, L. Belkoura, R. Strey, H. Rahn,\nS. Odenbach, and A. M. Schmidt, Macromolecules 44,\n2990 (2011).\n[11] L. Roeder, P. Bender, M. Kundt, A. Tsch ope, and A. M.\nSchmidt, Phys. Chem. Chem. Phys. 17, 1290 (2015).\n[12] D. Szab\u0013 o, G. Szeghy, and M. Zr\u0013 \u0010nyi, Macromolecules 31,\n6541 (1998).\n[13] T. Volkova, V. B ohm, T. Kaufhold, J. Popp, F. Becker,\nD. Borin, G. Stepanov, and K. Zimmermann, J. Magn.\nMagn. Mater. 431, 262 (2017).\n[14] M. M. Schmauch, S. R. Mishra, B. A. Evans, O. D. Velev,\nand J. B. Tracy, ACS Appl. Mater. Interfaces 9, 11895\n(2017).16\n[15] H. Deng, X. Gong, and L. Wang, Smart Mater. Struct.\n15, N111 (2006).\n[16] T. Sun, X. Gong, W. Jiang, J. Li, Z. Xu, and W. Li,\nPolym. Test. 27, 520 (2008).\n[17] N. Bock, A. Riminucci, C. Dionigi, A. Russo,\nA. Tampieri, E. Landi, V. Goranov, M. Marcacci, and\nV. Dediu, Acta Biomater. 6, 786 (2010).\n[18] X. Zhao, J. Kim, C. A. Cezar, N. Huebsch, K. Lee,\nK. Bouhadir, and D. J. Mooney, Proc. Natl. Acad. Sci.\nU.S.A. 108, 67 (2011).\n[19] R. M uller, M. Zhou, A. Dellith, T. Liebert, and\nT. Heinze, J. Magn. Magn. Mater. 431, 289 (2017).\n[20] R. Weeber, S. Kantorovich, and C. Holm, Soft Matter\n8, 9923 (2012).\n[21] A. Ryzhkov, P. Melenev, C. Holm, and Y. Raikher, J.\nMagn. Magn. Mater. 383, 277 (2015).\n[22] R. Weeber, S. Kantorovich, and C. Holm, J. Chem. Phys.\n143, 154901 (2015).\n[23] R. Weeber, S. Kantorovich, and C. Holm, J. Magn.\nMagn. Mater. 383, 262 (2015).\n[24] E. Jarkova, H. Pleiner, H.-W. M uller, and H. R. Brand,\nPhys. Rev. E 68, 041706 (2003).\n[25] S. Bohlius, H. R. Brand, and H. Pleiner, Phys. Rev. E\n70, 061411 (2004).\n[26] M. A. Annunziata, A. M. Menzel, and H. L owen, J.\nChem. Phys. 138, 204906 (2013).\n[27] J. J. Cerd\u0012 a, P. A. S\u0013 anchez, C. Holm, and T. Sintes, Soft\nMatter 9, 7185 (2013).\n[28] G. Pessot, P. Cremer, D. Y. Borin, S. Odenbach,\nH. L owen, and A. M. Menzel, J. Chem. Phys. 141,\n015005 (2014).\n[29] G. Pessot, R. Weeber, C. Holm, H. L owen, and A. M.\nMenzel, J. Phys.: Condens. Matter 27, 325105 (2015).\n[30] P. A. S\u0013 anchez, J. J. Cerd\u0012 a, T. M. Sintes, A. O. Ivanov,\nand S. S. Kantorovich, Soft Matter 11, 2963 (2015).\n[31] D. Ivaneyko, V. Toshchevikov, and M. Saphiannikova,\nSoft Matter 11, 7627 (2015).\n[32] A. M. Biller, O. V. Stolbov, and Y. L. Raikher, J. Appl.\nPhys. 116, 114904 (2014).\n[33] A. M. Biller, O. V. Stolbov, and Y. L. Raikher, Phys.\nRev. E 92, 023202 (2015).\n[34] M. Puljiz, S. Huang, G. K. Auernhammer, and A. M.\nMenzel, Phys. Rev. Lett. 117, 238003 (2016).\n[35] M. Puljiz and A. M. Menzel, Phys. Rev. E 95, 053002\n(2017).\n[36] A. M. Menzel, Soft Matter 13, 3373 (2017).\n[37] D. Heinrich, A. R. Go~ ni, A. Smessaert, S. H. L. Klapp,\nL. M. C. Cerioni, T. M. Os\u0013 an, D. J. Pusiol, and C. Thom-\nsen, Phys. Rev. Lett. 106, 208301 (2011).\n[38] M. Bloom, S. C. Bae, E. Luijten, and S. Granick, Nature\n491, 578 (2012).\n[39] C. E. Alvarez and S. H. L. Klapp, Soft Matter 9, 8761\n(2013).\n[40] J. Dobnikar, A. Snezhko, and A. Yethiraj, Soft Matter\n9, 3693 (2013).\n[41] S. H. Klapp, Curr. Opin. Colloid Interface Sci. 21, 76\n(2016).\n[42] H. An, S. J. Picken, and E. Mendes, Soft Matter 6, 4497\n(2010).\n[43] H.-N. An, B. Sun, S. J. Picken, and E. Mendes, J. Phys.\nChem. B 116, 4702 (2012).\n[44] M. Tarama, P. Cremer, D. Y. Borin, S. Odenbach,\nH. L owen, and A. M. Menzel, Phys. Rev. E 90, 042311\n(2014).[45] G. Pessot, H. L owen, and A. M. Menzel, J. Chem. Phys.\n145, 104904 (2016).\n[46] T. Nadzharyan, V. Sorokin, G. Stepanov, A. Bogolyubov,\nand E. Y. Kramarenko, Polymer 92, 179 (2016).\n[47] G. Pessot, M. Sch umann, T. Gundermann, S. Odenbach,\nH. L owen, and A. M. Menzel, J. Phys.: Condens. Matter\n30, 125101 (2018).\n[48] V. V. Sorokin, I. A. Belyaeva, M. Shamonin, and E. Y.\nKramarenko, Phys. Rev. E 95, 062501 (2017).\n[49] D. S. Wood and P. J. Camp, Phys. Rev. E 83, 011402\n(2011).\n[50] P. Melenev, Y. Raikher, G. Stepanov, V. Rusakov, and\nL. Polygalova, J. Intel. Mater. Syst. Struct. 22, 531\n(2011).\n[51] A. Y. Zubarev, Soft Matter 9, 4985 (2013).\n[52] D. Romeis, V. Toshchevikov, and M. Saphiannikova, Soft\nMatter 12, 9364 (2016).\n[53] A. Y. Zubarev, L. Y. Iskakova, and M. T. Lopez-Lopez,\nPhysica A 455, 98 (2016).\n[54] M. T. Lopez-Lopez, D. Y. Borin, and A. Y. Zubarev,\nPhys. Rev. E 96, 022605 (2017).\n[55] T. Gundermann, P. Cremer, H. L owen, A. M. Men-\nzel, and S. Odenbach, Smart Mater. Struct. 26, 045012\n(2017).\n[56] M. Sch umann and S. Odenbach, J. Magn. Magn. Mater.\n441, 88 (2017).\n[57] P. Cremer, H. L owen, and A. M. Menzel, Appl. Phys.\nLett. 107, 171903 (2015).\n[58] P. Cremer, H. L owen, and A. M. Menzel, Phys. Chem.\nChem. Phys. 18, 26670 (2016).\n[59] R. van Roij, Phys. Rev. Lett. 76, 3348 (1996).\n[60] B. W. Kwaadgras, M. W. J. Verdult, M. Dijkstra, and\nR. van Roij, J. Chem. Phys. 138, 104308 (2013).\n[61] A. Y. Zubarev, D. N. Chirikov, D. Y. Borin, and G. V.\nStepanov, Soft Matter 12, 6473 (2016).\n[62] P. M. Chaikin and T. C. Lubensky, Principles of Con-\ndensed Matter Physics (Cambridge University Press,\nCambridge, 2000).\n[63] J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258\n(1958).\n[64] P. W. Bates and P. C. Fife, SIAM J. Appl. Math. 53,\n990 (1993).\n[65] C. M. Elliott and D. A. French, IMA J. Appl. Math. 38,\n97 (1987).\n[66] R. L. Pego, Proc. Royal Soc. Lond. A 422, 261 (1989).\n[67] J. Bialk\u0013 e, H. L owen, and T. Speck, Europhys. Lett. 103,\n30008 (2013).\n[68] T. Speck, J. Bialk\u0013 e, A. M. Menzel, and H. L owen, Phys.\nRev. Lett. 112, 218304 (2014).\n[69] Y. Fily, S. Henkes, and M. C. Marchetti, Soft Matter\n10, 2132 (2014).\n[70] M. E. Cates, D. Marenduzzo, I. Pagonabarraga, and\nJ. Tailleur, Proc. Natl. Acad. Sci. U.S.A. 107, 11715\n(2010).\n[71] J. Stenhammar, A. Tiribocchi, R. J. Allen, D. Maren-\nduzzo, and M. E. Cates, Phys. Rev. Lett. 111, 145702\n(2013).\n[72] R. Wittkowski, A. Tiribocchi, J. Stenhammar, R. J.\nAllen, D. Marenduzzo, and M. E. Cates, Nat. Commun.\n5, 4351 (2014).\n[73] M. N. Popescu and S. Dietrich, Phys. Rev. E 69, 061602\n(2004).\n[74] S. Dietrich, M. N. Popescu, and M. Rauscher, J. Phys.:\nCondens. Matter 17, S577 (2005).17\n[75] K. H ollig, Trans. Am. Math. Soc. 278, 299 (1983).\n[76] G. I. Barenblatt, M. Bertsch, R. D. Passo, and M. Ughi,\nSIAM J. Math. Anal. 24, 1414 (1993).\n[77] T. P. Witelski, Appl. Math. Lett. 8, 27 (1995).\n[78] T. P. Witelski, Stud. Appl. Math. 97, 277 (1996).\n[79] J. Crank, The Mathematics of Di\u000busion , 2nd ed. (Oxford\nUniversity Press, London, 1975).\n[80] P. Cremer, M. Heinen, A. M. Menzel, and H. L owen, J.\nPhys.: Condens. Matter 29, 275102 (2017).\n[81] J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem.\nPhys. 54, 5237 (1971).\n[82] M. Doi and S. F. Edwards, The Theory of Polymer Dy-\nnamics (Clarendon Press, Oxford, 1986).\n[83] A. Novick-Cohen and R. L. Pego, Trans. Am. Math. Soc.\n324, 331 (1991).\n[84] See Electronic Supplementary Information (ESI) at [URL\nwill be inserted by publisher] for example movies illus-\ntrating the dynamic behaviors associated with the di\u000ber-\nent Scenarios I{IV.\n[85] L. Modica, Arch. Ration. Mech. Anal. 98, 123 (1987).[86] A. Braides, \u0000 -convergence for Beginners , Oxford Lecture\nSeries in Mathematics, Vol. 22 (Oxford University Press,\nOxford, 2002).\n[87] K. Sandomirski, E. Allahyarov, H. L owen, and S. U.\nEgelhaaf, Soft Matter 7, 8050 (2011).\n[88] S. Huang, G. Pessot, P. Cremer, R. Weeber, C. Holm,\nJ. Nowak, S. Odenbach, A. M. Menzel, and G. K. Auern-\nhammer, Soft Matter 12, 228 (2016).\n[89] A. Feld, R. Koll, L. S. Fruhner, M. Krutyeva,\nW. Pyckhout-Hintzen, C. Weiss, H. Heller, A. Weimer,\nC. Schmidtke, M.-S. Appavou, E. Kentzinger, J. Allgaier,\nand H. Weller, ACS Nano 11, 3767 (2017).\n[90] G. B. Whitham, Linear and Nonlinear Waves (John Wi-\nley & Sons, New York, 1999).\n[91] S. Patankar, Numerical Heat Transfer and Fluid Flow\n(Taylor & Francis, Boca Raton, 1980).\n[92] C. Hirsch, Numerical Computation of Internal and Ex-\nternal Flows: The Fundamentals of Computational Fluid\nDynamics (Butterworth-Heinemann, Oxford, 2007).18\nElectronic Supplementary Information\nSegun Goh,1Andreas M. Menzel,1and Hartmut L¨ owen1\n1Institut f¨ ur Theoretische Physik II: Weiche Materie,\nHeinrich-Heine-Universit¨ at D¨ usseldorf, D-40225 D¨ usseld orf, Germany\n(Dated: March 2, 2018)\nSUPPLEMENTARY MOVIES\nMovies I–IV present the particle-resolved simulation results for Scenarios I–I V. The composition of the movies is\ndescribed in Fig. S1.\n 1 2 3\n 0 20 40 60 80 100ri,i+1 / rc\ni 0.2 0.6 1 1.4\n-70  0  70ρi / ρc\nri / rct (Γ/k)−1 = 1100\nFIG. S1. A snapshot taken from one of the movies. In the second row, the w hole magnetic chain is presented, while in the first\nrow, enlarged figures for the region shaded in green in the second row are pl otted. Graphs on the left-hand side at the bottom\nshow the spring length distributions, while the density profiles are displayed on the right-hand side. Elapsed times from the\ninitial time of uniform configuration are indicated. The parameter values ar e the same as the ones used in Figs. 3–6 of the\nmain paper."    },    {        "title": "1712.04691v1.Studying_the_transfer_of_magnetic_helicity_in_solar_active_regions_with_the_connectivity_based_helicity_flux_density_method.pdf",        "content": "Draft version May28, 2022\nPreprint typeset using L ATEX style emulateapj v. 04 /20/08\nSTUDYING THE TRANSFER OF MAGNETIC HELICITY IN SOLAR ACTIVE REGIONS\nWITH THE CONNECTIVITY-BASED HELICITY FLUX DENSITY METHOD\nK. D almasse1,´E. Pariat2, G. V alori3, J. J ing4,andP. D´emoulin2\n1 CISL /HAO, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000, USA\n2 LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universit ´e, UPMC Univ. Paris 06, Univ. Paris Diderot, Sorbonne Paris cit ´e,\nF-92190 Meudon, France\n3 Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK and\n4 Space Weather Research Laboratory, Center for Solar Terrestrial Research, New Jersey Institute of Technology, 323 Martin Luther King Blvd., Newark, NJ\n07102-1982, USA\nDraft version May 28, 2022\nABSTRACT\nIn the solar corona, magnetic helicity slowly and continuously accumulates in response to plasma flows\ntangential to the photosphere and magnetic flux emergence through it. Analyzing this transfer of magnetic\nhelicity is key for identifying its role in the dynamics of active regions (ARs). The connectivity-based he-\nlicity flux density method was recently developed for studying the 2D and 3D transfer of magnetic helicity\nin ARs. The method takes into account the 3D nature of magnetic helicity by explicitly using knowledge of\nthe magnetic field connectivity, which allows it to faithfully track the photospheric flux of magnetic helicity.\nBecause the magnetic field is not measured in the solar corona, modeled 3D solutions obtained from force-\nfree magnetic field extrapolations must be used to derive the magnetic connectivity. Di \u000berent extrapolation\nmethods can lead to markedly di \u000berent 3D magnetic field connectivities, thus questioning the reliability of the\nconnectivity-based approach in observational applications. We address these concerns by applying this method\nto the isolated and internally complex AR 11158 with di \u000berent magnetic field extrapolation models. We show\nthat the connectivity-based calculations are robust to di \u000berent extrapolation methods, in particular with regards\nto identifying regions of opposite magnetic helicity flux. We conclude that the connectivity-based approach can\nbe reliably used in observational analyses and is a promising tool for studying the transfer of magnetic helicity\nin ARs and relate it to their flaring activity.\nSubject headings: magnetic fields - Sun: photosphere - Sun: corona - Sun: flares\n1.INTRODUCTION\nMagnetic helicity is a signed scalar quantity that measures\nthe three-dimensional complexity of a magnetic field in a\nvolume ( e.g., Finn & Antonsen 1985). Mo \u000batt (1969) and\nBerger & Field (1984) showed that magnetic helicity has a\nwell-defined geometrical interpretation in terms of the entan-\nglement, or braiding, of magnetic field lines. Magnetic he-\nlicity thus generalizes more local properties such as magnetic\ntwist and shear.\nThe emergence of twisted /sheared magnetic fields from the\nconvection zone into the solar corona ( e.g., Leka et al. 1996;\nMoreno-Insertis 1997; Longcope & Welsch 2000; D ´emoulin\net al. 2002b; Green et al. 2002; Georgoulis et al. 2009; Pevtsov\n2012; Poisson et al. 2015), and the stressing of the coronal\nmagnetic field by plasma flows along the photosphere ( e.g.,\nvan Ballegooijen & Martens 1989; Klimchuk & Sturrock\n1992; Chae et al. 2001; Moon et al. 2002; Liu & Schuck 2012;\nZhang et al. 2012; Vemareddy 2015), slowly and continuously\nbuild up magnetic helicity in the solar atmosphere. Because of\nits conservation property in highly conducting plasmas ( e.g.,\nWoltjer 1958; Taylor 1974, 1986; Berger 1984; Pariat et al.\n2015), magnetic helicity is thus believed to be a fundamen-\ntal component for understanding the dynamics of the coronal\nmagnetic field ( e.g., Zhang et al. 2006, 2008; Kazachenko\net al. 2012; Tziotziou et al. 2012; Romano et al. 2014).\nMagnetic helicity is hence at the heart of several MHD the-\nories of coronal processes including, but not limited to, coro-\nnal heating through the relaxation of braided magnetic fields\nElectronic address: dalmasse@ucar.edu(e.g., Heyvaerts & Priest 1984; Russell et al. 2015; Yeates\net al. 2015), the formation of filament channels through the\ninverse cascade of magnetic helicity ( e.g., Antiochos 2013;\nKnizhnik et al. 2015), the existence of CMEs as the mean for\nthe Sun to expel its magnetic helicity excess ( e.g., Rust 1994;\nLow 1996), and the production of very high-energy flares\nvia magnetic helicity annihilation (Linton et al. 2001; Ku-\nsano et al. 2004). Recently, Pariat et al. (2017) even showed\nthat specific quantities derived from magnetic helicity have a\nstrong potential for greatly improving the prediction of solar\neruptions.\nMethods to estimate magnetic helicity in the solar context\nare reviewed by Valori et al. (2016). Among these methods,\nanalyzing the temporal evolution of the helicity flux through\nthe photosphere provides valuable information about the he-\nlicity content of ARs and is one of the means for better un-\nderstanding the role of magnetic helicity in their dynamics\n(see review by e.g., D´emoulin & Pariat 2009, and references\ntherein). When an AR is followed from the beginning of its\nemergence, the temporal integration of the photospheric he-\nlicity flux gives an estimate of its coronal helicity ( e.g., Chae\n2001; Kusano et al. 2003; Mandrini et al. 2004; Jeong & Chae\n2007; LaBonte et al. 2007; Yang et al. 2009; Guo et al. 2013).\nOn the other hand, the photospheric distribution of helicity\nflux during the early stages of AR formation reflects the sub-\nphotospheric distribution of magnetic helicity in the associ-\nated emerging magnetic field ( e.g., Kusano et al. 2003; Chae\net al. 2004; Yamamoto et al. 2005; Pariat et al. 2006; Jing\net al. 2012; Park et al. 2013; Vemareddy & D ´emoulin 2017).\nThis, in turn, gives constraints on the processes generating the\nmagnetic field in the solar interior ( e.g., Kusano et al. 2002;arXiv:1712.04691v1  [astro-ph.SR]  13 Dec 20172 Dalmasse et al.\nPariat et al. 2007). Later on during the lifetime of an AR, the\ndistribution of the helicity flux allows to track where magnetic\nhelicity is being locally accumulated in response to additional\nmagnetic flux emergence and photospheric flows ( e.g., Chan-\ndra et al. 2010; Vemareddy et al. 2012b).\nStudying the distribution of the helicity flux in ARs is not\nstraightforward because it requires the use of a surface den-\nsity of a quantity which is inherently 3D and not local. While\nmagnetic helicity density per unit volume is an unphysical\nquantity, Russell et al. (2015) recently showed that it is pos-\nsible to construct and study a magnetic helicity density per\nunit surface from the recent developments of Yeates & Hornig\n(2011, 2013, 2014). Previously, Pariat et al. (2005) had shown\nthat it was possible to define a useful proxy of surface density\nof helicity flux by explicitly expressing magnetic helicity in\nterms of magnetic field lines linkage. Such an approach is\nachieved by including the connectivity of magnetic field lines\nin the definition of the total helicity flux, leading to the con-\nstruction of a so-called connectivity-based surface density of\nhelicity flux (further details are provided in Section 2.1).\nDalmasse et al. (2014) recently developed a method for the\npractical computation of the connectivity-based helicity flux\ndensity to be used in observational studies. Using analytical\ncase-studies and numerical simulations, they showed that the\nconnectivity-based calculations provide a reliable and faithful\nmapping of the helicity flux. In particular, the method is suc-\ncessful in revealing real mixed signals of helicity flux in mag-\nnetic configurations, as well as in relating the local transfer\nof magnetic helicity with the location of regions favorable to\nmagnetic reconnection. The former makes the method partic-\nularly interesting for testing the very high-energy flare model\nof Kusano et al. (2004) in observational surveys of solar ARs,\nwhile the latter provides a new way for analyzing the role of\nmagnetic helicity accumulation in flaring activity.\nFor analytical models and numerical MHD simulations, the\n3D magnetic field is known in the entire volume of the mod-\neled solar atmosphere and can be readily used to integrate\nmagnetic field lines. In observational studies, however, po-\nlarimetric measurements in the corona are not as numerous\nand routinely made as the photospheric and chromospheric\nones. And as the latters, coronal polarimetric measurements\nare also 2D and, thus, cannot lead to magnetic field data in\nthe full coronal volume without the use of some 3D model-\ning. On top of this, their inversion into magnetic field data\nis a very challenging task ( e.g., Rachmeler et al. 2012; Kra-\nmar et al. 2014; Plowman 2014; Dalmasse et al. 2016; Gibson\net al. 2016, and references therein). Hence, one must rely\non the approximate 3D solution of, e.g., nonlinear force-free\nfield (NLFFF) models ( e.g., Wheatland et al. 2000; Wiegel-\nmann 2004; Amari et al. 2006; Valori et al. 2007; Inoue et al.\n2012; Malanushenko et al. 2012) to extrapolate the coronal\nmagnetic field from the photospheric maps of the magnetic\nfield (vector magnetograms). Unfortunately, di \u000berent meth-\nods and assumptions can lead to markedly di \u000berent 3D mag-\nnetic field solutions. These strong di \u000berences between recon-\nstructed magnetic fields a \u000bect all subsequently derived quan-\ntities. For instance, DeRosa et al. (2009, 2015) reported varia-\ntions between extrapolation methods that can reach up to 30%\nin free magnetic energy and 200% in magnetic helicity.\nThe analyses of DeRosa et al. (2009, 2015) raise con-\ncerns about the reliability and relevance of the connectivity-\nbased helicity flux density approach in observational appli-\ncations. In this paper, we address these concerns by apply-\ning the connectivity-based method to observations of an ARwith di \u000berent magnetic field extrapolation models and imple-\nmentations. The selected AR is internally complex but ex-\nternally simple (i.e., no neighboring large-flux systems). The\nconnectivity-based helicity flux density method is reviewed\nin Section 2. Section 3 describes the dataset and the approach\ntaken to estimate uncertainties in the helicity flux intensity.\nThe magnetic field extrapolations are discussed in Section 4.\nSection 5 presents the results of our analysis. A discussion\nand interpretation of our results is provided in Section 6. Our\nconclusions are summarized in Section 7.\n2.METHOD\n2.1.Magnetic Helicity Flux Densities\nUnder ideal conditions, the transfer of magnetic helicity\nthrough the photosphere, S, and into the solar atmosphere is\n(e.g., D´emoulin et al. 2002a; Pariat et al. 2005)\ndH\ndt=Z\nSG\u0012(x) dS: (1)\nG\u0012(x) is a surface-density of magnetic helicity flux defined as\nG\u0012(x)=\u0000Bn(x)\n2\u0019Z\nS0d\u0012(x\u0000x0)\ndtBn(x0) dS0: (2)\nwhere\nd\u0012(x\u0000x0)\ndt=((x\u0000x0)\u0002(u\u0000u0))jn\njx\u0000x0j2; (3)\nis the relative rotation rate (or relative angular velocity) be-\ntween pairs of photospheric magnetic polarities located at x\nandx0and moving on the photospheric plane Swith flux-\ntransport velocity u=u(x) and u0=u(x0). The flux-transport\nvelocity uis (D ´emoulin & Berger 2003)\nu=vt\u0000vn\nBnBt: (4)\nwhere subscript “ t” and “ n” respectively denote the tangential\nand normal components of photospheric vector fields, vand\nBare the photospheric plasma velocity and magnetic fields.\nEquations (1) – (4) show that the total flux of magnetic he-\nlicity through the photosphere, S, can be expressed as the\nsummation of the net rotation of all pairs of photospheric el-\nementary magnetic polarities around each other, weighted by\ntheir magnetic flux. It further shows that, at any given time, t,\nthe total flux of magnetic helicity can be solely derived from\nphotospheric quantities, i.e., from a timeseries of vector mag-\nnetograms from which the flux-transport velocity can also be\nderived (Schuck 2008).\nThe surface-density of helicity flux, G\u0012, measures the varia-\ntion of magnetic helicity in an AR only from the relative mo-\ntions of its photospheric magnetic polarities. However, we\nrecall that magnetic helicity describes the global linkage of\nmagnetic field lines in the volume. Therefore, what e \u000bec-\ntively modifies the magnetic helicity of an AR is the rela-\ntive re-orientation of magnetic field lines, i.e., the variation\nof their mutual helicity, in response to the motions of their\nphotospheric footpoints. Such a global, 3D nature of mag-\nnetic helicity and its variation is not taken into account in the\ndefinition of G\u0012. As a consequence, G\u0012has a tendency to mis-\nrepresent the local variation of magnetic helicity in ARs by\nhiding the subtle e \u000bects of the mutual helicity variation be-\ntween magnetic field lines, and by overestimating the local\nhelicity flux when an AR is associated with opposite helicity\nfluxes ( e.g., Dalmasse et al. 2014).Studying the transfer of magnetic helicity in solar active regions 3\nPariat et al. (2005) showed that it is possible to remedy this\nissue by explicitly re-arranging those terms in Equation (1)\nthat are related to the connectivity of elementary magnetic\nflux tubes. That allows to express the total flux of magnetic\nhelicity in terms of the sum of the magnetic helicity variation\nof each individual elementary flux tube of the magnetic field,\nsuch that\ndH\ndt=Z\n\bdh\b\ndt\f\f\f\f\f\fcd\bc; (5)\nwhere h\bis the magnetic helicity of the elementary magnetic\nflux tube, c, and d \bcits elementary magnetic flux. h\bis the\nmagnetic helicity density and describes how any elementary\nflux tube is linked /winded around all other elementary flux\ntubes ( e.g., Berger 1988; Aly 2014; Yeates & Hornig 2014).\nIt can be shown that\ndh\b\ndt\f\f\f\f\f\fc=G\u0012(xc+)\njBn(xc+)j+G\u0012(xc\u0000)\njBn(xc\u0000)j(6)\n=˙\u0002B(xc+)\u0000˙\u0002B(xc\u0000); (7)\nwhere xc+(resp. xc\u0000) is the positive (resp. negative), pho-\ntospheric, magnetic polarity of the elementary magnetic flux\ntube c(see Figure 1 of Dalmasse et al. 2014, for a generic\nsketch of elementary flux tubes and their photospheric con-\nnectivity), and\n˙\u0002B(x)=\u00001\n2\u0019Z\nS0d\u0012(x\u0000x0)\ndtBn(x0) dS0: (8)\nNote that the advantage of using Equation (7) over Equa-\ntion (6) is to avoid artificial numerical singularities when\nBn(x) is very small.\nThen, Pariat et al. (2005) defined a new surface density\nof helicity flux by redistributing d h\b=dtat each photospheric\nfootpoint of the elementary magnetic flux tube, c, to map the\nphotospheric flux of magnetic helicity\nG\b(xc\u0006)=1\n2dh\b\ndt\f\f\f\f\f\fc\f\f\fBn(xc\u0006)\f\f\f: (9)\nNote that the factor 1 /2 in Equation (9) assumes that both\nphotospheric footpoints of an elementary flux tube contribute\nequally to the variation of its magnetic helicity (a more gen-\neral case is described in Pariat et al. 2005). Note also that the\ncalculations of d h\b=dtandG\brequire one significant addi-\ntional information as compared with G\u0012,i.e., the connectivity\nof the magnetic field.\nFinally, we stress here that both G\u0012andG\bareinstanta-\nneous estimations of injected helicity at particular location on\nthe photosphere, and not the total helicity content in the asso-\nciated field line.\n2.2.Numerical Method\nDalmasse et al. (2014) introduced a method to compute the\nconnectivity-based helicity flux density proxy, G\b. Using var-\nious analytical case studies and numerical MHD simulations,\nthey showed that G\bproperly tracks the site(s) of magnetic\nhelicity variations.\nTheir method is based on field line integration to derive\nthe magnetic connectivity required to compute G\b. For ob-\nservational studies, routine magnetic field measurements are\nmostly realized at the photospheric level ( cf.Section 1). We\nthus perform force-free field extrapolations to obtain the coro-\nnal magnetic field of the studied AR (see Section 4) and com-pute the photospheric distribution of magnetic helicity flux\nfrom Equations (7) and (9).\nEach pixel of the photospheric vector magnetogram is iden-\ntified as the cross-section of an elementary magnetic flux tube\nwith the photosphere. Each of these elementary flux tubes\nis associated with one magnetic field line that is integrated to\nobtain the connectivity. For any closed magnetic field line, we\nthus obtain a pair ( xc+;xc\u0000) of photospheric footpoints, where\nxc+is the positive magnetic polarity of the elementary flux\ntube and xc\u0000is its negative magnetic polarity at the photo-\nsphere ( z=0). We then introduce a slight modification to\nthe method proposed in Dalmasse et al. (2014). Instead of\ncomputing G\bfrom Equation (6), which can lead to artificial\nnumerical singularities when Bn(x) is very small, we com-\npute d h\b=dtandG\bfrom Equations (7) and (9). Hence, once\nthe conjugate footpoint of a field line is found, we compute\u0010˙\u0002B(xc+) ;˙\u0002B(xc\u0000)\u0011\nusing bilinear interpolation. Finally, field\nlines for which the conjugate footpoint reaches the top or lat-\neral boundaries are treated as open field lines and both d h\b=dt\nandG\bare simply set to zero. This is a second modification to\nthe method of Dalmasse et al. (2014) justified by the fact that,\nin this paper, we focus on comparing d h\b=dtandG\bcom-\nputed using di \u000berent magnetic field extrapolations, i.e., two\nquantities that are only defined for closed field lines.\n2.3.Metrics for Validation and Quantification of G \bMaps\nIn the connectivity-based helicity flux density approach,\nhalf the total helicity flux computed from G\u0012over the en-\nsemble of positive magnetic polarities is redistributed over\nthe ensemble of negative magnetic polarities, and vice-versa.\nFor this redistribution to be perfect, the total magnetic flux\nsummed over the magnetic polarities where G\bis computed\nmust vanish. If, for instance, the magnetic flux from the neg-\native magnetic polarities is smaller than that of the positive\nones, then a fraction of the total helicity flux computed from\nG\u0012over the positive magnetic polarities is not redistributed in\nthe negative ones. This means that part of the total helicity\nflux computed with G\u0012for the entire magnetic configuration\nis missing from that computed with G\b. We thus introduce a\nfirst metric to validate G\bmaps, i.e., the percentage of mag-\nnetic flux imbalance, \u001c\bimb:, for the closed magnetic flux where\nG\bis computed\n\u001c\bimb:=R\nScl:Bn(x) dS\nmin\u0012R\nScl:(Bn>0)Bn(x) dS;\f\f\f\fR\nScl:(Bn<0)Bn(x) dS\f\f\f\f\u0013:\n(10)\nwhereS\bcl:is the part of the photosphere associated with the\nclosed magnetic flux, \bcl:.\nBy definition, the fluxes measured by G\bare simply a re-\ndistribution of the fluxes measured by G\u0012at both footpoints\nof each elementary magnetic flux tube in the closed magnetic\nfield. Thus, the intensity of the magnetic helicity flux in one\npixel can be di \u000berent for G\bandG\u0012. However, the total flux\nof magnetic helicity integrated over the closed magnetic flux\nfrom G\bmust be the same as that integrated from G\u0012because\nEquation (5) is equal to Equation (1) for the closed magnetic\nfield. The second metric we defined for the validation of G\b\nmaps is thus\nCS\bcl:=R\nS\bcl:(G\b(x)\u0000G\u0012(x))dS\nR\nS\bcl:G\u0012(x) dS: (11)4 Dalmasse et al.\nIn theory, CS\bcl:should be strictly equal to 0. In practice, how-\never, its value depends on departures of \u001c\bimb:from 0. Accu-\nracy and validation of the G\bmap requires both that \u001c\bimb:and\nCS\bcl:be close to 0.\nFinally, we define two quantification indices to compare the\nsignal intensity between G\bandG\u0012within the closed mag-\nnetic field\nC+=R\nScl:(G\u0012>0)G\u0012(x) dS\nR\nScl:(G\b>0)G\b(x) dS; (12)\nC\u0000=R\nScl:(G\u0012<0)G\u0012(x) dS\nR\nScl:(G\b<0)G\b(x) dS: (13)\nC+andC\u0000respectively compare the total positive and total\nnegative magnetic helicity fluxes derived from G\u0012with those\nderived from G\bin the closed magnetic flux. Pariat et al.\n(2005) and Dalmasse et al. (2014) showed that the helicity\nflux density proxy G\u0012hides the true local helicity flux and\ntends to exhibit larger and spurious helicity flux intensities as\ncompared with the G\bproxy when opposite helicity fluxes are\npresent in an AR. C+andC\u0000allow to quantify the intensity\nof the spurious signals in G\u0012, thus providing an idea of the\nglobal improvement of the G\bmaps relatively to the G\u0012ones.\nIn particular, large departures of C\u0006from 1 are indicative of\nstrong spurious signals in G\u0012.\n3.OBSERV ATIONS AND ERROR ANALYSIS\n3.1.Data\nWe test the robustness of the connectivity-based helicity\nflux density method against di \u000berent magnetic field extrap-\nolation models of the internally complex AR 11158. This AR\nappeared on the solar disk at the heliographic coordinates S19\nE42 on 2011 February 10. The AR was the result of fast and\nstrong magnetic flux emergence that produced two large-scale\nbipoles, a northern and a southern one, in close proximity (see\nFigure 1; e.g., Schrijver et al. 2011). The complex, quadrupo-\nlar magnetic field of this AR produced several C- /M-/X-class\nflares and CMEs during its on-disk passage ( e.g., Toriumi\net al. 2014). A large fraction of this flaring activity was as-\nsociated with the collision between the negative magnetic po-\nlarity of the northern bipole, NN, and the positive magnetic\npolarity of the southern bipole, SP, which led to a strong and\ncontinuous shearing of their polarity inversion line. More de-\ntails on the configuration, evolution, and flaring activity of the\nAR can be found in e.g., Sun et al. (2012), Jiang et al. (2012),\nVemareddy et al. (2012a) and Inoue et al. (2013).\nThe coronal models of AR 11158 are computed using vec-\ntor magnetograms taken by the Helioseismic and Magnetic\nImager (HMI, e.g., Schou et al. 2012) onboard the So-\nlar Dynamics Observatory (SDO, e.g., Pesnell et al. 2012).\nSDO/HMI provides full-disk vector magnetograms of the Sun\nwith a pixel size of 0.500. For the purpose of this paper, we\nre-use part of the data from Dalmasse et al. (2013) who had\napplied the connectivity-based approach to AR 11158 with an\nNLFFF extrapolation without addressing the possible depen-\ndency of the results on the choice of extrapolation method. In\nparticular, vector magnetograms at 06:22 UT and 06:34 UT on\n2011 February 14 from the HMI-SHARP data series, HARP\nnumber 377 (Hoeksema et al. 2014) are re-used.\nThese two vector magnetograms are used to derive the\nphotospheric flux transport velocity field with the di \u000ber-\nential a \u000ene velocity estimator for vector magnetograms\nFig. 1.— Top: SDO/HMI photospheric vector-magnetogram at \u001806:28 UT.\nThe gray scale displays the vertical component, Bz(in Gauss), while the yel-\nlow/blue arrows show the transverse component of the magnetic field. Bot-\ntom: vertical magnetic field overplotted with the flux transport velocity field\n(green /orange arrows). Pink and cyan solid lines show Bz=\u0006500 Gauss\nisocontours.\n(DA VE4VM; Schuck 2008), using a window size of 19 pix-\nels as suggested by Liu et al. (2013). The computed flux\ntransport velocity field e \u000bectively represents an instantaneous\nflux transport velocity field at \u001806:28 UT. The two vector\nmagnetograms taken at 06:22 UT and 06:34 UT are averaged\nto construct an instantaneous vector magnetogram associated\nwith the flux transport velocity field at \u001806:28 UT. The con-\nstructed vector magnetic field is then used both to compute G\u0012\nand as the photospheric boundary condition for the di \u000berent\nFFF extrapolations presented Section 4. The vector magnetic\nfield and flux transport velocity field at \u001806:28 UT are shown\nin Figure 1.\n3.2.Estimation of Helicity-Flux Uncertainties\nAs part of testing the reliability of the connectivity-based\nhelicity flux density method, we also wish to evaluate un-\ncertainties in G\u0012andG\bmaps caused by those in the pho-\ntospheric magnetic field. In particular, our error analysis in-\ncludes both the e \u000bect of photon noise ( \u001910 Gauss) and sev-\neral sources of systematic errors. Hoeksema et al. (2014)\nrecently performed an extensive analysis of the uncertain-\nties in the measurements of the magnetic field strength by\nSDO/HMI. In particular, they focused on the uncertainty anal-\nysis for NOAA 11158. Among several sources of system-Studying the transfer of magnetic helicity in solar active regions 5\nFig. 2.— Selected closed field lines of the 3D extrapolated magnetic field for the potential field (labelled POT; left), the NLFFF from the magneto-frictional\nrelaxation method (labelled NLFFF-R; middle), and the NLFFF from the optimization method (labelled NLFFF-O; right). The magnetic field lines were integrated\nfrom the same -randomly selected- photospheric footpoints for all three extrapolations (the same colour is used for the same footpoint). The gray scale displays\nthe photospheric vertical magnetic field, Bz, with\u0006500 Gauss isocontours (purple and cyan solid lines).\natic errors, including those related with the inversion code,\nthey found that the dominant contribution of uncertainty in\nSDO/HMI magnetic field measurements is coming from the\ndaily variation of the radial velocity of the spacecraft along\nits geosynchronous orbit. Their analysis concludes that the\ntypical uncertainty in SDO /HMI measurements of the mag-\nnetic field strength is about 100 Gauss. The latter can be\neasily checked from the estimated field-strength error map of\nNOAA 11158 provided by the SDO /HMI pipeline and which\ncan be downloaded from JSOC1. We also want to compare\nthese errors with those related with the choice of magnetic\nfield extrapolation used to compute G\b.\nFor that purpose, we conduct a Monte Carlo experiment\nas proposed by Liu & Schuck (2012) for error estimation of\nthe total helicity flux. Random noise with a Gaussian dis-\ntribution having a width ( \u001b) of 100 Gauss is added to all\nthree components of the magnetic field for both vector mag-\nnetograms taken at 06:22 UT and 06:34 UT. 100 Gauss is the\n\u00191\u001buncertainty in the total magnetic field strength from\nHMI data estimated by Hoeksema et al. (2014) and further\nreported in Bobra et al. (2014). The uncertainty is then prop-\nagated through the chain of helicity flux density calculations\nto the flux transport velocity field at the photosphere derived\nfrom DA VE4VM, the corresponding vector magnetogram at\n\u001806:28 UT, and finally to the G\u0012andG\bmaps.\nAlthough they did not perform a full parametric analysis,\nWiegelmann et al. (2006) showed that extrapolations with\nthe optimization method were not significantly a \u000bected by\nmodest noise in the photospheric vector magnetogram. They\nfound that the preprocessing of the photospheric data towards\na more force-free state (see e.g., Section 4.2) strongly helps\nin that matter. This may be expected considering that the ran-\ndom noise on the photospheric vector magnetic field acts as\na source of non-force-free signals on high spatial frequencies\nwhile the preprocessing filters such signals out. Even if FFF\nreconstructions would likely be di \u000berently a \u000bected by noise\nin the photospheric data, its global e \u000bect on the extrapola-\ntions should be limited by the preprocessing stage. On the\nother hand, FFF extrapolations are more likely to be a \u000bected\nby more global e \u000bects including, but not limited to, large-\nscale non-force-free regions not suppressed by the prepro-\ncessing, magnetic flux imbalance, and lateral boundary con-\n1http://jsoc.stanford.edu /ajax/lookdata.htmlditions. Then, as far as G\bis concerned, the e \u000bect of random\nnoise on the extrapolation results is to introduce uncertainties\nin the connectivity of the magnetic field. In this regard, we\nbelieve that the choice of extrapolation method and prepro-\ncessing level is more fundamental and has a stronger impact\non the magnetic connectivity, and hence, on G\b. For these\nreasons, we decided not to propagate the noise to the FFF ex-\ntrapolations.\nThe noise propagation experiment is repeated 100 times,\nproducing 100 noise-added G\u0012andG\bmaps for all three FFF\nextrapolations considered in this paper. For each scalar quan-\ntity,F, the 1\u001bestimated error, \u001bF, is computed as the mean,\nover all the npixels of the map, of the root mean square of the\nN=100 noise-addedF-maps,Fi\nn:a:\u0010\nxj\u0011\n, compared with the\nno-noiseF-map,Fn:n:\u0010\nxj\u0011\n\u001bF=1\nnnX\nj=10BBBBB@1\nNNX\ni=1\u0010\nFi\nn:a:\u0010\nxj\u0011\n\u0000F n:n:\u0010\nxj\u0011\u001121CCCCCA1\n2\n: (14)\n4.FORCE-FREE MAGNETIC FIELD EXTRAPOLATIONS\nWe perform three force-free field extrapolations using dif-\nferent assumptions and methods: (1) the potential magnetic\nfield, (2) a nonlinear FFF (NLFFF) reconstruction using the\nmagneto-frictional method of Valori et al. (2010), and (3) an\nNLFFF generated with the optimization method of Wiegel-\nmann (2004). The extrapolations and setup used to produce\nthem are described hereafter.\nAt this stage, we wish to emphasize that many more ex-\ntrapolation assumptions, methods, and implementations, ex-\nist in the literature. These di \u000berent methods are further dis-\ntinguished in terms of the physical information that they ex-\ntract from the photospheric vector magnetograms and use as\nboundary conditions. For instance, some NLFFF codes use\nvector magnetograms as boundary conditions, as is the case of\nthe magneto-frictional relaxation implemented by, e.g., Val-\nori et al. (2010), or the optimization method implemented\nby,e.g., Wiegelmann & Inhester (2010). Others built on the\nGrad-Rubin approach (Grad & Rubin 1958) use the normal\ncomponent of the photospheric magnetic field and the force-\nfree parameter – derived from the vector magnetograms –\nas boundary conditions ( e.g., Amari et al. 2006; Wheatland\n2007). Recently, Malanushenko et al. (2012) also proposed6 Dalmasse et al.\na Quasi-Grad-Rubin method that only uses the photospheric\nnormal magnetic field, but combined with coronal loops fit-\nting to constrain the coronal distribution of the force-free pa-\nrameter. Finally, several codes haven been recently developed\nto perform a full MHD relaxation using photospheric vector\nmagnetograms, thus distinguishing them from NLFFF models\nthrough the inclusion of plasma forces in the reconstruction of\nthe 3D magnetic field of ARs ( e.g., Inoue & Morikawa 2011;\nJiang & Feng 2012; Zhu et al. 2013). All these di \u000berent codes\nand methods can be used to model the coronal magnetic field\nof ARs from which we can derive the magnetic connectiv-\nity required to use the connectivity-based helicity flux density\napproach to map the helicity flux in ARs.\nSeveral of these methods have been compared with each\nothers in e.g., DeRosa et al. (2009, 2015), including the two\nNLFFF methods used in this paper. From these studies, it\nappears that the three extrapolation methods considered here\ngenerally produce di \u000berences in field-lines distribution that\nare representative of the di \u000berences that can be expected be-\ntween the coronal magnetic fields reconstructed with other\nFFF methods. We thus expect the di \u000berences in the d h\b=dt\nandG\bcalculations presented in this paper to be representa-\ntive of the di \u000berences that would be obtained when computing\nthe connectivity-based helicity flux density with other extrap-\nolation methods. For this reason, we limit ourselves to the\nanalysis of helicity flux density calculations with the three ex-\ntrapolations described below.\nThe extrapolation presented hereafter are performed on a fi-\nnite field of view with open side and top boundaries, but with-\nout assuming magnetic flux balance. Magnetic flux is thus\nfree to leave the extrapolation domain as open-like magnetic\nfield. Such open-like magnetic field may represent truly open\nmagnetic field and /or connections with the very distant quiet\nSun and surrounding ARs. Helicity flux density calculations\nare not performed for open-like field lines because G\band\ndh\b=dtare not defined for open magnetic flux tubes ( cf.Sec-\ntion 2.2). For that reason, open field lines are not plotted in\nany of the field-line plots presented throughout the paper.\nWorking with a finite field of view may be a strong limita-\ntion to reconstruct the magnetic field of ARs. Not only is the\nentire photosphere always populated with quiet Sun magnetic\nflux, but there are also often more than one AR on the Sun at\na given time ( e.g., Schrijver et al. 2013). The e \u000bect of using a\nlimited field of view is to remove large-scale connections with\nthe distant quiet Sun and surrounding ARs that are outside\nthe field of view considered for the extrapolation. Such dis-\ntant connections may influence the results of the helicity flux\ndensity calculations with the connectivity-based method. To\ntest such an influence, one needs to compare the connectivity-\nbased helicity flux density computed from a global, full-Sun\nmagnetic field reconstruction in spherical geometry vs. from\na local magnetic field extrapolation. However, current global\nreconstructions of the solar magnetic field are limited by the\nfact that there is no full-Sun vector magnetograms at any sin-\ngle time, and hence, no proper data for the boundary condi-\ntions of full-Sun extrapolation codes. A proper analysis of the\ne\u000bect of distant ARs on the helicity flux density calculations\nwould therefore require numerical modeling. Such a study\nis not the goal of the present work. The vast majority of in-\nvestigations relying on NLFFF extrapolations are focusing on\nsingle ARs with the same type of limited field of view and in-\nherent hypotheses that are also used in the present manuscript.\nWe are presently testing the reliability of connectivity-based\nhelicity flux density calculations with regard to such NLFFFmodeling, i.e., we aim to determine if and to which extent dif-\nferent choices of NLFFF computation schemes applied to a\ncompact active region can impact the distribution of the helic-\nity flux density.\n4.1.Potential Field\nThe current-free magnetic field is the minimal-energy pos-\nsible state for the given distribution of magnetic field at the\nboundaries of the considered volume. In addition, the poten-\ntial field is often used as an initial state of numerical methods\nthat build the more complex NLFFF models (see review by,\ne.g., Wiegelmann & Sakurai 2012). It is therefore an impor-\ntant candidate to consider for testing the connectivity-based\nhelicity flux density method, despite its limitation in repro-\nducing nonlinear features of the coronal field.\nThe potential field can be directly computed using the po-\ntential theory and the reflection principle to solve the Laplace\nequation for the scalar potential in terms of the flux through\nthe photospheric boundary (Schmidt 1964). However, in or-\nder to speed up the calculation, such a method is actually used\nonly to compute the scalar potential on all six boundaries of\nthe considered volume. The magnetic scalar potential in the\nvolume is then computed solving the Laplace equation, sub-\njected to the obtained Dirichlet boundary conditions, using\na fast Helmholtz solver from the Intel Mathematical Kernel\nLibrary. For the required vertical component of the field on\nthe lower boundary, the same vertical component as for the\nNLFFF extrapolation described Section 4.2 is used. The po-\ntential field extrapolation is referred to as POT in the follow-\ning and selected field lines for this extrapolation are shown in\nthe leftmost panel of Figure 2.\n4.2.NLFFF from Magneto-Frictional Relaxation\nNLFFF extrapolations are models of the coronal magnetic\nfield that assume the corona to be static on the time scale of\ninterest, and to be dominated by magnetic forces that are dis-\ntributed in a way such that the resulting Lorentz force is every-\nwhere vanishing. Such assumptions are supposed to be valid\nin the entire volume of interest, boundaries included. The\nmagneto-frictional relaxation method implements numerical\nrelaxation and multi-grid techniques to solve the correspond-\ning equations (see Valori et al. 2007, 2010; DeRosa et al.\n2015, for more details).\nThe remapped and disambiguated vector magnetograms\nfrom the HMI-SHARP data series were interpolated to 100-\nresolution and averaged to construct the vector magnetogram\nat 06:28 UT ( cf.Section 3.1) used for the NLFFF extrapo-\nlation. Vector magnetograms are inferred from spectropolari-\nmetric measurements taken at photospheric heights, where the\nplasma is non-force-free. Therefore, in order to use the vector\nmagnetogram as a boundary condition for the NLFFF extrap-\nolation code, the forces acting on the magnetogram need to be\nreduced (preprocessing). To this purpose we use the method\nof Fuhrmann et al. (2007, 2011). In this application, only the\nhorizontal components of the field are preprocessed, yield-\ning a reduction of the forces from 0.035 to 0.002 in the non-\ndimensional units used in Metcalf et al. (2008). Since smooth-\ning is not necessarily facilitating the extrapolation (Valori\net al. 2013), no smoothing was applied. The resulting mag-\nnetogram was then extrapolated using the magneto-frictional\ncode into a volume of about 208 \u0002202\u0002145 Mm3.\nThe resulting extrapolated field has solenoidal errors that\ncan be quantified using the formula by Valori et al. (2013)Studying the transfer of magnetic helicity in solar active regions 7\nTABLE 1\nMetrics for validation and quantification of G\bmaps\nFFF model \u001c\bimb:CS\bcl:C+ C\u0000\nPOT 1 :2\u000210\u000034:2\u000210\u000041.32 1.71\nNLFFF-R \u00007:6\u000210\u000034:7\u000210\u000021.21 1.67\nNLFFF-O 2 :6\u000210\u000033:2\u000210\u000021.23 1.57\nNote. — The table presents the validation and quantification metrics for\nthe maps displayed in Figure 3. All metrics are dimensionless ratios defined\nby Equations (10) – (13).\ninto 9% of the total magnetic energy. The fraction of the total\ncurrent that is perpendicular to the field is \u001bJ=0:48 (Wheat-\nland et al. 2000), a rather high value that is not uncommon for\nextrapolation of HMI vector magnetograms with the magneto-\nfrictional method (see Valori et al. 2012, for an application to\nHinode /SP magnetogram with a much lower \u001bJ). Selected\nfield lines of the NLFFF obtained in this way, and referred to\nas NLFFF-R in the following, are shown in the central panel\nof Figure 2.\n4.3.NLFFF from Optimization Method\nFor the second NLFFF model considered in this paper , we\nuse the weighted optimization method (Wiegelmann 2004),\nwhich is an implementation and modification of the origi-\nnal optimization algorithm of Wheatland et al. (2000). The\noptimization method minimizes an integrated joint measure,\nwhich comprises the normalized Lorentz force, the magnetic\nfield divergence, and treatment of the measurement errors,\nover the computational domain (see Wiegelmann & Inhester\n2010; Wiegelmann et al. 2012, for more details).\nTo perform the extrapolation, the vector magnetogram (Fig-\nure 1) is first rebinned to 100per pixel and preprocessed to-\nwards the force-free condition using the method of Wiegel-\nmann et al. (2006). The extrapolation is finally performed\non a uniform grid of 256 \u0002256\u0002200 points covering \u0018\n185\u0002185\u0002144 Mm3. We find a solenoidal error of 1%\nof the total magnetic energy and a fraction of total current\nperpendicular to the magnetic field \u001bJ=0:20. These values\nare lower than for the NLFFF-R model (Section 4.2), which is\ndue both to di \u000berent preprocessing and extrapolation methods\nand strategies.\nIn the following, the NLFFF model built with the optimiza-\ntion method is referred to as NLFFF-O. A set of selected mag-\nnetic field lines is shown in the rightmost panel of Figure 2.\n5.RESULTS\nIn this section, we analyze the results from the connectivity-\nbased helicity flux density calculations. The validation of\nthe maps, error estimations, and qualitative comparisons are\nbriefly presented in Section 5.1. The one-to-one quantitive\ncomparisons are discussed in Section 5.2.\n5.1.Helicity-Flux Density Distribution\nTable 1 presents the validation metrics for the G\bmaps\ncomputed from the three FFF extrapolation models described\nSection 4. For all three G\bmaps, we obtain \u001c\bimb:below 1%\nandCS\bcl:below 5%. This allows us to verify that the mag-\nnetic flux over which the connectivity-based helicity flux den-\nsity is computed is very well balanced and that the calculation\nof dh\b=dtandG\bwell preserves the total helicity flux in that\nregion in the closed magnetic flux. Together, these numbers\nenable us to confirm the accuracy of the G\band d h\b=dtcal-\nculations discussed in the following.TABLE 2\nError estimations from Monte Carlo experiment\nG\u0012 G\b(POT) G\b(NLFFF-R) G\b(NLFFF-O)\n\u001b 3.7 2.6 2.2 3.0\nNote. — The errors are in units of 105Wb2m\u00002s\u00001,i.e.,\u001810\u0000100\ntimes smaller than the typical values displayed by the G\u0012andG\bmaps from\nFigure 3. See Section 3.2 for a description of error calculations.\nFigure 3 presents the surface density of helicity flux from\nthe purely photospheric proxy, G\u0012, and the connectivity-based\nproxy, G\b, computed from all three FFF extrapolations. The\nmean of the absolute value of the helicity flux signal in most\nof the AR ( i.e.,jBzj\u0015100 G) is 2:8\u0002106Wb2m\u00002s\u00001for the\nG\u0012map and 1:7\u0002106Wb2m\u00002s\u00001for the G\bmaps, while\na large fraction of the maps is associated with local helicity\nfluxes of 107Wb2m\u00002s\u00001. As shown in Table 2, the errors\nforG\u0012andG\bestimated from our Monte Carlo experiment\nare 3:7\u0002105Wb2m\u00002s\u00001forG\u0012and lower than 3\u0002105Wb2\nm\u00002s\u00001forG\b. The signal intensity of the surface density\nmaps in Figure 3 is thus well above the estimated noise level.\nFigure 3 shows that the largest di \u000berences in helicity flux\ndensity maps are between G\u0012and the three G\bmaps. In par-\nticular, the strongest di \u000berences are associated with magnetic\nflux systems that connect footpoints of opposite G\u0012signs, i.e.,\nfootpoints of NN connected to NP, footpoints of SN connected\nto SP, and footpoints of SN connected to NP. On the other\nhand, the G\u0012map is relatively similar to the three G\bmaps\nfor the flux system connecting NN to SP, because magnetic\nfield lines are connecting footpoints with similar values of\nG\u0012(x)=jBn(x)j. This is consistent with the work of Pariat et al.\n(2005) and Dalmasse et al. (2014) who showed that G\u0012hides\nthe true helicity flux signal when simultaneous opposite helic-\nity fluxes are present in a magnetic configuration. This e \u000bect\nis inherent to the definition of G\u0012that does not acknowledge\nthe fact that the variation of magnetic helicity in an elemen-\ntary magnetic flux tube comes from the motions of its two\nphotospheric footpoints with respect to the other elementary\nflux tubes of the entire magnetic configuration. As a result, the\ncomparison of the total positive helicity flux and total nega-\ntive helicity flux from G\u0012in the closed magnetic flux with the\nsame quantities computed for each one of the G\bmaps leads\nvalues of C+>1:2 and C\u0000>1:5 (see Table 1). Such values\nindicate that G\u0012is a\u000bected by moderate spurious positive sig-\nnals and rather high spurious negative helicity fluxes that are\ncorrected for by the use of G\b(cf.Section 2.3). Figure 3 thus\nhighlights the fact that the redistribution of helicity flux oper-\nated by G\bis crucial to the photospheric mapping of helicity\nflux in ARs, regardless of the coronal magnetic field model-\ning.\nAll three G\bmaps in Figure 3 are in very good qualita-\ntive agreement, showing (i) a negative helicity flux in SN, (ii)\na strong positive helicity flux in NN and SP, and (iii) strong\npositive and negative helicity fluxes with a strongly marked\ninterface in NP. The G\bmaps from the two NLFFF models\nare very similar. Setting aside the white signal in SN, which\nis due to open field-lines where G\bis not computed, and fo-\ncusing on the common areas where all three extrapolations\nhave closed magnetic flux, we find that the helicity flux den-\nsity map derived from the potential field is also similar to the\nG\bmaps derived from the two NLFFF models. The most no-\nticeable di \u000berence is in the south-east part of NN that does\nnot show any significant helicity flux for G\b(POT), contrary8 Dalmasse et al.\nFig. 3.— Surface densities of helicity flux for AR 11158 at \u001806:28 UT on 2011 /02/14 (in units of 107Wb2m\u00002s\u00001) computed from the purely photospheric\nproxy ( G\u0012; top left) and the connectivity-based proxy ( G\b) derived using the three FFF extrapolations (top right, and bottom row). Purple and cyan solid lines\nshow Bz=\u0006500 Gauss isocontours from the original photospheric vector magnetogram of SDO /HMI. The presence of strong white ( i.e., zero) signals in the left\npart of SN and north-right part of NP for all three G\bmaps is due to open-like magnetic flux where d h\b=dtandG\bare not defined, and hence, not computed.\nto both G\b(NLFFF-R) and G\b(NLFFF-O).\nFinally, the good qualitative agreement between the\nconnectivity-based helicity flux density calculations from the\nthree FFF extrapolations is further emphasized by the 3D rep-\nresentation of d h\b=dtin Figure 4. They all show that the inner\npart of the AR is dominated by strong positive helicity fluxes\nand is embedded within a magnetic field region dominated by\nstrong negative helicity fluxes. The core results of Dalmasse\net al. (2013) are thus confirmed with a weak dependance on\nthe extrapolation method.\n5.2.Quantitative Analysis\nWe now focus on the quantitative comparison of the helicity\nflux density calculations obtained from the three FFF models.\nWe restrict our analysis to the pixels of the G\bmaps that are\nabove the noise level (di \u000berent for each map) estimated from\nthe Monte Carlo experiment. The error levels are given in\nTable 2.\nFigure 5 displays the three maps of sign agreement. The re-\ngions where our analysis can be carried out (white and black)\nis mostly associated with the strong magnetic field of AR\n11158, i.e., wherejBzj&500 Gauss. These regions corre-\nspond to the area where most of the helicity flux (at least 88%\nof the total unsigned helicity flux) computed with G\bis com-\ning from. This is because the helicity flux intensity outside\nthese regions is below the noise level of the respective G\b\nmaps. Despite the presence of some relatively small areas of\ndisagreement (black patches), we find that these regions are\ndominated by agreement (white signal) over the sign of he-licity flux derived using di \u000berent FFF extrapolations. In par-\nticular, the percentage of surface area for which pairs of G\b\nmaps agree is always larger than 85%, which translates into\nmore than\u001984% in terms of magnetic flux. Therefore, the lo-\ncal sign of helicity flux computed from the connectivity-based\nhelicity flux density method is very robust to the di \u000berent FFF\nmodels and assumptions used for calculations.\nFigure 6 displays the linear correlation plots of G\bvalues in\neach pixel, for di \u000berent pairs of FFF extrapolations. For com-\nparisons between G\bfrom the potential field model and one\nof the two NLFFFs, the points are colored according to the\nstrength of the photospheric electric current density, jjzj, from\nthe di \u000berent preprocessed boundary employed in the NLFFF\nmodel under consideration. For the plot comparing G\bfrom\nthe two NLFFF extrapolations, the points are colored accord-\ning to (jjz(NLFFF-R)j\u0001jjz(NLFFF-O)j)1=2. Such a color coding\nwas introduced in order to investigate the dependency of the\nscatter on the electric current density of magnetic field lines\nwhere G\bis computed.\nIn each scatter plot, we find that the spatial distribution\nof points exhibits a clear ellipsoidal shape aligned along the\ny=xdiagonal line. Each one of these distributions display a\nmoderate dispersion. The three standard deviations computed\nfrom each scatter plot of Figure 6 are \u00142:6\u0002106Wb2m\u00002\ns\u00001, which is 5 to 10 times smaller than most of the signal in\nthe four main magnetic polarities. These standard deviations\nare\u001910 times larger than the G\berrors estimated from the\nMonte Carlo experiment (Table 2). As anticipated, it means\nthat, despite the substantial agreement on the sign of the in-Studying the transfer of magnetic helicity in solar active regions 9\nFig. 4.— 3D representation of the connectivity-based helicity flux density for each FFF extrapolation. The magnetic field lines were integrated from the same\n-randomly selected- photospheric footpoints for all three extrapolations. They are colored according to their d h\b=dtvalue. Purple and cyan solid lines show\nBz=\u0006500 Gauss isocontours from the FFF extrapolation.\nTABLE 3\nScatter plots dispersion vs .electric current density\njz ]0;0:2] ]0 :2;3:1] ]3 :1;15:6] ]15 :6;50:0]\n\u001bG\b[1:5;1:8] [1 :8;2:0] [2 :6;3:6] [3 :0;4:1]\nNote. — The interval for the standard deviation, \u001bG\b, is derived from the\nvalues obtained for the three scatter plots of Figure 6. The electric current\ndensity is in units of mA m\u00002and the dispersion in units of 106Wb2m\u00002\ns\u00001. From left to right, the five ranges of electric current density correspond\nto ]black; dark blue], ]dark blue; green], ]green; yellow], and ]yellow; red]\nof the color-scale in Figure 6.\njected helicity between di \u000berent extrapolations, a significant\nuncertainty in the connectivity-based calculations is coming\nfrom the choice of extrapolation model used to derive the field\nline connectivity.Figure 6 further shows that the ellipsoidal pattern of the\ndistribution of points is present independently of the electric\ncurrent density, i.e., of the color of the points. We also no-\ntice that the scattering of points away from the y=xline\npresents some relatively weak dependance on the electric cur-\nrent density of the field lines used for the connectivity-based\ncalculations. In particular, we find a dispersion in the range\n\u0019[1:5;2:0]\u0002106Wb2m\u00002s\u00001for black to green points, and\n\u0019[2:6;4:1]\u0002106Wb2m\u00002s\u00001for green to red points (finer de-\ntails are provided in Table 3). In addition, the number of pix-\nels with average-to-strong electric current density ( i.e., green\nto red points) is sensibly the same as the number of pixels\nwith very weak-to-average electric current density ( i.e., black\nto green points). This implies that the correlation coe \u000ecients,\ndisplayed in Figure 6 and discussed below, are not dominated\nby the di \u000berences in G\bvalues from nearly-potential mag-10 Dalmasse et al.\nFig. 5.— Maps of G\b-sign agreement between the surface-density maps ob-\ntained with the three FFF models. For each map, [white; black] =[agree;\ndisagree], while gray corresponds to pixels that are either associated with\nopen-like field-lines (where G\bis not computed) or where G\bis below the\nnoise level for at least one of the two models being compared. Purple and\ncyan solid lines show Bz=\u0006500 Gauss isocontours from the original photo-\nspheric magnetogram of SDO /HMI.\nnetic field lines.\nFor all three scatter plots, we find that the Pearson, cP, and\nSpearman, cS, correlation coe \u000ecients are such that cP\u00150:75\nandcS\u00150:72. We checked that these values are statistically\nsignificant by conducting a null hypothesis test (details and\nresults of this test are provided in Appendix A). We therefore\nconclude that the calculations of the connectivity-based helic-\nity flux density derived from di \u000berent FFF extrapolations are\nhighly correlated and consistent with each other.\nFor further comparison, we compute the vector correla-\ntion metric, Cvec, comparing the three 3D magnetic field\nextrapolations as defined by Equation (28) of Schrijver\net al. (2006). For that purpose, the POT and NLFFF-\nR 3D magnetic fields are interpolated on the same gridas the NLFFF-O (whose extrapolation domain is common\nto all three models) using trilinear interpolation. We find\nCvec(BNLFFF\u0000R;BPOT)=0:84,Cvec(BNLFFF\u0000O;BPOT)=0:90,\nandCvec(BNLFFF\u0000O;BNLFFF\u0000R)=0:90. Such values for the\nvector correlation metric of the magnetic fields are very high\neven though the 3D distributions of the magnetic field lines\nare relatively di \u000berent when comparing the three extrapola-\ntions, as inferred from Figure 2. We thus find that the vector\ncorrelation metric for the magnetic fields is higher than the\nPearson and Spearman correlation coe \u000ecients found when\ncomparing the helicity flux density calculations.\n6.DISCUSSION\nAt this point, we wish to emphasize that the robustness\nof the connectivity-based method against di \u000berent magnetic\nfield extrapolations does not mean that the extrapolations are\nvery much alike. On the contrary, all three extrapolations\nconsidered here produce 3D magnetic fields that are di \u000berent\nfrom each other as shown in Figure 2. So, what does it mean\nthat the connectivity-based helicity flux density calculations\nare robust against di \u000berent extrapolation methods?\nFirst of all, we remind the reader that the surface-density\nof helicity flux, G\b, only explicitly depends on the connectiv-\nity of magnetic field lines and not on their 3D geometry; the\nlatter only has an implicit e \u000bect on G\bby a\u000becting the mag-\nnetic connectivity. Secondly, expanding Equation (9) using\nEquation (7) leads to\nG\b(x+)=jBn(x+)j\n2\u0010˙\u0002B(x+)\u0000˙\u0002B(x\u0000)\u0011\n: (15)\nEquation (15) shows that, for a given footpoint x+, the helic-\nity flux density G\bfrom di \u000berent extrapolation models will\nbe exactly the same (1) if they have the exact same magnetic\nfield connectivity, or (2) if they have a di \u000berent connectivity,\nthe first extrapolation links x+tox1\u0000and the second links x+\ntox2\u0000,x1\u0000, but ˙\u0002B(x2\u0000)=˙\u0002B(x1\u0000). The same type of con-\nclusion can be drawn for G\b(x\u0000) by simply exchanging x+and\nx\u0000. Note that condition (2) relates to the spatial smoothness\nof˙\u0002B(x) and, by extension, G\u0012(x).\nIn general, G\bcan distinguish two magnetic fields that have\ndi\u000berent photospheric magnetic connectivities. This is ev-\nident from the connectivity-based flux density maps shown\nFigure 3 and the clear dispersion shown in the scatter plots\nof Figure 6. However, we argue that as long as two di \u000berent\n3D magnetic fields have, on average , a similar magnetic con-\nnectivity, then the G\bmaps computed from these magnetic\nfields should display a good overall agreement. Note that, al-\nthough not discussed, this was already observed by Dalmasse\net al. (2014) for the models analyzed in their Figures 8, 10,\n11, and 12. The strongest di \u000berences would then be expected\nin localized regions where the magnetic connectivity from the\nextrapolations is markedly di \u000berent, typically in the close sur-\nroundings of quasi-separatrix layers (QSLs) which are regions\nof strong gradients of the magnetic connectivity that are favor-\nable to magnetic reconnection ( e.g., D´emoulin et al. 1996;\nTitov et al. 2002; Aulanier et al. 2006; Janvier et al. 2013).\nThis is indeed what we find in our analysis of AR 11158, e.g.,\nat the interface of positive and negative helicity fluxes in NP\n(see Figure 3) that coincides with a large-scale QSL that sep-\narates field-lines connecting NP to NN and NP to SN.\nThen, we recall that G\bis only a 2D representation of the\nphysical, 3D, definition of local magnetic helicity variation,\ndh\b=dt. dh\b=dtis only defined for elementary magnetic fluxStudying the transfer of magnetic helicity in solar active regions 11\nFig. 6.— Scatter plots of pixel-to-pixel comparison of the surface-density of helicity flux. The black solid line shows the y=xline. “Pearson”, “Spearman”, and\n“Stand. dev.” respectively are the Pearson correlation coe \u000ecient, the Spearman correlation coe \u000ecient, and the standard deviation. The standard deviation is in\nunits of 107Wb2m\u00002s\u00001. From the left to the right panel, the color scale corresponds to jjz(NLFFF-R)j,jjz(NLFFF-O)j, andp\njjz(NLFFF-R)j\u0001jjz(NLFFF-O)j.\nNotice the color scale for the current density is not linear, but was instead chosen as k\u0001k1=4with saturation at 50 mA m\u00002for dynamic range optimization.\ntubes. Its 3D representation thus requires to plot individual\nfield lines. Since di \u000berent magnetic field extrapolation meth-\nods and assumptions generally produce 3D magnetic fields\nthat can di \u000ber significantly in the details of individual field\nlines ( e.g., DeRosa et al. 2009), then, di \u000berently from G\b,\ndh\b=dtcan always di \u000berentiate two magnetic fields. This is\nindeed what we see in Figure 4 where the three plots of d h\b=dt\nare easily distinguishable because of the di \u000berent 3D geom-\netry of magnetic field lines. Thus, the robustness of d h\b=dt\nagainst di \u000berent extrapolation models should be understood\nin terms of average or global distribution of helicity flux den-\nsity over the di \u000berent magnetic flux systems of an AR, and\nnot in terms of a one-to-one field-line and d h\b=dtcorrespon-\ndence. For AR 11158, this means looking at the AR in terms\nof the four flux systems NN-NP, NN-SP, SN-SP, and SN-NP.\nWhile the actual distribution of the magnetic field-lines and\ndh\b=dtin these four flux systems vary from one extrapolation\nto the other (see Figure 4), the sign and average helicity flux\nintensities agree very well, hence the robustness of the calcu-\nlations.\nOn the other hand, local di \u000berences exist in the\nconnectivity-based helicity flux density calculations per-\nformed with di \u000berent extrapolations. Such local di \u000berences\ncan be significant and are extremely important for physically\ninterpretating the local helicity flux, i.e., at the scale of a par-\nticular field line. This is illustrated in Figure 7 that shows\ndh\b=dtfor five field-lines that have been integrated from the\nsame photospheric starting footpoints for all three extrapola-\ntions of AR 11158. The connectivity and 3D geometry of\nthese field lines strongly di \u000ber from one extrapolation to the\nother, which results in di \u000berent helicity flux intensities and\nsigns. For instance, field-line 5 links SN to a small-scale pos-\nitive magnetic polarity on the north of SN with a strong neg-\native helicity flux for POT, while it links SN to NP with a\nmedium negative helicity flux for NLFFF-R, and links SN to\nSP with a medium positive helicity flux for NLFFF-O. When\ncomparing the three extrapolations, the five field-lines dis-\nplayed in Figure 7 are so di \u000berent in geometry and orientation\nwith respect to each other that the physical interpretation of\ntheir helicity flux density – based on field-lines reorientationin response to the motions of their photospheric footpoints ( cf.\nSection 5 and Figure 9 of Dalmasse et al. 2014), is entirely ex-\ntrapolation dependent.\nConsidering the current limitations of magnetic field ex-\ntrapolations, we conclude that the physical interpretation of\nthe connectivity-based helicity flux density calculations in ob-\nservational analyses will be robust at the scale of the di \u000berent\nflux systems forming an AR, but not necessarily at the ex-\ntremely local scale of individual magnetic field lines for which\ninterpretation of the signal should be taken with a lot of cau-\ntion.\nFinally, we recall that the analysis presented in this paper\nwas conducted using NLFFF extrapolations performed with a\nfinite field of view. As mentioned Section 4.1, this is a limita-\ntion since it disregards the e \u000bect of the distant quiet Sun and\nsurrounding ARs that are outside the field of view considered\nfor the extrapolation. Such e \u000bects would need another study\nwith large-scale magnetic field extrapolations, or even with\nfull-Sun numerical simulations.\n7.CONCLUSION\nThanks to the conservation properties of magnetic helic-\nity in the solar atmosphere, studying the photospheric flux\nof magnetic helicity appears to be a key element for improv-\ning our understanding of how this fundamental quantity af-\nfects the dynamics of solar active regions (ARs). For that pur-\npose, a connectivity-based helicity flux density method, built\nupon the work of Pariat et al. (2005), was recently developed\nand tested on various analytical case-studies and numerical\nmagnetohydrodynamics (MHD) simulations (Dalmasse et al.\n2014). The ability of this method to correctly capture the lo-\ncal transfer of magnetic helicity relies on its exploitation of\nthe connectivity of magnetic field lines, which enables it to\nembrace the 3D and global nature of magnetic helicity.\nFor the solar atmosphere, the application of the\nconnectivity-based helicity flux density method relies\non approximate 3D solutions obtained from force-free field\n(FFF) extrapolations of the photospheric magnetic field to\nderive the connectivity of magnetic field lines. In general,\nsuch FFF models provide reconstructed magnetic fields\nwhose 3D distribution strongly depends on the extrapola-12 Dalmasse et al.\nFig. 7.— 3D representation (top view) of the connectivity-based helicity flux\ndensity, as in Figure 4, illustrating local di \u000berences between d h\b=dtfrom the\ndi\u000berent FFF extrapolations. The magnetic field lines are labelled accord-\ning to the common photospheric footpoint from which they were integrated\nand which is indicated by the yellow disk. The color scale for the vertical\nmagnetic field (gray scale) and its isocontours are the same as in Figure 4.\ntion method used ( e.g., DeRosa et al. 2009, 2015). As a\nconsequence, the values of subsequently derived quantities,\nsuch as free magnetic energy and magnetic helicity, exhibit\nlarge variations from one FFF model to another. Since the\nmagnetic connectivity also depends on the 3D distribution\nof the extrapolated magnetic field, the connectivity-based\nhelicity flux density calculations may be strongly a \u000bected by\nthe choice of FFF reconstruction method. In this paper, we\naddressed this concern by applying the connectivity-based\napproach to solar observations with di \u000berent magnetic field\nextrapolation models and implementations.\nTo assess the reliability and relevance of the connectivity-\nbased helicity flux density method to solar observations, we\nconsidered the internally complex (several bipoles) and ex-\nternally simple (i.e., no neighboring large-flux system) AR11158 using the vector magnetogram data from SDO /HMI.\nThree FFF extrapolations, i.e., a potential field, a nonlin-\near FFF (NLFFF) extrapolation using the magneto-frictional\nmethod of Valori et al. (2010), and a second NLFFF from the\noptimization method of Wiegelmann (2004), were performed\nto reconstruct the coronal magnetic field of AR 11158 and ap-\nply the connectivity-based approach. Our analysis indicates\nthat the helicity flux density calculations derived from dif-\nferent FFF extrapolations are highly correlated (with Pearson\nand Spearman correlation coe \u000ecients larger than\u00190:72) and\nconsistent with each other, showing a very good agreement\nover identifying the local sign of helicity flux ( i.e., for more\nthan\u001985% of the surface where they were compared). We\nthus conclude that the connectivity-based helicity flux density\nmethod can be reliably used in observational analyses of ARs.\nThe results presented in this paper also enable us to propose\na procedure for estimating uncertainties in the connectivity-\nbased helicity flux density calculations applied to solar obser-\nvations, as follows:\n1. Perform a Monte Carlo experiment, as described in\nSection 3.2 and proposed by Liu & Schuck (2012), by\nadding random noise with a Gaussian distribution to the\nphotospheric vector magnetic fields used for computa-\ntion and propagate it through the chain of helicity flux\ndensity calculations. This allows to estimate uncertain-\nties related with magnetic field measurement errors for\nthe flux transport velocity, G\u0012, and d h\b=dtandG\bfrom\nthe NLFFF method chosen for the analysis.\n2. Apply the connectivity-based calculations with the\nNLFFF and the potential field to derive the standard\nerror from the comparison of G\bcomputed with each\nmagnetic field model. This allows to derive the error\ncontribution related with the uncertainty in the mag-\nnetic field connectivity due to the choice of magnetic\nfield extrapolation method.\n3. Sum the squared errors of G\bto estimate the overall\nuncertainty in helicity flux density calculations.\nThis procedure, and in particular step 2, is motivated by the\nfact that our analysis indicates that comparing the calculations\nwith the potential field and one of the two NLFFF models\ngives a standard error that is extremely close to the standard\nerror obtained from comparing the helicity flux density calcu-\nlations from the two NLFFF models. Computing the potential\nfield is relatively inexpensive as compared with computing an\nNLFFF model and, in fact, is already part of most NLFFF al-\ngorithms (see e.g., review by Wiegelmann & Sakurai 2012)\nthat use it as an initial state.\nThe reliability of the connectivity-based helicity flux den-\nsity calculations against di \u000berent FFF models o \u000bers several\ninteresting perspectives for analyzing the 2D and 3D transfer\nof magnetic helicity in solar ARs. During the early stages of\nAR formation, the connectivity-based method provides infor-\nmation on the distribution of magnetic helicity in the emerg-\ning magnetic field, which is an important constraint for mod-\nels of generation and transport of magnetic flux in the solar\nconvection zone ( e.g., Berger & Ruzmaikin 2000; Pariat et al.\n2007; Vemareddy & D ´emoulin 2017).\nOn the other hand, the study of the helicity flux distribution\nat later stages of ARs evolution allows to track the sites where\nmagnetic helicity is transferred to the corona, probing in this\nway the relationships between magnetic helicity accumulationStudying the transfer of magnetic helicity in solar active regions 13\nand the energetics of solar flares and coronal mass ejections\n(CMEs). The connectivity-based approach may further be\nused to test the very high-energy flare model of Kusano et al.\n(2004) based on magnetic helicity annihilation. Such a flare\nmodel requires the prior transfer and accumulation of mag-\nnetic helicity of opposite signs in di \u000berent magnetic flux sys-\ntems of an AR that would later reconnect together. Identifying\nAR candidates for hosting such a flare model requires reliable\nmaps that are not polluted by false helicity flux signals of op-\nposite signs. The present study shows that the connectivity-\nbased helicity flux density method is very well suited for that\npurpose.\nIn summary, the connectivity-based helicity flux density\nmethod is a very promising tool for helping us unveil the role\nof magnetic helicity in the dynamics of the solar corona.\nWe thank the anonymous referee for helpful comments that\nimproved the paper. K.D. thanks D. Nychka for his help on\nthe Monte Carlo experiment, and R. Centeno Elliott and A.\nGri˜n´on Mar ´ın for their insights on HMI noise characteriza-tion. K.D. acknowledges funding from the Advanced Study\nProgram, the High Altitude Observatory, and the Computa-\ntional and Information Systems Laboratory. E.P. acknowl-\nedges the support of the French Agence Nationale pour la\nRecherche through the HELISOL project, contract n\u000eANR-\n15-CE31-0001. G.V . acknowledges the support of the Lev-\nerhulme Trust Research Project Grant 2014-051. J.J. is sup-\nported by NASA grant NNX16AF72G and 80NSSC17K0016.\nThe data used here are courtesy of the NASA /SDO and the\nHMI science team. This work used the DA VE4VM code\nwritten and developed by P. Schuck at the Naval Research\nLaboratory. The helicity flux calculations were performed\npartly on the high-performance computing system Yellow-\nstone (ark: /85065 /d7wd3xhc) provided by NCAR’s Compu-\ntational and Information Systems Laboratory, sponsored by\nthe National Science Foundation, and partly on the multi-\nprocessors TRU64 computer of the LESIA. The National\nCenter for Atmospheric Research is sponsored by the Na-\ntional Science Foundation.\nREFERENCES\nAly, J.-J. 2014, in Journal of Physics Conference Series, V ol. 544, Journal of\nPhysics Conference Series, 012003\nAmari, T., Boulmezaoud, T. Z., & Aly, J. J. 2006, A&A, 446, 691\nAntiochos, S. K. 2013, ApJ, 772, 72\nAulanier, G., Pariat, E., D ´emoulin, P., & DeV ore, C. R. 2006, Sol. Phys.,\n238, 347\nBerger, M. A. 1984, Geophysical and Astrophysical Fluid Dynamics, 30, 79\nBerger, M. A. 1988, A&A, 201, 355\nBerger, M. A. & Field, G. B. 1984, Journal of Fluid Mechanics, 147, 133\nBerger, M. A. & Ruzmaikin, A. 2000, J. Geophys. Res., 105, 10481\nBobra, M. G., Sun, X., Hoeksema, J. T., et al. 2014, Sol. Phys., 289, 3549\nChae, J. 2001, ApJ, 560, L95\nChae, J., Moon, Y .-J., & Park, Y .-D. 2004, Sol. Phys., 223, 39\nChae, J., Wang, H., Qiu, J., et al. 2001, ApJ, 560, 476\nChandra, R., Pariat, E., Schmieder, B., Mandrini, C. H., & Uddin, W. 2010,\nSol. Phys., 261, 127\nCox, D. R. 2006, Principles of Statistical Inference (Cambridge University\nPress), iSBN 9780521685672\nDalmasse, K., Nychka, D., Gibson, S., Flyer, N., & Fan, Y . 2016, Frontiers\nin Astronomy and Space Sciences, 3, 24\nDalmasse, K., Pariat, E., D ´emoulin, P., & Aulanier, G. 2014, Sol. Phys., 289,\n107\nDalmasse, K., Pariat, E., Valori, G., D ´emoulin, P., & Green, L. M. 2013,\nA&A, 555, L6\nD´emoulin, P. & Berger, M. A. 2003, Sol. Phys., 215, 203\nD´emoulin, P., Mandrini, C. H., Van Driel-Gesztelyi, L., Lopez Fuentes,\nM. C., & Aulanier, G. 2002a, Sol. Phys., 207, 87\nD´emoulin, P., Mandrini, C. H., van Driel-Gesztelyi, L., et al. 2002b, A&A,\n382, 650\nD´emoulin, P. & Pariat, E. 2009, Advances in Space Research, 43, 1013\nD´emoulin, P., Priest, E. R., & Lonie, D. P. 1996, J. Geophys. Res., 101, 7631\nDeRosa, M. L., Schrijver, C. J., Barnes, G., et al. 2009, ApJ, 696, 1780\nDeRosa, M. L., Wheatland, M. S., Leka, K. D., et al. 2015, ApJ, 811, 107\nFinn, J. M. & Antonsen, T. M., J. 1985, Comments Plasma Phys. Controlled\nFusion, 9, 111\nFuhrmann, M., Seehafer, N., & Valori, G. 2007, A&A, 476, 349\nFuhrmann, M., Seehafer, N., Valori, G., & Wiegelmann, T. 2011, A&A, 526,\nA70\nGeorgoulis, M. K., Rust, D. M., Pevtsov, A. A., Bernasconi, P. N., &\nKuzanyan, K. M. 2009, ApJ, 705, L48\nGibson, S., Kucera, T., White, S., et al. 2016, Frontiers in Astronomy and\nSpace Sciences, 3, 8\nGrad, H. & Rubin, H. 1958, in Proc. of the Second UN International Atomic\nEnergy Conf. 31, Peaceful Uses of Atomic Energy\nGreen, L. M., L ´opez fuentes, M. C., Mandrini, C. H., et al. 2002, Sol. Phys.,\n208, 43\nGuo, Y ., Ding, M. D., Cheng, X., Zhao, J., & Pariat, E. 2013, ApJ, 779, 157\nHeyvaerts, J. & Priest, E. R. 1984, A&A, 137, 63\nHoeksema, J. T., Liu, Y ., Hayashi, K., et al. 2014, Sol. Phys., 289, 3483Inoue, S., Hayashi, K., Shiota, D., Magara, T., & Choe, G. S. 2013, ApJ,\n770, 79\nInoue, S., Magara, T., Watari, S., & Choe, G. S. 2012, ApJ, 747, 65\nInoue, S. & Morikawa, Y . 2011, Plasma and Fusion Research, 6, 2401067\nJanvier, M., Aulanier, G., Pariat, E., & D ´emoulin, P. 2013, A&A, 555, A77\nJeong, H. & Chae, J. 2007, ApJ, 671, 1022\nJiang, C. & Feng, X. 2012, ApJ, 749, 135\nJiang, Y ., Zheng, R., Yang, J., et al. 2012, ApJ, 744, 50\nJing, J., Park, S.-H., Liu, C., et al. 2012, ApJ, 752, L9\nKazachenko, M. D., Canfield, R. C., Longcope, D. W., & Qiu, J. 2012,\nSol. Phys., 277, 165\nKlimchuk, J. A. & Sturrock, P. A. 1992, ApJ, 385, 344\nKnizhnik, K. J., Antiochos, S. K., & DeV ore, C. R. 2015, ApJ, 809, 137\nKramar, M., Airapetian, V ., Miki ´c, Z., & Davila, J. 2014, Sol. Phys., 289,\n2927\nKusano, K., Maeshiro, T., Yokoyama, T., & Sakurai, T. 2002, ApJ, 577, 501\nKusano, K., Maeshiro, T., Yokoyama, T., & Sakurai, T. 2004, ApJ, 610, 537\nKusano, K., Yokoyama, T., Maeshiro, T., & Sakurai, T. 2003, Advances in\nSpace Research, 32, 1931\nLaBonte, B. J., Georgoulis, M. K., & Rust, D. M. 2007, ApJ, 671, 955\nLeka, K. D., Canfield, R. C., McClymont, A. N., & van Driel-Gesztelyi, L.\n1996, ApJ, 462, 547\nLinton, M. G., Dahlburg, R. B., & Antiochos, S. K. 2001, ApJ, 553, 905\nLiu, Y . & Schuck, P. W. 2012, ApJ, 761, 105\nLiu, Y ., Zhao, J., & Schuck, P. W. 2013, Sol. Phys., 287, 279\nLongcope, D. W. & Welsch, B. T. 2000, ApJ, 545, 1089\nLow, B. C. 1996, Sol. Phys., 167, 217\nMalanushenko, A., Schrijver, C. J., DeRosa, M. L., Wheatland, M. S., &\nGilchrist, S. A. 2012, ApJ, 756, 153\nMandrini, C. H., D ´emoulin, P., van Driel-Gesztelyi, L., et al. 2004, Ap&SS,\n290, 319\nMetcalf, T. R., De Rosa, M. L., Schrijver, C. J., et al. 2008, Sol. Phys., 247,\n269\nMo\u000batt, H. K. 1969, Journal of Fluid Mechanics, 35, 117\nMoon, Y .-J., Chae, J., Choe, G. S., et al. 2002, ApJ, 574, 1066\nMoore, D. S. & McCabe, G. P. 2003, Introduction to the Practice of\nStatistics (Freeman, W. H.), iSBN 9780716796572\nMoreno-Insertis, F. 1997, Mem. Soc. Astron. Italiana, 68, 429\nNeyman, J. & Pearson, E. S. 1933, Philosophical Transactions of the Royal\nSociety of London A: Mathematical, Physical and Engineering Sciences,\n231, 289\nPariat, E., D ´emoulin, P., & Berger, M. A. 2005, A&A, 439, 1191\nPariat, E., D ´emoulin, P., & Nindos, A. 2007, Advances in Space Research,\n39, 1706\nPariat, E., Leake, J. E., Valori, G., et al. 2017, A&A, 601, A125\nPariat, E., Nindos, A., D ´emoulin, P., & Berger, M. A. 2006, A&A, 452, 623\nPariat, E., Valori, G., D ´emoulin, P., & Dalmasse, K. 2015, A&A, 580, A128\nPark, S.-H., Kusano, K., Cho, K.-S., et al. 2013, ApJ, 778, 1314 Dalmasse et al.\nPesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, Sol. Phys., 275,\n3\nPevtsov, A. A. 2012, Astrophysics and Space Science Proceedings, 30, 83\nPlowman, J. 2014, ApJ, 792, 23\nPoisson, M., Mandrini, C. H., D ´emoulin, P., & L ´opez Fuentes, M. 2015,\nSol. Phys., 290, 727\nRachmeler, L. A., Casini, R., & Gibson, S. E. 2012, in Astronomical Society\nof the Pacific Conference Series, V ol. 463, Second ATST-EAST Meeting:\nMagnetic Fields from the Photosphere to the Corona., ed. T. R. Rimmele,\nA. Tritschler, F. W ¨oger, M. Collados Vera, H. Socas-Navarro,\nR. Schlichenmaier, M. Carlsson, T. Berger, A. Cadavid, P. R. Gilbert,\nP. R. Goode, & M. Kn ¨olker, 227\nRomano, P., Zuccarello, F. P., Guglielmino, S. L., & Zuccarello, F. 2014,\nApJ, 794, 118\nRussell, A. J. B., Yeates, A. R., Hornig, G., & Wilmot-Smith, A. L. 2015,\nPhysics of Plasmas, 22, 032106\nRust, D. M. 1994, Geophys. Res. Lett., 21, 241\nSchmidt, H. U. 1964, NASA Special Publication, 50, 107\nSchou, J., Borrero, J. M., Norton, A. A., et al. 2012, Sol. Phys., 275, 327\nSchrijver, C. J., Aulanier, G., Title, A. M., Pariat, E., & Delann ´ee, C. 2011,\nApJ, 738, 167\nSchrijver, C. J., De Rosa, M. L., Metcalf, T. R., et al. 2006, Sol. Phys., 235,\n161\nSchrijver, C. J., Title, A. M., Yeates, A. R., & DeRosa, M. L. 2013, ApJ,\n773, 93\nSchuck, P. W. 2008, ApJ, 683, 1134\nSun, X., Hoeksema, J. T., Liu, Y ., et al. 2012, ApJ, 748, 77\nTaylor, J. B. 1974, Physical Review Letters, 33, 1139\nTaylor, J. B. 1986, Reviews of Modern Physics, 58, 741\nTitov, V . S., Hornig, G., & D ´emoulin, P. 2002, J. Geophys. Res., 107, 1164\nToriumi, S., Iida, Y ., Kusano, K., Bamba, Y ., & Imada, S. 2014, Sol. Phys.,\n289, 3351\nTziotziou, K., Georgoulis, M. K., & Raouafi, N.-E. 2012, ApJ, 759, L4\nValori, G., D ´emoulin, P., Pariat, E., & Masson, S. 2013, A&A, 553, A38Valori, G., Green, L. M., D ´emoulin, P., et al. 2012, Sol. Phys., 278, 73\nValori, G., Kliem, B., & Fuhrmann, M. 2007, Sol. Phys., 245, 263\nValori, G., Kliem, B., T ¨or¨ok, T., & Titov, V . S. 2010, A&A, 519, A44\nValori, G., Pariat, E., Anfinogentov, S., et al. 2016, Space Sci. Rev., 201, 147\nvan Ballegooijen, A. A. & Martens, P. C. H. 1989, ApJ, 343, 971\nVemareddy, P. 2015, ApJ, 806, 245\nVemareddy, P., Ambastha, A., & Maurya, R. A. 2012a, ApJ, 761, 60\nVemareddy, P., Ambastha, A., Maurya, R. A., & Chae, J. 2012b, ApJ, 761,\n86\nVemareddy, P. & D ´emoulin, P. 2017, A&A, 597, A104\nWheatland, M. S. 2007, Sol. Phys., 245, 251\nWheatland, M. S., Sturrock, P. A., & Roumeliotis, G. 2000, ApJ, 540, 1150\nWiegelmann, T. 2004, Sol. Phys., 219, 87\nWiegelmann, T. & Inhester, B. 2010, A&A, 516, A107\nWiegelmann, T., Inhester, B., & Sakurai, T. 2006, Sol. Phys., 233, 215\nWiegelmann, T. & Sakurai, T. 2012, Living Reviews in Solar Physics, 9, 5\nWiegelmann, T., Thalmann, J. K., Inhester, B., et al. 2012, Sol. Phys., 281,\n37\nWoltjer, L. 1958, Proceedings of the National Academy of Science, 44, 489\nYamamoto, T. T., Kusano, K., Maeshiro, T., Yokoyama, T., & Sakurai, T.\n2005, ApJ, 624, 1072\nYang, S., Zhang, H., & B ¨uchner, J. 2009, A&A, 502, 333\nYeates, A. R. & Hornig, G. 2011, Physics of Plasmas, 18, 102118\nYeates, A. R. & Hornig, G. 2013, Physics of Plasmas, 20, 012102\nYeates, A. R. & Hornig, G. 2014, in Journal of Physics Conference Series,\nV ol. 544, Journal of Physics Conference Series, 012002\nYeates, A. R., Russell, A. J. B., & Hornig, G. 2015, Proceedings of the\nRoyal Society of London Series A, 471\nZhang, M., Flyer, N., & Low, B. C. 2006, ApJ, 644, 575\nZhang, Y ., Kitai, R., & Takizawa, K. 2012, ApJ, 751, 85\nZhang, Y ., Tan, B., & Yan, Y . 2008, ApJ, 682, L133\nZhu, X. S., Wang, H. N., Du, Z. L., & Fan, Y . L. 2013, ApJ, 768, 119\nAPPENDIX\nNULL HYPOTHESIS TESTING\nTo determine whether the values of correlation coe \u000ecients reported in Section 5.2 are statistically significant, we perform a null\nhypothesis test ( e.g., Neyman & Pearson 1933; Moore & McCabe 2003; Cox 2006). The null hypothesis states that an observed\nresult, or relationship between two variables, is due to random processes alone. The null hypothesis test is an argumentum ad\nabsurdum approach. The goal is to show that an observed relationship or result is very unlikely to occur under the null hypothesis,\nin which case the null hypothesis can be rejected and the alternative accepted. In the context of this paper, they can be formulated\nas follows\n- Null hypothesis: the values of G \bcomputed from di \u000berent FFF models are not correlated .\n- Alternative hypothesis: the values of G \bcomputed from di \u000berent FFF models are correlated .\nTo test this null hypothesis, we perform a permutation test. Let X=fx1;x2;:::;xngandY=fy1;y2;:::;yngbe two datasets.\nc(0)\nS=cS(X;Y) is the Spearman correlation coe \u000ecient of the two original datasets XandY, and for which we want to determine\nthe statistical significance. The permutation test consists in the following steps\n1. Create a new dataset Y(k)by randomly permuting the elements of Y; for instance, Y(k)=fy4;yn\u000010;:::;y2g.\n2. Compute the Spearman correlation coe \u000ecient, c(k)\nS=cS(X;Y(k)).\n3. Repeat steps (1) and (2) Ntimes, where Nis large (typically larger than 1000). This leads to Nsets of random permutations\nofY, and hence, N+1 Spearman correlation coe \u000ecients,fc(0)\nS;c(1)\nS;c(2)\nS;c(3)\nS;:::;c(N)\nSg.\nWe then determine the p-value of c(0)\nS,i.e., the probability of obtaining the Spearman correlation coe \u000ecient, c(0)\nS, between the two\noriginal datasets, XandY, if the null hypothesis were true. The p-value associated with c(0)\nSand estimated from the permutation\ntest is the fraction of c(k=f0;1;:::;ng)\nSthat are larger than the Spearman correlation coe \u000ecient from the two original datasets, c(0)\nS,i.e.,\np=m\nN+1; (A1)\nwhere mis the number of c(k=f0;1;:::;ng)\nSthat are\u0015c(0)\nS. The same method is applied with the Pearson correlation coe \u000ecient.\nWe reject the null hypothesis and accept the alternative if the estimated p-value for both the Pearson and Spearman correlation\ncoe\u000ecients is strictly smaller than 0 :001, i.e., the level below which we consider the correlation coe \u000ecients from the two original\ndataset to be statistically significant.Studying the transfer of magnetic helicity in solar active regions 15\nTABLE 4\nPermutation tests for the Spearman correlation coefficient of G\b(POT) vs.G\b(NLFFF-R)\nN \u0016S \u001bS c(0)\nSp\u0010\nc(0)\nS\u0011\n1048:8\u000210\u000051:0\u000210\u000020.72 1 :0\u000210\u00004\n105\u00006:3\u000210\u000067:4\u000210\u000030.72 1 :0\u000210\u00005\n106\u00005:4\u000210\u000067:1\u000210\u000030.72 1 :0\u000210\u00006\nNote. —\u0016Sand\u001bSare the mean and standard deviation of c(k=f0;1;:::;ng)\nS.c(0)\nSis the Spearman\ncorrelation coe \u000ecient computed from the original pair of datasets and p\u0010\nc(0)\nS\u0011\nits estimated p-\nvalue given the null hypothesis.\nThe results from the permutation tests are summarized in Table 4 for the Spearman correlation coe \u000ecient of G\b(POT) vs.\nG\b(NLFFF-R) only, because we obtained very similar results for both the Pearson and Spearman correlation coe \u000ecients of\nall three scatter plots from Figure 6. For all permutation tests reported in Table 4, the distribution of Spearman correlation\ncoe\u000ecients (not shown here), c(k=f0;1;:::;ng)\nS, exhibits a Gaussian-like profile with a mean, \u0016\u00190 (j\u0016j<10\u00004), and a very small\nstandard deviation, \u001b\u001410\u00002. We varied the number of random permutations and verified that the mean and standard deviation\nof the resulting distributions are not strongly dependent on the number of permutations as long as this number is large enough\n(typically N\u0015104). Table 4 shows that the p-value of c(0)\nSis at most 10\u00006(as taken from the test with N=106) for all permutation\ntests. This is only an upper bound for the p-value as suggested by their N\u00001dependency visible in Table 4 and the comparison\nbetween c(0)\nSand the standard deviation which places c(0)\nSat an\u0019100\u001bdistance from the mean in the tail of the distribution of\nc(k=f0;1;:::;ng)\nS. Note that the N\u00001dependency occurs because we obtain c(k)\nS<c(0)\nSfor all k>0 for all permutation tests, leading to\nm=1 in Equation (A1). The correlation coe \u000ecients reported in Figure 6 are thus statistically significant. We thus reject the null\nhypothesis and accept the alternative hypothesis. We therefore conclude that the calculations of the connectivity-based helicity\nflux density derived from di \u000berent FFF extrapolations are highly correlated and consistent with each other."    },    {        "title": "1401.1826v1.Generating_buoyant_magnetic_flux_ropes_in_solar_like_convective_dynamos.pdf",        "content": "arXiv:1401.1826v1  [astro-ph.SR]  8 Jan 2014Generating buoyant magnetic flux ropes in solar-like\nconvective dynamos\nNelson, N. J.1and Miesch, M. S.2\n1Los Alamos National Laboratory, P.O. Box 1663, T-2, MS-B283, Lo s Alamos, NM\n87545\n2High Altitude Observatory, National Center for Atmospheric Rese arch, 3080 Center\nGreen Dr., Boulder, CO 80301\nE-mail:njnelson@lanl.gov\nAbstract. Our Sun exhibits strong convective dynamo action which results in\nmagnetic flux bundles emerging through the stellar surface as magn etic spots. Global-\nscale dynamo action is believed to generate large-scale magnetic str uctures in the deep\nsolar interior through the interplay of convection, rotation, and s hear. Portions of\nthese large-scale magnetic structures are then believed to rise th rough the convective\nlayer, forming magnetic loops which then pierce the photosphere as sunspot pairs.\nPrevious global simulations of 3D MHD convection in rotating spherica l shells have\ndemonstrated mechanisms whereby large-scale magnetic wreaths can be generated in\nthe bulk of the convection zone. Our recent simulations have achiev ed sufficiently\nhigh levels of turbulence to permit portions of these wreaths to bec ome magnetically\nbuoyant and rise through the simulated convective layer through a combination of\nmagnetic buoyancy and advection by convective giant cells. These b uoyant magnetic\nloops are created in the bulk of the convective layer as strong Lore ntz force feedback in\nthe cores of the magnetic wreaths dampen small-scale convective m otions, permitting\nthe amplification of local magnetic energies to over 100 times the loca l kinetic energy.\nWhilethemagneticwreathsarelargelygeneratedtheshearingofax isymmetricpoloidal\nmagnetic fields by axisymmetric rotational shear (the Ω-effect), t he loops are amplified\nto their peak field strengths before beginning to rise by non-axisym metric processes.\nThis further extends and enhances a new paradigm for the genera tion of emergent\nmagnetic flux bundles, which we term turbulence-enabled magnetic b uoyancy.\nSubmitted to: Plasma Physics and Controlled Fusion\n1. Magnetic Spots on Sun-like Stars\nDynamo action in sun-like stars provides an important physical labor atory for\nunderstanding magnetic self-organization in highly turbulent syste ms. Solar and stellar\nmagnetic activity is dominated by the emergence and evolution of high ly concentrated\nflux ropes which emerge through the photosphere and into the atm osphere. Sunspots\nexhibit ordered collective behaviors in the form of the solar magnetic activity cycle with\nitswell-established observational featuressuch asthe11-year p olarityreversal timescale,Generating Buoyant Loops 2\nthe migration of active latitudes, Hale’s polarity law, and numerous ot hers (Hathaway,\n2010). These collective behaviors point to the global solar dynamo a s the source of the\nmagnetic loops which become sunspot pairs. The emergence and eje ction of magnetic\nflux also plays a key role in many dynamo models either by creating poloid al magnetic\nfield via the emergence and dispersal of magnetic flux (known as the Babcock-Leighton\nmechanism) orby regulatingthemagnetic helicity budget (Blackman a nd Brandenburg,\n2003). Similarcyclesofmagneticactivityandcollectivestarspotbeh aviorsarebeginning\nto be observed on a wide variety of sun-like stars (e.g., Berdyugina, 2005; Huber et al.,\n2012; Llama et al., 2012; Metcalfe et al., 2013).\nNumerical models have also made significant progress in examining com ponents\nof the dynamo and flux emergence processes that may operate in s un-like stars. The\nfundamental challenge to these models is the vast range of scale em ployed by processes\nof interest in stellar magnetism. This has led to three philosophical ap proaches.\nFirst, two-dimensional (2D) mean-field models seek to reduce the p roblem to the\nstudy axisymmetric fields and global behaviors. This demands the pa rameterization of\nessential three-dimensional (3D) effects. Mean-field models have been used extensively\nto examine the modes of large-scale dynamo action in the Sun (Charb onneau, 2010).\nMany mean-field models which can achieve realistic solar cycles and are used to\npredict the amplitude and timing of future solar cycles rely on the Bab cock-Leighton\nmechanism (e.g., Dikpati and Gilman, 2006; Choudhuri et al., 2007). T he Babcock-\nLeighton mechanism is a heavily parameterized version of buoyant ma gnetic transport\nwhichtransformstoroidalmagneticfieldsgeneratedthroughshe aramplificationnearthe\nbase of the solar convection zone into poloidal fields near the solar s urface. Attempts to\ninvestigate this mechanism in more realistic settings have met with mor e limited success\n(Durney, 1995; Miesch and Brown, 2012).\nThe second philosophical approach to modeling elements of convect ive dynamo\naction and flux emergence uses local 3D models which seek to captur e moderate or\nsmall scale behaviors that may be inaccessible to global scale models. In a “bottom-up”\napproach, local models often use Cartesian geometries and can be tailored to include\nonlytheneededphysics foragivenscaleandlocationintheSun. Loca lmodelshavebeen\nusedwithgreatsuccesstoinvestigatetheformationofbuoyantm agneticstructuresusing\nforced shear layers (Cline et al., 2003; Vasil and Brummell, 2009; Gu errero and K¨ apyl¨ a,\n2011) and the structure and evolution of photospheric active reg ions (Cheung et al.,\n2010; Rempel, 2011).\nLocal cartesian simulations have also demonstrated the spontane ous generation\nof coherent flux structures in turbulent MHD flows by mechanisms o ther than\ndifferential rotation. Of particular note is the negative effective ma gnetic pressure\ninstability (NEMPI) first proposed by Kleeorin et al. (1989) and rece ntly demonstrated\nin simulations of forced turbulence by Kemel et al. (2013). Here coh erent flux structures\nare generated through the suppression of turbulent pressure b y the magnetic field\nand the resulting compression and confinement of the coherent fie ld. Though similar\nprocesses may be occurring in our simulations, the coherent flux st ructures we describeGenerating Buoyant Loops 3\nhere exhibit a density deficit relative to their surroundings rather t han a density\nenhancement as expected from NEMPI. Furthermore, this densit y deficit produces a\nbuoyant acceleration which, together with advection, contribute s to the rise of the loops\nthrough the convection zone. As we stress below, both rotationa l shear and turbulence\nplay a role in the spontaneous formation of these structures, mak ing them in some sense\na hybrid of those seen, for example, by Guerrero and K¨ apyl¨ a (20 11) and Kemel et al.\n(2013).\nThepresenceofaphotosphereandrarifiedcoronacanalsopromo tethespontaneous\ngeneration of coherent magnetic loops with a vertical orientation a t the surface. Positive\nfeedback between radiative cooling and MHD induction can promote t he coalescence of\nvertical fields (Cheung et al., 2008; Stein and Nordlund, 2012) but it is not necessary;\nWarnecke et al. (2013) have demonstrated that even adibatic simu lations of forced\nturbulence can form bipolar magnetic structures in the presence o f a coronal envelope.\nThese results may have bearing on the ultimate fate of the rising loop s we describe\nhere. In particular, if magnetic flux on the Sun emerges as relatively diffuse loops such\nas the structures described here, then photospheric processe s may help them coalesce\ninto more concentrated bipolar active regions after emergence.\nThethirdphilosophical approachseeks toemployglobal3Dmodelsina “top-down”\nmanner. Global models use spherical geometry and realistic stratifi cation in an effort\nto achieve correct large-scale behavior and then include as broad a range of scales as\ncomputationally feasible. Global models require special attention to be devoted to the\nimpact of unresolved scales. This can be done through either explicit or implicit Large-\nEddy Simulation (LES) frameworks (see Grinstein et al., 2007). conv ective dynamo\nmodels have received strong recent attention as four different co des have yielded models\nwith global-scale toroidal magnetic structures, cycles of magnetic activity, and reversals\nin global magnetic polarity (Browning et al., 2006; Ghizaru et al., 2010; Gastine et al.,\n2012; K¨ apyl¨ a et al., 2012). A related approach uses 3D magnetoh ydrodynamic (MHD)\nor thin flux-tube models where magnetic structures are imposed an d then allowed to\nemerge through the convective layer (e.g., Fan, 2009; Weber et al., 2012; Jouve et al.,\n2013).\nChallenged and inspired by both theoretical and observational adv ances in\nunderstanding magnetic activity on sun-like stars, we have undert aken a series of 3D\nMHD models of global convective dynamo action in the deep interiors o f sun-like stars.\nThese models have shown that convective dynamos can create glob al-scale toroidal\nmagnetic wreaths in the bulk of the convection zone (Brown et al., 2 010) and that\nthese wreaths can lead to reversals in magnetic polarity and cycles o f magnetic activity\nas rotation is increased (Brown et al., 2011) or the level of turbulen ce is increased\n(Nelson et al., 2013). For our least diffusive simulations, portions of t hese wreaths\nhave become buoyant and risen coherently through our simulated d omain, forming\nbuoyant magnetic loops (Nelson et al., 2011). Collectively these loops exhibit statistical\nbehaviors that mimic those observed in sunspots (Nelson et al., 2014 ).\nIn studying the buoyant magnetic loops created by our simulations, we haveGenerating Buoyant Loops 4\npreviously focused on their properties at maximum radial extent, t he dynamics of\ntheir ascents, and the relation between the appearance of loops a nd the global-scale\naxisymmetric fields. Here we address the important question of how buoyant magnetic\nflux ropes are formed. In many ways our simulations are uniquely suit ed to address this\nquestion as previous models have used a forced shear-layer to gen erate their buoyant\nmagnetic structures or have inserted ad hocmagnetic structures by hand. Here we\ndirectly test the assumptions common to many previous models that buoyant magnetic\nfields are generated through axisymmetric toroidal shear acting o n poloidal magnetic\nfieldsandthatdissipationofthesestructuresisprimarilycausedby small-scaleturbulent\nadvective mixing (Fan, 2009; Guerrero and K¨ apyl¨ a, 2011; Jouve et al., 2013). Our\nanalysis shows that for a small number of sample buoyant magnetic lo ops, the creation\nmechanism is not primarily driven by the axisymmetric differential rota tion profile but\nrather depends heavily on local toroidal shear amplification. We also confirm that the\ndissipation mechanism is primarily due to small-scale turbulent advectio n.\n2. Simulation Overview\nHere we report on the results of 3D MHD simulations of turbulent con vective dynamo\naction in a rotating spherical shell spanning the bulk of the convect ion zone of a sun-\nlike star. These simulations have been conducted using the ASH code (Clune et al.,\n1999; Brun et al., 2004). ASH uses 1D stellar models to create a radia l profile which\nis used as the reference state and is kept fixed throughout the te mporal evolution of\nthe simulation. Convective motions and magnetic fields are treated a s perturbations\naround the background state. ASH uses spherical harmonics in th e horizontal directions\nand Chebyshev polynomials in the radius to achieve spectral accura cy over long time\nevolutions. Our domain extends from the base of the convection zo ne at 0.72R⊙to near\nthe photosphere at 0 .97R⊙with a density contrast of 25 between the upper and lower\nboundaries. This simulation has a dimensional bulk rotation rate of 12 40 nHz (thus a\nperiod of 9.3 days), or three times the current solar rate. In spite of the faster rotation\nrate, the results presented here may be largely revenant to solar behavior as the relevant\nnon-dimensional parameter which gauges the relative level of rota tional influence, the\nRossby number, is small both here and in the bulk of the solar interior .\nA number of simulations in this parameter regime have been conducte d to explore\nthe effects of viscous, thermal, and resistive diffusion on our solutio ns (Nelson et al.,\n2013). Here we will focus on one simulation, case S3. Case S3 uses a fi xed grid with 192\npoints in radius, 512 in latitude, and 1024 in longitude. Extensive discu ssions of case S3\nare provided in (Nelson et al., 2011, 2013, 2014). Case S3 uses a dy namic Smagorinsky\nsubgrid-scale model to achieve very low levels of explicit dissipation re lative to our\nother simulations.The dynamic Smagorinsky model was formulated to extrapolate the\nlocal unresolved turbulent dissipation using an assumption of scale- invariant behavior\nin the inertial range of the turbulence cascade (Germano et al., 199 1). Details of its\nimplementation in ASH can be found in Appendix A of (Nelson et al., 2013) . CaseGenerating Buoyant Loops 5\nFigure 1. (a) Snapshot of radial velocities vr(dark downflows, light upflows) at\n0.79R⊙for case S3 on a spherical surface shown in Mollweide projection. Co nvection\nis rotationally aligned at low latitudes and isotropic at high latitudes. (b ) Companion\nsnapshot of longitudinal magnetic field Bφat the same depth and instant in time. Two\nstrong wreath segments are present with a negative-polarity (blu e) wreath dominating\nthe northern hemisphere and a positive-polarity (red) wreath dom inating the southern\nhemisphere. (c) Companion snapshot of the local ratio of magnetic to kinetic energies\nshown on a logarithmic scale.\nD3b, which is otherwise identical to case S3 but uses a uniform enhan ced eddy viscosity\nhas roughly 50 times more viscous dissipation at mid-convection zone than case S3\n(Nelson et al., 2013). These references also contain additional info rmation on the\ndifferential rotation, magnetic activity cycles, and other feature s of case S3.\nCaseS3isaturbulentconvective dynamoinwhichstrongdynamoact ionisachieved\nas convective motions coupled with the bulk rotation of the system le ad to strong\ndifferential rotation. Figure 1a displays the convective radial veloc ities just below\nmid-convection zone. The convection at low-latitudes displays a str ong rotational\nalignment while the higher latitudes exhibit more isotropic convective p atterns. The\nstrong convective shear leads to the generation of magnetic fields with energies atGenerating Buoyant Loops 6\nFigure 2. Sample views of three buoyant magnetic loops near the apex of their rises\nthrough the convective layer. Loops are shown in 3D volume render ings of magnetic\nfield lines. For all three cases significant flux bundles coherently ext end from near the\nbase of the simulated convection zone at 0 .72R⊙to above 0 .90R⊙. These loops are\nshown to highlight the diverse magnetic morphologies found in this simu lation.\nor below the average local kinetic energies and with little large-scale o rganization.\nThe rotationally-aligned convective cells at low latitudes promote the equator-ward\ntransport of angular momentum, leading to strong differential rot ation. Differential\nrotation, in turn, provides shear which organizes and creates str ong toroidal magnetic\nfields - again at or near equipartition with the local kinetic energy. Fig ure 1b shows\nthe existence of strong magnetic wreaths of toroidal magnetic fie lds. The creation,\npersistence, and variability of these wreaths have been explored in a variety of settings\n(Brown et al., 2010, 2011; Nelson et al., 2013). These wreaths exhib it strong feedbacks\non the convective flows which generate them, leading to regions whe re the magnetic\nenergy density far exceeds the kinetic energy density. Figure 1c s hows the ratio of\nmagnetic to kinetic energy densities. The cores of the wreaths can achieve ratios above\n100. In these areas, Lorentz forces can suppress convective m otions. With convective\nmotions suppressed, turbulent mixing in the wreath cores is greatly reduced. We refer\nto this as reduced resolved turbulent dissipation.\nIn regions where coherent magnetic structures lead to greatly re duced levels of\nturbulent dissipation, portions of the wreaths can achieve magnet ic fields in the 30\nto 50 kG range. Some of these amplified sections can rise through a c ombination\nof magnetic buoyancy and favorable advection by convective giant cells and become\nbuoyant magnetic loops. Three such loops are shown in Figure 2. We h ave identifiedGenerating Buoyant Loops 7\nnearly200oftheseloopsincaseS3. Collectively theydisplay statistic al propertieswhich\nmimic observed statistical properties ofsolar active regions such a sHale’slaw, Joy’s law,\nthe hemispheric helicity rule, and active longitudes (Nelson et al., 2014 ). These buoyant\nmagnetic loops are flux ropes which are spontaneously generated b y convective dynamo\naction and rise in a self-consistent manner until they are either diss ipated or they reach\nthe upper boundary of our simulation, which in case S3 is impenetrable .\n3. The Formation of Buoyant Loops\nAs we have previously demonstrated, these buoyant magnetic loop s are coherent\nmagnetic structures that rise through our simulation domain via a co mbination of\nmagnetic buoyancy and advection by convective giant cells (Nelson e t al., 2011, 2013).\nAs they form in the convection zone, they are never in equilibrium but are constantly\nexperiencing advective forces from the surrounding convection, magnetic and thermal\nbuoyancy, and magnetic tension. In addition, they are continually b eing re-generated\nanddissipated by thesurrounding flows andour explicit restive diffus ion. Inmany ways,\ntheseloopsdefythetraditionalparadigmforbuoyantmagneticst ructureswhichassumes\naninitialequilibriumconfiguration(e.g.,Kersal´ e et al.,2007;Vasil an d Brummell,2009)\nor a well-organized and isolated initial magnetic topology (e.g., Fan, 20 09; Jouve et al.,\n2013).\nHere we will focus specifically on how the segments of the toroidal wr eaths that\nbecome buoyant magnetic loops are created by dynamo action. To d o this we examine\ntheevolutionofmagneticenergydensityinvolumes whichcontainaslic eoftheloop. We\ntreat this volume in a Lagrangiansense by tracking its motion. We beg in by considering\nthe MHD induction equation given by\n∂/vectorB\n∂t=∇×/parenleftBig\n/vector v×/vectorB/parenrightBig\n−∇×/parenleftBig\nη∇×/vectorB/parenrightBig\n. (1)\nThe first term on the right-hand side can be decomposed into three terms which\nrepresent advection −(/vector v·∇)/vectorB, shearing/parenleftBig\n/vectorB·∇/parenrightBig\n/vector v, and compression −/vectorB(∇·/vector v). The\ncompression term is further simplified by the anelastic approximation (see Brown et al.,\n2010; Nelson et al., 2013). The shear termcanfurther bedecompo sed into contributions\nfrom the axisymmetric shear associated with differential rotation a nd the smaller-\nscale shearing motions associated with convective flows, and into to roidal and poloidal\ncomponents. The evolution of the total magnetic energy density EMis given by taking\nthe dot product of /vectorB/4πwith Eqn. 1, which yields\n∂EM\n∂t=MPS+MMTS+MFTS+MAD+MRD+MAC, (2)\nwhere the production terms are\nMPS=/vectorBpol\n4π·/bracketleftBig/parenleftBig\n/vectorB·∇/parenrightBig\n/vector v/bracketrightBig\n, (3)\nMMTS=Bφˆφ\n4π·/bracketleftBig/parenleftBig\n/vectorB·∇/parenrightBig\n/angb∇acketleft/vector v/angb∇acket∇ight/bracketrightBig\n, (4)Generating Buoyant Loops 8\nMFTS=Bφˆφ\n4π·/bracketleftBig/parenleftBig\n/vectorB·∇/parenrightBig\n(/vector v−/angb∇acketleft/vector v/angb∇acket∇ight)/bracketrightBig\n, (5)\nMAD=−/vectorB\n4π·/bracketleftBig\n([/vector v−/vector vL]·∇)/vectorB/bracketrightBig\n, (6)\nMRD=−/vectorB\n4π·/bracketleftBig\n∇×/parenleftBig\nηt∇×/vectorB/parenrightBig/bracketrightBig\n, (7)\nMAC=vrB2\n4π∂ln ¯ρ\n∂r. (8)\nThese termsrepresent theproduction anddissipation ofmagnetic energydue topoloidal\nshear, mean toroidal shear, fluctuating toroidal shear, advect ion of magnetic energy into\nor out of the loop slice, resistive diffusion, and anelastic compression . Angle brackets\ndenote longitudinal means. The mean motion of the loop slice is given by /vector vL, thus if\nthe entire volume of the loop moved uniformly MADwould go to zero. The spherically-\nsymmetric background density is given by ¯ ρand the magnetic resistivity is given by ηt.\nNote that ηtin case S3 is computed using the dynamic Smagorinsky subgrid-scale m odel\n(see Nelson et al., 2013). We have chosen to divide the shear term int o three parts to\nexamine the relative contributions of toroidal shear from the conv ection and from the\ndifferential rotation, as well as the contribution of poloidal shear. In many mean-field\nmodels buoyant magnetic flux is assumed to be generated primarily by axisymmetric\ntoroidal shear associated with differential rotation.\nThese buoyant magnetic loops are most easily identified when they ar e at or near\ntheir maximum radial extent. Perhaps thegreatest difficulty in exam ining the formation\nof our buoyant magnetic loops is tracking the fluid volumes which beco me the tops of\nthese loops backward in time. We would like to trace the origin of the ma gnetic energy\nwhich arrives at the top of the domain. This is perhaps the greatest challenge and\nthe greatest insight which we can extract from this simulation, as it s elf-consistently\ngenerates buoyant magnetic loops from convective dynamo action .\nTo track slices of the loops from times when they can be identified as r adially\nextended coherent magnetic flux ropes to their origins has require d the development of\na novel tracking algorithm. We start with a slice over 2◦in longitude at top of the loop\nat timetn. We chose tnat a time near where the loop can be identified using magnetic\nfield lines and the top of the loopextends above 0 .90R⊙. The cross-section of the loop in\nradius and latitude Snis initially determined using magnetic field lines (similar to what\nis shown in Figure 2). We computed the total magnetic energy as En=/integraltext\nVnEMdV,\nwhereEMis the magnetic energy density and Vnis the volume define by the surface Sn\nin radius and latitude and is 2◦wide in longitude. We compute the volume-averaged\nmagnetic energy generation terms in Equation 2 and the volume-ave raged motion of the\nloop/vector vn\nL. Thecenter ofmagneticenergy /vectorCninradius, latitude, andlongitudeiscomputed\nas the geometrical center of the loop slice weighted by magnetic ene rgy density. We also\ncompute the magnetic energy generation terms in Equation 2 over t his volume. Thus\nwe know the change in the total magnetic energy of our volume at th e previous time\nstep.Generating Buoyant Loops 9\nFigure 3. Sample positions of the center of Loop #1 during its formation and ris e\nthrough the convection layer taken every 16 hours. The tracked volume begins at\n0.791R⊙and 5.7◦latitude, and ends 21.6 days later with the uppermost point. The\nfirst 14.1 days cover the formation of the loop. The final 7.5 days co ver the initial\nportion of the loop’s ascent through the convective layer. The mag netic topologies of\nthe first and last times shown here are displayed in Figure 4.\nThe center of the loop at time tn−1is then computed with a backward Euler time\nstep by/vector cn−1=/vectorCn−(tn−tn−1)/vector vn\nL. We use a surface in radius and latitude determined\nfrom a new field line tracing, however since we cannot reliably track ind ividual field\nlines without much finer temporal data, we generally find that this ne w surface does not\nenclose the correct total magnetic energy to account for the ma gnetic energy present at\nthe previous step plus any change due to the generation terms. We either either add or\nremove a uniform number of grid cells from the edge of the surface u ntil we obtain a\nvolume which contains the same total magnetic energy as was prese nt at the previous\nstep modulo changes from the production terms. This gives us a new loop profile in\nradius and latitude Sn−1. We compute the center of magnetic energy /vectorCn−1using the\nintegral method and compare it to the center we computed from th e backward Euler\ntime stepping. When the difference between the two centers is less t han 5% of the total\ndisplacement, we set the loop center to the average of the two and use the radius and\nlatitude profile determined from the modified magnetic field line contou r. This process\nis then repeated to obtain the loop center and profile at time tn−2.\nFor situations where the distance between the two centers is large r than 5% of the\ntotal displacement for that time step, we add a third method for de termining the loop\ncenter and shape. This third method is generally required once the t racked volume\nenters the lower convection zone where it can become difficult to dist inguish from the\nlarger magnetic wreath structure. The algorithm randomly places b etween 10 and 15\nsmall circular test regions over the loop profile Sn. Each test region is then stepped backGenerating Buoyant Loops 10\nFigure 4. Volume rendering of magnetic field lines passing through the tracking\nvolume for Loop #1 at the earliest and latest times tracked. Perspe ctive is looking\nsouth along the rotation axis with both frames rotated such that r adial position\nincreases upwards and longitude increases to the left. Color indicat es magnitude of the\nmagnetic field. In both cases the volume being tracked is roughly cen tered in longitude\nand outlined by the field lines.\nin time using the same method used for the entire slice. We compute a n ew profile in\nlatitudeand radiusby choosing aprofile that contains allof thetest regionsat time tn−1,\ncontains the correct total magnetic energy, and has the minimum d istance around its\nperimeter. We then use this profile to compute the center of magne tic energy using the\nintegral method, which we call /vectorCn−1. If this method produces a loop center of magnetic\nenergy that lies less than 5% of the total displacement from both /vector cn−1and/vectorCn−1then\nwe use the average of the three centers and the average of the m agnetic energy contour\nand the test region contour. If /vectorCn−1does not lie between /vector cn−1and/vectorCn−1then we deem\nthe method has failed at that step. When the method fails for three successive steps we\ndeclare that the tracking algorithm has lost the loop.\nFor this algorithm to work properly, we require the full 3D dataset o f the velocity,\nmagnetic field, and dynamic Smagorinsky viscosity at very high tempo ral cadences.\nTests of our tracking algorithm indicate that to successfully track a loop slice over more\nthan 20 days requires time steps of about five hours. There are on ly six loops for whichGenerating Buoyant Loops 11\nwe have sufficiently fine temporal data over their formation and rise . For one of these\nsix (#4) the tracking algorithm loses the loop after only 9 days even w ith our finest\ntemporal cadence. The other five (#1, #2, #3, #21, #22) are suc cessfully tracked for\nat least 20 days. Figure 3 shows the tracking of Loop #1 as an examp le. In the case of\nLoop #1, tracking was begun 5 days prior to its maximum radial exten t of 0.94R⊙. We\ndo find some sensitivity to the initial times, profiles and total magnet ic energies chosen.\nThis sensitivity to initial conditions is minimized when we begin tracking th e loop slices\nbefore they reach their maximum radial extent. As we can only curr ently perform this\nanalysis on a small number of loops, we are unable to make general st atements about\nall buoyant magnetic loops in our models.\nFigure 4 shows 3D volume renderings of field line tracings for the first and last\ntimes indicated in Figure 3. Tracking begins using the large bundle of fie ld lines seen in\nFigure 4(b) at approximately 55◦longitude. This volume is tracked through the path\nshown in Figure 3 backward in time to the magnetic topology shown in Fig ure 4(a).\nThe field that will become the buoyant loop is at that point heavily embe dded in the\nmuch larger-scale magnetic wreaths which are pervasive at this time and location below\nabout 0.82R⊙.\nOnce we have tracked the loop slices back as far as possible, we then examine the\nvolume-averaged values of the magnetic energy generation terms in Equation 2 at each\ntime-step. We identify when the maximum magnetic energy in the trac ked volume is\nachieved and label this tmax. Generally this occurs very close to the beginning of their\nrise toward the top of our domain. To assess the question of their h ow these loops\nare formed, we look at the generation of magnetic energy prior to tmax. For all five of\nthe loops which we tracked successfully at least 13 days of the trac king occurred prior\ntotmax. By examining the generation of magnetic energy immediately before tmaxwe\ncan probe whether the loops are primarily generated through a long , slow build-up of\nmagnetic energy or a rapid amplification on time-scales of about 10 da ys. For all five\nloops examined here, we find that between 72% and 87% of the magne tic energy present\nattmaxwas generated during the 13 to 17 days covered by our tracking alg orithm. This\nindicates that these loops are created primarily in a rapid amplification on very short\ntimescales rather than being the product of a slower build-up proce ss.\nFigure 5 shows the rise and expansion of a sample magnetic loop as det ermined\nby the tracking algorithm, along with the volume-averaged magnetic energy production\nterms in the loop slice at each time. We divide the time series into the for mation phase\nand the rise phase. The change between the two is deemed to occur when the loop\nbegins monotonically upwards movement. In the upper panel the blu e line shows the\nradial position of the loop center while the green region shows the ra dial extent of the\ntracked volume. In the formation phase the radial expansion of th e loop is primarily\ndue to increased accumulation of magnetic energy. In the rise phas e the expansion is\nlargely the result of the decreasing background pressure at highe r radial positions.\nA careful examination of the formation phase shows that magnetic energy is\nprimarily generated through the toroidal shear terms MFTSandMMTS, with poloidalGenerating Buoyant Loops 12\nFigure 5. The rise and magnetic energy generation terms as a function of time during\nthe formation and rise of Loop #1. The top panel shows the radial c enter (blue line)\nand radial extent (green shaded region). The rise is deemed to beg in once the loop\nbegins monotonically upward motion. The lower panel shows the aver age magnetic\nenergy generation rate broken down by the terms in Eqn. 2. During the formation\nstage energy is generated primarily by the fluctuating and mean tor oidal shear terms\n(MFTSandMPTS). During the rise the loop’s magnetic energy is primarily dissipated\nby the advection term ( MAD).\nshearMPSplaying a small role through most of the formation phase. All of thes e terms\nare highly variable with temporal standard deviations on the order o f their temporal\nmeans. The formation of Loop #1 is opposed primarily by restive diffus ion, which is\nsmall but persistent. Advection also provides a negative net contr ibution, but it is\nagain highly variable and becomes strongly positive for nearly two day s at the end of\nthe formation phase. The compression term predictably follows the radial motion of the\nloop as it moves slightly upward and then downward during this phase.\nThe rise phase exhibits some noticeably different behaviors. The fluc tuating\ntoroidal shear grows to large positive values as the loop moves radia lly upward while the\nmean toroidal shear term drops as the loop moves out of the region of peak differential\nrotation shear in the bottom of the convection zone before recov ering slightly during the\nlater stages of the rise phase. The poloidal shear term increases, again becoming largest\nin the later stages of the loop’s rise. The compression term becomes more negative as\nthe loop rises. The diffusion term is again far less variable than the oth er terms butGenerating Buoyant Loops 13\nFigure 6. Time- and volume-averaged magnetic energy generation rates for five\nbuoyant magnetic loops (#1, #2, #3, #21, and #22) tracked in the s ame manner\nas Loop #1 (see Figs. 3 and 5). Magnetic energy generation terms f ollow Eqn. 2. In\nthe formation phase the largest positive contribution comes from s mall-scale toroidal\nshearing of poloidal magnetic fields MFTS(green), follow by the differential rotation\nacting on poloidal magnetic fields MMTS(brown). In the formation stage this is\ngenerally opposed by resistive diffusion MRD(magenta). During the rise stage\nmagnetic energy is generated by MFTS(green) at even higher levels than in the\nformation state, while dissipation is dominated by small-scale advectio nMAD(yellow)\nwith contributions from MRD(magenta) and anelastic compression MAC(light blue).\nPoloidal shear MPS(dark purple) plays a small role in both the formation and rise\nphases.\nbecomes slowly more negative as the average resistivity increases w ith radial position.\nThe most striking change is the small-scale advection of magnetic ene rgy out of the\ntracked volume. As the loop rises and leaves the magnetically-domina ted region in and\naround the wreath, it encounters a significant increase in small-sca le motions which\ntend to remove strong fields and replace them with weaker, less-co herent fields from the\nsurrounding fluid. Over the first five days of the rise phase the adv ective term shows\nlittle change, but upon reaching roughly mid-convection zone which m arks the rough\nupper edge of the wreath, the advective term rapidly becomes mor e negative. By the\nlast time considered here, the advective term has become larger in m agnitude than any\nother term by more than a factor of two.\nWe conduct the same analysis of the production and dissipation of ma gnetic energy\non the other four loops for which we have successfully tracked the ir formation. Figure 6Generating Buoyant Loops 14\nshows the time- and volume-averaged values of each generation te rm for the formation\nand rise phases of these five loops. While there are significant variat ions in each loop,\nsome general trends emerge for these five samples. First, in all fiv e casesMFTSis\nprimarily responsible for the generation of magnetic energy in both t he formation and\nrise phases, which MMTSplaying a lesser but still significant role in the formation\nphase. Second, the loops are primarily opposed by resistive diffusion in the formation\nphase, which is augmented by MADin the rise phase. With these clear trends, these\nloops provide strong evidence in support of the essential role of tu rbulent amplification\nof magnetic fields in the creation of these buoyant loops and of reso lved turbulent\ndissipation in their eventual dissolution.\n4. Turbulent Convective Dynamos and Flux Emergence\nIn this paper, we have demonstrated that buoyant magnetic loops can be generated in\nsolar-likeconvective dynamos. Themagneticenergyintheseloopsis primarilygenerated\nby the shearing motions of moderate- to small-scale flows with the me an differential\nrotation shear playing a secondary role. This is in contrast to many d ynamo models\nwhich assume that buoyant magnetic structures are produced fr om global-scale toroidal\nmagnetic structures generated by the differential rotation profi le. In previous papers we\nhave shown that the axisymmetric shear from the differential rota tion plays a key role\nin the magnetic wreaths and activity cycles in case S3. It is therefor e notable that the\ngeneration of the buoyant loops analyzed here relies primarily on fluc tuating, turbulent\nshear rather than the axisymmetric differential rotation.\nThe abilitytocapturethespontaneous generationofbuoyant mag netic fluxropesin\nglobal convective dynamos is an important step forward in underst anding how coherent\nmagnetic structures are self-organized by turbulent dynamo act ion. Additionally, our\nuse of a dynamic Smagorinsky subgrid-scale model has lowered rest ive diffusion to the\npoint that resolved small-scale advection is responsible for most of t he dissipation of\nthese loops. While these loops experience far less turbulent flows th an they would in\nreal stellar interiors, it is heartening to see that these loops can su rvive when turbulent\nadvection becomes the dominant dissipation mechanism.\nIn a broader context, this work represents an important testing ground for numer-\nical models of magnetic flux ropes generated by turbulent dynamo a ction. Extensive\nsolar and stellar observations coupled with 3D models are making signifi cant progress\nin understanding the physical processes whereby stars achieve o rdered global behaviors\nand generate moderate-scale coherent flux ropes from highly cha otic turbulent driving.\nAcknowledgements\nWe thank Kyle Auguston, Chris Chronopoulos, Yuhong Fan, Nicholas Featherstone,\nBrandley Hindman, and Joyce Guzik for their suggestions and advice . This research isREFERENCES 15\npartly supported by NASA through Heliophysics Theory Program gr ants NNX08AI57G\nand NNX11AJ36G. Nelson is supported by a LANL Metropolis Fellowship . Work at\nLANL was done under the auspices of the National Nuclear Security Administration of\nthe U.S. Department of Engery at Los Alamos National Laboratory under Contract No.\nDE-AC52-06NA25396. Miesch is also supported by NASA SR&T grant NNH09AK14I.\nNCAR is sponsored by the National Science Foundation. The simulatio ns were carried\nout with NSF TeraGrid and XSEDE support of Ranger at TACC, and Kr aken at NICS,\nand with NASA HECC support of Pleiades. Field line tracings and volume r enderings\nshown in Figures 2 and 4 were produced using VAPOR (Clyne et al., 2007 ).\nReferences\nBerdyugina S 2005 Living Reviews in Solar Physics 2, 8.\nBlackman E G and Brandenburg A 2003 The Astrophysical Journal Letters 584, 99.\nBrownBP, Browning MK,BrunAS,Miesch MSandToomreJ2010 The Astrophysical\nJournal711(1), 424–438.\nBrownBP, Miesch MS, Browning MK,BrunASandToomreJ2011 The Astrophysical\nJournal731(1), 69.\nBrowning M K, Miesch M S, Brun A S and Toomre J 2006 The Astrophysical Journal\nLetters648, L157.\nBrun A S, Miesch M S and Toomre J 2004 The Astrophysical Journal 614, 1073.\nCharbonneau P 2010 Living Reviews in Solar Physics 7, 3.\nCheung M C M, Rempel M, Title A M and Sch¨ ussler M 2010 The Astrophysical Journal\n720(1), 233–244.\nCheung MCM,Sch¨ ussler M,TarbellTDandTitleAM2008 The Astrophysical Journal\n687(2), 1373.\nChoudhuri A, Chatterjee P and Jiang J 2007 Physical Review Letters 98(13), 131103.\nClineKS,Brummell NHandCattaneoF2003 The Astrophysical Journal 588,630–644.\nClune T, Elliott J, Miesch M S, Toomre J and Glatzmaier G A 1999 Parallel Computing\n25, 361–380.\nClyne J, Mininni P, Norton A and Rast M 2007 New Journal of Physics 9(8), 301–301.\nDikpati M and Gilman P A 2006 The Astrophysical Journal 649(1), 498–514.\nDurney B 1995 Solar Physics 160, 213–235.\nFan Y 2009 Living Reviews in Solar Physics 6, 4.\nGastine T, Duarte L and Wicht J 2012 Astronomy & Astrophysics 546, A19.\nGermanoM,PiomelliU,MoinPandCabotW1991 Physics of Fluids A: Fluid Dynamics\n3, 1760.\nGhizaru M, Charbonneau P and Smolarkiewicz P K 2010 The Astrophysical Journal\n715(2), L133–L137.REFERENCES 16\nGrinstein F F, Margolin L G and Rider W J, eds 2007 Implicit Large Eddy Simulation:\nComputing Turbulent Fluid Dynamics Cambridge University Press.\nGuerrero G and K¨ apyl¨ a P J 2011 Astronomy & Astrophysics 533, A40.\nHathaway D H 2010 Living Reviews in Solar Physics 7, 1.\nHuber D, Ireland M J, Bedding T R, Brand˜ ao I M, Piau L, Maestro V, W hite T R,\nBruntt H, Casagrande L, Molenda- ˙Zakowicz J, Aguirre V S, Sousa S G, Barclay\nT, Burke C J, Chaplin W J, Christensen-Dalsgaard J, Cunha M S, De Rid der J,\nFarrington C D, Frasca A, Garc´ ıa R A, Gilliland R L, Goldfinger P J, Hekk er S,\nKawalerSD,KjeldsenH,McAlisterHA,MetcalfeTS,MiglioA,MonteiroM JPFG,\nPinsonneault M H, Schaefer G H, Stello D, Stumpe M C, Sturmann J, St urmann L,\nten Brummelaar T A, Thompson M J, Turner N and Uytterhoeven K 20 12The\nAstrophysical Journal 760(1), 32.\nJouve L, Brun A S and Aulanier G 2013 The Astrophysical Journal 762, 4.\nK¨ apyl¨ a P J, Mantere M J and Brandenburg A 2012 The Astrophysical Journal\n755(1), L22.\nKemel K, Brandenburg A, Kleeorin N, Mitra D and Rogachevskii I 201 3Solar Physics\n287(1), 293.\nKersal´ e E, Hughes D and Tobias S M 2007 The Astrophysical Journal 663, 113–116.\nKleeorin N I, Rogachevskii I V and Ruzmaikin A A 1989 Soviet Astronomy Letters\n15, 639–645.\nLlama J, Jardine M, Mackay D H and Fares R 2012 Monthly Notices of the Royal\nAstronomical Society: Letters 422(1), L72–L76.\nMetcalfe T S, Buccino A P, Brown B P, Mathur S, Soderblom D R, Henry T J, Mauas\nP J D, Petrucci R, Hall J C and Basu S 2013 The Astrophysical Journal 763(2), L26.\nMiesch M S and Brown B P 2012 The Astrophysical Journal 746(2), L26.\nNelson N J, Brown B P, Brun A S, Miesch M S and Toomre J 2011 The Astrophysical\nJournal739(2), L38.\nNelson N J, Brown B P, Brun A S, Miesch M S and Toomre J 2013 The Astrophysical\nJournal762, 73.\nNelson N J, Brown B P, Brun A S, Miesch M S and Toomre J 2014 Solar Physics\n289(2), 441.\nRempel M 2011 The Astrophysical Journal 740(1), 15.\nStein R F and Nordlund ˚A 2012The Astrophysical Journal 753(1), L13.\nVasil G M and Brummell N H 2009 The Astrophysical Journal 690(1), 783–794.\nWarnecke J, Losada I R, Brandenburg A, Kleeorin N and Rogachevs kii I 2013 The\nAstrophysical Journal 777(2), L37.\nWeber M A, Fan Y and Miesch M S 2012 Solar Physics 287(1-2), 239–263."    },    {        "title": "2310.09043v1.Midpoint_geometric_integrators_for_inertial_magnetization_dynamics.pdf",        "content": "Midpoint geometric integrators for inertial magnetization dynamics\nM. d’Aquinoa,∗, S. Pernaa, C. Serpicoa\naDepartment of Electrical Engineering and Information Technology, University of Naples Federico II, Via Claudio\n21, I-80125, Naples, Italy\nAbstract\nWe consider the numerical solution of the inertial version of Landau-Lifshitz-Gilbert equation (iLLG), which\ndescribes high-frequency nutation on top of magnetization precession due to angular momentum relaxation.\nThe iLLG equation defines a higher-order nonlinear dynamical system with very different nature compared\nto the classical LLG equation, requiring twice as many degrees of freedom for space-time discretization.\nIt exhibits essential conservation properties, namely magnetization amplitude preservation, magnetization\nprojection conservation, and a balance equation for generalized free energy, leading to a Lyapunov structure\n(i.e. the free energy is a decreasing function of time) when the external magnetic field is constant in time.\nWe propose two second-order numerical schemes for integrating the iLLG dynamics over time, both based on\nimplicit midpoint rule. The first scheme unconditionally preserves all the conservation properties, making\nit the preferred choice for simulating inertial magnetization dynamics. However, it implies doubling the\nnumber of unknowns, necessitating significant changes in numerical micromagnetic codes and increasing\ncomputational costs especially for spatially inhomogeneous dynamics simulations. To address this issue, we\npresent a second time-stepping method that retains the same computational cost as the implicit midpoint\nrule for classical LLG dynamics while unconditionally preserving magnetization amplitude and projection.\nSpecial quasi-Newton techniques are developed for solving the nonlinear system of equations required at\neach time step due to the implicit nature of both time-steppings. The numerical schemes are validated on\nanalytical solution for macrospin terahertz frequency response and the effectiveness of the second scheme is\ndemonstrated with full micromagnetic simulation of inertial spin waves propagation in a magnetic thin-film.\nKeywords: magnetic inertia, terahertz spin nutation, micromagnetic simulations, inertial\nLandau-Lifshitz-Gilbert (iLLG) equation, implicit midpoint rule, numerical methods.\n1. Introduction\nThe study of ultra-fast magnetization processes has become increasingly important in recent years,\nparticularly for its potential applications to future generations of nanomagnetic and spintronic devices [1].\nSince the pioneering experiment by Beaurepaire et al. [2] that revealed subpicosecond spin dynamics, the\ninvestigation of ultra-fast magnetization processes has attracted the attention of many research groups,\nleading to a considerable body of research [3, 4, 5, 6, 7, 8, 9, 10].\nRecently, there have been exciting experimental developments in the direct detection of spin nutation in\nferromagnets in the terahertz range [11, 12]. This has confirmed the presence of inertial effects in magneti-\nzation dynamics, which were theoretically predicted several years ago [13, 14, 15]. Nutation-like magnetiza-\ntion motions in nanomagnets occurring at gigahertz frequencies under the action of time-harmonic applied\nexternal magnetic fields were also studied theoretically in past decades within the classical precessional\ndynamics[16].\n∗Corresponding author\nEmail addresses: mdaquino@unina.it (M. d’Aquino), salvatore.perna@unina.it (S. Perna), serpico@unina.it (C.\nSerpico)arXiv:2310.09043v1  [physics.comp-ph]  13 Oct 2023From a technological perspective, the observation of terahertz spin nutation opens up new possibilities\nfor exploiting novel ultra-fast regimes. For instance, it may be possible to use strong picosecond field pulses\nto drive ballistic magnetization switching into the inertial regime [17, 18, 19, 20, 21, 22]. This has important\nimplications for the development of ultra-fast magnetic devices, and it also has fundamental implications\nfor the physics of magnetism.\nFrom a theoretical point of view, inertial magnetization dynamics can be described by augmenting the\nclassical Landau-Lifshitz-Gilbert (LLG) precessional dynamics with a torque term modeling intrinsic angular\nmomentum relaxation [13, 14]. This approach has been successful in explaining the observed high-frequency\nspin nutation in uniformly-magnetized ferromagnetic samples [11], for which magnetization dynamics is\ngoverned by the following inertial version of the Landau-Lifshitz-Gilbert equation[13, 14]:\ndM\ndt=−γM×\u0012\nHeff−α\nγMsdM\ndt−τ2d2M\ndt2\u0013\n, (1)\nwhere M(t) is the magnetization vector field ( Msis the saturation magnetization of the material), Heffis\nthe magnetic effective field, αis the Gilbert damping, γis the absolute value of the gyromagnetic ratio and\nτdefines the time scale of inertial magnetic phenomena.\nHowever, when spatial changes of magnetization do occur in magnetic systems of nano- and micro-scale,\nthe description of spatially-inhomogeneous ultra-fast magnetization dynamics occurring at sub-picosecond\ntime scales becomes a challenging problem that requires appropriate extension of eq.(1) to take into account\nspace-varying vector fields in the region Ω occupied by the ferromagnetic body. This extension leads to the\nformulation of a novel equation where formally the total derivatives with respect to time become partial\nand the effective field is given by the variational derivative of the Gibbs-Landau free energy functional[23],\nresulting in the following:\n∂M\n∂t=−γM×\u0012\nHeff−α\nγMs∂M\n∂t−τ2∂2M\n∂t2\u0013\n, (2)\nwhere generally the natural (homogeneous Neumann) boundary conditions ∂M/∂n= 0 are inherited by\nthe classical LLG when no surface anisotropy is present at the body surface ∂Ω. Equation (2) reduces to\nthe purely precessional classical LLG equation when no inertia is considered (i.e. τ= 0). Nevertheless,\ndespite this apparent similarity, eq.(2) has profoundly different nature in that it has hyperbolic (wave-\nlike) character (instead of parabolic as the classical LLG equation) and admits the possibility of travelling\nsolutions (spin waves) with finite propagation speed[24]. For this reason, the iLLG dynamics deserves a\ndedicated investigation in his own rights.\nIn this respect, based on equations (1),(2), a number of theoretical studies have been proposed in the latest\nyears to characterize terahertz spin nutation[25, 26, 27, 28, 29, 30, 31, 32]. Most of these interesting studies\nrely on analytical approaches valid in idealized situations such as, for instance, analysis of magnetization\noscillations in single-domain particles (macrospin) or small-amplitude spin waves propagation in infinite\nmedia. Very recently, the possibility to observe propagation of ultra-short inertial spin waves in confined\nferromagnetic thin-films driven by ac terahertz fields has been also theoretically demonstrated[24]. These\nwaves exhibit behavior that deviates significantly from classical exchange spin-waves and can propagate at\na finite speed up to a limit of several thousands meters per second, which is comparable with the velocity\nof surface acoustic waves.\nWhile such phenomena occurring in confined micromagnetic systems mainly involve magnetization os-\ncillations around equilibria and can be investigated by analyzing the inertial LLG dynamics in the linear\nregime, no such possibility exists when far from equilibrium dynamics such as nonlinear oscillations[33],\nmagnetization switching[21] or even chaos[34] are considered.\nIn these situations where no analytical techniques can be applied, one has to resort to numerical sim-\nulation. In this respect, after the experimental evidence of the terahertz spin nutation[11], the study of\ninertial effects in magnetization dynamics is rapidly becoming an emergent field of research and, conse-\nquently, the need for accurate and efficient computational techniques that exploit the intrinsic properties of\nthe nutation dynamics beyond off-the-shelf time-stepping schemes is growing fast, too. Nonetheless, at the\n2present moment, very few works[35, 36] address ad-hoc numerical techniques for the inertial magnetization\ndynamics.\nIn this paper, after illustrating the general qualitative conservation properties of the continuous inertial\nmagnetization dynamics, we propose suitable time-integration schemes based on the implicit midpoint rule\ntechnique[37] for the numerical solution of the inertial LLG (iLLG) equation and their relevant properties\nare discussed. The midpoint rule is an unconditionally stable second order accurate scheme which preserves\nthe fundamental geometrical properties of the classical LLG dynamics[38]. The first time-stepping proposed\nhere is shown to preserve all relevant conservation properties of the iLLG dynamics unconditionally, i.e.\nregardless of the time step amplitude. Despite these remarkable properties, we show that, in general, the\nnumerical integration of inertial magnetization dynamics must address the issue of the higher order of the\ndynamical system that it describes, which implies dramatic changes of micromagnetic codes and results\nanyway in at least doubling the computational cost of the numerical scheme as compared to classical LLG\ndynamics. This has a huge impact when micromagnetic simulations with full spatial discretization on\nhundred thousands (or more) computational cells have to be performed, such as in the case of (sub)micron-\nsized magnetic systems. For this reason, we develop an additional efficient implementation of the midpoint\nrule technique for iLLG dynamics, based on suitable multistep method for the inertial term, which can be\nbuilt on the top of that associated with classical LLG dynamics and, therefore, retaining a computational\ncost with the same order of magnitude. The proposed techniques are first validated by computing the\nfrequency response of a magnetic thin-film modeled as single spin (macrospin) magnetized along the easy\ndirection and subject to out-of-plane ac field, and comparing the results with the analytical solution. Then,\nfull micromagnetic simulations of inertial spin wave propagation in a ferromagnetic nanodot are performed\nin order to demonstrate the accuracy and effectiveness of the second proposed time-stepping in reproducing\nspatially-inhomogeneous ultra-fast spin nutation dynamics.\n2. Inertial magnetization dynamics and qualitative properties\nThe starting point of the discussion is the inertial Landau-Lifshitz-Gilbert (iLLG) equation (2), expressed\nin dimensionless form[13, 24]:\n∂m\n∂t=−m×\u0012\nheff−α∂m\n∂t−ξ∂2m\n∂t2\u0013\n, (3)\nwhere m(r, t) is the magnetization unit-vector (normalized by the saturation magnetization Ms) at each\nlocation r∈Ω (Ω is the region occupied by the magnetic body), time is measured in units of ( γMs)−1\n(corresponding to 5.7 ps for γ= 2.21×105A−1s−1mandµ0Ms= 1T), αis the Gilbert (dimensionless and\npositive, typically in the order ∼10−3÷10−2) damping parameter, the parameter ξmeasures the strength\nof inertial effects in magnetization dynamics. It is worthwhile noting (see eq.(2)) that the dimensionless\nquantity ξcan be expressed as ξ= (γMsτ)2where τdetermines the physical time-scale of magnetic inertia,\nfor which previous works[13, 11, 21] assessed its order of magnitude as fractions of picosecond (this implies\nthat typically ξ∼10−2). Thus, the inertial effects in magnetization dynamics are governed by a quantity\nwith the same smallness as usual Gilbert damping α∼10−2. The effective field heff(r, t) is given by[23]:\nheff=−δg\nδm, (4)\nwhich takes into account interactions (exchange, anisotropy, magnetostatics, Zeeman) among magnetic\nmoments and is expressed as the variational derivative of the free energy functional (the dimensionless\nenergy is measured in units of µ0M2\nsV, with Vbeing the volume of region Ω)\ng(m,ha) =1\nVZ\nΩl2\nex\n2(∇m)2+fan−1\n2hm·m−ha·mdV , (5)\nwhere Aandlex=p\n(2A)/(µ0M2s) are the exchange stiffness constant and length, respectively, fanis the\nanisotropy energy density, hmis the magnetostatic (demagnetizing) field and ha(r, t) the external applied\nfield.\n3When the anisotropy is of uniaxial type, such that fan=κan[1−(m·ean)2] with κanandeanbeing the\nuniaxial anisotropy constant and unit-vector, respectively, the effective field can be expressed by the sum of\na linear operator Cacting on magnetization vector field plus the applied field:\nheff(r, t) =−Cm+ha, (6)\nwhere C=−l2\nex∇2+N+κanean⊗eanandNis the (symmetric-positive definite) demagnetizing operator\nsuch that:\nhm(r) =1\n4π∇∇ ·Z\nΩm(r′)\n|r−r′|dV=−Nm. (7)\nAs mentioned in the previous section, eq. (3) is usually complemented with the natural boundary conditions\n∂m/∂n= 0 at the body surface ∂Ω, which is typical when no surface anisotropy is considered. It can be\nshown that the operator Cwith the aforementioned boundary conditions is self-adjoint and positive-definite\nin the appropriate subspace of square-integrable vector fields[23].\nIt is also worth remarking that, for eq.(3), equilibrium magnetization fields are characterized by simul-\ntaneously vanishing time-derivatives of first and second order:\n∂m\n∂t=0,∂2m\n∂t2=0. (8)\nEquation (3) describes a nonlinear dynamical system of higher order compared to that associated with\nthe classical LLG equation (obtained by setting ξ= 0 in eq.(3)). In fact, by defining a new variable w\nresembling, in a purely formal fashion, the ’angular momentum’ of a point-particle of unitary mass, position\nvector mand velocity ∂m/∂tsuch that\nw=m×∂m\n∂t, (9)\none has:\n∂w\n∂t=m×∂2m\n∂t2. (10)\nFirst, by dot-multiplying both sides of eq.(3) by m, we observe that magnetization vector evolves on the\nunit-sphere |m|2= 1 since\nm·∂m\n∂t= 0. (11)\nThen, by cross-multiplying both sides of eq.(3) by m, one obtains:\nw=−m×(m×heff) +αm×w+ξm×∂w\n∂t. (12)\nBy performing further cross-multiplication of both sides of the latter equation by m, one ends up with:\nm×w= (m×heff)−αw−ξdw\ndt, (13)\nwhere the property m·∂w/∂t= 0 has been used.\nConsequently, iLLG eq.(3) can be rewritten as a set two coupled nonlinear equations for variables m\nandwas follows:\n∂m\n∂t=w×m, (14)\nξ∂w\n∂t=−m×w−αw+m×heff, (15)\nwhere eq.(14) comes from eq.(9) cross-multiplied by mcombined with the fact that |m|2= 1, and eq.(15)\nfrom eq.(13). We point out that, as a consequence of eq.(8) and the definition of wfrom eq.(9), equilibrium\nsolutions of eqs.(14)-(15) are such that:\n∂m\n∂t=0,∂w\n∂t=0. (16)\n4In this way, the implicit equation (3) has been transformed into a higher-order equation in standard explicit\nform, which is amenable of general considerations concerning the properties of the dynamical systems that\nit describes.\nTo this end, we now focus on the dynamical system expressed by eqs.(14)-(15) where the state variables\nm,ware considered independent of each other, remembering that it is equivalent to the original iLLG eq.(3)\nwhen eq.(9) holds. First of all, by dot-multiplying eq.(14) by m, one can immediately see that the motion\nof vector moccurs on the unit-sphere |m|= 1:\nm·∂m\n∂t= 0 ⇒ |m(r, t)|= 1∀r∈Ω, t≥t0, (17)\nprovided that mhas unit-amplitude at initial time t0. The latter will be referred to as magnetization\namplitude conservation property.\nNow, let us sum eq.(14) dot-multiplied by wand eq.(15) divided by ξand dot-multiplied by m. One\nhas:\nw·∂m\n∂t+m·∂w\n∂t=∂(w·m)\n∂t=−α\nξw·m. (18)\nThis means that, in any spatial location r∈Ω, the scalar product w·m, termed as ’angular momentum’\nprojection on magnetization, will have to decay exponentially to zero as follows:\nw(r, t)·m(r, t) =w(r, t0)·m(r, t0)e−α\nξt∀r∈Ω, t≥t0, (19)\nwhere the time decay constant is controlled by the ratio ξ/α > 0 between the intensities of damping and\ninertia. Thus, for t≫t0(practically t > t 0+ 5ξ/α), the ’angular momentum’ variable wis asymptotically\nconstrained to evolve on the manifold defined by w·m= 0. Interestingly, for zero damping α= 0, the\nlatter equation implies exact conservation of the product w·mat any time:\nw(r, t)·m(r, t) =w(r, t0)·m(r, t0)∀r∈Ω, t≥t0. (20)\nFrom equation (19) it is also worth noting that, for any value of α≥0 and initially vanishing magne-\ntization time-derivative ∂m/∂t(r, t0) = 0 at any location r∈Ω, which therefore implies w(r, t0) = 0, the\niLLG dynamics will occur such that the product w·mis always zero:\nw(r, t)·m(r, t) =w(r, t0)·m(r, t0) = 0 ∀r∈Ω, t≥t0. (21)\nFrom the above discussion, being that the inertial magnetization dynamics must fulfill the two constraints\n(17),(19), one can conclude that, in general, the dynamical system obtained by the iLLG eq.(3) and expressed\nby eqs.(14)-(15) has, in each spatial location r∈Ω, four independent state variables evolving on a four-\ndimensional state space. This means that the iLLG dynamics requires a double number of degrees of freedom\ncompared to the classical LLG for its description.\nFurthermore, eq.(3) admits an additional conservation property. In fact, by dot-multiplying eq.(15) by\nwand integrating over the region Ω, one has:\n1\nVZ\nΩξ\n2∂|w|2\n∂tdV=1\nVZ\nΩ−α|w|2+w·(m×heff)dV⇔ (22)\n1\nVZ\nΩξ\n2∂|w|2\n∂tdV=1\nVZ\nΩ−α|w|2+heff·∂m\n∂tdV. (23)\nBy using the fact that\ndg\ndt=1\nVZ\nΩδg\nδm·∂m\n∂t+δg\nδha·∂ha\n∂tdV=1\nVZ\nΩ−heff·∂m\n∂t−m·∂ha\n∂tdV , (24)\n5and remembering from (9) that |w|=|∂m/∂t|, one obtains the following energy balance equation:\nd\ndt\u0012\ng+1\nVZ\nΩξ\n2|w|2dV\u0013\n=−1\nVZ\nΩm·∂ha\n∂t−α|w|2dV⇔\nd\ndt \ng+1\nVZ\nΩξ\n2\f\f\f\f∂m\n∂t\f\f\f\f2\ndV!\n=−1\nVZ\nΩm·dha\ndt−α\f\f\f\f∂m\n∂t\f\f\f\f2\ndV . (25)\nThe latter equation can be put in a more compact form by defining the following generalized free energy:\n˜g(m,w,ha) =g(m,ha) +1\nVZ\nΩξ\n2|w|2dV=g(m,ha) +1\nVZ\nΩξ\n2\f\f\f\f∂m\n∂t\f\f\f\f2\ndV , (26)\nwhere the second term, in the framework of the purely formal mechanical analogy introduced before, can be\nseen as a sort of ’kinetic’ energy (see the last equality in eq.(26)) augmenting the classical micromagnetic\nfree energy interpreted as ’potential’ energy. Thus, the balance equation (25) becomes\nd˜g\ndt=−1\nVZ\nΩm·∂ha\n∂t−α\f\f\f\f∂m\n∂t\f\f\f\f2\ndV , (27)\nwhere the first term at the right-hand side describes energy pumping via time-varying external applied\nmagnetic field and the second term takes into account the intrinsic dissipation of magnetic materials.\nIt is apparent that, under the assumption of constant-in-time (even spatially-inhomogeneous) applied\nfield ( ∂ha/∂t= 0), the generalized free energy ˜ gmust be a decreasing function of time:\nd˜g\ndt=−1\nVZ\nΩα\f\f\f\f∂m\n∂t\f\f\f\f2\ndV≤0, (28)\nwhich reveals a Lyapunov structure for the iLLG in terms of the generalized free energy ˜ gsimilarly to\nwhat happens for the LLG dynamics in terms of then classical free energy g. This means, that, under\nthe above assumptions, the only possible attractors of the dynamics are stable equilibria (i.e. such that\n∂m/∂t= 0, ∂w/∂t= 0 and ˜ gis minimum).\nIn addition, in the absence of dissipation ( α= 0), one has the conservation property for the quantity ˜ g:\nd˜g\ndt=d\ndt \ng+1\nVZ\nΩξ\n2\f\f\f\f∂m\n∂t\f\f\f\f2\ndV!\n= 0 , (29)\nwhich is analogous to the conservation of ’total’ (potential + ’kinetic’) energy ˜ gin mechanical systems and\nhere strikingly expresses the conservative nature of the (lossless) spin nutation dynamics.\nWe remark that the balance equation (25),(27) could have been derived directly from eq.(3) by dot-\nmultiplying both sides by the quantity in parentheses and integrating over Ω.\nFinally, we observe that, in the absence of dissipation (i.e. α= 0), eqs.(14)-(15) admit three integrals of\nmotion:\n\n\n|m(r, t)|= 1∀r∈Ω, t≥t0 (30a)\nw(r, t)·m(r, t) =w(r, t0)·m(r, t0)∀r∈Ω, t≥t0 (30b)\n˜g=g+1\nVZ\nΩξ\n2\f\f\f\f∂m\n∂t\f\f\f\f2\ndV= ˜g0, t≥t0 (30c)\nthat we term amplitude, ’angular momentum’ projection on magnetization and ’total’ free energy conser-\nvation, respectively. The former two hold in a pointwise fashion, that is in any location and time instant\n(provided that they are fulfilled at initial time t0), while the last is an integral constraint on magnetization\nmotion (we remark that ˜ g(t0) = ˜g0is the initial ’total’ free energy).\n6The above conservation laws hold for spatially-inhomogeneous magnetization processes, but one can also\nconsider ’sufficiently small’ particles where the exchange interaction strongly penalizes spatial magnetization\ngradients and, thus, approximately treat them as uniformly-magnetized (macrospin) anisotropic particles,\nwhich eliminates the dependence on the spatial location rwithin the ferromagnet. This makes sense when\ndealing with magnetic nanosystems of dimensions in the order of the exchange length, such as those used as\nelementary cells for magnetic memories and other spintronic devices[1]. Under the assumption of spatially-\nuniform magnetization and anisotropy of uniaxial type, the free energy (5) has the simple expression[39]:\ng(m,ha) =1\n2Dxm2\nx+1\n2Dym2\ny+1\n2Dzm2\nz−m·ha, (31)\nwhere Dx, Dy, Dzare effective demagnetizing factors taking into account shape and crystalline anisotropy.\nThe aforementioned integrals of motion (30a)-(30c) become\n\n\n|m(t)|= 1∀, t≥t0, (32a)\nw(t)·m(t) =w(t0)·m(t0)∀, t≥t0, (32b)\n˜g=g+ξ\n2|w|2=g+ξ\n2\f\f\f\fdm\ndt\f\f\f\f2\n= ˜g0, t≥t0, (32c)\nwith ggiven by eq.(31) and will be instrumental in the validation of time-stepping techniques that we will\ndiscuss in the following sections.\n3. Spatially semi-discretized iLLG equation\nNow we proceed to the numerical discretization of the iLLG equation. In the following, we will refer\nto spatially semi-discretized equations on a collection of Nmesh points ( rj)N\nj=1associated with the related\ncomputational cells of volume Vj. This description is quite general and works both for finite-difference and\nfinite-element methods.\nWe will denote as m(t) = (m1, . . . ,mN)T,w(t) = (w1, . . . ,wN)T∈R3N(the notationTmeans matrix\ntranspose) the mesh vectors containing all cell vectors mj(t),wj(t)∈R3with j= 1, . . . , N .\nMoreover, we will use the operator notation for the cross-product for both cell and mesh vectors, namely:\nΛ(v)·w=v×w,Λ(v)·w= (v1×w1, . . . ,vN×wN)T, (33)\nmeaning that the latter operator is a skew-symmetric 3 N×3Nblock-diagonal operator that provides cross\nproduct of homologous cell vectors.\nThus, the semi-discretized iLLG equation will read as:\ndm\ndt= Λ(w)·m, (34)\nξdw\ndt=−Λ(m)·w−αw+ Λ(m)·heff, (35)\nwhere the discrete effective field heffis given by:\nheff(m, t) =−∂g\n∂m=−C·m(t) +ha(t), (36)\nthe symmetric positive-definite matrix Cplays the role of the effective field operator Candg(m,ha) is the\ndiscrete counterpart of the free energy defined by eq. (5):\ng(m,ha) =1\n2mT·C·m−hT\na·m. (37)\n7By using the same line of reasoning as for the continuous iLLG equation (14)-(15), one can derive the\nfollowing conservation properties:\n|mj(t)|=|mj(t0)| ∀t≥t0, j = 1, . . . , N , (38)\n(wj(t)·mj(t)) = (wj(t0)·mj(t0))e−α\nξt∀t≥t0, j = 1, . . . , N , (39)\nd\ndt \ng(m(t),ha(t)) +ξ\n2\f\f\f\fdm\ndt\f\f\f\f2!\n=d\ndt\u0012\ng(m(t),ha(t)) +ξ\n2|w|2\u0013\n=d˜g\ndt=−α\f\f\f\fdm\ndt\f\f\f\f2\n−m·dha\ndt,(40)\nwhere ˜ g(m,w,ha) =g(m,ha) +ξ\n2|w|2is the discrete ’total’ energy corresponding to ˜ gin the continuous\niLLG dynamics (see eq.(26)).\n4. Midpoint time-steppings for iLLG dynamics\nThe numerical solution of eqs.(34)-(35) with classical time-stepping techniques in general will corrupt the\nconservation properties (38)-(40) of semi-discretized inertial magnetization dynamics. Thus, such properties\nwill be fulfilled with an accuracy depending on the amplitude of the time-step ∆ t. For the classical purely\nprecessional LLG equation, it has been shown[38] that the implicit midpoint rule technique preserves the\nproperties of discrete magnetization dynamics regardless of the time-step. Here we propose two schemes\nbased on such technique for the iLLG spin nutation dynamics.\n4.1. Implicit midpoint rule (IMR)\nThe first is based on discretiztion of eqs.(34)-(35) at time tn+1\n2=tn+ ∆t/2 with the following second-\norder midpoint formulas:\nmn+1\n2=mn+1+mn\n2wn+1\n2=wn+1+wn\n2, (41)\nwhere mn,wndenote m(tn),w(tn), which leads to the following time-stepping for the j−th computational\ncell:\nmjn+1−mjn\n∆t=wn+1\n2\nj×mn+1\n2\nj , (42)\nξwn+1\nj−wn\nj\n∆t=−mn+1\n2\nj×wn+1\n2\nj−αwn+1\n2\nj\n+mn+1\n2\nj×heff,j(mn+1\n2, tn+1\n2), j = 1, . . . , N . (43)\nNow, by dot-multiplying the first equation by mn+1/2\nj , one can easily see that\n|mn+1\nj|2− |mn\nj|2= 0, j = 1, . . . , N , (44)\nmeaning that magnetization amplitude will be preserved unconditionally, namely independently of ∆ tin each\ncomputational cell. In addition, by dot-multiplying eq.(42) by wn+1\n2and eq.(43) by mn+1\n2and summing\nboth sides of the result, one immediately ends up with:\nwn+1\nj+wn\nj\n2·mn+1\nj−mn\nj\n∆t+mn+1\nj+mn\nj\n2·wn+1\nj−wn\nj\n∆t=−α\nξwn+1\n2\nj·mn+1\n2\nj, j = 1, . . . , N , (45)\nwhich expresses the reproduction of the property (39) in its mid-point time discretized version:\nwn+1\nj·mn+1\nj−wn\nj·mn\nj\n∆t=−α\nξwn+1\n2\nj·mn+1\n2\nj, j = 1, . . . , N . (46)\n8Remarkably enough, in the conservative case α= 0, the latter equation becomes:\nwn+1\nj·mn+1\nj−wn\nj·mn\nj\n∆t= 0 ⇒wn+1\nj·mn+1\nj=wn\nj·mn\nj, j = 1, . . . , N (47)\nwhich means that the ’angular momentum’ projection conservation property is fulfilled for any choice of the\ntime step ∆ t.\nNow let us consider the midpoint rule time-stepping for the mesh vectors:\nmn+1−mn\n∆t= Λ(wn+1\n2)·mn+1\n2, (48)\nξwn+1−wn\n∆t=−Λ(mn+1\n2)·wn+1\n2−αwn+1\n2+ Λ(mn+1\n2)·heff(mn+1\n2, tn+1\n2). (49)\nBy assuming constant applied field, dot-multiplying the second equation by wn+1/2and taking into\naccount eq.(36) and the symmetry of the matrix C, one obtains the following discretized energy balance:\nξ\n2|wn+1|2− |wn|2\n∆t=−α|wn+1\n2|2−gn+1−gn\n∆t⇔ (50)\n˜gn+1−˜gn\n∆t=−α|wn+1\n2|2,\nwhere ˜ gn= ˜g(mn), which implies that the total (discrete) energy ˜ gmust be either decreasing when α >0\nor being conserved when α= 0, both regardless of the time-step.\nEquations (42)-(43) represent a nonlinear system of 6 Ncoupled scalar equations, which must be solved\nat each time step. They can be regarded as the following two vector equations in u=mn+1,v=wn+1:\nF(u,v) =0,G(u,v) =0, (51)\nand can be solved by using Newton-Raphson iteration:\n\u0012uk+1\nvk+1\u0013\n=\u0012uk\nvk\u0013\n− ∂F\n∂u∂F\n∂v\n∂G\n∂u∂G\n∂v!−1\n·\u0012F(uk)\nG(vk)\u0013\n, (52)\nwhere the partial Jacobian matrices are given by:\n∂F\n∂u=I\n∆t−1\n4Λ(v+wn), (53)\n∂F\n∂v=1\n4Λ(u+mn), (54)\n∂G\n∂u=−1\n4Λ(v+wn) +1\n2Λ(u+mn)·C−Λ\u0014\nheff\u0012u+mn\n2\u0013\u0015\n, (55)\n∂G\n∂v=ξ\n∆tI+1\n4Λ(u+mn) +α\n2I, (56)\nand the linear operator notation Λ has been used for the cross product involving mesh vectors.\nThe above time-stepping has remarkable qualitative properties that reproduce those of the continuous\niLLG dynamics and, therefore, represents the preferred choice to realize inertial micromagnetic numerical\ncodes for the analysis of terahertz magnetization dynamics.\nHowever, it evidently requires to double the state variables and, consequently, the number of unknowns,\nowing to the introduction of the vector field walthough one is mainly interested to compute the dynamics\nof magnetization vector field m. This issue becomes even more pronounced when large-scale micromagnetic\nsimulations with full spatial discretization are considered, which would require dramatic modification of\n9numerical codes in order to introduce the auxiliary variable wand to solve a system of 6 Nnonlinear\ncoupled equations at each time-step. Moreover, we remark that in the latter situation, the 3 N×3Nmatrix\nCinvolved in the Newton iteration (see eq.(55)) is also fully-populated owing to the long-range nature of\nmagnetostatic interactions. Therefore, following what is done for the classical LLG equation[38], a quasi-\nNewton technique is required to solve the large nonlinear system, implemented by considering reasonable and\nsparse approximation of the matrix C(e.g. obtained retaining only exchange and anisotropy contributions\nCexandCan, respectively) and, in turn, of the full Jacobian defined by eqs.(53)-(56). Of course, the\ncomputational cost of such quasi-Newton method, involving the solution of several non-symmetric large\nlinear systems (e.g. by using GMRES methods[40]), will be at least double with respect to that associated\nwith LLG equation, posing a strong limit to the capability of solving large-scale iLLG dynamics.\n4.2. Implicit midpoint with multi-step inertial term (IMR-MS)\nFor these reasons, in order to obtain an alternative efficient numerical technique with minimum modifi-\ncation of existing micromagnetic codes, we propose a second time-stepping based on the implicit midpoint\nrule combined with a multi-step method for the inertial term. This technique is based on direct space-time\ndiscretization of eq.(3) at time tn+1\n2=tn+ ∆t/2:\nmn+1−mn\n∆t=−Λ(mn+1\n2)·\u0010\nheff\u0010\nmn+1\n2, tn+1\n2\u0011\n−αmn+1−mn\n∆t−ξd2m\ndt2\f\f\fn+1\n2\u0011\n. (57)\nThen, in order to retain the amplitude conservation property (44) of the aforementioned IMR scheme,\nwe use the first of midpoint formulas (41) in eq.(57) in a way that it is rewritten as system of 3 Nnonlinear\nequations in the unknowns mn+1:\nFn(mn+1) =0, (58)\nwhere Fn(y) :R3N→R3Nis the following vector function:\nFn(y) =\u0014\nI−αΛ\u0012y+mn\n2\u0013\u0015\u0000\ny−mn\u0001\n−∆tfn\u0012y+mn\n2\u0013\n−∆t ξd2m\ndt2\f\f\fn+1\n2\u0011\n, (59)\nand where\nfn(m) =−Λ(m)·heff\u0012\nm, tn+∆t\n2\u0013\n= Λ(m)·∂g\n∂m\u0012\nm,ha\u0012\ntn+∆t\n2\u0013\u0013\n(60)\nis the purely precessional term in the right-hand-side of the conservative semi-discretized iLLG equation\n(35) expressed by using the definition (36) of the discrete effective field.\nThen, we adopt a multi-step approach with a p−points backward formula for the second derivative in\nthe inertial term appearing in eqs.(57) and (59):\nd2m\ndt2\f\f\fn+1\n2≈1\n∆t2p≥3X\nk=1an+2−kmn+2−k= ∆2\np, (61)\nwhere the coefficients an+2−kare determined from truncation error analysis in order to control the accuracy\nof the approximation. This technique implies a slight modification of existing numerical codes based on\nimplicit midpoint rule time-stepping. In fact, once formula (61) is plugged into the time-stepping equation\n(57), the solution of the nonlinear coupled equations (58) can be obtained by using the Newton-Raphson\ntechnique[38] as follows:\ny0=mn,yk+1=yk+ ∆yk+1with Jn\nF(yk, tn)∆yk+1=−Fn(yk), (62)\nby simply considering the following augmented Jacobian matrix of the iteration:\nJF(u, t) =I\n∆t+α\n∆tΛ(mn) +ξ\n2∆t2a1Λ(u) +ξ\n2∆t2Λ p≥3X\nk=2an+2−kmn+2−k!\n−\n−ξ\n2∆t2Λ(u+mn)a1−1\n2Jf\u0012u+mn\n2, t+∆t\n2\u0013\n(63)\n10p order Coefficients scheme\n3O(∆t) a1= 1, a0=−2, a−1= 1 IMR-MS1\n4O(∆t2)a1= 3/2, a0=−7/2, a−1= 5/2, a−2=−1/2IMR-MS2\nTable 1: Table of coefficients for multi-step formula (61).\nwhere Jf(u, t) = Λ (u)·C+ Λ[heff(u, t)]. The linear system in eq.(62) is solved at each iteration kby\nconsidering the sparse approximation of the full matrix CasC≈Cex+Canin the Jacobian Jfand using\nthe GMRES method[40] until the residual ∥Fn(yk)∥decreases below a prescribed tolerance.\nNow, if one assumes that initially magnetization has zero velocity dm/dt(t= 0) = 0, which is reasonable\nin simulation of experimental situations, then at the first time step n= 1 one has mn+2−k= 0, k > 2. For\nthe subsequent steps n >1, one can use magnetization samples from the previous steps as mn+2−k, k > 2\nin eq.(57). The only cost of such operation is for the storage of p−2 magnetization vectors.\nIn this respect, the simplest choice is the classical p= 3 points formula:\nd2m\ndt2\f\f\fn+1\n2≈mn+1+mn−1−2mn\n∆t2= ∆2\np=3, (64)\nwhich, plugged into eq.(57), defines the IMR-MS1 scheme.\nHowever, an analysis of truncation error reveals that:\n∆2\np=3=d2m\ndt2\f\f\fn+1\n2−1\n2d3m\ndt3\f\f\fn+1\n2∆t+5\n24d4m\ndt4\f\f\fn+1\n2∆t2+. . . , (65)\nmeaning that the accuracy is just of first order O(∆t) (it would be O(∆t2) if the second derivative was\ncomputed at t=tn). Thus, in order to be consistent with discretization of other terms in eq.(57) at second\norder with respect to ∆ t, one can derive a more accurate formula using p= 4 points. To this end, we\ncompute a different second derivative formula:\n˜∆2\np=3=d2m\ndt2\f\f\fn+1\n2≈2mn+1−3mn+mn−2\n3∆t2, (66)\nwhich has truncation error such that:\n˜∆2\np=3=d2m\ndt2\f\f\fn+1\n2−5\n6d3m\ndt3\f\f\fn+1\n2∆t+13\n24d4m\ndt4\f\f\fn+1\n2∆t2+. . . . (67)\nNow we use Richardson extrapolation[41] to cancel O(∆t) order terms in the truncation and define the\nfollowing new difference formula (defining the IMR-MS2 scheme):\n∆2\np=4=5\n2∆2\np=3−3\n2˜∆2\np=3=3mn+1−7mn+ 5mn−1−mn−2\n2∆t2, (68)\nfor which the truncation error is O(∆t2):\n∆2\np=4=d2m\ndt2\f\f\fn+1\n2−7\n24d4m\ndt4\f\f\fn+1\n2∆t2+. . . . (69)\nThe coefficients for the above multi-step formulas with p= 3,4 are summarized in table 1.\nIn order to assess the order of accuracy of the proposed schemes, we consider the conservative iLLG\ndynamics ( α= 0) and numerically integrate eq.(3) using IMR, IMR-MS1, IMR-MS2 time-steppings for\ndifferent time step ∆ tand compare the results with a benchmark reference solution obtained by using\nstandard adaptive time step Dormand-Prince Runge-Kutta (RK45) scheme[42, 43]. Absolute tolerances are\nset to 10−14both for RK45 and for Newton iterations solving eq.(57) with IMR, IMR-MS1, IMR-MS2.\nIn the left panel of figure 1, we report the global truncation error ||∆m||arising from time-integration\nof iLLG eq.(3) in the interval [0,1] between the proposed schemes and the reference RK45 solution. A\n1110−410−310−210−110−710−610−510−410−310−210−1100\n∆t||∆m||\n  \nIMR\nIMR−MS1\nIMR−MS2\nO(∆t)\nO(∆t2)\n0 5 10 15 20 2510−2010−101001−|m|\n  \n0 5 10 15 20 2510−2010−10100∆g/g\n  \n0 5 10 15 20 2510−2010−10100\nt  [(γ Ms)−1]∆(w⋅ m)\n  RK45 IMR IMR−MS1 IMR−MS2Figure 1: Accuracy tests on conservative ( α= 0) iLLG dynamics. The values of parameters are Dx= 0,1, Dy= 0.2, Dz=\n0.7,ha= (0,0,0.1),m(t= 0) = (1,0,0), ξ= 0.03. (left) Global error ||∆m||att= 1 between IMR, IMR-MS1, IMR-MS2\nschemes and reference RK45 solution showing first-order O(∆t) behavior for IMR-MS1 and second-order O(∆t2) for IMR and\nIMR-MS2. (right) Conservation of properties of iLLG dynamics versus time (time step ∆ t= 0.001 for all IMR schemes).\nTop panel refers to amplitude 1 − |m|conservation, middle panel to relative error ∆˜ g/˜g= (˜g(t)−˜g(0))/˜g(0) in ’total’ free\nenergy conservation, bottom panel refers to error ∆( w·m) =w(t)·m(t)−w(0)·m(0) in ’angular momentum’ projection on\nmagnetization conservation. One can see that all IMR, IMR-MS1, IMR-MS2 schemes preserve amplitude |m|and projection\nw·mwith (double-precision) machine accuracy and only IMR also preserves energy. IMR-MS2 outperforms IMR-MS1 and\nRK45 in energy conservation.\nquick inspection of the figure confirms the expected first and second orders of accuracy for IMR-MS1 and\nIMR,IMR-MS2, respectively. Remarkably, IMR-MS2 has performance quite similar to the fully-implicit IMR\nwithout doubling the number of degrees of freedom. On the other hand, in the right panel of fig.1, one can\nlook at the conservation properties for the proposed schemes (with ∆ t= 0.001) and the reference solution. As\nexpected, all the three IMR schemes are able to preserve amplitude |m|and ’angular momentum’ projection\non magnetization w·mwith (double) machine precision, while only the fully-implicit IMR is able to do so\nfor the ’total’ energy ˜ g. Nevertheless, it is apparent (middle panel, blue and cyan solid lines) that IMR-MS2\nis able to guarantee the same precision as the RK45 concerning energy conservation while being significantly\nlower-order than RK45. For the evaluation of the variable wand energy ˜ gwhen considering IMR-MS\nschemes, we have used the central difference formula wn=mn×(mn+1−mn−1)/(2∆t).\n5. Numerical results\nIn order to validate the proposed techniques on physically relevant situations, we perform two different\nsimulations. The first describes the ultra-fast resonant spin nutation of a uniformly-magnetized thin-film\ndriven by ac terahertz appled field, similar to the experiment[11] that provided direct evidence of the\npresence of inertial effects. This will also be a basic testbed to compare the accuracy of the developed IMR\nand IMR-MS schemes. The second simulation will address the ultra-fast spatially-inhomogeneous dynamics\nof magnetization in a microscale nanodot excited with terahertz applied field and will demonstrate the\nefficiency of the IMR-MS time-stepping in full micromagnetic simulations.\n5.1. Nutation frequency response of a single-domain particle\nWe analyze the frequency response of a thin-film magnetized along the easy ydirection and subject\nto an out-of-plane ac field with small amplitude. In this situation, one can assume that macrospin iLLG\ndynamics occurs in the linear regime and analytical theory can be developed. To this end, let us assume\n12Figure 2: Time-domain linear relaxation of mzcomputed with IMR, IMR-MS with different time steps and compared with\nanalytical solution of linear iLLG eq.(70).\nthat the applied field is decomposed in a nonzero constant bias field plus a time-harmonic component\nha(t) =hdc+hac(t),|hac| ≪ |hdc|. We also assume that the free energy has the simple form (31) under\nthe macrospin approximation.\nThen, the iLLG eq.(3) can be linearized around an equilibrium m0such that m(t) =m0+ ∆m(t) in\nthe following way:\nd∆m\ndt=m0×\u0014\n(D+h0I)·∆m−αd∆m\ndt−ξd2∆m\ndt2\u0015\n, (70)\nwhere Ddenotes the diagonal matrix D= diag[ Dx, Dy, Dz],h0=heff(m0)·m0= (−D·m0+ha)·m0is\nthe projection of the equilibrium effective field on equilibrium magnetization.\nWe first observe that the dynamics fulfills the constraint m0·∆m= 0 (to the first order), therefore\nwe can consider only the dyanamics of the component ∆ m⊥of ∆mliving in the plane perpendicular to\nthe equilibrium m0. We also refer to the projection of the demag tensor DasD⊥. As a consequence, we\nwill deal with vectors having only two components associated with axes transverse to the equilibrium m0.\nWe observe that the skew-symmetric operator Λ is invertible (such that Λ ·Λ =−I) when restricted to\nthe plane orthogonal to m0and we express both ∆ m⊥,hacusing complex (phasor) domain as ∆ m⊥(t) =\n∆˜mejωt,hac=˜hacejωt. By using these formulas in eq.(70), one ends up with:\n∆˜m=\u0002\njωΛ(m0) + (D⊥+h0I) +jωαI −ξω2I\u0003−1\n| {z }\nχ(ω)˜hac, (71)\nwhich defines the magnetic susceptibility tensor χ(ω). When referred to principal axes, χ(ω) is a 2 ×2\nmatrix which can be easily computed.\nIt can be shown that the resonance frequencies associated with the above linear dynamical system are\nthe roots of the following fourth-degree polynomial:\nξ2ω4−2jαξω3−(α2+ξ(ω0y+ω0z) + 1) ω2+jα(ω0y+ω0z)ω+ω0yω0z= 0, (72)\nwhere ω0y=Dy−Dx+hdcandω0y=Dz−Dx+hdc. Equation (72) can be solved by using appropriate\nperturbation theory leading to the following resonance frequencies (computed in the conservative case when\n13Figure 3: Frequency response power spectrum |∆ ˜mz(ω)|2. detail of the nutation peak. The FMR frequency is around 18\nGHz whereas the nutation frequency is about 634 GHz. The respective linewidths are about 1 GHz and 27 GHz. Blue line\nis eq.(71), red dots and dashed line are analytical formulas (73),(75), black symbols are the result of numerical simulations of\niLLG dynamics with IMR-MS1.\nα= 0):\nωFMR≈ ±ωKp\n1 +ξ(ω0y+ω0z), ωK=√ω0yω0z, (73)\nωN≈ ±sp\n2ξ(ω0y+ω0z) + 1 + ξ(ω0y+ω0z) + 1\n2ξ2, (74)\nwhere ωK=p\n(Dy−Dx+hdc)(Dz−Dx+hdc) is the classical Kittel ferromagnetic resonance (FMR)\nfrequency. The former equation describes the influence of inertial effects on the FMR frequency, while the\nsecond formula gives the nutation resonance frequency (typically in the THz range). We observe that the\nabove formulas take into account the dependence on the external bias field through the parameters ω0y, ω0z.\nIt is also possible to determine closed-form expressions for the half-power (Full Width at Half Maximum,\nFWHM) linewidths:\n∆ωFMR≈α(ω0y+ω0z)−ξ(ω2\n0y+ 4ω0yω0z+ω2\n0z), (75)\n∆ωN≈α\nξ\"\n1 +1p\n2ξ(ω0y+ω0z) + 1#\n. (76)\nHere we consider an infinite thin-film ( Dx= 0, Dy= 0, Dz= 1) with material parameters: damping\nα= 0.023, saturation magnetization such that µ0Ms= 0.93T, inertial time scale τ= 1.26 ps.\nIn figure 2, we report the time-evolution of mzduring relaxation under zero bias field starting from an\ninitial state tilted in the x−zplane such that mx= 0.01, mz= 0.01. The analytical solution of eq.(70) is\nused to benchmark the proposed numerical techniques with different time step amplitudes. It is apparent\nthat IMR technique allows the use of the largest time steps (up to ∆ t= 0.01, around 25 samples per nutation\nperiod) yielding no significant loss of accuracy with respect to the analytical solution. One can also see that\nIMR-MS with first-order formula (64) (IMR-MS1) is accurate when ∆ t∼0.0001 (corresponding to 0.61 fs\nin physical units), while is not able to follow nutation dynamics after 5-6 periods (the period is 0.27) with\n50 times larger time step ∆ t= 0.005. Conversely, IMR-MS with second order formula (68) (IMR-MS2)\nperforms well with ∆ t= 0.005 (slightly above 50 samples per period) and provides a measured speedup of\n14about 30 times compared to the former. This occurs since the average number of Newton iterations remains\nof the same order of magnitude, namely 2 for IMR-MS1 with ∆ t= 0.0001 and 3 for IMR-MS2 with 50\ntimes larger ∆ t(the tolerance for Newton-Raphson iteration was set to 10−14). Finally, comparing IMR\nand IMR-MS2 methods, one can see that, despite correctly reproducing the nutation oscillation, IMR-MS2\nproduces a small phase-shift when the largest time step ∆ t= 0.01 is chosen.\nNext, by using eqs.(70), the frequency response power spectrum of the out-of-plane magnetization com-\nponent mzhas been computed under a bias field hax= 0.35T and ac field directed along y. The iLLG\nequation (3) has been solved numerically in order to determine the frequency response power spectrum.\nNamely, given the susceptibility χ(ω) in eq.(71) and the cross-power spectrum matrix (∆ ˜m)·(∆˜m)H=\n(χ(ω)·˜hac)·(χ(ω)·˜hac)H(Hmeans conjugate transpose) assuming that the only nonzero component of the\ninput field ˜hac= (˜hy,˜hz)Tis˜hy= 1 (Fourier Transform of a Dirac delta) and the output response is the\nout-of-plane magnetization ∆ ˜ mz, the output power spectrum is:\n|∆ ˜mz(ω)|2=|χzy(ω)|2, (77)\nwhich is reported in fig.3 and compared with analytical formulas (73),(75). In order to compute |χzy|2from\ntime-domain numerical simulations, we apply a sinusoidal field hy(t) at frequency ω0with sufficiently small\namplitude (in order to stay in the linear regime) and measure the steady-state oscillation of mz(t) after\nsufficiently long time so that the transient response has vanished. This has been performed by choosing a\nsimulated time T= 2ns and ac field amplitude equal to 0.06T. Then, the Fast Fourier Transform Mz(ω) =\nF[mz(t)] has been computed and the maximum of the power spectrum |Mz(ω)|2has been determined.\nThis procedure has been repeated for several values of ω0. The so determined points form samples of the\nfrequency response power spectrum |χzy(ω)|2.\nIn figure 3, numerical simulations performed with IMR-MS1 with ∆ tcorresponding to 1 fs are used to\ncompute the output power spectrum (black symbols). The results are in excellent agreement with analytical\ntheory.\n5.2. Spatially-inhomogeneous spin nutation driven by terahertz applied field\nIn order to perform efficient time-domain micromagnetic simulations of iLLG dynamics with full spatial\ndiscretization, the proposed IMR-MS2 time-stepping has been implemented in the finite-difference numerical\ncode MaGICo[38, 44], which retains the same computational cost as the simulation of classical precessional\nLLG dynamics while keeping important conservation properties as outlined above.\nTo validate the code, we explore typical spatio-temporal patterns of iLLG dynamics considering the\ntime-domain simulation of ultra-short inertial spin waves in a confined ferromagnetic thin-film. At tera-\nhertz frequencies, the behavior of small magnetization oscillations significantly deviates from the classical\ndescription of exchange-dominated spin-waves, in that an ultimate limiting propagation speed appears[24].\nThis difference is mostly due to the mathematical structure of eq.(3) compared with the classical LLG pre-\ncessional dynamics, i.e. the same equation where one sets ξ= 0. In fact, when inertial effects are taken into\naccount, the torque proportional to the second-order time-derivative transforms the classical LLG equation\ninto a wave-like equation with hyperbolic mathematical character. In this respect, on short time scales,\nfinite time delays are expected in magnetization response propagation far from local external excitation.\nThe considered sample is made of Cobalt and has a thin-film shape with square cross-section 200 ×200 nm2\nand thickness 5 nm. The ferromagnetic nanodot is initially at equilibrium, being saturated along the xaxis\nby a static field µ0Hax= 100 mT. The value of material parameters are γ= 2.211×105m A−1s−1,\nµ0Ms= 1.6 T,A= 13 pJ/m ( lex= 3.57 nm), τ= 0.653 ps ( ξ= 0.0338) and α= 0.005.\nThe applied field is a spatially-uniform sine wave step (turned on at t= 0) along the yaxis transverse\nto the equilibrium configuration with amplitude µ0Hay= 100 mT and frequency f= 1386 GHz. For the\nabove choice of parameters, this excitation frequency is slightly above the nutation resonance frequency and\ncorresponds to inertial spin waves with wavelength around 20 nanometers[24].\nThe numerical simulation of iLLG equation (3) is performed with a time-step of 25 fs, a 2 .5×2.5×5 nm3\ncomputational cell, which corresponds to discretize the thin-film into 80 ×80 square prims cell. In order\nto isolate short-wavelength spin wave propagation from the rest of the simulated spatial pattern, high-pass\n15Figure 4: Spatial profiles of short-wavelength magnetization out-of-plane component mzobtained by FFT high-pass filtering\nat time t= 1,10,30,49 ps. The magenta solid line is a guide for the eye to follow the spin wave oscillation profile along the x\ndirection.\nspatial filtering via two-dimensional Fast Fourier Transform is performed on magnetization components.\nThe simulated time is 100 ps and some snapshots of the magnetization out-of-plane component mz, taken\nat different time instants, are reported in figure 4.\nThe magnetization is initially at the equilibrium and mostly aligned with the static field along the x\ndirection except close to the square corners where there is the most pronounced deviation. Such a tilting acts\nas local excitation for inertial spin waves when the time-varying ac field step is applied[24]. In fact, as it can\nbe seen in the various panels of fig.4, two wavepacket with wavelength ≈20 nm propagate from the edges of\nthe nanodot toward its center after the application of the ac field, consistently showing a propagation with\nfinite speed ≈2000 m/s compatible with that predicted by the theory[24].\n6. Conclusion\nIn this paper, we have proposed second-order accurate and efficient numerical schemes for the time-\nintegration of the ultra-fast inertial magnetization dynamics. We have shown that the iLLG equation\ndescribes a higher-order dynamical system compared to the classical precessional dynamics which requires\nto double the degrees of freedom for its desription. We have derived the fundamental properties of the iLLG\ndynamics, namely conservation of magnetization amplitude and ’angular momentum’ projection, Lyapunov\nstructure and generalized free energy balance properties, and demonstrated that the proposed implicit\n16midpoint rule (IMR) time-stepping is able to correctly reproduce them unconditionally. Suitable Newton\ntechnique has been developed for the inversion of the nonlinearly coupled system of equations to be solved at\neach time-step. For large-scale micromagnetic simulations with full spatial discretization, efficient numerical\ntime-stepping schemes based on implicit midpoint rule combined with appropriate multi-step method for\nthe inertial term, termed IMR-MS of order 1 and 2, have been proposed. These schemes retain the same\ncomputational cost of the IMR for the classical LLG dynamics while providing conservation of magnetization\namplitude and accurate reproduction of the high frequency nutation oscillations. In particular, thanks to\nthe unconditional stability due to its implicit nature along with the second-order accuracy on the inertial\nterm, both the IMR and IMR-MS2 allow choosing moderately large time-steps only based on accuracy\nrequirements for the description of the nutation dynamics. The proposed techniques have been successfully\nvalidated against test cases of spatially-homogeneous and inhomogeneous magnetization iLLG dynamics\ndemonstrating their effectiveness. For these reasons, we believe that these numerical schemes can become a\nstandard de facto in the micromagnetic simulation of inertial magnetization dynamics in nano- and micro-\nscale magnetic systems.\nAcknowledgements\nM.d’A., S.P. and C.S. acknowledge support from the Italian Ministry of University and Research,\nPRIN2020 funding program, grant number 2020PY8KTC.\nReferences\n[1] B. Dieny, I. L. Prejbeanu, K. Garello, P. Gambardella, P. Freitas, R. Lehndorff, W. Raberg, U. Ebels, S. O. Demokritov,\nJ. Akerman, A. Deac, P. Pirro, C. Adelmann, A. Anane, A. V. Chumak, A. Hirohata, S. Mangin, S. O. Valenzuela,\nM. C. Onba¸ slı, M. d’Aquino, G. Prenat, G. Finocchio, L. Lopez-Diaz, R. Chantrell, O. Chubykalo-Fesenko, P. Bortolotti,\nOpportunities and challenges for spintronics in the microelectronics industry, Nature Electronics 3 (8) (2020) 446–459.\ndoi:10.1038/s41928-020-0461-5 .\nURL https://doi.org/10.1038/s41928-020-0461-5\n[2] E. Beaurepaire, J.-C. Merle, A. Daunois, J.-Y. Bigot, Ultrafast spin dynamics in ferromagnetic nickel, Physical Review\nLetters 76 (22) (1996) 4250–4253. doi:10.1103/physrevlett.76.4250 .\nURL https://doi.org/10.1103/physrevlett.76.4250\n[3] B. Koopmans, M. van Kampen, J. T. Kohlhepp, W. J. M. de Jonge, Ultrafast magneto-optics in nickel: Magnetism or\noptics?, Physical Review Letters 85 (4) (2000) 844–847. doi:10.1103/physrevlett.85.844 .\nURL https://doi.org/10.1103/physrevlett.85.844\n[4] C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast, K. Holldack, S. Khan, C. Lupulescu, E. F. Aziz, M. Wietstruk,\nH. A. D¨ urr, W. Eberhardt, Femtosecond modification of electron localization and transfer of angular momentum in nickel,\nNature Materials 6 (10) (2007) 740–743. doi:10.1038/nmat1985 .\nURL https://doi.org/10.1038/nmat1985\n[5] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, T. Rasing, All-optical magnetic recording\nwith circularly polarized light, Physical Review Letters 99 (4) (2007) 047601. doi:10.1103/physrevlett.99.047601 .\nURL https://doi.org/10.1103/physrevlett.99.047601\n[6] A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev, A. Kirilyuk, T. Rasing, Inertia-driven spin switching in\nantiferromagnets, Nature Physics 5 (10) (2009) 727–731. doi:10.1038/nphys1369 .\nURL https://doi.org/10.1038/nphys1369\n[7] A. Kirilyuk, A. V. Kimel, T. Rasing, Ultrafast optical manipulation of magnetic order, Reviews of Modern Physics 82 (3)\n(2010) 2731–2784. doi:10.1103/revmodphys.82.2731 .\nURL https://doi.org/10.1103/revmodphys.82.2731\n[8] C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad, Y. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski, K. Hono,\nY. Fainman, M. Aeschlimann, E. E. Fullerton, All-optical control of ferromagnetic thin films and nanostructures, Science\n345 (6202) (2014) 1337–1340. doi:10.1126/science.1253493 .\nURL https://doi.org/10.1126/science.1253493\n[9] C. Dornes, Y. Acremann, M. Savoini, M. Kubli, M. J. Neugebauer, E. Abreu, L. Huber, G. Lantz, C. A. F. Vaz, H. Lemke,\nE. M. Bothschafter, M. Porer, V. Esposito, L. Rettig, M. Buzzi, A. Alberca, Y. W. Windsor, P. Beaud, U. Staub,\nD. Zhu, S. Song, J. M. Glownia, S. L. Johnson, The ultrafast Einstein–de Haas effect, Nature 565 (7738) (2019) 209–212.\ndoi:10.1038/s41586-018-0822-7 .\nURL https://doi.org/10.1038/s41586-018-0822-7\n[10] M. Hudl, M. d’Aquino, M. Pancaldi, S.-H. Yang, M. G. Samant, S. S. Parkin, H. A. D¨ urr, C. Serpico, M. C. Hoffmann,\nS. Bonetti, Nonlinear magnetization dynamics driven by strong terahertz fields, Physical Review Letters 123 (19) (2019)\n197204. doi:10.1103/physrevlett.123.197204 .\nURL https://doi.org/10.1103/physrevlett.123.197204\n17[11] K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Z. Hagstr¨ om, S. S. P. K. Arekapudi, A. Semisalova, K. Lenz,\nB. Green, J.-C. Deinert, I. Ilyakov, M. Chen, M. Bawatna, V. Scalera, M. d’Aquino, C. Serpico, O. Hellwig, J.-\nE. Wegrowe, M. Gensch, S. Bonetti, Inertial spin dynamics in ferromagnets, Nature Physics 17 (2) (2020) 245–250.\ndoi:10.1038/s41567-020-01040-y .\nURL https://doi.org/10.1038/s41567-020-01040-y\n[12] V. Unikandanunni, R. Medapalli, M. Asa, E. Albisetti, D. Petti, R. Bertacco, E. E. Fullerton, S. Bonetti, Inertial spin\ndynamics in epitaxial cobalt films, Physical Review Letters 129 (2022) 237201. doi:10.1103/PhysRevLett.129.237201 .\nURL https://link.aps.org/doi/10.1103/PhysRevLett.129.237201\n[13] M.-C. Ciornei, J. M. Rub´ ı, J.-E. Wegrowe, Magnetization dynamics in the inertial regime: Nutation predicted at short\ntime scales, Physical Review B 83 (2) (2011) 020410(R). doi:10.1103/physrevb.83.020410 .\nURL https://doi.org/10.1103/physrevb.83.020410\n[14] E. Olive, Y. Lansac, J.-E. Wegrowe, Beyond ferromagnetic resonance: The inertial regime of the magnetization, Applied\nPhysics Letters 100 (19) (2012) 192407. doi:10.1063/1.4712056 .\nURL https://doi.org/10.1063/1.4712056\n[15] R. Mondal, M. Berritta, A. K. Nandy, P. M. Oppeneer, Relativistic theory of magnetic inertia in ultrafast spin dynamics,\nPhysical Review B 96 (2) (2017) 024425. doi:10.1103/physrevb.96.024425 .\nURL https://doi.org/10.1103/physrevb.96.024425\n[16] C. Serpico, M. d’Aquino, G. Bertotti, I. D. Mayergoyz, Quasiperiodic magnetization dynamics in uniformly magnetized\nparticles and films, Journal of Applied Physics 95 (11) (2004) 7052–7054. doi:10.1063/1.1689211 .\nURL http://aip.scitation.org/doi/10.1063/1.1689211\n[17] M. Bauer, J. Fassbender, B. Hillebrands, R. L. Stamps, Switching behavior of a stoner particle beyond the relaxation time\nlimit, Physical Review B 61 (5) (2000) 3410–3416. doi:10.1103/physrevb.61.3410 .\nURL https://doi.org/10.1103/physrevb.61.3410\n[18] G. Bertotti, I. Mayergoyz, C. Serpico, M. d’Aquino, Geometrical analysis of precessional switching and relaxation in\nuniformly magnetized bodies, IEEE Transactions on Magnetics 39 (5) (2003) 2501–2503. doi:10.1109/TMAG.2003.816453 .\nURL http://ieeexplore.ieee.org/document/1233123/\n[19] M. d’Aquino, W. Scholz, T. Schrefl, C. Serpico, J. Fidler, Numerical and analytical study of fast precessional switching,\nJournal of Applied Physics 95 (11) (2004) 7055–7057. doi:10.1063/1.1689910 .\nURL http://aip.scitation.org/doi/10.1063/1.1689910\n[20] T. Devolder, H. W. Schumacher, C. Chappert, Precessional Switching of Thin Nanomagnets with Uniaxial Anisotropy,\nSpringer Berlin Heidelberg, Berlin, Heidelberg, 2006, pp. 1–55. doi:10.1007/10938171\\_1 .\nURL https://doi.org/10.1007/10938171_1\n[21] K. Neeraj, M. Pancaldi, V. Scalera, S. Perna, M. d’Aquino, C. Serpico, S. Bonetti, Magnetization switching in the inertial\nregime, Physical Review B 105 (2022) 054415. doi:10.1103/PhysRevB.105.054415 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.105.054415\n[22] L. Winter, S. Großenbach, U. Nowak, L. R´ ozsa, Nutational switching in ferromagnets and antiferromagnets, Physical\nReview B 106 (21) (2022) 214403. doi:10.1103/physrevb.106.214403 .\nURL https://doi.org/10.1103/physrevb.106.214403\n[23] W. F. Brown, Micromagnetics, Interscience Publishers, 1963.\n[24] M. d 'Aquino, S. Perna, M. Pancaldi, R. Hertel, S. Bonetti, C. Serpico, Micromagnetic study of inertial spin waves in\nferromagnetic nanodots, Physical Review B 107 (14) (Apr. 2023). doi:10.1103/physrevb.107.144412 .\nURL https://doi.org/10.1103/physrevb.107.144412\n[25] T. Kikuchi, G. Tatara, Spin dynamics with inertia in metallic ferromagnets, Physical Review B 92 (18) (2015) 184410.\ndoi:10.1103/physrevb.92.184410 .\nURL https://doi.org/10.1103/physrevb.92.184410\n[26] S. Giordano, P.-M. D´ ejardin, Derivation of magnetic inertial effects from the classical mechanics of a circular current loop,\nPhysical Review B 102 (21) (2020) 214406. doi:10.1103/physrevb.102.214406 .\nURL https://doi.org/10.1103/physrevb.102.214406\n[27] I. Makhfudz, E. Olive, S. Nicolis, Nutation wave as a platform for ultrafast spin dynamics in ferromagnets, Applied Physics\nLetters 117 (13) (2020) 132403. doi:10.1063/5.0013062 .\nURL https://doi.org/10.1063/5.0013062\n[28] A. M. Lomonosov, V. V. Temnov, J.-E. Wegrowe, Anatomy of inertial magnons in ferromagnetic nanostructures, Physical\nReview B 104 (2021) 054425. doi:10.1103/PhysRevB.104.054425 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.104.054425\n[29] M. Cherkasskii, M. Farle, A. Semisalova, Dispersion relation of nutation surface spin waves in ferromagnets, Phys. Rev.\nB 103 (2021) 174435. doi:10.1103/PhysRevB.103.174435 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.103.174435\n[30] R. Mondal, L. R´ ozsa, Inertial spin waves in ferromagnets and antiferromagnets, Physical Review B 106 (13) (2022) 134422.\ndoi:10.1103/physrevb.106.134422 .\nURL https://doi.org/10.1103/physrevb.106.134422\n[31] S. V. Titov, W. J. Dowling, Y. P. Kalmykov, M. Cherkasskii, Nutation spin waves in ferromagnets, Physical Review B\n105 (21) (2022) 214414. doi:10.1103/physrevb.105.214414 .\nURL https://doi.org/10.1103/physrevb.105.214414\n[32] Z. Gareeva, K. Guslienko, Nutation excitations in the gyrotropic vortex dynamics in a circular magnetic nanodot, Nano-\nmaterials 13 (3) (2023) 461. doi:10.3390/nano13030461 .\n18URL https://doi.org/10.3390/nano13030461\n[33] P. E. Wigen, Nonlinear Phenomena and Chaos in Magnetic Materials, WORLD SCIENTIFIC, 1994. doi:10.1142/1686 .\nURL https://doi.org/10.1142/1686\n[34] E. A. Montoya, S. Perna, Y.-J. Chen, J. A. Katine, M. d’Aquino, C. Serpico, I. N. Krivorotov, Magnetization reversal\ndriven by low dimensional chaos in a nanoscale ferromagnet, Nature Communications 10 (1) (Feb. 2019). doi:10.1038/\ns41467-019-08444-2 .\nURL https://doi.org/10.1038/s41467-019-08444-2\n[35] M. Ruggeri, Numerical analysis of the landau–lifshitz–gilbert equation with inertial effects, ESAIM: Mathematical Mod-\nelling and Numerical Analysis 56 (4) (2022) 1199–1222. doi:10.1051/m2an/2022043 .\nURL https://doi.org/10.1051/m2an/2022043\n[36] P. Li, L. Yang, J. Lan, R. D. null, J. Chen, A second-order semi-implicit method for the inertial landau-lifshitz-gilbert\nequation, Numerical Mathematics: Theory, Methods and Applications 16 (1) (2023) 182–203. doi:10.4208/nmtma.\noa-2022-0080 .\nURL https://doi.org/10.4208/nmtma.oa-2022-0080\n[37] M. d’Aquino, C. Serpico, G. Miano, I. D. Mayergoyz, G. Bertotti, Numerical integration of Landau–Lifshitz–Gilbert\nequation based on the midpoint rule, Journal of Applied Physics 97 (10) (2005) 10E319. doi:10.1063/1.1858784 .\nURL http://aip.scitation.org/doi/10.1063/1.1858784\n[38] M. d’Aquino, C. Serpico, G. Miano, Geometrical integration of Landau–Lifshitz–Gilbert equation based on the mid-point\nrule, Journal of Computational Physics 209 (2) (2005) 730–753. doi:10.1016/j.jcp.2005.04.001 .\nURL https://doi.org/10.1016/j.jcp.2005.04.001\n[39] I. Mayergoyz, G. Bertotti, C. Serpico, Nonlinear Magnetization Dynamics in Nanosystems, Elsevier Science, 2009.\n[40] Y. Saad, M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,\nSIAM Journal on Scientific and Statistical Computing 7 (3) (1986) 856–869. doi:10.1137/0907058 .\nURL https://doi.org/10.1137/0907058\n[41] L. F. Richardson, IX. the approximate arithmetical solution by finite differences of physical problems involving differential\nequations, with an application to the stresses in a masonry dam, Philosophical Transactions of the Royal Society of\nLondon. Series A, Containing Papers of a Mathematical or Physical Character 210 (459-470) (1911) 307–357. doi:\n10.1098/rsta.1911.0009 .\nURL https://doi.org/10.1098/rsta.1911.0009\n[42] J. Dormand, P. Prince, A family of embedded runge-kutta formulae, Journal of Computational and Applied Mathematics\n6 (1) (1980) 19–26. doi:10.1016/0771-050x(80)90013-3 .\nURL https://doi.org/10.1016/0771-050x(80)90013-3\n[43] L. F. Shampine, M. W. Reichelt, The matlab ode suite, SIAM Journal on Scientific Computing 18 (1) (1997) 1–22.\narXiv:https://doi.org/10.1137/S1064827594276424 ,doi:10.1137/S1064827594276424 .\nURL https://doi.org/10.1137/S1064827594276424\n[44] M. d’Aquino, Magnetization Geometrical Integration Code (MaGICo), http://wpage.unina.it/mdaquino/index_file/\nMaGICo.html .\n19"    },    {        "title": "1001.4806v1.Spin_wave_instabilities_in_spin_transfer_driven_magnetization_dynamics.pdf",        "content": "arXiv:1001.4806v1  [cond-mat.mes-hall]  26 Jan 2010Spin-wave instabilities in spin-transfer-driven magneti zation dynamics\nG. Bertotti(a), R. Bonin(b), M. d’Aquino(c), C. Serpico(d), I. D. Mayergoyz(e)\n(a)INRIM - Istituto Nazionale di Ricerca Metrologica, 10135 To rino, Italy\n(b)Politecnico di Torino, sede di Verr` es, 11029 Aosta, Italy\n(c)Dip. Tecnologie, Universit` a ”Parthenope”, 80143 Napoli, Italy\n(d)Dip. Ing. Elettr., Universit` a ”Federico II”, 80125 Napoli , Italy\n(e)ECE Dept., UMIACS, AppEl Center, University of Maryland, Co llege Park MD 20742, USA\n(Dated: March 1, 2018)\nWe study the stability of magnetization precessions induce d in spin-transfer devices by the injec-\ntion of spin-polarized electric currents. Instability con ditions are derived by introducing a general-\nized, far-from-equilibrium interpretation of spin-waves . It is shown that instabilities are generated\nby distinct groups of magnetostatically coupled spin-wave s. Stability diagrams are constructed as a\nfunction of external magnetic field and injected spin-polar ized current. These diagrams show that\napplying larger fields and currents has a stabilizing effect o n magnetization precessions. Analytical\nresults are compared with numerical simulations of spin-tr ansfer-driven magnetization dynamics.\nPACS numbers: 75.60.Jk, 85.70Kh\nCurrents of spin-polarized electrons can induce large-\namplitude magnetization precessions at microwave fre-\nquencies in small-enough magnetic devices [1, 2]. There\nis mounting experimental evidence that these so-called\nspin-transfer phenomena do occur in nano-pillaror nano-\ncontact devices under current densities of the order of\n106−108A/cm2[3–6]. This discovery has boosted the\nalreadywidespreadinterestinthephysicsoftheinterplay\nbetween magnetism and electron transport, and has trig-\ngered efforts toward the promising development of new\ngenerations of microwave spin-transfer nano-oscillators.\nA spin-transfer device is a non-linear open system,\ndriven far-from-equilibrium by the action of the spin-\npolarized electric current. The excited magnetization\nprecessions represent strong excitations of the magnetic\nmedium, which in principle may giveriseto varioustypes\nof instability and eventually to transitions to chaotic dy-\nnamics. A parallel can be drawn with ferromagnetic-\nresonance Suhl’s instabilities [7], in which certain spin-\nwavescangetcoupledtotheuniformprecessionandstart\nto growto largenon-thermal amplitudes, thus destroying\nthe spatial uniformity of the original state.\nIn this Letter, we demonstrate that spin-wave insta-\nbilities may occur in spin-transfer-driven magnetization\ndynamics as well. However, the system is far from equi-\nlibrium and the classical notion of spin-waves fails. In-\ndeed, it is the large-amplitude magnetization precession\ninduced by spin transfer that plays the role of reference\nstate, and spin-waves only exist in a generalized, non-\nequilibrium sense, as small-amplitude perturbations of\nthat state [8–10]. This scenario emerges with clarity in\nthe time-dependent vector basis in which the reference\nmagnetization precession is stationary. The spin-wave\nequations in this basis are characterized by two features:\n(i) a well-defined dispersion relation ω(q;cosθ0), whose\nnon-equilibrium nature is revealed by its explicit depen-\ndence on the magnetization precession amplitude cos θ0;\nand (ii) the presence of time-periodic coupling terms dueto the magnetostatic fields generated by individual spin-\nwaves. This coupling leads to the appearance of narrow\ninstability tongues around the parametric resonance con-\nditionω(q;cosθ0)∼ω0, whereω0is the magnetization\nprecession angular frequency.\nSpin-wave instabilities occur only for particular com-\nbinations of external magnetic field and injected spin-\npolarized current. In addition, instabilities result in lim-\nitedspatialandtemporaldistortionswhichsomewhatob-\nscure but yet do not completely disrupt the precessional\ncharacterof the originalstate. This robustness of excited\nprecessions with respect to spin-wave instabilities has a\nprecise physical origin. Indeed, the discrete nature of the\nspin-wave spectrum caused by boundary conditions in\nsub-micrometer devices reduces the number of available\nspin-wave modes which can contribute to instabilities.\nOn the other hand, the strength of the magnetostatic ef-\nfects responsible for instabilities is drastically reduced,\ndue to the ultra-thin nature of spin-transfer devices, and\ninstability thresholds are consequently enhanced. Fi-\nnally, spin-transfer-driven precessions are characterized\nby large amplitudes and, as such, are less easily masked\nby the onset of non-uniform modes. In spin-transfer\nnano-oscillators, spin-wave instabilities are expected to\nresult in increased oscillator line-widths, a conclusion\nthat might explain some of the puzzling experimental\nresults obtained in this area [11].\nTo start the technical discussion, consider a ultra-thin\ndisk with negligible crystal anisotropy (e.g., permalloy).\nTypically, this disk will be the so-called free layer of a\nnanopillar spin-transfer device (see inset in Fig. 1). The\ndisk plane is parallel to the ( x,y) plane and is traversed\nby a flow of electrons with spin polarization along the ez\ndirection. The dimensionless equation for the dynamics\nof the normalized magnetization m(r,t) (|m|2= 1) in\nthe disk in the presence of spin transfer is [1, 12]:2\nFIG. 1: (Color online) Stability diagram in (h az,β/α) con-\ntrol plane for aultra-thin permalloy disk. System paramete rs:\nα= 0.02,d= 0.6,R= 23.6,N⊥= 0.02(lengths aremeasured\nin units of the exchange length lEX= 5.72 nm). Magnetiza-\ntion is parallel to spin-polarization in region P; anti-parallel\nto spin-polarization in region A; precessing around the spin-\npolarization axis in regions OandSW. Dashed line is an ex-\nample of line of constant precession amplitude (cos θ0= 0.5)\ncomputed from Eq.(2). Spin-wave instabilities occur in reg ion\nSW. Small framed area is shown in detail in Fig. 2. Inset:\ntypical geometry of a nanopillar spin-transfer device.\n∂m\n∂t−αm×∂m\n∂t= (1)\n−m×/parenleftbig\nhazez+hM+∇2m−βm×ez/parenrightbig\n.\nHere, the external magnetic field h azezand the magneto-\nstatic field hMare measured in units of the spontaneous\nmagnetization M s, time in units of ( γMs)−1(γis the\nabsolute value of the gyromagnetic ratio), and lengths\nin units of the exchange length. The external field is\nperpendicular to the disk plane, while the spin-transfer\ntorque is simply proportional to the sine of the angle be-\ntweenmandez. The parameter βis proportional to the\nspin-polarized current density (see [12] for the detailed\ndefinition), andintypicalsituationsitiscomparablewith\nthe damping constant α.\nWhenever |haz−β/α| ≤Nz−N⊥(NzandN⊥are the\ndisk demagnetizing factors, with Nz+2N⊥= 1), Eq.(1)\nadmits time-harmonic solutions m0(t), corresponding to\nspatially uniform precession of the magnetization around\nthez-axis[13](seeFig. 1). Theprecessionamplitudeand\nangular frequency are respectively equal to:\ncosθ0=haz−β/α\nNz−N⊥, ω0=β\nα. (2)\nTo study the stability of m0(t), consider the perturbed\nmotionm(r,t) =m0(t) +δm(r,t), with|δm(r,t)| ≪1.\nThe correspondingmagnetostaticfield will be: hM(r,t) =\n−Nzm0z−N⊥m0⊥+δhM(r,t), whereδhMrepresents themagnetostatic field generated by δm. Since we are inter-\nested in ultra-thin layers, we shall assume that δmdoes\nnot depend on z:δm(r,t) =δm(x,y,t).\nThe perturbation δmis orthogonal to m0(t) at all\ntimes, since the local magnetization magnitude |m|2= 1\nmust be preserved. Hence, it is natural to represent δm\nin the time-dependent vector basis ( e1(t),e2(t)) defined\nin the plane perpendicular to m0(t), withe2(t) paral-\nlel toez×m0(t) ande1(t) such that ( e1,e2,m0) form\na right-handed orthonormal basis. The perturbation can\nbe written as: δm(r,t) =δm1(r,t)e1(t)+δm2(r,t)e2(t).\nBylinearizingEq.(1)around m0(t)andaveragingthelin-\nearized equation over the layer thickness, one obtains the\nfollowing coupled differential equations in matrix form:\n/parenleftbigg\n1α\n−α1/parenrightbigg∂\n∂t/parenleftbigg\nδm1\nδm2/parenrightbigg\n=/parenleftbigg\n0 1\n−1 0/parenrightbigg/parenleftbigg\n/angb∇acketleftδhM/angb∇acket∇ight1\n/angb∇acketleftδhM/angb∇acket∇ight2/parenrightbigg\n+\n+/parenleftbigg0N⊥+∇2\n⊥\n−N⊥−∇2\n⊥0/parenrightbigg/parenleftbiggδm1\nδm2/parenrightbigg\n, (3)\nwhere∇2\n⊥=∂2/∂x2+∂2/∂y2, while/angb∇acketleft.../angb∇acket∇ightrepresents the\nzaverage over the thickness of the disk, and /angb∇acketleftδhM/angb∇acket∇ight1=\n/angb∇acketleftδhM/angb∇acket∇ight·e1(t),/angb∇acketleftδhM/angb∇acket∇ight2=/angb∇acketleftδhM/angb∇acket∇ight·e2(t).\nTo grasp the physical consequences of Eq.(3), consider\nthe plane-wave perturbation δm(r,t) =a(t)exp(iq·r)\nin an infinite layer ( N⊥= 0). The corresponding magne-\ntostatic field is [14]:\n/angb∇acketleftδhM/angb∇acket∇ight=−sqδmz−(1−sq)δmq;sq=1−exp(−qd)\nqd,\n(4)\nwhereδmz= (δm·ez)ezandδmq= (δm·eq)eq,eqbe-\ning the unit vector in the qdirection. The field −sqδmz\nis generated by the magnetic charges at the layer sur-\nface, whereas the field −(1−sq)δmqis due to volume\ncharges. By taking into account that ∇2\n⊥δm=−q2δm,\nδmz·e1(t) =δm1sin2θ0, andδmz·e2(t) = 0, one\nfinds from Eq.(3) that m0(t) is always stable with re-\nspect to the action of exchange forces and surface mag-\nnetic charges. Only volume charges can make the pre-\ncession unstable. This conclusion follows from the fact\nthat thez-axis, along which the surface-charge magneto-\nstatic field is directed, is a symmetry axis for the prob-\nlem. Surface-charge-driven instabilities may appear in\nnon-uniaxial systems.\nThe two-dimensional and uniaxial character of the\nproblem makes it natural to introduce polar coordinates\n(r,φ) in the disk plane, with the origin at the centre of\nthe disk. The natural boundary condition in polar coor-\ndinates is∂δm/∂r|r=R= 0, where Ris the disk radius.\nThe generic perturbation satisfying this boundary condi-\ntion consists of cylindrical spin-waves of the type:\nδm(r,φ,t) =+∞/summationdisplay\nn=−∞∞/summationdisplay\nk=0ank(t)Jn(qnkr) exp(inφ),(5)3\nwhereJn(z) is then-th order Bessel function. The wave-\nvector amplitude qnkis identified by two subscripts be-\ncause, for each n, it must satisfy the boundary condi-\ntion∂Jn(z)/∂z= 0 forz=qnkR, which has infinite\nsolutionsqn0,qn1,qn2,...of increasing amplitude. The\ncylindrical spin-waves Fnk(r,φ) =Jn(qnkr) exp(inφ) are\na complete orthogonal set of eigenfunctions of the ∇2\n⊥\noperator: ∇2\n⊥Fnk(r,φ) =−q2\nnkFnk(r,φ). The magne-\ntostatic field δhMcan be computed by applying Eq.(4)\nto the plane-wave integral representation: Fnk(r,φ) =\n1/(2πin)/integraltext2π\n0exp(iqnk·r) exp(inψ)dψ, where the polar\nrepresentation of randqnkisr= (r,φ) andqnk=\n(qnk,ψ), respectively. By following these steps, writing\nank(t) asank(t) =cnk,1(t)e1(t)+cnk,2(t)e2(t), and ne-\nglecting small terms proportional to N⊥, Eq.(3) is trans-\nformed into the following system of coupled equations:\ndcnk\ndt=Ankcnk+∞/summationdisplay\np=0∆+\nnk;p\n∆nkRn+2,p(t)cn+2,p(6)\n+∞/summationdisplay\np=0∆−\nnk;p\n∆nkR∗\nn−2,p(t)cn−2,p;cnk≡/parenleftbiggcnk,1\ncnk,2/parenrightbigg\n,\nwhere:\nAnk=1\n1+α2/parenleftbigg1−α\nα1/parenrightbigg/parenleftbigg0 −νnk\nνnk−κnksin2θ00/parenrightbigg\n,\n(7)\nRnk(t) = exp(2 iω0t)1−snk\n4× (8)\n×1\n1+α2/parenleftbigg1−α\nα1/parenrightbigg/parenleftbiggicosθ0−1\n−cos2θ0−icosθ0/parenrightbigg\n,\n∆±\nnk;p=/integraldisplayR\n0rJn(qnkr)Jn(qn±2,pr)dr , (9)\nand ∆ nk=/integraltextR\n0rJ2\nn(qnkr)dr,νnk=q2\nnk+ (1−snk)/2,\nκnk=−1 + 3(1−snk)/2,snkbeing the value of sqin\nEq.(4) forq=qnk.\nThe coupling terms proportional to Rnk(t) in Eq.(6)\nare the consequence of volume-charge magnetostatic ef-\nfects. They are all of the order of (1 −sq). One has that\n(1−sq)≪1 up toq∼1 in ultra-thin layers with d/lessorsimilar1\n(see Eq.(4)). If one neglects these terms altogether, one\nobtains a system of fully decoupled equations for indi-\nvidual cylindrical spin-waves, characterized by the dis-\npersion relation:\nω2(q;cosθ0) =/parenleftbigg\nq2+1−sq\n2/parenrightbigg\n× (10)\n×/parenleftbigg\nq2+sq+1−3sq\n2cos2θ0/parenrightbigg\n,which is obtained from Eq.(7) in the limit α→0. How-\never, the time-periodic coupling terms may give rise to\nparametric instabilities. Interestingly, these instabili-\nties are governed by a small number of dominant terms,\nwhich can be identified by using the asymptotic formula\nJn(z)∼/radicalbig\n2/πzcos(z−nπ/2−π/4) in the equation ex-\npressing boundary conditions. One obtains the estimate\nqnk≃π(2s+ 1)/4R, wheres=|n|+ 2k. When this\napproximate expression is used for qn±2,pin Eq.(9), one\nobtains:\nn≥2 : ∆±\nnk;p≃∆nkδp,k∓1,∆+\nn0;p≃0,\nn=±1 : ∆−\n1k;p= ∆+\n−1,k;p= ∆1kδpk,(11)\nn≤ −2 : ∆±\nnk;p≃∆nkδp,k±1,∆−\nn0;p≃0.\nThese relations have an important physical conse-\nquence, which is best appreciated by rewriting Eq.(5) in\nthe form:δm=/summationtext∞\ns=0δm(s), where:\nδm(s)=/summationdisplay\n|n|+2k=sank(t)Jn(qnkr) exp(inφ).(12)\nUnder the approximation (11), one finds from Eq.(6)\nthatδm(s1)is decoupled from δm(s2)for anys2/negationslash=s1.\nOn the other hand, for each s, the (s+ 1) cylindrical\nwaves (namely, n=s,s−2,...,−s+2,−s) involved in\nδm(s)form a one-dimensional chain, in the sense that\nonly neighboring waves in the above list are coupled.\nTheabsenceofcouplingbetweendistinct chainswouldbe\ncomplete if the approximation qnk≃π(2s+1)/4Rwere\nexact. In that case, all the cylindrical waves in δm(s)\nwould be characterized by exactly the same wave-vector\namplitude.\nInstabilities are governed by the multipliers of the one-\nperiod map [15] associated with the dynamics of δm(s).\nWehaveusedEqs.(6) and(11)tomakeanumericalstudy\nofthese multipliers for different chains, in orderto obtain\nthe instability pattern associatedwith each of them. The\nresults for a permalloy disk with radius R= 135 nm and\nthicknessd= 3.43 nm are shown in Fig. 1. The band ( O\n+SW) in between the PandAregions is where magne-\ntization precessionoccurs. Spin-waveinstabilities appear\nin region SW. Each chain δm(s)provides a distinct in-\nstability channel. In particular, chains s= 1,s= 2,\nands= 3 (s= 0 yields no instability at all) give rise to\nwell-separated instability regions that can be neatly re-\nsolved, as shown in Fig. 2(a). In general, the s-th chain\ngives rise to an instability tongue around the parametric\nresonance condition ω(q;cosθ0)∼ω0, whereω(q;cosθ0)\nis given by Eq.(10) and qis of the order of the wave-\nvector amplitudes involved in the chain (see dashed lines\nin Fig. 2(a)). According to parametric resonance theory,\nresonance occurs for ω=nω0/2,n= 1,2,.... The dom-\ninant, lowest-threshold resonance occurs for n= 1, that4\nFIG. 2: (Color online) (a): Magnification of Fig. 1. Labels\ns= 1,2,3 identify the perturbation chain responsible for the\ncorresponding instability tongue. The pair of dashed lines\naccompanying each of the s= 2 and s= 3 tongues (one\nline only for s= 1) represents the parametric resonance con-\nditionω(qnk;cosθ0) =ω0for the largest and smallest qnk\nin the chain. Horizontal line at β/α= 0.15 is line along\nwhich the computer simulations shown in (b) and (c) were\ncarried out. (b) and (c): Magnitude mavg\n⊥of average in-\nplane magnetization obtained from numerical integration o f\nEq.(1) under decreasing (b) and increasing (c) external mag -\nnetic field. Dashed line represents the prediction of Eq.(2)\nfor sinθ0. Snapshots illustrate magnetization patterns ap-\npearing just after the instability jumps. Vertical dotted l ines\nare guides for the eye to compare thresholds with theoretica l\npredictions obtained from (a).\nis, atω=ω0/2. However, one can see from Eq.(8) that\nthe parametric frequency is 2 ω0rather than ω0, which\nexplains why the resonancecondition is ω∼ω0. This pe-\nculiarity is the consequence of the rotational invariance\nofthe problem, and is expected to disappearin situations\nwith broken rotational symmetry. Interestingly, Fig. 1\nreveals that, for a given precession amplitude cos θ0(see\ndashed line), applying larger fields and currents has a\nstabilizing effect on the precession. Also, larger fields\nstabilize precessions of given frequency ω0=β/α.\nTo test the predictions of the theory, we have car-ried out computer simulations based on the numerical\nintegration of Eq.(1) by the methods discussed in Ref.\n[16]. Simulations were carried out by slowly varying\nthe external magnetic field under constant current. As\nshown in Fig. 2(b), at large fields the magnitude mavg\n⊥\nof the average in-plane magnetization is in full agree-\nment with the prediction of Eq.(2) for spatially uniform\nprecession. Then, under decreasing field mavg\n⊥exhibits\nwell-pronounced jumps, whose positions agree with the\ntheoretical instability thresholds for the s= 2 ands= 3\nchains within ten percent. Beyond these jumps, non-\nuniform modes appear in the dynamics (see Fig. 2(b)),\ncharacterizedbytwo-foldandthree-foldpatternsthatare\nconsistent with the symmetry of the cylindrical waves in-\nvolvedin the s= 2 ands= 3 chains, respectively. Agree-\nment with the theory is also confirmed by the hysteresis\nin the instability thresholds occurring under decreasing\nor increasing external field (Fig. 2(c)).\nThe stability of spin-transfer-driven magnetization\nprecessions has been studied in this Letter under the\nsimplest conditions, namely, uniaxial symmetry and pure\nsinθ0angulardependence ofthespin-torque. Severalfea-\ntures of physical interest have emerged: the role played\nby non-equilibrium spin-waves; the fact that instabilities\nare governed by distinct chains of magnetostatically cou-\npled spin-waves; and the fact that excited precessionsare\nnot completely disrupted but only somewhat obscured\nby spin-wave instabilities. Future work will be devoted\nto extending the present approach to more general, non-\nuniaxial geometries.\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] S. I. Kiselev et al., Nature 425, 380 (2003).\n[4] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and\nT. J. Silva, Phys. Rev. Lett. 92, 027201 (2004).\n[5] I. N. Krivorotov et al., Phys. Rev. B 76, 024418 (2007).\n[6] C. T. Boone et al., Phys. Rev. Lett. 103, 167601 (2009).\n[7] H. Suhl, J. Phys. Chem. Solids 1, 209 (1957).\n[8] G. Bertotti, I. D. Mayergoyz, and C. Serpico, Phys. Rev.\nLett.87, 217203 (2001).\n[9] A. Kashuba, Phys. Rev. Lett. 96, 047601 (2006).\n[10] D. A. Garanin and H. Kachkachi, Phys. Rev. B 80,\n014420 (2009).\n[11] Q. Mistral, et al., Appl. Phys. Lett. 88, 192507 (2006).\n[12] G. Bertotti et al., Phys. Rev. Lett. 94, 127206 (2005).\n[13] Y. B. Bazaliy, B. A. Jones, and S. C. Zhang, Phys. Rev.\nB69, 094421 (2004).\n[14] G. Bertotti, I. D. Mayergoyz, and C. Serpico, Nonlin-\near Magnetization Dynamics in Nanosystems (Elsevier,\nOxford, 2009), Sect. 8.5.\n[15] L. Perko, Differential Equations and Dynamical Systems\n(Springer, New York, 1996).\n[16] M. d’Aquino, C. Serpico, and G. Miano, J. Comput.\nPhys.209, 730 (2005)."    },    {        "title": "1505.04370v2.Equilibria__Dynamics_and_Current_Sheets_Formation_in_Magnetically_Confined_Coronae.pdf",        "content": "arXiv:1505.04370v2  [astro-ph.SR]  29 Oct 2015Draft version July 25, 2018\nPreprint typeset using L ATEX style emulateapj v. 01/23/15\nEQUILIBRIA, DYNAMICS AND CURRENT SHEETS FORMATION\nIN MAGNETICALLY CONFINED CORONAE\nA. F. Rappazzo\nAdvanced Heliophysics, Pasadena, CA 91106, USA; franco.ra ppazzo@gmail.com\nDraft version July 25, 2018\nABSTRACT\nThe dynamics of magnetic fields in closed regions of solar and stellar co ronae are investigated with a\nreducedmagnetohydrodynamic(MHD) model in the frameworkofP arkerscenarioforcoronalheating.\nA novel analysis of reduced MHD equilibria shows that their magnetic fi elds have an asymmetric\nstructure in the axial direction with variation length-scale zℓ∼ℓB0/b, whereB0is the intensity\nof the strong axial guide field, bthat of the orthogonal magnetic field component, and ℓthe scale\nofb. Equilibria are then quasi-invariant along the axial direction for varia tion scales larger than\napproximativelythelooplength zℓ/greaterorsimilarLz, andincreasinglymoreasymmetricforsmallervariationscales\nzℓ/lessorsimilarLz. Thecritical length zℓ∼Lzcorrespondsto the magnetic field intensity threshold b∼ℓB0/Lz.\nMagnetic fields stressed by photospheric motions cannot develop s trong axial asymmetries. Therefore\nfields with intensities below such threshold evolvequasi-statically, re adjusting to a nearby equilibrium,\nwithout developing nonlinear dynamics nor dissipating energy. But st ronger fields cannot access\ntheir corresponding asymmetric equilibria, hence they are out-of- equilibrium and develop nonlinear\ndynamics. The subsequent formation of current sheets and ener gy dissipation is necessary for the\nmagnetic field to relax to equilibrium, since dynamically accessible equilibr ia have variation scales\nlarger than the loop length zℓ/greaterorsimilarLz, with intensities smaller than the threshold b/lessorsimilarℓB0/Lz. The\ndynamical implications for magnetic fields of interest to solar and ste llar coronae are investigated\nnumerically and the impact on coronal physics discussed.\nKeywords: magnetohydrodynamics (MHD) — stars: activity — stars: solar-ty pe — Sun: corona —\nSun: magnetic topology — turbulence\n1.INTRODUCTION\nSolar observations show a close association between\nmagnetic field strength and coronal activity. In combi-\nnation with the ability of photospheric motions to stress\nthe field, these are the two key elements to understand\nthe observed coronal X-ray activity of the Sun, of all\nlate-type main sequence stars, and more in general of\nstars with a magnetized corona and an outer convective\nenvelope (G¨ udel 2004, 2009).\nIt has long been proposed that the work done by\nphotospheric motions on magnetic field line footpoints\ncan transform mechanical energy into magnetic energy\nand transfer it in the upper corona. The photospheric\n(horizontal) velocity can be split into irrotational and\nsolenoidal components. Only its solenoidal ( incompress-\nible) component has a non-vanishing vorticity and can\nthen twist the magnetic field lines, injecting magnetic\nenergy into the corona.\nGold & Hoyle (1960) conjectured that the magnetic\nfield would proceedthrough asequence offorce-freeequi-\nlibria while photospheric vortices twist the field lines,\nand the stored energy could subsequently be released\nwhen two flux tubes with similarly twisted field lines\ncome into contact with each other, or when the configu-\nration would become somehow unstable through an un-\ndetermined mechanism (Gold 1964). Sturrock & Uchida\n(1981) compute the energy flux into the corona due to\nthe work done by randomphotospheric vortical motions\non the magnetic field. They find that the correlation\ntimeof photospheric motions must be of the order of the\nobserved photospheric timescales (5-8 minutes) or longerto obtain an energy flux large enough to sustain an ac-\ntive corona, otherwise for shorter correlation times the\nresulting twisted field is too small. But the magnetic en-\nergy is still supposed to be stored in a force-free field in\nequilibrium, and no physical mechanism able to dissipate\nthis energy and heat the corona is envisioned.\nEnergy stored in a magnetic field in equilibrium, that\nsubsequently becomes unstable and releases its energy, is\nthe common thread of flare models (Shibata & Magara\n2011; Martin et al. 2012), with the processes leading to\nthe pre-flare magnetic energy storage and its subsequent\nfast release strongly debated. On the other hand this\npicture does not appear apt to describe the dynamics of\nthe long-lived slender X-ray bright loops, that in com-\nparison to a flare show little dynamics from their large-\nscales down to the smallest resolved scale ( ∼150km)\nof current state-of-the-art X-ray and EUV imagers on\nboard Hinode, SDO and Hi-C (Peter et al. 2013). While\nthe pre-flare magnetic structure is destroyed during the\nflare, the large-scale magnetic topology of the loops where\nthe basic heating occurs, and that make the corona shine\nsteadily in X-ray, is not strongly modified on compara-\nble timescales. This suggests that the energy deposition\nmust occur at very small scales yet observationally un-\nresolved. Furthermore the energy reservoir that supplies\ndissipation should consist of magnetic field fluctuations\n(with vanishing time-average, but non vanishing time-\naveraged rms) that adds up to the strong axial magnetic\nfield that defines the loop.\nParker (1972, 1988, 1994, 2012) was the first to sug-\ngest that, in contrast to previous quasi-static models,\nthe magnetic field brought about by photospheric vorti-2 Rappazzo\ncal motions would be in dynamical non-equilibrium in\nthe case of interest to coronal heating. Furthermore\nthe relaxationof this interlaced fields toward equilibrium\nwould necessarilyinvolvethe formation of current sheets .\nThe energy dissipation would then occur at small scales\nin the fashion ofsmall impulsive heating events, so-called\nnanoflares , a picture broadly used for the thermodynam-\nical modeling of the closed corona(Klimchuk 2006; Reale\n2014).\nUsing a simplified Cartesian model with a strong guide\nfieldthreadingacoronalloop,Parker(1972,1979)argued\nat first that a magnetic field could be in equilibrium only\nif it were invariant along z(the axial direction). Due\nto the complex and disordered nature of photospheric\nmotions the induced interlaced magnetic field would not\nbe invariant, and therefore not in equilibrium. Next,\ncounterexamples of magnetostatic equilibria that are not\ninvariant along zwere provided (Rosner & Knobloch\n1982; Bogoyavlenskij2000), and analytical investigations\n(van Ballegooijen 1985; Antiochos 1987; Cowley et al.\n1997) argued that smooth photospheric motions cannot\nlead to the formation of current sheets, whereas only a\ndiscontinuous velocity field can form discontinuities in\nthe coronal magnetic field.\nIn particular van Ballegooijen (1985) showed that the\nequilibria are the solutions of the two-dimensional (2D)\nEuler equation that in general are not z-invariant, thus\ninferring that the field would evolve quasi-statically, con-\ntinuously readjusting to a nearby force-free equilibrium\nwithout developing nonlinear dynamics nor forming cur-\nrent sheets. Reaching opposite conclusions Parker(1988,\n1994, 2000, 2012) pointed out that almost all field line\ntopologies relevant to the solar corona have a different\nstructure from the solutions of the Euler equation, so\nthat the magnetic field would be still in dynamical non-\nequilibrium.\nReduced magnetohydrodynamics (MHD) numerical\nsimulations with a continuous smooth velocity forcing at\nthe boundaries show that the dynamics can be seen as a\nparticular instance of magnetically dominated MHD tur-\nbulence (Dmitruk & G´ omez 1999; Dmitruk et al. 2003;\nRappazzo et al. 2007, 2008) as proposed by earlier 2D\nmodels (Einaudi et al. 1996; Dmitruk & G´ omez 1997),\nsuggesting that in the forced case the magnetic field\nis indynamical non-equilibrium rather than close to a\nquasi-static evolution. Similar dynamics are also dis-\nplayedbyboundarydrivensimulationsinthecoldplasma\nregime (Hendrix & van Hoven 1996) and in the fully\ncompressible MHD case (Galsgaard & Nordlund 1996;\nDahlburg et al. 2012). Furthermore Rappazzo & Velli\n(2011) have shown that while velocity fluctuations are\nmuch smaller than magnetic fluctuations, spectral en-\nergy fluxes toward smaller scales are akin to those of\na standard cascade with magnetic and kinetic ener-\ngies in equipartition, except for kinetic energy fluxes\nthat are negligible. This implies that at scales smaller\nthan those directly shuffled by photospheric motions, the\nsmallvelocity field is created and shaped by the unbal-\nanced Lorentz force of the out-of-equilibrium magnetic\nfield, that in turn creates small scales in the magnetic\nfield by distorting magnetic islands and pushing field\nlines together . Additionally Georgoulis et al. (1998);\nDmitruk et al. (1998) have established a link between\nboundary driven simulations and observed statistics ofcoronal activity. Indeed the bursts in dissipation dis-\nplayed by the system, that correspond to the formation\nand dissipation of current sheets, follow a power law be-\nhaviorintotalenergy,peakdissipationanddurationwith\nindexes not far from those determined observationally in\nX-rays.\nRecently Wilmot-Smith et al. (2009) have shown that\nthe relaxation of a slightly braided magnetic field (“pig-\ntail” braid) appears to evolve quasi-statically, with no\nformationof currentsheets, towardan equilibrium where\nonly large-scale current layers of thickness much larger\nthan the resolution scale are observed. This result would\nseem in contrast with Parker’s hypothesis, the results of\nthe forced numerical simulations discussed in the previ-\nous paragraph, and the recent results supporting the de-\nvelopment of finite time singularities in the cold plasma\nregime (Low 2013, 2015).\nTo get further insight on the dynamics of coronal mag-\nnetic fields, Rappazzo & Parker(2013) analyzed reduced\nMHD numerical simulations of the relaxation of initial\nmagnetic fields invariant along zand with different av-\neragetwists. They identified a critical intensity threshold\nfor the magnetic field. This is explained heuristically as\nduetoabalancebetweendifferent fieldlinetensionforces\nfor weak fields, while such a balance cannot be attained\nby stronger fields. The non-equilibrium of stronger fields\nstems from this force unbalance, and drives the relax-\nation forming current sheets and dissipating energy. On\nthe contrary weaker fields show little dynamics with no\nenergydissipation, confirmingthattheyareessentiallyin\nequilibrium. Such thresholdcanexplainqualitativelythe\nresult of Wilmot-Smith et al. (2009), although a quan-\ntitative comparison cannot be made because the inte-\ngrated equations (magneto-frictional relaxation vs. re-\nduced MHD) and initial topologies differ.\nThe magnetic intensity threshold found by\nRappazzo & Parker (2013) implies that a critical\ntwistexists above which dynamics develop, and below\nwhich the system remains very close to equilibrium.\nParker (1988) had conjectured that a critical twist is\nnecessary to explain the observationally inferred energy\nflux in active regions (Withbroe & Noyes 1977). In fact\nthe energy flux injected in the corona by photospheric\nmotions is the average Poynting flux ∝angb∇acketleftSz∝angb∇acket∇ight=B0∝angb∇acketleftuph·b∝angb∇acket∇ight\n(see Section 2.2, Equation [7]) that depends not only on\nthe photospheric velocity uphand the axial guide field\nB0, but also on the dynamic magnetic field component\nb, with stronger intensities corresponding to higher\naverage twists. Thus if nonlinear dynamics were to\ndevelop for weak field intensities, energy dissipation\nwould keep too low the value of b, and consequently\nthe flux ∝angb∇acketleftSz∝angb∇acket∇ight. This argument is further reinforced\nby the fact that current sheets thickness decreases at\nleast exponentially in time when nonlinear dynamics\ndevelop, reaching the Sweet-Parker thickness (Sweet\n1958; Parker 1957) on ideal timescales (about one\nAlfv´ en crossing time τA, Rappazzo & Parker 2013),\nthat current sheets are unstable to tearing modes with\n“ideal” growth rates (i.e., of order 1/ τA) already at\nthicknesses larger than Sweet-Parker (Pucci & Velli\n2014; Tenerani et al. 2015; Landi et al. 2015), and that\nmagnetic reconnection rates can be very fast in plasmas\nwith high Reynolds numbers (Lazarian & Vishniac\n1999; Loureiro et al. 2007; Lapenta 2008; Loureiro et al.3\n2009; Uzdensky et al. 2010; Huang & Bhattacharjee\n2010; Beresnyak 2013) and in the collisionless regime\n(Shay et al. 1999; Birn et al. 2001).\nThis paper is devoted to a more detailed discus-\nsion and analysis of the numerical simulations described\nby Rappazzo & Parker (2013), of additional simulations\nthat extend our previous work to initial conditions non-\ninvariant along z, and to a novel analysis of the struc-\nture of the reduced MHD equilibria, with the goal to\nshed light on the topics outlined in this introduction and\nadvance our understanding of coronal magnetic field dy-\nnamics, their relationship to dynamic non-equilibrium,\nMHD turbulence, quasi-static evolution, current sheets\nformation and activity of solar and stellar coronae.\nThe loop model along with initial and boundary condi-\ntionsforthe simulationsareintroducedin Section2. The\nstructure of the equilibria is analyzed in Section 3, and\nthe results of the numerical simulations are described in\nSection 4. Finally results and conclusions are summa-\nrized in Section 5, together with a discussion of their\nimpact on coronal physics.\n2.PHYSICAL MODEL\nA closed region of the solar corona is modeled, as in\nprevious work (Rappazzo et al. 2007), with a simplified\ncartesian geometry, uniform density ρand astrong and\nhomogeneous axial magnetic field B0=B0ˆ ezthread-\ning the system. Plasmas in such configurations are\nwell suited to be studied in the reduced MHD regime\n(Zank & Matthaeus 1992). Introducing the velocity and\nmagneticfieldpotentials ϕandψ, forwhich u=∇ϕ׈ ez,\nb=∇ψ׈ ez, vorticityω=−∇2\n⊥ϕ, and the current den-\nsityj=−∇2\n⊥ψ, the nondimensional reduced MHD equa-\ntions (Kadomtsev & Pogutse 1974; Strauss 1976) are:\n∂tψ= [ϕ,ψ]+B0∂zϕ+ηn∇2n\n⊥ψ, (1)\n∂tω= [j,ψ]−[ω,ϕ]+B0∂zj+νn∇2n\n⊥ω.(2)\nThe Poisson bracket of functions gandhis defined\nas [g,h] =∂xg∂yh−∂yg∂xh(e.g., [j,ψ] =b· ∇j),\nand Laplacian operators have only orthogonal compo-\nnents. To render the equations nondimensional the\nmagnetic fields are first expressed as Alfv´ en velocities\n(b→b/√4πρ), and then all velocities are normalized\nwithu∗= 1 kms−1, a typical value for photospheric mo-\ntions. The domain spans 0 ≤x,y≤L⊥and 0≤z≤Lz,\nwithL⊥= 1 andLz= 10. Magnetic field lines are\nline-tied to a motionless photosphere at the top and bot-\ntom plates ( z= 0 and 10), where a vanishing velocity\nu= 0 is in place. In the perpendicular ( x-y) direc-\ntions a pseudo-spectral scheme with periodic boundary\nconditions and isotropic truncation de-aliasing is used\n(2/3-rule, Canuto et al. 2006), while along za second-\norder finite difference scheme is implemented. The CFL\n(Courant-Friedrichs-Levy) condition is satisfied through\nan adaptive time-step. For a more detailed description of\nthe model and numerical code see Rappazzo et al. (2007,\n2008).\nDissipative simulations use hyper-diffusion (Biskamp\n2003), that effectively limits diffusion to the small scales,\nwithn= 4 andνn=ηn= (−1)n+1/Rn, withRn\ncorresponding to the Reynolds number for n= 1 (see\nRappazzo et al. 2008).2.1.Initial and boundary conditions\nSimulations are started at time t= 0 with a vanishing\nvelocityu= 0 everywhere, and a uniform and homoge-\nneous guide field B0. The orthogonal field bconsists of\nlarge-scale Fourier modes, set expanding the magnetic\npotential in the following way:\nψ0=b0/summationdisplay\nrsm(2Em)1\n2αrsmsin(krsm·x+2πξrsm)\nkrs/radicalBig/summationtext\nijα2\nijm(3)\nwithkrsm=2π\nL⊥(rˆ ex+sˆ ey)+2π\nLzmˆ ez,\nandkrs=2π\nL⊥/radicalbig\nr2+s2,\nwhere the coefficients αrsmandξrsmare two indepen-\ndent sets of random numbers uniformly distributed be-\ntween 0 and 1. The orthogonal wave-numbers ( r,s) are\nalways in the range 3 ≤(r2+s2)1/2≤4, while the\nparallel amplitudes Em(with/summationtext\nmEm= 1) are set to\ndistribute the energy in different ways in the axial direc-\ntion. Given the orthogonality of the base used in Eq. (3)\nthe normalization factors guarantee that for any choice\nof the amplitudes the rms of the magnetic field is set to\nb=∝angb∇acketleftb2\nx+b2\ny∝angb∇acket∇ight1/2=b0, while for total magnetic energy\nEM=b2\n0V/2/summationtext\nmEm, i.e., Emis the fraction of magnetic\nenergy in the parallelmodem. Two-dimensional (2D)\nconfigurations invariants along zare obtained consider-\ning the single mode m= 0 with E0= 1.\n2.2.Energetics\nFrom equations (1)-(2), with n= 1 and considering\nthe kinetic and magnetic Reynolds numbers equal, the\nfollowing energy equation can be obtained:\n∂\n∂t1\n2/parenleftbig\nu2+b2/parenrightbig\n=−∇·(S+F)−1\nR/parenleftbig\nj2+ω2/parenrightbig\n,(4)\nwhereS=B×(u×B) is the Poynting vector, and\nF= (p+u2/2)u−(ωu+jb)׈ ez/Ris an orthogonal\ntransport-related flux. Integrating equation (4) over the\nwhole box the energy ( E) equation is\n∂E\n∂t=S−1\nR/integraldisplay\nVdV/parenleftbig\nj2+ω2/parenrightbig\n, (5)\ni.e., as expected, the global energy balance depends on\nthecompetitionbetweentheenergyflowingintothecom-\nputational box from the photospheric boundaries Sand\nthe ohmic and viscous dissipation. Because in the x–\ny planes periodic boundary conditions are implemented,\nandFz= 0, the only relevant component of the flux\nvectors is that of the Poynting vector along the axial di-\nrectionSzthat, asB=B0ˆ ez+b, is given by\nSz=S·ˆ ez=−B0(u·b). (6)\nIndicating the photospheric velocity fields at the top and\nbottom boundaries z= 0 andLwithu0anduLfor the\nintegrated energy flux (i.e., the power) Swe obtain\nS=B0/integraldisplay\nz=Lda/parenleftbig\nuL·b/parenrightbig\n−B0/integraldisplay\nz=0da/parenleftbig\nu0·b/parenrightbig\n.(7)\nThe injected energy power is proportional to B0and de-\npends on the dot product of the photospheric velocities4 Rappazzo\nu0,Land the magnetic field bat the boundaries. But\nwhileu0,LandB0have fixed values (in our simplified\nmodel), the magnetic field bis determined by the linear\nor nonlinear dynamics developing in the computational\nbox.\nSince the magnetic field component bcan often be\nconsidered quasi-invariant along z(as described in the\nfollowing sections), as a shorthand we will indicate the\ndifference between the boundary velocities with uph=\nuL−u0, so that for quasi-invariant fields the Poynting\nflux can be approximated as ∝angb∇acketleftSz∝angb∇acket∇ight ∼S/ℓ2∼B0uphb.\n3.EQUILIBRIA AND THEIR DYNAMIC\nACCESSIBILITY: ANALYSIS\nAs discussed in the introduction, the properties of the\nequilibria of this system are pivotal to understand its dy-\nnamics (Parker 1972, 1988, 1994; van Ballegooijen 1985,\n1986), therefore their structure is analyzed here in detail.\nIt is shown that, depending on the ratio b0/B0ofthe rms\nof the orthogonal component to the guide magnetic field\nintensity, the equilibria can be approximately invariant\nalongzor strongly asymmetric . As shown in the follow-\ning this can explain why fields with a twist below a criti-\ncalvaluedonotformstrongcurrentsheets, whiletheydo\nat higher twists as conjectured by Parker (1988). Nev-\nertheless unlike commonly thought, such equilibria are\ngenerally not linearly unstable for most conditions rele-\nvant to coronal loops, since they arise from a balance of\nforces in an asymmetric and irregular topology .\nNeglecting velocity and diffusion terms, equilibria of\nEqs. (1)-(2) are given by B· ∇j= 0. Since the total\nmagnetic field Bis given by B=B0ˆ ez+b(x,y,z) with\nb·ˆ ez= 0, the equilibrium condition can be written as:\n∂j\n∂z=−b\nB0·∇j, (8)\nwhere the right-hand side term corresponds to the “2D\nperpendicular” Lorentz force component b·∇b, and the\nleft hand side to the “parallel” B0∂zbfield line tension\n(the labels refer to their derivative, but both components\nare orthogonal to B0, a more detailed discussion is in\nSection 4.1 prior to Equation [23]).\nAssigned bin an x-y plane, e.g., at the boundary z=0\nb(x,y,z= 0) =bbd(x,y), the integration of this equa-\ntion forz>0 allows to compute the corresponding equi-\nlibrium in the whole computational box 0 ≤z≤Lz.\nNow consider the 2D Euler equation (Euler 1761)\n∂ω\n∂t=−u·∇ω, (9)\nwith∇·u= 0. Introducing the velocitypotential φ, then\nu(x,y,t) =∇φ(x,y,t)׈ ez, and vorticity ω=−∇2φ.\nThe two equations (8) and (9) are identical under the\nmapping\n\n\nt→z,\nu→b\nB0,(10)\nand consequently ω→j/B0.\nThe related 2D Navier-Stokes equation is obtained by\nadding to the right hand side of Equation (9) the dissi-\npative term ν∇2ω, from which the 2D Euler equation isrecovered for ν= 0. The physics and solutions of the\n2D Euler and Navier-Stokes equations have been stud-\nied extensively theoretically, numerically and in the lab-\noratory, in the framework of 2D hydrodynamic turbu-\nlence (see reviews by Kraichnan & Montgomery 1980;\nTabeling 2002; Boffetta & Ecke 2012). Unlike the 3D\ncase, it has been shown that givena smooth initial condi-\ntionu0(x,y) at timet= 0 the 2D Euler equation admits\naunique and regular solution att>0, i.e., no finite time\nsingularity develops (Rose & Sulem 1978; Chemin 1993;\nBertozzi & Constantin 1993; Majda & Bertozzi 2001).\nIn 2D in addition to energy, also mean-square vortic-\nity (enstrophy ) is conserved. The coupled conservation\nconstraints have a strong impact on the dynamics that\ndiffers considerably from its 3D hydrodynamic counter-\npartandthe correspondingmagnetohydrodynamiccases.\nIn particular, indicating the Energy with E= (1/2)∝angb∇acketleftu2∝angb∇acket∇ight\n(the integrated square velocity), the enstrophy with Ω =\n(1/2)∝angb∇acketleftω2∝angb∇acket∇ight, and the palinstrophy with P= (1/2)∝angb∇acketleft|∇ω|2∝angb∇acket∇ight,\nthe following energy and enstrophy conservation equa-\ntions are obtained from the 2D Navier-Stokes equations\n(e.g., Boffetta & Ecke 2012):\ndE\ndt=−2νΩ =−ǫν(t),dΩ\ndt=−2νP. (11)\nSince all quantities ( E, Ω,P, andν) are positively de-\nfined, it follows that Ω can at most decrease. Therefore\nthe energy dissipation rate ǫνvanishes as viscosity tends\nto zero:\nlim\nν→0ǫν= 0. (12)\nThis result strongly differs from the 3D case where,\nin the K41 phenomenology introduced by Kolmogorov\n(1941), for a sufficiently small viscosity the energy dissi-\npation rate is approximately constant ǫν∼const and\nindependent from viscosity. This dissipative anomaly\nin 3D was first pointed out by Taylor (1935), and\nlater confirmed in laboratory experiments (Dryden 1943;\nSreenivasan 1984; Pearson et al. 2002) and by hy-\ndrodynamic numerical simulations (Sreenivasan 1998;\nKaneda et al. 2003).\nThus, in contrastto the 3D case, Equation (12) implies\nthat for small viscosities 2D turbulence is essentially un-\nable to dissipate energy at small scales. The viscous sink\nof energy is missing, because at any given time during\nthe decay of an initial large-scale velocity field, for a suf-\nficiently small value of ν, the dissipation is arbitrarily\nsmall.\nTherefore in two dimensions there cannot be a direct\nenergy cascade. As proposed by Kraichnan (1967), en-\nergy must flow toward the larger scales through an in-\nverse cascade . Thus during their evolution vortices (ve-\nlocity eddies) acquire increasingly larger scales, while it\nis enstrophy that develops a directcascade (Batchelor\n1969) with vorticity acquiring smaller scales. As can be\nseen from Equation (9) the convective derivative of ω\nvanishes as νtends to zero, i.e., vorticity is constant in\ntime at a point that moves with the fluid. The direct\nenstrophy cascade implies that an initial patch of vortic-\nity gets stretched in time to form filamentary structures,\nso that while ωis convected with the fluid its gradient\nincreases.\nMost numerical and laboratory experiments include a5\nsmall viscosity, but the regularity of the solutions of the\n2D Euler equation and the vanishing of the energy dissi-\npation rate as viscosity tends to zero for the 2D Navier-\nStokes equation allow to establish a clear connection be-\ntween the solutions of the 2D Euler and Navier-Stokes\nequations. Namely, the time evolution of the solutions\nof the same decay problem for the 2D Navier-Stokes and\nEuler equations is the same until dissipation sets in, i.e.,\nuntil sufficiently small scales are formed and dissipation\noccurs in the Navier-Stokes case.\nThe dual cascade picture has been investigated and\nsupported by 2D Navier-Stokes simulations of forced\nturbulence (e.g., Hossain et al. 1983; Frisch & Sulem\n1984; Sommeria 1986; Tabeling 2002; Boffetta & Ecke\n2012), of the decay of an initial condition consist-\ning of large-scale vortices (Matthaeus & Montgomery\n1980; Matthaeus et al. 1991), and by laboratory\nexperiments of 2D decays performed with soap\nfilms, (Gharib & Derango 1989; Belmonte et al. 1999;\nGreffier et al. 2002; Rivera et al. 2003).\nThe decaying (initial value problem, or ‘relaxation’)\ncase is particularly relevant to the analysis carried out\nin Section 3.1. Recently Mininni & Pouquet (2013) have\nperformed a number of numerical simulations of the de-\ncay of an initial condition with energy in a narrow band\nof wavenumbers (with length-scale ℓ, and velocity uℓ),\nwith sufficient resolution to allow the development of\nboth the inverse energy and direct enstrophy cascades.\nSince a decay is inherently non-steady, in order to com-\npare with Kolmogorov K41 phenomenology, they per-\nform several simulations where the initial condition has\nthe same velocity rms uℓ, but different random ampli-\ntudes. In this way different realizations are obtained, al-\nlowing them to perform ensamble averages that smooth\nout the fluctuations in a single realization, and can then\nbe compared more straightforwardly with the original\nK41 phenomenology (that as a matter of fact uses en-\nsambleaverages). Indeeda clear dual cascade is identified\nalso in the case of a decay . In particular at wavenumbers\nsmaller than the wavenumberof the initial condition, the\npeak of energy moves toward smaller wavenumbers with\ntime, and energy develops an Ek∼k−5/3spectrum, fol-\nlowing K41 phenomenology (that does not depend on\nthe direction of the cascade – inverse or direct – e.g.,\nsee Rose & Sulem 1978). The characteristic dynamic\ntimescale is given by the eddy turnover time\ntℓ=ℓ\nuℓ, (13)\nwhereℓanduℓinitially are the length-scale and velocity\nof the initial condition.\nIn K41 phenomenology the eddy turnover time (13) is\nthe typical timescale for an eddy of size ∼ℓto undergo\na significant distortion due to the relative motion of its\ncomponents, thus transferring (in the 2D case) its en-\nergy at larger scales, as schematically shown in Figure 1\n(left panel). The dimensional analysis of Equation (9)\nshows that tℓis also the timescale over which an initial\ncondition u0with energy at scales ∼ℓand∝angb∇acketleftu2\n0∝angb∇acket∇ight1/2=uℓ\nundergoes a significant distortion with ∝angb∇acketleft(δu)2∝angb∇acket∇ight1/2∼uℓ.\nFurthermore scales are defined as logarithmic bands of\nwavenumbers, e.g., kn∈(2n,2n+1], withn∈Nand\nscaleℓn∼ℓ⊥/kn= 2−nℓ⊥(the indexnwill be droppedhereafter). In fact a single wavenumber cannot repre-\nsent a scale (Aluie & Eyink 2010) since, from the un-\ncertainty principle, the associated Fourier mode is de-\nlocalized in space and does not give rise to a localized\nphysical structure such as an eddy, the building block of\nK41 phenomenology. Consequently in the time interval\ntℓ[Equation (13)] the energy of the system is transferred\napproximately from the scale ℓto the larger scale of dou-\nble size2ℓ.\nDue to the structure of the absolute statistical equi-\nlibria of the ideal truncated system (Kraichnan 1967;\nKraichnan & Montgomery 1980) the development of an\ninverse cascade is expected in both the 2D dissipative\n(Navier-Stokes) and ideal (Euler) cases. Generally the\ntime evolution of dissipative and ideal systems displays\noverall similar dynamics until energy reaches the small\nscales, when respectively thermalization and dissipation\nsets in. This is observed even when a direct energy cas-\ncade occurs, such as in 2D and 3D MHD, although spec-\ntral indices are observed to deviate from K41 in the ideal\ncases (e.g., Wan et al. 2009; Brachet et al. 2013). Since\nthe dynamics of the 2D Euler system may depart from\nK41 phenomenology, that has been developed and stud-\nied for the dissipative case, in the following the eddy\nturnover time tℓis then used as an estimate for the dy-\nnamical timescale of the 2D Euler equation solutions.\nThisisalsojustified bythe dimensionalanalysisofanini-\ntial condition consisting ofvorticesof scale ℓand velocity\nuℓ, that indicates tℓ∼ℓ/uℓas the order of magnitude of\nthe dynamically relevant timescale.\n3.1.Structure of Reduced MHD Equilibria and\nDynamics\nThis phenomenology can be applied to the reduced\nMHD equilibria (Equation [8]) using the mapping (10),\nas schematically shown in Figure 1. Eddies, that in\nthe hydrodynamic case are velocity vortices, correspond\nnow tomagnetic islands . Then, given a magnetic field\nbbd(x,y) at the boundary z= 0 with energyat scales ∼ℓ\n(i.e., a field structured in magnetic islands of scale ∼ℓ),\ntheunique and regular equilibrium solution beq(x,y,z)\nwithbeq(x,y,z= 0) =bbd(x,y) is characterized by an\nincreasingly stronger inverse cascade in the x-y planes\nfor higher values of z(see Figure 1, right panel), with\nthe orthogonal magnetic field length-scale ℓgetting pro-\ngressively larger up to doubling its value in the plane\nzℓ∼B0\nbbdℓ, (14)\nthe analogofthe eddy turnovertime, derivedfrom Equa-\ntion (13) using the mapping (10). If the magnetic field\nis characterized by a scale of order ℓatz= 0 it will have\nits energy at scale ∼2ℓatz=zℓ, corresponding respec-\ntively to magnetic islands of scales ℓand 2ℓin the x-y\nplanesz= 0 andz=zℓ.\nTherefore the equilibrium solution beq(x,y,z)is gen-\nerally asymmetric in the axial direction z, but it can be\nalmost invariant or have strong variations, depending on\nthe relative value of the length-scale zℓcompared to the\nloop length Lz. As long as the variation scale is larger\nthan the loop length ( zℓ>Lz) the field has a weak vari-\nation along z, but for increasingly smaller values of zℓ\n(zℓ<Lz) the field becomes progressivelymore asymmet-\nric along z due to the inverse cascade. Fixed the value6 Rappazzo\nFigure 1. Schematic of the inverse energy cascade for the solutions of the 2D Euler equation ( left panel ) and for reduced MHD equilibria\n(right panel ). In the hydrodynamic case ( left panel ) an eddy of size ℓdoubles its size in the eddy-turnover time tℓ∼ℓ/uℓ(Equation [13]).\nFor the structure of reduced MHD equilibria ( right panel ) this implies (via the mapping t→z,u→b/B0given by Equation (10)) that an\nequilibrium solution with magnetic islands of transverse s caleℓin the plane z= 0 will acquire an axial asymmetry along the z-direction\ncharacterized by the variation length-scale zℓ∼B0ℓ/bbd(Equation [14]), since magnetic islands in the plane z=zℓhave an approximately\ndouble transverse scale ( ∼2ℓ).\nof the guide field B0and the scale of the magnetic field\nℓ, the axial variation scale zℓisinversely proportional to\nthe magnetic field intensity at the boundary bbd: quasi-\ninvariant equilibria are then obtained for weak magnetic\nfieldsbbd, while increasinglystrongerfields result in more\nasymmetric equilibria.\nA strong asymmetry of the magnetic field balongz\nis generally not compatible with the dynamical solutions\nof the reduced MHD equations (1)-(2), as confirmed by\nnonlinear simulations (for a more detailed discussion of\nthe topicssummarizedin the presentand next paragraph\nsee Rappazzo et al. 2008). The derivatives along zap-\npear only in linear terms in Eqs. (1)-(2). Introducing the\nEls¨ asser variables z±=u±b, and neglecting nonlinear\nand diffusive terms, the remaining linear terms yield the\ntwo wave equations:\n∂tz±=±B0∂zz±,with ∇·z±= 0.(15)\nThus fluctuations propagate along zat the Alfv´ en speed\nB0, anda stronglyasymmetricfield cannot be generated,\nparticularly for the problem considered here with line-\ntying boundary conditions. For instance a velocity uph\nat the boundary z= 0 implies the “reflection” condition\nz−=−z++2uph(z−propagates inward, z+outward),\ni.e., generalized Alfv´ en waves are injected and propagate\nin the computational box, and only them= 0mode does\nnot propagate in the axial direction.\nConsidering an initial condition with only the guide\nfieldB=B0ˆ ez, vanishing orthogonal component b= 0,\nand a constant velocity uphat the photospheric bound-\nary that shuffles the magnetic field line footpoints, an or-\nthogonalcomponent ofthe magnetic field invariantalong\nzis generated and it grows linearly in time as\nb(x,y,z,t) =uph(x,y)t\nτA, (16)whereτA=Lz/B0is the Alfv´ en crossing time in the ax-\nial direction. Strictly speaking higher modes are present\ndepending on how the velocity uphis turned on, but\nthey represent a small contribution compared to Equa-\ntion (16) that is the strongly dominant term.\nThe solution (16) is obtained from Equations (15) (see\nRappazzo et al. 2008), thus neglecting nonlinear terms\nin the reduced MHD equations. This is justified as long\nasbis small. In fact Rappazzo & Parker (2013) have\nshown that for initial configurations with only the m= 0\nmode in the axial direction and a non-vanishing 2D or-\nthogonal Lorentz force component (with corresponding\ntermb·∇j∝negationslash= 0 in Equation [8]), nonlinearity is strongly\nsuppressed , i.e., the nonlinear terms can be neglected,\nfor magnetic fields with intensities below the magnetic\nthresholdb∼ℓB0/Lz. The magnetic field decays only\nfor larger magnetic fields b > ℓB 0/Lz, while for smaller\nvaluesthesystemdoesnotformsignificantcurrentsheets\nand energy is not dissipated.\nThe intensity threshold b∼ℓB0/Lzcorresponds to a\ncriticalequilibrium variation length-scale of the order of\nthe looplength zℓ∼Lz(Equation[14]). Since fieldswith\nonlym= 0 are invariant along zthe correspondent equi-\nlibria length-scale zℓcan be computed with bbd=b. For\nb<ℓB 0/Lzthe corresponding equilibrium, i.e., the equi-\nlibrium solution computed with bbd=bat the boundary,\nhaszℓ> Lzand it is quasi-invariant along z. Conse-\nquently magnetic fields with m= 0 and intensity below\nthe intensity threshold b∼ℓB0/Lzarevery close to their\ncorrespondingequilibrium solution. It is such close prox-\nimity to an equilibrium that suppresses nonlinearity , that\nat equilibrium is indeed entirely depleted.\nThe emerging phenomenology for the dynamics of ini-\ntially straight axial field lines shuffled by a velocity\nfielduphconstant or slowly changing in time (so that7\nthe induced magnetic field is quasi-invariant along z) is\ntherefore the following: since at first the induced mag-\nnetic field is small, the associated variation scale is large\nzℓ≫Lz, nonlinearities are suppressed and the magnetic\nfield grows as in Equation (16) until the variation length-\nscale becomes smaller than the loop length zℓ/lessorsimilarLz, when\nnonlinearity can develop leading to the formation of cur-\nrent sheets .\nThe decay of initial configurations with a parallel m=0\nmode are relevant for slow photospheric motions. But in\ngeneral the system has three characteristic timescales :\n1. the surface convective timescale τsc∼(ℓsc/2)/uph,\nessentially the typical lifetime of a granule, approx-\nimatelygivenbytheratioofhalfits length-scale ℓsc\nover the photospheric velocity uph(typical values\nfor the Sun are ℓsc∼103km,uph∼1km/s, and\nτsc∼5−8m),\n2. theAlfv´ en crossing time τA=Lz/B0, whereLz\nis the loop axial length and B0the Alfv´ en length\nassociated to the guide field, and\n3. thenonlinear timescale τnl, that is investigated\ntheoretically and numerically.\nFor typical X-ray bright loops Lz∼40×103km and\nB0∼2×103km/s, therefore the Alfv´ en crossing time\nτA=Lz/B0∼20s is much smaller than the photo-\nspheric timescale, with τA/τsc∼0.04. For these loops\nphotospheric motions are then characterized by a low\nfrequency , i.e.,\nτsc≫τA, (17)\nand a constant ( zero frequency ,τsc=∞) photospheric\nvelocity can be a good approximation, since such slow\nvariation of photospheric motions ( tsc/τA∼25) intro-\nduces only wavelenghtsmuch longerthan the loop length\nitself along z, and the resulting magnetic field (16) can\nbe considered invariant along z.\nNevertheless this condition can break down for longer\nsolar coronal loops and for loops on other active stars\nwith outer convective envelopes and magnetized coro-\nnae, that exhibit broad variations in magnetic field\nintensity and topologies (Donati & Landstreet 2009;\nReiners 2012), and photospheric motion properties\n(Ludwig et al. 2002; Beeck et al. 2013), for which τsc∼\nτA, orτsc<τA. In this case the resulting magnetic field\nwill have a more complex expression than Equation (16),\nand higher modes along zwill be present and contribute\nincreasingly more, the faster the convective timescale τsc\ncompared to the Alfv´ en crossing time τA.\nThereforethestructureofthemagneticfieldinducedin\ncoronal loops by photospheric granulation will be domi-\nnated by the m= 0 mode along z(i.e., the field is quasi-\ninvariant along z) for the typical X-ray bright solar loops\nfor whichτsc>> τA, while for longer loops (including\nloops on other active stars) higher modes ( m≥1) will\nbe increasinglymore important the smaller the timescale\nratioτsc/τA<1.\nIn both cases the variation scale zℓ∼ℓB0/bbd(Equa-\ntion [14]) measures the asymmetry of the equilibrium so-\nlution. In both cases, even in presence of higher modes\nm≥1, the dynamical solutions of the reduced MHD\nequations will not be asymmetric along z, in contrast to\nthe equilibria with zℓ<Lz.Table 1\nSimulations Summary\nRun z-modes B0 numerical grid Re4\nA m=0 1032D: 51221019\nB m=0 1033D: 5122×252 1019\nC m=1 1033D: 5122×252 1019\nD m=0–4 1033D: 5122×252 1019\nNote. — The second column indicates the parallel Fourier\nmodes used in the initial magnetic field (Equation [3]). B0is\nthe axial Alfv´ en velocity (same for all runs). The numerica l\ngridnx×ny×nzis indicated in the fourth column, it is three-\ndimensional for all runs except for run A that is two-dimensi onal\n(with gridnx×ny). The last column indicates the value of the\nhyperdiffusion coefficient Re4=−1/ν4=−1/η4used in Equa-\ntions (1)–(2). As described in the text, each run B–D is a coll ec-\ntive label for a set of simulations carried out with the param eters\nindicated in the table, but with initial conditions (Equati on [3])dif-\nfering for the values of the ratio b0/B0, the rms of the orthogonal\nmagnetic field over the guide field intensity.\nInthefollowingsectionsthedynamicsofconfigurations\nwith different initial conditions, with a single mode along\nz(m= 0 andm= 1), and with all modes 0 ≤m≤4\nexcited, are investigated through numerical simulations\n(see Table 1).\nThe 2Dphotospheric velocity uphmodels photospheric\nmotions. These have large scales (with ℓ∼103km)\nand are disordered, since they originate from turbulent\nconvection. Thus generally the magnetic field bwill\nhave a non-vanishing 2D Lorentz force component, i.e.,\nb·∇j∝negationslash= 0, since the opposite would imply that jshould\nbe constant on the field lines of b, a condition too sym-\nmetric to apply to turbulent convection(this could be re-\nalized, e.g., with an exactly 1D magnetic shear along one\ndirection, or with perfectly circular field lines). There-\nforein all simulations presented here the 2D Lorentz force\ncomponent of the magnetic field does not vanish, i.e.,\nb·∇j∝negationslash= 0.\nNotice that when b· ∇j= 0 we obtain ∂zj= 0 from\nthe equilibrium Equation (8), hence in this case the vari-\nation length-scale is formally infinite zℓ=∞and it is\nalways larger than the loop length. The inverse cascade\npicture of the 2D Euler equation described in section 3\nis indeed valid only for initial conditions that are out of\nequilibrium. If u·∇ω= 0 in Equation (9) then there is\nno time evolution and formally τℓ=∞(Equation [13]).\nReduced MHD equilibria with ∂zj= 0 (or equivalently\nb·∇j= 0) are those apt to describe classic linear insta-\nbilities (such as kink instabilities), and their dynamical\naccessibility will be discussed in the following sections.\n4.RESULTS\nThe results of the numerical simulations are presented\nin this section. All simulations, with the exception of\nRun A, implement line-tying boundary conditions with\nfield line footpoints rooted in a motionless plasma in the\nphotospheric-mimickingplanes z= 0andz=Lz. RunA\nimplements periodic boundary conditions in the axial di-\nrectionz, and since its initial condition is invariant along\nzthe dynamics will not introduce any variation along\nalong this direction ( ∂z= 0) and the simulation is re-8 Rappazzo\nstricted toa 2Dplane. In the orthogonaldirections(x–y)\nall runs use periodic boundary conditions.\nAll simulations consider the decay (or equivalently re-\nlaxation) of an initial magnetic configuration made of\nlarge-scale Fourier modes (as described in Section 2.1)\nwith non-vanishing 2D orthogonal Lorentz force compo-\nnent (b·∇j∝negationslash= 0). The guide field intensity is B0= 1000\nfor all simulations, corresponding to an Alfv´ en velocity\nof 1000km/s, a typical value for solar coronal loops.\nDissipative simulations with initial conditions made of\nsingle parallel modes are described in Sections 4.1 (m=0)\nand 4.2 (m=1), while in Section 4.3 the initial condition\nhas all modes with 0 ≤m≤4. A summary of the\nsimulations is shown in Table 1.\nThe initial magnetic fields used in these decaying sim-\nulations can be considered as “snapshots” of the coronal\nmagnetic field at different stages of its evolution, partic-\nularly for the problem of an initially straight field shuf-\nfled at its footpoints by photospheric motions [see Equa-\ntion (16)]. The different parallel modes are related to the\nphotospheric motions frequency, as discussed in the fol-\nlowing sections. Additionally the relaxation of magnetic\nfields is a topic of general and broad interest to solar\nphysics (e.g., Priest 2014).\n4.1.Runs A–B: single mode m=0\nThis section presents an extended analysis of the dis-\nsipative simulations carried out by Rappazzo & Parker\n(2013) that investigate the decay (or equivalently relax-\nation) of initial magnetic configurations with an out-of-\nequilibrium orthogonal magnetic field and parallel mode\nm=0 (runs A-B).\nTheinitial magneticfieldisinvariantalong zsince only\nthe mode m=0 is present. It has the same structure for\nall runs A-B, with same topology for the magnetic field\nlines ofb, as shown in Figure 4 at time t= 0, but dif-\nferent values of the orthogonal field intensity rms b0(the\nproportionality factor in equation [3]) are used. Respect\nto the guide field B0(same for all runs) the intensity is\nb0= 0.1B0for run A, while run B comprises a set of\nsimulations performed with same parameters except b0\nthat spans the range 0 .01≤b0/B0≤0.1.\nRuns A and B differ for the boundary conditions along\nz,periodic for run A and line-tied for runs B. In the\nperiodic case the invariance along zis preserved during\nthe time evolution, therefore for run A Equations (1)-(2)\nare integrated in a 2D x–y plane with ∂z= 0.\nPeriodicity along zis appropriate as a local approxi-\nmation for the central part ofverylong loops, where line-\ntying at the boundary has a weak influence. In partic-\nular line-tied forced simulations, with a velocity field at\nthephotospheric-mimickingboundaries,haveshownthat\ndynamics resembles those of periodic simulations for low\nvalues of the ratio fc=ℓscB0/Lzuph(Rappazzo et al.\n2007, 2008). Fixed the scale of granular cells ℓscand the\nphotospheric velocity rms uph, line-tying has a weaker\nimpact on longer loops (with larger Lz) or loops with\nweaker guide fields (smaller B0).\nSince the 2D run A is started with an out-of-\nequilibrium magnetic field ( b· ∇j∝negationslash= 0), it is therefore\nakin to 2D turbulence decay simulations (Hossain et al.\n1995; Galtier et al. 1997; Biskamp 2003) that use similar\ninitial conditions for the magnetic field, but addition-\nally have an initial velocity field in equipartition. TheFigure 2. Runs A–B (m=0): Total energy vs. time for line-tied\nsimulations with different values of b0/B0(runs B) and the 2D sim-\nulation (run A,with b0/B0= 10%). The inset shows in logarithmic\nscale total energy normalized with its inital value.\ndynamics are nevertheless similar and in the 2D case to-\ntal energy decays approximately as E∝t−1(Figure 2,\ninset), even if the initial velocity vanishes everywhere.\nUntil time t∼0.4τAtotal energy is conserved, but the\nnon-vanishing Lorentz force transfers ∼15% of magnetic\nenergyEMinto kinetic energy EK, henceforth leading\nto the formation of small-scales and current sheets, dis-\nsipating ∼90% of the initial magnetic energy, while ki-\nnetic energy remains much smaller than magnetic energy\nthroughout (Rappazzo & Parker 2013).\nIn Fourier space magnetic energy has initially only\nlarge-scale perpendicular modes k=3 and 4 (Figure 3,\ntop panel). As nonlinearity develops energy is progres-\nsively transferred toward both small ( direct) and large\nscales (inverse cascade ). In physical space the direct cas-\ncade gives rise to current sheets (Figure 4, top row) that\nenable dissipation through magnetic reconnection. As\ndissipation peaks at t∼1.7τA, the spectrum extends\nfully toward high wave-numbers exhibiting an approxi-\nmatek−5/3power-law. At the same time a substantial\nfraction of total energy has already been transferred to\nlarge-scalemodes k=1and2throughtheinversecascade,\nthat in physical space corresponds to the coalescence of\nmagnetic islands as magnetic reconnection occurs. As\ndynamics proceeds the energy transferred at the small\nscales is dissipated leading to the disappearance of cur-\nrent sheets and small-scales, so that the system finally\nrelaxes to a state with energy mostly at large scales\n(particularly in mode k=1), corresponding in physical\nspace to large-scale magnetic islands and large-scale cur-\nrent layers (Figure 4, top row, t∼210τA). The whole\nprocess is akin to Taylor relaxation (Taylor 1986) and\nself-organization leading to the formation of large-scale\nstructures (Hasegawa 1985).\nIn the 2D case (run A) solutions differing only for the\nvalue ofb0(the rms of the initial magnetic field in Equa-\ntion [3]) have a self-similar structure. Indicating with\nψ0(x,t) andϕ0(x,t) the solution of Equations (1)-(2)\nwith initial magnetic field rms b0, then the solutionswith\nb′\n0=σb0, and same random amplitudes in Equation (3),9\nare given by1:\nψ′(x,t) =σψ0(x,σt), ϕ′(x,t) =σϕ0(x,σt),(18)\nas can be verified by direct substitution. Consequently\nall these solutions have a similar structure and their time\nevolution differs only for the scaling factor σ. In partic-\nular if current sheets form for a certain value of b0, they\nwill always do for any value of b0at scaled times. Anal-\nogously energy will exhibit a power-law decay with the\nsame exponent becauseE′(t) =σ2E0(σt), implying that\nifE(t)∝t−αthenE′(t)∝t−α.\nWhen the same initial condition is used with line-tying\nboundary conditions, the system is no longer invariant\nalongz, as now the velocity must vanish at the top and\nbottom plates z= 0 andz=L, therefore the velocity\ncannot develop uniformly along z as in the periodic case.\nThe time evolution of total energy for line-tied simu-\nlations with different values of b0is shown in Figure 2.\nWhile the dynamics of the system with b0/B0= 10% is\nsimilar to the 2D case with energy dissipating ∼84% of\nits initial value, the behavior is increasingly different for\nlower values of b0, with progressively less energy getting\ndissipated. For b0/B0/lessorsimilar3% no significant energy dissi-\npation nor decay are observed. Additionally, also for the\ndecaying cases their dynamics are strongly suppressed\nonce energy crossesthis threshold. As shown in Fig. 2 no\nenergy decay is observed below E∼5×103, correspond-\ning to a ratio ∝angb∇acketleftb2∝angb∇acket∇ight1/2/B0∼3%. The inset in Fig. 2 shows\nthat energy decays with different power-law indices for\nlower values of b0/B0, hence time self-similarity is lost\nand the impact of line-tying on the dynamics is more\ncomplex than a simple delay as in the 2D case (Equa-\ntions [18]).\nMagnetic energy spectra (integrated along z, Figure 3)\nshow similar dynamics for the 2D ( top panel ) and the\n3D case with b0/B0= 10% ( middle panel ). In par-\nticular the spectra are fully extended toward the small\nscales as dissipation occurs, corresponding to the forma-\ntion and dissipation of current sheets in physical space\n(Figure 4, second row ). While similar behavior is ob-\nserved for magnetic field intensities b0>3%,below this\nthreshold the spectra do not extend to the high wave-\nnumbers where energy gets dissipated (Figure 3, bottom\npanel,b0/B0= 2%), i.e., current sheets do not thin be-\nlow the critical diffusive thickness that allows magnetic\nreconnection and energy dissipation to occur . As shown\nin Figure 4 at the peak of dissipation ( central column )\nboth current maxima and the number of current sheets\ndecrease for smaller b0/B0, and forb0/B0= 2% no cur-\nrent sheets are formed, but only a few ripples are visible\nin the magnetic field (enhanced in the current) and are\neventually dissipated on long timescales.\nFurthermore, spectra show that the inverse energy\ncascade decreases from the 2D to the 3D case with\nb0/B0= 10%, in fact while in the asymptotic state of the\n2D run (t= 210τA) most of the energy is in the k= 1\nmode and higher modes are much smaller, in the 3D case\nthe modek= 2 has the higher value. For 3D simulations\nwith smaller b0/B0the inverse cascade is progressively\n1Strictly speaking these self-similar solutions would requ ire the\nReynolds number to scale as R′=σR, but in the high-Reynolds\nregime the solutions of decaying turbulence do not depend on the\nReynolds number (Biskamp 2003; Galtier et al. 1997).Figure 3. Runs A–B (m=0): Magnetic energy spectra (integrated\nalongz) at selected times for the 2D run A with b0/B0= 10% ( top\npanel), and for run B simulations with b0/B0= 10% ( middle) and\nb0/B0= 2% (bottom). Energy is normalized with its initial value\nat time t=0, when energy is present only at perpendicular mod es\nk= 3 and 4 (diamond symbol). Spectra at the time of maximum\ndissipation for each simulation are drawn in a continuous li ne.\nfainter and does not occur for b0/B0/lessorsimilar3%, as shown in\nFigure 3 ( bottom panel ) forb0/B0= 2%, where no sig-\nnificant energy is found in perpendicular modes smaller\n(k≤2) than those present at time t= 0 (k= 3, 4).\nIn physical space (Figure 4) the inverse cascade cor-\nresponds to larger-scale magnetic islands in the asymp-\ntotic state. Since a larger fraction of magnetic energy\nis dissipated for higher values of b0/B0and in the 2D\ncase, more magnetic flux is reconnected, thus leading to\nincreased coalescence and larger magnetic islands. Ad-\nditionally the field line topology in the relaxed state is10 Rappazzo\nFigure 4. Runs A–B (m=0): Magnetic field lines of the orthogonal magnetic field c omponent band current density jat selected times.\nTop row shows snapshots from the 2D simulation (run A), while snapsh ots from three 3D simulations (run B) with different ratios b0/B0\nare shown in the remaining rows (in this case the mid-plane z= 5 is considered). The first column shows the initial condition at t= 0,\nsame for all except for the ratio b0/B0, thesecond column shows the fields at the time of maximum dissipation, while the third column\nshows the fields at a later time when the fields have relaxed and little if any energy dissipation occurs.\nsubstantially unaltered respect to the initial condition\nforb0/B0/lessorsimilar3% (cf. the first and last columns in Fig-\nure 4), with higher variations for higher values of b0/B0\nand most of all in the 2D case.\nA key difference distinguishes the 2D and 3D asymp-\ntotic topologies. Although all simulations are started\nwith a non-vanishing 2D Lorentz force component ( b·\n∇j∝negationslash= 0), the 2D simulation relaxes to an approximate\nequilibrium with b· ∇j= 0, but all 3D simulationsrelax to an orthogonal field with b· ∇j∝negationslash= 0, regard-\nless of how much energy is dissipated during the decay\n(none forb0/B0= 2%, and an 84% energy decay for the\nb0/B0= 10% run B). Further analysis is presented in the\nfollowingtounderstandthenatureoftheseequilibriaand\nhow they are approached.\nIn the 2D case the equilibrium condition (8) becomes\nsimplyb·∇j= 0. This requires the current density jto\nbe constantalongthe field linesof b(orequivalently that11\nthe isosurfaces of jare also isosurfaces of the magnetic\npotentialψ). This condition is generally satisfied only in\nhighly symmetric configurations such as one-dimensional\nmagnetic shears, e.g., b=f(y)ˆ ex, or rotationally invari-\nant fields as b=f(r)ˆ eθ(fis a generic function in carte-\nsian or cylindrical coordinates). In the 2D case the field\nlines are mostly circular in the asymptotic state (Fig-\nure 4) and the isosurfaces of the current density and of\nthe magnetic potential (the field lines of b) overlap. In\nthe 3D case the full equilibrium equation (8) has to be\nconsidered, and the fact that for run B with b0/B0<3%\nno significant dynamics occurs with the initial orthogo-\nnal magnetic field essentially unaltered even though its\n2D Lorentz force does not vanish, i.e., b·∇j∝negationslash= 0, implies\nthat∂zjincreases only slightly from its initial vanishing\nvalue.\nThe dynamical approach of the system to equilibrium\nis further investigated with the probability density func-\ntions (PDFs) of the equilibrium equation terms. These\nPDFs are histograms of the quantities of interest nor-\nmalized so that the resulting function f(q) multiplied by\nthe bin size ∆ qgives the fraction of points in the compu-\ntational box where the specific quantity qhas its value\nin the interval [ q−∆q/2,q+ ∆q/2], and the integral /integraltext\ndqf(q) = 1. PDFs have been computed for both the\nleft andright hand terms in the equilibrium equation (8),\nand for their difference, indicated with\nEq(x) =∂j\n∂z+b·∇j\nB0. (19)\nThis is a three-dimensionalscalarfunction that measures\nhow far the system is from equilibrium locally at point x,\nvanishing at equilibrium and with higher absolute values\nthe larger the departure from equilibrium. The PDFs\nhave been computed for all the simulations with m= 0\n(for the 2D case only the term b·∇j/B0is present), and\nthey all exhibit similar behavior. All these PDFs have\nvanishing averages, therefore their standard deviations\ncan be defined as:\nσz=/angbracketleftBigg/parenleftbigg∂j\n∂z/parenrightbigg2/angbracketrightBigg1/2\n, σ ⊥=/angbracketleftBigg/parenleftbiggb·∇j\nB0/parenrightbigg2/angbracketrightBigg1/2\n,(20)\nσeq=/angbracketleftBigg/parenleftbigg∂j\n∂z+b·∇j\nB0/parenrightbigg2/angbracketrightBigg1/2\n, (21)\nrespectively for the first and second terms and the whole\nEquation (19), labeled as parallel(σz),orthogonal (σ⊥)\nandtotal(σeq).\nIn Figure 5 ( top panel ) the PDFs of the equilibrium\nfunctionEq(x) are shown in a semi-log plot at selected\ntimes for run B with b0/B0= 10%. The abscissa is\nrescaled with the standard deviation of the PDFs to im-\nprove visualization, since they exhibit large variations in\ntime. The PDF of Eqis generally super-Gaussian (with\na peak around zero and “tails” farther out), particularly\nclose to maximum dissipation time ( t= 2τA), however\nthe centralpart appearscloserto a Gaussiandistribution\nat later times ( t= 10τA), and particularly in the final\nasymptotic stage ( t= 1018.2τA). Although long tails\nare present at most times, in all cases ∼95% of points\nlies within two standard deviations from zero (from 94%Figure 5. Runs A–B (m=0): Probability density functions\n(PDFs) of the reduced MHD equilibrium equation (8) at select ed\ntimes for the run B with b0/B0= 10% ( top panel ), shown in a\nsemi-log plot. To accommodate the large variation of the sta ndard\ndeviation the abscissa displays the values normalized with the stan-\ndard deviation at the corresponding time. The remaining pan els\ndisplay in logarithmic scale the standard deviations as a fu nction\nof time for the 2D simulation ( bottom panel ) and the 3D runs B\nwithb0/B0= 2% and 10% ( middle panels ). For the 3D cases the\northogonal ( σ⊥), parallel ( σz) and total ( σeq) standard deviations\nare shown.12 Rappazzo\natt= 0 to 96% at t= 10τA).\nAn appropriate quantitative measure of the distance of\nthe system from equilibrium is given by the standard de-\nviationofEq, shown in the mid panels of Figure 5 along\nwithσzandσ⊥for runs B with b0/B0= 2% and 10%.\nIn this case, since their averagesvanish, the standard de-\nviations are also the rms of the considered quantities. At\ntimet= 0 the derivative along zofjvanishes (∂zj= 0)\nso that initially σz= 0 for both runs, while σ⊥= 25.32\nand 633.09 respectively. But already after one Alfv´ en\ntimeτAthey both increase (substantially only for the\nrun withb0/B0= 10%) reaching similar values σz∼σ⊥,\nand subsequently continue to be very close while their\nvalues decrease. The rms of the equilibrium function Eq,\nthe standard deviation σeq, decreases with time, and it\nis asymptotically smaller than σzandσ⊥. As shown in\nFigure 5 all standard deviations decrease asymptotically\nlike a power-law with σz∼σ⊥∝t−α, where spectral\nindices are respectively α= 0.17 and 0.038 for the runs\nwithb0/B0= 10% and 2%, while the rms of Eqdecays\nat a faster rate with σeq∝t−β, whereβ= 1.45 and 1.25.\nThis implies that while the rms of ∂zjandb·∇j/B0\nhave about the same value and remain approximately\nconstant in the asymptotic state (when their power-law\ndecayoccurs), equilibrium is approached asthetwoterms\nin Equation (19) balance each other progressively more\nthroughout the computational box, thus leading to the\nrapid decrease of σeq.\nThe initial increase of the standard deviations is larger\nforb0/B0= 10% than for the 2% case, since for b0/B0=\n10%the system is fartherout ofequilibrium in the begin-\nning, with strongcurrentsformingquicklyandmaximum\ndissipation rate occurring at t∼1.7τA, when also stan-\ndard deviations approximately peak. Their subsequent\ndecrease up to t∼20τAis enhanced by the strong de-\ncay of magnetic energy and progressive disappearance of\ncurrent sheets, before approaching the asymptotic stage\naroundt∼200τA. Although the standard deviations\nfor run B with b0/B0= 2% have similar behavior to\nthe case with b0/B0= 10%, their variations are much\nsmaller, since the field just undergoes a slight adjust-\nment, without any significant energy decay (Figure 2),\ncurrentsheetsformationorsignificantchangeinthemag-\nnetic field topology (see bottom row in Figure 4).\nFor the 2D run A there is no parallel standard devia-\ntionσz. The rms of Eq,σeq, is equal to σ⊥, and simi-\nlarly to runs B its time evolution follows the power-law\nσeq∝t−βwithβ= 1.33, as shown in Figure 5 (bottom\npanel). It is worth mentioning that σeqinitially grows\nlarger respect to its initial value (= 633) for run B re-\nspect to run A with same b0/B0= 10%. The reaction of\nthe line-tied field sets the system further out of equilib-\nrium, an enhancement of nonlinearity that might favor\nthe development of singularities in the line-tied system\nrespect to the periodic case.\nFurther insight into the dynamics is gained analyzing\nthe probability density function of the cosine of the an-\ngleθbbetween band∇j\ncosθb=b·∇j\n|b||∇j|, (22)\nshown in Figure 6. At time t=0 the “2D perpendicu-\nlar” Lorentz force term does not vanish for all simula-Figure 6. Runs A–B (m=0): Probability density functions\n(PDFs) of the angle θbbetween the orthogonal magnetic field b\nand the current density gradient ∇jat selected times for the 2D\nrun A with b0/B0= 10% ( top panel ), and the 3D runs B with\nrespectively b0/B0= 10% ( middle) andb0/B0= 2% (bottom).\ntions (b· ∇j∝negationslash= 0). Since the orthogonal component of\nthe magnetic field differs for runs A–B only for a pro-\nportionality factor (Equation [3]), the PDF is the same\nfor all runs A–B (e.g., see mid-panel in Figure 6). Al-\nthough it is peaked around zero (corresponding to the\n2D equilibrium condition b·∇j= 0) it spreads out with\nsignificant values up to |cosθb|/lessorsimilar1/2, corresponding to\nan approximate 60◦angle around θb= 90◦. For the 2D\nrun A with b0/B0= 10% (top panel) the PDF initially\nspreads out during the strongest part of the nonlinear13\nstage (1/lessorsimilart/τA/lessorsimilar10) during which ∼90% of the initial\nenergy is dissipated, but afterward peaks progressively\nmore strongly around zero, corresponding to the equilib-\nrium condition for the orthogonal field b·∇j= 0, with\nrespectively 71% and 96% of the grid points in the vol-\nume in the region |cosθb|<0.1, spanning an angle of\n∼11.5◦aroundθb= 90◦, at times t=29.1 and 911 .6τA.\nThisconfirmsthat the systemapproachesasymptotically\nan equilibrium with b· ∇j= 0 corresponding in physi-\ncal space to increasingly circular field lines progressively\nmore coincident with the isosurfaces of j, as shown in\nFigure 4 (top row).\nIncontrast the picture is radically different for the line-\ntied simulations . As shown in the middle panel of Fig-\nure 6, for run B with b0/B0= 10%the PDF spreads\nincreasingly further out during the time evolution , flat-\ntening considerably already after one Alfv´ en time and\nremaining flat throughout the subsequent evolution , when\n∼80% of magnetic energy is dissipated, and in the fol-\nlowing asymptotic regime, with peaks forming in corre-\nspondence of alignment between the two fields ( θb∼0◦,\n180◦). Therefore, in contrast with the periodic case (2D\nrun A) the orthogonal magnetic field does not approach\nthe asymptotic equilibrium with b·∇j= 0 (that would\nimply also ∂zj= 0 in the 3D case, Equation [8]), in-\nstead in the 3D line-tied case the orthogonal component\nof the magnetic field remains with a non-vanishing “2D\nperpendicular” Lorentz force term (b·∇j∝negationslash= 0).\nFurthermore, if the initial magnetic field intensity\nis below the threshold set out by Rappazzo & Parker\n(2013), as shown in the bottom panel of Figure 6 for\nrun B with b0/B0= 2%, the PDF starts flattening out\nto some extent but then bounces back very close to its\ninitial profile, corresponding to a slight readjustment of\nthe magnetic field as shown in Figure 4 (bottom row),\nwith the orthogonal magnetic field preserving its non-\nvanishing perpendicular Lorentz force.\nThe results of the numerical simulations analyzed in\nthissectionareconsistentwiththeheuristicphenomenol-\nogy laid out by Rappazzo & Parker (2013) and the more\nrefined analysis of the structures of the equilibria ex-\npounded in Section 3. The asymmetry along zof the\nsolutions of the reduced MHD equilibrium equation (8)\ncan be estimated with the axial variation length-scale\nzℓ∼ℓB0/b(Equation [14], see also Figure 1), where ℓis\nthe perpendicular characteristic scale (in the x–yplane)\nof the magnetic field component b.\nAs discussed in Section 3.1, the dynamical solutions\nof the reduced MHD equations (1)-(2) generally cannot\nexhibit strong asymmetries along z, in particular when\ndriven from the photospheric boundaries. On the other\nhand the equilibria can be strongly asymmetric or quasi-\ninvariant along z, depending on the relative value of the\naxial variation scale zℓrespect to the loop axial length\nLz, with the critical length given byzℓ∼Lz. For small\nvalues ofbthe axial variation scale zℓis longer than the\nloop length Lzand the corresponding equilibrium solu-\ntion is quasi-invariant along z, while for larger values of\nbthe axial scale zℓis smaller than the loop length and\nthe equilibrium is more asymmetric along zthe larger\nthe magnetic field intensity b.\nAs shown in Figure 2 no substantial energy is dissi-\npated for 3D runs with b0/B0/lessorsimilar3%, and just mini-\nmal dynamics occurs as the field slightly readjusts (Fig-ures 3–6). Since the initial magnetic field (Equation [3])\nhas a perpendicular scale ℓ∼L⊥/3.87∼1/3.87 (the\naveraged wavenumber of the initial condition is 3.87\nandL⊥= 1), this threshold corresponds to a varia-\ntion length-scale for the initial magnetic field of about\nzℓ=ℓB0/b0/greaterorsimilar100/(3×3.87),i.e.,zℓ/greaterorsimilarLzsinceLz= 10.\nThereforefor b0/B0/lessorsimilar3%thecorrespondingequilibria,\ncomputed from Equation [8] with beq(z=0) =b0(z=0),\nas described in Section 3, have a large variation length-\nscalezℓ/greaterorsimilarLzand are therefore quasi-invariant along\nz. Since the initial condition has only the parallel m=0\nmode it is invariant along z, and therefore very close\nto the corresponding equilibrium, so that nonlinearity is\nstrongly depleted and only a slight readjustment of the\nfield occurs, with no significant energy dissipation. On\nthe contrary for initial conditions with b0/B0>3% the\ncorresponding equilibria have smaller variation length-\nscaleszℓ<Lz, hence the equilibrium solution is strongly\nasymmetric along z, differing substantially from the ini-\ntial condition that is then necessarily out-of-equilibrium,\nas shown by the subsequent dynamics and energy dissi-\npation.\nFurthermore, as shown in Figure 2, also in the cases\nwhen decay occurs, i.e., for initial conditions with\nb0/B0>3%, the magnetic field relaxes to a new equilib-\nrium that approximately satisfies the condition b/B0∼\n3% andzℓ∼Lz. But while the asymptotic energy of\nthe runs with b0/B0= 4%, 5% and 6% are approxi-\nmately the same, corresponding to a ratio b/B0∼3.3%,\nthe run with b0/B0= 10% relaxes to a slightly higher\nenergy with b/B0∼4%. On the other hand the run\nwithb0/B0= 10% has a stronger inverse cascade (Fig-\nure 3), with significantly larger magnetic islands in the\nasymptotic regime (Figure 4) and average wavenumber\n∼2.7 thus obtaining again zℓ∼Lz. When a strong in-\nverse cascade occurs the formation of larger perpendic-\nular scales ( ℓ) increases the value of the axial variation\nlength-scale zℓ∼ℓB0/b, thus attaining the equilibrium\nconditionzℓ∼Lzwith a larger value of b, and conse-\nquently a smaller dissipation of energy.\nThecritical variation length-scale zℓorigins from a\nbalance of forces because the reduced MHD equilib-\nrium equation (8) representsa balance between two force\nterms. In the reduced MHD limit (e.g., Montgomery\n1982) the Lorentz force splits into two terms with com-\nponents only in the orthogonal x–yplanes: the “perpen-\ndicular” ( b· ∇b) and “parallel” ( B0∂zb) field line ten-\nsions (plus the pressure term, determined through the\nincompressibility condition). The first term represents\nthe field line tension of the orthogonal component b, the\nonly one present in the 2D limit. The second is an addi-\ntional tension term due to the presence of the guide field\nB0, linked to the tension of the field lines of the total\nmagnetic field B0ˆ ez+b. An equilibrium is attained only\nwhen these two counteracting components of the Lorentz\nforce balance each other satisfying Equation (8). As out-\nlined in Rappazzo & Parker (2013) these two forces are\nof the same order of magnitude for the critical intensity\nb∼ℓB0\nLz, (23)\ncorresponding to the critical axial variation length-scale\nzℓ∼ℓB0/bℓ∼Lzas discussed in Section 3.1.14 Rappazzo\nFigure 7. Runs C(m=1) and D(m=0–4): Total energy vs. time for line-tied simulations wi th different values of b0/B0for run C with\nsingle parallel mode m=0 ( left panel ) and run D with m=1 ( right panel ). The 2D run A with b0/B0= 10% is added for comparison. The\ninsets show in logarithmic scale total energies normalized with their initial values.\nInitially also the 3D line-tied system starts to behave\nas in the 2D case, with the tension of perpendicular field\nlines creating an orthogonal velocity, that coupled with\nall others nonlinear terms are the only ones that can cas-\ncade energy and generate current sheets. But this dis-\nplacesthetotalline-tied(axiallydirected)fieldlines, that\nnow cannot be freely convected around as in the periodic\ncase because of the line-tying constraint at the bound-\naries, and is then counteracted by the enhanced axial\ntension that resists bending. Magnetic fields with inten-\nsities smaller than the threshold (23) a small variation of\nbalongz(corresponding to a variation scale larger than\nthe loop length Lz) is enough to reach an equilibrium,\nbut for intensities larger than of b/greaterorsimilarℓB0/Lzcurrent\nsheets must form and energy dissipate in order to reach\nthe physically accessible equilibria with b∼ℓB0/Lz,\nsince for larger magnetic field intensities the equilibria\nare strongly asymmetric along zand therefore physically\ninaccessible.\nThe analysis in this section has considered exclusively\ninitial conditions invariant along the z-direction, with\nonly the parallel mode m=0. In the following sections we\nextend it to include field variations along zwith higher\nparallel modes..\n4.2.Runs C: single mode m=1\nThe finite length of coronal loops renders the system\nakin to a resonant cavity . A forcing velocity with fre-\nquencyνat the photospheric boundary, e.g.,\nu(x,y,z=L,t) =uph(x,y)cos(ωt),(24)\nwhereω= 2πνis the angular frequency, will inject\nAlfv´ en waves at that frequency propagating in the ax-\nial direction of the loop. In general these waves, that\nare continuously injected and reflected at the boundaries\n(see Section 3.1, Equation [15]), will be out of phase and\ndecorrelated along the loop so that their sum will re-\nmain limited the whole time to values of the order of the\nforcing velocity at the boundary, with no growth in time\nfor the amplitude of the resulting velocity and magnetic\nfields. But for the resonant frequencies\nνn=nνA/2,withn∈N (25)andνA= 1/τA, the waves are in phase and they sum\ncoherently (Ionson 1985). Thus the magnetic and ve-\nlocity fields in the loop grow linearly in time similarly\nto the case with constant velocity (Equation [16]). For\ninstance, considering the boundary velocity (24) at the\nresonant frequency νnwithn≥1, the resulting fields\ngrow approximately as (Einaudi & Velli 1999; Rappazzo\n2006; Chiuderi & Velli 2015):\nb∼uphcos/parenleftbigg\nωnz\nvA/parenrightbigg\ncos(ωnt)t\nτA,(26)\nu∼uphsin/parenleftbigg\nωnz\nvA/parenrightbigg\nsin(ωnt)t\nτA.(27)\nIndeed a constant velocity can be regarded as the zero\nfrequency resonance n= 0 of the system, that differs\nfrom resonances with n≥1 because while the magnetic\nfield grows linearly in time, for n= 0 the velocity field\ndoes not grow and its value remains of the same order of\nmagnitude of the photospheric velocity ( u∼uph).\nAs mentioned in Section 3.1, for X-ray bright solar\ncoronalloops photosphericmotionshavealowfrequency,\ngiving rise to a coronal magnetic field dominated by the\nparallelm= 0 mode, and their low frequency can then\nbe approximated with zero. In general photospheric mo-\ntions characterized by a surface convective timescale τsc\nwill not have a single harmonic at the frequency 1 /τsc,\nratherthe amplitude of its Fouriertransformwill peak at\nthe frequency 1 /τscbut include many other harmonics.\nFor longer loops (with longer Alfv´ en crossing times\nτA=Lz/vAcomparable or longer than granulation\ntimescales), and for loopson other activestarswith mag-\nnetized coronae and outer convective envelopes, photo-\nsphericmotions can havefrequencies closerto resonances\nhigher than ν0= 0. Furthermore also when photospheric\nmotionshaveadominantlowfrequency,higherfrequency\nmodes will be present, although with smaller ampli-\ntudes (Nigro et al. 2008) that contribute considerably\nless heating (Milano et al. 1997). In all these cases pho-\ntospheric motions will give rise to parallel modes higher\nthan zero (m≥1) for the coronal magnetic field. These\nwill be the dominant modes when the photospheric fre-15\nFigure 8. Runs C(m=1) and D(m=0–4): Magnetic field lines of the orthogonal magnetic fiel d component band current density jin the\nmid-planez= 5 at selected times for runs C ( top row) and D ( bottom row ) withb0/B0= 10%. The left panels show the initial condition\natt= 0, the central panels show the fields at the time of maximum di ssipation, while the right panels show the fields at a later ti me when\nthey have relaxed (asymptotic regime).\nquency is resonant, and give a small contribution to the\nmagneticfieldwhenphotosphericmotionsfrequenciesare\ncloseto zero. Additionally higherparallelmodes canalso\nbe generated by nonlinear dynamics also when starting\nwith a zero parallel mode (Buchlin & Velli 2007), and\nby disturbances stemming from chromospheric dynamics\n(De Pontieu et al. 2007).\nIt is therefore of interest to consider initial conditions\nwith modes higher than m= 0, and in the numerical\nsimulations analyzed in this section the initial magnetic\nfield has only the parallel mode m= 1 (corresponding\nto the resonant frequency ν2=νA), while large-scale\nperpendicular modes are set as in previous simulations\nwith wavenumbers between 3 and 4 (Section 2.1, Equa-\ntion [3]). Thus unlike run B with m= 0, the out-of-\nequilibrium initial magnetic field now varies also along\nz, with both terms in the equilibrium equation (8) not\nvanishing ( ∂zj∝negationslash= 0,b· ∇j∝negationslash= 0). The magnetic field\nlines of the orthogonal component band current den-\nsityjare shown in Figure 8 (top row) at time t= 0\nin the mid-plane z= 5, while isosurfaces of the mag-\nnetic potential at t= 0 are shown in Figure 9 (central\ncolumn). In both figures the case with b0/B0= 10% is\nconsidered. This is one of a series of simulations collec-\ntively labeled as runs C, with same parameters for the\ninitial condition except the magnetic field intensity b0\n(the multiplicative factor in Equation [3]) that spans the\nrange 0.1%≤b0/B0≤10%.\nThe time evolution of total energy for runs C with\ndifferent values of b0is shown in Figure 7 (left panel).\nThe run with b0/B0= 10% has a similar behavior to\nrun B with same magnetic field intensity (cf. Figure 2).\nItsenergydecaysapproximatelyas E∝t−1, withcurrent\nsheets forming in physical space (Figure 8, top row) anddissipating ∼92% of the initial energy, a slightly higher\nvalue respect to the corresponding run B. Subsequently\nthe system relaxes to an asymptotic state with b/B0∼\n2.87%.\nThe analysis of the equilibria set forth in Section 3.1\nshows that the only dynamically accessible equilibria are\nthose with variation length-scale greater than approxi-\nmately the loop length zℓ/greaterorsimilarLz. These have structures\nvery elongated in the axial direction, and therefore dom-\ninated by the parallel mode m= 0. When initial condi-\ntions do not include the m= 0 mode, their higher modes\nwill necessarily have to transfer part of their energy to\nthe modem= 0 via nonlinear dynamics in order to relax\nto equilibrium. Furthermore most of the energy of the\nmodes with m≥1 that is not converted into the parallel\nzero mode must be dissipated for the relaxed state to be\nclose to a reduced MHD equilibrium with zℓ/greaterorsimilarLz, with\na predominant parallel zero mode.\nIn fact the isosurfaces of the magnetic potential ψin\nFigure9 showthat both runs B and C with b0/B0= 10%\n(respectively in the left and central columns) relax to a\nlower energy state with structures very elongated along\nz(the computational box has been rescaled for an im-\nproved visualization, but the axial length is ten times\nlongerthat the perpendicular cross section length), even\nthoughtheirinitialconditionsareradicallydifferent, con-\nsisting exclusively of mode m= 0 for run B and m= 1\nfor run C.\nConsequentlysimilardynamicswilloccurforallrunsC\nindependently from the value of b0/B0, as shown in Fig-\nure 7, because all of them do not have a mode m= 0\nin their initial magnetic field. Thus they all decay while\npart of the energy initially in the m= 1 mode is either\ntransferred to the m= 0 mode (and partially also to16 Rappazzo\nhigher order modes) or dissipated (including dissipation\nofthe higherordermodes generated duringthe nonlinear\ndynamics). On the contrary when the initial condition\nis made only of mode m= 0, the system is very close to\nan equilibrium for b0/B0/lessorsimilar3% (runs B, Figure 2) since\nthis condition implies zℓ/greaterorsimilarLzand the magnetic field is\nquasi-invariant along z, so that no substantial dynamics\noccur when b0/B0/lessorsimilar3%.\nFigure 7 shows that for runs C energy starts decaying\nat later times for smaller values of b0/B0. The longer\ntimescales for dissipation to occur and energy to start\ndecaying is consistent with the decrease of the strength\nof nonlinear interactions for lower values of b0/B0. For\ninstance the eddy turnover time increases as tℓ∼2ℓ/b0,\nsince no velocity is initially present and this soon be-\ncomes of the order of b0/2 (because for resonant fre-\nquencies velocity is in equipartition with the magnetic\nfield). Subsequently velocity strongly decreases once a\nzero mode is created and the system relaxes.\nClearly the fraction of magnetic energy dissipated dur-\ning the decay depends on several factors. The most im-\nportant of these is how much energy in transferred to the\nparallel zero mode, since all higher modes will be largely\ndissipated in order to reach an equilibrium with zℓ/greaterorsimilarLz.\nA detailed analysis of the energy fluxes between these\nmodes goes beyond the scope of the present paper, and\nmight be carried out in future work. Nevertheless it is\nclear from Figure 7 that runs C with 4% ≤b0/B0≤10%\ngenerate a zero mode quickly, while for b0/B0≤1% a\nlarge fraction of the initial energy is dissipated with only\na small fraction transferredto the zeromode. In all cases\nin the asymptotic stage, when the mode m= 0 is the\nstrongest mode, the relaxed magnetic field intensity bis\nbelow the stability threshold (23) with b/B0/lessorsimilar3%.\nInspiteofalltheaforementioneddifferences,thelonger\ndecaytimescales for runs C with lower values of b0/B0\nrendersimilarthe behavior of the system forcedby\nphotospheric motions for both cases with a velocity\nthat is constant in time (zero frequency) and a veloc-\nity with higher resonant frequencies. In fact considering\na straightened loop with initially only the guide field B0,\nif a constant velocity ( ν0= 0) is applied at the photo-\nspheric boundaries, the magnetic field will grow linearly\nin time initially (Equation [16]), because until the or-\nthogonal component of the magnetic field does not reach\nthe critical value b∼ℓB0/Lz(Equation [23]) the sys-\ntem is very close to equilibrium and nonlinear terms can\nbe neglected. The linear growth of the magnetic field is\nderived indeed from the linearized reduced MHD equa-\ntions (15).\nInsimilarfashionifthephotosphericvelocityfrequency\nis ahigher resonance νn=nνA/2, withn≥1, againnon-\nlineartermscanbeneglectedinitiallybecauseforlowval-\nues of the magnetic field intensity bthe decay timescales\nare much longer than the linear growth of the mag-\nnetic field, with the amplitude doubling every Alfv´ enic\ncrossing time τA. Equations (26)-(27) are obtained also\nfor the resonant frequencies from the linearized reduced\nMHD equations, analogouslyto the constant forcing case\n(Equations [15]-[16]). A statistical steady state will fi-\nnally be obtained when the energy flux injected in the\nsystem at the boundary by photospheric motions is bal-\nanced by a similar energy flux from the large toward the\nsmall scales (to form current sheets), in similar fashionto the constant velocity case (Rappazzo et al. 2007).\n4.3.Runs D: modes m=0–4\nIn this section we analyze numerical simulations\n(runs D) in which the initial magnetic field bincludes all\nparallel modes m∈[0,4], while large-scaleperpendicular\nmodes are set as in previous simulations with wavenum-\nbers between 3 and 4 (Section 2.1, Equation [3]). The\nparallel zero mode is the strongest, with higher parallel\nmodes having less energy. For all runs the fraction of\nmagnetic energy in the parallel mode m, indicated with\nEm, is set in the initial magnetic field (Equation [3]) so\nthatEm/E0= (m+ 1)−2.6, corresponding to a progres-\nsively smaller energy for higher modes. Explicitly Em/E0\n= 16.5%, 5.7%, 2.7%, 1.5% for m∈[1,4]. This spe-\ncific choice of values is arbitrary, but the presence of\nmultiple parallel modes of decreasing weight is chosen\nto represent the coronal field induced by photospheric\nmotions in which multiple frequencies are present. Al-\nthough to date there are no measurements of the spec-\ntrum of photospheric velocities, the presence of higher\nfrequency modes with decreasingly smaller amplitudes is\nexpected. Since smaller granules are expected to have\nfaster convective timescales this is partially confirmed\nby the recent detection of mini-granular structures with\ntheir size distributed as a power law with an approxi-\nmate Kolmogorov index ∼ −5/3 and dominant on scales\nsmaller than 600km (Abramenko et al. 2012).\nAs shown in Figure 7 (right panel) the dynamics are\nanalogous to those of run B with only the parallel mode\nm= 0 (cf. Figure 2). The dissipated energy decreases\nfor lower values of b0/B0, but unlike runs B a small but\ndiscernible energy dissipation occurs also for very small\nratios ofb0/B0. Each run D dissipates a larger fraction\nof magnetic energy respect to the corresponding run B\nwith sameb0/B0before reaching the asymptotic regime\n(cf. insets in Figures 7 and 2), because initially a fraction\nof the energy in runs D is in parallel modes higher than\nzero, and they will be dissipated during the decay so that\nthe system can relax to equilibrium (dominated by the\nmodem= 0 with variation length-scale zℓ/greaterorsimilarLz).\nDynamics in physical space are also similar to those of\nrun B, as shown in Figure 8 (bottom row) for b0/B0=\n10%, with current sheets forming and dissipating en-\nergy thus leading to a relaxed field with larger magnetic\nislands through an inverse cascade. The evolution of\nthe three-dimensional structure of the magnetic poten-\ntialψ, shown in Figure 9 (right column), is similar to\nthose of runs B and C. The relaxed magnetic potential\natt= 240τAhas a structure elongated along z, with a\nstrongm= 0 parallel mode similarly to runs B and C,\neven if its structure at time t= 0 is considerably more\ncomplex due to the presence of multiple parallel scales\n(m∈[0,4]).\n5.CONCLUSIONS AND DISCUSSION\nEquilibria and dynamics of the magnetically confined\nregions of solar and stellar atmospheres have been inves-\ntigated with a reduced MHD cartesian model to advance\nour understanding of the mechanism that powers the X-\nray activity of the Sun, late-type main sequence stars,\nand more in general of stars with a magnetized corona\nand an outer convective envelope.17\nFigure 9. Isosurfaces of the magnetic potential ψ(in yellow and transparent red and blue colors) for the three -dimensional simulations\nwith line-tied boundary conditions runs B, C and D, with resp ectively single parallel modes m= 0,m= 1 and all modes m∈[0,4], and\ninitial conditions with b0/B0= 10%. The elongated structure of ψalongzshows that in the asymptotic regime the fields relax into an\nequilibrium with a strong parallel m=0 mode not only for run B , that initially has only the mode m=0, but for all the initial conditions\nconsidered, including runs C that initially has only the par allel mode m=1 and run D that is started with many parallel mod es. Snapshots\ntimes, left to right, are respectively t= 516τA, 893τAand 240τA. The magnetic potential ψis defined in Section 2. The computational\nbox has been rescaled for an improved visualization, but the axial length is ten times longer that the perpendicular cros s section length.\nSince equilibria play a pivotal role in understanding\nthe dynamics of this system, their structure has been\nanalyzed in detail in Section 3. The mapping between\nthe solutions of the 2D Euler equation u2D(x,t) and re-\nduced MHD equilibria beq(x⊥,z) (Equation [10]: t→z,\nu2D→beq/B0) allows to formulate a heuristic quantita-\ntive analysis of the structure of the reduced MHD equi-\nlibria. The inverse cascade developed by the solutions of\nthe Euler equation in time, corresponds to an asymmet-\nric structure along zof the reduced MHD equilibria, as\npictorially summarized in Figure 1.\nIn similar fashion to 2D velocity vortices of scale ℓthat\ndouble their size in about one eddy turnover time tℓ∼\nℓ/uℓ(Equation [13]), a reduced MHD equilibrium with\northogonal magnetic field of intensity bbdat the bottom\nboundaryz= 0, and made of magnetic islands of scale ℓ,\nwill have progressively larger magnetic islands at larger\nz, doubling their transverse scale over the axial spatial\ndistance\nzℓ∼B0\nbbdℓ. (28)\nThis represents the parallel variation length-scale of\nthe equilibrium solution, and measures quantitatively its\nasymmetry . An equilibriumis stronglyasymmetricalong\nzif the variation scale is smaller than the loop lengthzℓ<Lz, but if the variation scale is greaterthan approx-\nimately the loop length zℓ/greaterorsimilarLzthen the equilibrium is\nquasi-invariant along z(at the loop scale).\nOn the other hand in reduced MHD any spatial vari-\nation of the physical fields along zis rapidly propagated\naway at the Alfv´ en speed B0(the fastest speed in the\nsystem) by the linear terms ∝B0∂zin Equations (1)-(2).\nTherefore the physical solutions cannot develop strongly\nasymmetric structures alongz, as confirmed by bound-\nary forced and decaying numerical simulations with\nline-tying boundary conditions and a strong guide field\n(e.g., Galsgaard & Nordlund 1996; Dmitruk & G´ omez\n1999; Rappazzo et al. 2008; Wilmot-Smith et al. 2011;\nDahlburg et al. 2012). This implies that physical solu-\ntions are close to an equilibrium or not depending on the\nintensity of the orthogonal component of the magnetic\nfieldb, thatregulatesthe valueofthe parallelequilibrium\nvariation scale zℓ(28) for a given guide field of intensity\nB0. Consequently the reduced MHD equilibria that can\nbe accessed dynamically are those quasi-invariant along z\nwith an orthogonal magnetic field component bfor which\nthe parallel length-scale is greater than approximately th e\nloop length zℓ/greaterorsimilarLz.\nThe simulations reported by Rappazzo & Parker\n(2013), analyzed in detail in Section 4.1 (runs A–B),\nare in agreement with this picture. They considered ini-18 Rappazzo\ntial magnetic fields b0invariant along z(i.e., only the\nparallel mode m= 0 is present initially and ∂j0= 0),\nbut with different intensities and non-vanishing Lorentz\nforce for the perpendicular magnetic field, i.e., b0·∇j0∝negationslash=\n0, and identified the magnetic field intensity threshold\nb∼ℓB0/Lz,corresponding to the critical variation scale\nzℓ∼Lz. Initial conditions with b0> ℓB0/Lz(zℓ< Lz)\nhave dynamics increasingly similar to 2D MHD turbu-\nlence decay for larger values of b0, with the orthogonal\nmagnetic field forming current sheets and dissipating en-\nergy that decays as a power-law in time with E∝t−α\n(α∼1 forb0/B0= 10%, Figure 2). As shown in\nEquation (18) in the 2D case the evolution of fields\nwith different initial intensities is self-similar in time\nwithb′(t) =b(t·b′\n0/b0)b′\n0/b0, implying that they all\ndecay with same power-law index and relax to asymp-\ntotic fields with intensities proportional to their initial\nvalueb′\n∞/b′\n0=b∞/b0. But in stark contrast with the 2D\ncasetheline-tied3Dsimulationsdecaywithprogressively\nshallower power-laws (Figure 2) for weaker initial mag-\nnetic fields with smaller ratios b0/B0, relaxing to asymp-\ntotic equilibria for which the ratio b∞/b0is not indepen-\ndent fromb0. Insteadb∞∼ℓB0/Lz, corresponding to\na variation scale approximately equal to the loop length\nzℓ∼Lz(as explained in Section 4.1 the orthogonal scale\nℓincreases for stronger magnetic field because an inverse\ncascade of magnetic energy occurs). Furthermore ini-\ntial magnetic fields with intensity below the threshold\nb0/lessorsimilarℓB0/Lz(zℓ/greaterorsimilarLz) show little dynamics with no\nsignificant decay nor current sheets formation and dissi-\npation.\nTherefore these simulations confirm numerically that\nthe dynamically accessible equilibria are those quasi-\ninvariant along z(i.e., with a dominant m= 0 mode)\nwith magnetic field intensity smaller than the thresh-\noldb/lessorsimilarℓB0/Lz, and corresponding parallel variation\nlength-scale larger than approximately the loop length\nzℓ/greaterorsimilarLz. The nature of this equilibria is radically dif-\nferent from the classic reduced MHD equilibria consid-\nered in plasma and solar physics in the framework of lin-\near instabilities (kink, tearing, etc.), that typically are\nstrictly invariant along z(∂z= 0) and in the reduced\nMHD framework have a vanishing orthogonal Lorenz\nforce component with b· ∇j= 0. This condition is\nsatisfied by very symmetric fields, e.g., a sheared field,\nor circular field lines (examples can be found in Parker\n1983; Longcope & Strauss 1993, and references therein).\nBut our initial magnetic fields (Section 2.1) have non-\nvanishing orthogonal Lorentz forces ( b·∇j∝negationslash= 0), a prop-\nerty that stems from the complexity and disorder of pho-\ntospheric motions. As shown in Figure 5 and 6 both\nterms in the equilibrium Equation (8) do not vanish as\nthe system relaxes to equilibrium , with their rms σzand\nσ⊥(Equation [20]) getting asymptotically equal, while\nthe rms of their sum σeq(Equation [21]) vanishes asymp-\ntotically as a power-law. Consequently the system does\nnot relax to a classic linearly unstable equilibrium with\n∂zj= 0 and b·∇j= 0, as confirmed by a visual inspec-\ntion of the orthogonal magnetic field bin Figure 4 (right\ncolumn).\nThe simulations (runs B) have very different dynamics\nwhether their initial parallel variation scales are larger\nor smaller than the critical length-scale zℓ∼Lz. For\nzℓ< Lz, withb > ℓB 0/Lz, the initial magnetic fieldis very far from the corresponding equilibrium, as this\nis too asymmetric along zand is therefore dynamically\ninaccessible. The only way for the out-of-equilibrium\nfield to reach an equilibrium is therefore to decay to a\nlower energy configuration with smaller buntil the criti-\ncal length scale zℓ∼Lzis reached, and this necessarily\nimplies the formation of current sheets and dissipation\nthrough nonlinear dynamics (a magnetically dominated\nnonlinear MHD turbulent cascade analyzed in depth in\nRappazzo & Velli 2011). On the contrary initial mag-\nnetic fields with zℓ/greaterorsimilarLz, for which b/lessorsimilarℓB0/Lz, are\nvery close to the corresponding equilibrium because they\nare both elongated along z. Thus the field simply read-\njusts to the close equilibrium with no significant nonlin-\near dynamics, current sheet formation nor dissipation, as\nshown in Figures 2–4 and particularly in Figures 5 and\n6. Strictly speaking also in this case a very small energy\ndissipation occurs, but it is negligible, does not involve\ntheformationofsignificantlystrongercurrents,andaddi-\ntionally nonlinearities are diminished in close proximity\nto equilibrium.\nThe quasi-static evolution of the magnetic field is then\nrestricted only to field intensities smaller than approxi-\nmatelyb/lessorsimilarℓB0/Lz, while stronger fields are necessarily\nout-of-equilibrium and develop turbulent dynamics with\nsubsequent current sheet formation and energy dissipa-\ntion.\nConsequently two distinct stages can be identified in\nthe dynamics of an initially uniform and strong axial\nmagnetic field B0ˆ ezshuffled at its footpoints by a con-\nstantorlowfrequencyphotosphericvelocity uph(seeSec-\ntion3.1foradiscussiononforcingfrequencies). Tomimic\nthe solenoidal component of the photospheric horizon-\ntal velocity (the irrotational component cannot twist the\nfield lines), the incompressible velocity at the boundary\nuphis made up of distorted vortices (see Rappazzo et al.\n2008, for a specific example) with uph·∇ωph∝negationslash= 0, given\nthe general complexity and disorder of photospheric mo-\ntions.At firstphotosphericmotionsgenerateanorthogo-\nnal coronal magnetic field component that grows linearly\nintimeandisamappingofthephotosphericvelocity,i.e.,\nb=upht/τA(Equation [16]). Until its intensity remains\nbelow the threshold b∼ℓB0/Lzthe field is essentially\nin equilibrium and nonlinearities do not develop , leading\nto its linear growth in time. In fact neglecting nonlinear\nterms in the reduced MHD equations, the linear growth\nfollows from the remaining linear terms (Equation [15])\nand boundary conditions. When the magnetic field in-\ntensity crosses the threshold the variation length-scale of\nthe corresponding equilibrium becomes smaller than the\nloop length zℓ/lessorsimilarLz. The structure of the equilibrium\nbecomes then too asymmetric along z, while the dynam-\nically induced magnetic field (Equation [16]) is quasi-\ninvariant along z. The magnetic field is then too distant\nfrom its corresponding equilibrium that cannot be ac-\ncessed dynamically unless the field intensity decreases.\nThe Lorentz force components that were in equilibrium\nduring the quasi-static stage now cannot reach a balance\nwith each other, the magnetic field is therefore in non-\nequilibrium and nonlinear dynamics develop.\nParker (1988) had conjectured a two-stage process\nto account for the inferred Poynting flux in active re-\ngions, estimated by Withbroe & Noyes (1977) at about19\nSz∼107erg cm−2s−1. In fact if current sheet for-\nmation and energy dissipation would be effective at an\nearlier stage, with too weak magnetic fields, then the\nPoynting flux would be too small to sustain the X-ray\nactivity of active regions. Reverting now to conven-\ntionalunits, the time and spaceaveragedPoyntingflux is\ngiven by ∝angb∇acketleftSz∝angb∇acket∇ight ∼S/ℓ2∼ρvA,/bardbluphvA,⊥(see Section 2.2,\ndensityρand Alfv´ en velocities are introduced from di-\nmensional calculations), where vA,/bardbl=B0/√4πρand\nvA,⊥=b/√4πρare the Alfv´ en velocities associated re-\nspectively to the guide field B0and the orthogonal mag-\nnetic field component b. Introducing the threshold mag-\nnetic field intensity b∼ℓB0/Lz(or the associated Alfv´ en\nvelocity) we obtain for the Poynting flux:\n∝angb∇acketleftSz∝angb∇acket∇ight ∼ρv2\nA,/bardbluphℓ\nLz=B2\n0uphℓ\n4πLz. (29)\nThis coincides with the strong guide field regime of\nthe scaling relation obtained by Rappazzo et al. (2008)\n(Equation [68] with α≫1) for boundary forced simula-\ntions, that yields for typical solar active region loops an\nenergy flux ∼1.6×106erg cm−2s−1, in the lower range\nof the constraint inferred by Withbroe & Noyes (1977).\nButrecentfullycompressibleMHDsimulationswithsim-\nilarsetup, that include the integrationofanenergyequa-\ntion with thermal conduction and energy losses provided\nby optically thin radiation, and in addition have den-\nsity stratification (with strong gradients from the chro-\nmosphere to the corona), exhibit Poynting fluxes of the\norderof∼107ergcm−2s−1, andmostimportantlyanX-\nray emission that matches the physical properties of the\nobserved radiation (Dahlburg et al. 2015, submitted).\nAdditionally numerical simulations with initial mag-\nnetic fields not invariant along zhave been carried out\n(runs C and D). Higher parallel modes can be present for\nboth the Sun and other active stars of interest. They can\nbe generated by the nonlinear dynamics even when pho-\ntospheric motions have a low frequency, or they can be\ndirectly excited by photospheric motions in long loops,\nthat on some stars have observationally inferred lengths\nof the order of several stellar radii (Favata et al. 2005;\nGetman et al. 2008; Peterson et al. 2010).\nRuns C include only the parallel mode m= 1, while\nfor runs D all modes m∈[0,4] are present. As in previ-\nously discussed runs B the initial magnetic field b0is not\nin equilibrium, now with both terms in the equilibrium\nEquation (8) non-vanishing ( b0·∇j0∝negationslash= 0 and∂zj0∝negationslash= 0).\nRemarkably these initial conditions decay to equilibria\nwith structures similar to those of runs B (whose initial\nmagnetic fields included only the mode m= 0), as shown\nin Figure 9. Independently from the structure of the ini-\ntial magnetic field , and from the specific modes that it\nincludes, the final equilibrium is always quasi-invariant\nalongzwith parallel variation length-scale larger than\napproximately the loop length zℓ/greaterorsimilarLz. Its structure\nis elongated along the axial direction, a strong m= 0\nparallel mode is present, and the magnetic field inten-\nsity is smaller than the threshold value b/lessorsimilarℓB0/Lz.\nThis further confirms the analysis of the equilibria per-\nformed in Section 3.1, i.e., that the reduced MHD equi-\nlibria dynamically accessible in a line-tied configuration\nare elongated along the axial direction with a domi-\nnantm= 0 parallel mode. It also implies that non-linear dynamics can transfer energy from higher parallel\nmodes to the mode m= 0 even when this is not initially\npresent (runs C), a process that can be of interest also in\nperiodic turbulence (Alexakis 2011; Schekochihin et al.\n2012), and has been conjectured to play a role in the\ndynamics that lead to the acceleration of the solar wind\n(Dmitruk et al. 2001).\nIn contrast to runs B, with initial conditions invari-\nant alongz, that decay only for magnetic field intensi-\nties above the threshold b∼ℓB0/Lz, in runs C a decay\nis always observed independently from the intensity of\nthe initial magnetic field (Figure 7). Since the accessi-\nble equilibria are quasi-invariant along zand the initial\nmagnetic field of runs C does not contain the m= 0\nmode but only the m= 1 mode, then it always decays.\nThe asymptotic stage is reached when the mode m= 0\nhas been generated and excess energy in higher modes\ndissipated. The intensity of the relaxed magnetic field\ndepends on the ability of nonlinear dynamics to trans-\nfer energy among higher parallel modes and from these\nto them= 0 mode, but the longer decay timescales\nfor lower values of b0/B0are consistent with a decrease\nof the strength of nonlinear interactions (e.g., the eddy\nturnover time decreases ad tℓ∼2ℓ/b0, see Section 4.2).\nThe longer nonlinear timescales render the effect of a\nhigh-frequency resonant photospheric forcing similar in\nmany aspects to that of a constant photospheric veloc-\nity (see Section 4.2 for a more complete discussion). In\nfact if a photospheric velocity with a higher resonant fre-\nquencyνn=nνA/2 withn≥1 is applied at the bound-\nary, nonlinear terms can be neglected initially because\nfor low values of the magnetic field intensity bthe de-\ncay timescales are much longer than the linear growth\nof the magnetic field. A statistical steady state will fi-\nnally be obtained when the energy flux injected in the\nsystem at the boundary by photospheric motions is bal-\nanced by a similar energy flux from the large toward the\nsmall scales (to form current sheets), in similar fashion\nto the constant velocity case (Rappazzo et al. 2007).\nWhen initial conditions include higher parallel modes\nm∈[0,4] and them= 0 mode has the largest amplitude\n(runs D, Section 4.3) the dynamics are very similar to\nrunsBwith onlythe m= 0modein theinitialconditions\n(cf. Figures2and7). Theexcessenergyinhigherparallel\nmodes is either dissipated or transferred to the m= 0\nmode and the relaxed fields have structures similar to\nruns B and C (Figure 9).\nTheparallelvariationscale zℓ∼ℓB0/bintroducedhere\n(Equation [28]) can be interpreted as a critical length or\ntwist. In fact given the magnetic field intensity b, sig-\nnificant nonlinear dynamics will develop only if the loop\nlength is longer than the variation scale Lz/greaterorsimilarzℓ. On the\nother hand, fixed the loop length Lznonlinear dynamics\nwill develop only if the variation scale is smaller than the\nloop length zℓ/lessorsimilarLz, or equivalently if the field intensity\nislargerthanthethreshold b/greaterorsimilarℓB0/Lz, thatcorresponds\nto an average twist larger than ∝angb∇acketleftΦ∝angb∇acket∇ight ∼Lzb/(ℓB0)/greaterorsimilarπ/3\n(this is only an estimate, since the orthogonal field does\nnot have cylindrical symmetry b∝negationslash=b(r), the twist should\nbe computed numerically for sample field lines).\nThe concept of a critical length or twist has been\ndeveloped in the study of several linear instabilities\nin coronal loops with line-tied boundary conditions,\nincluding kink and tearing instabilities (Raadu 1972;20 Rappazzo\nHood & Priest 1979, 1981; Einaudi & van Hoven 1981,\n1983;Velli & Hood1989;Velli et al.1990;Foote & Craig\n1990; Lionello et al. 1998; Huang & Zweibel 2009;\nHuang et al. 2010). It is found that the system is lin-\nearly unstable for a fixed magnetic field intensity only if\nthe loop length is larger than a critical value (for kink\nand other instabilities a critical twist can be used equiv-\nalently).\nIt is important to remark that although it is useful\nto regard the parallel variation scale zℓ(Equation [28])\nas a critical length, and that an average critical twist\ncan be defined, the dynamics that they help describe\narenot linear instabilities . In the configurations of in-\nterest to this paper the critical length or twist distin-\nguishtwodifferentdynamicregimesinwhichrespectively\nfor small field intensities bnonlinear dynamics are sup-\npressedandtheevolutioncanberegardedasquasi-static,\nwhileforstrongerintensitiesnonlineardynamicsdevelop.\nOf course the boundary between these two regimes is not\nsharp, but gradually asbis increased the “2D” Lorentz\nforceb·∇bcannot be balanced by the axial field line ten-\nsionB0∂zb, leading the system out of equilibrium . In\nparticular all the 3D line-tied magnetic fields that we\nhave considered lack the symmetries of linearly unstable\nconfigurations, and for all of them the orthogonal com-\nponent of the magnetic field is not symmetric and its 2D\nLorentz force component (for which b·∇j∝negationslash= 0) does not\nvanish at all times, from the initial condition to the re-\nlaxed asymptotic stage, as can be seen in Figures 4-6, 8\nand 9.\nThe reason for which the equilibria with zℓ/greaterorsimilarLzare\nnot linearly unstable is that they are close to each other.\nTherefore adding a perturbation to the magnetic field\nsimply changes slightly the corresponding equilibrium to\nwhich the field readjusts. For instance the initial condi-\ntion of run B with b0/B0= 2% is not exactly an equilib-\nrium (since ∂zj0= 0 and b0·∇j0∝negationslash= 0), therefore it can\nbe regarded as an equilibrium to which has been added\na small perturbation. But as shown particularly well\nin Figure 8 (bottom panel) no instability of sort is de-\ntected, rather the field undergoes a slight readjustment.\nClearly for a non-vanishing orthogonal magnetic field b\nin equilibrium, with a progressively smaller 2D Lorentz\nforce component for which b· ∇j→0 (withb∝negationslash= 0),\nthe field approaches a symmetric configuration that can\nbe linearly unstable, since from the equilibrium Equa-\ntion (8) also ∂zj→0 in this limit. But as previously dis-\ncussed the configurations of interest here are those with\nnon-vanishing orthogonal 2D Lorentz force component\nb·∇j∝negationslash= 0.\nThe relaxation of coronal magnetic fields has of-\nten been studied subsequently to a linear instabil-\nity, mostly kink modes. Particularly for the cases\nthat have a strong axial magnetic field the structure\nof the lower energy relaxed field appears elongated\nalongz(Mikic et al. 1990; Longcope & Strauss 1994;\nBaty & Heyvaerts 1996; Velli et al. 1997; Lionello et al.\n1998; Baty 2000; Gerrard et al. 2002; Browning et al.\n2008; Hood et al. 2009). Early boundary forced sim-\nulations have been performed by Ng & Bhattacharjee\n(1998), with similar setup as those of Rappazzo et al.\n(2008). But due to their low resolution they misinterpret\nasaninstability the dynamicsthat developasthe thresh-\noldb∼ℓB0/Lzis crossed, when the system graduallytransitions from quasi-static evolution to turbulent dy-\nnamics. Fromthe simulations(runsB) and the equilibria\nanalysis it is clear that the forces that are approxima-\ntively in balance below the threshold become gradually\nunbalanced for larger magnetic field intensities, leading\nto thedevelopment of nonlinear dynamics with no in-\ntermediate instability as would occur for instance with a\nkink mode, where the nonlinear stage would follow the\nlinear instability . The non-vanishing 2D Lorentz force\ntermb· ∇b(for which b· ∇j∝negationslash= 0), that in the 2D\ncase (run A) sets the system out-of-equilibrium and de-\nvelops turbulent nonlinear dynamics, can be balanced\nin the 3D line-tied runs B by the axial field line ten-\nsion termB0∂zbfor field intensities below the threshold\nb/lessorsimilarℓB0/Lz. For larger magnetic field intensities the\n2D Lorentz force term is stronger than its axial com-\nponent, hence the dynamics develop progressively more\nakin to the 2D case. Ultimately a force balance cannot\nbe reached because the corresponding equilibrium is too\nasymmetric along zand therefore dynamically inacces-\nsible, so that for larger intensities ba larger fraction of\nenergy must be necessarily dissipated for the system to\nbe able to relax and access a new equilibrium.\nThis two-stage process then provides a fully self-\nconsistent alternative to coronal heating models based\non instabilities. For instance, since resistive instabili-\nties are slow for macroscopically thick magnetic shears,\nDahlburg et al. (2005, 2009) obtain a shear intensity\nthresholdfordissipationsupposingthatnanoflareswould\noccur when photospheric motions shear the magnetic\nfield beyond a certain angle, when a secondary ideal\ninstability (triggered by the slow primary resistive in-\nstability) can develop thus accelerating the dynamics.\nOn the other hand, as discussed in this paper, as\nphotospheric motions disorderly twist the field lines,\nonce the magnetic field intensity is higher than the\nthresholdb/greaterorsimilarℓB0/Lzmagnetically dominated turbu-\nlent dynamics develop, forming current sheets that thin\ndown to the dissipative scales on fast Alfv´ en time-scales\n(Rappazzo & Parker 2013), while triggering the “ideal”\ntearing instability (see Introduction; Pucci & Velli 2014;\nLandi et al. 2015; Tenerani et al. 2015), and leading\nto dynamics similar to so-called plasmoid instability\n(Bulanov et al.1978;Biskamp1986;Loureiro et al.2007;\nLapenta 2008; Bhattacharjee et al. 2009).\nFinally, these simulations of decaying magnetic fields\nshow that, beyond the intensity threshold [Equa-\ntion (28)], current sheets form on fast ideal timescales\nbecause of the nonlinear dynamics that develop . This is\nin stark contrast with the frequent hypothesis of quasi-\nstatic evolution of the coronal magnetic field subject to\nfootpoint shuffling, that should continuously relax to a\nnearby equilibrium without forming current sheets (e.g.,\nvan Ballegooijen 1985). In the quasi-static scenario the\ncorona could be heated by the uniformly distributed\nsmall-scale current sheets created by the shredding of\nthe coronal magnetic field after many successive random\nwalk steps of its field lines footpoints (van Ballegooijen\n1986). But this mechanism would lead to current sheet\nformation on timescale longer than photospheric con-\nvection (several random walk steps would be required).\nWhile the relaxation simulations presented in this pa-\nper and in Rappazzo & Parker (2013) show that current\nsheets form on ideal Alfv´ en timescales (much faster than21\nconvective timescales), with the footpoints fixed at the\nphotospheric plates where no motions are in place (and\ntherefore no footpoint random walk occurs).\nThe author thanks Gene Parker and Marco Velli for\nhelpful and insightful discussions, and the anonymous\nrefereeforhisremarks. This researchhasbeensupported\nin part by NASA through a subcontract with the Jet\nPropulsion Laboratory, California Institute of Technol-\nogy, and NASA LWS grant number NNX15AB88G and\nNNX15AB89G.Computationalresourcessupportingthis\nwork were provided by the NASA High-End Comput-\ning (HEC) Program through the NASA Advanced Su-\npercomputing (NAS) Division at Ames Research Center.\nREFERENCES\nAbramenko, V. I., Yurchyshyn, V. B., Goode, P. R., Kitiashvi li,\nI. N., & Kosovichev, A. G. 2012, ApJL, 756, L27\nAlexakis, A. 2011, Phys. Rev. E, 84, 056330\nAluie, H., & Eyink, G. L. 2010, Physical Review Letters, 104,\n081101\nAntiochos, S. K. 1987, ApJ, 312, 886\nBatchelor, G. K. 1969, Physics of Fluids, 12, 233\nBaty, H. 2000, A&A, 360, 345\nBaty, H., & Heyvaerts, J. 1996, A&A, 308, 935\nBeeck, B., Cameron, R. H., Reiners, A., & Sch¨ ussler, M. 2013 ,\nA&A, 558, A48\nBelmonte, A., Goldburg, W. I., Kellay, H., et al. 1999, Physi cs of\nFluids, 11, 1196\nBeresnyak, A. 2013, ArXiv e-prints, arXiv:1301.7424\nBertozzi, A. L., & Constantin, P. 1993, Communications in\nMathematical Physics, 152, 19\nBhattacharjee, A., Huang, Y.-M., Yang, H., & Rogers, B. 2009 ,\nPhysics of Plasmas, 16, 112102\nBirn, J., Drake, J. F., Shay, M. A., et al. 2001, J. Geophys. Re s.,\n106, 3715\nBiskamp, D. 1986, Physics of Fluids, 29, 1520\nBiskamp, D. 2003, Magnetohydrodynamic Turbulence\n(Cambridge: Cambridge University Press)\nBoffetta, G., & Ecke, R. E. 2012, Annual Review of Fluid\nMechanics, 44, 427\nBogoyavlenskij, O. I. 2000, Physical Review Letters, 84, 19 14\nBrachet, M. E., Bustamante, M. D., Krstulovic, G., et al. 201 3,\nPhys. Rev. E, 87, 013110\nBrowning, P. K., Gerrard, C., Hood, A. W., Kevis, R., & van der\nLinden, R. A. M. 2008, A&A, 485, 837\nBuchlin, E., & Velli, M. 2007, ApJ, 662, 701\nBulanov, S. V., Syrovatskii, S. I., & Sakai, J. 1978, ZhETF Pi sma\nRedaktsiiu, 28, 193\nCanuto, C., Hussaini, M. Y., Quarteroni, A., & Zang, T. A. 200 6,\nSpectral Methods (Berlin: Springer-Verlag),\ndoi:10.1007/978-3-540-30726-6\nChemin, J.-Y. 1993, Annales scientifiques de l’ ´Ecole Normale\nSup´ erieure, 26, 517\nChiuderi, C., & Velli, M. 2015, Basics of Plasma Astrophysic s\n(Berlin: Springer-Verlag)\nCowley, S. C., Longcope, D. W., & Sudan, R. N. 1997,\nPhys. Rep., 283, 227\nDahlburg, R. B., Einaudi, G., Rappazzo, A. F., & Velli, M. 201 2,\nA&A, 544, L20\nDahlburg, R. B., Einaudi, G., Taylor, B., et al. 2015, submit ted,\nApJ\nDahlburg, R. B., Klimchuk, J. A., & Antiochos, S. K. 2005, ApJ ,\n622, 1191\nDahlburg, R. B., Liu, J.-H., Klimchuk, J. A., & Nigro, G. 2009 ,\nApJ, 704, 1059\nDe Pontieu, B., McIntosh, S. W., Carlsson, M., et al. 2007,\nScience, 318, 1574\nDmitruk, P., & G´ omez, D. O. 1997, ApJL, 484, L83\n—. 1999, ApJL, 527, L63\nDmitruk, P., G´ omez, D. O., & DeLuca, E. E. 1998, ApJ, 505, 974\nDmitruk, P., G´ omez, D. O., & Matthaeus, W. H. 2003, Physics o f\nPlasmas, 10, 3584\nDmitruk, P., Matthaeus, W. H., Milano, L. J., & Oughton, S.\n2001, Physics of Plasmas, 8, 2377\nDonati, J.-F., & Landstreet, J. D. 2009, ARA&A, 47, 333\nDryden, H. L. 1943, Quarterly of Applied Mathematics, 1, 7\nEinaudi, G., & van Hoven, G. 1981, Physics of Fluids, 24, 1092\n—. 1983, Sol. Phys., 88, 163Einaudi, G., & Velli, M. 1999, Physics of Plasmas, 6, 4146\nEinaudi, G., Velli, M., Politano, H., & Pouquet, A. 1996, ApJ L,\n457, L113\nEuler, L. 1761, Novi Commentarii Acad. Sci. Petropolitanae , 6,\n271\nFavata, F., Flaccomio, E., Reale, F., et al. 2005, ApJS, 160, 469\nFoote, B. J., & Craig, I. J. D. 1990, ApJ, 350, 437\nFrisch, U., & Sulem, P. L. 1984, Physics of Fluids, 27, 1921\nGalsgaard, K., & Nordlund, ˚A. 1996, J. Geophys. Res., 101, 13445\nGaltier, S., Politano, H., & Pouquet, A. 1997, Physical Revi ew\nLetters, 79, 2807\nGeorgoulis, M. K., Velli, M., & Einaudi, G. 1998, ApJ, 497, 95 7\nGerrard, C. L., Arber, T. D., & Hood, A. W. 2002, A&A, 387, 687\nGetman, K. V., Feigelson, E. D., Micela, G., et al. 2008, ApJ,\n688, 437\nGharib, M., & Derango, P. 1989, Physica D Nonlinear\nPhenomena, 37, 406\nGold, T. 1964, NASA Special Publication, 50, 389\nGold, T., & Hoyle, F. 1960, MNRAS, 120, 89\nGreffier, O., Amarouchene, Y., & Kellay, H. 2002, Physical\nReview Letters, 88, 194101\nG¨ udel, M. 2004, A&A Rev., 12, 71\nG¨ udel, M. 2009, in Lecture Notes in Physics, Berlin Springe r\nVerlag, Vol. 778, Turbulence in Space Plasmas, ed. P. Cargil l &\nL. Vlahos, 269\nHasegawa, A. 1985, Advances in Physics, 34, 1\nHendrix, D. L., & van Hoven, G. 1996, ApJ, 467, 887\nHood, A. W., Browning, P. K., & van der Linden, R. A. M. 2009,\nA&A, 506, 913\nHood, A. W., & Priest, E. R. 1979, Sol. Phys., 64, 303\n—. 1981, Geophysical and Astrophysical Fluid Dynamics, 17, 297\nHossain, M., Matthaeus, W. H., & Montgomery, D. 1983, Journa l\nof Plasma Physics, 30, 479\nHossain, M., Gray, P. C., Pontius, Jr., D. H., Matthaeus, W. H .,\n& Oughton, S. 1995, Physics of Fluids, 7, 2886\nHuang, Y.-M., & Bhattacharjee, A. 2010, Physics of Plasmas, 17,\n062104\nHuang, Y.-M., Bhattacharjee, A., & Zweibel, E. G. 2010, Phys ics\nof Plasmas, 17, 055707\nHuang, Y.-M., & Zweibel, E. G. 2009, Physics of Plasmas, 16,\n042102\nIonson, J. A. 1985, A&A, 146, 199\nKadomtsev, B. B., & Pogutse, O. P. 1974, Soviet Journal of\nExperimental and Theoretical Physics, 38, 283\nKaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K., & Uno, A .\n2003, Physics of Fluids, 15, L21\nKlimchuk, J. A. 2006, Sol. Phys., 234, 41\nKolmogorov, A. 1941, Akademiia Nauk SSSR Doklady, 30, 301\nKraichnan, R. H. 1967, Physics of Fluids, 10, 1417\nKraichnan, R. H., & Montgomery, D. 1980, Reports on Progress\nin Physics, 43, 547\nLandi, S., Del Zanna, L., Papini, E., Pucci, F., & Velli, M. 20 15,\nApJ, 806, 131\nLapenta, G. 2008, Physical Review Letters, 100, 235001\nLazarian, A., & Vishniac, E. T. 1999, ApJ, 517, 700\nLionello, R., Velli, M., Einaudi, G., & Miki´ c, Z. 1998, ApJ, 494,\n840\nLongcope, D. W., & Strauss, H. R. 1993, Physics of Fluids B, 5,\n2858\n—. 1994, ApJ, 426, 742\nLoureiro, N. F., Schekochihin, A. A., & Cowley, S. C. 2007,\nPhysics of Plasmas, 14, 100703\nLoureiro, N. F., Uzdensky, D. A., Schekochihin, A. A., Cowle y,\nS. C., & Yousef, T. A. 2009, MNRAS, 399, L146\nLow, B. C. 2013, ApJ, 768, 7\n—. 2015, Science China Physics, Mechanics, and Astronomy, 5 8, 2\nLudwig, H.-G., Allard, F., & Hauschildt, P. H. 2002, A&A, 395 ,\n99\nMajda, A. J., & Bertozzi, A. L. 2001, Vorticity and\nIncompressible Flow (Cambridge: Cambridge University Pre ss)\nMartin, S. F., Panasenco, O., Berger, M. A., et al. 2012, in\nAstronomical Society of the Pacific Conference Series, Vol. 463,\nSecond ATST-EAST Meeting: Magnetic Fields from the\nPhotosphere to the Corona., ed. T. R. Rimmele, A. Tritschler ,\nF. W¨ oger, M. Collados Vera, H. Socas-Navarro,\nR. Schlichenmaier, M. Carlsson, T. Berger, A. Cadavid, P. R.\nGilbert, P. R. Goode, & M. Kn¨ olker, 157\nMatthaeus, W. H., & Montgomery, D. 1980, Annals of the New\nYork Academy of Sciences, 357, 203\nMatthaeus, W. H., Stribling, W. T., Martinez, D., Oughton, S ., &\nMontgomery, D. 1991, Physical Review Letters, 66, 2731\nMikic, Z., Schnack, D. D., & van Hoven, G. 1990, ApJ, 361, 690\nMilano, L. J., G´ omez, D. O., & Martens, P. C. H. 1997, ApJ, 490 ,\n442\nMininni, P. D., & Pouquet, A. 2013, Phys. Rev. E, 87, 033002\nMontgomery, D. 1982, Physica Scripta Volume T, 2, 83\nNg, C. S., & Bhattacharjee, A. 1998, Physics of Plasmas, 5, 40 2822 Rappazzo\nNigro, G., Malara, F., & Veltri, P. 2008, ApJ, 685, 606\nParker, E. N. 1957, J. Geophys. Res., 62, 509\n—. 1972, ApJ, 174, 499\n—. 1979, Cosmical magnetic fields: Their origin and their act ivity\n(New York: Oxford University Press)\n—. 1983, ApJ, 264, 635\n—. 1988, ApJ, 330, 474\n—. 1994, Spontaneous current sheets in magnetic fields (New\nYork: Oxford University Press)\n—. 2000, Physical Review Letters, 85, 4405\n—. 2012, Plasma Physics and Controlled Fusion, 54, 124028\nPearson, B. R., Krogstad, P.- ˚A., & van de Water, W. 2002,\nPhysics of Fluids, 14, 1288\nPeter, H., Bingert, S., Klimchuk, J. A., et al. 2013, A&A, 556 ,\nA104\nPeterson, W. M., Mutel, R. L., G¨ udel, M., & Goss, W. M. 2010,\nNature, 463, 207\nPriest, E. 2014, Magnetohydrodynamics of the Sun (Cambridg e:\nCambridge University Press)\nPucci, F., & Velli, M. 2014, ApJL, 780, L19\nRaadu, M. A. 1972, Sol. Phys., 22, 425\nRappazzo, A. F. 2006, PhD thesis, Universit` a di Pisa\nRappazzo, A. F., & Parker, E. N. 2013, ApJL, 773, L2\nRappazzo, A. F., & Velli, M. 2011, Phys. Rev. E, 83, 065401\nRappazzo, A. F., Velli, M., Einaudi, G., & Dahlburg, R. B. 200 7,\nApJL, 657, L47\n—. 2008, ApJ, 677, 1348\nReale, F. 2014, Living Reviews in Solar Physics, 11, 4\nReiners, A. 2012, Living Reviews in Solar Physics, 9, 1\nRivera, M. K., Daniel, W. B., Chen, S. Y., & Ecke, R. E. 2003,\nPhysical Review Letters, 90, 104502\nRose, H. A., & Sulem, P. L. 1978, Journal de Physique France,\n39, 441\nRosner, R., & Knobloch, E. 1982, ApJ, 262, 349Schekochihin, A. A., Nazarenko, S. V., & Yousef, T. A. 2012,\nPhys. Rev. E, 85, 036406\nShay, M. A., Drake, J. F., Rogers, B. N., & Denton, R. E. 1999,\nGeophys. Res. Lett., 26, 2163\nShibata, K., & Magara, T. 2011, Living Reviews in Solar Physi cs,\n8, 6\nSommeria, J. 1986, Journal of Fluid Mechanics, 170, 139\nSreenivasan, K. R. 1984, Physics of Fluids, 27, 1048\n—. 1998, Physics of Fluids, 10, 528\nStrauss, H. R. 1976, Physics of Fluids, 19, 134\nSturrock, P. A., & Uchida, Y. 1981, ApJ, 246, 331\nSweet, P. A. 1958, in IAU Symposium, Vol. 6, Electromagnetic\nPhenomena in Cosmical Physics, ed. B. Lehnert, 123\nTabeling, P. 2002, Phys. Rep., 362, 1\nTaylor, G. I. 1935, Royal Society of London Proceedings Seri es A,\n151, 421\nTaylor, J. B. 1986, Reviews of Modern Physics, 58, 741\nTenerani, A., Rappazzo, A. F., Velli, M., & Pucci, F. 2015, Ap J,\n801, 145\nUzdensky, D. A., Loureiro, N. F., & Schekochihin, A. A. 2010,\nPhysical Review Letters, 105, 235002\nvan Ballegooijen, A. A. 1985, ApJ, 298, 421\n—. 1986, ApJ, 311, 1001\nVelli, M., & Hood, A. W. 1989, Sol. Phys., 119, 107\nVelli, M., Hood, A. W., & Einaudi, G. 1990, ApJ, 350, 428\nVelli, M., Lionello, R., & Einaudi, G. 1997, Sol. Phys., 172, 257\nWan, M., Oughton, S., Servidio, S., & Matthaeus, W. H. 2009,\nPhysics of Plasmas, 16, 080703\nWilmot-Smith, A. L., Hornig, G., & Pontin, D. I. 2009, ApJ, 69 6,\n1339\nWilmot-Smith, A. L., Pontin, D. I., Yeates, A. R., & Hornig, G .\n2011, A&A, 536, A67\nWithbroe, G. L., & Noyes, R. W. 1977, ARA&A, 15, 363\nZank, G. P., & Matthaeus, W. H. 1992, Journal of Plasma\nPhysics, 48, 85"    },    {        "title": "0903.2416v1.Spin_torque_driven_magnetization_dynamics_in_a_nanocontact_setup_for_low_external_fields__numerical_simulation_study.pdf",        "content": "Spin-torque driven magnetization dynamics in a nanocontact setup for low \nexternal fields: numerical simulation study  \n \nD.V. Berkov, N.L. Gorn \nInnovent Technology Development e.V., \nPruessingstr. 27B, D-07745 Jena, Germany \n \n \nAbstract \nWe present numerical simulation studies of the steady-state magnetization dynamics driven \nby a spin-polarized current in a point contact geometry for the case of a relatively large con-\ntact diameter (D = 80 nm) and small external field (H = 30 Oe). We show, that under these \nconditions the magnetization dynamics is qualitatively different from the dynamics observed \nfor small contacts in large external fields. In particular, the ‘bullet’ mode with a homogeneous \nmode core, which was the dominating localized mode for small contacts, is not found here. \nInstead, all localized oscillation modes observed in simulations correspond to different motion \nkinds of vortex-antivortex (V-AV) pairs. These kinds include rotational and translational \nmotion of pairs with the V-AV distance d ~ D and creation/annihilation of much smaller \n(satellite) V-AV pairs. We also show that for the geometry studied here the Oersted field has a \nqualitative effect on the magnetization dynamics of a 'free' layer. This effect offers a possibi-\nlity to control magnetization dynamics by a suitable electric contact setup, optimized to pro-\nduce a desired Oersted field. Finally, we demonstrate that when the magnetization dynamics \nof the 'fixed' layer (induced only by the stray field interaction with the 'free' layer) is taken \ninto account, the threshold current for the oscillation onset is drastically reduced and new \ntypes of localized modes appear. In conclusion, we show that our simulations reproduce \nsemiquantitatively several important features of the magnetization dynamics in a point contact \nsystem for low external fields reported experimentally. \n \n \n \nPACS numbers: 85.75.-d, 75.75.+a, 75.40.Gb, 75.40.Mg  \n \n  2 I. INTRODUCTION \nMagnetization dynamics induced in thin multilayer elements by a spin polarized current \n(SPC) is currently one of the most intensively studied topics in the solid state magnetism. \nAfter theoretical predictions [1, 2]  and first experimental confirmations of this phenomenon \n[3] it was quickly realized that SPC-induced magnetization excitations and switching \nrepresent not only a very interesting phenomenon from the fundamental point of view (see \nreview papers [4, 5, 6, 7], but is also are a very promising candidate for numerous device \napplications (for recent reviews see [8, 9, 10, 11]). \nAmong various experimental geometries used to study the spin torque induced magnetization \ndynamics, the so called point contact setup (where the current is injected into a multilayer ele-\nment with lateral sizes in the mkm-region via a contact with the diameter Dc ~ 10 - 100 nm) is \none of the most interesting designs: It offers a large variety of magnetization oscillation \nmodes, depending on the contact size, applied field strength and direction and magnetic \nmaterials used to compose a multilayer element [12, 13, 14, 15]. Moreover, one of the desired \napplications of the SPC-induced dynamics is the construction of dc-fed microwave \ngenerators, which should output enough power to be applicable in real technical devices. For \nthis application the point contact setup is especially interesting: there exist - at least in \nprinciple - a possibility to synchronize magnetization oscillations induced by several point \ncontacts attached to the same multilayer [16, 17], thus greatly increasing the output power due \nto the constructive wave interference.  \nFor all these reasons magnetization dynamics in point contact devices has been the subject of \nan intensive research during the last few years. Already the first reliable experimental obser-\nvation of magnetization oscillation in this geometry [12] posed an intriguing question about \nthe nature of the observed oscillation mode. Namely, the measured oscillation frequency, \nbeing below the homogeneous FMR frequency for the magnetic layer studied in [12], could \nnot correspond to the propagating wave mode predicted for such devices by Slonczewski [18]. \nNumerical simulations have shown that in such systems at least one localized mode type \ncould exist [19] (in addition to the propagating wave), which was independently identified by \nthe analytical theory [20] as a non-linear ‘bullet’. Further theoretical and numerical studies \nhave proven [21, 22] that this ‘bullet’ was indeed the mode observed in the pioneering paper \n[12].  \nDetailed numerical simulations [22, 23] have suggested, that a much more complicated \nlocalized modes, consisting of vortex-antivortex pairs, can exist in point contact devices. The \nfrequency of these modes, according to our simulation predictions, should be much lower than \nfor the ‘bullet’ mode (not to mention the propagating wave mode), and, should be also nearly \ncurrent-independent [22]. Both features would make these modes very interesting from the \npoint of view of technical application, because such modes would expand the frequency range \nof SPC-based microwave generators and offer a stability of the generated frequency with \nrespect to current strength fluctuations.  Further experimental studies have indeed shown, that \nmagnetization oscillations observed in the nanocontact setup in very weak in-plane [14, 24] or \nout-of-plane fields [14, 25] can not correspond to the ‘bullet’ mode. Due to their very low \nfrequency f ~ 100 - 500 MHz (what is really low for the SPC-induced dynamics), the oscilla-\ntions observed in Ref. [14, 25] were attributed to magnetization dynamics governed by the \nmotion of a single vortex (for a strong out-of-plane field in [25]) or vortex-antivortex pairs (in \nvery weak fields [24, 26]). \nEspecially interesting in this context are magnetization oscillations observed in point-contact \ndevices with relatively large contact diameter (~ 100 nm) and in weak external fields. In such \nconditions, due to the absence of the stabilizing influence of the external field and relatively \nlarge area flooded by a spin polarized current, strongly inhomogenous magnetization excita-\ntions can occur. Indeed, a qualitatively new oscillation mode was found experimentally in \n[14], where for a point contact with the diameter \u0001 60 - 80 nm, attached to an extended  3 Co90Fe10/Cu/Ni80Fe20 multilayer, microwave oscillations with extremely low frequency (down \nto ~ 100 MHz) and very weak frequency dependence on the current strength were observed. It \nwas suggested that the observed dynamics can be explained by the generation and movement \nof a magnetic vortex, but no supporting simulations or analytical theory were reported in [14].  \nFor all the reasons explained above, we have performed systematic numerical studies of the \nSPC-induced magnetization excitations for the case, when a point contact with a relatively \nlarge diameter is attached to an extended multilayer and the applied field is very weak (some \npreliminary results of this research have been briefly reported in the last part of our overview \n[26]). The paper is organized as follows. In Sec. II we describe in detail the simulation \nmethodology, geometry and magnetic parameters of the simulated system. Sec. III contains \nthe description of our main results: we start with the single layer system without the Oersted \nfield (subsection III.A), proceed with the demonstration of the Oersted field effects (part \nIII.B) and finish our presentation with the analysis of magnetodipolar interaction effects, \nwhen the ‘hard’ magnetic layer is included into the simulated system. Sec. IV contains the \ncomparison of our results with experimental data and numerical simulations of other groups. \n \nII. SIMULATED SYSTEM AND SIMULATION METHODOLOGY \nAs mentioned in the Introduction, in this paper we intend to study the influence of various \nphysical factors on the magnetization precession induced in the point contact geometry by a \nspin-polarized current (SPC) flowing perpendicular to the magnetic multilayer plane. In \ncontrast to most previous numerical simulation studies [21, 22, 23, 25, 27], we focus our \nattention on the case where (i) the diameter of the point contact Dc is relatively large and (ii) \nthe external field Hext is small compared to the saturation magnetization of the material. As \nexpected from general arguments and confirmed by recent experiments of the NIST group \n[14], for this case one can expect a qualitatively different magnetization dynamics as \ncompared to systems with  smaller point contacts and high external fields. In particular, both \nthe larger value of the contact diameter and the smaller strength of the external field should \nallow for more complicated magnetization configurations, leading to an even richer set of \nnon-linear localized modes than that reported in [21, 22, 25]. For this reason and keeping in \nmind at least a qualitative comparison with the experiments performed in [14], we have \nchosen the value Dc = 80 nm for the contact diameter and Hext = 30 Oe for all result sets \npresented below. \nThis relatively large value of Dc allowed us to choose larger lateral size of the discretization \ncell than in Ref. [22]: in the present study we use the mesh with the in-plane cell size 5 x 5 \nnm2. In order to understand the influence of the interlayer interaction on the SPC-induced \nmagnetization dynamics, we have studied both a single layer system and the complete \nmagnetic trilayer consisting of materials as used in [14]. For the single layer system magnetic \nparameters corresponding to Ni80Fe20 (Permalloy) (saturation magnetization MS = 640 G, \nexchange stiffness constant A = 1\u000210-6 erg/cm, negligible magnetocrystalline anisotropy) and \nthe layer thickness hfree = 5 nm were used. For the trilayer system we have adopted the \nparameters of the ’fixed’ (hard) layer as for Co90Fe10, namely MS = 1500 G [28], A = 2\u000210-6 \nerg/cm (various sources give the values of the CoFe exchange stiffness in the range A = (1 - 3) \n\u000210-6 erg/cm), the layer thickness hfix = 20 nm and the spacer thickness (distance between two \nmagnetic layers) hCu = 4 nm [14]. Both magnetic layers were not discretized further into \nsublayers; it was checked that such a discretization, leading to a large increase of the \ncomputational time, did not significantly affect the results.  \nIn order to avoid the artificial influence of the system borders, we have used periodic bounda-\nry conditions (PBC). To suppress the spin wave propagation between different PBC-replica, \nwe employ the damping parameter which increased towards the simulated area borders as \ndescribed in [22, 23]. In the present study the spatially dependent damping coefficient was \nchosen in the form [ ] 0 0 dec () 1tanh(( )/ ) r rR R l l l = +D×+ - , where r denotes the distance to  4 the point contact center, l0 = 0.02 is the ‘native’ material damping constant, R0 = 1000 nm, \nRdec = 100 nm and the damping increase parameter is Dl = 1.0. It has turned out that the usage \nof this damping profile and the lateral size of the simulated area L x L = 2000 x 2000 nm2 is \nsufficient to fully suppress the above mentioned artificial spin wave interference arising due \nto PBC.   \nWe have also studied the influence of the Oersted field of the dc-current flowing within the \ncontact area (see Sec. III.B). Unfortunately, the electric current distribution in the \nexperimental setup is not known exactly, so we could not compute the corresponding field \ndirectly from this distribution. For this reason, we had to adopt another strategy to study the \nOersted field effect, which is explained at the beginning of Sec. III.B. \nThe magnetization dynamics itself was simulated using our commercially available \nmicromagnetic package (see [29] for implementation details) with the extensions allowing to \nuse (i) the site-dependent damping constant as explained above and (ii) the site-dependent \ncurrent density in order to mimic the current flowing through the point contact area only. \nThermal fluctuations were neglected (T = 0). Spin torque acting on the free layer only was \nincluded by adding the Slonczewski torque term st J[ [ ]] a G= ´ ´ M Mp to the ‘normal’ \nLandau-Lifshitz-Gilbert (LLG) equation. The amplitude of this torque aJ is proportional to \ncurrent strength I and its spin polarization degree P and depends also on the magnetic layer \nthickness h, the contact area 2\nc c S Rp= , and magnetization MS (see, e.g., [30]): \nJ 2\nS c 2||IPa\neM hS××=\n××\u0001      (1) \n The spin polarization direction of electrons p in the dc-current flowing through the contact \narea was chosen to be opposite to the applied field direction Hext for a single layer system and \nopposite to the local instantaneous magnetization direction of the fixed layer for a trilayer \nsystem. The reason for this choice is the following: the magnetization of a free layer in real \nexperiments is supposed to be excited by spin-polarized electrons reflected from the fixed \nmagnetic layer (of a trilayer system) towards the free one.  \n \nIII. RESULTS AND DISCUSSION \n \nA. Magnetization dynamics of the single-layer system in the absence of the Oersted field \nIn order to understand the influence of various physical factors on the magnetization dyna-\nmics separately, we proceed in a usual way, ‘switching on’ these factors in turn, thus isolating \ncorresponding effects. Hence we start our study with the simplest system including the free \nmagnetic layer only and neglecting the Oersted field of the spin-polarized current. \nOscillation power spectra for this system (Py layer with the thickness hPy = 5 nm, subject to a \nspin-polarized current flowing through the point contact  with the diameter Dc = 80 nm, \nplaced into the in-plane external field Hext = 30 Oe) are shown in Fig. 1. This figure displays \nthe oscillation spectra of the mz magnetization component, whereby the x-axis is chosen along \nthe external field direction, and the z-axis is directed in the film plane perpendicular to Hext.  \n1. Propagating wave mode \nThe first mode observed after the oscillation onset is the ‘normal’ propagating wave mode W1 \n(Slonczewski mode) predicted in [18]; the index in the notation W1 is necessary to distinguish \nthis mode in the single-layer system from analogous propagating wave modes in other \nsystems considered below. It can be seen that this mode exists in the relatively broad current \nrange (IW \u0001 14 - 20 mA) and its frequency decreases continuously with growing current (we \npoint out for clarity that the discrete character of the f(I)-dependence for this mode seen in  5 Fig. 1 is an image artifact arising due to the discrete set of currents used in simulations and \nrapid decrease of the frequency with current). \nI, mAf, GHz\nL1\nL2W(1)\n \nFig. 1. Spectral power of the mz-component (in-plane component perpendicular to the external field) as \nthe function of the current strength in the single layer system without the Oersted field. Arrows indicate \nthe position of modes analyzed below in more detail (see Fig. 2, 3, 4)  The discrete character of the f(I) \ndependence of the W1 mode is an image artifact due to the discrete set of simulated currents and rapid \nfrequency decrease with current for this mode. \nThe threshold current value for the oscillations onset is Ith \u0001 14 mA and the corresponding \nthreshold oscillation frequency is fth \u0001 7.05 GHz. It is instructive to compare these values to \nthe analytical calculations for the Slonczewski mode [18, 20]. We remind that these \ncalculations predict the threshold current  \n 0\nth 0 2\nc1.86()()DHI H\nRs×» +G  (2) \nwhere the factor B S c /(2|| ) gP eMhS s m= ×× [20] is related to our spin torque amplitude aJ as \nS J I M a s g= ×. The first term in (2) describes the (dominant) energy loss due to spin-wave \nemission by the point contact area and is proportional to the spin wave dispersion D(H0). For \nthe field-in-plane geometry this dispersion reads  \n0 S\n0\nS 0 0 S2 2()\n( 4 )H M ADHM HH Mp g\np+= ×\n+    (3) \nThe energy losses within the point contact area due to the Gilbert damping are given by the \nsecond term in (2): 0 0 S () ( 2 ) H H M gl p G =× + ; for the situation studied here these damping \nlosses are much smaller than those due to first term.  \nFor the system geometry (h = 5 nm, Rc = 40 nm), material parameters (MS = 640 G, A = 1\u000210-6 \nerg/cm, l = 0.02), current polarization P = 0.4 and external field H0 = 30 Oe used in our \nsimulations, the threshold current value calculated from this analytical theory is an\nth37 I»  mA. \nThis result is about 2.5 times larger than the simulated value (sim\nth14 I» mA).  6 Analytical prediction for the oscillation frequency [31] an 2\nth 0 0Dk w w= +  contains the \nhomogeneous FMR frequency, which for the in-plane-field is 0 0 0 S ( 4 ) HH M w p = + and the \nwave vector of the excited circular spin wave, which was found to be [18, 20] k0 = 1.2/Rc. In \ncontrast to the large discrepancy for the threshold currents, this analytical result leads to the \noscillation frequency an an\nth th/2 7.85 f w p = »  GHz, the value only slightly above the frequency \nsim\nth 7.05 f=  GHz  observed numerically. \nA possible reason for the large disagreement between the analytically predicted and \nnumerically simulated threshold currents could be the approximations made by the derivation \nof Eq. (2) for an\nthI: first, it was obtained for the perpendicularly magnetized point contact, and \nsecond, it was assumed that the group velocity of emitted spin waves is isotropic with respect \nto the propagation direction. The partial adjustment of the Eq. (2) to our case of an in-plane \nmagnetized contact could be achieved by using the expression (3) for the spin wave dispersi-\non in case of an in-plane magnetization. However, the dependence of the spin wave velocity \non the propagation direction could not be taken into account. We note that in our case the \npoint contact diameter is relatively large, so that the wave vectors of the emitted spin waves k \n~ 1/Rc are relatively small. Hence the wave group velocity substantially depends on the angle \nbetween the wave propagation direction and external field (see gray-scale maps in Fig. 2). \nThis high anisotropy of the group velocity could lead to significant discrepancies between \nanalytical calculations and numerical data.  \nAt the same time the expression for threshold frequency an 2\nth 0 0Dk w w= +  uses - besides the in-\nplane spin wave dispersion factor D - only the wave vector k0 ~ 1/Rc, which exact value relies \nmainly on the geometrical consideration (circular shape of the current-flooded area). Hence \nthe analytical prediction for the oscillation frequency should be more reliable, leading to a \nmuch better agreement between analytical theory and numerical simulations, as found above. \nThese our arguments are supported, in particular, by analogous comparisons for the nanocon-\ntact with a smaller radius (Rc = 20 nm) in [21, 22], where the anisotropy of the group velocity \nwas much lower due to larger spin wave vectors. The agreement between analytically \nobtained and numerically simulated threshold currents for this case was, indeed,  decisively \nbetter (see [21, 22] for details). \nWith increasing current the propagating wave mode demonstrates the strong downward \nfrequency shift due to the growing oscillation amplitude. The frequency decreases from its \ninitial value fth \u0001 7 GHz at Ith \u0001 14 mA to f \u0001 5 GHz reached for the current I \u0001 20 mA, where \nthe transition to localized modes occurs (see below). This frequency decrease with the \ngrowing current is nearly linear (we remind, that the jumps on the oscillation power plot in \nFig. 1 are solely due to the discrete set of the current values used in simulations). This \nfrequency decrease with increasing current due to the growing oscillation amplitude is a non-\nlinear effect which is well understood theoretically [20, 32] and hence will not be further \ndiscussed here. \nConcluding this discussion of the propagating wave mode, we would like to emphasize the \nmodulation of the main wave profile by a ‘secondary’ wave with the vector k \u0001 2k0 \napproximately twice as large as the ‘main’ wave vector k0, which can be clearly seen at the \ngray-scale maps displayed in Fig. 2. This modulation is due to the another non-linear effect \narising due to the conservation of the local moment magnitude 2 2 2 2\nS x y z M M M M + + = . This \ncondition leads to the contribution of the spin wave with the frequency corresponding to Mx-\noscillations to the wave pattern of the Mz-projection. The effect becomes more pronounced \nwith the increase of the magnetization oscillation amplitude; the picture shown in Fig. 2 \ncorresponds to I = 16 mA, where the oscillation amplitude of the Mz-projection under the \npoint contact is close to its maximal value max max\nS/ 1z z m M M = ».  7 t, ns\n2.0 2.2 2.4 2.6 2.8 3.0-1.0-0.50.00.51.0\n mX(t)\nmZ(t)\n2 x 2 mkm\n \nFig. 2. Magnetization time dependencies and snapshots of the magnetization configurations as gray-scale \nimages of  the mz-component for the propagating (Slonczewski) mode W1 in the single layer system when \nthe Oersted field is neglected. \n2. Localized modes \nWhen the current strength exceeds the next critical value loc\nth20 I»  mA, a large frequency \njump down to f \u0001 1.2 GHz occurs. This frequency is well below the homogeneous FMR \nfrequency 0 0 0 S (/2) ( 4 )1.4 f HH M gp p = + »  GHz for the film studied here, so that magne-\ntization oscillations after the jump (above loc\nthI) should correspond to a localized mode. The \nspatiotemporal analysis of magnetization configurations reveals that all the modes occurring \nfor loc\nth II>  are indeed localized. Dynamical processes responsible for observed magnetization \noscillations (see figures below) are qualitatively different for various modes. In many cases \nthese processes are also highly irregular, so we describe and discuss below only those \nlocalized modes which are generated by relatively simple magnetization dynamics.  \nMagnetization oscillations for the first such mode, which appears after the transition from the \npropagating wave mode W1 to localized oscillations, are shown in Fig. 3. First of all we point \nout, that this first localized mode (L1-mode) has a completely different nature compared to the \nspin-wave ‘bullet’ observed in systems with the relatively small point contact diameter (Rc = \n20 nm) [20, 21, 22, 23]. We remind, that this localized ‘bullet’ mode has a relatively \nhomogenous core magnetization structure, whereby the magnetization oscillation amplitude \ndecreases exponentially with the distance from the contact center. In contrast to the ‘bullet’ \nmode, in our system the magnetization structure of the 1st localized mode is highly \ninhomogeneous: magnetization oscillations are caused by an appearance and rotation of a \nvortex-antivortex pair. This qualitative difference between the two systems studied here and \nin [20, 21, 22] is due to the small contact diameter Rc = 20 nm and the large external field Hext \n= 2 kOe used in the paers cited above: both these factors strongly favor a homogeneous \nmagnetization configuration of the localized mode core found in [20, 21, 22, 23]. In our case, \nwhere the point contact diameter is twice as large and the external field is nearly absent (Hext \n= 30 Oe), a formation of more complicated excitations - vortex-antivortex (V-AV) pairs - \nbecomes possible.  \n  8 t, ns\n10 11 12 13 14 15 16mX(t)\n-0.4-0.20.00.20.4\nt, ns\n10 11 12 13 14 15 16mZ(t)\n-0.4-0.20.00.20.4a\ncc\n \nb\nc400x 400nm2\n400x 400nm2DDDDt= 0.1nsDDDDt= 0.15ns\n \nFig. 3. Magnetization time dependencies (a) and snapshots of the magnetization configurations ((b) and (c)) \nfor the first localized mode L1 in the single layer system for HOe = 0. Panel (b) shows the rotation process of \nthe main vortex-antivortex pair (images of the in-plane magnetization orientations) together with the arrow \nplot of a typical V-AV configuration. Panel (c) illustrates the creation, propagation and decay of a satellite V-\nAV pair (gray-scale images of the out-of-plane magnetization orientations my ). \nAs it can be seen from magnetization maps and arrow plots presented in Fig. 3, two major \nprocesses contribute to the magnetization dynamics of the L1-mode in our system: (i) rotation \nof a vortex-antivortex pair with a relatively large V-AV distance (Fig. 3b) and (ii) generation \nand subsequent translational motion of a much smaller V-AV (Fig. 3c). \nTo start the analysis of these two processes, we first point out, that due to the periodic boun-\ndary conditions applied to the system and the homogeneous starting magnetization state, mag-\nnetic excitations with non-zero vorticity should appear in pairs in order to guarantee the con-\nservation of the total topological charge [33, 34]. This situation is qualitatively different from  9 those observed for finite small nanoelements (like nanodisks studied numerically in [25]), \nwhere due to the open boundaries and the out-of-plane external field the formation of a single \nvortex was possible. \nBoth rotational and translational motions of a V-AV pair mentioned above have been studied \nby Komineas and Papanicolaou [34, 35]. They have shown that a V-AV pair should rotate \nwhen vortex and antivortex have opposite polarities and undergo a translational motion when \nthe polarities of a vortex and antivortex are the same. Results of our simulations agree with \nthis analytical statement: as it can be seen from the gray-scale maps in Fig. 3c, polarities of \nvortex and antivortex are opposite for the large rotating V-AV pair, but their polarities \ncoincide within the small translationally moving V-AV pair.  \nThe next step is obviously the quantitative comparison of the rotation frequency and the \ntranslational velocity of a V-AV pair predicted in [34, 35] with our simulation results. In [34] \nit was shown, that a V-AV pair with vortex and antivortex having opposite polarities \npossesses a zero linear momentum (so that the pair center does not move), but a non-zero \nangular momentum L, so that such a pair should rotate (L increases as ~ d2 with the V-AV \nseparation distance d). The angular velocity of this rotation wwww  can be determined from the \ncondition, that for this velocity the extended energy functional F = E - wwww L in the rotating \ncoordinate system should possess a stationary point. The total magnetic energy of a V-AV \npair in this functional E = Eex + Ean consists - in the approximations used in [34] (no external \nfield, large V-AV separation) - from the sum of exchange Eex and the anisotropy Ean energies \nof the vortex and antivortex. In the notation of Komineas Ean includes also the demagnetizing \nenergy in the thin-film geometry.  \nThe form of the extended energy functional F together with the scaling arguments similar to \nthose used in the nonlinear dynamics of continuous media enables the determination of the \nrotation frequency f of the V-AV dipole. Taking into account that the explanation given in the \nPRL-paper [34] was necessarily very brief and that scaling arguments mentioned above are \nnot commonly familiar to the micromagnetic community, we present the basic line of these \narguments here in order to make our paper self-contained. The main idea is, that the stationary \npoint of the extended energy functional F = E – wwww L should ‚survive’ the rescaling of spatial \nvariables, which by itself does not change the physics of the system. In 2D systems (thin film \nlimit) the exchange energy Eex is invariant with respect to such a rescaling, because Eex is a \n2D integral over a square of the magnetization gradient. In contrast, Ean is not invariant with \nrespect to this transformation, being in the simplest case a 2D integral over the 2nd power of \nthe magnetization components themselves. Hence the energy functional F = E – wwww L = Eex + \nEan – wwww L may have a stationary point which is stable with respect to rescaling only when Ean \n– wwww L = 0. Using this relation together with the expressions for the vortex anisotropy energy \nEan and the dependence of the angular momentum L on the V-AV separation d, Komineas \n[34] derived the rotation frequency for the V-AV pair, which in non-reduced units reads \n 2\nS22Af\nMdg\np=××       (4) \nHere g is the gyromagnetic ratio, A - exchange stiffness and MS - saturation magnetization of \nthe material. An additional factor of 2 (compared to the Eq. (11) in [34]) takes into account \nthe magnetostatic interaction between vortex and antivortex, as discussed in the concluding \npart of [34]. \nSubstituting into Eq. (4) magnetic parameter values used in our simulations (MS = 640 G, A = \n1\u000210-6 erg/cm) and the average value of the vortex-antivortex distance d \u0001 75 nm determined \nfrom the simulated magnetization configuration like those shown in Fig. 3, we obtain the \nanalytical value of the rotation frequency an\nrot0.62 f»  GHz.   10 The rotation period of the V-AV pair determined from the inspection of the simulated magne-\ntization configurations turns out to be Trot \u0001 1.60 ns, which results in the simulated value of \nthe rotation frequency sim\nrot0.62 f» GHz. Such an excellent coincidence between analytically \ncomputed and simulated values of f is somewhat unexpected for the case studied here: neither \nthe deformation of vortex and antivortex structures due to the V-AV interaction, nor the \nchanging of the V-AV separation during the simulated pair rotation due to the presence of an \nexternal field, nor the process of the formation and emission of the 2nd V-AV pair (see below) \nare included into the analytical theory. So such a good agreement between simulations and \nanalytical theory means, that (i) the approximation of a large V-AV separation adopted in [34] \nworks in our case fairly well, (ii) the external field is weak enough not to disturb the free V-\nAV pair rotation and (iii) the process of a generation and emission of the satellite V-AV pair \nis very fast compared to the rotation period of the main V-AV pair. \nNow we turn our attention to the second type of the V-AV pair dynamics - translational moti-\non, also observed for the 1st localized mode L1. Fig. 3c demonstrates that during the rotation \nof the main (large) V-AV pair another (much smaller) V-AV pair consisting of a vortex and \nantivortex with the same polarities is generated. Such a composite object has zero angular \nmomentum (so that such a pair does not rotate) [35], but a non-zero linear momentum P, \nwhich results in a translational motion of this V-AV pair (the so called Kelvin motion, well \nknown from the fluid dynamics). This translational motion can be clearly seen in Fig. 3c and \nits velocity can be compared to the analytical result from [35]. \nAnalytical estimation made in [35] is based for the translational V-AV motion on the \nextended energy functional F = E – vP in a translationally moving coordinate system, where v \nis the linear velocity of the V-AV pair. The same scaling arguments concerning the stability \nof the stationary point of this functional with respect to the rescaling of spatial variables apply \nto this case also. Together with the expression of the anisotropy energy for the V-AV pair \nthese arguments lead to the following estimation for the pair velocity in the limit of large V-\nAV separations: \n \nS22AvMdg×\u0002        (5) \n(here an additional factor of 2 results from the same V-AV interaction as explained in the text \nafter the Eq. (4)). Substituting the same material parameter values and the V-AV separation d \n\u0001 25 nm determined from the simulated magnetization configuration shown in Fig. 3c into the \nEq. (5), we obtain the analytical estimation van ~ 4.4\u0002104 cm/s. The pair velocity for this \nsystem, measured from simulated magnetization configurations, is vsim \u0001 6.3\u0002104 cm/s. The \nsignificant difference between analytically calculated and simulated values of this V-AV pair \nis most probably due to the fact that for such closely placed vortex and antivortex, the \ndeformation of their structures due to their mutual interaction plays a noticeable role (the V-\nAV separation is only about several exchange lengths 2\nex S (/2 )6.23 l A Mp = »  nm). \nAn important circumstance is that this V-AV pair is gradually destroyed during its \ntranslational motion due to the presence of the finite energy dissipation, which could not be \ntaken into account by the analytical theory [35]. This gradual decay is in contrast to the \nsteady-state rotation of the large V-AV pair with the opposite V-AV polarities, which is due \nto the constant energy supply via the spin-polarized current. We also note that the formation \nand emission of this small V-AV pair is an important mechanism of the magnetic energy \nirradiation out of the point contact area. \nWhen the current is increased further (I > 26 mA, see Fig. 1), a variety of more complicated \nand partially irregular localized modes appear. The overall trend is the increase of the number \nof V-AV pairs generated and annihilated per unit time. We postpone the detailed discussion of  11 the intriguing magnetization dynamics in this current region to future publications and discuss \nonly the regular localized mode L2, observed for high current values (40-50 mA).  \nt, ns\n5 6 7 8 9 10mX(t)\n-0.8-0.40.0\nt, ns\n5 6 7 8 9 10mZ(t)\n-1.0-0.50.00.51.0a\nb c\n \n400x 400nm2\nb\nc\nDDDDt= 0.2nsDDDDt= 0.02ns\n400x 400nm2\n \nFig. 4. Magnetization time dependencies (a) and gray-scale snapshots ((b) and (c)) of the out-of-plane mag-\nnetization component my  for the second localized mode L2 (single layer, HOe = 0). Panel (b) shows the \ncreation process of two V-AV pairs and an in-plane arrow plot of a typical V-AV quadrupole configuration. \nPanel (c) displays the propagation and decay of the two V-AV pairs. Time intervals corresponding to the \nimage rows of (b) and (c) are marked with vertical lines on mz(t)- plot in the panel (a). \nSuch a strong current leads to the periodical creation/annihilation of a vortex-antivortex \nquadrupole, consisting of two vortices and two antivortices all having the same polarity (Fig. \n4), and located symmetrically with respect to the point contact center. The formation of this \nV-AV quadrupole starts with the appearance of a ring-shaped magnetization structure (see the \n1st gray-scale map in Fig. 4b), which evolves very fast into two V-AV pairs which form a \nnearly symmetrical V-AV quadrupole. From the point of view of non-linear excitation \ndynamics such a quadrupole represents the next possible stable excitation kind (after a single \nV-AV pairs) for systems where the total topological charge is initially zero and should be  12 conserved. The whole process of the quadrupole formation shown in detail in Fig. 4b is very \nfast, taking only ~ 50 ps.  \nThe quadrupole built this way does not rotate (because, as mentioned earlier, all its vortices \nand antivortices have the same polarity), so it can disappear only relatively slowly via the \nsymmetry breaking with respect to the V-AV distances and Kelvin motion of the V-AV pairs \narising after this symmetry breaking. The lifetime of the V-AV quadrupole limited by this \nprocess is ~ 1 ns as it can be seen from the magnetization time dependencies (Fig. 4a) and \ngray scale maps of the out-of-plane magnetization projections (Fig. 4c). The two solitons, \neach representing one V-AV pair, propagate in opposite directions and decay due to the \n‘normal’ energy damping. These solitons are responsible for the energy emission out of the \npoint contact area for this localized mode. \nWe note once more, that in all cases considered above the oscillation frequency of the mx- \ncomponent is two times larger than that of the mz-component, as it can be clearly seen from \nFig. 2a, 3a and 4a. This means, that by experimental observations of these modes using the \nGMR-effect, the strong second harmonics should be present not only due to the non-sinusoi-\ndal character of mz-oscillations, but also due to the mixing of signals from mx- and mz-compo-\nnents. This mixing unavoidably happens when the magnetization of the fixed layer deviates \n(even slightly) from the external field direction, e.g., due to the influence of a random magne-\ntocrystalline anisotropy of this fixed layer. As mentioned above, for Co90Fe10 often used in \nsuch experiments [12, 14], the cubic anisotropy constant Kcub = – 5.6\u0002105 erg/cm3 is not small \n[28], so that even in nanocrystalline materials noticeable deviations of the magnetization from \nits average direction (along the external field) can be expected. \nB. Magnetization dynamics for the single-layer system: Influence of the Oersted field \nIn this subsection we discuss the influence of the Oersted field, induced by the spin-polarized \ndc-current, on the magnetization dynamics. When a quantitative comparison with real \nexperiments made on relatively large point contacts and in small external fields is aimed, one \ncannot neglect the influence of the Oersted field, because for this situation it is not small \ncompared to the external field. Hence the corresponding influence can crucially change the \nmagnetization dynamics, especially taking into account that the Oersted field of a thin wire is \nstrongly inhomogeneous. \nUnfortunately, it is not possible to compute the Oersted field quantitatively, because neither \nthe experimental geometrical setup nor the current distribution in the contact leads are known \nexactly. For this reason we have adopted the following strategy in order to study the Oersted \nfield effect. We are interested in the Oersted field acting on the free layer magnetization \nwithin the area directly under the point contact and in its vicinity. To compute the Oersted \nfield for this region, we make use of the two circumstances. First, for a standard symmetrical \nlead setup, the Oersted field created by the current flowing within a particular lead far away \nfrom the point contact, is mostly cancelled by the corresponding field of the symmetrically \nlocated lead, where the current flows in the opposite direction. Second, in the lead region \nwhich is attached to the contact, the current flows from the lead into the point contact wire, \nwhich diameter is much smaller than the lateral lead size. So in a good qualitative approxi-\nmation, the current flow within this lead region is similar to the water stream flowing from the \nlarge basin into a narrow drain channel. Hence for any particular current flow line there exist \na symmetrically located flow line with the opposite current direction, so that the overall \nOersted field of the lead area adjacent to the contact is also largely cancelled out. \nThe above arguments, which have been presented and quantitatively elaborated by Miltat \n[36], lead to the conclusion that the major contribution to the Oersted field under the point \ncontact is due to the current flowing in the point contact itself. Still, there exist a possibility to \nchange the Oersted field varying the height of the contact wire, so, in order to study the \nOersted field effect, we proceed in the following way. First we calculate the field Hinf(I, r)  13 generated by a current I in an infinitely long conductor with the diameter equal to that of the \npoint contact (r denotes the distance from the contact center). Then, taking into account that \nthe real contact wire has a finite length, we assume that the actual Oersted field has the same \ncircular symmetry, but is weaker than Hinf(r). To take account of this weakening, we \nintroduce the weakening coefficient k (0 1k££), compute the Oersted field as HOe(r) = \nk \u0002Hinf(r) and study the magnetization dynamics for several values of this coefficient k . \nHOe= 0\n HOe= 0.1 Hlong\nHOe= 0.5 Hlong\nI, mA I, mA I, mAf, GHz\nW(2)\nL3L4\nL1L2W(1)\n \nFig. 5. Spectral power of the mz-component vs current strength in the single-layer system for different \nstrengths of the Oersted field computed as explained in the text. Arrows show the positions of modes \nanalyzed in detail below. The discrete character of the f(I) dependence for the W1 , W2 and L4 modes is an \nimage artifact (see caption to Fig. 1). \nWe have found several qualitative effects of the Oersted field. To explain these effects in \ndetail, we concentrate ourselves on two ‘limiting’ cases: relatively small (k  = 0.1) and large \nOersted fields (k  = 0.5). Corresponding oscillation power maps in standard current-frequency \ncoordinates are shown in Fig. 5 (middle and right panel) in comparison with the case when \nHOe is absent (left panel, the same as in Fig. 1). \nThese oscillation power plots allow to identify the following qualitative effects of the Oersted \nfield on the magnetization dynamics:  \n(i) ‘On average’, the oscillation spectra become more regular, especially for large Oersted \nfields (see Fig. 5c, k  = 0.5), where well defined spectra with sharp peaks for nearly all current \nvalues are observed. This effect is due to the stabilizing influence of the Oersted field, which - \nat least for large k -values - is strong enough to guarantee the existence of well defined \nmodes, preventing the system from sliding into a quasichaotic behavior. \n(ii) The current region, where the propagating wave mode exists, narrows with increasing k. \nThis contraction can be explained by a strongly non-homogeneous character of the Oersted \nfield in the geometry under study, which induce a non-homogeneous and asymmetrical \nequilibrium magnetization configuration (in the absence of SPC-induced oscillations). Such a \nconfiguration obviously suppresses the oscillation mode represented by the wave propagating \ncircularly out of the point contact area. \n(iii). Localized modes containing V-AV pairs also behaves themselves qualitatively different \nin presence of the Oersted field, because this circular field acts differently on a vortex and an \nantivortex, thus disturbing a free rotation of a V-AV pair (see discussion below). \nTo study the magnetization dynamics in presence of the Oersted field in more detail, we \nconsider first the case of the weak field HOe=0.1\u0002Hinf. For small currents we observe again the \npropagating mode W2 similar to that found in the absence of HOe. As explained above, the \ncurrent region where this mode exists is narrower than for W1 (DIW2 = 15-17.5 mA for k = 0.1, \ninstead of DIW1 = 15-20 mA for HOe = 0). Due to the abovementioned asymmetry of the \nunderlying equilibrium magnetization state caused by the Oersted field (and also due the \noverlapping of  the Oersted field with the homogeneous external field H0 = 30 Oe used in all  14 our simulations presented here), the group velocity of a spin wave significantly depends on \nthe propagation direction. Hence magnetization pattern for the W2-mode becomes circularly \nasymmetric (see gray-scale maps in Fig. 6).  \nt, ns\n6.0 6.1 6.2 6.3 6.4 6.5mX(t)\n-1.0-0.50.00.51.0\nmZ(t)\n1 x 1 mkm1\n2\n3\n1 2 3\n \nFig. 6. The same as in Fig. 2 for the propagating mode in the single layer system for HOe = 0.1Hinf. \nBy higher currents the propagating wave mode vanishes, and the localized modes appear. In \npresence of the Oersted field these modes differ qualitatively from localized modes when HOe \nis neglected. In particular, it is instructive to compare the localized mode L2 arising at the high \ncurrent I = 48 mA at HOe = 0 (see Fig. 4) with the localized mode L3 appearing for the same \ncurrent, but for HOe=0.1\u0002Hinf  (shown in Fig. 7). From gray-scale maps and arrow plots shown \nin Fig. 7 it is clear, that in contrast to the quadrupole V-AV mode in the absence of HOe, the \nmagnetization configuration of the L3-mode at higher currents even in presence of a relatively \nweak Oersted field (k = 0.1) contains most of the time only a single V-AV pair. This fact can \nbe again attributed to the strong circular asymmetry of the magnetization state in presence of \nHOe, so that the formation of a highly symmetric V-AV quadrupole becomes impossible. \nMagnetization dynamics of this new L3 mode is also completely different. Due to the same \nreason - circularly asymmetric equilibrium magnetization configuration - the V-AV pair can \nnot rotate free around its center even when vortex and antivortex have opposite polarities. \nMagnetization dynamics of the L3-mode is thus governed by the oscillation of the vortex \nposition, whereby the antivortex remains nearly immobile. During these oscillations the \nvortex generates a 2nd V-AV pair with a very small V-AV distance and the same polarities, \nwhich are, however, opposite to the polarity of the initial vortex. The antivortex from this new \nsmall V-AV pair annihilates with the initial vortex (irradiating a burst of spin waves), thus \nleaving a single vortex with the polarity opposite to the initial vortex, so that effectively the \npolarity of the vortex in the initial large V-AV pair is reversed. Then this new vortex starts to \nmove in the opposite direction, generates a new small V-AV pair and the process in repeated. \nA very similar process was reported as a mechanism to change the vortex polarity during a \nmotion of a single vortex in [37].  15 mX(t)\nt, ns\n10.0 10.5 11.0 11.5 12.0-1.0-0.50.00.51.0\nmZ(t)a\nb\n \n400x 400nm2b\nDDDDt= 0.05ns\n \nFig. 7. (a) Magnetization time dependencies for the high-current localized mode L3 in the single layer system \nfor HOe = 0.1Hinf. (b) Oscillations of the main V-AV pair, accompanied by the creation and annihilation of a \nsatellite V-AV pair (snapshots of the out-of-plane magnetization component my and in-plane arrow plots). \nWhen the Oersted field coefficient k is increased further, the propagating mode region \ncontracts even more (right panel in Fig. 5 with HOe=0.5\u0002Hinf). Increasing the current for this k \nvalue, we still observe after a W-mode a relatively narrow region of irregular dynamics, after \nwhich a transition to a localized mode L4 similar to that shown in Fig. 7 takes place. Due to \nthe larger Oersted field and stronger deformation of the magnetization configuration the (also \nimmobile) antivortex of this L4-mode is located farther from the point contact center than for \nthe L3-mode at the same current (Fig. 8b). Magnetization dynamics of the L4-mode is gover-\nned by the same process of the vortex oscillation accompanied by the creation-annihilation of \na small V-AV pair as for the L3-mode. An important point is that the large Oersted field leads \nfor this case to a strong (and approximately linear) increase of the mode frequency with the \ncurrent strength, because the oscillation frequency of the vortex in the potential well created \nby the Oersted field near the point contact center increases with the current strength. \nConcluding this subsection, we would like to emphasize the following important point: by \nchanging the electric setup (design of the point contact wiring) the strength of the Oersted \nfield can be made different for the same current strength - at least up to some extent. As it can \nbe seen from our simulations, such changing of the Oersted field strength can be used to \ncontrol the dominating oscillation mode of the point contact device.  16 t, ns\n2.0 2.5 3.0 3.5 4.0mX(t)\n-1.0-0.50.00.51.0\nmZ(t)a\n \nb\n4 x 4mkm2\n \nFig. 8. (a)  Magnetization time dependencies for the high-current localized mode L4 in the single layer system \nfor the large Oersted field HOe = 0.5Hinf; (b) Image of the in-plane magnetization orientation for the whole \nsimulation area and the enlarged in-plane arrow plot of the V-AV pair. \n \nC. Magnetization dynamics for the CoFe/Cu/Py trilayer system: Influence of the ‚hard‘ \nmagnetic layer \nThe next important question for the SPC-induced magnetization dynamics is the influence of \nthe ‘fixed’ magnetic layer on magnetization oscillations of the ‘free’ layer. Up to our know-\nledge, this influence was not studied yet for the point contact setup in the extended thin film \ngeometry, so the results presented below are especially interesting. We remind that the ‘fixed’ \nlayer parameters have been chosen to imitate the Co90Fe10 underlayer used in the system \nstudied experimentally in [14]: MS = 1500 G, A = 2\u000210-6 erg/cm, layer thickness hfix = 20 nm \nand the spacer thickness (distance between free and fixed magnetic layers) hCu = 4 nm. \nMagnetization dynamics of the fixed layer induced by the magnetodipolar interaction between \nthe free and fixed layers was fully taken into account. Further, we assumed that the fixed layer \nthickness is large enough to neglect the spin torque effect on this layer. \nIn principle, one should also keep in mind that the magnetocrystalline anisotropy of Co90Fe10 \nis not small (cubic anisotropy with Kcub = – 5.6\u0002105 erg/cm3) and hence, generally speaking, \ncan not be neglected. However, the influence of this anisotropy for a polycrystalline Co90Fe10 \nmaterial as used in [14] is partially ‘averaged out’ [38] due to the very small grain size (~10-\n20 nm). Both for this reason and lacking the exact knowledge about the grain size and texture \nof magnetic layers studied experimentally in [14], we present here only the results where the \nmagnetocrystalline anisotropy of the fixed layer is neglected. Our preliminary studies show \nthat when this random anisotropy is taken into account, a substantial dependence of the results \non the grain size and film texture is observed, so that one should possess a quantitative \ninformation about these parameters in order to make a meaningful quantitative comparison of \nsimulated and experimental data.  17 In order to study solely the effect of the magnetodipolar interaction between the fixed and free \nlayers, we have also neglected the influence of the Oersted field in simulations which results \nare presented in this subsection. \nI(mA) I(mA)f(GHz)a b\n \nFig. 9. Comparison of spectral power maps of the mz-component for the single layer (a) and trilayer (b) sys-\ntems (HOe = 0). The strong reduction of the current strength for the oscillation onset and for the transition to \nthe localized modes due to the influence of the hard magnetic layer can be clearly seen (the discrete character \nof the f(I) dependence for W-modes is due to the same reasons as explained in Fig. 1). \nThe first very important result of our simulations is that the magnetodipolar interlayer interac-\ntion leads to the drastic reduction of the threshold current Ith for the magnetization oscillations \nonset: from the comparison of oscillation power plots in Fig. 9 it follows, that the presence of \nthe hard magnetic underlayer reduces the threshold current from Ith \u0001 13 mA for the single \nlayer system to \u0001 8 mA for the trilayer. The current where the transition from the propagating \nto localized modes occurs, is reduced from Iloc \u0001 20 mA for the single Py layer system to Iloc \u0001 \n12 mA (i.e., it nearly halves) when the presence of the CoFe underlayer is taken into account. \nThe most probable qualitative explanation of this effect is the following: when the \nmagnetization of the free layer deviates slightly from its in-plane orientation (under the \ninfluence of the spin torque within the point contact area), a stray field is generated. \nStraightforward geometrical consideration shows that this stray field causes the deviation of \nthe fixed layer magnetization in the direction opposite to that of the free layer. This, in turn, \nresults in the stronger deviation of the free layer magnetization due to the influence of the \nfixed layer stray field, thus leading to the positive feedback between the magnetization \ndynamics of the free and fixed layer. Such a positive feedback leads to the decrease of the \nthreshold current for the oscillation onset. An additional decrease of the transition current \nfrom the propagating to the localized modes can be explained by the fact that magnetodipolar \ninterlayer interaction field is strongly inhomogeneous, thus favoring the appearance of the \nlocalized modes. From the quantitative point of view, however, such a large decrease of the \nthreshold current due to the magnetodipolar interlayer interaction is surprising. \nNow we proceed to the discussion of the hard magnetic layer influence on various oscillation \nmodes. As it can be seen from Fig. 9b, for the trilayer system the first appearing mode (for \ncurrents slightly higher than Ith \u0001 8 mA) is also the propagating one. The pattern of the spatial \nwave propagation for this mode is asymmetric (see Fig. 10), what in this case is due to the \ninhomogeneous magnetodipolar field created by the hard magnetic layer (we remind that the \nOersted field is not included into the trilayer simulations in order to isolate the interlayer \ninteraction effect). The frequency at the oscillation onset is substantially lower than for a \nsingle-layer system (~ 6 GHz instead of ~ 7 GHz), showing that the interlayer interaction \nshould be also taken into account, if the quantitative comparison between simulated and  18 experimental frequencies is aimed. When the current increases further, the oscillation \nfrequency decreases almost linearly with current, as for a single-layer system.  \nt, ns\n3.2 3.3 3.4 3.5 3.6mX(t)\n-1.0-0.50.00.51.0\nmZ(t)\n1 x 1 mkm \nFig. 10. The same as in Fig. 2 for the propagating (Slonczewski) mode in the trilayer system for HOe = 0. \nAt the current value loc\nth12.3 I»  mA we find a transition to the first localized mode of our \ntrilayer (L5-mode) with a very low (for the SPC-induced magnetization dynamics) oscillation \nfrequency ~ 380 MHz. Similar to the first localized mode L1 of a single-layer system, the \nmain dynamic process for this mode is the rotation of a V-AV pair with opposite polarities of \nthe vortex and antivortex (Fig. 11b). During this rotation the V-AV distance slowly changes \ndue to the interaction with the hard magnetic layer and presence of a small constant external \nfield. However, in contrast to the L1-mode for a single layer case, here we observe during the \nrotation of this main V-AV pair the creation of a small (satellite) V-AV pair with opposite \nvortex and antivortex polarities. For this reason this small pair does not propagate (like the \nsatellite pair for the L1-mode), but its antivortex immediately annihilates with the vortex of the \nmain pair, emitting a burst of spin waves (Fig. 11c), similar to localized modes found in the \npresence of the Oersted field (see Fig. 7b and 8). This very fast creation-annihilation process \nmanifests itself in a small cusp on the time dependence of the mz-projection and a large peak \non mx(t) (Fig. 11a). Due to the extremely anharmonic time dependencies of both in-plane \nmagnetization projections the oscillation power spectrum of this mode contains very strong \nhigher harmonics clearly visible in Fig. 9b. \nFor the L5-mode it is also possible to compare the rotation frequency of the V-AV pair with \nthat calculated from the theory of Komineas [34]. The typical V-AV distance deduced from \nsimulations for this mode is d \u0001 100 nm, what results in the analytically calculated (see Eq. \n(4)) frequency fan \u0001 0.36 GHz. Simulated frequency for this case is fsim \u0001 0.38 GHz. Again, we \nobtain a very good agreement between theory and simulations, although the theory includes \nneither the magnetodipolar interlayer interaction, nor the process of the V-AV creation-\nannihilation. As mentioned above, the latter process probably has almost no effect on the \nrotation frequency of the main V-AV pair, because (compare time scales in Fig. 11b and 11c) \nthe satellite V-AV creation-annihilation happens very fast compared to the rotation period of \nthe main V-AV pair. However, it is not clear why the effect of the hard layer stray field on the \nrotation of the main V-AV pair is apparently also rather small. We would also like to mention, \nthat in order to enable a more meaningful comparison, one should use not the ‘typical’ V-AV \ndistance, but the inverse square of this distance 21/d\u0001\u0002, averaged over the rotation period.  19 t, ns\n4 5 6 7 8 9 10mX(t)\n-1.0-0.50.00.51.0\nmZ(t)\na\nb\ncc\nb\n400 x 400 nm\n400 x 400 nmDDDDt = 0.5 ns\nDDDDt = 0.05 ns\n \nFig. 11. Magnetization time dependencies (a) and snapshots of the magnetization configurations ((b) and (c)) \nfor low-current localized mode L5 in the trilayer system (HOe = 0). Panel (b) shows the rotation of the main \nvortex-antivortex pair (images of the out-of-plane magnetization component) and panel (c) illustrates the cre-\nation and annihilation of a satellite V-AV pair accompanied by the emitting of a spin wave burst. Time inter-\nvals corresponding to the image rows of (b) and (c) are marked with vertical lines on the plot in the panel (a). \nWhen the current increases above \u0001 16 mA, magnetization dynamics becomes less regular, \nwhich manifests itself in a quasicontinuous power spectrum up to the current value \u0001 19 mA,  \nwhere the next regular dynamic mode (L6) appears. The major spectral peak of this mode has \napproximately the same frequency f \u0001 0.38 GHz as for L5, but the analysis of the \nmagnetization configurations reveals, that the complete period of mz-oscillations Tz  \u0001 7.8 ns \ncorresponds to an ever lower frequency fz \u0001 0.13 GHz; corresponding relatively weak spectral \nband can be also seen in Fig. 9b. \nThis high-current mode is the most complicated among regular modes studied here and com-\nbines all the processes analyzed above (see Fig. 12). Its formation starts from the nearly \nhomogeneous magnetization deviation under the point contact area, which evolves very \nrapidly into a V-AV pair with the same polarities of the vortex and antivortex (Fig. 12, image \nrow A). This process is up to some extent similar to the formation of the high-current mode L2 \nfor a single layer system (Fig. 4), but in the single layer case two V-AV pairs were formed. \nDuring the next stage (row B in the same figure) the V-AV distance in this pair increases, and \nthe pair orientation is slightly changed, what is possible due to the magnetodipolar field of the \nhard layer (as mentioned above, the V-AV pair with the same V-AV polarities could move \nonly translationally in the absence of external fields). At the third stage (row C in Fig. 12) the \nsmaller V-AV pair is formed near the vortex of the main pair. The polarities of vortex and \nantivortex in this satellite pair are the same, but opposite relative to the V-AV polarities of the \nmain pair. For this reason the antivortex of the new satellite pair annihilates very fast with the \nvortex of the main pair, emitting a burst of spin waves, as explained above.  20 t, ns\n5 6 7 8 9mX(t)\n-1.0-0.50.00.51.0A\nBC\nDE\nmZ(t)\n \nImage sizes300x 300nm\nDDDDt= 0.05ns\nDDDDt= 0.25nsA\nB\nDDDDt= 0.05nsC\nDDDDt= 0.05nsE\nDDDDt= 0.40nsDV AV\nV\nAV\nV\nAV\nVAV\n \nFig. 12. The same as in Fig. 11 for the high-current localized mode L5 in the trilayer system (HOe = 0). Panels \nA-E show various dynamic processes constituting this very complicated mode; see text for details. Corres-\nponding time intervals are marked on the plot in the panel (a). \nAs a result of this annihilation, the V-AV pair with nearly the same V-AV distance as at the \nvery beginning of the process, but opposite polarities of vortex and antivortex, is left. Due to \nopposite V-AV polarities, this pair starts to rotate (row D in Fig. 12), and the distance \nbetween vortex and antivortex continuously decreases (probably also due to the influence of  21 the underlayer stray field), until they annihilate (row E). The nearly homogeneous magnetiza-\ntion deviation under the point contact area left after this annihilation is directed opposite to the \ninitial deviation at the start of the process. This means that the images displayed in the rows \nA-E in Fig. 12 correspond to a half of the complete magnetization dynamic period of this \nmode.  \nWe would also like to emphasize, that for all localized modes found in the system when the \nOersted field was neglected (L1, L2, L5 and L6), the mode frequency  was nearly independent on \nthe current strength, although some of these modes existed in a quite large current region \n(e.g., modes L5 and L6). The most probable explanation of this interesting feature is that the \nmode frequency is determined by the rotation frequency of the V-AV pair(s) constituting the \nmode. This rotation frequency, in turn, is governed by the V-AV separation within the pair, \nwhich in our system is defined mainly by the point contact diameter flooded by the current (in \nthe absence of the Oersted field) and additionally - by the stray field of the underlayer (for the \ntrilayer system). For this reason the mode frequency does not change noticeably with the \ncurrent strength. The increasing amount of energy pumped into the system when the current \nstrength grows is probably ‘consumed’ during the process of the creation-annihilation of \nsatellite V-AV pairs discussed above. Indeed, a micromagnetic analysis has shown that the \nmagnetization configuration of these pairs depends on the currents strength. \nWhen the Oersted field is included, and it is large enough, the antivortex becomes immobile, \nand the mode frequency is determined by the oscillation of the vortex position within the \npotential well created by the Oersted field. For this reason the mode frequency increases with \ncurrent, because larger currents create stronger Oersted fields. This frequency increase is \nespecially pronounced for large k  (panel for k = 0.5 in Fig. 5), \nD. Comparison with experimental results \nAt present there exist only very few experimental studies (partially supported by numerical \nsimulations) of the SPC-driven microwave oscillations in the point contact geometry, where it \nis proven - or at least suggested with a high plausibility degree - that this dynamics is gover-\nned by the vortex/antivortex motion [14, 24, 25]. Since we have chosen our simulation \nparameters according to the device characteristics from [14], we shall mainly compare our \nresults with those reported in this paper.  \nFirst we note that we have simulated the in-plane field geometry, so that our results should be \ncompared with the experimental data reported in the first part and in Fig.1 of [14]. It also \nfollows from our simulations that for this particular setup, the strong influence of the inter-\nlayer interaction on the power spectra of the microwave oscillations can not be neglected (see \nSec. III.C above). Hence only simulated data obtained for the complete trilayer system (see \nFig. 9) can be used for a meaningful comparison with the experiment. \nThis comparison shows that our simulation results for the first localized mode of the trilayer \n(mode L5, see Fig. 11) could reproduce semiquantitatively several important features of the \nreal experiment. First, the current region where this mode is observed experimentally (DIexp \u0001 \n6 - 12 mA) (see Fig.1 in [14]) is close to the region where our L5 mode is found  numerically \n(DIsim \u0001 12 - 16 mA). The frequency of the experimentally observed microwave oscillations \nfexp \u0001 100 - 220 MHz [14] is of the same order of magnitude as the simulated frequency fsim \u0001 \n380 MHz. Next, the weak dependence of the experimentally measured frequency on the \ncurrent strength corresponds fairly well to our observation that the frequency of the localized \nmodes for a trilayer is nearly current-independent. Finally, the strongly non-sinusoidal \ncharacter of simulated magnetization oscillations is in accordance with the presence of several \nhigher harmonics in the experiment [14]. \nThe comparison of experiment and simulations could be more meaningful, if a better charac-\nterization of the experimentally studied system would be available. The problem is not only  22 that several important geometric parameters (e.g.,  the point contact diameter) are not known \nexactly. We have also found, that the threshold current Ith for the oscillation onset depends on \nthe average characteristics of the crystallographic structure of the hard layer (grain size and \ntexture), and, what is less evident - that Ith is also different for various particular realizations \nof a random grain structure with the same average parameters. This difference is caused by \nsubstantial variations of the equilibrium magnetization configuration of the hard layer for \nvarious random grain structure realizations. The oscillation frequency itself and even the type \nof the first localized mode also depend on the crystallographic structure of the hard layer; it is \nworth noting here that the fundamental frequency of, e.g., our L6-mode (f6 \u0001 130 MHz) lies \nwithin the experimentally measured frequency range.  \nFurther, we have observed, that when  the Oersted field is taken into account (even strongly \nweakened with respect to a maximal possible field of an infinitely long nanowire), the type of \nthe first localized mode could be changed. The Oersted field also causes the increase of the \noscillation frequency with current similar to the frequency behavior observed experimentally. \nThere exist, however, important issues, where we observe a qualitative disagreement with ex-\nperimental findings. The most important one is the presence of several well defined different \noscillation modes in the simulated dynamics, whereby experimentally only one oscillation \nmode was found (this statement is based mainly on the absence of any jumps on the current \ndependencies of the oscillation frequency and power). The absence of other localized modes \nat currents higher than the ‘switch-off’ current for the 1st localized mode in the experiment \ncan be in principle explained as follows: according to [14], when the current is increased, the \nwell-defined spectral peak evolves into “a broader band spectral output at larger current”. \nThis observation is consistent with the spectrum evolution found in our simulations, as \ndemonstrated by the transition from a spectrum consisting of sharp spectral lines to a broad-\nband spectrum at I \u0001 16 mA in Fig. 9b. The next localized mode could be suppressed in a real \nexperiment, because it emerges at much larger currents, when the sample heating and/or spin \ntorque fluctuations due to the high spin current density prevents the formation of a well-\ndefined oscillation mode. \nThe absence of the propagating mode (W-mode in our notation) is more difficult to explain. In \nprinciple, one can speculate that the propagating mode vanishes due to the presence of the \nOersted field, which reduces the current region DIW where the W-mode exists (see Fig. 5). \nHowever, we note that significant reduction of the current interval DIW requires high Oersted \nfields - with the magnitude close to that achieved for a very long contact wire. Taking into \naccount, that in real experiments the length (height) of the cylindrical wire forming the point \ncontact is usually of the same order of magnitude as the contact diameter, this explanation is \nquestionable. Still, it can be fully excluded only when one will be able to perform simulations \nwith the Oersted field computed according to the real electrical setup. \nTo provide another explanation why the W-mode is absent in the experiment [14], we would \nlike to remind that the same discrepancy was found for systems with a small point contact \ndiameter in high external fields. Simulations have predicted that for such systems, when the \ncurrent increases, the propagating mode should emerge first [21, 22], and the localized \n(‘bullet’) mode should appear at higher currents [20]). However, experimentally only a \nlocalized mode was found [12], which was later unambiguously identified as the ‘bullet’ \nmode [21, 22, 26]. The probable explanation of this contradiction was based on the theory of \nSlavin et al. [20], where it was shown that the threshold current L\nthI for the 'bullet' mode is \nsmaller thanW\nthIfor the W-mode, but the ‘bullet’ mode has the finite (and not even small) \nmagnetization oscillation amplitude already at its threshold. For these reason the 'bullet' mode \nwas not observed in simulations made by increasing current and at T = 0. These arguments \nwere supported by numerical simulations, where it was shown that (i) when the current \ndecreases, the ‘bullet’ mode still exists for currents smaller than W\nthI [21, 26] and (ii) the  23 average energy of the localized mode is smaller at the transition current W L®  for the \nincreasing current [22] (so that the W-mode is actually metastable). Hence it was suggested, \nthat in systems with a small contact diameter in high external fields, the energy required to \nexcite the localized mode with a finite oscillation amplitude is supplied by thermal fluctua-\ntions. These fluctuations excite the L-mode at its threshold current L\nthI, which is smaller than \nW\nthI for the W-mode, so that the latter did not emerge at all. Whether this explanation is appli-\ncable to our case, where the current flooded area is much larger - so that both thermal energy \nand the energy for a V-AV pair formation are higher - should be a subject of further studies. \nAnother problem which we have encountered is the reproduction of the experimentally obser-\nved hysteretic behavior of magnetization oscillations when the current was first increased to \nthe value where the well-defined mode disappeared and then decreased to zero [14]. We \ncould not reliably confirm this observation in our simulations, because we have found that the \nhysteretic behavior simulated numerically depends on the rate with which the current is \ndecreased (this was not the case for systems with smaller point contact diameter in high exter-\nnal fields). Due to the computer time limitations we had to decrease the current strength from \nits maximum to the value for which we intended to study the magnetization dynamics within \ntred ~ 10 ns. Such interval was not enough to achieve results stable with respect to further inc-\nrease of tred. Experimentally, where the current is reduced within a macroscopic time interval, \nthe corresponding problem obviously does not exist. \nComparison with other reports on the SPC-induced magnetization dynamics in the point con-\ntact geometry [24, 25] is possible only from a qualitative point of view, because systems \nstudied in these papers considerably differ from the system investigated by us. The paper of \nMistral et al. [25] also deals with the magnetization dynamics for the point-contact injection \nin an extended thin film sample, but the contact diameter used there is much larger (d \u0001 200 \nnm) and the applied field is out-of-plane and relatively strong (Hperp \u0001 2000 G). According to \nsimulations performed in [25] this field leads to a formation of a single vortex already in the \nabsence of any current, i.e., in the equilibrium magnetization configuration. When the spin-\npolarized dc-current is applied to this configuration, the vortex is driven out of the point \ncontact area and its precession around this area governs the magnetization dynamics observed \nin simulations and (most probably) experimentally. Because the magnetization dynamics is \ndominated by a single vortex motion, results of [25] can not be directly compared to ours. \nIn a very recently published paper Finocchio et al [24] have studied magnetization oscillations \nin a multilayer nanopillar device with the elliptical cross-section and relatively small lateral \ndimensions 250 x 150 nm2. The current was injected into the nanopillar via a small point \ncontact with the diameter up to d \u0001 40 nm. Experimentally magnetization oscillations with the \nfrequency f \u0001 0.8 GHz were observed for zero and small applied fields, whereby the \noscillation frequency was found to be nearly current-independent. Supporting numerical \nsimulations [24] have shown that under these experimental conditions the creation and \nsubsequent rotation of a single V-AV pair (analogous to our simplest localized modes \ndiscussed in detail above) can take place with a frequency quite close to the values measured \nexperimentally. The simulated rotation frequency of such a pair also was almost independent \non the current, in a quailtative agreement with our findings. However, a more detailed compa-\nrison of our data with the results from [24] is not really meaningful: the small lateral size of \nthe nanopillar device studied in [24] is important not only by itself (strongly changing, e.g., \nconditions for the existence of a propagating mode), but also because it leads to the large \ninfluence of the shape anisotropy (stray field of the nanopillar borders), which may \nqualitatively affect the magnetization dynamics of localized modes. Here we would like only \nto mention that the absence of other dynamic modes and processes of the creation-annihilation \nof satellite V-AV pairs (found by us for an extended thin film geometry) is most probably due \nto this small lateral size of the nanopillar studied in [24]. \n  24 IV. CONCLUSION \nWe have presented a detailed study of the magnetization dynamics induced by a spin polari-\nzed current injected via a point contact into an extended magnetic multilayer for the case, \nwhen the point contact diameter (Dc = 80 nm) is relatively large compared to systems studied \npreviously [12, 13, 20, 21, 22, 23] and the in-plane external field (H0 = 30 Oe) is very small. \nUnder these conditions the system exhibits a rich variety of well-defined oscillation modes, \nwhich can be divided into propagating and localized ones.  \nThe frequency of propagating modes in the simplest case (when the Oersted field and \nmagnetodipolar interaction between the ‘free’ and ‘fixed’ layers are neglected) can be \nsatisfactory described by the Slonczewski theory [18, 31]. However, the agreement between \nsimulated and analytically predicted threshold currents is decisively worse than for the point \ncontact with the much smaller diameter (Dc = 40 nm, see [21, 22, 31]). We assume that this is \ndue to the strong anisotropy of the group velocity of spin waves, emitted out of the point \ncontact area for the low external field and large point contact diameter (both factors lead to \nrelatively low oscillation frequencies). Inclusion of the Oersted field and/or interlayer \ninteraction narrows the current region for the existence of propagating modes and results in an \nasymmetric wave propagation pattern. This asymmetry is due not only to the influence of the \nOersted and/or ‘fixed’ layer stray fields on the spin wave propagation itself, but also due to \nthe deformation of the equilibrium magnetization configuration of the ‘free’ layer by these \nfields. \nWhen the Oersted field is neglected, localized modes for the system studied here are governed \nby the rotational and translational movement of V-AV pairs, in contrast to the small contact \ndiameter case, where the dominating mode was the non-linear ‘bullet’ [20, 21, 22]. The \nsimulated rotation frequency (for pairs with opposite polarities of V and AV) and translational \nmotion velocity (for pairs with the same polarities of V and AV) for the steady-state motion \nof the V-AV pairs are in a good quantitative agreement with the theory of Komineas et al [34, \n35], which employs the scaling arguments familiar from the non-linear dynamics. However, \nthe actual dynamic modes involve much more complicated processes, in particular, the \ncreation/annihilation of additional satellite V-AV pairs (which seem to play an important role \nfor the energy emission out of the point contact area) and creation/annihilation of the V-AV \nquadrupoles. These processes obviously require further investigation to achieve their deeper \nunderstanding. \nWe have also shown that the inclusion of the Oersted field can lead to qualitative changes of \nmagnetization oscillation modes. In particular, for sufficiently large Oersted fields, the dyna-\nmics of V-AV localized modes are dominated by the oscillation of vortex in the potential well \ncreated by the Oersted field, whereby the antivortex is nearly immobile. This offers (in prin-\nciple) a possibility to control the dominating magnetization dynamic mode by adjusting the \nelectric contact setup, which is responsible for the Oersted field strength and configuration. \nFinally, we have demonstrated that the magnetodipolar interlayer interaction is qualitatively \nimportant for the understanding of the magnetization dynamics in point contact systems at \nlow external fields, leading both to a strong decrease of the threshold current for the \noscillation onset and to qualitative changes in the observed magnetization oscillation modes.  25 References: \n[1] J.C. Slonczewski, Current-driven excitation of magnetic multilayers, J. Magn. Magn. Mat., 159 (1996) L1 \n[2] L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev., B54 \n(1996) 9353 \n[3] J.Z. Sun, Current-driven magnetic switching in manganite trilayer junctions, J. Magn. Magn. Mat., 202 \n(1999) 157; M. Tsoi, A.G.M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, P. Wyder, Excitation of a \nmagnetic multilayer by an electric current, Phys. Rev. Lett., 80 (1998) 4281 \n[4] Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, B.I. Halperin, Nonlocal magnetization dynamics in \nferromagnetic heterostructures, Rev. Mod. Phys., 77 (2005) 1375;  \n[5] M.D. Stiles, J. Miltat, Spin Transfer Torque and Dynamics, in: Spin Dynamics in Confined Magnetic \nStructures III, Springer Series Topics in Applied Physics 101, Springer-Verlag Berlin, Heidelberg 2006. \n[6] M.D. Stiles, Theory of Spin-transfer Torque, in: Handbook of Magnetism and Advanced Magnetic Materials, H. \nKronmüller and S. Parkin (Eds), Vol 2: Micromagnetism, 2007, JohnWiley & Sons, Ltd, Chichester, UK \n[7] D.C. Ralph, M.D. Stiles, Spin transfer torques, J. Magn. Magn. Mat., 320 (2008) 1190 \n[8] W. H. Rippard, M. R. Pufall, Microwave Generation in Magnetic Multilayers and Nanostructures, in: \nHandbook of Magnetism and Advanced Magnetic Materials, H. Kronmüller and S. Parkin (Eds), Vol 2: \nMicromagnetism, 2007, JohnWiley & Sons, Ltd, Chichester, UK \n[9] F. B. Mancoff, S. Kaka, Microwave Excitations in Spin Momentum Transfer Devices, in: Handbook of \nMagnetism and Advanced Magnetic Materials, H. Kronmüller and S. Parkin (Eds), Vol 5: Spintronics and \nMagnetoelectronics, 2007, John Wiley & Sons, Ltd, Chichester, UK \n[10] T.J. Silva, W.H. Rippard, Developments in nano-oscillators based upon spin-transfer point-contact \ndevices J. Magn. Magn. Mat., 320 (2008) 1260 \n[11] J.A. Katine, E. E. Fullerton, Device implications of spin-transfer torques J. Magn. Magn. Mat., 320 \n(2008) 1217 \n[12] W.H. Rippard, M.R. Pufall, S. Kaka, S.E. Russek, T.J. Silva, Direct-current induced dynamics in \nCo90Fe10/Ni80Fe20 point contacts, Phys. Rev. Lett.,  92 (2004) 027201 \n[13] W.H. Rippard, M.R. Pufall, S. Kaka, T.J. Silva, S.E. Russek, Current-driven microwave dynamics in \nmagnetic point contacts as a function of applied field angle, Phys. Rev., B70  (2004) 100406(R) \n[14] M. R. Pufall, W. H. Rippard, M. L. Schneider, S. E. Russek, Low-field current-hysteretic oscillations in \nspin-transfer nanocontacts, Phys. Rev., B75  (2007) 140404(R) \n[15] W. H. Rippard, M. R. Pufall, M. L. Schneider, K. Garello, S. E. Russek, Spin transfer precessional \ndynamics in Co60Fe20B20 nanocontacts, J. Appl. Phys., 103 (2008) 053914 \n[16] F. B. Mancoff, N. D. Rizzo, B. N. Engel, S. Tehrani, Phase-locking in double-point-contact spin-transfer \ndevices, Nature, 437, 393 (2005) \n[17] S. Kaka, M.R. Pufall1 W.H. Rippard, T.J. Silva, S.E. Russek,  J. A. Katine, Mutual phase-locking of \nmicrowave spin torque nano-oscillators, Nature, 437, 389 (2005) \n[18] J.C. Slonczewski, Excitation of spin waves by an electric current, J. Magn. Magn. Mat., 195  (1999) 261 \n[19] D.V. Berkov, Micromagnetic simulations of the magnetization dynamics in nanostructures with special \napplications to spin injection, J. Magn. Magn. Mat., 300 (2006) 159-163 \n[20] A. Slavin, V. Tiberkevich, Spin Wave Mode Excited by Spin-Polarized Current in a Magnetic \nNanocontact is a Standing Self-Localized Wave Bullet, Phys. Rev. Lett., 95 (2005) 237201 \n[21] G. Consolo, B. Azzerboni, G. Gerhart, G.A. Melkov, V. Tiberkevich, A.N. Slavin, Excitation of self-\nlocalized spin-wave \"bullets\" by spin-polarized current in in-plane magnetized magnetic nano-contacts: a \nmicromagnetic study, Phys. Rev., B76 (2007) 144410 \n[22] D.V. Berkov, N.L. Gorn,  Magnetization oscillations induced by a spin-polarized current in a point-\ncontact geometry: Mode hopping and nonlinear damping effects, Phys. Rev., B76 (2007) 144414 \n[23] D.V. Berkov, N.L. Gorn, Micromagnetic simulations of the magnetization precession induced by a spin-\npolarized current in a point-contact geometry (Invited), J. Appl. Phys., 99 (2006) 08Q701 \n[24] G. Finocchio, O. Ozatay, L. Torres, R. A. Buhrman, D. C. Ralph, B. Azzerboni,  Spin-torque-induced \nrotational dynamics of a magnetic vortex dipole, Phys. Rev., B78 (2008) 174408  26 [25] Q. Mistral, M. van Kampen, G. Hrkac, Joo-Von Kim, T. Devolder, P. Crozat, C. Chappert, L. Lagae, T. \nSchrefl, Current-Driven Vortex Oscillations in Metallic Nanocontacts, Phys. Rev. Lett., 100 (2008) 257201 \n[26] D.V. Berkov, N.L. Gorn, Non-linear magnetization dynamics in nanodevices induced by a spin-\npolarized current: micromagnetic simulation, J. Phys. D: Appl. Phys., 41 (2008) 164013 \n[27] D.V. Berkov, J. Miltat,  Spin-torque driven magnetization dynamics: Micromagnetic modeling, J. Magn. \nMagn. Mat., 320 (2008) 1238 \n[28] J. Pelzl, R. Meckenstock, D. Spoddig, F. Schreiber, J. Pflaum, Z. Frait, Spin-orbit coupling effects on g-\nvalue and damping factor of the ferromagnetic resonance in Co and Fe films, J. Phys.: Cond. Matt., 15 \n(2003) S451 \n[29] D.V. Berkov, N.L. Gorn, MicroMagus - package for micromagnetic simulations, \nhttp:\\\\www.micromagus.de \n[30] I. N. Krivorotov, D. V. Berkov, N. L. Gorn, N. C. Emley, J. C. Sankey, D. C. Ralph, and R. A. \nBuhrman, Large-amplitude coherent spin waves excited by spin-polarized current in nanoscale spin valves, \nPhys. Rev., B76 (2007) 024418 \n[31] A.N. Slavin, P. Kabos, Approximate Theory of Microwave Generation in a Current-Driven Magnetic \nNanocontact Magnetized in an Arbitrary Direction, IEEE Trans. Magn.,  MAG-41 (2005)  1264 \n[32] A. Slavin, V. Tiberkevich, Excitation of Spin Waves by Spin-Polarized Current in Magnetic Nano-\nStructures, IEEE Trans. Magn.,  MAG-44 (2008)  1916 \n[33] O. Tchernyshyov and G.-W. Chern, Fractional Vortices and Composite DomainWalls in Flat \nNanomagnets, Phys. Rev. Lett., 95 (2005) 197204 \n[34] S. Komineas, Rotating Vortex Dipoles in Ferromagnets, Phys. Rev. Lett., 99 (2007) 117202 \n[35] S. Komineas, N. Papanicolaou, Dynamics of vortex-antivortex pairs in ferromagnets, cond-\nmat/0712.3684 \n[36] J. Miltat, Spin Transfer: Towards a fair estimate of the Oersted field in pillar-like geometries, Talk on \nInternational Colloquium on Ultrafast Magnetization Processes, Irsee, Germany, Sept. 2008 (invited) \n[37] R. Hertel, C.M. Schneider, Exchange Explosions: Magnetization Dynamics during Vortex-Antivortex \nAnnihilation, Phys. Rev. Lett., 97 (2006) 177202 \n[38] K. Suzuki, G. Herzer, Soft Magnetic Nanostructures and Applications, in: Advanced Nanomagnetic \nStructures, D. Sellmyer and R. Skomski (Eds), Springer 2006, 365 \n \n \n "    },    {        "title": "2202.09732v1.Magneto_optical_Spectroscopy_with_RAMBO__A_Table_Top_30_T_Magnet.pdf",        "content": "Journal of the Physical Society of Japan SPECIAL TOPICS\nMagneto-optical Spectroscopy with RAMBO: A Table-Top 30 T Magnet\nFuyang Tay,1;2*Andrey Baydin,1;3 †Fumiya Katsutani,1and Junichiro Kono1;3;4;5\n1Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA\n2Applied Physics Graduate Program, Smalley-Curl Institute, Rice University, Houston, Texas, 77005, USA\n3Smalley-Curl Institute, Rice University, Houston, Texas, 77005, USA\n4Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA\n5Department of Materials Science and NanoEngineering, Rice University, Houston, Texas 77005, USA\nOptically probing materials in high magnetic fields can provide enlightening insight into field-modified electronic\nstates and phases, while optically driving materials in high magnetic fields can induce novel nonequilibrium many-body\ndynamics of spin and charge carriers. While there are high-field magnets compatible with standard optical spectroscopy\nmethods, they are generally bulky and have limited optical access, which prohibit performing state-of-the-art ultrafast\nand/or nonlinear optical experiments. The Rice Advanced Magnet with Broadband Optics (RAMBO), a unique 30-\nT pulsed mini-coil magnet system with direct optical access, has enabled previously challenging experiments using\nfemtosecond optical pulses, including time-domain terahertz spectroscopy, in cutting-edge materials placed in strong\nmagnetic fields. Here, we review recent experimental advances made possible by the first-generation RAMBO setup.\nAfter summarizing technological aspects of combining optical spectroscopic techniques with the mini-coil magnet, we\ndescribe results of magneto-optical studies of a wide variety of materials, providing new insight into the states and\ndynamics of four types of quasiparticles in solids – excitons, plasmons, magnons, and phonons – in high magnetic fields.\n1. Introduction\nCutting-edge materials such as topological insulators,1)\nlow-dimensional materials,2)and strongly correlated materi-\nals3)generally show unusual properties in the presence of an\nexternal magnetic field ( B), which breaks the time-reversal\nsymmetry of the system. Therefore, optical spectroscopy with\naccess to a strong applied magnetic field is an indispensable\ntool for studying such materials.\nNumerous e \u000borts have been made over the past decades\nto generate higher and higher magnetic fields for use in\nmagneto-optical spectroscopy experiments, both in university\nand national laboratory settings. Currently, fields B.15 T can\nbe readily produced by commercial superconducting magnets\nwith windows, whereas access to higher fields is typically\nlimited to special facilities in national laboratories.4–10)Mag-\nnetic fields utilized for condensed matter research can be ei-\nther static or pulsed. Pulsed magnets can generally produce\nhigher magnetic fields11)and even peak fields B>100 T can\nbe generated for magneto-optical experiments.12–17)\nAlthough some of these magnet systems are designed with\noptical access, there are limitations on the types of optical\nmeasurements that can be performed in high magnetic fields.\nThe primary issue is that optical access is usually available\nthrough optical fibers due to the large size of the magnet sys-\ntems. This raises challenges in optical alignment, and thus,\nthe optical setup cannot be easily modified for di \u000berent types\nof optical measurements, e.g., photoluminescence (PL) spec-\ntroscopy, magneto-optic Kerr e \u000bect measurements, and tera-\nhertz time-domain spectroscopy (THz-TDS). The dispersion\nand/or pulse broadening that occurs for light propagating in\nthe optical fibers severely restricts possible ultrafast opti-\ncal measurements, polarization-sensitive measurements, and\nexperiments involving broadband radiation. Additionally, in\n*ft13@rice.edu\n†baydin@rice.edupulsed high-field magnets, it takes the magnet a long time to\ncool down after each shot, which results in the wait time be-\ntween successive magnetic shots ranging from several min-\nutes to more than an hour.\nHere, we review magneto-optical studies performed\nwith the Rice Advanced Magnet with Broadband Optics\n(RAMBO), a unique table-top, mini-coil pulsed magnet sys-\ntem.18)The RAMBO magnet can generate a magnetic field\npulse with a peak field up to 30 T, and the sample cryostat pro-\nvides temperature control between 12 K and 300 K. RAMBO\nhas direct optical access through optical windows, which al-\nlows ultrafast optical spectroscopy experiments with mini-\nmal pulse broadening. The portability and small footprint of\nRAMBO permit its convenient incorporation into any optical\nspectroscopy setup.\nThis review article is organized as follows. First, in Sec-\ntion 2, we introduce the configuration of RAMBO and several\noptical spectroscopies that have been used with it. Next, in\nSection 3, we discuss representative magneto-optical studies\nof various excitations in materials such as excitons, plasmons,\nmagnons, and phonons that have been successfully performed\nwith the RAMBO system. Finally, in Section 4, we conclude\nthis review with an outlook for the future of RAMBO and\nhigh-field magneto-optical studies of condensed matter.\n2. Experimental Methods\n2.1 The magnet system\nThe RAMBO system consists of two cryostats – one cryo-\nstat for keeping the magnet cold, and one cryostat for control-\nling the temperature of the sample under study. The magnet\ncryostat (shown on the left in Fig. 1a), which contains the\nmagnet coil, is always filled with liquid nitrogen to ensure\nthat the magnet is cold before each shot as well as to rapidly\ncool down the magnet after each shot. The sample cryostat\n(shown on the right in Fig. 1a) is a commercial liquid helium\nflow cryostat (Cryo Industries, Inc., CFM-1738-102).\n1arXiv:2202.09732v1  [physics.optics]  20 Feb 2022J. Phys. Soc. Jpn. SPECIAL TOPICS\n012345\nTime (ms)-30-1501530B (T)\n1.8 2 2.229.2529.529.7530(a)\n(b)\nFig. 1. (a) Schematic diagram of the RAMBO system. Illustration by\nTanyia Johnson. (b) Magnetic field pulse time profile with a peak field of\n+30 T (blue) and\u000030 T (red). The inset displays an expanded view of the\nmagnetic field profile around the peak, showing a magnetic field variation of\n\u00180:5 T (or\u00182 %) in a time duration of \u00180:5 ms.\nThe sample is attached to the tip of a tapered sapphire pipe\nin such a way that it is located at the center of the magnet bore.\nThe bottom of the sapphire pipe was attached to the cryo-\nstat cold finger. The lowest achievable temperature is \u001812 K,\nmeasured through a temperature sensor attached to the tip of\nthe sapphire pipe close to the sample temperature. A heater is\nmounted on the cold finger to vary the sample temperature be-\ntween 12 K and 300 K. The two cryostats are connected with\na vacuum pipe, and they share the same vacuum space. Op-\ntical windows are held on both ends of the magnet system,\nenabling optical experiments in a transmission geometry.\nTwo coils have been constructed for RAMBO. Detailed in-\nformation on the first coil is provided in a previous publica-\ntion.18)The second-generation magnet coil is made by wind-\ning a 1 mm\u00021.5 mm AgCu wire around the metal bore. The\ncoil is composed of 13 layers, and each layer consists of 14-\n15 turns. The total wire length is approximately 16.5 m. The\ninductance of the coil at 77 K is 411.7 \u0016H, which is close to\nthe previous coil.18)\nA pickup coil is wrapped around the sapphire pipe near the\nsample to measure the generated magnetic field pulse. Fig-\nure 1b shows a measured temporal profile of a magnetic fieldpulse with a peak field of \u000630 T and a pulse duration of about\n4.7 ms. The magnetic field reaches its maximum at about 2 ms\nafter the capacitor bank discharge process is initiated by the\ntrigger pulse. The inset in Fig. 1b shows that the magnetic\nfield value only changes by about 2 % within 500 \u0016s around\nthe peak. Therefore, it is reasonable to assume that the sam-\nple experiences a constant magnetic field at the peak value for\nthe duration of 500 \u0016s. A versatile magnet control unit is used\nto synchronize the magnetic field pulse with other instruments\nsuch as the output from the amplified Ti:sapphire laser.\nThe maximum sample size is restricted by the diameter of\nthe magnet bore, which is 12 mm. The numerical aperture of\nthe sapphire pipe is 0.03.18)Nevertheless, the cylindrical sap-\nphire pipe has been replaced by a conical sapphire pipe with\nan outer diameter of 8 mm at the side to mount the sample and\nan outer diameter of 12.6 mm at the side contacted to the cold\nfinger, in order to have an even larger numerical aperture.\n2.2 Time-resolved photoluminescence spectroscopy\n123906-4 Noe et al. Rev. Sci. Instrum. 84, 123906 (2013)\nFIG. 3. Illustration of the optical Kerr gate setup for time-resolved photoluminescence experiments. Light blue arrows indicate the direction of th e light. After\nexcitation through a window on the helium flow cryostat side with the output of an amplified Ti:sapphire laser, the photoluminescence is collected thro ugh the\nwindow on the magnet cryostat side and focused onto a Kerr medium located between two crossed polarizers where it overlaps in space with a gate pulse. Th e\ngate pulse causes transient birefringence in the medium so the polarization of the photoluminescence that passes the medium during the medium’s resp onse to\nthe gate is rotated and can then partially pass through the second polarizer. After making a series of magnet shots with an incrementally changing time delay\nbetween the excitation and gate pulses, the time-resolved photoluminescence intensity can be mapped as a function of time and wavelength.\nresponse to the gate pulse, the Kerr medium acts as a wave-\nplate rotating the polarization of the PL, allowing some of\nthe PL to pass the second polarizer into the collection. In this\nway, the time-resolved PL can be mapped out as a function ofwavelength and time.\n2. Time-integrated photoluminescence and\nabsorption spectroscopy\nThe time-integrated PL is collected with the same\noptical setup, except that the crossed polarizers and the Kerr\nmedium are removed and the gate beam is blocked. Because\nthe PL emission from our sample occurs on a time scale ofns or less, the magnetic field is essentially held constant dur-\ning the excitation and emission processes. Under this scheme,\nthe time-integrated PL is collected upon a single laser shotand single magnetic field shot. In addition to the strong exci-\ntation measurements, we use a laser diode (World Star Tech,\nTECiRL-15G-780) centered at 780 nm for weak excitation PLmeasurements. The optical layout for the weak excitation PL\nmeasurements is the same as the time-integrated PL measure-\nments described above, except a laser diode is used insteadof the amplified Ti:sapphire laser. The laser diode is mod-\nulated to be turned on for 400 µs at the peak of the mag-\nnetic field. The magnetic field variation at the peak of thefield for this time period is ∼4 parts in 300, less than 2% [see\nFig.2(b)].For transmission measurements, we use a LED (Thor-\nlabs, Inc., LED880L-50) centered at 880 nm with emissionfrom 800 nm to 970 nm; the optical layout for the trans-\nmission measurements is not shown. We take two transmis-sion measurements while the sample is located in the mag-\nnet: one at a high magnetic field, T(B)\nInMag, and the other at\nB=0,T(B=0)InMag. Then, we take two more measurements\nusing the helium flow cryostat without connecting the mag-net cryostat. One transmission measurement with the sam-\nple placed over an aperture where the sample is at the sametemperature as it was in the magnet, T(B=0), and another\nmeasurement of the incident light without the sample in thelight path, T\nInc. The reference spectrum, TRef, is calculated\nfromTIncby assuming a constant wavelength dependence for\nthe reflection loss for GaAs at 12.5 K and the sapphire platethat the sample was mounted on. The transmittance is calcu-\nlated by T(B)=\nT(B=0)\nTRefT(B)InMag\nT(B=0) InMag. Finally, we calculate the\nabsorbance, A(B), as A(B)=−log (T(B)).\nIII. EXPERIMENTAL RESULTS AND DISCUSSION\nFigure 4displays the absorbance and weak excitation PL\nspectra as a function of magnetic field using the LED and laser\ndiode, respectively. At 0 T, the absorbance shows a stair-step\nprofile with excitonic peaks located near the subband edges,\ntypical of a quasi-2D quantum well sample.50The subband\nedges can be identified as the E 1H1located at 1.325 eV , the\nFig. 2. Illustration of the time-resolved photoluminescance spectroscopy\nsystem using the optical Kerr gating method. The output from the amplified\nTi:sapphire laser is split into an excitation beam and a gate beam. The light\nblue arrows indicate the propagation direction of the excitation beam. After\nexcitation, the PL emitted by the sample is collected and focused onto a Kerr\nmedium placed between a pair of crossed polarizers. The gate beam induces\ntransient birefringence in the Kerr medium so that the PL that passes through\nthe medium simultaneously is rotated and can then partially pass through\nthe second polarizer. A delay line is used to change the time delay between\nthe excitation beam and the gate beam to map out the emission dynamics.\nAdapted from Ref. 18.\nThe RAMBO system has been incorporated into a time-\nresolved PL spectroscopy setup, which is shown in Fig. 2, to\nmeasure ultrafast light emission dynamics at high magnetic\nfields.18)The output of the Ti:sapphire regenerative amplifier\n(Clark-MXR, Inc., CPA-2001) centered at 775 nm with 150 fs\npulse duration, 1 kHz repetition rate, and pulse energy up to\n5\u0016J is split into an excitation beam and a gate beam. The ex-\ncitation beam is focused onto the sample inside the cryostat\nwith a spot size of about 500 \u0016m. A chopper is used to reduce\nthe repetition rate of the excitation beam to 50 Hz. The mag-\nnet control unit is synchronized with the pulsed laser so that\nthe excitation pulse hits the sample at the peak of the magnetic\nfield pulse. As discussed before, the magnetic field variation\nis negligible for transient phenomena of the order of picosec-\nonds excited by a femtosecond pulse.\n2J. Phys. Soc. Jpn. SPECIAL TOPICS\nTwo o \u000b-axis parabolic mirrors are used to collect and refo-\ncus the emission onto a Kerr medium, toluene. Two crossed\npolarizers are placed before and after the toluene. Without a\ngate pulse, the PL is blocked by the crossed polarizers. Con-\nversely, the Kerr medium rotates the polarization of the PL\nwhen a gate pulse with a polarization of 45\u000ewith respect to\nthe crossed polarizers hits the Kerr medium simultaneously,\nallowing some of the PL to pass the second polarizer into the\ndetector scheme. The path distance of the gate pulse is con-\ntrolled by a delay line to map out the PL dynamics. After the\nKerr medium, a long-pass filter is used to eliminate the gate\npulse. The PL is focused onto an optical fiber and measured\nwith a silicon charge-coupled device (CCD) camera attached\nto a grating spectrometer.\n2.3 Time-integrated photoluminescence spectroscopy\nThe time-integrated PL spectroscopy setup is the same as\nthe time-resolved PL setup depicted in Fig. 2 except that the\ncrossed polarizers, the Kerr medium (toluene), and the gate\nbeam are removed. In addition to the strong pulsed excitation\nsource, a laser diode (World Star Tech, TECiRL-15G-780)\nand a continuous-wave Ar ion laser (Melles Griot) are used as\nlight sources for di \u000berent measurements. Both light sources\nare modulated to be turned on for a short duration ( <500\u0016s)\nat the peak of the magnetic pulse.\n2.4 Transmission spectroscopy\nTwo light sources are used for transmission measurements,\na light-emitting diode (LED) (Thorlabs, Inc., LED880L-50)\nand a supercontinuum laser (EXU6, NKT Photonics). The\nsupercontinuum laser generates pulses with a repetition rate\nof 80 MHz, a pulse duration of >1 ps, and a pulse energy of\naround 0.01 nJ. Light transmits through the sample inside the\ncryostat, and a spectrometer is used to detect and analyze the\ntransmitted light. The light beams are modulated to study the\ntransmission of the sample only at the peak magnetic field. A\nquarter-wave plate is used to create circularly polarized light\nfor some measurements.\n27\n6DPSOH3ULVP\n6DSSKLUH 3ULVP\n\u000bD\f\n\u000bE\f\u000bF\f &RLO\n&RLO\nFigure 3.8 : (a) An illustration of the sample for a V oigt configuration and (b) an image of\nthe fabricated sample. (c) The schematic illustration of the V oigt configuration, showing\nthe light propagation.\nrate was 80 MHz. When this light source was modulated with the chopper operated in\nthe way shown in Fig. 3.5, each 500-µs pulse contained about 40,000 pulses. The pulse-\nto-pulse fluctuations were averaged well by this enormous number of pulses. The light\nwas circularly-polarized by a combination of a polarizer and a quarter-wave plate. Thepolarization was changed by rotating the quarter-wave plate. The cryostats were placed\non a high-load z-stage (MLJ150, Thorlabs) installed on an xy-stage. The position needed\nto be adjusted after reaching the target temperature since the inside parts, especially the\ncopper cold finger, were shrunk or expanded as the temperature changed. The transmitted\nlight was focused onto an optic fiber, which delivered the light to a grating monochromator\n(IsoPlane SCT 320, Teledyne Princeton Instruments) equipped with a CCD camera (ProEM\nHS:512BX3, Teledyne Princeton Instruments). For measuring photoexcited emission spec-tra, a continuous-wave laser (Ar ion laser, Melles Griot) and a regenerative amplifier laser\n(CPA2001, Clark MXR) were employed for CW excitation and pulsed excitation, respec-\ntively. Suitable long-pass filters were placed in front of the collection fiber to eliminate theexcitation laser.\nFig. 3. (a) Illustration of the sample holder for transmission measurements\nin the V oigt configuration. The sample is sandwiched between a pair of prisms\nand attached to a sapphire substrate. (b) Image of the fabricated sample.\n(c) Schematic diagram of the V oigt configuration setup. The blue solid lines\nrepresent the tapered sapphire pipe, and the red solid line indicates the light\npropagation path.\nMeasurements in the V oigt geometry can also be made with\nthe RAMBO system. A pair of prisms can be used to sand-\nwich the sample, as shown in Fig. 3a. The light beam passes\nthrough the sample along the normal direction after the firsttotal internal reflection, and then it is reflected back by the\nsecond prism to leave the cryostat. The magnetic field is par-\nallel to the sample surface and perpendicular to the light prop-\nagation direction.\n2.5 Single-shot terahertz time-domain spectroscopy\nA broadband terahertz (THz) pulse (0.25-1.6 THz) is gener-\nated either in a Mg-doped stoichiometric LiNbO 3crystal with\nthe tilted-pulse-front excitation method19)or in a ZnTe crystal\nby optical rectification for a broader bandwidth. The emitter\ncrystal is pumped by the output of an amplified Ti:sapphire\nlaser (Clark-MXR, Inc., CPA-2001) centered at 775 nm with\n150 fs pulse duration and 1 kHz repetition rate. The generated\nTHz pulse is focused onto the sample mounted on the sap-\nphire pipe with a pair of parabolic mirrors.\nThe transmitted THz radiation is detected through electro-\noptic sampling20, 21)with a (110) ZnTe detection crystal. In\nconventional THz-TDS, a THz waveform is recorded in a\npoint-by-point manner with the step-scan method,22)where\nthe measurement is repeated for di \u000berent time delays between\nthe THz pulse and the gate pulse arriving at the detection crys-\ntal. In this way, only one data point of the THz waveform can\nbe collected in a magnet shot. However, the wait time between\nshots of our magnet system is on the order of several minutes\nfor shots up to 30 T. Therefore, this method is time-consuming\nand ine \u000ecient to obtain THz waveforms.\nL1 \nIris reflective \nechelon \nPBS L2 \nL3 \nL4 WP \nEO \ncrystal L5 polarizer \nCL L6 \nCCD \nL7 L8 \nPM \nTHz path  \nafter sample \n in magnet IR probe \npath \nQWP pulse front \n before and  \nafter echelon \nreflective \nechelon \nFig. 1. Schematic diagram of the single-shot detection scheme. Two 10 ×telescopes, L1-L4,\nare used to expand the optical gate beam so that a relatively uniform intensity profile reflects\noffof the echelon mirror. After encoding time delay information onto the intensity profile\nof the gate beam with the echelon optic, the intensity profile at the plane of the echelon\nsurface is imaged onto the electro-optic sampling crystal after reflecting off of a pellicle\nbeam splitter, PBS, overlapping with the THz beam and finally onto a CCD camera with\nimage relay optics, L5-L8. A Wollaston prism, WP, is used to separate the two orthogonal\npolarizations of the elliptically polarized gate beam after a quarter-wave plate, QWP, and a\ncylindrical lens, CL, is used to focus the beam in one direction so that both polarizations\ncan be measured with a single CCD camera. Note that the Wollaston prism and the resulting\nseparated beams are shown rotated 90◦with respect to the actual orientation.\n2. Experimental setup\nWe use an amplified Ti:sapphire laser system (Clark-MXR, Inc. CPA-2001) to generate laser\npulses centered at 775 nm with 1 kHz repetition rate, 150 fs pulse duration, and 1 mJ pulse\nenergy. A beam splitter is placed in the path of the beam to reflect 20% of the power for optically\npumping a sample and transmit 80% of the power for the THz generation and detection portions.\nTwo mirrors are placed on a 500 mm linear stage in the path of the portion of the beam for THz\ngeneration and detection to provide a maximum optical delay up to 3.33 ns between the optical\npump and the THz probe pulses. A second beam splitter is placed in the path after the delay stage\nto reflect 10% of the remaining power for single-shot THz detection and transmit 90% of the\nremaining power for THz generation. A second linear delay stage after this beam splitter is used\nto adjust the timing of the gate pulse with respect to the THz pulse. The THz generation section\nutilizes the tilted-pulse-front excitation method [30] for generating THz pulses in Mg-doped\nstoichiometric LiNbO 3or the more common ZnTe generation for a greater THz bandwidth. The\n                                                                                            Vol. 24, No. 26 | 26 Dec 2016 | OPTICS EXPRESS 30331 \nFig. 4. Schematic diagram of the detection scheme of the single-shot THz-\nTDS. L1-L4 are lenses to expand the infrared gate beam so that the trans-\nmitted gate beam has a relatively uniform intensity profile. Time delay infor-\nmation is encoded onto the intensity profile of the gate beam upon reflection\nfrom the echelon mirror. The gate beam overlaps with the THz beam transmit-\nted through the sample by using a pellicle beamsplitter, PBS. The beams are\nfocused onto an electro-optic sampling crystal, EO crystal. A quarter-wave\nplate, QWP, a cylindrical lens, CL, and a Wollaston prism, WP, are used to\ndetect the birefringence in the EO crystal induced by the electric field of the\nTHz beam. Adapted from Ref. 23.\nDi\u000berent rapid scanning methods have been demonstrated\nto overcome this issue.24–27)We employ the single-shot THz\ndetection technique in our THz-TDS setup.23, 28–30)A large\nreflective echelon mirror is utilized to encode time delay in-\nformation on a single gate pulse.23, 29, 31–33)The size of the\ncustomized echelon mirror is 20 mm \u000220 mm, and it consists\n3J. Phys. Soc. Jpn. SPECIAL TOPICS\nof 1000 steps of 20 \u0016m width and a step height of 5 \u0016m. As\nshown in the inset of Fig. 4, the reflected gate beam exhibits\na stair-step wavefront profile with the step height correspond-\ning to an incremental delay of \u001833 fs. The initial 150 fs gate\npulse becomes a\u001833 ps pulse upon reflection from the ech-\nelon mirror. Two 10 \u0002telescopes are used to expand the gate\nbeam before the echelon mirror so that the intensity profile of\nthe beam that reflects o \u000bof the echelon mirror is relatively\nuniform. A quarter-wave plate, a Wollaston prism, and a sili-\ncon CCD camera are placed at the path of the gate pulse after\nthe detection crystal to measure the induced birefringence in\nthe crystal that is proportional to the electric field amplitude\nof the THz pulse.20, 21)\nA pair of wire-grid polarizers is placed before and after the\nsample to ensure that the THz pulse is linearly polarized. The\nsecond polarizer after the sample can be rotated by 90\u000efor\nFaraday and Kerr rotation measurements. It should be noted\nthat both Faraday and Kerr rotation signals can be extracted\nin a transmission configuration.34–37)The THz pulse that di-\nrectly transmits through the sample contains only the Faraday\nrotation signal. Nevertheless, the back reflection pulse expe-\nriences both Faraday and Kerr rotation due to additional re-\nflection events at the interfaces. Therefore, the Kerr rotation\nsignal can be extracted by subtracting the Faraday rotation\nfrom the polarization rotation of the back reflection pulse.\nFor optical-pump /THz-probe spectroscopy experiments, an\nadditional beam splitter is used to split the near-infrared pulse\ninto the pump beam for sample excitation and the beam for\nTHz generation and detection.23)The optical delay between\nthe optical pump and the THz probe pulses is controlled by a\nlinear delay stage.\n3. Magneto-optical Studies of Excitons, Plasmons,\nMagnons, and Phonons with RAMBO\n3.1 Excitons\nAn exciton is a bound state of an electron in the conduc-\ntion band and a hole in the valence band due to the Coulomb\ninteraction. It usually forms when a material is excited by\na photon of energy higher than the band gap. The envelope\nwavefunction of the exciton is analogous to that of a hydro-\ngen atom. This section only considers Wannier-Mott excitons\nin semiconductors. Owing to the large dielectric constant in\nsemiconductors and smaller reduced mass of the exciton, the\nWannier-Mott exciton has a Bohr radius much larger than the\nlattice spacing and a small binding energy ( \u00181-100 meV).\nAn example of the energy states for a quantum well (QW)\nwithout considering Coulomb interactions is illustrated in\nFig. 5. Two electron subbands and three hole subbands are\nconfined in the QW. A notation is used to denote the transition\nbetween two bands, for instance, E 1H1represents the transi-\ntion between the electron 1 band and the heavy hole 1 band.\nWhen an external magnetic field is applied along the growth\ndirection, the states split due to Landau quantization. Neand\nNhdenote the electron and hole Landau quantum numbers,\nrespectively.\nThe magnetic field, B, dependence of the energies of ex-\ncitonic states is critically dependent on key exciton parame-\nters such as the reduced e \u000bective mass, \u0016\u0003, binding energy,\ne\u000bective Bohr radius, and e \u000bective g-factor, g\u0003. Therefore,\nmagneto-optical spectroscopy has been commonly used to\ndetermine these important materials parameters for various\n13\n(OHFWURQ\u0014(OHFWURQ\u0015\n+HDY\\\u0003KROH\u0014\n/LJKW\u0003KROH\u0014\n+HDY\\\u0003KROH\u0015(\n\u000b\u0014\u000f\u0014\f\u000b\u0015\u000f\u0015\f\n\u000b\u0013\u000f\u0013\f\n]\n(\u0014+\u0014\n(\u0014/\u0014\n(\u0015+\u0015\n%1K \u0003\u0013\u0003\n\u0014\n\u00151H \u0003\u0013\u0014\n&RQGXFWLRQ\u0003%DQG\u0015\n9DOHQFH\u0003EDQG\u000bD\f\u0003% \u0003\u0013 \u000bE\f\u0003%\u0003\u0013\nFigure 2.2 : The illustration of the Landau levels for a quantum well.Fig. 5. (a) Illustration of the electron energy states for a QW at B=0.\nTwo electron subbands and three hole subbands are considered in this case.\n(b) The electron and hole subbands split into multiple Landau levels (LL)\nwhen a magnetic field is applied perpendicular to the QW.\nsemiconductors.\nApproximate solutions for excitonic levels can be obtained\nin the low- Blimit or the high- Blimit. A dimensionless pa-\nrameter,\r=16\u00192~3\"2B=\u0016\u00032e3,38)is introduced to define the\nlow-Band high- Bregimes where ~is the reduced planck con-\nstant,\"is the permittivity, and eis the electronic charge. In the\nlow-Bregime (\r\u001c1), the magnetoexciton transition energy,\nE, can be approximately expressed as38, 39)\nE(B)=E(B=0)+\u001bB2\u0006g\u0003\u0016BB; (1)\nwhere\u0016Bis the Bohr magneton and \u001bis the diamagnetic-shift\ncoe\u000ecient. The second term is the diamagnetic term that is\nproportional to B2and the last term is the energy splitting term\ndue to the Zeeman e \u000bect, which can be probed by circularly\npolarized light. On the other hand, in the high- Bregime (\r\u001d\n1),Eexhibits a linear Bdependence and it can be fit by the\nequation below38, 39)\nE(B)=Eg+ \nN+1\n2!~eB\n\u0016\u0003\u0006g\u0003\u0016BB; (2)\nwhere EgandNare the band gap and Landau quantum num-\nber, respectively.\n3.1.1 Magnetoexcitons in InGaAs /GaAs quantum wells\nThe InGaAs /GaAs sample was grown by molecular beam\nepitaxy and contained 15 periods of 15-nm-wide InGaAs\nQWs with 8-nm-wide GaAs barriers. Figures 6a and 6b show\nthe ratios of the intensity of transmitted light at a finite mag-\nnetic field, I(B), to the intensity of transmitted light at zero\nmagnetic field, I(0), using light with di \u000berent circular polar-\nizations (\u001b+and\u001b\u0000). The supercontinuum laser was used as\nthe light source. The magnetic field dependence of magne-\ntoexcitonic transition energies from E 1H1(0,0) to E 1H1(3,3)\nare mapped out, and \u0016\u0003is calculated to be (0.065 \u00060.05) m0\naccording to Eq. (2).\nThe sample was also measured in the V oigt geometry where\nthe applied magnetic field was parallel to the quantum well\nplane. The energy shifts of the E 1H1(0,0) transition from the\nFaraday and V oigt configuration measurements are plotted in\nFig. 6c. The energy shift in the V oigt geometry is smaller than\nthat in the Faraday geometry because the size of the exciton\nperpendicular to the QW is smaller.\n4J. Phys. Soc. Jpn. SPECIAL TOPICS\nFigure 6d displays the result of magneto-PL measure-\nments using the Ar ion laser at a wavelength of 488 nm.\nThe E 1H1(0,0) exciton peak at 0 T in the PL measurement\nis 6 meV lower than that observed in the absorption spectrum,\nwhich is likely due to alloy disorder in the QW. The PL inten-\nsity and energy increase as Bincreases. The emission from\nthe E 1H1(1,1) state is shown in the inset.\n30\n\u000bD\f \u000bE\f\n\u000bF\f\n\u000bG\f \u000bH\f\n\u000b\u0014\u000f\u0014\f\n\u000b\u0015\u000f\u0015\f\n\u000b\u0016\u000f\u0016\f\u000b\u0013\u000f\u0013\f\n\u000b\u0014\u000f\u0014\f\u000b\u0015\u000f\u0015\f\u000b\u0016\u000f\u0016\f\n\u000b\u0013\u000f\u0013\f\u000b\u0014\u000f\u0014\f\n\u000b\u0015\u000f\u0015\f\n\u000b\u0016\u000f\u0016\f\u000b\u0013\u000f\u0013\f\n\u000b\u0014\u000f\u0014\f\u000b\u0015\u000f\u0015\f\u000b\u0016\u000f\u0016\f\n\u000b\u0013\u000f\u0013\f\u0014\u0011\u0016\n\u0014\u0011\u0015\u0014\u0011\u0014\n\u0014\u0011\u0013\n\u0013\u0011\u001c\u0013\u0011\u001b\n\u0013\u0011\u001a,\u000b%\f\u0012,\u000b\u0013\f\n\u0014\u0011\u0017\u0015 \u0014\u0011\u0017\u0013 \u0014\u0011\u0016\u001b \u0014\u0011\u0016\u0019 \u0014\u0011\u0016\u0017 \u0014\u0011\u0016\u0015 \u0014\u0011\u0016\u0013\n(QHUJ\\\u0003\u000bH9\f\u000b\u0014\u000f\u0014\f\u000b\u0015\u000f\u0015\fı\u0010\n\u000b\u0013\u000f\u0013\f\u0010\u0014\u00157\n\u000e\u0014\u00157\u0014\u0011\u0015\n\u0014\u0011\u0014\n\u0014\u0011\u0013\n\u0013\u0011\u001c\u0013\u0011\u001b,\u000b%\f\u0012,\u000b\u0013\f\n\u0014\u0011\u0017\u0015 \u0014\u0011\u0017\u0013 \u0014\u0011\u0016\u001b \u0014\u0011\u0016\u0019 \u0014\u0011\u0016\u0017 \u0014\u0011\u0016\u0015 \u0014\u0011\u0016\u0013\n(QHUJ\\\u0003\u000bH9\f\u000b\u0014\u000f\u0014\f\u000b\u0015\u000f\u0015\f\u000b\u0016\u000f\u0016\fı\u0010\n\u000b\u0013\u000f\u0013\f\u000e\u001a7\n\u0010\u001a7\u0013\u0011\u0018\n\u0013\u0011\u0017\n\u0013\u0011\u0016\n\u0013\u0011\u0015\n\u0013\u0011\u0014$WWHQXDQFH\n\u0014\u0011\u0017\u0015 \u0014\u0011\u0017\u0013 \u0014\u0011\u0016\u001b \u0014\u0011\u0016\u0019 \u0014\u0011\u0016\u0017 \u0014\u0011\u0016\u0015 \u0014\u0011\u0016\u0013\n(QHUJ\\\u0003\u000bH9\f,Q*D$V\u0012*D$V\u00034:\n\u0014\u0015.(\u0014+\u0014\n(\u0014/\u0014\nFigure 4.1 : (a) An attenuation spectrum at 12 K, the ratios of the intensity at the magnetic\nfieldI(B)to the intensity without magnetic field I(0)at (b)±7 T and (c) ±12 T probed by\ntheσ−polarization, and magnetic-field-dependent I(B)/I(0)spectral map probed by the\n(d)σ+and the (e) σ−polarizations, respectively, for the InGaAs quantum well sample.32\n\u000bD\f\n \u000bF\f\n\u000bE\f\u0017\n\u0016\u0015\n\u0014\n\u00131RUPDOL]HG\u0003,QWHQVLW\\\n\u0014\u0011\u0016\u0019 \u0014\u0011\u0016\u0018 \u0014\u0011\u0016\u0017 \u0014\u0011\u0016\u0016 \u0014\u0011\u0016\u0015 \u0014\u0011\u0016\u0014 \u0014\u0011\u0016\u0013\n(QHUJ\\\u0003\u000bH9\f,Q*D$V\u0012*D$V\u00034:\n&:\u0003H[FLWDWLRQ\u0016\u00137\n\u00137\u0014\u0011\u0017\u0017\n\u0014\u0011\u0017\u0015\n\u0014\u0011\u0017\u0013\n\u0014\u0011\u0016\u001b\n\u0014\u0011\u0016\u0019\n\u0014\u0011\u0016\u0017\n\u0014\u0011\u0016\u0015 (QHUJ\\\u0003\u000bH9\f\n\u0010\u0016\u0013 \u0010\u0015\u0013 \u0010\u0014\u0013 \u0013\u0014\u0013 \u0015\u0013 \u0016\u0013\n0DJQHWLF\u0003ILHOG\u0003\u000b7\f\u000b\u0013\u000f\u0013\f\u000b\u0014\u000f\u0014\f\u000b\u0015\u000f\u0015\f \u000b\u0016\u000f\u0016\f\n\u000b\u0013\u000f\u0013\f\u000b\u0014\u000f\u0014\f$EV\u0011\u0003ı\u000e\n$EV\u0011\u0003ı\u0010\n3/\nFigure 4.2 : PL spectra for the InGaAs quantum well sample excited by a continuous-wave\nlaser at (a) 0 T and (b) 30 T. (b) A magnetic-field-dependent PL map from −30 T to +30 T.\n(c) Photon energies of the observed peaks as a function of magnetic field. Black open\nand solid triangles denote the peak positions extreacted from transmission spectra probed\nby the σ+and the σ−polarizations, respectively. Black dotted lines denote the transitions\nobserved for σ+polarization. Red solid squares denote the peak positions extracted from\nthe photoluminescence spectra.35\n\u000bD\f \u000bE\f \u0014\u0011\u0013\n\u0013\u0011\u001c\u0013\u0011\u001b\n\u0013\u0011\u001a\u0013\u0011\u0019\n\u0013\u0011\u0018\n\u0013\u0011\u0017\n\u0013\u0011\u0016$WWHQXDQFH\n\u0014\u0011\u0017\u001b\u0014\u0011\u0017\u0019\u0014\u0011\u0017\u0017\u0014\u0011\u0017\u0015\u0014\u0011\u0017\u0013\u0014\u0011\u0016\u001b\u0014\u0011\u0016\u0019\u0014\u0011\u0016\u0017\u0014\u0011\u0016\u0015\n(QHUJ\\\u0003\u000bH9\f\u0013\u00037\u0014\u0013\u0011\u0018\u00037\u0015\u0013\u0011\u0015\u00037\u0016\u0013\u0011\u0013\u00037(\u0014+\u0014\n(\u0014/\u0014(\u0015+\u0015,Q*D$V\u0012*D$V\u00034:\n\u001a\u001a\u0003.\u0003\u00039RLJW\u0015\u0013\n\u0014\u0018\n\u0014\u0013\n\u0018\n\u0013\n\u0010\u0016\u0013\u0010\u0015\u0013\u0010\u0014\u0013 \u0013\u0014\u0013\u0015\u0013\u0016\u0013\n0DJQHWLF\u0003)LHOG\u0003\u000b7\f,Q*D$V\u0012*D$V\u00034:\n(\u0014+\u0014\u000b\u0013\u000f\u0013\f\nı\u0010\u000b)DUDGD\\\f\nı\u000e\u000b)DUDGD\\\f\n9RLJW\nFigure 4.5 : (a) Magnetoabsorption spectra for the InGaAs quantum well sample measured\nin the V oigt geometry. (b) Photon energies of the E 1H1(0,0)peak measured in the Faraday\ngeometry probed by σ+(black open triangles) and the σ−(black solid triangles) polariza-\ntions and measured in the V oigt geometry (red open circle).\nrelationships, and the coherence between dipoles develops spontaneously. This leads to a\nburst of radiation after a finite delay time. Figure 4.3(a) shows SF spectra at representativemagnetic fields. Together with the emission from the E\n1H1(0,0)transition, emission from\nthe E 1H1(1,1)and E 1H1(2,2)transitions are observed and noted in the graph by the arrow\nlines. Figure 4.3(b) shows the magnetic-field dependent SF map from −30 T to +30 T. The\nemission peaks from the E 1H1(3,3)and E 1H1(4,4)are resolved and noted by dashed lines.\nThe energies of emission peaks are displayed in Figure 4.3(c). Figure 4.4 summarizes the\npeak energies obtained in transmission, PL and SF.\nFigure 4.5(a) shows absorption spectra in magnetic fields taken in the V oigt geometry.\nFigure 4.5(b) shows the peak energies as a function of magnetic field for both the Faraday\nand V oigt geometries. The magnetic field was applied parallel to the quantum well plane.\nBecause the size of the exciton perpendicular to the well is smaller, the degree of the shift\nis smaller than the case in the Faraday geometry.\nFinally, as another example, we performed the magnetoPL experiments on a GaAs/\nAlGaAs quantum well sample as demonstrated in Fig. 4.6. The full width at half maximum\n(FWHM) of the peak at 30 T is ∼3 meV , which is much sharper than that in the InGaAs\nMagnetic Field (T)Magnetic Field (T) Energy shift (meV)\nMagnetic Field (T)Energy (eV)\nEnergy (eV)Magnetic field (T)Energy (eV)\nI(B)/I(0) Intensity (a.u.)(a)\nPL(b)\n(c) (d)\nFig. 6. Magnetic-field-dependent I(B)=I(0) spectral map probed by (a) \u001b+\nand (b)\u001b\u0000polarizations, respectively. (c) Energy shifts of the E 1H1(0,0)\npeak measured in the Faraday geometry using probe beams with \u001b+and\u001b\u0000\npolarization and measured in the V oigt geometry. (d) Magneto-PL spectrum\nfrom\u000030 T to 30 T. The inset displays an expanded view of the PL map.\n3.1.2 Magnetoexcitons in InSe\nA\u001850\u0016m thick InSe thin film was prepared by cleaving\na single crystal. The magnetic field was applied along the\nc-axis of InSe. Figure 7a shows magnetoabsorption spectra\nup to 30 T probed by the \u001b+and\u001b\u0000polarizations at 85 K.\nThe lowest exciton peak is observed at 1.321 eV at 0 T. The\npeak shifts with increasing magnetic field and a second peak\nemerges at high B. The peaks at B,0 are attributed to N=0\nandN=1 transition. The N=0 peak energies are plotted\nin Fig. 7c and fit by Eq. (1). The g\u0003and\u001bare obtained as\n1:06\u00060:04 and 4:08\u000210\u00003meV/T2, respectively.\n3.1.3 Superfluorescence from In 0:2Ga0:8As quantum wells\nWhen numerous two-level dipoles are initially prepared\nin an incoherent state and confined in a small volume, they\nspontaneously develop macroscopic coherence from vacuum\nfluctuations, leading to a burst of radiation after a finite de-\nlay time. This phenomenon is known as superfluorescence\n(SF),40–44)and it was first observed in atomic and molecu-\nlar systems.45–47)The observations of SF in semiconductors\nhad been challenging owing to the fast scattering rates of car-\nriers. Noe and co-workers have demonstrated that in a high\nmagnetic field perpendicular to a QW, incoherent electron-\nhole pairs excited by an ultrafast optical pulse spontaneously\n38\n\u000bD\f \u000bE\f\n\u000bF\f\u0014\u0011\u0016\u0015\u0019\n\u0014\u0011\u0016\u0015\u0018\n\u0014\u0011\u0016\u0015\u0017\n\u0014\u0011\u0016\u0015\u0016\n\u0014\u0011\u0016\u0015\u0015\n\u0014\u0011\u0016\u0015\u0014(QHUJ\\\u0003\u000bH9\f\n\u0016\u0013\u0015\u0018\u0015\u0013\u0014\u0018\u0014\u0013\u0018\u0013\n0DJQHWLF\u0003ILHOG\u0003\u000b7\f˪\u000e\n˪\u0010,Q6H\n\u001b\u0018.\n1 \u0003\u0013\n\u0014\u0011\u0016\u0017\u0019\n\u0014\u0011\u0016\u0017\u0017\u0014\u0011\u0016\u0017\u0015\u0014\u0011\u0016\u0017\u0013\u0014\u0011\u0016\u0016\u001b\u0014\u0011\u0016\u0016\u0019\n\u0014\u0011\u0016\u0016\u0017\n\u0014\u0011\u0016\u0016\u0015(QHUJ\\\u0003\u000bH9\f\n\u0019\u0013\u0018\u0013\u0017\u0013\u0016\u0013\u0015\u0013\u0014\u0013\u0013\n0DJQHWLF\u0003ILHOG\u0003\u000b7\f,Q6H\n\u0017.\n1 \u0003\u0013\nı \u0003\u0017\u0011\u0016\u0016î\u0014\u0013\u0010\u0016PH9\u00127\u0015\u0013\u0011\u0015\u001a\u0018\u0003PH9\u00127\na%\u0015a%\u0014\u0011\u0019\u0013\n\u0014\u0011\u0018\u0018\u0014\u0011\u0018\u0013\n\u0014\u0011\u0017\u0018\n\u0014\u0011\u0017\u0013\n\u0014\u0011\u0016\u0018 (QHUJ\\\u0003\u000bH9\f\n\u0019\u0013\u0018\u0013\u0017\u0013\u0016\u0013\u0015\u0013\u0014\u0013\u0013\n0DJQHWLF\u0003ILHOG\u0003\u000b7\f1 \u0003\u00141 \u0003\u00151 \u0003\u00161 \u0003\u0017\n1 \u0003\u0013,Q6H\n\u0017.\n/DQGDX\u0003OHYHOV1 \u0003\u0018 1 \u0003\u0019\nFigure 4.8 : Photon energy of the absorption peaks in InSe. (a) The N=0 peak probed by\nσ+(black) and the σ−(red) polarizations up to 30 T at 85 K. (b) The N=0 up to 65 T at\n4 K. (c) All observed transitions at 4 K.(b)37\n\u000bD\f \u000bE\f\n\u0018\n\u0017\n\u0016\n\u0015\n\u00142'\n\u0014\u0011\u0017\u0013 \u0014\u0011\u0016\u001b \u0014\u0011\u0016\u0019 \u0014\u0011\u0016\u0017 \u0014\u0011\u0016\u0015 \u0014\u0011\u0016\u0013\n(QHUJ\\\u0003\u000bH9\f˪\u000e\n˪\u0010\u00177\u001b7\u0014\u00157\u0014\u00197\u0015\u00137\u0015\u00187\u0016\u00137\n\u00137\n,Q6H\n\u001b\u0018\u0003.1 \u0003\u0013\n\u0014\n\u0018\n\u0017\n\u0016\n\u00152'\n\u0014\u0011\u0019\u0013\u0014\u0011\u0018\u0018\u0014\u0011\u0018\u0013\u0014\u0011\u0017\u0018\u0014\u0011\u0017\u0013\u0014\u0011\u0016\u0018\u0014\u0011\u0016\u0013\n(QHUJ\\\u0003\u000bH9\f\u00137\u0014\u0014\u0011\u00137\u0014\u001c\u0011\u001b7\u0015\u001a\u0011\u00147\u0016\u001a\u0011\u001b7\u0017\u0019\u0011\u001b7\u0018\u0018\u0011\u00187\u0019\u0018\u0011\u00137\n,Q6H\n\u0017\u0003.\u0014\u0015 1 \u0003\u0013 \u0016\n\u0017 \u0016G\u0013$WWHQXDQFH$WWHQXDQFH\nFigure 4.7 : (a) Magnetoabsorption spectra for InSe (a) up to 30 T probed by σ+(red) and\ntheσ−(black) polarizations at 85 K, and (b) up to 65 T probed by unpolarized light at 4 K\nin InSe.\nquantum well sample (∼ 9 meV). It is because the GaAs well has better crystalline quality\nthan the InGaAs well. Using Eq. (4.1), the reduced effective mass µ∗was calculated to be\n(0.084 ±0.02)m 0.\n4.2 InSe\nA thin film of InSe was prepared by cleaving a single crystal. The typical film thickness was\nmeasured by a profilometer to be as around 50 µm. Magnetoabsorption spectra were taken\nby two different systems, the 30 T table-top pulsed magnet system (RAMBO) in Chapter 3and a 65 T fiber-coupled pulsed magnet system at the Los Alamos National Laboratory in\nNew Mexico. The characteristics of the latter system were as follows: The magnetic field\npulse duration was around 80 ms, and the spectra were taken every 2.2 ms. A Xenon flash\nlamp was employed as a light source, and the light beam was unpolarized. In both systems,\nthe magnetic field was applied along the c-axis of InSe (B /bardblc).(a)\nMagnetic field (T)01.3211.3221.32610T4T8T12T16T20T25T30T5\n51.30 1.32 1.34 1.36 1.38 1.40\n10 15 20 25 301.3231.3241.325Energy (eV)Energy (eV)234AttenuanceN = 0\nN = 1Fig. 7. (a) Attenuation spectra of InSe up to 30 T at 85 K (b) The peak\nenergy of N=0 transitions extracted from (a). The dashed lines are fits to\nthe data.\nform a macroscopic dipole and cooperatively interact with\neach other to emit an intense SF burst.48–52)The high Bin-\ncreased the dipole moment as well as the density of states and\nsuppressed scattering.53, 54)\nFigure 8a displays absorbance spectra for an In 0:2Ga0:8As\nQW sample at various magnetic fields, taken with the\nRAMBO system. The LED was used as the light source for\nthe measurements. At 0 T, the peaks at 1.325 eV , 1.4 eV , and\n1.442 eV are attributed to E 1H1, E1L1, and E 2H2transitions,\nrespectively.18)The quasi-2D density of states are quantized\ninto multiple discrete Landau levels when an external mag-\nnetic field is applied perpendicular to the QW. The peak fre-\nquencies in the absorption spectra increase with increasing\nmagnetic field as the energy separation between LL becomes\nlarger at higher fields.\nFigures 8b and 8c display time-intagrated PL spectra\nfor center- and edge-collected emission averaged over four\nsingle-shot measurements. For the edge-emission measure-\nments, a right-angle microprism was attached to the edge of\nthe sample to redirect the in-plane emission to escape the\ncryostat.18, 48)The peaks in the PL spectra correspond to the\nLL transitions. The peaks shift higher in frequency with in-\ncreasing B, which are consistent with the absorption spectra.\n5J. Phys. Soc. Jpn. SPECIAL TOPICS\n123906-5 Noe et al. Rev. Sci. Instrum. 84, 123906 (2013)Absorbance (norm. units)\n1.45 1.40 1.35 1.30\nEnergy (eV)0 T2 T4 T6 T8 T10 T12 T14 T16 T18 T20 T22.5 T25 T27.5 T\nIntensity (norm. units)\n1.40 1.35 1.30\nEnergy (eV)0 T2 T4 T6 T8 T10 T12 T14 T16 T18 T20 T22.5 T25 T27.5 TAbsorbance PL(a) (b)\nFIG. 4. (a) Absorbance spectra taken using a light emitting diode, and (b)\nweak excitation photoluminescence spectra using a laser diode, for undoped\nInGaAs quantum wells as a function of magnetic field at 12.5 K.\nE1L1located at 1.4 eV , and the E 2H2located at 1.442 eV . The\nlarge separation, 75 meV , between the E 1H1and E 1L1edges is\ndue to strain51between the In 0.2Ga0.8As and the GaAs layers.\nWith increasing magnetic field, Landau quantization causes\nthe quasi-2D density of states to evolve into a series of deltafunctions reminiscent of a quasi-0D density of states as the\nelectron motion becomes fully quantized with the application\nof a high magnetic field perpendicular to the quantum wells.\nThe weak excitation PL spectra show a single peak of emis-\nsion near the band edge that increases in strength and shiftsto higher energy with increasing magnetic field (diamagnetic\nshift\n52,53).\nFigure 5displays the results of the time-integrated PL\nspectra for both center- and edge-collected emission taken at∼13 K with 5 µJ excitation pulse energy using the amplified\nTi:sapphire laser. At each magnetic field and for each emis-\nsion direction, four single-shot measurements were taken. The\ndisplayed results are in each case the average spectrum of the\nfour measurements. The spectra are normalized with respectto the collected light by the CCD for the edge-collected emis-\nsion at 30 T and no geometrical considerations were used\nregarding the collection direction, center, or edge. The peaklocated at ∼1.50 eV in the center-collected data is emission\nfrom the GaAs barriers and/or substrate. In both sets of spec-tra, we see multiple Landau level (LL) peaks, which increase\nin separation with increasing magnetic field. In the center col-\nlected spectra, the emission strength increases steadily for all1.50 1.45 1.40 1.35 1.30\nEnergy (eV)x5\nx10x2x2\nx2\nx2\nx10\nx10\nx10 6 T7 T8 T9 T10 T12 T14 T16 T18 T20 T25 T30 TIntensity (norm. units)\n1.50 1.45 1.40 1.35 1.30\nEnergy (eV)x100\nx100\nx100\nx100\nx100\nx100\nx100\nx100\nx100 6 T7 T8 T9 T10 T12 T14 T16 T18 T(a) (b) Center Edge\nFIG. 5. Time-integrated photoluminescence spectra upon intense excitation\nusing an amplified Ti:sapphire laser for both (a) center- and (b) edge-emissionwith 5 µJ excitation pulse energy at 13 K.\nof the LL transitions arising out of the E 1H1transition with in-\ncreasing magnetic field. However, in the edge-collected spec-\ntra, we see a dramatic increase in emission strength from 6 to\n30 T for the 00 LL. The dramatic increase in intensity between\nthe center- and edge-emission illustrates the fact that weare observing stimulated emission, or superfluorescence,\n54–59\nfrom a dense electron-hole plasma for the in-plane direc-tion. The intensity of the edge-collected emission would beless than the center-collected emission if both were typi-\ncal spontaneous emission because of the geometry of the\ncollection.\nFigure 6displays the result of the time-resolved PL map\nat 10 T and at 19 K for the edge-collected emission. Afterlaunching a series of magnet pulses, we partially created amap showing a burst of emission from the 11 LL energy. Tak-\ning vertical and horizontal slices at the peak of the burst, we\ncan determine the pulse duration to be ∼10 ps and spectral\nwidth to be ∼5 meV . In our previous measurements at the\nNational High Magnetic Field Laboratory, our temporal reso-lution for time-resolved PL measurements was limited to 20\nps due to dispersion in the graded-index fiber that was used\nfor collection.\n56–59Here, we are able to place an upper limit\nfor the SF pulse as the temporal resolution is limited by the\nKerr medium, which, for toluene, is 1 ps.60(a)123906-5 Noe et al. Rev. Sci. Instrum. 84, 123906 (2013)Absorbance (norm. units)\n1.45 1.40 1.35 1.30\nEnergy (eV)0 T2 T4 T6 T8 T10 T12 T14 T16 T18 T20 T22.5 T25 T27.5 T\nIntensity (norm. units)\n1.40 1.35 1.30\nEnergy (eV)0 T2 T4 T6 T8 T10 T12 T14 T16 T18 T20 T22.5 T25 T27.5 TAbsorbance PL(a) (b)\nFIG. 4. (a) Absorbance spectra taken using a light emitting diode, and (b)\nweak excitation photoluminescence spectra using a laser diode, for undoped\nInGaAs quantum wells as a function of magnetic field at 12.5 K.\nE1L1located at 1.4 eV , and the E 2H2located at 1.442 eV . The\nlarge separation, 75 meV , between the E 1H1and E 1L1edges is\ndue to strain51between the In 0.2Ga0.8As and the GaAs layers.\nWith increasing magnetic field, Landau quantization causes\nthe quasi-2D density of states to evolve into a series of deltafunctions reminiscent of a quasi-0D density of states as the\nelectron motion becomes fully quantized with the application\nof a high magnetic field perpendicular to the quantum wells.\nThe weak excitation PL spectra show a single peak of emis-\nsion near the band edge that increases in strength and shiftsto higher energy with increasing magnetic field (diamagnetic\nshift\n52,53).\nFigure 5displays the results of the time-integrated PL\nspectra for both center- and edge-collected emission taken at∼13 K with 5 µJ excitation pulse energy using the amplified\nTi:sapphire laser. At each magnetic field and for each emis-\nsion direction, four single-shot measurements were taken. The\ndisplayed results are in each case the average spectrum of the\nfour measurements. The spectra are normalized with respectto the collected light by the CCD for the edge-collected emis-\nsion at 30 T and no geometrical considerations were used\nregarding the collection direction, center, or edge. The peaklocated at ∼1.50 eV in the center-collected data is emission\nfrom the GaAs barriers and/or substrate. In both sets of spec-tra, we see multiple Landau level (LL) peaks, which increase\nin separation with increasing magnetic field. In the center col-\nlected spectra, the emission strength increases steadily for all1.50 1.45 1.40 1.35 1.30\nEnergy (eV)x5\nx10x2x2\nx2\nx2\nx10\nx10\nx10 6 T7 T8 T9 T10 T12 T14 T16 T18 T20 T25 T30 TIntensity (norm. units)\n1.50 1.45 1.40 1.35 1.30\nEnergy (eV)x100\nx100\nx100\nx100\nx100\nx100\nx100\nx100\nx100 6 T7 T8 T9 T10 T12 T14 T16 T18 T(a) (b) Center Edge\nFIG. 5. Time-integrated photoluminescence spectra upon intense excitation\nusing an amplified Ti:sapphire laser for both (a) center- and (b) edge-emissionwith 5 µJ excitation pulse energy at 13 K.\nof the LL transitions arising out of the E 1H1transition with in-\ncreasing magnetic field. However, in the edge-collected spec-\ntra, we see a dramatic increase in emission strength from 6 to\n30 T for the 00 LL. The dramatic increase in intensity between\nthe center- and edge-emission illustrates the fact that we\nare observing stimulated emission, or superfluorescence,54–59\nfrom a dense electron-hole plasma for the in-plane direc-\ntion. The intensity of the edge-collected emission would beless than the center-collected emission if both were typi-\ncal spontaneous emission because of the geometry of the\ncollection.\nFigure 6displays the result of the time-resolved PL map\nat 10 T and at 19 K for the edge-collected emission. Afterlaunching a series of magnet pulses, we partially created amap showing a burst of emission from the 11 LL energy. Tak-\ning vertical and horizontal slices at the peak of the burst, we\ncan determine the pulse duration to be ∼10 ps and spectral\nwidth to be ∼5 meV . In our previous measurements at the\nNational High Magnetic Field Laboratory, our temporal reso-lution for time-resolved PL measurements was limited to 20\nps due to dispersion in the graded-index fiber that was used\nfor collection.\n56–59Here, we are able to place an upper limit\nfor the SF pulse as the temporal resolution is limited by the\nKerr medium, which, for toluene, is 1 ps.60(b) (c)\nFig. 8. (a) Absorbance spectra for In 0:2Ga0:8As QW taken at various mag-\nnetic fields up to 27.5 T at 12.5 K. The traces are vertically o \u000bset. Time-\nintegrated magneto-PL spectra for both (b) center- and (c) edge-emission at\n13 K. Adapted from Ref. 18.\n123906-6 Noe et al. Rev. Sci. Instrum. 84, 123906 (2013)\n1.0\n0.5\n0.0\n120 110 100 90 80\nTime (ps)1.355\n1.350\n1.345\n1.340Energy (eV)\n110 105 100 95 90\nTime (ps)\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0Intensity (a.u.)\n1.0\n0.5\n0.0Intensity (a.u.)\n1.360 1.350 1.340\nEnergy (eV)(a)\n(b) (c)\nFIG. 6. Superfluorescent burst of radiation from a highly excited InGaAs\nquantum well sample at 10 T measured using the mini-coil magnet. A\nzoomed version (a) shows the details of the data quality, and we show spectral(b) and temporal (c) slices of the data.\nIV. CONCLUSION\nWe have developed a unique mini-coil magnet system for\nnonlinear and ultrafast optical spectroscopy studies of mate-\nrials. Using this system, we have extended our most recentstudies on superfluorescence from a high-density electron-\nhole plasma in semiconductor quantum wells\n56–59to higher\nmagnetic field strengths and with better temporal resolutionfor the time-resolved PL results by developing this system.\nFrom a more general perspective, this unique magneto-\noptical spectroscopy system will open doors to many newtypes of experiments in condensed matter systems at high\nmagnetic fields. Depending on the data acquisition speed,experiments that require the magnetic field dependence can\nbe swept from 0 to 30 T within a single magnet pulse, and\nthen repeated to improve the signal-to-noise ratio by averag-ing. The optical access via interchangeable windows allows\nus to introduce a variety of wavelengths, and, importantly,\nthe application most suited for this magnet will be time-domain terahertz spectroscopy\n7–15because of the compact\ndesign and direct optical access to the sample. Further-more, the direct optical access allows polarization-sensitivemeasurements\n14,61without the complications that arise with\noptical fibers. Finally, the mini-coil design can be reproducedby other researchers around the world and incorporated into\nsetups that already use expensive ultrafast laser systems or\nother sophisticated optical systems, greatly expanding theavailability of high magnetic fields for condensed matter and\nmaterials research.\nACKNOWLEDGMENTS\nWe acknowledge support from the National Science\nFoundation (through Grant No. DMR-1310138), the Depart-\nment of Energy (through Grant No. DE-FG02-06ER46308),\nand the Robert A. Welch Foundation (through Grant\nNo. C-1509). We thank G. S. Solomon for providing us with\nthe InGaAs quantum well sample used in this study.\n1P. Kner, W. Schafer, R. Lovenich, and D. S. Chemla, Phys. Rev. Lett. 81,\n5386 (1998).\n2N. A. Fromer, C. E. Lai, D. S. Chemla, I. E. Perakis, D. Driscoll, and\nA. C. Gossard, Phys. Rev. Lett. 89, 067401 (2002).\n3K. M. Dani, J. Tignon, M. Breit, D. S. Chemla, E. G. Kavousanaki, and\nI. E. Perakis, Phys. Rev. Lett. 97, 057401 (2006).\n4J. Kono, A. H. Chin, A. P. Mitchell, T. Takahashi, and H. Akiyama, Appl.\nPhys. Lett. 75, 1119 (1999).\n5M. A. Zudov, A. P. Mitchell, A. H. Chin, and J. Kono, J. Appl. Phys. 94,\n3271 (2003).\n6G. A. Khodaparast, D. C. Larrabee, J. Kono, D. S. King, S. J. Chung, and\nM. B. Santos, Phys. Rev. B 67, 035307 (2003).\n7X. Wang, D. J. Hilton, L. Ren, D. M. Mittleman, J. Kono, and J. L. Reno,\nOpt. Lett. 32, 1845 (2007).\n8Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, andR. Shimano, Phys. Rev. Lett. 104, 256802 (2010).\n9X. Wang, A. A. Belyanin, S. A. Crooker, D. M. Mittleman, and J. Kono,\nNat. Phys. 6, 126 (2010).\n10X. Wang, D. J. Hilton, J. L. Reno, D. M. Mittleman, and J. Kono, Opt.\nExpress 18, 12354 (2010).\n11Y. Takahashi, R. Shimano, Y. Kaneko, H. Murakawa, and Y. Tokura, Nat.\nPhys. 8, 121 (2011).\n12T. Arikawa, X. Wang, D. J. Hilton, J. L. Reno, W. Pan, and J. Kono, Phys.\nRev. B 84, 241307(R) (2011).\n13S. Bordács, I. Kézmárki, D. Szaller, L. Demkó, N. Kida, H. Murakawa,Y. Onose, R. Shimano, T. R ˝o˝om, U. Nagel, S. Miyahara, N. Furukawa, and\nY. Tokura, Nat. Phys. 8, 734 (2012).\n14T. Arikawa, X. Wang, A. A. Belyanin, and J. Kono, Opt. Express 20, 19484\n(2012).\n15R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino,T. Morimoto, and H. Aoki, Nat. Commun. 4, 1841 (2013).\n16E. Kojima, R. Shimano, Y. Hashimoto, S. Katsumoto, Y. Iye, and M.Kuwata-Gonokami, Phys. Rev. B 68, 193203 (2003).\n17J. Wang, C. Sun, J. Kono, A. Oiwa, H. Munekata, L. Cywinski, and L. J.Sham, Phys. Rev. Lett. 95, 167401 (2005).\n18J. Wang, C. Sun, Y. Hashimoto, J. Kono, G. A. Khodaparast, L. Cywin-\nski, L. J. Sham, G. D. Sanders, C. J. Stanton, and H. Munekata, J. Phys.:\nCondens. Matter 18, R501 (2006).\n19Y. Hashimoto, S. Kobayashi, and H. Munekata, Phys. Rev. Lett. 100,\n067202 (2008).\n20J. Wang, L. Cywinski, C. Sun, J. Kono, H. Munekata, and L. J. Sham, Phys.\nRev. B 77, 235308 (2008).\n21K. C. Hall, J. P. Zahn, A. Gamouras, S. March, J. L. Robb, X. Liu, and J.\nK. Furdyna, Appl. Phys. Lett. 93, 032504 (2008).\n22E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Phys. Rev. Lett.\n76, 4250 (1996).\n23For a review, see, e.g., G. Zhang, W. Hübner, E. Beaurepaire, and J.-Y.Bigot, in Spin Dynamics in Confined Magnetic Structures I , edited by B.\nHillebrands and K. Ounadjela (Springer, Berlin, 2002), pp. 245–288.\n24B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fahnle,T. Roth, M. Cinchetti, and M. Aeschlimann, Nat. Mater. 9, 259 (2010).\n25A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010).\n26A. I. Lobad, R. D. Averitt, and A. J. Taylor, Phys. Rev. B 63, 060410(R)\n(2001).\n27S. A. McGill, R. I. Miller, O. N. Torrens, A. Mamchik, I.-W. Chen, andJ. M. Kikkawa, Phys. Rev. Lett. 93, 047402 (2004).\n28A. B. Sushkov, M. Mostovoy, R. V. Aguilar, S.-W. Cheong, and\nH. D. Drew, J. Phys.: Condens. Matter 20, 434210 (2008).\n29D. Talbayev, S. A. Trugman, A. V. Balatsky, T. Kimura, A. J. Taylor, and\nFig. 9. (a) Time-resolved PL map of emission from the 11 LL transition at\n10 T and at 19 K for the edge-collected emission. (b) Spectral and (c) tempo-\nral profile of the superfluorescent burst of radiation. The red points denote the\nexperimental data, while solid lines are fits to the data. Adapted from Ref. 18.\nIn the center-emission spectra, the emission strength of all\ntransitions increase steadily with increasing B. By contrast,\nthe edge-collected emission strength of the 00 LL transition\nwith the lowest peak energy increases dramatically from 6 T\nto 30 T. The drastic increase in the edge-emission as com-\npared to the center-emission indicates that the observed edge-\nemission is SF.Figure 9 shows the edge-collected emission intensity of the\ntime-resolved PL measurement as a function of time and en-\nergy at 19 K and at 10 T. The pulse duration and spectral width\nof the SF pulse are determined to be \u001810 ps and\u00185 meV , re-\nspectively. The temporal resolution of the setup was limited\nby the Kerr medium, which is \u00181 ps for toluene.55)\n3.2 Plasmons\nTHz spectroscopy is sensitive to free electron dynamics,\nespecially through collective excitation of an electron gas, or\nplasmons. In the presence of a magnetic field, a plasmon exci-\ntation becomes a magnetoplasmon, whose frequency is given\nby the cyclotron frequency, !c=eB=m\u0003, where m\u0003is the ef-\nfective mass of the electrons, in the case of a free electron\ngas with no confinement. In addition, polarization-dependent\nTHz spectroscopy allows one to determine both the diagonal\nand o \u000b-diagonal elements of the conductivity tensor through\nFaraday and Kerr rotations.34, 37, 56–61)\n3.2.1 Cyclotron resonance in photoexcited Si\nThe RAMBO system has been used for measuring the cy-\nclotron resonance of photoexcited carriers in a single crys-\ntal of intrinsic Si at high magnetic fields.23)The Si sample\nwas excited by near-infrared pulses and then probed by THz\npulses after a delay of 100 ps. The free carriers excited by the\npump beam move in circular orbits due to the external mag-\nnetic field at the cyclotron frequency, !c, which lies within the\nTHz frequency range in the magnetic field range of RAMBO.\nFigures 10a and 10b show the relative THz transmission of\nthe Si sample as a function of magnetic field up to 30 T at 10 K\nwith LiNbO 3generation. The dips in the transmittance spec-\ntra correspond to the cyclotron resonance. The dip frequency\nis extracted and plotted in Fig. 10c. The measurements taken\nat 10 K and 83 K used, respectively, LiNbO 3and ZnTe crys-\ntals to generate THz pulses. From the resonance frequencies,\ntogether with the relation !c=eB=m\u0003, two e \u000bective masses,\nm\u0003=0:19m0andm\u0003=0:39m0, were obtained, corresponding\nto the light holes and heavy holes, respectively, in Si.\n3.2.2 Faraday and Kerr e \u000bects in Bi 1\u0000xSbx\nThe RAMBO system, together with a 10-T supercon-\nducting magnet system,62–69)have been used to study\nthe semimetal-to-topological-insulator transition in Bi 1\u0000xSbx\nfilms. The optical conductivity, polarization rotation angle,\n\u0012(!), and ellipticity change, \u0011(!), of the Faraday and Kerr\nrotations were extracted from the measurements.\nFigure 11a shows the phase diagram of Bi 1\u0000xSbxalloys\nas a function of x, where the transition happens at x=\n0:07.70–74)The Bi 1\u0000xSbxfilms were grown on silicon sub-\nstrates by molecular beam epitaxy.75, 76)The magnetic field\ndependence of \u0012Fat 0.7 THz for Bi 1\u0000xSbxfilms with di \u000ber-\nentxexhibits a trend that is consistent with the phase dia-\ngram. For semimetallic films (sample 1-2), \u0012Fexhibits a dip\nbelow zero within 10 T. By contrast, \u0012Fbecomes positive and\nincreases with increasing Bfor topological insulating films\n(sample 3-5). No dip is observed and the change of \u0012Ftraces\nwith increasing xis monotonic.\nFigures 11c and 11d show \u0012Fand\u0011Fspectra for sample\n1 (x=0) at various magnetic fields. A resonance feature that\nshifts to higher frequencies with increasing Bis observed in\nboth spectra. Based on a bulk band model with realistic band\n6J. Phys. Soc. Jpn. SPECIAL TOPICS\n6\n4\n2\n0Electric Field (a.u.)\n8 6 4 2 0 -2\nTime (ps)30 T\n0 T16 T\nNo Pump6\n420Electric Field (a.u.)\n8 6 4 2 0 -2\nTime (ps)30 T\n16 T\nNo Pump0 T(a) (b)\nFig. 5. Measured THz waveforms for both the LiNbO 3,a), and ZnTe, b), generation. At 0 T\nwe measure the transmitted THz waveform with and without optically pumping the silicon\nsample. At high magnetic field, the silicon sample is optically pumped. Data was taken at\n10 K for a)and 83 K for b).\n10\n8\n6\n4\n2\n0Power With Pump @ B / Power Without Pump at B=0\n2.01.51.00.5\nFrequency (THz)30 T\n27.5 T\n25 T\n22.5 T\n20 T\n18 T\n14 T16 T\n12 T11\n10\n9\n8\n7\n6\n5\n4\n3\n2\n1\n0Power With Pump @ B / Power Without Pump at B=0\n1.0 0.2\nFrequency (THz)0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T7\n6\n5\n4\n3\n2\n1\n0Power With Pump @ B / Power Without Pump at B=0\n1.0 0.2\nFrequency (THz)0 T5 T10 T16 T20 T25 T30 T(a) (b) (c)\nFig. 6. Magnetic field dependence of the relative THz transmission for optically pumped\nsilicon. Data shown in a)and b)were taken with LiNbO 3generation, and the sample\ntemperature was 10 K. Data shown in c)was taken with ZnTe generation, and the sample\ntemperature was 83 K. Cyclotron resonance lines can be seen and the dashed lines are a\nguide to the eye for the heavier mass feature (blue) and the lighter mass feature (red). All\ntraces are offset linearly with respect to the incremental increase in magnetic field.\nthe Lorentz force. The cyclotron frequency, ωc=eB/m∗, is determined by e, the electronic\ncharge, B, the applied magnetic field, and m∗, the effective mass. Figure 5 shows the measured\nwaveforms after passing through the Si sample upon optical pumping at different magnetic fields,\nas well as the THz waveform at 0 T with no optical pump for comparison. The SNR is sufficient\nto clearly see a change in the electric field in the time-domain with applied magnetic field.\nFigure 6 shows the relative THz transmission versus magnetic field. The relative transmission\ncalculation uses the Fourier transform of the 0 T data without optical pumping as the reference\nfor the data shown. At 0 T, a Drude-like response is observed in the frequency domain, where free\ncarrier absorption increases with decreasing frequency. With increasing the magnetic field to 10 T,two dips become clearly evident in the relative transmission corresponding to cyclotron resonance\nabsorption of the photoexcited carriers. With further increase in magnetic field above 10 T, the\ncyclotron resonance feature corresponding to the lower-mass carrier leaves the bandwidth of the\nLiNbO 3generated THz radiation, while the feature corresponding to the higher-mass carrier\n                                                                                            Vol. 24, No. 26 | 26 Dec 2016 | OPTICS EXPRESS 30335 \n2.5\n2.0\n1.5\n1.0\n0.5\n0.0Frequency (THz)\n30 25 20 15 10 5 0\nMagnetic Field (T) Light (10 K)\n Heavy (10 K)\n Heavy (83 K)\nFig. 7. Cyclotron resonance center frequency vs. magnetic field. Results from data taken\nat 10 K and 83 K with both generation schemes is combined to identify two features with\nfrequency linear with applied magnetic field.\nremains clearly visible up to 20 T with only the tail of the feature remaining at 25 T and 30 T.\nUsing ZnTe instead for THz generation allowed us to track the heavier mass feature up to 30 T.\nIt should be noted that the data taken with LiNbO 3and ZnTe generation were taken at different\ntemperatures, 10 K and 83 K, respectively. At higher temperatures, cyclotron resonance lines are\ngenerally broader due to increased scattering with phonons making them more difficult to clearly\nresolve.\nAfter fitting the center frequency of the cyclotron resonance vs. magnetic field (Fig. 7) with a\nline and calculating the effective mass, we determine the carrier effective mass, m∗=0.19m0for\nthe lighter carrier and m∗=0.39m0for the heavier feature, where m0=9.11×10−31kg. The\nlower-mass value matches very well with literature value for one electron effective mass, whereas\nthe higher-mass value is slightly below the literature value [31, 32] for the heavier electron when\nthe magnetic field is in the [100] orientation of Si.\n4. Conclusion\nWe have demonstrated optical-pump/THz-probe measurements in bulk, intrinsic silicon in\nmagnetic fields up to 30 T by developing a single-shot THz-TDS system around a minicoil\npulsed magnet. The single-shot measurement faithfully reproduces the THz electric field when\ncompared to the commonly used step scan technique. To our knowledge, these results mark the\nhighest magnetic field reported for THz-TDS measurements. Unlike rapid scanning techniques\nbased on ECOPS or ASOPS, this single-shot technique could potentially be used to perform\nTHz-TDS measurements in ultrahigh magnetic field where typical magnetic field pulse durations\nare on the order of µs where the variation of the magnetic field during the measurement of the\nTHz pulse would remain negligible. For ms duration pulses, the efficiency of the data taking\nprocess could be improved by an order of magnitude relative to these measurements with the use\nof a high frame rate camera operating at ∼1,000 frames per second, the same repetition rate as\nour 1 kHz laser. The magnetic field dependence could be taken with a single pulse instead of\nachieving one magnetic field data point at the peak of the magnetic field. This improvement is\nvitally important for the purpose of varying other parameters in addition to the magnetic field\nsuch as the sample temperature, optical-pump/THz-probe time delay, or optical pump power, as\nthe majority of the time performing these measurements is spent waiting for the magnet coil to\ncool between magnet shots.\n                                                                                            Vol. 24, No. 26 | 26 Dec 2016 | OPTICS EXPRESS 30336 (a)\n(c)(b)\nFig. 10. The relative THz transmission of intrinsic Si at 100 ps after optical\nexcitation at various magnetic fields up to (a) 10 T and (b) 30 T, respectively.\nThe data was taken at 10 K with LiNbO 3generation. (c) Extracted cyclotron\nfrequencies versus magnetic field. The data taken at 83 K used ZnTe genera-\ntion. The solid lines are linear fits to the data. Adapted from Ref. 23.\nparameters, the authors concluded that the feature is mainly\ncaused by bulk hole cyclotron resonance.37, 77, 78)\nFigures 12a and 12b show, respectively, \u0012F(!;B) and\n\u0012K(!;B) of the Bi 0:9Sb0:1film versus frequency and Bup\nto 30 T at a temperature of 21 K. At a fixed B, both\u0012Fand\n\u0012Kspectra are featureless. Nevertheless, their values increase\nwith increasing Band finally saturate when Bis above 15 T.\nIn contrast to the semimetallic film, a surface band model was\nused to analyze the experimental data.37, 74, 79)By summing\nup the contributions from the ¯\u0000-point electron pocket, the ¯M-\npoint electron pocket, and the hole pocket, 2D color maps\nof\u0012Fand\u0012Kwere constructed and shown as Figs. 12c and\n12d. The theoretical calculations are in agreement with the\nexperimental results as they reproduce the saturation behav-\nior of Faraday and Kerr rotations at high B. These results sug-\nXINWEI LI et al. PHYSICAL REVIEW B 100, 115145 (2019)\n( ygrenE a.u.)Semimetal\n(SM)Topological Insulator\n(TI)\nT\nLs\nLa\n= 0.04 = 0.1\nFIG. 1. Schematic phase diagram of Bi 1−xSbxalloys. The en-\nergies of different band edges at high symmetry points (T and L\npoints) are plotted as a function of Sb composition x. The five arrows\npointing to the horizontal axis mark the nominal values of xof the\nfive film samples we studied. See Table Ifor more details of the\ncharacteristics of the samples.\nIn this paper, we describe results of THz Faraday and\nKerr rotation spectroscopy measurements on thin films of\nBi1−xSbx. This alloy system is known to show a semimetal\n(SM)-to-TI transition as a function of x, and its phase diagram\n(Fig. 1) is well established [ 11,14,35–37], although some\ncontroversy remains as to whether surface states in the SMregion are topologically nontrivial [ 38]. We performed mea-\nsurements using a single-shot THz time-domain polarimetrysetup combined with a 30-T pulsed magnet system [ 39–41].\nThe wide tunability of the band structure of Bi\n1−xSbxwith x\nmakes this material system suitable for identifying and distin-guishing the uniquely different magneto-optical responses ofsamples in the topologically trivial and nontrivial phases.\nA THz beam normally incident on the sample surface in\nthe Faraday geometry exhibited polarization rotations due tothe field-induced Hall conductivities of the surface and/orbulk carriers. Faraday rotation spectra for the SM films had apronounced dip, which blue-shifted with the magnetic field,while the Faraday rotations in the TI films were positiveand spectrally featureless, increasing and then saturating withincreasing magnetic field. Using a theoretical model incor-porating realistic band parameters and the Kubo formula forcalculating the optical conductivity, we found that the opticalHall signals in the SM (TI) samples can be attributed tocarriers in the bulk (surface) bands. The model suggested thatthe magneto-optical signal from the SM films was dominatedby the cyclotron resonance of bulk high-mobility holes, whilethat from the TI film resulted from the summed contributionsof multiple electron and hole pockets associated with thesurface bands.\nII. SAMPLES AND METHODS\nA. Bi 1−xSbxfilms\nWe studied Bi 1−xSbxfilms on silicon substrates grown\nby molecular beam epitaxy using the methods described inRefs. [15 ,42]. Five samples with different xvalues were\nstudied; see Table I. The nominal xvalues were 0, 0.04,\n0.08, 0.1, and 0.15, respectively, while the thickness, t, was\nnominally 40 nm for all films. According to Fig. 1, samples\n1 and 2 were in the SM regime while samples 3–5 werein the TI regime. We used a combination of structural and\nchemical characterization methods to precisely determine the\nactual values of x,t, and crystal orientation of the films;\nin situ reflection high-energy electron diffraction (RHEED)\npatterns determined the crystal orientation, while ex situ x-\nray diffraction (XRD), x-ray fluorescence (XRF), and atomicforce microscopy (AFM) experiments provided informationon the crystal structure, chemical composition, and film thick-ness, respectively. The obtained parameters of the samples aresummarized in Table I.\nFor sample 1 (x =0), RHEED determined that the film\norientation was /angbracketleft001/angbracketright. No Sb was incorporated in this sample,\nso that chemical analysis was not needed. AFM determinedthat t=68 nm. For sample 2 (nominal x=0.04), XRF and\nXRD measurements were performed. An obtained XRF spec-trum was fit with a model built in the measurement software,using xand tas adjustable parameters, and the parameters that\ngave the best fit were x=0.03 and t=60 nm. XRD showed a\ndominating diffraction peak due to the (001) plane of the film,\nconfirming that the crystal orientation was /angbracketleft001/angbracketright.\nFor sample 4 (nominal x=0.10), fitting analysis on an\nXRF spectrum allowed us to determine x=0.136 and t=\n54 nm. An XRD curve showed diffraction peaks from the\n(001) and (012) planes, suggesting that the film possibly hadsome spatial inhomogeneity in terms of orientation; how-ever, the ARPES data shown in Fig. 2(a) confirmed that\nthe film area on which our magneto-optical measurementswere performed was dominated by the /angbracketleft001/angbracketrightorientation.\nTransport measurements were also performed on the film; seeFig.2(b) for the resistance-temperature ( R-T) characteristic.\nThe increasing Rwith decreasing Tin the 120 K <T<\n250 K region can be attributed to the decreasing number ofthermal carriers in the insulating bulk, while the subsequentdecrease of Rin the T<120 K region is likely due to surface\nTABLE I. Characteristics of the five Bi 1−xSbxthin film samples studied. The Sb content ( x), film thickness ( t), and crystal orientation of\nthe films are shown. The experimental methods used to determine the parameters are indicated in the parentheses. SM =semimetal. TI =\ntopological insulator.\nSample No. Nominal x Actual x Thickness t(nm) Orientation Character\n1 0 0 68 (AFM) /angbracketleft001/angbracketright (RHEED) SM\n2 0.04 0.03 (XRF) 60 (XRF) /angbracketleft001/angbracketright (XRD) SM\n3 0.08 0 .03<x<0.136 70 (AFM) N/A TI\n4 0.10 0.136 (XRF) 54 (XRF) /angbracketleft001/angbracketright and/angbracketleft012/angbracketright(XRD) /angbracketleft001/angbracketright (ARPES) TI\n5 0.15 x>0.136 77 (AFM) /angbracketleft001/angbracketright (RHEED) TI\n115145-2TERAHERTZ FARADAY AND KERR ROTATION … PHYSICAL REVIEW B 100, 115145 (2019)\nFIG. 6. (a) Faraday rotation and (b) Faraday ellipticity spectra\nfor sample 1 at a temperature of 2 K at different magnetic fields up\nto 10 T. Curves at different magnetic fields are vertically offset for\nclarity, and the baselines for the different spectra are indicated bydashed lines.\nthe two images spatially separated on the CMOS camera by\nthe Wollaston prism gave the time-domain THz electric fieldsignal.\nIII. RESULTS\nA. Semimetallic samples: samples 1 and 2\nFigures 6(a) and6(b) display Faraday rotation (θ F) and\nFaraday ellipticity ( ηF) spectra, respectively, for sample 1\n(x=0) at T=2 K in Bup to 10 T. The curves are inten-\ntionally offset vertically for clarity, and the zero baselines areshown as dashed colored lines. A resonance feature that shiftshigher in frequency with increasing Bis clearly observed. We\nthen calculated the real and imaginary parts of the optical Hallconductivity (σ\nxy) of the sample by taking into account the\nreference signal; see Eq. ( 6). Figure 7shows the calculated\nσxyspectra. The signs of the real and imaginary parts of σxy\nare both opposite to that obtained from the standard 2DEGsample shown in Fig. 4. This indicates that holes are the major\ncontributors to the magneto-optical signal.\nFigures 8(a) and8(b) display θ\nFandηFspectra, respec-\ntively, for sample 2 at T=2 K in Bup to 10 T. The major res-\nonance feature that shifts with Bin a similar manner to that in\nsample 1 is observed, except that the linewidth is broader. Thissuggests that the major contributors to the magneto-opticalsignal have the same carrier origin as in the Bi film, and theSb incorporation reduces the carrier mobility. Figures 9(a)\nand9(b) show Kerr rotation (θ\nK) and Kerr ellipticity (η K)\nspectra, respectively, obtained for the same sample under thesame conditions as in Fig. 8. The resonance feature induces aFIG. 7. (a) Re( σxy) and (b) Im( σxy) spectra for sample 1 at\na temperature of 2 K at different magnetic fields up to 10 T.\nCurves at different magnetic fields are vertically offset for clarity,and the baselines for the different spectra are indicated by dashed\nlines.\nFIG. 8. (a) Faraday rotation and (b) Faraday ellipticity spectra\nfor sample 2 at a temperature of 2 K at different magnetic fields up\nto 10 T. Curves at different magnetic fields are vertically offset for\nclarity, and the baselines for the different spectra are indicated by\ndashed lines.\n115145-5XINWEI LI et al. PHYSICAL REVIEW B 100, 115145 (2019)\nFIG. 12. Magneto-optical response of sample 4 (nominal x=\n0.1) up to 30 T. (a) Faraday rotation and (b) Kerr rotation maps versus\nTHz frequency and magnetic field. (c) Faraday rotation and (d) Kerr\nrotation versus magnetic field at a fixed THz frequency of 0.7 THz,\ncorresponding to the cuts marked by the red dashed lines in (a) and(b). The solid red lines in (c) and (d) are calculated curves using the\ntheoretical model described later in the Discussions section.\nIV . DISCUSSION\nA. Theory of magneto-optical response of Bi\n1−xSbxin the\nsemimetallic regime\nIn order to understand the THz magneto-optical response\nof Bi 1−xSbxfilms in the SM regime, we developed a detailed\ntheoretical model. We took into account the bulk bands of\nBi, which allowed us to determine the origin of the experi-\n-0.10-0.050.000.05θF (rad)\n10 8 6 4 2 0\nMagnetic field (T)Sample 1\nSample 2Sample 3Sample 4Sample 5@ 0.7 THz\nFIG. 13. Magneto-optical response of all samples. θFis plotted\nvsBatT=2 K for a fixed THz frequency (0.7 THz). See Table Ifor\nthexvalues for the samples.mentally observed magneto-optical signal for sample 1. The\nreason for choosing the Bi sample instead of Bi 0.96Sb0.04for\nthis analysis is because the Bi sample showed a much sharper\nresonance feature.\nThe bulk band structure we considered is schematically\ndepicted in Fig. 14[59,60]. There is an indirect negative band\ngap between the valence-band maximum at the Tpoint and\nthe conduction-band minima at the three equivalent Lpoints\n(later we refer to these as a single point, L). For a (001) Bi\nfilm, the hole pocket at the Tpoint has an isotropic in-plane\neffective mass and a parabolic dispersion relation, while the\nbands at the Lpoint host Dirac electrons with a hyperbolic\ndispersion, but a small gap 2/Delta1 exists at the Lpoint.\nThe Hamiltonian for the T-point holes and the L-point\nelectrons are\nHh=E0\nh−¯h2/parenleftbig\nk2\nx+k2\ny/parenrightbig\n2Mc−¯h2k2\nz\n2Mz, (7)\nHe=\n/Delta1 0 i¯hvzkz i¯hv(kx−iky)\n0 /Delta1 i¯hv(kx+iky) −i¯hvzkz\n−i¯hvzkz −i¯hv(kx−iky) −/Delta1 0\n−i¯hv(kx+iky) i¯hvzkz 0 −/Delta1\n,\n(8)\nwhere E0\nh=38.5 meV is the T-point band edge offset, krep-\nresents the wave vector, Mc=0.0677 m0and Mz=0.721 m0\nare, respectively, the in-plane and out-of-plane hole effective\nmasses, m0is the free electron mass, /Delta1=7.65 meV is half of\ntheL-point gap, and vandvzare the in-plane and out-of-plane\nelectron Dirac velocities, respectively. Because our film hasa finite thickness, we considered the quantum confinementeffect on k\nzalong the growth direction for both the hole\nand electron pockets. As schematically shown in Fig. 14,\nmany hole subbands described by discrete kz’s with quantum\nnumber Nhare above the Fermi energy EF, while only two\nelectron subbands (described by quantum number Ne) are\nfilled at the Lpoint; the electrons are more strongly influenced\nby quantum confinement than the holes because of the lighterelectron mass.\nThe key for the magneto-optical response of carriers is to\ncalculate the optical conductivity tensor given by the Kuboformula:\nσ\nαβ(ω)=i¯h\nS/summationdisplay\nm,nfm−fn\nEm−En/angbracketleft/Psi1m|ˆjα|/Psi1n/angbracketright/angbracketleft/Psi1 n|ˆjβ|/Psi1m/angbracketright\n¯hω+Em−En+iγ,(9)\nwhere αandβtake choices between xand y,Sis the\nsample area, fm(fn),Em(En), and |/Psi1m/angbracketright(|/Psi1 n/angbracketright) are, respec-\ntively, the occupation factor calculated by the Fermi-Diracdistribution function, energy, and eigenfunction of the mth\n(nth) eigenstate of the system Hamiltonian, ˆj\nαand ˆjβare\ncurrent operators, and γis the scattering rate responsible\nfor transition line broadening. We calculated the conductivitytensors contributed by each pocket and later added all theircontributions.\nIn a magnetic field, the Landau-level eigenenergies for the\nhole and electron pockets are\nE\nh,n=Eh\n0−/parenleftbigg\nn+1\n2/parenrightbigg¯heB\nMc−¯h2k2\nz\n2Mz+shgµBB, (10)\n115145-8(a)\n(b)(c) (d)Fig. 11. Magneto-optical response of Bi 1\u0000xSbxfilms. (a) Schematic phase\ndiagram of Bi 1\u0000xSbx. The five arrows pointing to the horizontal axis mark\nthe doping concentration of the five samples in (b): sample 1 ( x=0), sample\n2 (x=0:04), sample 3 ( x=0:08), sample 4 ( x=0:1) and sample 5 ( x=\n0:15). The solid lines represent the energies of di \u000berent band edges at high\nsymmetry points ( TandLpoints). (b) \u0012Fat 0.7 THz and at T=2 K for all\nsamples. (c) \u0012Fand (d)\u0011Fspectra at various magnetic fields at 2 K. Adapted\nfrom Ref. 37.\nTERAHERTZ FARADAY AND KERR ROTATION … PHYSICAL REVIEW B 100, 115145 (2019)\nFIG. 20. Calculated magneto-optical response of Bi 0.9Sb0.1at\nT=21 K in Bup to 30 T. (a) Faraday rotation and (b) Kerr rotation\nmaps versus THz frequency and magnetic field. The cuts marked by\nthe red dashed lines are plotted together with experimental data in\nFigs. 12(c) and12(d).\nconductivity as a sum of contributions from all hole pockets\nis obtained as\nσ(h)(ω)=6/summationdisplay\ni=1σ(hi)(ω) (41)\n=3/parenleftBigg\nσ(h1)\nxx(ω)+σ(h1)\nyy(ω)2 σ(h1)\nxy(ω)\n−2σ(h1)\nxy(ω) σ(h1)\nxx(ω)+σ(h1)\nyy(ω)/parenrightBigg\n.(42)\nFinally, the procedure for calculating the conductivity ten-\nsor for the ¯M-point electron pocket, σ(e,¯M)\nxx and σ(e,¯M)\nxy, is\nsimilar to that of the hole pocket.\nWe summed up the contributions from all pockets to obtain\nthe total conductivity tensor elements, σtot\nxxandσtot\nxy, and cal-\nculated the Faraday and Kerr rotations following the processdiscussed in Sec. II B. All band parameters are either given in\nthe literature or can be obtained through fits to predeterminedband structures. The three free parameters we can tune tomatch the theoretical results with experimental data are the\n¯/Gamma1-point electron scattering rate γ\n¯/Gamma1\ne, the hole scattering rate\nγh, and the ¯M-point electron scattering rate γ¯M\ne. We found\nthatγ¯/Gamma1\ne=10 meV, γh=60 meV, and γ¯M\ne=10.5 meV give\nthe best fit with the experimental data, as shown by thepolarization rotation curves at 0.7 THz in Figs. 12(c) and\n12(d) . Maps of calculated Faraday and Kerr rotations as\na function of THz frequency and magnetic field using the\nFIG. 21. Calculated magneto-optical response of the nominal\nBi0.9Sb0.1film up to 180 T. (a) Faraday rotation and (b) Kerr rotation\nversus magnetic field at a fixed THz frequency of 0.7 THz. The\ntotal rotation signal is plotted together with the separate contributions\nfrom the three carrier pockets.\noptimized parameters are shown in Fig. 20. These results can\nbe compared to the experimental maps shown in Figs. 12(a)\nand12(b), and there is agreement between the experimental\nand theoretical results. The onset of polarization-rotation sat-\nuration is predicted by theory to blueshift with magnetic fieldbecause of the blueshifting cyclotron resonance energies of allcarrier species.\nThe three optimized scattering rates, γ\n¯/Gamma1\ne,γh, and γ¯M\ne, are all\nlarger than the bandwidth of our THz probe, suggesting thatthe surface carriers do not have high enough mobility for theircyclotron resonance peaks to appear in our measurements. Inaddition, γ\nhis much larger than γ¯/Gamma1\neandγ¯M\ne. This observation\nis not surprising as we examine the linewidths of surfacebands measured by ARPES in previous studies [18 ], but its\neffect is that the contribution of surface holes in the THzpolarization rotation signal is negligibly small compared tothat of the surface electrons.\nWe provide some additional comments on the saturation\nbehavior of Faraday and Kerr rotations in the B>15 T region\nin Figs. 12(c) and12(d). It might be tempting to explain\nthese features as the quantized optical Hall effect observed\nin several recent studies on other TI systems [ 32,33,45].\nHowever, as shown in the calculation of θ\nFandθKin a much\nwider magnetic field range in Fig. 21, we found that the\n115145-13(d)XINWEI LI et al. PHYSICAL REVIEW B 100, 115145 (2019)\nFIG. 12. Magneto-optical response of sample 4 (nominal x=\n0.1) up to 30 T. (a) Faraday rotation and (b) Kerr rotation maps versus\nTHz frequency and magnetic field. (c) Faraday rotation and (d) Kerr\nrotation versus magnetic field at a fixed THz frequency of 0.7 THz,\ncorresponding to the cuts marked by the red dashed lines in (a) and(b). The solid red lines in (c) and (d) are calculated curves using the\ntheoretical model described later in the Discussions section.\nIV . DISCUSSION\nA. Theory of magneto-optical response of Bi\n1−xSbxin the\nsemimetallic regime\nIn order to understand the THz magneto-optical response\nof Bi 1−xSbxfilms in the SM regime, we developed a detailed\ntheoretical model. We took into account the bulk bands of\nBi, which allowed us to determine the origin of the experi-\n-0.10-0.050.000.05θF (rad)\n10 8 6 4 2 0\nMagnetic field (T)Sample 1\nSample 2Sample 3Sample 4Sample 5@ 0.7 THz\nFIG. 13. Magneto-optical response of all samples. θFis plotted\nvsBatT=2 K for a fixed THz frequency (0.7 THz). See Table Ifor\nthexvalues for the samples.mentally observed magneto-optical signal for sample 1. Thereason for choosing the Bi sample instead of Bi\n0.96Sb0.04for\nthis analysis is because the Bi sample showed a much sharperresonance feature.\nThe bulk band structure we considered is schematically\ndepicted in Fig. 14[59,60]. There is an indirect negative band\ngap between the valence-band maximum at the Tpoint and\nthe conduction-band minima at the three equivalent Lpoints\n(later we refer to these as a single point, L). For a (001) Bi\nfilm, the hole pocket at the Tpoint has an isotropic in-plane\neffective mass and a parabolic dispersion relation, while thebands at the Lpoint host Dirac electrons with a hyperbolic\ndispersion, but a small gap 2/Delta1 exists at the Lpoint.\nThe Hamiltonian for the T-point holes and the L-point\nelectrons are\nH\nh=E0\nh−¯h2/parenleftbig\nk2\nx+k2\ny/parenrightbig\n2Mc−¯h2k2\nz\n2Mz, (7)\nHe=\n/Delta1 0 i¯hvzkz i¯hv(kx−iky)\n0 /Delta1 i¯hv(kx+iky) −i¯hvzkz\n−i¯hvzkz −i¯hv(kx−iky) −/Delta1 0\n−i¯hv(kx+iky) i¯hvzkz 0 −/Delta1\n,\n(8)\nwhere E0\nh=38.5 meV is the T-point band edge offset, krep-\nresents the wave vector, Mc=0.0677 m0and Mz=0.721 m0\nare, respectively, the in-plane and out-of-plane hole effective\nmasses, m0is the free electron mass, /Delta1=7.65 meV is half of\ntheL-point gap, and vandvzare the in-plane and out-of-plane\nelectron Dirac velocities, respectively. Because our film hasa finite thickness, we considered the quantum confinementeffect on k\nzalong the growth direction for both the hole\nand electron pockets. As schematically shown in Fig. 14,\nmany hole subbands described by discrete kz’s with quantum\nnumber Nhare above the Fermi energy EF, while only two\nelectron subbands (described by quantum number Ne) are\nfilled at the Lpoint; the electrons are more strongly influenced\nby quantum confinement than the holes because of the lighterelectron mass.\nThe key for the magneto-optical response of carriers is to\ncalculate the optical conductivity tensor given by the Kuboformula:\nσ\nαβ(ω)=i¯h\nS/summationdisplay\nm,nfm−fn\nEm−En/angbracketleft/Psi1m|ˆjα|/Psi1n/angbracketright/angbracketleft/Psi1 n|ˆjβ|/Psi1m/angbracketright\n¯hω+Em−En+iγ,(9)\nwhere αandβtake choices between xand y,Sis the\nsample area, fm(fn),Em(En), and |/Psi1m/angbracketright(|/Psi1 n/angbracketright) are, respec-\ntively, the occupation factor calculated by the Fermi-Diracdistribution function, energy, and eigenfunction of the mth\n(nth) eigenstate of the system Hamiltonian, ˆj\nαand ˆjβare\ncurrent operators, and γis the scattering rate responsible\nfor transition line broadening. We calculated the conductivitytensors contributed by each pocket and later added all theircontributions.\nIn a magnetic field, the Landau-level eigenenergies for the\nhole and electron pockets are\nE\nh,n=Eh\n0−/parenleftbigg\nn+1\n2/parenrightbigg¯heB\nMc−¯h2k2\nz\n2Mz+shgµBB, (10)\n115145-8XINWEI LI et al. PHYSICAL REVIEW B 100, 115145 (2019)\nFIG. 12. Magneto-optical response of sample 4 (nominal x=\n0.1) up to 30 T. (a) Faraday rotation and (b) Kerr rotation maps versus\nTHz frequency and magnetic field. (c) Faraday rotation and (d) Kerr\nrotation versus magnetic field at a fixed THz frequency of 0.7 THz,\ncorresponding to the cuts marked by the red dashed lines in (a) and(b). The solid red lines in (c) and (d) are calculated curves using the\ntheoretical model described later in the Discussions section.\nIV . DISCUSSION\nA. Theory of magneto-optical response of Bi\n1−xSbxin the\nsemimetallic regime\nIn order to understand the THz magneto-optical response\nof Bi 1−xSbxfilms in the SM regime, we developed a detailed\ntheoretical model. We took into account the bulk bands of\nBi, which allowed us to determine the origin of the experi-\n-0.10-0.050.000.05θF (rad)\n10 8 6 4 2 0\nMagnetic field (T)Sample 1\nSample 2Sample 3Sample 4Sample 5@ 0.7 THz\nFIG. 13. Magneto-optical response of all samples. θFis plotted\nvsBatT=2 K for a fixed THz frequency (0.7 THz). See Table Ifor\nthexvalues for the samples.mentally observed magneto-optical signal for sample 1. Thereason for choosing the Bi sample instead of Bi\n0.96Sb0.04for\nthis analysis is because the Bi sample showed a much sharperresonance feature.\nThe bulk band structure we considered is schematically\ndepicted in Fig. 14[59,60]. There is an indirect negative band\ngap between the valence-band maximum at the Tpoint and\nthe conduction-band minima at the three equivalent Lpoints\n(later we refer to these as a single point, L). For a (001) Bi\nfilm, the hole pocket at the Tpoint has an isotropic in-plane\neffective mass and a parabolic dispersion relation, while thebands at the Lpoint host Dirac electrons with a hyperbolic\ndispersion, but a small gap 2/Delta1 exists at the Lpoint.\nThe Hamiltonian for the T-point holes and the L-point\nelectrons are\nH\nh=E0\nh−¯h2/parenleftbig\nk2\nx+k2\ny/parenrightbig\n2Mc−¯h2k2\nz\n2Mz, (7)\nHe=\n/Delta1 0 i¯hvzkz i¯hv(kx−iky)\n0 /Delta1 i¯hv(kx+iky) −i¯hvzkz\n−i¯hvzkz −i¯hv(kx−iky) −/Delta1 0\n−i¯hv(kx+iky) i¯hvzkz 0 −/Delta1\n,\n(8)\nwhere E0\nh=38.5 meV is the T-point band edge offset, krep-\nresents the wave vector, Mc=0.0677 m0and Mz=0.721 m0\nare, respectively, the in-plane and out-of-plane hole effective\nmasses, m0is the free electron mass, /Delta1=7.65 meV is half of\ntheL-point gap, and vandvzare the in-plane and out-of-plane\nelectron Dirac velocities, respectively. Because our film hasa finite thickness, we considered the quantum confinementeffect on k\nzalong the growth direction for both the hole\nand electron pockets. As schematically shown in Fig. 14,\nmany hole subbands described by discrete kz’s with quantum\nnumber Nhare above the Fermi energy EF, while only two\nelectron subbands (described by quantum number Ne) are\nfilled at the Lpoint; the electrons are more strongly influenced\nby quantum confinement than the holes because of the lighterelectron mass.\nThe key for the magneto-optical response of carriers is to\ncalculate the optical conductivity tensor given by the Kuboformula:\nσ\nαβ(ω)=i¯h\nS/summationdisplay\nm,nfm−fn\nEm−En/angbracketleft/Psi1m|ˆjα|/Psi1n/angbracketright/angbracketleft/Psi1 n|ˆjβ|/Psi1m/angbracketright\n¯hω+Em−En+iγ,(9)\nwhere αandβtake choices between xand y,Sis the\nsample area, fm(fn),Em(En), and |/Psi1m/angbracketright(|/Psi1 n/angbracketright) are, respec-\ntively, the occupation factor calculated by the Fermi-Diracdistribution function, energy, and eigenfunction of the mth\n(nth) eigenstate of the system Hamiltonian, ˆj\nαand ˆjβare\ncurrent operators, and γis the scattering rate responsible\nfor transition line broadening. We calculated the conductivitytensors contributed by each pocket and later added all theircontributions.\nIn a magnetic field, the Landau-level eigenenergies for the\nhole and electron pockets are\nE\nh,n=Eh\n0−/parenleftbigg\nn+1\n2/parenrightbigg¯heB\nMc−¯h2k2\nz\n2Mz+shgµBB, (10)\n115145-8(c)Experiment Theory\nFig. 12. Experimental result of (a) \u0012Fand (b)\u0012Kversus frequency and mag-\nnetic field for sample 4. Theoretical calculation of (c) \u0012Fand (d)\u0012Kspectra\nat fields between 0 and 30 T. Adapted from Ref. 37.\ngest that the combined e \u000bort of the THz magnetospectroscopy\nmeasurements and the detailed theoretical analysis can be em-\nployed to study surface and bulk carrier contributions to the\noptical conductivity spectra of topological materials.\n3.3 Magnons\n3.3.1 Ultrastrong magnon-magnon coupling in YFeO 3\nAntiferromagnetic materials host spin waves (magnons)\nwith typical frequencies in the THz range. Makihara et al.\nhave studied ultrastrong magnon–magnon coupling domi-\nnated by antiresonant interactions in YFeO 3,80)a canted\nantiferromagnet that supports quasi-ferromagnetic (qFM)\nand quasi-antiferromagnetic (qAFM) magnon modes. These\nmagnon modes can be probed with THz radiation if the THz\nmagnetic field component is perpendicular to the spin orien-\ntation. Figure 13a shows a geometry where an external mag-\n7J. Phys. Soc. Jpn. SPECIAL TOPICS\nFig. 13. (a) Schematic of THz magnetospectroscopy studies of YFeO 3in\na tilted magnetic field. H DCwas applied in the b-cplane at angle \u0012with re-\nspect to the c-axis, with kTHzkHDCandHTHzpolarized in the b-cplane.\n(b) Transmitted THz waveform for \u0012=20\u000eat 12.60 T displaying beating\nin the time-domain and two peaks in the frequency domain corresponding\nto the simultaneous excitation of both magnon modes in YFeO 3. (c) Exper-\nimentally measured magnon frequencies for \u0012=0\u000e;60\u000eversus H DC(black\ndots) with calculated resonance magnon frequencies (solid red lines) and de-\ncoupled qFM and qAFM magnon frequencies (black dashed-dotted lines).\nThe UM frequency becomes lower than the qAFM frequency at \u0012=90\u000e, in-\ndicating a dominant VBSS compared to the vacuum Rabi splitting-induced\nshifts. (d) Normalized co-rotating ( jg1j=!0, blue dotted line) and counter-\nrotating (jg2j=!00, red solid line) coupling strengths displaying ultrastrong\nmagnon–magnon coupling and dominance of the counter-rotating terms. !0\nis the frequency at which the qFM and qAFM modes cross. (e) Theoretical\nillustration of the qFM mode, qAFM mode, lower mode (LM), UM, and co-\nrotating coupled magnon frequencies that are obtained by setting g2=0, for\n\u0012=60\u000e. The vacuum Bloch–Siegert shifts (VBSSs) are highlighted by the\nshaded area. Adapted from Ref. 80\nnetic field, H DC, is applied at an angle \u0012to the c-axis. When\nselection rules are satisfied, both magnon modes can be mea-\nsured as beating in the time domain and peaks in the fre-\nquency domain, as shown in Fig. 13b.\nFigure 13c shows an example of the evolution of qFM and\nqAFM modes in an external magnetic field. At \u0012=0, only the\nqFM mode is experimentally accessible, while at \u0012=60\u000ebothmagnon modes can be traced as a function of magnetic field.\nInterestingly, these two modes exhibit anticrossing behaviors,\nwhich is an indication of strong coupling /hybridization.81–83)\nTo explain the hybridization between the qFM and qAFM\nmodes, the authors developed a microscopic model, which in-\ncludes interactions between spins in the two sublattices, their\nsymmetric exchange and antisymmetric exchange, the single-\nion anisotropies, and the Zeeman interaction. The system\nwas e \u000bectively described by two coupling strengths g1(co-\nrotating coupling strength) and g2(counter-rotating coupling\nstrength). Figure 13(d) shows normalized coupling strengths\nas a function of angle \u0012, where g2>g1. This led to the ob-\nservation of a dominant vacuum Bloch-Siegert shift (VBSS)\nfor the upper mode (UM), which is unique to the anisotropic\nHopfield Hamiltonian. Figure 13e summarizes results of nu-\nmerical calculations demonstrating VBSSs at \u0012=60\u000e. In ad-\ndition, quantum fluctuation suppression of up to 5.9 dB was\ntheoretically showed, indicating two-mode squeezed vacuum\nmagnonic ground state in this system.80)\n3.4 Phonons\nUsually, phonons are insensitive to magnetic fields. Un-\nlike spectral features related to electrons, which respond to\nmagnetic fields through their orbital and spin magnetic mo-\nments, phonon-related features typically remain independent\nof magnetic fields unless there is strong electron-phonon in-\nteraction. Recently, a strong magnetic response of a transverse\noptical (TO) phonon mode in a thin film of lead telluride\n(PbTe) has been observed using THz time-domain magne-\ntospectroscopy.84)PbTe, one of the most widely used ther-\nmoelectric materials, is known to have a soft lattice, hosting\nanharmonic phonons. The displacements of the TO phonon,\nschematically shown in Fig. 14a, can become circular under\nthe application of a high magnetic field (see Fig. 14b).\nFig. 14. Schematic of the PbTe crystal structure with real-space TO lattice\ndisplacements (a) without and (b) with a magnetic field applied perpendicular\nto the lattice plane. Blue (red) spheres represent Te (Pb) ions. (c) Magnetic\ncircular dichroism at 9 T as a function of temperature. The dashed line is a\nguide to the eye. (d) Magnetic field induced frequency shift of the TO phonon.\nThe dashed lines are fits to the data. Adapted from Ref. 84.\n8J. Phys. Soc. Jpn. SPECIAL TOPICS\nThe physical picture that emerged via comparison of ex-\nperimental data and theoretical modelling is that the magnetic\nfield exerted a Lorentz force directly on the ions that form the\nlattice, inducing ionic cyclotron motion, which created chiral\nphonons. Figure 14c shows magnetic circular dichroism of\nthe TO phonon as a function of temperature for a 9 T applied\nmagnetic field, which means that the response to right- and\nleft-hand polarized light di \u000bers. Figure 14d summarizes the\nR- and L-handed TO phonon frequency as a function of mag-\nnetic field, together with fits (dashed lines). This dependence\nhaving linear and quadratic terms with respect to the mag-\nnetic field indicates a large phononic Zeeman splitting and\nthe existence of a phononic diamagnetic shift. The authors\nconcluded that the magnetic phonon moment comes from TO\nphonon anharmonicity and morphic e \u000bects due to high mag-\nnetic fields.84)\n4. Summary and Outlook\nOverall, since the development of the RAMBO system,\nwe have been able to spectroscopically probe a number of\nnew physical phenomena in high magnetic fields, includ-\ning superfluorescence in InGaAs quantum wells,18)Faraday\nand Kerr rotations in a semimetal and topological insulator\ncompound,37)ultrastrong magnon-magnon coupling in a rare-\nearth orthoferrite,80)and large magnetic moments of phonons\nin PbTe.84)The easy optical access of RAMBO allows one to\noptically probe various collective excitations in di \u000berent en-\nergy scales in condensed matter systems by using di \u000berent\nlight sources. Recently, similar table-top high field magnets\nhave been developed by other groups.24, 26, 85)The growing in-\nterest in the development of such setups is a testament to the\npromise that more exciting research studies of modern materi-\nals under extreme conditions can be performed in university-\nlevel laboratories.\nWhile the RAMBO system has been well-established for\nperforming advanced magneto-optical spectroscopy experi-\nments on materials in high magnetic fields, it can be further\nimproved in multiple aspects. First, the mini-coil can be re-\ndesigned to support even larger magnetic fields without sac-\nrificing the bore diameter. Second, lower temperatures below\n10 K are highly desirable for some condensed matter samples\nthat show a variety of low-temperature phases. Third, more\nelectrical connections can be introduced for measuring optical\nand transport properties simultaneously. Finally, combination\nof RAMBO with even more optical setups covering di \u000berent\nspectral ranges will be beneficial for studying interplay be-\ntween di \u000berent energy-scale excitations.\nIn conclusion, we reviewed the technical specifications of\nthe first-generation RAMBO system and discussed several re-\nsearch advances enabled by RAMBO. As RAMBO-like sys-\ntems become available to more research groups, we expect to\nsee further new discoveries in materials physics. We believe\nthat, with further development of advanced pulsed magnets,\nnext generation RAMBO systems promise to push the fron-\ntiers of materials research in high magnetic fields even further.\n1) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).\n2) M. O. Goerbig, Rev. Mod. Phys. 83, 1193 (2011).\n3) E. Morosan, D. Natelson, A. H. Nevidomskyy, and Q. Si, Adv. Mater.\n24, 4896 (2012).4) F. Herlach and N. Miura, High Magnetic Fields: Science and Technol-\nogy: Magnet Technology and Experimental Techniques (World Scien-\ntific, Hackensack, NJ, 2003), V ol. 1.\n5) J. E. Crow, D. M. Parkin, H. J. Schneider-Muntau, and N. S. Sullivan,\nPhys. B: Condens. Matter 216, 146 (1996).\n6) J. Singleton, C. H. Mielke, A. Migliori, G. S. Boebinger, and A. H.\nLacerda, Phys. B: Condens. Matter 346–347 , 614 (2004).\n7) T. Kiyoshi, A. Sato, H. Nagai, F. Matsumoto, M. Kosuge, M. Yuyama,\nS. Nimori, T. Asano, K. Itoh, S. Matsumoto, G. Kido, and K. Watanabe,\nJ. Phys.: Conf. Ser. 51, 651 (2006).\n8) J. Wosnitza, A. D. Bianchi, J. Freudenberger, J. Haase, T. Her-\nrmannsd ¨orfer, N. Kozlova, L. Schultz, Y . Skourski, S. Zherlitsyn, and\nS. A. Zvyagin, J. Magn. Magn. Mater. 310, 2728 (2007).\n9) L. Li, T. Peng, H. F. Ding, X. T. Han, Z. C. Xia, T. H. Ding, J. F. Wang,\nJ. F. Xie, S. L. Wang, Y . Huang, X. Z. Duan, K. L. Yao, F. Herlach,\nJ. Vanacken, and Y . Pan, J. Low Temp. Phys. 159, 374 (2010).\n10) F. Debray and P. Frings, C. R. Phys. 14, 2 (2013).\n11) N. Miura and F. Herlach, Pulsed and Ultrastrong Magnetic Fields , ed.\nF. Herlach (Springer, Berlin, 1985), V ol. 57, pp. 247–350.\n12) J. Kono, N. Miura, S. Takeyama, H. Yokoi, N. Fujimori,\nY . Nishibayashi, T. Nakajima, K. Tsuji, and M. Yamanaka, Phys-\nica B 184, 178 (1993).\n13) J. Kono and N. Miura, Cyclotron Resonance in High Magnetic Fields ,\ned. N. Miura and F. Herlach (World Scientific, Singapore, 2006), pp.\n61–90.\n14) L. G. Booshehri, C. H. Mielke, D. G. Rickel, S. A. Crooker, Q. Zhang,\nL. Ren, E. H. H ´aroz, A. Rustagi, C. J. Stanton, Z. Jin, Z. Sun, Z. Yan,\nJ. M. Tour, and J. Kono, Phys. Rev. B 85, 205407 (2012).\n15) M. Jaime, R. Daou, S. A. Crooker, F. Weickert, A. Uchida, A. E.\nFeiguin, C. D. Batista, H. A. Dabkowska, and B. D. Gaulin, PNAS 109,\n12404 (2012).\n16) S. Wang, Z. Wu, S. Jiang, G. Wang, R. Huang, and T. Peng, IEEE Trans.\nAppl. Supercond. 30, 1 (2020).\n17) D. Nakamura, A. Ikeda, H. Sawabe, Y . H. Matsuda, and S. Takeyama,\nRev. Sci. Instrum. 89, 095106 (2018).\n18) G. T. Noe, H. Nojiri, J. Lee, G. L. Woods, J. L ´eotin, and J. Kono, Rev.\nSci. Instrum. 84, 123906 (2013).\n19) J. Hebling, K.-L. Yeh, M. C. Ho \u000bmann, B. Bartal, and K. A. Nelson, J.\nOpt. Soc. Am. B 25, B6 (2008).\n20) Y .-S. Lee, Principles of Terahertz Science and Technology (Springer,\nNew York, NY, 2009).\n21) R. P. Prasankumar and A. J. Taylor, Optical Techniques for Solid-State\nMaterials Characterization (CRC Press, 2012).\n22) D. Molter, G. Torosyan, G. Ballon, L. Drigo, R. Beigang, and J. L ´eotin,\nOpt. Express 20, 5993 (2012).\n23) G. T. Noe, I. Katayama, F. Katsutani, J. J. Allred, J. A. Horowitz, D. M.\nSullivan, Q. Zhang, F. Sekiguchi, G. L. Woods, M. C. Ho \u000bmann, H. No-\njiri, J. Takeda, and J. Kono, Opt. Express 24, 30328 (2016).\n24) D. Molter, F. Ellrich, T. Weinland, S. George, M. Goiran, F. Keilmann,\nR. Beigang, and J. L ´eotin, Opt. Express 18, 26163 (2010).\n25) G. T. Noe, Q. Zhang, J. Lee, E. Kato, G. L. Woods, H. Nojiri, and\nJ. Kono, Appl. Opt 53, 5850 (2014).\n26) B. F. Spencer, W. F. Smith, M. T. Hibberd, P. Dawson, M. Beck, A. Bar-\ntels, I. Guiney, C. J. Humphreys, and D. M. Graham, Appl. Phys. Lett.\n108, 212101 (2016).\n27) A. Baydin, T. Makihara, N. M. Peraca, and J. Kono, Front. Optoelec-\ntron. 14, 110 (2021).\n28) B. Yellampalle, K. Y . Kim, G. Rodriguez, J. H. Glownia, and A. J. Tay-\nlor, Opt. Express 15, 1376 (2007).\n29) Y . Minami, Y . Hayashi, J. Takeda, and I. Katayama, Appl. Phys. Lett.\n103, 051103 (2013).\n30) S. M. Teo, B. K. Ofori-Okai, C. A. Werley, and K. A. Nelson, Rev. Sci.\nInstrum. 86, 051301 (2015).\n31) I. Katayama, H. Sakaibara, and J. Takeda, Jpn. J. Appl. Phys. 50, 102701\n(2011).\n32) Y . Minami, K. Horiuchi, K. Masuda, J. Takeda, and I. Katayama, Appl.\nPhys. Lett. 107, 171104 (2015).\n33) G. Mead, I. Katayama, J. Takeda, and G. A. Blake, Rev. Sci. Instrum.\n90, 053107 (2019).\n34) R. Vald ´es Aguilar, A. V . Stier, W. Liu, L. S. Bilbro, D. K. George,\nN. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage,\nPhys. Rev. Lett. 108, 087403 (2012).\n35) R. Shimano, G. Yumoto, J. Y . Yoo, R. Matsunaga, S. Tanabe, H. Hibino,\n9J. Phys. Soc. Jpn. SPECIAL TOPICS\nT. Morimoto, and H. Aoki, Nat. Commun. 4, 1841 (2013).\n36) L. Wu, M. Salehi, N. Koirala, J. Moon, S. Oh, and N. P. Armitage,\nScience 354, 1124 (2016).\n37) X. Li, K. Yoshioka, M. Xie, G. T. Noe, W. Lee, N. Marquez Peraca,\nW. Gao, T. Hagiwara, Ø. S. Handegård, L.-W. Nien, T. Nagao, M. Kita-\njima, H. Nojiri, C.-K. Shih, A. H. MacDonald, I. Katayama, J. Takeda,\nG. A. Fiete, and J. Kono, Phys. Rev. B 100, 115145 (2019).\n38) K. Cong, G. T. Noe, and J. Kono, Excitons in Magnetic Fields (Elsevier,\nOxford, 2018), pp. 63–81.\n39) N. Miura, Physics of Semiconductors in High Magnetic Fields (Oxford\nUniversity Press, Oxford, 2008).\n40) R. H. Dicke, Phys. Rev. 93, 99 (1954).\n41) R. Bonifacio and L. A. Lugiato, Phys. Rev. A 11, 1507 (1975).\n42) R. Bonifacio and L. A. Lugiato, Phys. Rev. A 12, 587 (1975).\n43) Q. H. F. Vrehen, M. F. H. Schuurmans, and D. Polder, Nature 285, 70\n(1980).\n44) M. F. H. Schuurmans, Q. H. F. Vrehen, D. Polder, and H. M. Gibbs, Su-\nperfluorescence , ed. D. Bates and B. Bederson (Academic Press, Jan-\nuary 1982), V ol. 17, pp. 167–228.\n45) N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, Phys.\nRev. Lett. 30, 309 (1973).\n46) M. Gross, C. Fabre, P. Pillet, and S. Haroche, Phys. Rev. Lett. 36, 1035\n(1976).\n47) H. M. Gibbs, Q. H. F. Vrehen, and H. M. J. Hikspoors, Phys. Rev. Lett.\n39, 547 (1977).\n48) G. T. Noe, J.-H. Kim, J. Lee, Y . Wang, A. K. W ´ojcik, S. A. McGill,\nD. H. Reitze, A. A. Belyanin, and J. Kono, Nat. Phys. 8, 219 (2012).\n49) G. T. Noe, J.-H. Kim, J. Lee, Y .-D. Jho, Y . Wang, A. W ´ojcik, S. McGill,\nD. Reitze, A. Belyanin, and J. Kono, Fortschr. Phys. 61, 393 (2013).\n50) J.-H. Kim, J. Lee, G. T. Noe, Y . Wang, A. K. W ´ojcik, S. A. McGill,\nD. H. Reitze, A. A. Belyanin, and J. Kono, Phys. Rev. B 87, 045304\n(2013).\n51) J.-H. Kim, G. T. Noe, S. A. McGill, Y . Wang, A. K. W ´ojcik, A. A.\nBelyanin, and J. Kono, Sci. Rep. 3, 3283 (2013).\n52) K. Cong, Q. Zhang, Y . Wang, G. T. Noe, A. Belyanin, and J. Kono, J.\nOpt. Soc. Am. B 33, C80 (2016).\n53) A. A. Belyanin, V . V . Kocharovsky, and V . V . Kocharovsky, Solid State\nCommun. 80, 243 (1991).\n54) A. A. Belyanin, V . V . Kocharovsky, and V . V . Kocharovsky, Quantum\nSemiclass. Opt. 9, 1 (1997).\n55) S. Chen, M. Okano, B. Zhang, M. Yoshita, H. Akiyama, and Y . Kane-\nmitsu, Appl. Phys. Lett. 101, 191108 (2012).\n56) G. S. Jenkins, A. B. Sushkov, D. C. Schmadel, N. P. Butch, P. Syers,\nJ. Paglione, and H. D. Drew, Phys. Rev. B 82, 125120 (2010).\n57) A. D. LaForge, A. Frenzel, B. C. Pursley, T. Lin, X. Liu, J. Shi, and\nD. N. Basov, Phys. Rev. B 81, 125120 (2010).\n58) J. N. Hancock, J. L. M. van Mechelen, A. B. Kuzmenko,\nD. van der Marel, C. Br ¨une, E. G. Novik, G. V . Astakhov, H. Buhmann,\nand L. W. Molenkamp, Phys. Rev. Lett. 107, 136803 (2011).\n59) L. Wu, W.-K. Tse, M. Brahlek, C. M. Morris, R. V . Aguilar, N. Koirala,\nS. Oh, and N. P. Armitage, Phys. Rev. Lett. 115, 217602 (2015).\n60) V . Dziom, A. Shuvaev, A. Pimenov, G. V . Astakhov, C. Ames, K. Ben-\ndias, J. B ¨ottcher, G. Tkachov, E. M. Hankiewicz, C. Br ¨une, H. Buh-\nmann, and L. W. Molenkamp, Nat. Commun. 8, 15197 (2017).\n61) B. Cheng, P. Taylor, P. Folkes, C. Rong, and N. P. Armitage, Phys. Rev.\nLett. 122, 097401 (2019).\n62) X. Wang, D. J. Hilton, L. Ren, D. M. Mittleman, J. Kono, and J. L.\nReno, Opt. Lett. 32, 1845 (2007).\n63) X. Wang, A. A. Belyanin, S. A. Crooker, D. M. Mittleman, and J. Kono,\nNat. Phys. 6, 126 (2010).\n64) T. Arikawa, X. Wang, D. J. Hilton, J. L. Reno, W. Pan, and J. Kono,\nPhys. Rev. B 84, 241307 (2011).\n65) T. Arikawa, X. Wang, A. A. Belyanin, and J. Kono, Opt. Express 20,\n19484 (2012).\n66) Q. Zhang, T. Arikawa, E. Kato, J. L. Reno, W. Pan, J. D. Watson, M. J.\nManfra, M. A. Zudov, M. Tokman, M. Erukhimova, A. Belyanin, and\nJ. Kono, Phys. Rev. Lett. 113, 047601 (2014).\n67) Q. Zhang, M. Lou, X. Li, J. L. Reno, W. Pan, J. D. Watson, M. J. Manfra,\nand J. Kono, Nat. Phys. 12, 1005 (2016).\n68) X. Li, M. Bamba, Q. Zhang, S. Fallahi, G. C. Gardner, W. Gao, M. Lou,\nK. Yoshioka, M. J. Manfra, and J. Kono, Nat. Photonics 12, 324 (2018).\n69) X. Li, M. Bamba, N. Yuan, Q. Zhang, Y . Zhao, M. Xiang, K. Xu, Z. Jin,\nW. Ren, G. Ma, S. Cao, D. Turchinovich, and J. Kono, Science 361, 794(2018).\n70) D. Hsieh, D. Qian, L. Wray, Y . Xia, Y . S. Hor, R. J. Cava, and M. Z.\nHasan, Nature 452, 970 (2008).\n71) D. Hsieh, Y . Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder,\nL. Patthey, J. G. Checkelsky, N. P. Ong, A. V . Fedorov, H. Lin, A. Bansil,\nD. Grauer, Y . S. Hor, R. J. Cava, and M. Z. Hasan, Nature 460, 1101\n(2009).\n72) L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).\n73) J. C. Y . Teo, L. Fu, and C. L. Kane, Phys. Rev. B 78, 045426 (2008).\n74) H.-J. Zhang, C.-X. Liu, X.-L. Qi, X.-Y . Deng, X. Dai, S.-C. Zhang, and\nZ. Fang, Phys. Rev. B 80, 085307 (2009).\n75) T. Hirahara, Y . Sakamoto, Y . Saisyu, H. Miyazaki, S. Kimura, T. Okuda,\nI. Matsuda, S. Murakami, and S. Hasegawa, Phys. Rev. B 81, 165422\n(2010).\n76) I. Katayama, H. Kawakami, T. Hagiwara, Y . Arashida, Y . Minami, L.-\nW. Nien, O. S. Handegard, T. Nagao, M. Kitajima, and J. Takeda, Phys.\nRev. B 98, 214302 (2018).\n77) P. J. de Visser, J. Levallois, M. K. Tran, J.-M. Poumirol, I. O. Nedoliuk,\nJ. Teyssier, C. Uher, D. van der Marel, and A. B. Kuzmenko, Phys. Rev.\nLett. 117, 017402 (2016).\n78) Z. Zhu, B. Fauqu ´e, Y . Fuseya, and K. Behnia, Phys. Rev. B 84, 115137\n(2011).\n79) H. M. Benia, C. Straßer, K. Kern, and C. R. Ast, Phys. Rev. B 91, 161406\n(2015).\n80) T. Makihara, K. Hayashida, G. T. Noe, X. Li, N. Marquez Peraca,\nX. Ma, Z. Jin, W. Ren, G. Ma, I. Katayama, J. Takeda, H. Nojiri,\nD. Turchinovich, S. Cao, M. Bamba, and J. Kono, Nat. Commun. 12,\n3115 (2021).\n81) A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and\nF. Nori, Nat. Rev. Phys. 1, 19 (2019).\n82) P. Forn-D ´ıaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Rev. Mod.\nPhys. 91, 025005 (2019).\n83) N. M. Peraca, A. Baydin, W. Gao, M. Bamba, and J. Kono, Ultra-\nstrong Light–Matter Coupling in Semiconductors , ed. S. T. Cundi \u000band\nM. Kira (Elsevier, Amsterdam, 2020), V ol. 105, Chap. 3, pp. 89–151.\n84) A. Baydin, F. G. G. Hernandez, M. Rodriguez-Vega, A. K. Okazaki,\nF. Tay, G. T. Noe, I. Katayama, J. Takeda, H. Nojiri, P. H. O. Rappl,\nE. Abramof, G. A. Fiete, and J. Kono, arXiv:2107.07616 (2021).\n85) K. W. Post, A. Legros, D. G. Rickel, J. Singleton, R. D. McDonald,\nX. He, I. Bo ˇzovi´c, X. Xu, X. Shi, N. P. Armitage, and S. A. Crooker,\nPhys. Rev. B 103, 134515 (2021).\nFuyang Tay is a graduate student in\nthe Applied Physics Graduate Program\nat Rice University. He received his B.S.\ndegree in Physics from Nanyang Tech-\nnological University in 2018. His current\nresearch interests include ultrastrong\nlight-matter coupling and ultrafast\nphenomena in condensed matter.\nAndrey Baydin is a J. Evans Attwell-\nWelch Postdoctoral Research Fellow\nin the Smalley-Curl Institute at Rice\nUniversity. He obtained his Ph.D. degree\nin Physics from Vanderbilt University,\nUSA in May 2018. His current research\ninterests include ultrafast spectroscopy\nof quantum materials and light-matter\ninteraction in the ultrastrong coupling\nregime.\n10J. Phys. Soc. Jpn. SPECIAL TOPICS\nFumiya Katsutani received his Bach-\nelor’s degree in electrical engineering\nfrom National Institution for Academic\nDegrees and University Evaluation in\nJapan in 2013 and his Master’s degree\nin electrical engineering from Osaka\nUniversity in 2014. He obtained his\nPh.D. degree in Electrical and Computer\nEngineering from Rice University in\n2020. Since 2020, he has been an optical\nengineer & application scientist at Shimadzu Corporation.\nJunichiro Kono is Karl F. Hasselmann\nChair in Engineering, Professor in the\nDepartments of Electrical & Computer\nEngineering, Physics & Astronomy, and\nMaterials Science & Nanoengineering,\nand Chair of Applied Physics at Rice\nUniversity. He received his B.S. and\nM.S. degrees in applied physics from the\nUniversity of Tokyo in 1990 and 1992,\nrespectively, and completed his Ph.D. in\nphysics at the State University of New\nYork at Bu \u000balo in 1995. He was a post-\ndoctoral research associate at the University of California Santa Barbara from\n1995-1997, and the W. W. Hansen Experimental Physics Laboratory Fellow\nin the Department of Physics at Stanford University from 1997-2000. His\ncurrent research interests include quantum optics in condensed matter, ultra-\nstrong light-matter coupling, and terahertz science and technology.\n11"    },    {        "title": "1111.0397v2.Dynamics_of_the_magnetic_flux_penetration_into_type_II_superconductors.pdf",        "content": "Dynamics of the magnetic \rux penetration into type II superconductors\nN. A. Taylanov\nNational University of Uzbekistan\nAbstract\nThe magnetic \rux penetration dynamics of type-II superconductors in the \rux \row regime is studied\nby analytically solving the nonlinear di\u000busion equation for the magnetic \rux induction, assuming\nthat an applied \feld parallel to the surface of the sample and using a power-law dependence of\nthe di\u000berential resistivity on the magnetic \feld induction. An exact solution of nonlinear di\u000busion\nequation for the magnetic induction is obtained using a well known self-similar technique.\nKey words : superconductors, nonlinear di\u000busion, \rux \row, \rux creep.\nx1. Introduction\nTheoretical investigations of the magnetic \rux penetration dynam-\nics into superconductors in a various regimes with a various current-\nvoltage characteristics is one of key problems of electrodynamics of\nsuperconductors. Mathematical problem of theoretical study the dy-\nnamics of evolution and penetration of magnetic \rux into the sample\nin the viscous \rux \row regime can be formulated on the basis of a\nnonlinear di\u000busion-like equation [1-3] for the magnetic \feld induction\nin a superconductor [4-14]. The dynamics of space-time evolution of\nthe magnetic \rux penetration into type-II superconductors, where the\n\rux lines are parallel to the surface of the sample for the viscous \rux\n\row regime with a nonlinear relationship between the \feld and current\ndensity in type II superconductors has been studied by many authors\n[4-7]. The magnetic \rux penetration problem was theoretically stud-\nied for the particular case, where the \rux \row resistivity independent\nof the magnetic \feld by authors [5]. Similar problem has been con-\nsidered in [6] for the semi-in\fnite sample in parallel geometry. The\nsituation, where \rux \row resistivity depends linearly on the magnetic\n\feld induction was considered analytically in [4]. Analogical problem\nfor the creep regime with a nonlinear relationship between the current\nand \feld has been considered in [8-14]. The magnetic \rux penetration\ninto the superconductor sample, where the \rux lines are perpendicular\nto the surface of the sample is described by a non-local nonlinear dif-\nfusion equation [7]. This problem has been exactly solved by Briksin\nand Dorogovstev [7] for the case thin \flm geometry in the \rux \row\nregime of a type-II superconductors.\nx2. Objectives\nIn the present paper the magnetic \rux penetration dynamics of\ntype-II superconductors in the \rux \row regime is studied by analyt-\nically solving the nonlinear di\u000busion equation for the magnetic \rux\ninduction, assuming that an applied \feld parallel to the surface of the\nsample and using a power-law dependence of the di\u000berential resistivity\non the magnetic \feld induction. An exact solution of nonlinear di\u000bu-\nsion equation for the magnetic induction ~B(r;t) is obtained by using a\nwell known self-similar technique. We study the problem in the frame-\nwork of a macroscopic approach, in which all lengths scales are larger\nthan the \rux-line spacing; thus, the superconductor is considered as\nan uniform medium.\nx3. Formulation of the problem\nBean [15] has proposed the critical state model which is success-\nfully used to describe magnetic properties of type II superconductors.\nAccording to this model, the distribution of the magnetic \rux density\n~Band the transport current density ~jinside a superconductor is given\nby a solution of the equation\nrot~B=\u00160~j: (1)\nWhen the penetrated magnetic \rux changes with time, an electric \feld\n~E(r;t) is generated inside the sample according to Faraday's lawrot~E=d~B\ndt: (2)\nIn the \rux \row regime the electric \feld ~E(r;t) induced by the moving\nvortices is related with the local current density ~j(r;t) by the nonlinear\nOhm's law\n~E=\u001a~j: (3)\nIn combining the relation (3) with Maxwell's equation (2), we obtain\na nonlinear di\u000busion equation for the magnetic \rux induction ~B(r;t)\nin the following form\nd~B\ndt=1\n\u00160rh\n\u001a(B)r~Bi\n: (4)\nFormally, this di\u000berential equation is simply a nonlinear di\u000busion equa-\ntion with a di\u000busion coe\u000ecient depending on magnetic induction B.\nThe parabolic type di\u000busion equation (4) allows to obtain a time and\nspace distribution of the magnetic induction pro\fle in a superconduc-\ntor sample. It is evident that the space-time structure of the solution\nof the di\u000busion equation (4) is determined by the character of depen-\ndence of the di\u000berential resistivity coe\u000ecient \u001aon the magnetic \feld\ninductionB. Usually, in real experimental situation [16], the di\u000beren-\ntial resistivity \u001agrows with an increase of magnetic \feld induction\n\u001a=~B\u001e0\n\u0011c2=\u001an~B\nHc2; (5)\nwhere\u001anis the di\u000berential resistivity in the normal state; \u0011is the\nviscous coe\u000ecient, \u001e0=\u0019hc= 2eis the magnetic \rux quantum, Hc2\nis the upper critical \feld of superconductor [16]. In the case, when\nthe di\u000berential resistivity \u001ais a linear function of the magnetic \feld\ninductionBan exact solution of the di\u000busion equation (4) can be\neasily obtained by using the well-known scaling methods [1, 2]. For\nthe complex dependence of \u001a(B) it can be use by empirical power-law\ndependence \u001a(B) =Bn, where n is the positive constant parameter.\nx4. Basic equations\nWe formulate the general equation governing the dynamics of the\nmagnetic \feld induction in a superconductor sample. We study the\nevolution of the magnetic penetration process in a simple geometry -\nsuperconducting semi-in\fnitive sample x\u00150. We assume that the ex-\nternal magnetic \feld induction Beis parallel to the z-axis. When the\nmagnetic \feld with the \rux density Be(t) is applied in the direction\nof the z-axis, the transport current ~j(r;t) and the electric \feld ~E(r;t)\nare induced inside the slab along the y-axis. For this geometry, the\nspatial and temporal evolution of magnetic \feld induction ~B(r;t) is de-\nscribed by the following nonlinear di\u000busion equation in the generalized\ndimensionless form [7]\ndb\ndt=d\ndx\u0014\nbn\u0014db\ndx\u0015q\u0015\n; (6)\n1arXiv:1111.0397v2  [cond-mat.supr-con]  4 Nov 2011where we have introduced the dimensionless parameters b=B=Be,\nj=j=jc,x0=x=x 0,t0=t=\u001cand variables x0=Be=\u00160jcis the mag-\nnetic \feld penetration depth in a Bean model; \u001c=\u001anj2\nc\u00160=B2\neis the\nrelaxation di\u000busion time; q is the positive constant parameter.\nThe di\u000busion equation (6) can be integrated analytically subject\nto appropriate initial and boundary conditions in the center of the\nsample and on the sample's edges. We consider the case, when the\nmagnetic \feld applied to sample increases with time according to a\npower law with the exponent of \u000b\u00150\nb(0;t) =b0t\u000b(7)\nBoundary condition (7) is equivalent to a linear increase in the mag-\nnetic \feld with time, which corresponds to a real experimental situa-\ntion. As can be easily seen that the case \u000b= 0 describes a constant\napplied magnetic \feld at the surface of the sample, while the case\n\u000b= 1 corresponds to linearly increasing applied \feld, respectively.\nThe other boundary condition follows from the continuity of the \rux\nat the free boundary x=xp\nb(xp;t) = 0; (8)\nwherexpis the dimensionless position of the front of the magnetic\n\feld. The \rux conservation condition for the magnetic \feld induction\ncan be formulated in the following integral form\nZ\nb(x;0)dx= 1: (9)\nIt should be noted that the nonlinear di\u000busion equation (6), completed\nby the boundary conditions for magnetic induction, totally determines\nthe problem of the space-time distribution of the magnetic \rux pen-\netration into superconductor sample in the \rux \row regime with a\npower-law dependence of di\u000berential resistivity on the magnetic \feld\ninduction. Solution of this equation gives a complete description of\nthe time and space evolution of the magnetic \rux in a sample.\nx5. Scaling solution\nIn the following analysis we derive an evolution equation for the\nmagnetic induction pro\fle and formulate a similarity solution for the\nb(x, t). As can be shown that the nonlinear di\u000busion equation (6) can\nbe solved exactly using well known scaling methods [1, 2]. At long\ntimes we present a solution of the nonlinear di\u000busion equation for the\nmagnetic induction (6) in the following scaling form\nb(x;t) =t\u000bf\u0010x\nt\f\u0011\n: (10)\nThe similarity exponents \u000band\fare of primary physical importance\nsince the parameter \u000brepresents the rate of decay of the magnetic\ninduction b(x, t), while the parameter \fis the rate of spread of the\nspace distribution as time goes on. Inserting this scaling form into\ndi\u000berential equation (6) and comparing powers of t in all terms, we\nget the following relationship for the exponents \u000band\f\n\u000b+ 1 =\u000b(n+q) +\f(1 +q):\nUsing the condition of the \rux conservation (9) we obtain\n\u000b=\f=\u00001\nn+ 2q\nwhich suggests the existence of self-similar solutions in the form\nb(z) =t\u00001=(n+2q)f(z); z =x=t1=(n+2q): (11)\nSubstituting this scaling solution (11) into the governing equation (6)\nyields an ordinary di\u000berential equation for the scaling function f(z) in\nthe form\n(n+ 2q)d\ndz\u0014\nfn\u0012df\ndz\u0013q\u0015\n+zdf\ndz+f= 0: (12)\nThe boundary conditions for the function f(z) now become\nf(0) = 1; f(z0) = 0: (13)The above equation (12), depending on the initial and the boundary\nconditions describes a scaling|like behavior magnetic \rux front with\na time|dependent velocity in the sample. After a further integra-\ntion and applying the boundary conditions (13) we get the following\nsolution of the problem\nf(z) =f(z0)\"\n1\u0000\u0012z\nz0\u00131+q#1=(n+q\u00001)\n: (14)\nf(z0) =\"\nn+q\u00001\n1 +q\u00121\nn+ 2q\u00131=q#q=(n+q\u00001)\nz(q+1)=(n+q\u00001)\n0:\nThe position of the front z0can now be found by substituting the\nsolution (14) into the integral condition (9) and it is given by\nz(n+2q)=(n+q\u00001)\n0=\"\nn+q\u00001\n1 +q\u00121\nn+ 2q\u00131=q#q=(n+q\u00001)\n1\nq+ 1\u0000\u0012n+q\nn+q\u00001+1\n2\u0013\n\u0000\u0012n+q\nn+q\u00001\u0013\n\u0000\u00121\nq+ 1\u0013:(15)\nIt is convenient to write the self-similar solution (14) in terms of a\nprimitive variables, as\nb(x;t) =b0(t)\"\n1\u0000\u0012x\nxp\u00131+q#1=(n+q\u00001)\n: (16)\nb0(t) =t\u00001=(n+2q)2\n4n+q\u00001\n1 +q \nz(q+1)=q\n0\nn+ 2q!1=q3\n5q=(n+q\u00001)\n:\nThis solution describes the propagation of the magnetic \feld into the\nsample, the magnetic induction being localized in the domain between\nthe surface x=0 and the \rux front xp. This solution is positive in the\nplanex2\np> x2and is zero outside of it. Note, that only the x > 0\nandt>0 quarter of the plane is presented, because of it has physical\nrelevance. The penetrating \rux front position x=xp(t) as a function\nof time can be described by the relation\nv\u0018dxp\ndt\u0018t\u0000(2q+n\u00001)=(n+2q):\nFig.1. The pro\fle of the magnetic \rux front velocity at di\u000berent\nvalues of n=3, 7, 11.\nThe velocity of the magnetic \rux front decreases rapidly as the\nmagnetic \rux propagates (Fig1).\n2x6. Particular case\nThe spatial and temporal pro\fles of magnetic \rux penetration in\nthe sample depends on the set of three independent parameters, n, q\nand\u000b. It is of interest to consider the nonlinear di\u000busion equation for\nthe magnetic induction for di\u000berent values of the exponents n, q and\n\u000b. For a given parameter set n, q and \u000bthe form of the scaling func-\ntion f(z) can be obtained by solving the nonlinear di\u000busion equation\n(6) analytically by a self-similar technique. We solve the nonlinear dif-\nfusion equation analytically to provide expressions for the time-space\nevolution of the magnetic induction for di\u000berent values of exponents n,\nq and\u000b. Next, we systematically analyze the e\u000bect of di\u000berent values\nof exponents on the shape of the magnetic \rux front in the sample.\nVarying the parameters of the equation, we may observe a various\nshapes of the magnetic \rux front in the sample. A similar approach\nhas been presented in Ref. [7] within the framework of non-linear \rux\ndi\u000busion in transverse geometry. As can be shown below that di\u000berent\nvalues exponents n and q generate di\u000berent space{time magnetic \rux\nfronts. Below we consider a few more practically relevant examples for\nwhich the magnetic \rux front has a di\u000berent shape depending on the\ndi\u000berent values of exponents n and q.\nx6. 1. Case q= 1\nLet us \frst consider the most interesting case q=1. In this par-\nticular case the spatial and temporal evolution of the magnetic \rux\ninduction is totally determined by the parameters n and \u000b. In the\nfollowing analysis we derive an evolution equation for the magnetic\ninduction pro\fle and apply the scalings of the previous section to for-\nmulate a similarity solution for the b(x, t). For this particular case\nnonlinear di\u000busion equation (6) can be solved exactly using the scaling\nmethod. Thus, based on the scalings described in the previous section,\nwe get the following relation for the exponents\n\u000b=\f=\u00001\nn+ 2:\nThe last relation suggests the existence of solution to equation (6) of\nthe form\nb(x;t) =t\u00001=(n+2)f(z); z =x=t1=(n+2): (17)\nSubstituting the similarity solution (17) into the governing equation\n(6) yields an ordinary di\u000berential equation for the scaling function f(z)\n(2 +n)d\ndz\u0012\nfndf\ndz\u0013\n+zdf\ndz+f= 0: (18)\nIntegrating the equation (18) by parts and applying the boundary con-\nditions (13) give\nf(z) =\u0014n\n2(n+ 2)z2\n0\u00151=n\u0014\n1\u0000z2\nz2\n0\u00151=n\n; (19)\nwhich is the explicit form of the similarity solution, which we have been\nseeking. The position of the front z0can now be found by substituting\nthe last solution into the integral condition (9), so we have\n\u0014n\n2(n+ 2)z2\n0\u00151=nZz0\n0\u0014\n1\u0000z2\nz2\n0\u00151=n\ndz= 1;\nBy using the following transformation\nz=z0sin!;\nand after integrating we obtain\nz(n+2)=n\n0\u0014n\n2(n+ 2)\u00151=n\n=2p\u0019\u0000\u00123\n2+1\nn\u0013\n\u0000\u0012\n1 +1\nn\u0013:\nIt is convenient to write the self-similar solution (19) in terms of a\nprimitive variables, as\nb(x;t) =b0\"\n1\u0000x2\nx2p#1=n\n; (20)where\nb0=\u0014n\n2(n+ 2)z2\n0\u00151=n\nt\u00001=(n+2):\nEquation (20) constitutes an exact solution of the nonlinear \rux-\ndi\u000busion equation for the situation, when q=1. As can be seen the\nsolution (20) describes the propagation of the \rux pro\fle inside the\nsample. The pro\fle of the the normalized \rux density b(x;t) for this\ncase is shown schematically in \fgure 2.\nFig.2. The distribution of the normalized \rux density b(x;t) at\ndi\u000berent times t=0.1, 0.2, 0.3 for n=1, q=1.\nThe penetrating \rux front position x=xp(t) as a function of time\ncan be described by the relation\nxp=z0t1=(n+2):\nThe velocity of penetration of a magnetic \rux into a superconductor\ncan be naturally determined from the last relation\nv\u0018t\u0000(n+1)=(n+2):\nInterestingly, that the normalized current density j(x;t) in the region,\n0<x<xpcan be presented using the equations\nj(x;t) =\u00001\ncdb\ndx;\nAfter a simple analytical calculation, we can easily obtain the space\nand time pro\fles of the normalized current density j(x;t) in the fol-\nlowing form\nj(x;t) =2b0\ncnx2p\"\n1\u0000\u0012x\nxp\u00132#1=n\u00001\nx;\nx6. 2. Case n= 0\nLet us consider the case n= 0. For this particular case the long-\ntime asymptotic behavior of the magnetic induction has the scaling\nform\nb(x;t) =f(z); z =xt\u00001=(q+1): (21)\nSubstituting the scaling solution into the governing equation (6)\nyields an ordinary di\u000berential equation for for the distribution f(z) in\nthe form\n(q+ 1)d\ndz\u0014df\ndz\u0015q\n+zdf\ndz= 0:\nIntegrating the last equation and applying the boundary conditions\n(13) we get the following solution of the problem\n3j(z) =\u0000df\ndz=\u0014q\u00001\n2q(q+ 1)(z2\n0\u0000z2)\u00151=(q\u00001)\n; (22)\nwhere the position of the front z0can be found by substituting the\nsolution (22) into the integral condition (9), and it is given by the\nfollowing expression\nz0=2\n66642qq+ 1\nq\u000010\nBB@p\u0019\n2\u0000\u0012q\nq\u00001\u0013\n\u0000\u00123\n2+1\nq\u00001\u00131\nCCA1\u0000q3\n77751=(q+1)\n:\nThe scaling solution (22) can be written in terms of the primitive vari-\nables as\nj(x;t) =1\nt1=(q+1)\u0014q\u00001\n2q(q+ 1)\u00151=(q\u00001)\nz2=(q\u00001)\n0\"\n1\u0000x2\nx2p#1=(q\u00001)\n:\nThe the \rux front can be approximately given as xp=z0t1=(q+1).\nThe velocity of penetration of a current density into a superconductor\nis determined from the relation\nv=dxp\ndt\u0018t\u0000q=(q+1):\nWe note that, an analogous problem has also been studied in [10, 11]\nin connection with the magnetic relaxation of a superconducting slab\nin the \rux creep regime in the framework of an approximate power-law\ndependence of the electric \feld E on the current density j. The authors\n[11] showed that in the case logarithmic barriers the relaxation process\ncauses the system self-organize into critical state. Supposing that the\nhomogeneous magnetic induction B0is induced by a constant mag-\nnetic \feld, they found the expression for the magnetization moment\nin the limit of n\u001d1. It has been shown by Koziol [12] that a pinning\npotential depending logarithmicaly on current density leads to a sim-\nilar nonlinear di\u000busion equation for the spatiotemporal evolution of\nthe \rux density with a a power-law current-voltage characteristic. An\napproximate and exact solutions of the di\u000busion problem have been\nderived assuming that an external magnetic \feld directed to parallel\nof the surface of a sample. The scaling relation between the character-\nistic relaxation time, magnetic \feld and a sample size has been found.\nSimilar problem has been considered by Gilchrist [8, 9]. The \rux creep\nproblem has been solved by Wang et.all., [13] and for an exponential\nmodel by authors [14], numerically.\nx6. 3. Case n= 1\nLet us now consider the case n=1. In this particular case the\nspatial and temporal evolution of the magnetic \rux induction is de-\ntermined by the parameters q and \u000b. In the following analysis we\nderive an evolution equation for the magnetic induction pro\fle for the\ncase n=1 and apply the scalings of the previous section to formulate\na similarity solution for the b(x, t). For this particular case nonlinear\ndi\u000busion equation (6) can be solved exactly using the scaling method.\nThus, based on the scalings described in the previous section, we get\nthe following relation for the exponents\n\u000b=\f=\u00001\n2q+ 1:\nThe last relation suggests the existence of solutions of the form\nb(x;t) =t\u00001=(2q+1)f(z); z =x=t1=(2q+1): (23)\nSubstituting the similarity solution (23) into the governing equa-\ntion (6) yields an ordinary di\u000berential equation for the function f(z)\n(2q+ 1)d\ndz\u0012\nf\u0012df\ndz\u0013q\u0013\n+zdf\ndz+f= 0: (24)\nIntegrating the equation (24) by parts and applying the boundary con-\nditions givef(z) = \u0012q\nq+ 1\u0013qzq+1\n0\n2q+ 1!1=q\"\n1\u0000\u0012z\nz0\u0013(q+1)=q#\n; (25)\nwhich is the explicit form of the similarity solution we have been seek-\ning. The position of the front can now be found by substituting the\nlast solution into the integral condition (9), and we have\nz0= [q1=(q+1)(2q+ 1)])(q+1)=(2q+1):\nIt is convenient to write the self-similar solution (25) in terms of a\nprimitive variables, as\nb(x;t) =b0\"\n1\u0000x2\nx2p#1=q\n; (26)\nwhere\nb(x;t) =t\u00001=(2q+1) \u0012q\nq+ 1\u0013qzq+1\n0\n2q+ 1!1=q\n:\nEquation (26) describes an exact solution of the nonlinear \rux-\ndi\u000busion equation for the situation, when n=1. As can be seen the\nsolution (26) describes the propagation of the \rux pro\fle into the sam-\nple. The the \rux front can be approximately given as xp=t1=(2q+1).\nThe velocity of penetration of a magnetic \rux induction front into a\nsuperconductor is determined from relation\nv=dxp\ndt\u0018t\u00002q=(2q+1):\nThe velocity of the magnetic \rux front decreases rapidly as the mag-\nnetic \rux propagates.\nx6. 4. Case n= 1; q = 1\nLet us now consider the case, where n= 1; q = 1. For this\nparticular case the nonlinear di\u000busion equation for the distribution\nmagnetic \rux admits an exact self-similar solution with a sharp front\nmoving with a constant velocity. The scaling analysis show that the\ndi\u000busion equation can be solved for the following values of parameters\n1).\u000b=0,\f=1/2; 2). \u000b=1/3,\f=2/3; 3). \u000b=1,\f=1; 4).\u000b=1/3,\n\f=1/3. Let us consider a solution of the nonlinear di\u000busion problem\nfor the exponents \u000b=1/3,\f=1/3. In this case the nonlinear di\u000busion\nequation (6) admits an exact solution [4] with the similarity variable\nb(x;t) =t\u00001=3f(xt\u00001=3): (27)\nBy substituting the solution (27) into di\u000busion equation (6) we obtain\nan ordinary di\u000berential equation for the scaling function f(z) in the\nform\n3d\ndz\u0012\nfdf\ndz\u0013\n+zdf\ndz+f= 0: (28)\nIntegrating the equation (28) and using the boundary conditions for\nthe magnetic induction f(z) we get a simple equation\n3df\ndz+z= 0: (29)\nSeparating variables and integrating equation (29) gives an implicit\nsolution in the form\nf=1\n6(z2\n0\u0000z2); (30)\nwhere the integration constant is determined by integral relation (9)\nand has the form\nz0=\u00129\n2\u00131=3\n:\nThus, we have solution for the magnetic induction in a primitive vari-\nables\n4b(x;t) =1\n6t1=3\"\u00129\n2\u00132=3\n\u0000x2\nt2=3#\n: (31)\nThe expression (31) describes an exact solution of the nonlinear \rux-\ndi\u000busion equation for the situation, when n= 1; q = 1. As can be\nseen the solution (31) describes the propagation of the \rux pro\fle into\nthe sample. The pro\fles of the the normalized \rux density b(x;t) and\nelectric \feld density e(x;t) for this case is shown schematically in the\n\fgures 3a-3b.\nFigs.3a and 3b. The distributions of the normalized \rux density\nb(x;t) and electric \feld density e(x;t) at di\u000berent times t=0.1, 0.2,\n0.3 for n=1, q=1.\nIt is evident from the solution (31) that the shock wave like mag-\nnetic \rux propagates with wave front\nxp=z0t1=3: (32)\nIt can be shown that the sharp \rux front moves with the time\ndependent velocity\nv=dxp\ndt\u0018z0t\u00002=3: (33)\nThe case\u000b=1/3,\f=1/3 implies that the \rux front asymptotically ex-\npands according to power-law xp/t1=3and magnetic \rux penetration\npro\fle decreases as b/t\u00001=3. Similar problem for the case penetra-\ntion of magnetic \feld into superconductors has been studied by Bass\n[4]. An analogous theoretical result in the viscous \row mode for vor-\ntices was obtained in [5] using the model of the critical state for oxide\nhigh-Tcsuperconductors.Finally, in the case \u000b=0,\f=1/2 the \rux front position has the\nform ofxp/t1=2and an approximate analytical solution for the case\ncan be easily found solving the di\u000busion equation with the help of sim-\nilarity variable b(x;t) =f(z), wherez=f(x=t1=2). In the vicinity of\nthe front (x/xp) the magnetic \rux pro\fle is given by the following\nsimple relation b/p\n2(x\u0000xp). In this case, which corresponds to a\nconstant external magnetic \feld the \rux front position xpgrows with\ntime asxp/t1=2for long times.\nx6. 5. Case n= 2; q = 1\nThe scaling analysis show that the di\u000busion equation can be solved\nfor the following values of parameters 1) \u000b=1/4,\f=1/4; 2) \u000b=1/4,\n\f=1/6. The solution to di\u000busion equation for this case can be pre-\nsented in the following form\nb(x;t) =t\u00001=4f(x=t1=4): (34)\nSubstituting the solution (34) into the partial di\u000berential equation (6)\nand integrating with the boundary conditions we can see that magnetic\n\rux pro\fle has an exact solution in the form\nf=z0\n2\u0014\n1\u0000z2\nz2\n0\u00151=2\n;\nwhere the integration constant has the form\nz0=2p\u0019:\nThe solution for the magnetic induction in a primitive variables has\nthe form\nb(x;t) =1p\u0019t1=4\u0014\n1\u0000\u0019\n4x2\nt1=2\u00151=2\n: (35)\nThe expression (35) describes an exact solution of the nonlinear \rux-\ndi\u000busion equation for the situation, when n= 2; q = 1. As can be\nseen the solution (35) describes the propagation of the \rux pro\fle into\nthe sample. The evolution of the self-simulating process of magnetic\n\feld penetration into a superconductor is shown schematically in the\n\fgures 4a-4b.\n5Figs.4a and 4b. The distributions of the normalized \rux density\nb(x;t) and electric \feld density e(x;t) at di\u000berent times t=0.1, 0.2,\n0.3 forn= 2; q = 1.\nIn this case the front coordinate is xp=t1=4. A main result is that\nthe magnetic \rux will move as a shock front with velocity v=t3=4.\nSimilar problem for the case penetration of magnetic \feld into super-\nconductors has been studied in [5].\nx6. 6. Case n= 0; q = 1\nIn this case the di\u000busion equation can be solved in a di\u000berent val-\nues of\u000band\f, in particular 1). \u000b=0,\f=1/2; 2).\u000b=-1/2,\f=-1/2;\n3).\u000b=1,\f=1. Let us consider the case \u000b=0,\f=-1/2. In this case the\ndi\u000berential equation is linear and admits an exact analytical solution.\nAt long times we present a solution of the nonlinear di\u000busion equation\nfor the magnetic induction (6) in the following scaling form\nb(x;t) =t\u000bfhx\nt\fi\n: (36)\nInserting the scaling form (36) into di\u000berential equation (6) we have\nt\u000b+2\fd2f\ndz2=t\u000b\u00001\u0012\n\u000bf+\fzdf\ndz\u0013\n: (37)\nComparing powers of t in all terms in (37), we get the following rela-\ntionship for the exponents \u000band\f\n\u000b=\f=\u00001\n2;\nThen the problem admits a similarity solution of the form\nb=t\u00001=2f(xt\u00001=2): (38)\nSubstituting the expression (38) into the nonlinear di\u000busion equation\n(6) gives an ordinary di\u000berential equation for the function similarity\nfunction f(z), namely\n2d2f\ndz2+zdf\ndz+f= 0: (39)\nIntegrating (39) twice and applying the boundary conditions we have\na self-similar solution in the form\nb(x;t) =1p\n4\u0019texp\"\n\u0000x2\nx2p#\n: (40)\nwhere\nxp=p\n4t:\nThe last formulae describes an exact solution of the nonlinear \rux-\ndi\u000busion equation for the situation, when n= 2; q = 1. As can be\nseen this solution describes the propagation of the \rux pro\fle into the\nsample. The pro\fle of the self-simulating process of magnetic \feld\npenetration into a superconductor for this case is shown schematically\nin the \fgures 5a-5b.\nFigs.5a and 5b. The distributions of the normalized \rux density\nb(x;t) and electric \feld density e(x;t) at di\u000berent times t=0.1, 0.2,\n0.3 forn= 0; q = 1 .\nSimilar problem has been solved in [6] using an extended critical\nstate model for type-II superconductors. Assuming that the critical\ncurrent density and resistivity are constants a one-dimensional di\u000bu-\nsion equation for the magnetic \feld were solved analytically for the\ncase when an external magnetic \feld is increased with time according\nto a power law with the exponent of one- half.\nx7. Flux creep\nLet us consider the magnetic \rux penetration process for the \rux creep\nregime. According to Kim-Anderson theory [17, 18] the thermally\nactivated \rux motion in superconductor sample is described by an\nArrhenius-type expression\nv=v0exp[\u0000U=kT ]; (41)\nwherev0is the resistivity at T=0, Uis an activation energy for ther-\nmally activated \rux jumps, and determines the vortex pinning; Tis the\ntemperature and kis the Boltzmann constant. The activation energy\nU=U(~j;~B;T) depends on temperature T, magnetic \feld induction\n~Band current density ~j. For the simple case it can be presented by\nKim-Anderson [17] formulae\nU(j) =U0\u0012\n1\u0000j\njc\u0013\n; (42)\nhereU0is the characteristic scale of the activation energy and jc=\njc(B) is the critical current density. In the \rux creep state the e\u000bective\n6activation energy Ugrows logarithmically [19] with decreasing current\ndensity as\nU(j) =U0ln\u0012jc\nj\u0013n\n; (43)\nwhere the exponent ndepends upon the \rux creep regime. The ex-\npression (43) gives a quite realistic description for activation barriers\nin a wide range of temperatures and magnetic \felds. The power law\ncharacteristic dependence for U(~j) has been observed in numerous ex-\nperiments, and it has been used extensively in recent theoretical stud-\nies for the \feld and current distributions in superconductors [19, 20].\nEquation (43) is equivalent to a power law for the \rux-\row resistivity\nv=v0\f\f\f\fj\njc\f\f\f\fn\n: (44)\nIn this case the phenomenological relation ~E(~j) may be chosen in the\npower-law form\n~E=v0jBj\f\f\f\f\f~j\njc\f\f\f\f\fn\n\u0001~j: (45)\nIfn= 1 the last equation reduces to Ohm's law, describing the nor-\nmal or \rux-\row state. For in\fnitely large n, the equation describes\nthe Bean critical state model j=jc[15]. When 1 <n<1, the last\nequation describes nonlinear \rux creep.\nIn such case the analytic solution of the nonlinear creep equation\ncan be constructed by choosing the critical current dependence on\nthe magnetic \feld. Many models have been for the functional form\nofjc(B). For the critical current we adopt the power-law model [21],\nwhich can be applied over a relatively wide magnetic \feld range except\nin the high \feld region near the upper critical \feld\njc(B) =j0\u0012B0\nB\u0013\r\n: (46)\nwherej0andB0are the characteristic values of the current density\nand magnetic \feld induction; \ris the dimensionless pinning parameter,\nusually 0<\r < 1. If we assume \r=0, the above model reduces to the\nBean-London model [15]. This model is applicable to the case where\njccan be regarded approximately \feld independent. This power-law\nmodel for the critical current has also been used by other groups [14].\nAnother possible decay law would be exponential, which has often been\nused to take into account its decrease with the magnetic \feld [22]\nj=j0exp\u0012\n\u0000B\nB0\u0013\n;\nwhereB0is a phenomenological parameter related to the pinning abil-\nity: the smaller it is, the more drastic is the decrease of the critical\ncurrent with \feld. The numerical methods have been applied to re-\nsolve the \rux di\u000busion equation, employing the exponential critical\nstate model [14].\nNext, based on the power law and exponential models, we shall\nstudy the distribution of the magnetic induction, current density and\nmagnetization of superconductors.\nx7.1. Power law model\nFor the one-dimensional geometry the spatial and temporal evolution\nof magnetic \feld induction ~B(r;t) is described by the following non-\nlinear di\u000busion equation [10] in the dimensionless form\ndb\ndt=d\ndx\"\nb\rn+1\f\f\f\fdb\ndx\f\f\f\fn\u00001db\ndx#\n; (47)\nwhere we have introduced the dimensionless variables and parameters\nb=B\nB0; xp=\u00160j0\nB0x; t =t\n\u001c; j =j\nj0;\nb=\u000f=E\nv0B0; B 0=\u00160j0v0\u001c:The solution of above parabolic type di\u000busion equation allows to ob-\ntain the time and space distribution of the magnetic induction pro\fle\nin the considered sample. Inserting the scaling form (10) into di\u000beren-\ntial equation (47) and comparing powers of t in all terms, we get the\nfollowing relationship for the exponents \u000band\f\n\u000b+ 1 =\f+\u000b(\rn+ 1) +\u000bn+\fn: (48)\nUsing the condition of the \rux conservation we obtain\n\u000b=\f=1\n(2n+n\r+ 1)\nwhich suggests the existence of self-similar solutions in the form\nb(x;t) =t\u00001=(2n+n\r+1)f(z); z =x=t1=(2n+n\r+1): (49)\nSubstituting this scaling solution (49) into the governing equation (47)\nyields an ordinary di\u000berential equation for the scaling function f(z) in\nthe form\nd\ndz\u0014\nf\rn+1\f\f\f\fdf\ndz\f\f\f\fn\u0015\n+1\n(2n+n\r+ 1)d\ndz\u0012\nzdf\ndz\u0013\n= 0: (50)\nThe solution of the equation (50) must satisfy the following boundary\nconditions for the scaling function f(z)\nf(0) = 1; f(z0) = 0: (51)\nAfter further integrating of equation (50) and applying the boundary\nconditions we get the following solution of the problem\nf(z) =f(z0)\"\n1\u0000\u0012z\nz0\u0013(n+1)=n#1=(\r+1)\n(52)\nf(z0) =2\n4n(\r+ 1)\n(n+ 1) \nz(n+1)\n0\n2n+n\r+ 1!1=n3\n51=(\r+1)\n:\nThe position of the front z0can now be found by substituting the\nsolution (52) into the integral condition (9) and it is given by\nz(2n+n\r+1)=(\r+1)\n0=n\nn+ 1F\u0012\r+ 2\n\r+ 1+1\n2\u0013\n\u0000\u0012\r+ 2\n\r+ 1\u0013\n\u0000\u0012n\nn+ 1\u0013\n\"\nn(\r+ 1)\n(n+ 1)\u00121\n2n+n\r+ 1\u00131=n#1=(\r+1):\nThe last solution can be presented as\nb(x;t) =b0\"\n1\u0000\u0012x\nxp\u0013(n+1)=n#1=(\r+1)\n(53)\nwhere\nb0(0;t) =t\u00001=(2n+n\r+1)2\n4n(\r+ 1)\n(n+ 1) \nz(n+1)\n0\n2n+n\r+ 1!1=n3\n51=(\r+1)\n:\nThis solution describes the propagation of the magnetic \feld into the\nsample, the magnetic induction being localized in the domain between\nthe surface x=0 and the \rux front xp. The \rux front can be approxi-\nmately given as xp=s0t1=(2n+n\r+1). The velocity of penetration of\nthe magnetic \rux induction front into the superconductor sample is\ndetermined from the relation\nv\u0018t\u0000n(2+\r)=(2n+n\r+1): (54)\nThe velocity of the magnetic \rux front decreases rapidly as the mag-\nnetic \rux propagates.\nLet us consider the most interesting case n=1. In this particular\ncase the spatial and temporal evolution of the magnetic \rux induction\nis totally determined by the parameters \r,\u000band\f. In the following\nanalysis we derive an evolution equation for the magnetic induction\npro\fle for the case n=1 and apply the scalings of the previous section\nto formulate a similarity solution for the b(x, t). Based on the scalings\n7described in the previous section, we get the following relation for the\nexponents\n\u000b=\f=1\n\r+ 3: (55)\nThe last relation suggests the existence of solutions of the form\nb(x;t) =t\u00001=(\r+3)f(z); z =x=t1=(\r+3): (56)\nSubstituting the similarity function (56) into the governing equation\n(47) yields the following solution of the problem\nf(z) =\u0014z2\n0\n2(\r+ 1)\n(\r+ 3)\u00151=(\r+1)\"\n1\u0000\u0012z\nz0\u00132#1=(\r+1)\n; (57)\nwhich is the explicit form of the similarity solution we have been seek-\ning. The position of the front has the form\nz(\r+3)=(\r+1)\n0=\u00142\n1(\r+ 3)\n(\r+ 1)\u00151=(\r+1)2p\u0019G\u00121\n1 +\r+3\n2\u0013\nG\u00121\n1 +\r+ 1\u0013: (58)\nor\nb(x;t) =b0\"\n1\u0000x2\nx2p#1=(\r+1)\n; (59)\nwhere\nb0=\u0014z2\n0\n2(\r+ 1)\n(\r+ 3)\u00151=(\r+1)\nt\u00001=(\r+3):\nEquation (59) constitutes an exact solution of the nonlinear \rux-\ndi\u000busion equation for the situation, when n=1. The evolution of the\nself-simulating process of magnetic \feld penetration into a supercon-\nductor is shown schematically in \fgure 6a and 6b.\nFig.6a and 6b. The distributions of the normalized \rux density\nb(x;t) at time t=1, for \r=1, 2.\nx7.2. Exponential model n=1\nLet us consider a solution to the nonlinear di\u000busion equation for\nthe exponential model, assuming that n=1. The nonlinear di\u000berential\nequation (47), describing evolution of the magnetic \rux penetrated\ninto the superconductor sample for the exponential model can be trans-\nformed to the following form\ndb\ndt=d\ndx\u0014\nebdb\ndx\u0015\n: (60)\nIt is remarkable that one of explicit solution of the nonlinear di\u000busion\nequation (60) can be obtained by using the method of separation of\nvariables. According to the method of separation of variables, we look\nfor the solution of (60) in the form\nb(x;t) =a(x)g(t): (61)\nSubstituting this expression for b(x, t) into the partial di\u000berential\nequation (60) and separating variables we obtain the following equa-\ntions for the distributions of the functions g(t) and a(x)\ne\u0000\rgdg\ndt=\u0015; (62)\nd\ndx\u0012\ne\rada\ndx\u0013\n=\u0015: (63)\nIntegrating the di\u000berential equations (62) and (63) with respect to t\nand x, respectively and using the boundary conditions we obtain\ne\u0000\rb=\u0015(t\u0000tp); (64)\ne\ra=\u0015\n2[(x2\u0000x2\np)]: (65)\nwheretpis a constant, the parameter xpis de\fned by using the inte-\ngral relation (9). Combining the equations (64) and (65) according to\nrelation (61) we get\n\u001e=1\n\rln\"\n(x2\u0000x2\np)\n2(tp\u0000t)#\n: (66)\nx8. Magnetization\nLet us consider the magnetization pro\fle of the sample which is\ngiven by the following relation\n\u0000\u00160M(t) =b0\u00001\ndZd\n0b(x)dx:\n8Substituting here the scaling law, Eq. (15) one obtains\n\u0000\u00160M(t) =b0\u0014\n1\u0000xp(t)\nd n\u0015\n;\nwhere d is the distance of the \rux front penetrating into the sample;\n nis a numerical factor which is determined by the average value of\nthe scaling function f(z) in the region 0 <xp<d, so\n n=1\nz0Zz0\n0f(z)dz:\nExpressing the time dependence explicitly, the magnetization can be\nwritten as\n\u0000\u00160M(t) =b0\"\n1\u0000 n\u0012t\nt\u0003\u0013\f#\n;\nfor 0< t < t\u0003, wheret\u0003= (d=z0)\f. Di\u000berentiating the last equation\nand using an explicit solution for xp(t) for the case \r=\u00001=none\nobtains the expression to the magnetic relaxation rate\ndM\ndlnt\u00191\nn+ 1\u0012t\nt\u0003\u00131=(n+1)\n:\nAs the parameter n increases the pro\fle of the relaxation rate is seen\nto become more linear. At low temperature limit when n\u001d1 the\nrelaxation rate varies linearly with time. The time dependence of the\nmagnetic relaxation rate is shown in \fgure 7.\nFig.7. The time dependence of the magnetic relaxation rate for n=3,\n5, 11.\nIt has been shown by Vinokur [11] that at low temperatures the\nrelationship between the time and relaxation rate is a linear over a\nvery long time period in the early stage of the \rux penetration. A\nsuch perfect linear relationship characterizes that the system behaves\na self-organized criticality.\nConclusion\nThus, the problem of magnetic \rux penetration into the half-space\nsuperconductor sample is studied in the \rux \row regime in parallel ge-\nometry assuming that an external magnetic \feld increasing with time\nin accordance with the power law (7). Assuming that the \rux \row\nresistivity as a power-law function of the magnetic \feld induction we\nfound an exact analytical solution for the nonlinear local magnetic \rux\ndi\u000busion equation. The obtained solution describes space-time distri-\nbution of the magnetic induction in the sample. It was shown thatthe magnetic \rux density pro\fles at di\u000berent times follow the di\u000berent\nscaling law. Similar scaling pro\fles can be found for the electric \feld\nand current density pro\fles, too. It was also shown that the spatial and\ntemporal pro\fles of magnetic \rux penetration in the sample depends\non the set of three independent parameters, n, q and \u000b. For a given\nparameter set n, q and \u000bthe form of the scaling function b(x, t) was\nobtained by solving the nonlinear di\u000busion equation for the magnetic\n\feld induction analytically by a self-similar technique. We analyzed\nthe e\u000bect of di\u000berent values of exponents on the propagation of the\n\rux front in the sample. Varying the parameters of the equation, we\nhave observed a various shapes of the magnetic \rux front. Finally, we\nhave discussed the \rux creep problem, brie\ry.\nAcknowledgements\nThis study was supported by the NATO Reintegration Fellowship\nGrant and Volkswagen Foundation Grant. We are thankful for helpful\ndiscussions with Professor A. S. Rakhmatov. Part of the simulation\nwork herein was carried on in the Condensed Matter Physics at the\nAbdus Salam International Centre for Theoretical Physics.\nReferences\n1. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon,\nOxford, 1987.\n2. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdjumov, and A. S.\nStepanenko, Peaking Regimes for Quasilinear Parabolic Equa-\ntions, Nauka, Moskow, 1987.\n3. D. G. Aranson, J.L. Vazquez, Phys. Rev. Lett., 72, 1994.\n4. F. Bass, B. Ya. Shapiro, I. Shapiro, M. Shvartser, Physica C\n297, 269, 1998.\n5. I. B. Krasnyuk, Technical Physics, 52, 2007.\n6. V. Meerovich, M. Sinder and V. Sokolovsky, Supercond. Sci.\nTechnol., 9, 1996.\n7. V. V. Bryksin, S. N. Dorogovstev, Physica C 215, 1993.\n8. J. Gilchrist, C. J. van der Beek, Physica C 231, 1994.\n9. J. Gilchrist, Physica C 291, 1997.\n10. D. V. Shantsev, Y. M. Galperin, and T. H. Johansen\narXiv:cond-mat/0108049 v1, 2001.\n11. V. V. Vinokur, M. V. Feigelman, and V. B. Geshkenbein, Phys.\nRev. Lett., 67, 915, 1991.\n12. Z. Koziol and E. P. Chatel, IEEE Trans. Magn., 30, 1169, 1994.\n13. W. Wang and J. Dong, Phys. Rev. B 49, 698, 1994.\n14. M. Holiastou, M. Pissas, D. Niarchos, P. Haibach, U. Frey and\nH. Adrian, Supercond. Sci. Technol., 11, 1998.\n15. C. P. Bean, Phys. Rev. Lett. 8, 250, 1962; Rev. Mod. Phys.,\n36, 31, 1964.\n16. A. M. Campbell and J. E. Evetts, Critical Currents in Super-\nconductors (Taylor and Francis, London, 1972), Moscow, 1975.\n17. P. W. Anderson , Y.B. Kim Rev. Mod. Phys., 36. 1964.\n18. P. W. Anderson, Phys. Rev. Lett., 309, 317, 1962.\n19. E. Zeldov, N. M. Amer, G. Koren, A. Gupta, R. J. Gambino,\nand M. W. McElfresh, Phys. Rev. Lett., 62, 3093, 1989.\n20. P. H. Kes, J. Aarts, J. van der Berg, C.J. van der Beek, and\nJ.A. Mydosh, Supercond. Sci. Technol., 1, 242, 1989.\n21. F. Irie and K. Yamafuji, J. Phys. Soc. Jpn., 23, 255, 1967.\n22. W. A. Fietz, M. R. Beasley, J. Silcox and W. W. Webb, Phys.\nRev., 136 A335, 1964.\n9"    },    {        "title": "1507.06729v1.Coherent_zero_field_magnetization_resonance_in_a_dipolar_spin_1_Bose_Einstein_condensate.pdf",        "content": "arXiv:1507.06729v1  [cond-mat.quant-gas]  24 Jul 2015Coherent zero-field magnetization resonance in a dipolar sp in-1 Bose-Einstein\ncondensate\nWenxian Zhang,1,2S. Yi,3M. S. Chapman,4and J. Q. You2\n1School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China\n2Beijing Computational Science Research Center, Beijing 10 0084, China\n3Institute of Theoretical Physics, Chinese Academy of Scien ces, Beijing 100190, China\n4School of Physics, Georgia Institute of Technology, Atlant a, Georgia 30332-0430, USA\n(Dated: September 28, 2018)\nWith current magnetic field shielding and high precision det ection in dipolar spinor Bose-Einstein\ncondensates, it is possible to experimentally detect the lo w or zero field nonsecular dipolar dynam-\nics. Here we analytically investigate the zero-field nonsec ular magnetic dipolar interaction effect,\nwith an emphasis on magnetization dynamics in a spin-1 Bose- Einstein condensate under the sin-\ngle spatial mode approximation within the mean field theory. Due to the biaxial nature of the\ndipolar interaction, a novel resonance occurs in the conden sate magnetization oscillation, contrast\nto the previous assumption of a conserved magnetization in s trong magnetic fields. Furthermore,\nwe propose a dynamical-decoupling detection method for suc h a resonance, which cancels the stray\nmagnetic fields in experiments but restores the magnetizati on dynamics. Our results shed new lights\non the dipolar systems and may find potential applications be yond cold atoms.\nPACS numbers: 03.75.Kk, 03.75.Mn, 67.85.Fg\nI. INTRODUCTION\nMagnetic dipolar interactions exist in a wide variety of\nphysical systems, including nuclear spins [1], atoms and\nmolecules [2, 3], condensed matter [4], chemical [5] and\nbiological systems [6]. Due to the relative weakness of\nthe magnetic dipolar interaction to the Zeeman effect of\nan external magnetic field, the nonsecular interaction in\na dipolar system, which varies temporally with Larmor\nfrequency and breaks the rotational invariance, was usu-\nally neglected in most experiments over decades. Such\nan approximation results in the famous magnetization\nconservation during the time evolution [1].\nRecently, with the precision measurement of an ultra-\nweak or zero magnetic field, the investigation of the non-\nsecular dipolar interaction effects becomes revived in the\nareas of zero-field nuclear magnetic resonance and dipo-\nlarspinorBose-Einsteincondensates[7–11]. Particularly,\nthe achievements in spinor Bose-Einstein condensates\n(BECs) provide a highly tunable and controllable sys-\ntem where the spin interactions, including the magnetic\ndipolar interaction, can be accurately engineered [3, 12–\n19]. Such dipolar spinor condensates make them an ideal\ntestbed to extensively explore the full magnetic dipolar\ninteraction effect, including not only the secular part but\nalso the nonsecular one which breaks the rotational sym-\nmetry and definitely brings new physics. Recent experi-\nments by Pasquiou et al.investigated the incoherent de-\nmagnetization process in spin-352Cr condensates in an\nultralow magnetic field [10]. However, the coherent dy-\nnamics induced by the nonsecular dipolar terms in cold\natoms has yet been touched.\nIn this paper, we investigate the effect of the magnetic\ndipolar interaction in a spin-1 BEC, focusing on the co-\nherent magnetization dynamics induced by the nonsecu-\nlar dipolar interaction in low or zero field (See Fig. 2).By adopting mean field theory and single spatial mode\napproximation(SMA) [20–22], we analytically obtain the\nmagnetization dynamics of the spin-1 BEC. Contrast to\nthe usual assumption of magnetization conservation, we\nfind a novel resonance in the magnetization oscillation\nin the dipolar condensate. With large-scale numerical\ncalculations, we also explore the spin dynamics beyond\nthe SMA, which agrees well with the analytical predic-\ntion. Furthermore, a practical control protocol utilizing\ndynamical decoupling techniques is proposed to liberate\nthe nonsecular part of the magnetic dipolar interaction\nby canceling the stray magnetic fields. Our results shed\nnew light on the dipolar systems, in particular on the\nresonant magnetization dynamics induced by the non-\nsecular part of the dipolar interaction, and open a door\nto understanding the complex spin texture observed in\ndipolar spinor BECs [23–27].\nThe paper is organized as follows. In Sec. II, we intro-\nduce the mean field Hamiltonian describing the dipolar\nspin-1 condensate at zero magnetic field. In Sec. III,\nwe present the equation of motion for the condensate\nmagnetization and the analytical solution. We then de-\nsigndynamicaldecouplingpulsesequencetosuppressthe\nZeeman effect of a nonzero magnetic field and perform\nnumerical simulation with parameters in practical exper-\niments beyond SMA in Secs. IV and V, respectively. We\nconclude in Sec. VI.\nII. SYSTEM HAMILTONIAN\nIn the mean field theory and under the SMA, a dipo-\nlar spin-1 Bose condensate is described by the Hamilto-\nnian [13, 28, 29],\nH=cf2+ds(3f2\nz−f2)+3dn(f2\ny−f2\nx),2\nwhere the spin exchange interaction strength is c=\n(c2/2)/integraltextdrρ2(r) withρ(r) =|φ(r)|2being the density\nof the mode function φ(r). The spin-exchange interac-\ntion coefficient c2= 4π/planckover2pi12(a2−a0)/(3M) withMbeing\nthe atom’s mass and a0,2thes-wave scattering length\nof two spin-1 atoms in the symmetric channel of the\ntotal spin 0 and 2, respectively. The condensate spin\nf2=f2\nx+f2\ny+f2\nz, wherefx,y,z=/an}b∇acketle{t/vectorξ|Fx,y,z|/vectorξ/an}b∇acket∇i}htwith|/vectorξ/an}b∇acket∇i}ht=\n(ξ+,ξ0,ξ−)Tbeing the spin wave function and Fx,y,zthe\nspin-1 matrices. Note that we have neglected the con-\nstant terms under the SMA, contributed by the kinetic\nenergy, the trapping potential, and the spin-independent\ninteraction proportional to c0= 4π/planckover2pi12(2a2+a0)/(3M).\nThe dipolar interaction includes two parts, the secu-\nlar part with ds= (cd/4)/integraltext\ndrdr′|r−r′|−3ρ(r)ρ(r′)(1−\n3cos2θe) and the nonsecular part with dn=\n(cd/4)/integraltext\ndrdr′|r−r′|−3ρ(r)ρ(r′)sin2θeei2ϕe, whereθe\nandϕeare respectively the polar and azimuthal an-\ngles of ( r−r′). The dipolar interaction coefficient is\ncd=µ0µ2\nBg2\nF/(4π) withµ0being the vacuum permeabil-\nity,µBthe Bohr magneton, and gFthe Land´ eg-factor\nfor an electron. For87Rb atoms, cd/|c2| ≈0.09, and\nfor23Na atoms, cd/|c2| ≈0.007. Note that dn= 0 if\nρ(r) is axially symmetric. In general, the value of ds,n\ncan be positive, 0, or negative, depending on the specific\nshape of the mode function [29, 30]. In a large or mod-\nerate magnetic field, the secular part survives due to the\nconservation of fz, but the nonsecular part, which flips\nthe spin, is strongly suppressed by the Zeeman effect and\nusually neglected. In Sec. IV, we will propose a dynam-\nical decoupling method to cancel the Zeeman effect and\nto “restore” the suppressed nonsecular part.\nThe original Hamiltonian can be rewritten as\nH=A0f2+Axf2\nx+Azf2\nz (1)\nwhereA0=c−ds+3dn,Ax=−6dn, andAz= 3(ds−dn).\nWehaveusedtherelationthat f2\ny=f2−f2\nx−f2\nz. Forsuch\naspinsystem,besidestheconservationofthetotalenergy\nand the spin fof the condensate (thus the conservation\nof the isotropic energy E0=A0f2), we notice that the\nsum of the rest two anisotropic terms, which we define as\nExz=Axf2\nx+Azf2\nz, is also conserved. The conservation\nofExzdefines the spin trajectories in the x-zplane into\nthe following two categories, depending on the values of\nAxandAz(besides the trivial one, AxAz= 0 where fz\norfxis constant): (I) AxAz>0, the trajectory in the\nx-zplane is an ellipse (or a circle for Ax=Az); (II)\nAxAz<0, the trajectory is a hyperbola (or a parabola\nforAx=−Az). Of course, fx,zmust be smaller than\n(fy/ne}ationslash= 0) or equal to f(fy= 0).\nThe iso-energy contour plots of Exzon the surface of\nthe spin sphere for categories (I) and (II) are presented\nin Fig. 1. For both categories the oscillations of fzeither\npasses zero or not, depending on the initial value of fz.\nAs shown in the figure, there is an alternative way to\nclassify the spin dynamics: the x-,y-, orz-region, where\nthe spins rotates around the corresponding x-,y-, orz-\naxis. Actually, the trajectory of the spin behaves like\nFIG. 1. (Color online) The iso-energy contour plot of the\nanisotropic energy Exzin unit of |c|on the surface of a spin\nsphere (a) for category (I) with Ax= 0.5 andAz= 1 and (b)\nfor category (II) with Ax=−1 andAz= 1. The black lines\nmark the boundary between the zandyregions in (a) and\nbetween the zandxregions in (b).\nnothing but a nonlinear rigid pendulum, which oscillates\naroundx-,y-, orz- axis correspondingly.\nIII. ANALYTICAL RESULTS OF\nMAGNETIZATION RESONANCE UNDER SMA\nBy treating the condensate spin as a classical spin,\nwhich rotates in an effective magnetic field ( bx,by,bz) =\n(2Axfx,0,2Azfz), we obtain the following equation of\nmotion for the magnetization, ˙fz≡dfz/dt= 2Axfxfy.\nByutilizingagaintherelations f2\ny=f2−f2\nx−f2\nzandf2\nx=\n(Exz−Azf2\nz)/Ax, we find a closed equation of motion for\nfz,\n˙f2\nz= 4(Azf2\nz−Exz)/bracketleftbig\n(Ax−Az)f2\nz+Exz−Axf2/bracketrightbig\n,(2)\nwherefandExzare determined by the initial state.\nThe analytical solution for fzis an inverse function of\nthe elliptic integral of the first kind F[·,·],\nt=t0+1\n2/radicalbig\nAz(Axf2−Exz)F/bracketleftbigg\nsin−1/parenleftbiggfz\nx/parenrightbigg\n,x2\ny2/bracketrightbigg\n,(3)\nwherex2=Exz/Azandy2= (Axf2−Exz)/(Ax−Az).\nThe value of x(y) is determined by fx(y)= 0 (i.e., ˙fz=\n0).\nThe period of the oscillation is\nT≡/contintegraldisplay\n˙fz(t)−1dfz\n=2/radicalbig\nAz(Axf2−Exz)/vextendsingle/vextendsingle/vextendsingle/vextendsingleF/bracketleftbigg\nsin−1/parenleftBigy\nx/parenrightBig\n,x2\ny2/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle(4)\nif the oscillation is in the x-region,\nT=2/radicalbig\nAz(Axf2−Exz)K/parenleftbiggx2\ny2/parenrightbigg\nif in they-region, or\nT=2/radicalbig\nAz(Axf2−Exz)/vextendsingle/vextendsingle/vextendsingle/vextendsingleF/bracketleftbigg\nsin−1/parenleftBigy\nx/parenrightBig\n,x2\ny2/bracketrightbigg\n−K/parenleftbiggx2\ny2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle3\nFIG. 2. (Color online) Periodic spin dynamics induced by the\nmagnetic dipolarinteraction. Oscillation ofthemagnetiz ation\nfzwith an initial rotation angle (a) θ= 15o(below the critical\nvalueθc) and (d) θ= 75o(aboveθc), where the spin oscillates\naroundz- andx-axis, respectively. (b) Dependence of the\noscillation period Ton the initial angle θ. Solid lines are\nthe SMA prediction and circles are the numerical results wit h\ncoupled Gross-Pitaevskii equations. A resonance occurs at θc,\nmarkedbytheblackarrow in(b), wherethespinevolvesalong\nneitherx- norz- axis. (c) Typical iso-energy spin trajectories\nof the condensate on the surface of the spin sphere. The red\nand blue arrows show the spin evolution direction.\nif in thez-region, where K(·) is the complete elliptic in-\ntegral of the first kind. The function K(k) is nearly a\nconstantπ/2 aroundk= 0 and diverges rapidly if k→1.\nIn addition, K(k) =F(π/2,k), andK(1/k)/k1/2also di-\nverges ifk→0.\nThe analytical solution for fzshows oscillatory and\nperiodic motion. This is in sharp contrastto the previous\nunderstanding of the conservation of the magnetization\n(in large magnetic fields), because the nonsecular terms\nbreak the rotational symmetry around z-axis. As shown\nin Fig. 1, the amplitude ofthe oscillation of fzis|y|if the\ncondensate spin is in the x-region,|x|if in they-region,\nor|x−y|if in thez-region. Remarkably, the period T\ndiverges if x= 0 (y= 0), i.e., the initial spin state is\nset on the boundary between the x-region (y-region) and\nthez-region(see Fig. 1). This divergenceindicates that a\nresonance occurs by changing the initial spin state across\ntheboundary. Suchaninterestingresonancehasnotbeen\nrevealed before, because of the neglect of the nonsecular\nterms, and is obviously due to the competition between\nthe secular and nonsecular dipolar terms.\nFor a clear view, we present in Fig. 2 the analytical\nresults for the category (II) with ds/|c|= 0.225 and\ndn/|c|= 0.0644 (|c|= 1.3 Hz is calculated for a numeri-\ncally simulated spin-1 BEC). Two typical oscillations of\nfzin thez-region and x-region are shown in Fig. 2(a)\nand (d). Other trajectories are shown in Fig. 2(c) and\nthe periods ofthe oscillationswith respect to initial polarangleθ(f= 1 and zero azimuthal angle) of the conden-\nsate spin are shown in Fig. 2(b). We observe a clear res-\nonance signature at θc, which is determined by Exz= 0,\ni.e., cos2θc=Ax/(Ax−Az). Similarly, for the category\n(I)wealsoobservearesonanceinthe fzoscillationperiod\nand the critical angle cos2θc=Ax/Az, which is obtained\nfromAxf2−Exz= 0 [31].\nThe resonance in the oscillation period can be under-\nstood physically by treating the dipolar spin-1 BEC as\na two-axis rigid nonlinear pendulum (see Fig. 2). For\na single-axis pendulum, i.e., either Ax= 0 orAz= 0,\nthe condensate spin oscillates around an axis, zorx,\nwith a constant spin component along this axis. But\nin categories (I) and (II), neither of Ax,zis zero, so the\nevolution of the condensate spin is not purely around a\nsingle axis but a more complicated oscillation between\nthe two axes, due to the competition of the two nonlin-\near terms of Axf2\nxandAzf2\nz. In general, by setting the\ncondensatespin initially closerto anaxis, e.g., z-axis, the\nevolution of the condensate spin is around this axis since\nthe corresponding term ( Azf2\nz) is dominant. However,\nthere exists a clear boundary where the two-axis terms\nare balanced, e.g, Axf2\nx=−Azf2\nzin Fig. 1(b), and the\ncondensate spin evolves in a third direction and reaches\nto a dead end ( fx,z= 0). The oscillation never completes\nand the period becomes infinite. Thus a resonance peak\nappears in the spin oscillation period.\nAn alternative way of understanding the resonance\nbehavior is to treat the condensate in a semiclassical\nmanner. In a single particle picture, the energy level\nof an atom in the |mF=±1/an}b∇acket∇i}htstate is shifted by an\namount of Azdue to the term Azf2\nz. Similarly, the\ntermAxf2\nxnot only shifts oppositely the energy level\nof|mF=±1/an}b∇acket∇i}ht(ifAxAz<0), but also couples them. As\nshown in Fig. 2 and 1(b), when the resonant condition\nAxf2\nx+Azf2\nz= 0 is satisfied and the condensate reaches\nits steady state ( fx,z= 0), the total shift of the energy\nlevels of |mF=±1/an}b∇acket∇i}htbecome zero, i.e., the energy levels\nof|mF=±1/an}b∇acket∇i}htare resonant to that of |mF= 0/an}b∇acket∇i}ht.\nIV. SUPPRESSING THE ZEEMAN EFFECT OF\nAN EXTERNAL STRONG MAGNETIC FIELD\nInside a magnetic shield room, the stray magnetic field\nhave been reduced to as low as 0.1 mG [10, 11]. However,\neven in such a low magnetic field, the Zeeman effect still\noverwhelms the dipolar-interaction-induced spin dynam-\nics along the z-axis, since the Zeeman energy µBB∼140\nHz forB= 0.1 mG is much larger than the dipolar inter-\naction energy, typically in the order of 0 .1 Hz at a con-\ndensate density of 1014cm−3. As shown in Figs. 3(a),\nthe oscillation amplitude of fzforB= 0.1 mG (dash-\ndotted line and the inset) is much smaller than that for\nB= 0 (dotted line). The consequence is that the magne-\ntization dynamics due to the dipolar interaction is rather\nchallenging to observe in a practical experiment because\nthe stray magnetic field effects dominate the magnetiza-4\n0 1 20.750.80.850.90.95fz(a)\nTime (sec)2.4 2.45 2.50.8640.8650.866\n10−310−210−100.51\nτ (sec)Normalized amplitude(b)\nFIG. 3. (a) Coherent magnetization dynamics of a dipolar\nspin-1 BEC in a magnetic field. B= 0 (dotted line), B= 0.1\nmG (dash-dotted line), B= 0.1 mG with control pulse delay\nτ= 0.004 s (solid line), and B= 0.1 mG with τ= 0.04 s\n(dashed line). The initial condition is the same as in Fig. 2\nwithθ= 30o. Other parameters are the same as in Fig. 2.\nThe Zeeman effect of the nonzero magnetic field suppresses\nthefzdynamics (dash-dotted line and the inset for a zoom-\nin view), but the fast and periodic application of πxpulses\nrestores the dipolar fzdynamics. (b) Dependence of the nor-\nmalized amplitude of the dipolar magnetization oscillatio ns\non the control pulse delay τatB= 0.1 mG (solid line) and\nB= 1 mG (dashed line).\ntion dynamics.\nHowever, the resonant dipolar magnetization dynam-\nicscanberevealedundercurrentexperimentalconditions\nif we employ dynamical decoupling techniques to cancel\nthe Zeeman effect [1, 28, 32]. In particular, by applying\nfrequentπxpulse, whichrotatesthe condensatespin 180o\nalongx-axis, the Zeeman effect is eliminated while leav-\ning the dipolar interaction intact to the leading order.\nWe assume that the control pulse πxis an instantaneous\n(hard) pulse and the delay between two adjacent pulses\nisτ(Appendix A). The numerical simulation results are\npresented in Fig. 3. As shown in Fig. 3(a), in the limit\nof smallτ(solid line), the dipolar magnetization dynam-\nics is restored; in the limit of large τ(dashed line), the\nZeemaneffect isnotwellsuppressedandthedipolarmag-\nnetization oscillation amplitude is small. Under control\npulses, the dependence of the normalized magnetization\noscillation (which is the ratio of the magnetization oscil-\nlation amplitude at B= 0.1 mG under control pulses to\nthe free magnetization oscillation at B= 0) on the pulsedelayτis shown in Fig. 3(b). Obviously, the Zeeman ef-\nfect is canceled and the magnetization oscillation is well\nrestored at B= 0.1 mG ifτis smaller than 0 .02 seconds,\nwhich is easily realizable in experiments (Appendix A).\nV. NUMERICAL RESULTS WITH A LARGE\nNUMBER OF ATOMS\nOur previous analysis are based on the SMA, which\nmay not be valid in some experiments for spin-1 BECs,\nparticularly for a condensate with a large number of\natoms [17, 33, 34]. It is an open question whether\nthe SMA results remain valid under practical experi-\nmental conditions, where the number of87Rb atoms\nis 104and the trap frequencies are {ωx,ωy,ωz}=\n(2π×){90,140,200}Hz. Forthistrapgeometry,theSMA\nwith a three-dimension Gaussian wave function predicts\nthatds= 0.215 anddn= 0.066, while the numerical re-\nsults for a fully polarized ground state (all atoms are in\n|+ 1/an}b∇acket∇i}htstate) areds= 0.225 anddn= 0.064. It shows\nobviously that the SMA is slightly violated.\nWe start the numerical simulation by tilting the fully\npolarized ground state away from the z-axis by a polar\nangleθin thex-zplane. In this way, the subsequent\nmagnetization evolution is solely due to the dipolar in-\nteraction since the condensate magnetization would re-\nmain constant in the absence of the anisotropic dipolar\ninteraction. The magnetization dynamics of the conden-\nsate is obtained by numerically solving the three coupled\nGross-Pitaevskii equations in the given trap[28, 34, 35]\ni/planckover2pi1∂ψα\n∂t= [T+Vext+c0n]ψα+Beff·Fαβψβ,(5)\nwhere the kinetic energy is T=−/planckover2pi12∇2/(2M) with\nMbeing the atom mass of87Rb, the trapping poten-\ntial isVext(x,y,z) =M(ω2\nxx2+ω2\nyy2+ω2\nzz2)/2, and\nthe total number density is n=/summationtext\nαψ∗\nαψαwithψα\n(α=±1,0) being the three components of the con-\ndensate wave function. The effective field originat-\ning from the spin-exchange and dipolar interactions is\nBeff=c2S+cd/integraltext\ndr′{S(r′)−3[S(r′)·e]e}/|r−r′|3,where\nS=ψ∗\nαFαβψβis the spin density with Fthe atom spin-1\nmatrix, and eis the unit vector along r−r′. The spin-\nexchange interaction coefficient c2and the dipolar inter-\naction coefficient cdare given previously. We solve these\ncoupled equations using the operator splitting method,\nwherethe term involvingthe integraloperator Beffis cal-\nculated with convolution theorem and fast Fourier trans-\nform.\nWe present in Figs. 2(a), 2(b), and 2(d) the numeri-\ncal results, as well as the SMA predictions with ds,nob-\ntained numerically from the fully polarized ground state.\nWe observe a pretty good agreement of the spin oscilla-\ntions between the numerical results and the SMA predic-\ntion, due to the fact that the three components of the\nground state of the ferromagnetically interacting spin-1\ncondensate share the same spatial wave function [21, 22].5\nHowever, we find slight mismatches of the oscillation am-\nplitude in Fig. 2(a) and the resonance peak position in\nFig. 2(b). These discrepancies might be due to the trap\nanisotropy: the atomsspins aremorelikely toalign along\nthe loosely trapped x- ory-axis so that the total dipolar\nenergy is lower, which means that the z-region is smaller\nthan the SMA results. Consequently, the boundary be-\ntweenthex-regionand the z-regionis shifted to asmaller\nθc, which is what we observe in Fig. 2(b), and the lower\nvalue offzin Fig. 2(a) becomes smaller.\nVI. CONCLUSION\nThe weak dipolar interaction effect, particularly the\neffect of the nonsecular part, in spin-1 Bose conden-\nsates such as87Rb is experimentally challenging to ob-\nserve [34, 36]. Our proposal provides a practical way to\ndetect the condensate magnetization oscillation induced\nby the nonsecular dipolar interaction, where a resonance\nemerges in the oscillation period. Numerical results with\nexperimental parameters, such as the large number of\natoms and the suppression of Zeeman effect of the stray\nmagnetic field with dynamical decoupling method, con-\nfirm that the resonant behavior of the coherent magne-\ntization dynamics is detectable experimentally. Our re-\nsults point to a new direction for future investigations\non many dipolar effects, including the competition with\nthe short-range contact interaction, the quantum dipolar\neffects which are of great interests for quantum metrol-\nogy and next-generation-precision magnetometers based\non spinor Bosecondensates[11, 37–41], and the structure\ndeterminationin chemicalandbiologicaldipolarsystems.\nACKNOWLEDGMENTS\nW.Z. thanks V. V. Dobrovitski for inspiring discus-\nsions on the suppression of the magnetic field Zeeman\neffect. This work is supported by National Basic Re-\nsearch Program of China Grant Nos. 2013CB922003and2014CB921401, National Natural Science Foundation of\nChina under Grant Nos. 11275139, 11434011, 11121403,\nand 91421102, National Science Foundation (US) (Grant\nNo. 1208828), NSAF Grant No. U1330201, and the Fun-\ndamental Research Funds for the Central Universities.\nAppendix A: Dynamical decoupling\nFor a spin-1 condensate in a magnetic field, H=\nωzSz+V, whereωzis the Zeeman splitting and Vis the\nweak two-boday interaction (including the spin exchange\nand dipolar interaction). Suppose the control pulse πx\nis an instantaneous (hard) pulse and the delay between\ntwo adjacent pulses is τ. Note that the Vis unaffected\nbutSzbecomes −Szby theπxpulses. Let us consider\nthe evolution after 2 pulses [1],\nU(t= 2τ) =U(τ)UπxU(τ)Uπx.\nSinceUπx= exp( −iπSx) =U†\nπxandU(τ) =\nexp[−iτ(ωzSz+V)]≈exp(−iτωzSz) exp(−iτV) for\nsmallτ(i.e.,||H||τ≪1), we find that\nU(t= 2τ) =U(τ)UπxU(τ)U†\nπx\n≈U(τ)exp(−iτV)Uπxexp(−iτωzSz)U†\nπx\n≈exp(−i2τV). (A1)\nAfter periodically applying 2 Kpulses, the evolution\noperator is\nU(t= 2Kτ) = [U(τ)Uπx]2K\n≈exp(−itV) (A2)\nto the leading order of τ. Clearly, the effect of the mag-\nnetic field is removed if τ→0.\nApossiblewayto realizeexperimentallythe hardpulse\nis by applying a square pulse of constant B1, e.g. 10 mG,\nalongxdirection within a pulse width w∼10−4s. The\npulse width is much smaller than the typical pulse delay\nτ∼10−2s and the dipolar dynamics time scale ∼10\ns [31].\n[1] C. P. Slichter, Principles of Magnetic Resonance\n(Springer-Verlag, New York, 1992).\n[2] M. A. Baranov, Phys. Rep., 464, 71 (2008).\n[3] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and\nT. Pfau, Rep. Prog. Phys., 72(2009).\n[4] P. Teixeira, J. Tavares, and M. da Gama, J. Phys. Cond.\nMat.,12, R411 (2000).\n[5] J. Prestegard, C. Bougault, andA. Kishore, Chem. Rev.,\n104, 3519 (2004).\n[6] M. Blackledge, Prog. Nucl. Magn. Reson. Spectrosc., 46,\n23 (2005).\n[7] M. P. Ledbetter, T. Theis, J. W. Blanchard, H. Ring,\nP.Ganssle, S.Appelt, B.Bl¨ umich, A.Pines, andD.Bud-\nker, Phys. Rev. Lett., 107, 107601 (2011).[8] T. Theis, P.Ganssle, G. Kervern, S.Knappe, J. Kitching,\nM. P. Ledbetter, D. Budker, and A. Pines, Nat. Phys,\n7, 571 (2011).\n[9] D. Sheng, S. Li, N. Dural, and M. V. Romalis, Phys.\nRev. Lett., 110, 160802 (2013).\n[10] B. Pasquiou, E. Mar´ echal, G. Bismut, P. Pedri,\nL. Vernac, O. Gorceix, and B. Laburthe-Tolra, Phys.\nRev. Lett., 106, 255303 (2011).\n[11] Y. Eto, H. Ikeda, H. Suzuki, S. Hasegawa, Y. Tomiyama,\nS. Sekine, M. Sadgrove, and T. Hirano, Phys. Rev. A,\n88, 031602 (2013).\n[12] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys., 85,\n1191 (2013).\n[13] S. Yi, L. You, and H. Pu, Phys. Rev. Lett., 93, 0404036\n(2004).\n[14] R. Cheng, J. Liang, and Y. Zhang, J. Phys. B-Atom\nMol. Opt. Phys., 38, 2569 (2005).\n[15] A. Widera, F. Gerbier, S. F¨ olling, T. Gericke, O. Mande l,\nand I. Bloch, Phys. Rev. Lett., 95, 190405 (2005).\n[16] B. Sun, W.X.Zhang, S. Yi, M. S.Chapman, andL. You,\nPhys. Rev. Lett., 97, 123201 (2006).\n[17] S. Yi and H. Pu, Phys. Rev. Lett., 97, 020401 (2006).\n[18] Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett.,\n96, 080405 (2006).\n[19] Y. Kawaguchi, H. Saito, K. Kudo, and M. Ueda, Phys.\nRev. A, 82, 043627 (2010).\n[20] H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P.\nBigelow, Phys. Rev. A, 60, 1463 (1999).\n[21] S. Yi, O. E. M¨ ustecaplıo˘ glu, C. P. Sun, and L. You,\nPhys. Rev. A, 66, 011601(R) (2002).\n[22] W. Zhang, S. Yi, and L. You, New J. Phys., 5, 77 (2003).\n[23] M. Chang, Q. Qin, W. Zhang, L. You, and M. Chapman,\nNature Phys., 1, 111 (2005).\n[24] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore ,\nand D. M. Stamper-Kurn, Nature (London), 443, 312\n(2006).\n[25] R. W. Cherng and E. Demler, Phys. Rev. Lett., 103,\n185301 (2009).\n[26] J. Kronj¨ ager, C. Becker, P. Soltan-Panahi, K. Bongs,\nand K. Sengstock, Phys. Rev. Lett., 105, 090402 (2010).\n[27] Y. Eto, H. Saito, and T. Hirano, Phys. Rev. Lett., 112,\n185301 (2014).\n[28] B.-Y. Ning, S. Yi, J. Zhuang, J. Q. You, and W. Zhang,\nPhys. Rev. A, 85, 053646 (2012).\n[29] Y. Huang, Y. Zhang, R. L¨ u, X. Wang, and S. Yi, Phys.\nRev. A, 86, 043625 (2012).[30] In general, ds,n= 0 (dn= 0) if the mode function has\na spherical (an axial) symmetry. For a Gaussian mode\nfunction with characteristic lengths qx,y,zinx,y,zdirec-\ntions,ds>0 (<0) ifqx,y< qz(qx,y> qz), anddn>0\n(<0) ifqx< qy(qx> qy).\n[31] The lifetime of a spin-1 BEC longer than 15 seconds has\nbeen observed in experimental groups of M. S. Chapman\n(Georgia Institute of Technology) and RuquanWang(In-\nstitute of Physics, Chinese Academy of Science) (private\ncommunications).\n[32] B.-Y. Ning, J. Zhuang, J. You, and W. Zhang, Phys.\nRev. A, 84, 013606 (2011).\n[33] M. Vengalattore, S. R. Leslie, J. Guzman, and D. M.\nStamper-Kurn, Phys. Rev. Lett., 100, 170403 (2008).\n[34] Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett.,\n98, 110406 (2007).\n[35] J.-N. Zhang, L. He, H. Pu, C.-P. Sun, and S. Yi, Phys.\nRev. A, 79, 033615 (2009).\n[36] K. Gawryluk, K. Bongs, and M. Brewczyk, Phys. Rev.\nLett.,106, 140403 (2011).\n[37] E. M. Bookjans, C. D. Hamley, and M. S. Chapman,\nPhys. Rev. Lett., 107, 210406 (2011).\n[38] C. D. Hamley, C. S. Gerving, T. M. Hoang, E. M. Book-\njans, and M. S. Chapman, Nat. Phys., 8, 305 (2012).\n[39] C. S. Gerving, T. M. Hoang, B. J. Land, M. Anquez,\nC. D. Hamley, and M. S. Chapman, Nat. Comm., 3\n(2012).\n[40] T. M. Hoang, C. S. Gerving, B. J. Land, M. Anquez,\nC. D. Hamley, and M. S. Chapman, Phys. Rev. Lett.,\n111, 090403 (2013).\n[41] M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman,\nL. E. Sadler, and D. M. Stamper-Kurn,Phys. Rev. Lett.,\n98, 200801 (2007)."    },    {        "title": "1001.2682v1.Control_of_the_chirality_and_polarity_of_magnetic_vortices_in_triangular_nanodots.pdf",        "content": " 1Control of the chirality and polarity of magnetic \nvortices in triangular nanodots \nM. Jaafar1, , R. Yanes1, D. Perez de Lara2, O. Chubykalo-Fesenko1, A. Asenjo1*, E.M. Gonzalez2, \nJ.V. Anguita3, M. Vazquez1 and J.L. Vicent2  \n1Instituto de Ciencia de Material es de Madrid, CSIC, Sor Juana Inés de la Cruz, 3, Cantoblanco, \n28049, Madrid, Spain \n2 Departamento Física de Materiales, Faculta d CC. Físicas, Universidad Complutense, 28040 \nMadrid, Spain \n3Instituto de Microelectrónica de Madrid, CN M-CSIC, Isaac Newton 8, Tres Cantos, 28760 Madrid, \nSpain \n \nAUTHOR EMAIL ADDRESS (aasenjo@icmm.csic.es) \n \nABSTRACT \nMagnetic vortex dynamics in lithographically prepar ed nanodots is currently a subject of intensive \nresearch, particularly after recent demonstration that the vortex polarity can be controlled by in-plane \nmagnetic field. This has stimulated the proposal s of non-volatile vortex magnetic random access \nmemories. In this work, we demons trate that triangular na nodots offer a real alte rnative where vortex \nchirality, in addition to polarity, ca n be controlled. In the static regime, we show that vortex chirality \ncan be tailored by applyi ng in-plane magnetic field, which is experimentally imaged by means of \nVariable-Field Magnetic Force Microscopy. In a ddition, the polarity can be also controlled by \napplying a suitable out-of-plane magnetic field co mponent. The experiment and simulations show \nthat to control the vortex polarity, the out-of-plane field component, in this particular case, should be \nhigher than the in-plane nucleati on field. Micromagnetic simulations  in the dynamical regime show \nthat the magnetic vortex polarity can be changed with short-duration magne tic field pulses, while \nlonger pulses change the vortex chirality.   \n  2KEYWORDS (Word Style “BG_Keywords”). Magnetic Vortex, Micromagnetic simulations, Magnetic Force \nMicroscopy, Magnetic nanostructures, Reversal Magnetization.   3I. INTRODUCTION \nLithographed magnetic nanostructures offer an abunda nt possibility for tail oring their properties \nthrough the suitable choice of their geometry that dete rmines the strength of shape anisotropies with \nmagnetostatic origin1. The nanostructuring techniques have ope ned a road for the discovery of new \nphysical phenomena at the nanoscale with important consequences for nanotechnological \napplications. Magnetic nanostructures present as va luable candidates for th e development of high \ndensity storage media, high speed magnetic random access memories, Nanoelectromechanical \nsystems (NEMS) and magnetic  sensors and logic devices2. Arrays of nanoelements exhibit different \nmagnetic behaviour as a function of th eir size, aspect ratio and separations3,4. Particularly, the \nground state magnetic configuration of  a nanodot evolves from a single domain state to a vortex state \nby increasing its size5,6. Since its first observation by mean s of the X-ray magnetic circular \ndichroism technique7,8, the experimental study of vortex dynamics has become possible and has \nimmediately attracted a lot of attention with the aim to control this process.  \nThe vortex ground state exists in a wide range of nanodot sizes from several nanometers to \nmicrons depending on the interplay between the magnetocrystalline anis otropy, exchange and \nmagnetostatic energies5,9. It is characterized by the polarity  -the up or down direction of the vortex \ncore magnetization- and the chirality  -clockwise or counter-clock wise magnetization rotation. The \nresulting four possible states ar e independent and, consequently, th e vortex state could store the \ninformation of four magnetic bits . It is known that the energy barri er separating the two states with \nopposite polarities is a Bloch point w ith an excess of exchange energy10 and consequently, the vortex \nis remarkably stable against thermal fluctuations. At the same time, it has been recently discovered \nthat the vortex polarity could be switched with very small perpe ndicular field and current pulses8,11,12 \nvia the mechanism of the vor tex-antivortex pair creation13,14,15. This has strongly stimulated the idea \nof the use of the vortex state in non-vol atile high density magnetic storage media11,12,16 and vortex \nmagnetic random access memories (VRAM)17. All previously re ported analysis on the switching of \nthe polarity of the vortex core is devoted to the ci rcular dots where the chirality is not visible by \nMagnetic Force Microscopy (MFM) and cannot be co ntrolled. Alternatively, it has been reported \nthat in circular dots with truncated edges a nd other certainly complex engineered defects18-25 the  4nucleation point of the vortex could be controlled. This allows a possibility of additional control of \nvortex chirality. Here we show th at in a simple triangular dot geometry both vortex polarity and \nchirality can be tailored by suitable application of magnetic fields. Notice that the use of this quite \nsimple triangular geometry provide s an additional advantage of breaking the circular symmetry so \nmaking the chirality and the polarity direct ions well distinguishable by MFM technique26. This also \noffers a possibility of signal codification.  \nIn the present paper, we discuss several possibil ities aiming to achieve a complete control of both \nvortex polarity and chirality in triangular dots. We illustrate the concept of the vortex states control \nby applying in-plane and out-of plane magnetic fields in a static regime. Complementary \nmicromagnetic simulations reveal the mechanisms of chirality and polarity control by applying field \npulses. Additionally, the micromagnetic simulations provide us the information on the minimum \nfield strength and duration requirements. \n \nII. VORTEX CONFIGURATION C ONTROL BY IN-PLANE FIELD \nA. Experimental results \nIn this work, triangular Ni nanostructures have been prepared by nanolithography techniques. \nTheir topography was characterized by SEM (Sca nning Electron Microscopy) and AFM (Atomic \nForce Microscope), while the magnetic behaviou r was studied by VFMFM (Variable Field Magnetic \nForce Microscopy). Square arrangement of Ni(111) triangles have been fabr icated by electron beam \nlithography and magnetron sputtering techniques 27. The triangular dots, 50 nm  thick, have a lateral \nsize of 500 nm. The lattice parame ter of the array is 800 nm.  To study and control the magnetic \nbehaviour of the sample we have used a comme rcial AFM/MFM system from Nanotec Electrónica \nS.L which has been conveniently modified so that magnetic fields can be a pplied in the course of \nMFM operation. Our MFM includes a PLL (Phase Lock Loop) system that keeps constant the phase \nof the cantilever oscillation and where the magnetic  signal corresponds to the frequency shift of the \ncantilever oscillation. Out-of-plane and in-pla ne magnetic fields up to 1.5 kOe and 2 kOe, \nrespectively, can be applied preserving the necessa ry high mechanical stab ility of the microscope28.  5The probes are commercial Si cantilevers (N anosensors PPP –FMR, k=1.5 N/m and f = 75 KHz) \ncoated by a CoCr sputtered thin film. Before each experiment the probes are magnetized along their \npyramid axis. The thickness of the coating has been  carefully selected (25 nm) in order to prevent \nthe influence of the stray field of the tip on th e magnetic state of the sample. The MFM contrast \nobtained with those homemade prob es is relatively low (5Hz, which corresponds to a force gradient \nof about 10-4 N/m) but enough to identify incontestabl y (except in few cases ) the chirality and \npolarity of the vortex. The behaviour of this sort of tips under an externally applied magnetic field \nhas been thoroughly analysed recently29.  \nThe sample was initially demagnetized i.e., the a rray of nanotriangles presents the four possible \nvortex state configurations randomly  distributed as observed in Fig 1 (b). In this MFM image, we \ncan distinguish the MFM contrast corresponding to  three in-plane domains with clockwise or \ncounter clockwise rotation (as indicated by the a rrows) and the core of the vortex with the \nmagnetization pointing in up or down direction (black  or white contrast resp ectively). In previous \nwork [see ref 26], we have confirmed the lack of interactions between nei ghbouring triangles in this \narray. \n   \nFigure 1.   (Color online) (a) AFM and (b) MFM images of a Ni nanostructure arrangement after demagnetizing the \nsample. Black and white spots in the triangles of Fig 1b denote the core polarity pointing in up or down direction, \nrespectively. The arrows represent the chirality of the closure flux.  \n \n800nm\n800nm 6In Figure 2 series, in situ  magnetic field is applied along in -plane direction to induce a well \ndefined chirality in each structure.  At the initial state (Fig 2a), th e four triangles present the same \nvortex polarity and random chiralities. When the ma gnetic field is applied in  +x direction the vortex \ncore moves towards the base or th e top of the triangle regarding thei r clockwise or counter clockwise \nchirality respectively26 (Fig 2b). An increase of the field larg e enough to nearly satu rate the magnetic \nstructures results in  a single dipolar contrast (Fig 2c). No tice that upon decreasing the value of the \napplied field, the vortex nucleates al ways in the base of the triangle s (Fig 2d). However, they present \nrandom vortex polarities since no out-of-plane field is applied. Finally, at zero field the resulting \nchirality of each nanotriangle is determined by th e direction of the previously saturated magnetic \nstate (see Fig 2f). In Figure 3 we present an im age corresponding to a larger  region than that of \nFig 2f indicating that the vorte x core chirality has been ch anged everywhere. The observed \nmagnetisation-remagnetization process and the corre sponding MFM images are in agreement with \nstatic micromagnetic simulations, as was reported in Ref. 26.  \n \nFigure 2.  (Color online) Series of MFM imag es obtained at (a) 0 Oe, (b) +150 Oe, (c) +200 Oe, (d) + 100 Oe, (e) \n+50 Oe, (f) 0 Oe. The round arrows in the Figure schematically  show the vortex chirality direction while the straight (d) (e) (f) \nH = 0 Oe (a) (b) (c) \n400nm \nH = 0 Oe H  7ones – the direction of the core movement.  Straight large arrow outside the MFM images in dicates the direction of the \napplied magnetic field.   \n \nFigure 3.  (Color online) Image obtained after saturating the sample by applying an in-plane magnetic field in +x \ndirection (see arrow). This image is subsequent to the se ries in Figure 2. This image is a zoom-out of Fig  2f. \n \nTo establish the reproducibility of the method, th e magnetic field is now applied in the opposite \ndirection to produce the counter cl ockwise chirality of the nanostruc tures. Fig 4a corresponds to the \nMFM image obtained after satura ting the sample under a magnetic field applied along the -x \ndirection. Note that whil e the chirality of the clos ure flux is opposite  to that observe d after saturating \nalong the +x direction, th e vortex core is again randomly dist ributed in up and down directions.  \nAs a result of our observation, the vortex chirality can be controlled by an in-plane field producing \narbitrary vortex polarity.  \n1.3µm\n800nm  \nH  8 \nFigure 4.  (Color online) MFM image obtained after saturating the sa mple in -x direction to control the counter clockwise \nchirality of the closure flux.  \n \nB. Micromagnetic Simulations \nTo understand the micromagnetic mechanism and the minimum requirement for the control of \nvortex chirality and polarity we perform microm agnetic simulations. For the simulations, Ni \ntriangles of the same dimensions as the experiment al ones were discretized in finite elements. The \nfinite element Magpar code30 has been used for the micromagnetic simulations considering the \nfollowing Ni parameters: the anis otropy constant K= -5 x 103 (J/m3), the exchange stiffness \nA= 3.4 x 10 –12 (J/m), the saturation magnetisation value μ0Ms= 0.61 (T).   \n-400 -300 -200 -100 0 100 200 300 400-1.0-0.50.00.51.0\nHEXT(Oe)Mx/Ms\n \n 600nm   9Figure 5.   (Color online) Descending branch of the simulated hysteresis cycle for one isolated dot (solid line) and four \nnanodots separated by 300 nm ( ○) and 100 nm ( Δ). The in-plane magnetic field H EXT is parallel to the base of the \ntriangles. \n \nFig. 5 presents modelled hysteresis cycles for a single magnetic nanodot ( ) and 4 nanodots \nseparated by 300 nm ( ○) (as in the experimental  situation) and 100nm ( Δ). The in-plane magnetic \nfield H EXT is parallel to the base of the triangles. As we  clearly see, the interact ions in our system are \nnegligible (the differences may be  attributed to different nanodot di scretizations). We conclude that \nfor this separation the stray fields from the tr iangular corners have little effect on the vortex \nnucleation field. Consequently from now on we will focus our modelling on single dots only. \nThe use of variable-field MFM has enabled the experimental demonstration that chirality can be \ncontrolled in the static regime, that is, saturating the sample in a given direction by applying the field \nfor sufficiently long time. To present a complete pict ure of the vortex states control, we should also \nconsider the case of fast pulse fields which is al so more interesting from the application point of \nview. In what follows, we will investigate the vortex  configuration as a func tion of the applied field \nduration, understanding that the long pulse duration corre sponds to that of the st atic control. For this \npurpose, the external in-plane field of the maximum value, Hmax is supposed to have the risetime 0.5 \nns, duration, tH , and the decay time 0.5ns.  \nAs observed experimentall y, long field pulses durations produce re versal of the core chirality. The \nmechanism is illustrated in Fig 6 and is the same in  the dynamical and in the static case. The control \nof the core chirality is provided by the triangul ar geometry of the dot. Due to the energy \nminimization, the vortex is always created on the tria ngular base (as observed also by variable-field \nMFM). During the first part of the process the core polarisation is reversed due to creation of the \nvortex-antivortex pair. The new created vortex with  the opposite polarisation is expelled through the \ntriangle corner opposite to the triangle base. Befo re this happens, the magne tisation in the triangle \nstarts to divide into two domains - precursor of the new chirality. It is known that the vortex core \nmoves perpendicular to the field and the directi on of the motion depends on the vortex chirality 9. \nThus, there is a coupling between th e direction of the chirality and the direction of motion which in  10our case is always from the triangular base to th e opposite tip. Therefore, unlike circular and squared \ndots, the chirality in triangular na nodots is completely determined by  the applied field direction. For \nboth applied field directions the vortex is crea ted on the triangular base  and moves towards its \ncentre, where it is stabilized at zero fields.  \nThus, the mechanism of the vortex chirality switchi ng is the expulsion of th e vortex core from the \ndot and the creation of the new vorte x with the chirality compatible w ith the applied field direction.  \nIn other words, application of the field in +x dire ction selects clockwise chirality and visa versus as \nshown in the above experimental results (Fig 2, Fig3 and Fig4). Un like the static case of MFM \nimaging, in the particular case of Fig 6, the fiel d is switched off before the original vortex is \nexpelled.  \nThe process of the chirality change occurs in timescale of the order of  5 ns. For stronger and \nlonger field pulses multiple vortex-antivortex pair s creation can be observe d and the final core \npolarisation becomes random, sim ilar to static MFM images.  \n  11\n(a)\n(f) (e)(c) (b)\n(d)\n(g)(i) (h)\n(a)\n(f) (e)(c) (b)\n(d)\n(g)(i) (h)\n \nFigure 6.  (Color online) Dynamical configurations during the vortex chirality reversal under applied field \nHmax=250 Oe, duration t H=3 ns and field rise and decay ti mes 0.5ns at subsequent timeshots.  (a) t=0, the original vortex \nwith core up and clockwise chirality is in the nanodot centre.  (b) t=0.56ns, the vortex-antiv ortex pair is created. (c) t= \n1.675 ns, the original vortex has annihilated with the antivortex. The new core-down vortex is moving perpendicular to \nthe field direction towards the triangle corner. (d) t=2.68ns and (e) t= 3.9ns, the vortex continues to move towards the triangle corner. The 90-degree domain wall (an onset of the c onter clockwise chirality) is created (f) t=4.1ns. The domain \nwall gives birth to two vortices with the opposite polarisations and an anti-vortex (g) t=4.5ns The vortex (core down)-antivortex pair is annihilated. (h) t=4.83 ns, the vortex moves perpendicular to the field towards the nanotriangle centre. \n(i) t=5.63 ns, the vortex is stabilised in the nanodot centre.   \n \nWe have observed that for shor t duration field pulses only vorte x core polarity is changed. The \nmechanism of core polarisation sw itching is the same to that re ported earlier for circular dots13,14,15. \nThis mechanism is illustrated in Fig 7. Namely, a new vortex-antivortex pair is created with the H  12subsequent annihilation of the original vortex and anti-vortex (see Fig 7b). The remaining new \nvortex with changed polarisation performs a circular  motion around the dot centre (Fig 7 c-e) until \nthis motion is damped out (Fig 7f). Note that the process of the core polarisation reversal occurs in \nthe timescale less than 0.5ns, af ter this moment the applied field may be switched off.  \n \nFigure 7.  (Color online) Dynamical configurations during the vortex polarisation reversal under applied field \nHmax=250 Oe and duration t H=1ns at subsequent timeshots (a) t=0 ns Original vortex with the polarisation up is in the \nnanodot centre. (b) t=0.5ns Creation of the vortex-antivortex pair  (c) t=1.5 ns The original vortex has disappeared, the \nnew vortex with the polarisation down  starts to move arou nd the nanodot centre in the clock-wise direction. (d) t=2.7ns \nand (e) t=3.63ns Continuation of the vortex gyration  (f) t=4.64 ns. The gyroscopic motion is damped out and the vortex \nis stabilised in the dot centre. \n \nIn Fig 8 we have presented the diagram showing th e results of the applica tion of external fields \napplied along –x direction of different durations a nd strengths. Notice that in some region of the \nparameters the polarisation of th e new vortex is determined by the polarisation of the previous one \nand is opposite to it due to the magnetostatic energy minimization. However, the magnetostatic \n(a) (b) (c) \n(d) (e) (f) \nH 13coupling coming from the vortices cores is negligible and the field parameters range where we \nobserved that both the polarity  and the chirality could be  controlled is small.  \n100 150 200 250 300 350 400024681012  Field pulse duration tH (ns)\nHAPP(Oe)(C=1; P=1)\n(C=1; P=-1)\n(C=-1; P=1)\n(C=-1; P=-1)no reversal\npolarity control\nchirality control\n \nFigure 8.  (Color online) Diagram showing the result of the appli cation of the external field with the maximum strength \nHmax, field pulse duration t H and field rise and decay times 0.5ns. The in itial vortex state is core-up polarisation (P=1) \nand clockwise chirality (C=1). The final states are indicated  by different symbols corresponding to clockwise (C=1) or \ncounter clockwise (C=-1) chiralities and polarisation up (P=1 ) or down (P=-1). \n \nTo explore new possibilities of chirality and po larity vortex control we propose to combine in-\nplane and out-of-plane magnetic fields.   \n III. CHIRALITY AND POLARITY C ONTROL BY A COMBINED FIELD \nA. Experimental results \nThe polarity of the vortex can be selected by appl ying an external out-of-p lane field as shown in \nFig 9a. Vortex is quite a stable object and demagne tising field from the thin film geometry is quite \nstrong, consequently, strong perpendicular fields  are necessary (around 700 Oe) to change the \npolarity of the vortex. However, such perpendicula r fields will leave chiral ity undefined as shown in \nthe MFM image in Fig 9a. It could be expected th at with a combined action of out-of-plane and a \nvariable in-plane fields, one can  independently contro l vortex chirality and polarity. Experimental \nevidences have been found of such vortex confi guration control. MFM image in Fig 9b has been  14acquired in remanent state after applying a magnetic  field of 1500 Oe at 30º from the surface plane. \nBoth chirality and polarity were determined for all individual triangles th rough the combined action \nof static in-plane and ou t-of-plane fields applied out of the MF M system. It is re asonable to suppose \nthat since the vortex appears on th e triangular base with arbitrary po larity, a small component of the \nperpendicular field could help to  control vortex polarity. However, experimenta lly it has been found \nthat the required field is not small. For exampl e, the magnetic field prod uced by the MFM tip (about \n200 Oe) is not sufficient to stabilize the nucleated vortex core with a desire d direction as deduced \nfrom the results shown in Fig3 and Fig 4. \n \nFigure 9.  (Color online) (a) MFM image obtained after saturating the sample in out-of-plane direction to control the \nvortex polarity. (b) MFM image obtained after applying a combined in-plane and out-of-plane field producing the \nclockwise configured nanostructures with vortex cores pointing down.  \n \nB. Micromagnetic Simulations \nThe experimental results suggest  that one can help the creati on of the vortex with specific \npolarisation applying a magnetic fiel d with an out-of-plane component . To investigate this idea, we \nhave performed micromagnetic simulations of hyste resis cycles with applied fields at different \nangles for the case of four magnetic nanodots. The micromagnetic simulations clarify us the vortex \nbehaviour.  \n 600nm\n(b) (a) \n  380nm   15Fig.10 represents simulated hysteresis cycles (corresponding to the magnetization along the \napplied field direction) in a system of 4 nanodots a nd shows that the vortex nucleation field increases \nwith the out-of-plane field angl es. Since we use finite element method, each of our  triangles is \ndifferent, emulating the experi mental situation. The results show that in the remanence \ncorresponding to Fig.11 (in-plane field), three of our triangle s appeared with vortex core \nconfigurations “up” and one “down”. Saturating our sample with field applied at 5o out-of-plane and \nhaving a negative component we have  obtained at the remanence two vortices with core “up” and \ntwo with core “down”. It  was necessary to apply a field with at least 35o out-of-plane to obtain all \nthe vortices in the config uration core “down”.   \n0 125 250 375 500 625 7500.000.250.500.751.00MH/Ms\nHEXT(Oe) θ=0º\n θ=5º\n θ=10º\n θ=15º\n θ=30º\n θ=35º\n \nFigure 10.   (Color online) Part of the descen ding branch of the simulated hysteres is cycles with fields applied at \ndifferent out-of-plane angles, θ,  for a system of four dots separated by 300 nm. \n   \nTo understand the situation, we pres ent in Fig.11 the simulated MFM im ages (a, c, d and f) and the \ncolor map plot of the M z component (b and e) before and after the vortex nucleation for applied \nfields at 5o (a-c) and 35o (d-f). The simulated MFM images (Fig.11 a and d) show that before the \nvortex polarity formation the chirality is already pr esent, as was also mentioned above. Because of \nthis, the local field on the triangular base is alrea dy parallel to it and, therefore, the action of the \nperpendicular external field is redu ced by the dipolar field. Consequent ly, the overall internal field is \nnot sufficient to produce the desired polarisation. A dditionally it is worth to mention that in the  16upper left triangle the original  vortex was created with the polarisation down but dynamically \nchanged it, as it is seen in Fig.11 (c). \n \nFigure 11.   (Color online) Simulated images of the four  triangular nanodots with applied field at 5o (upper rows, a-c) \nand 35o (lower rows, d-f). The left and middle columns are images  before the vortex nucleation, (a and d) represent the \nsimulated MFM images and (b and e) are the corresponding Mz component. The right column presents the simulated \nMFM images after the vortex nucleation. \n \nFor the field applied at 35o the nucleation field is larger than that for 5o, so that it’s perpendicular \ncomponent and the overall balance of  the total field at th is moment is negative,  allowing the creation \nof the vortex polarisation in the desi red direction.  To summarise, the simulations show that in order \nto achieve the vortex polarisation co ntrol, at the moment and in th e place of the vor tex creation the \noverall total field should be pointed in the desired direction and this could be achieved only with out-\nof-plane field components of about 430 Oe. \n \n  17IV. CONCLUSIONS \nIn conclusion, by means of variable field MFM and micromagnetic simulations we have shown \nthat triangular nanodots offer a possibility to s imultaneously control magnetic vortex chirality and \npolarity. The four magnetic states  are perfectly visible by Magneti c Force Microscope offering the \npossibility for codification. \nWe have studied theoretical and experimentally the effect of in-plane magnetic fields applied \nparallel to the triangle base. In the static case, we observe the change of the vortex chirality due to \nthe vortex annihilation and the coupling of the vor tex core direction moti on and the applied field \ndirection.  \nWe have shown that in a dynamic regime, by vary ing the field pulse durat ion it is possible to \ncontrol both vortex chirality and pola rity: with short field pulses it is possible to change the vortex \npolarity while with longer pulses the chirality can be changed. The mechanism of the vortex chirality \ncontrol in this conditions is the same as that in the static case. At the same time, the mechanism for \nthe change of the vortex polarisation for short fi eld pulses is due to the vortex-antivortex pair \ncreation. Thus, by triggering pulses (in-plane directio n) with different durati ons one can control the \nfour vortex states. \nAlternatively, the change of the vortex polaris ation can be produced by applying additionally a \nperpendicular field component. Since the vortex chirality is formed before its polarity, the \nperpendicular applied field action on the triangle base is reduced by the magnetostatic field. The \nmicromagnetic simulations show that for a complete polarity control a minimum strength of the out-\nof-plane component (approx. 430 Oe) is necessar y. The corresponding experimental value could be \nlarger due to the magnetostatic shape anisot ropy arising from the thin film geometry. \nFor the future applications, the use of dynamically  induced polarity control has some advantages \nas compared to that of the static fields. The co ntrol of the vortex states  makes the triangular dots \nunique candidates for the applications in non-vol atile magnetic storage such as vortex MRAM \nmemories. \n  18ACKNOWLEDGMENTS:  The author s acknowledge the financia l support from the Spanish \nMinisterio Ciencia e In novación through the projects NAN2004-09087, NAN2004-09183-C10-04, \nFIS2005-07392, FIS2008-06249, MAT2007-66719-C03- 01, MAT2007-65420-C02-01, Consolider \nCSD2007-00010, CS2008-023 and CAM gran ts S-0505/ESP/0337, 505/MAT/0194 and CAM-\nCSIC200660M046, and Fondo Social Europeo. M.Jaaf ar and R.Yanes wish to thank CAM and \nCSIC, respectively for the financial support.   \n \n  \nREFERENCES: \n1 J.I. Martin, J. Nogues, K. Liu, J.L. Vicent, I.K. Schuller, J.Magn.Magn.Mater. 256 , 449 (2003) \n2 B.Azzerboni, L.Pareti, G.As ti “Magnetic Nanostructures in Modern Technology: Spintronics, Magnetic MEMS and \nRecording” , Soringer-Verlag, New York, (2007). \n3 R.P. Cowburn, J. Phys.D: Apply.Phys.  33, R1-R16 (2000) \n4 J.L. Costa-Kramer , J.I. Martin, J.L. Menendez, A.Cebolla da, J.V.Anguita, F. Briones and J.L. Vicent,  Appl. Phys. \nLett. 76, 3091 (2000). \n5 H.Hoffman and F.Steinbauer, J. Appl. Phys.  92, 5463 (2002) \n6 K.L. Metlov and K.Y. Guslienko, J.Magn.Magn.Mater. 242-245  , 1015 (2002)   \n7 H. Stoll, A. Puzic, B. van Waeyenberge, P. Fischer, J. Ra abe, M. Buess, T. Haug, R. Hö llinger, C. Back, D. Weiss, and \nG. Denbeaux , Appl. Phys. Lett. 84, 3328–3330 (2004). \n8 B.Van Wayenberge, A.Puzic, H.Stoll,  K.W.Chou, T.Tyliszczak, R.Hertel, M. Farle, H.Brukl, K.Rott, G.Reiss, \nL.Neudecker, D.Weiss, C.H.Back and G.Shutz, Nature (London) 444, 461 (2006). \n9 W.Scoltz, K.Yu.Guslienko, V.Novosad, D.Suess, T.Schr efl, R.W.Chantrell and J.Fidler, J.Magn.Magn.Mater. 266,  \n155 (2003). \n10 A.Thiaville, J.M.Garcia, R.Dittrich, J. Miltat  and T.Schrefl, Phys. Rev. B  67, 094410 (2003). \n11 K.Yamada, S.Kasai, Y.Nakatani, K.Kobayashi, H.Kohno, A.Thiaville and T.Ono, Nature Mater. 6, 270 - 273  (2007). \n12 R.P.Cowburn, Nature Mater. 6, 255 (2007). \n13 K.S.Lee, K.Y.Guslienko, J.Y.Lee and S.-K.Kim, Phys Rev. B 76, 174410 (2007). \n14 R.Hertel, S.Gliga, M.Fahnle and  C.M.Schenider, Phys Rev Lett 98, 117201 (2007). \n15 K.Y.Guslienko, J.-Y.Lee and S.K.Kim, Phys Rev Lett  100, 027203 (2008). \n16 J.Thomas, Nature Nanotechn.,  2, 206 (2007). \n17 S.-K.Kim, K.-S-Lee, Y.-S.Yu, Y.-S.Choi, Appl. Phys. Lett. 92, 022509 (2008). \n18 M. Schneider, H. Hoffmann, and J. Zweck, Appl. Phys. Lett. 79, 3113 (2001)  \n19 Kuo-Ming Wu, Jia-Feng Wang, Yin-Hao Wu, Ching-Ming Lee, Jong-Ching Wu, and Lance Horng, J. Appl. Phys. \n103, 07F314 (2008)  \n20 T. Taniuchi, M. Oshima, H. Akinaga, K. Ono, J. Appl. Phys. 97, 10J904 (2005) \n21 P. Vavassori, R. Bovolenta, V. Metlushko , B. Ilic, J. Appl. Phys. 99, 053902 (2006) \n22 F. Giesen, J. Podbielski, B. Botters and D. Grundler, Phys. Rev. B 75, 184428 (2007) \n23 P. Vavassori, O. Donzelli, M. Grimsditch,  V. Metlushko, and B. Ilic, J. Appl. Phys. 101, 023902  (2007) \n24 P. Vavassori, D. Bisero, V. Bonanni, A. Busato, M. Grimsd itch, K. M. Lebecki, V. Metlushko, and B. Ilic, Phys. Rev. \nB 78, 174403 (2008) \n25 S. Y. H. Lua, S. S. Kushvaha, Y. H. Wu, K.  L. Teo, and T. C. Chong, Appl. Phys. Lett.  93 , 12250 (2008) \n26 M. Jaafar R. Yanes, A. Asenjo, O. Chubykalo-Fese nko, M. Vázquez, E. M. González and J. L. Vicent, \nNanotechnology 19, 285717 (2008). \n27 J.I. Martin, Y. Jaccard Y, A. Hoffma nn, J.Nogues, J.M. George, J.L.Vicent  and I.K. Schuller, J. Appl. Phys. 84, 411 \n(1998).  \n28 M. Jaafar, J Gómez-Herrero, A Gil, P Ares , M Vázquez and A Asen jo, Ultramicroscopy, 109 , 693 (2009)   \n29 M.  Jaafar, A. Asenjo and M. Vázquez, IEEE Nano  7 , 245(2008) \n30 http://magnet.atp.tuwien.ac.at./scholz/magpar \n "    },    {        "title": "1309.4676v2.Tunable_non_equilibrium_dynamics__field_quenches_in_spin_ice.pdf",        "content": "Tunable non-equilibrium dynamics: field quenches in spin ice\nS. Mostame1;3, C. Castelnovo2, R. Moessner3, and S. L. Sondhi4\n1Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138, USA\n2TCM group, Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, United Kingdom\n3Max Planck Institute for the Physics of Complex Systems, N ¨othnitzer Str. 38, D-01187 Dresden, Germany and\n4Department of Physics, Princeton University, Princeton, NJ 08544, USA\n(Dated: May 25, 2022)\nWe present non-equilibrium physics in spin ice as a novel setting which combines kinematic constraints,\nemergent topological defects, and magnetic long range Coulomb interactions. In spin ice, magnetic frustration\nleads to highly degenerate yet locally constrained ground states. Together, they form a highly unusual magnetic\nstate – a “Coulomb phase” – whose excitations are pointlike defects – magnetic monopoles – in the absence\nof which effectively no dynamics is possible. Hence, when they are sparse at low temperature, dynamics be-\ncomes very sluggish. When quenching the system from a monopole-rich to a monopole-poor state, a wealth of\ndynamical phenomena occur the exposition of which is the subject of this article. Most notably, we find reac-\ntion diffusion behaviour, slow dynamics due to kinematic constraints, as well as a regime corresponding to the\ndeposition of interacting dimers on a honeycomb lattice. We also identify new potential avenues for detecting\nthe magnetic monopoles in a regime of slow-moving monopoles. The interest in this model system is further\nenhanced by its large degree of tunability, and the ease of probing it in experiment: with varying magnetic fields\nat different temperatures, geometric properties – including even the effective dimensionality of the system – can\nbe varied. By monitoring magnetisation, spin correlations or zero-field Nuclear Magnetic Resonance, the dy-\nnamical properties of the system can be extracted in considerable detail. This establishes spin ice as a laboratory\nof choice for the study of tunable, slow dynamics.\nI. INTRODUCTION\nNature and origin of unusual – in particular, slow – dynam-\nics in disorder-free systems1are amongst the most fascinating\naspects of disciplines as diverse as the physics of structural\nglasses and polymers2, chemical reactions and biological mat-\nter3. Kinetically constrained models, following the original\nidea by Fredrickson and Andersen4, represent one paradigm\nin which unusual dynamics is generated by short-distance in-\ngredients alone without disorder5. Another is provided by\nreaction-diffusion systems, in which spatial and temporal fluc-\ntuations feed off each other to provide a wide variety of dy-\nnamical phenomena6especially due to the slow decay of long\nwavelength fluctuations.\nSpin ice systems1allow to combine both aspects – thanks\nto the nature of their emergent topological excitations, which\ntake the form of magnetic monopoles2,5with long range\nCoulomb interactions. The ground state correlations in these\nlocalised spin systems lead to kinematic constraints in the\nreaction-diffusion behaviour of these mobile excitations8,11.\nUnderstanding the dynamics of spin ice systems, and in par-\nticular proposing new ways to probe their out-of-equilibrium\nproperties, is of direct experimental relevance. For instance,\nmodelling the emergent excitations near equilibrium12,13al-\nlowed to gain insight on the observed spin freezing at low tem-\nperatures14,15, and to explain in part the time scales measured\nin zero-field NMR16. Despite the fast paced progress, sev-\neral open questions remain unanswered in particular concern-\ning the behaviour of spin ice materials far from equilibrium,\nas evidenced for instance by recent low-temperature magnetic\nrelaxation experiments17–19.\nIn this article, we study the strongly out-of-equilibrium\nbehaviour following a quench from a monopole-rich to a\nmonopole-poor regime by means of varying an applied mag-netic field. We uncover a wide range of dynamical regimes\nand we provide a theoretical understanding of their origin.\nWe show that the initial evolution maps onto a deposition\nof dimers on a honeycomb lattice, in presence of long range\nCoulomb interactions between the ‘vacancies’, i.e., the uncov-\nered sites. The long time behaviour can instead be understood\nas a dynamical arrest due to the appearance of field-induced\nenergy barriers to monopole motion. This regime can be seen\nas similar to conventional spin ice, but with a monopole hop-\nping time exponentially sensitive to temperature: we have a\nCoulomb liquid in which time can pass arbitrarily slowly.\nWe discuss how to probe these phenomena in experiment,\nshowing how by monitoring magnetisation, spin correlations\nor zero-field Nuclear Magnetic Resonance (NMR), the dy-\nnamical properties of the system can be extracted in detail.\nOverall, this richness and versatility establishes spin ice as\na laboratory of choice for the study of slow dynamics arising\nfrom an interplay of frustration (local constraints), topolog-\nical defects (monopoles) and magnetic long-range Coulomb\ninteractions.\nII. PHASE DIAGRAM AND QUENCH PROTOCOLS\nSpin ice materials consist of magnetic rare earth ions ar-\nranged on a corner-sharing tetrahedral (pyrochlore) lattice.\nTheir magnetic moments are well-modeled by classical Ising\nspins constrained to point in the local [111] direction (either\ninto or out of their tetrahedra)1, as illustrated in the top left\npanel of Fig. 2 (see also the Suppl. Info.).\nThe frustration of the magnetic interactions manifests it-\nself in the fact that there are not just a handful of ground\nstates – an unfrustrated Ising ferromagnet, for instance, has\nonly two ground states. Rather, spin ice has an exponentiallyarXiv:1309.4676v2  [cond-mat.str-el]  10 Jan 20142\nspin icesaturated ice (SatIce)\nkagome ice (KagIce)/f_irst order transition\ncritical endpointparamagnet\ncrossover\n0.1 0.2 0.3 0.4 0.50.2\n0.8 0.7 0.61.2\n1\n0.8\n0.6\n0.4\ntemperature (K)magnetic /f_ield (Tesla) crossover\nFIG. 1. Sketch of the equilibrium phases of spin ice in presence of\na[111] magnetic field. The blue vertical arrows denote examples of\nfield quenches discussed in this work.\nlarge number of ground states, namely all those configurations\nwhich obey the ice rule that each tetrahedron has two spins\npointing into it, and two out (2in-2out).\nAt any finite temperature, there will be excitations in the\nform of tetrahedra which violate the ice rule, having three\nspins pointing in and one out or vice versa. Crucially, these\ndefects can be thought of as magnetic monopoles which are\ndeconfined and free to carry their magnetic charge across the\nsystem5. The energy cost \u0001associated with the creation of a\nmonopole is of the order of a few degrees Kelvin for spin ice\nmaterials Dy 2Ti2O7and Ho 2Ti2O7, leading to an exponential\nsuppression of their density for T\u001c\u0001.\nWe refer the reader interested in the detailed background to\nRef.2,20. Here, we emphasize that spin ice is the only experi-\nmentally accessible magnetic system that realises a Coulomb\nphase described by a low-energy emergent gauge field. The\nnew phenomena we describe here can ultimately be traced\nback to this fundamentally new feature.\nA. The equilibrium phase diagram of spin ice\nFour distinct regimes are encountered in spin ice in pres-\nence of a field along a [111] crystallographic direction, as dis-\nplayed in Fig. 11. In order to understand this phase diagram, it\nis convenient to divide the spin lattice (pyrochlore, or corner-\nsharing tetrahedral lattice) into alternating kagome and trian-\ngular layers, as illustrated in Fig. 2. Further details on spin ice\nsystems as well as on their phase diagram in a 111field can\nbe found in the Suppl. Info.\nIn the limit of strong fields, in the saturated (SatIce) regime,\nall the spins point along the field direction, while respecting\nthe local easy axes (see Fig. 2, top middle panel). The ice\nrules are violated everywhere and each tetrahedron hosts a\nmonopole; the monopoles form an “ionic crystal”.\nAs the field strength is reduced, violations of the ice rules\nare no longer sufficiently offset by a gain in Zeeman energy\nand a regime where most tetrahedra obey the ice rule is re-\ncovered for T\u001c\u0001. This necessarily requires some of thespins to point against the applied field. At intermediate field\nstrengths, they will dominantly be the spins in the kagome\nplanes, as their Zeeman energy is smaller by a factor of 3com-\npared to the spins in the triangular planes. This leads to an ex-\ntensively degenerate regime known as kagome ice (KagIce),\nillustrated in the bottom middle panel in Fig. 2.\nAt low field strengths, the kagome ice regime becomes\nentropically unstable to the ‘conventional’ spin ice regime ,\nnamely the ensemble of all configurations satisfying the ice\nrules irrespective of the polarisation of the triangular spins\n(top right panel in Fig. 2). All of these regimes crossover at\nsufficiently large temperatures into a conventional paramag-\nnetic regime.\nThese different regimes appear in the field-temperature\nphase diagram of spin ice as illustrated in Fig. 1. Note that\nonly the KagIce and SatIce phases are separated at low tem-\nperatures by an actual (first order) phase transition21,22(see\nSuppl. Info. for further details).\nB. Field quench protocols and non-equilibrium phenomena\nIn this article we discuss field quenches from SatIce to Kag-\nIce, as exemplified by the blue vertical arrows in Fig. 1. We fo-\ncus on field quenches at (constant) temperature T.0:6K\u001c\n\u0001, as the system appears to relax straightforwardly to equi-\nlibrium at higher temperatures (blue dash-shaded region in\nFig. 1) on time scales of the order of a typical microscopic\nspin-flip time scale.\nWe are interested here mostly in the low-temperature dy-\nnamics that follows quenches across the first order phase tran-\nsition in Fig. 1. As we shall see in the following, these\nquenches are characterised by interesting slow relaxation phe-\nnomena and dynamical arrest. By comparison, quenches at\nhigher temperatures T\u00180:5\u00000:6K appear to be fast and\nfeatureless. The presence of a critical point in the phase di-\nagram at these higher temperatures raises a number of sepa-\nrately interesting questions and ideas for further work, such\nas the possibility of quenching to / across a critical point, and\ntesting whether the critical correlations bear any signatures in\nthe resulting dynamical behaviour. Whilst interesting in their\nown right, these issues are beyond the scope of the present\nwork.\nThe dynamics of spin ice at T.5K appears to be based\non incoherent spin reversals, well modelled by a Monte Carlo\n(MC) single-spin flip dynamics13. From AC susceptibility\nmeasurements15, a Monte Carlo time step corresponds to ap-\nproximately 1 ms in Dy 2Ti2O7, which we use to convert MC\ntime into physical time in this work.\nWe used Monte Carlo simulations with single spin flip dy-\nnamics, with Dy 2Ti2O7spin ice parameters as in Ref.23. We\nemployed Ewald summation techniques24to treat the long\nrange dipolar interactions, and the Waiting Time Method25\nto access long simulation times at low temperatures (see also\nRef.8).\nObservables that we monitor after a field quench are\nmonopole density and magnetisation, as illustrated in Fig. 3.\nIn addition to gaining insight on non-equilibrium dynamical3\n[111]  direction\n(a)\n(b)\n(c)\n(d)\n(e)\n(f )\nFIG. 2. (a) Pyrochlore lattice formed by the rare earth ions, highlighting the layered structure of alternating triangular and kagome planes in\nthe[111] crystallographic direction; the four spins in the bottom left tetrahedron illustrate the local easy axes and the 2in-2out ice rule. (b)\nSaturated ice state (SatIce), where all the spins have a positive projection along the [111] field direction (d); each tetrahedron hosts a monopole\n(positive 3in-1out shaded in red, and negative 3out-1in shaded in blue) forming a crystal of “magnetic ions”. If we consider for instance the\ntwo topmost tetrahedra, they host a positive (3in-1out) and a negative (1in-3out) monopole, respectively; the pair is non-contractible , since\nreversing the intervening spin does not annihilate the monopoles but rather produces an even higher energy excitation (a pair of 4in and 4out\ntetrahedra, i.e., a pair of doubly charged monopoles). This is to be compared for instance, with flipping a spin belonging to one of the kagome\nlayers in this configuration, which results in the annihilation of two adjacent monopoles and corresponding de-excitation of the tetrahedra\ndown to their 2in-2out lowest energy state. (c) Generic low-energy spin ice configuration, where the spins satisfy the local 2in-2out ice rules.\n(e) Configuration belonging to the kagome ice phase (KagIce), intermediate between SatIce and spin ice, where all the triangular spins are\npolarised in the field direction while the kagome spins arrange themselves in order to satisfy the ice rules. This requires every triangle in the\nkagome planes to have exactly one spin with a negative projection in the field direction; the positions of these anti-aligned spins can be mapped\nto a dimer covering of the dual honeycomb lattice, as illustrated in (f), with a top view down the [111] axis.\nphenomena in spin ice – in particular on the role of the emer-\ngent magnetic monopoles – our aim is to investigate whether\na protocol can be devised that allows to prepare spin ice sam-\nples in a low-temperature yet monopole-rich state by means\nof a rapid reduction of an applied [111] magnetic field. At-\ntaining such state would be a significant step towards a more\ndirect experimental detection of the magnetic monopoles, for\ninstance using zero-field NMR techniques26, as we discuss to-\nwards the end of the article.\nThe use of field quenches instead of thermal quenches8of-\nfers two practical advantages: (i) a fast variation in the exter-\nnally applied magnetic field is experimentally less challeng-\ning than performing a controlled thermal quench; and (ii) in\nquenches from SatIce to KagIce, the monopole density is to an\nexcellent approximation proportional to the magnetisation of\nthe sample, a thermodynamic quantity, and therefore directly\nand simply accessible in experiments.\nIn order to make contact with a broader range of experimen-tal settings, we briefly review towards the end of the paper\nalternative experimental protocols, such as quenches to zero\nfield, as well as slow “quenches” (i.e., finite-rate field ramps).\nFor completeness, these are discussed in greater detail in the\nSupporting Information.\nWe point out that in the following every reference to the\nmagnetic field is intended as the externally applied field H.\nFurthermore, we note that Monte Carlo simulations are devoid\nof demagnetisation corrections, which ought to be taken into\naccount when comparing to possible experimental results.\nIII. SATURATION TO KAGOME ICE\nAs illustrated in Fig. 2, KagIce comprises the hardcore\ndimer coverings of the honeycomb lattice. The equilibrium\ndensity of monopoles in this regime is exponentially small at\nlow temperatures, and they correspond to monomers in the4\n−1 1 3 5 7 9 1310 10 10 10 10 10 10\ntime (MC step)monopole density per tetrahedron\ntime (MC step)monopole density per tetrahedron\ntime (MC step)monopole density per tetrahedron\n1110−410−310−210−110010\n−1 1 3 5 7 1310 10 10 10 10 101110−410−310−210−110010\n−1 1 3 5 7 9 1310 10 10 10 10 10 101110−410−310−210−110010\n910\nFIG. 3. Monopole density (thick lines), density of triangular spins in the direction of the initial magnetization (thin dot-dashed lines),\nand density of noncontractible pairs (thin solid lines) from Monte Carlo simulations for a system of size L= 8 (8192 spins), fields H=\n0:2;0:4;0:6Tesla (left, middle, and right panel, respectively) and temperatures T= 0:1;0:15;0:2;0:3;0:4;0:5K (red, blue, green, magenta,\ncyan, yellow). Note that the thick lines represent the total monopole density, inclusive of those involved in noncontractible pairs; therefore the\nthick lines are always higher or at most equal to the thin lines of the same colour. All triangular spins are initially polarised in the direction of\nthe the applied field and their initial density is 1. For sufficiently low temperatures with respect to the strength of the applied field, their density\nbecomes again approximately one at long times (i.e., in equilibrium). At intermediate times, some of the triangular spins reverse, as shown by\nthe dip in the density; the latter has been magnified by a factor of 100 and 1000 in the left and middle panels, respectively, for visualization\npurposes. In the right panel the density of triangular spins in the direction of the applied field remains 1throughout the simulations, as expected\nfor fieldsH > 0:42Tesla (see main text and SI). In this case the triangular spins remain heavily polarised throughout the quench, and the\nmonopole motion is effectively two-dimensional. Notice that at the longest times, practically all monopoles are in non-contractible pairs.\n(Here and in later figures the black dotted horizontal line indicates the density threshold of one monopole in the entire Monte Carlo system.)\ndimer language. SatIce instead corresponds to an ionic crys-\ntal of monopoles, i.e., a configuration without any dimers.\nMicroscopically, in SatIce pairs of neighbouring positive and\nnegative monopoles on the same kagome plane annihilate by\ninverting the intervening spin, which is equivalent to the depo-\nsition of a dimer on the corresponding bond in the honeycomb\nlattice.\nField quenches from SatIce to KagIce can thus be under-\nstood as a stochastic deposition process of dimers, with non-\nvanishing desorption probability, in presence of long range\n(3D) Coulomb interactions between the monomers (i.e., the\nmonopoles). Eventually the dimers become fully packed, up\nto an exponentially small concentration of thermally excited\nmonomers at equilibrium.\nThe physics of dimer deposition processes with only short\nrange interactions was recently considered in the experimen-\ntal context of molecular random tilings27, which were shown\nto exhibit a variety of different (stochastic) dynamical regimes\nakin to kinetically constrained models1,28. In the following we\nshow how the presence of long range interactions, as well as\nadditional kinematic constraints in the motion of monopoles\ndue to the underlying spin configuration, give rise to remark-\nably rich behaviour.\nA. Short-times: dimer deposition\nThe initial spin flip events following a quench from SatIce\nto KagIce lower the energy of spin interactions by removing\nmonopoles, but they increase the Zeeman energy. As the lat-\nter is proportional to the quench field, at weak fields, all flips\nare initially downhill in energy and thus take place on a time\nscale of the order of 1 MC step, independently of tempera-ture. A mean-field equation capturing the initial decay of the\nmonopole density is\nd\u001a(t)\ndt=\u00003\n\u001c0\u001a2(t)\u001a(t= 0) = 1 (1)\n\u001a(t) =1\n1 + 3(t=\u001c0)(2)\nwhere\u001ais the monopole density per tetrahedron and the factor\n3 appears because each monopole can equally annihilate with\nthree of its neighbours. This is in good agreement with the\nsimulation results in Fig. 4.\nFor quench fields above a threshold of 0:3Tesla, the initial\ndecay becomes activated, as the Zeeman energy for reversing\na kagome spin 20\u0016BH=3'4:48HK exceeds the spin-spin\ninteraction energy gain of about \u00001:34K (Dy 2Ti2O7param-\neters)32. For fields beyond this threshold, the initial decay is\ndescribed by a dimer deposition process with field-dependent\nenergy barriers.\nThe near-neighbour monopole pair annihilation continues\nuntil no suitable pairs are left. In dimer deposition language,\nthis corresponds to the random packing limit where it is no\nlonger possible to deposit futher dimers without first rearrang-\ning those already present on the lattice.\nFor a wide range of fields and temperatures, the initial de-\ncay takes approximately 10MC steps (see Fig. 3) and reaches\na monopole density in the range 0.05 - 0.08, depending on\nfield and temperature.\nWe compare this value to that obtained from numerical sim-\nulations of random depositions of dimers on the honeycomb\nlattice, where we measure the density of vacancies at the end\nof each deposition process when it is no longer possible to add\na (hard-core) dimer to the system without rearranging the ones\nalready deposited. The density of leftover vacancies in the5\n−110010\ntime (MC step)monopole density per tetrahedron\n−110010\n−210−110010110\nFIG. 4. Comparison of the initial monopole density decay in simu-\nlations with the analytical mean field result in Eq. (2) (black solid\nline). The Monte Carlo simulations are for an L= 8 system,\nwithH= 0:2Tesla (left panel) and H= 0:3Tesla (right panel),\nT= 0:1;0:2;0:3;0:4;0:5;0:6K (red, blue, green, magenta, cyan,\nand yellow curves).\nnon-interacting simulations equals 0:121\u00060:003. It is inter-\nesting to notice that this value is at least 50% larger than that\nfollowing field quenches from SatIce to KagIce. We find that\nthe difference is due to the combination of a non-vanishing\ndesorption probability and long range Coulomb attraction be-\ntween oppositely charged monopoles, which are present in the\nspin ice quenches and not in the random deposition process.\nB. Long times: from noncontractible pairs to\ndynamically-obstructed monopole diffusion\nAt the end of the initial decay, no neighbouring monopoles\nare left in any of the kagome planes. The majority of\nmonopoles are ‘isolated’ and only a minority forms (noncon-\ntractible) pairs across adjacent planes (3).\nTwo oppositely-charged monopoles on neighbouring tetra-\nhedra form a noncontractible pair whenever inverting the di-\nrection of the shared spin leads to an even higher energy con-figuration with the two tetrahedra in the 4in and 4out state,\nrespectively8. For instance, every monopole pair across a tri-\nangular spin in SatIce is noncontractible (see Fig. 2). Indeed,\nin KagIce, where the triangular spins are polarised in the field\ndirection, noncontractible pairs can only form between pos-\nitive and negative monopoles belonging to adjacent kagome\nplanes.\nFor small values of the applied field ( H\u00180:2Tesla),\nwe see from Fig. 3 that the decay in the density of the iso-\nlated monopoles is fast in comparison to the decay of noncon-\ntractible pairs, and eventually only the latter are left to deter-\nmine the long time behaviour of the system. This regime can\nbe understood in analogy to thermal quenches8(see Suppl.\nInfo.).\nWhereas isolated monopoles are essentially free to\nmove, noncontractible pairs have an activation barrier to\nmove/decay, and thus their lifetime increases exponentially\nwith inverse temperature. This explains why the isolated\nmonopoles decay faster than noncontractible pairs; it also ex-\nplains the mechanism responsible for the long time decay in\nthe overall monopole density of the system. Whereas lattice-\nscale kinematic constraints control the formation (or possibly\nthe survival from the initial SatIce state) of noncontractible\npairs, it is their decay that ultimately controls the long time\nbehaviour of the monopole density.\nThe life time of a noncontractible pair is determined by the\nenergy barriers of the allowed decay processes (namely, pro-\ncesses where the two charges separate and annihilate one an-\nother elsewhere on the lattice; or they may annihilate with\nother charges in the system). For quenches to zero field, the\nenergy barriers are solely due to spin-spin interactions and the\ncorresponding decay occurs via a process, identified in ther-\nmal quenches8, whereby the noncontractible pair recombines\nby hopping the long way around a hexagon containing the ob-\nstructing spin.\nThe presence of a small applied field does not alter the pic-\nture qualitatively, but it introduces two important differences.\nFirstly, the mechanism and barrier to the decay of noncon-\ntractible pairs is altered. Once the majority of the triangu-\nlar spins are polarised, it is straightforward to see that the\nhexagonal processes invoked in Ref. 8 are no longer avail-\nable: decay must proceed by separation of the monopoles until\nthey meet oppositely charged ones from other noncontractible\npairs. This process incurs a Coulomb energy barrier for sep-\narating the positive and negative charges in a noncontractible\npair, which grows with the distance to the next monopole in\nthe plane. Moreover, the presence of an applied magnetic field\nalso affects the value of the barrier to separation, as the Zee-\nman energy for spin flips needs to be taken into consideration\nalong with the Coulomb attraction of the oppositely charged\nmonopoles.\nSecondly, isolated monopoles hopping within kagome\nplanes exhibit activated behaviour of their own due to the Zee-\nman energy (see below for details). This explains the opening\nof an intermediate time window between the end of the initial\ndecay, discussed in the previous section, and the final decay\ncontrolled by noncontractible pairs. This time window is ab-\nsent in thermal quenches8as well as in field quenches down6\nto zero field (see Suppl. Info.).\nThis intermediate-time regime is governed by diffusion-\nannihilation processes of isolated monopoles across larger and\nlarger distances as their density is reduced. Depending on the\nvalue of the field and temperature, our simulations show dif-\nferent behaviours following the initial decay, including an ap-\nparent power-law at low fields (see left panel in Fig. 3). This\nregime is controlled by “finite time” processes rather than by\nany asymptotic behaviour. Albeit challenging to model in the\nabsence of an asymptotic regime, it is a novel and interesting\nexample of a reaction-diffusion process in presence of long\nrange Coulomb interactions and kinematic constraints that is\nexperimentally accessible in spin ice materials.\nAs the field strength increases, the decay of the isolated\nmonopoles gets comparatively slower, until for H\u00180:6Tesla\nthey remain the majority with respect to noncontractible pairs\nat all time. In this regime, a new mechanism controls the long\ntime behaviour: the system remains always in a Coulomb liq-\nuid phase, rather than condensing its monopoles into noncon-\ntractible pairs. It is the isolated monopoles themselves that, by\nbecoming exceedingly slow, determine the long time decay of\nthe system.\nIn order to better understand and confirm this scenario, let\nus look in detail at how such slowing down of the monopole\ntime scales takes place. Consider the motion of a monopole\nwithin a kagome layer (the illustrative Fig. S3 provided in the\nSuppl. Info. may be of help here). Ordinary monopole mo-\ntion involves alternatively flipping pairs of spins with negative\nand positive projection onto the applied field. This yields a\nperiodic energy landscape due to the Zeeman energy change\n\u00064:48Hfor the kagome spins, where His measured in Tesla\nand the energy is measured in degrees Kelvin. As the field\nincreases from zero, this barrier progressively slows down the\nmonopoles.\nFor sufficiently large applied fields, another process be-\ncomes energetically more convenient: A new monopole-\nantimonopole pair is first excited neighbouring the existing\nmonopole, which then annihilates with the opposite mem-\nber of the pair ( pair-assisted hopping – see figure in the\nSupplementary Material). The energy barrier for this pro-\ncess is \u0001s\u0000Enn+E2n\u00004:48H'4:45\u00004:48H, where\n\u0001s'5:64K is the cost of flipping a spin in spin ice\nandEnn\u0000E2n'1:19K is the Coulomb energy differ-\nence between two monopoles at nearest-neighbour vs second-\nneighbour distance (Dy 2Ti2O7parameters). The new process\nbecomes energetically more favourable than the ordinary Zee-\nman barrier if H&0:5Tesla. Notice that the sign of the Zee-\nman contribution in the new process is reversed: if the fields\nwere increased significantly beyond the threshold, monopole\ndiffusion would once again become a fast process. However,\nthis is forbidden by an intervening first order phase transition5\nwhich already affects field quenches to H > 0:7Tesla.\nThe largest quench field we could study was H= 0:6Tesla.\nAt this value, the two processes above give rise to barriers of\n2:7K and 1:8K, respectively. These values ought to be cor-\nrected for and broadened by quadrupolar terms that are miss-\ning in the monopole picture used in the estimates5. Indeed, the\nlong time decay of the curves in Fig. 3, collapses upon rescal-\n010\n1010\n010\n−1010−510010defect density per tetrahedron\nrescaled time ( t/Exp[ 2.4/ T ])−4−2\n1010\n−4−2FIG. 5. Collapse of the long time decay of the monopole density\n(thick lines) and of the noncontractible monopole density (thin lines)\nafter rescaling the time axis by a factor exp(\u00002:4=T). The Monte\nCarlo simulations are for L= 6and10(left and right panels, respec-\ntively), with H= 0:6Tesla andT= 0:13;0:15;0:18K (red, blue,\nand green curves). The good quality of the collapse indicates that the\nsimulated systems are large enough for the energy scale of 2:4K not\nto exibit appreciable system size depedence.\ning the time axis by a Boltzmann factor exp(\u00002:4=T), as il-\nlustrated in Fig. 5. The corresponding value of the activation\nbarrier, 2:4K, is in reasonable agreement with the estimates\nfor the barrier to isolated monopole motion.\nWe stress that for H&0:5Tesla, the system enters a\nlong-lived Coulomb liquid phase with an enhanced (out-of-\nequilibrium) density of exceedingly slow monopoles. This is\na regime that may be experimentally promising to probe and\ndetect single monopoles, as we discuss farther below.\nIV . EXPERIMENTAL PROBES\nWe have argued that different quench regimes lead to the\nappearance of a variety of non-equilibrium dynamical phe-\nnomena. The next question to ask is how best to access such7\nquantities experimentally. On the one hand, a solid state sys-\ntem like spin ice does not permit direct imaging of the mag-\nnetic state in the bulk, unlike two-dimensional systems such\nas artificial spin ices29. On the other hand, we have access to\na number of well-developed probes; this being a spin system,\nnatural options are magnetisation and NMR.\nA. Magnetisation\nOne of the advantages of using [111] field quenches from\nSatIce to KagIce is the fact that we have direct access to the\nmonopole density \u001aper tetrahedron by measuring the mag-\nnetisationMin the [111] direction (in units of \u0016B=Dy):\nthe equation M(\u001a) = (5\u001a+ 10)=3is extremely accurate for\nH&0:42Tesla, and it remains very useful also at lower field\nstrengths, with deviations due to reversed triangular spins of\nless than 1% down toH\u00180:2Tesla (see Supporting Infor-\nmation), for T.0:6K.\nB. Nuclear magnetic resonance\nA promising approach to measure the monopole density in\nspin ice samples follows from the zero-field NMR measure-\nments pioneered by the Takigawa group26,30. This technique\nuses Dy 2Ti2O7samples with NMR active17O nuclei at the\ncentres of the tetrahedra. The large internal fields induced by\nthe rare earth moments at these locations ( \u00184:5Tesla) result\nin a peak in the NMR signal from the17O ions in zero external\nfield.\nWhen a monopole is present in a tetrahedron, the violation\nof the ice rules results in a sizeable reduction of the field at\nthe centre of a tetrahedron ( \u00183:5Tesla). This in turn ought to\ngive rise to an additional shifted peak in the NMR signal. The\nrelative intensity of the two peaks can be used for instance to\nextract the magnetic monopole density in the sample26.\nOne of the challenging aspects of the zero-field NMR ap-\nproach to detect monopoles is the trade-off between having\na large number of monopoles to enhance the signal and hav-\ning sufficiently long persistence times (the time a monopole\nspends in a given tetrahedron) to reduce the noise in the signal.\nHigh-temperatures cannot be used to increase the monopole\npopulation because they lead to fast magnetic fluctuations. On\nthe other hand, the system cannot be allowed to thermalise at\nlow temperature as this would result in an exponentially small\n– thus undetectable – monopole density.\nThe field quenches discussed in this paper can be used to\nprepare a spin ice sample in a strongly-out-of-equilibrium\nstate that offers great potential to NMR measurements. In-\ndeed, the long-time regime at low temperatures ( T < 0:2K)\nhas a sizeable density \u001a\u00180:01of long-lived, static\nmonopoles (see Fig. 3).\nIn Fig. 6 we show the density of monopoles \u001a(t;\u001c)that\nare present in the system at a given time tand that have re-\nmained in their current tetrahedron for longer than \u001c(persis-\ntence time). Clearly, \u001a\u0011\u001a(t;0)and\u001a(t;\u001c) = 0 ,8t < \u001c .\nmonopole density per tetrahedron\ntime (MC steps)010\n210110010−110\n−210\n−310\n310410510610FIG. 6. Density \u001a(t;\u001c)of monopoles present in the system at a\ngiven timet, whose persistence time in their current tetrahedron is\nlarger than\u001c. This is shown for a system of size L= 6 and a\nquench toH= 0:6Tesla atT= 0:2K. The different colours cor-\nrespond to different choices for \u001cwhile the curves are presented as\na function of t(horizontal axis): \u001c= 0:1;1;10;102;103;104MC\nsteps (red, blue, green, magenta, cyan, and yellow). Note that the\nlargest value of the persistence time \u001c= 104MC steps is com-\nparable with the characteristic time scale of the long-time decay\nexp(2:4=T)\u0018105MC steps, which explains the slight departure\nof the yellow curve from the others. It is clear that it is thus possible\nto realise monopole densities of the order of 1%which persist on the\nscale of minutes!\nFig. 6 confirms that the long time decay of the monopole den-\nsity is indeed due to long persistence times of the residual\n(essentially static) monopoles rather than mobile monopoles\nwhich are somehow unable to annihilate each other.\nC. Alternative experimental protocols\nIn practice, depending on details of the set-up, other proto-\ncols may be easier to implement than sudden quenches to non\nzero field values. For instance, it may be more practical to re-\nmove a field altogether – quenching to SpinIce rather than to\nKagIce – or to lower the field gradually to allow time to dissi-\npate the field energies. We present here only a brief summary\nof the results. Details can be found in the relevant sections of\nthe Supporting Information.\nWe find that zero-field quenches lack the intermediate-time\nregime between the end of the dimer deposition process and\nthe onset of the long time decay controlled by noncontractible\npairs (this intermediate-time regime is evident for instance in\nFig. 3). Once again, one concludes that zero field quenches\ncan be used to prepare spin ice samples in a metastable state\nrich in monopoles forming noncontractible pairs, albeit the\nprocess appears to be about one order of magnitude less effi-\ncient than SatIce to KagIce quenches with respect to the re-\nsulting density.8\nReplacing an instantaneous field quench (on the scale of the\nmicroscopic time \u001c0= 1 MC step\u00181ms) with a constant-\nrate field sweep clearly has a profound effect on the short time\nbehaviour – it corresponds to an annealed dimer deposition\nprocess, which would itself be an interesting topic for further\nstudy. However, shortly after the field reaches its final value,\nMC simulations show that the evolution of the monopole den-\nsity agrees quantitatively with the behaviour following a field\nquench to the same field value.\nTherefore, any experimental limitations to achieve fast vari-\nations of applied magnetic fields affect only the behaviour of\nthe system while the field is varied, whereas the main features\ndiscussed above occurring on a longer time scale remain ex-\nperimentally accessible.\nACKNOWLEDGMENTS\nThis work was supported in part by EPSRC grants\nEP/G049394/1 and EP/K028960/1 (C.C.). This work was\nin part supported by the Helmholtz Virtual Institute “New\nStates of Matter and Their Excitations”. C. Castelnovo ac-\nknowledges hospitality and travel support from the MPIPKSin Dresden. We are grateful to M. Takigawa and K. Kitagawa,\nR. Kremer, S. Zherlytsin and to T. Fennell for discussions on\nexperimental probes of field quenches in spin ice.\nV . CONCLUSIONS\nSpin ice offers a realization of several paradigmatic\nconcepts in non-equilibrium dynamics: dimer absorption,\nCoulombic reaction-diffusion physics and kinetically con-\nstrained slow dynamics. There is an unusually high degree of\ntuneability: the timescale of the elementary dynamical move\nthrough a Zeeman energy barrier; the dimensionality of the\nfinal state (d = 2 KagIce vs d = 3 SpinIce); and the relative\nimportance of dimer desorption compared to Coulomb inter-\nactions. The additional availability of a range of experimental\nprobes thus allows broad and detailed experimental studies.\nWith the advent of incipient thermal ensembles of artificial\nspin ice, where the constituent degrees of freedom can even\nbe imaged individually in real space, there even appears an\nentirely new setting for the study of these phenomena on the\nhorizon31.\n1Stinchcombe R (2001) Stochastic non-equilibrium systems. Adv\nPhys 50: 431-496.\n2Angell C A (1995) Formation of Glasses from Liquids and\nBiopolymers. Science 267: 1924-1935.\n38th Tohwa University International Symposium, Fukuoka, Japan,\nNovember 1998. Slow Dynamics in Complex Systems , eds\nTokuyama M and Oppenheim I (Springer), AIP Conference Pro-\nceedings, V ol. 469 (2004).\n4Fredrickson G H, Andersen H C (1984) Kinetic Ising-model of\nthe glass-transition. Phys. Rev. Lett. 53: 1244-1247.\n5Ritort F, Sollich P (2003) Glassy dynamics of kinetically con-\nstrained models. Adv Phys 52: 219-342.\n6Toussaint D, and Wilczek F (1983) Particle-antiparticle annihila-\ntion in diffusive motion. J Chem Phys 78: 2642-2647.\n7Bramwell S T and Gingras M J P (2001) Spin ice state in frustrated\nmagnetic pyrochlore materials. Science 294: 1495-1501.\n8Castelnovo C, Moessner R, and Sondhi S L (2008) Magnetic\nmonopoles in spin ice. Nature 451: 42-45.\n9Castelnovo C, Moessner R, and Sondhi S L (2012) Spin Ice, Frac-\ntionalization, and Topological Order. Annu Rev Condens Matter\nPhys 3: 35-55.\n10Castelnovo C, Moessner R, and Sondhi S L (2010) Thermal\nquenches in spin ice. Phys. Rev. Lett. 104: 107201-4.\n11Levis D and Cugliandolo L F (2012) Out of equilibrium dynamics\nin the bidimensional spin-ice model. EPL 97: 30002-6; Levis D\nand Cugliandolo L F (2013) Defects dynamics following thermal\nquenches in square spin-ice. Phys. Rev. B 87: 214302-14.\n12Ryzhkin I A (2005) Magnetic Relaxation in Rare-Earth Oxide Py-\nrochlores. JETP 101: 481-486.\n13Jaubert L D C and Holdsworth P C W (2009) Signature of mag-\nnetic monopole and Dirac string dynamics in spin ice. Nat Phys\n5: 258-261.\n14Matsuhira K, Hinatsu Y , Tenya K, and Sakakibara T (2000) Low\ntemperature magnetic properties of frustrated pyrochlore ferro-magnets Ho 2Sn2O7and Ho 2Ti2O7.J Phys: Condens Matter 12:\nL649-656.\n15Snyder J et al. (2004) Low-temperature spin freezing in the\nDy2Ti2O7spin ice. Phys. Rev. B 69: 064414-6.\n16Henley C L (2013) NMR relaxation in spin ice due to diffusing\nemergent monopoles. arXiv:1210.8137 .\n17Matsuhira K, et al. (2011) Spin Dynamics at Very Low Tempera-\nture in Spin Ice Dy 2Ti2O7.J Phys Soc Japan 80: 123711-4\n18Yaraskavitch L R, et al. (2012) Spin dynamics in the frozen\nstate of the dipolar spin ice material Dy 2Ti2O7.Phys. Rev. B 85:\n020410(R)-5.\n19Revell H M, et al. (2012) Evidence of impurity and boundary ef-\nfects on magnetic monopole dynamics in spin ice. Nat Phys 9:\n34-47.\n20Henley C L (2010) The Coulomb phase in frustrated systems. Ann\nRev Condens Matter Phys 1: 179-210.\n21Hiroi Z, Matsuhira K, Takagi S, Tayama T, and Sakakibara\nT (2003) Specific heat of kagome ice in the pyrochlore oxide\nDy2Ti2O7.J Phys Soc Japan 72: 411-418.\n22Sakakibara T, Tayama T, Hiroi Z, Matsuhira K, and Takagi S\n(2003) Observation of a liquid-gas-type transition in the py-\nrochlore spin ice compound Dy 2Ti2O7in a magnetic field. Phys.\nRev. Lett. 90: 207205-4.\n23Melko R G and Gingras M J P (2004) Monte Carlo studies of\nthe dipolar spin ice model. J Phys: Condens Matter 16: R1277-\nR1319.\n24de Leeuw S W, Perram J W, and Smith E R (1980) Simulation of\nelectrostatic systems in periodic boundary conditions. I: Lattice\nsums and simulation results. Proc R Soc London Ser A 373: 27-\n56.\n25Dall J and Sibani P (2001) Faster Monte Carlo simulations at low\ntemperatures. The waiting time method. Computer Phys Comm\n141: 260-267.\n26Sala G, et al. (2012) Magnetic Coulomb Fields of Monopoles in9\nSpin Ice and Their Signatures in the Internal Field Distribution.\nPhys. Rev. Lett. 108: 217203-4.\n27Blunt M O, et al. (2008) Random tiling and topological defects in\na two-dimensional molecular network. Science 322: 1077-1081.\n28Garrahan J P, Stannard A, Blunt M O, and Beton P H (2009)\nMolecular random tilings as glasses. Proc Natl Acad Sci USA 106:\n15209-15213.\n29For reviews, see Heyderman L J and Stamps R L (2013) Artificial\nferroic systems: novel functionality from structure, interactions\nand dynamics. J Phys: Condens Matter 25, 363201; Nisoli C,\nMoessner R, and Schiffer P (2013) Artificial spin ice: Designing\nand imaging magnetic frustration. Rev Mod Phys 85, 1473-1490.\n30Kitagawa K and Takigawa M, poster at the ESF-HFM meeting\n“Topics In the Frustration of Pyrochlore Magnets ” in Abingdon,\nUK (2009).\n31Porro J M, Bedoya-Pinto A, Berger A, and Vavassori P (2013)\nExploring thermally induced states in square artificial spin-ice ar-\nrays. New J Phys 15: 055012-12; Zhang S, et al. (2013) Crystal-\nlites of magnetic charges in artificial spin ice. Nature 500: 553-\n557; Farhan A, et al. (2013) Exploring hyper-cubic energy land-\nscapes in thermally active finite artificial spin-ice systems. Nat\nPhys 9: 375-382; Farhan A, et al. (2013) Direct Observation of\nThermal Relaxation in Artificial Spin Ice. Phys. Rev. Lett. 111:\n057204-5.\n32The energy gain of \u00001:34K for reversing a kagome spin in Sat-\nIce was obtained by extrapolating MC values to infinite system\nsize. It is in reasonable agreement with the value (2\u000b\u00001)Enn\u0000\n2\u0001' \u0000 1:73K that obtains from the effective description in\nterms of magnetic monopoles, where \u000b= 1:64is the Madelung\nconstant of the Zincblende structure (an ionic diamond lattice),\n\u0001'4:35K is the energy cost of a monopole and Enn'3:06K\nis the energy of two neighbouring monopoles (Dy 2Ti2O7param-\neters).10\nSupplementary Information\nVI. DIPOLAR SPIN ICE MODEL\nWe present here a brief summary of the model and proper-\nties of spin ice. For a more thorough review, see for instance\nRef. 1 and Ref. 2.\nSpin ice models and materials are systems where the mag-\nnetic degrees of freedom are localised on a lattice of corner\nsharing tetrahedra (pyrochlore lattice), illustrated in Fig. S1.\nGiven a choice of [111] direction (i.e., one of the major diag-\nFIG. S1. Illustration of pyrochlore lattice of a spin ice system (the\nspins are drawn only in the bottom left tetrahedron for convenience).\nGiven a choice of [111] direction (i.e., one of the major diagonals of\nthe cube), the pyrochlore lattice can be seen as a layered structure of\nalternating triangular (yellow) and kagome (green) planes of spins.\nonals of the cube), the pyrochlore lattice can be seen as a lay-\nered structure of alternating triangular and kagome planes of\nspins. This is particularly important in the present manuscript\nwhere we will study the behaviour of the system in presence of\na magnetic field applied in the [111] direction, which results\nin a crucial difference between the corresponding triangular\nand kagome spins.\nThe largest energy scale in spin ice systems is a single ion\nanisotropy that forces the spins to lie along the axis connect-\ning the centre of a tetrahedron to the corresponding vertex, as\nillustrated in Fig. S2. This energy scale is usually so large\ncompared to interactions as well as the relevant temperature\nand field energy scales that one can model the spins as strictly\neasy-axis (Ising).\nThe spins interact via exchange and dipolar coupling, which\nin these systems happen to be of approximately the same mag-\nFIG. S2. Illustration of a spin ice tetrahedron, demonstrating the 4\ninequivalent local easy axes. The configuration in the left panel cor-\nresponds to one of the lowest energy (2in-2out) states of the Hamil-\ntonian in Eq. (3). The configuration in the right panel corresponds to\na monopole (3in-1out or, in this case, 3out-1in); this is the lowest en-\nergy excitation in the system. Higher energy excitations are obtained\nby arranging all four spins pointing into or out of a tetrahedron.\nnitude at nearest-neighbour distance. The model used in this\npaper is known as the dipolar spin ice model3. If we label\nthe sitesiof the three-dimensional pyrochlore lattice, and the\nlocal easy-axis ^ei, each spin can be represented as a classi-\ncal vector~ \u0016i=\u0016Si^ei, where\u0016is the size of the magnetic\nmoment and Si=\u00061.\nThe thermodynamic properties of spin ice are described\nwith good accuracy by a Hamiltonian energy that encom-\npasses a uniform nearest-neighbour exchange term of strength\nJ\u00181\u00003K and long-range dipolar interactions,\nH=J\n3X\nhijiSiSj+Dr3\nnn\n2X\nij\u0014^ei\u0001^ej\njrijj3\u00003 (^ei\u0001rij) (^ei\u0001rij)\njrijj5\u0015\n(3)\nwherernn\u00183:5˚A is the pyrocholore lattice constant, and\nD=\u00160\u00162=(4\u0019r3\nnn)\u00181:41K.\nDue to a peculiar interplay between interactions, lattice ge-\nometry and local easy-axis anisotropy, it is not possible to\nminimise simultaneously each term in the Hamiltonian and\nthe system is frustrated. To a first approximation, the energy is\nminimised by configurations where each tetrahedron has two\nspins pointing in and two pointing out (2in-2out, Fig. S2)4.\nThese configuration are in one-to-one correspondence with\nproton disorder in (cubic) water ice, hence the name spin ice\nand the fact that the 2in-2out rules are referred to as ice rules .\nThere are many ways to fulfill the ice rule condition on\nthe pyrochlore lattice, which result in an extensive degener-\nacy. The ensamble of these configurations is neither ordered\nnor trivially disordered; it is perhaps best understood as an in-\nstance of classical topological order, whereby the system de-\nvelops an additional (gauge) symmetry at low temperatures2.\nExcitations over these low energy states take the form of\ndefective tetrahedra with one spin pointing in and three out,\nor vice versa (3out-1in, Fig. S2). In recent years it was shown\nthat these defects behave as Coulomb-interacting point like\nmagnetic charges (i.e., a Coulomb liquid) free to move in\nthree dimensions5.\nAt closer inspection, the 2in-2out configurations are not ex-\nactly degenerate (in presence of dipolar interactions). Their\nsplitting is however much smaller than the strength of the in-\nteractionsJ; D , and a conventional ordering transition is not11\nobserved in the system down to a correspondingly small tem-\nperature (only numerical evidence of this transition is avail-\nable to date, though see Ref. 6 for some preliminary experi-\nmental results). Therefore, these systems exhibit a relatively\nbroad temperature range between the cost of a monopole ex-\ncitation,T\u0018J;D, down to the transition temperature Tc\u001c\nJ;D, in which their behaviour is captured at a coarse grained\nlevel by a Coulomb liquid of magnetic charges. This descrip-\ntion has indeed gone a long way in allowing us to understand\nthe behaviour observed experimentally in spin ice materials.\nVII. PHASE DIAGRAM IN A [111] MAGNETIC FIELD\nThe effect of an external magnetic field applied on a spin\nice system along one of the [111] axes is not the same on all\nspin sublattices. As it is evident from Fig. S1 and Fig. S2,\nthe field is aligned with the local easy axis of one spin per\ntetrahedron and canted with respect to the other three spins,\nwhich thus incur a lesser Zeeman coupling. In the alternating\nkagome-triangular layer interpretation of the pyrochlore lat-\ntice, the parallel spins live on the triangular planes, whereas\nthe canted spins live on the kagome planes (see Fig. S1).\nWhen the field energy is stronger than spin-spin interac-\ntions, the lowest energy state where each spin has a positive\ncomponent in the direction of the field has each tetrahedron\neither in a 1in-3out or in a 3in-1out configuration (an example\nof such configuration is shown in the right panel of Fig. S2).\nThis configuration corresponds to a perfect ionic crystal of\nmagnetic charges living at the centres of the tetrahedra (which\nform a bipartite diamond lattice).\nAs the field is reduced, the interactions favour the restora-\ntion of the 2in-2out ice rules. However, since one out of four\nspins is more strongly Zeeman coupled to the field, the ice\nrules will be restored predominantly via re-arrangement of the\ncanted spins. Namely, the spins in the triangular planes re-\nmain essentially fully polarised whereas one every three spins\nin the kagome layer gets reversed to point opposite to the field\ndirection (left panel in Fig. S2).\nThese new configurations that occur at intermediate field\nvalues are not as disordered as generic 2in-2out spin ice, yet\nthey are only partially polarised and they remain extensively\ndegenerate. They are referred to as kagome ice states. These\nstates are, up to thermal fluctuations, devoid of monopole ex-\ncitations.\nIt is interesting to re-interpret the role of the [111] applied\nfield in terms of the Coulomb liquid picture of the excitations\nof the spin ice ground states. From vanishing to intermediate\nfields, the system is able to respond without violating the ice\nrules, and a continuous crossover is observed from spin ice\nto kagome ice. When the field is further increased, it begins\nto compete with the ice rules, as it favours an ionic crystal\nof magnetic charged. Namely, the applied field in the spin\nlanguage translates into a staggered chemical potential in the\nmonopole picture.\nThe phase diagram of a liquid of Coulomb interacting\ncharges as a function of temperature and staggered chemical\npotential is the well-known liquid-gas phase diagram. Thisis indeed a rather distinctive phase diagram, characteristic of\nitinerant systems and long range interactions, which is quite at\nodds (unprecedented!) with the behaviour of a localised spin\nsystem. The experimental observation of such phase diagram\nin spin ice materials was indeed one of the first smoking guns\nthat the theoretical proposal for emergent magnetic monopole\nexcitations in these systems was indeed correct5.\nDetails about this phase diagram can be found in Ref. 5\n(Fig. 4) and it is schematically represented in Fig. 1 in the\nmain text. At low temperatures, the high-field polarised state\n(saturated ice) is separated from the intermediate field kagome\nice state by a first order transition. In the monopole lan-\nguage, this transition connects a low-density (kagome) to a\nhigh-density (saturated) monopole phase. In the spin lan-\nguage, the order parameter describing the transition is the\nmagnetisation, which jumps discontinuously from a low to\na high moment. As discussed in the main text, up to ther-\nmal fluctuations in the triangular spins (which remain essen-\ntially polarised throughout the transition), the monopole den-\nsity and the [111] magnetisation of the system are directly re-\nlated (M= (5\u001a+ 10)=3).\nAt some finite temperature, the line of first order points ter-\nminates at a characteristic critical end point (Coulombic crit-\nicality), and at higher temperatures the transition is replaced\nby a continuous crossover from kagome to saturated ice.\nVIII. FIELD QUENCHES IN NEAREST-NEIGHBOUR\nSPIN ICE\nIn view of the advent of “exchange spin ice” materials,\nwhere the Coulomb interactions between monopoles is rela-\ntively weak7on account of the super exchange between neigh-\nbouring spins dominating over their dipolar interactions, it is\ninstructive to study the long time behaviour of the monopole\ndensity in field quenches in the simpler context of nearest-\nneighbour spin ice. There, the lack of dipolar (or further range\nexchange) interactions implies that the monopoles are trivially\ndeconfined and no metastable bound states can form. For field\nquenches down to zero field, this removes all possible activa-\ntion barriers and the monopole density is controlled by the\nasymptotics of reaction diffusion processes (see Ref.8).\nThe situation is different in field quenches to finite fields,\nwhere there exist Zeeman energy barriers to monopole mo-\ntion which thus affect the long time decay of the mononopole\ndensity. In the discussion hereafter we shall assume for con-\nvenience that the triangular spins in the system are fully po-\nlarized, which is indeed the relevant scenario for the long time\ndecay of the monopole density.\nStraightforward hopping of a monopole is object to an al-\nternating energy landscape due to the Zeeman energy to flip\na kagome spin, 4:48H(see main text and Fig. S3). As this\nenergy scale grows with increasing field, an alternative pro-\ncess becomes energetically preferable: the creation of a new\nmonopole pair neighbouring the existing monopole, with sub-\nsequent annihilation of the latter with its opposite member\nof the pair. The energy scale for this process in nearest-\nneighbour spin ice is 4Je\u000b\nnn\u00004:48H, which decreases with12\nnndE = 4.48 H\ndE = 4 J eff − 4.48 H\nFIG. S3. Schematic representation of monopole motion in a kagome plane, both via ordinary hopping (top) and via pair-assisted hopping\n(bottom). Both processes result in a negative monopole being transferred from a downward-pointing tetrahedron (left) to one of the four\nnearest downward-pointing tetrahedra (right). The same holds for a positive monopole in an upward-pointing tetrahedron. The two processes\nencompass two spins being flipped each, and face different energy barriers dEwith opposite field dependence on the first spin flip. The figure\nshows the value of the barriers in the nearest-neighbour spin ice model. In the main text we discuss how they are modified in presence of\nlong-range dipolar interactions; the field dependence however remains unchanged. The second spin flip that completes the transfer always\ndecreases the energy of the system. The tails of the green arrows indicate the spin flipped in going from one panel to the next. Only the spins\nin the front three tetrahedra are drawn for convenience. The triangular spins remain polarised throughout.\nincreasing field. The two processes are illustrated schemati-\ncally in Fig. S3.\nFor the typical value of Je\u000b\nnn= 1:11K9, the two energy\nscales cross over at Hth'0:5Tesla. ForH <H thwe expect\nthat the long time decay of the monopole density is controlled\nby the energy barrier 4:48H; forH >H thwe expect that the\nrelevant barrier is instead 4Je\u000b\nnn\u00004:48H. Fig. S4 confirms\nthat the corresponding Boltzmann factors lead to a very good\ncollapse of the long time behaviour of the monopole density in\nnearest-neighbour spin ice. Note that the very good collapse\nof the data from nearest-neighbour simulations suggests that\ntheentropic long range Coulomb interactions that are present\neven in nearest-neighbour spin ice do not play a significant\nrole in the long time behaviour of the monopole density at\nthese temperatures.\nThe introduction of dipolar interactions in the system\nchanges the pair-assisted energy scale (even for an isolatedmonopole), due to the Coulomb interaction between the three\ncharges in the intermediate stage. In addition to the en-\nergy cost of creating two new monopoles, we need to take\ninto account two Coulomb interaction terms between opposite\ncharges at nearest-neighbour distance and one term between\nlike charges at second neighbour distance. An estimate of the\nresulting barrier for typical Dy 2Ti2O7parameters is given in\nthe main text. The dependence on the magnetic field by con-\ntrast remains unchanged.\nIX. THE TRIANGULAR SPINS\nMuch of our interpretation of the behaviour of spin ice fol-\nlowing a field quench from SatIce to KagIce in this paper re-\nlies on the triangular spins being almost fully polarised. We\nhave verified this assumption explicitly in our numerical sim-13\n010\n1010\n010\n−1010−510010defect density per tetrahedron\nrescaled time−4−2\n1010\n−2−1\n−1510\nFIG. S4. Density of monopoles for different fields and temperatures\nfrom the numerical simulations of nearest neighbour spin ice with\nJe\u000b\nnn= 1:11K andL= 6. The time axis has been rescaled by a\nfactor exp[\u0000(4:48H)=T], forH= 0:4Tesla (top panel); and by\na factor exp[\u0000(4Je\u000b\nnn\u00004:48H)=T], forH= 0:6Tesla (bottom\npanel). The different colour curves correspond to different values\nof the temperature, T= 0:5;0:3;0:2;0:15;0:1;0:08;0:06K (red,\nblue, green, magenta, cyan, yellow and black). Note the excellent\ncollapse of the final long-time decay over approximately 15 orders\nof magnitude!\nulations (see Fig. 3 in the main text and Fig. S5). Even at\nlow fields, only a small fraction of triangular spins is tem-\nporarily reversed during the out of equilibrium relaxation. In\nall regimes considered in this manuscript (with the excep-\ntion of quenches down to zero field, to be discussed next), to\nwithin 1% the magnetisation tracks the monopole density as\nM(\u001a) = (5\u001a+ 10)=3, in units of\u0016B=Dy: the magnetisation\nis an excellent proxy for the monopole density.\nIn the following, we discuss what can be learnt from the\nbehaviour of the triangular planes alone. First of all, for fields\nH&0:42Tesla, no reversed triangular spin can be present in\nthe system for any length of time ( &1MC step), as it is unsta-\nble to flipping and creating two monopoles: the Zeeman split-\nting of a triangular spin 20\u0016BH=k B'13:44HK exceeds\nthe energy cost of monopole pair creation, \u0001s'5:64K for\nDy2Ti2O7.\nWhenH.0:42Tesla, reversed triangular spins can be\nin a (long-lived) metastable state. Our simulations show that\nthis occurs mostly towards the end of the initial decay and\nthroughout the intermediate decay (see Fig. 3); the densitiesremain negligibly small even at very low fields: less than 1%\nof the triangular spins are reversed at any given time during,\nsay, anH= 0:3Tesla field quench (see Fig. S5).\nFor0:42&H&0:32Tesla, while it costs energy to\nflip a reversed triangular spin and create two monopoles, the\nsystem can then readily lower its energy by moving the two\nmonopoles one step each. In this field range, the Coulomb\ncost to separate the monopoles is offset by the Zeeman energy\ngain so that the final state has lower energy than the starting\none before we flipped the triangular spin. This process in-\ncurs an energy barrier of approximately \u0001s\u000020\u0016BH=k B'\n5:64\u000013:44H'0:3K atH= 0:4Tesla (estimated in the\nCoulomb liquid approximation), which is in reasonable agree-\nment with the value 0:62K obtained by collapsing the long\ntime behaviour in the right panel of Fig. S5.\nNaturally, misaligned triangular spins have another relax-\nation channel, whereby a monopole moves from one kagome\nplane to the next, by flipping the triangular spin without incur-\nring an energy barrier. This process is controlled by the time\nscale for a monopole to come by, set by (among other items)\nthe monopole hopping barrier within a kagome plane (see\nmain text). For H= 0:4Tesla, said barrier is 4:48H'1:8K,\nwhich is clearly less efficient than the former (and indeed in\nworse agreement with the data).\nIt is perhaps more interesting to investigate the triangular\nspin relaxation at lower fields, H.0:32Tesla, yet large\nenough so that the equilibrium density of reversed triangu-\nlar spins is negligibly small at the temperatures of interest\n(H&0:2Tesla).\nIn this case, flipping a triangular spin and separating the two\nresulting monopoles leads to an overall increase in the energy\nof the system unless the monopoles (which live on separate\nkagome planes) eventually meet other opposite monopoles\nand annihilate. This process likely incurs a large barrier due\nto the Coulomb energy that needs to be overcome in order to\nseparate oppositely charged monopoles to large distances.\nIt would be therefore natural to expect that triangular spin\nrelaxation for 0:32&H&0:2Tesla proceeds via stray\nmonopole motion, which faces a rather low barrier to hop-\nping within kagome planes ( 4:48H.1:3K forH.\n0:3Tesla). However, contrary to the field range 0:42&H&\n0:32Tesla, where the triangular spins relax whilst plenty of\nfree monopoles are available in the system, we clearly see in\nFig. 3 that the triangular spin relaxation for 0:32&H&\n0:2Tesla takes place in a regime where essentially all remain-\ning monopoles are frozen into noncontractible pairs! Indeed,\nin order to collapse the long time decay of the triangular spin\ndensity in the left and middle panel of Fig. S5 we ought to in-\nvoke energy barriers of the order of 3\u00004K, which are entirely\ninconsistent with free monopole hopping barriers .1:3K.\nIn this regime, it appears that separating monopole pairs\nto large distances is unavoidable, whether the pair originates\nfrom the reversal of a triangular spin or from an existing non-\ncontractible pair frozen in the system. Estimating the corre-\nsponding energy barrier in this case is a tall order, as it de-\npends on the required separation distance (which is in turn\nrelated to the density of reversed spins and noncontractible\npairs). Moreover, in order to compute the Coulomb energy14\n−16 −14 −12 −10 −8 −6 −210 10 10 10 10 10 10\nrescaled time rescaled time\nrescaled time−410defect density per tetrahedron\ndefect density per tetrahedron\ndefect density per tetrahedron024681012141618x 10−3\n010\n−14 −12 −10 −8 −6 −410 10 10 10 10 10−210010012345678x 10−3\n−210010210410012345678x 10−4\n910\nFIG. S5. Density of reversed triangular spins for different fields and temperatures from the numerical simulations. The Monte Carlo simu-\nlations are for an L= 8 system, with H= 0:2;0:3;0:4Tesla (left, middle and right panel, respectively), T= 0:1;0:2;0:3;0:4;0:5;0:6K\n(red, blue, green, magenta, cyan, and yellow curves). The time axis in each panel has been rescaled by a factor exp[\u0000dE=T ], with\ndE= 4:0;3:2;0:62K, from left to right.\ncost to separate two charges to said distance, one needs to\ntake into account possible screening effects that are expected\nin a Coulomb liquid10.\nAlthough this is beyond the scope of the present\nmanuscript, it is intriguing to speculate that this scenario\nmight in fact allow to directly probe the long range nature\nof the Coulomb interations by measuring the energy scale\nneeded to collapse the long-time triangular spin density de-\ncay. Although less straightforward to measure than the overall\nmagnetisation of the sample, it may be possible to access ex-\nperimentally the magnetisation of the triangular planes alone\nusing neutron scattering measurements in the non-spin-flip\nchannel in the KagIce regime. NMR experiments can also\nin principle pick out the triangular sites, as they are symmetry\ndistinct from the kagome ones in a field.\nHere we limit ourselves to one further observation. If the\nmonopole pair to be separated originates from a triangular\nspin flip, one expects a negative Zeeman energy contribution,\nwhereas if the monopoles come from a noncontractible pair,\nthe Zeeman contribution is positive. In Fig. S5 (left and mid-\ndle panel) we observe a larger barrier for H= 0:2Tesla\n(4:0K) than forH= 0:3Tesla ( 3:2K), which suggests that\nthe former process is the one responsible for the long time\ndecay of the reversed triangular spin density.\nX. ZERO-FIELD QUENCHES TO SPIN ICE\nIn this section we discuss the case of quenches down to\nzero field, driving the system from the SatIce directly to the\nspin ice regime. This is experimentally relevant because the\ncharacterisation of the behaviour of the system following the\nquench can then be done using measurements in zero field,\na scenario that is for instance more suitable to using NMR\ntechniques.\nThe simulation results in Fig. S6 show that the\nintermediate-time regime, where the noncontractible pairs are\nonly a small fraction of the total monopole density, is essen-\ntially absent in quenches from SatIce to spin ice.\ntime (MC step)monopole density per tetrahedron\n−1 1 3 5 710 10 10 10 10−410−310−210−110010\n910FIG. S6. Monopole density (thick lines), density of triangular spins\npointing in the direction of the initial magnetization (thin dot-dashed\nlines), and density of noncontractible pairs (thin solid lines) from\nMonte Carlo simulations for a system of size L= 8, fieldH= 0and\ntemperatures T= 0:1;0:2;0:3;0:4;0:5;0:6K (red, blue, green, ma-\ngenta, cyan, and yellow). All triangular spins are initially polarised\nin the direction of the the applied field and their initial density is 1.\nAs time progresses, some of the triangular spins reverse, as shown\nby the decrease in the density (magnified by a factor of 10 for visual-\nization purposes). The lower the temperature, the larger the density\nof polarised triangular spins that remain ‘frozen in’ in the system.\nThe overall behaviour resembles closely the one following\na thermal quench8. The two protocols share the feature of be-\ning quenches from a monopole-rich phase to a spin ice phase\nwith low equilibrium monopole density. The only difference\nis that in one case the high-monopole-density phase is a fully-\nordered ionic crystal of monopoles while in the other it is a\ndisordered Coulomb liquid.?\nOnce again, the long time behaviour is controlled by non-\ncontractible pairs. However, since the quench is to spin ice\nrather than to kagome ice, the triangular spins are not fully15\npolarised in the final state (indeed, there is no longer a dis-\ntinction between kagome and triangular spins in zero field).\nAs a result, the noncontractible pairs are allowed to decay by\ndirect annihilation of their positive and negative monopoles,\nsay, after they hop around a hexagonal loop on the lattice.\nThis process has been thoroughly described in Ref.8and has\na system-size- independent energy barrier due to the cost of\nseparating two Coulomb interacting monopoles up to third-\nneighbour distance.\nWe verified that the Poissonian decay model proposed in\nRef.8is in qualitative agreement with the long-time behaviour\nof the monopole density following a field quench to zero field.\nHowever, a detailed comparison of thermal and zero field\nquenches would require an extensive campaign of low tem-\nperature MC simulations which we leave for future study.\nXI. FINITE-RATE FIELD RAMPS\nFast changes in the applied magnetic field are experimen-\ntally more manageable than correspondingly fast changes in\ntemperature. For example, they are not limited by the sample\nheat capacity and thermal conductivity/contact issues. More-\nover, spin ice samples are insulators and their inductance is\nnegligible.\nHowever, a sudden change in magnetic field in an experi-\nmental setting designed to keep the sample at sub-Kelvin tem-\nperatures is nonetheless a tall order. Magnetic field sweeps\nwith a superconducting magnet (as in a typical NMR setting)\nusually achieve no more than 0:8Tesla/min. Opening the cir-\ncuit in a solenoid would yield a faster field change, but of\ncourse this could generate large amounts of heat that the re-\nfrigerator is then unable to dispense with quickly enough.\nOther techniques that promise to reach larger sweep rates\ninclude physically moving the sample relative to the magnet\n(a permanent magnet can be used given the relatively small\nfields involved, H.1Tesla); or using magnetic field pulses\ninstead of stepping the magnetic field.\nEither way, rates large enough to approximate a quench\nsudden on the time scale of a single spin flip, \u00181ms, will\nrequire substantial experimental effort. It is therefore impor-\ntant to run field-sweep simulations and compare what part of\nthe out-of-equilibrium phenomenology discussed in this paper\npersists for lower sweep rates.\nIn the left panel of Fig. S7 we compare the monopole\ndensity as a function of time from three different protocols:\n(i) an instantaneous quench from SatIce to H= 0:5Tesla;\n(ii) a constant rate sweep where the field is lowered from\nH= 2 Tesla at time t= 0 down toH= 0:5Tesla at time\nt= 103MC steps (i.e., approximately 1s in real time), and\nit is held constant thereafter; and (iii) a constant rate sweep\nwhere the field reaches its final value H= 0:5Tesla at time\nt= 6 104MC steps (i.e., approximately 1minute in real\ntime). Similar results arise for other values of the final ap-\nplied field. It is clear that the long-time regime remains ac-\ncessible in field sweep measurements. As a matter of fact,the field sweep curves merge with their quench counterpart\npromptly after the field has reached its final value. There-\n010\n510610710defect density per tetrahedron\ntime (MC step)1010\n−3−2\n41010−1\n10−4010\n1010\n−3−210−1\n10−4defect density per tetrahedron\n010210410 10610810−2\nFIG. S7. Top panel: Monopole density as a function of time in sim-\nulations of a system of size L= 6 with initial and final field values\nofH= 2 Tesla andH= 0:5Tesla, respectively, from three sets of\ndata (from left to right): (i) field quench; (ii) field sweep at constant\nrate over 103MC steps; and (iii) field sweep at constant rate over\n6 104MC steps. Different colours correspond to different temper-\natures,T= 0:2;0:3;0:4;0:5K (blue, green, magenta, and cyan).\nBottom panel: Monopole density as a function of time in field sweep\nsimulations from H= 1:0Tesla down to H= 0Tesla at the constant\nrate of 0:8Tesla/min, for T= 0:2;0:1;0:09;0:08;0:07;0:05K (red,\nblue, green, magenta, cyan, and yellow). The vertical dashed line in-\ndicates the time where the field reaches its final value H= 0Tesla.\nfore, sufficiently fast field-sweeps can still be used to prepare\nspin ice in a quasi-static monopole-rich phase (equivalent to\na frozen, over-ionised electrolyte), where direct detection of\nthe monopoles might be within reach of zero-field NMR mea-\nsurements.\nIn the right panel of Fig. S7 we show the behaviour of the\nmonopole density in field sweeps down to zero field (at the\nconstant rate of 0:8Tesla/min, starting from H= 1 Tesla).\nRemarkably, even for such slow field sweeps, it is only a mat-\nter of reaching a sufficiently low temperature ( T.1K) be-\nfore the system enters a long-lived metastable state where the\ndensity of monopoles remains much larger than in thermal\nequilibrium.16\n1Bramwell S T and Gingras M J P (2001) Spin ice state in frustrated\nmagnetic pyrochlore materials. Science 294: 1495-1501.\n2Castelnovo C, Moessner R, and Sondhi S L (2012) Spin Ice, Frac-\ntionalization, and Topological Order. Annu Rev Condens Matter\nPhys 3: 35-55.\n3Siddarthan R, et al. (1999) Ising Pyrochlore Magnets: Low-\nTemperature Properties, “Ice Rules”, and beyond. Phys. Rev. Lett.\n83: 1854-1857.\n4Isakov S V , Moessner R, and Sondhi S L (2005) Why Spin Ice\nObeys the Ice Rules. Phys. Rev. Lett. 95: 217201-4.\n5Castelnovo C, Moessner R, and Sondhi S L (2008) Magnetic\nmonopoles in spin ice. Nature 451: 42-45.6Pomaranski D, et al. (2013) Absence of Pauling’s residual entropy\nin thermally equilibrated Dy 2Ti2O7.Nat Phys 9: 353-356.\n7Kimura K, et al. (2013) Quantum fluctuations in spin-ice-like\nPr2Zr2O7.Nat Comm 4: 1934-6.\n8Castelnovo C, Moessner R, and Sondhi S L (2010) Thermal\nquenches in spin ice. Phys. Rev. Lett. 104: 107201-4.\n9den Hertog B C and Gingras M J P (2000) Dipolar interactions\nand origin of spin ice in ising pyrochlore magnets. Phys Rev Lett\n84, 3430-3433.\n10Castelnvo C, Moessner R, Sondhi S L (2011) Debye-H ¨uckel the-\nory for spin ice at low temperature. Phys. Rev. B 84: 144435-14"    },    {        "title": "1105.4763v1.Low_dimensionality_and_predictability_of_solar_wind_and_global_magnetosphere_during_magnetic_storms.pdf",        "content": "JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,\nLow-dimensionality and predictability of solar wind and global\nmagnetosphere during magnetic storms\nT.\u0014Zivkovi\u0013 c,1and K. Rypdal,1\nAbstract. The storm index SYM-H, the solar wind velocity v, and interplanetary mag-\nnetic \feld Bzshow no signatures of low-dimensional dynamics in quiet periods, but tests\nfor determinism in the time series indicate that SYM-H exhibits a signi\fcant low-dimensional\ncomponent during storm time, suggesting that self-organization takes place during mag-\nnetic storms. Even though our analysis yields no discernible change in determinism dur-\ning magnetic storms for the solar wind parameters, there are signi\fcant enhancement\nof the predictability and exponents measuring persistence. Thus, magnetic storms are\ntypically preceded by an increase in the persistence of the solar wind dynamics, and this\nincrease is also present in the magnetospheric response to the solar wind.\n1. Introduction\nUnder the in\ruence of the solar wind, the magnetosphere\nresides in a complex, non-equilibrium state. The plasma\nparticles have non-Maxwellian velocity distribution, MHD\nturbulence is present everywhere, and intermittent energy\ntransport known as bursty-bulk \rows occurs as well [ An-\ngelopoulos et al. , 1999]. The magnetospheric response to\nparticular solar events constitutes an essential aspect of\nspace weather while the response to solar variability in gen-\neral is often referred to as space climate [Watkins , 2002].\nTheoretical approaches to space climate involve concepts\nand methods from stochastic processes, nonlinear dynamics\nand chaos, turbulence, self-organized criticality, and phase\ntransitions.\nSelf-organization can lead to low-dimensional behavior in\nthe magnetosphere [ Klimas et al. , 1996; Vassiliadis et al. ,\n1990; Sharma et al. , 1993]. However, power-law dependence\nobserved in the Fourier spectra of the auroral electrojet\n(AE) index is a typical signature of high dimensional col-\nored noise indicating multi-scale dynamics of the magneto-\nsphere. In order to reconcile low-dimensional, deterministic\nbehavior with high-dimensionality, Chang [1998] proposed\nthat a high-dimensional system near self-organized critical-\nity (SOC) [ Bak et al. , 1987] can be characterized by a few\nparameters whose evolution is governed by a small number\nof nonlinear equations. Some magnetospheric models, like\nthe one presented in Chapman et al. [1998], are based on the\nSOC-concept. Here a system tunes itself to criticality and\nthe energy transport across scales is mediated by avalanches\nwhich are power-law distributed in size and duration.\nOn the other hand, it was suggested in Sitnov et al.\n[2001] that the substorm dynamics can be described as a\nnon-equilibrium phase transition; i.e. as a system tuned ex-\nternally to criticality. Here, a power-law relation is given,\nwith characteristic exponent close to the input-output crit-\nical exponent in a second-order phase transition. In fact, it\nis claimed in Sharma et al. [2003] that the global features of\nthe magnetosphere correspond to a \frst order phase transi-\ntion whereas multi-scale processes correspond to the second-\norder phase transitions.\nThe existence of metastable states in the magnetosphere,\nwhere intermittent signatures might be due to dynamical\n1Department of Physics and Technology, University of\nTroms\u001c, Norway.\nCopyright 2021 by the American Geophysical Union.\n0148-0227/21/$9.00phase transitions among these states, was suggested by Con-\nsolini and Chang [2001], and forced and/or self-organized\ncriticality (FSOC) induced by the solar wind was introduced\nas a conceptual description of magnetospheric dynamics.\nThe concept of intermittent criticality was suggested by Bal-\nasis et al. [2006] who asserted that during intense magnetic\nstorms the system develops long-range correlations, which\nfurther indicates a transition from a less orderly to a more\norderly state. Here, substorms might be the agents by which\nlonger correlations are established. This concept implies a\ntime-dependent variation in the activity as the critical point\nis approached, in contrast to SOC.\nIn the present paper we investigate determinism and pre-\ndictability of observables characterizing the state of the\nmagnetosphere during geomagnetic storms as well as dur-\ning its quiet condition, but the emphasis is on the evolu-\ntion of these properties over the course of major magnetic\nstorms. The measure of determinism employed here in-\ncreases if the system dynamics is dominated by modes gov-\nerned by low-dimensional dynamics. Hence, the determin-\nism in most cases is a measure of low-dimensionality. For a\nlow-dimensional, chaotic system the predictability measure\nincreases when the largest Lyapunov exponent increases,\nand hence it is really a measure of un-predictability. For\na high-dimensional or stochastic system it is related to the\ndegree of persistence in time series representing the dynam-\nics. High persistence means high predictability.\nOne of the most useful data tools for probing the mag-\nnetosphere during substorm conditions is the AE minute\nindex which is de\fned as the di\u000berence between the AU in-\ndex, which measures the eastward electrojet current in the\nauroral zone, and the AL index, which measures the west-\nward electrojet current, and is usually derived from 12 mag-\nnetometers positioned under the auroral oval [ Davies and\nSugiura , 1966]. The auroral electrojet, however, does not\nrespond strongly to the speci\fc modi\fcations of the mag-\nnetosphere that occur during magnetic storms. A typical\nstorm characteristic, however, is a change in the intensity of\nthe symmetric part of the ring current that encircles Earth\nat altitudes ranging from about 3 to 8 Earth radii, and is\nproportional to the total energy in the drifting particles that\nform this current system [ Gonzalez et al. , 1994]. The indices\nDstand SYM-H indices are both designed for the study of\nstorm dynamics. These indices contain contribution from\nthe magnetopause current, the partial and symmetric ring\ncurrent, the substorm current wedge, the magnetotail cur-\nrents, and induced currents on the Earth's surface. They are\nderived from similar data sources, but SYM-H has the dis-\ntinct advantage of having 1-min time resolution compared\nto the 1-hour time resolution of Dst.Wanliss [2006] has\n1arXiv:1105.4763v1  [physics.space-ph]  24 May 2011X - 2 TATJANA ZIVKOVIC, KRISTOFFER RYPDAL: MAGNETOSPHERE DURING STORMS\nrecommended that the SYM-H index be used as a de facto\nhigh-resolution Dstindex. The analysis of these indices are\ncentral to this study. We particularly focus on SYM-H and\nSYM-H?which is derived from the SYM-H when the con-\ntribution of the magnetopause current is excluded.\nThe typical magnetic storm consists of the initial phase,\nwhen the horizontal magnetic \feld suddenly increases and\nstays elevated for several hours, the main phase where this\ncomponent is depressed for one to several hours, and the re-\ncovery phase which also lasts several hours. The initial phase\nhas been associated with northward directed IMF (little en-\nergy enters the magnetosphere), but it has been discovered\nthat this phase is not essential for the storm to occur Akasofu\n[1965]. In order to de\fne a storm, we follow the approach of\nLoewe and Pr olss [1997], where the Dstminimum is a com-\nmon reference epoch, the main-phase decrease is su\u000eciently\nsteep, and the recovery phase is also de\fned.\n1.1. Data acquisition\nThe SYM-H index data are downloaded from World Data\nCenter, with 1-min resolution. We also use minute data\nfor the interplanetary magnetic \feld (IMF) component Bz,\nminute data for the solar wind bulk velocity valong the\nSun-Earth axis, as well as \row pressure which is given in nT.\nThese data are retrieved from the OMNI satellite database\nand are given in the GSE coordinate system. Gaps of miss-\ning data in Bz,vand \row pressure are linearly interpolated\nfrom the data which are not missing, while SYM-H data are\nanalyzed for the entire period. The same result for the Bz\nandvis obtained when gaps of missing data are excluded\nfrom the analysis.\nData for the period from January 2000 till December 2005\nis used to compute general properties of the magnetosphere.\nIn order to analyze storm conditions all the indices are ana-\nlyzed during ten intense magnetic storms. Analyzed storms\noccurred on 6 April 2000, 15 July 2000, 12 August 2000, 31\nMarch 2001, 21 October 2001, 28 October 2001, 6 November\n2001, 7 September 2002, 29 October 2003, and 20 Novem-\nber 2003. These storms are characterized with Dstminimum\nwhich is in the range between -150 nT to -422 nT.\nThe remainder of the paper is organized as follows: sec-\ntion 2 describes the data analysis methods employed. Sec-\ntion 3 presents analysis results discerning general statistical\nscaling properties of global magnetospheric dynamics using\nminute data over several years and data generated by a nu-\nmerical model which produces realizations of a fractional\nOrnstein-Uhlenbeck (fO-U) process. In particular we study\nhow determinism and predictability of the geomagnetic and\nsolar wind observables change over the course of magnetic\nstorms. Section 4 is reserved for discussion of results and\nsection 5 for conclusions.\n2. Methods\n2.1. Recurrence-plot analysis\nThe recurrence plot is a powerful tool for the visualiza-\ntion of recurrences of phase-space trajectories. It is very\nuseful since it can be applied to non-stationary as well as\nshort time series [ Eckmann et al. , 1987], and this is the na-\nture of data we use to explore magnetic storms. Prior to\nconstructing a recurrence plot the common procedure is to\nreconstruct phase space from the time-series x(t) of length\nNby time-delay embedding [ Takens , 1981].\nSuppose the physical system at hand is a deterministic\ndynamical system describing the evolution of a state vector\nz(t) in a phase space of dimension p, i.e.zevolves according\nto an autonomous system of 1st order ordinary di\u000berential\nequations;\ndz\ndt=f(z);f:Rp!Rp(1)and that an observed time series x(t) is generated by the\nmeasurement function g:Rp!R ,\nx(t) =g(z(t)): (2)\nAssume that the dynamics takes place on an invariant set\n(an attractor)A\u0012Rpin phase space, and that this set has\nbox-counting fractal dimension d. Since the dynamical sys-\ntem uniquely de\fnes the entire phase-space trajectory once\nthe state z(t) at a particular time tis given, we can de\fne\nuniquely an m-dimensional measurement function,\ng:A!Rm;g(z) = (x(t);x(t+\u001c);:::;x (t+ (m\u00001)\u001c)):\n(3)\nwhere the vector components are given by equation (2), and\n\u001cis a time delay of our choice. If the invariant set Ais\ncompact (closed and bounded), gis a smooth function and\nm> 2d, the map given by equation (3) is a topological em-\nbedding (a one-to-one continuous map) between AandRm.\nThe condition m> 2dcan be thought of as a condition for\nthe image g(A) not to intersect itself, i.e. to avoid that two\ndi\u000berent states on the attractor Aare mapped to the same\npoint in the m-dimensional embedding space Rm. If such\nan embedding is achieved, the trajectory x(t) =g(z) (where\ng(z) is given by equation (3)) in the embedding space is a\ncomplete mathematical representation of the dynamics on\n0 500 1000 1500 2000−6−4−20246\ntBza)\n200 600 10001400 1800200600100014001800b)\nt (min)\nt (min)(min)\nFigure 1. Bzduring quiet condition on September 5,\n2001. a)Bztime series, b) Recurrence plot of the time\nseries shown in (a).TATJANA ZIVKOVIC, KRISTOFFER RYPDAL: MAGNETOSPHERE DURING STORMS X - 3\nthe attractor. Note that the dimension pof the original\nphase space is irrelevant for the reconstruction of the em-\nbedding space. The important thing is the dimension dof\nthe invariant setAon which the dynamics unfolds.\nThere are practical constraints on useful choices of the\ntime delay \u001c. If\u001cis much smaller than the autocorrelation\ntime the image of Abecomes essentially one-dimensional. If\n\u001cis much larger than the autocorrelation time, noise may de-\nstroy the deterministic connection between the components\nofx(t), such that our assumption that z(t) determines x(t)\nwill fail in practice. A common choice of \u001chas been the\n\frst minimum of the autocorrelation function, but it has\nbeen shown that better results are achieved by selecting the\ntime delay as the \frst minimum in the average mutual in-\nformation function, which can be percieved as a nonlinear\nautocorrelation function [ Abarbanel , 1996]. Here we use the\naverage mutual information function to calculate the value\nof\u001c.\nThe recurrence-plot analysis deals with the trajectories in\nthe embedding space. If the original time series x(t) hasN\nelements, we have a time series of N\u0000(m\u00001)\u001cvectors x(t)\nfort= 1;2;:::;N\u0000(m\u00001)\u001c. This time series constitutes\nthe trajectory in the reconstructed embedding space.\nThe next step is to construct a [ N\u0000(m\u00001)\u001c]\u0002[(N\u0000(m\u0000\n1)\u001c] matrixRi;jconsisting of elements 0 and 1. The matrix\nelement (i;j) is 1 if the distance is kxi\u0000xjk\u0014\u000fin the\nreconstructed space, and otherwise it is 0. The recurrence\nplot is simply a plot where the points ( i;j) for which the\n0 500 1000 1500 2000−40−30−20−10010203040\ntBz\n200600100014001800b)\n600 200 10001400 1800time (min)a)\ntime (min)\nFigure 2. Bzduring the strong storm on 6 April 2000.\na)Bztime series, b) Recurrence plot of the time series\nshown in (a).\na)\n110\n-10\n5\n-5\n10\n-10\n-10\n-10-1010\n10\n102020\n20-20\n-20\n20\n-2010\n-10-20\ntime (days)h\nh\nh\nh\nh\nh\nh\nh\nh\nh\nR1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n1-300-200-1000\ntime (days)b)Figure 3. a) Intrinsic mode functions obtained by EMD\nforBzfor the magnetic storm on 6 April 2000, b) Dstfor\nthe same event.\ncorresponding matrix element is 1 is marked by a dot. For a\ndeterministic system the radius \u000fis typically chosen as 10%\nof the diameter of the reconstructed attractor, but varies for\ndi\u000berent sets of data. For a non-stationary stochastic pro-\ncess like a Brownian motion there is no bounded attractor\nfor the dynamics, and the diameter is limited by the length\nof the data record. The \frst example of recurrence plot is\nshown in Figure 1, obtained from the Bzwhen no storm is\npresent on 5 September 2001. In Figure 2, the recurrence\nplot is shown for the Bzfor the strong storm on 6 April\n2000. In both cases, embedding dimension is m= 1 and\n\u000f\u00180:4, which corresponds to 10% of the data range.\n2.2. Empirical mode decomposition\nThe empirical mode decomposition (EMD) method, de-\nveloped in Huang et al. [1998] is very useful on non-\nstationary and nonlinear time series. EMD method can give\na change of frequency in any moment of time (instantaneous\nfrequency) and a change of amplitude in the system. How-\never, in order to properly de\fne instantaneous frequency, a\ntime series should have the same number of zero crossings\nand extrema (or they can di\u000ber at most by one), and a lo-\ncal mean should be close to zero. The original time series\nusually does not have these characteristics and should be\ndecomposed into intrinsic mode functions (IF) for which in-\nstantaneous frequency can be de\fned. Decomposition can\nbe obtained through the so-called sifting process. This is an\nadaptive process derived from the data and can be brie\ry\ndescribed as follows: All local maxima and minima in the\ntime series s(t) are found, and all local maxima and min-\nima are \ftted by cubic spline and these \fts de\fne the upper\n(lower) envelope of the time series. Then the mean of theX - 4 TATJANA ZIVKOVIC, KRISTOFFER RYPDAL: MAGNETOSPHERE DURING STORMS\nupper and lower envelope m(t) is de\fned, and the di\u000berence\nbetween the time series and this mean represents the \frst\nIF,h(t) =s(t)\u0000m(t), if instantaneous frequency can be ob-\ntained, de\fned by some stopping criterion. If not, the proce-\ndure is repeated (now starting from h(t) instead of s(t)) until\nthe \frst IF is produced. Higher IFs are obtained by sub-\ntracting the \frst IF from the time series s(t) and the entire\npreviously mentioned procedure is repeated until a residual,\nusually a monotonic function, is left. We use a stopping cri-\nterion de\fned by Rilling et al. [2003], where \u0011(t)< \u0012 1on\n1\u0000\rfraction of the IF, and \u0011(t)< \u0012 2, on the remaining\nfraction of the IF. Here \u0011=m(t)=a(t),a(t) is the IF ampli-\ntude, and\r= 0:05,\u00121= 0:05, and\u00122= 0:5. By the above\nde\fnitions, IFs are complete in the sense that their summa-\ntion gives the original time series: s(t) =PM\n1h(t) +R(t)\nwhereMis the number of IFs and Ris a residual. In Figure\n3a we show the IFs from EMD performed on the IMF Bz\nduring a magnetic storm on 6 April 2000 (whose time series\nis plotted in Figure 2a), while in Figure 3b the Dstindex\nfor the same storm is shown.\nIn order to study stochastic behavior of a time series\nby means of EMD analysis, we refer to Wu and Huang\n[2004] who studied characteristics of white noise using the\nEMD method. They derived for white noise the relation-\nship logEm=\u0000logTm, whereEmandTmrepresents em-\npirical variance and mean period for the m'th IF. Here,\nEm= (1=N)PN\nt=1h(t)2, whereh(t) is the m'th IF and Tm\nis the ratio of the m'th IF length to the number of its zero\ncrossings. Franzke [2009] analyzed telecommunication in-\ndices and noticed a resemblance to autoregressive processes\nof the \frst order AR(1), which are stochastic and linear pro-\ncesses. For such processes log Em=\u0010logTm. For fractional\nGaussian noise processes ( H < 1) and fractional Brown-\nian motions ( H > 1) we have the connection \u0010= 2H\u00002,\nwhereHis the Hurst exponent, as shown by Flandrin and\nGon\u0018 calv \u0012es[2004]. A useful feature of the EMD analysis is\nthe possibility of extraction of trends in the time series [ Wu\net al. , 2007], because the slowest IF components should often\n2000 4000 6000 8000−20−15−10−505101520\ntx(t)a)\nb)\nFigure 4. a) Time series representing one component\nof the numerical solution of the Lorenz system. b) Av-\nerage displacement vectors Vjin each box visited by a\n2-dimensional projection of the m= 3 -dimensional em-\nbedding space reconstructed from the time series in (a).be interpreted as trends. This is an advantage compared to\nthe standard variogram or rescaled-range techniques [ Beran ,\n1994], whose estimation of the scaling exponents is biased\nby the trend.\n2.3. A test for determinism\nIn this paper we employ a simple test for determinism,\ndeveloped by Kaplan and Glass [1992], where the following\nhypothesis is tested: When a system is deterministic, the\norientation of the trajectory (its tangent) is a function of\nthe position in the phase space. Further, this means that\nthe tangent vectors of a trajectory which recurs to the same\nsmall \\box\" in phase space, will have the same directions\nsince these are uniquely determined by the position in phase\nspace. On the other hand, trajectories in a stochastic sys-\ntem have directions which do not depend uniquely on the\nposition and are equally probable in any direction. This\ntest works only for continuous \rows, and is not applicable\nto maps since consecutive points on the orbit may be very\nseparated in the phase space. For \rows, the trajectory orien-\ntation is de\fned by a vector of a unit length, whose direction\nis given by the displacement between the point where tra-\njectory enters the box jto the point where the trajectory\nexits the same box. The displacement in m-dimensional\nembedding space is given from the time-delay embedding\nreconstruction:\n\u0001x(t) = [x(t+b)\u0000x(t);x(t+\u001c+b)\u0000x(t+\u001c);:::;\nx(t+ (m\u00001)\u001c+b)\u0000x(t+ (m\u00001)\u001c)]; (4)\nwherebis the time the trajectory spends inside a box. The\norientation vector for the kth pass through box jis the unit\nvector uk;j= \u0001xk;j(t)=j\u0001xk;j(t)j. The estimated averaged\ndisplacement vector in the box is\nVj=1\nnjnjX\nk=1uk;j; (5)\n2000 4000 6000 8000345678\ntx(t)\nb)a)\nFigure 5. a) Time series representing the numerical\nsolution of the equation for the f-OU process. b) Av-\nerage displacement vectors Vjin each box visited by a\n2-dimensional projection of the m= 8 -dimensional em-\nbedding space reconstructed from the time series in (a).TATJANA ZIVKOVIC, KRISTOFFER RYPDAL: MAGNETOSPHERE DURING STORMS X - 5\nwherenjis the number of passes of the trajectory through\nboxj. If the dynamics is deterministic, the embedding di-\nmension is su\u000eciently high, and in the limit of vanishingly\nsmall box size, the trajectory directions should be aligned\nandVj\u0011jVjj= 1. In the case of \fnite box size, Vjwill not\ndepend very much on the number of passes nj, andVjwill\nconverge to 1 as nj!1 . In contrast, for the trajectory\nof a random process, where the direction of the next step is\ncompletely independent of the past, Vjwill decrease with nj\nasVj\u0018n\u00001=2\nj. In our analysis we will choose the linear box\ndimension equal to the mean distance a phase-space point\nmoves in one time step and set b= 1 time step in equation\n(4).\nIn Figure 4b we show displacement vectors Vjaveraged\nover the passes through the box j, for a three-dimensional\nembedding of the Lorenz attractor, whose time series is\nshown in Figure 4a; in Figure 5b the same is shown for a\nrandom process, in this case a fractional Ornstein-Uhlenbeck\n(fO-U) process. These model systems will be used through-\nout this paper as archetypes of low-dimensional and stochas-\ntic systems, respectively. The Lorenz system has the form\ndx=dt =a(y\u0000x) (6)\ndy=dt =\u0000xz+cx\u0000y\ndz=dt =xy\u0000bz;\nwith standard coe\u000ecient values a= 10,b= 8=3, andc= 28,\nwhich give rise to a chaotic \row. The fO-U process is de-\nscribed by the stochastic equation:\ndSt=\u0015(\u0016\u0000St) +\u001bdW t; (7)\nwheredWtis a fractional Gaussian noise with Hurst expo-\nnentH[Beran , 1994]. The drift ( \u0015and\u0016) and di\u000busion\n(\u001b) parameters are \ftted by the least square regression to\nthe time series of the SYM-H storm index. This will be\nexplained in more detail in section 3.2.\nThe degree of determinism of the dynamics can be as-\nsessed by exploring the dependence of Vjonnj. In practice,\nthis can be done by computation of the averaged displace-\nment vector\nLn\u0011hVjinj=n; (8)\nwhere the average is done over all boxes with same number n\nof trajectory passes. As shown in Kaplan and Glass [1993],\nthe average displacement of npasses in m-dimensional phase\nspace for the Brownian motion is\nRn=1pn(2\nm)1=2\u0000[(m+ 1)=2]\n\u0000(m=2); (9)\nwhere \u0000 is the gamma function. The deviation in hVjibe-\ntween a given time series and the Brownian motion can be\ncharacterized by a single number given by the weighted av-\nerage over all boxes of the quantity,\n\u0003(\u001c)\u00111P\njnjX\njnjhVji2(\u001c)\u0000R2\nnj\n1\u0000R2nj; (10)\nwhere we have explicitly highlighted that the averaged dis-\nplacementhVji(\u001c) of the trajectory in the reconstructed\nphase space depends on the time-lag \u001c. For a completely\ndeterministic signal we have \u0003 = 1, and for a completely\nrandom signal \u0003 = 0.\nAll systems described by the laws of classical (non-\nquantum) physics are deterministic in the sense that they\nare described by equations that have unique solutions if the\n0 10 20 30 4000.20.40.60.81\nnLna)\n0 20 40 60 80 10000.20.40.60.81\nτΛb)Figure 6. a)Ln: square symbols are derived from nu-\nmerical solutions of the Lorenz system, and triangles from\nthese solutions after randomization of phases of Fourier\ncoe\u000ecients. b) \u0003( \u001c): diamonds from Lorenz system, and\ntriangles after randomization of phases.\ninitial state is completely speci\fed. In this sense it seems\nmeaningless to provide tests for determinism. The test de-\nscribed in this section is really a test of low dimensionality .\nThe test is performed by means of a time-delay embedding,\nfor embedding dimension mup to a maximum value M,\nwhereMis limited by practical constraints. High Mre-\nquires longer time series in order to achieve adequate statis-\ntics. A test that fails to characterize the system as determin-\nistic form\u0014Min reality only tells us that the embedding\ndimension is too small, i.e. the number of degrees of free-\ndomdof the system exceeds M=2. Such systems will in the\nfollowing be characterized as random, or stochastic.\nIn Figure 6a, we plot Lnversusnfor a time series gener-\nated as a numerical solution of the Lorenz system. Here we\nusem= 3,b= 1 ,\u001c= 14 and the box size is of the order of\naverage distance a phase-space point moves during one time-\nstep. In the same plot we also show the same characteristic\nfor the surrogate time series generated by randomizing the\nphases of the Fourier coe\u000ecients of the original time series.\nThis procedure does not change the power spectrum or auto-\ncovariance, but destroys correlation between phases due to\nnonlinear dynamics. For low-dimensional, nonlinear systems\nsuch randomization will change Ln, as is demonstrated for\nthe Lorenz system in Figure 6a. We also calculate \u0003 ver-\nsus\u001cfor these time series and plot the results in Figure 6b.\nAgain, \u0003(\u001c) for the original and surrogate time series are\nsigni\fcantly di\u000berent.\nFor the numerically generated fO-U process, where m=\n8,b= 1 and\u001c= 20, we observe in Figure 7 that Lnand\n\u0003(\u001c) for the original and surrogate time series do not di\u000ber,\ndemonstrating that these quantities are insensitive to ran-\ndomization of phases of Fourier coe\u000ecients if the process is\ngenerated by a linear stochastic equation.X - 6 TATJANA ZIVKOVIC, KRISTOFFER RYPDAL: MAGNETOSPHERE DURING STORMS\nOne should pay attention to the nature of the exper-\nimental data used in the test of determinism. For low-\ndimensional data contaminated by low-amplitude noise or\nBrownian motions, the analysis results will depend on the\nbox size, but the problem is solved by choosing it su\u000eciently\nlarge. For a low-dimensional system represented by an at-\ntractor of dimension dthe results may also depend on the\nchoice of embedding dimension m. The estimated determin-\nismLntends to increase with increasing muntil it stabilizes\n0 10 20 30 4000.20.40.60.81\nnLna)\n0 10 20 30 40 5000.050.10.150.20.25\nτΛb)\nFigure 7. a)Ln: square symbols are derived from nu-\nmerical solutions of the fO-U stochastic equation, and tri-\nangles from these solutions after randomization of phases\nof Fourier coe\u000ecients. b) \u0003( \u001c): squares from fO-U equa-\ntion, and triangles after randomization of phases.\n2 4 6 8 100.20.30.40.50.60.70.80.91\nmL3\nlorenz\nFO−U\ntSYM−H\nFigure 8. L3as a function of embedding dimension m\nfor solution of Lorenz equations (triangles), fO-U process\n(squares), and tSYM-H (circles).atLn\u00191 asmapproaches 2 d. For a random signal there\nis no such dependence on embedding dimension, as demon-\nstrated by example in Figure 8. Here we plot the determin-\nismL3(Lnwhenn= 3) versus embedding dimension mfor\nthe Lorenz and fO-U time series. For comparison we also\nplot this for transformed SYM-H, tSYM-H, during magnetic\nstorm times (the transformation and reasons for it are ex-\nplained in section 3). It converges to a value less than 1, and\nfor embedding dimensions higher than for the Lorenz time\nseries. This indicates that this geomagnetic index during\nmagnetic storms exhibit both a random and a deterministic\ncomponent, and that the dimensionality of this component\nis higher than for the Lorenz system.\n2.4. A test for predictability\nIn this subsection we develop an analysis which is based\non the diagonal line structures of the recurrence plot. In our\nstudy we use the average inverse diagonal line length:\n\u0000\u0011hl\u00001i=X\nll\u00001P(l)=X\nlP(l); (11)\nwhereP(l) is a histogram over diagonal lengths:\nP(l) =NX\ni;j=1(1\u0000Ri\u00001;j\u00001)(1\u0000Ri+l;j+l)l\u00001Y\nk=0Ri+k;j+k:\nFor a low-dimensional, chaotic deterministic system (for\nwhich the embedding dimension is su\u000ecient to unfold the\nattractor) \u0000 is an analog to the largest Lyapunov exponent,\nand is a measure of the degree of unpredictability .\nFor stochastic systems, the recurrence plots do not have\nidenti\fable diagonal lines, but rather consists of a pattern\nof dark rectangles of varying size, as observed in Figure 1.\nFor embedding dimension m= 1 such a dark rectangle cor-\nresponds to time intervals I1= (t1;t1+ \u0001t1) on the hori-\nzontal axis and I2= (t2;t2+ \u0001t2) on the vertical axis, for\nwhich the signal x(t) is inside the same \u000f-interval whenever\ntis included in either I1orI2. In this case the length of\nunbroken diagonal lines lis a characteristic measure of the\nlinear size of the corresponding rectangle, and the PDF P(l)\na measure of the distribution of residence times of the tra-\njectory inside \u000f-intervals. For selfsimilar stochastic processes\nsuch as fractional Brownian motions P(l) can be computed\nanalytically, and \u0000 computed as function of the selfsmilar-\nity exponent h. Since the residence time linside an\u000f-box\nincreases as the smoothness of the trajectory increases (in-\ncreasingh), we should \fnd that \u0000( h) is a monotonically\n0 0.5 1 1.5 2\nx 106−20020\ntime (min)δ (SYM−H)a)\n0 0.5 1 1.5 2\nx 106−10010\ntime (min)δ (tSYM−H)b)\nFigure 9. a) Increments for the SYM-H index. b) In-\ncrements for the transformed signal tSYM-H.TATJANA ZIVKOVIC, KRISTOFFER RYPDAL: MAGNETOSPHERE DURING STORMS X - 7\ndecreasing function of h. In section 3.4 we compute \u0000( h)\nnumerically for a synthetically generated fO-U process and\nthus demonstrate this relationship between \u0000 and h. Hence\nboth \u0000 and hcan serve as measures of predictability, but \u0000 is\nmore general, because it is not restricted to selfsimilar pro-\ncesses or processes with stationary increments, and applies\nto low-dimensional chaotic as well as stochastic systems.\n3. Results\nInRypdal and Rypdal [2010] it is shown that the \ructu-\nation amplitude (or more precisely; the one-timestep incre-\nment) \u0001y(t) of the AE index is on the average proportional\nto the instantaneous value y(t) of the index. This gives\nrise to a special kind of intermittency associated with mul-\ntiplicative noises, and leads to a non-stationary time series\nof increments. However, the time series \u0001 y(t)=y(t) is sta-\ntionary, implying that the stochastic process x(t) = logy(t)\nhas stationary increments. Thus, a signal with stationary\nincrements, which still can exhibit a multifractal intermit-\ntency, can be constructed by considering the logarithm of\nthe AE index. Similar properties pertain to the SYM-H\nindex, although in these cases we have to add a constant\nc1before taking the logarithm, i.e. x(t) = log (c1+c2y(t))\nhas stationary increments. Using the procedure described\ninRypdal and Rypdal [2010] the estimated coe\u000ecients are\nc1= 0:7725 andc2= 0:0397. In Figure 9a we show the in-\ncrements for the original SYM-H data, while in Figure 9b we\nshow the increments for the transformed signal x(t), which\nin the following will be denoted tSYM-H.\n3.1. Scaling of storm- and solar wind parameters\nIn this section, we employ EMD and variogram analy-\nsis to tSYM-H, IMF Bzand solar wind \row speed v. The\nEMD analysis is used to compute intrinsic mode functions\n(IF) for time intervals of 50000 minutes using data for the\nentire period from January 2000 till December 2005. The\nempirical variance estimates Eversus mean period Tfor\neach IF component in tSYM-H, Bz, andvare shown as log-\nlog plots in Figure 10a. In section 2.2 we mentioned that\nFlandrin and Gon\u0018 calv \u0012es[2004] has demonstrated that for\nfractional Gaussian noise the slope \u0010is equal to\u0010= 2H\u00002,\nwhereHis the Hurst exponent. This estimate for the slope\nseems valid for our data as well, as is shown in the \fgure\nfrom comparison with the variogram, even though the time\nseries on scales up to 104minutes are non-stationary pro-\ncesses having the character of fractional Brownian motions\n[Rypdal and Rypdal , 2010]. The results from the two dif-\nferent methods shown in Figures 10a and 10b are roughly\nconsistent, using the relations h=H\u00001 and\u0010= 2H\u00002,\nwhich implies 2 h=\u0010. In practice, we have calculated \u0010from\nEMD as a function of 2 hfor fractional Gaussian noises and\nmotions with self-similarity exponent h, and have derived a\nrelation\u0010= 0:94 (2h) + 0:1143.\nThe variogram represent a second order structure func-\ntion:\n\rk=1\n(N\u0000k)N\u0000tX\nn=1(sn+k\u0000sn)2; (12)\nwhich scales with a time-lag kas\rk=k2h,his denoted as\nselfsimilarity exponent, and sis a time series. Note that a\nHurst exponent H > 1 implies that the process is a nonsta-\ntionary motion, and if the process is selfsimilar, the selfsim-\nilarity exponent is h=H\u00001. In our terminology a white\nnoise process has Hurst exponent H= 0:5 and a Brownian\nmotion has H= 1:5.\nFrom Figure 10a we observe three di\u000berent scaling\nregimes for tSYM-H. For time scales less than a few hundred\nminutes it scales like an uncorrelated motion ( h\u00190:5). Ontime scales from a few hours to a week it scales as an an-\ntipersistent motion ( h\u00190:25\u00000:35 depending on analysis\nmethod), and on longer time scales than a week it is close\nto a stationary pink noise ( h\u00190). Similar behavior was ob-\nserved for log AEinRypdal and Rypdal [2010], but there the\nbreak between non-stationary motion and stationary noise\n(wherehchanges from h>0 toh\u00190) occurs already after\nabout 100 minutes, indicating the di\u000berent time scales in-\nvolved in ring current (storm) dynamics and electrojet (sub-\nstorm) dynamics.\nResults for vindicate a regime with antipersistent motion\n(h= 0:25) up to a few hundred minutes, and then an uncor-\nrelated or weakly persistent motion ( h= 0:5) up to a week.\nOn longer time scales than this the variogram indicates that\nthe process is stationary.\nThe exponent hforBzcan not be estimated from the\nvariogram since it is di\u000ecult to obtain a linear \ft to the con-\ncave curve in Figure 10b. The concavity is less pronounced\nin the curve derived from the EMD method in Figure 10a\nand\u0010= 0:47, corresponding to an antipersistent motion\n(h= 0:23), can be estimated on time scales up to a few\nhundred minutes. Bzbecomes stationary already after a\nfew hundred minutes, which is similar to the behavior in\nlogAE, as pointed out by Rypdal and Rypdal [2010]. In Ryp-\ndal and Rypdal [2011] the concavity of the variogram follows\nfrom modelling Bzas a (multifractal) Ornstein-Uhlenbeck\nprocess with a strong damping term that con\fnes the mo-\ntion on time scales longer than 100 minutes. Accounting for\n10010210410610−410−2100102104106\nk (min)γb)100105 10−610−410−2100102104\nTEa)\n10 102 4\n(min)ζ=1.09ζ=0.69ζ=0.25ζ=0.47ζ=0.50ζ=1.09\n2h=0.562h=0.95\n2h=12h=0.52k\nFigure 10. a) The empirical variance estimates Ever-\nsus mean period Tfor each IF component in tSYM-H,\nBz, andvshown as log-log plots. b) The variogram \rk\nshown in log-log plot. In both panels stars are for tSYM-\nH, diamonds for IMF Bz, and triangles for v. Note that\na generalization of the result \u0010= 2H\u00002 in Flandrin and\nGon\u0018 calv \u0012es[2004] yields 2 h=\u0010.X - 8 TATJANA ZIVKOVIC, KRISTOFFER RYPDAL: MAGNETOSPHERE DURING STORMS\nthis con\fnement the \\true\" selfsimilarity exponent of the\nstochastic term turns out to be h\u00190:5. Thus the antiper-\nsistence derived from the EMD analysis may be a spurious\ne\u000bect from this con\fnement. The conclusion in Rypdal and\nRypdal [2011] is that Bzand logAEbehave as uncorrelated\nmotions up to the scales of a few hours and become sta-\ntionary on scales longer than this. Moreover, the stochastic\nterm modelling the two signals share the same multifractal\nspectrum. In comparison, tSYM-H and vare non-stationary\nmotions on scales up to a week before they reach the sta-\ntionary regime.\n3.2. Change of determinism during storm times\nIn Figure 11 we show Lnand \u0003(\u001c) for tSYM-H and its\nsurrogate time-series with randomized phases of Fourier co-\ne\u000ecients. We observe that Lnand \u0003(\u001c) for the surrogate\ntime series does not deviate from those computed from the\noriginal tSYM-H, indicating that the dynamics of tSYM-H\nis not low-dimensional and nonlinear. The same results are\nobtained for IMF Bzand \row speed v(not shown here).\nIn the following analysis we test for determinism in tSYM-\nH,Bzandvfor ten intense storms. The reference point in\nour analysis is the storm's main phase, and then we analyze\nall the data spanning the time interval three days before and\nthree days after the storm in tSYM-H, Bzandv. We com-\nputeLnforn= 6 with a time resolution of 12 hours. The\nchoice ofn= 6 from the Ln-curve is a compromise between\nclear separation between low-dimensional and stochastic dy-\nnamics and small error bars (which increase with increasing\nn). In order to improve statistics for L6, we compute de-\nterminism using data from all ten storms. This means that\n0 10 20 3000.20.40.60.81\nnLn\n0 20 40 60 80 10000.10.20.30.4\nτΛa)\nb)\nFigure 11. a)Ln: square symbols are for tSYM-H, and\ntriangles are for this signal after randomization of phases\nof Fourier coe\u000ecients. b) \u0003( \u001c): squares is for tSYM-H,\nand triangles are after randomization of phases.eachL6is computed over 12 hours interval over 10 storms,\nwhich gives 12\u000160\u000110 = 7200 points. As a reference, we com-\nputeL6for the fO-U process, whose coe\u000ecients are \ftted\nby the least square regression to the SYM-H index during\ninvestigated storms. In all computations, we use embedding\ndimensionm= 7, time-delay \u001c= 20, andb= 1. In Figure\n12a we plot L6for tSYM-H Bz,v, and fO-U, and in Figure\n12b theDstindex averaged over all ten storms is plotted,\nsince this index shows precisely when the storm takes place.\nWe can observe that L6is essentially the same for Bz,vand\nfO-U, and stays approximately constant during the course\nof a storm. However, L6for tSYM-H increases during storm\ntime. In order to demonstrate that the change in L6is sig-\nni\fcant, we plot in Figure 13 Lnfor tSYM-H, where the\ntriangles are the mean of Lncomputed 3 days before and\nafter the storm for ten di\u000berent storms. These curves rep-\nresent non-storm conditions. The upper curve (squares) is\nthe mean over all ten storms computed at the time of the\nDstminimum, i.e. it represents the Ln-curve around storm\nonset.\nIn addition, we test determinism for the quantity SYM-\nH?=0.77 SYM-H-11.9p\nPdyn[Kozyra and Liemohn , 2003],\nwherePdynis the Solar wind's dynamic pressure. SYM-\nH?is a corrected index where the e\u000bect of the magne-\ntopause current due to Pdynis subtracted, and thus rep-\nresents the ring-current contribution to SYM-H. In order\nto obtain stationary increments we analyze a transformed\nindex; tSYM-H?=log(c1+c2SYM-H?), wherec1= 1:7694,\nandc2= 0:0292. Since some data points are missing in\n−3−2−1 01230.350.450.550.65\ndays before/after the stormL6\n−3−2−1 0123−200−150−100−500Dstb)a)\ndays before/after the storm\nFigure 12. a)L6for IMFBz(squares),v(triangles),\ntSYM-H (diamonds), fO-U (stars) computed before, dur-\ning and after storm onset. The values for L6are com-\nputed using 12 hour intervals and are averaged over ten\ndi\u000berent storms. b) The Dstindex averaged over the ten\nstorms.TATJANA ZIVKOVIC, KRISTOFFER RYPDAL: MAGNETOSPHERE DURING STORMS X - 9\nPdyn, we have made a linear interpolation over the missing\npoints. It seems that the interpolation decreases determin-\nism in tSYM-H?, and for reference we compute determinism\nfor the interpolated tSYM-H, where the interpolation is done\nover the same points as in tSYM-H?, even though tSYM-H\ndoes not have missing points. In Figure 14 we plot L6for\ntSYM-H?, tSYM-H, andp\n(Pdyn). We see that the deter-\nminism in tSYM-H?is lower than that in tSYM-H, but it\nstill increases during the storm. On the other handp\n(Pdyn)\nshows no change in determinism during storm events.\n3.3. Discussion of results on determinism\nThe determinism (as measured by L6) of the storm in-\ndex tSYM-H and tSYM-H?has been shown to exhibit a\npronounced increase at storm time. A rather trivial expla-\nnation of this enhancement would be that it is caused by the\n2468101200.20.40.60.81\nnLn\nFigure 13. Lnfor tSYM-H, where the triangles are\nthe mean of Lncomputed 3 days before and after the\nstorm for ten di\u000berent storms. These curves represent\nnon-storm conditions. The upper curve (squares) is the\nmean over all ten storms computed at the time of the Dst\nminimum, i.e. it represents the Ln-curve around storm\nonset. Many curves are terminated for nmax<12 be-\ncause there were no boxes with more than nmaxpassages\nof the phase-space trajectory.\n−2 −1 0 1 20.350.450.550.65\ndays before/after the stormL6\nFigure 14. L6averaged over ten storms for time series\nwhere missing points have been interpolated; tSYM-H\n(diamonds), tSYM-H?(stars),p\n(Pdyn) (triangles)\\trend\" incurred by the wedge-shaped drop and recovery of\nthe storm indices associated with a magnetic storm. We test\nthis hypothesis by superposing such a wedge-shaped pulse\nto an fO-U process and compute L6. Next, we take tSYM-H\nfor ten storms and for each set of data subtract the wedge-\nshaped pulse (computed by a moving-average smoothing).\nThe residual signal represents the \\detrended\" \ructuations.\nThe result is shown in Figure 15 and reveals that the trend\nin fO-U process has no discernible in\ruence on the determin-\nism during the storm while, on the other hand, we observe\nthat the enhancement of L6around storm time persists in\nthe \\detrended\" \ructuations. This result suggests that the\nincreasing determinism during storms is a result of an en-\nhanced low-dimensional component in the storm indices. As\nmentioned in section 2.3 for low-dimensional dynamics, non-\nlinearity may be important for the measure of determinism.\nFor a nonlinear, low-dimensional system the destruction of\nnonlinear coupling by randomizing phases of Fourier coef-\n\fcients will in general reduce the determinism, while for\na linear, stochastic process we will observe no such e\u000bect.\nBut what role will nonlinearity play if it is introduced in the\n−3−2−101230.350.40.450.5\ndays before/after the stormL6\nFigure 15. L6: triangles are derived from an fO-U pro-\ncess with a \\storm trend\" imposed, diamonds are derived\nfrom the \\detrended\" tSYM-H.\n-0.4 0 0.4 0.8 1.2-5-3-113x 10-4\ny\nM (y, \nδt)\nFigure 16. The drift term in the fO-U equation com-\nputed from tSYM-H. The smooth solid curve is a six-\norder polynomial \ft.X - 10 TATJANA ZIVKOVIC, KRISTOFFER RYPDAL: MAGNETOSPHERE DURING STORMS\ndeterministic terms of a stochastic equation? The determin-\nistic term in the fractional Langevin equation representing\nthe fO-U is a linear damping term. However, the best rep-\nresentation of the damping/drift term in an fO-U model for\ntSYM-H is not linear. Following Rypdal and Rypdal [2011],\nify=tSYM-H, the drift term is given as the conditional\nprobability density \u000ey(t;\u000et) given that y(t) =y:\nM(y;\u000et) =E[\u000ey(t;\u000et)jy(t) =y]: (13)\nIn fO-UM(y;\u000et) is a linear function of y, but a polyno-\nmial \ft to drift term derived from tSYM-H data requires a\nsixth order polynomial, con\frming the nonlinearity of the\ntSYM-H process. This is shown in Figure 16. Next, we\ntest determinism for the nonlinear fO-U process, whose scal-\ning exponent his estimated from the variogram of tSYM-\nH, and where m= 8 and\u001c= 10 is used. Figure 17a\nshowsLnfor numerical realizations of this process compared\nwith the same analysis after randomization of the phases of\nthe Fourier coe\u000ecients. The result reveals that the non-\nlinear fO-U process is not more deterministic than its ran-\ndomized version. Next, we form the composite time series\nx=xL+ 1:85xn, wherexLis the solution of the Lorenz\nsystem and xnis the nonlinear fO-U process, both signals\nwith zero mean and unit variance. Again, embedding di-\nmensionm= 8 is used. Now the Ln-curve is lowered when\nthe phases are randomized, as shown in Figure 17b, which\ncon\frms our conjecture that determinism is a measure of\nlow-dimensionality.\n2 4 6 8 100.20.40.60.81a)\nnLn\n2 4 6 8 100.50.60.70.80.91b)\nnLn\nFigure 17. Ln. a) diamonds are derived from numerical\nsolutions of the nonlinear fO-U. Triangles are from these\nsolutions after randomization of phases of Fourier coe\u000e-\ncients. b) diamonds are derived from numerical solutions\nof the nonlinear fO-U with a solution of the xcomponent\nof the Lorenz system superposed. Triangles are from the\nlatter signals after randomization of phases.3.4. Change of predictability during magnetic storms\nEven though we deal with a predominantly stochastic sys-\ntem, its correlation and the degree of predictability changes\nin time, and our hypothesis is that abrupt transitions in the\ndynamics take place during events like magnetic storms and\nsubstorms. We therefore employ recurrence plot quanti\fca-\ntion analysis as a tool for detection of these transitions.\nWe compute the average inverse diagonal line length\n\u0000\u0011hl\u00001ias de\fned in equation (11), but the same results\ncan be drawn from other quantities that can be derived from\nthe recurrence plot [ Marwan et al. , 2007]. \u0000 can be used as a\nproxy for the positive Lyapunov exponent in a system with\nchaotic dynamics, and is sensitive to the transition from reg-\nular to chaotic behavior, as can be shown heuristically for\nthe case of the Lorenz system, where we use a= 10,b= 8=3,\nandcis varied from 20 to 40, such that transient behavior\nis obtained. For c= 24:74 a Hopf bifurcation occurs, which\ncorresponds to the onset of chaotic \rows. In Figure 18a we\nplot a bifurcation diagram for the xcomponent of the Lorenz\nsystem as a function of the parameter c, while in Figure 18b\nwe show \u0000 for the xcomponent again as a function of the\n20 25 30 35 40−50050\ncxa)\n20 25 30 35 4000.51\ncΓb)\nFigure 18. a)xcomponent of the Lorenz system as a\nfunction of the parameter c. b) \u0000\u0011hl\u00001ias a function\nof the parameter c.\n−3−2−1 0120.20.30.40.50.60.7\ndays before/after the stormΓ\nFigure 19. a) \u0000 for tSYM-H (stars), tSYM-H?(circles),\nBz(squares),v(upward triangles), detrended tSYM-H\n(downward triangles) averaged over ten storms. Error\nbars represent standard deviation based on data from\nthese ten storms. Time origin is de\fned by the minimum\nof the average Dstindex for the ten storms.TATJANA ZIVKOVIC, KRISTOFFER RYPDAL: MAGNETOSPHERE DURING STORMS X - 11\nparameter c. Similar results have been obtained from the\nlongest diagonal length, when applied to the logistic map\n[Trulla et al. , 1996].\nIn the following analysis, we use embedding dimension\nm= 1, because the results do not seem dependent on m\nand because, in the case of stochastic or high-dimensional\ndynamics, a topological embedding cannot be achieved for\nany reasonable embedding dimension. This fact demon-\nstrates the robustness of the recurrence-plot analysis, which\nresponds to changes in the dynamics of the system even if\nit is a stochastic or high-dimensional system for which no\nproper phase-space reconstruction is possible.\nSince reduction in \u0000 means increase of predictability it\nmay also be a signature of higher persistence in a stochastic\nsignal. This motivates plotting \u0000 and 2 h(computed as a\nlinear \ft from the variogram over the time scales up to 12\nhours) for solar wind parameters and magnetic indices. Fig-\nure 19 shows \u0000 for tSYM-H, Bz,v, tSYM-H?and detrended\ntSYM-H averaged over 10 magnetic storms. Figure 20 shows\nthe same for 2 h, but detrended tSYM-H is not shown since\nits 2hchanges insigni\fcantly during the course of the storm.\nWe observe that the increase in the predictability and per-\nsistence does not occur simultaneously for all observables.\nWhileBz, tSYM-H, detrended tSYM-H and tSYM-H?get\n−3−2−1 0120.40.60.811.21.4\ndays before/after the storm2h\nFigure 20. a) 2hfor tSYM-H (stars), tSYM-H?(circles),\nBz(squares),v(triangles) averaged over the same storms\nas in the previous \fgure.\n0.20.30.40.5\n0 0.2 0.4 0.6 0.8 1Γ\nh\nFigure 21. \u0000 vs.hcomputed from numerical realizations\nof the fO-U process.the most predictable during or after the main phase of the\nstorm, solar wind's \row speed becomes the most predictable\nprior to the storm's main phase. From a hundred realiza-\ntions of the fO-U process generated numerically with the\ncoe\u000ecients in the stochastic equation \ftted to model the\ntSYM-H signal, we \fnd \u0000 = 0 :4 and 2h= 1, in good agree-\nment with the results obtained from the tSYM-H time se-\nries. The general relationship between \u0000 and hcan also be\nexplored through numerical realizations of fO-U processes.\nFigure 21 shows \u0000 computed for varying has a mean value\nof 100 realizations of such a process for each h. For persis-\ntent motions ( h>0:5) there is a linear dependence between\n\u0000 andh, and a best \ft yields\n\u0000\u00190:72\u00000:57h: (14)\nThis analysis shows the importance of \u0000 as a universal\nmeasure for predictability: in low-dimensional systems it\nis a proxy for the Lyapunov exponent, while for persistent\nstochastic motions it is a measure of persistence through\nequation (14).\n4. Conclusions\nThe storm index SYM-H and the solar wind observables\n(\row velocity vand IMFBz) show no clear signatures of\nlow-dimensional dynamics during quiet periods. However,\nlow-dimensionality increases in SYM-H and SYM-H?during\nstorm times, indicating that self-organization of the mag-\nnetosphere takes place during magnetic storms. This con-\nclusion is drawn from the study of ten intense, magnetic\nstorms in the period from 2000-2003. Even though our\nanalysis shows no discernible change in determinism dur-\ning magnetic storms for solar wind parameters, there is an\nenhancement of the predictability of the solar wind observ-\nables as well as the geomagnetic storm indices during major\nstorms. We interpret this as an increase in the persistence\nof the stochastic components of the signals. The increased\npersistence in the solar wind \row v, prior to the storm's\nmain phase could indicate that vis more important driver\nthanBzduring magnetic storms. This is consistent with\na reexamination of the solar wind-magnetosphere coupling\nfunctions done by Newell et al. [2006], who found that the\nmost optimal function is of the form v2Bsin4(\u0012=2)2=3, where\n\u0012= arctan(By=Bz). Also, it has been shown in Pulkkinen\net al. [2007] through numerical simulation, that increased v\nchanges the magnetospheric response from a steadily con-\nvecting state to highly variable in both space and time.\nIt has been shown in Nose et al. [2001] that the plasma\nsheet is the dominant source for the ring current based on\nthe similarity in composition of the inner plasma sheet and\nring current regions. During the main phase of the storm,\nions from the plasma sheet are \rowing to the inner mag-\nnetosphere on the open drift paths and then move to the\ndayside magnetopause. In this storm phase the ring current\nis highly asymmetric, as was experimentally shown by ener-\ngetic neutral atom imaging (see Kozyra and Liemohn [2003]\nand references therein). During the recovery phase, ions\nfrom the plasma sheet are trapped on closed drift paths,\nand form the symmetric ring current. Therefore, the in-\ncrease in determinism of the ring-current (SYM-H?) during\nstorms implies increased determinism in the plasma sheet as\nwell.\nA magnetic storm is a coherent global phenomenon in-\nvesting a vast region of the inner magnetosphere, and imply-\ning large scale correlation. The counterpart of this increase\nof coherence is the reduction of the spontaneous incoherent\nshort time scale \ructuations. Consequently, one should ex-\npect a reduction of the free degrees of freedom which impliesX - 12 TATJANA ZIVKOVIC, KRISTOFFER RYPDAL: MAGNETOSPHERE DURING STORMS\nan increase of determinism, i.e. the possible emergence of a\nlow-dimensionality.\nAnalysis of predictability shows signi\fcant di\u000berences be-\ntweenBzon one hand, and vand SYM-H on the other.\nWhile the former is a non-stationary, slightly anti-persistent\nmotion up to time scales of approximately 100 minutes, and\na pink noise on longer time scales, the latter are slightly per-\nsistent motions on scales up to several days and noises on\nlonger time scales. These di\u000berences indicate the di\u000berent\nrole the solar wind Bzand the velocity vplay in driving\nthe substorm and storm current systems; Bzis important\nin substorm dynamics which will be studied in a separate\npaper, while vis a major driver of storms.\nAcknowledgments. Recurrence plot and its quantities are\ncomputed by means of the Matlab package downloaded from\nhttp://www.agnld.uni-potsdam.de/~marwan/toolbox/.\nThe authors acknowledge illuminating discussions with M.\nRypdal and B. Kozelov. The authors would like to thank the Ky-\noto World Data Center for Dstand SYM-H index, and CDAweb\nfor allowing access to the plasma and magnetic \feld data of the\nOMNI source. Also, comments of two anonymous referees are\nhighly appreciated.\nReferences\nAbarbanel, H. (1996), Analysis of observed chaotic data, Institute\nfor nonlinear science, Springer, New York.\nAkasofu, S. I. (1965), The development of geomagnetic storms\nwithout a preceding enhacement of the solar plasma pressure,\nPlanet. Space Sci., 13 , 297.\nAngelopoulos, V., T. Mukai, S. Kokobun (1999), Evidence for in-\ntermittency in Earth's plasma sheet and implications for self-\norganized criticality, Phys. Plasmas, 6 , 4161.\nBak, P., C. Tang, and K. Wiesenfeld (1987), Self-organized criti-\ncality: An explanation of 1/f noise, Phys. Rev. Lett., 59 , 381.\nBalasis, G., I. A. Daglis, P. Kapiris, M, Mandea, D. Vassiliadis, K.\nEftaxias (2006), From pre-storm activity to magnetic storms:\na transition described in terms of fractal dynamics, Ann. Geo-\nphys, 24 , 3557.\nBeran, J . (1994), Statistics for long-memory processes, Mono-\ngraphs on statistics and applied probability , Chapman&\nHall/CRC, Boca Raton.\nChang, T. (1998), Self-organized criticality, multi-fractal spectra,\nand intermittent merging of coherent structures in the magne-\ntotail, Astrophysics and space science ,264, 303.\nChapman, S.C., N. Watkins, R. O. Dendy, P. Helander, and G.\nRowlands (1998), A simple avalanche model as an analogue\nfor magnetospheric activity, Geophys. Res. Lett., 25 , 2397.\nConsolini, G., and T. S. Chang (2001), Magnetic \feld topology\nand criticality in geotail dynamics: relevance to substorm phe-\nnomena, Space Sci. Rev., 95 , 309.\nDavies, T. N, and M. Sugiura (1966), Auroral electrojet activity\nindex AE and its universal time variations, J. Geophys. Res.,\n71, 785.\nEckmann, J. P, S. O. Kamphorst and D. Ruelle (1987), Europhys.\nLett. 5 , 973.\nFlandrin, P., and P. Gon\u0018 calv\u0012 es (2004), Empirical mode decom-\nposition as data-driven wavelet- like expansion, International\nJournal of Wavelets, Multiresolution and Information Pro-\ncessing, 2 , 1.\nFranzke, C. (2009), Multi-scale analysis of teleconnection indices:\nclimate noise and nonlinear trend analysis, Nonlin. Processes\nGeophys., 16 , 65.\nGonzalez, W. D., J. A. Joselyn, Y. Kamide, H. W. Kroehl, G.\nRostoker, B. T. Tsurutani, and V. M. Vasyliunas (1994), What\nis a geomagnetic storm?, J. Geophys. Res., 99 , 5771.\nHuang, N.E, Z. Seng, S. R. Long, M. C. Wu, H. H. Shih, Q.\nZheng, N. Yen, C. C. Thung, H. H. Liu (1998), The empirical\nmode decomposition and the Hilbert spectrum for nonlinear\nand non-stationary time series analysis, Proc. R. Soc. Lond. A\n454, 903.\nKaplan, D. T., and L. Glass (1992), Direct test for determinism\nin a time series, Phys. Rev. Lett. ,68, 427.Kaplan, D. T., and L. Glass (1993), Coarse-grained embedding\nof time series: random walks, Gaussian random processes, and\ndeterministic chaos, Physica D ,64, 431.\nKlimas, A., D. Vassiliadis, D. N. Baker, and D. A. Roberts (1996),\nThe organized nonlinear dynamics of the magnetosphere, J.\nGeophys. Res., 101 , 13, 089-13,113.\nLoewe, C. A., and G. W. Pr olss (1997), Clasi\u000ecation and mean\nbehavior of magnetic storms, J. Geophys. Res., 102 , 14,209.\nMarwan, N., M. C. Romano, M. Thiel and J. K} urths (2007),\nRecurrence plots for the analysis of complex systems Physics\nReports 438 , 237.\nNewell, P. T., T. Sotirelis, K. Liou, C. I. Meng, and\nF. J. Rich (2006), Cusp latitude and the optimal solar\nwind coupling function, J. Geophys. Res. ,111, A09207,\ndoi:10.1029/2006JA011731.\nNose, M., S. Ohtani, K. Takahashi, A. T. Y. Lui., R. W. McEntire,\nD. J. Williams, S. P. Christon, and K. Yumoto (2001), Ion com-\nposition of the near-Earth plasma sheet in storm and quiet in-\ntervals: Geotail/ EPIC measurements, J. Geophys. Res. ,106,\n8391.\nPulkkinen, T. I., C. C. Goodrich, and J. G. Lyon (2007), So-\nlar wind electric \feld driving of magnetospheric activity: Is\nit velocity or magnetic \feld, Geophys. Res. Lett. ,34, L21101,\ndoi:10.1029/2007/GL031011.\nRilling, G., P. Flandrin, and P. Goncalves (2003), On empiri-\ncal mode decomposition and its algorithms, IEEE-EURASIP\nWorkshop on Nonlinear signal and image processing NSIP-03,\nGRADO(I).\nRypdal, M., and K. Rypdal (2010), Stochastic modeling of the\nAE index and its relation to \ructuations in Bzof the IMF on\ntime scales shorter than substorm duration, J. Geophys. Res.,\n115, A11216.\nRypdal, M., and K. Rypdal (2011), Discerning a linkage be-\ntween solar wind turbulence and ionospheric dissipation by\na method of con\fned multifractal motions, J. Geophys. Res.,\n116, A02202.\nKozyra, J. U., and M. W. Liemohn (2003), Ring current energy\ninput and decay, Sp. Sci. Rev. ,109, 105.\nSharma, A. S., D. Vassiliadis, and K. Papadopoulos (1993), Re-\nconstruction of low-dimensional magnetospheric dynamics by\nsingular spectrum analysis, Geophys. Res. Lett. ,20, 335.\nSharma, A. S., A. Y. Ukhorskiy, and M. Sitnov (2003), Mod-\neling the magnetosphere using time series data, Geophysical\nMonograph ,142, 231.\nSitnov, M. I., A. S. Sharma, and K. Papadopoulos (2001), Mod-\neling substorm dynamics of the magnetosphere: From self-\norganization and self-organized criticality to nonequlibrium\nphase transitions, Phys. Rev. E, 65 , 016116.\nTakens, F. (1981), Detecting strange attractors in \ruid turbu-\nlence, in: Dynamical Systems and Turbulence , edited by D.\nRand, and L. S. Young, Springer, Berlin.\nTrulla, L. L, A. Guliani, J. P. Zbilut, and C. L. Webber, Jr. (1996),\nRecurrence quanti\fcation analysis of the logisitc equation with\ntransients, Phys. Lett. A 223 , 255.\nVassiliadis, D. V, A. S. Sharma, T. E. Eastman, and K. Pa-\npadopoulos (1990), Low-dimensional chaos in magnetospheric\nactivity from AE time series, Geophys. Res. Lett., 17 , 1841.\nWanliss, J. A., and K. M. Showalter (2006), High-resolution\nglobal storm index: Dst versus SYM-H, J. Geophys. Res.,\n111, A02202.\nWatkins, N. W., (2002), Scaling in the space climatology of the\nauroral indices: is SOC the only possible description, Nonlin.\nProcesses in Geophys., 9 , 389.\nWu, B. Z, and N. E. Huang (2004), A study of the character-\nistics of white noise using the empirical mode decomposition\nmethod, Proc. R. Soc. Lond. A, 460, 1597.\nWu, B. Z, N. E. Huang, S. R. Long, C.K. Peng (2007), On\nthe trend, detrending, and variability of nonlinear and non-\nstationary time series, PNAS, 104, 38.\nT.\u0014Zivkovi\u0013 c, Department of physics and Technology, Univer-\nsity of Troms\u001c, Prestvannveien 40, 9037 Troms\u001c, Norway (tat-\njana.zivkovic@uit.no)\nK. Rypdal, Department of physics and Technology, University\nof Troms\u001c, Prestvannveien 40, 9037 Troms\u001c, Norway\n||||||||||-"    },    {        "title": "1710.09623v1.Dynamical_spin_accumulation_in_large_spin_magnetic_molecules.pdf",        "content": "Dynamical spin accumulation in large-spin magnetic molecules\nAnna P lomi\u0013 nska,1,\u0003Ireneusz Weymann,1,yand Maciej Misiorny2, 1,z\n1Faculty of Physics, Adam Mickiewicz University, 61-614 Pozna\u0013 n, Poland\n2Department of Microtechnology and Nanoscience MC2,\nChalmers University of Technology, SE-412 96 G oteborg, Sweden\n(Dated: November 10, 2021)\nThe frequency-dependent transport through a nano-device containing a large-spin magnetic\nmolecule is studied theoretically in the Kondo regime. Speci\fcally, the e\u000bect of magnetic anisotropy\non dynamical spin accumulation is of primary interest. Such accumulation arises due to \fnite o\u000b-\ndiagonal in spin components of the dynamical conductance. Here, employing the Kubo formalism\nand the numerical renormalization group (NRG) method, we demonstrate that the dynamical trans-\nport properties strongly depend on magnetic con\fguration of the device and intrinsic parameters\nof the molecule. Speci\fcally, the e\u000bect of dynamical spin accumulation is found to be greatly af-\nfected by the type of magnetic anisotropy exhibited by the molecule, and it develops for frequencies\ncorresponding to the Kondo temperature. For the parallel magnetic con\fguration of the device,\nthe presence of dynamical spin accumulation is conditioned by the interplay of ferromagnetic-lead-\ninduced exchange \feld and the Kondo correlations.\nI. INTRODUCTION\nOver the past two decades, nano-devices involving\nindividual spin impurities strongly tunnel-coupled to\nleads have proven to be an excellent test-bed for study-\ning quantum many-body e\u000bects in electronic transport,\namong which the Kondo e\u000bect is one of the most promi-\nnent ones [1{3]. In principle, the role of such a spin\nimpurity can be played by any system that either in-\nherently exhibits spin or is capable to accommodate\na single conduction electron, which has been experi-\nmentally demonstrated for various nanoscopic structures,\nsuch as, quantum dots [4{7], magnetic adatoms [8{12]\nor molecules [13{22]. A proper understanding of the ef-\nfect of charge and spin correlations on electronic trans-\nport is especially sought for devices based on large-\nspin (S >1=2) impurities. Importantly, such systems\nare a suitable platform for applications in emerging tech-\nnologies for storage and processing information [23, 24],\nwhose aim is to utilize magnetic properties of single\natoms [25{27] or molecules [28{31]. The key property\nof a large spin to serve as a base for a memory de-\nvice is the uniaxial magnetic anisotropy that introduces\nan energy barrier for spin reversal [32]. The uniaxial\ncomponent of magnetic anisotropy is, however, often ac-\ncompanied by the transverse one [33] that, allowing for\nthe under-barrier transitions [34], has the parasitic ef-\nfect on the spin stability. In the stationary transport\nregime, the interplay between the Kondo correlations and\nthe magnetic anisotropy of a spin impurity has been pre-\ndicted to signi\fcantly a\u000bect transport characteristics of\na device. Speci\fcally, this interplay leads to a number\nof spectroscopic features ranging from the current sup-\npression due to the spin reversal barrier [35{38] to some\n\u0003anna.plominska@amu.edu.pl\nyweymann@amu.edu.pl\nzmisiorny@amu.edu.plmore intricate, Berry's phase-related e\u000bects originating\nfrom the quantum tunneling of spin [39{45].\nIn the present paper, on the other hand, we address\nthe dynamical aspect of spin-dependent transport\nthrough magnetic molecules in the Kondo regime.\nWhereas this problem has been studied for spin one-half\nimpurities [46{55], it has only recently attracted some\nattention in the context of large-spin impurities [56].\nIn general, by analyzing the dynamical response of a sys-\ntem to an external time-dependent bias one obtains a di-\nrect access to the \ructuations in the system, which is en-\nsured by the \ructuation-dissipation theorem [57] linking\nthe dynamical conductance of the system with its noise\npower spectral density.\nHere, we speci\fcally focus on the in\ruence of mag-\nnetic anisotropy on the e\u000bect of dynamical spin accu-\nmulation , which can be attributed to the non-zero o\u000b-\ndiagonal in spin component of frequency-dependent con-\nductance [48, 50]. The physical meaning of such accu-\nmulation can be better understood if one imagines that\nit actually corresponds to the situation when, for in-\nstance, one injects electrons of given spin orientation\nbut detects the current of opposite spin direction. For\nthis purpose, we consider a magnetic molecule as an ex-\nemplar of a large-spin impurity. Formed by a single\nconducting orbital exchange-coupled to an anisotropic\nmagnetic core, such a model of a molecule captures\ne\u000bects both due to charging and magnetic anisotropy.\nThe dynamical linear-response transport characteristics\nof the system are obtained using a combination of the\nKubo approach [56, 57] and the numerical renormaliza-\ntion group (NRG) method [58, 59].\nWe show that the e\u000bect of dynamical spin accumu-\nlation strongly depends on the intrinsic parameters of\nthe molecule and the magnetic con\fguration of the de-\nvice. For the antiparallel magnetic con\fguration and\nin the case of an easy-axis type of uniaxial magnetic\nanisotropy, the spin accumulation becomes generally sup-\npressed, however, it can be restored if a transversearXiv:1710.09623v1  [cond-mat.mes-hall]  26 Oct 20172\nelectrodeLeft\nelectrodeRight\nGate electrodee\nmagnetic\ncoreorbital\nlevelMagnetic molecule\nxyz\nFigure 1. Schematic depiction of a large-spin magnetic mole-\ncule embedded in a magnetic tunnel junction. For a detailed\ndescription of the model system see Sec. II.\nanisotropy component is also present. This is contrary to\nthe case of an easy-plane type of anisotropy, where a pro-\nnounced dynamical spin accumulation can develop both\nin the absence and presence of transverse anisotropy.\nFor all considered cases, we \fnd that a local maximum\nin the dynamical spin accumulation emerges for energy\nscale corresponding to the Kondo temperature, with the\nheight dependent on the strength of Kondo correlations.\nFurthermore, for the parallel magnetic con\fguration of\nthe device, we demonstrate that the presence and mag-\nnitude of dynamical spin accumulation is conditioned\nby the interplay of ferromagnetic-lead proximity-induced\nquadrupolar exchange \feld and the correlations respon-\nsible for the formation of the Kondo e\u000bect.\nThe paper is organized as follows: In Sec. II an ac-\ncount of basic premises and assumptions of the model\nis provided, while in Sec. III the theoretical framework\nused in calculations of the dynamical conductance is out-\nlined. Numerical results are discussed in Sec. IV, which\nbegins with a detailed review of energy reference scales\nand model parameters (Sec. IV A). The analysis of the re-\nsults we start for an anisotropic molecule, and next, we\ninclude stepwise the uniaxial (Sec. IV B 1) and trans-\nverse (Sec. IV B 2) component of magnetic anisotropy\ninto the picture. The e\u000bects of the spin polarization\nand magnetic con\fguration of electrodes are discussed\nin Sec. IV C and Sec. IV D, respectively. In Sec. IV E\nthe transport behavior in the case of the antiferromag-\nnetic coupling between the molecule's core spin and\nthe orbital level is discussed. Finally, the main conclu-\nsions are presented in Sec. V\nII. MODEL SYSTEM\nIn order to study the e\u000bect of dynamical spin accu-\nmulation in the case of a large-spin impurity, we employ\nhere the model system that consists of a large-spin mag-\nnetic molecule embedded in the magnetic tunnel junc-\ntion, see Fig. 1. Speci\fcally, the magnetic molecule is\nrepresented as a large internal spin ^S(referred to alsoas magnetic core), with S > 1=2, coupled viaexchange\ninteraction Jto a single orbital level (OL). It is as-\nsumed that the molecule is tunnel-coupled to ferromag-\nnetic electrodes of the junctions only through the OL,\nwhich essentially means that transport of electrons across\nthe junction takes place exclusively through this orbital\n[60, 61]. Moreover, spin-dependent electron tunneling\nprocesses between the OL and electrodes lead to broad-\nening of the former, and this broadening is described\nby the spin-dependent hybridization function \u0000q\n\u001bwith\nq=L(eft);R(ight).\nThe full Hamiltonian ^Htotalcharacterizing the system\nunder consideration has the following form\n^Htotal=^HOL+^Hcore+^HOL-core +^Hel+^Htun:(1)\nHere, the \frst three terms are related to the magnetic\nmolecule. In particular, ^HOLaccounts for the key prop-\nerties of the conducting OL and it reads as\n^HOL=\"X\n\u001b^n\u001b+U^n\"^n#; (2)\nwith the \frst term representing the contribution due to\noccupation of the OL by an electron of spin \u001band en-\nergy\", and the second term including the Coulomb inter-\nactionUthat arises in the situation when two electrons\nof opposite spins reside in the OL. The relevant occupa-\ntion operator ^ n\u001b= ^cy\n\u001b^c\u001bis de\fned in terms of electron\ncreation (^cy\n\u001b) and annihilation (^ c\u001b) operators for the OL.\nWe note that application of a voltage to the gate elec-\ntrode allows for tuning the OL energy \". Furthermore,\nthe second term of Hamiltonian (1) describes magnetic\nanisotropy of the molecule's magnetic core within the\ngiant-spin approach [32],\n^Hcore=\u0000D^S2\nz+E\u0000^S2\nx\u0000^S2\ny\u0001\n; (3)\nwithDandEdenoting the uniaxial and transverse mag-\nnetic anisotropy constants, respectively. Finally, the ex-\nchange interaction between the magnetic core e\u000bective\nspin ^Sand the spin of a single electron occupying the or-\nbital ^s= (1=2)P\n\u001b\u001b0^\u001b\u001b\u001b0^cy\n\u001b^c\u001b0, where ^\u001b\u0011(^\u001bx;^\u001by;^\u001bz)\nstands for the Pauli spin operator, is given by\n^HOL-core =\u0000J^s\u0001^S; (4)\nwith theJ-coupling being ferromagnetic (FM) forJ >0\nand antiferromagnetic (AFM) for J <0.\nFerromagnetic electrodes of the junction are approxi-\nmated as reservoirs of non-interacting and spin-polarized\nelectrons and described by the Hamiltonian\n^Hel=X\nq\u001bZW\n\u0000Wd\u000f\u000f^aqy\n\u001b(\u000f)^aq\n\u001b(\u000f); (5)\nwhere ^aqy\n\u001b(\u000f)\u0002\n^aq\n\u001b(\u000f)\u0003\nis the operator responsible for cre-\nation [annihilation] of a spin- \u001belectron in the qth elec-\ntrode, and Wdenotes the conduction band half-width.3\nMoreover, only the case of a collinear relative orientation\nof the spin moments of electrodes, that is, the parallel (P)\nand antiparallel (AP) magnetic con\fguration, is consid-\nered. It is also assumed that the orientation of these\nspin moments is collinear with the principal axis of the\nmolecule corresponding to the uniaxial component of its\nmagnetic anisotropy.\nUltimately, the single electron tunneling processes be-\ntween the OL and electrodes are captured by the last\nterm of Hamiltonian (1),\n^Htun=X\nq\u001br\n\u0000q\n\u001b\n\u0019ZW\n\u0000Wd\u000fh\n^aqy\n\u001b(\u000f)^c\u001b+ ^cy\n\u001b^aq\n\u001b(\u000f)i\n:(6)\nLet us introduce the total broadening \u0000q= \u0000q\n\"+ \u0000q\n#of\nthe OL due to its tunnel-coupling to the qth electrode,\nand de\fne the spin polarization coe\u000ecient for theqth\nelectrode as pq=\u0000\n\u0000q\n\"\u0000\u0000q\n#\u0001\n=\u0000\n\u0000q\n\"+ \u0000q\n#\u0001\n. Next, assuming\nthat both electrodes are made of the same material ( pL=\npR\u0011p) and that the OL is tunnel-coupled symmetrically\nto both electrodes (\u0000L= \u0000R\u0011\u0000), one can parametrize\nthe hybridization functions as follows: \u0000L\n\"(#)= \u0000R\n\"(#)=\n(\u0000=2)(1\u0006p) for the parallel magnetic con\fguration, and\n\u0000L\n\"(#)= \u0000R\n#(\")= (\u0000=2)(1\u0006p) for the antiparallel one.\nIII. DYNAMICAL SYSTEM RESPONSE\nSince the main goal is to analyze the e\u000bect of dy-\nnamical spin accumulation, below we outline a deriva-\ntion of the frequency-dependent conductance (admit-\ntance) in terms of relevant spectral functions. In the\nnext step, these functions will be calculated with the help\nof the Wilson's numerical renormalization group (NRG)\nmethod [58, 59, 62, 63].\nTo begin with, let us assume that an external bias volt-\nageVL(R)(t) modulated periodically in time is applied\nto the ferromagnetic electrodes. To take into account\nthe e\u000bect of such a time-dependent bias, the full Hamil-\ntonian (1) of the system becomes modi\fed by adding a\nnew term [47, 48, 51, 56],\n^Hbias=X\nq\u001b^Qq\n\u001bVq(t); (7)\nwith the operator ^Qq\n\u001bdescribing the spin- \u001bcomponent\nof charge induced in the qth electrode de\fned as\n^Qq\n\u001b=\u0000jejZW\n\u0000Wd\u000f^aqy\n\u001b(\u000f)^aq\n\u001b(\u000f): (8)\nTo calculate the current \rowing through the system of\na large-spin magnetic molecule, we use the Kubo formula\nIq(t)\u0011h^Iq(t)i=X\nq0\u001b\u001b0Z\ndt0Gqq0\n\u001b\u001b0(t\u0000t0)Vq0(t0);(9)whereGqq0\n\u001b\u001b0(t\u0000t0) stands for the time-dependent conduc-\ntance and takes the following form\nGqq0\n\u001b\u001b0(t\u0000t0) =\u0000i\n~\u0012(t\u0000t0)\n[^Iq\n\u001b(t);^Qq0\n\u001b0(t0)]\u000b\n; (10)\nwith the current, ^Iq\n\u001b(t) = d ^Qq\n\u001b(t)=dt, and charge, ^Qq0\n\u001b0(t0),\noperators given in the interaction picture, and h:::i\ndenoting the quantum-statistical average. Next, after\nFourier-transforming Eq. (10) and performing laborious,\nalbeit straightforward calculations, one can \fnd a gen-\neral expression for the frequency-dependent (dynami-\ncal) conductance Gqq0\n\u001b\u001b0(!) |for a detailed derivation see,\ne.g., Ref. [56]. At this point, let us focus on the cur-\nrent response IR(!) in the right electrode, and use that\nthis current is invariant under an overall potential shift\nby\u0000VR(!), which yields\nIR(!) =X\n\u001b\u001b0GRL\n\u001b\u001b0(!)\u0002\nVL(!)\u0000VR(!)\u0003\n: (11)\nThus, taking into consideration only the real part of\ntheright-left component of the dynamical conductance,\nGc(!)\u0011P\n\u001b\u001b0\u0002\nReGRL\n\u001b\u001b0(!)\u0003c;for the parallel ( c= P)\nand antiparallel ( c= AP) magnetic con\fguration of the\njunction one obtains\nGc(!) =X\n\u001b\u001b0Gc\n\u001b\u001b0(!); (12)\nwith the spin-resolved components Gc\n\u001b\u001b0(!) of the form\nGc\n\u001b\u001b0(!) =G0\n2\u000bc\n\u001b\u001a\n\u000e\u001b\u001b0\u0002\ngOL\n\u001b(!)]c\n+1\n2\fc\n\u001b\u001b0\u0002\ngI\n\u001b\u001b0(!)\u0003c\u001b\n:(13)\nThe factors \u000bc\n\u001band\fc\n\u001b\u001b0in the equation above depend\nonly on the magnetic con\fguration and the spin polar-\nization coe\u000ecient pof electrodes,\n\u000bP\n\u001b= 1 +\u0011\u001bpand\u000bAP\n\u001b= 1\u0000p2; (14)\nwith\u0011\"(#)=\u00061,\n\fP\n\u001b\u001b0=r1 +\u0011\u001b0p\n1 +\u0011\u001bpand\fAP\n\u001b\u001b0=1 +\u0011\u001bp\n1 +\u0011\u001b0p: (15)\nFurthermore, in Eq. (13), G0\u00112e2=his the conduc-\ntance quantum, while gOL\n\u001b(!) and gI\n\u001b\u001b0(!) represent two\ndi\u000berent (dimensionless) contributions to the conduc-\ntance [48],\ngOL\n\u001b(!) =1\n2!Z\nd!0AOL\n\u001b(!0)\nA0\u0002\nf(!0\u0000!)\u0000f(!0+!)\u0003\n;(16)\nand\ngI\n\u001b\u001b0(!) =\u00001\n!\u0001AI\n\u001b\u001b0(!)\n~\u001aA0: (17)4\nImportantly, the physical origin of each of these contri-\nbutions can be deduced from analysis of the two spectral\nfunctionsAOL\n\u001b(!) andAI\n\u001b\u001b0(!) occurring in Eqs. (16)\nand (17), respectively. In particular, the former spec-\ntral function describes the orbital level (OL), and it is\nde\fned as\nAOL\n\u001b(!)\u0011\u00001\n\u0019Imhhc\u001bjcy\n\u001biir\n!: (18)\nThe latter, on the other hand, is given by\nAI\n\u001b\u001b0(!)\u0011\u00001\n\u0019Imhh^I\u001bj^Iy\n\u001b0iir\n!; (19)\nand this spectral function is associated with the dimen-\nsionless current operator\n^I\u001b\u0011^cy\n\u001b^\t\u001b\u0000^\ty\n\u001b^c\u001b: (20)\nHere, the \feld operator ^\t\u001bcorresponds essentially to the\neven linear combination of electrode operators,\n^\t\u001b=p\u001aZ\nd\u000fh\n\u0003L\n\u001b^aL\n\u001b(\u000f) + \u0003R\n\u001b^aR\n\u001b(\u000f)i\n(21)\nwith \u0003q\n\u001b=p\n\u0000q\n\u001b=(\u0000L\u001b+ \u0000R\u001b) and\u001a= 1=(2W) being the\ndensity of states of a conduction band. Finally, the scal-\ning factorA0= 1=(\u0019\u0000) in Eqs. (16)-(17) denotes the\nspectral function of a single-level quantum dot (or in\nother words, that of the OL disconnected from the inter-\nnal spin,J= 0) at!= 0 and for nonmagnetic electrodes.\nThe function f(!) in Eq. (16) is the Fermi-Dirac distri-\nbution,f(!) =\b\n1+exp[ ~!=k BT]\t\u00001, withTbeing tem-\nperature and kBstanding for the Boltzmann constant.\nAt this point, we would like to emphasize that one\nof the main quantities of interest in this paper is associ-\nated with the o\u000b-diagonal in spin components of the spin-\nresolved dynamical conductance. In particular, Gc\n\"#(!)\nandGc\n#\"(!) take into account the spin correlations be-\ntween the spin-up and spin-down channels and their \f-\nnite values can be associated with the e\u000bect of dynamical\nspin accumulation that can build up in the molecule at\n\fnite driving frequencies ![48, 50].\nAs one can see from Eqs. (13) and (16)-(17), in or-\nder to calculate the spin-resolved components Gc\n\u001b\u001b0(!)\nof the dynamical conductance, one needs to know \frst\nthe spectral functions: AOL\n\u001b(!), Eq. (18), and AI\n\u001b\u001b0(!),\nEq. (19). In the present work, these functions are de-\nrived using the NRG method [58, 59, 62, 63]. The idea\nof NRG is based on the logarithmic discretization of the\nconduction band with the discretization parameter \u0003. In\nthe next step, such a discretized model is mapped onto\na semi-in\fnite chain with exponentially decaying hop-\npings. Importantly, the \frst site of semi-in\fnite chain\nis coupled to a spin impurity. To obtain the follow-\ning results, we used the discretization parameter \u0003 = 2,\nand we kept Nk= 2560 states during calculations. The\nhigh accuracy of calculations was achieved by averag-\ning the spectral data over Nz= 4 di\u000berent discretizationmeshes [64]. Moreover, when discussing the behavior\nof the conductance in the zero-frequency limit, we will\npresent the data obtained from the full density-matrix\nNRG approach [62, 65] by assuming T=W = 10\u000012, which\nis much smaller than the other energy scales considered\nthroughout this paper.\nIV. NUMERICAL RESULTS AND DISCUSSION\nThe main goal of this paper is to investigate the e\u000bect\nof dynamical spin accumulation in large-spin magnetic\nmolecules, and in particular, to discuss how this e\u000bect is\na\u000bected when the molecular spin is subject to magnetic\nanisotropy. For this purpose, we consider the model of\na magnetic molecule introduced in Sec. II with the mag-\nnetic core characterized by spin S= 2. To conduct a\nsystematic analysis of the problem, and to understand\nhow the dynamical spin accumulation manifests itself in\nfrequency-dependent transport characteristics, \frst we\nwill address the simplest example of a spin-isotropic\nmolecule (D= 0 andE= 0). Next, we will discuss in\nSec. IV B 1 how dynamical spin accumulation changes\nwhen the uniaxial component of magnetic anisotropy be-\ncomes gradually involved ( D6= 0 andE= 0). Finally,\nalso the transverse component ( D6= 0 andE6= 0) will\nbe included in Sec. IV B 2 to establish the complete pic-\nture of the problem.\nA. Energy reference scales and model parameters\nIn the regime of strong tunnel coupling of a molecule\nto electrodes, which is of key interest here, transport\nproperties of the system are determined by strong charge\nand spin correlations. As a result, one can generally ex-\npect the Kondo e\u000bect to play a dominant role as soon as\nthe OL is occupied by a single electron and temperature\nis lower than some characteristic energy scale, referred to\nas the Kondo temperature TK[2]. In general, TKcan de-\npend on di\u000berent parameters of a system under consider-\nation. For instance, in the case of a single-level quantum\ndot (the Anderson impurity) attached to ferromagnetic\nelectrodesTK, in the vicinity of the particle-hole symme-\ntry point (\"=U\u0019\u00000:5), is determined by the Coulomb\ninteraction U, the broadening of the level \u0000 and the spin\npolarization of electrodes p[66, 67]. On the other hand,\nin large-spin systems the Kondo temperature can be ad-\nditionally a\u000bected by other parameters of the model, such\nas, the spin length Sor magnetic anisotropy constants D\nandE[35, 39, 40, 44]. Thus, in the following calculations\nwe use the Kondo temperature TKfor a bare OL ( J= 0)\ntunnel-coupled to nonmagnetic electrodes ( p= 0), given\nin energy units ( kB\u00111), as a consistent energy reference\nscale insensitive to magnetic properties of the molecule.\nSuch a choice of TKcorresponds in fact to the Kondo\ntemperature of a single-level quantum dot, and hence-\nforth we will refer to this temperature as T0\nK. Speci\fcally,5\nthe temperature dependence of zero-frequency normal-\nized linear conductance G(!= 0;T)=G(!= 0;T= 0)\nat the particle-hole symmetry point ( \"=U=\u00000:5) is\nused for estimation of T0\nKfrom the following condi-\ntionG(!= 0;T=T0\nK)=G(!= 0;T= 0) = 1=2.\nIn this paper, all the results are obtained for T= 0.\nWith regard to the magnetic con\fguration of the junc-\ntion, the main discussion is carried out for the case of the\nantiparallel orientation of spin moments in electrodes. In\nsuch a con\fguration the e\u000bective spintronic dipolar [67]\nand quadrupolar [68] exchange \felds are generally absent,\nwhich allows us to analyze how the dynamical spin accu-\nmulation is a\u000bected exclusively by the intrinsic molecular\nmagnetic anisotropy. Later on, we will also include the\nspintronic contribution to magnetic anisotropy by switch-\ning the junction into the parallel magnetic con\fguration.\nNote that throughout the paper a molecule is assumed\nto be electrostatically tuned viaa gate electrode to the\nparticle-hole symmetric point \"=U=\u00000:5, so that in the\nparallel magnetic con\fguration the dipolar exchange \feld\ndoes not arise [37]. In this way, we can consistently ex-\nclude any e\u000bects stemming from the presence of a mag-\nnetic \feld, either real or e\u000bective, which are not the sub-\nject of the present analysis.\nMoreover, the Coulomb energy is chosen U=W = 0:4,\nwith the half-width of conduction band Wserving here\nas the energy unit (that is, W\u00111), whereas the mag-\nnitude of the constant Jdescribing the exchange cou-\npling between the OL and the electrodes is taken to be\njJj=W= 0:0045. Depending on whether Jispositive or\nnegative , one can expect either the underscreened [69{72]\nortwo-stage [73{75] Kondo e\u000bect, respectively, to arise\nin the system [76]. Here, we focus on the case of J >0,\nthat is, on the ferromagnetic type of the J-coupling, and\nthe key di\u000berences occurring for the antiferromagnetic\ncoupling (J < 0) will be addressed only at the end, in\nSec. IV E. Finally, the broadening of the OL due to tun-\nneling of electrons to/from external electrodes is assumed\nto be \u0000=U= 0:1, while the spin polarization of electrodes\nisp= 0:5, unless stated otherwise. Consequently, for the\nparameters assumed above one \fnds the reference Kondo\ntemperature to be T0\nK=W= 0:002.\nB. The e\u000bect of magnetic anisotropy on the\ndynamical spin accumulation\nBefore we proceed to a discussion of how the dy-\nnamical spin accumulation is a\u000bected by the presence\nof magnetic anisotropy, it may be instructive to focus\n\frst brie\ry on frequency-dependence transport features\nof a spin-isotropic molecule1. This case is illustrated\n1Note that in order to enable qualitative comparison of the present\nresults with the previous studies for a single-level quantum dot\n(QD) [47, 50, 51], in Fig. 2 dotted lines representing the latterwith the solid line in Fig. 2, where the dynamical con-\nductanceGAP(!) for the antiparallel magnetic con\fg-\nuration of the junction is shown as a function of fre-\nquency!. SinceGAP(!) can be in general resolved into\nfour spin components GAP\n\u001b\u001b0(!), see Eq. (12), in Fig. 2\napart from the total conductance GAP(!) [plotted in\n(a,e)] we also present the spin-diagonal GAP\n\u001b\u001b(!) [in (b,f)]\nand o\u000b-diagonal GAP\n\u001b\u001b(!) [in (c,g)] contributions, with the\nnotation\u001bto be read as\"\u0011# and#\u0011\" . In particular,\nhere only the components for \u001b=\"are shown, because\nin the antiparallel magnetic con\fguration the following\nsymmetries hold:2\nGAP\n\"\"(!)=GAP\n##(!) andGAP\n#\"(!)=(1\u0000p)2\n(1+p)2GAP\n\"#(!):\nMoreover, since the dynamical spin accumulation essen-\ntially leads to enhancement of imbalance in the number\nof electrons with opposite spin orientations transferred\nacross the junction viaa molecule, we introduce here the\nfrequency-dependent parameter P(!) characterizing the\nspin polarization of the current injected into a drain elec-\ntrode de\fned as:\nP(!)\u0011I\"(!)\u0000I#(!)\nI\"(!) +I#(!): (22)\nIn the situation under consideration, the role of a drain\nis played by the right electrode, I\u001b(!)\u0011IR\n\u001b(!), Eq. (11),\nandI\u001b(!)/G\u001b\u001b(!) +G\u001b\u001b(!), so that\nP(!) =P0(!) +Pdsa(!): (23)\nImportantly, the current spin polarization parameter\nP(!) consists of two terms: the \frst representing the\ndiagonal in spin contribution to the conductance,\nP0(!) =G\"\"(!)\u0000G##(!)P\n\u001b\u001b0G\u001b\u001b0(!); (24)\nand the second arising exclusively due to the dynamical\nspin accumulation,\nPdsa(!) =G\"#(!)\u0000G#\"(!)P\n\u001b\u001b0G\u001b\u001b0(!); (25)\ncase have been added. However, in the main text we do not\ndiscuss this case.\n2The origin of the two symmetries can be explained by noting that\nthe tunnel-coupling of a molecule to two electrodes, Eq. (6), can\nbe e\u000bectively reduced to a single channel problem by applying\nan appropriate unitary transformation [77]. Interestingly, in the\ncase of antiparallel magnetic con\fguration of the junction the\nnew e\u000bective spin-dependent tunnel coupling becomes actually\nindependent of the spin polarization pof electrodes. As a re-\nsult, calculations of the spectral functions AOL\n\u001b(!), Eq. (18), and\nAI\n\u001b\u001b0(!), Eq. (19), proceed as if electrodes were nonmagnetic , so\nthat eventually the values of these functions do not depend on\nspin indices.6\n00.20.40.60.8\n00.10.20.30.4\n00.10.20.3\n00.10.20.30.40.50.6\n10\u0000610\u0000410\u0000210010\u0000610\u0000410\u00002100\nConductance GAP(!)=G0\nQD\njDj=T0\nK= 0\njDj=T0\nK= 10\u00004\njDj=T0\nK= 10\u00002\njDj=T0\nK= 0.5 00.20.40.60.8 (a)Uniaxial magnetic anisotropy:\neasy-axis type ( D>0)\n(e)easy-plane type ( D<0)GAP\n\"\"(!)=G0\n00.10.20.30.4\n(b) (f)GAP\n\"#(!)=G0\n00.10.20.3(c) (g)Spin polarization PAP(!)\nFrequency !=T0\nK00.10.20.30.40.50.6\n10\u0000610\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000610\u0000410\u00002100(h)\nFigure 2. The e\u000bect of the uniaxial component of\nmagnetic anisotropy ( D6= 0 and E= 0) on the frequency-\ndependent conductance of a large-spin magnetic molecule\nshown for the FM J-coupling and the junction in the antipar-\nallel (AP) magnetic con\fguration. Left [right ]column corre-\nsponds to the molecule exhibiting the easy-axis (D > 0) [easy-\nplane (D < 0)] type of magnetic anisotropy. Bottom pan-\nels(d,h) present the current spin polarization PAP(!) which\narises here solely due to the dynamical spin accumulation,\ni.e.,PAP(!) =PAP\ndsa(!), Eq. (25). The solid lines represent\nthe case of a spin-isotropic molecule, while the thin dotted\nlines are for a single-level quantum dot (QD), i.e., for J= 0.\nThe vertical dashed lines indicate the corresponding excita-\ntion energies, for details see the main text. The scaling fac-\ntorG0stands for the conductance quantum. For a discussion\nof parameters assumed in calculations see Sec. IV A.\nwhich vanishes in the limit of !!0 [50]. Recall\nthat the spin-resolved components G\u001b\u001b0(!) of conduc-\ntance are given by Eq. (13). One can, thus, imme-\ndiately conclude that in the antiparallel magnetic con-\n\fgurationPAP\n0(!) = 0, and consequently, no spin po-\nlarization of the current occurs in the stationary case,PAP(!= 0) = 0, whereas for \fnite-frequency transport\nPAP(!) =PAP\ndsa(!) |namely, the spin polarization of\nthe current is here a purely dynamical e\u000bect.\nFirst of all, we recall that the conductance of a\nspin-isotropic system in the zero-frequency limit ap-\nproaches lim !!0GAP(!) = (1\u0000p2)G0, andGAP(!= 0)\nconsists solely of the diagonal-in-spin components,\nthat is,GAP(!= 0) =P\n\u001bGAP\n\u001b\u001b(!= 0), with the o\u000b-\ndiagonal components being identically equal to zero,\nGAP\n\u001b\u001b(!= 0) = 0. As the driving frequency !gets\nlarger, one observes a monotonic decrease in GAP(!), see\nFig. 2(a). However, a closer examination of spin compo-\nnents of the conductance reveals that whereas GAP\n\"\"(!)\ndecreases in value as well, GAP\n\"#(!) actually follows the\nopposite trend and exhibit a maximum. The broad max-\nimum inGAP\n\"#(!) appears approximately at the frequency\ncorresponding to the Kondo temperature TK[56]. As one\nmay notice, here TK\u001cT0\nK, which stems from the fact\nthatTKbecomes suppressed with the increase of J[76].\nSuch a \fnite-frequency feature in the o\u000b-diagonal in spin\ncomponents of GAP(!) is a hallmark of the dynamical\nspin accumulation occurring in the system. Moreover,\nunlikeGAP\n\"#(!), the spin polarization PAP(!) of the cur-\nrent increases monotonically with !until!\u0019T0\nK, where\na local maximum occurs, see Fig. 2(d).\nOn the other hand, in the limit of large frequen-\ncies,!&T0\nK, one can already point out that the ef-\nfect of magnetic anisotropy is expected to be negligi-\nble (ifjDj<T0\nK). This stems from the fact that by\nfurther raising frequency !, one in fact increases the\namount of energy pumped into the system. As a re-\nsult, excitations between states belonging to the two\nspin multiplets S+ 1=2 andS\u00001=2, arising due to\ntheJ-coupling, take place, and they are observed as\na small resonance in GAP(!) at frequency !\u0019J, indi-\ncated by the arrow in Fig. 2(a). Interestingly, such res-\nonant transitions enhance the diagonal-in-spin conduc-\ntance, Fig. 2(b), whereas they have a detrimental e\u000bect\non the dynamical spin accumulation, leading to a local\nminimum in GAP\n\"#(!), Fig. 2(c), and consequently, also\nin the current spin polarization PAP(!), Fig. 2(d). Fi-\nnally, with the further increase of !one expects that\nexcitations related to the Coulomb interaction should be-\ncome the key factor determining the behavior of conduc-\ntance [51, 56]. Since the purpose of the present work is to\nstudy the e\u000bects due to magnetic anisotropy, which take\nplace at frequencies corresponding to energy scales set by\nmagnetic anisotropy constants DandE, Eq. (3), such a\nlarge-frequency regime ( !\u0019U) is not relevant and, thus,\nit will not be considered here.\n1. Uniaxial magnetic anisotropy\nThe situation changes when the molecular spin starts\nenergetically preferring some speci\fc spatial orienta-\ntion(s), which essentially means that the spin is sub-7\nject to magnetic anisotropy described generally by the\nHamiltonian (3). Let us \frst analyze the case when such\na preferable orientation of the spin is related to a spe-\nci\fc axis, customarily associated with the zquantiza-\ntion axis, see Fig. 1, and represented by the term \u0000D^S2\nz\nin Eq. (3). One can immediately see that depending\non the sign of the uniaxial magnetic anisotropy con-\nstantD, the energy of the system becomes minimized if\nthe molecular spin is oriented either along (for D> 0)\nor perpendicular to (for D< 0) thezaxis. The for-\nmer case is often referred to as the `easy-axis' type of\nmagnetic anisotropy, while the latter one as the `easy-\nplane' type. For systems characterized by a half-integer\nlarge spin ( S >1=2), as the one considered here, it is a\nknown, experimentally observed fact that their Kondo-\ndominated zero-frequency linear current response re-\nmains una\u000bected by the uniaxial magnetic anisotropy\nifD< 0 [10, 11, 78, 79], whereas for D> 0 the trans-\nport becomes suppressed [18, 19, 22, 80].\nThe dynamical conductance for the case of uniaxial\nmagnetic anisotropy of the easy-axis type ( D> 0) is pre-\nsented in the left column of Fig. 2. Several distinctive\nfeatures in GAP(!) that evolve with the increase of D\ncan be immediately spotted in Fig. 2(a). To begin with,\nthe zero-frequency conductance GAP(!= 0) becomes re-\nduced and GAP(!) remains frequency-independent as !\ngrows. Once the frequency reaches !\u0019D, a small\npeak forms, and for even larger values of !the conduc-\ntance gets diminished. Comparing the diagonal-in-spin\ncomponent GAP\n\"\"(!) in Fig. 2(b) with the o\u000b-diagonal\noneGAP\n\"#(!) in Fig. 2(c), one can conclude that the en-\nhancement of the conductance at !\u0019Dcan be fully at-\ntributed to the e\u000bect of the dynamical spin accumulation.\nImportantly, it can be noticed that, unlike in the spin-\nisotropic case where the spin accumulation, PAP(!)6= 0,\npersists over a wide range of frequencies, now the e\u000bect\narises only above some threshold frequency !\u0003. More-\nover, for!&!\u0003the diagonal component GAP\n\"\"(!) de-\ncreases monotonically (until !\u0019T0\nK), whereas GAP\n\"#(!)\n\frst builds up rapidly and then changes only insigni\f-\ncantly up to the limit of !&T0\nK. As a result, the cur-\nrent spin polarization PAP(!), similarly as in the spin-\nisotropic case, increases steadily for larger and lager !,\nthough the achievable values of PAP(!) are appreciably\nsmaller than for a spin-isotropic molecule.\nThe presence of the threshold frequency !\u0003above\nwhich the dynamical spin accumulation takes place\ncan be understood by considering the mechanism un-\nderlying the spin-exchange (Kondo) processes respon-\nsible for \ripping the spin orientation of an electron\nin the molecular OL. To gain an intuitive picture\nof such processes, it is instructive to analyze the\neigenstates of a free-standing molecule, described by\nthe Hamiltonian ^Hmol=^HOL+^Hcore+^HOL-core;see\nEqs. (2)-(4), which participate in transport. Since\nthe results are obtained at T= 0, it su\u000eces to con-\nsider only the states of lowest energy. For J >0,\nthe ground state doublet of the S+1=2 spin multiplet hasthe formjStot\nz=\u00065=2i\u0011j\u0006 1=2iOL\nj\u0006 2icore;with\njsz(Sz)iOL(core) denoting the spin state of the OL (mag-\nnetic core). Because the spin-exchange processes, oc-\ncurring in the OL due to its strong hybridization to\nelectrodes, can just lead to \ripping of the OL spin,\nj\u00001=2iOL$j1=2iOL, without a\u000becting the state of the\ninternal spinjSzicore, no direct transitions between the\ndoublet ground states are possible. In fact, any pair\nof molecular states jStot\nz;1iandjStot\nz;2ican support the\nspin-exchange processes due to tunneling of electrons\nonly ifjStot\nz;1\u0000Stot\nz;2j= 1, which basically stems from\nthe fact that angular momentum exchanged between\nthe tunneling current and the molecule has to be con-\nserved. It means that the only allowed transitions\nfrom the statesjStot\nz=\u00005=2iandjStot\nz= 5=2ican be\nthose to the \frst excited doublet states3jStot\nz=\u00003=2i\nandjStot\nz= 3=2i, respectively. Importantly, these states\nare separated from the ground state doublet by an en-\nergy gap \u0001. As a result, the spin exchange processes,\nwhich underlie the dynamical spin accumulation, be-\ncome active if the energy pumped into the molecule\nby means of periodic driving external potential sat-\nis\fes the condition !&!\u0003\nD>0\u0019\u0001. In the limit of\nD\u001cJand at the particle-hole symmetry point ( \"=U=\n\u00000:5), one can thus estimate [38]: !\u0003\nD>0\u0019KSDwith\nKS= 2S(2S\u00001)=(2S+ 1). Those excitation energies\nare marked in left column of Fig. 2 with vertical dashed\nlines. As can be seen, the agreement between this esti-\nmate and numerical data is quite satisfactory. It proves\nthat the dynamical spin accumulation builds up in the\nmolecule for frequencies !&!\u0003\nD>0.\nThe picture developed above changes signi\fcantly if\na molecule is characterized by the uniaxial magnetic\nanisotropy of the easy-plane type ( D< 0), see the right\ncolumn of Fig. 2. In particular, the major di\u000berence is\nthat in such a situation at frequencies !.Dthe conduc-\ntanceGAP(!) [see Fig. 2(e)] always reaches the limit of\nunitary transport, that is, GAP(!) = (1\u0000p2)G0, which\nis a signature of the Kondo e\u000bect. It is clear from\nFigs. 2(f)-(g) that this e\u000bect is not related to the dynam-\nical spin accumulation, as at low frequencies ( !\u001cD)\ntransport is fully determined only by the conductance\ncomponents diagonal in spin, GAP(!)\u0019P\n\u001bGAP\n\u001b\u001b(!).\nFurthermore, it can be noticed that the components o\u000b-\ndiagonal in spin are now characterized by smaller thresh-\nold frequencies !\u0003, and that their magnitudes are larger,\ncompare Fig. 2(g) with Fig. 2(c). This, in turn, af-\nfects values of the current spin polarization PAP(!),\nwhich in the present case exceed those for a spin-isotropic\nmolecule, see Fig. 2(h). However, in other respects,\nthe behavior of the relevant quantities under discussion,\nshown in Figs. 2(e)-(h), qualitatively resembles that ob-\n3Note that the states jStot\nz=\u00063=2iconstituting the \frst excited\ndoublet of the S+ 1=2 spin multiplet have the from of a linear\ncombination of the following states:\b\nj\u00061=2iOL\nj\u0006 1icore;\nj\u00071=2iOL\nj\u0006 2icore\t\n:8\nserved forD> 0.\nTo understand the origin of the Kondo e\u000bect re-\nvival we again invoke the spectrum of a free-standing\nmolecule. Since the molecular spin is character-\nized by the uniaxial magnetic anisotropy of the easy-\nplane type, it means that the ground state dou-\nblet of the S+ 1=2 spin multiplet is formed by\nthe states with the lowest Stot\nzcomponent, namely,\njStot\nz=\u00061=2i. These states arise as superpositions\nof states\b\nj\u00061=2iOL\nj0icore;j\u00071=2iOL\nj\u0006 1icore\t\n,\nfrom which it is clear that the ground state doublet\ncan now support the electron spin exchange processes\nin the OL |the mechanism underlying the Kondo e\u000bect.\nThe e\u000bective exchange interaction between the molecule\nand the leads is conditioned by the excitation energies be-\ntween the ground state doublet and the empty and fully-\noccupied orbital-level molecular states, which basically\ndepends on all model parameters in a nontrivial fashion.\nConsequently, it is a tedious task to provide a simple an-\nalytical formula for the energy scale !\u0003. Instead, let us\njust conclude from the inspection of frequency-dependent\ntransport characteristics shown in the right column of\nFig. 2 that the Kondo temperature is of the order of the\nmagnetic anisotropy constant, TK\u0018jDj, while the en-\nergy scale!\u0003\nD<0is slightly smaller than TKand grows\nlinearly withjDj. This behavior can be clearly seen in\nthe dynamical spin accumulation shown in Fig. 2(g). The\nonset ofGAP\n\"#(!) moves to larger frequencies with increas-\ningjDjand in the limit of very large magnetic anisotropy\nthe system's dynamical behavior approaches the quan-\ntum dot case, indicated by the thin dotted line.\nFrom the above discussion one can already formulate\nsome more universal statements concerning the behav-\nior of the dynamical spin accumulation. It is clear that\nthis e\u000bect is most e\u000bective when the spin-exchange pro-\ncesses are relevant. This happens for frequencies corre-\nsponding to the energy scale responsible for the forma-\ntion of the Kondo state. Thus, one can observe that\nthe maximum in GAP\n\u001b\u001b(!) develops for some resonant fre-\nquency!r, which is of the order of the Kondo temper-\nature,!r\u0019TK. On the other hand, the width of this\nmaximum depends strongly on the frequency range of\nthe slope when the conductance as a function of !in-\ncreases due to the Kondo e\u000bect |note that we discuss\nthe behavior on logarithmic scale. This is why a broad\nmaximum can be observed for spin-isotropic molecules,\nwhile for \fnite magnetic anisotropy the frequency range\nof enhanced spin accumulation is much reduced. When\nthe conductance GAP(!) reaches a plateau with lower-\ning!, the spin-\rip processes become quenched and a\nmany-body delocalized screened-spin state is formed be-\ntween the molecule's spin and the spins of conduction\nelectrons. The electrons, when tunneling through the\njunction, experience then only a phase shift and spin-\n\rip processes are suppressed [2]. As a consequence, the\no\u000b-diagonal components of frequency-dependent conduc-\ntance get suppressed and the e\u000bect of dynamical spin ac-\ncumulation disappears. The energy scale when this hap-pens is, in turn, described by !\u0003, which corresponds to\nthe onset of dynamical spin accumulation with increasing\nthe driving frequency !.\nAs far as the height of the maximum in GAP\n\u001b\u001b(!) is\nconcerned, one can see that if the value of zero-frequency\nconductance is smaller than its maximum value, which\ne\u000bectively means that the Kondo e\u000bect cannot fully de-\nvelop in the system, the magnitude of GAP\n\u001b\u001b(!) gets re-\nduced. This can be especially seen for the easy-axis type\nof magnetic anisotropy presented in the left column of\nFig. 2. On the other hand, for magnetic molecules with\nanisotropy of the easy-plane type, the ground state is al-\nways a spin doublet, so that at low frequencies the Kondo\ne\u000bect can fully develop and, consequently, while the po-\nsition of maximum in GAP\n\u001b\u001b(!) depends on D, its maxi-\nmum value does not. In fact, the maximum value of the\ndynamical spin accumulation is then comparable to the\nquantum dot case, see the right column of Fig. 2.\n2. Uniaxial and transverse magnetic anisotropy\nThe uniaxial component of magnetic anisotropy along\nthezaxis is often accompanied by the transverse one, de-\nscribed by the term E\u0000^S2\nx\u0000^S2\ny\u0001\nin Eq. (3). In essence, it\ncaptures the e\u000bect of breaking the rotational symmetry\naround the zaxis, which, in other words, means that\nthe internal spin has a tendency to align along some\ndirections with respect to the plane perpendicular to\nthezaxis. In particular, for E > 0 the energy of the spin\nis minimized if it is oriented along the yaxis. The signif-\nicance of this transverse term of magnetic anisotropy lies\nin the fact that such a term leads to mixing of the axial\nspin statesjSzicore, which can be easily seen if one intro-\nduces in Eq. (3) the ladder operators ^S\u0006\u0011^Sx\u0006i^Sy:\nGenerally, this mixing is at the foundation of many im-\nportant e\u000bects in\ruencing transport, such as, the quan-\ntum tunneling of spin [39, 44], or the Berry-phase block-\nade [41{43].\nFigure 3 illustrates how inclusion of the transverse\nmagnetic anisotropy ( D6= 0 andE6= 0) a\u000bects the \fnite-\nfrequency conductance and the dynamical spin accumula-\ntion in the case of the easy-axis ( D> 0, left column) and\neasy-plane ( D< 0, right column) type of uniaxial mag-\nnetic anisotropy. Let us \frst focus on the case of D> 0.\nThe \frst noticeable di\u000berence, as compared with the case\nofD> 0 andE= 0 [see Figs. 2(a)-(d)], is that one ob-\nserves the revival of the Kondo e\u000bect for su\u000eciently low\nfrequencies. Such a restoration of transport occurs as a\nconsequence of the mixing caused by the second term of\nHamiltonian (3), because now each molecular spin state\ne\u000bectively becomes a superposition of all possible OL\nelectronic spin and internal spin states [44]. This, in turn,\nmeans that the spin exchange processes leading to tran-\nsitions between the states of the ground state doublet are\npermitted. Furthermore, for !\u0019!ra pronounced max-\nimum in the dynamical conductance GAP(!) is visible,\nFig. 3(a), and it stems from the dynamical spin accu-9\n00.20.40.60.8\n00.10.20.30.4\n00.10.20.3\n00.10.20.30.40.50.6\n10\u0000610\u0000410\u0000210010\u0000610\u0000410\u00002100\nConductance GAP(!)/G0\nE=jDj= 0\nE=jDj= 1=10\nE=jDj= 1=5\nE=jDj= 1=3 00.20.40.60.8\n(a)Uniaxial andtransverse magnetic anisotropy:\neasy-axis type ( D>0)\n(e)easy-plane type ( D<0)GAP\n\"\"(!)/G0\n00.10.20.30.4 (b) (f)GAP\n\"#(!)/G0\n00.10.20.3(c) (g)Spin polarization PAP(!)\nFrequency !=T0\nK00.10.20.30.40.50.6\n10\u0000610\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000610\u0000410\u00002100(h)\nFigure 3. Analogous to Fig. 2 except that now also the ef-\nfect of the transverse component of magnetic anisotropy Eis\nincluded for a selected value of uniaxial magnetic anisotropy\nconstantjDj=T0\nK= 10\u00002. Here, the transverse constant Eis\nalways assumed positive. Note that to enable an easy com-\nparison with results in Fig. 2, the scales are kept identical\nas in Fig. 2, and the \fnely dashed line representing the case\nofE= 0 is added to serve as the reference line.\nmulationGAP\n\"#(!), which at this particular frequency !r\nexhibits a sharp resonance, as one can see in Fig. 3(c).\nWe also note that at !ra kink develops in the conduc-\ntance component diagonal in spin GAP\n\"\"(!), Fig. 3(b),\nwhile spin polarization PAP(!) exhibits a local maxi-\nmum, Fig. 3(d).\nThe behavior of dynamical transport properties in the\npresence of transverse component of magnetic anisotropy\ncan be understood by invoking the discussion in the pre-\nvious section. Generally, one can again notice that the\n10\u0000810\u0000710\u0000610\u0000510\u0000410\u00003\n10\u0000210\u0000110\u0000610\u0000510\u0000410\u0000310\u0000210\u00001100\n10\u0000210\u00001\nResonance frequency !r=T0\nK\nTransverse anisotropy E=Dantiparallel (AP)\n10\u0000810\u0000710\u0000610\u0000510\u0000410\u00003\n10\u0000210\u00001(a)\nindepend. of p\nE=D=1=3\n!r=!r(E=D=1=3)\nTransverse anisotropy E=Dp= 0.25\np= 0.5\np= 0.7510\u0000610\u0000510\u0000410\u0000310\u0000210\u00001100\n10\u0000210\u00001(b)\nparallel (P)Figure 4. (a) Dependence of the position !rof the resonance\nin the conductance component o\u000b-diagonal in spin, G\u001b\u001b(!),\non the transverse magnetic anisotropy constant E. For the\nantiparallel (AP) magnetic con\fguration (squares) the data\npoint for E=D = 1=3 (marked by a \fnely dashed vertical line)\ncorresponds to the resonance indicated in Fig. 3(c) by the ar-\nrow. Note that for this magnetic con\fguration !ris indepen-\ndent of the spin polarization p, see Fig. 5(g), and that lines\nconnecting the data points serve as a guide for eyes. The data\npoints at E=D = 1=3 for the parallel (P) magnetic con\fgura-\ntion represent the position of relevant peaks in Fig. 7(g). (b)\nData points shown in (a) rescaled by !r(E=D = 1=3) to high-\nlight the change of the slope occurring in the parallel magnetic\ncon\fguration. Here, D=T0\nK= 10\u00002and other parameters as\nin Fig. 3.\nvalue of!rcoincides with the Kondo temperature, that\nis, the dynamical spin accumulation exhibits a maximum\nat!=!r\u0019TK. Such a feature arises for all nonzero\nvalues of anisotropy Econsidered in the \fgure, and im-\nportantly, the position of this feature depends strongly\non the transverse anisotropy component |small changes\ninElead to a large shift of the resonance frequency !r.\nThis is directly related to a strong dependence of the\nKondo temperature on the model parameters and, in\nparticular, \fnite transverse anisotropy [44]. An explicit,\nnumerically determined dependence of TKonEcan be\nseen in Fig. 4, which illustrates how the position of the\nresonance in the dynamical spin accumulation GAP\n\u001b\u001b(!)\nevolves when the transverse anisotropy parameter Eis\nmodi\fed. Clearly, small changes in Eresult in large\nmodi\fcation of !rand, thus, TK. From the slope of\nthe calculated curve, see squares in Fig. 4, we estimate\nthat!r/(E=D )\u00003. We also note that the position of\nthis curve is insensitive to the spin polarization pof elec-\ntrodes. Moreover, it can be observed in Fig. 3(c) that\nthe width of the peak in GAP\n\"#(!) andPAP(!) |plotted\non a logarithmic scale| hardly depends on E. One can\nconclude, thus, that the width of the resonance in dy-\nnamical spin accumulation, which occurs at !r\u0019TK, is\nalso approximately given by the Kondo temperature.\nOn the contrary, in the case of the uniaxial magnetic\nanisotropy of the easy-plane type ( D< 0), presented\nin the right column of Fig. 3, the dynamical conduc-\ntanceGAP(!) is modi\fed more subtly. Now, one ob-\nserves the well-developed Kondo e\u000bect [Figs. 3(e)-(f)] and\na pronounced maximum both in the dynamical spin ac-10\ncumulation GAP\n\"#(!) [Fig. 3(g)] and in the current spin\npolarizationPAP(!) [Fig. 3(h)], already visible in the\nabsence of transverse component of magnetic anisotropy.\nThe increase of Eresults only in a small reduction of the\nthreshold frequency !\u0003at whichGAP\n\"#(!) starts building\nup, and at which also the suppression of the Kondo ef-\nfect takes place |in other words, the raise of Eleads to\na slight decrease of the Kondo temperature. Moreover,\nfor largerEthe maximum in GAP\n\"#(!) gets broader and\neventually a small dip on the top of it develops. Inter-\nestingly, at the frequency where this dip is observed, one\ncan also notice a local maximum in GAP\n\"\"(!), see Fig. 3(f).\nC. In\ruence of the spin polarization of electrodes\nIn order to explore further the subtle interplay between\nthe Kondo e\u000bect and the dynamical spin accumulation,\nhere we consider the e\u000bect of spin polarization of elec-\ntrodes. For this purpose, in Fig. 5 and Fig. 6 we show\nhow the dynamical transport response of the system un-\nder investigation changes for di\u000berent values of the spin\npolarization parameter pin the case of D> 0 (Fig. 5)\nandD< 0 (Fig. 6), respectively. The left (right) column\nin those \fgures corresponds to the case of zero (\fnite)\ntransverse magnetic anisotropy constant E.\nLet us \frst consider the case of D> 0 shown in Fig. 5.\nGenerally, as expected for the Kondo e\u000bect, with the in-\ncrease ofpthe low-frequency conductance GAP(!.!\u0003),\nthat is, below the threshold frequency !\u0003for the dynam-\nical spin accumulation to kick in, becomes suppressed,\nsee Figs. 5(a,e). This behavior stems from the fact that\nin electrodes characterized by a large degree of spin po-\nlarization, there is a great imbalance between the num-\nbers of spin-majority and spin-minority electrons to be\ninvolved in the spin exchange processes leading to the\nKondo e\u000bect. In consequence, the larger the imbalance\nis, the less e\u000bective these processes become, and the more\nthe Kondo e\u000bect becomes suppressed. The dependence\nof the zero-frequency conductance GAP(!= 0) in the an-\ntiparallel magnetic con\fguration on the spin polarization\npof electrodes is presented in the inset to Fig. 5(c) and\nit can be described by a simple formula, GAP(!= 0)\u0011\nGAP(!= 0;p) = (1\u0000p2)GAP(!= 0;p= 0).\nThe situation changes qualitatively for frequen-\ncies!&!\u0003, where the o\u000b-diagonal-in-spin component\nof conductance GAP\n\"#(!), Figs. 5(c,g), starts contributing\nsigni\fcantly, so that the dynamical spin accumulation\nemerges as the dominant e\u000bect. Importantly, although\none can notice that GAP\n\"#(!) does not vanish even when\nthe electrodes are non-magnetic ( p= 0), no spin polar-\nization of the current is observed in such a case, that is,\nPAP(!) = 0, as one can see in Figs. 5(d,h). Moreover,\nthe behavior of the dynamical conductance GAP(!) is\nthen primarily governed by its diagonal-in-spin compo-\nnentGAP\n\"\"(!) |compare in Figs. 5 panels (a,e) with (b,f).\nOn the other hand, in the opposite limit of strongly spin-\npolarized electrodes ( p>0:5), whereGAP\n\"\"(!) gets pro-\n00.20.40.60.81\n00.10.20.30.40.5\n00.10.20.30.40.5\n00.20.40.60.81\n00.25 0.50.75 1\n00.20.40.60.81\n10\u0000410\u0000210010\u0000410\u00002100\nConductance GAP(!)/G0\np= 0\np= 0.25\np= 0.5\np= 0.75\np= 0.95\n00.20.40.60.81(a)Easy axis ( D>0)\nWithout transverse\nanisotropy ( E=D= 0)\n(e)With transverse\nanisotropy ( E=D= 1=3)GAP\n\"\"(!)/G0\n00.10.20.30.40.5(b)\n(f)GAP\n\"#(!)/G0\n00.10.20.30.40.5(c)GAP(!=0)=G0\nSpin polarization p00.20.40.60.81\n00.25 0.50.75 1E=D= 0E=D= 1=3 (g)Spin polarization PAP(!)\nFrequency !=T0\nK00.20.40.60.81\n10\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000410\u00002100(h)Figure 5. Evolution of the frequency-dependent conduc-\ntance GAP(!) in (a,e), its spin components GAP\n\"\"(!) in (b,f)\nandGAP\n\"#(!) in (c,g), as well as the current spin polariza-\ntionPAP(!) in (d,h) presented as a function of the spin-\npolarization coe\u000ecient pof electrodes for D=T0\nK= 10\u00002(the\nuniaxial magnetic anisotropy of the easy-axis type). Left\n(right )column corresponds to the case without (with) the\ntransverse component of the magnetic anisotropy included,\nthat is, for E= 0 ( E=D=3). The inset in (c) presents the\ndependence of the zero-frequency conductance GAP(!= 0) on\nthe spin polarization pin the case of E= 0 and E=D = 1=3.\nOther parameters are the same as in Fig. 2.\ngressively suppressed for large p, the features associated\nwith the dynamical spin accumulation GAP\n\"#(!) become in\nfact increasingly visible in the total conductance GAP(!)\nwithin the entire range of frequencies !|compare in\nFigs. 5 panels (a,e) with (c,g). This observation illus-\ntrates the key generic di\u000berence between the response of11\n00.20.40.60.81\n00.10.20.30.40.5\n00.10.20.30.40.5\n00.20.40.60.81\n10\u0000410\u0000210010\u0000410\u00002100\nConductance GAP(!)/G0\np= 0\np= 0.25\np= 0.5\np= 0.75\np= 0.95\n00.20.40.60.81\n(a)Easy plane ( D<0)\nWithout transverse\nanisotropy ( E=jDj= 0)\n(e)With transverse\nanisotropy ( E=jDj= 1=3)GAP\n\"\"(!)/G0\n00.10.20.30.40.5\n(b) (f)GAP\n\"#(!)/G0\n00.10.20.30.40.5(c) (g)Spin polarization PAP(!)\nFrequency !=T0\nK00.20.40.60.81\n10\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000410\u00002100(h)\nFigure 6. Analogous to Fig. 5 except that here the case of the\nuniaxial magnetic anisotropy of the easy-plane type ( D=T0\nK=\n\u000010\u00002) is considered. Note that the dependence of the zero-\nfrequency dynamical conductance GAP(!= 0) on pis the\nsame regardless of whether the transverse component of mag-\nnetic anisotropy is present or not, and it is given by the curve\nforE=D = 1=3 shown in the inset to Fig. 5(c).\nthe Kondo e\u000bect and the dynamical spin accumulation\nto a large spin polarization of electrodes. Speci\fcally,\nunlike the Kondo e\u000bect, the dynamical spin accumula-\ntion is augmented with the increase of p. This is a direct\nconsequence of the fact that GAP\n\"#(!) is associated with\nmajority spin bands of both leads, while the diagonal-in-\nspin components GAP\n\u001b\u001b(!) depend on both majority and\nminority spin bands.\nIn addition, one can note that while the low ( !\u001c!\u0003)\nand high-frequency ( !\u001d!\u0003) behavior of the dynamicalconductance and its spin-resolved contributions is qual-\nitatively similar in the case of E= 0 and \fnite E, huge\ndi\u000berences occur when !\u0019!\u0003. ForE= 0, the dynamical\nspin accumulation starts growing for !&!\u0003to reach a\nplateau, whereas for \fnite E,GAP\n\"#(!) increases to form a\nstrong maximum, the height of which grows with increas-\ningp. To understand this di\u000berence let us recall that in\nthe absence of transverse magnetic anisotropy the Kondo\ne\u000bect develops only partially. This implies that dynam-\nical spin accumulation has a moderate, relatively wide\nin frequencies, maximum. On the other hand, for \fnite\ntransverse magnetic anisotropy the Kondo resonance can\nfully develop with a clear sharp maximum in dynamical\nspin accumulation at !r\u0019TK, which becomes greatly en-\nhanced with increasing spin polarization p. In fact, in the\nlimit of half-metallic leads ( p= 1), the dynamical con-\nductance would be exclusively due to the e\u000bect of dynam-\nical spin accumulation, that is, GAP(!) =P\n\u001bGAP\n\u001b\u001b(!).\nWe also note that while the spin-resolved components\nof the frequency-dependent conductance strongly depend\nonp, the characteristic energy scales, !\u0003,TKand conse-\nquently!r, hardly do so, see Fig. 5.\nThe above discussion is also relevant to the case of\neasy-plane type of magnetic anisotropy ( D< 0), which\nis shown in Fig. 6. With raising the spin polarization,\nthe diagonal-in-spin components of the dynamical con-\nductance become suppressed [Figs. 6(b,f)], while the o\u000b-\ndiagonal component GAP\n\"#(!) increases [Figs. 6(c,g)], and\nfor su\u000eciently large pgives a dominant contribution to\nthe total conductance. In such a situation, the interplay\nof these two contributions results in a non-monotonic fre-\nquency dependence of GAP(!), see Figs. 6(a,e), which\nexhibits a local maximum due to dynamical spin ac-\ncumulation. Similar to the case of easy-axis magnetic\nanisotropy presented in Fig. 5, large spin polarization p\nof the leads induces large spin polarization of the cur-\nrent, see Figs. 6(d,h). As far as the e\u000bects related to the\ntransverse component of magnetic anisotropy are con-\ncerned, \fnite Eresults mainly in quantitative changes in\nthe dynamical response of the system, cf. the left and\nright column of Fig. 6, manifesting as a small reduction\nof the Kondo temperature and, consequently, the energy\nscale!\u0003. However, a qualitative di\u000berence can be still\nobserved in the dynamical spin accumulation, which in\nthe case ofE=jDj=3 exhibits a small dip at intermediate\nfrequencies, see Fig. 6(g) for !=T0\nK\u001910\u00002.\nD. Parallel magnetic con\fguration of the junction\nUp to this point, the discussion has been concentrated\non the situation of the junction in the antiparallel mag-\nnetic con\fguration, which allowed us to exclude from the\npicture some subtle e\u000bects due to the spintronic e\u000bective\nexchange \felds. To acquire a complete understanding of\nhow the magnetic con\fguration of the junction a\u000bects\nthe process of dynamical spin accumulation, we now also\nconsider the parallel magnetic con\fguration of the junc-12\n00.20.40.60.81\n00.10.20.30.40.5\n00.050.10.15\n00.20.40.60.81\n00.25 0.50.75 1\n00.20.40.60.81\n10\u0000810\u0000610\u0000410\u0000210010\u0000810\u0000610\u0000410\u00002100\nConductance GP(!)=G0\np= 0\np= 0.25\np= 0.5\np= 0.75\np= 0.95\n00.20.40.60.81(a)Easy axis ( D>0)\nWithout transverse\nanisotropy ( E=D= 0)\n(e)With transverse\nanisotropy ( E=D= 1=3)GP\n\"\"(!)=G0\n00.10.20.30.40.5\n(b) (f)GP\n\"#(!)=G0\n00.050.10.15(c)\nGP(!=0)=G0\nSpin polarization p00.20.40.60.81\n00.25 0.50.75 1E=D= 0E=D= 1=3 (g)Spin polarization PP(!)\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000810\u0000610\u0000410\u00002100(h)\nFigure 7. Analogous to Fig. 5 but now the parallel (P)\nmagnetic con\fguration of the junction is shown. Note that at\npresent GP\n\"#(!) =GP\n#\"(!), and thus,Pdsa(!) = 0, so that the\ncurrent spin polarization occurs only due to the diagonal-in-\nspin terms GP\n\"\"(!)6=GP\n##(!), that is,PP(!)\u0011PP\n0(!). Note\nthat although, for the sake of consistency, we plot here the\nsame set of pvalues as in previous \fgures, the frequency range\nhas been extended here to include lower values of !. More-\nover, the long-dashed line is identical to that in Fig. 5 and\nit serves as the reference line. Recall that D=T0\nK= 10\u00002and\nthe other parameters are the same as in Fig. 2.\ntion.\nFor this purpose, we present in Figs. 7 and 8 how the\ndynamical transport characteristics of the system depend\non the spin polarization pof electrodes for the parallel\nmagnetic con\fguration. To begin with, let us \frst focus\non the case of the easy-axis type of uniaxial magnetic\n00.20.40.60.81\n00.10.20.30.40.5\n00.050.10.15\n00.20.40.60.81\n10\u0000810\u0000610\u0000410\u0000210010\u0000810\u0000610\u0000410\u00002100\nConductance GP(!)=G0\np= 0\np= 0.25\np= 0.5\np= 0.75\np= 0.9500.20.40.60.81\n(a)Easy plane ( D<0)\nWithout transverse\nanisotropy ( E=jDj= 0)\n(e)With transverse\nanisotropy ( E=jDj= 1=3)GP\n\"\"(!)=G0\n00.10.20.30.40.5\n(b) (f)GP\n\"#(!)=G0\n00.050.10.15(c) (g)Spin polarization PP(!)\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000810\u0000610\u0000410\u00002100(h)Figure 8. Analogous to Fig. 7 except that here the case of the\nuniaxial magnetic anisotropy of the easy-plane type ( D=T0\nK=\n\u000010\u00002) is considered. The other parameters are the same as\nin Fig. 2.\nanisotropy ( D> 0) shown in Fig. 7. One can see that\nif only the uniaxial component of magnetic anisotropy\nis present the dynamical conductance GP(!) does not\ndi\u000ber qualitatively from the antiparallel case, compare\nFig. 7(a) with Fig. 5(a). Nevertheless, two key quan-\ntitative di\u000berences can be spotted immediately: First,\nthe di\u000berent values of conductance in the zero-frequency\nlimit,GAP(!= 0)>GP(!= 0) for 0 <p.0:85 [com-\npare the insets in Fig. 5(c) and Fig. 7(c)], and second,\nthe behavior of the resonance around the threshold fre-\nquency!\u0003which marks the onset of the dynamical spin\naccumulation, as discussed in Sec. IV B 1. Speci\fcally,\nwith the increase of the spin polarization pof electrodes13\nthis resonance becomes shifted towards larger frequencies\nand its magnitude gets attenuated.\nThe origin of the suppression of GP(!= 0) and the\nshift can be explained by taking into account the fact\nthat in the present con\fguration the spintronic ef-\nfective exchange \felds can arise. Since we consider\nthat the molecule is tuned to the particle-hole symme-\ntry point ( \"=U=\u00000:5), one expects actually only the\nquadrupolar \feld [68], which essentially provides an addi-\ntional uniaxial contribution \u0000Ds\u0000^Sz+^sz\u00012(withDs>0)\nto the magnetic anisotropy Hamiltonian (3). As a result,\nthe energy separation \u0001 /D+Dsbetween the states\nparticipating in transport, that is, the ground state dou-\nblet and \frst excited doublet in the S+ 1=2 spin multi-\nplet, increases. This, in turn, translates into the larger\nthreshold frequency !\u0003\nD>0and also means that the spin\nexchange processes leading to the Kondo e\u000bect at low\nfrequencies are more subdued, as compared with the an-\ntiparallel case where Ds= 0.\nOn the other hand, the suppression of the features oc-\ncurring for !&!\u0003has its roots in the response of the\ndynamical spin accumulation GP\n\"#(!) to increasing p, see\nFig. 7(c). Importantly, this response is strikingly dif-\nferent from that for the antiparallel magnetic con\fgu-\nration in Fig. 5(c). First of all, it should be noted\nthat now one \fnds GP\n\"#(!) =GP\n#\"(!), which straightfor-\nwardly leads to the conclusion that the dynamical spin\naccumulation does not contribute to the spin polariza-\ntion of the current, PP\ndsa(!) = 0. In fact, the current\nspin polarization PP(!) shown in Fig. 7(d) is exclusively\ndue to the di\u000berence between the diagonal-in-spin com-\nponentsGP\n\"\"(!) andGP\n##(!), that is,PP(!)\u0011PP\n0(!).\nMoreover, the intensity of GP\n\"#(!) is signi\fcantly reduced\nwith respect to GAP\n\"#(!), and it exhibits the opposite be-\nhavior with the increase of p, namely, in the parallel mag-\nnetic con\fguration the dynamical spin accumulation is\ndiminished for large p. This behavior can be understood\nby realizing that in the antiparallel con\fguration the\no\u000b-diagonal conductance GAP\n\"#(!) is associated with the\nmajority-spin subbands of both leads. Consequently, in-\ncreasing the spin polarization results in an enhancement\nof dynamical spin accumulation. On the other hand, in\nthe parallel con\fguration, the o\u000b-diagonal components\ndepend on both majority and minority spin subbands of\nboth leads, such that the minority spin channel provides\na bottleneck for GP\n\"#(!). Consequently, in the parallel\nmagnetic con\fguration the e\u000bect of dynamical spin accu-\nmulation becomes suppressed with increasing p, contrary\nto the case of antiparallel con\fguration. Note also that in\nthe limit of half-metallic leads ( p!1),GP\n\"#(!) would be\nfully suppressed and the total conductance would be ex-\nclusively given by the majority spin diagonal component\nof the conductance, while in the antiparallel con\fgura-\ntion all components would disappear, except for GAP\n\"#(!),\nnamely, the total conductance would be only due to the\ndynamical spin accumulation.\nLet us now take the transverse component of mag-netic anisotropy into consideration, see the right column\nof Fig. 7. Comparing with the case of the antiparallel\nmagnetic con\fguration shown in Fig. 5(e), one observes\nin Fig. 7(e) that the suppression of the Kondo e\u000bect oc-\ncurring for large pproceeds now in a qualitatively di\u000ber-\nent manner. With the increase of the spin polarization,\nthe Kondo temperature TKinitially decreases whereas\nthe zero-frequency conductance GP(!= 0) =G0remains\nuna\u000bected, and only above some threshold value of p\nalsoGP(!= 0) becomes diminished |for a detailed evo-\nlution ofGP(!= 0) as a function of psee the inset in\nFig. 7(c). In consequence, as long as the Kondo e\u000bect\ndominates transport, the current injected into the right\nelectrode does not display the spin polarization, that is,\nPP(!) = 0 for!.TK, see Fig. 7(h). Furthermore, we\nnotice that in the large frequency regime !&!\u0003, the\ntransport features due to the dynamical spin accumu-\nlation behave identically to those discussed in the sit-\nuation without the transverse component of magnetic\nanisotropy. On the other hand, in the opposite limit it\ncan be seen that, unlike for the antiparallel con\fguration,\nthe resonance arising in GP\n\"#(!) shifts its position !rto-\nwards smaller frequencies when pbecomes larger. This\nresults from the dependence of the Kondo temperature\nonp, which decreases as praises.\nThe explicit dependence of !ron the transverse mag-\nnetic anisotropy is presented in Fig. 4 for a couple of\nselected values of spin polarization. It can be clearly\nseen that the resonance frequency, and, thus, also the\nKondo temperature, strongly depends now both on the\nvalue ofEand the spin polarization p. Interestingly, the\nslope of the dependence of !ronE=D di\u000bers slightly\nfrom the case of antiparallel con\fguration, and it is also\nsensitive to a change of p, see Fig. 4(b). Note that the\ndata points (squares) in Fig. 4 corresponding to the an-\ntiparallel magnetic con\fguration are actually valid also\nfor the nonmagnetic junction ( p= 0). One can see that\nwith increasing p, the dependence of the Kondo tem-\nperature on Ebecomes sharper. Moreover, because the\nKondo temperature greatly depends on spin polarization,\nforp&0:75, the Kondo e\u000bect actually develops at ex-\ntremely small energy scales, which are completely not\nrelevant from the experimental point of view.\nThe behavior of dynamical system's response changes\nif one considers the uniaxial magnetic anisotropy of the\neasy-plane type ( D< 0), see Fig. 8. It can be seen that\nin such a case, already without the transverse compo-\nnent of magnetic anisotropy, a signi\fcant qualitative dif-\nference in evolution of the dynamical conductance as a\nfunction of parises between the antiparallel (left column\nin Fig. 6) and parallel (left column in Fig. 8) magnetic\ncon\fguration of the junction. One can notice that, unlike\nfor the antiparallel case, the transition between the highly\nconducting state for small pand the weakly conducting\nstate for large poccurring for !.!\u0003is rather abrupt.\nFurthermore, also when including the transverse com-\nponent of magnetic anisotropy (right column in Fig. 8),\nthis transition proceeds in a qualitatively di\u000berent man-14\n00.20.40.60.81\n0 0.2 0.4 0.6 0.8 100.040.080.12\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\n0 0.2 0.4 0.6 0.8 100.040.080.12\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\n10\u0000810\u0000610\u0000410\u0000210000.20.40.60.81\n10\u0000810\u0000610\u0000410\u0000210000.20.40.60.81\n10\u0000810\u0000610\u0000410\u0000210000.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100\nConductance GP(!)=G0\nSpin polarization p!= 000.20.40.60.81\n0 0.2 0.4 0.6 0.8 1(c)\nGP\n\"#(!)=G0\nSpin polarization p00.040.080.12\n0 0.2 0.4 0.6 0.8 1(d)\nConductance GP(!)=G0\nSpin polarization p00.20.40.60.81\n0 0.2 0.4 0.6 0.8 1(g)\nGP\n\"#(!)=G0\nSpin polarization p00.040.080.12\n0 0.2 0.4 0.6 0.8 1(h)0 0.2 0.4 0.6 0.8 1Conductance GP(!)=G0Spin polarization p\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(a)Uniaxial magnetic anisotropy of the easy-plane type ( D<0)\nWithout transverse anisotropy ( E=jDj= 0)!=T0\nK= 10\u00006\n!=T0\nK= 10\u00004\n!=T0\nK= 10\u00003\n!=T0\nK= 10\u000010 0.04 0.08 0.12GP\n\"#(!)=G0Spin polarization p\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(b)0 0.2 0.4 0.6 0.8 1Conductance GP(!)=G0Spin polarization p\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(e)With transverse anisotropy ( E=jDj= 1=3)\n0 0.04 0.08 0.12GP\n\"#(!)=G0Spin polarization p\nFrequency !=T0\nK00.20.40.60.81\n10\u0000810\u0000610\u0000410\u00002100(f)\nFigure 9. Evolution of the dynamical conductance GP(!) and its o\u000b-diagonal-in-spin component GP\n\"#(!) as functions of the\nspin polarization pof electrodes for the uniaxial magnetic anisotropy of the easy-plane type ( D=T0\nK=\u000010\u00002).Left panels [(a)-\n(d)] correspond to the case without the transverse component of magnetic anisotropy ( E= 0), whereas right panels [(e)-(f)] to\nthe case when the transverse component is included ( E=jDj= 1=3).Bottom row presents cross-sections of the respective map\nplots in the top row for chosen values of frequencies !marked in (a), with the solid line standing for the zero-frequency limit.\nThe other parameters are the same as in Fig. 2.\nner as compared to the antiparallel case (right column\nin Fig. 6). Importantly, it can be observed that above\nsome threshold value of pthe dynamical spin accumula-\ntionGP\n\"#(!) shown in Fig. 8(g) changes its character and\nit develops a resonance typical to the easy-axis type of\nuniaxial magnetic anisotropy, see Fig. 7(g).\nIt is worth noting that the di\u000berence in the behavior\nof the dynamical conductance observed in the parallel\nand antiparallel magnetic con\fgurations on the value of\nthe spin polarization of electrodes has its origin in com-\npletely di\u000berent mechanisms governing the suppression\nof the Kondo e\u000bect. In the case of antiparallel con\fg-\nuration, the system behaves e\u000bectively as if coupled to\nnonmagnetic leads and the e\u000bect of spin polarization en-\nters only through the prefactors in appropriate formu-\nlas. In the parallel con\fguration, on the other hand,\nthe e\u000bective couplings do depend on spin, which results\nin nontrivial spin-resolved molecule-bath renormalization\ne\u000bects that give rise to \fnite local exchange \felds. As a\nconsequence, the interplay between the degree of spin\npolarization, which conditions the strength of exchange\n\felds, and the electronic correlations driving the Kondo\ne\u000bect, gives rise to large qualitative di\u000berences, as com-\npared to the case of antiparallel con\fguration.In order to gain a better insight into how the dynam-\nical conductance evolves with increasing the spin po-\nlarizationp, in Fig. 9 we analyze in detail the depen-\ndence of the total dynamical conductance GP(!) and\nits o\u000b-diagonal-in-spin component GP\n\"#(!) onp. Indeed,\nwe \fnd for E= 0 [Figs. 9(a)-(d)] that in the paral-\nlel magnetic con\fguration the transport response of the\nsystem is very sensitive to the change of p, with the\nsuppression of the Kondo e\u000bect taking place suddenly\natp0\u00190:5 [Fig. 9(a)] and the value of GP(!) altering\nforp > p 0only slightly, see the relevant cross-sections\ngiven by the dashed lines in Fig. 9(c). As discussed above,\nthe dynamical spin accumulation [Fig. 9(b)] arises only\nfor!&!\u0003, but at present the threshold energy !\u0003de-\npends onpnon-monotonically. Namely, it \frst decreases\naspgrows, while above p0the opposite trend is visible,\nandGP\n\"#(!) becomes quickly attenuated. In the left col-\numn of Fig. 10 we present how the total dynamical con-\nductanceGP(!) [Fig. 10(a)] and its o\u000b-diagonal-in-spin\ncomponent GP\n\"#(!) [Fig. 10(b)] vary in paroundp0. One\ncan see that, in fact, GP(!) does not change smoothly\nduring the transition between p.p0andp&p0and two\ncharacteristic features arise |also two pronounced peaks\nare visible in GP\n\"#(!). Again, such a behavior can be at-15\n00.20.40.60.81\n00.040.080.12\n0.485 0.49 0.495 0.5 0.38 0.4 0.42 0.44\nConductance GP(!)=G0\n!=T0\nK= 10\u00008\n!=T0\nK= 10\u00007\n!=T0\nK= 10\u00006\n!=T0\nK= 10\u0000500.20.40.60.81\n(a)Easy plane ( D<0)\nWithout transverse\nanisotropy ( E=jDj= 0)\n(c)With transverse\nanisotropy ( E=jDj= 1=3)GP\n\"#(!)=G0\nSpin polarization p00.040.080.12\n0.485 0.49 0.495 0.5(b)\nSpin polarization p0.38 0.4 0.42 0.44(d)\nFigure 10. (a)-(b) [(c)-(d)] Cross-sections of the map plots\nshown in the top row of Fig. 9 for selected values of frequen-\ncies!and resolved around the features marked by the arrow\nin Fig. 9(a) [(e)].\ntributed to the presence of the spintronic component Ds\nof magnetic anisotropy. Speci\fcally, recall that unlike the\nintrinsic uniaxial component D, which only a\u000bects the\nspin ^Szof the molecular magnetic core [see Eq. (3)], the\nspintronic component Dshas the e\u000bect on the total spin\nof the molecule, ^Sz+ ^sz. Importantly, it leads to such a\nsituation that di\u000berent doublets within the S+ 1=2 spin\nmultiplet e\u000bectively respond in a somewhat dissimilar\nway to varying of p. SinceDs>0, with the increase of p\nthe e\u000bect of intrinsic uniaxial anisotropy ( D< 0) gets re-\nduced, the system undergoes a transition from the ground\nstate doubletjStot\nz=\u00061=2ifor smallptojStot\nz=\u00065=2i\nfor largep. This transition, however, is not direct\nand it proceeds viathe doubletjStot\nz=\u00063=2i, namely:\nThe \frst (left one) of two additional features visible in\nFigs. 10(a)-(b) appears when the doublets jStot\nz=\u00061=2i\nandjStot\nz=\u00063=2iare degenerate, and the latter doublet\nbecomes the new ground state. With the further increase\nofp, at some other critical value pthis doublet becomes\neventually degenerate with jStot\nz=\u00065=2i, which results\nin formation of the right feature in Figs. 10(a)-(b). For\neven larger p, the spintronic component Dsstarts dom-\ninating over the intrinsic one Dand the dynamical spin\naccumulation GP\n\"#(!) displays characteristics of the uni-\naxial magnetic anisotropy of the easy-axis type, compare\nalso Fig. 8(c) with Fig. 7(c). Moreover, since in this limit\nthe ground state is jStot\nz=\u00065=2i, a signi\fcant suppres-\nsion of the dynamical spin accumulation occurs.\nWe note that the range of spin polarization values \u0001 p,\nfor which this transition occurs, is conditioned by the in-terplay of the quadrupolar exchange \feld and the Kondo\ntemperature. More speci\fcally, the Kondo e\u000bect be-\ncomes approximately suppressed once D+Ds&TK. As\ncan be seen in Fig. 9(a) for p<p 0, the Kondo temper-\nature is of the order of TK=T0\nK\u001910\u00003. On the other\nhand, the dependence of Dson the spin polarization\ncan be approximated by Ds/\u00002p2=U[68]. This implies\nthat \u0001p\u0019pTKU=\u0000\u00190:02, which agrees reasonably well\nwith the numerical data shown in Fig. 10(a).\nThe e\u000bect of the spintronic component of magnetic\nanisotropy is even more pronounced if the molecule ex-\nhibits also the intrinsic transverse component of mag-\nnetic anisotropy ( E6= 0). The corresponding dynamical\ntransport quantities for this case are shown in Figs. 9(e)-\n(h). First of all, the presence of non-zero Eresults in\npersistence of the Kondo e\u000bect to large values of p, as\ncompared to the case of E= 0 discussed above. One can\nsee that larger values of spin polarization are needed to\nsuppress the Kondo peak in GP(!). One can also observe\nthat forp>p 0the dynamical spin accumulation GP\n\"#(!)\n[shown in Fig. 9(f)] displays a pronounced maximum\nfor frequencies corresponding to the Kondo temperature.\nThis resonance is a clear signature of the e\u000bective uni-\naxial magnetic anisotropy of the easy-axis type, as dis-\ncussed in Sec. IV B 2 [see especially Fig. 3(c)]. More-\nover, analogously to the case of E= 0 [see Figs. 10(a)-\n(b)],GP\n\"#(!) is characterized by two maxima at p\u00190:39\nandp\u00190:44, which translate into a non-monotonic de-\npendence of GP(!) on the spin polarization at low fre-\nquencies!, see Figs. 9(e,g) and also the magni\fcation\nof the relevant range of pin Figs. 10(c)-(d). Noticeably,\nthese features are a pure dynamical e\u000bect and they dis-\nappear in the zero-frequency limit, as illustrated by the\nsolid line in Figs. 9(g,h). The occurrence of these two\nmaxima can be understood by invoking exactly the same\narguments as those used above to explain the origin of\nthe two resonances in GP\n\"#(!) forE= 0. The only dif-\nference is associated with the increased separation of the\ntwo present resonances. It occurs as a result of mod-\ni\fcation of states and energies of the molecule due to\nthe presence of the transverse (second) term in Hamilto-\nnian (3), which are further renormalized non-trivially by\nspin-dependent electron tunneling processes.\nE. Antiferromagnetic coupling between the OL and\nthe magnetic core\nFinally, in this section we discuss the main results in\nthe case when the coupling between the molecule's mag-\nnetic core and the orbital level is of the antiferromag-\nnetic (AFM) type ( J <0). For this purpose, in Fig. 11\nwe show the frequency-dependent transport coe\u000ecients\nin the antiparallel con\fguration calculated for uniaxial\nmagnetic anisotropy of the easy-axis type ( D> 0) when\nthe transverse component of magnetic anisotropy is ab-\nsent (E= 0, left column) and \fnite ( E6= 0, right col-\numn), respectively.16\n00.20.40.60.8\n00.10.20.30.4\n00.10.20.3\n00.20.40.60.8\n10\u0000810\u0000610\u0000410\u0000210010\u0000810\u0000610\u0000410\u00002100\nConductance GAP(!)=G0\nQD\njDj=T0\nK= 0\njDj=T0\nK= 10\u00004\njDj=T0\nK= 10\u00002\njDj=T0\nK= 0.5\n00.20.40.60.8\n(a)Uniaxial magnetic anisotropy of easy-axis type ( D>0)\nWithout transverse\nanisotropy ( E=D= 0)\nE=D= 0\nE=D= 1=5\nE=D= 1=3(e)With transverse\nanisotropy ( E=D= 1=3)GAP\n\"\"(!)=G0\n00.10.20.30.4\n(b) (f)GAP\n\"#(!)=G0\n00.10.20.3(c) (g)Spin polarization PAP(!)\nFrequency !=T0\nK00.20.40.60.8\n10\u0000810\u0000610\u0000410\u00002100(d)\nFrequency !=T0\nK10\u0000810\u0000610\u0000410\u00002100(h)\nFigure 11. The e\u000bect of magnetic anisotropy on the fre-\nquency-dependent conductance of a large-spin magnetic\nmolecule shown for the AFM J-coupling ( J=T0\nK=\u00002:25) and\nthe junction in the antiparallel (AP) magnetic con\fgura-\ntion. Note that only the uniaxial magnetic anisotropy of\nthe easy-axis plane ( D > 0) is presented. Left column [(a)-\n(d)] corresponds to the molecule exhibiting exclusively the\nuniaxial component of magnetic anisotropy ( E= 0), whereas\nright column [(e)-(h)] shows the e\u000bect of the transverse com-\nponent for D=T0\nK= 10\u00002. All other parameters are the same\nas in Fig. 2.\nWe recall that a spin-isotropic molecule ( D= 0) with\nthe AFMJ-coupling generically exhibits the two-stage\nKondo e\u000bect, so that for frequencies !smaller than the\nenergy scale characteristic of the second stage of screen-\ningT\u0003\nK,!.TK, the value of conductance is expected\nto drop signi\fcantly, see the solid line in the left columnof Fig. 11. Such a suppression of conductance GAP(!)\n[Fig. 11(a)], and also of the dynamical spin accumula-\ntionGAP\n\u001b\u001b(!) [Fig. 11(c)], is a direct consequence of the\nAFM coupling between the spin of an electron in the or-\nbital level and the molecule's core spin. This coupling\nsurpasses the AFM interaction between the OL's spin\nand spins of conduction electrons that gives rise to the\nKondo resonance. Note that for the assumed value of\nexchange coupling J, the temperature T\u0003\nKis relatively\nhigh, so that the conductance only slightly increases with\nlowering!due to the \frst-stage Kondo e\u000bect and then\nimmediately drops down, which results in relatively low\nmaximum around !\u0019TK.\nAdding the uniaxial component of magnetic anisotropy\ndoes not a\u000bect signi\fcantly the low-frequency results, as\nthe dynamical conductance and its spin-resolved compo-\nnents are suppressed already for !.TK, sinceT\u0003\nKandTK\nare in fact of a similar order. Moreover, similarly as\nin the case of the FM J-coupling, see Figs. 2(a)-(d), a\nthreshold frequency !\u0003arises above which the dynami-\ncal spin accumulation GAP\n\u001b\u001b(!) is observed, as shown in\nFig. 11(c). Even though the low-frequency transport is\nat present substantially reduced, the values of the spin\npolarization of the current injected into a drain electrode\nachieved for !>!\u0003can be still quite large and become\nsuppressed with increasing D, see Fig. 11(d).\nWhen the molecule also possesses transverse compo-\nnent of magnetic anisotropy, the second stage of Kondo\nscreening can become suppressed. This is seen in\nFigs. 11(e)-(f) where \fnite Erestores the Kondo peak\nat low frequencies. As expected, since now a pronounced\nKondo resonance is established for !.TK, the dynami-\ncal spin accumulation exhibits a maximum with its height\nbeing of the same order as in the quantum dot case, see\nFig. 11(g). A maximum at the same frequency is also\nobserved in the spin polarization [Fig. 11(h)].\nFinally, we note that for the uniaxial magnetic\nanisotropy of the easy-plane type ( D< 0), and also when\nthe magnetic con\fguration of the device is parallel, one\n\fnds a qualitatively similar behavior to the case of ferro-\nmagnetic exchange interaction discussed in previous sec-\ntions. However, due to the dominance of the second-stage\nKondo e\u000bect the transport is generally suppressed.\nV. CONCLUSIONS\nIn this paper we have analyzed the dynamical trans-\nport properties of magnetic molecules coupled to ferro-\nmagnetic electrodes in the Kondo regime. The molecule\nwas modeled by a LUMO level, directly coupled to ex-\nternal leads and additionally coupled through a ferro-\nmagnetic exchange interaction to the core spin of the\nmolecule. We have focused on the e\u000bect of dynami-\ncal spin accumulation, which can be associated with the\no\u000b-diagonal-in-spin component of the dynamical conduc-\ntance. We have in particular addressed the question of\nhow the dynamical spin accumulation becomes a\u000bected17\nby the presence of uniaxial, either of easy plane or easy\naxis type, and transverse anisotropy of the molecule. Our\nconsiderations have been performed in the linear response\nregime by using the Kubo formula, while all the dynam-\nical response functions were determined by using the nu-\nmerical renormalization group method.\nWe have generally shown that, in the case of antipar-\nallel magnetic con\fguration of the device, the dynam-\nical spin accumulation can develop for frequencies cor-\nresponding to the energy scale responsible for the for-\nmation of the Kondo e\u000bect, since then the spin-exchange\nprocesses are most e\u000bective. A local maximum in G\u001b\u001b(!)\nthus develops for resonant frequency !r, which is of the\norder ofTK. The width of this local maximum depends\nin turn on the energy scale where the Kondo state is\nformed. Consequently, while for spin isotropic molecules\ndynamical spin accumulation exhibits a broad maximum,\nin the presence of magnetic anisotropy the width of this\nmaximum becomes reduced. Another important energy\nscale describing the behavior of the dynamical spin accu-\nmulation denoted by !\u0003corresponds to the frequency be-\nlow which the conductance reaches a plateau. Then, the\nspin-\rip processes become quenched, such that G\u001b\u001b(!)\nvanishes. On the other hand, the height of the maxi-\nmum in dynamical spin accumulation turned out to de-\npend strongly on the magnitude of the Kondo e\u000bect. We\nhave shown that if the Kondo resonance develops fully\nfor!<T K, the spin accumulation exhibits a local max-\nimum, the height of which is of the same order as in\nthe case of quantum dots. However, when due to the\npresence of magnetic anisotropy the Kondo e\u000bect cannot\nfully emerge, the e\u000bect of dynamical spin accumulation\ncorrespondingly becomes reduced.\nWhen the junction is in the parallel magnetic con\fg-\nuration, the situation drastically changes, since then the\nspin-resolved renormalization of the molecule comes into\nplay. As a result, dynamical transport properties, and\nalso the dynamical spin accumulation, are conditioned\nby the interplay of the quadrupolar exchange \feld and\nKondo correlations. For the out-of-plane type of mag-\nnetic anisotropy we have predicted a large dynamical\nspin accumulation if the transverse anisotropy is present.\nOn the other hand, for the easy plane type of magnetic\nanisotropy, we have shown that, with increasing the spin\npolarization of the leads, there is a transition from a spin\ndoublet to higher-spin state, which results in a suppres-\nsion of the Kondo resonance. Interestingly, the suppres-\nsion of the low-frequency conductance occurs in a non-monotonic fashion, which is an indication of a highly non-\ntrivial competition between the intrinsic and spintronic\ncontributions to the magnetic anisotropy of the molecule.\nIn addition, we have also performed the analysis as-\nsuming that the exchange coupling between the orbital\nlevel and magnetic core is of antiferromagnetic type.\nThen, in the absence of magnetic anisotropy and for\nnonmagnetic leads, the system exhibits two-stage Kondo\ne\u000bect. We have shown that \fnite magnetic anisotropy\nof the molecule leads to the suppression of both the\nfrequency-dependent conductance and the e\u000bect of dy-\nnamical spin accumulation.\nFinally, we recall that all the considerations presented\nin this paper were performed for the particle-hole symme-\ntry point of the model. However, it is worth emphasizing\nthat the results discussed in the case of the antiparal-\nlel magnetic con\fguration are qualitatively valid also out\nof the particle-hole symmetry point, provided the sys-\ntem is in the Kondo regime (the orbital level is singly\noccupied). On the contrary, in the case of the paral-\nlel magnetic con\fguration, detuning from the symmetry\npoint results in a development of the e\u000bective dipolar ex-\nchange \feld. Because this \feld acts in a similar fashion to\nexternal magnetic \feld and splits the orbital level, most\nof the presented e\u000bects will become suppressed once the\nmagnitude of exchange \feld surpasses the relevant energy\nscales for considered phenomena. Last but not least, we\nwant to notice that, since the position of the orbital level\ncan be tuned by a gate voltage, the e\u000bects predicted in\nthis paper can be tested in near future with present-day\nexperimental techniques.\nACKNOWLEDGMENTS\nWork supported by the Polish Ministry of Science and\nEducation as Iuventus Plus project (IP2014 030973) in\nyears 2015-2017 (A.P., M.M.) and the Polish National\nScience Centre from funds awarded through the decision\nNo. DEC-2013/10/E/ST3/00213 (I.W.). M.M. also ac-\nknowledges \fnancial support from the Polish Ministry\nof Science and Education through a young scientist fel-\nlowship (0066/E- 336/9/2014) and from the Knut and\nAlice Wallenberg Foundation. Part of calculations was\nperformed at Pozna\u0013 n Supercomputing and Networking\nCenter.\n[1] G. D. Scott and D. Natelson, \\Kondo resonances in\nmolecular devices,\" ACS Nano 4, 3560{3579 (2010).\n[2] A. C. Hewson, The Kondo problem to heavy fermions\n(Cambridge University Press, Cambridge, 1997).\n[3] P. Coleman, Introduction to Many-Body Physics (Cam-\nbridge University Press, Cambridge, 2015).[4] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu,\nD. Abusch-Magder, U. Meirav, and M. A. Kastner, \\The\nKondo e\u000bect in a single-electron transistor,\" Nature 391,\n156{159 (1998).\n[5] S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwen-\nhoven, \\A Tunable Kondo E\u000bect in Quantum Dots,\" Sci-\nence281, 540{544 (1998).18\n[6] S. Sasaki, S. De Franceschi, J. M. Elzerman, W. G.\nvan der Wiel, M. Eto, S. Tarucha, and L. P. Kouwen-\nhoven, \\Kondo e\u000bect in an integer-spin quantum dot,\"\nNature 405, 764{767 (2000).\n[7] J. Nyg\u0017 ard, D. H. Cobden, and P. E. Lindelof, \\Kondo\nphysics in carbon nanotubes,\" Nature 408, 342{346\n(2000).\n[8] J. Li, W. D. Schneider, R. Berndt, and B. Delley, \\Kondo\nscattering observed at a single magnetic impurity,\" Phys.\nRev. Lett. 80, 2893{2896 (1998).\n[9] N. Knorr, M. A. Schneider, L. Diekh oner, P. Wahl, and\nK. Kern, \\Kondo E\u000bect of Single Co Adatoms on Cu\nSurfaces,\" Phys. Rev. Lett. 88, 096804 (2002).\n[10] A. F. Otte, M. Ternes, K. von Bergmann, S. Loth,\nH. Brune, C. P. Lutz, C. F. Hirjibehedin, and A. J.\nHeinrich, \\The role of magnetic anisotropy in the Kondo\ne\u000bect,\" Nat. Phys. 4, 847{850 (2008).\n[11] M. Ternes, A. J. Heinrich, and W.-D. Schneider, \\Spec-\ntroscopic manifestations of the Kondo e\u000bect on single\nadatoms,\" J. Phys.: Condens. Matter 21, 053001 (2008).\n[12] P. Jacobson, T. Herden, M. Muenks, G. Laskin,\nO. Brovko, V. Stepanyuk, M. Ternes, and K. Kern,\n\\Quantum engineering of spin and anisotropy in mag-\nnetic molecular junctions,\" Nat. Commun. 6, 9536\n(2015).\n[13] W. Liang, M. P. Shores, M. Bockrath, J. R. Long, and\nH. Park, \\Kondo resonance in a single-molecule transis-\ntor,\" Nature 417, 725{729 (2002).\n[14] J. Park, A. N. Pasupathy, J. I. Goldsmith, C. Chang,\nY. Yaish, J. R. Petta, M. Rinkoski, J. P. Sethna, H. D.\nAbru~ na, P. L. McEuen, and D. C. Ralph, \\Coulomb\nblockade and the Kondo e\u000bect in single-atom transis-\ntors,\" Nature 417, 722{725 (2002).\n[15] S. Kubatkin, A. Danilov, M. Hjort, J. Cornil, J.-\nL. Br\u0013 edas, Ni. Stuhr-Hansen, P. Hedeg\u0017 ard, and\nT. Bj\u001crnholm, \\Single-electron transistor of a single or-\nganic molecule with access to several redox states,\" Na-\nture425, 698{701 (2003).\n[16] A. Zhao, Q. Li, L. Chen, H. Xiang, W. Wang, S. Pan,\nB. Wang, X. Xiao, J. Yang, J. G. Hou, and Q. Zhu,\n\\Controlling the Kondo E\u000bect of an Adsorbed Magnetic\nIon Through Its Chemical Bonding,\" Science 309, 1542{\n1544 (2005).\n[17] N. Roch, S. Florens, V. Bouchiat, W. Wernsdorfer, and\nF. Balestro, \\Quantum phase transition in a single-\nmolecule quantum dot,\" Nature 453, 633{637 (2008).\n[18] J. J. Parks, A. R. Champagne, T. A. Costi, W. W. Shum,\nA. N. Pasupathy, E. Neuscamman, S. Flores-Torres, P. S.\nCornaglia, A. A. Aligia, C. A. Balseiro, G. K.-L. Chan,\nH. D. Abru~ na, and D. C. Ralph, \\Mechanical control\nof spin states in spin-1 molecules and the underscreened\nKondo e\u000bect,\" Science 328, 1370{1373 (2010).\n[19] A. S. Zyazin, H. S. J. van der Zant, M. R. Wegewijs,\nand A. Cornia, \\High-spin and magnetic anisotropy sig-\nnatures in three-terminal transport through a single\nmolecule,\" Synth. Met. 161, 591{597 (2011).\n[20] T. Choi, M. Badal, S. Loth, J.-W. Yoo, C. P. Lutz,\nA. J. Heinrich, A. J. Epstein, D. G. Stroud, and J. A.\nGupta, \\Magnetism in single metalloorganic complexes\nformed by atom manipulation,\" Nano Lett. 14, 1196{\n1201 (2014).\n[21] R. Frisenda, R. Gaudenzi, C. Franco, M. Mas-Torrent,\nC. Rovira, J. Veciana, I. Alcon, S. T. Bromley,\nE. Burzur\u0013 \u0010, and H. S. J. van der Zant, \\Kondo E\u000bect in aNeutral and Stable All Organic Radical Single Molecule\nBreak Junction,\" Nano Lett. 15, 3109{3114 (2015).\n[22] L. Liu, K. Yang, Y. Jiang, B. Song, W. Xiao, S. Song,\nS. Du, M. Ouyang, W. A. Hofer, A. H. Castro Neto, and\nH.-J. Gao, \\Revealing the Atomic Site-Dependent g Fac-\ntor within a Single Magnetic Molecule via the Extended\nKondo E\u000bect,\" Phys. Rev. Lett. 114, 126601 (2015).\n[23] L. Bogani and W. Wernsdorfer, \\Molecular spintronics\nusing single-molecule magnets,\" Nat. Mater. 7, 179{186\n(2008).\n[24] J. Bartolom\u0013 e, F. Luis, and J. F. Fern\u0013 andez, eds., Molec-\nular magnets: Physics and Application , NanoScience and\nTechnology (Springer, Heidelberg, 2014).\n[25] A. A. Khajetoorians and A. J. Heinrich, \\Toward single-\natom memory,\" Science 352, 296{297 (2016).\n[26] F. Donati, S. Rusponi, S. Stepanow, C. W ackerlin,\nA. Singha, L. Persichetti, R. Baltic, K. Diller, F. Patthey,\nE. Fernandes, J. Dreiser, \u0014Z.\u0014Sljivan\u0014 canin, K. Kummer,\nC. Nistor, P. Gambardella, and H. Brune, \\Magnetic re-\nmanence in single atoms,\" Science 352, 318{321 (2016).\n[27] F. D. Natterer, K. Yang, W. Paul, P. Willke, T. Choi,\nT. Greber, A. J. Heinrich, and C. P. Lutz, \\Reading\nand writing single-atom magnets,\" Nature 543, 226{228\n(2017).\n[28] M. Mannini, F. Pineider, P. Sainctavit, C. Danieli,\nE. Otero, C. Sciancalepore, A. M. Talarico, M. A. Ar-\nrio, A. Cornia, D. Gatteschi, and R. Sessoli, \\Magnetic\nmemory of a single-molecule quantum magnet wired to a\ngold surface,\" Nat. Mater. 8, 194{197 (2009).\n[29] R. Vincent, S. Klyatskaya, M. Ruben, W. Wernsdorfer,\nand F. Balestro, \\Electronic read-out of a single nuclear\nspin using a molecular spin transistor,\" Nature 488, 357{\n360 (2012).\n[30] A. Cornia and P. Seneor, \\Spintronics: The molecular\nway,\" Nature Mater. 16, 505{506 (2017).\n[31] C. A. P. Goodwin, F. Ortu, D. Reta, N. F. Chilton, and\nD. P. Mills, \\Molecular magnetic hysteresis at 60 kelvin\nin dysprosocenium,\" Nature 548, 439{442 (2017).\n[32] D. Gatteschi, R. Sessoli, and J. Villain, Molecular nano-\nmagnets (Oxford University Press, New York, 2006).\n[33] M. Misiorny, E. Burzur\u0013 \u0010, R. Gaudenzi, K. Park, M. Lei-\njnse, M. R. Wegewijs, J. Paaske, A. Cornia, and H. S. J.\nvan der Zant, \\Probing transverse magnetic anisotropy\nby electronic transport through a single-molecule mag-\nnet,\" Phys. Rev. B 91, 035442 (2015).\n[34] M. Mannini, F. Pineider, C. Danieli, F. Totti, L. So-\nrace, Ph. Sainctavit, M.-A. Arrio, E. Otero, L. Joly, J. C.\nCezar, A. Cornia, and R. Sessoli, \\Quantum tunnelling\nof the magnetization in a monolayer of oriented single-\nmolecule magnets,\" Nature 468, 417{421 (2010).\n[35] R. \u0014Zitko, R. Peters, and Th. Pruschke, \\Properties of\nanisotropic magnetic impurities on surfaces,\" Phys. Rev.\nB78, 224404 (2008).\n[36] R. \u0014Zitko and Th. Pruschke, \\Many-particle e\u000bects in ad-\nsorbed magnetic atoms with easy-axis anisotropy: The\ncase of Fe on the CuN/Cu (100) surface,\" New J. Phys.\n12, 063040 (2010).\n[37] M. Misiorny, I. Weymann, and J. Barna\u0013 s, \\Interplay of\nthe Kondo E\u000bect and Spin-Polarized Transport in Mag-\nnetic Molecules, Adatoms, and Quantum Dots,\" Phys.\nRev. Lett. 106, 126602 (2011).\n[38] M. Misiorny, I. Weymann, and J. Barna\u0013 s, \\In\ru-\nence of magnetic anisotropy on the Kondo e\u000bect and\nspin-polarized transport through magnetic molecules,19\nadatoms, and quantum dots,\" Phys. Rev. B 84, 035445\n(2011).\n[39] C. Romeike, M. R. Wegewijs, W. Hofstetter, and\nH. Schoeller, \\Quantum-Tunneling-Induced Kondo Ef-\nfect in Single Molecular Magnets,\" Phys. Rev. Lett. 96,\n196601 (2006).\n[40] C. Romeike, M. R. Wegewijs, W. Hofstetter, and\nH. Schoeller, \\Kondo-transport spectroscopy of single\nmolecule magnets,\" Phys. Rev. Lett. 97, 206601 (2006),\nsee also: Phys. Rev. Lett 106, 019902 (2011).\n[41] M. N. Leuenberger and E. R. Mucciolo, \\Berry-Phase\nOscillations of the Kondo E\u000bect in Single-Molecule Mag-\nnets,\" Phys. Rev. Lett. 97, 126601 (2006).\n[42] G. Gonz\u0013 alez and M. N. Leuenberger, \\Berry-phase block-\nade in single-molecule magnets,\" Phys. Rev. Lett. 98,\n256804 (2007).\n[43] G. Gonz\u0013 alez, M. N. Leuenberger, and E. R. Mucci-\nolo, \\Kondo e\u000bect in single-molecule magnet transistors,\"\nPhys. Rev. B 78, 054445 (2008).\n[44] M. Misiorny and I. Weymann, \\Transverse anisotropy\ne\u000bects on spin-resolved transport through large-spin\nmolecules,\" Phys. Rev. B 90, 235409 (2014).\n[45] J. I. Romero, E. Vernek, G. B. Martins, and E. R. Muc-\nciolo, \\Magnetic \feld modulated Kondo e\u000bect in a single-\nmagnetic-ion molecule,\" Phys. Rev. B 90, 195417 (2014).\n[46] M. Sindel, W. Hofstetter, J. von Delft, and M. Kin-\ndermann, \\Frequency-Dependent Transport through a\nQuantum Dot in the Kondo Regime,\" Phys. Rev. Lett.\n94, 196602 (2005).\n[47] A. I. T\u0013 oth, L. Borda, J. von Delft, and G. Zar\u0013 and, \\Dy-\nnamical conductance in the two-channel Kondo regime of\na double dot system,\" Phys. Rev. B 76, 155318 (2007).\n[48] C. P. Moca, I. Weymann, and G. Zar\u0013 and, \\Theory of\nfrequency-dependent spin current noise through corre-\nlated quantum dots,\" Phys. Rev. B 81, 241305 (2010).\n[49] C. P. Moca, P. Simon, C. H. Chung, and G. Zar\u0013 and,\n\\Nonequilibrium frequency-dependent noise through a\nquantum dot: A real-time functional renormalization\ngroup approach,\" Phys. Rev. B 83, 201303 (2011).\n[50] C. P. Moca, I. Weymann, and G. Zarand, \\Theory of\nac spin current noise and spin conductance through a\nquantum dot in the Kondo regime: Equilibrium case,\"\nPhys. Rev. B 84, 235441 (2011).\n[51] I. Weymann and C. P. Moca, \\Frequency-dependent con-\nductance of Kondo quantum dots coupled to ferromag-\nnetic leads,\" J. Appl. Phys. 109, 07C704 (2011).\n[52] C. P. Moca, P. Simon, Chung-Hou Chung, and\nG. Zar\u0013 and, \\Finite-frequency-dependent noise of a quan-\ntum dot in a magnetic \feld,\" Phys. Rev. B 89, 155138\n(2014).\n[53] R. Chirla and C. P. Moca, \\Finite-frequency thermoelec-\ntric response in strongly correlated quantum dots,\" Phys.\nRev. B 89, 045132 (2014).\n[54] B. Hemingway, S. Herbert, Mi. Melloch, and A. Kogan,\n\\Dynamic response of a spin-1/2 Kondo singlet,\" Phys.\nRev. B 90, 125151 (2014).\n[55] L. G. G. V. Dias da Silva and R. de Sousa, \\Spin versus\ncharge noise from kondo traps,\" Phys. Rev. B 92, 085123\n(2015).\n[56] A. P lomi\u0013 nska, M. Misiorny, and I. Weymann, \\Spin-\nresolved dynamical conductance of a correlated large-spin\nmagnetic molecule,\" Phys. Rev. B 95, 155446 (2017).\n[57] R. Kubo, M. Toda, and N. Hashitsume, Statistical\nphysics II: Nonequilibrium statistical mechanics , 1st ed.,Springer Series in Solid-State Sciences, Vol. 31 (Springer,\nHeidelberg, 1986).\n[58] K. G. Wilson, \\The renormalization group: Critical phe-\nnomena and the kondo problem,\" Rev. Mod. Phys. 47,\n773 (1975).\n[59] R. Bulla, T. A. Costi, and Th. Pruschke, \\Numerical\nrenormalization group method for quantum impurity sys-\ntems,\" Rev. Mod. Phys. 80, 395 (2008).\n[60] F. Elste and C. Timm, \\Transport through anisotropic\nmagnetic molecules with partially ferromagnetic leads:\nSpin-charge conversion and negative di\u000berential conduc-\ntance,\" Phys. Rev. B 73, 235305 (2006).\n[61] M. Misiorny, I. Weymann, and J. Barna\u0013 s, \\Spin e\u000bects in\ntransport through single-molecule magnets in the sequen-\ntail and cotunneling regimes,\" Phys. Rev. B 79, 224420\n(2009).\n[62] O. Legeza, C. P. Moca, A. I. T\u0013 oth, I. Weymann, and\nG. Zar\u0013 and, \\Manual for the \rexible DM-NRG code,\"\narXiv:0809.3143v1 (2008), (the open access Budapest\ncode is available at http://www.phy.bme.hu//~dmnrg/.\n[63] A. I. T\u0013 oth, C. P. Moca, O. Legeza, and G. Zar\u0013 and, \\Den-\nsity matrix numerical renormalization group for non-\nAbelian symmetries,\" Phys. Rev. B 78, 245109 (2008).\n[64] W. C. Oliveira and L. N. Oliveira, \\Generalized numer-\nical renormalization-group method to calculate the ther-\nmodynamical properties of impurities in metals,\" Phys.\nRev. B 49, 11986 (1994).\n[65] A. Weichselbaum and J. von Delft, \\Sum-rule conserv-\ning spectral functions from the numerical renormalization\ngroup,\" Phys. Rev. Lett. 99, 76402 (2007).\n[66] F. D. M. Haldane, \\Scaling Theory of the Asymmetric\nAnderson Model,\" Phys. Rev. Lett. 40, 416 (1978).\n[67] J. Martinek, Y. Utsumi, H. Imamura, J. Barna\u0013 s,\nS. Maekawa, J. K onig, and G. Sch on, \\Kondo e\u000bect\nin quantum dots coupled to ferromagnetic leads,\" Phys.\nRev. Lett. 91, 127203 (2003).\n[68] M. Misiorny, M. Hell, and M. R. Wegewijs, \\Spintronic\nmagnetic anisotropy,\" Nat. Phys. 9, 801{805 (2013).\n[69] P. Coleman and C. P\u0013 epin, \\Singular Fermi liquid behav-\nior in the underscreened Kondo model,\" Phys. Rev. B\n68, 220405 (2003).\n[70] A. Posazhennikova and P. Coleman, \\Anomalous con-\nductance of a spin-1 quantum dot,\" Phys. Rev. Lett. 94,\n036802 (2005).\n[71] W. Koller, A. C. Hewson, and D. Meyer, \\Singular\ndynamics of underscreened magnetic impurity models,\"\nPhys. Rev. B 72, 045117 (2005).\n[72] M. Misiorny, I. Weymann, and J. Barna\u0013 s, \\Under-\nscreened Kondo e\u000bect in S= 1 magnetic quantum dots:\nExchange, anisotropy, and temperature e\u000bects,\" Phys.\nRev. B 86, 245415 (2012).\n[73] A. Posazhennikova, B. Bayani, and P. Coleman, \\Con-\nductance of a spin-1 quantum dot: The two-stage Kondo\ne\u000bect,\" Phys. Rev. B 75, 245329 (2007).\n[74] R. \u0014Zitko, \\Kondo screening in high-spin side-coupled two-\nimpurity clusters,\" J. Phys.: Condens. Matter 22, 026002\n(2009).\n[75] K. P. W\u0013 ojcik and I. Weymann, \\Two-stage Kondo ef-\nfect in T-shaped double quantum dots with ferromagnetic\nleads,\" Phys. Rev. B 91, 134422 (2015).\n[76] M. Misiorny, I. Weymann, and J. Barna\u0013 s, \\Tempera-\nture dependence of electronic transport through molec-\nular magnets in the Kondo regime,\" Phys. Rev. B 86,20\n035417 (2012).\n[77] L. I. Glazman and M. E. Raikh, \\Resonant Kondo trans-\nparency of a barrier with quasilocal impurity states,\"\nJETP. Lett. 47, 452{455 (1988).\n[78] M. Ternes, \\Spin excitations and correlations in scanning\ntunneling spectroscopy,\" New J. Phys. 17, 063016 (2015).\n[79] A. A. Khajetoorians, M. Valentyuk, M. Steinbrecher,\nT. Schlenk, A. Shick, J. Kolorenc, A. I. Lichtenstein,T. O. Wehling, R. Wiesendanger, and J. Wiebe, \\Tuning\nemergent magnetism in a Hund's impurity,\" Nat. Nan-\notechnol. 10, 958{964 (2015).\n[80] J. J. Parks, A. R. Champagne, G. R. Hutchison,\nS. Flores-Torres, H. D. Abru~ na, and D. C. Ralph, \\Tun-\ning the Kondo E\u000bect with a Mechanically Controllable\nBreak Junction,\" Phys. Rev. Lett. 99, 026601 (2007)."    },    {        "title": "1605.05211v1.Direct_observation_of_dynamic_modes_excited_in_a_magnetic_insulator_by_pure_spin_current.pdf",        "content": " 1Direct observation of dynamic mode s excited in a magnetic insulator \nby pure spin current \n \nV. E. Demidov1*, M. Evelt1, V. Bessonov2, S. O. Demokritov1,2, J. L. Prieto3, M. \nMuñoz4\n, J. Ben Youssef5, V. V. Naletov6,7, G. de Loubens6, O. Klein8, M. Collet9, P. \nBortolotti9, V. Cros9 and A. Anane9 \n1Institute for Applied Physics and Center for Nanotechnology, University of Muenster,  \n48149 Muenster, Germany \n2M.N. Miheev Institute of Metal Physics of Ural Branch of Russian Academy of \nSciences, Yekaterinburg 620041, Russia. \n3Instituto de Sistemas Optoelectr ónicos y Microtecnologa (UPM), Ciudad \nUniversitaria, Madrid 28040, Spain \n4IMM-Instituto de Microelectrónica de Madrid (CNM-CSIC), PTM, E-28760 Tres \nCantos, Madrid, Spain \n5Laboratoire de Magnétisme de Bretagne CNRS,  Université de Br etagne Occidentale, \n29285 Brest, France \n6Service de Physique de l’ État Condensé, CEA, CNRS, Université Paris-Saclay, CEA \nSaclay, 91191 Gif-sur-Yvette, France \n7Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation \n8INAC-SPINTEC, CEA/CNRS and Univ. Gre noble Alpes, 38000 Grenoble, France \n9Unité Mixte de Physique CNRS, Thales, Un iv. Paris Sud, Université Paris-Saclay, \n91767 Palaiseau, France \n \n  2Abstract: \nExcitation of magnetization dynamics by pur e spin currents has been recently \nrecognized as an enabling mechanism for spintronics and magnonics, which allows \nimplementation of spin-torque devices based on low-damping insulating magnetic \nmaterials. Here we report the first spat ially-resolved study of the dynamic modes \nexcited by pure spin current in nanometer-thick microscopic insulating Yttrium Iron \nGarnet disks. We show that these modes exhibit nonlinear self-b roadening preventing \nthe formation of the self-localized magnetic bullet, which plays a crucial  role in the \nstabilization of the single-mode magnetization oscillations in all-metallic systems. This peculiarity associated with the efficien t nonlinear mode coupling in low-damping \nmaterials can be among the main factors govern ing the interaction of  pure spin currents \nwith the dynamic magnetization in high-quality magnetic insulators. \n \n  3One of the decisive advantages of pure spin currents for the emerging \ntechnologies, such as spintronics1-3 and magnonics4-6, is the possibility to excite the \nmagnetization dynamics in insulating magnetic  materials by the spin-transfer torque7,8. \nThe breakthrough idea that the spin-transfer to rque can be utilized to  induce spin waves \nin magnetic insulators suggested in the pioneering work by Kajiwara et al. (Ref. 7) had \nan enormous impact on the developments  in the physics of magnetism and revived \ninterest to magnetic insulators, such as Y ttrium Iron Garnet (YIG).  The main advantage \nof this material is the unmatched small dynamic magnetic damping, which is expected \nto result in more efficient operation of spin-torque devices by enabling significant \nreduction of the density of the driving current  necessary for the onset of current-induced \nauto-oscillations. Although th is advantage should greatly simplify the implementation \nof YIG-based spin-torque device s, it took many years of inte nse research to achieve the \nspin-current induced excitation of cohere nt magnetization dynamics in this material9-10.  \nIn contrast, significantly mo re rapid progress has been achieved in the studies of \nall-metallic spin-current driven systems, which has led in the recent years to the \ndemonstration of a large variety of nano-de vices exhibiting highl y coherent magnetic \noscillations driven by the pure spin current11-18. This progress was gr eatly facilitated by \nthe flexible geometry of spin-current driven  systems enabling the direct magneto-optical \nimaging of the current-induced dynamics.  In particular, the spatially-resolved \nmeasurements have shown that the significant role in the stabilization of the single-mode current-induced oscillati ons is played by the nonlin ear spatial self-localization \neffects resulting in the forma tion of the so-called bullet mode\n11,19, which is a two-\ndimensional standing analogue of a spin-wave soliton20.  \nSince the dynamic nonlinearity of the spin system largely determines the \nbehaviors of the dynamic magnetization under th e influence of the spin current, the \npeculiarities of the nonlinear response of  low-damping YIG can be among the most  4important factors affecting the current -induced excitation of the magnetization \ndynamics in this material. In fact, it is  long-known that the low damping in YIG \nfacilitates the nonlinear couplin g between dynamic magnetic modes21. This \nphenomenon was found to strongly affect the formation of spin-wave solitons22 and \nresult in a suppression of the nonlin ear spatial self-loc alization phenomena23, which \nplay a crucial role in spin -torque driven nano-systems17.  \nHere we study experimentally the eff ects of the dynamic nonlinearity on the \nspatial characteristics of dynamic modes exci ted by the pure spin current in microscopic \nYIG disks. By using the direct spatially-reso lved imaging of the modes, we show that \nthe dynamic magnetization exhibits strong spatia l localization at the onset of the auto-\noscillations. However, just above the onset , the amplitude of the auto-oscillations \nrapidly saturates and the excited mode experiences a spatial broadening. This \nphenomenon is opposite to the nonlinear self -localization observed in all-metallic \nsystems and can be the main factor determ ining the response of the magnetization in \nlow-damping materials to the spin current.  We  attribute these behaviors to the efficient \nnonlinear mode coupling, which prevents the growth of the amplitudes of the dynamic \nmagnetization to the level necessary for th e onset of nonlinear se lf-localization and \nformation of the bullet mode. \n \nResults \nTest devices.  The schematic of our experiment is shown in Fig. 1.  The studied \ndevice consists of a YIG(20 nm)/Pt(8 nm) disk with a 2 m diameter  (details about the \npreparation of the YIG films can be found in  Ref. 24). The Pt la yer is electrically \ncontacted by using two Au(80 nm) electrodes with a 1 m wide gap between them. By  5applying an electric voltage betw een the electrodes, we inject the dc electric current into \nthe Pt layer. Because of the large difference in the electric conductance of the electrodes \nand the Pt film, the current is injected into  the Pt layer only in the electrode gap area. \nDue to the spin-Hall effect25,26, the in-plane electri cal current in the Pt film is converted \ninto a transverse spin accumulation. The asso ciated pure spin current is flowing in the \nout-of-plane direction and is injected into the YIG film7 resulting in a spin-transfer \ntorque (STT)27,28 on its magnetization M. The device is magnetized by the in-plane \nstatic magnetic field H0 applied perpendicular to the di rection of the dc current flow. \nFor positive currents, as defined in Fig. 1, the polarization of the spin current corresponds to the STT compensating the dyna mic damping in the magnetic film. For \nsufficiently large dc currents, the damping becomes completely compensated, which results in the excitation of magnetic auto-oscillations\n10. \n \nMagnetooptical measurements. To observe the oscillations induced by the spin \ncurrent we use micro-focus Br illouin light scattering (BLS)29 spectroscopy, which \nenables the spectrally- and the spatially -resolved measurements of the dynamic \nmagnetization. We focus probing laser light  into a diffraction-limited spot on the \nsurface of YIG (Fig. 1) and analyze the spectru m of light inelastically  scattered from the \nmagnetization oscillations. The resulting BL S signal at the selected frequency is \nproportional to the intensity of the dynamic magnetization at this frequency, at the \nlocation of the laser spot. The wavelength of the laser is chosen to be 473 nm, which \nprovides high sensitivity of the method for m easurements with ultra-thin YIG. The \npower of the probing light is  as low as 0.05 mW, which gua rantees negligible laser-\ninduced heating of the sample.  6In Fig. 2a we show BLS spectra recorded by placing the probing laser spot in the \nmiddle of the YIG disk for different values of the dc current I. The spectrum shown by \nsolid black symbols recorded for I = 0 characterizes the magnetic  excitations existing at \nroom temperature due to the thermal fluctuat ions. Similar to the all-metallic systems11, \nthe application of the dc current results in the gradual enhancement of the magnetic \nfluctuations followed at I = IC  8 mA by an abrupt emergen ce of a narrow intense peak \n(blue squares in Fig. 2a) marking the onset of  the current-induced auto-oscillations of \nthe YIG magnetization. The intensity of the au to-oscillation peak exceeds that of the \nthermal fluctuations at I = 0 by about two orders of ma gnitude.  Further increase in I \nresults in the lowering of the central frequency of the peak and in its spectral broadening \n(red diamonds in Fig. 2a).  \nTo obtain information about the dynamical  modes contributing to the current-\ninduced auto-oscillations, we perform two-dimensional mapping of the dynamic \nmagnetization in the YIG disk by rastering the probing laser spot in the two lateral \ndirections. In Fig. 2b we show a typical spatial map obtained for I = IC+1 mA. As seen \nfrom these data, the auto-oscillations are strongly localized in the y-direction in the area \ncorresponding to the gap between the electrode s, whose edges are marked in Fig. 2b by \nthe horizontal lines. This localization can be attributed to the formation of an effective confining potential due to the joint effects of the spin- and elec trical currents, as \ndiscussed below. In contrast, there is no pr onounced localization of the auto-oscillations \nin the x-direction. They clearly extend over the entire active device area, where the \ninjection of the spin current takes place.  \nFigure 3 illustrates the spatial localization of the auto-oscillations for different \nvalues of the dc current. x-profiles of the oscillation in tensity recorded at different I \n(Fig. 3a) show that the auto-oscillations do not always occupy the entire active device  7area. At currents close to IC, they are strongly localized in the middle of the gap. \nHowever, with the increase in I, they rapidly expand in the x direction. This process is \ncharacterized in detail in Fig. 3b, which show s the current dependence of the full width \nat half maximum (FWHM) of the x-profiles. This width lin early increases from 0.5 m \nto about 1.4 m for I varying from IC to IC+1 mA and then saturate s at the value of 1.5 \nm.  \nWe emphasize that these behaviors are in contradiction with the typical \nmanifestations of the dynamic magnetic nonlinea rity, which is expected  to result in the \nspatial self-localizati on of the intense oscillations and lead to a formation of the spin-\nwave bullet19 experimentally observed in all-metal lic systems driven by the pure spin \ncurrent11,17. The absence of the self-localization in  YIG can be attributed to a nonlinear \nlimitation of the amplitude of the auto-oscillations. Indeed, if this amplitude is limited by any other nonlinear phenomenon, the self -localization mechanism requiring the \ndynamic magnetization to be comparable with the static magnetization can be \nsuppressed. \nThis scenario is supported by the data of Fig. 3c. The maximum intensity of the \nauto-oscillations (up triangles in Fig. 3c) saturates immediately after the onset and then \nstays nearly constant over the entire used range of I. In contrast, the integral intensity \n(down triangles in Fig. 3c) exhi bits a gradual increase up to I = 9.2 mA. This indicates \nthat the amplitude of the dynamic mode excite d at the onset cannot grow above a certain \nlevel and that the further increase in I results in the energy flow into other dynamical \nmodes leading to the formation of a str ongly nonlinear spatially-extended mode, which \ncannot be treated as a linear combin ation of the eigenmodes of the system\n23. These \nbehaviors can be attributed to  the efficient nonlinear mode  coupling stimulated by the  8small dynamic damping in YIG. We emph asize that the same mechanism was \npreviously found to result in the formation of traveling dark solitons in a medium \ncharacterized by the attractiv e nonlinearity, where the se lf-localization effects are \nexpected to lead to the fo rmation of bright solitons22. \nAccording to the described scenario, the current-induced auto-oscillations are also \nexpected to demonstrate a spatial broadening in the y-direction. The corresponding \nexperimental data are shown in Fig. 4. We note that the y-profiles of the auto-\noscillations (Fig. 4a) are mostly determined  by the confining potential, which prevents \nthe extension of the dynami c excitations outside the re gion of the gap between the \nelectrodes. Because of the limite d spatial resolution of the used technique, which can be \nestimated as 240-250 nm, one cannot quantit atively analyze the broadening of the y-\nprofiles with the increase in the dc current. However, from Fig. 4b it is seen, that the y-\nwidth also tends to increase with increasing I.  \n \nDiscussion \nFinally, we address the nature of the lo calized dynamic mode, wh ich is excited in \nthe YIG disk at I = IC before the nonlinear broadening takes place. To understand why \nthis mode shows a strong localization at  the center of the disk, we perform \nferromagnetic-resonance (FMR) measurements ba sed on the excitation of the system by \na dynamic magnetic field created by an additional 5 m wide stripe antenna aligned \nparallel to the gap between the electrodes (s ee also Supplementary Figure 1). In Figs. 5a \nand 5b we show the spatial maps of the field-driven FMR mode recorded by BLS \nwithout any dc current in Pt and at I = 7 mA < IC, respectively. The maps clearly \ndemonstrate that the FMR mode initially spread over the entire disk becomes localized  9at its center when the dc curr ent is applied. We emphasize that the measurements were \nperformed at sufficiently small microwav e powers to exclude any nonlinear self-\nlocalization effects30. Therefore, the observed localizati on can only be attributed to the \njoint effect of the Oersted field of th e dc current, the local reduction of the \nmagnetization due to the Joule heating, and the partial compensation of the dynamic \ndamping by the pure spin current injected into the YIG film.  \nTo prove this assumption we perform micromagnetic simulations using the \nsoftware package MuMax3 (Ref. 31) (see Me thods for details). The obtained spatial \nmaps of the intensity of the dynamic magne tization in the FMR mode are shown in \nFigs. 5c and 5d for the cases of I  = 0 and 7 mA, respectively. As seen from the \ncomparison with the experimental data (Fi g. 5a and 5b), the simulations reproduce well \nthe localization behavior and clear ly show that the effects of the driving current result in \na shrinking of the FMR mode in both lateral directions resulting in the formation of a \nstrongly localized linear mode of the system. It  is this mode that is most probably first \nexcited by the pure spin at the onset of th e auto-oscillations (s ee corresponding profiles \nin Figs. 3a and 4a). It is to be noticed that these current-induced  localization effects \ncounteract the nonlinear broadening. Therefore,  in a properly designed system, they can \nbe utilized to suppress this detrimental phenomenon. \nIn conclusion, we demonstrate that th e dynamic nonlinearity of the low-damping \nYIG films leads to a nonlinear self-broad ening of the current -induced oscillations \ninstead of the nonlinear self -localization typical for me tallic ferromagnetic films \ncharacterized by a relatively large dynami c damping. Since the self-localization \nmechanisms play a crucial role for the stab ilization of the singl e-mode current-induced \nauto-oscillations, in order to achieve highly-coherent oscillat ions in the YIG films, it is \nnecessary to utilize additiona l confining mechanisms, such as the patterning of the film  10into a nano-size element or creation of an additional confining potential. Our results \nshine light on the physical mechanisms responsible for the peculiarities of the interaction of magnetization in YIG with pure spin currents and create a base for implementation of efficient insulato r-based spin-torque nano-devices. \n \nMethods \nSample fabrication.   20 nm thick YIG films were grown by the pulsed laser deposition \non Gadolinium Gallium Garnet (GGG) (111) substrates\n24. The 8 nm thick layer of Pt \nwas deposited using dc magnetron sputteri ng. The dynamical properties of bare YIG \nfilms and YIG/Pt bilayers have been de termined from broadband FMR measurements. \nThe YIG/Pt microdiscs and the Au(80 nm)/ Ti(20 nm) electrodes were defined by e-\nbeam lithography. The system was insulated by a 300 nm thick SiO 2 layer, and a \nbroadband 5 m wide microwave antenna made of 250 nm thick Au was defined on top \nof the system using optical lithography. \nMagneto-optical measurements.   Micro-focus  BLS measurements were performed by \nfocusing light produced by a co ntinuous-wave single-frequenc y laser operating at the \nwavelength of 473 nm into a diffraction-limite d spot. The light scat tered from magnetic \noscillations was analyzed by a six-pass Fabry-Perot interferometer TFP-1 (JRS \nScientific Instruments, Switzerland) to ob tain information about  the BLS intensity \nproportional to the square of the amplitude of the dynamic magnetization at the location \nof the probing spot. By raster ing the spot over the surface of the sample using a closed-\nloop piezo-scanner, two-dimensional maps of the dynamic magnetizat ion were recorded \nwith the spatial step size of 100 nm. The positioning system  was stabilized by custom-\ndesigned software-controlled active feedback, providing long-term spatial stability of \nbetter than 50 nm. All measurements we re performed at room temperature. \nSimulations. The micromagnetic simulations were performed by using the \nsoftware package MuMax3 (Ref. 31). The co mputational domain with dimensions of \n220.02 m3 was discretized into 10 1010 nm3 cells. The FMR mode was excited by  11a 50 ps wide pulse of the out-of-plane magne tic field. The spatia l map of the dynamic \nmagnetization was reconstructed based on the Fourier analysis  of the dynamic response \nof the magnetization. Standard valu e for the exchange stiffness of 3.66 10-12 J/m was \nused, while the value of th e saturation magnetization 4 M0=2.2 kG and the Gilbert \ndamping constant of 2 10-3 were determined from the FMR measurements. For the case \nof the disk free from the influence of the driving current, the  saturation magnetization \nM,  the damping constant , and the static field H0=1000 Oe were considered to be \nuniform across the disk area. To take into acco unt the effects of the driving current, the \nparameters M, , and H0 were reduced in the central rectangular region corresponding \nto the area of the gap between the electrodes. The reduction of M was taken from the \nresults of the FMR measurements (Supplem entary Figure 1), the reduction of  was \ncalculated assuming the linear dependence of effective damping on the current strength, \nand the reduction of the field was cal culated by using the Ampere’s law. \n   12References \n \n1. Maekawa, S. Concepts in Spin Electronics (Oxford University Press, Oxford, 2006). \n2. Ralph, D. C. & Stiles, M. D. Spin transfer torques. J. Magn. Magn. Mater. 320, \n1190–1216 (2008). \n3. Locatelli, N., Cros, V., & Grollier , J., Spin-torque building blocks. Nature Mater.  13, \n11-20 (2014). \n4. Kruglyak, V. V., Demokritov, S.  O., & Grundler, D. Magnonics. J. Phys. D: Appl. \nPhys.  43, 264001 (2010). \n5. Urazhdin, S. et al. , Nanomagnonic devices based on the spin-transfer torque. Nature \nNanotech.  9, 509-513 (2014). \n6. Chumak, A.V., Vasyuchka, V.I., Serga, A.A. & Hillebrands, B. Magnon spintronics. \nNature Phys.  11, 453-461 (2015). \n7. Kajiwara, Y. et al. Transmission of electrical signals  by spin-wave interconversion in \na magnetic insulator. Nature 464, 262–266 (2010). \n8. Xiao J.,  Bauer, G.E.W, Spin-Wave Ex citation in Magnetic Insulators by Spin-\nTransfer Torque. Phys. Rev. Lett. 108, 217204 (2012) \n9. Hamadeh,  A. et al. , Full Control of the Spin-Wave Damping in a Magnetic Insulator \nUsing Spin-Orbit Torque, Phys. Rev. Lett. 113, 197203 (2014). \n10. Collet, M., et al. ,  Generation of coherent spin-wave modes in Yttrium Iron Garnet \nmicrodiscs by spin-orbit torque. Nat. Commun.  7, 10377 (2016). \n11. Demidov, V. E. et al.  Magnetic nano-oscillator driven by pure spin current. Nature \nMater.  11, 1028-1031 (2012).  1312. Liu, L., Pai, C.-F., Ralph, D. C., & Buhrma n, R. A. Magnetic Os cillations Driven by \nthe Spin Hall Effect in 3-Termin al Magnetic Tunnel Junction Devices. Phys. Rev. \nLett. 109, 186602 (2012). \n13. Liu, R. H., Lim, W. L., & Urazhdin, S. Spectral Characteristics of the Microwave \nEmission by the Spin Hall Nano-Oscillator. Phys. Rev. Lett.  110, 147601 (2013). \n14. Demidov, V. E., Urazhdin, S., Zholud, A ., Sadovnikov, A. V. & Demokritov, S. O. \nNanoconstriction-based Spin-Hall nano-oscillator. Appl. Phys. Lett.  105, 172410 \n(2014). \n15. Duan, Z. et al.  Nanowire spin torque oscillato r driven by spin orbit torques. Nat. \nCommun.  5, 5616 (2014). \n16. Demidov, V. E. et al.  Spin-current nano-oscillator ba sed on nonlocal spin injection. \nSci. Rep.  5, 8578 (2015). \n17. Yang, L. et al. , Reduction of phase noise in nanow ire spin orbit torque oscillators. \nSci. Rep.  5, 16942 (2015). \n18. Demidov, V. E. et al.   Excitation of coherent propa gating spin waves by pure spin \ncurrents. Nat. Commun.  7, 10446 (2016).  \n19. Slavin, A. & Tiberkevich, V. Spin wave mo de excited by spin-polarized current in a \nmagnetic nanocontact is a standing self-localized wave bullet. Phys. Rev. Lett.   95, \n237201 (2005). \n20. Kalinikos, B. A., Kovshikov, N. G., & Slavin, A. N., Envelope solitons and \nmodulation instability of dipole-exchange magnetization waves in yttrium iron garnet \nfilms. Sov. Phys. JETP  67, 303-312 (1988). \n21. Suhl, H. The theory of ferromagne tic resonance at high signal powers. Journal of \nPhysics and Chemistry of Solids  1, 209-227 (1957).  1422. Scott, M. M., Kostylev, M.P., Kalinikos, B.A., & Patton, C.E., Excitation of bright \nand dark envelope solitons for magnetost atic waves with attractive nonlinearity. \nPhys. Rev. B 71, 174440 (2005). \n23. Demidov, V. E., Hansen, U.-H., & Demokr itov, S. O., Spin-wave eigenmodes of a \nsaturated magnetic square at  different precession angles, Phys. Rev. Lett.  98, 157203 \n(2007). \n24. d’Allivy Kelly, O. et al.  Inverse spin hall effect in nanometer-thick yttrium iron \ngarnet/Pt system. Appl. Phys. Lett.  103, 082408 (2013). \n25. Dyakonov, M. I. & Perel, V. I. Possibility of orienting electron spins with current. \nSov. Phys. JETP Lett.  13, 467-469 (1971).  \n26. Hirsch, J.E. Spin Hall Effect. Phys. Rev. Lett.  83, 1834-1837 (1999). \n27. Slonczewski, J. C. Current-drive n excitation of magnetic multilayers. J. Magn. \nMagn. Mater.  159, L1–L7 (1996). \n28. Berger, L. Emission of spin waves by a ma gnetic multilayer trav ersed by a current. \nPhys. Rev. B  54, 9353–9358 (1996). \n29. Demidov, V. E. & Demokritov S. O. Ma gnonic waveguides studied by microfocus \nBrillouin light scattering. IEEE Trans. Mag.  51, 0800215 (2015). \n30. Jungfleisch, M. B. et al.  Large Spin-Wave Bullet in a Ferrimagnetic Insulator \nDriven by the Spin Hall Effect. Phys. Rev. Lett.  116, 057601 (2016). \n31. Vansteenkiste, A., Leliaert, J., Dvornik, M., Helsen, M., Garcia-Sanchez, F., & Van \nWaeyenberge, B., The design and verification of mumax3, AIP Advances  4, 107133 \n(2014). \n * Corresponding author:  Correspondence and requests for materials should be \naddressed to V.E.D. ( demidov@uni-muenster.de ).  15 \nAcknowledgements:  We acknowledge E. Jacquet, R. Lebourgeois, R. Bernard and A. \nH. Molpeceres for their contribution to samp le growth, and O. d’Allivy Kelly and A. \nFert for fruitful discussion. This resear ch was partially supported by the Deutsche \nForschungsgemeinschaft, the ANR Grant Tr inidad (ASTRID 2012 program), and the \nprogram Megagrant № 14.Z50.31.0025 of the Russian Ministry of Education and \nScience. M.C. acknowledges DGA for fina ncial support. V. V. N. acknowledges \nsupport from the Competitive Growth of KFU. \n \nAuthor Contributions:  VED, ME and VB performed BLS measurements and data \nanalysis.  ME additionally performed micromagnetic simulations. AA and VC \nsupervised the growth of the YIG films. JBY performed the magnetic characterizations. \nMC, VVN, JLP, MM, GdL and AA designed, nanofabricated, and characterized the \nsamples. VVN, PB, JBY, VED, SOD, VC , AA, GdL and OK in itiated and conducted \nthe project. All authors co-wrote and discussed the manuscript.  \n \nAdditional Information:  The authors have no compe ting financial interests.  16Figure legends \n \nFigure 1. Experimental layout. YIG(20 nm)/Pt(8 nm) disk with the diameter of 2 m \nis electrically contacted by using tw o Au(80 nm) electrodes with a 1 m wide gap \nbetween them. The in-plane dc electrical current in the Pt fi lm is converted  into the out-\nof-plane spin current by the spin-Hall effect. The spin current is injected into the YIG film and exerts the spin-transfer torque on its magnetization M resulting in the \nexcitation of magnetic auto-oscillations. Th e excited oscillations are detected by the \nmagneto-optical technique utilizing probing  laser light focused through the sample \nsubstrate.  \nFigure 2. Spectral and spatial characte ristics of the auto-oscillations.  a, BLS spectra \nrecorded by placing the probing  laser spot in the middle of the YIG disk for different \nstrength of the dc current I: black solid symbols – I = 0, blue open squares – I = I\nC  8 \nmA, red open diamonds – I = IC+1 mA. The intensity of the peak for I = IC is by about \ntwo orders of magnitude larg er compared to that for I = 0. Lines are the guides for the \neye. The spectral linewidth of the peaks is determined by the limited spectral resolution of the measurement apparatus. The data were recorded at H\n0 = 1000 Oe. b, Typical \nspatial map of the intensity of current-induced magnetic auto-oscillations in the YIG disk. The contours of the disk and the edges of the electrodes are shown by white lines. \nThe map was recorded for I = I\nC+1 mA and H0 = 1000 Oe.  \nFigure 3.  Dependence of the spatial localization of the auto-oscillations on the dc \ncurrent.  a, Spatial profiles of the oscillation in tensity in the direction along the gap \nbetween the electrodes recorded for different dc currents, as labeled. b, Current  17dependence of the full width at half ma ximum (FWHM) of the spatial profiles. c, \nCurrent dependences of the maximum intensity of the auto-oscillations detected in the center of the gap and that of the spatially-integ rated intensity. The data were recorded at \nH\n0 = 1000 Oe. Symbols are experimental data , curves are guides for the eye. \nFigure 4. Spatial localization of the au to-oscillations across the gap between the \nelectrodes.  a, Spatial profiles of the oscillation intensity recorded for different dc \ncurrents, as labeled. b, Current dependence of the full width at half maximum (FWHM) \nof the spatial profiles. The data were recorded at H0 = 1000 Oe. Symbols are \nexperimental data, curves  are guides for the eye. \nFigure 5. Spatial characteristics of the fi eld-driven ferromagnetic resonance mode. \na, and b, Spatial maps of the FMR mode measured by BLS at I = 0 and 7 mA, \nrespectively. The contours of the disk and the edges of the electrodes are shown by \nwhite lines. The maps were recorded at H0 = 1000 Oe. c, and d, Normalized spatial \nmaps of the intensity of the dynamic magne tization in the FMR mode obtained from \nmicromagnetic simulations. c corresponds to I = 0 and d corresponds to I = 7 mA.  \n   18\n\n \n  Fig. 1 \nFig. 2 \nFig. 3 Fig. 4 x, m-1 0 1BLS intensity, a.u.\n0.00.51.0\nICa\nbIC+2 mA IC+1 mA\n89 1 0FWHM, m\n0.51.01.5\nIC\nCurrent, mA89 1 0Maximum BLS \nintensity, a.u.\n0123\nIntegral BLS \nintensity, a.u.\n01c\nBLS Intensity, a.u.0.00.51.0\nx, m-1 0 1y,m\n-101c\nx, m-1 0 1-101d\n(mz/mzmax)2\n0.00.51.0-1 0 1y,m\n-101a\n-1 0 1-101bFrequency, GHz3456BLS Intensity, a.u.\n0.00.51.01.5\nIC\nx, m-1 0 1y,m\n-101a\nb\nBLS Intensity, a.u. 0.00.51.0Thermal (x50)IC+1 mA\nH0\ny, m-1 0 1BLS intensity, a.u.\n0.00.51.0\nICaIC+2 mA IC+1 mA\nCurrent, mA89 1 0FWHM, m\n0.40.50.6b\nFig. 5  19 \nSupplementary figures. \n \nSupplementary Figure 1. BLS measurements of the ferromagnetic resonance in the \nYIG disk.  In order to measure the ferromagnetic resonance (FMR) in the YIG disk, we \ntransmit a microwave current  through an additional 5 m wide stripe antenna, which \ncreates a nearly unif orm in-plane dynamic magnetic fiel d across the YI G disk.  By \nvarying the frequency of the excitation si gnal in the antenna and measuring the BLS \nintensity, we record the FMR curves and obtain the FMR frequency. Additionally, the \nelectrical dc current I<IC is applied through the Pt layer to analyze its influence on the \nFMR. The FMR frequency obtai ned for different currents is shown by up-triangles. The \nobserved variation of the fre quency originates from the Oers ted field of the current and \nthe reduction of the static magnetization M caused by the Joule heating of the sample. \nWe estimate the Oersted field from the Ampere’s law and calculate the current \ndependence of the magnetization by using the Kittel formula for the FMR frequency. \nThe results shown by the down-triangles indicate that M is reduced by about 8% at I=7 \nmA. The obtained dependence is nearly symmet rical with respect to the change of the \nsign of the current, in agreem ent with the expectations for the effects of the Joule \nheating. The data were obtained at H0=1000 Oe. Lines are guides for the eye. \nCurrent, mA-8 -6 -4 -2 0 2 4 6 8Frequency, GHz\n4.74.84.95.0\n4M, G\n200021002200"    },    {        "title": "1812.07610v1.Dynamical_response_of_the_Ising_model_to_the_amplitude_modulated_time_dependent_magnetic_field.pdf",        "content": "arXiv:1812.07610v1  [cond-mat.stat-mech]  18 Dec 2018Dynamical response of the Ising model to the amplitude modul ated time dependent\nmagnetic field\n¨Umit Akıncı1\nDepartment of Physics, Dokuz Eyl¨ ul University, TR-35160 I zmir, Turkey\nAbstract\nThe dynamical Ising model under the effect of the amplitude modulat ed time dependent periodic\nmagnetic field has been solved by using EFT with Glauber type of stoch astic process. Severalcases\nwith amplitude modulation have been investigated. It has been shown that, amplitude modulation\ncould display dynamical phase transition on the magnetic system.\nKeywords: Dynamic Ising model; amplitude modulated field, dynamical p hase tran-\nsition\n1 Introduction\nDynamical response of the spin system to the time dependent magn etic field has attract attention\nboth theoretically and experimentally. Relation between the relaxat ion time of the spin system\nand period of the driving periodic external magnetic field determines the dynamic phase of the\nsystem. Dynamic phase transition (DPT) in these systems come fro m the competition between\nthese two time scales [1].\nTheoretically DPT in spin systems first observed within the mean field a pproximation (MFA)\n[2] for the spin-1 /2 Ising model. After that time, DPT in the spin-1 /2 Ising model has been\nwidely studied within the several techniques such as MFA [3], Monte Ca rlo simulation (MC) [4],\neffective field theory (EFT) [5]. Dynamical character was mostly inclu ded within the Glauber-type\nstochastic process [6] to these techniques.\nExperimentally, DPT can be observed in several systems such as, u ltrathin Co films [7],\nFe/Au(001) films [8], epitaxial Fe/GaAs(001) thin films [9], fcc Co(001 ), and fcc NiFe/Cu/Co(001)\nlayers [10] and Fe/InAs(001) ultrathin films [11].\nTheoreticalefforts continuing with different type of spin systems. Higher spin Isingmodels such\nas spin-1 Blume-Capel model [12] and spin-3 /2 Ising model [13] have been studied as well as spin\nsystems with quenched disorder; random fields [14], random crysta l fields [15], bond dilution [16]\nand site dilution [17]. Besides dynamical properties of the Ising model on different geometries have\nbeen studied. Nanotube [18], nanowire [19], thin film [20] and Bethe latt ice [21] geometries are\namong them. Although not as common as Ising model, dynamical prop erties of the other models\ndo exist. Heisenberg model [22] and Ashkin-Teller [23] model have be en studied by means of MC\nand EFT with Glauber type of stochastic process, respectively.\nOn the other hand, special attention was paid on the different type s of external time dependent\nmagnetic fields. These different types of driving fields are promising d iverse dynamical phase\ntransition and hysteresis properties. Ising ferromagnets under the effect of the pulsed [24, 25] and\nrandomly varied external time dependent fields [26] have been stud ied within the MC. For the\nsystem with pulsed field, it has been shown that, ratio of time width of the response (i.e. time\ndependent magnetization) of the system to the time width of the pu lse field diverges at the critical\ntemperature of the static system [25]. Besides, ratio between the response magnetization peak\nheight and the pulse height gives peak near this temperature [25]. Fo r the randomly varied field it\nhas been shown that, if the interval of the distribution of which ran dom field chosen large enough,\ndynamically disordered phase can be created [26]. These results hav e been obtained within the\nMC simulation and by solving MFA equations created with Glauber type o f stochastic process.\nRandomly varied (fast switching) external field problem was also stu died with slightly different\napproach [27, 28]. Besides, effect of the polarized electromagnetic wave [29] and field which have\ngradient [30] inspected by the same methods. Other examples of th is category are, standing wave\n[31] and propagating wave type of fields [32, 33].\n1umit.akinci@deu.edu.tr\n1Indeed dynamical spin system under the different type of fields des erves more attention, due\nto the rich dynamical behavior and possible experimental application s. For this aim this work\nis devoted to the exploration of the dynamical properties of the Is ing model which drives with\nexternal periodic magnetic field with amplitude modulation. Amplitude m odulation is mostly\nused in communication systems and it is not a difficult task to obtain mod ulated signal. The\nformulation is EFT with Glauber type of stochastic process [6]. The pa per is organized as follows:\nIn Sec. 2 we briefly present the model and formulation. The results and discussions are presented\nin Sec. 3, and finally Sec. 4 contains our conclusions.\n2 Model and Formulation\nLetthe periodicmagneticfieldwith frequency ω H0cosωtactsonthe Isingspin system. Amplitude\nmodulated magnetic field can be defined by\nH(t) = [H0+r(t)]cos(ωt), (1)\nwherer(t) is the time dependent field. Here, the term H0+r(t) is called as modulating field, while\nH(t) given by (1) as modulated field. Choosing modulating field as sinusoida l form with frequency\nωmis named as tone modulation in communication literature[34]. Then the t otal magnetic field is\ngiven by\nH(t) = [H0+H1cos(ωmt)]cos(ωt). (2)\nThe (Fourier) spectrum of this field consists of frequencies ±ω,ω±ωm,−ω±ωm.\nAlthough, the period of the magnetic field is not necessary for findin g the time series of the\nmagnetization, in order to calculate the dynamical order paramete r we need the period of the\nmagnetic field. In general, let f(x) andg(x) be the periodic functions with periods pandq,\nrespectively. The period of product (or sum) of these two functio ns is defined by r=ap=bq(the\nsmallest common multiple), where aandbare the positive integers. Note that rneed not to be\nthe smallest period, but this could not change the value of the dynam ical order parameter.\nThe Hamiltonian of the dynamical Ising model is given by\nH=−J/summationdisplay\n<i,j>SiSj−H(t)/summationdisplay\niSi, (3)\nwhere the first summation is over the nearest neighbors of the latt ice, while the second one is over\nall the lattice sites. Here, Siis thezcomponent of the spin variables at a site i,Jis the exchange\ninteraction and H(t) is the external magnetic field which is given by Eq. (2).\nGlauber-type stochastic process [6] can be used for investigatin g dynamic properties of the\nconsidered system. In general, in the Glauber type of stochastic p rocess (as done in Ref. [2] for\nthe mean field approximation) the thermal average (denoted with /angb∇acketleft/angb∇acket∇ight) of a spin variable Si, which\ncan take values ±1 can be given as\nθd/angb∇acketleftSi/angb∇acket∇ight\ndt=−/angb∇acketleftSi/angb∇acket∇ight+/angbracketleftbiggTriSiexp(−βHi)\nTriexp(−βHi)/angbracketrightbigg\n. (4)\nHere,θis the single spin flip rate per unit time, β= 1/(kBT),kBandTdenote the Boltzmann\nconstant and temperature, respectively. Tristands for the trace operation over the site i. AlsoHi\ndenotes the part of the Hamiltonian of the system related to the sit ei, which is given by,\nHi=−Si\nJz/summationdisplay\nj=1Sj+H(t)\n=−Si(hi+H(t)) (5)\nwherezis the number of nearest neighbor sites and hiis the local field that represents the nearest\nneighbor interactions of a site i. Inserting Eq. (5) into Eq. (4) yields,\nθd/angb∇acketleftSi/angb∇acket∇ight\ndt=−/angb∇acketleftSi/angb∇acket∇ight+/angbracketleftbiggTriSiexp(βSi(hi+H(t)))\nTriexp(βSi(hi+H(t)))/angbracketrightbigg\n. (6)\n2Performing trace operation over degrees of freedom Si=±1\nθd/angb∇acketleftSi/angb∇acket∇ight\ndt=−/angb∇acketleftSi/angb∇acket∇ight+/angb∇acketlefttanh(β(hi+H(t))))/angb∇acket∇ight (7)\ncan be obtained. Since the effect of the exponential differential op erator on any function F(x) is\ndefined by\nexp (a∇)F(x) =F(x+a), (8)\nfor any constant aand∇=d/dx, Eq. (7) can be written by within differential operator technique\n[35] as\nθd/angb∇acketleftSi/angb∇acket∇ight\ndt=−/angb∇acketleftSi/angb∇acket∇ight+/angb∇acketleftexp (hi∇)/angb∇acket∇ightf(x+H(t))|x=0, (9)\nwhere\nf(x) = tanh( βx). (10)\nThe last term in Eq. (9) can be evaluated within the decoupling approx imation and this will yield\nθdm\ndt=−m+z/summationdisplay\nn=0Anmn, (11)\nwherem=/angb∇acketleftSi/angb∇acket∇ightand\nAn=1\n2z/parenleftbiggz\nn/parenrightbiggz−n/summationdisplay\nr=1n/summationdisplay\ns=0/parenleftbiggz−n\nr/parenrightbigg/parenleftbiggn\ns/parenrightbigg\n(−1)sf[(z−2r−2s)J] (12)\nThis first order differential equation can be solved by standard met hods such as the Runge-Kutta\nmethod [36].\nDynamical order parameter of the system can be defined as integr ation of the time dependent\nmagnetization over one period ( P) of the magnetic field,\nQ=1\nP/contintegraldisplay\nm(t)dt. (13)\nSince the determination of the period of the modulated magnetic field (defined in Eq. (2)) will\nyield multiples of the period ( nP, wherenis integer), integral in Eq. (13) will be taken over the\ntimenP. In this case Pwill be replaced by nPin Eq. (13).\n3 Results and Discussion\nThe dimensionless quantities given by\nτ=kBT\nJ,h0=H0\nJ,h1=H1\nJ,h(t) =H(t)\nJ. (14)\nwill be used throughout the work. Our investigation will be focused o n square ( z= 4) lattice. We\nsetθ= 1 throughout our numerical calculations.\n3.1 Unmodulated case\nIn order to construct the foundation of our discussion about the amplitude modulation, let us\nreview unmodulated case briefly, i.e. H1= 0.0 in Eq. (2). We can see several time series of\nthe magnetization for different values of the Hamiltonian parameter s and temperature in Fig. 1.\nOn each figure the values of these parameters have been denoted . Time series are obtained by\nRunge-Kutta method which was used for the solution of Eq. (11). I n Fig. 1 (a) the system is in a\ndynamically disordered state, the value of the dynamical order par ameterQ= 0.0, magnetization\noscillates around the zero value. When the value of the frequency in creases, the system becomes\n3unable to follow the magnetic field, then the system becomes ordere d. This situation can be seen\nin Fig. 1 (b). Also, we can see by comparing Fig. 1 (a) with (c) that, de creasing amplitude of the\nfield has similar effect. Decreasing amplitude means, decreasing ener gy supplied to the system by\nmagnetic field, thus the spin-spin interaction overcome this effect. Lastly, decreasing temperature,\ndue to the decreasing thermal fluctuations may result in ordered p hase. This last observation can\nbe seen by comparing Fig. 1 (a) with (d). These are very well known r esults in the literature.\n-1-0.5 0 0.5 1\n 500  520  540  560  580  600m\ntω=0.5 h0=0.5 τ=3.0\n(a)H(t)\nm(t) Q=0.00 -1-0.5 0 0.5 1\n 500  520  540  560  580  600m\ntω=0.5 h0=0.5 τ=3.0\n(a)H(t)\nm(t) Q=0.00 -1-0.5 0 0.5 1\n 500  520  540  560  580  600m\ntω=1.5 h0=0.5 τ=3.0\n(b)H(t)\nm(t) Q=0.26 -1-0.5 0 0.5 1\n 500  520  540  560  580  600m\ntω=1.5 h0=0.5 τ=3.0\n(b)H(t)\nm(t) Q=0.26\n-1-0.5 0 0.5 1\n 500  520  540  560  580  600m\ntω=0.5 h0=0.2 τ=3.0\n(c)H(t)\nm(t) Q=0.32 -1-0.5 0 0.5 1\n 500  520  540  560  580  600m\ntω=0.5 h0=0.2 τ=3.0\n(c)H(t)\nm(t) Q=0.32 -1-0.5 0 0.5 1\n 500  520  540  560  580  600m\ntω=0.5 h0=0.5 τ=2.5\n(d)H(t)\nm(t) Q=0.73 -1-0.5 0 0.5 1\n 500  520  540  560  580  600m\ntω=0.5 h0=0.5 τ=2.5\n(d)H(t)\nm(t) Q=0.73\nFigure 1: Time series of the magnetization for the unmodulated field ( h1= 0.0) with selected\nvalues of frequency, amplitude and temperature.\n3.2 Modulated case with h0= 0.0\nSimpler case of modulation given by Eq. (2), is the case with H0= 0.0, named as multiplier\nmodulation. The (Fourier) spectrum of this field consists of freque nciesω±ωm,−ω±ωm. Since the\naim is to determine the effect of the modulation, we have fixed the fre quency of the unmodulated\nfield asω= 1.0 andh1= 0.5. Note that, due to the fact that field consist of product of two\ncosines, it will be enough to inspect the modulation frequency range 0.0< ωm<1.0. The effect\nof the modulation can be seen in Fig. 2, where are time series of the fie ld and magnetization for\nseveral values of the modulation frequency ωmhas been plotted. The value of the dynamical order\nparameter Qis given in each case. As seen in Fig. 2, changing the frequency of the modulation\nchanges the behavior of the magnetization drastically. For lower va lues of the ωm, both of field and\nmagnetization have wave packet like behavior (see Fig. 2 (a) and (b) ). System is in an ordered\nstate with the value of dynamical order parameter Q= 0.77. When the value of the modulation\nfrequency rises, the view of both field and magnetization changes w hile the value of the order\nparameter is the same (see Fig. 2 (c)). If the value of the ωmstill increases, the value of the order\nparameter starts to decrease (compare Fig. 2 (c) with (d)) and d rop to zero (see Fig. 2 (f)).The\nmodulation could create dynamical phase transition for fixed values of amplitude and temperature.\nFor a closer look at the variation of the dynamical order parameter with the modulation fre-\n4quency we depict it for several values of temperature and amplitud e, which can be seen in Fig. 3.\nAt first sight symmetry about the ωm= 1.0 takes attention. In general this symmetry is about\ntheωm=ω(remember that we have fixed the value of the modulation frequenc y asωm= 1.0).\nAs seen in Fig. 3 abrupt change of value of dynamical order paramet er to the value of zero,\nleave place to smooth change when the temperature rises (compar e curves related to the τ= 1.5\nwithτ= 1.8 in Fig. 3 (a) or curves related to the τ= 1.0 withτ= 1.5 in Fig. 3 (b)). Second\neffect of the rising temperature is to widen the plateau of Q= 0.0.\n3.3 Modulated case with h0>0.0\nThis case includes fields that satisfy H0>0.0. We can see from (2) that, the field term h0cos(ωt)\nsurvives, regardless of the choice of ωmandh1.\nIn order to determine the behavior of the magnetization with modula tion frequency we depict\nthe time series of the applied modulated field and magnetization for se lected values of modulation\nfrequency, as can be seen in Fig. 4. Other parameter values are fix ed ash0= 0.25,h1= 0.25,ω=\n1.00 andτ= 2.50. As seen in Fig. 4, rising modulation frequency drastically change t he behavior\nof magnetization in time, as in the case of modulation with h0= 0.0. Again, for lower frequencies,\nwave packet like behavior takes attention. Rising frequency chang es this behavior. In contrast\nto the case h0= 0.0, modulation with h0>0.0 could not create dynamically disordered phase\n(compare Fig. 2 (e), (f) with Fig. 4 (e), (f)). This is due to surviving termh0cos(ωt) regardless\nof the value of modulation frequency.\n4 Conclusion\nThe dynamical Ising model under the effect of the amplitude modulat ed time dependent periodic\nmagneticfield hasbeen solvedbyusingEFTwith Glaubertype ofstoch asticprocess. SinceDPT in\ntime dependent periodicmagneticfield iswellknown, valuesofsomeHa miltonianparametersfixed,\nsuch as frequency and temperature. Detailed investigation on time series constructed with several\namplitudemodulationcasehasbeencarriedout. Ithasbeenshownt hatinthemultiplieramplitude\nmodulation case, rising modulation frequency could create DPT. The modulation frequency and\nfrequency of the magnetic field has inverse effect on dynamical pha ses of the system. While rising\nmodulation frequency could create dynamically disordered phase (a s shown in this work), rising\nfrequencydragthesystemtotheorderedphase. Thisisbecause ofthefactthat, sincethefrequency\ngets higher, magnetic system could not follow the driving field. Thus w e can say that when the\nmodulation frequency gets higher, magnetic system has the chanc e to follow the driving modulated\nmagnetic field.\nWe hope that the results obtained in this work may be beneficial form both theoretical and\nexperimental point of view.\nAcknowledgements\nThe author would like to thank to S ¸ebnem Se¸ ckin U˘ gurlu from Doku z Eyl¨ ul University for valuable\ndiscussions on frequency and amplitude modulation.\nReferences\n[1] B.K. Chakrabarti, M. Acharyya, Rev. Modern Phys. 71 (1999) 8 47.\n[2] T. Tom´ e, M.J. de Oliveira, Phys. Rev. A 41 (1990) 4251.\n[3] M. Acharyya, Phys. Rev. E 58 (1998) 179.\n[4] M. Acharyya, Phys. Rev. E 59 (1999) 218.\n[5] B. Deviren, O. Canko, M. Keskin, Chinese Physics B 19 (2010) 050 518.\n5[6] R.J. Glauber, Journal of Mathematical Physics 4 (1963) 294.\n[7] Q. Jiang, H.-N. Yang, G.-C. Wang, Physical Review B 52 (1995) 149 11.\n[8] Y.-L. He, G.-C. Wang, Physical Review Letters 70 (1993) 2336.\n[9] W.Y. Lee, B.-Ch.Choi, J.Lee, C.C.Yao, Y.B.Xu, D.G. Hasko,J.A.C.B land, AppliedPhysics\nLetters 74 (1999) 1609.\n[10] W.Y. Lee, B.-Ch. Choi, Y.B. Xu, J.A.C. Bland, Physical Review B 60 (1 999) 10216.\n[11] T.A. Moore, J. Rothman, Y.B. Xu, J.A.C. Blanda, Journalof Applie d Physics 89 (2001) 7018.\n[12] X.L. Shi, G.Z. Wei, Physica A 391 (2012) 29\n[13] X.L. Shi, Y. Qi, Journal of Magnetism and Magnetic Materials 393 ( 2015) 204\n[14]¨U Akıncı, Y. Yuksel, H. Polat, Physical Review E 83 (2011) 061103\n[15] E. Vatansever, H. Polat, Journal of Magnetism and Magnetic M aterials 332 (2013) 28\n[16] E. Vatansever, ¨U Akıncı, Y. Yuksel, H. Polat, Journal of Magnetism and Magnetic Mat erials\n329 (2013) 14\n[17]¨U Akıncı, Y. Yuksel, E. Vatansever, H. Polat, Physica A 391 (2012) 5 810\n[18] B. Deviren, Y. Sener, M. Keskin Physica A 392 (2013) 3969\n[19] E. Kantar, M. Ertas Journal of Superconductivity and Novel Magnetism 29 (2016) 781\n[20] B.O. Akta¸ s, ¨U. Akıncı, H. Polat Thin Solid Films 562 (2014) 680\n[21] S.A. Deviren, E. Albayrak Physica A 390 (2011) 3283\n[22] M. Acharyya, Physical Review E 69 (2004) 027105\n[23]¨U Akıncı, Physica A 469 (2017) 740\n[24] M. Acharyya, B.K. Chakrabarti Physical Review B 52 (1995) 65 50\n[25] M. Acharyya, J.K. Bhattacharjee, B.K. Chakrabarti Physica l Review E 55 (1997) 2392\n[26] M. Acharyya, Physical Review E 58 (1998) 174\n[27] J. Hausmann, P. Rujan, J. Phys. A: Math. Gen. 32 (1999) 61.\n[28] J. Hausmann, P. Rujan, J. Phys. A: Math. Gen. 32 (1999) 75.\n[29] M. Acharyya, Acta Physica Polonica B 45 (2014) 1027\n[30] A. Dhar, M. Acharyya Communications in Theoretical Physics 66 (2016) 563\nMaterials 426 (2017) 53\n[31] A. Halder, M. Acharyya Journal of Magnetism and Magnetic Mat erials 420 (2016) 290\n[32] M. Acharyya Journal of Magnetism and Magnetic Materials 382 ( 2015) 206\n[33] M. Acharyya Journal of Physics: Conference Series 638 (201 5) 012008\n[34] B. P. Lathi, Z. Ding, Modern Digital and Analog communication Sys tems (Fourth edition,\nOxford University Press, USA 2010)\n[35] R. Honmura, T. Kaneyoshi J. Phys. C 12, 3979 (1979).\n[36] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran\n90 (Third edition, Cambridge University Press, USA 2007)\n6-1-0.5 0 0.5 1\n 500  600  700  800  900  1000  1100  1200m\nt(a)ωm=0.01Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000  1100  1200m\nt(a)ωm=0.01Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000  1100  1200m\nt(b)ωm=0.05Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000  1100  1200m\nt(b)ωm=0.05Q=0.77H(t)\nm(t)\n-1-0.5 0 0.5 1\n 500  550  600  650  700m\nt(c)ωm=0.40Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  550  600  650  700m\nt(c)ωm=0.40Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  550  600  650  700m\nt(d)ωm=0.80Q=0.75H(t)\nm(t)-1-0.5 0 0.5 1\n 500  550  600  650  700m\nt(d)ωm=0.80Q=0.75H(t)\nm(t)\n-1-0.5 0 0.5 1\n 500  600  700  800  900  1000m\nt(e)ωm=0.94Q=0.68H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000m\nt(e)ωm=0.94Q=0.68H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000m\nt(f)ωm=0.95Q=0.00H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000m\nt(f)ωm=0.95Q=0.00H(t)\nm(t)\nFigure 2: Time seriesof the magnetizationfor the modulated field with selected values of frequency\nωm. Other parameter values fixed as h0= 0.00,h1= 0.50,ω= 1.00 andτ= 2.50\n7 0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Q\nωm(a)ω=1.0\nh0=0.0 h1=0.5\nτ=0.5\nτ=1.0\nτ=1.5\nτ=1.8  0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Q\nωm(a)ω=1.0\nh0=0.0 h1=0.5\nτ=0.5\nτ=1.0\nτ=1.5\nτ=1.8  0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Q\nωm(a)ω=1.0\nh0=0.0 h1=0.5\nτ=0.5\nτ=1.0\nτ=1.5\nτ=1.8  0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Q\nωm(a)ω=1.0\nh0=0.0 h1=0.5\nτ=0.5\nτ=1.0\nτ=1.5\nτ=1.8  0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Q\nωm(a)ω=1.0\nh0=0.0 h1=0.5\nτ=0.5\nτ=1.0\nτ=1.5\nτ=1.8  0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Q\nωm(b)ω=1.0\nh0=0.0 h1=1.0\nτ=0.5\nτ=1.0\nτ=1.5\nτ=1.8  0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Q\nωm(b)ω=1.0\nh0=0.0 h1=1.0\nτ=0.5\nτ=1.0\nτ=1.5\nτ=1.8  0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Q\nωm(b)ω=1.0\nh0=0.0 h1=1.0\nτ=0.5\nτ=1.0\nτ=1.5\nτ=1.8  0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Q\nωm(b)ω=1.0\nh0=0.0 h1=1.0\nτ=0.5\nτ=1.0\nτ=1.5\nτ=1.8  0 0.2 0.4 0.6 0.8 1\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Q\nωm(b)ω=1.0\nh0=0.0 h1=1.0\nτ=0.5\nτ=1.0\nτ=1.5\nτ=1.8\nFigure 3: Variation of the dynamical order parameter with the modu lation frequency for the\nparameter values of ω= 1.0,h0= 0.0 and for chosen values of the temperature for amplitudes (a)\nh1= 0.5 and (b) h1= 1.0\n8-1-0.5 0 0.5 1\n 500  600  700  800  900  1000  1100  1200m\nt(a)ωm=0.01Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000  1100  1200m\nt(a)ωm=0.01Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000  1100  1200m\nt(b)ωm=0.05Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000  1100  1200m\nt(b)ωm=0.05Q=0.77H(t)\nm(t)\n-1-0.5 0 0.5 1\n 500  550  600  650  700m\nt(c)ωm=0.40Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  550  600  650  700m\nt(c)ωm=0.40Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  550  600  650  700m\nt(d)ωm=0.80Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  550  600  650  700m\nt(d)ωm=0.80Q=0.77H(t)\nm(t)\n-1-0.5 0 0.5 1\n 500  600  700  800  900  1000m\nt(e)ωm=0.94Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000m\nt(e)ωm=0.94Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000m\nt(f)ωm=0.95Q=0.77H(t)\nm(t)-1-0.5 0 0.5 1\n 500  600  700  800  900  1000m\nt(f)ωm=0.95Q=0.77H(t)\nm(t)\nFigure 4: Time seriesof the magnetizationfor the modulated field with selected values of frequency\nωm. Other parameter values fixed as h0= 0.25,h1= 0.25,ω= 1.00 andτ= 2.50\n9"    },    {        "title": "2305.16898v1.Magnetized_dusty_plasma__On_issues_of_its_complexity_and_magnetization_of_charged_dust_particles.pdf",        "content": "Magnetized dusty plasma: On issues of its complexity and magnetization of\ncharged dust particles\nMangilal Choudhary\nDepartment of Physics and Astrophysics, University of Delhi, Delhi-110007, India\nIt is possible to excite various linear and non-linear low-frequency modes in dusty plasma which is an admixture\nof electrons, ions, gas atoms, and negatively charged solid particles. The experimental as well as theoretical\nstudy of these low-frequency dynamical modes in dusty plasma is very complex because of the involvement of\ndynamics of electrons, ions, and neutrals. If the external magnetic field is introduced to dusty plasma then\nthe dynamics of it will be more complex. The complexity of magnetized dusty plasma where plasma species\nare magnetized is discussed by keeping the experimental observations in magnetized dusty plasma devices\nin mind. The requirement of theoretical modeling, as well as computation experiments in understanding\nthe dynamics of dusty plasma in the presence of a strong magnetic field, is highlighted in the context of\nexperimental findings. The major challenges to magnetizing charged massive particles in experiments and\nsome expected solutions are discussed in this report.\nKeywords: Dusty plasma, Magnetized dusty plasma,\nDust rotational motion, Dust vortices, Dust void\nI. INTRODUCTION\nAn ionized gas is assumed to be a plasma state (the\nfourth state of matter) if it qualifies certain conditions1.\nThis is a fact that plasma is a complex dynamical state\nhowever it becomes a more complex system if solid par-\nticles of sized nm to µmare added to it. In the back-\nground of charged species (electrons and ions), these solid\nparticles undergo various charging processes and acquire\neither negative or positive charges on their surface. In\nlow-temperature plasma ( Te>> T i), dust grains acquire\nnegative charges up to 10−16to 10−14C due to the collec-\ntion of more energetic electrons than the slowly moving\nions2,3. These well seperated negatively charged solid\nparticles float in the plasma medium and can modify the\ncharacteristics response of ambient plasma4–6with exter-\nnal or internal electromagnetic perturbation. However,\nthese charged massive particles always settle down on\nthe wall of the experimental device because of the finite\ngravitational force acting on them. A strong electric field\ncan be created using various discharge configurations7–12\nfor providing confinement to charged particles against\ngravity in plasma. Once the density of dust grains in the\nelectrostatic confining potential well crosses a threshold\ndensity then long-range coulomb interaction among nega-\ntively charged dust particles turns the dust grain medium\nto respond collectively with internal or external pertur-\nbation.\nIn last more than 25 years, a wide spectrum of experi-\nmental and theoretical studies has been devoted to ex-\nploring the collective response of dusty plasma in the\nform of various dynamical structures/phenomena such\nas dust acoustic waves9,10,13–17, rotational and vortex\nmotion18–25, crystallization and phase transition26–29, in-\nstability driven modes30–34and voids35–37. The major\nobjectives of these theoretical and experimental studies\nof dusty plasma were to understand the naturally occur-ring dusty plasmas, control the impurities in semicon-\nductor industries, explore the role of impurities on char-\nacteristics of plasma, establish it as a model system to\nunderstand the complex physical systems, etc. After a\nwide range of dusty plasma studies, still there are some\nopen questions: Could we understand all astrophysical\nsystems (plasma with dust impurities) with existing ex-\nperimental results or available theoretical models? Do\nwe need to explore the dusty plasma in the presence of a\nmagnetic field? Would be there a finite role of magnetic\nfield on the cloud of charged dust grains? Getting the\nanswers to such open questions is only possible if dedi-\ncated experimental and theoretical studies of magnetized\ndusty plasma will be conducted.\nIn recent years, magnetized dusty plasma has been a pop-\nular research topic among the dusty plasma community\nworldwide. The dynamics of dust grain medium or well-\nseparated dust grains strongly depends on the charac-\nteristics of background plasma that can be modified in\nthe presence of the external magnetic field. It is expected\nthat the complexity of dusty plasma increases in the pres-\nence of external magnetic field. The magnetization of\nelectrons is possible at a low magnetic field (B <0.05T)\nas compared to massive charged dust particles (B >2 T).\nThe study of dusty plasma in the presence of an external\nmagnetic field (magnetized dusty plasma) is categorized\nas weakly and strongly magnetized dusty plasma. A wide\nrange of theoretical as well experimental work on weakly\nmagnetized dusty plasma where only electrons are mag-\nnetized is conducted by various research groups38–40. It is\na fact that strong magnetic field is required for the study\nof strongly magnetized dusty plasma. There are few de-\nvices based on permanent magnet41and superconducting\nelectromagnet42–44for introducing the strong magnetic\nfield (few Tesla) to dusty plasma. The researchers are\ncontinuously putting effort to explore the effect of mag-\nnetic field on dusty plasma phenomena such as excitation\nof waves, charging mechanism, rotational and vortex mo-\ntion, crystallization and melting, etc. However, there are\nmany challenges in understanding the experimentally ob-\nserved dynamics of dusty plasma because of its complexarXiv:2305.16898v1  [physics.plasm-ph]  26 May 20232\nnature.\nA detailed discussion on a few issues regarding under-\nstanding the experimental results and producing strongly\nmagnetized dusty plasma is given in subsequent sections.\nThe complexity of magnetized dusty plasma is discussed\nin Sec. II. The major challenges in achieving magneti-\nzation conditions of negatively charged dust grains in rf\ndischarges at strong B-field are discussed in Sec. III. The\nproposed suggestions to get magnetized dusty plasma\n(dust particles will be magnetized) are given in Sec. IV.\nThe concluding remark is given in Sec. V.\nII. COMPLEXITY OF MAGNETIZED DUSTY PLASMA\nDusty plasma is considered a complex dynamical sys-\ntem. It becomes a more complex system once the exter-\nnal magnetic field is applied. The complexity of weakly\nor strongly magnetized dusty plasma in terms of various\nexperimentally observed physical phenomena is discussed\nin respective subsections.\n1. Dust charging\nIn low-temperature weakly ionized plasma, charging\nof solid dust particles mainly depends on the collection\ncurrents of plasma species3,45,46. The magnitude of\nelectron current ( Ie) and ion current ( Ii) to dust surface\nstrongly depends on their density and average energy at\nthe fixed neutral background. The charging currents are\nexpected to change once the magnetic field is introduced\nto dusty plasma. As the magnetic field is applied, the\ngyro-radius of electrons ( rge=mevte/eB) and of ions\n(rgi=mivti/eB) decreases with increasing the strength\nof external B-field. The electrons are magnetized at a\nlower magnetic field (B <0.05 T) but a higher B-field\n(B>0.1 T) is required to magnetize ions in typical\ndusty plasma experiments44. In magnetized plasma,\nthe currents IeandIito the dust surface are expected\nto change due to the following reasons: (a) magnetic\nfield reduces the loss of energetic electrons from the\nplasma region to the wall of the experimental chamber.\nThe confinement of energetic electrons may change the\naverage energy or electron temperature of bulk plasma\nelectrons44. This change in electron temperature may\nalter the charging currents. (b) Magnetic field (weak)\nmay increase the confinement of electrons44which could\nbe a cause of the change in the charging mechanism.\n(c) Magnetic field modifies the collisional frequency of\nelectrons/ions with background neutrals. In this way,\nthe ionization rate may get modified in the presence of\nthe magnetic field. Modification in the plasma forma-\ntion rate or ionization process will affect the charging\ncurrents. (d) In magnetized plasma, net electron current\n(Ie) is assumed to be the sum of two possible currents\nalong B-field ( Ie∥) and transverse to B-field ( Ie⊥). The\nIe⊥strongly depends on the strength of the externalB-field which definitely will affect the charging currents.\n(e) Once ions are magnetized then net ion current to the\ndust surface will have two components along B-field ( Ii∥)\nand transverse to B-field ( Ii⊥. The change in ion current\nmay increase or decrease the dust charge at strong B-\nfiled1,47. (f) It is also expected to observe a finite effect of\ncollisions (background neutral density) on the charging\ncurrents in the presence of the magnetic field due to the\nIe⊥andIi⊥dependence on collisional relaxation time1,44.\nIt should be noted that the estimation of accurate charge\non dust grains in weakly/strongly magnetized plasma is\nnecessary to understand the dynamics/characteristics\nof dusty plasma. In the last few years, theoretical\nas well as experimental works have been carried out\nto estimate the charge on dust grains in magnetized\nplasma. Melzer et al.48adopted normal mode analysis\nof the dust cluster in the magnetized rf plasma for\nextracting charge on nm to micron-sized dust grains. At\nthe same time, Choudhary et al.44estimated charge on\nmagnetic and non-magnetic spherical probes (large dust\ngrains) in magnetized rf discharge. The theoretical study\nand computational experiments have been performed\nto understand the charging mechanism in magnetized\nplasma background47,49–53However, there are inconsis-\ntencies in the numerically estimated and experimentally\nobserved values of the dust charge in weakly magnetized\ndusty plasma. The available theoretical models to\nunderstand the charging mechanism of non-magnetic\n(magnetic) solid spherical particles in the magnetized\nplasma (weakly and strongly) are incomplete. Therefore,\nan adequate theoretical model and novel experimental\ntechniques are required to estimate the accurate charge\non non-magnetic and magnetic solid particles in the\nbackground magnetized plasma at different strengths of\nmagnetic fields.\n2. Dust-acoustic waves\nThe excitation of low-frequency modes in dusty plasma\nis a result of its collective response in the presence\nof internal/external electromagnetic perturbation. The\nstudy of dust acoustic waves (low-frequency modes)\nhas a more than 25 years old history7and is still\na vibrant research topic for researchers. The propa-\ngation characteristics of dust acoustic modes in low-\ntemperature direct current (DC) and radio-frequency\n(rf) plasma9–11,54–58were explained based on the avail-\nable theoretical models14,46,59,60. Recently, Choudhary et\nal.61and Melzer et al.62conducted experimental studies\non self-excited dust acoustic waves (DAWs) in the pres-\nence of external magnetic field and observed the damp-\ning as well modification in propagation parameters at the\nhigher magnetic field. The role of external magnetic field\non propagating DAW in capacitative coupled discharge61\nis shown in Fig. 1. The propagating waves are getting\ndamped with increasing the strength of external B-field.3\n \ng \n1 mm \n B = 0 T \nB = 0.08 T \nB = 0.15 T \nFIG. 1. Propagation of dust acoustic waves at different\nstrengths of external magnetic fields. Details of experimental\nconfiguration and parameters are available in ref.61\nThus we do not see any wave modes in dusty plasma\nmedium at strong B-field (see Fig. 1).\nThe excitation and propagation characteristics of dust\nacoustic waves strongly depend on the background elec-\ntric field, neutrals density, dust charge, plasma density,\netc. Since the magnetic field modifies plasma parame-\nters, the propagation characteristics of DAWs get modi-\nfied with the application of external B-field. Salimullah\net al.63performed a theoretical study of DAWs in the\nmagnetized plasma. However, this proposed theoretical\nmodel could not explain the modification and damping of\nwaves which were observed experimentally in magnetized\ndusty plasma devices61,62. The interpretation of experi-\nmentally observed results on modification and damping\nof DAWs in the magnetized plasma could only be pos-\nsible if an adequate theoretical model will be developed.3. Rotational motion and vortex formation\nIn the absence of magnetic field, dusty plasma\nshows rotational motion as well as vortex flow25,64–66.\nThe collective dynamics of 2-dimensional (2D) dust\ngrains medium in the presence of magnetic field where\nmedium exhibits rotational motion is studied by many\nresearchers18,67–70. In presence of magnetic field, 3-\ndimensional dusty plasma exhibits vortex motion and\npossibility to form counter-rotating vortices in a ver-\ntical plane71,72. There are available theoretical mod-\nels to explain the onset of rotational motion73and vor-\ntex flow25,74–76in unmagnetized and magnetized dusty\nplasma. Choudhary et al.19carried out the study of\n2D annular dusty plasma in the presence of a strong\nmagnetic field. The dust grains rotate anti-clockwise\n(uni-directional) in the presence of B-field as shown in\nFig. 2(a). It is expected (as per experimental configura-\ntion) to observe the opposite flow of dust grains in the\nannular region in the presence of B-field. However, the\nopposite flow of dust grains in the annular region in the\npresence of B-field is only possible with specific discharge\nconditions. A typical PIV image of an annular dusty\nplasma at a magnetic field (B = 0.4 T) where dust grains\nrotate in opposite directions is shown in Fig. 2(b). Strong\ntheoretical support is required to obtain the specific dis-\ncharge conditions for generating the unidirectional or op-\nposite flow of dust particles in annular magnetized dusty\nplasma.\nAt a strong magnetic field, dust grains in an annular re-\ngion try to be in a pair and flow together up to a certain\ndistance. A still image of annular dusty plasma is shown\nin Fig. 3(a) in which encircled regions (I and II) high-\nlight the isolated and pair of dust particles, respectively.\nThree images at different times are superimposed to see\nthe motion of isolated particles and dust pairs in the B-\nfield. Encircled regions (I and II) in Fig. 3(b) show the\nelongated tracks of isolated dust particles and dust pairs\nduring the rotational motion. It is required to develop\na theoretical model for understanding such complex phe-\nnomena of annular dusty plasma in the presence of strong\nmagnetic field.\nThe recent experimental study of 3-dimensional dusty\nplasma in a strong magnetic field71(B>0.1 T) demon-\nstrates its complexity due to the formation of rotating\ndust torus (vortices in the vertical plane) and rotation\nof dust particles in the horizontal plane. A typical im-\nage of dusty plasma in the vertical and horizontal plane\nis shown in Fig. 4. The dust grains are confined in a\nbowl-shaped potential well and the dynamics of grains\ncan be tracked by analyzing images either in a vertical\nplane (see Fig. 4(a)) or horizontal plane (see Fig. 4(b)).\nIt is observed that the dynamics of dust grains become\nmore complex in the presence of B-field if the volume\nof the medium is increased. A large volume (3D) dusty\nplasma can be created by using smaller-sized dust parti-\ncles in the experiment. The dynamics of 3D dusty plasma\nat different B-fields are demonstrated by PIV images of4\n-1.4-1.2-1-0.8-0.6-0.4-0.20\nX-Position (cm)-1.4-1.2-1-0.8-0.6-0.4-0.20\nY-Position (cm)\n00.20.40.60.811.21.41.61.82mm/s\n(a)\n-1.4-1.2-1-0.8-0.6-0.4-0.2\nX-Position (cm)-1.5-1-0.5\nY-Position (cm)\n00.050.10.150.20.25mm/s\n(b)\nFIG. 2. (a)PIV image (average of 50 images) of annular dusty plasma flow (uni-directional) between two conducting (aluminum)\nrings (inner and outer diameter of the first ring are 5 mm and 12 mm respectively, same for the second ring are 30 mm and 50\nrespectively). The diameter of dust particles is 6.28 µm and argon pressure during experiments was 30 Pa. The peak-to-peak\nrf voltage was 55 V and the applied B-field was 0.2 T. (b) PIV image of annular dusty plasma flow (opposite-rotation) between\ntwo non-conducting (Teflon) rings (inner and outer diameter of the first ring are 20 mm and 30 mm, the same for the second\nring are 40 mm and 50 mm respectively). The diameter of dust particles is 6.28 µm and argon pressure during experiments was\n30 Pa. The peak-to-peak voltage was 55 V and the applied B-field was 0.4 T. More details about the experimental configuration\nto create annular dusty plasma are given in ref.19\n \nI \nII \nY \nX \n(a) \nI \nII \nY \nX \n(b) \nFIG. 3. (a) A dusty plasma image in a horizontal plane that shows the pairing of dust grains in an annular region at a strong\nmagnetic field (B = 0.6 T). There is no pairing of dust particles in the encircled region I and a clear pairing of two dust particles\nin encircled region II. (b) A composite image of three still images taken at different times (dt = 80 ms). The elongated dust\ntracks in the encircled region I represent the motion of single dust grains. Various elongated pairs in encircled region II clearly\nindicate the motion of pairs up to a certain distance with time. The experiments were performed in argon gas at 30 Pa. The\nannular dust medium was created in between two conducting (aluminum) rings (the inner and outer diameters of the first ring\nare 5 mm and 12 mm respectively, same for the second ring are 30 mm and 50 respectively). The diameter of dust particles is\n6.28µm and peak-to-peak rf voltage was 55 V\nthe medium in a horizontal plane at different positions in\nvertical directions (see Fig: 5). The topmost dust layers\nshow only rotational motion in the azimuthal direction.\nIn between the topmost and bottom layers, dusty plasma\nexhibits mixed motion (rotation in the horizontal plane\nand vortex flow in the vertical plane) in the presence of a\nstrong B-field. The direction of vortex flow (torus in 3Ddusty plasma) changes (anti-clockwise to clockwise) as we\nmove towards the bottom layer, as displayed in Fig: 5.\nThe results (piv images) indicate the formation of multi-\nple counter-rotating tori in the large-volume magnetized\ndusty plasma. The formation and direction of vortex flow\nalong with the rotational motion can be explained with\nthe help of an appropriate futuristic theoretical model.5\n \nTopmost layer  \ng \nBottom layer \nElectrode  \n(a) Dusty plasma in a vertical plane \n(b)Dusty plasma in a horizontal \nplane\n \nDust grains layer at a given vertical location plane\nFIG. 4. Typical dusty plasma in a vertical (Y–Z) plane (b) in horizontal (X–Y) plane. The dust grains are confined in a\npotential well created by placing an addition of either conducting or non-conducting ring of an appropriate size and thickness.\nThe 3D dusty plasma looks like a bowl shape in the vertical plane, as denoted by the boundary. The width of dusty plasma in\nthe vertical direction depends on the depth of the potential well and the size of dust particles\n4. Void formation\nA dust-free region (void) in dusty plasma is formed\neither by internal instabilities77or external electrostatic\nperturbation35. If a floating or negatively biased cylindri-\ncal probe (rod) is inserted into the dusty plasma medium,\na dust-free region is formed around it35–37. The stud-\nies on void formation in dusty plasma were carried out\neither in DC discharge or RF discharge. In unmagne-\ntized equilibrium dusty plasma, dust particles exhibit\nrandom motion about their equilibrium position. The\nformation of void around a negatively biased or floating\nprobe (rod) is the result of competition between inward\nion drag force and outward electric field on dust grains\nnear the probe35,78The experimental results of the dust-\nfree region around negatively biased or floating objects\n(probe/rod) in unmagnetized dusty plasma can be ex-\nplained by the available theoretical model35,78. In the\npresence of the magnetic field, dust particles rotate in a\nhorizontal plane (in the case of 2D dusty plasma) due\nto the azimuthal flow of ions19,67,70. The flowing ions\nset dust particles into rotational motion in a plane per-\npendicular to the magnetic field. A dust-free region is\nexpected to form around a negatively charged/floating\nprobe (rod) if it is inserted into the rotating dusty\nplasma. However, the available theoretical models for\nunmagnetized dusty plasma35,78may need some correc-\ntions due to the following reasons: (a) The ion flow in\nthe radial electric field of charged probe (rod) will be\nmodified in the presence of B-field. Only the radial ve-\nlocity component of ions will be responsible to compen-\nsate the electric force acting on dust particles. (b) Theelectric and ion drag forces acting on dust grains may get\nchanged due to the variation of dust charge79in the pres-\nence of B-field. Hence, the characteristics of void may get\nchanged in the magnetized dusty plasma. Thus the the-\nory of void formation in weakly or strongly magnetized\ndusty plasma would be very complex.\nThe interaction of a positively biased probe (cylindri-\ncal/spherical) with 3-dimensional dusty plasma in the\npresence of B-field will be very complex. A typical image\nof dust grain medium in the horizontal plane around a\ncylindrical probe (1 mm diameter) at B = 0.7 T is shown\nin Fig. 6. Around the positively biased probe (+70 V),\nwe observed a dust-free region in rotating dusty plasma\n(plane perpendicular to B-field) and vortex motion of\ndust particles near the probe in the vertical plane (along\nthe axis of the probe). The complexity in understand-\ning such experimental results of magnetized dusty plasma\ncan be resolved by developing promising theoretical mod-\nels and computational experiments.\nCrystallization and melting of dusty plasma\nThe study of dusty plasma crystals in various discharge\nconfigurations has been a hot research topic in the dusty\nplasma because of its wide scope in understanding medi-\nums at kinetic level26–28,80–82. Apart from neutral pres-\nsure and input power, the external magnetic field can also\nalter the characteristics of dust-plasma crystal83. The\ndust grains in a crystalline state perform random motion\nabout their equilibrium position in the absence of an ex-\nternal B-field but they have rotational motion if the mag-6\n-1.4 -1.2 -1-0.8 -0.6 -0.4\nX-Position (cm)-1.4-1.2-1-0.8-0.6-0.4\nY-Position (cm)\n00.511.5mm/s\n(a)\n-1.4 -1.2 -1-0.8 -0.6 -0.4\nX-Position (cm)-1.4-1.2-1-0.8-0.6-0.4\nY-Position (cm)\n00.511.5mm/s\n(b)\n-1.4 -1.2 -1 -0.8 -0.6\nX-Position (cm)-1.4-1.3-1.2-1.1-1-0.9-0.8-0.7-0.6-0.5\nY-Position (cm)\n00.511.5mm/s\n(c)\n-1.4 -1.2 -1 -0.8 -0.6\nX-Position (cm)-1.4-1.3-1.2-1.1-1-0.9-0.8-0.7-0.6-0.5\nY-Position (cm)\n00.511.522.5mm/s\n(d)\n-1.4-1.3-1.2-1.1-1-0.9-0.8-0.7\nX-Position (cm)-1.4-1.3-1.2-1.1-1-0.9-0.8-0.7\nY-Position (cm)\n00.511.522.5mm/s\n(e)\n-1.4-1.3-1.2-1.1 -1-0.9\nX-Position (cm)\n-1.4-1.3-1.2-1.1-1-0.9\nY-Position (cm)\n00.511.522.5mm/s\n(f)\nFIG. 5. (a)PIV images of dusty plasma (obtained after averaging 50 images) flow in a horizontal plane at different vertical\nlocations of confined dust grains in potential well (a) Topmost dust grains (b) 2 mm below the topmost dust grains plane (c)\n4 mm lower the topmost dust grains plane (d) 6 mm lower the topmost dust grains plane (e) 8 mm lower the topmost dust\ngrains plane (f) 12 mm lower the topmost dust grains bed/layer. The diameter of dust particles is 2 µm and argon pressure\nwas set at 35 Pa. The peak-to-peak rf voltage was 52 V and the applied magnetic field was 0.15 T7\n \n1 mm \nProbe \nI \nII \nIII \nFIG. 6. A composite image of five still images at different\ntime intervals (dt = 120 ms) in the presence of a positively\nbiased (+70 V) cylindrical probe (in a vertical direction) in a\n3-dimensional magnetized dusty plasma. Three different re-\ngions are marked in the image. Region I represents the dust\ngrain medium (2D image) far away from the biased probe,\nregion II represents the dust-free region (void) and region III\nindicates the outward motion of dust grains which is a sig-\nnature of the 3D motion of dust particles. The experiments\nwere performed in argon plasma at a pressure of 40 Pa. The\npeak-to-peak voltage on the electrode is 55 V and the exter-\nnal applied B-field is 0.7 T.\nnetic field is introduced. A typical still image (crystalline\nstate) and superimposed image of five consecutive images\nat different times of dusty plasma crystal in the magnetic\nfield of strength 0.5 T is depicted in Fig. 7. The dust\ngrain medium can be in a crystalline state (solid) or liq-\nuid state or gaseous state while it has rotational motion.\nWe know that coupling constant (strength) and screening\nlength26,28,80decide the state of dusty plasma. Both pa-\nrameters of dust grain medium are strongly dependent on\nbackground plasma parameters, therefore, the external\nmagnetic field definitely can modify the characteristics\nof dusty plasma crystal. However, there is a lack of the-\noretical model/computational simulation to understand\nthe dusty plasma states in the presence strong magnetic\nfield.\nIII. CHALLENGES IN MAGNETIZING THE CHARGED\nDUST PARTICLES IN STRONG B-FIELD\nIn recent years, dedicated experimental devices are\nbuilt to study the strongly magnetized dusty plasma\nwhere massive charged particles (nm to µm) can be\nmagnetized41–44. However, there are many challenges to\nmagnetizing the solid-charged dust particles in the am-\nbient plasma background after application of a strong B-\nfield. Thomas et al.84suggested the strength of magnetic\nfield (few Tesla) to magnetize the charged dust particles\n(nm to µm) in the plasma background. The magnetic\nfield, mass of dust particles, the number density of neu-\ntrals (gas pressure), velocity of dust particles, and charge\n \n(a) \nY \nx \n(b) \nY \nx FIG. 7. (a) Image shows the crystalline (solid) state of\ndusty plasma at B = 0 T (b) A composite image of seven\nstill images at different time intervals (dt = 160 ms) in pres-\nence of B-field (B = 0.5 T). The dust grains were confined\nby a smaller aluminum ring electrode (10 mm inner and 20\nmm outer diameter, 2 mm width) on the lower powered rf\nelectrode. The size of dust grains was 6.28 µm and argon\npressure was 30 Pa for this experimental result\nof dust particles are major factors that directly affect the\ncondition of dust magnetization84,85. The magnetization\nof dust particles (100 nm to 10 µm) of low mass density\n(1 to 1.5 gm/ cm3) and low thermal velocity (10 mm to\n20 mm/s) is only possible at the high magnetic field (1 <\nB<5 T) in moderate collisional plasma. It should be\nnoted that theoretical estimation of dust magnetization84\nin this range of size and velocity is only possible if plasma\n(dust) parameters remain constant with increasing exter-\nnal B-field. In experimental magnetized dusty plasma,\nthe charging mechanism as well as forces acting on dust\ngrains are strongly dependent on B-field44,48. There are\na few major challenges in achieving the magnetization\nconditions84for 100 nm to 10 µm particles in ground-\nbased dusty plasma experiments.\n•The charging mechanism of dust particles strongly\ndepends on the external magnetic field. The charge\naccumulated on dust grains is expected to change\nwhich needs to be addressed in magnetization con-\nditions.\n•Dust grain medium is in equilibrium at low power\nand moderate pressure. Low pressure and high\npower make dusty plasma highly unstable resulting\nin the excitation of waves61. The moderate pressure\n(p<50 Pa) and power (P <10 W) plasma does not\nremain homogeneous but become inhomogeneous\ndue to the formation of filamentary structures86in\nthe presence of B-field. It is possible to keep plasma\nhomogeneous (no filamentary structures) at high\npressure but a much strong B-field will be required\nto magnetize the same-sized charged dust grains.8\n•Confinement of dust grains at a very high mag-\nnetic field ( >2 Tesla) is also one of the con-\ncerns in the experiment. The confinement po-\ntential is created above the lower powered elec-\ntrode (in rf discharge) by placing an additional\nconducting/non-conducting ring of appropriate\nwidth and thickness71. The electric potential dis-\ntribution of the power electrode as well the ring\nelectrode is expected to change with increasing the\nmagnetic field that can modify the confinement po-\ntential for charged dust particles at strong B-field\nin rf discharges.\n•The azimuthal flow of ions and electrons in the\npresence of dusty plasma can drive the dust par-\nticles in the same direction67. The angular fre-\nquency of dust rotation depends on the strength of\nthe magnetic field. The azimuthal flow of electrons\nor ions keeps dust particles in the rotational motion\nin the presence of B-field. It would be difficult to\nget a stationary equilibrium dust cloud where dust\ngrains only experience the Lorentz force and mag-\nnetization of these massive dust particles in plasma\nwill be happened. However, there are challenges in\nkeeping dust grains in random motion instead of ro-\ntational motion in the presence of a strong B-field.\nIV. THE REQUIRED FUTURISTIC STEPS TO GET\nMAGNETIZED DUSTY PLASMA\nMagnetization of charged dust particles in plasma de-\nvices with the application of a very high B-field is a chal-\nlenging task due to the major concerns which were dis-\ncussed in the earlier section. However, more research\nwork in the field of dusty plasma (experimental and the-\noretical) with a strong B-field may help to achieve the\nconditions of magnetization of charged particles. A few\nsteps are suggested based on experimental findings in\nmagnetized dusty plasma devices and published works\nin such devices.\n•As per the experimental study, negative charges\non dust particles are reduced at higher magnetic\nfield44,48. It is also reported that magnetic particles\ncan have higher charges on their surface than non-\nmagnetic particles of the same size in magnetized\nplasma background44. The charges on the dust sur-\nface depend on its surface area. The surface area\nof spherical particles increases with increasing size\nbut same time the mass of particles also increases.\nTo overcome this issue, hollow magnetic spheri-\ncal particles can be used in place of non-magnetic\nsolid particles. In this way, the strength of external\nmagnetic field can be reduced to achieve the mag-\nnetization condition at moderate power and pres-\nsure. The moderate power and pressure may help\nin getting homogeneous plasma (without filamen-\ntary structures) at a comparatively low B-field.•A large radius aluminum ring on the lower elec-\ntrode in rf discharges which is used to create a\npotential well for charged particles provides poor\nconfinement at a high magnetic field (B >1 T).\nThe metal ring is in contact with the lower elec-\ntrode (in rf discharge) therefore the potential (volt-\nage) on it will be the self-biasing voltage of the\nlower-powered electrode. The self-bias voltage on\nthe ring, as well as of the lower electrode, may\nget changed significantly due to external B-field.\nHowever, the non-conducting (Teflon/nylon) ring\nalways remains in floating condition (floating po-\ntential) which could provide a better confinement\nto charged dust particles in strongly magnetized\nplasma. Thus, dusty plasma experiments with\nsmaller-sized non-conducting confinement rings of\nappropriate thickness should be performed.\n•Experimental study of Chaubey et al.87,88in after-\nglow dusty plasma demonstrates the large positive\ncharges on dust surface and holding them above the\nlower powered (biased) electrode without plasma\nbackground. If the levitated positively charged\ndust grains are exposed to the strong external mag-\nnetic field (few Tesla) then Lorentz force may act on\nthem and start to gyrate (rotate) about B-field lines\nin the plane. The plasma density is very low (neg-\nligible) in afterglow discharge and charges on dust\ngrains are frozen which may help in avoiding the\nissues such as the formation of filamentary struc-\nture, azimuthal flow of electrons and ions, charging\nmechanism of dust grains, etc. arising in normal\nmagnetized rf discharges. However, some dedicated\nexperiments in such after-glow discharge configura-\ntion are required to demonstrate the proposed idea\nfor producing the magnetized dusty plasma.\nV. SUMMARY\nIn this report, we have discussed the complexity in\nunderstanding the experimental results of dusty plasma\nin the presence of magnetic field. The few experimen-\ntally observed results such as dust-acoustic waves, void\nformation, rotational motion of particles, crystalliza-\ntion of dust grains, etc. in magnetized dusty plasma\ndevice at Justus-Liebig University Giessen, Germany\nare discussed. The complexity in explaining the ex-\nperimentally observed results of dusty plasma (weakly\nor strongly magnetized) without appropriate theoretical\nmodels/computational experiments is highlighted. The\nproblems faced in achieving magnetization conditions like\nthe formation of filamentary structures, poor particle\nconfinement, and getting stationary dust grain medium\nat strong B-field are mentioned. Does it possible to come\nout from some of these challenges in achieving magne-\ntization conditions? To get the answer of questions, a\nfew ideas based on experimental knowledge of the sub-9\nject are proposed. These suggested steps may work in\nachieving the magnetization conditions of charged dust\nparticles in rf discharges at strong B-field. However, the\nhighlighted complexity and issues in achieving magne-\ntized dusty plasma may motivate researchers to work on\nthese important research areas.\nVI. ACKNOWLEDGEMENT\nAuthor is thankful to Prof. Markus H Thoma for\nproviding experimental facility at I-Institute of Physics,\nJustus-Liebig University Giessen, Germany.\n1F. F. Chen, Introduction to plasma physics and controlled fusion\n(Springer US, 1984).\n2A. Barkan, N. D’Angelo, and R. L. Merlino, “Charging of dust\ngrains in a plasma,” Phys. Rev. Lett. 73, 3093–3096 (1994).\n3J. Goree, “Charging of particles in a plasma,” Plasma Sources\nScience and Technology 3, 400–406 (1994).\n4Y. Nakamura, “Experiments on ion-acoustic shock waves in a\ndusty plasma,” Physics of Plasmas 9, 440–445 (2002).\n5Y. Nakamura and A. Sarma, “Observation of ion-acoustic soli-\ntary waves in a dusty plasma,” Physics of Plasmas 8, 3921–3926\n(2001).\n6N. Bilik, R. Anthony, B. A. Merritt, E. S. Aydil, and U. R.\nKortshagen, “Langmuir probe measurements of electron energy\nprobability functions in dusty plasmas,” Journal of Physics D:\nApplied Physics 48, 105204 (2015).\n7R. L. Merlino, “25 years of dust acoustic waves,” J. Plasma\nPhysics 80, 773–786 (2014).\n8L. G. D’yachkov, O. F. Petrov, and V. E. Fortov, “Dusty plasma\nstructures in magnetic dc discharges,” Contributions to Plasma\nPhysics 49, 134–147.\n9P. Bandyopadhyay, G. Prasad, A. Sen, and P. K. Kaw, “Experi-\nmental study of nonlinear dust acoustic solitary waves in a dusty\nplasma,” Phys. Rev. Lett. 101, 065006 (2008).\n10M. Choudhary, S. Mukherjee, and P. Bandyopadhyay, “Prop-\nagation characteristics of dust–acoustic waves in presence of a\nfloating cylindrical object in the dc discharge plasma,” Physics\nof Plasmas 23, 083705 (2016).\n11M. Choudhary, S. Mukherjee, and P. Bandyopadhyay, “Trans-\nport and trapping of dust particles in a potential well created\nby inductively coupled diffused plasmas,” Rev. Sci. Instrum. 87,\n053505 (2016).\n12N. Chaubey and J. Goree, “Preservation of a dust crystal as it\nfalls in an afterglow plasma,” Frontiers in Physics 10(2022).\n13M. Schwabe, M. Rubin-Zuzic, S. Zhdanov, H. M. Thomas, and\nG. E. Morfill, “Highly resolved self-excited density waves in a\ncomplex plasma,” Phys. Rev. Lett. 99, 095002 (2007).\n14P. K. Shukla and A. A. Mamun, “Dust-acoustic shocks in a\nstrongly coupled dusty plasma,” IEEE Trans. Plasma Sci. 29,\n221–225 (2001).\n15C. Thompson, A. Barkan, N. D’Angelo, and R. L. Merlino, “Dust\nacoustic waves in a direct current glow discharge,” Phys. Plasmas\n4, 2331–2335 (1997).\n16T. M. Flanagan and J. Goree, “Observation of the spatial growth\nof self-excited dust-density waves,” Phys. Plasmas 17, 123702\n(2010).\n17V. I. Molotkov, O. F. Petrov, M. Y. Pustyl’nik, V. M. Torchinskii,\nV. E. Fortov, and A. G. Khrapak, “Dusty plasma of a dc glow\ndischarge: methods of investigation and characteristic features of\nbehavior,” High Temperature 42, 827–841 (2004).\n18V. Y. Karasev, E. S. Dzlieva, S. I. Pavlov, M. A. Ermolenko, L. A.\nNovikov, and S. A. Maiorov, “The dynamics of dust structures\nunder magnetic field in stratified glow discharge,” Contributions\nto Plasma Physics 56, 197–203 (2016).19M. Choudhary, R. Bergert, S. Moritz, S. Mitic, and M. H. Thoma,\n“Rotational properties of annulus dusty plasma in a strong mag-\nnetic field,” Contributions to Plasma Physics 61, e202000110\n(2021).\n20T. Bockwoldt, O. Arp, K. O. Menzel, and A. Piel, “On the origin\nof dust vortices in complex plasmas under microgravity condi-\ntions,” Phys. Plasmas 21, 103703 (2014).\n21M. Choudhary, S. Mukherjee, and P. Bandyopadhyay, “Experi-\nmental observation of self excited co-rotating multiple vortices\nin a dusty plasma with inhomogeneous plasma background,”\nPhysics of Plasmas 24, 033703 (2017).\n22V. Singh Dharodi, S. Kumar Tiwari, and A. Das, “Visco-elastic\nfluid simulations of coherent structures in strongly coupled dusty\nplasma medium,” Physics of Plasmas 21, 073705 (2014).\n23M. Choudhary, S. Mukherjee, and P. Bandyopadhyay, “Collective\ndynamics of large aspect ratio dusty plasma in an inhomogeneous\nplasma background: Formation of the co-rotating vortex series,”\nPhysics of Plasmas 25, 023704 (2018).\n24M. Laishram, D. Sharma, P. K. Chattopdhyay, and P. K. Kaw,\n“Nonlinear effects in the bounded dust-vortex flow in plasma,”\nPhys. Rev. E 95, 033204 (2017).\n25K.-B. Chai and P. M. Bellan, “Vortex motion of dust particles due\nto non-conservative ion drag force in a plasma,” Phys. Plasmas\n23, 023701 (2016).\n26H. Thomas, G. E. Morfill, V. Demmel, J. Goree, B. Feuerbacher,\nand D. M¨ ohlmann, “Plasma crystal: Coulomb crystallization in\na dusty plasma,” Phys. Rev. Lett. 73, 652–655 (1994).\n27J. H. Chu and L. I, “Direct observation of coulomb crystals and\nliquids in strongly coupled rf dusty plasmas,” Phys. Rev. Lett.\n72, 4009–4012 (1994).\n28M. G. Hariprasad, P. Bandyopadhyay, G. Arora, and A. Sen,\n“Experimental observation of a dusty plasma crystal in the cath-\node sheath of a DC glow discharge plasma,” Physics of Plasmas\n25, 123704 (2018).\n29A. Melzer, A. Homann, and A. Piel, “Experimental investigation\nof the melting transition of the plasma crystal,” Phys. Rev. E\n53, 2757–2766 (1996).\n30S. K. Tiwari, A. Das, D. Angom, B. G. Patel, and P. Kaw,\n“Kelvin-helmholtz instability in a strongly coupled dusty plasma\nmedium,” Physics of Plasmas 19, 073703 (2012).\n31V. S. Dharodi, B. Patel, and A. Das, “Kelvin–helmholtz instabil-\nity in strongly coupled dusty plasma with rotational shear flows\nand tracer transport,” Journal of Plasma Physics 88, 905880103\n(2022).\n32B. M. Veeresha, A. Das, and A. Sen, “Rayleigh–taylor instability\ndriven nonlinear vortices in dusty plasmas,” Phys. Plasmas 12,\n044506 (2005).\n33K. Avinash and A. Sen, “Rayleigh-taylor instability in dusty\nplasma experiment,” Physics of Plasmas 22, 083707 (2015).\n34V. S. Dharodi and A. Das, .\n35E. Thomas, Jr., K. Avinash, and R. L. Merlino, “Probe induced\nvoids in a dusty plasma,” Phys. Plasmas 11, 1770–1774 (2004).\n36B. J. Harris, L. S. Matthews, and T. W. Hyde, “Dusty plasma\ncavities: Probe-induced and natural,” Phys. Rev. E 91, 063105\n(2015).\n37S. Sarkar, M. Mondal, M. Bose, and S. Mukherjee, “Observation\nof external control and formation of a void in cogenerated dusty\nplasma,” Plasma Sources Sci. Technol. 24, 035007 (2015).\n38A. R. Abdirakhmanov, S. K. Kodanova, and T. S. Ramazanov,\n“Dynamics of dust particles in the glow discharge stratum\nin crossed e ×b field,” Contributions to Plasma Physics n/a,\ne202200148 (2022).\n39V. Y. Karasev, E. S. Dzlieva, S. I. Pavlov, M. A. Ermolenko, L. A.\nNovikov, and S. A. Maiorov, “The dynamics of dust structures\nunder magnetic field in stratified glow discharge,” Contributions\nto Plasma Physics 56, 197–203 (2016).\n40T. Trottenberg, D. Block, and A. Piel, “Dust confinement\nand dust-acoustic waves in weakly magnetized anodic plasmas,”\nPhys. Plasmas 13, 042105 (2006).10\n41E. S. Dzlieva, L. G. Dyachkov, V. Y. Karasev, L. A. Novikov, and\nS. I. Pavlov, “Dusty plasma under conditions of glow discharge\nin magnetic field of up to 2.5 t,” Plasma Physics Reports 49,\n10–14 (2023).\n42E. Thomas, U. Konopka, D. Artis, B. Lynch, S. Leblanc,\nS. Adams, R. L. Merlino, and M. Rosenberg, “The magnetized\ndusty plasma experiment (mdpx),” Journal of Plasma Physics\n81(2015).\n43B. Tadsen, F. Greiner, and A. Piel, “Probing a dusty magnetized\nplasma with self-excited dust-density waves,” Phys. Rev. E 97,\n033203 (2018).\n44M. Choudhary, R. Bergert, S. Mitic, and M. H. Thoma, “Com-\nparative study of the surface potential of magnetic and non-\nmagnetic spherical objects in a magnetized radio-frequency dis-\ncharge,” Journal of Plasma Physics 86, 905860508 (2020).\n45A. Barkan, N. D’Angelo, and R. L. Merlino, “Charging of dust\ngrains in a plasma,” Phys. Rev. Lett. 73, 3093–3096 (1994).\n46P. K. Shukla, “A survey of dusty plasma physics,” Physics of\nPlasmas 8, 1791–1803 (2001).\n47V. N. Tsytovich, N. Sato, and G. E. Morfill, “Note on the charg-\ning and spinning of dust particles in complex plasmas in a strong\nmagnetic field,” New Journal of Physics 5, 43–43 (2003).\n48A. Melzer, H. Kr¨ uger, S. Sch¨ utt, and M. Mulsow, “Finite dust\nclusters under strong magnetic fields,” Physics of Plasmas 26,\n093702 (2019).\n49D. Lange, “Floating surface potential of spherical dust grains in\nmagnetized plasmas,” Journal of Plasma Physics 82, 905820101\n(2016).\n50L. Patacchini, I. H. Hutchinson, and G. Lapenta, “Electron col-\nlection by a negatively charged sphere in a collisionless magne-\ntoplasma,” Physics of Plasmas 14, 062111 (2007).\n51T. Yukihiro, K. Gakushi, Y. Takatoshi, and I. Osamu, “Charg-\ning of dust particles in magnetic field,” J. Plasma Fusion Res.\nSERIES 8, 273–276 (2009).\n52S. K. Kodanova, N. K. Bastykova, T. S. Ramazanov, G. N. Nig-\nmetova, S. A. Maiorov, and Z. A. Moldabekov, “Charging of a\ndust particle in a magnetized gas discharge plasma,” IEEE Trans-\nactions on Plasma Science 47, 3052–3056 (2019).\n53H. Davari, B. Farokhi, and M. Ali Asgarian, “Particle simula-\ntion of the strong magnetic field effect on dust particle charging\nprocess,” Scientific Reports 13, 197–203 (2023).\n54A. Barkan, R. L. Merlino, and N. D’Angelo, “Laboratory obser-\nvation of the dust-acoustic wave mode,” Phys. Plasmas 2, 3563–\n3565 (1995).\n55H. Thomas, G. E. Morfill, V. Demmel, J. Goree, B. Feuerbacher,\nand D. M¨ ohlmann, “Plasma crystal: Coulomb crystallization in\na dusty plasma,” Phys. Rev. Lett. 73, 652–655 (1994).\n56V. E. Fortov, A. D. Usachev, A. V. Zobnin, V. I. Molotkov, and\nO. F. Petrov, “Dust-acoustic wave instability at the diffuse edge\nof radio frequency inductive low-pressure gas discharge plasma,”\nPhys. Plasmas 10, 1199–1208 (2003).\n57J. Pramanik, G. Prasad, A. Sen, and P. K. Kaw, “Experimental\nobservations of transverse shear waves in strongly coupled dusty\nplasmas,” Phys. Rev. Lett. 88, 175001 (2002).\n58A. Homann, A. Melzer, S. Peters, R. Madani, and A. Piel, “Laser-\nexcited dust lattice waves in plasma crystals,” Physics Letters A\n242, 173 – 180 (1998).\n59N. N. Rao, P. K. Shukla, and M. Y. Yu, “Dust-acoustic waves in\ndusty plasmas,” Planet. Space Sci. 38, 543–546 (1990).\n60B. Farokhi, P. K. Shukla, N. L. Tsintsadze, and D. D. Tskhakaya,\n“Dust lattice waves in a plasma crystal,” Phys. Plasmas 7, 814–\n818 (2000).\n61M. Choudhary, R. Bergert, S. Mitic, and M. H. Thoma, “In-\nfluence of external magnetic field on dust acoustic waves in a\ncapacitive rf discharge,” Contributions to Plasma Physics 60,\ne201900115 (2020).\n62A. Melzer, H. Kr¨ uger, S. Sch¨ utt, and D. Maier, “Dust-density\nwaves in radio-frequency discharges under magnetic fields,”\nPhysics of Plasmas 27, 033704 (2020).63M. Salimullah and M. Salahuddin, “Dust-acoustic waves in a\nmagnetized dusty plasma,” Physics of Plasmas 5, 828–829 (1998).\n64A. Agarwal and G. Prasad, “Spontaneous dust mass rotation in\nan unmagnetized dusty plasma,” Physics Letters A 309, 103 –108\n(2003).\n65M. Kaur, S. Bose, P. K. Chattopadhyay, D. Sharma, J. Ghosh,\nY. C. Saxena, and E. Thomas, “Generation of multiple toroidal\ndust vortices by a non-monotonic density gradient in a direct cur-\nrent glow discharge plasma,” Phys. Plasmas 22, 093702 (2015).\n66S. Mitic, R. S¨ utterlin, A. V. I. H. H¨ ofner, M. H. Thoma, S. Zh-\ndanov, and G. E. Morfill, “Convective dust clouds driven by ther-\nmal creep in a complex plasma,” Phys. Rev. Lett. 101, 235001\n(2008).\n67U. Konopka, D. Samsonov, A. V. Ivlev, J. Goree, V. Steinberg,\nand G. E. Morfill, “Rigid and differential plasma crystal rotation\ninduced by magnetic fields,” Phys. Rev. E 61, 1890–1898 (2000).\n68H. Feng, L. Yan-Hong, C. Zhao-Yang, W. Long, and Y. Mao-Fu,\n“Cluster rotation in an unmagnetized dusty plasma,” Chinese\nPhysics Letters 30, 115201 (2013).\n69F. M. H. Cheung, N. J. Prior, L. W. Mitchell, A. A. Samarian,\nand B. W. James, “Rotation of coulomb crystals in a magne-\ntized inductively coupled complex plasma,” IEEE Transactions\non Plasma Science 31, 112–118 (2003).\n70O. Ishihara and N. Sato, “On the rotation of a dust particulate\nin an ion flow in a magnetic field,” IEEE Transactions on Plasma\nScience 29, 179–181 (2001).\n71M. Choudhary, R. Bergert, S. Mitic, and M. H. Thoma, “Three-\ndimensional dusty plasma in a strong magnetic field: Observation\nof rotating dust tori,” Physics of Plasmas 27, 063701 (2020).\n72D. A. Law, W. H. Steel, B. M. Annaratone, and J. E. Allen,\n“Probe-induced particle circulation in a plasma crystal,” Phys.\nRev. Lett. 80, 4189–4192 (1998).\n73P. K. Kaw, K. Nishikawa, and N. Sato, “Rotation in collisional\nstrongly coupled dusty plasmas in a magnetic field,” Physics of\nPlasmas 9, 387–390 (2002).\n74O. Vaulina, A. Samarian, A. Nefedov, and V. Fortov, “Self-\nexcited motion of dust particles in a inhomogeneous plasma,”\nPhysics Letters A 289, 240–244 (2001).\n75M. Laishram, D. Sharma, and P. K. Kaw, “Dynamics of a con-\nfined dusty fluid in a sheared ion flow,” Phys. Plasmas 21, 073703\n(2014).\n76M. R. Akdim and W. J. Goedheer, “Modeling of self-excited dust\nvortices in complex plasmas under microgravity,” Phys. Rev. E\n67, 056405 (2003).\n77M. Klindworth, A. Piel, A. Melzer, U. Konopka, H. Rothermel,\nK. Tarantik, and G. E. Morfill, “Dust-free regions around lang-\nmuir probes in complex plasmas under microgravity,” Phys. Rev.\nLett. 93, 195002 (2004).\n78J. Goree, G. E. Morfill, V. N. Tsytovich, and S. V. Vladimirov,\n“Theory of dust voids in plasmas,” Phys Rev E Stat Phys Plas-\nmas Fluids Relat Interdiscip Topics. 59, 7055–67 (1999).\n79L. M. Vasilyak, S. P. Vetchinin, and D. N. Polyakov, “Effect of\nionization on void formation in an rf discharge under micrograv-\nity conditions,” Plasma Physics Reports 49, 290–295 (2023).\n80Z. Donk´ o, P. Hartmann, and G. J. Kalman, “Two-dimensional\ndusty plasma crystals and liquids,” Journal of Physics: Confer-\nence Series 162, 012016 (2009).\n81S. Arumugam, P. Bandyopadhyay, S. Singh, M. G. Hariprasad,\nD. Rathod, G. Arora, and A. Sen, “Dpex-ii: a new dusty plasma\ndevice capable of producing large sized dc coulomb crystals,”\nPlasma Sources Science and Technology 30, 085003 (2021).\n82N. Chaubey and J. Goree, “Preservation of a dust crystal as it\nfalls in an afterglow plasma,” Frontiers in Physics 10, 879092\n(2022).\n83S. Jaiswal, T. Hall, S. LeBlanc, R. Mukherjee, and E. Thomas,\n“Effect of magnetic field on the phase transition in a dusty\nplasma,” Physics of Plasmas 24, 113703 (2017).\n84E. Thomas, R. L. Merlino, and M. Rosenberg, “Magnetized dusty\nplasmas: the next frontier for complex plasma research,” Plasma\nPhysics and Controlled Fusion 54, 124034 (2012).11\n85A. Melzer, H. Kr¨ uger, D. Maier, and S. Sch¨ utt, “Physics of mag-\nnetized dusty plasmas,” Reviews of Modern Plasma Physics 5,\n11 (2021).\n86M. Menati, E. Thomas, and M. J. Kushner, “Filamentation of\ncapacitively coupled plasmas in large magnetic fields,” Physics\nof Plasmas 26, 063515 (2019).87N. Chaubey, J. Goree, S. J. Lanham, and M. J. Kushner, “Pos-\nitive charging of grains in an afterglow plasma is enhanced by\nions drifting in an electric field,” Physics of Plasmas 28, 103702\n(2021).\n88N. Chaubey and J. Goree, “Coulomb expansion of a thin dust\ncloud observed experimentally under afterglow plasma condi-\ntions,” Physics of Plasmas 29, 113705 (2022)."    },    {        "title": "0907.3094v2.Current_induced_dynamics_in_non_collinear_dual_spin_valves.pdf",        "content": "arXiv:0907.3094v2  [cond-mat.mes-hall]  25 Sep 2009APS/123-QED\nCurrent-induced dynamics in non-collinear dual spin-valv es\nP. Bal´ aˇ z,1,∗M. Gmitra,2and J. Barna´ s1,3\n1Department of Physics, Adam Mickiewicz University, Umulto wska 85, 61-614 Pozna´ n, Poland\n2Institute of Phys., P. J. ˇSaf´ arik University, Park Angelinum 9, 040 01 Koˇ sice, Slov ak Republic\n3Institute of Molecular Physics, Polish Academy of Sciences Smoluchowskiego 17, 60-179 Pozna´ n, Poland\n(Dated: October 22, 2018)\nSpin-transfer torque and current induced spin dynamics in s pin-valve nanopillars with the free\nmagnetic layer located between two magnetic films of fixed mag netic moments is considered theo-\nretically. The spin-transfer torque in the limit of diffusiv e spin transport is calculated as a function\nof magnetic configuration. It is shown that non-collinear ma gnetic configuration of the outermost\nmagnetic layers has a strong influence on the spin torque and s pin dynamics of the central free\nlayer. Employing macrospin simulations we make some predic tions on the free layer spin dynamics\nin spin valves composed of various magnetic layers. We also p resent a formula for critical current\nin non-collinear magnetic configurations, which shows that the magnitude of critical current can be\nseveral times smaller than that in typical single spin valve s.\nPACS numbers:\nI. INTRODUCTION\nMagnetic multilayers are ones of the most relevant ele-\nments in the developmentofcutting-edgetechnologies. If\ncurrent flows along the axis of a metallic hybrid nanopil-\nlar structure, a spin accumulation builds up in the vicin-\nity of the normal-metal/ferromagnet interface. More-\nover, the current produces spin-transfer torque (STT)\nwhich acts on magnetic moments of the ferromagnetic\nlayers1,2. As a consequence, magnetization of a particu-\nlar layer may be driven to an oscillation mode3, or can\nbe switched between possible stable states4. However,\ncurrent-induced control of magnetic moments requires a\nrather high current density. Therefore, reduction of the\ncritical current density remains the most challenging re-\nquirementfrom the pointofviewofpossibleapplications.\nSince the critical currents are related to STT, which in\nthe diffusive transport regime is proportional to spin ac-\ncumulation at the normal-metal/ferromagnet interface,\none may alternatively rise a question of possible ways to\nenhance the spin accumulation.\nOne of the possibilities to decrease the critical current\ndensity in metallic structures is the geometry proposed\nby L. Berger5, in which the free magnetic layer is located\nbetween two pinned magnetic layers of opposite magne-\ntizations. Indeed, in such a dual spin valve (DSV) geom-\netry both interfaces of the free layer can generate STT,\nand this may decrease the critical current several times5.\nTo examine the influence of an additional magnetic\nlayer on the free layer’s spin dynamics one needs to find\nfirst the STT acting on the central layer for arbitrary di-\nrectionofitsmagnetizationvector. Todothis, weemploy\nthe spin-dependent diffusive transport approach6, based\non Valet-Fert description7. In this paper we present a\ncomprehensive survey of STT-induced effects in DSVs\nfor generally non-collinear magnetic configuration of the\noutermost magnetic films. We also study asymmetric\nexchange-biased DSV, in which magnetic moment of one\nof the outer magnetic layers is fixed to an antiferromag-netic layer due to exchange anisotropy. We show that a\nnon-standard (wavy-like) angular dependence of STT –\noriginally predicted for asymmetric spin valves6,8– can\nalso occur in DSV geometry. Such a non-standard an-\ngular dependence of the torque is of some importance\nfrom the application point of view, as it allows to induce\nsteady-state precessional modes without external mag-\nnetic field8,10. Furthermore, we examine current-induced\nspin dynamics within the macrospin model, and derive a\nformulaforcriticalcurrentswhich destabilizetrivialfixed\npointsofthecentralspin’sdynamicsinnon-collinearcon-\nfigurations of the outermost magnetizations.\nIn section II we describe the model and introduce ba-\nsic formula for STT and spin dynamics in DSVs. Then,\nin section III we study dynamics in a symmetric DSV\ngeometry, where we compare STT and spin dynamics in\nDSVs and single spin valves (SSVs). Finally, in section\nIV we analyze an exchange-biased DSV structure with\nnoncollinear magnetic moments of the outermost mag-\nnetic layers. Finally, we conclude in section V.\nII. MODEL\nWe consider a multilayer nanopillar structure,\nFL/NL/FC/NR/FR,consistingofthreeferromagnetic(F)\nlayers separated by normal-metal (N) layers; see Fig. 1.\nSpin moment of the central layer, F C, is free to rotate,\nwhile the right F Rand left F Lferromagnetic layers are\nmuch thicker and their net spin moments are assumed to\nbe fixed for current densities of interest. Fixing of these\nmoments can be achieved either by sufficiently strong\ncoercieve fields, or by exchange anisotropy at interfaces\nwith antiferromagnetic materials.\nIn the Landau-Lifshitz-Gilbert phenomenological de-\nscription, the dynamics of a unit vector along the net\nspin moment ˆsof the central (free) magnetic layer is de-2\nFIG. 1: Schematic of a dual spin valve with fixed magnetic\nmoments of the outer magnetic layers, F Land F R, and free\ncentral magnetic layer, F C, separated by normal-metal layers\nNLand N R. Here, ˆSL,ˆSR, andˆsare unit vectors along the\nspin moments of the F L, FR, and F Clayers, respectively.\nscribed by the equation\ndˆs\ndt+αˆs×dˆs\ndt=Γ, (1)\nwhereαis the Gilbert damping parameter. The right-\nhandsiderepresentsthetorquesduetoeffectivemagnetic\nfield and spin-transfer,\nΓ=−|γg|µ0ˆs×Heff+|γg|\nMsdτ, (2)\nwhereγgis the gyromagneticratio, µ0is the vacuum per-\nmeability, Msstands for the saturation magnetization of\nthe free layer, and ddenotes thickness of the free layer.\nConsidering the thin ferromagnetic free layer of ellipti-\ncal cross-section, the effective magnetic field Heffcan be\nwritten as\nHeff=−Hextˆez−Hani(ˆs·ˆez)ˆez+Hdem,(3)\nand includes applied external magnetic field ( Hext), uni-\naxial anisotropy field ( Hani), and the demagnetization\nfield (Hdem), where ˆezis the unit vector along the z-axis\n(easy axis), compare scheme in Fig. 1. The demagnetiza-\ntion field can be written in the form Hdem= (ˆs·¯N)Ms=\n(Hdxsx,Hdysy,Hdzsz), where ¯Nis a diagonal demagne-\ntization tensor.\nSpin-transfer torque\nGenerally, STT acting on a magnetic layer is deter-\nmined by the electron spin angular momentum absorbedfrom conduction electrons within a few interfacial atomic\nlayersof the ferromagnet11. Thus, the STT acting on the\ncentral layer F Ccan be calculated as τ= (/planckover2pi1/2)(jL\n⊥−jR\n⊥),\nwherejL\n⊥andjR\n⊥are the spin current components per-\npendicular to magnetic moment of the free layer and\ncalculated at the corresponding left and right normal-\nmetal/ferromagnet interfaces. The torque consists of in-\nplaneτ/bardbl(in the plane formed by magnetic moments of\nthe two interacting films) and out-of-plane τ⊥(normal to\nthis plane) parts, τ=τ/bardbl+τ⊥, which have the following\nforms\nτ/bardbl=Iˆs×/bracketleftBig\nˆs×/parenleftBig\naLˆSL−aRˆSR/parenrightBig/bracketrightBig\n,(4a)\nτ⊥=Iˆs×/parenleftBig\nbLˆSL−bRˆSR/parenrightBig\n, (4b)\nwhereIis the current density, and ˆSLandˆSRare the\nunit vectors pointing along the fixed net-spins of the F L\nand FRlayers, respectively. The parameters aL,aR,bL,\nandbRdepend, generally, on the magnetic configuration\nand material composition of the system, and have been\ncalculated in the diffusive transport limit6. According to\nfirst-principles calculations of the mixing conductance12,\nthe out-of-plane torque in metallic structures is about\ntwoordersofmagnitudesmallerthanthein-planetorque.\nAlthough this component has a minor influence on crit-\nical currents, it may influence dynamical regimes, so we\ninclude it for completeness in our numerical calculations.\nLet us consider now how the STT affects spin\ndynamics of the free magnetic layer. We rewrite\nEq.(1) in spherical coordinates ( φ,θ), which obey ˆs=\n(cosφsinθ,sinφsinθ,cosθ); see Fig. 1. Defining unit\nbase vectors, ˆeφ= (ˆez׈s)/sinθandˆeθ=ˆeφ׈s, Eq.(1)\ncan be rewritten as\nd\ndt/parenleftbigg\nφ\nθ/parenrightbigg\n=1\n1+α2/parenleftbigg\nsin−1θ−αsin−1θ\nα 1/parenrightbigg/parenleftbigg\nvφ\nvθ/parenrightbigg\n,(5)\nwhere the overall torques vθandvφ, changing the angles\nθandφ, respectively, read\nvθ=Γ·ˆeθ=−|γg|µ0(Hdx−Hdy)cosφsinφsinθ+|γg|\nMsdτθ, (6a)\nvφ=Γ·ˆeφ=−|γg|µ0/bracketleftbig\nHext+(Hani+Hdxcos2φ+Hdysin2φ−Hdz)cosθ/bracketrightbig\nsinθ+|γg|\nMsdτφ. (6b)\nThe first terms in Eqs. (6) describe the torques due to demagnetiz ation and anisotropy fields of the free layer,3\nwhileτθ=τ/bardbl\nθ+τ⊥\nθandτφ=τ/bardbl\nφ+τ⊥\nφarethecorresponding\ncomponents of the current-induced torque, which origi-\nnate from τ/bardblandτ⊥.\nAs we have already mentioned above, the main con-\ntribution to STT comes from τ/bardbl. Assuming now that\nmagnetic moment of the left magnetic layer is fixed\nalong the z-axis,ˆSL= (0,0,1), and magnetic moment\nof the right magnetic layer is rotated by an angle Ω from\nthez-axis and fixed in the layer’s plane (see Fig. 1),\nˆSR= (0,sinΩ,cosΩ), one finds\nτ/bardbl\nθ= (aL−aRcosΩ)Isinθ+aRIsinΩsinφcosθ,(7a)\nτ/bardbl\nφ=aRIcosφsinΩ. (7b)\nThe component τ/bardbl\nθconsists of two terms. The first one is\nanalogous to the term which describes STT in a SSV.\nHowever, its amplitude is now modulated due to the\npresence of F R. In turn, the second term in Eq. (7a)\nis nonzero only in noncollinear magnetic configurations.\nFrom Eq. (7b) follows that τ/bardbl\nφis also nonzero in non-\ncollinear configurations, and only if the magnetization\npoints out of the layer’s plane ( φ/negationslash=π/2). When mag-\nnetic moments of the outer magnetic layers are parallel\n(Ω = 0), τ/bardbl\nθ= (aL−aR)Isinθ. For a symmetric DSV,\naL(θ) =aR(θ), and hence STT acting on ˆsvanishes. On\nthe other hand, in the antiparallel configuration of ˆSL\nandˆSR(Ω =π), the maximal spin torque enhancement\ncan be achieved, τ/bardbl\nθ= (aL+aR)Isinθ.\nSimilar analysis of τ⊥leads to the following formulas\nforτ⊥\nθandτ⊥\nφ:\nτ⊥\nθ=bRIcosφsinΩ, (8a)\nτ⊥\nφ=−(bL−bRcosΩ)Isinθ+bRcosθsinφsinΩ.(8b)\nThus, if the outer magnetic moments are collinear, τ⊥\nθ=\n0, whileτ⊥\nφreduces to τ⊥\nφ=−(bL−bR)Isinθfor Ω = 0,\nandτ⊥\nφ=−(bL+bR)Isinθfor Ω =π. Hence, in symmet-\nric spin valves, where bL=bR,τ⊥\nφvanishes in the parallel\nconfiguration of the outermost magnetic moments and is\nenhanced in the antiparallel configuration.\nIn the following sections we investigate STT and its\neffects on critical current and spin dynamics. We start\nfrom symmetric spin valves in antiparallel magnetic con-\nfiguration (Ω = π). Then, we proceed with asymmetric\nexchange-biased dual spin valves.\nIII. SYMMETRIC DSVS\nLet us consider first symmetric DSVs with antiparallel\norientation of magnetic moments of the outermost ferro-\nmagneticfilms: ˆSL= (0,0,1)andˆSR= (0,0,−1). Asin-\ndicatedbyEqs.(7)and(8), suchaconfigurationmaylead\nto enhancement of STT in comparison to that in SSVs.\nThus, let us analyze first STT in two types of structures:\nthe double spin valve F(20)/Cu(10)/F(8)/Cu(10)/F(20)in the antiparallel configuration, and the corresponding\nsingle spin valve F(20)/Cu(10)/F(8). The numbers in\nbrackets correspond to layer thicknesses in nanometers.\nIn Fig. 2 we show the angular dependence of STT for\nDSVs and SSVs, when the vector ˆschanges its orien-\ntation, described by angle θ, in the layer plane ( φ=\nπ/2). The magnetic layers made of Permalloy, Ni 80Fe20\n(Fig. 2a), and of Cobalt (Fig. 2b) are considered. Due\nto the additional fixed layer (F R), STT in DSVs is\nabout twice as large as in SSVs, which is consistent\nwith Berger’s predictions2. Additionally, the angular de-\npendence of STT acting on the free layer in Co/Cu/Co\nspin valves is more asymmetric than in Py/Cu/Py. This\nasymmetry, however, disappears in Co/Cu/Co/Cu/Co\nDSVsduetosuperpositionofthecontributionsfromboth\nfixed magnetic layers to the STT.\n 0 0.1 0.2 0.3 0.4\n 0 0.25 0.5 0.75 1 \nθ / πτθ(a)\n 0 0.25 0.5 0.75 100.050.10.150.2 \nθ / π(b)\nFIG. 2: Spin transfer torque τθin symmetric DSVs,\nF(20)/Cu(10)/F(8)/Cu(10)/F(20), in the antiparallel con -\nfiguration, Ω = π, (solid lines), and STT in SSVs,\nF(20)/Cu(10)/F(8) (dashed lines), where (a) F = Permal-\nloy, (b) F = Cobalt. STT is shown in the units of /planckover2pi1|I|/|e|,\nand calculated for φ=π/2. The material parameters as in\nRef.13.\nThe enhancement of STT in dual spin valves may lead\nto two important improvements of the spin dynamics:\nreductionofthecriticalcurrentneededtotriggerthespin\ndynamics, and decrease of the switching time. The latter\nis defined as the time needed to switch the magnetization\nfrom one stable position to the opposite one. The fixed\npoints of the dynamics of ˆsare given by the equations\nvθ= 0 and vφ= 0. If Ω = π, they are satisfied for θ= 0,\nandθ=π. Employing the ’zero-trace’stability condition\nof the linearized Landau-Lifshitz-Gilbert equation14, we\nfind the critical current destabilizing the initial ( θ= 0)\nstate in the form\nI0\nc,DSV=αµ0Msd\na0\nL+a0\nR/parenleftbigg\nHext+Hani+Hdx+Hdy\n2−Hdz/parenrightbigg\n,\n(9)\nwherea0\nLanda0\nRare calculated for θ→0. Addition-\nally, we have omitted here the terms resulting from τ⊥\nbecause of their small contribution to the critical cur-\nrent. Equation (9) is analogous to the expression for\ncritical current in SSV13. According to our calculations,\nthe critical current in DSVs with F = Cobalt is 6-times\nsmaller than in SSVs, as reported by Berger5. However,\nif F = Permalloy, the introduction of a second fixed mag-\nnetic layer reduces the critical current only by a factor\nof 2. This difference arises from the dependence of spin4\naccumulation on spin-flip length which is about 10-times\nlonger in Cobalt than in Permalloy6.\nTo estimate the switching time in DSVs as well as in\nSSVs we employ the single-domain macrospin approx-\nimation to the central layer. Equations (1), including\nEq. (3) and Eq. (4), completely describe the dynamics\nof central layer’s spin, ˆs. In our simulations we assumed\na constant current of density I. The positive current\n(I >0) is defined for electrons flowing from F Rtowards\nFL(current then flows from F Ltowards F R); opposite\ncurrent is negative. Apart from this, the demagnetiza-\ntion field of the free layer of elliptical cross-section with\nthe axes’ lengths 130nm and 60nm has been assumed,\nwhile external magnetic field was excluded, Hext= 0.\nFor each value of the current density, from Eq. (1) we\nfound evolution of ˆsstarting from an initial state until ˆs\nswitched to the opposite state (provided such a switch-\ning was admitted). In all simulation, the spin was ini-\ntiallyslightlytiltedinthelayerplanefromtheorientation\nˆs= (0,0,1), assuming θ0= 1◦andφ0=π/2. A success-\nful switching, with the switching time ts, is counted when\nsz(ts)<−0.99, where sz(t) is the exponentially weighted\nmoving average15,sz(t) =η sz(t) + (1−η)sz(t−∆t),\n∆tis the integration step, and the weighting parame-\nterη= 0.1. The moving average szis calculated only\nin time intervals, where sz(t) remains continuously be-\nlow the value of −0.9; otherwise sz(t) =sz(t). Fig. 3\ncompares the switching times in DSVs and in the corre-\nsponding SSVs. In both cases shown in Fig. 3, a con-\nsiderable reduction of the switching time is observed in\nDSVs. Similarly as for the critical current, the reduction\nof current required for switching in DSVs with Cobalt\nlayers is larger from that in DSVs with Permalloy layers.\n02468\n01234tsw [ns]\nI / I0(a)\n     \n0.51.52.53.5 \nI / I0(b)\nFIG. 3: Switching time in DSVs\nF(20)/Cu(10)/F(8)/Cu(10)/F(20) in the antiparallel mag-\nnetic configuration, Ω = π, (solid lines), and in SSVs\nF(20)/Cu(10)/F(8) (dashed lines), where (a) F = Permalloy,\n(b) F = Cobalt. The switching time is shown as a function\nof the normalized current density I/I0, withI0= 108Acm−2.\nThe other parameters as in Fig.2.\nIV. EXCHANGE-BIASED DSV\nLet us consider now an asymmetric exchange-biased\nDSV structure with an antiferromagnetic layer IrMn ad-\njacent to the F Rlayer in order to pin its magneticmoment in a required orientation, i.e. the structure\nCo(20)/Cu(10)/Py(4)/Cu(4)/Co(10)/IrMn(8). The left\nmagnetic layer, F L= Co(20), is assumed to be thick\nenough so its magnetic moment is fixed, ˆSL=ˆez. In\nturn, magnetic moment of the right ferromagnetic layer,\nFR= Co(10), is fixed in the layer plane at a certain angle\nΩ with respect to ˆSLdue to the exchange-bias coupling\nto IrMn.\nFrom Eqs. (6), (7), and (8) follows, that in a general\ncase (Ω/negationslash= 0), the points θ= 0 and θ=πare no more\nsolutions of the conditions for fixed points, vθ= 0 and\nvφ= 0,becauseoftheadditionaltermsinSTT,whichap-\npear in non-collinear situations (discussed in section II).\nThese additional terms lead to a nontrivial θ-dependence\nof STT, and to a shift of the fixed points out of the\ncollinear positions.\nTo analyze this effect in more details, let us consider\nfirst STT assuming ˆsin the layer plane ( φ=π/2). Ac-\ncording to Eq. (7b) and Eq. (8b), and to the fact that the\nparameters bare much smaller than a, the component τφ\nis verysmall. In Fig. 4(a) we show the secondcomponent\nof the torque, namely τθ, as a function of the angle θand\nfor different values of the angle Ω. The configurations\nwhereτθ= 0 are presented by the contour in the base\nplane of Fig. 4(a).\nIn the whole range of the angle Ω, two ’trivial’ zero\npoints are present. Additionally, for small angles close\nto Ω = 0, two additional zero points occur. The ap-\npearance of these additional zero points closely resem-\nbles non-standard wavy-like STT angular dependence,\nwhich has been already reported in single spin valves\nwith magnetic layers made of different materials6,8,16.\nSuch a θ-dependence dictates that both zero-current\nfixed points ( θ= 0,π) become simultaneously stabi-\nlized (destabilized) for positive (negative) current. This\nbehavior is of special importance for stabilization of\nthe collinear configurations, and for microwave genera-\ntion driven by current only (without external magnetic\nfield)6,8,17. We note, that in contrast to SSVs, the wavy-\nlikeθ-dependence in exchange-biased DSVs is related\nrather to asymmetric geometry of the multilayer than\nto bulk and interface spin asymmetries. This trend is\ndepicted in Fig. 4(b), where we show variation of STT\nfor different thicknesses of F Rat Ω = 0. The wavy-like\ntorque angular dependence appears for the thickness of\nFRmarkedly different from that of F L.\nMaking use of the above described STT calculations,\nextended to arbitrary orientation of ˆs, we performed nu-\nmerical simulations of spin dynamics induced by a con-\nstant current in zero external magnetic field ( Hext= 0).\nThe sample cross-section was assumed in the form of an\nellipse with the axes’ lengths 130nm and 60nm. As be-\nfore, the numerical analysis has been performed within\nthe macrospin framework, integrating equation of mo-\ntion, Eq.(1). Inthe simulationweanalyzedthelong-term\ncurrent-induced spin dynamics started from the initial\nstate corresponding to θ0= 1◦,φ0=π/2.\nAs one might expect, the current-induced spin dynam-5\nθ / π\nΩ / π 0\n 0.5\n 1\n 1.5\n 2 0 0.25 0.5 0.75 1(a)\nθ / π\nΩ / π-0.2-0.1 0 0.1 0.2\nτθ\n 0\n 0.5\n 1\n 1.5\n 2 0 0.25 0.5 0.75 1τθ(a)-0.04-0.02 0 0.02\n 0 0.25 0.5 0.75 1 \nθ / πτθ4 nm\n6 nm\n8 nm\n20 nm(b)\nFIG. 4: (a) STT acting on the central magnetic layer\nin Co(20)/Cu(10)/Py(4)/Cu(4)/Co(10)/IrMn(8) exchange-\nbiased DSV as a function of the angle θ, calculated for\nΩ =kπ/4, k= 0,1,2,3,4. The contour plot in the base\nplane corresponds to zeros of τθ. (b) Wavy-like STT angular\ndependence in exchange biased DSV for Ω = 0, calculated for\ndifferent thicknesses of the F Rlayer. STT is shown in the\nunits of/planckover2pi1|I|/|e|. The other parameters as in Fig.2.\nics of the free layer depends on the angle Ω, current den-\nsityI, and on current direction. To designate different\nregimes of STT-induced spin dynamics, we constructed\na dynamical phase diagram as a function of current and\nthe angle Ω. The diagram shows the average value /angbracketleftsz/angbracketright\nof thezcomponent of the free layer net-spin in a sta-\nble dynamical regime (see Fig. 5a), as well as its disper-\nsionD(sz) =/radicalBig\n/angbracketlefts2z/angbracketright−/angbracketleftsz/angbracketright2(see Fig. 5b). The average\nvalue/angbracketleftsz/angbracketrightprovides an information on the spin orienta-\ntion, whereas the dispersion distinguishes between static\nstates (for which D= 0) and steady precessional regimes\n(whereD >0), in which the zcomponent is involved.\nFor each point in the diagram, a separate run from the\ninitial biased state φ0=π/2 andθ0= 1◦was performed.\nIn the/angbracketleftsz/angbracketrightdiagram, Fig. 5(a), one can distinguish three\nspecific regions. Region (i) covers parameters for which\na weak dynamics is induced only: ˆsfinishes in the equi-\nlibrium stable point which is very close to ˆs= (0,0,1).\nAs the angle Ω increases, STT becomes strong enough to\ncauseswitching,asobservedintheregion(ii). Thehigher\nthe angle Ω, the smaller is the critical current needed for\ndestabilization of the initial state. For smaller Ω, the\ncurrent-induced dynamics occurs for currents flowing in\nthe opposite direction, see the region (iii). This behav-\nior is caused by different sign of STT in the initial state.\nFrom/angbracketleftsz/angbracketrightwe conclude, that neither of the previously\nmentioned stable states is reached. To elucidate the dy-\nnamics in region (iii), we constructed the corresponding\nmap ofD(sz), Fig. 5(b). This map reveals three different\nmodes of current-induced dynamics. For small current\namplitudes, in-plane precession (IPP) around the initial\nstable position is observed. The precessional angle rises 0 0.25 0.5 0.75  1\n -3-2-1 0 1 2 3I / I0\n-1-0.5 0 0.5 1\nsz average(a)\n(i) (ii)\n(iii)\n00.10.20.30.40.5\nΩ / π-3-2-10I / I0\n 0 0.2 0.4 0.6 0.8\nsz dispersion(b)\n(c)\n(d)OPP\nSSIPP\n-1-0.500.51-1-0.500.510.50.751sz\nsxsyszIPP\n-0.55-0.5-0.45-1-0.500.51 -1-0.500.51sz\nsxsysz OPP\n29.0529.0629.0729.0829.09\n0510R [fΩ m2]\ntime [ns]29.129.229.329.4\n0 5 10 \ntime [ns]\nFIG. 5: (Color online) Dynamical phase diagram for\nCo(20)/Cu(10)/Py(4)/Cu(4)/Co(10)/IrMn(8) exchange-\nbiased double spin valve as a function of the angle Ω and\nnormalized current density I/I0(withI0= 108Acm−2): (a)\naverage value of the szspin component; (b) dispersion of\ntheszspin component; (c) typical precessional orbits; (d)\nresistance oscillations associated with the IPP for Ω = 0 .4π\nandI=−1.3I0(left part), and with the OPP for Ω = 0 .1π\nandI=−1.3I0(right part). The other parameters as in\nFig.3.\nwith increasing current amplitude. Above a certain criti-\ncalvalue of I, the precessionsturn toout-of-planepreces-\nsions (OPPs). In a certain range of Ω, the OPPs collapse\nto a static state (SS), where the spin ˆsremains in an\nout-of-plane position close to ±ˆex.\nAsˆSRdeparts from the collinear orientation, the crit-\nical current needed for destabilization of the initial state\nincreases. This growth is mostly pronounced close to\nΩ =π/2. To describe the critical current, we analyzed\nEq. (5) with respect to the stability of ˆsin the upper po-6\nsition. Assuming that even in noncollinear configuration\nthe stable position of ˆsis close to θ= 0, we have lin-\nearized Eq. (5) around this point for arbitrary Ω. Then,\nfor the critical current needed for destabilization of the\nconsidered stable state we find\nI0\nc,EBDSV≃αµ0Msd/parenleftBig\nHani+Hdx+Hdy\n2−Hdz/parenrightBig\na0\nL(Ω)−a0\nR(Ω)cosΩ,(10)\nwherea0\nL(Ω) and a0\nR(Ω) are calculated (for each config-\nuration) assuming θ→0. Comparison of Eq. (10) with\nthe results of numerical simulations is shown in Fig. 5.\nWhen considering the opposite stable point as the initial\nstate, we need to take aLandaRforθ→π. Clearly, to\ndestabilize the θ=πstate one needs current of opposite\ndirection.\nCurrent-induced oscillations are usually observed ex-\nperimentally viathe magnetoresistance effect3. When\nelectric current is constant, then magnetic oscillations\ncause the corresponding resistance oscillations, which in\nturn lead to voltage oscillations. The later are measured\ndirectly in experiments. In Fig. 5(d) we show oscilla-\ntions in the system resistance associated with the IPP\n(left) and OPP (right). As the amplitude of the oscilla-\ntions correspondingto the OPP mode is sufficiently large\nto be measured experimentally, the amplitude associated\nwith the IPP mode is relatively small. This is the reason,\nwhy IPP mode is usually not seen in experiments.\nV. DISCUSSION AND CONCLUSIONS\nWehavecalculatedSTTinmetallicdualspinvalvesfor\narbitrary magnetic configuration of the system, but with\nmagnetic moments of the outer magnetic films fixed in\ntheirplaneseitherbylargecoercievefieldsorbyexchange\nanisotropy. In the case of symmetric DSV structures, we\nfound a considerableenhancement ofSTT in the antipar-\nallel magnetic configuration. This torque enhancement\nleads to reduction of the critical current for switching as\nwell as to reduction of the switching time. The switch-ing improvement has been found to be dependent on the\nspin-flip lengths in the magnetic and nonmagnetic lay-\ners. According to our numerical simulations, an ultrafast\nsubnanosecond current-induced switching processes can\noccurin DSVs with antiparallelmagnetic momentsofthe\noutermost magnetic films.\nIn exchange-biased spin valves we have identified con-\nditions which can lead to various types of spin dynam-\nics. For Ω /lessorsimilarπ/2, the negative current excites the central\nmagneticlayer,whileforΩ /greaterorsimilarπ/2,oppositecurrentdirec-\ntion is needed. We have evaluated parameters for which\nswitchingtoanewstablestateortoaprecessionalregime\nappears. This is especially interesting from the applica-\ntion point of view. However, the static state (SS) should\nbe treated carefully. Such a state is often connected with\nthewavy-like angular dependence of STT13,16,17. The\nSS becomes stable in the framework of macrospin model,\nwhen STT disappearsin acertain noncollinearconfigura-\ntion. However, experimentally such a state has not been\nobserved so far16. It has been shown in more realistic\nmicromagnetic approach, that this static state may cor-\nrespond to out-of-plane precessions18. The reason of this\nis the finite strength of exchange coupling in magnetic\nfilm, which does not fully comply with the macrospin\napproximation in some cases.\nFinally, we have derived an approximate formula for\nthe critical current in exchange biased DSVs, valid for\nnon-collinear magnetic configurations. This formula,\nEq.(10), is in good agreement with the numerical sim-\nulations.\nAcknowledgment\nThe work has been supported by the the EU\nthrough the Marie Curie Training network SPIN-\nSWITCH (MRTN-CT-2006-035327). M.G. alsoacknowl-\nedges support VEGA under Grant No. 1/0128/08, and\nJ.B. acknowledges support from the Ministry of Science\nandHigherEducationasaresearchprojectinyears2006-\n2009.\n∗Electronic address: balaz@amu.edu.pl\n1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n2L. Berger, Phys. Rev. B 54, 9353 (1996).\n3S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature\n425, 380 (2003).\n4J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n5L. Berger, J. Appl. Phys. 93, 7693 (2003).\n6J. Barna´ s, A. Fert, M. Gmitra, I. Weymann, and\nV. K. Dugaev, Phys. Rev. B 72, 024426 (2005).\n7T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).\n8M. Gmitra and J. Barna´ s, Phys. Rev. Lett. 96, 207205\n(2006).9A. A. Kovalev, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B66, 224424 (2002).\n10O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. Deranlot,\nF. Petroff, G. Faini, J. Barna´ s, and A. Fert, Nature Phys.\n3, 492 (2007).\n11M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n12K. Xia, P. J. Kelly, G. E. W. Bauer, A. Brataas, and\nI. Turek, Phys. Rev. B 65, 220401(R) (2002).\n13M. Gmitra and J. Barna´ s, Appl. Phys. Lett. 89, 223121\n(2006).\n14S. Wiggins, Introduction to Applied Nonlinear dynamical\nSysystems and Chaos (Springer-Verlag, 1990).\n15S. W. Roberts, Technometrics 42, 97 (2000).7\n16O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. Deranlot,\nF. Petroff, G. Faini, J. Barna´ s, and A. Fert, Phys. Rev. B\n77, 174403 (2008).\n17P. Balaz, M. Gmitra, and J. Barna´ s, Phys. Rev. B 79,144301 (2009).\n18E. Jaromirska, P. Bal´ aˇ z, L. L´ opez D´ ıaz, J. Barna´ s, to be\npublished."    },    {        "title": "1803.00017v2.Roles_of_chiral_renormalization_on_magnetization_dynamics_in_chiral_magnets.pdf",        "content": "Roles of chiral renormalization on magnetization dynamics in chiral magnets\nKyoung-Whan Kim,1,\u0003Hyun-Woo Lee,2,yKyung-Jin Lee,3,4Karin Everschor-Sitte,5Olena Gomonay,5,6and Jairo Sinova5,7\n1Institut für Physik, Johannes Gutenberg-Universität Mainz, Mainz 55128, Germany\n2Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea\n3Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea\n4KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea\n5Institut für Physik, Johannes Gutenberg Universität Mainz, Mainz 55128, Germany\n6National Technical University of Ukraine “KPI,\" Kyiv 03056, Ukraine\n7Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, 162 53 Praha 6, Czech Republic\n(Dated: July 28, 2021)\nInmetallicferromagnets,theinteractionbetweenlocalmagneticmomentsandconductionelectronsrenormal-\nizesparametersoftheLandau-Lifshitz-GilbertequationsuchasthegyromagneticratioandtheGilbertdamping,\nand makes them dependent on the magnetic configurations. Although the effects of the renormalization for\nnonchiral ferromagnets are usually minor and hardly detectable, we show that the renormalization does play a\ncrucial role for chiral magnets. Here the renormalization is chiral and as such we predict experimentally identi-\nfiableeffectsonthephenomenologyofmagnetizationdynamics. Inparticular,ourtheoryfortheself-consistent\nmagnetization dynamics of chiral magnets allows for a concise interpretation of domain wall creep motion. We\nalso argue that the conventional creep theory of the domain wall motion, which assumes Markovian dynamics,\nneedscriticalreexaminationsincethegyromagneticratiomakesthemotionnon-Markovian. Thenon-Markovian\nnature of the domain wall dynamics is experimentally checkable by the chirality of the renormalization.\nRenormalization is a useful concept to understand interac-\ntion effects between a physical system and its environment.\nIn metallic ferromagnets, magnetic moments experience such\nrenormalization due to their coupling to conduction electrons\nthrough exchange interactions. Spin magnetohydrodynamic\ntheory [1–3] examines the renormalization of dynamical pa-\nrameters in the Landau-Lifshitz-Gilbert (LLG) equation as\nfollows. Magnetization dynamics exerts a spin motive force\n(SMF) [4, 5] on conduction electrons, and the resulting spin\ncurrentgeneratesspin-transfertorque(STT)[6–8]thataffects\nthemagnetizationdynamics itself. Thisself-feedbackofmag-\nnetizationdynamics[9]renormalizestheGilbertdampingand\nthe gyromagnetic ratio. However, its consequences rarely go\nbeyond quantitative corrections in nonchiral systems [10–14]\nand are commonly ignored.\nChiralmagnetsareferromagnetsthatpreferaparticularchi-\nrality of magnetic texture due to spin-orbit coupling (SOC)\nand broken inversion symmetry. Examples include ferro-\nmagnets in contact with heavy metals, such as Pt [15] and\nthose with noncentrosymmetric crystal structures [16]. Mag-\nnetization dynamics in chiral magnets are usually described\nby generalizing the conventional LLG equation to include\nthe chiral counterpart of the exchange interaction called the\nDzyaloshinskii-Moriya interaction (DMI) [17–19] and that of\nSTTcalledspin-orbittorque(SOT)[20–23]. Thisdescription\nis incomplete, however, since it ignores the renormalization\nby the self-feedback of magnetization dynamics. Although\nthe renormalization in chiral magnets has been demonstrated\ntheoretically for a few specific models [24–27], most experi-\nmentalanalysesofchiralmagnetsdonottakeintoaccountthe\nrenormalization effect.\nInthiswork,wedemonstratethattherenormalizationinchi-\nralmagnetsshouldbechiralregardlessofmicroscopicdetails\nand these effects should be nonnegligible in chiral magnets\nwith large SOT observed in many experiments [21–23, 28–30]. Unlike in nonchiral systems, the chiral renormalization\ngenerates experimentally identifiable effects by altering the\nphenomenology of magnetization dynamics. This provides a\nuseful tool to experimentally access underlying physics. We\nillustratethiswiththefield-drivenmagneticdomainwall(DW)\nmotion with a controllable chirality by an external magnetic\nfield [31, 32]. We find that not only is the steady state DW\nvelocity chiral due to the chiral damping [25], but also the\neffective mass of the DW [33] is chiral due to the chiral gy-\nromagnetic ratio. The chiral gyromagnetic ratio also signifi-\ncantly affects the DW creep motion, which is one of the tech-\nniques to measure the strength of the DMI [32]. We argue\nthat the chiral gyromagnetic ratio is the main mechanism for\nthe non-energetic chiral DW creep velocity [34], contrary to\nthe previous attribution to the chiral damping [25, 34]. We\nalso highlight the importance of the tilting angle excitation\nand its delayed feedback to the DW motion. This has been\nignoredinthetraditionalcreeptheory[35,36]foralongtime,\nsince its effects merely alter the velocity prefactor which is\nindistinguishable from other contributions, such as the impu-\nrity correlation length [37]. However, in chiral magnets, it is\ndistinguishablebymeasuringtheDWvelocityasa functionof\nchirality (not a single value).\nTo get deep insight into the chiral renormalization, we\nadopt the self-feedback mechanism of magnetization dynam-\nics through conduction electrons and develop a general, con-\ncise, and unified theory for chiral magnets. There are several\nprevious reports on the anisotropic or chiral renormalization\nof the magnetic damping [24–26, 38] and the gyromagnetic\nratio[27,38,39]intheRashbamodel[40]. Tounifyandgen-\neralize the previous works, we start from the general Onsager\nreciprocityrelationandpredictallthecoreresultsoftheprevi-\nous reports. Our theory can be generalized to situations with\nanyphenomenologicalspintorqueexpression,whichcaneven\nbedeterminedbysymmetryanalysisandexperimentswithoutarXiv:1803.00017v2  [cond-mat.mes-hall]  14 Mar 20182\nMagnetization under\nchiral self-feedback Effective  equation of \nmotion for magnetization SOT \nchiral \nSMF\nLLG ( γ, α)\nchiral LLG ( ζγ , G )(a) (b) \nFIG.1. (a)Magnetizationdynamicsdescribedbytheunrenormalized\nLLG equation. The dynamics of magnetization and that of electrons\narecoupledtoeachotherbytheexchangeinteraction. (b)Aftertracing\nout the electron degree of freedom, the gyromagnetic ratio ( \u0010\r) and\nthe magnetic damping ( G) are chirally renormalized [Eq. (1)].\nknowing its microscopic mechanism. We provide a tabular\npicture(SeeTableIbelow)forphysicalunderstandingofeach\ncontribution to the chiral renormalization. Furthermore, one\ncan utilize the generality of the Onsager relation to include\nmagnon excitations [26], thermal spin torques [41], and even\nmechanical vibrations [42] in our theory.\nToexaminetheconsequencesofthechiralrenormalization,\nwestartfromthefollowingrenormalizedLLGequation,which\nwe derive in the later part of this paper,\n(\u0010\r)\u00001\u0001@tm=\u0000m\u0002He\u000b+\r\u00001m\u0002G\u0001@tm+\r\u00001Text;(1)\nwhere mis the unit vector along magnetization, \ris the un-\nrenormalized gyromagnetic ratio, He\u000bis the effective mag-\nneticfield,and Textreferstospintorqueinducedbyanexternal\ncurrent.\u0010andG,whicharegenerallytensorsandfunctionsof\nmand its gradients, address respectively the renormalization\nofthegyromagneticratioandthemagneticdamping,depicted\nin Fig. 1. If the renormalization is neglected, Eq. (1) reduces\nto the conventional LLG equation with \u0010= 1andG=\u000b,\nwhere\u000bistheunrenormalizedGilbertdamping. Otherwise \u0010\nandGare dependent on the chirality of magnetic texture. At\nthe end of this paper, we show that the chiral renormalization\nis completely fixed once the expressions of STT and SOT are\ngiven.\nWe first examine implications of the chiral renormaliza-\ntion on a few exemplary types of field-driven DW dynamics\n(Fig.2). Westartfrom He\u000b=H0+Hext+Hth,where H0is\nthe energetic contribution (without an external field), Hext=\n(Hx;0;Hz)is the external field, and Hthis a thermal fluctu-\nation field. We use the DW profile m(x) = (sin\u001esech[(x\u0000\nX)=\u0015];cos\u001esech[(x\u0000X)=\u0015];tanh[(x\u0000X)=\u0015])whereX,\n\u001eand\u0015aretheposition,thetiltingangle,andthewidthofthe\nDW, respectively. Taking Xand\u001eas the collective coordi-\nxyz\nφφ\nHxHz v(   )FIG. 2. Chiral dynamics of a DW between domains with m=\u0007^z\n(red and blue respectively). The DW chirality is characterized by\nthe DW tilting angle \u001e[the positivity (negativity) of \u001ecorresponds\nto the left-handed (right-handed) chirality], and can be controlled\nby an in-plane field ( Hx). The DW motion is driven by an applied\nfield (Hz). Measuring the DW velocity as a function of \u001e(orHx),\nthe difference between v(\u001e)andv(\u0000\u001e)gives the information of the\nchiral renormalization.\nnates, Eq. (1) gives\n\u000bX\ne\u000b\n\u0015dX\ndt+1\n\u0010e\u000bd\u001e\ndt=FX+\u0018X; (2a)\n\u00001\n\u0010e\u000bdX\ndt+\u000b\u001e\ne\u000b\u0015d\u001e\ndt=F\u001e+\u0018\u001e; (2b)\nwhereFX=\u001e= (\r=2)R\n(H0+Hext)\u0001(@X=\u001em)dxrefertothe\nforce onXand\u001e.\u0018X=\u001e= (\r=2)R\nHth\u0001(@X=\u001em)dxis the\nthermal force on Xand\u001e.\nTheeffectivedamping \u000bX=\u001e\ne\u000bandthegyromagneticratio \u0010e\u000b\nare given by\n\u000bX\ne\u000b=\u0015\n2Z\n(@Xm\u0001G\u0001@Xm)dx; (3a)\n\u000b\u001e\ne\u000b=1\n2\u0015Z\n(@\u001em\u0001G\u0001@\u001em)dx; (3b)\n\u0010\u00001\ne\u000b=1\n2Z\u0002\n(m\u0002@\u001em)\u0001\u0010\u00001\u0001@Xm\u0003\ndx:(3c)\nNote that without the chiral renormalization, Eq. (2) reduces\ntotheThieleequations[43]with \u000bX=\u001e\ne\u000b=\u000band\u0010e\u000b= 1. We\nemphasizethat \u000bX=\u001e\ne\u000band\u0010e\u000bdependonthetiltingangle \u001eand\nthus on the chirality of the DW. Figure 3 shows the \u001edepen-\ndenciesoftheseparameters. Theasymmetricdependenceson\n\u001econfirmtheirchiraldependences. Notethat,evenforpurely\nfield-drivenDWmotion,thechiraldependencesoftheparam-\netersaredeterminedbytheexpressionof current-induced spin\ntorque.\nWe first consider the steady-state dynamics of DW in the\nflow regime, where the effects of the pinning and the thermal\nforces are negligible. Then, translational symmetry along\nXguarantees the absence of contribution from H0toFX,\nthus only the external field contribution survives in the right-\nhand side of Eq. (2a), FX+\u0018X\u0019\u0000\rHz. In a steady state\n(d\u001e=dt = 0), Eq. (2a) gives the DW velocity as\nv\row=\u0000\r\u0015\n\u000bX\ne\u000bHz; (4)3\nα\u0001\u0002\u0002\u0001ϕα\u0001\u0002\u0002\u0001\u0001\nα\u0001\u0002\u0002ϕϕα\u0001\u0002\u0002ϕ\u0001\nζ\u0001\u0002\u0002\u0003ϕζ\u0001\u0002\u0002\u0003\u0001\n\u0001\u0002\u0003 \u0001\u0002\u0004 \u0001\u0002\u0001 \u0001\u0002\u0004 \u0001\u0002\u0003\u0001\u0002\u0005\u0001\u0002\u0004\u0001\u0002\u0006\u0001\u0002\u0001\u0001\u0002\u0006\u0001\u0002\u0004\u0001\u0002\u0005\nϕ π\u0007\b\tRighthandedchirality Left handedchirality\nFIG. 3. The effective dynamical parameters, \u000bX\ne\u000b(the red, solid\ncurve),\u000b\u001e\ne\u000b(the red, dashed curve), and \u0010\u00001\ne\u000b(the blue curve), as a\nfunction of the DW tilting angle \u001e. We take the phenomenological\nexpression of spin torque in magnetic bilayers [21–23, 30], which is\na typical example with large SOT: T= (\r~=2eMs)f(js\u0001r)m\u0000\n\f1m\u0002(js\u0001r)m+kSO(^z\u0002js)\u0002m\u0000\f2kSOm\u0002[(^z\u0002js)\u0002\nm]g, where each term refers to the adiabatic STT [44], nonadiabatic\nSTT[45,46],fieldlikeSOT[47,48],anddampinglikeSOT[30,49–\n51], induced by the spin current js. Here,Ms= 1000 emu =cm3is\nthesaturationmagnetization, e>0isthe(negative)electroncharge,\n^zistheinterfacenormaldirection, kSO= 1:3 (nm)\u00001characterizes\nthestrengthoftheSOT.Wetake \f1= 0:05,\f2= 5,\u0015= 8 nm,and\nthe electrical conductivity \u001b\u00001\n0= 6\u0016\ncm. The parameters are on\nthe order of the typical values for Pt/Co systems [28, 45, 52].\nwhichisinverselyproportionaltothechiraldamping \u000bX\ne\u000beval-\nuatedatthesteady-statetiltingangle \u001eeqforwhichd\u001e=dt = 0.\nAs\u001eeqcanbemodulatedby Hx,themeasurementof v\rowasa\nfunction ofHxprovides a direct test of the chiral dependence\non\u000bX\ne\u000b.\nAs an experimental method to probe the chiral dependence\nof\u0010e\u000b, we propose the measurement of the DW mass, called\nthe Döring mass [33]. It can be performed by examining\nthe response of DW under a potential trap to an oscillating\nfieldHz[53]. Unlike v\row,\u001eis not stationary for this case,\nand dynamics of it is coupled to that of X. Such coupled\ndynamicsof \u001eandXmakes\u0010e\u000brelevant. IntheSupplemental\nMaterial[54],weintegrateoutthecoupledequations[Eq.(2)]\nto obtain the effective Döring mass,\nmDW=1\n\u00102\ne\u000b2MsS\n\rjF0\n\u001e(\u001eeq)j; (5)\nwhereSis the cross-sectional area of the DW. Here, \u0010e\u000brep-\nresents a measurement of its value for \u001e=\u001eeq, which can be\nvariedbyHx.mDWprovidesanexperimentalwaytomeasure\nthe chiral dependence of \u0010e\u000b.\nIn the creep regime of the DW where the driving field is\nmuch weaker than the DW pinning effects, the implication of\nthechiralrenormalizationgobeyondmerelychiralcorrections\ntotheDWvelocity. TherecentcontroversiesonthechiralDW\ncreep speed vcreepmeasured from various experiments [32,\n34, 55, 56] require more theoretical examinations. Typically,vcreepis believed to follow the Arrhenius-like law vcreep =\nv0exp(\u0000\u0014H\u0000\u0016\nz=kBT)[35, 36], where v0is a prefactor, \u0016is\nthe creep exponent typically chosen to be 1/4 [57], and \u0014is\na parameter proportional to the DW energy density. Based\non the observation that the DMI affects \u0014, an experiment [32]\nattributedthechiraldependenceof vcreeptotheDMI.However\nlater experiments [34, 55, 56] found features that cannot be\nexplained by the DMI. In particular, Ref. [34] claimed that\nthe chiral dependence of vcreepis an indication of the chiral\ndamping [25], based on the observation v0/(\u000bX\ne\u000b)\u00001. On\nthe other hand, our analysis shows that the explanation of the\nchirality dependence may demand more fundamental change\nto the creep law, which assumes the dynamics of \u001eto be\nessentially decoupled from that of Xand thus irrelevant for\nvcreep. As a previous experiment on the DW creep motion in\na diluted semiconductor [58] argued the coupled dynamics of\n\u001eandXto be important, it is not a prioriclear whether the\nassumptionofdecoupling Xand\u001eholdsinthecreepregime.\nWe consider the coupling between the dynamics of Xand\n\u001easfollows. Afterthedynamicsof Xexcites\u001e,thedynamics\nof\u001eresults in a feedback to Xwith a delay time \u001c. Since the\ndynamics at a time tis affected by its velocity at past t\u0000\u001c,\nit is non-Markovian. The traditional creep theory takes the\nMarkovian limit ( \u001c!0), thus\u001e=\u001eeqat any instantaneous\ntime,decoupledfromthedynamicsof X. Toshowthecrucial\nroleofafinitefeedbacktime \u001c,wecalculatetheescaperateof\nthe DW over a barrier, which is known to be proportional to\nv0[37] and apply the Kramer’s theory [59] for barrier escape\nand its variations for non-Markovian systems [60, 61]. After\nsomealgebraintheSupplementalMaterial[54],Eq.(2)gives\nv0/\u001a(\u000bX\ne\u000b)\u00001\u001c\u00170\u001c\u00102\ne\u000b\u000bX\ne\u000b\u000b\u001e\ne\u000b(Markovian );\n\u0010e\u000b\u001c\u00170&\u00102\ne\u000b\u000bX\ne\u000b\u000b\u001e\ne\u000b(non-Markovian );\n(6)\nwhere\u00170is called the reactive frequency [61] and is on the\norder of 2\u0019times the attempt frequency ( \u00191 GHz[37]). We\nemphasizethatthetworegimesshowverydifferentbehaviorin\nthesenseofunderlyingphysicsaswellasphenomenology. The\nvalidityoftheMarkovianassumptiondependsonthetimescale\nof\u001ccomparedto \u00102\ne\u000b\u000bX\ne\u000b\u000b\u001e\ne\u000b. Sincethedampingissmall,itis\nnotguaranteedforoursituationtobeintheMarkovianregime.\nIndeed,wedemonstrateintheSupplementalMaterial[54]that\nthesecondregime(non-Markovian)inEq.(6)ismorerelevant\nwithrealisticvalues,thusthechiralityof v0mainlyoriginates\nfrom the gyromagnetic ratio, not the damping [34]. One can\nmeasure the chiral dependence of \u000bX\ne\u000band\u0010e\u000bfrom the flow\nregime[Eqs.(4)and(5)]andcomparetheirchiraldependences\ntothecreepregimetoobservethenon-Markoviannatureofthe\nDWdynamics. Thisadvantageoriginatesfromthepossibility\nthatonecanmeasuretheDWvelocityasa functionofchirality,\nin contrast to nonchiral magnets where one measures the DW\nvelocity as a single value.\nSo far, we present the role of the chiral renormalization for\ngiven renormalized tensors Gand\u0010. To provide underlying\nphysical insight into it, we present a analytic derivation of\nEq. (1) in general situations. We start from the LLG equation4\n\r\u00001@tm=\u0000m\u0002He\u000b+\r\u00001\u000bm\u0002@tm+\r\u00001Tandreferto\nthe scenario illustrated in Fig. 1. Note that There includes a\ncontribution from an internally generated SMF ( Tint) as well\nas that from an external current [ Textin Eq. (1)]. We write\ndown the spin torque in a general current-linear form T=\n\u0000(\r~=2eMs)m\u0002P\nuAu(m)js;u,whereurunsoverx;y;z.\nHere the spin current jsis split into an internally generated\nSMF [4, 5] js;intand the external current js;ext. The former\nisproportionalto @tm,thusitrenormalizesthegyromagnetic\nratio and the damping. The latter generates Textin Eq. (1).\nTheexpressionof js;intisgivenbytheOnsagerreciprocityof\nSTT and SMF [62]: js;int;u=\u0000(\u001b0~=2e)Au(\u0000m)\u0001@tm,\nwhere\u001b0is the charge conductivity [63]. Substituting this to\nTint= (\r~=2eMs)m\u0002P\nuAu(m)js;int;ugivestheeffective\nLLGequation \r\u00001@tm=\u0000m\u0002He\u000b+\r\u00001m\u0002A\u0001@tm+\n\r\u00001Text, whereA=\u000b+\u0011P\nuAu(m)\nAu(\u0000m),\u0011=\n\r~2\u001b0=4e2Msand\nisthedirecttensorproduct. Asaresult,\nTintis taken care of by renormalizing \u000bintoAin the LLG\nequation.\nTherenormalizeddampingandgyromagneticratioaregiven\nby separating different contributions of Awith different time\nreversal properties. A damping contribution is required to\nbe dissipative (odd in time reversal), whereas a gyromagnetic\nterm should be reactive (even in time reversal). Separating\nthese gives Eq. (1) where G= (A+AT)=2and\u0010\u00001=\n1\u0000m\u0002(A\u0000AT)=2. Theparticularchoicefortheadiabatic\nSTTandthenonadiabaticSTT Au(m) =m\u0002@um+\f@um\nreproduces the renormalized LLG equation for nonchiral sys-\ntems [1–3]. When one uses Au(m)for a particular chiral\nsystem, Eq. (1) produces the effective LLG equation for it, as\nreported by a numerical study for a one-dimensional Rashba\nmodel [27].\nIn chiral magnets, it is known that spin torque includes two\nmore contributions called fieldlike SOT [47, 48] and damp-\ninglikeSOT[30,49–51]. Thecharacterizationoffieldlikeand\ndampinglikeSOTisregardlessofthechoiceofSOC,sinceitis\ndetermined by the time reversal characteristic. Since Au(m)\nconsistsoffourcontributions,thereare16contributionsinthe\nfeedback tensor \u0001A=\u0011P\nuAu(m)\nAu(\u0000m)for each\nu. We tabulate all terms of \u0001Ain Table I. The contributions\nSTT:Ax(m)\nSMF:\nAx(\u0000m)Adiabatic\nm\u0002@xmNonadiabatic\n\f1@xmFLT\nkSOm\u0002(^y\u0002m)DLT\n\f2kSO^y\u0002m\nm\u0002@xmG\u0010\u00001G\u0010\u00001\n\u0000\f1@xm\u0010\u00001G\u0010\u00001G\nkSOm\u0002(^y\u0002m)G\u0010\u00001G\u0010\u00001\n\u0000\f2kSO^y\u0002m\u0010\u00001G\u0010\u00001G\nTABLE I. Example characterization of contributions in Ax(m)\nAx(\u0000m). Counting orders of gradients and mgives the conven-\ntional (white), chiral (lighter gray), or anisotropic (darker gray) con-\ntributions to the gyromagnetic ratio ( \u0010\u00001) or the damping (G). The\nform of the fieldlike SOT (FLT) and dampinglike SOT (DLT) are\ntaken from magnetic bilayers [30, 47–51] for illustration, but the\ncharacterization procedure works generally.withthewhitebackgroundarezerothorderinSOCbutsecond\norder in gradient and are the conventional nonlocal contribu-\ntions [3, 65]. Those with the lighter gray background are first\norder in gradient and chiral [27]. Those with the darker gray\ncolor are zeroth order in gradient and anisotropic [66]. In\nthisway,ourtheoryprovidesaunifiedpictureontheprevious\nworks. Whether a term contributes to \u0010\u00001orGis deter-\nmined by the order in m. After a direct product of STT and\nSMF, a term even (odd) in mgivesG(\u0010\u00001), since it gives\na time irreversible (reversible) contribution appearing in the\nLLGequationas m\u0002A\u0001@tm. Thesameanalysiswithsimple\norder countings works for any Au(m). It holds even if our\ntheory is generalized to other physics, such as magnons [26],\nthermal effects [41], and mechanical effects [42].\nAsanexampleofapplicationsofTableI,weadoptthespin-\nHall-effectdrivenSOT[21,67,68],wherealargedampinglike\nSOTarises. FromTableI,onecanimmediatelyfigureoutthat\nthecombinationofthedampinglikeSOTandtheconventional\nSMF(themosttoprightcell)givesachiralgyromagneticratio\ncontribution. As another example, one notices that the com-\nbination of the dampinglike SOT and its Onsager counterpart\n(the fourth term in the SMF) gives an anisotropic damping\ncontribution. Note that the Onsager counter part of the spin-\nHall-effect driven SOT is the inverse spin Hall effect driven\nby spin pumping current generated by the magnetization dy-\nnamics. In this way, Table I provides useful insight for each\ncontribution.\nTable I also allows for making the general conclusion that\nthe magnitude of the chiral gyromagnetic ratio is determined\nbythesizeofthedampinglikeSOT( \f2)andthatofthenona-\ndiabatic STT ( \f1). This is an important observation since\nmany experiments on magnetic bilayers and topological in-\nsulators [21–23, 30] shows a large dampinglike SOT. This\nconclusionisregardlessofthemicroscopicdetailsoftheSOT,\nbecauseadampinglikecontributionissolelydeterminedbyits\ntime-reversal property.\nTo summarize, we demonstrate that the chiralities of the\ngyromagnetic ratio and Gilbert damping have significant im-\nplicationswhichgofurtherbeyondmerelythechangeinmag-\nnetization dynamics. The chirality plays an important role in\ninvestigating underlying physics because physical quantities,\nwhich were formerly treated as constants, can now be con-\ntrolled through their chiral dependence. An example is the\nnon-Markovian character of the DW creep motion, which is\ndifficult to be verified in nonchiral systems. From the non-\nMarkovian nature of the DW creep motion, we conclude that\nthe non-energetic origin of chiral DW creep originates from\nthe chiral gyromagnetic ratio rather than the chiral damping.\nWealsoprovideageneral,concise,andunifiedtheoryoftheir\nchiralities, which provide useful insight on the self-feedback\nof magnetization.\nWe acknowledge M. D. Stiles, Y. Tserkovnyak, A. Thiav-\nille, S.-W. Lee, V. Amin, and D.-S. Han for fruitful discus-\nsion. This work is supported by the Alexander von Humboldt\nFoundation, the ERC Synergy Grant SC2 (No. 610115), the\nTransregionalCollaborativeResearchCenter(SFB/TRR)1735\nSPIN+X, and the German Research Foundation (DFG) (No.\nEV 196/2-1 and No. SI 1720/2-1). K.W.K also acknowl-\nedgessupportbyBasicScienceResearchProgramthroughthe\nNational Research Foundation of Korea (NRF) funded by the\nMinistry of Education (2016R1A6A3A03008831). H.W.L.\nwassupportedbyNRF(2011-0030046). K.J.Lwassupported\nby NRF (2015M3D1A1070465, 2017R1A2B2006119).\n\u0003kyokim@uni-mainz.de\nyhwl@postech.ac.kr\n[1] Y. Tserkovnyak and C. H. Wong, Phys. Rev. B 79, 014402\n(2009).\n[2] C. Wong and Y. Tserkovnyak, Phys. Rev. B 80, 184411 (2009).\n[3] S. Zhang and S. S.-L. Zhang, Phys. Rev. Lett. 102, 086601\n(2009).\n[4] G. E. Volovik, J. Phys. C 20, L83 (1987).\n[5] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601\n(2007).\n[6] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[7] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[8] D.C.RalphandM.D.Stiles,J.Magn.Magn.Mater. 320,1190\n(2008).\n[9] K.-J.Lee,M.D.Stiles,H.-W.Lee,J.-H.Moon,K.-W.Kim,and\nS.-W. Lee, Phys. Rep. 531, 89 (2013).\n[10] S.-I.Kim,J.-H.Moon,W.Kim,andK.-J.Lee,Curr.Appl.Phys.\n11, 61 (2011).\n[11] K.-W.Kim,J.-H.Moon,K.-J.Lee,andH.-W.Lee,Phys.Rev.B\n84, 054462 (2011).\n[12] J.-V. Kim, Phys. Rev. B 92, 014418 (2015).\n[13] H. Y. Yuan, Z. Yuan, K. Xia, and X. R. Wang, Phys. Rev. B 94,\n064415 (2016).\n[14] R. Cheng, J.-G. Zhu, and D. Xiao, J. Phys. D: Appl. Phys. 49,\n434001 (2016).\n[15] H.Yang,A.Thiaville,S.Rohart,A.Fert,andM.Chshiev,Phys.\nRev. Lett. 115, 267210 (2015).\n[16] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A.\nNeubauer, R. Georgii, and P. Böni, Science 323, 915 (2009).\n[17] I. J. Dzyaloshinsky, Phys. Chem. Solids 4, 241 (1958).\n[18] T. Moriya, Phys. Rev. 120, 91 (1960).\n[19] A. Fert and P.M. Levy, Phys. Rev. Lett. 44, 1538 (1980).\n[20] I. M. Miron, K. Garello1, G. Gaudin, P.-J. Zermatten, M. V.\nCostache1,S.Auffret,S.Bandiera,B.Rodmacq,A.Schuhl,and\nP. Gambardella, Nature (London) 476, 189 (2011).\n[21] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A.\nBuhrman, Science 336, 555 (2012).\n[22] S.Emori,U.Bauer,S.-M.Ahn,E.Martinez,andG.S.D.Beach,\nNature Mater. 12, 611 (2013).\n[23] K.-S. Ryu, L. Thomas, S.-H. Yang, and S. S. P. Parkin, Nat.\nNanotechnol. 8, 527 (2013).\n[24] K.-W. Kim, J.-H. Moon, K.-J. Lee, and H.-W. Lee, Phys. Rev.\nLett.108, 217202 (2012).\n[25] C. A. Akosa, I. M. Miron, G. Gaudin, and A. Manchon, Phys.\nRev. B 93, 214429 (2016).\n[26] U. Güngördü and A. A. Kovalev, Phys. Rev. B 94, 020405(R)\n(2016).\n[27] F. Freimuth, S. Blügel, and Y. Mokrousov, Phys. Rev. B 96,\n104418 (2017).\n[28] I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-Prejbeanu,\nS. Auffret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonfim, A.Schuhl, and G. Gaudin, Nature Mater. 10, 419 (2011).\n[29] A.R.Mellnik,J.S.Lee,A.Richardella,J.L.Grab,P.J.Mintun,\nM. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth,\nand D. C. Ralph, Nature (London) 511, 449 (2014).\n[30] H.Kurebayashi,J.Sinova,D.Fang,A.C.Irvine,T.D.Skinner,\nJ. Wunderlich, V. Novák, R. P. Campion, B. L. Gallagher, E.\nK. Vehstedt, L. P. Zârbo, K. Výborný, A. J. Ferguson, and T.\nJungwirth, Nature Nanotechnol. 9, 211 (2014).\n[31] A. Thiaville, S. Rohart, E. Jué, V. Cros, and A. Fert, Europhys.\nLett.100, 57002 (2012).\n[32] S.-G.Je,D.-H.Kim,S.-C.Yoo,B.-C.Min,K.-J.Lee,andS.-B.\nChoe, Phys. Rev. B 88, 214401 (2013).\n[33] V. W. Döring, Z. Naturforsch. A 3A, 373 (1948).\n[34] E. Jué, C. K. Safeer, M. Drouard, A. Lopez, P. Balint, L. Buda-\nPrejbeanu,O.Boulle,S.Auffret, A.Schuhl,A.Manchon,I.M.\nMiron, and Gilles Gaudin, Nat. Mater. 15, 272 (2015).\n[35] S.Lemerle,J.Ferré,C.Chappert,V.Mathet,T.Giamarchi,and\nP. Le Doussal, Phys. Rev. Lett. 80, 849 (1998).\n[36] P. Chauve, T. Giamarchi, and P. Le Doussal, Phys. Rev. B 62,\n6241 (2000)\n[37] J.Gorchon,S.Bustingorry,J.Ferré,V.Jeudy,A.B.Kolton,and\nT. Giamarchi, Phys. Rev. Lett. 113, 027205 (2014).\n[38] G. Tatara, N. Nakabayashi, and K.-J. Lee, Phys. Rev. B 87,\n054403 (2013).\n[39] E.vanderBijlandR.A.Duine,Phys.Rev.B 86,094406(2012).\n[40] Y.A.BychkovandE.I.Rashba,Pis’maZh.Eksp.Teor.Fiz. 39,\n66 (1984) [JETP Lett. 39, 78 (1984)].\n[41] M. Hatami, G. E. W. Bauer, Q. Zhang, and P. J. Kelly, Phys.\nRev. Lett. 99, 066603 (2007).\n[42] M.Matsuo,J.Ieda,E.Saitoh,andS.Maekawa,Phys.Rev.Lett.\n106, 076601 (2011).\n[43] A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n[44] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004).\n[45] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[46] K.-W. Kim, K.-J. Lee, H.-W. Lee, and M. D. Stiles, Phys. Rev.\nB92, 224426 (2015).\n[47] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008).\n[48] A. Matos-Abiague and R. L. Rodríguez-Suárez, Phys. Rev. B\n80, 094424 (2009).\n[49] X.WangandA.Manchon,Phys.Rev.Lett. 108,117201(2012).\n[50] K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee, Phys.\nRev. B 85, 180404(R) (2012).\n[51] D. A. Pesin and A. H. MacDonald, Phys. Rev. B 86, 014416\n(2012).\n[52] K.-W. Kim, H.-W. Lee, K.-J. Lee, and M. D. Stiles, Phys. Rev.\nLett.111, 216601 (2013).\n[53] E. Saitoh, H. Miyajima, T. Yamaoka, and G. Tatara, Nature\n(London), 432, 203 (2004).\n[54] See Supplemental Material below for detailed information on\nnon-Markovian nature of the DW dynamics.\n[55] R. Lavrijsen, D. M. F. Hartmann, A. van den Brink, Y. yin, B.\nBarcones,R.A.Duine,M.A.Verheijen,H.J.M.Swagten,and\nB. Koopmans, Phys. Rev. B 91, 104414 (2015).\n[56] A. L. Balk, K.-W. Kim, D. T. Pierce, M. D. Stiles, J. Unguris,\nand S. M. Stavis, Phys. Rev. Lett. 119, 077205 (2017).\n[57] F.Cayssol,D.Ravelosona,C.Chappert,J.Ferré,andJ.P.Jamet,\nPhys. Rev. Lett, 92, 107202 (2004).\n[58] M. Yamanouchi, J. Ieda, F. Matsukura, S. E. Barnes, S.\nMaekawa, and H. Ohno, Science 317, 1726 (2007).\n[59] H. A. Kramers, Physica (Amsterdam) 7, 284 (1940).\n[60] R. F. Grote and J. T. Hynes, J. Chem. Phys. 73, 2715 (1980).\n[61] E. Pollak, H. Grabert, and P. Hänggi, J. Chem. Phys. 91, 4073\n(1989).\n[62] Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77, 134407(2008).\n[63] To use the Onsager relation, one needs to assume a slow dy-\nnamicsofmagnetization,thusthecoefficientinfrontof @tmis\nquasistatic.Therefore,relativisticcorrections[64],forexample,\ncannot be considered in our theory.\n[64] R.Mondal,M.Berritta,A.K.Nandy,andP.M.Oppeneer,Phys.\nRev. B 96, 024425 (2017).\n[65] W.Wang,M.Dvornik,M-A.Bisotti,D.Chernyshenko,M.Beg,\nM.Albert,A.Vansteenkiste,B.V.Waeyenberge,A.N.Kuchko,\nV.V.Kruglyak,andH.Fangohr,Phys.Rev.B 92,054430(2015).\n[66] K.M.D.HalsandA.Brataas,Phys.Rev.B 89,064426(2014).\n[67] S.-M. Seo, K.-W. Kim, J. Ryu, H.-W. Lee, and K.-J. Lee, Appl.\nPhys. Lett. 101, 022405 (2012).\n[68] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T.\nJungwirth, Rev. Mod. Phys. 87, 1213 (2015).Supplementary Materials for\n“Roles of chiral renormalization of magnetization dynamic s in chiral magnets\"\nKyoung-Whan Kim,1Hyun-Woo Lee,2Kyung-Jin Lee,3, 4Karin Everschor-Sitte,1Olena Gomonay,1, 5and Jairo Sinova1, 6\n1Institut für Physik, Johannes Gutenberg Universität Mainz, Mainz 5512 8, Germany\n2PCTP and Department of Physics, Pohang University of Science and Te chnology, Pohang 37673, Korea\n3Department of Materials Science and Engineering, Korea University, S eoul 02841, Korea\n4KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea\n5National Technical University of Ukraine “KPI\", Kyiv 03056, Ukraine\n6Institute of Physics, Academy of Sciences of the Czech Republic, Cuk rovarnická 10, 162 53 Praha 6 Czech Republic\nI. THE NON-MARKOVIAN NATURE OF THE DW DYNAMICS\nA. Integrating out φ\nIn the linear response regime, we may take Fφ≈ −|F′\nφ(φeq)|(φ−φeq)and the dynamical coefficients ζeffandαX/φ\neffto be\nevaluated at φ=φeq. Without loss of generality, we assume the initial conditio nX(0) = 0 andφ(0) = φeq. We then define the\nLaplace transforms L[f(t)](s) =/integraltext∞\n0e−stf(t)dt. We denote L[X] =QandL[φ−φeq] =P. Then the Laplace transform of\nEq. (2) in the main text is\nsαX\neff\nλQ+s\nζeffP=L[FX] +L[ξX], (S1a)\n−s\nζeffQ+sαφ\neffλP=−|F′\nφ(φeq)|P+L[ξφ], (S1b)\nEliminating Pin Eq. (S1) gives\n1\nζ2\neffs2\n|F′\nφ(φeq)|+sαφ\neffλQ+sαX\neff\nλQ=−γHz\ns+L[Fpin] +/parenleftBigg\nL[ξX]−s\nζeffL[ξφ]\nb+sαφ\neffλ/parenrightBigg\n, (S2)\nwhich is an equation of Xonly. Taking the inverse Laplace transform, we obtain the fo llowing non-Markovian equation:\n1\nλ/integraldisplayt\n0f(t−u)X′(u)du=FX+˜ξX(t). (S3)\nHere f(t)is a feedback function from φ, whose explicit form is\nf(t) =L−1/bracketleftBigg\nαX\neff+1\nζ2\neffαφ\neffsτ\n1 +sτ/bracketrightBigg\n=/parenleftBigg\nαX\neff+1\nζ2\neffαφ\neff/parenrightBigg\nδ(t)−1\nζ2\neffαφ\neffτe−t/τΘ(t), (S4)\nandτ=αφ\neffλ/|F′\nφ(φeq)|is the relaxation time of φdegree of freedom. The correlation relation for the effectiv e thermal\nfluctuation field ˜ξX(t)is given by the fluctuation-dissipation theorem ∝angbracketleft˜ξX(t)˜ξX(t′)∝angbracketright ∝Tf(|t−t′|)where Tis the temperature.\nThe noise is ‘colored’ in the sense that it is no longer a white random noise.\nB. Order-of-magnitude estimation of τ\nTo estimate the order of magnitude of τ, we use the fact that the magntude of |Fφ|is determined by the DMI or the hard axis\nanisotropy: |F′\nφ(φeq)| ≈γλ(π/2)×(2H⊥orDλ−1). We take the DMI field Dλ−1being 30 mT [1] for a rough estimation.\nThen, |F′\nφ(φeq)|/λ≈γ×30 mT ≈5 GHz , so that τ=αφ\neffλ/|F′\nφ(φeq)| ≈αφ\neff×0.2 ns, which is small compared to the time\nscale of the dynamics of X.2\nC. First order approximation - chiral mass correction\nSince τis small, compared to the times scale of the dynamics of X, we may expand f(t)byτ, in the sense of the gradient\nexpansion in time space. Then, f(t)≈ L[αX\neff+ (1/ζ2\neffαφ\neff)sτ] =αX\neffδ(t) + (τ/ζ2\neffαφ\neff)δ′(t). Putting this into Eq. (S3) gives\nτ\nζ2\neffαφ\neff1\nλd2X\ndt+αX\neff\nλdX\ndt=FX+˜ξX(t), (S5)\nwhere the first term represents a massive term. To obtain the D W mass, we need to find the factor which makes FXhave the\ndimension of force. Note that the force generated by pushing the DW is calculated by Ms/integraltext\nHeff·∂Xmd3x= (2MsS/γ)FX.\nTherefore, the mass is defined by multiplying the factor 2MsS/γ,\nmDW=1\nζ2\neff2MsSτ\nγαφ\neffλ, (S6)\nwhich is equivalent to Eq. (5) in the main text.\nD. Higher order contributions - chiral creep\nTo calculate v0, one needs to solve a barrier escape problem. For an energy ba rrierEb, Kramer [2] derived the thermal escapes\nrate\nΓ =ν\n2π/radicalBigg\n|F′(Xm)|\n|F′(XM)|e−Eb/kBT, (S7)\nwhere F′(Xm)andF′(XM)are the derivatives of the force (second derivatives of the p inning energy landscape) evaluated at\nthe potential well and the saddle point respectively. νis called the reactive frequency [3] which we calculate belo w. Then, v0\nis proportional to Γ. According to the Kramer’s theory, ν∝1/αX\nefffor a high damping and Markovian limit, which was also\nconfirmed by the functional renormalization group techniqu e [4].\nHowever, we generalize this result to a non-Markovian situa tion [Eq. (S3)]. To do this, we apply the theory of escape rate for\na non-Markovian equation of motion [3, 5], based on which, th e reactive frequency νcorresponding to Eq. (S3) is given by the\npositive root of the following algebraic equation:\n1\nλνL[f(t)](ν) =|F′(XM)|, (S8)\nwhose exact solution can be calculated from Eq. (S4). As a res ult,\nν=2ν0\n(1−τν0) +/radicalBig\n(1 +τν0)2+ 4τν0/ζ2\neffαX\neffαφ\neff≈\n\nν0∝1\nαX\neffν0τ≪ζ2\neffαX\neffαφ\neff,\nζeff/radicalBigg\nν0αX\neffαφ\neff\nτ∝ζeffν0τ/greaterorsimilarζ2\neffαX\neffαφ\neff,(S9)\nwhere ν0=λ|F′(XM)|/αX\neffis the reactive frequency for τ= 0. In the second limit, we assume that the damping parameters\nare small, thus the last term in the denominator in Eq. (S9) do minates the other terms in the denominator. The two limits sh ows\ncompletely different dependences of νon the dynamical parameters. Therefore, it is important to d etermine the relevant regime.\nAssuming Fpinis random, |F′(XM)| ≈ |F′(Xm)|in Eq. (S7) gives ν0/2πto be the typical attempt frequency ≈1 GHz [6].\nFrom τ≈αφ\neff×0.2 ns estimated above, we obtain τν0≈αφ\neffwhich is an order of magnitude larger than ζ2\neffαX\neffαφ\neff, thus the\nsecond regime in Eq. (S9) is more relevant, contrary to the tr aditional creep theory just taking τ= 0.\n[1] S.-G. Je, D.-H. Kim, S.-C. You, B.-C. Min, K.-J. Lee, and S.- B. Choe, Phys. Rev. B 88, 214401 (2013).\n[2] H. A. Kramers, Physica (Amsterdam) 7, 284 (1940).\n[3] E. Pollak, H. Grabert, and P. Hänggi, J. Chem. Phys. 91, 4073 (1989).\n[4] P. Chauve, T. Giamarchi, and P. Le Doussal, Phys. Rev. B 62, 6241 (2000)\n[5] R. F. Grote and J. T. Hynes, J. Chem. Phys. 73, 2715 (1980).\n[6] K. Gorchon, S. Bustingorry, J. Ferré, V. Jeudy, A. B. Kolton, a nd T. Giamarchi, Phys. Rev. Lett. 113, 027205 (2014)."    },    {        "title": "1605.05643v2.Standing_magnetic_wave_on_Ising_ferromagnet__Nonequilibrium_phase_transition.pdf",        "content": "arXiv:1605.05643v2  [cond-mat.stat-mech]  16 Jul 2016Standing magnetic wave on Ising ferromagnet:\nNonequilibrium phase transition.\nAjayHalder⋆and MuktishAcharyya†\nDepartment of Physics,\nPresidency University\n86/1 College Street, Kolkata-73, India\n⋆ajay.rs@presiuniv .ac.in\n†muktish .physics@presiuniv .ac.in\nAbstract\nThe dynamical response of an Ising ferromagnet to a plane pol arised standing\nmagnetic field wave is modelled and studied here by Monte Carlo simulation in\ntwo dimensions. The amplitude of standing magnetic wave is m odulated along the\ndirection x. We have detected two main dynamical phases namely, pinnedand\noscillating spin clusters . Depending on the value of field amplitude the system\nis found to undergo a phase transition from oscillating spin cluster to pinned as\nthe system is cooled down. The time averaged magnetisation o ver a full cycle of\nmagnetic field oscillations is definedas the dynamic order parameter . Thetransition\nis detected by studying the temperature dependences of the v ariance of the dynamic\norder parameter, the derivative of the dynamic order parame ter and the dynamic\nspecific heat. The dependence of the transition temperature on the magnetic field\namplitude and on the wavelength of the magnetic field wave is s tudied at a single\nfrequency. A comprehensive phase boundary is drawn in the pl ane described by the\ntemperatureand field amplitude for two different wavelengths of the magnetic wave.\nThe variation of instantaneous line magnetisation during a period of magnetic field\noscillation for standing wave mode is compared to those for t he propagating wave\nmode. Also the probability that a spin at any site, flips, is ca lculated. The above\nmentioned variations and the probability of spin flip clearl y distinguish between the\ndynamical phases formed by propagating magnetic wave and by standing magnetic\nwave in an Ising ferromagnet.\nKeywords: Standing wave, Ising Model, Metropolis rate, Mon te-Carlo Simu-\nlation.\n1I. Introduction\nThe study of the dynamical response of a thermodynamical system has become an\nactive field of research [1, 2] in recent years. Ferromagnetic sys tem is one of some im-\nportant systems whose response to various kinds of driving force in equilibrium as well\nas in non-equilibrium situations hold the key attention of many resear chers for a long\ntime. A ferromagnetic system responses in a unique way to a time dependent magnetic\nfield and studies of such dynamical responses revealed many interesting facts of some\ndynamical behaviour of the system. The nonequilibrium dynamic phas e transition and\nthe hysteretic response are the main characteristic features of the ferromagnetic system\ndriven by time dependent magnetic field. Some observations or stud ies regarding – (i)\ndivergences of dynamic specific heat and relaxation time near trans ition point [3, 4], (ii)\ndivergence of the relevant length scale near transition point [5], (iii)studies regarding\nexistence of tricritical point [6, 7], (iv)its relation with stochastic resonance [6], and the\nhysteresis loss [8] etc., establish that the dynamic phase transition is similar in many as-\npects to the well known equilibrium thermodynamic phase transition. This fact is further\nsupported by some experimental findings like– (i)detection of dynamic phase transition\nin the ultra-thin Cofilm onCu(001) system by surface magneto-optic Kerr effect [9, 10],\n(ii)direct excitation of propagating spin waves by focussed ultra shor t optical pulse [11],\n(iii)the transient behaviour of dynamically ordered phase in uniaxial cob alt film [12] etc.\nThe surface and bulk transition [13] are found to be in different unive rsality class in the\ndynamic transition of Ising ferromagnet driven by oscillating magnet ic field. The surface\ncritical behaviour is observed to differ from that of the bulk in these studies [13].\nApart from the Ising model, nonequilibrium dynamic phase transition h as also been\nobserved in other magnetic models. The off-axial dynamic phase tra nsition has been\nobserved in the anisotropic classical Heisenberg model [14] and in th e XY model [15].\nThe multiple (surface and bulk) dynamic transition has been observe d [16] in the classical\nHeisenberg model. The dynamic transition has also been observed in t he kinetic spin-\n3/2 Blume-Capel model [17] and in the Blume-Emery-Griffith model [18 ]. To study the\ndynamical phase transition in mixed spin systems also took much atte ntion in modern\nresearch[19, 20, 21, 22, 23].\nMainly, sinusoidally oscillating or randomly varying magnetic field, which a reuniform\nover the space (lattice) at any instant of time has been used to stu dy the nonequilib-\nrium dynamical phase transition and other characteristic behaviou r in Ising magnets\nand in various other magnetic models. The outcome of the above men tioned studies of\nthe nonequilibrium phase transition has prompted the researchers towards the situations\nwheremagneticexcitationsarealso varied in space atanyparticularinstantoftime. Prop-\nagating magnetic field wave is an example of such spatially as well as tem porally varied\nmagnetic field applied to the ferromagnetic system. This kind of varia tion of magnetic\nfield is closely related to the situation where an electromagnetic wave passes through a\n2magnetic system. Actually, the varying ( in time as well as in space ) magnetic field wave\ncoupled with the spins of the ferromagnetic system affects the dyn amic nature of the\nsystem.\nThe nonequilibrium phase transition in Ising ferromagnet swept by pr opagating mag-\nnetic field wave is studied[24]. Similar observations are obtained in the r andom field Ising\nmodel (RFIM) swept by propagating magnetic field wave [25]. A pinned phase and a\nphase of coherent motion of spin clusters have been observed. In RFIM the nonequilib-\nrium phase transition has been studied at zero temperature and is t uned by quenched\nrandom (field) disorder [25].\nPinned phase and propagating phase (phase of coherent motion of spin clusters) are\nalso observed in the two dimensional Ising ferromagnet swept by pr opagating magnetic\nfield wave [26]. The transition is detected by studying the variance of the dynamic order\nparameter, the derivative of the dynamic order parameter, and the dynamic specific heat\nwhich show sharp peak or dip near transition temperature. In the propagating phase\nspin clusters form a definite pattern which move coherently with the magnetic field wave,\nwhereas in the pinned phase the spin clusters do not move coherent ly in time. The\ndynamic phase transition is observed to depend upon the amplitude a nd wave length\nof the propagating magnetic wave. The phase boundary is found to shrinktowards the\nlow temperature for shorter wavelengths. The relevant length sc ale also diverges near\nthe transition. A dynamic symmetry breaking breathing and spreading transitions [27]\nare also recently found in Ising ferromagnet irradiated by spherica l magnetic wave. The\nnonequilibrium behaviour of the random field Ising ferromagnet, at z ero temperature,\ndriven by standing magnetic field wave [28] has been studied recently by Monte Carlo\nsimulation in two dimensions using uniform, bimodal and Gaussain distrib utions of the\nquenched random fields. Depending on the values of the amplitude of standing magnetic\nfield wave and the strength of quenched random field three distinct nonequilibrium phases\nnamely, pinned, oscillating spin clusters and random are observed. T hese phases, though\nhave similarities, are different from those found in case of propagat ing magnetic wave.\nThere has been much amount of studies done in the nonequilibrium dyn amic phase\ntransition using the Ising ferromagnet and still considerable amoun t of work is going on\nto understand other characteristic behaviours related to such d ynamic phase transitions.\nBut in all these studies in a two dimensional Ising ferromagnet bound ary conditions has\nbeen kept periodic, to preserve the translational invariance.\nItwouldbeinterestingtoknowhowtheIsingferromagnet,drivenb ystandingmagnetic\nwave behaves atfinitetemperatures andhowthedifference withth epropagatingmagnetic\nwave can be characterised and quantified. How does the boundary affect the dynamic\nphase transitions?\nIn the present study we have shown the effects of Standing magne tic wave on Ising\nferromagnet. The paper is organised as follows: The model and the MC simulation\n3technique are discussed in Sec. II, the numerical results are repo rted in Sec. III and the\npaper ends with a summary in Sec. IV.\nII. Model and Simulation\nThe Hamiltonian( time dependent )ofa two dimensional Ising ferromagnet, having uni-\nform nearest neighbour spin-spin interaction in presence of an ext ernal standing magnetic\nwave is represented by,\nH(t) =−JΣΣ′sz(x,y,t)sz(x′,y′,t)−Σhz(x,y,t)sz(x,y,t) (1)\nwheresz(x,y,t) is the Ising spin variable (±1) at lattice site ( x,y) at time t. The sum-\nmation Σ′extends over the nearest neighbour sites ( x′,y′) of given site ( x,y).J(>0)\nis theferromagnetic Spin-Spin interaction strength between the nearest neighbours. It is\nconsidered to be uniform over the whole lattice for simplicity. hz(x,y,t) is themagnetic\nfieldat site (x,y) at time t, which has the following form of Standing wave,\nhz(x,y,t) =h0sin(2πft)cos(2πx/λ) (2)\nTheh0,fandλrepresentrespectively the field amplitude, the frequencyandthe wavelength\nof the standing magnetic wave. Here, the magnetic wave is assumed aslinearly polarised\nalongthe direction parallel tothe spins ( sz). The modulationinamplitude ofthe standing\nmagnetic field wave is considered along the x direction only. It is worth y to mention that\nthe magnetic field wave considered here is externally applied magnetic field wave and it\nhas no connection with the usual spin wave formed in real ferromag nets.\nAnL×Lsquare lattice of Ising spins is considered with open boundary conditions\napplied at both directions. Also antinodes of standing magnetic field w ave are taken at\nx-boundaries. Monte Carlo Metropolis single spin flip algorithm is used for simulation\nof the dynamics. The initial spin configuration corresponds to high t emperature random\ndisordered state in which 50% of the lattice sites have spin state (+1 ) and the other 50%\nhave (−1). Any spin chosen, randomly, at site ( x,y) is updated with the Metropolis\nprobability [29] at temperature T, given by,\nW(sz→ −sz) =Min[exp(−∆E/kBT),1] (3)\nwhere ∆Eis the change in energy due to spin flip and kBis the Boltzman constant. L2\nrandom updating of spin states in an L×Lsquare lattice constitute the unit time step\ncalledMonte Carlo Step per Spin (MCSS). The values of the applied magnetic field and\nthe temperature are measured in the units of JandJ/kB, respectively. Any dynamical\nstate is reached by cooling the system slowly in small steps, from the high-temperature\nstate, which is the dynamically disordered state. The values of differ ent dynamical pa-\nrameters at any temperature are calculated after the system ac hieved steady state and\n4initial transient states are discarded. The system is kept at const ant temperature for a\nsufficiently long time and the average values of those parameters ar e taken throughout\nthe time for consideration of the steady state dynamical behaviou r.\n5III. Results\nIn the present study a square lattice of size ( L= 100) is considered. The frequency of\nstanding wave is taken through the study as f= 0.01MCSS−1. Different field amplitude\n(h0) and wavelength ( λ) of the standing magnetic field wave are considered to study the\ndependence of transition temperature as dependent on these pa rameters. Total length\nof simulation is 2 ×105MCSS for each temperature value. The steady state dynamical\nbehaviour is studied here after discarding initial (5 ×104MCSS) transient data for each\ntemperature value. The measured quantities are thus obtained by averaging over 15 ×104\nMCSS. Since, f= 0.01MCSS−1, a full cycle requires 100 MCSS. So, in15 ×104MCSS,\nwe have 15 ×102no. of cycles. All dynamical quantities are calculated by averaging o ver\n15×102cycles. Temperature is cooled in small steps of 0 .05J/kB, i.e. ∆T= 0.05J/kB,\nhere. This particular choice is a compromise between the computatio nal time and the\nprecision in measuring the transition temperature.\nTwo distinct phases namely, PinnedandOscillating spin clusters are identified in the\nsteady state. The pinned phase is such a phase where all the spins a re almost paral-\nlel and remain parallel (along a fixed direction either upward or downw ard) due to the\nsmall value of the probability of spin flip. The pinned state is formed be low a certain\ntransition temperature called the dynamic transition temperature , whereas the oscillating\nspin clusters phase is formed above this temperature. In the low te mperature and for\nsmall values of the amplitude of standing magnetic field wave, the pro balibility of spin\nflip becomes very small, which leads to the dynamical pinned phase . These phases are\nshown in fig.1. In the pinned phase most of the spins are in some prefe rred direction i.e.\neither upward or downward but in oscillating spin clusters phase appr oximately half of\nthe total spins are up and the others are down. The oscillating spin c lusters phase has a\ndefinite pattern of spins forming bands parallel to yaxis. Alternate bands of up and down\nspins having the bandwidth λ/2 are formed in the dynamically disordered phase. The\nband of up spins becomes a band of down spins, after a time 1 /2fand it bcemoes again a\nband of up spins after a further time interval 1 /2f. In this way the spin band oscillates,\nforming a standing wave, instead of showing a propagation observe d in earlier studies[24].\nFor sufficiently high values of temperature and the amplitude of the s tanding magnetic\nfield wave, due to the higher rate of spin flip, the system of spins effe ctively follow the\nspatio-temporal variation of applied magnetic standing wave, even tually leading to an\noscillating spin bands phase.\nThe dynamic order parameter Qfor such transition is defined as the time averaged\nmagnetisation per site over a full cycle of the standing magn etic field oscillations , i.e.\nQ=f\nL/contintegraldisplay/integraldisplay\nm(x,t)dxdt.\n6Herem(x,t), defined as,\nm(x,t) =1\nL/integraldisplay\ns(x,y,t)dy,\nis theaverage instantaneous line magnetisation per site at lattice coordinate ( x),s(x,y,t)\nbeing the instantaneous spin variable at lattice point ( x,y). In the pinned phase, the\norder parameter Qhas non-zero value because of arrangement of spins throughout the\nwhole lattice whereas in the oscillating spin-clusters phase it is zero be cause spin-clusters\nare arranged in alternate values ( ±1). So, as temperature decreases Qbecomes non-zero\n(at lower temperature) from a zero value (at higher temperature ) defining the dynamic\ntransition.\nThetemperature variations of the dynamic order parameter Q, the derivative of Qi.e.\ndQ\ndT, theL2×variance of Qi.e.L2/an}bracketle{t(δQ)2/an}bracketri}htand the dynamic specific heat C=dE\ndT, where\nE=f/contintegraldisplay\n{−JΣΣ′sz(x,y,t)sz(x′,y′,t)}dt\nis the average dynamic cooperative energy per spin state of the sy stem (without consid-\nering the field energy). All the above mentioned dynamical quantitie s are studied for two\ndifferent values of field amplitude ( h0= 0.6 & 1.0) and two different values of wavelength\n(λ= 25 & 50) of the standing magnetic field wave (see fig.2 and fig.3). The derivatives\nare calculated numerically using three-point central difference for mula [30]. All these\nquantities are calculated statisically over 1500 different samples (i.e. cycles of standing\nwave). Transition is detected by the sharp peaks (for L2/an}bracketle{t(δQ)2/an}bracketri}htandC) or dip (fordQ\ndT)\nin the temperature variations of the corresponding quantities.\nIt is evident from all the figuresthat the transition temperature decreases with in-\ncrease in the field amplitude. The nature of transition looks similar to t hat observed in\nthe case of propagating magnetic field wave [26]. But there are differ ences in different\nphases formed in both the cases. As can be seen in fig.4 that the inst antaneous line mag-\nnetisation at lattice sites ( x), which lie between any two consecutive nodes of standing\nwave, oscillates coherently with different amplitudes. Whereas it pro pagates along with\nthe propagating wave. At nodes of standing magnetic wave, the am plitude of oscillation\nof instantaneous line magnetisation is minimum (zero) and at antinode s it is maximum.\nThus, the instantaneous line magnetisation at different positions ( x) forms loops between\nany two consecutive nodes of standing magnetic field wave. In such a case of standing\nmagnetic wave, spins inside a loop oscillate coherently where two near est loops oscillate\nin opposite phase. At loop boundaries i.e. at nodes spins feel minimum e ffect of the\nmagnetic field and thus their dynamics are more thermally driven. On t he other hand\nthe dynamics of the spins at the antinodes are governed by the field . This is shown in\nfig.5. The probability that a spin, at any site ( x) in the lattice will flip, depends on\nthe temperature and the local magnetic field strength. Since at no des the magnetic field\n7strength is minimum the probability of spin flip is high. The peaks at node s (fig.5) show\nthat the average probability of spin flip over a full period of magnetic oscillation is quite\nlarge at these sites as compared to other lattice sites. Again in case of Ising ferromagnet,\ndriven by propagating magnetic field wave, the probability of spin flip is quite low at all\nlattice sites, since they all feel the same magnetic field strength ov er a full period. This\ncharacteristic difference distinguishes between the dynamical pha ses in Ising ferromagnet\ndriven by standing magnetic wave and propagating magnetic wave.\nNow collecting all the values of the transition temperatures Tdcorresponding to differ-\nent values of the magnetic field amplitudes h0for a particular wavelength λ, a comprehen-\nsive phase boundary may be drawn. Fig.6. shows the phase boundar y for two different\nwavelengths λ= 25 & 50 .As can be seen from the diagrams that the phase boundary\nshrinks towards the low field and low temperature values for shorte r wavelength, which is\nconsistent with the results obtained previously with propagating an d standing magnetic\nwave using periodic boundary conditions.\nIV. Summary:\nThe dynamical response of a two dimensional Ising ferromagnet, h avingopen bound-\naries, to the standing magnetic field wave is modelled and studied here. Mon teCarlo\ntechnique is used for simulating the observed result. In steady sta te, two distinct phases;\nnamelypinnedandoscillating spin clusters are observed. The pinned phase, with asym-\nmetric and static arrangement of spins is the dynamically ordered ph ase having non-zero\nvalue of average magnetisation. The oscillating spin clusters phase c onsists of many par-\nallel band shaped spin clusters and has zero average magnetisation . As the system is\ncooled from high temperature to low temperature, the dynamic ord er parameter becomes\nnon-zero below a certain transition temperature. The dynamic tra nsition seems to be of\ncontinuous nature and the dynamic transition temperature ( Td) depends on the values\nof the amplitude ( h0) and the wavelength ( λ) of the standing magnetic field wave at a\nsingle frequency. It should be mentioned here that the earlier stud ies[6], on the dynamic\ntransition, in the Ising ferromagnet driven by oscillating (in time but u niform over space)\nmagnetic field, reported the presence of discontinuos transition a nd located a tricritical\npoint on the phase diagram. Later on, the studies[7] on the distribu tion of dynamic order\nparameter with much improved statistics showed the absence of an y discontinuous tran-\nsition. So, to identify any tricritical behaviour (or discontinuous tr ansition, if any), one\nshould study this with much improved statistics, which is beyond the s cope of our com-\nputational facilities. Here, we do not make any such comment on the presence/absence\nof any tricritical point.\nPhase boundaries are drawn for two different wavelengths in Tdversush0plane. The\nphase boundary is observed to shrink towards asymetric phase fo r shorter wavelength.\n8Unlike the propagating phase where instantaneous line magnetisatio n oscillates with the\nsameamplitudeatalllatticesites, thesameoscillateswithdifferentam plitudesatdifferent\nlattice sites along the standing wave. The spins flip more frequently a t the nodes of the\nstanding wave. With open boundary condition applied to the lattice th e system achieved\nsteady state after longer time as compared to periodic boundaries applied to the lattice\n[26]. Apart from this, the nature of transition is found similar to the e arlier studies with\nperiodic boundaries applied in the case of propagating wave.\nIt would be interesting to see the effects of Standing wave on the hig hly anisotropic\nferromagnetic thin film (Co/Ni system) experimentally by time resolv ed magneto-optic\nKerr (TRMOKE) effect.\nControlling the dynamics of a group of spins by external magnetic fie ld having a\nspatio-temporal variation is quite important in the branch of spintr onics, magnonics in\nmodern condensed matter physics[31]. This present study is a simple statistical mechan-\nical approach of achieving various dynamical modes of Ising ferrom agnet irradiated by\na standing magnetic wave, just to have a preliminary notion about th e behaviour of a\nferromagnetic sample placed in intense optical pattern.\n9V. References:\n1. B. K. Chakrabarti and M. Acharyya, Rev. Mod. Phys. 71(1999) 847\n2. M. Acharyya, Int. J. Mod. Phys. C 16(2005) 1631\n3. M. Acharyya, Phys. Rev. E 56, 2407 (1997).\n4. M. Acharyya, Physica A 235, 469 (1997).\n5. S. W. Sides, P. A. Rikvold, M. A. Novotny, Phys. Rev. Lett. 81, 834 (1998).\n6. M. Acharyya, Phys. Rev. E 59, 218 (1999).\n7. G. Korniss, P. A. Rikvold, M. A. Novotny, Phys. Rev. E 66, 056127 (2002)\n8. M. Acharyya, Phys. Rev. E 58, 179 (1998).\n9. Q. Jiang, H. N. Yang, G. C. Wang, Phys. Rev. B 52, 14911 (1995)\n10. Q. Jiang, H. N. Yang, G. C. Wang, J. Appl. Phys. 79, 5122 (1996).\n11. Y. Au et al., Phys. Rev. Lett. 110, 097201 (2013).\n12. A. Berger et al.,Phys. Rev. Lett. 111, 190602 (2013).\n13. H. Park and M. Pleimling, Phys. Rev. Lett. 109(2012) 175703\n14. M. Acharyya, Int. J. Mod. Phys. C 14, 49 (2003).\n15. H. Jung, M. J. Grimson, C. K. Hall, Phys. Rev. B 67, 094411 (2003).\n16. H. Jung, M. J. Grimson, C. K. Hall, Phys. Rev. E 68, 046115 (2003).\n17. M. Keskin, O. Canko, B. Deviren, Phys. Rev. E 74, 011110 (2006).\n18. U. Temizer, E. Kantar, M. Keskin, O. Canko, J. Magn. Magn. Mater. 320, 1787 (2008).\n19. M. Ertas, B. Deviren and M. Keskin, Phys. Rev. E ,86(2012) 051110\n20. U. Temizer, J. Magn. Magn. Mater. 372(2014) 47\n21. E. Vatansever, A. Akinci and H. Polat, J. Magn. Magn. Mater. ,389(2015) 40\n22. M. Ertas and M. Keskin, Physica A ,437(2015) 430\n23. X. Shi, L. Wang, J. Zhao, X. Xu, J. Magn. Magn. Mater. ,410(2016) 181\n24. M. Acharyya, Phys. Scr. 84, 035009 (2011).\n25. M. Acharyya, J. Magn. Magn. Mater. 334, 11 (2013).\n26. M. Acharyya, Acta Physica Polonica B 45, 1027 (2014).\n27. M. Acharyya, J. Magn. Magn. Mater. 354, 349 (2014)\n28. M. Acharyya, J. Magn. Magn. Mater. 394, 410 (2015).\n29. K. Binder and D. W. Heermann, Monte-Carlo Simulation in Statistical Physics , Springer Series in\nSolid State Sciences, Springer, New York, 1997.\n30. C. F. Gerald, P. O. Weatley, Applied Numerical Analysis , Reading, MA: Addison-Wesley, 2006; J.\nB. Scarborough, Numerical Mathematical Analysis , Oxford: IBH, 1930.\n31. S. D. Bader and S. S. P. Parkin, Annual Reviews in Condensed Matter Physics ,1(2010) 71-88\n100102030405060708090100\n0102030405060708090100y\nx(a)\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···\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··························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lattice morphology for frequency f= 0.01 & field amplitude h0= 0.6. (a) oscillating\nspin clusters phase (temp. T= 2.0), (b) Pinned phase (temp. T= 1.5). In these figures ( ·) symbol\ndenotes the up spin states.\n1100.10.20.30.40.50.60.70.80.91\n00.511.522.533.5Q\nT(a)\n• •• • • • • •• • • • •• • • •• ••• ••••••••••••••••••••••• •• •• • • • • • • • • • • • • • • • •\n∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗\n-5-4.5-4-3.5-3-2.5-2-1.5-1-0.500.5\n00.511.522.533.5dQ\ndT\nT(b)• • • • • • • • • • • •• ••• • ••• • •••••••\n•\n•••••••••••••••• •• • • • • • • • • • • • • • • • ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗∗∗ ∗ ∗ ∗ ∗ ∗∗∗∗ ∗∗∗∗∗∗\n∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗\n050100150200250\n00.511.522.533.5L2/an}bracketle{t(δQ)2/an}bracketri}ht\nT(c)\n• • • • • •• • • • • • • • • •• • •• •••••••••••\n•\n•\n••••• •• • • • • • • • • • • • • • • • • • • • • • • • • ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗∗∗∗∗∗∗∗∗∗∗\n∗\n∗∗∗∗∗∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 00.20.40.60.811.21.4\n00.511.522.533.5Cv\nT(d)\n•••••••••• • • •••••••••••••••••••\n•\n•\n•\n••••••••••••••••• •• • • • • • • • • •∗∗∗∗∗∗ ∗ ∗ ∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗\n∗\n∗\n∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗ ∗ ∗ ∗ ∗ ∗\nFIG.2.Temperature ( T) variations of (a)Q,(b)dQ\ndT,(c)L2/an}bracketle{t(δQ)2/an}bracketri}htand(d)Cfor two different values\nof standing magnetic field amplitude h0. HereQis the order parameter, Lis the lattice size and Cis\nthe specific heat. Symbols ( •) & (∗) represent h0= 0.6 &h0= 1.0 respectively. The frequency and the\nwavelength of the standing wave are respectively 0 .01MCSS−1and 25 lattice units. The size of the\nlattice is 100 ×100.\n1200.10.20.30.40.50.60.70.80.91\n00.511.522.533.5Q\nT(a)\n•• • • • • • • •• • •• • •• ••• ••••••••••••••••••••••••••• • •• • • • • • • • • • • • • • • •\n∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗\n-4-3.5-3-2.5-2-1.5-1-0.50\n00.511.522.533.5dQ\ndT\nT(b)• • • • • • • • •••• • • •• •••••••••••\n•\n•\n•••••••••••••••••• • •• • • • • • • • • • • • • ∗∗∗ ∗∗ ∗ ∗∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗\n∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗ ∗ ∗ ∗ ∗ ∗\n050100150200250300350400\n00.511.522.533.5L2/an}bracketle{t(δQ)2/an}bracketri}ht\nT(c)\n• • • • • • • • • •• • • • • • •• • ••••••••••••\n•\n••••• •• • •• • • • • • • • • • • • • • • • • • • • • • • ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗∗∗∗∗∗∗∗∗∗\n∗\n∗∗∗∗∗∗∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 00.10.20.30.40.50.60.70.80.9\n00.511.522.533.5Cv\nT(d)\n•••••••••••••••••••••••••••••••\n••\n•\n•\n•\n•••••••••••••••• •• • • • • • • • •∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗ ∗ ∗ ∗ ∗\nFIG.3.Temperature ( T) variations of (a)Q,(b)dQ\ndT,(c)L2/an}bracketle{t(δQ)2/an}bracketri}htand(d)Cvfor two different values\nof standing magnetic field amplitude h0. HereQis the order parameter, Lis the lattice size and Cvis\nthe specific heat. Symbols ( •) & (∗) represent h0= 0.6 &h0= 1.0 respectively. The frequency and the\nwavelength of the standing wave are respectively 0 .01MCSS−1and 50 lattice units. The size of the\nlattice is 100 ×100.\n13-1-0.8-0.6-0.4-0.200.20.40.60.81\n0102030405060708090100m(x,t)\nx(a)\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⋄\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◦\n◦◦\n◦\n◦\n◦\n◦◦◦◦◦◦◦◦◦◦◦◦\n◦◦◦◦\n◦\n◦◦\n◦◦◦\n◦\n◦\n◦◦◦ ◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦\n◦\n◦\n◦◦\n◦◦\n◦\n◦\n◦\n◦\n◦◦\n-1-0.500.51\n0102030405060708090100m(x,t)\nx(b)\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∗\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◦◦◦◦\nFIG.4. Periodic variation of instantaneous line magnetisation m(x,t) at different lattice sites ( x); (a)\nfor standing wave , (b)for propagating wave in disordered phase, temperature T= 2.00 in units of J/kB.\nDifferent symbols represent different times (+) at 199900 MCSS, ( ⋄) at 199925 MCSS, ( ∗) at 199950\nMCSS, (◦) at 199975 MCSS, where time period of magnetic field oscillation is 100 M CSS. The standing\nwave is along x axis and the propagating wave propagates along xaxis. Here the values field amplitude\nh0, frequency fand wavelength λare 0.6J,0.01MCSS−1& 25lattice units respectively.\n140.10.120.140.160.180.20.220.240.26\n0102030405060708090100Ps(x)\nx•\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∗∗∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗∗ ∗ ∗ ∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗∗∗ ∗∗ ∗∗ ∗∗ ∗ ∗∗∗ ∗ ∗ ∗ ∗∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗∗∗∗∗\nFIG.5.Probability Ps(x) ofspin flips at different lattice sites along xaxis for standingmagnetic wave( •)\nand for propagating magnetic wave ( ∗) respectively in disordered phase, temperature T= 2.00 in units\nofJ/kB. The standing wave is along xaxis and the propagating wave propagates along xaxis. Here the\nvalues of field amplitude h0, frequency fand wavelength λare 0.6J,0.01MCSS−1& 25lattice units\nrespectively.\n00.511.522.53\n0 0.5 1 1.5 2 2.5h0\nTd(Q/ne}ationslash= 0)(Q= 0)\n••••••••••••••∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\n∗\nFIG.6. Phase diagram (dynamic transition temperature Tdvs. field amplitude h0) for two different\nwavelength ( λ= 25 (•) & 50 (∗)) of the standing magnetic field wave. The frequency of the stand ing\nwave isf= 0.01MCSS−1.\n15"    },    {        "title": "1808.03005v1.Effect_of_Magnetic_Field_Strength_on_Solar_Differential_Rotation_and_Meridional_Circulation.pdf",        "content": "E\u000bect of Magnetic Field Strength on Solar Di\u000berential Rotation\nand Meridional Circulation\nS.Imada ,1M.Fujiyama1\nABSTRACT\nWe studied the solar surface \rows (di\u000berential rotation and meridional circu-\nlation) using a magnetic element feature tracking technique by which the surface\nvelocity is obtained using magnetic \feld data. We used the line-of-sight mag-\nnetograms obtained by the Helioseismic and Magnetic Imager aboard the Solar\nDynamics Observatory from 01 May 2010 to 16 August 2017 (Carrington ro-\ntations 2096 to 2193) and tracked the magnetic element features every hour.\nUsing our method, we estimated the di\u000berential rotation velocity pro\fle. We\nfound rotation velocities of \u001830 and -170 m s\u00001at latitudes of 0\u000eand 60\u000ein\nthe Carrington rotation frame, respectively. Our results are consistent with pre-\nvious results obtained by other methods, such as direct Doppler, time-distance\nhelioseismology, or cross correlation analyses. We also estimated the meridional\ncirculation velocity pro\fle and found that it peaked at \u001812 m s\u00001at a latitude of\n45\u000e, which is also consistent with previous results. The dependence of the surface\n\row velocity on the magnetic \feld strength was also studied. In our analysis, the\nmagnetic elements having stronger and weaker magnetic \felds largely represent\nthe characteristics of the active region remnants and solar magnetic networks,\nrespectively. We found that magnetic elements having a strong (weak) magnetic\n\feld show faster (slower) rotation speed. On the other hand, magnetic elements\nhaving a strong (weak) magnetic \feld show slower (faster) meridional circulation\nvelocity. These results might be related to the Sun's internal dynamics.\nSubject headings: Sun: photosphere|Sun: \row|Sun: solar cycle\n1. INTRODUCTION\nThe 11-year variation in solar activity is an important source of decadal variation in the\nsolar-terrestrial environment. The solar cycle variation is attributed to a dynamo occurring\n1Institute for Space-Earth Environmental Research (ISEE), Nagoya University, Furo-cho, Chikusa-ku,\nNagoya 464-8601, JapanarXiv:1808.03005v1  [astro-ph.SR]  9 Aug 2018{ 2 {\nin the solar interior. Therefore, over the last few decades, considerable e\u000bort has been made\nto understand how the solar dynamo action occurs in the solar interior. To date, several\ntheories have been proposed on the basis of both mean-\feld dynamo theory (e.g., Jouve\net al. 2008) and global three-dimensional magnetohydrodynamic simulations (e.g., Hotta\net al. 2016). Although solar dynamo theory is still a matter of debate, the prediction of\nsolar cycle activity has been intensively discussed recently in the context of space weather.\nThe development of prediction schemes for the next solar cycle is a key to long-term space\nweather study, which is closely related to solar \rare and coronal mass ejections (e.g., Tsuneta\net al. 1992; Imada et al. 2007, 2011, 2013). The relationship between the polar magnetic\n\feld at solar minimum and the activity of the next solar cycle has received much attention\nin recent years. Many researchers currently believe that the polar magnetic \feld at solar\nminimum is one of the best predictors of the next solar cycle (e.g., Svalgaard et al. 2005).\nTo estimate the polar magnetic \feld, the surface \rux transport (SFT) model has often been\nused, and several studies have succeeded in estimating the polar magnetic \felds (see Jiang\net al. 2014; Iijima et al. 2017, and references therein). The SFT model requires several\nparameters such as the meridional circulation, the di\u000berential rotation, and the turbulent\ndi\u000busion. These parameters have also been observationally investigated (e.g., Hathaway &\nRightmire 2011).\nDi\u000berential rotation, in which the solar surface rotates di\u000berentially depending on lat-\nitude, has long been discussed (see Schroter 1985, and references therein). The rotation\nrate has been studied using Doppler measurements at the solar surface (e.g., Ulrich et al.\n1988). The rotation rate has also been derived from the motion of magnetic features such\nas sunspots (e.g., Howard et al. 1984) and magnetic elements (e.g., Komm et al. 1993).\nSmall-scale coronal features, (EUV bright points, Braj et al. 2001), (X-ray bright points,\nHara 2009), also show di\u000berential rotation, although large-scale coronal features (for ex-\nample, coronal holes) rotate almost rigidly. The di\u000berential rotation pro\fle determined by\ncoronal small-scale structures seems to correspond to the pro\fle obtained from photospheric\nmagnetic \felds.\nThe meridional \row has also been discussed for a long time, and the poleward \row at\nlow and intermediate latitudes is well established (e.g., Hathaway & Upton 2014). Because\nthis \row is two to three orders of magnitude slower than the rotational \row, its basic entire\nstructure is di\u000ecult to observe and is still controversial. It has been discussed whether or\nnot the meridional \row varies with latitude, depth, and time. These variations in the \row\nare believed to be strongly related to the presence of sunspots; thus, the solar cycle is also\nbelieved to a\u000bect the meridional \row.\nTo date, various studies have investigated the basic structure of di\u000berential rotation and{ 3 {\nmeridional circulation. However, few studies have focused on the in\ruence of the magnetic\n\feld strength on the solar surface \row. We studied the solar surface \row, di\u000berential rotation,\nand meridional circulation using a magnetic element feature tracking technique in which the\nsurface velocity is obtained using magnetic \feld data. To investigate the in\ruence of the\nmagnetic \feld strength on the solar surface \row, we also analyzed the relationship between\nthe average magnetic \feld strength inside magnetic elements and the surface \row velocity.\n2. DATA AND OBSERVATIONS\nWe use a series of line-of-sight magnetograms obtained from the Helioseismic and Mag-\nnetic Imager (HMI) aboard the Solar Dynamics Observatory (SDO; Pesnell et al. 2012)\nfrom 01 May 2010 to 16 August 2017 (from CR2096 to CR2193, \u0018100 Carrington rotations)\nat a cadence of 1 hr. The analyzed period corresponds roughly to the \frst half of solar cycle\n24. We also calibrate the absolute value of the magnetograms using the method of Liu et\nal. (2012). The line-of-sight magnetic \feld is assumed to be largely radial, so we divide\nthe magnetic \feld strength at each image pixel by the cosine of the heliographic angle from\nthe disk center to minimize the apparent variations in \feld strength with longitude from\nthe central meridian. Each full-disk magnetogram is mapped onto heliographic coordinates\nusing the equidistant cylindrical projection (e.g., Komm et al. 1993; Hathaway & Rightmire\n2011). The resolution of the projected map is 0.1\u000e, and the range of the projection is \u000690\u000e\nfor the central meridional distance and latitude. To avoid a small signal-to-noise ratio for\nthe magnetogram, we only use distances from the center of less than 75\u000e. Furthermore, we\nalso use the Sun's rotation axis correction as described by Hathaway & Rightmire (2011), in\nwhich they found that the accepted position of the Sun's rotation axis is in error by \u00180.08,\nas was noted previously by Howard et al. (1984) and Beck & Giles (2005).\nThe magnetic element feature tracking technique has been discussed and is now well\nestablished (e.g., Iida et al. 2012; Lamb 2017). To detect the magnetic element features,\nwe use a clumping method (e.g., Parnel et al. 2009) to identify each magnetic element\nhaving a magnetic strength exceeding a given threshold. The adopted threshold of 40 G was\nobtained by \ftting the histogram of the magnetic strength. The magnetic element features\nare selected when the total magnetic \rux inside the magnetic element (\b) is larger than\n1019Mx. These threshold values were obtained from an evaluation of the noise level in the\nMichelson Doppler Imager (MDI) on the Solar and Heliospheric Observatory (SOHO; Parnel\net al. 2009). It is well known that magnetic elements near sunspot behave di\u000berently. Thus,\nmagnetic elements close to sunspots were masked out in a previous study (Komm et al.\n1993). We de\fne magnetic elements having a total magnetic \rux larger than 1021Mx as{ 4 {\nsunspots and mask out magnetic elements less than 100 Mm from these de\fned sunspots.\nAfter the magnetic elements are detected, we track their motions. The travel distance of\nmagnetic elements in a 1-hr interval is roughly 0.4\u000e{0.7\u000e, because the solar rotation speed is\n10{15 deg day\u00001. Therefore, we identify the same magnetic elements between two images to\nbe within -0.1\u000e{1.0\u000ein the longitudinal direction and -0.3\u000e{0.3\u000ein the latitudinal direction.\nThe detection of the merging and splitting of magnetic elements is generally di\u000ecult and\nhas a high degree of uncertainty (e.g., Schriver et al. 1997; Iida et al. 2012). To avoid this\nuncertainty, we track only the elements for which the total magnetic \rux changes little. We\nde\fne the \rux change ratio (FCR) as jlog10(\b 2/\b1)j, where \b 1and \b 2are the magnetic\n\rux of the magnetic elements in the previous and following maps, respectively. The magnetic\nelements are tracked when FCR <0.1. When there are several candidates, we select the\nelement that has the lowest FCR value.\nFigure 1 shows an example of our detection of magnetic elements. The full-disk magne-\ntogram on 9 August 2010, mapped to heliographic coordinates using the equidistant cylin-\ndrical projection, is shown in Figure 1a. We detect and track only the magnetic elements\ninside the white dashed line in Figure 1a, which corresponds to 75\u000e. A total of 400 magnetic\nelements are detected by our method (Figure 1b). We identify 10 sunspots and mask out\nthe magnetic elements close to them using the above criteria. Finally, we track 309 magnetic\nelement features on this map (Figure 1c). As seen in Figure 1, we identify the magnetic\nelements in both of the quiet sun and the active region remnant. Therefore, the magnetic\nelements having stronger and weaker magnetic \felds largely represent the characteristics of\nthe active region remnants and solar magnetic networks, respectively. We con\frmed that\nour results, which are discussed below, are not sensitive to the detection criteria discussed\nabove.\n3. RESULTS\nFigure 2 shows the average di\u000berential rotation pro\fle derived from the entire data set\nfrom 01 May 2010 to 16 August 2017 ( \u0018100 CR). The velocities are taken relative to the\nCarrington frame of reference, which has a sidereal rotation rate of 14.184 deg day\u00001. The\naverage di\u000berential rotation pro\fle is well \ftted by the following equation:\nf(\u0012) = (a+bsin2(\u0012) +csin4(\u0012)) cos(\u0012); (1)\nwhere\na= 32:2 m s\u00001; (2)\nb=\u0000262:6 m s\u00001; (3){ 5 {\nc=\u0000304:2 m s\u00001: (4)\nFor comparison, we also added the \ftted curve of Hathaway & Rightmire (2011). We found\nrotation velocities of \u001830 and -170 m s\u00001at latitudes of 0\u000eand 60\u000e, respectively. As shown\nin Figure 2, we can see a weak north-south asymmetry in the deviation of the measured\npro\fle from the symmetric pro\fle (given by the red \ftted line). The di\u000berential rotation\nwas slightly stronger in the north than in the south. Flattening of the pro\fle at the equator\nis also observed, which has been discussed in past studies (e.g., Snodgrass 1983).\nThe meridional \row is generally more di\u000ecult to measure and has larger uncertainties\nthan the rotational \row, because the meridional \row is two to three orders of magnitude\nslower than the rotational \row. The average meridional \row pro\fle for the entire data set is\nshown in Figure 3. The pro\fle at low latitude is well represented with just two antisymmetric\nterms illustrated in the following equation:\nf(\u0012) = (dsin(\u0012) +esin3(\u0012)) cos(\u0012); (5)\nwhere\nd= 30:1 m s\u00001; (6)\ne=\u000026:4 m s\u00001: (7)\nHowever, the average meridional \row pro\fle at a high latitude shows substantial di\u000berences\nfrom the \ftted curve. The peak poleward meridional \row velocity is \u001812 m s\u00001at a latitude\nof 45\u000e. Our average meridional \row pro\fle shows substantially di\u000berent \rows in the north\nand in the south. The \row velocity is faster in the north and peaks at a higher latitude than\nin the south. The \row in the south appears to nearly vanish at the extreme southern limit\nof our measurements, whereas the \row in the north is still poleward with a speed of about\n\u00185 m s\u00001at the northern limit.\nWe also study the in\ruence of the magnetic \feld strength on the solar di\u000berential ro-\ntation and meridional circulation. Figure 4 shows the magnetic \feld strength dependence\nof the solar surface \row analyzed using the entire data set. The vertical axes in Figure\n4a and b show the residuals of the di\u000berential rotation speed and meridional \row from the\naverage at each latitude, respectively. The horizontal axes show the average magnetic \feld\nstrength inside the magnetic element. Note that the \feld strengths that we are interested in\nare not the intrinsic \feld strengths of small-scale magnetic elements, which are likely to be\nkilogauss, because we average the \feld strengths inside the magnetic elements. The data at\neach magnetic \feld strength are averaged, and the standard errors are shown (black lines:\naveraged with 10 data points). We can see that magnetic elements having a strong (weak)\nmagnetic \feld show faster (slower) rotation speeds. On the other hand, magnetic elements\nhaving a strong (weak) magnetic \feld show slower (faster) meridional circulation velocities.{ 6 {\n4. Discussion and Summary\nWe studied the di\u000berential rotation and meridional circulation using a magnetic element\nfeature tracking technique and magnetic \feld data from 01 May 2010 to 16 August 2017\n(approximately half of cycle 24). Using our method, we derived the di\u000berential rotation\nvelocity pro\fle and found rotation velocities of \u001830 and -170 m s\u00001at latitudes of 0\u000eand 60\u000e\nin the Carrington rotation frame, respectively. We also estimated the meridional circulation\nvelocity pro\fle and found that it peaked at \u001812 m s\u00001at 45\u000e. The dependence of the\nsurface \row velocity on the magnetic \feld strength was also studied. We found that magnetic\nelements having a strong (weak) magnetic \feld showed faster (slower) rotation speeds. On\nthe other hand, magnetic elements having a strong (weak) magnetic \feld showed slower\n(faster) meridional circulation velocities.\nLet us discuss the validity of our results by comparison with past observations. We\nstudied the di\u000berential rotation velocity in cycle 24. The angular rotation rate is nearly\nidentical to that found by Hathaway & Rightmire (2011) for the time interval from 1996 to\n2010 (mainly cycle 23) using di\u000berent methods (a cross-correlation technique). The dashed\nline in Figure 2 shows the di\u000berential rotation result of Hathaway & Rightmire (2011). The\nweak north-south asymmetry of the di\u000berential rotation and the \rattening of the pro\fle at\nthe equator seem to be the same as those in our results. We also studied the meridional\ncirculation velocity in cycle 24. The meridional circulation is also similar to that found\nby Hathaway & Rightmire (2011) for the time interval from 1996 to 2010 using di\u000berent\nmethods. Our meridional \row is slightly slower than the meridional \row discussed by Komm\net al. (1993) for the time interval from 1975 to 1991. Hathaway & Upton (2014) also\ndiscussed the meridional circulation in solar cycle 24 using HMI data from the time interval\nof April 2010 to July 2013. Their results also seem to be consistent with our results at\nnot only low latitudes but also high latitudes. Lamb (2017) also discussed the di\u000berential\nrotation and meridional circulation velocity using a magnetic feature tracking technique.\nTheir results also seem to be consistent with ours, although their meridional circulation\nvelocity at the peak latitude (45\u000e) is slightly faster than ours (16.7 m s\u00001). This small\ndiscrepancy might arise from the di\u000berence in the observation period and the details of the\ntracking method. Therefore, we can conclude that our derived velocities are reasonable and\nconsistent with past observations.\nOne of our interesting \fndings is the e\u000bect of the magnetic \feld strength on the solar\ndi\u000berential rotation and meridional circulation. As mentioned in Section 2, we identify the\nmagnetic elements in both of the quiet sun and the active region remnant. The magnetic\nelements having stronger and weaker magnetics \feld represent the characteristics of the\nactive region remnants and solar magnetic networks, respectively. Therefore, it is plausible{ 7 {\nthat the magnetic \feld strength dependence on solar di\u000berential rotation and meridional\ncirculation corresponds to the velocity di\u000berence between the active region remnant and the\nsolar magnetic network.\nDikpati et al. (2010) pointed out that the surface \row velocities obtained by magnetic\nelements feature tracking are a\u000bected by the surface turbulent magnetic di\u000busion. They\ndiscussed that the di\u000busive transport of magnetic elements away from the active latitude\nproduce an arti\fcial outward \row from the active latitude. Hathaway & Rightmire (2011)\nquantitatively estimated its e\u000bect from the observation and concluded that the di\u000busive\ntransport e\u000bect must be limited. They also discussed the impact of the in\row toward active\nlatitude on the meridional \row and found that the in\row is one of the origins of the long-term\nvariation of meridional \row pro\fle. We also checked the temporal variation of di\u000berential\nrotation speed and meridional \row and con\frmed the presence of the in\row toward the\nactive latitude and the torsional oscillation of Cycle 24 (not shown here). These signals\nseem to be weaker compared to those of Hathaway & Rightmire (2011) (Cycle 23). It is\nplausible that the poleward \rows of active region remnants decelerated by the in\row toward\nthe active latitude. The active region remnants is preferentially located at a higher latitude\nthan the active latitude, where the in\row toward the active latitude is opposite direction\nto the meridional \row. This might increase the magnetic \feld dependence of meridional\ncirculation. The impact of the in\row on the di\u000berential rotation speed should be limited,\nbecause the in\row accelerates and decelerates the di\u000berential rotation to the same degree.\nThe dependence of magnetic \feld strength on solar di\u000berential rotation and meridional\ncirculation might be related to the Sun's internal dynamics. The internal rotation speed\nand meridional \row of the sun have been studied by helioseismology. Thompson et al.\n(2003) studied the internal rotation speed of the sun and discussed the presence of a near-\nsurface shear layer located in approximately the outer 5% of the solar radius. According\nto their study, the internal rotation speed increases linearly with depth. If we assume that\nmagnetic elements having a stronger magnetic \feld represent deeper regions in the solar\ninterior, the trend of our results is consistent with that of the helioseismology observation.\nThe same can be said of the meridional circulation. Chen & Zhao (2017) studied the\ninternal meridional \row pro\fles. They found that the \row velocity decreases linearly by\n30% with depth around 0.975 R sunand is almost constant from 0.975 to 0.95R sun. The\ntrend of the magnetic \feld dependence on the meridional circulation in our analysis is also\nconsistent with the helioseismic observation. If we compare our results with those of the\nhelioseismology study, we can speculate that magnetic elements having \felds of 250 G, 100\nG, and 80 G might correspond to the \rows at 0.95, 0.975, and 1.0 R sun, respectively. We\nbelieve that our \fndings are useful for understanding the magnetic \feld transport at the\nsolar surface and the structure of the near-surface shear layer.{ 8 {\nWe would like to thank the referee for the useful comments which helped improving\nthe manuscript. The authors thank H. Iijima, H. Hotta, and Y. Iida for fruitful discussions.\nThis work was partially supported by the Grant-in-Aid for 17K14401 and 15H05816. This\nwork was also partially supported by ISEE CICR International Workshop program, and the\nauthors thank all members of the workshop. The Solar Dynamics Observatory is part of\nNASA's Living with a Star program.\nREFERENCES\nBeck, J.G., & Giles, P., 2005, ApJ, 621, L153\nBrajsa, R., et al. 2001, A&A, 374, 309\nChen, R., & Zhao, J., 2017, ApJ, 849, 144\nDikpati, M., et al. 2010, ApJ, 722, 774\nHara, H., 2009, ApJ, 697, 980\nHathaway, D.H., & Rightmire, L., 2011, ApJ, 729, 80\nHathaway, D.H., & Upton, L., 2014, J. Geophys. Res., 119, 3316\nHotta, H., et al. 2016, Sci, 351, 1427\nHoward, R., et al. 1984, ApJ, 283, 337\nIida, Y., et al. 2012, ApJ, 752, 149\nIijima, H., et al. 2017, A&A, 607, 2\nImada, S., et al. 2007, PASJ, 59, S793\nImada, S. et al. 2011b, ApJ, 743, 57\nImada, S., Aoki, K., Hara, H., Watanabe, T., Harra, L. K., & Shimizu, T. 2013, Astrophys.\nJ., 776, L11\nJiang, J., et al. 2014, Space Sci. Rev., 186, 491\nJouve, L., et al. 2008, A&A, 483, 949\nKomm, R.W., et al. 1993, Sol. Phys., 147, 27{ 9 {\nLamb, D.A., 2017, ApJ, 836, 10\nLiu, Y., et al. 2012, Sol. Phys., 279, 295\nParnell, C.E., et al. 2009, Sol. Phys., 698, 75\nPesnell, W.D., et al. 2012, Sol. Phys., 275, 3\nSchrijver, C.J., et al. 1997, ApJ, 487, 424\nSchroter, P.H., 1985, Sol. Phys., 100, 141\nSnodgrass, H.B., 1983, Sol. Phys., 270, 288\nSvalgaard, L., et al. 2005, Geophys. Res. Lett., 32, L01104\nThompson, M. J., et al. 2012, ARA&A, 275, 41\nTsuneta, S., et al. 1992, PASJ, 44, 63\nUlrich, R.K., et al. 1988, Sol. Phys., 117, 291\nThis preprint was prepared with the AAS L ATEX macros v5.2.{ 10 {\n-90 -60 -30   0  30  60  90-90-60-30  0 30 60 90 LAT \n-90 -60 -30   0  30  60  90-90-60-30  0 30 60 902010.08.09 09:10:25 \n-90 -60 -30   0  30  60  90\nCMD\n-90 -60 -30   0  30  60  90-90 -60 -30   0  30  60  90\n -90 -60 -30   0  30  60  90OtherAround sunspot(negative)Around sunspot(positive)Sunspot(negative)Sunspot(positive)(a) (b) (c)\nFig. 1.| (a) Full-disk magnetogram mapped onto heliographic coordinates using equidistant\ncylindrical projection. (b) Magnetic element features detected by a clumping method. (c)\nRemoval of magnetic elements that are close to sunspots. Red and blue asterisks represent\npositive and negative sunspots, respectively. Orange and sky-blue crosses are masked-out\nmagnetic elements.\nCR2096 to CR2193 (2010.05.01 - 2017.08.16)\n-90 -60 -30 0 +30 +60 +90\nLatitude [deg]-200-1000100\n-200-1000100Prograde Velocity [m/s] H&R 2011\nThis Study\nFig. 2.| Average pro\fle of di\u000berential rotation derived from the entire data set from 01\nMay 2010 to 16 August 2017 ( \u0018100 CR, half of cycle 24). Solid line: the \ftted curve of this\nstudy, dashed line: the \ftted curve of Hathaway & Rightmire 2011. The prograde velocity\nvalues are taken relative to the Carrington rotation frame.{ 11 {\nCR2096 to CR2193 (2010.05.01 - 2017.08.16)\n-90 -60 -30 0 +30 +60 +90\nLatitude [deg]-30-20-10010 20 30 Northward Flow [m/s] H&R 2011\nThis Study\nStandard Error\nFig. 3.| Average pro\fle of meridional circulation derived from the entire data set from 01\nMay 2010 to 16 August 2017 ( \u0018100 CR, half of cycle 24). Solid line: the \ftted curve of this\nstudy, dashed line: the \ftted curve of Hathaway & Rightmire 2011.{ 12 {\n0 100 200 300\nMagnetic Strength [G]-60-40-20020 40 60 Residual Rotational Verocity [m/s] Standard Error\nAveraged (10 points)\n0 100 200 300\nMagnetic Strength[G] -10-5 0510 Residual Poleward Flow [m/s] Standard Error\nAveraged (10 points)(b) Meridional Flow (a) Differential Rotation\nFig. 4.| E\u000bect of magnetic \feld strength on (a) solar di\u000berential rotation and (b) meridional\ncirculation. The vertical axes in Figure 4a and b show the residuals of the di\u000berential rotation\nspeed and meridional \row from the average at each latitude, respectively. The horizontal\naxes show the average magnetic \feld strength inside the magnetic element."    },    {        "title": "1112.0022v1.Relation_of_Astrophysical_Turbulence_and_Magnetic_Reconnection.pdf",        "content": "arXiv:1112.0022v1  [astro-ph.GA]  30 Nov 2011Relation of Astrophysical Turbulence and Magnetic Reconne ction\nA. Lazarian,1Gregory L. Eyink,2and E. T. Vishniac3\n1)Department of Astronomy, University of Wisconsin, 475 Nort h Charter Street, Madison, WI 53706,\nUSA\n2)Department of Applied Mathematics & Statistics, The Johns H opkins University University, Baltimore, MD 21218,\nUSAa)\n3)Department of Physics and Astronomy, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4M1,\nCanada\n(Dated: 16 October 2018)\nAstrophysicalfluidsaregenericallyturbulentandthismustbetake nintoaccountformosttransportprocesses.\nWe discuss how the preexisting turbulence modifies magnetic reconn ection and how magnetic reconnection\naffects the MHD turbulent cascade. We show the intrinsic interdepe ndence and interrelation of magnetic\nturbulence and magnetic reconnection, in particular, that strong magnetic turbulence in 3D requires recon-\nnection and 3D magnetic turbulence entails fast reconnection. We f ollow the approach in Eyink, Lazarian\n& Vishniac 2011 to show that the expressions of fast magnetic reco nnection in Lazarian & Vishniac 1999\ncan be recovered if Richardson diffusion of turbulent flows is used ins tead of ordinary Ohmic diffusion. This\ndoes not revive, however, the concept of magnetic turbulent diffu sion which assumes that magnetic fields can\nbe mixed up in a passive way down to a very small dissipation scales. On t he contrary, we are dealing the\nreconnection of dynamically important magnetic field bundles which st rongly resist bending and have well\ndefined mean direction weakly perturbed by turbulence. We argue t hat in the presence of turbulence the\nvery concept of flux-freezing requires modification. The diffusion t hat arises from magnetic turbulence can be\ncalled reconnection diffusion as it based on reconnection of magnetic field lines. The reconnection diffusion\nhas important implications for the continuous transport processe s in magnetized plasmas and for star forma-\ntion. In addition, fast magnetic reconnection in turbulent media indu ces the First order Fermi acceleration of\nenergetic particles, can explain solar flares and gamma ray bursts. However, the most dramatic consequence\nof these developments is the fact that the standard flux freezing concept must be radically modified in the\npresence of turbulence.\nI. PURPOSE AND OUTLINE\nThe purpose of this short paper is to discuss processes\nthat govern the change of the magnetic field topology in\nastrophysical fluids. We claim that it is incorrect to ig-\nnore ubiquitous astrophysical turbulence while studying\nmagnetized reconnection. We also show the intrinsic and\nvery deep relation between magnetic reconnection and\nturbulence.\nIn what follows, we discuss the problem of astrophysi-\ncal reconnection and point out the difference that exists\nin terms of reconnection between astrophysical systems\nand their present day numerical models ( §2), reveal the\nrelation between magnetic turbulence and magnetic re-\nconnection in §3, proceed to the discussion of the mag-\nnetic turbulence and magnetic field wandering in §4. We\nrelate the model of fast turbulent reconnection to the\nRicharson diffusion in §5 and discuss flux freezing in as-\ntrophysical systems in §6. The connection with previous\nstudies stressing the role of turbulence in reconnection is\nprovided in §7. We discuss astrophysical implications of\nour work in §8 and the summary is presented in §9.\na)also Department of Physics & AstronomyII. ASTROPHYSICAL RECONNECTION VERSUS\nNUMERICAL RECONNECTION\nIt is generally believed that a magnetic field embed-\nded in a highly conductive fluid preserves its topology\nfor all time due to the magnetic fields being frozen-in1,2.\nAlthough ionized astrophysical objects, like stars and\ngalacticdisks, arealmostperfectlyconducting,theyshow\nindications of changes in topology, “magnetic reconnec-\ntion”, on dynamical time scales3. Reconnection can be\nobserved directly in the solar corona4, but can also be in-\nferred from the existence of large-scale dynamo activity\ninside stellar interiors5,6. Solar flares7andγ-ray bursts\n(see8,9) are usually associated with magnetic reconnec-\ntion. A lot of previous work has concentrated on show-\ning how reconnection can be rapid in plasmas with very\nsmallcollisionalrates10,11, which substantiallyconstrains\nastrophysicalapplicationsofthe correspondingreconnec-\ntion models.\nA theory of magnetic reconnection is necessary to un-\nderstand whether reconnection is represented correctly\nin numerical simulations. One should keep in mind that\nreconnection is fast in computer simulations due to high\nnumerical diffusivity. Therefore, if there are situations\nwhere magnetic fields reconnect slowly, numerical sim-\nulations do not adequately reproduce the astrophysical\nreality. This means that if collisionless reconnection is\nindeed the only way to make reconnection fast, then the\nnumerical simulations of many astrophysical processes,2\nincluding those in interstellar media, which is collisional\nat the relevant scales, arein error. At the same time, it is\nnot possible to conclude that reconnection must always\nbe fast on the empirical grounds, as solar flares require\nperiods of flux accumulation time, which correspond to\nslow reconnection.\nTo understand the difference between reconnection in\nastrophysical situations and in numerical simulations,\noneshouldrecallthatthedimensionlesscombinationthat\ncontrols the resistive reconnection rate is the Lundquist\nnumber12, definedas S=LxVA/λ,whereLxisthelength\nof the reconnection layer, VAis the Alfv´ en velocity, and\nλ=ηc2/4πis Ohmic diffusivity. Because of the huge as-\ntrophysical length-scales Lxinvolved, the astrophysical\nLundquist numbers are also huge, e.g. for the ISM they\nare about 1016, while present-day MHD simulations cor-\nrespond to S <104. As the numerical efforts scale as L4\nx,\nwhereLxis the size of the box, it is feasible neither at\npresent nor in the foreseeable future to have simulations\nwith realistically Lundquist numbers.\nIII. TURBULENCE AND MAGNETIC RECONNECTION\nWhile astrophysical fluids show a wide variety of prop-\nerties in terms of their collisionality, degree of ioniza-\ntion, temperature etc., they share a common property,\nnamely, most of the fluids are turbulent. The turbulent\nstate of the fluids arises from large Reynolds numbers\nRe≡LV/ν, whereLis the scale of the flow, Vis it ve-\nlocityand νis the viscosity, associatedwith astrophysical\nmedia. Note, that the large magnitude of Reis mostly\nthe consequence of the large astrophysical scales Lin-\nvolved as well as the fact that (the field-perpendicular)\nviscosity is constrained by the presence of magnetic field.\nObservations of the interstellar medium reveal a Kol-\nmogorov spectrum of electron density fluctuations (see\nRef. 13 and 14) as well as steeper spectral slopes of su-\npersonic velocity fluctuations (see Ref. 15 for a review).\nMeasurement of the solar wind fluctuations also reveal\nturbulence power spectrum16). Ubiquitous non-thermal\nbroadening of spectral lines as well as measures obtained\nby other techniques (see Ref 17) confirm that turbulence\nis present everywhere we test for its existence. As turbu-\nlence is known to change many processes, in particular\nthe process of diffusion, the natural question is how it\naffects magnetic reconnection.\nTo deal with strong, dynamically important magnetic\nfields Lazarian & Vishniac18[henceforth LV99] proposed\na model of fast reconnection in the presence of sub-\nAlfv´ enic turbulence. It is important to stress that un-\nlike laboratory controlled settings, in astrophysical situ-\nations turbulence is preexisting, arising usually from the\nprocesses different from reconnection itself. We claim\nfurther in the paper that any modeling of astrophysical\nreconnection should account for the fact that magnetic\nreconnection takes place in the turbulent environment.\nLV99 identified stochastic wandering of the magneticfield-lines as the most critical property of MHD turbu-\nlence which permits fast reconnection. As we discuss\nmore fully below, this line-wandering widens the out-\nflow region and alleviates the controlling constraint of\nmass conservation. The LV99 model has been success-\nfully tested recently in Ref.19 (see also higher resolu-\ntion results in Ref. 20). This model is radically differ-\nent from its predecessors which also appealed to the ef-\nfects of turbulence (see more comparisons in section 7).\nFor instance, unlike Ref. 21 and 22 the model does not\nappeal to changes of the microscopic properties of the\nplasma. The nearest progenitor to LV99 was the work of\nMatthaeus & Lamkin23,24, who studied the problem nu-\nmerically in 2D MHD and who suggested that magnetic\nreconnection may be fast due to a number of turbulence\neffects, e.g. multiple X points and turbulent EMF. How-\never, Ref. 23 and 24 did not realize the key role of played\nby magnetic field-line wandering, and did not obtain a\nquantitative prediction for the reconnection rate, as did\nLV99.\nIV. MODEL OF MHD TURBULENCE AND MAGNETIC\nFIELD WANDERING\nWandering of magnetic field lines in LV99 model\nmostly depends on the Alfvenic component of magnetic\nperturbations. The exact scaling of the component, at\nleast within the currently discussed turbulence models\nproviding spectral indexes of −5/3 or−3/2 is not impor-\ntant, and the theoretical a possibility offast reconnection\nwas demonstrated in LV99 for turbulence with a wide\nrange of spectral indexes and anisotropies. At the mo-\nment, we feel that the theory by Goldreich & Sridhar25\n(henceforth GS95) is the most trusted one26. Therefore\nin what follows we present our estimates using this the-\nory.\nThe GS95 theory was formulated originally for a situ-\nation when the energy injection happened at the Alfven\nvelocity vA. More general relations applicable to sub-\nAlfvenic turbulence were obtained in LV99, and we use\nthose below. In particular, eddies in MHD turbulence\nbecome anisotropic and the key relation between the the\nparallel and perpendicular scales of eddies in the Alfenic\nturbulence is given by the so-calledcritical balancewhich\ncan be written in terms of parallel ℓ/bardbland perpendicular\nto magnetic field ℓ⊥scales of the eddies (LV99):\nℓ/bardbl≈Li/parenleftbiggℓ⊥\nLi/parenrightbigg2/3\nM−4/3\nA (1)\nwhereLiis the isotropic injection scale of the turbulence\nandMA≡uL/vAis the Alfv´ en Mach number of motions\nat the injection scale. Note, that the critical balance con-\ndition is only satisfied in the system of reference oriented\nwith respect to the local magnetic field, which is different\nfrom the usual global system of reference related to the\nmean magnetic field. Thus we avoid using wave-vectors\nin characterizing the parallel and perpendicular scales27.3\nIn terms of perpendicular to magnetic field motions\nthe scaling of Alfvenic turbulence is Kolmogorov-type:\nδuℓ≈uL/parenleftbiggℓ⊥\nLi/parenrightbigg1/3\nM1/3\nA. (2)\nwhich may be interpreted as a Kolmogorov cascade of\nmixing motions, which are not constrained by magnetic\nfields.\nThe scaling relations for Alfvenic turbulence allow us\nto calculate the rate of magnetic field spreading. A bun-\ndle of field lines confined within a region of width yat\nsome particular point will spread out perpendicular to\nthe mean magnetic field direction as one moves in either\ndirection following the local magnetic field lines. The\nrate of field line diffusion is given approximately by\nd/angbracketlefty2/angbracketright\ndx∼/angbracketlefty2/angbracketright\nλ/bardbl, (3)\nwhereλ−1\n/bardbl≈ℓ−1\n/bardbl,ℓ/bardblis the parallel scale chosen so that\nthe correspondingverticalscale, ℓ⊥, is∼ /angbracketlefty2/angbracketright1/2, andxis\nthe distance along an axis parallel to the mean magnetic\nfield. Therefore, using equation (1) one gets\nd/angbracketlefty2/angbracketright\ndx∼Li/parenleftbigg/angbracketlefty2/angbracketright\nL2\ni/parenrightbigg2/3/parenleftbigguL\nvA/parenrightbigg4/3\n(4)\nwherewehavesubstituted /angbracketlefty2/angbracketright1/2forℓ⊥. Thisexpression\nforthediffusioncoefficientwillonlyapplywhen yissmall\nenough for us to use the strong turbulence scaling rela-\ntions, or in other wordswhen /angbracketlefty2/angbracketright< L2\ni(uL/vA)4. Larger\nbundles will diffuse at a maximum rate of Li(uL/vA)4.\nFor/angbracketlefty2/angbracketrightsmall equation (4) implies that a given field line\nwill wanderperpendicular to the mean field line direction\nby an average amount\n/angbracketlefty2/angbracketright1/2≈x3/2\nL1/2\ni/parenleftbigguL\nvA/parenrightbigg2\n(5)\nin a distance x. The fact that the rms perpendicular\ndisplacement growsfaster than xis significant. It implies\nthat if we consider a reconnection zone, a given magnetic\nflux element that wanders out of the zone has only a\nsmall probability of wandering back into it. This finding\nplays an important role both for magnetic reconnection\nin LV99 and heat transfer (see Ref. 28 and 29).\nV. RICHARDSON DIFFUSION AND LV99 MODEL\nThe advantage of the classical Sweet-Parker scheme of\nreconnection is that it naturally follows from the idea\nof Ohmic diffusion. Indeed, mass conservation requires\nthat the inflow of matter through the scale of the contact\nregionLxbe equal to the outflow of matter through the\ndiffusion layer ∆, i.e.\nvrec=vA∆\nLx. (6)\nFIG. 1. Sweet-Parker laminar reconnection versus LV99 tur-\nbulent reconnection. Unlike the Sweet-Parker reconnectio n\nmodel , in the LV99 model the outflow is limited by magnetic\nfield wandering rather than Ohmic diffusivity. From Lazarian\net al.30.\nFIG. 2. Simulations of turbulent reconnection by Kowal et\nal.19. Magnetic field lines are visualized along with the fluc-\ntuations of current density. Substantial changes of the mag -\nnetic field directions are due to reconnection, as the drivin g is\nsubAlfvenic and the turbulent perturbations of magnetic fie ld\nare of low amplitude.\nThe mean-square vertical distance that a magnetic\nfield-line can diffuse by resistivity in time tis\n/angbracketlefty2(t)/angbracketright ∼λt. (7)\nThe field lines are advected out of the sides of the recon-\nnection layerof length Lxat a velocity of order vA.Thus,\nthe time that the lines can spend in the resistive layer is\nthe Alfv´ en crossing time tA=Lx/vA.Thus, field lines\ncan only be merged that are separated by a distance\n∆ =/radicalbig\n/angbracketlefty2(tA)/angbracketright ∼/radicalbig\nλtA=Lx/√\nS, (8)\nwhereSis Lundquist number. Combining Eqs. (6) and\n(8) one gets the famous Sweet-Parker reconnection rate,\nvrec=vA/√\nS.4\nIn LV99 magnetic field wandering determines the scale\nof the outflow ∆ (see Figure 1). Using expressions from\nthe earlier section one can obtain (LV99):\nVrec< vAmin/bracketleftBigg/parenleftbiggLx\nLi/parenrightbigg1/2\n,/parenleftbiggLi\nLx/parenrightbigg1/2/bracketrightBigg\nM2\nA.(9)\nThis limit on the reconnection speed is fast, both in the\nsense that it does not depend on the resistivity, and in\nthe sense that it represents a large fraction of the Alfv´ en\nspeed. To prove that Eq. (9) indeed constitutes the re-\nconnection rate LV99 goes through a thorough job of\nconsidering all other possible bottlenecks for the recon-\nnection and shows that they provide higher reconnection\nspeed. It may be that because of the complexity of the\nargument, the LV99 theory was considered with caution\nby an appreciable part of the community till the time\nwhen it was successfully tested in Ref. 19 (see Figure 2).\nBelow we provide a new derivation of the LV99 recon-\nnection rates which makes apparent that the LV99 model\nis a natural generalization of the laminar Sweet-Parker\nmodel to flows with background turbulence. The new\nargument in Eyink, Lazarian & Vishniac?(henceforth\nELV11) is based on the concept of Richardson diffusion.\nIt is known in hydrodynamic turbulence that the combi-\nnation of small scale diffusion and large scale shear gives\nrise to Richardson diffusion, where the mean square sep-\naration between two particles grows as t3once the rms\nseparation exceeds the viscous damping scale. A similar\nphenomenon occurs in MHD turbulence. In both cases\nthe separation at late times is independent of the mi-\ncroscopic transport coefficients. Although the plasma is\nconstrained to move along magnetic field lines, the com-\nbination of turbulence and ohmic dissipation produces a\nmacroscopicregionofpointsthat are”downstream”from\nthe same initial volume, even in the limit of vanishing re-\nsistivity.\nRichardson diffusion (see Ref. 31) implies the mean\nsquared separation of particles /angbracketleft|x1(t)−x2(t)|2/angbracketright ≈ǫt3,\nwheretis time,ǫis the energy cascading rate and /angbracketleft/angbracketrightde-\nnote an ensemble averaging. For subAlfvenic turbulence\nǫ≈u4\nL/(vALi) (see LV99) and therefore analogously to\nEq. (8) one can write\n∆≈/radicalBig\nǫt3\nA≈L(L/Li)1/2M2\nA (10)\nwhere it is assumed that L < L i. Combining Eqs. (6)\nand (10) one gets\nvrec,LV99≈vA(L/Li)1/2M2\nA. (11)\nin the limit of L < L i. Analogous considerations allow\nto recover the LV99 expression for L > L i, which differs\nfrom Eq. (11) by the change of the power 1 /2 to−1/2.\nThese results coincide with those given by Eq. (9).\nIt is important to stress that Richardson diffusion ul-\ntimately leads to diffusion over the entire width of large\nscale eddies once the plasma has moved the length of onesuch eddy. The precise scaling exponents for the turbu-\nlent cascade does not affect this result, and all of the\nalternative scalings considered in LV99 yield the same\nbehavior.\nItisalsoimportanttoemphasizethattheoriginalLV99\nargument made no essential use of averaging over turbu-\nlent ensembles. The stochastic line-wandering which is\nthe essence of their argument holds in every realization\nof the flow, at each instant of time. The “spontaneous\nstochasticity” of field lines is not a statistical result in\nthe usual sense of turbulence theory and does not arise\nfrom ensemble-averaging. The only use of ensembles in\nLV99 is to get a measure of the “typical” wandering dis-\ntance ∆ of the field lines (in an rms sense). If one looks\nat different ensemble members of the turbulent flow, or\nat different single-time snapshots of the steady reconnec-\ntion state, then ∆ will fluctuate. Thus, the reconnection\nrate will also fluctuate a considerable amount over en-\nsemble members or over time. E.g. see Figs. 12-14 in\nRef. 19. But it will be “fast” in each realization and at\neach instant of time, because the mass outflow constraint\nis lifted by the large wandering of field lines.\nThe LV99 theory, therefore, does not involve “turbu-\nlent resistivity” or “turbulent magnetic diffusivity” as\nthis is usually understood. This is ordinarily meant to\nbe an enhanced diffusivity experienced by the ensemble-\naveraged magnetic field /angbracketleftB/angbracketright. However, it is only if one\nassumessomescale-separationbetweenthe meanand the\nfluctuations that the effect of the fluctuations can be le-\ngitimately described as an enhanced diffusivity?. In re-\nalistic turbulent flows, with no scale-separation,this phe-\nnomenological description as an effective diffusivity can\nbe wildly inaccurate. LV99 makes no appeal to such con-\ncepts and, indeed, never considers the ensemble-average\nfield/angbracketleftB/angbracketrightat all.\nVI. VIOLATION OF FLUX FREEZING AND\nRECONNECTION DIFFUSION\nWhile the derivation of the LV99 expressions in the\nprevious section may look trivial, appealing to the con-\ncept of Richardson diffusion, in reality the justification of\nthe treatment is rooted in fundamental progressachieved\nrecently in understanding the concept of frozen-in field-\nlines for turbulent MHD plasmas32(see also Ref. 33 ?\n, 34).\nItisclearthatinthepresenceofmagneticreconnection\noccurring densely in space the magnetic field lines can-\nnot preserve their identity in turbulent flows. In fact,\nfield-wandering is a process that is difficult to under-\nstand within the standard concept of flux freezing. Note\nthat line-wandering implies that every space point is a\nnexus of infinitely many distinct lines. If magnetic field\nlines behaved like elastic threads which cannot change\ntheir topology, turbulence would create a system of unre-\nsolvedmagneticknotsdrainingtheenergytosmallscales.\nTherefore, instead of fluid-type MHD one would get vis-5\ncoelastic dynamics like Jello or rubber.\nThe high speed of reconnection given by equation (9)\nnaturally leads to a question of self-consistency. Is it\nreasonable to take the turbulent cascade suggested in\nGS95 when field lines in adjacent eddies are capable of\nreconnecting? It turns out that in this context, our es-\ntimate for Vrec,global is just fast enough to be interest-\ning. We note that when considering the intersection of\nnearly parallel field lines in adjacent eddies the acceler-\nation of plasma from the reconnection layer due to the\npressure gradient is not ℓ−1\n/bardblv2\nA, but rather ( ℓ3\n/bardbl/ℓ2\n⊥)−1v2\nA,\nsince only the energy of the component of the magnetic\nfield which is not shared is available to drive the outflow.\nOn the other hand, the characteristic length contraction\nof a given field line due to reconnection between adjacent\neddies is only ( ℓ/bardbl/ℓ2\n⊥)−1. This gives an effective ejection\nrate ofvA/ℓ/bardbl. Since the width of the diffusion layer over\na length ℓ/bardblis justℓ⊥, we can replace equation (9) with\nvrec,eddy≈vAℓ⊥\nℓ/bardbl. (12)\nThe associated reconnection rate is just\nτ−1\nreconnect ∼vA/ℓ/bardbl, (13)\nwhich in GS95 is just the nonlinear cascade rate on the\nscaleℓ/bardbl. Note, that this result is general and does not in-\nvolve assuming that GS95 model of turbulence is correct.\nIt, however, assuresthat the turbulent reconnectionrates\nare high enough to avoid the formation of unresolved\nmagnetic field knots, i.e. unresolved field line intersec-\ntions, in the GS95 and similar models of 3D turbulence.\nAnother consequence of magnetic reconnection is the\ndiffusion of plasma between different magnetic lines in\nturbulentflow. Thisprocessallowsconvectiveheattrans-\nfer as described in Ref. ?as well as magnetic field\nremoval from molecular clouds and accretion disks35,36.\nThe astrophysical diffusion enabled by reconnection of\nmagnetic field lines gives rise to a concept of reconnec-\ntion diffusion , which for many processes, e.g. for various\nstages of star formation may be more efficient than the\nambipolar diffusion. Below we briefly discuss this new\nconcept (see also Ref. 20, 35, and 37).\nThe common wisdom based on the notion of nearly\nperfect flux freezingfor astrophysicalmagneticfields sug-\ngests that mixing of plasma entrained on different flux\ntubes is impossible. However, it is easy to see that this\nis not true in the presence of LV99 reconnection. First\nof all, the process can connect magnetic field lines with\ndifferent plasma loading, inducing diffusion and mixing\nalong the newly emerging magnetic field lines. Then, in\nturbulent fluid the magnetic field lines can be shredded\nand mixed by eddies of smaller and smaller scales. These\ntwo processes are illustrated by separate figures below,\nbut, in reality, these two processes in turbulent magne-\ntized fluid take place simultaneously.\nFigure 3 illustrates the concept of reconnection diffu-\nsionfortwoflux tubeswhichhavethe sametotalpressure\nFIG. 3. Reconnection diffusion: process 1. Illustration of t he\nmixing of matter due to reconnection as two flux tubes of\ndifferent magnetic field strength interact.\nFIG. 4. Reconnection diffusion: process 2. Illustration of t he\nmixing of matter due to reconnection as two flux tubes of\ndifferent magnetic field strength interact.\nPtot=Pplasma+Pmagn. In the absence of turbulence the\ntwo flux tubes are in pressure equilibrium and the en-\ntrained plasma stays on the flux tubes. In the presence\nof 3D turbulence flux tubes can reconnect (minimizing\nthe energy of the Z-component of magnetic field) which\nallows plasma flow from one flux tube to another. This\nprocess is an illustration of a multi-scale process taking\nplace in realistic turbulent flows.\nFigure 4 illustrates the concept of reconnection dif-\nfusion when the magnetic pressure in the flux tubes is\nthe same. This is, for instance, is the case of heat ad-\nvection by turbulence. The mixing is happening as new\nmagnetic flux tubes areconstantly formed from magnetic\nflux tubes that belong to different eddies. In the figure\ntwo adjacent eddies are shown and the process is lim-\nited to the effects of eddies of a single scale. It is clear\nthat plasmas which was originally entrained over differ-\nent flux tubes gets into contact along the new emerg-\ning flux tubes. The process similar to the depicted one6\ntakes place at different scales down to the scale of the\nsmallest eddies. Molecular diffusivity then takes over. In\nthe case of heat transfer small scale molecular diffusivity\nwill ensure that the temperature along the newly formed\nmagnetic flux tubes is the same.\nNaturally, the process illustrated by Figure 4 also hap-\npens when the pressure of plasmas along magnetic flux\ntubes is different. LV99 process of magnetic field recon-\nnection ensures that bothmagnetic field and plasmas are\ndiffusing at the turbulent diffusion rate. We also note\nthat in the presence of forcing, e.g. gravitational forces\nacting upon the conducting gas, the diffusion will be ac-\ncompanied by the removalof magnetic field from the cen-\nterofthegravitationalpotentialduetothemagneticfield\nbouyancy. It is important to understand that the process\nin Figure 4 is limited only by the velocity of the eddies.\nTherefore supersonic turbulence can induce supersonic\nmixing.\nReconnection diffusion is due to eddies that are per-\npendicular to the localdirection of magnetic field. This\ndirection, in general, does not coincide with the mean\nmagnetic field direction. Therefore in the lab system of\nreference related to the mean magnetic field the diffusion\nof magnetic field and plasmas will happen both parallel\nand perpendicular to the mean magnetic field direction.\nVII. RELEVANT WORK ON RECONNECTION RATES\nHaving discussed the LV99 model and new arguments\nsupporting it, it is worth considering recent work on al-\nternative approaches to calculating reconnection rates.\nOver the last decade, more traditional approaches to re-\nconnection have changed considerably. At the time of\nits introduction, the models competing with LV99 were\nmodifications of the single X-point collisionless reconnec-\ntion scheme first introduced by Petschek38. Those mod-\nels had point-wise localized reconnection regions which\nwere stabilized via plasma effects so that the outflow\nopened up on larger scales. Such configurations would\nbe difficult to realize in the presence of random forcing,\nwhich would be expected to collapse the reconnection\nlayer. Moreover, Refs.39 argued that observations of so-\nlar flares were inconsistent with single X-point reconnec-\ntion.\nIn response to these objections, more recent models of\ncollisionless reconnection have acquired several features\nin common with the LV99 model. In particular, they\nhave moved to consideration of volume filling reconnec-\ntion, (although it is not clear how this volume filling is\nachieved in the presence of a single reconnection layer\n(see Ref. 11)). While much of the discussion still centers\naround magnetic islands produced by reconnection, in\nthreedimensionsthese islandsareexpected to evolveinto\ncontracting 3D loops or ropes40, which is broadly similar\nto what is depicted in Figure 1, at least in the sense of\nintroducing stochasticity to the reconnection zone.\nThe departure from the concept of laminar reconnec-tion and the introduction of magnetic stochasticity is\nalso apparent in a number of the recent papers appeal-\ning to the tearing mode instability to drive fast recon-\nnection (see Refs. 41 and 42)43LV99 showed that the\nlinear growth of tearing modes is insufficient to obtain\nfast reconnection. More recent work is based on the idea\nthat the non-linear growth of magnetic islands due to\nmergers provides large scale growth rates larger than the\ntearing mode linear growth rates on these scales. A sit-\nuation where the non-linear growth is faster than the\nlinear one is rather unusual and requires further inves-\ntigation (see Ref. 44.) Since tearing modes exist even\nin a collisional fluid, this may open another channel of\nreconnection in such fluids. This reconnection, as we dis-\ncuss below, should not be “too fast” to account for the\nobservational data.\nIf local turbulence is driven by release of energy from\nthe magnetic field, it may result in a runaway turbulent\nreconnection process which may be relevant to some nu-\nmerical simulations46,47. Alternatively, if tearing modes\nbegin by driving relatively slow reconnection then a sim-\nilar runaway might result48.\nIn anycase, in most astrophysicalsituations one has to\ndeal with the pre-existing turbulence, which is the conse-\nquence of high Reynolds number of astrophysical fluids.\nSuch turbulence may modify or suppress instabilities, in-\ncluding the tearing mode instability. In this paper we\nhave shown that it, by itself, induces fast reconnection\non dynamical time scales.\nVIII. ASTROPHYSICAL IMPLICATIONS\nFast magnetic reconnection induces numerious astro-\nphysical implications. In the sections above we have dis-\ncussed a new concept of reconnection diffusion that is\nlikely to dominate diffusion of magnetic fields and plas-\nmas in various astrophysical environments from accre-\ntion disks to intracluster medium (see e.g. Refs. 49 and\n50). The role of reconnection diffusion may be different.\nFor instance, in circumstellar accretion disks and cores of\nmolecular clouds it removes magnetic flux, while in more\ndiffuse interstellar medium it mostly provides good mix-\ning destroying the correlation of magnetic field strength\nwith plasma density. Turbulent transport of heat, as well\nas impurities and dust is happening due to the reconnec-\ntion diffusion.\nAt the same time, the LV99 magnetic reconnection\ninduces a number of new effects. For instance, parti-\ncles trapped over magnetic field lines get accelerated via\nFirst order Fermi process35,51. This acceleration is also\nreported in a situation when loops of magnetic field are\nformed via tearing reconnection11,52. The physics is the\nsame, namely, in both cases magnetic fields shrink and\nthe particles entrained overmagnetic field lines get accel-\nerated. This provides a likely explanation of the origin of\nthe anomalous cosmic rays measured by Voyagers53,54as\nwell as the anisotropy of cosmic rays in the direction of7\nheliotail55reported by different groups. We would like to\nstress that the acceleration in 3D and 2D is found to be\ndifferent56and therefore results of 2D numerical experi-\nments dealing with particle acceleration should be taken\nwith grain of salt.\nNaturally, the acceleration of energetic particles and\nreconnection diffusion do not encompass all the possi-\nble applications of the LV99 model of reconnection. For\ninstance, Lazarian et al8proposed a way of explaining\ngamma ray bursts appealing to turbulent reconnection.\nThis idea was further elaborated in a high impact pa-\nper by Zhang & Yan9In general, bursts and flares are\nthe natural consequence of the LV99 model. If magnetic\nfields are originallylaminar, the low reconnection rate al-\nlows the accummulation of magnetic flux. As turbulence\nincreases, for example, due to the outflow, the reconnec-\ntion rate increases, making the outflow more turbulent.\nThis induces positive feedback resulting in what can be\ntermed “reconnection instability”.\nWhatever the particular astrophysical consequences of\nLV99 model, the most striking is the fact that in turbu-\nlent fluids the basic idea of flux freezing is dramatically\nmodified in turbulent fluids. This shakes the foundations\noftheastrophysicalMHD, asastrophysicalfluidsareusu-\nally turbulent.\nIX. SUMMARY\nOur main points can be summarized as follows:\nAstrophysicalfluids areturbulent and turbulence must\nbe accounted in the models of astrophysical magnetic re-\nconnection.\nTurbulence and magnetic reconnection are two inter-\ndependent processes: turbulence makes magnetic recon-\nnection fast, but magnetic reconnection is required for\nthe turbulence to evolve in a self-similar fashion.\nPlasmaeffects maybeimportant forlocalreconnection\neffects, but this in most cases will not affect the resulting\nglobal reconnection rates.\nACKNOWLEDGMENTS\nWe thank A. Bhattacharjee, P. H. Diamond, and\nA. Pouquet for some useful discussions. AL is sup-\nported by the NSF grant AST 0808118, NASA grant\nNNX09AH78G and the Center for Magnetic Self Organi-\nzation. AL acknowledges Humboldt Award at the Uni-\nversities of Cologne and Bochum and a Fellowship at the\nInternational Institute of Physics (Brazil). GE was par-\ntiallysupportedbyNSFgrantsAST0428325andCDI-II:\nCMMI 0941530. The work of ETV is supported by the\nNational Science and Engineering Research Council of\nCanada.\n1H. Alfv´ en, Ark. Mat., Astron. o. Fys. 29B, 1 (1942).\n2E. N. Parker, Cosmic Magnetic Fields (Oxford University Press,\n1979).3E. N. Parker, ApJ 162, 665 (1970).\n4T. Yokoyama and K. Shibata, Nature 375, 42 (1995).\n5E. N. Parker, ApJ 408, 707 (1993).\n6M. Ossendrijver, AandA Rev. 11, 287 (2003).\n7P. A. Sturrock, Nature 211, 695 (1966).\n8A. Lazarian, V. Petrosian, H. Yan, and J. Cho, Beaming and\nJets in Gamma Ray Bursts (2003), proceedings, volume 45.\n9B. Zhang and H. Yan, ApJ 726, 90 (2011).\n10J. F. Drake, Nature 410, 525 (2001).\n11J. F. Drake, M. Swisdak, H. Che, and M. A. Shay, Nature 443,\n553 (2006).\n12The magnetic Reynolds number, which is the ratio of the mag-\nnetic field decay time to the eddy turnover time, is defined usi ng\nthe injection velocity vlas a characteristic speed instead of the\nAlfv´ en speed VA, which is taken in the Lundquist number.\n13R. B. J. Armstrong, J. W. and S. R. Spangler, ApJ 443, 209\n(1995).\n14A. Chepurnov and A. Lazarian, ApJ 710, 853 (2010).\n15A. Lazarian, Space Science Reivews 143, 357 (2009),\narXiv:0811.0839.\n16R. J. Leamon, C. W. Smith, N. F. Ness, W. H. Matthaeus, and\nH. K. Wong, Journal Geophis. Res. 103, 4775 (1998).\n17B. Burkhart, S. Stanimirovi´ c, A. Lazarian, and G. Kowal, Ap J\n708, 1204 (2010).\n18A. Lazarian and E. T. Vishniac, ApJ 517, 700 (1999).\n19G. Kowal, A. Lazarian, E. T. Vishniac, and K. Otmianowska-\nMazur, ApJ 700, 63 (2009).\n20A. Lazarian, R. Santos-Lima, and E. de Gouveia Dal Pino,\ninNumerical Modeling of Space Plasma Flows, Astronum-2009 ,\nAstronomical Society of the Pacific Conference Series, Vol. 429,\nedited by N. V. Pogorelov, E. Audit, & G. P. Zank (2010) pp.\n113–+, arXiv:1003.2640 [astro-ph.GA].\n21T. W. Speiser, Plan. and Space Sci. 18, 613 (1970).\n22A. R. Jacobson and R. W. Moses, Phys. Rev. A. 29, 3335 (1984).\n23W. H. Matthaeus and S. L. Lamkin, Phys. Fluids, 28 28, 303\n(1985).\n24W.H.Matthaeus and S.L.Lamkin,Phys.Fluids 29,2513 (1986).\n25P. Goldreich and S. Sridhar, ApJ 438, 763 (1995).\n26It is discussed in Beresnyak & Lazarian (2010) that the prese nt\nday numerical simulations may not be sufficient to reveal the\nactual slope of Alfvenic cascade. In Ref.?the slope of −5/3\nwas obtained in the largest reduced MHD simulations employi ng\nhyperdiffusion.\n27The wavevectors were used in GS95 where the distinction of lo cal\nand global reference frames was not done and in a number of\nlater papers where wavevectors were used mostly due to histo ric\nreasons. The first paper to discuss local system of reference was\nLV99.\n28R. Narayan and M. V. Medvedev, ApJL 562, L129 (2001).\n29A. Lazarian, ”ApJL” 645, L25 (2006), arXiv:astro-ph/0608045.\n30A. Lazarian, E. T. Vishniac, and J. Cho, ApJ 603, 180 (2004).\n31A. Kupiainen, Ann. Henri Poincar´ e 4, Suppl. 2 , S713 (2003).\n32G. L. Eyink, Phys. Rev. E 83, 056405 (2011).\n33G. L. Eyink and H. Aluie, Physica D 223, 82 (2006).\n34G. L. Eyink, J. Math. Phys. 50, 083102 (2009).\n35A.Lazarian,in Magnetic Fields in the Universe: From Laboratory and Stars t o Primordial Structures. ,\nAmerican Institute of Physics Conference Series, Vol. 784, edited\nby E. M. de Gouveia dal Pino, G. Lugones, & A. Lazarian\n(2005) pp. 42–53, arXiv:astro-ph/0505574.\n36R. Santos-Lima, A. Lazarian, E. M. de Gouveia Dal Pino, and\nJ. Cho, ApJ 714, 442 (2010).\n37A. Lazarian, ArXiv e-prints (2011),\narXiv:1108.2280 [astro-ph.GA].\n38H. Petschek, AAS-NASA Symposium (NASA SP-50) (NASA\nSP-50), 425 (1964).\n39A. Ciaravella and J. C. Raymond, ApJ 686, 1372 (2008).\n40W. Daughton, V. Roytershteyn, B. J. Albright, K. Bowers,\nL. Yin, and H. Karimabadi, (2008), aGU Fall Meeting Ab-\nstracts.\n41N. F. Loureiro, D. A. Uzdensky, A. A. Schekochihin, S. C. Cow-8\nley, and T. A. Yousef, MNRAS 399, L146 (2009).\n42A. Bhattacharjee, Y.-M. Huang, H. Yang, and B. Rogers, Phys.\nPlasmas 16, 112102 (2009).\n43The idea of appealing to the tearing mode as a means of enhanc-\ning the reconnection speed can be traced back to Refs.?.\n44P. H. Diamond, R. D. Hazeltine, Z. G. An, B. A. Carreras, and\nH. R. Hicks, Physics of Fluids 27, 1449 (1984).\n45R. Sych, V. M. Nakariakov, M. Karlicky, and S. Anfinogentov,\nAandA505, 791 (2009).\n46G. Lapenta, Phys. Rev. Lett. 100, 235001 (2008).\n47L. Bettarini and G. Lapenta, A & A 518, A57 (2010).\n48.\n49A. C. Fabian, J. S. Sanders, R. J. R. Williams, A. Lazarian,\nG.J.Ferland, andR.M.Johnstone, ”MNRAS” 417, 172 (2011),\narXiv:1105.1735 [astro-ph.GA].50R. Santos-Lima, E. M. de Gouveia Dal Pino, and A. Lazarian,\nArXiv e-prints (2011), arXiv:1109.3716 [astro-ph.GA].\n51E. M. de Gouveia dal Pino and A. Lazarian,\n”A&A” 441, 845 (2005).\n52J. F. Drake, M. Swisdak, T. D. Phan, P. A. Cassak, M. A. Shay,\nS. T. Lepri, R. P. Lin, E. Quataert, and T. H. Zurbuchen,\nJournal of Geophysical Research (Space Physics) 114, A05111 (2009).\n53A. Lazarian and M. Opher, ”ApJ” 703, 8 (2009),\narXiv:0905.1120 [astro-ph.EP].\n54J. F. Drake, M. Opher, M. Swisdak, and J. N. Chamoun,\n”ApJL” 709, 963 (2010), arXiv:0911.3098 [astro-ph.SR].\n55A. Lazarian and P. Desiati, ”ApJ” 722, 188 (2010),\narXiv:1008.1981 [astro-ph.CO].\n56G. Kowal, E. M. de Gouveia Dal Pino, and A. Lazarian,\n”ApJ”735, 102 (2011), arXiv:1103.2984 [astro-ph.HE]."    },    {        "title": "0810.5017v1.Anomalous_magnetization_behavior_in_Ce_Fe_Si_2.pdf",        "content": "Anomalous magnetization behavior in Ce(Fe,Si) 2 \n \nArabinda Haldar1, K. G. Suresh1,* and A. K. Nigam2\n \n1Magnetic Materials Laboratory, Department of Physics  \nIndian Institute of Technol ogy Bombay, Mumbai- 400076, India \n2Tata Institute of Fundamental Research,  \nHomi Bhabha Road, Mumbai- 400005, India \n \n \nAbstract \n \nWe report the effect of Si doping on  the magnetization behavior of CeFe 2. It is \nfound that Si stabilizes the dynami c antiferromagnetic state of CeFe 2. Multi-step \nmagnetization behavior, unusua l relaxation effect, ther mal and magnetic history \ndependence, which are signatures of ma rtensitic scenario, are found to be \npresent in this system. We also show  that one can induce the magnetization \nsteps with the help of appropriate  measurement protocol. Detailed \nmagnetization relaxation studies have been carried out to understand the \ndynamics of magnetic phase transition.         \n \n             --------------------------------------------------- ------------- \n*Corresponding author (email: suresh@phy.iitb.ac.in ) \n   \n I. Introduction \n \nMultiple magnetization steps seen in a few intermetallics compounds and a large number \nof oxides has drawn a lot of attention r ecently. It is well known that structural \nheterogeneities arising from the magnetic he terogeneities presents a martensitic-like \nscenario in these systems a nd is responsible for the anomalous magnetic behavior. A lot \nof research activity has been made to expl ore the origin of the martenistic scenario, \nespecially in the inte rmetallics. This featur e is found in single crystalline samples, thin \nfilms and even in polycrys talline compounds. This observa tion is quiet unusual in \npolycrystalline samples and attr acts a lots of attention. Various explanations had been put \nforward with detailed experimental results lik e relief of strain across the phase separated \nregion [1-4], burst like growth of ferromagne tic phase (FM) with increasing field [5, 6] \nand quenched disorder [7]. \n CeFe\n2 is unique among the seri es of R (rare earth)-Fe 2 compounds. It has low saturation \nmoment (M s) / 3 . 2Bμf.u. (M s= / 9 . 2Bμf.u. of LuFe 2), anomalously low Curie \ntemperature (T C); 230K (T C=610K & 545K for LuFe 2 and YFe 2 respectively) [8, 9]. It is \nto be mentioned here that this system show s two phase nature when selected elements \n(Ru, Re, Al, Co, Ir, Os and Ga) is substituted at  the iron site [10, 11].  Most interestingly \nsharp jumps in magnetization have been obser ved in magnetization in Ru, Re [3] and Ga \n[4] doped compounds. Unusual steps in low temperature magnetization isotherms in \ncertain doped CeFe 2 compounds [3, 4] proved that th e system can be a promising \nmaterial to study the underlying physics in phase separated systems.     In this report we present a detailed study on temperature, field, time dependence on Si \ndoped CeFe 2 with various Si concentr ations. Effect of zero fiel d cooling (ZFC) and field \ncooling (FC) the compound during the measurement is demonstrated. We have also \nstudied relaxation phenomenon in detail to unde rstand the dynamics of the steps observed \nin M(H) isotherms. The presence of induction time to occur these steps has been discussed.     \nII. Experimental Details  \n \nPolycrystalline compounds Ce(Fe 1-xSix)2 with x=0.01, 0.025 & 0.05 were  prepared by arc \nmelting of the constituent elements: Ce (99.9%), Fe(99.999%) and Si(99.999%).  The \ndetails of preparation have been reported elsewhere [4].  The structural analysis was \nperformed by taking room temperat ure x-ray diffraction using Cu-K α radiation. The \ndiffractograms were refined by th e Rietveld analysis using Fullprof  suite program. \nMagnetization measurements in the temperatur e range of 1.8- 300 K and in fields up to \n90 kOe were carried out in Physical Prope rty Measurement System (PPMS, Quantum \nDesign Model 6500) which has a vibrating sa mple magnetometer (VSM) attachment. \nMagnetization measurements have been taken both in zero field cooled (ZFC) and field \ncooled (FC) modes.  \n     III. Results and Discussions  \n15 30 45 60 75 90 Expt.  Theo.  Bragg Position  Diff.  \n X=0.01 \n  \n2θ (deg.)\n X=0.025 \n Ce(Fe1-xSix)2\n X=0.05\n Intensity (arb. units)\n \nFIG. 1. Room temperature x-ray diffractograms for Ce(Fe 1-xSix)2 compounds using Cu-\nKα radiation.  \n \nX-ray diffraction using Cu-K α radiation was taken on the samples with x=0.01, 0.025 & \n0.05, compounds at room temperature.  In fig.  1 open circles represents experimental \npoints and the red line shows the fitted line ob tained from Rietveld refinement of the \ndiffractograms. As can be seen, all the compoun ds have formed in single phase. Rietveld \nrefinement confirms the MgCu 2 type cubic structure with the space group of m Fd3 o f  \nall these compounds.  0 75 150 225 3000.00.20.40.60.81.01.2\nx=0.05\nx=0.025\nx=0.01\n  M (μB/f.u.)\nT (K)H=500Oex=0(a)\n \n0 75 150 225 300 3750.00.51.01.52.0\n50kOe\n20kOe\n0.5kOe\n  M (μB/f.u.)\nT (K)x=0.05(b)\n \nFIG. 2. (a) Temperature varia tion of magnetization of Ce(Fe 1-xSix)2 compounds. (b) \nTemperature dependence of magnetization of Ce(Fe 1.95Si0.05)2 compound at different \napplied magnetic fields. In both the cases data have been taken during warming the \nsample in both ZFC (closed point s) and FC (open points) mode. \nFig. 2a shows the variation of magnetization with temperat ure of the all the compounds \nin a field of 500 Oe. M vs. T plot for CeFe 2 [4] has been shown for better comparison. \nWith substitution of Si, the fluctuating an tiferromagnetic (AFM) state present in the parent compound gets stabilized for x=0.025 & 0. 05, which is similar to our recent result \nwith Ga substitution [4]. For x=0.01 substitution Curie temperature decreases by 10 K \ncompared to parent compound (T C=230K) and with more substitution we have order-\norder (FM-AFM) transition at 41K (x=0.025)  and 65K (x=0.05). The Curie temperature \ndecreases monotonically with Si. As can be observed fr om the figure, AFM ground state \nis not fully stabilized in x=0.025 compounds whereas it is fully stabilized at x=0.10. \nFrom the ZFC and FCW data, there is no cons iderable difference observed between these \ntwo modes for x=0 and 0.1, while x=0. 025 compound shows some difference at low \ntemperatures. But x=0.05 compound shows a huge difference between ZFC and FC data \nat low temperatures. This reflects the streng th of AFM induced by Si. The fact that the \ndifference is large even in fields as hi gh as 20kOe implies that the AFM is very \ndominant.  \n0123450.00.51.01.52.02.53.0M (μΒ/f.u.)\nH (kOe)\n  \n x=0 @T=2K\n x=0.01 @T=5K(a)\n -75 -50 -25 0 25 50 75-3-2-10123(b)\nT=2Kx=0.0255\n432\n  M (μB/f.u.)\nH (kOe)1\n \n-90 -60 -30 0 30 60 90-2-1012(c)\n5\n432\n1\n  M (μB/f.u.)\nH (kOe)T=3Kx=0.05\n \nFIG. 3. Isothermal magnetization curves for (a) Ce(Fe 1.99Si0.01)2 at T=5K, (b) \nCe(Fe 1.985Si0.025)2 at T=2K, (c) Ce(Fe 1.95Si0.05)2 compound at T=3K. Arrows show the \nfield (Sweep rate: 100Oe/sec) directions in which measurem ents have been carried out. \n \nMagnetization isotherms have been taken on a ll the samples at very low temperatures up \nto maximum field of 90kOe . Fig. 3a shows the M vs. H plots for x=0.01 compound along \nwith that of the parent compound. Five l oop magnetization data have been taken for x=0.025 and x=0.05 substituted compounds (Fig. 3b and c). The field sweep rate during \nall these measurements in fig. 3 was 100 Oe /sec. and the sample was zero field cooled \nfrom 325K. M(H) behaviors of x=0 & 0.01 compounds are similar to a typical ferromagnet. Interestingly a metamagnetic tran sition from AFM to FM phase is observed \nfor higher Si substituted (x=0.025 & 0. 05) compounds which was expected after \nobserving the temperature dependence of magnetization data of  these compounds. M \nvs. \nT data also shows that the unstable AFM pha se gets partially st abilized in x=0.025 \ncompounds and it almost fully stable with broad AFM ground state near low temperature \nregion for x=0.05 compound. At T ≤3K both the compounds (x=0.025 & 0.05) are more \nor less in AFM dominating ground state. Application of exte rnally applied field favors \nthe final state to be ferromagnetic. Depending on the strength of AFM, the metamagnetic transition occurs with increase in field. The metamagnetic transition in x=0.025 starts at a \nsmall field (~ 4kOe) and then the moment increases very rapi dly. Then up to a field of \nabout 18kOe moment does not increases much. Further increment in  field assists the \nsystem to transfer into a fully ferromagnetic phase. Moment does not increase considerably up to 32kOe in x=0.05 com pound and a sudden jump in magnetization is \nobserved at ~ 32kOe, aligning th e moments parallel to the dir ection of applied field (Fig. \n3c).  02 0 4 0 6 00.00.51.01.52.0\n  M (μB/f.u.)\nH (kOe) ZFC\n FC@20kOex=0.05\nT=2.5K(a)\n \n0 2 04 06 08 00.00.51.01.52.0\nx=0.05\n  M (μB/f.u.)\nH (kOe) ZFC\n FC@60kOeT=2K(b)\n \nFIG. 4. Isothermal magnetization curve at (a) T=2.5K and (b) T=2K for Ce(Fe 1.95Si0.05)2 \ncompound. Measurements have been performe d in both increasing an d decreasing field \n(sweep rate=100Oe/sec) and the sample was cooled from 325K in zero and 60kOe (a) & \n20kOe (b) field. \n \nWhen we further decrease the temperature from 3K, one can observe dramatic change in \nmagnetization behavior at the AFM - FM transi tion (Fig. 4). An extremely sharp step is found in M(H) at T=2.5K  at a critical field (H C) 42.6kOe while the value of H C is \n38.8kOe at T=2K, for x=0.05n compound. Mome nt does not increase much beyond this \ncritical field and a plateau region in M(H) is observed which is followed by a sudden \njump in moment value. Th e height of the step ( MΔ) is 0.5 and 0.64 . . /u fBμ  at T=2.5 \nand 2K respectively. We have found that by lowering the temperature, sharper steps can \nbe realized in this compound and the critical field for appearing the steps is less at T=2K \ncompared to T=2.5K. This critical field is  not a unique quantity and depends on the \nextrinsic measurement protocol which will be shown in the following paragraph. This kind of step behavior was also observed in Ce(Fe\n1.975Ga0.025)2 compound [4] and Ru & Re \ndoped compounds [3]. A similar behavior is al so found in other systems like manganites \nand Gd 5Ge4 [12]. The sharpness of the steps in magne tization in the present case is quite \nsurprising since it is polycrysta lline in nature. This is a ne w sort of metamagnetism which \nhave been often explained by relief of strain across phase separated region [1-4] or burst like growth of FM phase with increasing fi eld [5,6] or quenched disorder [7]. The \npresence of sharp step in magnetization indicates that this may be a martensitic type behavior across the order-or der (AFM–FM) transition. \n The data has been also taken when the samp le was cooled in a field (both higher and \nlower than H\nC). While cooling the sample in 20kOe ( < HC=42.6kOe) field to T=2.5K, \nthe low field magnetization got enhanced and a smooth metamagnetic transition occurs across AFM-FM transformation rather than a sharp step as found in ZFC. Even a cooling field 60kOe (  H\nC=38.8kOe) at T=2K can not convert th e system considerably to its FM \nphase but results a smooth metamagnetic transition. Comparing the situation in the case >of Ga doping [4], it is seen that the ground st ate AFM coupling is stronger in the case of \nSi doping.  \n0.00.51.01.52.0\n0 2 04 06 08 00.00.51.01.52.00.00.51.01.52.0\n(b)M (μB/f.u.)\nH (kOe)10Oe/secT=1.9K\n  \n(c)Interrupted \nsweepT=1.9K\n  (a)\n  III II I\n  \nT=2K\n100Oe/sec\n \nFIG. 5. Isothermal magnetization curves showing the dependence of measurement \nprocedure. Field sweep rates are (a) 100Oe/sec, (b) 10Oe/sec and (c) interrupted sweep. \n \n \nThe formation of FM phase by this disc ontinuous phase transformation requires the \nnucleation of the new phase in highly locali zed regions (special sites) on which the \nheterogeneous nucleation takes place. The sm all doping concentration inside the matrix can act as one of these special  sites. Nucleation also occurs  at the grain boundaries, grain \nedges and grain corners in a polycrystalline compound. As shown in above figure, stage I \nis the incubation period in which the AFM phase is metastable. New clusters of very \nsmall sizes, which are precursor s to the final stable FM phase continuously form and \ndecompose in the matrix. The distribution of these small clusters evolves with time to \nproduce larger clusters which are more stable and therefore less  likely to revert back to \nthe matrix. It is observed that when we decrease magnetic field magnetization does not \nfollow the same path. This may be understood by assuming that some of the largest of \nthese clusters evolve into stable nuclei of FM phase and remain in the system permanently and continue to grow with drivi ng force. In stage II, the distribution of the \nFM clusters has built up into a quasi-steady state. But dramatic obs ervations were found \nacross this transition producing extremely sharp steps to reach the quasi-steady state. In stage III, rate of nucleation is decreased a nd formation of new stable phase is almost \ncomplete.  We have also shown here that the steps can be induced or can be masked by the extrinsic \nmeasurement protocol. It can be seen here that  definition of critical  field totally depends \non the extrinsic factors. In above Fig.5, M(H) data has been taken in three different ways; \n(a) 100 Oe/sec at T=2K, (b) 10 Oe/sec at T=1. 9K and (c) interrupted sweep at T=1.9K \nwhere the scan was delayed for 2 hrs each at different fields. A sharp step observed in \ncase (a) at 2K, becomes smooth at slower sw eep rate even when the temperature is \nreduced from 2 to 1.9 K. Six small and sharp steps are observed in the interrupted sweep.  The absence of steps in slow field sweep ra te can be explained from the martensitic picture. A slow change in driving forces (magnetic field) assists the system to transform \nslowly and progressively from a distorted phase to another more ordered phase. During \nthis process the system finds enough time to  overcome the elastic energy smoothly across \nthe two phases which blocks the transition up to a certain field. The influence of the \nexternal driving force confirms our presumption that this system posses a martensitic type behavior.   Many doped CeFe\n2 systems [10, 11] are found to posses a structural distortion across first \norder AFM-FM transition. We can assume that when clusters form in the matrix phase, \nan elastic-misfit strain energy is ge nerated because of volume or/and shape \nincompatibilities between the cluster and the ma trix. This strain energy acts as a barrier to \nthe nucleation [13]. The applic ation of magnetic field at lo w temperature favors the FM \nphase, in a way, a new crystal structure. Externally applied magnetic field favors FM \nfraction within the matrix and system releases  its strain energy across the transition. It \ncan be assumed here that depe nding on the experimental cond ition this release of strain \nenergy may occur in a “military” fashion. As  a consequence moments suddenly jump to \ncertain magnetization value producing ultra sharp steps across this order-order transition.  \n \n      \n  25 30 35 40 45 500.40.60.81.01.21.4\n0 2 04 06 08 00.00.51.01.52.0\n  M (μB/f.u.)\nH (kOe)}\n  \nx=0.05M (μB/f.u.)\nH (kOe)T=1.9K\ninterrupted sweep\n(a)\n \n0 2000 4000 6000 80001.001.021.041.061.081.101.121.14\n  \n42kOe40kOe\n41kOe\n38kOe\n39kOe37kOeM(t)/M(t=0)\nt (sec.)(b)\n \n \nFIG. 6. (a) Field dependence of magnetization at T=1.9K for Ce(Fe 1.95Si0.05)2 compound. \nDuring sweeping from 0 to 80kOe the following fields: 37, 38, 39, 40, 41 and 42 kOe; \nwere held for 2hrs each. Field sweep rate wa s 100Oe/sec. (b) Variations of normalized \nmagnetization with time at different holding fi elds. Arrow shows the jump in normalized \nmagnetization for the holding field of 37kOe. \n \n The interrupted sweep and growth of FM pha se during those times are demonstrated in \nFig. 6. In the interrupted sweep the field sweep rate was 100 Oe/sec. and the following \nfields (near about the metamagnetic transiti on region); 37, 38, 39, 40, 41 & 42kOe, were \nheld constant for two hrs. Interestingly six steps are found (Fig.  6a) at each of the above \nmentioned six fields indicating that there exis ts an induction period to  appear these steps. \nThis type of behavior can be compared with  the report by Wu et al. [14]. Magnetic \nrelaxation at different constant  fields can be well described by a stretched exponential of \nthis type: . Where }] ) / ( exp{ 1 )][ ( ) ( [ ) ( ) (0 0ατt H M H M H M H Mt − − − + =∞ τ is the \ncharacteristic relaxation time and α is called stretching parameter that can range between \n0 and 1. From our fitting we have obtained the value of the stretching parameter is almost \n0.5 and system having a characteristic relaxation ) (τ in a range of 1300 to 2700 sec. \ndepending on the applied field. System rela xes to its actual magnetization value for a \nparticular field during these holding times. The growth of the FM phase at different fields \nhas been shown in Fig. 6b. A jump in normalized magnetization is observed for a holding \nfield of 37kOe (arrow in Fig. 6b). The moments are relaxed to different final values for \ndifferent holding fields. As a consequence the step sizes in M(H) depend on the holding \nfields and also on the wait time (her e 2 hrs. for each  holding field).  38 39 40 41 420.0200.0240.0280.0320.0360.040\n  Magnetic viscosity, S\nH (kOe)T=1.9K\n \nFIG. 7. Field dependence of magnetic vi scosity for x=0.05 compound at T=1.9K.  \n \nMagnetic viscosity has been calculated using the relation: \n)}] (log { ) 1 (10 0 t d dM M S × = . At 2K the metamagnetic type transition starts around 36 \nkOe and up to nearly 52 kOe there is a huge change in magnetization is observed (see Fig. 4b).  The change of magnetic viscosity (S ) across this region has been shown in Fig. \n7. An increment S with increasing the field can  be clearly observed fr om this figure. At \nhigher field (above 41kOe), S seem s to have almost saturated value. While there is a huge \njump of S is found around the field 39-41 kO e region. We conclude here that the \nobserved variation of S is cons istent with the field and can attributed to the change in \nfraction of antiferro and ferromagnetic phases.    0 1 02 03 04 05 00.00.51.01.52.02.5\n5 1 01 52 02.02.12.22.3\n  M (μB/f.u.)\nH (kOe)T=50Kx=0.025\n  M (μB/f.u.)\nH (kOe) 10K\n 30K\n 50K(a)\n \n \n0 1 02 03 04 05 00.00.51.01.52.0\nx=0.0570K\n50K\n30K10K\n \n  \nH(kOe)M (μΒ/f.u.)(b)\n \n \nFIG. 8. Isothermal magnetization at vari ous higher temperatures for (a) Ce(Fe 1.985Si0.025)2 \nand (b) Ce(Fe 1.95Si0.05)2 compounds. Area of the hysteresis loops getting diminished with \nincrement in temperature.  \n  \nFig. 8 shows the high temperature behavior  of magnetization isot herms of both x=0.025 \nand 0.05 compounds. Arrows indicate increasing and decreasing field. In both the cases with increase of temperature area of the hysteresis loops gets  decreased. Area of the loop \ncan be made to zero above 50K in x=0.025 compound and 70K for x=0.05 compound.  \n \nIV. Conclusions  \nDetailed magnetization measurements ha ve been performed in Si doped CeFe 2 \npolycrystalline compounds. This work presen ts the appearance of the steps in the \nmagnetization isotherms, which can be indu ced by extrinsic experimental protocol. \nPresence of incubation time and variation of magnetic viscosity has been discussed to \nunderstand the dynamics of the system. Shar p steps across field induced AFM to FM \nphase have been attributed to the martens itic behavior during transformation. Multiple \nsteps can be achieved with proper relaxati on protocol. Reproducibility of the steps and \nstrong dependence on external parameters open up lots of opportunity to study the \ndynamics in these kinds of systems. This study also shows that there are several similarities between Ga and Si doping in CeFe\n2 with regard to the martensitic scenario \nand magnetization behavior.   \nV. References  \n1R. Mahendiran, A. Maignan, S. Hebert, C. Martin, M. Hervieu, B. Raveau, J.F. Mitchell, \nand P. Schiffer, Phys. Rev. Lett. 89, 286602 (2002). \n \n2V. Hardy, S. Majumdar, S. J. Crowe, M. R. Lees, D. McK. Paul, L. Herve, A. Maignan, \nS. Hebert, C. Martin, C. Yaicle, M. Hervieu,  and B. Raveau, Phys. Rev. B 69, 020407(R) \n(2004). \n \n3S. B. Roy, M. K. Chattopadhyay, and P. Chaddah, Phys. Rev. B 71, 174413 (2005). \n 4 Arabinda Haldar, K. G. Suresh and A.  K. Nigam, Phys. Rev. B (In Press). \n \n5V. Hardy, A. Maignan, S. Hebert, C. Yaicle , C. Martin, M. Hervieu, M. R. Lees, G. \nRowlands, D. M. Paul, and B. Raveau, Phys. Rev. B 68, 220402(R) (2003). \n \n6V. Hardy, S. Hébert, A. Maignan, C. Martin, M. Hervieu, and B. Raveau, J. Magn. \nMagn. Mater. 264, 183 (2003) . \n  \n7L. M. Fisher , A. V. Kalinov , I. F. Voloshin , N. A. Babushkina, D. I. Khomskii, Y. Zhang \nand T. T. M. Palstra, Phys. Rev. B, 70, 212411 (2004). \n \n8Olle Eriksson, Lars Nordstrom, M.S.S. Brooks, Borje Johansson, Phys. Rev. Lett., 60, \n2523 (1988). \n \n9C. Giorgetti, S. Pizzini, E. Dartyge, A. Font aine, F. Baudelet, C. Brouder, P. Bauer, G. \nKrill, S. Miraglia, D. Fruchart and J. P Kappler, Phys. Rev. B 48, 12732(1993). \n \n10S. B. Roy and B. R. Coles, J. Phys.: Condens. Matter 1, 419 (1989). \n \n11S. B. Roy and B. R. Coles, Phys. Rev. B 39, 9360 (1990). \n \n12E.M. Levin, K.A. Gschneidner, Jr.,  and V.K. Pecharsky, Phys. Rev. B 65, 214427 \n(2002). \n \n13Robert W. Balluffi, Samuel M. Allen and W. Craig Carter, Kinetics of Materials  (John \nWiely & Sons, Inc.) \n14T. Wu and J. F. Mitchell , Phys. Rev. B, 69, 100405(R) (2004). "    },    {        "title": "1307.1509v1.Magnetic_properties_of_the_nucleon_in_a_uniform_background_field.pdf",        "content": "arXiv:1307.1509v1  [hep-lat]  5 Jul 2013Magnetic properties of the nucleon in a uniform background fi eld\nThomas Primer, Waseem Kamleh, and Derek Leinweber\nSpecial Research Centre for the Subatomic Structure of Matt er,\nSchool of Chemistry and Physics, University of Adelaide, SA 5005, Australia∗\nMatthias Burkardt\nDepartment of Physics, New Mexico State University, Las Cru ces, NM 88003-8001, USA\n(Dated: August 20, 2018)\nWe present results for the magnetic moment and magnetic pola risability of the neutron and the\nmagnetic moment of the proton. These results are calculated using the uniform background field\nmethod on 323×64 dynamical QCD lattices provided by the PACS-CS collabora tion as part of the\nILDG. We use a uniform background magnetic field quantised by the periodic spatial volume. We\ninvestigate ways to improve the effective energy plots used t o calculate magnetic polarisabilities,\nincluding the use of correlation matrix techniques with var ious source smearings.\nPACS numbers: 12.38.gc,13.40.em,13.40.-f\nKeywords: magnetic moment, magnetic polarisability, latt ice QCD, background field\nI. INTRODUCTION\nThe magnetic moment and magnetic polarisability are\nfundamental properties of a particle that describe its re-\nsponsetoanexternalmagneticfield. Developingtheabil-\nitytocalculatethesepropertiesviathefirstprinciplesap-\nproach of lattice QCD is important. There are two well\nknown techniques for calculating magnetic moments on\nthe lattice. One is the three-point function method [1–3],\nwhich is used to calculate baryon electromagnetic form\nfactors that can be converted into magnetic moments by\nperforming an extrapolation to zero momentum. The\nother is the background field method [4–11], which uses\na phase factor on the gauge links to induce an external\nfield across the whole lattice. This external field causes\nan energyshift from which the magneticmoment and po-\nlarisability can be derived by making use of the following\nenergy-field relation [4, 12],\nE(B) =MN−/vector µ·/vectorB+e|B|\n2MN−4π\n2βB2+O(B3),(1)\ndefining/vector µasthemagneticmomentand βasthemagnetic\npolarisability. We note the term e|B|/2MNis the ground\nstate Landau energy. In principle, there is a tower of\nenergy levels with energy, (2 n+ 1)e|B|/2MNforn=\n0,1,2,....\nWhen deriving the background field method on a peri-\nodic lattice there arises a quantisation condition which\nlimits the available choices of magnetic field strength\nbased on the size of the lattice [5]. If the lattice is too\nsmall the field will be large and higher order terms in the\nenergy relationof Eq. (1) will begin to dominate [7]. Pre-\nvious calculations have avoided this problem by using a\nDirichlet boundary condition in a spatial dimension and\na linearised form of the phase factor, which allows for an\n∗thomas.primer@adelaide.edu.auarbitrary choice of field strength [11]. Others have used\nthe exponential phase, but instead ofcorrectingthe value\nof the field at the boundary they put the quark origin at\nthe centre of the lattice and hope that the boundary is\nfar enough away for the effects of the discontinuity to be\nsmall [6]. Using either of these methods introduces finite\nvolume errors which can be hard to predict. Our calcula-\ntion is the first to use periodic boundary conditions and\nthe quantised exponential phase factor, creating a uni-\nform magnetic field everywhere. We present results for\nboth the magnetic moment and the magnetic polarisabil-\nity of the neutron. For the proton we present only mag-\nnetic moment results because the Landau levels interfere\nwith polarisability calculations for charged particles.\nII. BACKGROUND FIELD METHOD\nWe make use of the background field method to sim-\nulate a constant magnetic field along one axis [5]. The\ntechnique is formulated on the lattice by first consider-\ning the continuum case, where the covariant derivative\nis modified by the addition of a minimal electromagnetic\ncoupling,\nDµ=∂µ+gGµ+qAµ, (2)\nwhereAµis the electromagnetic four-potential and qis\nthe charge on the fermion field. On the lattice this is\nequivalent to multiplying the usual gauge links by a sim-\nple phase factor\nU(B)\nµ(x) = exp(iaqAµ(x)). (3)\nTo obtain a uniform magnetic field along the z-axis we\nnote that /vectorB=/vector∇×/vectorA,and hence\nBz=∂xAy−∂yAx. (4)\nNote that this equation does not specify the gauge po-\ntential uniquely, there are multiple valid choices of Aµ2\nthat give rise to the same field. We choose Ax=−By\nto produce a constant magnetic field of magnitude Bin\nthezdirection. The resulting field can be checked by\nexamining a single plaquette in the ( µ,ν) = (x,y) plane,\nwhich is related to the magnetic field through the field\nstrength tensor,\n/squareµν(x) = exp/parenleftbig\niqa2Fµν(x)/parenrightbig\n, (5)\nwhich is exact for a constant background field because\nall higher order terms involve a second or higher order\nderivative. For a general plaquette at coordinates x,y\nthe result is,\nexp(−iaqBy)exp(iaqB(y+a)) = exp( ia2qB),(6)\ngiving the desired field over most of the lattice. However\non a finite lattice (0 ≤x/a≤Nx−1),(0≤y/a≤Ny−1)\nthere is a discontinuity at the boundary due to the pe-\nriodic boundary conditions. In order to fix this problem\nwe make use of the ∂xAyterm from equation (4), giving\nAythe following values,\nAy(x,y) =/braceleftBigg\n0,fory/a < N y−1,\nNyBx,fory/a=Ny−1..(7)\nThis ensures that we now get the required value at the\ny/a=Ny−1 boundary.\nThere is then the issue of the double boundary, x/a=\nNx−1andy/a=Ny−1,wheretheplaquetteonlyhasthe\nrequired value under the condition exp( −ia2qBNxNy) =\n1. This gives rise to the quantisation condition which\nlimits the choices of magnetic field strength based on the\nlattice size,\nqBa2=2πn\nNxNy, (8)\nwherenis an integer specifying the field strength in mul-\ntiples of the minimum field strength quantum.\nIII. SIMULATION DETAILS\nThese calculations use the 2+1 flavour dynamical-\nfermion configurations provided by the PACS-CS group\n[13]throughtheILDG[14]. Theseare323×64latticesus-\ning acloverfermion action andIwasakigaugeaction with\nβ= 1.9 and physical lattice spacing a= 0.0907(13) fm.\nWe use four values of the light quark hopping parameter,\nκud= 0.13700, 0.13727, 0.13754, 0.13770, corresponding\nto the pion masses mπ= 702, 572, 413, 293 MeV. The\nlattice spacing for each mass was set using the Sommer\nscale with r0= 0.49 fm. The size of the ensemble was\n320 for the two lighter masses and 400 for the heavier\nones.\nIn order to get correlation functions at four different\nmagnetic field strengths we calculated propagators at\nsix non-zero field strengths, qBa2=+0.0061, −0.0123,+0.0184, +0.0245, −0.0368,−0.0492. These correspond\nton= +1,−2,+3,+4,−6 and−8 in Eq. (8). Using the\nrelationships qd=−e/3 andqu= 2e/3 to combine up\nand down quark propagators with the appropriate field\nstrengths resulted in hadrons in fields of strength eB=\n−0.087, +0.174, −0.261,−0.345 GeV2at the physical\nlattice spacing. Unless specified otherwise we used the\ninterpolating field χ1= (uTCγ5d)uwith 100 sweeps of\nGaussian smearing at the source. We put the origin of\nthe electromagnetic gauge field at the same lattice site\nas the quark origin to ensure that the smeared source\nmaintains good overlap with the ground states.\nIt should be noted that the configurations are dy-\nnamical only in the QCD sense, there was no magnetic\nfield included when they were generated. The back-\nground field can be put on the sea quarks by perform-\ning a separate HMC calculation for each field strength,\nbut this is obviously very computationally expensive. It\nalso destroys the correlations between the different field\nstrengths which would lead to much larger errors in the\nenergy shifts used to calculate moments and polarisabil-\nities. While techniques for a re-weighting of configura-\ntions in order to correct for the background field are un-\nderexploration[15], these havenot been employedin this\nwork. Because these effects are proportional to SU(3)\nflavour symmetry breaking in the vacuum we anticipate\nthat the corrections due to the effect of the background\nfield on the sea quarks will be small.\nWe also performed an initial calculation using\nquenched gauge configurations. These were 323×40\nlattices using a FLIC fermion action and Symanzik im-\nproved gauge action at β= 4.52. There were 192\nconfigurations at seven quark masses, corresponding to\nmπ= 0.8400, 0.7745, 0.6929, 0.6261, 0.5399, 0.4353,\n0.2751 GeV. The lattice spacing was a= 0.128 fm\nand like the dynamical calculation, boundary conditions\nwere periodic for the spatial dimensions and fixed for\nthe time boundary. We used fields corresponding to\nn= 1,−2,4,−8 in the quantisation condition to save\non computation.\nIV. MAGNETIC MOMENT\nA. Formalism\nWhen a charge or system of charges with angular mo-\nmentum is placed in an external magnetic field it is en-\nergetically favourable to have its axis either aligned or\nanti-aligned with the direction of the field. The tendency\nofthe system to align with the field is proportionalto the\nmagnetic moment of the system and the strength of the\nfield.\nWe calculate zero-momentum projected correlation\nfunctions containing spin-up and spin-down components\nin the (1,1) and (2,2) positions of the Dirac matrix re-\nspectively. For a magnetic field aligned to the axis of the\nspin we see the magnetic moment manifest as a shift in3\nthe energy which has the same magnitude, but opposite\nsign, for spin-up and spin-down.\nWe make use of the sign difference in the energy shift\nbetween spin-up and spin-down in order to isolate the\nmagnetic moment term from the expansion ofthe energy.\nTaking the difference of the spins,\nδE(B) =1\n2(E↑(B)−E↓(B)) =−µB. (9)\nIn addition to the bare mass and polarisability term,\nthis difference also cancels out the Landau energy term\ne|B|/2MN. For the neutron this term should be zero be-\ncause it is proportional to the net charge. However for\nthe proton, even though taking the difference cancels out\nthe term, it can still affect the results. This is because\nwe use a standard projection to zero momentum in our\ncorrelationfunctions, but when Landaulevelsarepresent\none obtains a superposition of Landau states. There are\nproposed techniques for dealing with the Landau levels\n[16], but we have found that the effect on the magnetic\nmoment results is small, and defer this issue to a subse-\nquent investigation.\nIn terms of correlation functions there are multiple\nvalid ways of taking the spin-difference, for example fit-\nting the energy and then taking the difference or com-\nbining correlation functions and then fitting. By taking\na combination of correlation functions before fitting for\ntheenergythestatisticalerrorisgreatlyreducedandpro-\nvidesstrongconstraintsonthe fitregime. Thisisbecause\ntheerrorsarehighlycorrelatedbetweenthezeroandnon-\nzero field correlation functions, meaning the fluctuations\ndo not change significantly due to the field. The com-\nbination required for isolating the moment term can be\nwritten as,\nδE(B,t) =1\n2/parenleftbigg\nln/parenleftbiggG↑(B,t)\nG↑(0,t)G↓(0,t)\nG↓(B,t)/parenrightbigg/parenrightbigg\nfit.(10)\nThe inclusion of the bare correlation functions without\na magnetic field in this expression is not strictly neces-\nsary, but it is useful in correcting for the small statistical\ndifference between spin-up and spin-down zero-fieldener-\ngies and making the zerofield point zero by construction.\nWealsodefinespin-uptomeanalignedwiththemagnetic\nfield and spin-down to mean anti-aligned to the field so\nthat we can treat all the fields as positive in our discus-\nsion.\nB. Results\nFigure 1 shows the energy shift from the difference of\nspin-up and spin-down nucleons for the heaviest quark\nmass at all four non-zero magnetic field strengths. Both\nthe proton and the neutron show a good linear progres-\nsion over the field strengths as expected, with excellent\nplateaus. There is very little excited state contribution\ntothe energyshifts inevidence. Theneutroneffectiveen-\nergy is generally slightly smoother than the proton, withFIG. 1. Spin-difference energy shift of Eq. (10) for the heavi -\nest quark mass at all four field strengths. The proton values\nare given by the squares and the neutron values by the cir-\ncles. The shifts increase in magnitude with the strength of\nthe field.\nFIG. 2. Fits of the spin-difference energy shift to the field\nstrength at each quark mass for the proton. The solid line is\na purely linear fit to just the first two points and the dashed\nline is a linear plus cubic fit to all four points.\nsimilar results for the other quark masses. The larger\nerrors in the proton energies may be due to the effect\nof the Landau levels in the momentum projection. At\nthe two higher field strengths there tends to be a small\ndrift in the value over time, with the true plateau per-\nhaps only occurring at around time slice 23 or 24. This\nleads to a slight difference in the value of the energy shift\ndepending on the choice of fit window, however this has\nlittle effect on the magnetic moment result for reasons\ndescribed below.\nFigures 2 and 3 show the spin-difference energy shifts\nplotted against the magnetic field strength. These are fit\nto a linear coefficient which gives the magnetic moment.\nIn order to fit the largest field strength, and to a lesser\nextent the second largest, we had to include a cubic term\nin the fit. With the cubic term included all four data\npoints are fit easily. The cubic term is able to absorb\nsome variation in the energy shifts at the higher field4\nFIG. 3. Fits of the spin-difference energy shift to the field\nstrength at each quark mass for the neutron. The solid line is\na purely linear fit to just the first two points and the dashed\nline is a linear plus cubic fit to all four points.\nstrengths, which is why the drift in the effective energy\nshown in Fig. 1 doesn’t significantly affect the resulting\nmagnetic moment value.\nThis is seen in Table I, which gives values of the neu-\ntron magnetic moment for a number of fit windows. The\nsame window is used at every field strength in order to\nmaintain consistency and prevent introducing systematic\nerrors. The values agree well within errors for all but\nthe earliest fit window, suggesting that time slice 19 is\nslightly too early to fit due to excited state contamina-\ntion. This shows that the first two points are the main\ndriversofthelinearcoefficientandthereforethemagnetic\nmoment value. We also performed a purely linear fit to\nonly the first two points and found that the linear coeffi-\ncients agreed well within errors. Since the fit is naturally\nconstrained to go through zero, two non-zero field points\nare enough to give us confidence that our field strengths\nare small enough to make the higher order contributions\nnegligible.\nFigures 4 and 5 show the proton and neutron magnetic\nmoment results, compared with a three point function\ncalculation for reference [18]. Here we used values taken\nfrom fit window 20-22 because this window had good χ2\nper degree of freedom and small errors. The magnetic\nmoment is reported in units of nuclear magnetons, which\nare reached by,\nµ=−δE\neB/bracketleftbigge\n2MN/bracketrightbigg\n2MN (11)\nwhere we have started with Eq. (9) and introduced the\nelementary charge esince we actually fit the energy shift\nagainsteB, then bringin twice the physicalnucleonmass\nMNin order to get the nuclear magneton ( µN=e/planckover2pi1\n2MN),\ngiven that we are using natural units ( c=/planckover2pi1= 1).\nThe results compare favourably. The lines are chiral\nfits to the dynamical results using the approach from\n[19], and guide the anticipated trajectory to the physical\npoint. The reason the extrapolated values are smallerFIG. 4. Neutron magnetic moment as a function of pion mass\nsquared. The left most point gives the experimental value\n[17]. The dashed line is a chiral extrapolation of the dynami -\ncal points.\nFIG. 5. Proton magnetic moment as a function of pion mass\nsquared. The left most point gives the experimental value\n[17]. The dashed line is a chiral extrapolation of the dynami -\ncal points.\nin magnitude than the experimental values is expected\nto come from finite volume effects [20] as those have not\nbeen examined here.\nTABLE I. Magnetic moment values for the neutron at each κ\nvalue for a variety of fit windows.\nwindow 0.13700 0.13727 0.13754 0.13770\n19-21 -1.187(12) -1.300(13) -1.420(16) -1.486(36)\n20-22 -1.194(11) -1.317(15) -1.462(22) -1.483(30)\n21-23 -1.198(13) -1.338(20) -1.454(27) -1.500(40)\n22-24 -1.201(15) -1.343(25) -1.454(32) -1.508(49)\n20-24 -1.199(10) -1.321(15) -1.462(20) -1.485(31)5\nV. MAGNETIC POLARISABILITY\nA. Formalism\nThe magnetic polarisability is a measure of the defor-\nmation of a non-pointlike particle when it is placed in\na magnetic field. This deformation causes a change in\nthe energy which we can measure using the background\nfield method. The effect of the magnetic polarisability is\nsecond order in B. This means that at the “small” field\nstrengths we are using the effect is much smaller than\nthat due to the magnetic moment, which can make it\nhard to measure. It also makes it more important to use\nthe full exponential phase factor, since the errors intro-\nduced by the linearised form are also at order B2[21].\nTo extract the polarisability from the energy we take\nthe average of spin-up and spin-down energy shifts to re-\nmove the magnetic moment term and explicitly subtract\nthe zero-field mass. The spin-averaged energy shift is\nδEβ(B) =1\n2/parenleftbig\n(E↑(B)−E↑(0))+(E↓(B)−E↓(0))/parenrightbig\n=e|B|\n2MN−4π\n2βB2.\nThis leaves us with the polarisability term, but also with\nanother term due to the Landau energy. This energy\narises from the quantisation of orbits for charged par-\nticles in magnetic fields and can’t be isolated from the\nrelevant polarisability term. As a result it is difficult\nto calculate magnetic polarisabilities of charged parti-\ncles because there is not only the ground state Landau\nenergy but also a tower of Landau levels with energy\n(2n+ 1)e|B|\n2MN. The need for small field strengths makes\nthe Landau level problem even worse because it means\ntheLandaulevelsareclosertogether, whichmakesittake\nlonger in Euclidean time for the levels above the ground\nstate to be exponentially suppressed [16].\nThe influence of the Landau levels on the proton is\nreadily apparent in Figure 6, which shows the spin-\naverage of the energy shift due to the field. Since the ex-\nperimental value ofthe magnetic polarisabilityis approx-\nimately the same for the proton and neutron we would\nexpectthistolooksimilartotheneutronresultsinFig.7.\nInstead we see much largererrorsand no consistenttrend\nacross the field strengths. Due to the large and unpre-\ndictable systematic errors caused by this effect we are\nnot presenting values for the magnetic polarisability of\nthe proton in this first exploratory investigation. Fortu-\nnately for a neutral particle like the neutron the Landau\nterm is zero and can be ignored.\nAs with the magnetic moment we construct ratios of\ncorrelation functions which we then fit for an effective\nenergy,\nδEβ(B,t) =1\n2/parenleftbigg\nln/parenleftbiggG↑(B,t)\nG↑(0,t)G↓(B,t)\nG↓(0,t)/parenrightbigg/parenrightbigg\nfit.(12)\nIn this case the zero-field correlators are necessary to re-\nmove the bare neutron mass. Combining the correlationFIG. 6. Proton spin-averaged energy shift for the heaviest\nquark mass at all field strengths. The top line is the smallest\nfield strength, with the other three agreeing well within err ors\nfor most of the relevant time frame.\nFIG. 7. Neutron spin-averaged energy shift for the heaviest\nquark mass at all four field strengths. The magnitude of the\nshift increases with the field strength.\nfunctions before fitting is especially important in the po-\nlarisability case because the energy shift is smaller than\nthe errors on the zero-field mass. This means that if the\ncorrelated errors were not allowed to cancel before the fit\nwe would not see a clear signal for the shift due to the\npolarisability.\nB. Results\nFigure 7 gives the spin-averaged effective energy shift\nfor the heaviest quark mass considered. Unlike the spin-\ndifference case the plateau behaviour is quite poor, with\na fairly constant downward slope that begins to plateau\nonly after significant noise is appearing. Only in the\ncase of the smallest field strength does something like\na real plateau appear before the signal is lost to noise.\nThe situation is very similar at other quark masses, with\nthe errors getting larger and the noise coming earlier at\nlighter quark masses, as seen in Figure 8 for the light-6\nFIG. 8. Neutron spin-averaged energy shift for the lightest\nquark mass at all four field strengths. The magnitude of the\nshift increases with the field strength.\nFIG. 9. Spin-averaged effective mass for the neutron at κ=\n0.13727. The top line (squares) is for zero magnetic field and\nthe bottom line (circles) is with the smallest field strength\nconsidered.\nest quark mass considered. The plots show that at each\nfield strength the energy shift starts at approximately\nzero and grows with Euclidean time. Typically the lack\nof a plateau in an effective energy plot is due to the pres-\nence of excited state energies in addition to the ground\nstate. Figure 9, illustrating the bare effective mass, does\nreveal a systematic drift in the energy suggesting some\nimprovement in the interpolating field may be possible.\nIn order to check for excited state overlap and to try\nand improve the plateau behaviour we looked at different\nsources. We experimented with a number of different\nsource smearings, trying 16 and 35 sweeps in addition to\nour usual 100. We also tried a point source on the basis\nthat it should have no bias towards any shape and may\ntherefore reach the required form more quickly. Figure\n10 shows the energy shift due to the field for all smearing\nchoices at the heaviest quark mass and the smallest field.\nThe three smeared sources have different behaviour but\nagree well within errors by time slice 24, just before the\nsignal is lost to noise. The point source has large excitedstate contributions and approaches agreement with the\nother sources as the signal is lost.\nWe notice from these plots that the best plateau be-\nhaviour comes from 16 sweeps of smearing for spin-up\nand 100 sweeps of smearing for spin-down. To take ad-\nvantage of this we constructed a spin-average from the\nspin-up correlationfunction with 16 sweeps and the spin-\ndowncorrelationfunction with 100sweeps. We found the\nimproved plateau behaviour shown in Figure 11, leading\nus to investigate further the possibility of combining dif-\nferent source smearings.\nThe variational method as implemented in Ref. [22]\ninvolves using an n×ncorrelation matrix Gij(t) con-\nstructed from different source and sink smearing levels\nto solve a pair of eigenvalue equations. The right and\nleft eigenvectors uα\njandvα\nican then by used to project\nout energy eigenstates αto effectively isolate the n−1\nlowest energy states,\nGα(t) =vα\niGij(t)uα\nj (13)\nCombining correlation matrix techniques with the back-\nground field method introduces new considerations. Nor-\nmally the eigenvector analysis is performed on spin-\naveraged correlation functions because spin-up and spin-\ndown are equal up to statistics. For a background field\nFIG. 10. Energy shift at the smallest field strength. For spin -\nup we have from top to bottom: 100, 35, 16 sweeps of smear-\ning and point source, for spin-down the order is reversed.7\nFIG. 11. Spin-averaged effective energy shift for the heavie st\nquark mass at the smallest background field. The dashed\nlines are for 16 and 100 sweeps of smearing and the solid line\nis from the combination of spin-up with 16 sweeps and spin-\ndown with 100 sweeps.\ncalculation we need to consider spin-up and spin-down\nseparately, with each field strength getting its own eigen-\nvector equation to solve. This is because the Hilbert\nspace changes and one must first isolate the state before\ncombining it with states from other Hilbert spaces. In\notherwords,the eigenvectors uα\niandvβ\njarefield andspin\ndependent and a recurrence relation leading to a gener-\nalised eigenvalue equation cannot be written for combi-\nnations of spins and fields. After solving the eigenvalue\nequations we construct the same ratio as in Eq. (12) but\nusing the projected correlation functions from Eq. (13).\nWe first performed a variational analysis using a 2 ×2\ncorrelation matrix made from 16 and 100 sweeps of\nsmearing. The resultant spin-average energy shift us-\ning the ground state projected correlation functions was\napproximately equal to the original 16 and 100 sweeps\ncorrelationfunctions. The plateau behaviour wasnot no-\nticeably better than using either of the smearings alone.\nWe also tried other various combinations of sources with\ndifferent smearings including 16, 35 and 100 sweeps as\nwell as different interpolating fields in 3 ×3 and 4 ×4\ncorrelation matrices. None of these combinations was\nfound to result in a statistically significant improvement\nin the plateau behaviour.\nOne hypothesis is that the neutron is not as free from\nLandauleveleffectsaswebelieved. Theideaisthat there\nis enough play in the extended structure of the neutron\nto allow the uandd-quarks to respond to the external\nfield with a non-trivial Landau energy. Since the neutron\nhasanon-zerochargeradiusitmayhaveaLandauenergy\nbut with asmall effective charge. This wouldlead to very\nclosely spaced Landau levels which could decay smoothly\nin the manner we observe.\nSince we could not achieve good plateaus for fitting,\nwe extracted energy shifts by simply taking the value at\nthe point just before noise dominates the signal. Because\nall the different smearings agree at this point we are con-FIG. 12. The spin-averaged energy shifts as a function of the\nbackground field strength for the neutron. The solid line is\na pure quadratic fit to the first two points while the dashed\nline is a fit of all the points to a quadratic plus quartic.\nTABLE II. Summary of the main results.\nmπ a µnµp βn\nκ(MeV) (fm) ( µN) (µN) (10−4fm3)\n0.13700 702 0.1022(15) −1.19(1) 1.86(3) 1.08(8)\n0.13727 572 0.1009(15) −1.32(2) 2.01(3) 1.17(14)\n0.13754 413 0.0961(13) −1.46(2) 2.23(4) 0.98(10)\n0.13770 293 0.0951(13) −1.48(3) 2.27(5) 1.20(13)\nfident the excited states have been suppressed.\nFigure 12 shows fits of these spin-averaged energy\nshifts as a function of the field strength. In addition\nto the quadratic polarisability term we required a quar-\ntic term in order to fit the higher field strengths. This\nhigher order term is small at the heaviest mass but starts\nto become significant at the lighter masses.\nFigure 13 shows our neutron magnetic polarisability\nresults with a comparison between our quenched and dy-\nnamical calculations. The quenched and dynamical re-\nsults agree well within errors. The dashed line shows a\nfit of the dynamical results to\nβ=a+b/mπ+cln(mπ)+dm2\nπ,\nwhere the values of coefficients bandcwere set from val-\nues calculated in χPT [23] and aanddwere fit freely.\nThe extrapolated value of 1 .8±0.2×10−4fm3is well\nwithin the error bar of the experimental value, which is\nlarge due to how difficult the measurement is to perform.\nWith an improved calculation at near physical pion mass\nwe could soon expect a lattice result that verifies the chi-\nral curvature and sets a challenge for the measurement\nof the experimental value for the neutron magnetic po-\nlarisability.8\nFIG. 13. Neutron magnetic polarisability vs pion mass. The\nred point illustrates the experimental value [17]. The line\nrepresents a fit of the dynamical points using χPT.\nVI. CONCLUSION\nWe have performed the first calculations of the mag-\nnetic moment and magnetic polarisability of the neutron\nin a uniform background field. Results for the magnetic\nmomentareveryclearandagreewellwithpreviouscalcu-\nlations. The approach can be used in a precision manner\nto directly determine the magnetic moment without the\nneed for extrapolating form factors in Q2.Magnetic polarisability calculations have proved more\ndifficult due to late appearing plateaus. We have calcu-\nlated results that agree with our previous quenched re-\nsults and have an excellent approach to the experimental\nvalue. The extrapolated value at the physical pion mass\nis 1.8±0.2×10−4fm3. Here the uncertainty is statistical\nonly. Future studies in chiral effective field theory and in\nlattice QCD are needed to quantify and correct for the\nsystematic errors associated with the finite volume of the\nlattice and problems associated with the isolation of the\nground state energy shift.\nFurther study is required to improve our understand-\ning of the physical effects associated with the magnetic\npolarisability calculation. It is important to determine\nthe role of uandd-quark sectors within the neutron and\nelucidate their contributionsto the Landauenergyofthis\nneutral baryon.\nACKNOWLEDGMENTS\nWe are grateful for the generosity of the PACS-CS col-\nlaboration for providing the gauge configurations used in\nthis study. The contributions of the ILDG in making\nthe configurations accessible is also appreciated. This re-\nsearch was undertaken with the assistance of resources\nat the NCI National Facility in Canberra, Australia,\nand the iVEC facilities at the University of Western\nAustralia. These resources were provided through the\nNational Computational Merit Allocation Scheme, sup-\nported by the Australian Government. This research is\nsupported by the Australian Research Council.\n[1] G. Martinelli and C. T. Sachrajda,\nNucl.Phys. B316, 355 (1989).\n[2] T. Draper, R. Woloshyn, and K.-F. Liu,\nPhys.Lett. B234, 121 (1990).\n[3] D. B. Leinweber, R. Woloshyn, and T. Draper,\nPhys.Rev. D43, 1659 (1991).\n[4] Bernard, Draper, and Olynyk,\nPhys.Rev.Lett. 49, 1076 (1982).\n[5] Smit and Vink, Nucl.Phys. B286, 485 (1987).\n[6] H. Rubinstein, S. Solomon, and\nT. Wittlich, Nucl.Phys. B457, 577 (1995),\narXiv:hep-lat/9501001 [hep-lat].\n[7] Burkardt, Leinweber, and\nJin, Phys.Lett. B385, 52 (1996),\narXiv:hep-ph/9604450 [hep-ph].\n[8] Zhou, Lee, Wilcox, and Christensen,\nNucl.Phys.Proc.Suppl. 119, 272 (2003),\narXiv:hep-lat/0209128 [hep-lat].\n[9] Lee, Kelly, Zhou, and\nWilcox, Phys.Lett. B627, 71 (2005),\narXiv:hep-lat/0509067 [hep-lat].\n[10] Lee, Zhou, Wilcox, and Chris-\ntensen, Phys.Rev. D73, 034503 (2006),\narXiv:hep-lat/0509065 [hep-lat].[11] Lee, Moerschbacher, and Wilcox,\nPhys.Rev. D78, 094502 (2008),\narXiv:0807.4150 [hep-lat].\n[12] H. Fiebig, W. Wilcox, and R. Woloshyn,\nNucl.Phys. B324, 47 (1989).\n[13] Aoki, et al. (PACS-CS Collabora-\ntion), Phys.Rev. D79, 034503 (2009),\narXiv:0807.1661 [hep-lat].\n[14] M. G. Beckett, etal., Comput. Phys.Commun. 182, 1208\n(2001).\n[15] W. Freeman, A. Alexandru, F. Lee, and M. Lujan, PoS\nInternational Lattice Conference 2012 (2012).\n[16] B. Tiburzi and S. Vayl, (2012),\narXiv:1210.4464 [hep-lat].\n[17] Beringer, et al. (Particle Data Group), Phys.Rev. D86,\n010001 (2012).\n[18] Boinepalli et al., Phys.Rev. D74, 093005 (2006),\narXiv:hep-lat/0604022 [hep-lat].\n[19] D. B. Leinweber and A. W.\nThomas, Phys.Rev. D62, 074505 (2000),\narXiv:hep-lat/9912052 [hep-lat].\n[20] J. Hall, D. Leinweber, and\nR. Young, Phys.Rev. D85, 094502 (2012),\narXiv:1201.6114 [hep-lat].9\n[21] Guerrero, Wilcox, and Christensen, PoS LAT-\nTICE2008 , 150 (2008), arXiv:0901.3296 [hep-lat].\n[22] M. Mahbub, W. Kamleh, D. Leinweber, P. Moran, and\nA. Williams, AIP Conf.Proc. 1432, 261 (2012).[23] V. Bernard, N. Kaiser, U. G. Meissner,\nand A. Schmidt, Z.Phys. A348, 317 (1994),\narXiv:hep-ph/9311354 [hep-ph]."    },    {        "title": "1301.3071v2.Polarised_Electromagnetic_wave_propagation_through_the_ferromagnet__Phase_boundary_of_dynamic_phase_transition.pdf",        "content": "arXiv:1301.3071v2  [cond-mat.stat-mech]  9 Feb 2014Polarised electromagnetic wave\npropagation through the\nferromagnet: Phase boundary\nof dynamic phase transition\nMuktish Acharyya\nDepartment of Physics, Presidency University,\n86/1 College Street, Calcutta-700073, India\nE-mail:muktish.physics@presiuniv.ac.in\nAbstract: The dynamical responses of ferromagnet to the propagating elec -\ntromagnetic field wave passing through it are modelled and studied he re by\nMonte Carlo simulation in two dimensional Ising model. Here, the electr o-\nmagnetic wave is linearly polarised in such a way that the direction of ma g-\nnetic field is parallel to that of the magnetic spins. The coherent spin -cluster\npropagating mode is observed. The time average magnetisation ove r the full\ncycle (time) of the field defines the order parameter of the dynamic phase\ntransition. Depending on the value of the temperature and the amp litude\nof the propagating magnetic field wave, a dynamical phase transitio n is ob-\nserved. The transition is detected by studying the temperature d ependences\nofthevarianceofthedynamicorder parameter, thederivative of thedynamic\norder parameter and the dynamic specific heat. The phase bounda ry of the\ndynamic transitions are drawn for two different values of the wave le ngth\nof the propagating magnetic field wave. The phase boundary is obse rved to\nshrink (inward) for shorter wavelength of the EM wave. The signat ure of the\ndivergence of the relevant length scale is observed at the transitio n point.\nPACS Nos:05.50.+q, 05.70.Ln, 75.30.Ds, 75.30.Kz, 75.40.G b\nKeywords: Ising ferromagnet, Monte Carlo simulation, Pola rised\nelectromagnetic wave, Dynamic phase transition\n1I. Introduction.\nThedyamical response ofIsing ferromagnettoatimedependent m agnetic\nfield has become an active field of research [1]. The hysteretic respo nses\nand the nonequilibrium dynamic phase transitions are two main points o f\nattention. The scaling behaviour [2] of hysteresis loop area with the am-\nplitude, frequency of the sinusoidally oscillating magnetic field is the ma in\noutcome of the research. Another interesting aspect is the none quilibrium\ndynamic phase transition which has produced variety of interesting results\nand prompted the researchers to take continuous attention in th is field. His-\ntorically, some important observations like (i) divergences of dynam ic specific\nheat and relaxation time near the transition point [3], (ii) divergence o f the\nrelevant length-scale near the transition point [4], (iii) studies regar ding the\nexistence of tricritical point [5, 6], (iv) the relation with the stochas tic reso-\nnance [5] and the hysteretic loss [7], enriched the field and establishe d that\nthe dynamic transition has similarity to the well-known equilibrium therm o-\ndynamic phase transition. Very recently, a surface dynamic phase transition\n[8] is observed in kinetic Ising ferromagnet driven by oscillating magne tic\nfield. The dynamic phase transition was detected also experimentally [9] in\nultrathin Co film on Cu(001) system by surface magneto-optic Kerr effect.\nThe direct excitation of propagating spin waves by focused ultrash ort optical\npulses are investigated recently [10]. The transient behaviour of th e dynam-\nically ordered phase in uniaxial Cobalt film is also studied experimentally\n[11].\nThis dynamic phase transition is also observed in other magnetic mode ls.\nThe off-axial dynamic phase transition was observed [12] in the aniso tropic\nclassical Heisenberg model and in the XY model [13]. The multiple (surf ace\nand bulk) dynamic transition was observed [14] in the classical Heisen berg\nmodel. The multiple dynamic transition was found[15] also in the Heisen-\nberg ferromagnet driven by polarised magnetic field. The dynamic tr an-\nsition was observed [16] in the kinetic spin-3/2 Blume-Capel model an d in\nthe Blume-Emery-Griffith model[17] by meanfield calculations. The dyn amic\nphasetransitionwasstudiedbyMonteCarlosimulation[18]andbymea nfield\ncalculation[19] in Ising metamagnet.\nIt may be noted here, that all the studies mentioned so far, were d one\nby sinusoidally oscillating magnetic field which was uniform over the spac e\n(lattice) at any instant of time. In those studies, the spatio-temp oral varia-\ntion of applied magnetic field was not considered. One such spatio-te mporal\n2variation of applied magnetic field would be the propagating magnetic fi eld\nwave. In reality, if the electromagnetic wave passes through the f erromagnet,\nthe varying (with space and time) magnetic field coupled with the spin, will\naffect the dynamic natureof thesystem. Here also dynamic transit ion will be\nobserved. Very recently, it is reported briefly [20] that propagat ing magnetic\nfield wave would lead to dynamical phase transition in Ising ferromagn et. A\npinned phase and a phase of coherent motion of spin-clusters were observed\nrecently [21] in random field Ising model, swept by propagating magne tic\nfield wave. Here the nonequilibrium dynamic phase transition is athermal\nand tuned by quenched random (field) disorder. A rich dynamical ph ase\nboundary (with four different phases) was also drawn. A dynamic sy mmetry\nbreaking breathing and spreading transitions [22] are also found re cently in\nferromagnetic film irradiated by spherical electromagnetic wave.\nHere, in this paper, the nonequilibrium dynamic phase transition is stu d-\niedextensively intwo dimensional Isingferromagnetswept bypolaris edprop-\nagating electromagnetic field wave. The technique employed here is M onte\nCarlo (MC) simulation. The phase boundary of the dynamical phase t ran-\nsition is drawn in this study. The paper is organised as follows: The mod el\nand the MC simulation technique are discussed in section-II, the num erical\nresults are reported in section-III and the paper ends with a summ ary, in\nsection-IV.\nII. Model and Simulation.\nTheHamiltonian(timedependent)representingthetwodimensional Ising\nferromagnet (having uniform nearest neighbour interaction) in pr esence of\na polarised propagating electromagnetic field wave (having spatio-t emporal\nvariation) can be written as,\nH(t) =−JΣs(x,y,t)s(x′,y′,t)−Σh(x,y,t)s(x,y,t). (1)\nThes(x,y,t) represents the Ising spin variable ( ±1) at lattice site ( x,y)\nat timeton a square lattice of linear size L.J(>0) is the ferromagnetic\n(taken here as uniform) interaction strength. The summation in th e first\nterm represents the Ising spin-spin interaction and is carried over the nearest\nneighbours only. The h(x,y,t) is the value of the magnetic field (at point\n(x,y) and at any time t) of the propagating electromagnetic wave. It may\nbe noted here that the electromagnetic wave is linearly polarised in su ch a\n3way that the direction of magnetic field is parallel to that of the spins . For\na propagating magnetic field wave h(x,y,t) takes the form\nh(x,y,t) =h0cos(2πft−2πy/λ) (2)\nTheh0,fandλrepresent the amplitude, frequency and the wavelength\nrespectively of the propagating electromagnetic field wave which pr opagates\nalong the y-direction. In the present simulation, a L×Lsquare lattice is\nconsidered. The boundary condition, used here, is periodic in both t he (x\nandy) directions. The initial ( t= 0) configuration, is chosen as the half of\nthe total number (selected randomly) of spins are up ( s(x,y,t= 0) = +1).\nThis configuration of spins, corresponds to the high temperature disordered\nphase. The spins are updated randomly (a site ( x,y) is chosen at random)\nand spin flip occurs (at temperature T) according to the Metropolis rate[23]\n(W)\nW(s→ −s) = Min[exp( −∆E/kBT),1], (3)\nwhere∆Eisthechangeinenergyduetothespinflipand kBistheBoltzmann\nconstant. L2such random updates of spins constitutes the unit time step\nhere and is called Monte Carlo Step per spin (MCSS). Here, the value o f\nmagnetic field ismeasured inthe unit of J. Andthe temperature ismeasured\nin the unit of J/kB. The dynamical steady state is reached by cooling the\nsystem slowly in small step ( δT= 0.02 here) of temperature, from the high\ntemperature, dynamically disordered configuration. This particula r choice is\nacompromisebetweenthecomputationaltimeandtheprecisioninme asuring\nthe transition temperature. The frequency of the propagating m agnetic field\nwave was taken f= 0.01 throughout the study. The total length of the\nsimulation is2 ×105MCSS andfirst 105MCSS transient datawere discarded.\nThe data are taken by averaging over 105MCSS. In some cases, near the\ntransition points, averaging was done over 2 ×105MCSS, after discarding\ninitial 2×105MCSS. Since the frequency of the propagating field is f= 0.01,\nthe complete cycle of the field requires 100 MCSS. So, in 105MCSS, 103\nnumbers of cycles of the propagating field are present. The time av eraged\ndata over the full cycle (100 MCSS) of the propagating field are fur ther\naveraged over 1000 cycles.\n4III. Results.\nIn this study, a square lattice of size L= 100 is considered. The steady\nstate dynamical behaviours of the spins are studied here. The amp litude,\nfrequency and the wavelength of the propagating wave are taken h0= 0.6,\nf= 0.01 andλ= 25 respectively. The magnetic field is propagating along\nthe y-direction (vertically upward in the graphs). The temperatur e of the\nsystem is taken T= 1.50. The configuration of the spins, at any instant of\ntimet= 100100 MCS, are shown in Fig-1(a). Here, it is noted that, the\nclusters of spins are formed in strips and these strips move cohere ntly as\ntime goes on. The propagation of the spin-strips are clear in Fig-1(b ), where\nthe snapshot was taken at instant t= 100125 MCSS. The similar study is\nperformed at a lower temperature T= 1.26 (with all other parameters of the\npropagating field remain same). Here, the spin clusters are observ ed to be\nformed in such shapes which are not like the strips (as observed in th e case of\nhigher temperature T= 1.50 , mentioned above). This is shown in Fig-1(c),\nat any instant t= 100100 MCSS. These irregularly shaped spin-clusters are\nobserved to propagate (along the direction of propagating magne tic field),\nwhich is clear from Fig-1(d) (for t= 100125 MCSS).\nTo show the propagations of these spin-clusters, the instantane ous line\nmagnetisation m(y,t) = (/integraltexts(x,y,t)dx/L) was plotted against yat any par-\nticular instant t= 100100 MCSS. This is shown in Fig-2(a) (compare with\nFig-1(a)). The periodic variation of m(y,t) alongy-direction is found. This\nwas observed to propagate (see Fig-2(b) and compare with Fig-1( b) when\nshown at different time t= 100125MCSS. It may benoted here, that the line\nmagnetisation is periodic (with y) at any instant of time t. This is also peri-\nodic in time tat any position y. The oscillation is symmetric about m(y) = 0\nline(forhigher temperature T= 1.50). Here, thetimeaverage magnetisation\nover a full cycle of the propagating field is Q=f\nL/integraltext /integraltextm(y,t)dydt, becomes\nzero (due to symmetric oscillation about m(y) = 0 line). This corresponds\nto a dynamically symmetric phase.\nNow, for lower temperature T= 1.26, the spatio-temporal periodicity, of\nthe line magnetisation, is lost. The symmetric-oscillation (about m(y) = 0\nline) is lost here. This corresponds to a dynamically symmetry-broke n phase.\nAs a consequence, the time averaged magnetisation over a full cyc le of the\npropagating field, becomes nonzero. These are shown in Fig-2(c) a nd Fig-\n2(d) (may be compared with fig-1(c) and fig-1(d) respectively). B ut the\nspin-clusters were observed to propagate in this case. So, as the temperature\n5decreases, Qbecomes nonzero (lower temperature) from a zero value (higher\ntemperature). This Qdefines the order parameter of the dynamic phase\ntransition.\nThe temperature variations of the dynamic order parameter Q, its vari-\nance<(δQ)2>are studied. The dynamic energy is E=f/contintegraltextH(t)dtand the\ndynamic specific heat is C=dE\ndT. The derivatives are calculated numerically\nby using the three points central difference formula[24]. All these q uanti-\nties are calculated statistically over 1000 different samples. The tem perature\nvariations of Q,dQ\ndT,<(δQ)2>andCare studied for two different values of\nthe amplitude of the propagating electromagnetic field wave and are shown\nin Fig-3. As the temperature decreases, Qstarts to grow from zero and near\nthe transition point it becomes nonzero. Near the transition tempe ratures,\nthe<(δQ)2>andCshow sharp peak anddQ\ndTshow a sharp dip. From the\nfigure it is also evident that the transition occurs at lower temperat ure (Td)\nfor higher values of the field amplitude ( h0). In this case, for λ= 25, the\ntransitions occur at Td= 1.88 andTd= 1.29 forh0= 0.3 andh0= 0.6 re-\nspectively. These values of the transition temperatures are obta ined fromthe\nposition of sharp dips of thedQ\ndTand corresponding sharp peaks of <(δQ)2>\nandCshown in Fig-3. Collecting all the values of the transition temper-\natures (Td) (depending on the values of h0), the comprehensive dynamical\nphase boundary is obtained.\nThis dynamic transition temperature ( Td) was observed to depend on the\nwave length ( λ) of the propagating magnetic field wave. The temperature\ndependences of Q,dQ\ndT,<(δQ)2>andCare studied and shown in Fig-4,\nfor two different values of λ(= 25 and 50). From the figure, it is clear that,\ntransition occurs at higher temperature (with same h0) for higher value of\nthe wavelength ( λ). To be precise, for h0= 0.3, the transitions occur at\nTd= 1.88 andTd= 1.94 forλ= 25 and λ= 50 respectively. Here also, the\nvalues of the transition temperatures are obtained from the posit ion of sharp\ndips of thedQ\ndTand corresponding sharp peaks of <(δQ)2>andC(shown\nin Fig-4). So, the dynamical phase boundary should shift depending on the\nvalue ofλ.\nIn Fig-5, the dynamical phase boundaries are drawn for two differe nt\nvaluesof λ(=25and50),intheplaneformedby Tdandh0. Itisobservedthat\nthe boundary shrinks inward (region of lower Tandh0) as the wavelength\nof the propagating magnetic field decreases.\nThe dynamic phase transition, mentioned above, is associated with t he\n6divergences of relevant length scale. For this reason, the L2<(δQ2)>\nis studied as the function of temperature T. Is is found that the peak of\nL2<(δQ)2>(observed at Td) increases as Lincreases. This is shown in\nFig-6. This result is quite conclusive to say that there exists the dive rging\nlength scale associated with the dynamic phase transition. It may be noted\nhere, that this method was successfully employed [4] to show the div erging\nlength scale, associated with the dynamic transition, in Ising ferrom agnet\ndriven by oscillating (but not propagating) magnetic field.\nIV. Summary.\nThe dynamical responses of a ferromagnet to a polarised electrom agnetic\nwave are modelled and studied here by Monte Carlo simulation in two di-\nmensional Ising ferromagnet. In the steady state, the coheren t motion (in\npropagating mode) of spin clusters was observed. The time averag e magneti-\nsation over the full cycle of the propagating EM wave is a measure of the\norderparameterinthedynamicphasetransitionobserved here. T hedynamic\nphasetransitionobservedhereseemstobeofcontinuoustypean dfoundtobe\ndependent ontheamplitudeandthewave lengthofthepropagating polarised\nEM wave. Hence, a phase boundary (transition temperature as a f unction\nof the amplitude) is drawn for two different values of the wavelength s of EM\nwave. The phase boundary is found to shrink (towards the lower va lues of\nthe temperature and amplitude of field) for shorter wavelength.\nThesignatureofthedivergence ofrelevant lengthscalenearthet ransition\nis also observed here. This observation in the case of dynamic trans ition is\nanalogous to that observed in equilibrium critical phenomenon revea ling the\ngrowthofcritical correlation. Itwouldbeinteresting toknowtheu niversality\nclass of this dynamic phase transition. To know the universality class , one\nhas to estimate precisely the critical exponents, through a syste matic study\nof scaling analysis.\nAcknowledgments: The library facilities provided by the University of\nCalcutta is gratefully acknowledged.\n7References\n1. B. K. Chakrabarti, M. Acharyya, Rev. Mod. Phys. , 71 (1999) 847; See\nalso, M. Acharyya, Int. J. Mod. Phys. C, 16 (2005) 1631\n2. M. Acharyya and B. K. Chakrabarti, Phys. Rev. B 52 (1995) 6550;\nSee also, M. Acharyya and B. K. Chakrabarti, in Annual reviews of\ncomputational physics , Ed. D. Stauffer, (World Scientific, Singapore),\nVol.-1, (1994) 107\n3. M. Acharyya, Phys. Rev. E 56 (1997) 2407\n4. S. W. Sides, P. A. Rikvold and M. A. Novotny, , Phys. Rev. Lett. 81\n(1998) 834\n5. M. Acharyya, Phys. Rev. E 59 (1999) 218\n6. G. Korniss, P. A. Rikvold, M. A. Novotny, Phys. Rev. E, 66 (2002)\n056127\n7. M. Acharyya, Phys. Rev. E , 58 (1998) 179\n8. H. Park and M. Pleimling, Phys Rev Lett 109 (2012) 175703.\n9. O. Jiang, H. N. Yang, G. C. Wang, Phys Rev B 52 (1995) 14911; Q.\nJiang, H. N. Yang and G. C. Wang, J. Appl. Phys. 79 (1996) 5122.\n10. Y. Au et al, Phys. Rev. Lett. ,110(2013) 097201\n11. A. Berger et al, Phys. Rev. Lett 111(2013) 190602\n12. M. Acharyya, Int. J. Mod. Phys. C 14 (2003) 49.\n13. H. Jung, M. J. Grimson, C. K. Hall, Phys Rev B 67 (2003) 094411.\n14. H. Jung, M. J. Grimson, C. K. Hall , Phys Rev E 68 (2003) 046115.\n15. M. Acharyya, Phys. Rev. E. 69 (2004) 027105\n16. M. Keskin, O. Canko, B. Deviren, Phys. Rev. E 74 (2006) 011110\n817. U. Temizer, E. Kantar, M. Keskin, O. Canko, J. Magn. Magn. Mater.\n320 (2008) 1787\n18. M. Acharyya, J. Magn. Magn. Mater. , 323 (2011) 2872\n19. M. Keskin, O. Canko, M. Kirak, Phys. Stat. Solidi B, 244 (2007) 3775;\nB. Deviren, M. Keskin, Phys. Lett. A 374 (2010) 3119\n20. M. Acharyya, Physica Scripta , 84 (2011) 035009\n21. M. Acharyya, J. Magn. Magn. Mater. , 334 (2013) 11\n22. M. Acharyya, J. Magn. Magn. Mater. , 354 (2014) 349.\n23. K. Binder and D. W. Heermann, 1997, Monte Carlo Simulation in\nStatistical Physics (Springer Series in Solid State Sciences) (New Yo rk:\nSpringer)\n24. C. F. Gerald and P. O. Weatley, 2006, Applied Numerical Analysis\n(Reading, MA: Addison-Wesley); J. B. Scarborough, 1930, Numer ical\nMathematical Analysis (Oxford: IBH)\n90204060(a)80100\n0204060801000204060(b)80100\n020406080100\n0204060(c)80100\n0204060801000204060(d)80100\n020406080100\nFig-1.The motion of spin-clusters of down spins (shown by black dots),\nswept by propagating magnetic field wave, for different values of (a ) Time\n= 100100 MCSS, T=1.5 and h0= 0.6 (b) Time = 100125 MCSS, T= 1.5\nandh0= 0.6 (c) Time = 100100 MCSS, T= 1.26 andh0= 0.6 (d) Time =\n100125 MCSS, T= 1.26 andh0= 0.6.\n10-1-0.500.51\n0102030405060708090100(a)\ny∗∗∗∗\n∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n∗\n∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n∗\n∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••\n-1-0.500.51\n0102030405060708090100(b)\ny∗∗∗∗∗∗∗\n∗\n∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n∗\n∗\n∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••\n-1-0.500.51\n0102030405060708090100(c)\ny∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••\n-1-0.500.51\n0102030405060708090100(d)\ny∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••\nFig-2.Thepropagationoffield ( •)andthelinemagnetisation( ∗)forvarious\nvalues of (a) Time = 100100 MCSS, T=1.5 and h0= 0.6 (b) Time = 100125\nMCSS,T= 1.5 andh0= 0.6 (c) Time = 100100 MCSS, T= 1.26 and\nh0= 0.6 (d) Time = 100125 MCSS, T= 1.26 andh0= 0.6.\n1100.40.8\n11.41.82.2Q\nT(a)\n∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗∗∗ ∗∗ ∗∗ ∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗ ∗∗∗ ∗∗ ∗∗ ∗ ∗∗ ∗∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗\n• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • ••• ••• •••• ••• ••• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • •• • • • • • • • • •• • • •• •• •• •• •• ••••••••••••••••••••••••••• •• • • •• • • • • •• • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • •\n-35-155\n11.41.82.2dQ\ndT\nT(b)∗ ∗ ∗ ∗ ∗∗∗ ∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗∗ ∗ ∗ ∗∗∗ ∗∗ ∗∗ ∗ ∗∗ ∗∗∗ ∗ ∗ ∗ ∗ ∗∗∗ ∗ ∗ ∗∗∗∗∗ ∗∗∗∗∗ ∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗ ∗∗∗∗∗∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ • • • • • • • • • • • • • • • • • • • • ••• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••• • • • • • • • • • • • • ••• • • • • • • • • • • • • • • • • • • ••• • • • ••• • • • • • • • • • • • • • • ••• • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • •• •••• •• •••••• •••••• •\n•\n••••••••••• •••• • •• • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •\n00.020.040.06\n11.41.82.2< δQ2>\nT(c)\n∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ ∗∗∗ ∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • •• • •• •••••••••••••••••\n•\n•••••••• •••• • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •\n0246\n11.41.82.2C\nT(d)\n∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗ ∗∗•••••••••••••••••••••• •••• •••• • • ••••\nFig-3.Thetemperature(T) dependences ofthe(a) Q, (b)dQ\ndT, (c)<(δQ)2>\nand (d)C, for two different values of h0forpropagating magnetic field wave\nhavingf= 0.01 andλ= 25. In each figure, h0= 0.3(∗) andh0= 0.6(•).\n1200.40.8\n1.51.71.92.1Q (a)\nT∗ ∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗ ∗∗ ∗\n••••••••••••••••••••••••••••••• •• •• •\n-10-6-2\n1.51.71.92.1dQ\ndT(b)\nT∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗∗∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗ ∗∗ ∗ ∗ ∗ ••••••••••••\n•\n•••••••••••••••• •• •• • • •\n00.010.020.030.04\n1.51.71.92.1<(δQ)2> (c)\nT∗ ∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n∗∗∗∗∗∗∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ • •••••••••••••\n•\n••••••• • • •• • • • • • • • • • •\n0.20.81.42\n1.51.71.92.1C (d)\nT∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ••••••••••••••••••••••••••••••••••••\nFig-4.Thetemperature(T) dependences ofthe(a) Q, (b)dQ\ndT, (c)<(δQ)2>\nand (d)C, for two different values of λforpropagating magnetic field wave\nhavingf= 0.01 andh0= 0.3. In each figure, λ= 25(∗) andλ= 50(•).\n130.20.40.60.81.01.21.41.61.8\n0.511.52h0\nTQ/negationslash= 0Q= 0\n∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n••••••••••••••••\nFig-5.The phase diagram for dynamic phase transition by propagating\nmagnetic field wave for two different values of wavelengths, λ= 25(•) and\nλ= 50(∗). Here,f= 0.01.\n140100200300400500\n0.81.01.21.41.6L2<(δq)2>\nTL= 50(∗)L= 100(•)\n∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗∗∗∗∗∗∗\n∗\n∗∗∗∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ • • • • • • • •••••••••\n•••••• • • • • • • • • • • • • • • • • • •\nFig-6.The plot of temperature ( T) versusL2<(δQ)2>for different system\nsizes (L). Here,h0= 0.6,λ= 25 and f= 0.01.\n15"    },    {        "title": "1803.07188v1.Magnetic_susceptibility__nanorheology__and_magnetoviscosity_of_magnetic_nanoparticles_in_viscoelastic_environments.pdf",        "content": "Magnetic susceptibility, nanorheology, and magnetoviscosity of magnetic nanoparticles\nin viscoelastic environments\nPatrick Ilg, Apostolos E.A.S. Evangelopoulos\nSchool of Mathematical, Physical Sciences and Computational Sciences, University of Reading, Reading, RG6 6AX, UK\n(Dated: August 6, 2021)\nWhile magnetic nanoparticles suspended in Newtonian solvents (ferro\ruids) have been intensively\nstudied in recent years, the e\u000bects of viscoelasticity of the surrounding medium on the nanoparticle\ndynamics are much less understood. Here we investigate a mesoscopic model for the orientational dy-\nnamics of isolated magnetic nanoparticles subject to external \felds, viscous and viscoelastic friction\nas well as the corresponding random torques. We solve the model analytically in the overdamped\nlimit for weak viscoelasticity. By comparison to Brownian Dynamics simulations we establish the\nlimits of validity of the analytical solution. We \fnd that viscoelasticity does not only slow down\nthe magnetization relaxation, shift the peak of the imaginary magnetic susceptibility \u001f00to lower\nfrequencies and increase the magnetoviscosity, it also leads to non-exponential relaxation and a\nbroadening of \u001f00. The model we study also allows to test a recent proposal for using magnetic\nsusceptibility measurements as a nanorheological tool using a variant of the Germant-DiMarzio-\nBishop relation. We \fnd for the present model and certain parameter ranges that the relation of\nthe magnetic susceptibility to the shear modulus is satis\fed to a good approximation.\nPACS numbers: 75.75.Jn, 75.50.Mn, 83.80.Hj, 05.40.JcarXiv:1803.07188v1  [cond-mat.soft]  19 Mar 20182\nI. INTRODUCTION\nMagnetic nanoparticles suspended in Newtonian solvents are known as ferro\ruids [1]. Since their conception in the\n1960s, they have attracted considerable attention, due to their interesting properties, most notably the magnetoviscous\ne\u000bect [2], and the breadth of applications [3]. Ferro\ruids are reasonably well understood today, at least for not too\nstrong interactions [4, 5]. More recently, new magnetic materials like soft magnetic elastomers and ferrogels have been\nsynthesized [6{10]. These new materials combine properties of both traditional elastomers and ferro\ruids, because\nthe suspending medium is neither a purely elastic solid nor a Newtonian liquid. Magnetic nanoparticles are also\nused more and more often in a number of biomedical applications such as drug delivery and magnetic hyperthermia\n[11, 12], magnetorelaxometry [13], and biosensors [14]. In these instances, magnetic nanoparticles are embedded in\nmore complex, non-Newtonian environments. Magnetorelaxometry is an example of a powerful biomedical application,\nwhere analyzing the rotational di\u000busion of magnetic nanoparticles is used as a diagnostic tool [13]. Another example is\nthe rotational microrheology of magnetic endosomes that has been employed to study the local viscoelasticity in cells\n[15]. However, for an accurate interpretation of all these experimental data a detailed knowledge of the magnetization\ndynamics in a complex environment is needed.\nThe need for a better understanding of viscoelastic e\u000bects on the dynamics of magnetic colloids has prompted a\nnumber of experimental and theoretical investigations. Barrera et al. [16] were one of the \frst to measure the dynamic\nsusceptibility of magnetic nanoparticles suspended in \ruids of varying viscoelasticity, which was tuned via progressive\ngelation of a gelatin solution. Recent experiments along similar lines have been performed and analyzed in terms\nof di\u000berent, mostly phenomenological models [17]. While these models are found to \ft the experimental results on\nthe imaginary part of the dynamic susceptibility reasonably well, they lead to di\u000berent conclusions about an e\u000bective\nshear modulus. Similarly, experimental results on magneto-optical transmission on Nickel-nanorods suspended in\ndi\u000berent viscoelastic liquids were interpreted within Kelvin-Voigt and Maxwell models, which, however, resulted in\ncalculated shear moduli di\u000berent from the macroscopic measured ones [18]. A careful study on dynamic susceptibility\nmeasurements has shown that for various polymeric solvents a variant of the Germant-DiMarzio-Bishop model can be\nused to infer the complex shear modulus [19]. Such nanorheological measurements are of great practical use as well\nas theoretical interest. However, their theoretical foundation in the present context remains unclear.\nDespite a number of recent e\u000borts, the dynamics of magnetic nanocolloids in viscoelastic environments remains\nconsiderably less well-studied than ferro\ruids. Macroscopic approachs in terms of nonequilibrium thermodynamics\nestablished the hydrodynamic equation of ferrogels [20, 21]. Some \frst steps in coarse graining a molecular model of\npolymer chains permanently attached to magnetic particles are presented in [22]. Mesoscopic models for the rotational\ndynamics of single-domain magnetic nanocolloids have been investigated for some time (see [23{25] and references\ntherein), where viscoelasticity e\u000bects are modeled in terms of non-Markovian Langevin or Brownian dynamics. More\nrecently, the need for modeling the viscoelastic medium by a generalized Maxwell model [26] together with a proper\nthree-dimensional treatment of rotations has been recognized [27].\nHere, we study the same model for the three-dimensional rotational dynamics of single-domain magnetic nanopar-\nticles in a generalized Maxwell \ruid as proposed in Ref. [27]. We solve the model analytically in case of weak\nviscoelasticity. Brownian dynamics simulations establish the range of validity of the analytical result and provide nu-\nmerical results for the full range of viscoelasticity e\u000bects. Our results show that the Germant-DiMarzio-Bishop model\nworks surprisingly well for certain parameter ranges, which provides a theoretical justi\fcation for the nanorheological\nmeasurements advocated in Ref. [19].\nThe paper is organized as follows. The model that we study here is presented in Sect. II, including the overdamped\nlimit as well as a description of its numerical solution via Brownian Dynamics simulations. The analytical as well as\nnumerical results are presented in Sect. III. Finally, some conclusions are o\u000bered in Sect. IV.\nII. KINETIC MODEL OF MAGNETIC NANOPARTICLES IN VISCOUS AND VISCOELASTIC MEDIA\nConsider an ensemble of statistically independent, non-interacting colloidal magnetic nanoparticles (i.e. the ultra-\ndilute limit). Let the orientation of the particle be described by the three-dimensional unit vector u(u2= 1). For\ngiven angular velocity !, the orientational motion is given by _u=!\u0002u. The equation of motion is given by the\nangular momentum balance I_!=T, whereIis the moment of inertia of the particle and Tthe sum of all torques\nacting on the particle.\nHere, we want to consider torques exerted by an external magnetic \feld Has well as torques exerted by the\nsurrounding viscous and viscoelastic medium. In particular, we make the following assumptions: (i) rigid dipole\napproximation; i.e. the dipole moment remains always parallel to the particle orientation, m=muwheremdenotes\nthe magnetic moment of a single colloidal particle. With this assumption we neglect internal N\u0013 eel relaxation, so we\nare restricted to large enough particles and not too high frequencies [1]. Under this assumption, the torque due to an3\nexternal magnetic \feld His given by TH=\u0000L\b =mu\u0002H, where \b =\u0000mu\u0001His the Zeeman potential energy\nandL=u\u0002@=@uthe rotational operator. We also assume (ii) that we can model the torque due to the viscous\nsolvent by rotational friction \u0018= 8\u0019\u0011sa3and corresponding white noise with thermal energy kBT, where\u0011sis the\nsolvent viscosity and athe hydrodynamic radius of the particle. (iii) Furthermore, we assume that we can model the\nviscoelastic contribution by the retarded friction \u0010(t) and corresponding random torque R. (iv) Finally, we assume\ndilute conditions such that inter-particle interactions are negligible. Also e\u000bects due to hydrodynamic memory are\nassumed to be irrelevant, since we will later consider long enough time scales.\nUnder these assumptions we arrive at the generalized Langevin equation for the rotational motion\nI_!(t) =mu(t)\u0002H(t)\u0000\u0018[!(t)\u0000\n(t)]\u0000Zt\n0dt0\u0010(t\u0000t0)[!(t0)\u0000\n(t0)] +p\n2kBT\u0018_Wt+R(t): (1)\nThe torque due to friction is only experienced if the particle's angular velocity !di\u000bers from the angular velocity\n\nof the surrounding medium. White noise associated with the viscous solvent is modeled by the three-dimensional\nWiener process Wt. The viscoelastic contribution exerts not only a friction torque described by the retarded rotational\nfriction\u0010(t) but also a random torque RwithhRi= 0. The \ructuation-dissipation theorem (FDT) requires that\nthese torques are related by hR(t)R(t0)i=kBTI\u0010(t\u0000t0) with Ithe three-dimensional unit matrix [28].\nIn the following, we consider the viscoelastic contribution as a single-mode Maxwell model where the memory kernel\ncan be expressed as \u0010(t) = (\u00100=\u001cM)e\u0000t=\u001cM, where\u001cMis the relaxation time and \u00100the rotational friction coe\u000ecient\nof the particle in the viscoelastic medium. In this case, we conclude from FDT that Ris exponentially correlated,\ni.e. a three-dimensional Ornstein-Uhlenbeck process, hR(t)R(t0)i=kBT\u00100=\u001cMexp [\u0000(t\u0000t0)=\u001cM]I, i.e.Robeys the\nstochastic di\u000berential equation\ndR(t) =\u00001\n\u001cMR(t)dt+BRdWR\nt; (2)\nwithBR=p2kT\u0010 0=\u001cM. Here, WR\ntis another three-dimensional Wiener process, statistically independent of Wt.\nIn the limit \u001cM!0, we obtain \u0010(t) =\u00100\u000e(t), i.e. a Newtonian bath and the system reduces to the corresponding\nmodel of ferro\ruids [5] with an e\u000bective rotational friction coe\u000ecient \u0018+\u00100. The Maxwell model has frequently been\nemployed to model the e\u000bect of viscoelastic media on colloid dynamics [24]. For polymer solutions, however, it has\nbeen argued that a generalized Maxwell (also termed Je\u000brey's) model that we employ here is more appropriate to\nproperly describe the high-frequency behavior [26, 29].\nA. Extended variable formalism\nThe stochastic integro-di\u000berential equation (1) with (2) can be converted into a system of stochastic di\u000berential\nequations by introducing the auxiliary variable\nz(t) =\u0000Zt\n0dt0\u0010(t\u0000t0)[!(t0)\u0000\n(t0)] +R(t); (3)\nwhich can be interpreted as the e\u000bective torque due to friction and noise of the viscoelastic environment.\nFor the exponential memory kernel, the time derivative is particularly simple and we arrive at the following system\nof stochastic di\u000berential equations\ndu= d!\u0002u (4)\nId!= [\u0000L\b\u0000\u0018(!\u0000\n) +z]dt+p\n2kBT\u0018dW (5)\ndz=\u00001\n\u001cM[z+\u00100(!\u0000\n)]dt+BRdWR: (6)\nDe\fne the probability density F(u;!;z;t). ThenFobeys the Fokker-Planck equation corresponding to the stochas-\ntic di\u000berential equations (4-6)\n@\n@tF=\u0000L\u0001(!F)\u00001\nI@\n@!\u0001[(\u0000L\b\u0000\u0018(!\u0000\n) +z)F] +kBT\u0018\nI2@2\n@!2F\n\u0000@\n@z\u0001[(\u00001\n\u001cMz\u0000\u00100\n\u001cM(!\u0000\n))F] +kBT\u00100\n\u001c2\nM@2\n@z2F: (7)4\nIn the absence of \row, \n= 0, the stationary solution to Eq. (7) is given by the Boltzmann distribution\nF0(u;!;z) =Z\u00001exp [\u0000\f\b(u)\u00001\n2\fI!2\u0000z2\n2z2eq]; (8)\nwith normalization constant Z,\f= 1=(kBT), andz2\neq=\u00100=(\f\u001cM). The equilibrium probability density factorizes\nand therefore the equilibrium magnetization is given by the Langevin function and is independent of the (retarded)\nfriction, as it should. Furthermore, the equilibrium Gaussian \ructuations of the angular velocity !and torques zare\nspeci\fed byh!\u000b!\fi0= (\fI)\u00001\u000e\u000b\fandhz\u000bz\fi0=z2\neq\u000e\u000b\f.\nB. Overdamped limit\nBy construction, the model presented above exhibits three relaxation times: the inertial relaxation time \u001cI=I=\u0018,\nthe Brownian relaxation time of rotational di\u000busion due to the viscous solvent \u001cD=\u0018=(2kBT), and the relaxation\ntime of the viscoelastic contribution \u001cM.\nFor colloidal particles in general and magnetic nanoparticles in particular, the usual condition \u001cI\u001c\u001cD;\u001cMholds\n[30]. Therefore, we consider in the following the overdamped limit \u001cI!0, i.e. Eq. (1) for I_!!0. Note, however, that\nmeasurements of the magnetic susceptibility have shown the in\ruence of inertia e\u000bects at su\u000eciently high frequencies\n[31]. In the overdamped limit, we can eliminate the angular velocity !from Eq. (5) as independent variable and\narrive at the reduced set of stochastic di\u000berential equations\ndu= [\n\u00001\n\u0018L\b +1\n\u0018z]\u0002udt+BdW\u0002u (9)\ndz= [\u00001 +q\n\u001cMz+q\n\u001cML\b]dt+BR(dWR\u0000pqdW); (10)\nwhereB=p\n2kBT=\u0018 andq=\u00100=\u0018the ratio of the friction coe\u000ecients. Note that taking the overdamped limit\nleads to the appearance of correlated noise in the auxiliary variable z. The Fokker-Planck equation for the reduced\nprobability density f(u;z;t) corresponding to Eqs. (9) and (10) reads [32]\n@\n@tf=\u0000L\u0001[(\n\u00001\n\u0018L\b +1\n\u0018z)f] +kT\n\u0018L2f\n\u0000@\n@z\u0001[(\u00001 +q\n\u001cMz+q\n\u001cML\b)f] +kBT\u00100\n\u001c2\nM(1 +q)@2\n@z2f\u0000q2kBT\n\u001cM@\n@z\u0001Lf: (11)\nEquation (11) agrees with the corresponding Fokker-Planck equation in [27]. We note that in the limit q!0, we\nrecover the kinetic model for (non-interacting) ferro\ruids [5]. In the absence of \row, \n= 0, we \fnd the equilibrium\nBoltzmann distribution\nf0(u;z) =Z\u00001\n1exp [\u0000\f\b(u)\u0000z2=(2z2\neq)] (12)\nas stationary solution to Eq. (11). Note that f0can also be obtained from F0, Eq. (8), when integrated over !, as it\nshould be and Z1is the corresponding normalization constant. We emphasize that stationary properties should not\nbe a\u000bected by taking the overdamped limit. However, this is not the necessarily the case in some earlier works where\nonly a viscoelastic bath is present and additional manipulations need to be invoked [24].\nFor later use, we introduce the Fokker-Planck operator Lfrom Eq. (11) via @f=@t =Lf. Separating the e\u000bects\nof internal dynamics L0, the external \feld Lhand external \row L\nallows the decomposition L=L0+Lh+L\n.\nFurthermore, we de\fne the adjoint operators LybyR\ndudzALf=R\ndudzfLyAfor an arbitrary function A=A(u;z).\nUsing integration by parts we derive from (11) the explicit form of the adjoint operators,\n\u001cDLy\n0A=1\n2L2A\u00001 +q\n\u001c\u0003\nMz\u0003\u0001@\n@z\u0003A+\u000fz\u0003\u0001LA\n+1 +q\n\u001c\u0003\nM@2\n@z\u00032A\u00002\u000f@\n@z\u0003\u0001LA (13)\n\u001cDLy\nh(t)A=(u\u0002h(t))\u0001[1\n2LA\u0000\u000f@\n@z\u0003A] (14)\n\u001cDLy\n\n(t)A=\n\u0003(t)\u0001LA; (15)5\nwhere we introduced the dimensionless quantities z\u0003=z=zeq;\n\u0003=\u001cD\n, the ratio of relaxation times \u001c\u0003\nM=\u001cM=\u001cD,\nh=mH=(kBT) and\u000f=p\nq=(2\u001c\u0003\nM) =\u001cDzeq=\u0018. Note that the parameter \f= 4\u000f2was used in Ref. [27] and interpreted\nas \\springiness\".\nC. Brownian Dynamics simulations\nFor comparison to analytical results, we also perform Brownian dynamics simulations of the model equations.\nThereby we avoid certain assumptions detailed below that we employ for the analytical calculations.\nWe have implemented the stochastic di\u000berential equations (9) and (10) using a \frst-order Euler scheme as well as\na second order Heun scheme [32]. For the current purpose, both schemes give identical results for small enough time\nsteps. For simplicity, we here only describe the Euler scheme.\nUsing the dimensionless quantities introduced above with t\u0003=t=\u001cD, we de\fne the dimensionless increment of the\nangular velocity \u0001 !\u0003=\u001cD\u0001!by\n\u0001!\u0003= (\u00001\n2u\u0002h+\u000fz\u0003)\u0001t\u0003+ \u0001W\u0003(16)\nwhere \u0001 W\u0003denotes a dimensionless three-dimensional Wiener increment, i.e. a three-dimensional Gaussian random\nvariable with zero mean and variance \u0001 t\u0003. With the help of \u0001 !\u0003, the increments of the variables uandz\u0003over a\nshort time step \u0001 t\u0003are given by\n\u0001u= (\n\u0003\u0001t\u0003+ \u0001!\u0003)\u0002u (17)\n\u0001z\u0003=\u00001\n\u001c\u0003\nMz\u0003\u0001t\u0003\u00002\u000f\u0001!\u0003+q\n2=\u001c\u0003\nM\u0001W\u0003\u0003(18)\nwhere \u0001 W\u0003\u0003is another three-dimensional Gaussian random variable with zero mean and variance \u0001 t\u0003, statistically\nindependent of \u0001 W\u0003.\nWe found \u0001 t\u0003= 10\u00003to be small enough to obtain identical results to the corresponding Heun scheme for the\nobservables of interest. An ensemble of N= 5\u0002105independent realizations of uandz\u0003was usually used in order\nto obtain reliable estimates for mean values.\nIII. RESULTS\nA. Short-time rotational di\u000busion\nSince the orientation uis restricted to the three-dimensional unit sphere, rotational di\u000busion is de\fned for short\ntimes only from the relation\nh(u(t)\u0000u(0))2i0= 4Dt; t\u001c1=(2D) (19)\nwith the rotational di\u000busion coe\u000ecient D. Averagesh\u000fi0are taken with respect to the equilibrium initial ensemble f0,\nEq. (12). In terms of the auto-correlation function C(t) =hu(t)\u0001u(0)i0, the mean-squared orientational displacement\ncan be expressed as h(u(t)\u0000u(0))2i0= 2(1\u0000C(t)). The short-time behavior of C(t) can be computed from the Taylor\nseriesC(t) = 1 + _C(0)t+1\n2C(0)t2+O(t3), where we used the fact that C(0) =hu2i0= 1. The \frst and second order\nterms are calculated from _C(0) =h[Lyu]\u0001ui0and C(0) =h[(Ly)2u]\u0001ui0, respectively. Some details of the calculation\ncan be found in appendix A. In the absence of external \row we \fnd\nh(u(t)\u0000u(0))2i0=2t\n\u001cD\u0000\u0012\n1 + 2\u000f2+1\n2hL1(h)\u0013\u0012t\n\u001cD\u00132\n+O(t3); (20)\nwhereL1(h) = coth(h)\u00001=hdenotes the Langevin function. Interestingly, the \frst order term is independent of the\nviscoelastic bath and external magnetic \feld that both contribute only from the second order on. Thus, in view of\nthe de\fnition (19), we conclude that D= 1=(2\u001cD) is identical to the viscous di\u000busion on short time scales t\u001c\u001cD.6\nB. Magnetization relaxation\nIn magnetorelaxometry, the relaxation of the magnetization is analyzed after a strong ordering \feld is switched o\u000b\n[13]. The present treatment does not include internal N\u0013 eel relaxation that is important for a proper interpretation\nof the corresponding experimental results. Nevertheless, we here provide a detailed analysis of the e\u000bect of medium\nviscoelasticity on the magnetization relaxation for magnetically hard nanoparticles. Our results would therefore be\nuseful for separating out viscoelasticity e\u000bects due to the biological environment from the magnetization relaxation\nsignal.\nAssume that \n= 0 and a strong ordering magnetic \feld has been applied along the z-direction for a su\u000eciently\nlong time so that the Boltzmann equilibrium (12) is established. At time t= 0, the external \feld is switched o\u000b\ninstantaneously. Then, the reduced magnetization M=M sat=huziobeys the ordinary di\u000berential equationd\ndthuzi=\nhLy\n0uziwith initial condition huzi(0) = 1. De\fne a0=huzi;a1=hozi;a2=hqziwitho\u000b=\u0000\u000f\u000b\f\ru\fz\u0003\n\r,q\u000b=\n\u000f\u000b\f\r\u000f\u0016\f\u0015u\u0015z\u0003\n\rz\u0003\n\u0016. With these quantities, the \frst members of the moment hierarchy read\n\u001cD_a0=\u0000a0+\u000fa1 (21)\n\u001cD_a1=\u0000\u00151a1+\u000fa2+ 4\u000fa0 (22)\n\u001cD_a2=\u0000\u00152a2+\u000fa3+ 6\u000fa1\u00004\n\u001c\u0003\nMa0\u00008\u000f2a0; (23)\nwhere\u0015n= 1 +n=\u001c\u0003\nM+ 2n\u000f2. Note that only in the limit \u000f!0, i.e. in the absence of viscoelastic contributions, the\nmoment system truncates at the \frst order. In general, we are faced with an in\fnite hierarchy for which an exact\nsolution is unknown.\nTo make further progress analytically, we look for solutions as power series in \u000f,\nak(t) =1X\nn=0\u000fna(n)\nk(t): (24)\nInserting the expansion (24) into Eqs. (21)-(23) and matching equal orders of \u000fwe \fnd forO(\u000f0) a single-exponential\ndecay,a(0)\n0(t) =e\u0000t=\u001cD, corresponding to the purely viscous limit of the model. Matching also next orders in \u000fwe\n\fnd (see Appendix B for some details of the calculations)\nhuzi(t) =e\u0000t=\u001cD\u001a\n1 +q\u0014t\n\u001cD+\u001c\u0003\nM\u0010\n\u00001 +e\u0000t=\u001cM\u0011\u0015\u001b\n+O(\u000f3): (25)\nIt is interesting to note that the magnetization relaxation from Eq. (25) is not simply given by a superposition of\ntwo exponentials. In other words, the viscous and viscoelastic contributions to the relaxation can not be considered\nindependent. Figures 1 and 2 show the magnetization relaxation huzi(t) on a semi-logarithmic scale. The analytical\nformula (25) is compared with results from Brownian dynamics simulations of the model. Deviations from the single-\nexponential behavior of the purely viscous model are obvious. Increasing viscoelastic contributions slows down the\nmagnetization relaxation more and more. The approximate formula (25) provides an accurate description for the\nwhole relaxation process when q.0:2, whereas for larger values of qonly the early stages of the relaxation are\ncaptured correctly by Eq. (25) while for late stages the magnetization is underestimated.\nC. Magnetic susceptibility\nWe are interested in the linear response of the system to an externally applied weak magnetic \feld H(t). When the\ndimensionless magnetic \feld h(t) =mH(t)=(kBT) is small, a \frst order perturbation expansion gives\nf(t) =feq+1\n\u001cDZt\n0dt0e(t\u0000t0)L0(u+\u001cD\n\u0018u\u0002z)\u0001h(t0)feq+O(h2) (26)\nwhere we assumed that the system was initially in equilibrium, f(0) =feq, i.e. by Eq. (12) for \b = 0. For simplicity\nof notation, we here suppress the arguments ( u;z) of the probability density fandfeq. The induced magnetization\nM(t) =nmR\ndudz uf(t) is therefore linearly related to the applied \feld when his small enough, M\u000b(t) =Rt\n0dt0\u001f\u000b\f(t\u0000\nt0)H\f(t0) with\u001f\u000b\f(t) =nm2\nkT\u001cDhu\u000betL0(u+\u001c\n\u0018u\u0002z)\fieq. With the help of the adjoint operator Ly\n0de\fned in Eq. (13)\nand with _u=Ly\n0u, the susceptibility tensor can be written as \u001f\u000b\f(t) =\u0000nm2\nkT\u001cDd\ndthu\u000b(t)u\f(0)ieq. Since the system7\n0 2 4 6 810-310-210-1100\nFIG. 1. Magnetization relaxation M(t)=Msat=huzi(t) after switching o\u000b a strong ordering \feld. Parameters are chosen as\nq= 0:2 and\u001c\u0003\nMvarying from \u001c\u0003\nM= 0:5 to 5 as indicated in the legend. Symbols and solid lines correspond to simulation and\nanalytical results from Eq. (25), respectively. For comparison, the solid line is the Debye law exp [ \u0000t=\u001cD].\n10-210-110000.20.40.60.81\n(a)\n0 2 4 6 810-310-210-1100\n(b)\nFIG. 2. Magnetization relaxation M(t)=Msat=huzi(t) as in Fig. 1 but for parameters \u001c\u0003\nM= 1 andq= 0:2;0:4;1 from bottom\nto top. Panel (a) and (b) show the same data but on di\u000berent axis scales.\nis isotropic in the absence of an external \feld, \u001f\u000b\f(t) =\u001f(t)\u000e\u000b\f. With the one-sided Fourier-transform, ~ \u001f(!) =R1\n0dt0\u001f(t)e\u0000i!t, the frequency-dependent complex susceptibility takes the usual form [28]\n~\u001f(!) =\u001f0\u0000\u001f0i!Z1\n0dthu(t)\u0001u(0)ieqe\u0000i!t(27)\nwhere\u001f0=nm2=(3kBT) is the static (zero-frequency) susceptibility.\nFor smallq, we \fnd that C(t) =hu(t)\u0001u(0)ieqobeys the same di\u000berential equation (21)-(23) with the same initial con-\ndition and therefore C(t) is also given by Eq (25). Note that this is a special case of the general \ructuation-dissipation\nrelation between relaxation and correlation functions [28]. From the Laplace transform ~C(s) =R1\n0dtC(t)e\u0000stthe\ncomplex susceptibility can be obtained via ~ \u001f(!) =\u001f0[1\u0000i!~C(i!)] as\n~\u001f(!)=\u001f0=1 +iq\u001cM!\n1 +i\u001cD!\u0000iq\u001c\u0003\nM\u001c1!\n1 +i\u001c1!\u0000iq\u001cD!\n(1 +i\u001cD!)2(28)\nwhere 1=\u001c1= 1=\u001cD+ 1=\u001cMis the e\u000bective relaxation time of the combined viscous and viscoelastic e\u000bect. Introducing8\nreal and imaginary part, ~ \u001f=\u001f0\u0000i\u001f00, we \fnd\n\u001f0(!)=\u001f0=1 +q\u001c\u0003\nM(\u001cD!)2\n1 + (\u001cD!)2\u0000q\u001c\u0003\nM(\u001c1!)2\n1 + (\u001c1!)2\u00002q(\u001cD!)2\n[1 + (\u001cD!)2]2(29)\nand\n\u001f00(!)=\u001f0=\u001cD!(1\u0000q\u001c\u0003\nM)\n1 + (\u001cD!)2+q\u001c\u0003\nM(\u001c1!)\n1 + (\u001c1!)2+q\u001cD!(1\u0000(\u001cD!)2)\n[1 + (\u001cD!)2]2(30)\nFrom Eq. (30) we \fnd that the imaginary part \u001f00is no longer given by a single Lorentzian as in the Debye model.\nIn qualitative agreement with experimental observations [17, 19], the location of the loss peak moves towards lower\nfrequencies as the in\ruence of viscoelasticity increases. At the same time, the height of the peak decreases and the\nwidth increases. All these features are seen in the experiments [17, 19] and are described by Eq. (30). The same\nconclusions have been reached in Ref. [27] with the help of an e\u000bective \feld approximation and numerical solutions\nof the moment system (21)-(23). We compare the analytical formula in Eqs. (29) and (30) to results of Brownian\ndynamics simulations shown in Fig. 3. We \fnd that results are relatively insensitive to the precise value of \u001c\u0003\nMbetween\n0:5 and 2:0, whereas corresponding variation in the value of qleads to signi\fcant changes in the susceptibilities. The\nanalytical formulae (29), (30) we \fnd to be accurate for q.0:5.\n10-210-110010110200.20.40.60.81\n(a)\n10-210-110010110200.20.40.60.81\n(b)\nFIG. 3. Real and imaginary part of the complex susceptibility as a function of dimensionless frequency \u001cD!of the applied\nmagnetic \feld. Left panel shows results for q= 0:5 and di\u000berent values of \u001cM, whereas in the right panel the value \u001c\u0003\nM= 1:0 was\n\fxed and di\u000berent values for the parameter qwere chosen. Symbols and dashed lines correspond to simulation and analytical\nresults from Eqs. (29, 30), respectively. For comparison, the solid black line shows the Debye susceptibility.\nD. Magnetic Nanorheology\nIn Ref. [19], Roeben et al. suggest to transfer the Germant-DiMarzio-Bishop (GDB) model for the dielectric per-\nmittivity to the magnetic susceptibility, thereby relating the magnetic to the mechanical response,\n~\u001f(!)\u0000\u001f1\n\u001f0\u0000\u001f1=1\n1 +i!~\u001ce\u000b(!)=1\n1 +K~G(!): (31)\nIn the last equation, we introduced the complex shear modulus ~Gfrom ~G(!) =i!~\u0011(!) and related the complex\nviscosity ~\u0011to the e\u000bective relaxation time via ~ \u001ce\u000b(!) =~\u0018(!)=(2kBT) =K~\u0011(!), whereK= 4\u0019a3=(kBT).\nThe idea of magnetic nanorheology proposed in [19] in this context is to use measurements of the magnetic sus-\nceptibilities ~ \u001f=\u001f0\u0000i\u001f00in order to infer information on mechanical properties of the environment where the\nnanoparticles perform their rotational relaxation. Reformulating Eq. (31), we \fnd K~G(!) =1\n\u001f\u0003(!)\u00001, where9\n\u001f\u0003=~\u001f(!)\u0000\u001f1\n\u001f0\u0000\u001f1=\u001f0\nN\u0000i\u001f00\nNand\u001f0\nN= (\u001f0\u0000\u001f1)=(\u001f0\u0000\u001f1) and\u001f00\nN=\u001f00=(\u001f0\u0000\u001f1). De\fning the storage\n(G0) and loss modulus ( G00),~G=G0+iG00, we arrive at\nG0\nGDB(!) =1\nK\u0014\u001f0\nN\n(\u001f0\nN)2+ (\u001f00\nN)2\u00001\u0015\n(32)\nG00\nGDB(!) =1\nK\u001f00\nN\n(\u001f0\nN)2+ (\u001f00\nN)2: (33)\nNote that Eqs. (32) and (33) correct typos in the corresponding Eqs. (8) and (9) of Ref. [19]. In the Debye limit,\n\u001f0\nN=\u001f0=\u001f0= 1=[1 + (\u001cD!)2] and\u001f00\nN=\u001f00=\u001f0=\u001cD!=[1 + (\u001cD!)2], we \fnd from Eqs. (32) and (33) the known result\nG0= 0 andG00=\u001cD!=K =\u0011s!of a Newtonian \ruid.\nFor the model we study here, we worked out the magnetic susceptibilities in Eqs. (29) and (30) to leading order in\nthe ratio of friction coe\u000ecients q. Inserting these expressions into (32) and (33), noting that \u001f1= 0, we arrive at the\nGDB model for the storage and loss modulus,\nG0\nGDB(!) =\u0011s!q\u001cM!\n(1 +\u001cM)2+ (\u001cM!)2+O(q2) (34)\nG00\nGDB(!) =\u0011s!\u0014\n1 +q1 +\u001c\u0003\nM\n(1 +\u001c\u0003\nM)2+ (\u001cM!)2\u0015\n+O(q2): (35)\nOn the other hand, we can directly test Eqs. (34) and (35) by comparing them to the true mechanical modulus\ncorresponding to the model under study. Since we assumed two independent frictional torques for the viscous and\nviscoelastic contribution we have ~\u0018e\u000b(!) =\u0018+~\u0010(!). The retarded friction of the viscoelastic contribution was taken\nto be of the form \u0010(t) = (\u00100=\u001cM)e\u0000t=\u001cMwhich leads to ~\u0010(!) =\u00100=[1+i!\u001cM] and therefore the e\u000bective relaxation time\n~\u001ce\u000b(!) =~\u0018(!)\n2kT=\u001cD\u0014\n1 +q\n1 +i\u001cM!\u0015\n(36)\nUsing the above de\fnition of the complex modulus ~G(!) =i!~\u001ce\u000b(!)=K, we \fnd that the e\u000bective storage and loss\nmodulus of the environment is given by\nG0(!) =\u0011s!q\u001cM!\n1 + (\u001cM!)2(37)\nG00(!) =\u0011s!\u0014\n1 +q\n1 + (\u001cM!)2\u0015\n(38)\nThese expressions for the generalized Maxwell (sometimes called Je\u000brey's) model are similar but di\u000berent from\nEqs. (34), (35). We note that viscoelasticity e\u000bects vanish for \u001cM!0 and we recover from both expressions the\nresults for a Newtonian solvent with and additional factor 1 + qto account for the increased friction. For a gen-\neral viscoelastic contribution, both expressions converge in the high-frequency limit to those of a Newtonian \ruid as\nlim!!1G00(!) =G1[1 +c=(\u001cM!)2], whereG1=\u0011s!is the in\fnite-frequency shear modulus and c= 1 +qand\nc=q(1+\u001c\u0003\nM) for the present model and the GDB assumption, respectively. Viscoelasticity leads to a non-zero value of\nthe high-frequency storage modulus lim !!1G0(!) =q=(K\u001c\u0003\nM). Higher order terms in qappear in the GDB expression\nbut not in Eq. (37). For low frequencies we \fnd lim !!0G00(!) =\u0011s![1 +bq]!0 whereb= 1 andb= 1=(1 +\u001c\u0003\nM)\nfor the present model and the GDB assumption, respectively. The storage modulus vanishes quadratically for low\nfrequencies, lim !!0G0(!) = (q=K)\u001c\u0003\nMb2(\u001cD!)2. For low values of qfor which Eqs. (34), (35) apply, good agreement is\nfound between the two expressions. This is especially true for the loss modulus, whereas the storage modulus shows a\nsomewhat larger discrepancy (not shown). Figure 4 shows a comparison of G00(!) as given by Eqs. (38) and (35) for\nlarger values of q. Due to the relative insensitivity of the susceptibilites on \u001c\u0003\nMfor intermediate values of q, also the\nvalue ofG00is only weakly a\u000bected and can be reliably estimated from the GDB model in this regime (see Fig. 4a).\nWe \fnd, however, some discrepancies for intermediate frequencies when the parameter qincreases. But overall, we\n\fnd that the GDB model estimates the loss modulus for the present range of parameters quite well.\nE. Magnetoviscosity\nThe magnetoviscous e\u000bect, i.e. the change of apparent viscosity due to an externally applied magnetic \feld, is not\nonly of great theoretical interest but plays also an important role for various applications [5]. How is the magnetovis-\ncous e\u000bect altered when the carrier medium is viscoelastic?10\n10-210-110010110210-210-1100101(a)\n10-210010210-210-1100101(b)\nFIG. 4. Left and right panel show the dimensionless loss modulus G00(!)\u001cD=\u0011sas a function of \u001cD!for the same conditions\nas in Fig. 3. Symbols are results for the GDB assumption, Eq. (35), using the numerical results for the susceptibilites. Lines\ncorrespond to the generalised Maxwell model, Eq. (38).\nTo address this question, consider the situation where the system is exposed to a constant magnetic \feld H.\nIn addition, a steady shear \row V= _\ryexis applied, so that the vorticity is constant and \n=\u0000( _\r=2)ez. The\nnonequilibrium stationary state attained under the action of a constant magnetic \feld and \nis not known in general.\nFor weak \row,j\u001cD\nj=\u001cD_\r=2\u001c1, we make the following ansatz\nf(u;z) =f0(u;z)[1 + ( u\u0000hui0)\u0001a+z\u0001b+O( _\r2)] (39)\nwheref0(u;z) is the equilibrium probability density (12) in the presence of a magnetic \feld but in the absence of \row.\nThe ansatz (39) satis\fes the normalisation conditionR\ndudzf(u;z) = 1. The unknown vectors a;bare independent\nofuandzand are assumed to be \frst order in the \row rate and therefore small. Otherwise positivity of fis not\nguaranteed. As similar procedure has been used successfully for structure-forming ferro\ruids [33].\nWith the ansatz (39), we can calculate arbitrary moments in terms of a;b, e.g.\nhui=L1^h+ (L2\u0000L2\n1)^h^h\u0001a+L1\nha (40)\nwhere h=h^hwithh2=h2and^hthe unit vector in the direction of the external \feld. Here we use Lj(h) =hPj(u\u0001^h)i0,\nwithL1(h) = coth(h)\u00001=hthe Langevin function introduced above and L2(h) = 1\u00003L1(h)=h. We also \fndhzi=z2\neqb\nandhoi=\u0000L1zeq^h\u0002b.\nFrom the Fokker-Planck equation, we can derive the following time evolution equations for the lowest order moments\n\u001cDd\ndthui=\u0000hui+1\n2(h\u0000huui\u0001h) +\u000fhoi+\u001cD\n\u0002hui (41)\n\u001cMd\ndthzi=\u0000(1 +q)hzi\u0000qkBThui\u0002h (42)\nIn the stationary state, the left hand side is zero. Expressing the moments with the help of the ansatz (39) we arrive\nat an algebraic system of equations for the unknown aandb. Solving this system of equations (see Appendix C) and\ninserting the result into (40) we \fnd the steady-state orientation due to an external \feld and weak \row as\nhui=L1^h+L2\n1\nh\n2\u0010\n2+L2\n3\u0000qL2\n1\n1+p\u0011\u001cD\n\u0002^h: (43)\nForq!0, Eq. (43) reduces to the known result of Shliomis model of ferro\ruids [5]. The magnetization component\nperpendicular to the \feld direction leads to a viscous torque that manifests itself in the rotational viscosity [1, 5]\n\u0011rot=M?H\n2 _\r=MsatkBT\u001cD\n2mhu?ih\n\u001cD_\r= (1=2)nkBT\u001cDhu?ih\n\u001cD_\r(44)11\nUsing\u001cD=\u0018=(2kT) we \fndnkBT\u001cD=n\u0018=2 = 4n\u0019\u0011 sa3= 3\u001e\u0011swith\u001e=n(4=3)\u0019a3the volume fraction. With these\nrelations we arrive at the following expression for the rotational viscosity\n\u0011rot=3\n2\u0011s\u001e3L2\n1\n2 +L2\u0014\n1\u00003qL2\n1\n(1 +q)(2 +L2)\u0015\u00001\n: (45)\nTherefore, we \fnd that to \frst order in qand in the \row rate the magnetoviscosity is independent of \u001cMand depends\nonly onq. The maximum viscosity contribution is\n\u00111\nrot= lim\nh!1\u0011rot=3\n2\u0011s\u001e(1 +q): (46)\nThus, the maximum viscosity increase due to a viscoelastic bath is lager by a factor 1 + qthen the pure viscous case\nsimply by the increased rotational friction that the colloid experiences.\nFor vanishing magnetic \feld we \fnd \u0011rot(h= 0) = 0, i.e. no viscous contribution in the absence of a magnetic \feld.\nFor weak \felds h\u001c1, we \fnd that the rotational viscosity increases as\n\u0011rot=3\n2\u0011s\u001e\u0012h2\n6\u0000h4\n36(1 +q)+O(h6)\u0013\n(47)\nFigure 5 shows the rotational viscosity as a function of the dimensionless applied magnetic \feld h. We \fnd that the\nrotational viscosity increases proportionally to the additional viscoelastic friction. The value of \u001c\u0003\nM, on the contrary,\nhas little e\u000bect on the steady state rotational viscosity in this parameter range. For comparison, we also performed\nBrownian dynamics simulations of Eqs. (17), (18) with \u001cD_\r= 0:1. We veri\fed that identical results are obtained for\nlower values of \u001cD_\r. The numerical results from Brownian dynamics simulations are in good agreement with Eq. (45)\nfor small values of q. For larger values of q, the analytical result (45) signi\fcantly underestimates the true value of the\nrotational viscosity. Also the dependence of \u0011roton the viscoelastic relaxation time \u001cMis not captured by Eq. (45).\nFrom Fig. 6, we clearly see that increasing \u001c\u0003\nMleads to a corresponding decrease of \u0011rot. The same qualitative trend\nwas found in Ref. [25] where only two-dimensional rotations in a purely viscoelastic bath were considered.\n246810121400.20.40.60.811.2(a)\n246810121400.511.52\n(b)\nFIG. 5. Scaled rotational viscosity \u0011rot=(3\u0011s\u001e=2) as a function of the Langevin parameter h. Symbols denote the results of\nBrownian dynamics simulations and dashed lines are the analytical results (45). The solid black line is the result for vanishing\nviscoelasticity. Panel (a): Fixed value q= 0:2 and\u001c\u0003\nMvarying from 0 :5 up to 5. Panel (b): From bottom to top q= 0:2;0:4;1:0\nwith\u001c\u0003\nM= 1:0.\nIV. CONCLUSIONS\nIn the present contribution, we study a microscopic model of the non-Markovian dynamics of magnetic nanocol-\nloids in a viscoelastic environment that can be described by the combination of a Newtonian and viscoelastic medium12\n246810121400.511.52\nFIG. 6. Same as right panel of Fig. 5 but for q= 1.\nproposed in Ref. [27]. When viewed as an extension of the basic model of ferro\ruid dynamics, the additional viscoelas-\nticity e\u000bects are described in this model by two dimensionless parameters: (i) the ratio q=\u00100=\u0018of friction coe\u000ecients\nof the nanocolloid in the viscoelastic and in the viscous component and (ii) the corresponding ratio \u001c\u0003\nM=\u001cM=\u001cDof\nrelaxation times. Due to the inherent non-linearity of three-dimensional rotational motion, we are only able to \fnd\nanalytical solutions of the model for weak viscoelasticity, i.e. small values of q. We also test the analytical results\nagainst Brownian Dynamics simulations.\nWe \fnd that viscoelasticity leads to a slowing down of the magnetization relaxation compared to the purely viscous\ncase, showing a non-exponential decay that is mainly controlled by the ratio qof friction coe\u000ecients and rather\ninsensitive to the precise value of scaled relaxation times \u001c\u0003\nM. Consequently, the magnetic susceptibility deviates from\nthe Debye law, where the peak of \u001f00moves towards lower frequencies, while broadening and the amplitude decreasing\nas viscoelastic e\u000bects increase. These \fndings are in qualitative agreement with experimental results [17, 19]. We also\ntest a recent proposal put forward in Ref. [19] to use measurements of the magnetic susceptibilities \u001f0and\u001f00to infer\nmechanical properties of the surrounding medium via the Germant-DiMarzio-Bishop relation. For the present model,\nwe \fnd that the GDB relation is satis\fed to a good approximation, at least for the parameter range investigated here.\nFinally, we work out the in\ruence of viscoelasticity on the magnetoviscous e\u000bect. The increase of rotational friction\ndue to the viscoelastic contribution leads to a corresponding increase in the maximum rotational viscosity. Besides\nthis, increasing the relaxation time of the viscoelastic component relative to the viscous one leads to a decrease of the\nrotational viscosity. A similar reduction of the magnetoviscosity with increasing \u001c\u0003\nMwas found in [25] for a simpli\fed\nmodel.\nIt will be interesting to compare the present model more quantitatively with experimental results on magnetic\nsusceptibility, nanorheology and magnetoviscosity. Such comparisons would on the one hand allow for more reliable\ninterpretation of the experimental results and on the other hand stimulate improvements over the current model\nformulation.\nACKNOWLEDGMENTS\nValuable discussions with Frank Ludwig, Annette Schmidt, Andreas Tsch ope are gratefully acknowledged. We also\ngratefully acknowledge support from the German Science Foundation (DFG) within the priority program SPP1681\n\\Field controlled particle matrix interactions: synthesis multiscale modelling and application of magnetic hybrid\nmaterials\" under grant no. IL 122/1-1. PI was also supported by a EU FP7{MC{CIG Grant No. 631233.13\nAppendix A: Short-time dynamics\nThe time derivative of the correlation function C(t) =hu(t)\u0001u(0)i0can be expressed as _C(t) =h[Lyu(t)]\u0001u(0)i0.\nWe \fnd from the de\fnition (13) and (14) of the adjoint operator that\n\u001cDLyu=\u0000u+\u000fo+1\n2(h\u0000uu\u0001h) +\n\u0002u; (A1)\nwhereo\u000b=\u0000\u000f\u000b\f\ru\fz\u0003\n\r. Therefore, in the absence of external \row, \u001cD_C(0) =\u0000C(0) +\u000fho\u0001ui0=\u00001 since averages\nare taken with respect to the equilibrium state (12) for which ho\u0001ui0= 0.\nThe next order term is found from C(t) =h[(Ly)2u(t)]\u0001u(0)i0. Applying Lyto Eq. (A1), we need\n\u001cDLy\n0o=\u0000\u001cD\n\u001c1o+\u000fq+ 4\u000fu\n\u001cDLy\nho=1\n2(z\u0003\u0001u)u\u0002h\u00001\n2z\u0003\u0001(u\u0002h)\u0000\u000f(h\u0000uu\u0001h)\n\u001cDLyuu\u0001h=\u00002uu\u0001h+\u000f(u(o\u0001h) +o(u\u0001h))\n+1\n2(uh2\u00002u(u\u0001h)2+h(u\u0001h))\nwhereq\u000b=\u000f\u000b\f\r\u000f\u0016\f\u0015u\u0015z\u0003\n\rz\u0003\n\u0016. Inserting these expressions into Cand taking equilibrium averages for which hq\u0001ui0=\u00002,\nwe arrive at Eq. (20).\nAppendix B: Moment equations for relaxational dynamics\nWe start with the moment system (21)-(23) and insert the moment expansion (24). Matching equal orders of \u000fwe\n\fnd forO(\u000f0):\n\u001cD_a(0)\n0=\u0000a(0)\n0 (B1)\n\u001cD_a(0)\n1=\u0000\u00151a(0)\n1 (B2)\n\u001cD_a(0)\n2=\u0000\u00152a(0)\n2\u00004\n\u001c\u0003\nMa(0)\n0 (B3)\nFrom Eq (B1) we \fnd\na(0)\n0(t) =a(0)\n0(0)e\u0000t=\u001cD=e\u0000t=\u001cD(B4)\ndue to the initial condition a(0)\n0(0) =huzi(0) = 1. This is the familiar case where viscoelastic e\u000bects are absent, q= 0.\nFurthermore, a(0)\n1(t) = 0 if we start with the equilibrium for a strong \feld and Gaussian in z,a(0)\n1(0) = 0. Inserting\na(0)\n0(t) =e\u0000t=\u001cDin Eq. (B3), we \fnd\na(0)\n2(t) =\u00002e\u0000t=\u001cD+ (qz(0) + 2)e\u0000t=\u001c2=\u00002e\u0000t=\u001cD(B5)\nwhere we used\na(0)\n2(0) =hqzi(0) =hz\u0003\n3(z\u0003\u0001u)i0\u0000hu(z\u0003)2i0=h(z\u0003\n3)2i0\u0000h(z\u0003)2i0=\u00002\nfrom equilibrium Gaussian averages with Eq. (12).\nNow look at the \frst order terms:\n\u001cD_a(1)\n0=\u0000a(1)\n0+a(0)\n1 (B6)\n\u001cD_a(1)\n1=\u0000\u00151a(1)\n1+a(0)\n2+ 4a(0)\n0 (B7)\nSincea(0)\n1= 0 we have also a(1)\n0= 0 and therefore no correction to the magnetization dynamics to \frst order in \u000f.\nThe solution to a(1)\n1reads\na(1)\n1(t) = 2\u001c\u0003\nM[e\u0000t=\u001cD\u0000e\u0000t=\u001c1] (B8)14\nwherea(1)\n1(0) = 0.\nNow, \fnally, we can compute the correction to the simple exponential magnetization decay from\n\u001cD_a(2)\n0=\u0000a(2)\n0+a(1)\n1 (B9)\nand \fnd the relaxation to second order in \u000fgiven by Eq. (25).\nAppendix C: Flow-induced deviation of orientation\nWith the ansatz (39), we can calculate arbitrary moments in terms of a;b, e.g.\nhui=hui0+ (huui0\u0000hui0hui0)\u0001a+huzi0\u0001b (C1)\nwhere h=h^hwith ^hthe unit vector in the direction of the external \feld. The \frst and second moment of the\norientations are calculated from\nhuui=huui0+ (huuui0\u0000huui0hui0)\u0001a+huuzi0\u0001b\nhuui0=L2^h^h+L1\nhI\nhuuui0=L3^h^h^h+L2\nh(^hI)sym\n)huui=L2^h^h+L1\nhI+ (L3\u0000L2L1)(^h\u0001a)^h^h+L2\u0000L2\n1\nh(^h\u0001a)I+L2\nh(a^h+^ha)\nwhereL1;L2have been introduced in the main text and L3(h) =L1(h)\u00005L2(h)=h.\nFor later use, we also provide\nhuui\u0001^h= (L2+L1=h)^h+ (L3\u0000L2L1+2L2\u0000L2\n1\nh)ak^h+L2\nha\nwhere we de\fned ak=a\u0001^h.\nFrom Eq (42), we \fnd\n0 =\u0000(1 +q)z2\neqb\u0000qkBTL1a\u0002^h (C2)\n)b=\u0000qkBTL1\n(1 +q)z2eqa\u0002^h (C3)\n)hoi=qkBTL2\n1\n(1 +q)zeq^h\u0002(a\u0002^h) (C4)\nInserting these results into (41) we \fnd the condition for the stationary state to be\n0 =\u0000L1^h\u0000(L2\u0000L2\n1)ak^h\u0000L1\nha\n+h\n2\u0012\n[1\u0000L2\u0000L1=h]\u0000(L3\u0000L2L1+2L2\u0000L2\n1\nh)ak\u0013\n^h\u0000L2\n2a\n+\u000fqkBTL2\n1\n(1 +q)zeq^h\u0002(a\u0002^h) +\u001cDL1\n\u0002^h (C5)\nNote that ^h\u0002(a\u0002^h) =a\u0000ak^h. Scalar multiplication of the above equation by ^hyields a linear equation for ak:\n0 =\u0014\n\u0000(L2\u0000L2\n1+L1\nh)\u0000h\n2\u0012\nL3\u0000L2L1+3L2\u0000L2\n1\nh\u0013\u0015\nak (C6)\nand therefore ak= 0 since 1\u0000L2\u0000L1=h= 2L1=hand therefore\u0000L1+h\n2[1\u0000L2\u0000L1=h] = 0.15\nThus, we know that a=ak^h+a?=a?witha?\u0001^h= 0. Applying the orthogonal projector I\u0000^h^hto Eq (C5) we\narrive at\n0 =\u0000L1\nha?\u0000L2\n2a?+\u000fqkBTL2\n1\n(1 +q)zeqa?+\u001cDL1\n\u0002^h\n=\u0012\n\u00002 +L2\n6+\u000fqkBTL2\n1\n(1 +q)zeq\u0013\na?+\u001cDL1\n\u0002^h\n)a?=L1\n2+L2\n6\u0000\u000fqkBTL2\n1\n(1+q)zeq\u001cD\n\u0002^h\nInserting this result into (40) we \fnd the mean orientation due to \feld and weak \row as given in Eq. (43).\n[1] R. E. Rosensweig, Ferrohydrodynamics (Cambridge University Press, Cambridge, 1985).\n[2] M. Kr oger, P. Ilg, and S. Hess, J. Phys.: Condens. Matter 15, S1403 (2003).\n[3] I. Torres-D\u0013 \u0010az and C. Rinaldi, Soft Matter 10, 8584 (2014).\n[4] C. Holm and J. J. Weis, Curr. Opin. Colloid Interface Sci. 10, 133 (2005).\n[5] P. Ilg and S. Odenbach, in Colloidal Magnetic Fluids: Basics, Development and Applications of Ferro\ruids , Lecture Notes\nin Phys., Vol. 763, edited by S. Odenbach (Springer, Berlin, 2008).\n[6] M. Zr\u0013 \u0010nyi, Colloid Polym. Sci 278, 98 (2000).\n[7] R. Messing, N. Frickel, L. Belkoura, R. Strey, H. Rahn, S. Odenbach, and A. M. Schmidt, Macromolecules 44, 2990 (2011).\n[8] P. Bender, A. G unther, A. Tsch ope, and R. Birringer, J. Magn. Magn. Mater. 323, 2055 (2011).\n[9] N. N. Reddy, Y. M. Mohan, K. Varaprasad, S. Ravindra, P. A. Joy, and K. M. Raju, J. Appl. Polym. Sci. 122, 1364\n(2011).\n[10] P. Ilg, Soft Matter 9, 3465 (2013).\n[11] R. Hergt, S. Dutz, R. M uller, and M. Zeisberger, J. Phys.: Condens. Matter 18, S2919 (2006).\n[12] C. Alexiou, R. Jurgons, C. Seliger, O. Brunke, H. Iro, and S. Odenbach, Anticancer Research 27, 2019 (2007).\n[13] F. Wiekhorst, U. Steinho\u000b, D. Eberbeck, and L. Trahms, Pharmaceutical research 29, 1189 (2012).\n[14] J. B. Haun, T. J. Yoon, H. Lee, and R. Weissleder, Wiley Interdisciplinary Reviews: Nanomedicine and Nanobiotechnology\n2, 291 (2010).\n[15] C. Wilhelm, F. Gazeau, and J.-C. Bacri, Phys. Rev. E 67, 061908 (2003).\n[16] C. Barrera, V. Flori\u0013 an-Algarin, A. Acevedo, and C. Rinaldi, Soft Matter 6, 3662 (2010).\n[17] H. Remmer, E. Roeben, A. M. Schmidt, M. Schilling, and F. Ludwig, Journal of Magnetism and Magnetic Materials 427,\n331 (2017).\n[18] A. Tsch ope, K. Birster, B. Trapp, P. Bender, and R. Birringer, Journal Of Applied Physics 116, 184305 (2014).\n[19] E. Roeben, L. Roeder, S. Teusch, M. E\u000bertz, U. K. Deiters, and A. M. Schmidt, Colloid and Polymer Science 292, 2013\n(2014).\n[20] E. Jarkova, H. Pleiner, H.-W. M uller, and H. Brand, Phys. Rev. E 68, 041706 (2003).\n[21] A. Attaran, J. Brummund, and T. Wallmersperger, Journal of Intelligent Material Systems and Structures 28, 1358\n(2017).\n[22] G. Pessot, R. Weeber, C. Holm, H. L owen, and A. M. Menzel, J. Phys.: Condens. Matter 27, 325105 (2015).\n[23] V. S. Volkov and A. I. Leonov, Physical Review E 64, 051113 (2001).\n[24] Y. L. Raikher and V. V. Rusakov, Colloid Journal 67, 610 (2005).\n[25] Y. L. Raikher and V. V. Rusakov, Colloid Journal 70, 77 (2008).\n[26] Y. L. Raikher, V. V. Rusakov, and R. Perzynski, Soft Matter 9, 10857 (2013).\n[27] V. V. Rusakov and Y. L. Raikher, Journal of Chemical Physics 147(2017).\n[28] G. Mazenko, Nonequilibrium Statistical Mechanics (Wiley-VCH, Weinheim, 2006).\n[29] M. Grimm, S. Jeney, and T. Franosch, Soft Matter 7, 2076 (2011).\n[30] J. K. G. Dhont, An introduction to dynamics of colloids , Studies in interface science (Elsevier, Amsterdam, 1996).\n[31] P. C. Fannin, S. W. Charles, and T. Relihan, J. Phys. D: Appl. Phys. 28, 1765 (1995).\n[32] H. C. Ottinger, Stochastic Processes in Polymeric Fluids (Springer, Berlin, 1996).\n[33] P. Ilg, M. Kr oger, S. Hess, and A. Y. Zubarev, Phys. Rev. E 67, 061401 (2003)."    },    {        "title": "2301.03256v1.X_ray_detected_ferromagnetic_resonance_techniques_for_the_study_of_magnetization_dynamics.pdf",        "content": "X-ray detected ferromagnetic resonance techniques\nfor the study of magnetization dynamics\nGerrit van der Laan1and Thorsten Hesjedal2\n1Diamond Light Source, Harwell Science and Innovation Campus, Didcot, OX11 0DE, United Kingdom\n2Department of Physics, Clarendon Laboratory, University of Oxford, Oxford, OX1 3PU, United Kingdom\n(Dated: January 10, 2023)\nElement-speci\fc spectroscopies using synchrotron-radiation can provide unique insights into\nmaterials properties. The recently developed technique of X-ray detected ferromagnetic resonance\n(XFMR) allows studying the magnetization dynamics of magnetic spin structures. Magnetic\nsensitivity in XFMR is obtained from the X-ray magnetic circular dichroism (XMCD) e\u000bect, where\nthe phase of the magnetization precession of each magnetic layer with respect to the exciting\nradio frequency is obtained using stroboscopic probing of the spin precession. Measurement of\nboth amplitude and phase response in the magnetic layers as a function of bias \feld can give a\nclear signature of spin-transfer torque (STT) coupling between ferromagnetic layers due to spin\npumping. Over the last few years, there have been new developments utilizing X-ray scattering\ntechniques to reveal the precessional magnetization dynamics of ordered spin structures in the GHz\nfrequency range. The techniques of di\u000braction and re\rectometry ferromagnetic resonance (DFMR\nand RFMR) provide novel ways for the probing of the dynamics of chiral and multilayered magnetic\nmaterials, thereby opening up new pathways for the development of high-density and low-energy\nconsumption data processing solutions.\nKeywords: FMR, XMCD, X-ray scattering, X-ray re\rectivity, spin structures\nI. INTRODUCTION\nMagnetization dynamics is at the heart of high fre-\nquency magnetic nanoscale devices based on spin waves,\nspin pumping, and spin-torque oscillators in the GHz\nfrequency range. Traditionally, ferromagnetic resonance\n(FMR) has been a work horse technique to determine the\nfundamental parameters for magnetic resonance and re-\nlaxation in thin \flms. The recent growing complexity of\nmany modern magnetic materials and devices requires\nthe development of advanced measurement techniques\nthat more directly reveal the microscopic origin of the\ndynamical magnetic interactions that are at play.\nThe novel techniques of X-ray detected FMR (XFMR)\nenables studying the magnetization dynamics of indi-\nvidual layers, where element-speci\fc magnetic contrast\nis obtained using the X-ray magnetic circular dichroism\n(XMCD) e\u000bect [1]. Not only can the FMR signal be mon-\nitored in X-ray absorption, it can also be done in X-ray\ndi\u000braction and re\rectivity, using techniques termed as\nDFMR and RFMR, respectively [2]. In these X-ray mea-\nsurements, time-resolved FMR gives both the amplitude\nand phase of the spin precession for the di\u000berent chemical\nelements, and hence di\u000berent layers, in the sample. The\nchallenge of such measurements lays in the fact that the\nprecession cone angle is small ( <1\u000e) and that the preces-\nsion frequency is on the order of GHz. The solution is\nto use lock-in techniques and to detect the phase of the\nprecession stroboscopically by using the time structure\nof the X-ray pulses from the synchrotron ( \u0018500 MHz,\ni.e., corresponding to a period between the pulses of 2\nns). The radio frequency (RF) \feld applied to drive the\nspin precession is synchronized with the X-ray pulses us-\ning the clock of the synchrotron. Therefore, each X-raypulse measures the magnetization cone at precisely the\nsame point in the precession cycle. Hence, XFMR com-\nbines the techniques of FMR and XMCD. Thus, the spin\nprecession along the bias \feld is pumped by the RF \feld\nto generate the magnetic resonance (i.e., FMR), whose\namplitude and phase is probed by the synchronized X-\nray pulses using the XMCD e\u000bect. The time dependence\nis recorded using a delay line to vary the phase of the RF\nsignal with respect to the X-ray pulses.\nDuring the last few years, many XFMR studies either\nin time-averaged or time-resolved mode have been re-\nported [3{52]. The \frst element-speci\fc and time-depen-\ndent measurement of the magnetization dynamics using\npump-probe XMCD was reported by Bailey et al. [3] on a\npermalloy (Py = Ni 80Fe20) thin \flm, where the moments\non the Ni and Fe sites were found to precess together at\nall frequencies, and by Arena et al. [4] on a Py/Cu/CoZr\ntrilayer, where at resonance, a weak ferromagnetic cou-\npling was found in the phase and amplitude response of\nindividual layers across resonance.\nThe amplitude and phase response of the magnetic\nprobe layer measured by XFMR provides a signature for\neither static exchange interaction in strongly exchange-\ncoupled bilayers or spin-transfer torque (STT) coupling\ndue to spin pumping. Marcham et al. [25] \frst evidenced\nSTT in a CoFe/Cu/Py spin valve using XFMR where the\n\feld dependence of the \fxed layer phase showed a clear\nsignature of STT due to spin pumping. Using XFMR,\nBaker et al. [28] reported a strong anisotropy of the spin\npumping, providing new opportunities for device appli-\ncations.\nPreviously, time-resolved XFMR has been reviewed in\ngreat detail in Ref. [1]. Here, we present a timely up-\ndate, especially emphasizing the newly developed time-arXiv:2301.03256v1  [cond-mat.mtrl-sci]  9 Jan 20232\nresolved FMR techniques in X-ray re\rectivity and di\u000brac-\ntion.\nThe outline of the remainder of this paper is as follows.\nSec. II gives a brief theoretical background of magnetiza-\ntion dynamics and STT. Sec. III describes the experimen-\ntal setup, conditions, and considerations for the various\nXFMR techniques. Sec. IV highlights several recent ex-\namples of XFMR, DFMR, and RFMR experiments and\nmentions their scienti\fc impact. Finally, conclusions are\ndrawn in Sec. V.\nII. BACKGROUND ON FMR AND STT\nA. Ferromagnetic resonance (FMR)\nBefore presenting the experimental details and show-\ncasing several recent examples, we will brie\ry introduce\nsome relevant background material.\nFMR arises when the energy levels of a quantized sys-\ntem of electronic moments are Zeeman split by a uniform\nmagnetic \feld and the system absorbs energy from an os-\ncillating magnetic \feld [53]. A resonance occurs when the\ntransverse AC \feld is applied at the Larmor frequency\ncorresponding to the energy di\u000berence between the mag-\nnetic levels, i.e., ~!= \u0001E. The spin precession in a\nsingle-domain magnetic material can be described with\nthe equation of motion, the so-called Landau-Lifshitz-\nGilbert (LLG) equation,\n_ m=\u0000\rm\u0002He\u000b+\u000b(m\u0002_ m); (1)\nwhere the e\u000bective \feld He\u000b=\u0000@F(M)=@Mis ob-\ntained by minimization of the free energy Fwith re-\nspect to the magnetization M. The free energy contains\nterms such as the exchange, Dzyaloshinskii-Moriya, de-\nmagnetization, magnetocrystalline anisotropy, magneto-\nstatic, external Zeeman \feld, and elastic energy. Further,\n_ m=\u000em=\u000et; the reduced magnetization is m=M=Ms,\nwhereMs=jmjis the saturation magnetization; and\n\r=g\u0016B=~is the gyromagnetic ratio, where \u0016Bis the\nBohr magneton and gis the Land\u0013 e (spectroscopic split-\nting) g-factor. The dimensionless damping parameter\n\u000b\u001c1 (typically 10\u00003{10\u00002for 3dmetals) determines\nthe width of the resonance absorption peak.\nThe \frst right-hand term in Eq. (1) corresponds to\nthe torque due to the e\u000bective \feld He\u000b. In a classical\npicture,\u001c=dS=dtequates to the time change in an-\ngular momentum S, which leads to the spin precession.\nThe second right-hand term corresponds to the damp-\ning, which can also be written in the form of the Gilbert\ndamping term\u0000\u000b\r(m\u0002m\u0002He\u000b). Both torque and\ndamping are vectorially sketched in Fig. 1(a). Without\nexternal RF excitation, the magnetization would relax to\nthe steady state given by Brown's equation, m\u0002He\u000b= 0.\nLinearization of the LLG equation gives the relation\nbetween the frequency \u00170(or circular frequency !0) and\n\feld, which in the form of the Kittel equation is writtenas [1]\n2\u0019\u00170\u0011!0=\rp\nHe\u000bBe\u000b=\rp\nHe\u000b(Ms+He\u000b):(2)\nB. Spin-transfer torque (STT)\nThe layer selectivity of XFMR makes this technique\na unique probe to investigate STT and related spin cur-\nrents in multi-layered spin valves [1]. STT is the e\u000bect\nin which the spin direction in a magnetic layer can be\nmodi\fed using a spin-polarized current [54, 55].\nSpin pumping occurs when the precessing magnetiza-\ntion vector generated by FMR in a ferromagnetic (FM)\nlayer emits a pure spin current into an adjacent normal\nmetal (NM) layer [56]. Traditionally, spin currents have\nbeen probed using indirect measurements. For instance,\nin the metals through which they \row they can create an\nelectrical voltage drop perpendicular to the spin current\ndirection, or a torque that bends the magnetization di-\nrection. However, such indirect measurements are often\nambiguous because they are also in\ruenced by other fac-\ntors, such as magnetic proximity e\u000bects at the interface.\nSTT gives an extra term in the LLG equation, which\nis (anti)-parallel to the (anti)-damping (see Fig. 1(a)).\nAccording to Slonczewski [54], the adiabatic torque is\n\u001cs=\u000bsm\u0002_ m, where\u000bsis the STT damping. The\nspin current pumped across a FM/NM interface due to\nprecession is [56]\nIs=~\n4\u0019g\"#\ne\u000bm\u0002_ m; (3)\nwhereg\"#\ne\u000bis the e\u000bective spin-mixing conductance. The\nspin pumping depends critically on the FM/NM inter-\nface (the material-dependent g\"#\ne\u000b) and the spin di\u000busion\nlength in the NM.\nFor two FM layers iandjwith di\u000berent resonance fre-\nquencies and coupled by both spin pumping (dynamic ex-\nchange coupling) and static exchange coupling, the cou-\npled LLG equations are\n_ mi=\u0000\rmi\u0002He\u000b;i+\u000b0\nimi\u0002_ mi\n+\u000bs\nimi\u0002(_ mi\u0000_ mj) +Aexmj\u0001mi;(4)\nand equivalently when exchanging i$j, where miis\nthe magnetization direction, He\u000b;ithe e\u000bective \feld, \u000b0\ni\nthe Gilbert damping, and \u000bs\nithe STT damping in layer\ni. The spin pumping induced coupling is determined by\n\u000bs\niand the static exchange coupling by Aex.\nIII. EXPERIMENTAL\nXFMR provides an element-speci\fc and time-resolved\nmeasurement of the precessional dynamics of each FM\nlayer on a ps time scale, where the spin precession in-\nduced by a driving RF signal is detected using the XMCD3\nHeffh(t) m(t)Sample\nCPW(b)\nm×Heffz\nyHeff\nm(t)m ×m ×Heff\nSTT\nx(a)\nFIG. 1. (a) Precession, damping, and spin transfer torque\n(STT) in FMR. The precession m\u0002He\u000baround the e\u000bective\n\feldHe\u000bis damped by the Gilbert term m\u0002m\u0002He\u000b. The\nspin-transfer torque is parallel (antiparallel) to the Gilbert\ndamping term, and can enhance (oppose) the latter depend-\ning on the direction of the spin current. (b) Schematics of\nthe sample geometry for XFMR. The sample (red disk) is\nmounted on the signal line (the central strip) of a coplanar\nwaveguide (CPW). The magnetization m(t) precesses about\nHe\u000b, driven by the in-plane continuous RF \feld h(t) in the\nCPW. The cone angle of precession is exaggerated for clar-\nity; its typical magnitude is \u00181\u000e. Circularly polarized X-ray\npulses from the synchrotron impinge at an grazing incidence\nangle on the sample in transverse geometry in order to enable\nstroboscopic detection of the oscillatory component of m(t)\nat variable phase delays.\ne\u000bect [57]. However, before performing the XFMR ex-\nperiment, the static magnetization of the samples has\nto be precharacterized with standard techniques, such as\nsuperconducting quantum interference device (SQUID)\nmagnetometry to measure the hysteresis loops along the\neasy and hard direction, followed by standard FMR mea-\nsurements.\nA. VNA-FMR\nVector network analyzers (VNA)-FMR measurements\nare used to characterize the magnetic resonances in order\nto judge whether these are suitable and intense enough\nfor the XFMR measurements at the synchrotron. VNA-\nFMR is a broadband FMR technique, where the sample is\nmounted onto a coplanar waveguide (CPW) and driven\nby an external RF \feld, while under a static magnetic\nbias \feld. Measurement of the S-parameters of the sam-\nple results in a frequency-\feld map, where the resonances\nappear in the form of Kittel curves (Eq. (2)). The angu-\nlar dependence of the resonances gives information about\nthe magnetic anisotropy [53]. It allows us to chose the\nbest azimuthal angle of the applied \feld with respect to\nthe crystallographic axes to separate the magnetic res-\nonances at the optimal distance for detecting STT [28].\nHence, at a given RF frequency, this gives us the cor-\nresponding \feld values of the resonances in the XFMR\nexperiments. Conventional FMR will normally probe the\nwhole thickness of a thin \flm since the skin depth of, e.g.,metallic iron at 10 GHz, is on the order a micron.\nB. XMCD\nAt the synchrotron, \frst the static XMCD is measured\nby sweeping the photon energy across the absorption\nedge of the magnetic elements. This allows us to se-\nlect the \fxed photon energies suitable for XFMR. The\nstatic XMCD is obtained from the di\u000berence between\ntwo X-ray absorption spectra recorded with the helic-\nity vector of the circularly polarized X-rays parallel and\nantiparallel, respectively, to the applied magnetic \feld\n[58]. The XMCD signal is proportional to the projection\nof the helicity vector, which is along the incident beam\ndirection ^k, onto the sample magnetization M, hence\nIXMCD/^k\u0001M.\nThe XMCD at the soft X-ray absorption edges, such\nas the Fe, Co, and Ni L2;3, is very strong [59], which\nhelps to compensate for the small changes in magneti-\nzation direction due to the limited cone angle ( <1\u000e) of\nthe precession in XFMR. The X-ray penetration length,\nwhich limits the sampling depth, is in the nm range, e.g.,\nfor pure Fe it is\u001820 nm at the Fe L3maximum at\u0018707\neV [57]. By tuning the photon energy away from the\nabsorption maximum the penetration length can be in-\ncreased (\u0018600 nm below the edge at 700 eV). Note that\nthe length scale of the probing depth is well matched to\nthe thickness of the magnetic layers in spin valves. The\ntypical lateral spot size of the X-ray beam on the sample\nis 200\u000220\u0016m2, again well suited for small devices.\nC. Time-resolved measurements\nThe measurement of the projected magnetic moment\nin XFMR does not require to take the di\u000berence between\nopposite circular polarizations as done in XMCD. In-\nstead, a change in the projection of the magnetization\nprecession is measured using a \fxed circular polarization.\nXFMR can be measured in two distinctly di\u000berent ge-\nometries, namely ( i) time-averaged in longitudinal ge-\nometry [12, 20] or ( ii) time-resolved in transverse geom-\netry [8, 25, 28]. In longitudinal geometry, a shortening\nof the magnetization vector along the z-axis (parallel to\nthe X-ray beam direction) leads to a di\u000berence \u0001 Mz=\nMs(1\u0000cos\u0012)\u00191\n2Ms\u00122, where\u0012is the small cone angle of\nthe magnetization precession. The time-averaged XFMR\nrequires no synchronization with the synchrotron clock,\ntherefore it can be done at an arbitrary frequency, but\nit needs a larger RF power which can lead to nonlinear\ne\u000bects.\nOnly measurements in transverse geometry give access\nto the precessional phase. This geometry is depicted in\nFig. 1(b). The transverse component of the magnetiza-\ntion precession will give a sinusoidal variation on top of\nthe static X-ray absorption signal. With the incident X-\nray beam perpendicular to the bias \feld, the oscillating4\ncomponent of the magnetization precession is measured\nwith a magnitude jMyj=Mssin\u0012\u0019Ms\u0012. Thus, for a\ntypical cone angle of \u0012\u00191\u000e, the transverse geometry\ngives a signal that is larger by a factor of \u0018200 com-\npared to the longitudinal geometry. Due to the shape\nanisotropy of the \flm, the precession is strongly ellipti-\ncal, often with a larger in-plane amplitude. This favors\na measurement geometry with the X-rays at grazing in-\ncidence. A good compromise is an X-ray incidence angle\nof\u001835\u000ewith respect to the plane of the sample, which\nensures that the signal is sensitive to the larger in-plane\ncomponent of the magnetization precession.\nUsing a vector magnet system, such as the portable oc-\ntupole magnet system (POMS) at Diamond [57], where\nthe \feld can be applied in any direction, permits a simple\nchange of the \feld from ( i) parallel to the photon direc-\ntion, as needed for static XMCD scans, to ( ii) orthogonal\nto both X-ray beam and RF \feld direction, as required\nfor time-resolved XFMR.\nThe detection of the X-ray absorption can be done by\neither X-ray transmission [27, 31], \ruorescence yield [23],\nor X-ray scattering or re\rectivity [2, 35, 36, 60, 61]. How-\never, RF plays havoc with total-electron yield. In the\ncase of transmission, the incident X-rays impinge on the\nsample through a hole in the signal line of the CPW.\nAfter passing through the sample, the transmitted X-\nrays are detected with X-ray excited optical luminescence\n(XEOL) emerging from the MgO or sapphire (Al 2O3)\nsubstrate using a photodiode placed behind the sample.\nNote that not all substrates, such as non-transparent ones\nlike Si, are suitable for XEOL detection [62].\nTime resolution is established by using the periodic\nX-ray pulses from the synchrotron (normally operating\nin multibunch or hybrid mode). To enable stroboscopic\nprobing, the RF driving \feld is taken as a harmonic of\nthe X-ray pulse frequency, hence the resonance is driven\nat multiples of the master oscillator clock of the stor-\nage ring. These harmonics are generated using an RF\ncomb generator (Atlantic Microwave) driven by the mas-\nter oscillator clock, which has a frequency of 499.65 MHz\n(at the DLS, ALS, and BESSY synchrotron). This cor-\nresponds to\u00182 ns intervals between consecutive X-ray\npulses, which have a pulse width of \u001835 ps (at DLS and\nBESSY) or\u001870 ps (at ALS). The desired frequency is\nselected using \flters and ampli\fers to drive a narrow\nband, high power (25{30 dBm) RF \feld to the CPW.\nA programmable delay line (Colby Instruments) enables\nphase shifting of the RF oscillation with respect to the\nX-ray pulses with a step resolution of \u00180.5 ps. De-\npending on the speci\fc technique either the transmitted,\ndi\u000bracted, or re\rected X-rays are measured using a pho-\ntodiode. Fig. 2 show a schematic representation of the\nsetup for DFMR; for XFMR and RFMR the electronics\nis very similar. The signal is obtained using a lock-in\nampli\fer (LIA) by switching the signal at a given audio\nfrequency. There are two usual modulation modes. In\namplitude modulation the LIA measures the di\u000berence\nbetween signals obtained with the RF signal on and o\u000b.\nFIG. 2. Schematic of the setup for DFMR measurements in\nthe RASOR di\u000bractometer at the Diamond Light Source. The\nsample is placed on the CPW, which is mounted on the cold\n\fnger inside the di\u000bractometer. Incident circularly or linearly\npolarized X-rays are scattered o\u000b the sample and detected via\na photodiode in a #-2#geometry. A variable magnetic \feld\nis applied in the scattering plane via a pair of permanent\nmagnets whose distance can be controlled externally. An RF\nsignal is fed to the CPW to drive the ferromagnetic resonance\nin the magnetic sample. As the synchrotron gives X-ray pulses\nat a frequency of \u0018500 MHz, a comb generator is used to\nproduce higher harmonics, which are selected and fed to the\nCPW. To probe the time dependence of the scattered X-ray\nintensity, a tunable delay line is used, which shifts the phase\nbetween the pump (the RF signal) and the probe (the pulsed\nX-rays). (Adapted from Ref. [63]).\nIn 180\u000e-phase modulation, the LIA measures the di\u000ber-\nence between signals obtained with the RF of opposite\nphase.\nD. XFMR\nIn order to record the time-resolved XAS signal with\ncircular polarization at \fxed photon energy, the RF fre-\nquency is locked to a multiple of the synchrotron clock.\nThen at \fxed angles and for given temperature, this\nleaves two free scanning parameters, namely the mag-\nnetic bias \feld strength and the delay time between X-ray\npulses and RF \feld.\nMagnetic \feld scans record the signal by sweeping the\nbias \feld at a constant delay time. The signal contains\nboth real and imaginary parts of the magnetic suscepti-\nbility, whose relative contributions strongly change across\nresonance. By measuring two \feld scans, which di\u000ber by5\n90\u000ein phase (obtained using the corresponding time de-\nlays), and \ftting these scans simultaneously using the\nKramers-Kronig relation, gives a good apprehension of\nthe \feld dependence of the resonances [40].\nDelay scans record the signal for each of the magnetic\nlayers at constant bias \feld by sweeping the delay time.\nAs an example, Fig. 3(a) shows a series delay scans over\ntwo periods of the phase taken at di\u000berent bias \felds (40{\n200 mT) across the Co resonance in a magnetic tunnel\njunction (MTJ), in more detail discussed in Sec. IV A.\nThe solid lines represent sinusoidal \fts to the experimen-\ntal data (dots), from which the amplitude and relative\nphase of the magnetization precession can be extracted.\nA sinusoidal function of the form\nS(t) =Xsin(2\u0019\u0017t) +Ycos(2\u0019\u0017t); (5)\nis \ftted to the delay scan, where tis the time delay and\n\u0017the frequency of the RF. This procedure is repeated\nfor various \feld strengths and directions. By extracting\nthe coe\u000ecients XandYin Eq. (5) from the delay scans,\nthe amplitude Aand phase of the oscillations can be\ndetermined using the relationships\nA=p\nX2+Y2;  = 2 arctan\u0012Y\nA+X\u0013\n:(6)\nXFMR precessional plots are assembled by combining\nthe amplitudes and phases extracted from the delay scans\nmeasured over a range of bias \felds. This gives the \feld\ndependence of the amplitude and phase for each element\n(e.g., for Co and Ni in Fig. 3(b)), from which the type\nof coupling between layers can be assessed. By normaliz-\ning the XFMR signal to the static XMCD, the amplitude\nof the signal can be obtained per atom for each chemi-\ncal element in the sample. This enables a quantitative\ndecomposition of the resonance features [31].\nThe static coupling (i.e., exchange interaction) and dy-\nnamic coupling (i.e., spin pumping) give a very di\u000ber-\nent XFMR response, as can be understood from Eq. (4).\nConsider a pump layer FM1 that is free to rotate, and a\nprobe layer FM2 that is pinned. Using XFMR at a \fxed\nfrequency, we scan the \feld across the entire resonance.\nAt resonance, FM1 will show a symmetric peak for the\namplitude, while the phase is 90\u000edelayed with respect\nto the RF driving \feld. Across the entire resonance, the\nphase will change by 180\u000e. To investigate the type of\ncoupling between both layers we measure the XFMR re-\nsponse of FM2 at the resonance condition of FM1.\nFor static exchange coupling, E=\u0000Aexm1\u0001m2, so\nthatHe\u000b;2/m1. This means that the e\u000bective \feld in\nthe second layer is aligned along the magnetization of the\n\frst layer. Then the \feld dependent precession of FM2\nwill show a dispersive (bipolar) peak in the amplitude\nand a symmetric (unipolar) peak in the phase.\nOn the other hand, for dynamic exchange coupling\nHe\u000b;2/_ m2=\u0000i!m1. The magnetic \feld is imagi-\nnary, resulting in a 90\u000ephase change. In this case, the\n\feld dependent precession of FM2 will show a unipolar\npeak in the amplitude and bipolar peak in the phase.This behavior means that XFMR can distinguish be-\ntween static and dynamic coupling by their amplitude\nand phase signature in the precessional plot, and thus\ndetermine the relative contribution of these couplings.\nThis has previously been utilized for, e.g., exchange cou-\npled layers [4, 31], spin values [25], MgO magnetic tunnel\njunctions [29, 45], topological insulators [27, 30, 41], spin\nvalve with\u000e-layer [26], Heusler alloys [40], NiO antiferro-\nmagnetic interlayer [42], exchange springs [44], and \u000b-Sn\nthin \flms [47].\nE. DFMR and RFMR\nDFMR and RFMR measurements have been per-\nformed in the RASOR soft X-ray di\u000bractometer on beam-\nline I10 at the Diamond Light Source [57] (see setup in\nFig. 2). Incident X-rays with wavevector kiilluminate\nthe sample, while the scattered beam ( ks) is detected us-\ning a photodiode. The scattering geometry is con\fgured\nto probe the sample at certain di\u000braction or specular re-\n\rectivity conditions. The sample in the di\u000bractometer is\nmounted on a CPW that is connected to a liquid He cryo-\nstat arm which can reach temperatures down to 12 K. A\nbias magnetic \feld is applied by two permanent magnets,\nwhich can be positioned to vary both the \feld strength\nup to 200 mT and the orientation within the scatter-\ning plane. Perpendicular to the bias \feld, a transverse\nRF \feld around the central conductor of the CPW is\ngenerated, which excites the magnetization dynamics in\nthe system. In contrast to conventional XFMR measure-\nments, where the sample is mounted \rip-chip onto the\nCPW, in the scattering geometry the sample is mounted\nface up to allow for the X-ray beam to probe its sur-\nface. To ensure good coupling between the CPW and the\nprobed top surface, the sample must either be thinned,\nor in the case of multilayers, grown on a thin substrate\nof the order of 100 \u0016m.\nIn DFMR, where the detector is aligned to a Bragg\npeak or magnetic scattering peak, the stroboscopic sig-\nnal is used to measure delay scans for di\u000berent linear or\ncircular polarization of the incident X-rays, to give infor-\nmation about the periodic spin structure.\nIn RFMR where the photo diode detector accepts the\nre\rected beam the stroboscopic signal is used to measure\ndelay scans for di\u000berent values of the scattering vector\nQz, to obtain depth information. An advantage of RFMR\nover DFMR is that it can be done on thin \flms and\nmultilayers, and no single crystals are needed.\nIV. X-RAY BASED FMR EXAMPLES\nA. XFMR of spin-current mediated exchange\ncoupling in MgO-based MTJs\nMagnetic tunnel junctions composed of ferromagnetic\nlayers which are mutually interacting through a nonmag-6\nnetic spacer layer are at the core of magnetic sensor and\nmemory devices. G ladczuk et al. [45] used layer-resolved\nXFMR to investigate the coupling between the magnetic\nlayers of a Co/MgO/Py MTJ. Two magnetic resonance\npeaks were observed for both magnetic layers, as probed\nat the Co and Ni L3X-ray absorption edges.\nFigure 3 shows XFMR delay scans for the Co layer in\nthe Co/MgO/Py MTJ at 80 K continuously driven at 4\nGHz. The curves in Fig. 3(a) show a strong increase in\namplitude as well as a large phase shift across the reso-\nnance at\u001890 mT. The amplitude and phase of the pre-\ncession, which are extracted using Eq. (6), are shown in\nFig. 3(b) and (c), respectively, for both Co (orange) and\nNi (blue) as a function of the bias \feld. The amplitude\ncurves show that the Ni resonance originating from the\nPy layer around \u0018120 mT is strongly coupled with the\nCo layer. On the other hand, the Co resonance around\n\u001890 mT is only weakly present in the Py layer. Instead\nof plotting amplitude Aand phase , one can also plot\nthe FMR signal in the ( X,Y)-plane as a function of \feld\n[45]. Since the sine and cosine functions in Eq. (5) are\northogonal, the estimators of XandYare given by pro-\njections to orthogonal subspaces.\nA theoretical model based on the Landau-Lifshitz-Gil-\nbert-Slonczewski equation (Eq. (4)) was developed, in-\ncluding exchange coupling and spin pumping between the\nmagnetic layers. Fits to the experimental data were car-\nried out, both with and without a spin pumping term,\nand the goodness of the \ft was compared using a likeli-\nhood ratio test. This rigorous statistical approach pro-\nvided an unambiguous proof of the existence of interlayer\ncoupling mediated by spin pumping through MgO [45].\nIt was also found that spin pumping is more e\u000bective\nat lower temperatures, which agrees with the theoretical\nunderstanding.\nB. XFMR of coherent spin currents in\nantiferromagnetic NiO\nAntiferromagnets have recently gained large interest in\nthe \feld of spintronics, as they allow for faster and more\nrobust memory operation than present technologies and\nas they can carry spin current over long distances. How-\never, many fundamental physics questions about these\nmaterials regarding their spin transport properties still\nremain unanswered [64]. A spin current generated by\nspin pumping should have a single wave mode, carrying\nthe coherent magnetization excitation. In contrast, spin\ncurrents generated by thermal gradients produce incoher-\nent currents with a continuum of spin excitation modes.\nThe magnetic excitations in antiferromagnets typically\nhave THz frequencies, while the resonant excitation of\nthe ferromagnetic injector is in the GHz range. Con-\nventional spin pumping experiments measure only the\ntime-averaged DC component of the spin current, i.e.,\nthey cannot distinguish between GHz and THz frequen-\ncies, which is needed to determine how the spin current\nFIG. 3. Time resolved precession. (a) Series of XFMR de-\nlay scans for the Co layer in a Co/MgO/Py MTJ at 80 K\ncontinuously driven at 4 GHz. As expected, the period of\nthe precession is 250 ps, and the delay scan covers two peri-\nods. For clarity, the data points (circles) obtained at di\u000berent\nmagnetic \feld values (between 40 and 200 mT) are shifted\nby a constant o\u000bset and have been di\u000berently colored. The\ndrawn lines represent the \ftted sinusoidal functions. Their\namplitude and phase as a function of magnetic \feld strength\nis plotted in panels (b) and (c), respectively, for both the Co\n(orange) and Py (blue) layers. (Adapted from G ladczuk et\nal.[45]).\npropagates. Alternative techniques such as XFMR are\nneeded to measure the time-varying AC spin current.\nDabrowski et al. [42] used XFMR to study the coher-\nent spin current propagation in a device with three layers,\nwhere the top (injector) and bottom (sink) layers were\nferromagnetic NiFe and FeCo, respectively, and the mid-\ndle layer was epitaxial NiO (001). The phase and am-7\nFIG. 4. DFMR delay scans of the structural and magnetic\npeaks as a function of linear polarization angle. Measure-\nments of (a,b) the anisotropic mode B at 6 GHz and (c,d)\nthe isotropic mode A at 2 GHz. The results for the magnetic\npeak and the structural (0,0,3) peak are shown in the left\nand the right column, respectively. The magnetic resonance\nmodes are probed with linearly polarized light for the range\nof incident polarization angles \u0011between 0{180\u000e. (Adapted\nfrom Ref. [60]).\nplitude of the magnetization precession within adjoining\nsource and sink FM layers were detected, from which\nthe injection and transmission of pure AC spin current\nthrough NiO can be inferred. It was found that magne-\ntization modes in the FM layers oscillate in phase. Fur-\nthermore, the e\u000eciency of the spin transfer varied with\nthe thickness of the antiferromagnet, with a maximal ef-\n\fciency for a 2-nm-thick layer. These results indicate\nthat a spin current propagates coherently through the\nantiferromagnetic NiO layer. The AC spin current is en-\nhanced for NiO thicknesses of less than 6 nm, both with\nand without a nonmagnetic spacer layer inserted into the\nstack, in a manner consistent with previously reported\nexperimental measurements of DC spin current and the-\noretical studies [65]. The XFMR results show that the\npropagation of spin current through NiO layers is medi-\nated by evanescent antiferromagnetic spin wave modes at\nGHz frequencies, rather than THz frequency magnons.\nC. DFMR for mode-resolved detection of\nmagnetization dynamics\nRecent scienti\fc interest has shifted towards more\ncomplex magnetically ordered materials, which are\npromising for high-density and low-energy consumption\ndevices. These systems contain chiral magnetic phases\nsuch as helical, conical, or skyrmion spin structures,\noriginating from the Dzyaloshinskii-Moriya interaction(DMI) found in noncentrosymmetric bulk materials, as\nwell as in systems where symmetry breaking occurs at\na ferromagnetic/heavy metal interface. Such spin struc-\ntures are much more complex than simple ferromagnetic\nstructures, especially their dynamic behavior is so far ill-\nunderstood.\nThe periodic structure of magnetically ordered sys-\ntems can be probed by resonant elastic X-ray scattering\n(REXS), making use of interference e\u000bects from the reg-\nularly repeating magnetization density variations. This\nleads to pure magnetic X-ray scattering peaks which give\ninformation about the static magnetic structure. Analy-\nsis of these magnetic peaks in REXS measurements using\nsynchrotron radiation has led to signi\fcant progress in\nthe understanding of chiral magnetic systems [66, 67].\nIn a pioneering DFMR experiment, Burn et al. [60]\ninvestigated the complex dynamic behavior of the chi-\nral spin structure in Y-type hexaferrite Ba 2Mg2Fe12O22.\nVNA-FMR measurements of this material showed a \feld-\nfrequency map containing two ferromagnetic resonance\nmodes. While mode A is isotropic, i.e., its \feld value is\nindependent of the direction of the applied \feld, mode B\nis anisotropic, showing greater absorption at increasingly\nhigher \felds as the \feld direction rotates out-of-plane.\nREXS at the Fe L2;3absorption edge was used to char-\nacterize the static magnetic structure of the hexaferrite\nand to determine its \feld dependence. Static REXS mea-\nsurements along (0,0, `) in zero \feld show a (0,0,3) struc-\ntural peak decorated with two incommensurate magnetic\nsatellites.\nThe DFMR signal was measured by pointing the pho-\ntodiode at the scattered beam, selecting either the struc-\ntural or the magnetic satellite peak (Fig. 2). Delay scans\nwere measured as a function of applied \feld using linearly\npolarized X-rays. Sinusoidal \fts to the measured data en-\nables the extraction of amplitude and phase. Fig. 4 shows\nthe delay scans of the structural and magnetic peaks of\nthe Y-type hexaferrite for variable incident linear polar-\nization angles \u0011. The panels (a,b) in the top row refer to\nthe anisotropic mode B at 6 GHz, and the panels (c,d)\nin the bottom to the isotropic mode A at 2 GHz. The\nleft and right column refer to the results for the mag-\nnetic peak and the structural (0,0,3) peak, respectively.\nThe results were compared to computer simulations of\nthe Y-type hexaferrite to obtain insight in the periodic\nspin structure of this material.\nA second example of the use of DFMR for mode re-\nsolved detection concerns the dynamic behavior of topo-\nlogical spin textures and chiral magnets, which is an area\nof signi\fcant interest and key to the development of fast\nand e\u000ecient spintronics devices. DFMR measurements\nby Burn et al. [63] revealed how the time-dependence\nof the magnetization dynamics relate to the complex\nspin texture in the well-known chiral magnetic system\nCu2OSeO 3. Using polarized soft X-rays, the dynamic\nexcitations in all three dimensions were probed, which\nrevealed phase shifts that were previously undetectable\nand indistinguishable using conventional FMR.8\nFIG. 5. (a) Pseudo-3D plot of the RFMR signal and its pro-\njection showing the dynamic contribution to the re\rectivity\nfor a [CoFeB/MgO/Ta] 4multilayer as a function of pump-\nprobe time delay. The measurements were carried out with\nleft-circularly polarized X-rays at the Fe L3resonance (707.7\neV) and in an out-of-plane \feld of 29 mT using RF excitation\nat 2 GHz. The various delay curves are shown for di\u000berent Qz,\nranging between 0 and 0.6 nm\u00001. The color scale represents\nthe normalized intensity for each delay scan, highlighting the\nsinusoidal dependence and the shift in phase as Qzis varied\nwhen the intensity is small. (b) Static and dynamic re\rectiv-\nity, and (c) phase of the dynamic re\rectivity as a function of\nQz. The phase point size is scaled by the strength of the dy-\nnamic signal amplitude, and the blue-shaded regions indicate\nwhere the 180\u000ephase shifts have been subtracted to reveal the\notherwise smooth phase variation. (Adapted from Ref. [61]).\nD. RFMR on a [CoFeB/MgO/Ta] 4multilayer\nX-ray re\rectivity with the photon energy tuned to the\nabsorption edge has become a valuable tool for character-\nizing the depth-dependent structure of layered materials.\nThe X-ray re\rectivity is measured as a function of the\nscattering vector Qz=ks\u0000ki= (4\u0019=\u0015) sin#, where ki\n(ks) is the ingoing (outgoing) wavevector of the X-rays\nwith incident angle #and wavelength \u0015. The scattering\nlength density, which gives the scattering strength of the\nchemical and magnetic species within the depth pro\fle of\nthe \flm, is obtained through \ftting the re\rectivity data.\nBurn et al. [61] revealed the depth dependence of the\nmagnetization dynamics in a [CoFeB/MgO/Ta] 4multi-layer system. The structural depth pro\fle was charac-\nterized through static X-ray re\rectometry. The dynamic\nre\rectivity was probed with stroboscopic DFMR using\nan out-of-plane saturating \feld of HBias= 29 mT and\nan RF \feld generated by the CPW beneath the sample.\nThe RF \feld was phase-locked to the fourth harmonic of\nthe\u0018500 MHz synchrotron master clock at 2 GHz. The\ntime dependence of the re\rectivity during precession was\nmapped out as a function of the time delay between the\nRF pump and X-ray probe. Fig. 5(a) shows a color map\nof the sinusoidal variation in the re\rected signal with a\n500 ps period, corresponding to the 2 GHz excitation.\nThe amplitude and phase of the dynamic signal are ex-\ntracted by \ftting the sinusoidal delay scans. The ampli-\ntude is plotted in Fig. 5(b) alongside the static re\rectivity\nfor the di\u000berent values of Qz, ranging between 0 and 0.6\nnm\u00001, and the phase in Fig. 5(c).\nBoth the static intensity and the amplitude of the\ndynamic signal in Fig. 5(b) show re\rectivity fringes re-\nsulting from interference e\u000bects arising from the layered\nchemical and magnetic structure. Additional minima are\nobserved in the dynamic case. The phase of the dynamic\nsignal in Fig. 5(c) shows variations with two contribu-\ntions. Firstly, abrupt 180\u000ephase jumps occur, coincid-\ning with minima in the amplitude of the dynamic signal.\nThese 180\u000ejumps correspond to inversion of the sign of\nthe XMCD signal measured at di\u000berent scattering con-\nditions. In addition, there are smoother variations in\nthe phase, which can be attributed to variations in the\nmagnetization dynamics occurring between the magnetic\nlayers in the multilayered structure.\nTo reveal the depth dependent magnetization dynam-\nics, the experimental results were compared with model-\ning of the dynamic behavior In all layers, the magnetiza-\ntion precesses about a nominal static state when excited\nby an RF \feld. It was shown that inclusion of a small,\nbut signi\fcant phase lag of 5\u000ebetween the four layers\nis necessary to explain the observed change in phase of\nthe dynamic signal. In contrast, a single slab of mag-\nnetic thin \flm material shows a coherent precession of\nthe magnetization as a function of depth.\nWith RFMR, the dynamics from di\u000berent layers con-\ntaining the same element can be explored, and this tech-\nnique has the potential to study the dynamics of inter-\nfacial layers and proximity e\u000bects in complex thin \flm\nand multilayer materials for future magnetic memory and\nprocessing device applications.\nV. CONCLUSIONS\nAlthough conventional FMR is a powerful technique\nto study magnetic resonances in thin \flms and multi-\nlayers, the measured response corresponds to an aver-\nage over the entire magnetic structure of the sample. In\ncontrast, X-ray based FMR techniques allow for time-\nresolved measurements of the magnetization dynamics,\nand, in addition, o\u000ber the bene\fts of XMCD, such as9\nelement-, site-, and shell-speci\fcity [57]. The time reso-\nlution is achieved by stroboscopic probing using higher\nharmonics (1-10 GHz) of the synchrotron master clock.\nXFMR can be used to study spin-transfer torque, dipo-\nlar \feld strength, magneto-crystalline anisotropy, inter-\nlayer exchange coupling, gyromagnetic ratio and damp-\ning constants. It can be applied to study the behav-\nior of spintronics systems, e.g., spin pumping in mag-\nnetic multilayers, heterostructures, spin valves, MTJ, etc.\nThe amplitude and phase of the magnetic resonances\nextracted from the \feld-dependence of the precessional\nplots enable us to distinguish between static and dy-\nnamic exchange coupling and to quantify their relative\ncontributions. Apart from measuring the signal in ab-\nsorption, XFMR can also be detected in di\u000braction and\nre\rectivity; each of these techniques bringing unique ad-vantages. DFMR reveals the dynamical spin modes at\nthe probed magnetic wavevectors, and RFMR gives the\ndepth-resolved dynamics in magnetic multilayers. Future\nXFMR studies can be envisaged to investigate vortex dy-\nnamics, spatial resolution imaging, and X-ray hologra-\nphy.\nVI. ACKNOWLEDGMENTS\nThe XFMR experiments were carried out on beamline\nI10 at the Diamond Light Source (Oxfordshire, United\nKingdom). We like to acknowledge valuable collabo-\nrations with Alex A. Baker, David M. Burn, Maciej\nDabrowski, Adriana I. Figueroa, Lukasz Gladczuk, and\nRobert J. Hicken.\n||||||||{\n[1] G. van der Laan, J. Electron Spectrosc. Relat. Phenom.\n220 (2017) 137{146.\n[2] D. M. Burn, S. L. Zhang, G. van der Laan, T. Hesjedal,\nAIP Advances 11 (2021) 015327.\n[3] W. E. Bailey, L. Cheng, D. J. Keavney, C.-C. Kao,\nE. Vescovo, D. A. Arena, Phys. Rev. B 70 (2004) 172403.\n[4] D. A. Arena, E. Vescovo, C. C. Kao, Y. Guan, W. E.\nBailey, Phys. Rev. B 74 (2006) 064409.\n[5] Y. Guan, W. E. Bailey, C.-C. Kao, E. Vescovo, D. A.\nArena., J. Appl. Phys. 99 (2006) 08J305.\n[6] D. A. Arena, E. Vescovo, C.-C. Kao, Y. Guan, W. E.\nBailey, J. Appl. Phys. 101 (2007) 09C109.\n[7] Y. Guan, W. E. Bailey, E. Vescovo, C.-C. Kao, D. A.\nArena, J. Magn. Magn. Mater. 312 (2007) 437{378.\n[8] D. A. Arena, Y. Ding, E. Vescovo, S. Zohar, Y. Guan,\nW. E. Bailey, Rev. Sci. Instrum. 80 (2009) 083903.\n[9] W. Bailey, C. Cheng, R. Knut, O. Karis, S. Au\u000bret, S. Zo-\nhar, D. Keavney, P. Warnicke, J.-S. Lee, D. A. Arena,\nNat. Commun. 4 (2013) 2025.\n[10] P. Warnicke, R. Knut, E. Wahlstr om, O. Karis, W. E.\nBailey, D. A. Arena, J. Appl. Phys. 113 (2013) 033904.\n[11] P. Warnicke, E. Stavitski, J.-S. Lee, A. Yang, Z. Chen,\nX. Zuo, S. Zohar, W. E. Bailey, V. G. Harris, D. A.\nArena, Phys. Rev. B 92 (2015) 104402.\n[12] J. Goulon, A. Rogalev, F. Wilhelm, N. Jaouen,\nC. Goulon-Ginet, G. Goujon, J. Ben Youssef, M. V. In-\ndendom, JETP Lett. 82 (2005) 696{701.\n[13] J. Goulon, A. Rogalev, F. Wilhelm, N. Jaouen,\nC. Goulon-Ginet, C. Brouder, Eur. Phys. J. B 53 (2006)\n169{184.\n[14] J. Goulon, A. Rogalev, F. Wilhelm, C. Goulon-Ginet,\nG. Goujon, J. Synchrotron Rad. 14 (2007) 257{271.\n[15] A. Rogalev, J. Goulon, F. Wilhelm, C. Brouder,\nA. Yaresko, J. Ben Youssef, M. V. Indenbom, J. Magn.\nMagn. Mater. 321 (2009) 3945{3962.\n[16] J. Goulon, A. Rogalev, F. Wilhelm, G. Goujon,\nC. Brouder, A. Yaresko, J. Ben Youssef, M. V. Inden-\nbom, J. Magn. Magn. Mater. 322 (2010) 2308{2329.\n[17] J. Goulon, A. Rogalev, G. Goujon, F. Wilhelm, J. Ben\nYoussef, C. Gros, J.-M. Barbe, R. Guilard, Int. J. Mol.\nSci. 12 (2011) 8797{8835.[18] J. Goulon, C. Brouder, A. Rogalev, G. Goujon, F. Wil-\nhelm, J. Magn. Magn. Mater. 366 (2014) 1{23.\n[19] G. Boero, S. Rusponi, P. Bencok, R. S. Popovic,\nH. Brune, P. Gambardella, Appl. Phys. Lett. 87 (2005)\n152503.\n[20] G. Boero, S. Mouaziz, S. Rusponi, P. Bencok, F. Nolting,\nS. Stepanow, P. Gambardella, New J. Phys. 10 (2008)\n013011.\n[21] G. Boero, S. Rusponi, P. Bencok, R. Meckenstock, J.-\nM. Thiele, F. Nolting, P. Gambardella, Phys. Rev. B 79\n(2009) 224425.\n[22] G. Boero, S. Rusponi, J. Kavich, A. Lodi Rizzini, C. Pi-\namonteze, F. Nolting, C. Tieg, J.-U. Thiele, P. Gam-\nbardella, Rev. Sci. Instrum. 80 (2009) 123902.\n[23] M. K. Marcham, P. S. Keatley, A. Neudert, R. J. Hicken,\nS. A. Cavill, L. R. Shelford, G. van der Laan, N. D.\nTelling, J. R. Childress, J. A. Katine, P. Shafer, E. Aren-\nholz, J. Appl. Phys. 109 (2011) 07D353.\n[24] M. K. Marcham, W. Yu, P. S. Keatley, L. R. Shelford,\nP. Shafer, S. A. Cavill, H. Qing, A. Neudert, J. R. Chil-\ndress, J. A. Katine, E. Arenholz, N. D. Telling, G. van der\nLaan, , R. J. Hicken, Appl. Phys. Lett. 102 (2013) 062418.\n[25] M. K. Marcham, L. R. Shelford, S. A. Cavill, P. S. Keat-\nley, W. Yu, P. Shafer, A. Neudert, J. R. Childress, J. A.\nKatine, E. Arenholz, N. D. Telling, G. van der Laan, R. J.\nHicken, Phys. Rev. B 87 (2013) 180403(R).\n[26] J. Li, L. R. Shelford, P. Shafer, A. Tan, J. X. Deng, P. S.\nKeatley, C. Hwang, E. Arenholz, G. van der Laan, R. J.\nHicken, Z. Q. Qiu, Phys. Rev. Lett. 117 (2016) 076602.\n[27] A. A. Baker, A. I. Figueroa, L. J. Collins-McIntyre,\nG. van der Laan, T. Hesjedal, Sci. Rep. 5 (2015) 7907.\n[28] A. A. Baker, A. I. Figueroa, C. J. Love, S. A. Cavill,\nT. Hesjedal, G. van der Laan, Phys. Rev. Lett. 116 (2016)\n047201.\n[29] A. A. Baker, A. I. Figueroa, D. Pingstone, V. K. Lazarov,\nG. van der Laan, T. Hesjedal, Sci. Rep. 6 (2016) 35582.\n[30] A. I. Figueroa, A. A. Baker, L. J. Collins-McIntyre,\nT. Hesjedal, G. van der Laan, J. Magn. Magn. Mater.\n400 (2016) 178.\n[31] G. B. G. Stenning, L. R. Shelford, S. A. Cavill, F. Ho\u000b-\nmann, M. Haertinger, T. Hesjedal, G. Woltersdorf, G. J.10\nBowden, S. A. Gregory, C. H. Back, P. A. J. de Groot,\nG. van der Laan, New J. Phys. 17 (2015) 013019.\n[32] P. Klaer, F. Ho\u000bmann, G. Woltersdorf, E. Arbelo Jorge,\nM. Jourdan, C. H. Back, H. J. Elmers, J. Phys. D: Appl.\nPhys. 44 (2011) 425004.\n[33] T. Martin, G. Woltersdorf, C. Stamm, H. A. D urr,\nR. Mattheis, C. H. Back, G. Bayreuther, J. Appl. Phys.\n103 (2008) 07B112.\n[34] T. Martin, G. Woltersdorf, C. Stamm, H. A. D urr,\nR. Mattheis, C. H. Back, G. Bayreuther, J. Appl. Phys.\n105 (2009) 07D310.\n[35] R. Salikhov, R. Abrudan, F. Br ussing, S. Buschhorn,\nM. Ewerlin, D. Mishra, F. Radu, I. A. Garifullin,\nH. Zabel, Appl. Phys. Lett. 99 (2011) 092509.\n[36] R. Salikhov, R. Abrudan, F. Br ussing, K. Gross, C. Luo,\nK. Westerholt, H. Zabel, F. Radu, I. A. Garifullin, Phys.\nRev. B 86 (2012) 144422.\n[37] K. W. Chou, A. Puzic, H. Stoll, G. Sch utz, B. Van\nWaeyenberge, T. Tyliszczak, K. Rott, G. Reiss,\nH. Br uckl, I. Neudecker, D. Weiss, C. H. Back, J. Appl.\nPhys. 99 (2006) 08F305.\n[38] K. Ollefs, R. Meckenstock, D. Spoddig, F. M. R omer,\nC. Hassel, C. Sch oppner, V. Ney, M. Farle, A. Ney, J.\nAppl. Phys. 117 (2015) 223906.\n[39] S. Bonetti, R. Kukreja, Z. Chen, D.Spoddig, K. Ollefs,\nC. Sch oppner, R. Meckenstock, A. Ney, J. Pinto,\nR. Houanche, J. Frisch, J. St ohr, H. A. D urr, H. Ohldag,\nRev. Sci. Instrum. 86 (2015) 093703.\n[40] C. J. Durrant, L. R. Shelford, R. A. J. Valkass, R. J.\nHicken, A. I. Figueroa, A. A. Baker, L. Du\u000by, G. van\nder Laan, P. Shafer, E. Arenholz, C. Klewe, S. A. Cavill,\nJ. R. Childress, J. A. Katine, Phys. Rev. B 96 (2017)\n144421.\n[41] A. A. Baker, A. I. Figueroa, T. Hesjedal, G. van der Laan,\nJ. Magn. Magn. Mater. 473 (2019) 470{476.\n[42] M. Dabrowski, T. Nakano, D. Burn, A. Frisk, D. G. New-\nman, C. Klewe, Q. Li, M. Yang, P. Shafer, E. Arenholz,\nT. Hesjedal, G. van der Laan, Z. Q. Qiu, R. J. Hicken,\nPhys. Rev. Lett. 124 (2020) 217201.\n[43] C. Klewe, Q. Li, M. Yang, A. T. N'Diaye, D. M. Burn,\nT. Hesjedal, A. I. Figueroa, C. Hwang, J. Li, R. J. Hicken,\nP. Shafer, E. Arenholz, G. van der Laan, Z. Qiu, Syn-\nchrotron Radiat. News 33 (2020) 12{19.\n[44] M. Dabrowski, A. Frisk, D. M. Burn, D. G. Newman,\nC. Klewe, A. T. N'Diaye, P. Shafer, G. J. Bowden,\nT. Hesjedal, G. van der Laan, G. Hrkac, R. J. Hicken,\nACS Appl. Mater. Interfaces 12 (2020) 52116{52124.\n[45] L. Gladczuk, L. Gladczuk, P. Dluzewski, K. Lasek,\nP. Aleshkevych, D. M. Burn, G. van der Laan, T. Hes-\njedal, Phys. Rev. B 103 (2021) 064416.[46] M. Dabrowski, R. J. Hicken, A. Frisk, D. G. Newman,\nC. Klewe, A. N'Diaye, P. Shafer, G. van der Laan, T. Hes-\njedal, G. J. Bowden, New J. Phys. 23 (2021) 023017.\n[47] L. Gladczuk, L. Gladczuk, P. Dluzewski, G. van der\nLaan, T. Hesjedal, Phys. Status Solidi RRL 15 (2021)\n2100137.\n[48] X. Yang, J.-F. Cao, J.-Q. Li, F.-Y. Zhu, R. Yu, J. He, Z.-\nL. Zhao, Y. Wang, R.-Z. Tai, Nucl. Sci. Tech. 33 (2022)\n63.\n[49] C. Klewe, S. Emori, Q. Li, M. Yang, B. A. Gray, H.-\nM. Jeon, B. M. Howe, Y. Suzuki, Z. Q. Qiu, P. Shafer,\nE. Arenholz, New J. Phys. 24 (2022) 013030.\n[50] Y. Lim, S. Wu, D. A. Smith, C. Klewe, P. Shafer, S. Emo,\narXiv 2208.07294 (2022).\n[51] S. Emori, C. Klewe, J.-M. Schmalhorst, J. Krieft,\nP. Shafer, Y. Lim, D. A. Smith, A. Sapkota, A. Srivas-\ntava, C. Mewes, Z. Jiang, B. Khodadadi, H. Elmkharram,\nJ. J. Heremans, E. Arenholz, G. Reiss, T. Mewes, Nano\nLett. 20 (2020) 7828{7834.\n[52] Y. Pogoryelov, M. Pereiro, S. Jana, A. Kumar,\nS. Akansel, M. Ranjbar, D. Thonig, D. Primetzhofer,\nP. Svedlindh, J. Akerman, O. Eriksson, O. Karis, D. A.\nArena, Phys. Rev. B 101 (2020) 054401.\n[53] A. H. Morrish, The Physical Principles of Magnetism,\nJohn Wiley & Sons, New York, 1965.\n[54] J. C. Slonczewski, J. Magn. Magn. Mater 159 (1996) L1{\nL7.\n[55] L. Berger, Phys. Rev. B 54 (1996) 9353{9358.\n[56] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, B. I.\nHalperin, Rev. Mod. Phys. 77 (2005) 1375.\n[57] G. van der Laan, A. I. Figueroa, Coord. Chem. Rev. 277-\n278 (2014) 95{129.\n[58] G. van der Laan, J. Phys.: Conf. Ser. 430 (2013) 012127.\n[59] G. van der Laan, B. T. Thole., Phys. Rev. B 43 (1991)\n13401{13411.\n[60] D. M. Burn, S. Zhang, K. Zhai, Y. Chai, Y. Sun, G. van\nder Laan, T. Hesjedal, Nano Lett. 20 (2020) 345{352.\n[61] D. M. Burn, S. L. Zhang, G. Q. Yu, Y. Guang, H. J.\nChen, X. P. Qiu, G. van der Laan, T. Hesjedal, Phys.\nRev. Lett. 125 (2020) 137201.\n[62] C. A. F. Vaz, C. Mouta\fs, M. Buzzi, J. Raabe, J. Elec-\ntron Spectrosc. Relat. Phenom. 189 (2013) 1{4.\n[63] D. M. Burn, S. L. Zhang, G. van der Laan, T. Hesjedal,\nPhys. Rev. B 106 (2022) 174409.\n[64] H. Reichlova, R. Schlitz, S. T. B. Goennenwein, Physics\n13 (2020) 83.\n[65] R. Khymyn, I. Lisenkov, V. S. Tiberkevich, A. N. Slavin,\nB. A. Ivanov, Phys. Rev. B 93 (2016) 224421.\n[66] G. van der Laan, C. R. Physique 9 (2008) 570{584.\n[67] S. L. Zhang, A. Bauer, H. Berger, C. P\reiderer, G. van\nder Laan, T. Hesjedal, Phys. Rev. B 93 (2016) 214420."    },    {        "title": "1602.03319v1.The_inverse_thermal_spin_orbit_torque_and_the_relation_of_the_Dzyaloshinskii_Moriya_interaction_to_ground_state_energy_currents.pdf",        "content": "The inverse thermal spin-orbit torque and the\nrelation of the Dzyaloshinskii-Moriya interaction to\nground-state energy currents\nFrank Freimuth, Stefan Bl ugel and Yuriy Mokrousov\nPeter Gr unberg Institut and Institute for Advanced Simulation, Forschungszentrum\nJ ulich and JARA, 52425 J ulich, Germany\nE-mail: f.freimuth@fz-juelich.de\nAbstract. Using the Kubo linear-response formalism we derive expressions to\ncalculate the heat current generated by magnetization dynamics in magnets with\nbroken inversion symmetry and spin-orbit interaction (SOI). The e\u000bect of producing\nheat currents by magnetization dynamics constitutes the Onsager reciprocal of the\nthermal spin-orbit torque (TSOT), i.e., the generation of torques on the magnetization\ndue to temperature gradients. We \fnd that the energy current driven by magnetization\ndynamics contains a contribution from the Dzyaloshinskii-Moriya interaction (DMI),\nwhich needs to be subtracted from the Kubo linear response of the energy current in\norder to extract the heat current. We show that the expressions of the DMI coe\u000ecient\ncan be derived elegantly from the DMI energy current. Guided by formal analogies\nbetween the Berry phase theory of DMI on the one hand and the modern theory of\norbital magnetization on the other hand we are led to an interpretation of the latter\nin terms of energy currents as well. Based on ab-initio calculations we investigate the\nheat current driven by magnetization dynamics in Mn/W(001) magnetic bilayers. We\npredict that fast domain walls drive strong ITSOT heat currents.\nPACS numbers: 72.25.Ba, 72.25.Mk, 71.70.Ej, 75.70.TjarXiv:1602.03319v1  [cond-mat.mes-hall]  10 Feb 2016The inverse thermal spin-orbit torque and its relation to DMI 2\n1. Introduction\nThe interaction of heat current with electron spins is at the heart of spin caloritronics [1].\nIt leads to thermal spin-transfer torques (STTs) on the magnetization in spin valves,\nmagnetic tunnel junctions, and domain walls when a temperature gradient is applied [2,\n3, 4, 5, 6, 7, 8]. While the thermal STT does not require spin-orbit interaction (SOI),\nit only exists in noncollinear magnets. In spin valves and magnetic tunnel junctions\nthis noncollinearity arises when the magnetizations of the free and \fxed layers are not\nparallel, while in domain walls it arises from the continuous rotation of magnetization\nacross the wall.\nIn the presence of SOI electric currents and heat currents can generate torques\nalso in collinear magnets: In ferromagnets with broken inversion symmetry the so-\ncalled spin-orbit torque (SOT) acts on the magnetization when an electric current is\napplied (Figure 1a) [9, 10, 11, 12, 13, 14, 15, 16, 17] . The inverse spin-orbit torque\n(ISOT) consists in the production of an electric current due to magnetization dynamics\n(Figure 1b) [18, 19, 20]. The application of a temperature gradient results in the thermal\nspin-orbit torque (TSOT) (Figure 1c) [21]. TSOT and SOT are related by a Mott-like\nexpression [22].\nIn this work we discuss the inverse e\u000bect of TSOT, i.e., the generation of heat\ncurrent due to magnetization dynamics in ferromagnets with broken inversion symmetry\nand SOI (Figure 1d). We refer to this e\u000bect as inverse thermal spin-orbit torque\n(ITSOT). While the SOT is given directly by the linear response of the torque to\nan applied electric \feld [16], expressions for the ITSOT are more di\u000ecult to derive\nbecause the energy current obtained from the Kubo formalism contains also a ground-\nstate contribution that does not contribute to the heat current. Analogous di\u000eculties\nare known from the case of the inverse anomalous Nernst e\u000bect, i.e., the generation of\na heat current transverse to an applied electric \feld E[23]. In this case the energy\ncurrent obtained from the Kubo formalism contains besides the heat current also the\nmaterial-dependent part \u0000E\u0002Morbof the Poynting vector, where Morbis the orbital\nmagnetization. This energy magnetization does not contribute to the heat current and\nneeds to be subtracted from the Kubo linear response [23, 24, 25].\nWhen inversion symmetry is broken in magnets with SOI the expansion of the free\nenergyFin terms of the magnetization direction ^n(r) and its gradients contains a term\nlinear in the gradients of magnetization, the so-called Dzyaloshinskii-Moriya interation\n(DMI) [26, 27]:\nFDMI(r) =X\njDj(^n(r))\u0001\u0012\n^n(r)\u0002@^n(r)\n@rj\u0013\n; (1)\nwhereris the position and the index jruns over the three cartesian directions, i.e.,\nr1=x;r2=y;r3=z. The DMI coe\u000ecients Djcan be expressed in terms of\nmixed Berry phases [22, 28]. DMI does not only a\u000bect the magnetic structure by\nenergetically favoring spirals of a certain handedness but also enters spin caloritronics\ne\u000bects [29, 30]. Here, we will show that DMI gives rise to the ground-state energy currentThe inverse thermal spin-orbit torque and its relation to DMI 3\nFigure 1. Family of SOT-related e\u000bects in a Mn/W magnetic bilayer with broken\nstructural inversion symmetry. (a) SOT: An applied electric \feld Egenerates a torque\n\u001con the magnetization. ^nis the magnetization direction. (b) ISOT: Magnetization\ndynamics@^n=@tdrives an electric current J. (c) TSOT: The application of a\ntemperature gradient rTgenerates a torque \u001c. (d) ITSOT: Magnetization dynamics\ndrives a heat current JQ.\nJDMI\nj=\u0000Dj\u0001\u0000^n\u0002@^n\n@t\u0001\nwhen magnetization precesses. This DMI energy current needs\nto be subtracted from the linear response of the energy current in order to obtain the\nITSOT heat current.\nThis work is structured as follows. In section 2 we show that magnetization\ndynamics drives a ground-state energy current associated with DMI and we highlight\nits formal similarities with the material-dependent part of the Poynting vector. In\nsection 3 we develop the theory of ITSOT. We derive the energy current based on the\nKubo linear-response formalism and subtract JDMIin order to extract the heat current.\nIn section 4 we show that the expressions of DMI and orbital magnetization can also be\nderived elegantly by equating the energy currents obtained from linear response theory\ntoJDMIand\u0000E\u0002Morb, respectively. In section 5 we present ab-initio calculations of\nTSOT and ITSOT in Mn/W(001) magnetic bilayers.The inverse thermal spin-orbit torque and its relation to DMI 4\n2. Ground-state energy current associated with the Dzyaloshinskii-Moriya\ninteraction\nTo be concrete, we consider a \rat cycloidal spin spiral propagating along the xdirection.\nThe magnetization direction is given by\n^nc(r) =^nc(x) =0\nB@sin(qx)\n0\ncos(qx)1\nCA; (2)\nwhereqis the spin-spiral wavenumber, i.e., the inverse wavelength of the spin spiral\nmultiplied by 2 \u0019. The free energy contribution FDMI(r) given in (1) simpli\fes for the\nspin spiral of (2) as follows:\nFDMI(r) =FDMI(x) =Dx(^nc(x))\u0001\u0014\n^nc(x)\u0002@^nc(x)\n@x\u0015\n=\n=qDx(^nc(x))\u0001^ey=qDxy(^nc(x));(3)\nwhere ^eyis the unit vector pointing in ydirection and we de\fned Dij(^n) =Di(^n)\u0001^ej.\nWhether Dxyis nonzero or not depends on crystal symmetry. The tensor Dij(^n) is\naxial and of second rank like the SOT torkance tensor [22]. Additionally, it is even\nunder magnetization reversal, i.e., Dij(^n) =Dij(\u0000^n). Therefore, Dij(^n) has the same\nsymmetry properties as the even SOT torkance [16]. According to (3) the cycloidal spiral\nof (2) is a\u000bected by DMI if Dxyis nonzero. This is the case e.g. for magnetic bilayers\nsuch as Mn/W(001) and Co/Pt(111) (the interface normal points in zdirection), where\nalso the component tyxof the even SOT torkance tensor is nonzero [16, 22, 31, 32].\nWe consider now a Neel-type domain wall that moves with velocity w < 0 inx\ndirection. The magnetization direction at time t0= 0, which we denote by ^n0(x), is\nillustrated in Figure 2a. ^n0(x) can be interpreted as a modi\fcation of ^nc(x) ((2)), where\ntheq-vector depends on position:\n^n0(x) =0\nB@sin(q(x)x)\n0\ncos(q(x)x)1\nCA: (4)\nSince the domain wall moves with velocity w, the magnetization direction ^n(x;t) at\npositionxand timetis given by\n^n(x;t) =^n0(x\u0000wt): (5)\nIn Figure 2 we discuss the magnetization direction at position x0at the three times\nt0= 0,t1> t 0andt2> t 1. At time t0= 0 the domain wall is far away from\nx0. Therefore, the magnetization is collinear at x0andFDMI(x0;t0) = 0. At time\nt1the domain wall starts to arrive at x0. Consequently, the magnetization gradient\n@^n(x0;t1)=@x 0becomes nonzero and thus FDMI(x0;t1)6= 0. Due to the motion of theThe inverse thermal spin-orbit torque and its relation to DMI 5\n(a)\n(b)\n(c)Position x x0\nt1\nt2Time t\nt0\nFigure 2. Illustration of a Neel-type domain wall that moves into the negative\nxdirection. Arrows represent the magnetization direction ^n(x;t). ^n(x0;t) is\nhighlighted by oval boxes. (a) ^n(x;t0) =^n0(x) is locally collinear at x0and therefore\nFDMI(x0;t0) = 0. (b) ^n(x;t1) =^n0(x\u0000wt1) starts to become noncollinear at x0and\nthereforeFDMI(x0;t1)6= 0. (c) ^n(x;t2) is strongly noncollinear at x0.\ndomain wall the DMI contribution FDMI(x;t) to the free energy is time dependent:\nHow much DMI free energy is stored at a given position in the magnetic structure\nis determined by the local gradient of magnetization, which moves together with the\nmagnetic structure. The partial derivative of FDMI(x;t) with respect to time is given\nby\n@FDMI(x;t)\n@t=\n=Dx(^n0(x\u0000wt))\u0001\u0014\n^n0(x\u0000wt)\u0002@2^n0(x\u0000wt)\n@x@t\u0015\n+\n+@Dx(^n0(x\u0000wt))\n@t\u0001\u0014\n^n0(x\u0000wt)\u0002@^n0(x\u0000wt)\n@x\u0015\n=\n=Dx(^n0(x\u0000wt))\u0001\u0014\n^n0(x\u0000wt)\u0002@2^n0(x\u0000wt)\n@x@t\u0015\n+\n+@Dx(^n0(x\u0000wt))\n@x\u0001\u0014\n^n0(x\u0000wt)\u0002@^n0(x\u0000wt)\n@t\u0015\n=\n=@\n@x\u001a\nDx(^n0(x\u0000wt))\u0001\u0014\n^n0(x\u0000wt)\u0002@^n0(x\u0000wt)\n@t\u0015\u001b\n=\n=\u0000@\n@xJDMI\nx;(6)The inverse thermal spin-orbit torque and its relation to DMI 6\nwhereJDMI\nxin the last line is the xcomponent of the DMI energy current density\nJDMI=\u0000X\nij^ejDji(^n)\u0014\n^ei\u0001\u0012\n^n\u0002@^n\n@t\u0013\u0015\n=\u0000D(^n)\u0012\n^n\u0002@^n\n@t\u0013\n:(7)\nBy considering additionally spirals propagating in yandzdirection we \fnd that the\ngeneral form of (6) is the continuity equation\n@FDMI\n@t+r\u0001JDMI= 0 (8)\nof the DMI energy current JDMI. According to (7) and (8) the energy current JDMI\nis driven by magnetization dynamics and its sources and sinks signal the respective\ndecrease and increase of DMI energy density. When we compute the energy current\ndriven by magnetization dynamics in section 3 we therefore need to be aware that this\nenergy current contains JDMIin addition to the ITSOT heat current that we wish to\ndetermine. Thus, we need to subtract JDMIfrom the energy current in order to extract\nthe ITSOT heat current.\nIt is reassuring to verify that the material-dependent part Jorb=\u0000E\u0002Morbof\nthe Poynting vector, which needs to be subtracted from the energy current to obtain the\nheat current in the case of the inverse anomalous Nernst e\u000bect [23], can be identi\fed\nby arguments analogous to the above. We sketch this in the following. The energy\ndensity due to the interaction between orbital magnetization Morband magnetic \feld\nBis given by\nForb(r;t) =\u0000Morb(r;t)\u0001B(r;t): (9)\nWe assume that the magnetic \feld is of the form\nB(r;t) =B0(x\u0000wt)^ez; (10)\ni.e., the magnetic \feld at time tcan be obtained from the magnetic \feld at time t0= 0\nby shifting it by wt, as illustrated in Figure 3. Additionally, we assume that the orbital\nmagnetization is of the same form, i.e., Morb(r;t) =Morb\n0(x\u0000wt)^ez. Consequently,\nalsoForb(r;t) =Forb\n0(x\u0000wt).B(r;t) can be expressed as B(r;t) =r\u0002A(r;t) in\nterms of the vector potential\nA(r;t) =^eyZx\u0000wt\n0B0(x0)dx0: (11)\nDue to the motion of the pro\fle of B(r;t) the energy density in (9) changes as a functionThe inverse thermal spin-orbit torque and its relation to DMI 7\nof time. The partial derivative of Forb(r;t) with respect to time is\n@Forb\n@t=\u0000@Morb\n@t\u0001B\u0000Morb\u0001@B\n@t\n=w@Morb\n@x\u0001[r\u0002A] +Morb\u0001[r\u0002E]\n=w@Morb\n@x\u0001^ez@Ay\n@x+Morb\u0001[r\u0002E]\n=\u0000@Morb\n@x\u0001^ez@Ay\n@t+Morb\u0001[r\u0002E]\n=\u0000E\u0001[r\u0002Morb] +Morb\u0001[r\u0002E]\n=r\u0001[E\u0002Morb];(12)\nwhere we used the Maxwell equation r\u0002E+@B\n@t= 0 andE=\u0000@A\n@tvalid in Weyl's\ntemporal gauge with scalar potential set to zero. Thus,\n@Forb\n@t+r\u0001Jorb= 0 (13)\nwith\nJorb=\u0000E\u0002Morb; (14)\nas expected.\nIn the following we discuss several additional formal analogies and similarities\nbetween DMI, classical electrodynamics and orbital magnetization. We introduce the\ntensors C(r) and \u0016C(r) with elements\nCij(r) =^ei\u0001\u0014\n^n(r)\u0002@^n(r)\n@rj\u0015\n(15)\nand\n\u0016Cij(r) =@^ni(r)\n@rj(16)\nto quantify the noncollinearity of ^n(r).Cand \u0016Care related through the matrix\nK(^n) =0\nB@0\u0000^n3^n2\n^n3 0\u0000^n1\n\u0000^n2^n1 01\nCA (17)\nasC=K\u0016C. The free energy FDMI(r) can be expressed in terms of CandDas\nfollows:\nFDMI(r) =X\njDj(r)\u0001\u0014\n^n(r)\u0002@^n(r)\n@rj\u0015\n=X\nijDji(r)^ei\u0001\u0014\n^n(r)\u0002@^n(r)\n@rj\u0015\n=X\nijDji(r)Cij(r) = Tr[D(r)C(r)] =\n= Tr[D(r)K(^n(r))\u0016C(r)] = Tr[ \u0016D(r)\u0016C(r)];(18)The inverse thermal spin-orbit torque and its relation to DMI 8\n(a)\n(b)\n(c)t1\nt2\nt3Time tPosition x x0\nFigure 3. Illustration of a magnetic \feld ramp that moves into the negative x\ndirection. Arrows represent the magnetic \feld B0(x\u0000wt)^ezat position xand timet.\nB0(x) describes a ramp that increases linearly with x. The magnetic \feld at position\nx0is highlighted by an oval box. (a) Snapshot at time t1. (b) At time t2the magnetic\n\feld at position x0has increased because the ramp has moved to the left since t1.\nConsequently, also the energy density Forb\n0(x0\u0000wt2) is now di\u000berent. (c) At time t3\nthe magnetic \feld at position x0has increased further.\nwhere we de\fned \u0016D=DK. Similarly, JDMIin (7) can be expressed in terms of \u0016Das\nJDMI=\u0000D\u0012\n^n\u0002@^n\n@t\u0013\n=\u0000\u0016D@^n\n@t: (19)\nThe energy density Forb=\u0000Morb\u0001(r\u0002A) in (9) involves the curl of the vector potential\nA, while the material-dependent part of the Poynting vector, i.e., Jorb=\u0000E\u0002Morb=\n@A\n@t\u0002Morb, involves the time-derivative of A. Similarly, the spatial derivatives @^n=@ri\nenterFDMIin (18) via the tensor \u0016Cwhile the temporal derivative @^n=@tentersJDMI\nin (19). Thus, in the theory of DMI the magnetization direction ^nplays the role of an\ne\u000bective vector potential.\nThe curl of orbital magnetization constitutes a bound current Jmag=r\u0002Morb\nthat does not contribute to electronic transport. Hence it needs to be subtracted from\nthe linear response electric current driven by gradients in temperature or chemical\npotential in order to obtain the measurable electric current [23]. Similarly, the spatial\nderivatives \u001cbound\nj =P\ni@\n@riDij=r\u0001[D^ej] that result from the presence of gradients in\ntemperature or chemical potential constitute torques that are not measurable and need\nto be subtracted from the total linear response to temperature or chemical potential\ngradients in order to obtain the measurable torque [22]. Table 1 summarizes the formal\nanalogies and similarities between the orbital magnetization and DMI.The inverse thermal spin-orbit torque and its relation to DMI 9\nTable 1. Formal analogies between the theories of orbital magnetization (OM) and\nDzyaloshinskii-Moriya interaction (DMI). The vector potential Ais assumed to satisfy\nWeyl's temporal gauge, hence the scalar potential is set to zero.\nOM DMI\n'vector potential' A ^n\n'magnetic' \feld B=r\u0002A \u0016Cij=@^ni\n@rj\nenergy density Forb=\u0000Morb\u0001BFDMI= Tr[ \u0016D\u0016C]\n'electric' \feld E=\u0000@A\n@t@^n\n@t\nenergy current Jorb=\u0000Morb\u0002@A\n@tJDMI=\u0000\u0016D@^n\n@t\nbound property Jmag=r\u0002Morb\u001cbound\nj =r\u0001[D^ej]\n3. Inverse thermal spin-orbit torque (ITSOT)\nIn ferromagnets with broken inversion symmetry and SOI, a gradient in temperature\nTleads to a torque \u001con the magnetization, the so-called thermal spin-orbit torque\n(TSOT) [22, 21]:\n\u001c=\u0000\frT: (20)\nThe inverse thermal spin-orbit torque (ITSOT) consists in the generation of heat\ncurrent by magnetization dynamics in ferromagnets with broken inversion symmetry\nand SOI. The e\u000bect of magnetization dynamics can be described by the time-dependent\nperturbation \u000eHto the Hamiltonian H[16]\n\u000eH=sin(!t)\n!\u0014\n^n\u0002@^n\n@t\u0015\n\u0001T; (21)\nwhereT(r) =m\u0002^n\nxc(r) is the torque operator. \nxc(r) =1\n2\u0016B\u0002\nVe\u000b\nminority (r)\u0000Ve\u000b\nmajority (r)\u0003\nis the exchange \feld, i.e., the di\u000berence between the potentials of minority and majority\nelectrons.m=\u0000\u0016B\u001bis the spin magnetic moment operator, \u0016Bis the Bohr magneton\nand\u001b= (\u001bx;\u001by;\u001bz)Tis the vector of Pauli spin matrices. The energy current JE\ndriven by magnetization dynamics is thus given by\nJE=\u0000B(^n)\u0014\n^n\u0002@^n\n@t\u0015\n; (22)\nwhere the tensor Bwith elements\nBij(^n) = lim\n!!0ImGR\nJE\ni;Tj(~!;^n)\n~!(23)\ndescribes the Kubo linear response of the energy current operator\nJE=1\n2V[(H\u0000\u0016)v+v(H\u0000\u0016)] (24)The inverse thermal spin-orbit torque and its relation to DMI 10\nto magnetization dynamics. \u0016is the chemical potential, vis the velocity operator and\nthe retarded energy-current torque correlation-function is given by\nGR\nJE\ni;Tj(~!;^n) =\u0000i1Z\n0dtei!t\n[JE\ni(t);Tj(0)]\u0000\u000b\n: (25)\nIn (23) we take the limit frequency !!0, which is justi\fed when the frequency is\nsmall compared to the inverse lifetime of electronic states, which is satis\fed for magnetic\nbilayers at room temperature and frequency !=(2\u0019) in the GHz range.\nWithin the independent particle approximation (23) becomes Bij=BI(a)\nij+BI(b)\nij+\nBII\nij, with\nBI(a)\nij=1\nhZ1\n\u00001dEdf(E)\ndETr\nJE\niGR(E)TjGA(E)\u000b\nBI(b)\nij=\u00001\nhZ1\n\u00001dEdf(E)\ndERe Tr\nJE\niGR(E)TjGR(E)\u000b\nBII\nij=\u00001\nhZ1\n\u00001dEf(E) Re Tr\u001c\nJE\niGR(E)TjdGR(E)\ndE\n\u0000JE\nidGR(E)\ndETjGR(E)\u001d\n;(26)\nwhereGR(E) andGA(E) are the retarded and advanced single-particle Green functions,\nrespectively. f(E) is the Fermi function. Bcontains scattering-independent intrinsic\ncontributions and, in the presence of disorder, additional disorder-driven contributions.\nThe intrinsic Berry-curvature contribution is given by\nBint\nij=2~\nNX\nknX\nm6=nfknImh knjTjj kmih kmjJE\nij kni\n(Ekm\u0000Ekn)2\n=1\nNVX\nknfkn[Aknji\u0000(Ekn\u0000\u0016)Bknji];(27)\nwhere\nAknij=~X\nm6=nIm\u0014h knjTij kmih kmjvjj kni\nEkm\u0000Ekn\u0015\n(28)\nand\nBknij=\u00002~X\nm6=nIm\u0014h knjTij kmih kmjvjj kni\n(Ekm\u0000Ekn)2\u0015\n(29)\nandj kniare the Bloch wavefunctions with corresponding band energies Ekn,fkn=\nf(Ekn), andNis the number of kpoints.\nAs discussed in section 2 we subtract JDMI((7)) from JEin order to obtain the\nheat current JQ:\nJQ=JE\u0000JDMI=\u0000~\f\u0014\n^n\u0002@^n\n@t\u0015\n; (30)The inverse thermal spin-orbit torque and its relation to DMI 11\nwith\n~\f=B\u0000D: (31)\nInserting the Berry-curvature expression of DMI [22, 28]\nDij=1\nNVX\nkn\u001a\nfknAknji+1\n\fln\u0002\n1+e\u0000\f(Ekn\u0000\u0016)\u0003\nBknji\u001b\n; (32)\nwe obtain for the intrinsic contribution\n~\fint\nij=Bint\nij\u0000Dij=\n=1\nNVX\nkn\u001a\nfkn[Aknji\u0000(Ekn\u0000\u0016)Bknji]\n\u0000\u0014\nfknAknji+1\n\fln\u0002\n1+e\u0000\f(Ekn\u0000\u0016)\u0003\nBknji\u0015\u001b\n=\u00001\nNVX\nknBknji\u001a\nfkn(Ekn\u0000\u0016)+\n+1\n\fln\u0002\n1 +e\u0000\f(Ekn\u0000\u0016)\u0003\u001b\n;(33)\nwhere\f= (kBT)\u00001. Using\nfkn(Ekn\u0000\u0016) +1\n\fln\u0002\n1 +e\u0000\f(Ekn\u0000\u0016)\u0003\n=\n=\u0000Z\u0016\n\u00001dEf0(Ekn+\u0016\u0000E)(Ekn\u0000E) =\n=\u0000Z\u0016\n\u00001dEZ1\n\u00001dE0f0(E0+\u0016\u0000E)(E0\u0000E)\u000e(E0\u0000Ekn) =\n=\u0000Z1\n\u00001dE0f0(E0)(E0\u0000\u0016)\u0002(E0\u0000Ekn);(34)\nwhere \u0002 is the Heaviside unit step function, we can rewrite (33) as\n~\fint\nij(^n) =\u00001\neVZ1\n\u00001dEf0(E)(E\u0000\u0016)tint\nji(^n;E): (35)\nHere,\ntint\nij(^n;E) =\u0000e\nNX\nkn\u0002(E\u0000E kn)Bknij (36)\nis the intrinsic SOT torkance tensor [16, 22] at zero temperature as a function of Fermi\nenergyEande=jejis the elementary positive charge.\nThe intrinsic TSOT and ITSOT are even in magnetization, i.e., ~\fint\nij(^n) =~\fint\nij(\u0000^n).\n(26) contains an additional contribution which is odd in magnetization, i.e., ~\fodd\nij(^n) =\n\u0000~\fodd\nij(\u0000^n), and which is given by\n~\fodd\nij(^n) =1\neVZ1\n\u00001dEf0(E)(E\u0000\u0016)todd\nji(^n;E); (37)The inverse thermal spin-orbit torque and its relation to DMI 12\nwheretodd\nji(^n;E) is the odd contribution to the SOT torkance tensor as a function of\nFermi energy [16]. The total ~\fij(^n) coe\u000ecient, i.e., the sum of all contributions, is\nrelated to the total torkance tji(\u0000^n;E) for magnetization in \u0000^ndirection by\n~\fij(^n) =\u00001\neVZ1\n\u00001dEf0(E)(E\u0000\u0016)tji(\u0000^n;E); (38)\nwhich contains (35) and (37) as special cases.\nIt is instructive to verify that the ITSOT described by (38) is the Onsager-reciprocal\nof the TSOT ((20)), where [22]\n\fij(^n) =1\neZ1\n\u00001dEf0(E)(E\u0000\u0016)\nTtij(^n;E): (39)\nComparison of (38) and (39) yields\n\f(^n) =\u0000V\nT[~\f(\u0000^n)]T(40)\nand thus 0\nB@\u0000JQ\n\u001c=V1\nCA=0\nB@T\u0015(^n) ~\f(^n)\n[~\f(\u0000^n)]T\u0000\u0003(^n)1\nCA0\nB@rT\nT\n^n\u0002@^n\n@t1\nCA; (41)\nwhere\u0015is the thermal conductivity tensor and \u0003describes Gilbert damping and\ngyromagnetic ratio [18]. As expected, the response matrix\nA(^n) =0\nB@T\u0015(^n) ~\f(^n)\n[~\f(\u0000^n)]T\u0000\u0003(^n)1\nCA (42)\nsatis\fes the Onsager symmetry A(^n) = [A(\u0000^n)]T.\n(38) and (30) are the central result of this section. Together, these two equations\nprovide the recipe to compute the heat current JQdriven by magnetization dynamics\n@^n=@t. We discuss applications in section 5.\n4. Using the ground-state energy currents to derive expressions for DMI\nand orbital magnetization\nThe expression (32) for the DMI-spiralization tensor Dwas derived both from\nsemiclassics [28] and static quantum mechanical perturbation theory [22]. Alternatively,\ntheT= 0 expression of Dcan also be obtained elegantly and easily by invoking the third\nlaw of thermodynamics: For T!0 the ITSOT must vanish, ~\f!0, because otherwise\nwe could pump heat at zero temperature and thereby violate Nernst's theorem. Hence,\nD!Baccording to (31). In other words, at T= 0 the energy current density JEinThe inverse thermal spin-orbit torque and its relation to DMI 13\n(22) is identical to the DMI energy current density JDMI=\u0000D\u0000^n\u0002@^n\n@t\u0001\nbecause the\nheat current is zero. Thus, at T= 0 we obtain from (27)\nDij=Bint\nij=1\nNVX\nknfkn[Aknji\u0000(Ekn\u0000\u0016)Bknji]; (43)\nwhich agrees with (32) at T= 0.\nSimilarly, we can derive the T= 0 expression of orbital magnetization from the\nenergy current Jorb=\u0000E\u0002Morbdiscussed in (14): For T!0 the inverse anomalous\nNernst e\u000bect (i.e., the generation of a transverse heat current by an applied electric\n\feld) has to vanish according to the third law of thermodynamics. Hence, the energy\ncurrent driven by an applied electric \feld at T= 0 does not contain any heat current\nand is therefore identical to Jorb. We introduce the tensor Rto describe the linear\nresponse of the energy current Jto an applied electric \feld E, i.e.,J=RE. We\ndescribe the e\u000bect of the electric \feld by the vector potential A=\u0000Esin(!t)=!and\ntake the limit !!0 later. The Hamiltonian density describing the interaction between\nelectric current density Jand vector potential is \u0000J\u0001A, from which we obtain the\ntime-dependent perturbation\n\u000eH=\u0000sin(!t)\n!eE\u0001v: (44)\nIntroducing the retarded energy-current velocity correlation-function\nGR\nJE\ni;vj(~!) =\u0000i1Z\n0dtei!t\n[JE\ni(t);vj(0)]\u0000\u000b\n(45)\nwe can write the elements of the tensor Ras\nRij=elim\n!!0ImGR\nJE\ni;vj(~!)\n~!: (46)\nThis allows us to determine JorbasJorb=RintE, where the intrinsic Berry-curvature\ncontribution to the response tensor Ris given by\nRint\nij=\u00002e~\nNX\nknfknX\nm6=nImhuknjJE\nijukmihukmjvjjukni\n(Ekm\u0000Ekn)2\n=1\nNVX\nknfkn[Mknij\u0000(Ekn\u0000\u0016)Nknij];(47)\nwith\nMknij=e~X\nm6=nImhuknjvijukmihukmjvjjukni\nEkn\u0000Ekm(48)\nand\nNknij= 2e~X\nm6=nImhuknjvijukmihukmjvjjukni\n(Ekm\u0000Ekn)2: (49)The inverse thermal spin-orbit torque and its relation to DMI 14\nFromMorb\u0002E=RintEwe obtain\nMorb=\u00001\n2^ek\u000fkijRint\nij: (50)\nIt is straightforward to verify that Morbgiven by (50) agrees to the T= 0 expressions\nfor orbital magnetization derived from quantum mechanical perturbation theory [33],\nfrom semiclassics [23], and within the Wannier representation [34, 35].\nCombining the third law of thermodynamics with the continuity equations (8) and\n(13) provides thus an elegant way to derive expressions for DandMorbatT= 0. We\ncan extend these derivations to T >0 if we postulate that the linear response to thermal\ngradients is described by Mott-like expressions. In the case of the TSOT this Mott-like\nexpression is (39), while it is [23, 36, 37]\n\u000bxy=1\neZ1\n\u00001dEf0(E)E\u0000\u0016\nT\u001bxy(E) (51)\nin the case of the anomalous Nernst e\u000bect, where \u001bxy(E) is the zero-temperature\nanomalous Hall conductivity as a function of Fermi energy Eand the anomalous Nernst\ncurrent due to a temperature gradient in ydirection is jx=\u0000\u000bxy@T=@y . While (39)\nand (51) were, respectively, derived in the previous section and in [23], we now instead\nconsider it an axiom that within the range of validity of the independent particle\napproximation the linear response to thermal gradients is always described by Mott-\nlike expressions. Thereby, the derivation in the present section becomes independent\nfrom the derivation in the preceding section. Applying the Onsager reciprocity principle\nto (39) and (51) we \fnd that the ITSOT and the inverse anomalous Nernst e\u000bect are,\nrespectively, described by (38) and by\nJQ\ny=T\u000bxyEx: (52)\nEmploying the general identity (34) (but in contrast to section 3 we now use it\nbackwards) we obtain\n~\fint\nij=\u00001\nNVX\nknBknji\u001a\nfkn(Ekn\u0000\u0016)+\n+1\n\fln\u0002\n1 +e\u0000\f(Ekn\u0000\u0016)\u0003\u001b (53)\nfrom (35) and, similarly, (52) can be written as\nJQ\ny=\u00001\nNVX\nknNknyx\u001a\nfkn(Ekn\u0000\u0016)+\n+1\n\fln\u0002\n1 +e\u0000\f(Ekn\u0000\u0016)\u0003\u001b\nEx:(54)\nThe \fnite-Texpressions of DandMorbare now easily obtained, respectively, by\nsubtracting the ITSOT heat current given by (53) from the energy current (27) andThe inverse thermal spin-orbit torque and its relation to DMI 15\nby subtracting the heat current (54) from Jy=Rint\nyxEx. This leads to (32) for the\nDMI spiralization tensor and to\nMorb\nz=1\nNVX\nknfkn\u001a\nMknyx+1\n\fNknyxln\u0002\n1+e\u0000\f(Ekn\u0000\u0016)\u0003\u001b\n(55)\nfor the orbital magnetization. (55) agrees to the \fnite- Texpressions of Morb\nzderived\nelsewhere [33, 23].\n5. Ab-initio calculations\nWe investigate TSOT and ITSOT in a Mn/W(001) magnetic bilayer composed of one\nmonolayer of Mn deposited on 9 layers of W(001). The ground state of this system\nis magnetically noncollinear and can be described by the cycloidal spin spiral (2) [31].\nBased on phenomenological grounds [38, 19] we can expand torkance as well as TSOT\nand ITSOT coe\u000ecients locally at a given point in space in terms of ^nand \u0016C:\ntij(^n;\u0016C) =X\nkt(1;0)\nijk^nk+X\nklt(0;1)\nijkl\u0016Ckl+\n+X\nklmt(1;1)\nijklm^nk\u0016Clm+X\nkt(2;0)\nijkl^nk^nl+\u0001\u0001\u0001:(56)\nThe coe\u000ecients t(1;0)\nijk,t(0;1)\nijkl,t(1;1)\nijklm,. . . in this expansion can be extracted from\nmagnetically collinear calculations. Analogous expansions of the TSOT and ITSOT\ncoe\u000ecients are of the same form. Here, we consider only t(1;0)\nijkandt(2;0)\nijkl, which give rise\nto the following contribution to the torque \u001c:\n\u001c=todd\nxx(^ez)^n\u0002(E\u0002^ez)+\n+teven\nyx(^ez)^n\u0002[^n\u0002(E\u0002^ez)];(57)\nwhere we used that for magnetization direction ^nalongzit follows from symmetry\nconsiderations that txx=tyy,txy=\u0000tyx,teven\nxx= 0 andtodd\nyx= 0. The SOT in\nthis system has already been discussed by us [16]. In order to obtain TSOT and\nITSOT, we calculate the torkance for the magnetically collinear ferromagnetic state with\nmagnetization direction ^nset alongzas a function of Fermi energy and use (39) and (38)\nto determine the TSOT and ITSOT coe\u000ecients \fand ~\f, respectively. Computational\ndetails of the density-functional theory calculation of the electronic structure as well as\ntechnical details of the torkance calculation are given in [16]. The torkance calculation\nis performed with the help of Wannier functions [39, 40] and a quasiparticle broadening\nof \u0000 = 25 meV is applied.\nDue to symmetry it su\u000eces to discuss the TSOT coe\u000ecients \feven\nyxand\fodd\nxx, which\nare shown in Figure 4 as a function of temperature. For small temperatures we \fnd\n\fij/Tas expected from\n\fij'\u0000\u00192k2\nBT\n3e@tij\n@\u0016; (58)The inverse thermal spin-orbit torque and its relation to DMI 16\n0 100 200 300 400 500\nTemperature [K]-505101520β [µeVa0/K]βyxeven\nβxxodd\nFigure 4. Thermal torkance \fvs. temperature of a Mn/W(001) magnetic bilayer\nfor magnetization in zdirection. Solid line: Even component \feven\nyx of the thermal\ntorkance. Dashed line: Odd component \fodd\nxxof the thermal torkance. \fis plotted in\nunits of\u0016eVa 0=K= 8:478\u000210\u000036Jm/K, where a0is Bohr's radius.\nwhich is obtained from (39) using the Sommerfeld expansion. Slightly above 100K both\n\feven\nyxand\fodd\nxxstop following the linear behavior of the low temperature expansion\n(58): After reaching a maximum both \feven\nyxand\fodd\nxxdecrease and \fnally change\nsign. At T= 300K the thermal torkances are \feven\nyx = 5:24\u000210\u000035Jm/K and\n\fodd\nxx=\u00003:21\u000210\u000036Jm/K. Thermal torkances of comparable magnitude have been\ndetermined in calculations on FePt/Pt magnetic bilayers [21].\nUsing (40) and the volume of the unit cell of V= 1:58\u000210\u000028m3to convert\nthe TSOT coe\u000ecients into ITSOT coe\u000ecients, we obtain ~\feven\nyx=\u000099:49\u0016J/m2and\n~\fodd\nxx=\u00006:09\u0016J/m2atT= 300K. When the magnetization precesses around the zaxis\nin ferromagnetic resonance (this situation is sketched in Figure 1d) with frequency !\nand cone angle \u0012according to\n^n(t) = [sin(\u0012) cos(!t);sin(\u0012) sin(!t);cos(\u0012)]T; (59)\nthe following ITSOT heat current is obtained from (30) in the limit of small \u0012:\nJQ\nx=!\u0012h\n~\fodd\nxxcos(!t)\u0000~\feven\nyxsin(!t)i\nJQ\ny=!\u0012h\n~\feven\nyxcos(!t) +~\fodd\nxxsin(!t)i\n;(60)\nwhere we made use of ~\fxx=~\fyy=~\fodd\nxxand\u0000~\fxy=~\fyx=~\feven\nyx, which follows from\nsymmetry considerations. Using the ITSOT coe\u000ecients ~\feven\nyxand~\fodd\nxxdetermined aboveThe inverse thermal spin-orbit torque and its relation to DMI 17\natT= 300K we can determine the amplitudes of JQ\nxandJQ\ny. Assuming a cone angle\nof 1\u000eand a frequency of != 2\u0019\u00015GHz we \fnd that the amplitude of the oscillating heat\ncurrent density JQ\nxis\n!\u0012r\u0010\n~\feven\nyx\u00112\n+\u0010\n~\fodd\nxx\u00112\n\u001955kW\nm2: (61)\nThe heat current density JQ\nyhas the same amplitude. We can use the thermal\nconductivity of bulk W of \u0015xx=174 W/(Km) [41] at T=300 K to estimate\nthe temperature gradient needed to drive a heat current of this magnitude:\n(55kW/m2)/\u0015xx=316 K/m. The thickness of the Mn/W(001) \flm is 1.58 nm.\nThe amplitude of the heat current per length \rowing in xdirection is thus\n55 kW/m2\u00011.58 nm\u001987\u0016W/m. These estimates suggest that JQis measurable in\nferromagnetic resonance experiments.\nAccording to (60) the heat current can be made larger by increasing the cone angle.\nHowever, in ferromagnetic resonance experiments the cone angle \u0012is small. Therefore,\nwe estimate the heat current driven by a \rat cycloidal spin spiral that moves with\nvelocitywinxdirection. Its magnetization direction is given by\n^nc(r;t) =^nc(x;t) =0\nBBBB@sin(qx\u0000wt)\n0\ncos(qx\u0000wt)1\nCCCCA: (62)\nWith ^nc(r;t)\u0002@^nc(r;t)=@t=wq^eywe get\nJQ\nx=\u0000~\feven\nxywq (63)\nfrom (30), i.e., a constant-in-time heat current in xdirection. Using ~\feven\nxy= 99:49\u0016J/m2\ndetermined above and a spin-spiral wavelength of 2.3nm [31] we obtain a heat current\ndensity of JQ\nx=-270kW/m2for a spin spiral moving with a speed of w=1ms\u00001. This\nestimate suggests that fast domain walls moving at a speed of the order of 100ms\u00001\ndrive signi\fcant heat currents that correspond to temperature gradients of the order of\n0.1K/(\u0016m).\n6. Summary\nMagnetization dynamics drives heat currents in magnets with broken inversion\nsymmetry and SOI. This e\u000bect is the inverse of the thermal spin-orbit torque. We\nuse the Kubo linear-response formalism to derive equations suitable to calculate the\ninverse thermal spin-orbit torque (ITSOT) from \frst principles. We \fnd that a ground-\nstate energy current associated with the Dzyaloshinskii-Moriya interaction (DMI) is\ndriven by magnetization dynamics and needs to be subtracted from the linear responseThe inverse thermal spin-orbit torque and its relation to DMI 18\nof the energy current in order to extract the heat current. We show that the ground-\nstate energy currents obtained from the Kubo linear-response formalism can also be\nused to derive expressions for DMI and for orbital magnetization. The ITSOT extends\nthe picture of phenomena associated with the coupling of spin to electrical currents\nand heat currents in magnets with broken inversion symmetry and SOI. Based on ab-\ninitio calculations we estimate the heat currents driven by magnetization precession and\nmoving spin-spirals in Mn/W(001) magnetic bilayers. Our estimates suggest that fast\ndomain walls in magnetic bilayers drive signi\fcant heat currents.\nAcknowledgments\nWe gratefully acknowledge computing time on the supercomputers of J ulich Super-\ncomputing Center and RWTH Aachen University as well as \fnancial support from the\nprogramme SPP 1538 Spin Caloric Transport of the Deutsche Forschungsgemeinschaft.\n[1] Bauer G E W, Saitoh E and van Wees B J 2012 Nature materials 11391\n[2] Chico J, Etz C, Bergqvist L, Eriksson O, Fransson J, Delin A and Bergman A 2014 Phys. Rev. B\n90(1) 014434\n[3] Yuan Z, Wang S and Xia K 2010 Solid state communications 150548\n[4] Kim S K and Tserkovnyak Y 2015 Phys. Rev. B 92(2) 020410\n[5] Hatami M, Bauer G E W, Zhang Q and Kelly P J 2007 Phys. Rev. Lett. 99(6) 066603\n[6] Jia X, Xia K and Bauer G E W 2011 Phys. Rev. Lett. 107(17) 176603\n[7] Yu H, Granville S, Yu D P and Ansermet J P 2010 Phys. Rev. Lett. 104(14) 146601\n[8] Leutenantsmeyer J C, Walter M, Zbarsky V, M unzenberg M, Gareev R, Rott K, Thomas A, Reiss\nG, Peretzki P, Schuhmann H, Seibt M, Czerner M and Heiliger C 2013 SPIN 031350002\n[9] Manchon A and Zhang S 2008 Phys. Rev. B 78(21) 212405\n[10] Garate I and MacDonald A H 2009 Phys. Rev. B 80(13) 134403\n[11] Chernyshov A, Overby M, Liu X, Furdyna J K, Lyanda-Geller Y and Rokhinson L P 2009 Nature\nPhys. 5656\n[12] Liu L, Lee O J, Gudmundsen T J, Ralph D C and Buhrman R A 2012 Phys. Rev. Lett. 109(9)\n096602\n[13] Garello K, Miron I M, Avci C O, Freimuth F, Mokrousov Y, Bl ugel S, Au\u000bret S, Boulle O, Gaudin\nG and Gambardella P 2013 Nature Nanotech. 8587\n[14] Haney P M, Lee H W, Lee K J, Manchon A and Stiles M D 2013 Phys. Rev. B 87(17) 174411\n[15] Kurebayashi H, Sinova J, Fang D, Irvine A C, Skinner T D, Wunderlich J, Novak V, Campion\nR P, Gallagher B L, Vehstedt E K, Zarbo L P, Vyborny K, Ferguson A J and Jungwirth T 2014\nNature nanotechnology 9211\n[16] Freimuth F, Bl ugel S and Mokrousov Y 2014 Phys. Rev. B 90(17) 174423\n[17] Manchon A, Koo H C, Nitta J, Frolov S M and Duine R A 2015 Nature materials 14871\n[18] Freimuth F, Bl ugel S and Mokrousov Y 2015 Phys. Rev. B 92(6) 064415\n[19] Hals K M D and Brataas A 2015 Phys. Rev. B 91(21) 214401\n[20] Ciccarelli C, Hals K M D, Irvine A, Novak V, Tserkovnyak Y, Kurebayashi H, Brataas A and\nFerguson A 2014 Nature nanotechnology 1050\n[21] G\u0013 eranton G, Freimuth F, Bl ugel S and Mokrousov Y 2015 Phys. Rev. B 91(1) 014417\n[22] Freimuth F, Bl ugel S and Mokrousov Y 2014 Journal of physics: Condensed matter 26104202\n[23] Xiao D, Yao Y, Fang Z and Niu Q 2006 Phys. Rev. Lett. 97(2) 026603\n[24] Cooper N R, Halperin B I and Ruzin I M 1997 Phys. Rev. B 55(4) 2344\n[25] Qin T, Niu Q and Shi J 2011 Phys. Rev. Lett. 107(23) 236601The inverse thermal spin-orbit torque and its relation to DMI 19\n[26] Moriya T 1960 Phys. Rev. 120(1) 91\n[27] Dzyaloshinsky I 1958 Journal of Physics and Chemistry of Solids 4241\n[28] Freimuth F, Bamler R, Mokrousov Y and Rosch A 2013 Phys. Rev. B 88(21) 214409\n[29] Kovalev A A and G ung ord u U 2015 epl10967008\n[30] Manchon A, Ndiaye P B, Moon J H, Lee H W and Lee K J 2014 Phys. Rev. B 90(22) 224403\n[31] Ferriani P, von Bergmann K, Vedmedenko E Y, Heinze S, Bode M, Heide M, Bihlmayer G, Bl ugel\nS and Wiesendanger R 2008 Phys. Rev. Lett. 101(2) 027201\n[32] Yang H, Thiaville A, Rohart S, Fert A and Chshiev M 2015 Phys. Rev. Lett. 115(26) 267210\n[33] Shi J, Vignale G, Xiao D and Niu Q 2007 Phys. Rev. Lett. 99(19) 197202\n[34] Thonhauser T, Ceresoli D, Vanderbilt D and Resta R 2005 Phys. Rev. Lett. 95(13) 137205\n[35] Ceresoli D, Thonhauser T, Vanderbilt D and Resta R 2006 Phys. Rev. B 74(2) 024408\n[36] Weischenberg J, Freimuth F, Bl ugel S and Mokrousov Y 2013 Phys. Rev. B 87(6) 060406\n[37] Wimmer S, K odderitzsch D and Ebert H 2014 Phys. Rev. B 89(16) 161101\n[38] van der Bijl E and Duine R A 2012 Phys. Rev. B 86(9) 094406\n[39] Mosto\f A A, Yates J R, Lee Y S, Souza I, Vanderbilt D and Marzari N 2008 Computer Physics\nCommunications 178685\n[40] Freimuth F, Mokrousov Y, Wortmann D, Heinze S and Bl ugel S 2008 Phys. Rev. B 78035120\n[41] Ho C Y, Powell R W and Liley P E 1972 Journal of physical and chemical reference data 1279"    },    {        "title": "1502.06482v3.Torsion_induced_effects_in_magnetic_nanowires.pdf",        "content": "Torsion induced e\u000bects in magnetic nanowires\nDenis D. Sheka,1,\u0003Volodymyr P. Kravchuk,2,yKostiantyn V. Yershov,2, 3,zand Yuri Gaididei2,x\n1Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine\n2Bogolyubov Institute for Theoretical Physics, 03143 Kyiv, Ukraine\n3National University of \\Kyiv-Mohyla Academy\", 04655 Kyiv, Ukraine\n(Dated: June 13, 2015)\nMagnetic helix wire is one of the most simple magnetic systems which manifest properties of both\ncurvature and torsion. There exist two equilibrium states in the helix wire with easy-tangential\nanisotropy: a quasi-tangential magnetization distribution in case of relatively small curvatures and\ntorsions, and an onion state in opposite case. In the last case the magnetization is close to tangential\none, deviations are caused by the torsion and curvature. Possible equilibrium magnetization states\nin the helix magnet with di\u000berent anisotropy directions are studied theoretically. The torsion also\nessentially in\ruences the spin-wave dynamics, acting as an e\u000bective magnetic \feld. Originated\nfrom the curvature induced e\u000bective Dzyaloshinskii interaction, this magnetic \feld leads to the\ncoupling between the helix chirality and the magnetochirality, it breaks mirror symmetry in spin-\nwave spectrum. All analytical predictions on magnetization statics an dynamics are well con\frmed\nby the direct spin lattice simulations.\nPACS numbers: 75.30.Et, 75.75.-c, 75.78.-n\nI. INTRODUCTION\nDuring the past few years there is a growing inter-\nest to curvature e\u000bects in physics of nanomagnetism.\nA crucial aspect of the interest is caused by recent\nachievements in nanotechnologies of \rexible, stretchable\nand printable magnetoelectronics (see Ref. 1 and refer-\nences therein). E\u000bects of the curvature on the magne-\ntization structure in nanomagnetic particles of nontriv-\nial geometry were studied for cylinders,2,3torus,4, half-\nspheres5, spherical shells,6hemispherical caps,7,8cylin-\ndrical capped nanomembranes,9, cone shells10,11, and\nparaboloidal shells12. Chiral and curvature e\u000bects with\naccount of the nonlocal dipolar interaction were discussed\nfor cylinder nanotubes.13{15\nVery recently we have developed fully three dimen-\nsional (3D) approach for studying statics and dynamics\nof thin magnetic shells and wires of arbitrary shape.10,11\nThis approach gives a possibility to derive the energy for\narbitrary curves and surfaces and arbitrary magnetiza-\ntion vector \felds on the assumption that the anisotropy\ncontribution greatly exceeds the dipolar and other weak\ninteractions, i.e. for hard magnets. We have shown11\nthat due to the curvature two additional e\u000bective mag-\nnetic interactions originate from the exchange term: (i)\ncurvature induced e\u000bective anisotropy, which is bilinear\nwith respect to curvature and torsion and (ii) curvature\ninduced e\u000bective Dzyaloshinskii interaction, which is lin-\near with respect to curvature and torsion. This novel\napproach open doors for studying several perspective di-\nrections in nanomagnets, including topologically induced\npatterns6,16and magnetochiral e\u000bects11,16.\nThe simplest system which displays both the prop-\nerties of the curvature and torsion is a helix wire,\nwhich is characterized by constant curvature and tor-\nsion. The interest to such a geometry is motivated by\nrecent experiments on rolled{up ferromagnetic microhe-lix coils.17,18Depending on the anisotropy direction dif-\nferent arti\fcial complex helimagnetic{like con\fgurations\nwere experimentally realized: hollow{bar{, corkscrew{,\nand radial{magnetized 3D micro{helix coils.17Rolled\nmagnetic structures are now widely discussed in the\ncontext of possible application in \rexible and stretch-\nable magnetoelectronic devices,19in particular, rolled-\nup GMR sensors,20for magneto\ruidic applications, spin-\nwave \flters,21,22and microrobots23. Helix coil magnetic\nstructures have the potential to be used a variety of\nbioapplication areas, such as in medical procedures, cell\nbiology, or lab{on{a{chip.24\nIn the current study we apply our theory11aimed to\ndescribe magnetization statics and linear dynamics in the\nhelix wire. We analyze equilibrium states for di\u000berent\ntypes of magnetocrystalline anisotropy. The equilibrium\nstate is determined by the relationship between the cur-\nvature, torsion and the anisotropy strength: we describe\npossible magnetization distributions analytically. For\nthree types of anisotropy (easy-tangential, easy-normal\nand easy-binormal) we compute phase diagrams of pos-\nsible equilibrium states. In each of these cases the equi-\nlibrium state is either onion one (high curvatures) or\nanisotropy-aligned state (for small ones), these results\nare summarized in Fig. 4. For example, in the most\ninteresting case of easy-tangential anisotropy a quasi-\ntangential magnetization distribution appears for strong\nenough anisotropy, see Fig. 3(a,b). We show that pure\ntangential magnetization distribution is impossible. The\ndeviation from the tangential state is determined by the\nthe curvature and torsion; besides there exists the cou-\npling between the helix chirality and the magnetochiral-\nity of magnetization distribution.\nWe study the problem of spin wave dynamics in the\nhelix wire. Our analysis shows that the curvature and\ntorsion act on magnons in two ways: besides the stan-\ndard potential scattering of magnons, there appears anarXiv:1502.06482v3  [cond-mat.mes-hall]  13 Jun 20152\ne\u000bective torsion induced magnetic \feld. The vector po-\ntential of e\u000bective \feld is mainly determined by the prod-\nuct of the torsion and the magnetochirality. The origin\nof this \feld is the curvature induced e\u000bective Dzyaloshin-\nskii interaction.11Finally, the torsion breaks the symme-\ntry of spin wave spectrum with respect to the direction\nof spin wave propagation, see Fig. 5. This e\u000bect is com-\npletely analogous to the e\u000bect of asymmetry of magnon\ndispersion due to the natural Dzyaloshinskii interaction\nin magnetic \flms.25{27\nThe paper is organized as follows. In Sec. II we intro-\nduce the model of the curved wire and discuss di\u000berent\nanisotropy-aligned states. The model of the helix wire\nappears in Sec. III. Equilibrium magnetization distribu-\ntions are describe analytically for the easy{tangential he-\nlix wire: the quasi-tangential state (see Sec. III A) and\nthe onion one (see Sec. III B). The phase diagram of en-\nergetically preferable states appears in Sec. III C. The\nproblem of spin-wave dynamics is discussed in Sec. IV.\nIn Sec. V we study statics and linear dynamics for he-\nlix wires with other anisotropy orientations: the easy-\nnormal anisotropy (Sec. V A) and easy-binormal one\n(Sec. V B). We verify our theory by numerical simulations\nof the helix-shaped chain of discrete magnetic moments\nin Sec. VI. In Section VII we present \fnal remarks and\ndiscuss possible perspectives and generalizations, in par-\nticular, how to take into account magnetostatics e\u000bects.\nSome details about the computation of the onion state\nare presented in Appendix A.\nII. THE MODEL OF A CURVED WIRE\nWe consider a model of a curved cylindrical wire. Let\n\r(s) be a 1D curve embedded in 3D space R3withs\nbeing the arc length coordinate. It is convenient to use\nFrenet{Serret reference frame with basic vectors e\u000b:\net=\r0;en=e0\nt\nje0tj;eb=et\u0002en (1)\nwithetbeing the tangent, enbeing the normal, and\nebbeing binormal to \r. Here and below the prime\ndenotes the derivative with respect to the arc length s\nand Greek indices \u000b;\fnumerate curvilinear coordinates\n(TNB{coordinate) and curvilinear components of vector\n\felds. The relation between e0\n\u000bande\u000bis determined by\nFrenet{Serret formulas:\ne0\n\u000b=F\u000b\fe\f;kF\u000b\fk=\r\r\r\r\r\r0\u00140\n\u0000\u00140\u001c\n0\u0000\u001c0\r\r\r\r\r\r: (2)\nHere\u0014and\u001care the curvature and torsion of the wire,\nrespectively.\nThe wire of a \fnite thickness hcan be de\fned as the\nfollowing space domain\nr(s;u;v ) =\r(s) +uen+veb; (3)whereuandvare coordinates within the wire cross sec-\ntion (juj;jvj.h).\nLet us describe the magnetic properties of the wire.\nThe magnetic energy of the wire can collect di\u000berent\ncontributions such as energies of exchange interaction,\nanisotropy, and dipolar one. We start our analysis with\nthe case of a hard magnet where the anisotropy contri-\nbution greatly exceeds the dipolar and other weak inter-\nactions. For such hard magnets a quality factor28\nQ\u0011K\n2\u0019M2s(4)\nis supposed to be large; here K > 0 is the constant of\nmagnetocrystalline anisotropy and Msis the saturation\nmagnetization.\nWe assume the magnetization spatial one-\ndimensionality, which can be formalized as m=m(s;t).\nThis assumption is appropriate for the cases when\nthe thickness hdoes not exceed the characteristic\nmagnetic length w=p\nA=KwithAbeing the exchange\nconstant. The wire thickness is also supposed to be\nsmall in comparison with radii of curvature and torsion.\nTherefore our model provides an adequate picture under\nthe following assumptions:\nh.w\u001c1\n\u0014;1\n\u001c; Q\u001d1: (5)\nThat is why in the current study one can restrict our-\nself to the consideration of Heisenberg magnets with the\nenergy\nE=ASZ\nds\u0000\nEex+Ean\u0001\n;\nEex=\u0000m\u0001r2m; Ean=\u0000(m\u0001ean)2\nw2(6)\nwitheanbeing the unit vector along the anisotropy axis,\nandSbeing the cross-section area.\nTypically, orientation of the anisotropy axis eanis de-\ntermined by the wire geometry, e.g. it can be tangential\nto the wire,17which means in general complicated spatial\ndependence due to the curvilinear geometry. Therefore\nit is convenient to represent the energy of the magnet\nin the curvilinear reference frame (1), where Eanhas a\nsimplest form. For an arbitrary thin wire the exchange\nenergy density can be presented as follows11\nEex=E0\nex+EA\nex+ED\nex;E0\nex=jm0j2;\nEA\nex=K\u000b\fm\u000bm\f;ED\nex=F\u000b\f\u0000\nm\u000bm0\n\f\u0000m0\n\u000bm\f\u0001\n:(7)\nHere the \frst term E0\nexdescribes the common isotropic\npart of exchange expression which has the same form as\nfor the straight wire. The second term EA\nexdescribes an\ne\u000bective anisotropy interaction, where the components of\nthe tensorK\u000b\f=F\u000b\u0017F\f\u0017are bilinear with respect to the\ncurvature\u0014and the torsion \u001c. This term is similar to the\n\\geometrical potential\".29Note that a curvature caused3\nAnisotropy Anisotropy Magnetization states\ntype axis in a helix wire\nean Equilibrium states Orientation\naccording\nto Ref. 17\nEasy{tangential et quasi-tangential\nand onioncorkscrew\nEasy{normal en normal and onion radial\nEasy{binormal eb quasi-binormal\nand onionhollow{bar\nTABLE I. Types of equilibrium magnetization states for var-\nious uniaxial anisotropies in a helix-shaped magnetic wire.\n\\geometric\" e\u000bective magnetic \feld was considered re-\ncently for curved magnonic waveguides.30The last term\nED\nexin the exchange energy functional is the curvature\ninduced e\u000bective Dzyaloshinskii interaction, which is lin-\near with respect to curvature and torsion. We will see\nbelow that this e\u000bective interaction causes an e\u000bective\nmagnetic \feld; namely this interaction is responsible for\nthe magnetochiral e\u000bects.\nWe consider three types of curvilinear uniaxial\nanisotropy which correspond to possible curvilinear di-\nrections (1), see Table I: (i) an easy{tangential anisotropy\ncorresponds to the anisotropy axis eandirected along et,\nwhere the anisotropy interaction tries to orient the mag-\nnetization along the curve. Note that in soft magnets\nsuch kind of anisotropy appears e\u000bectively as a shape\nanisotropy caused by the dipolar interaction.31(ii) An\neasy{normal anisotropy is determined by the normal vec-\ntoren. (iii) An easy{binormal anisotropy direction cor-\nresponds to the binormal basic vector eb.\nAll three types of anisotropic magnets can be real-\nized experimentally: In straight nanostrips/nanowires\nthe anisotropy can have well{de\fned uniaxial directions,\ne.g., in-plane along the strip, in-plane perpendicularly\nto the strip, or out-of-plane, which corresponds to the\nuniformly magnetized samples in the corresponding di-\nrection. Using the coiling process,17it is possible to\nobtain 3D microhelix coil strips with di\u000berent magne-\ntization orientation: corkscrew-, radial-, and hollow-bar-\nmagnetized, see Table I to get a link between anisotropy\ntype and the magnetization orientation.\nFor the further analysis it is convenient to introduce\nthe angular parametrization of the magnetization unit\nvectormusing the local Frenet{Serret reference frame:\nm= sin\u0012cos\u001eet+ sin\u0012sin\u001een+ cos\u0012eb;(8)\nwhere angular variables \u0012and\u001edepend on both spatial\nand temporal coordinates. Then the energy density (7)\nreads:11\nEex= [\u00120\u0000\u001csin\u001e]2+ [sin\u0012(\u001e0+\u0014)\u0000\u001ccos\u0012cos\u001e]2;(9a)\nEET\nan=\u0000sin2\u0012cos2\u001e\nw2;EEN\nan=\u0000sin2\u0012sin2\u001e\nw2;\nEEB\nan=\u0000cos2\u0012\nw2: (9b)\n2RPm\neteneb\nm eteneb\nm\netenebφθ\n(a)(b)\n0xyFIG. 1. (Color online) Schematics of the helix wire of the\nradiusRand the pitch P. (a) Arrangement of the curvilinear\nFrenet{Serret reference frame ( et;en;eb) from the front view.\n(b) Arrangement of the magnetization angles \u0012and\u001ewith\nrespect to the magnetization unit vector m.\nHere EET\nan,EEN\nan, and EEB\nandenotes anisotropy energy\ndensities of easy{tangential, easy{normal, and easy{\nbinormal types, respectively.\nThe magnetization dynamics follows the Landau{\nLifshitz equation. In terms of angular variables \u0012and\n\u001ethese equations read\nMs\n\r0sin\u0012@t\u001e=\u000eE\n\u000e\u0012;\u0000Ms\n\r0sin\u0012@t\u0012=\u000eE\n\u000e\u001e(10)\nwithMsbeing the saturation magnetization and \r0being\nthe gyromagnetic ratio.\nIII. EQUILIBRIUM MAGNETIZATION STATES\nOF A HELIX WIRE WITH EASY{TANGENTIAL\nANISOTROPY\nWe study the curvilinear e\u000bects using the helix geom-\netry, which is the simplest geometry which manifests the\nproperties of both curvature and torsion. A typical pa-\nrameterization of the helix wire reads\n\r(\u001f) =^xRcos\u001f+^yRsin\u001f+^zp\u001f; (11a)\nwhereRis the helix radius, p=P=(2\u0019) withPbeing the\npitch of the helix, and \u001fis azimuthal angle of a cylindrical\nframe of reference with ^z-axis aligned along the helix\naxis, see Fig. 1. The helix has the constant curvature4\n\u0014=R=(R2+p2) and the torsion \u001c=p=(R2+p2). For\nthe further analysis it is instructive to rewrite (11a) as\na function of the arc length sand in terms of curvature\nand torsion\n\r(s) =^x\u0014s2\n0cos\u0012s\ns0\u0013\n+^y\u0014s2\n0sin\u0012s\ns0\u0013\n+^zs0\u001cs;\ns0=1p\n\u00142+\u001c2:(11b)\nOne has to notice a one-to-one correspondence between\n(R;p){parametrization (11a) and ( \u0014;\u001c){one (11b).\nIn order to derive the explicit form of Landau{Lifshitz\nequations, we substitute the energy functional (9) into\nthe Landau{Lifshitz equations (10):\n\u0000Ms\n2\r0Asin\u0012@t\u001e=\u001ccos\u001e\u0000\n\u0014cos 2\u0012\u00002@s\u001esin2\u0012\u0001\n+@ss\u0012\u0000sin\u0012cos\u0012h\n(\u0014+@s\u001e)2\u0000\u001c2cos2\u001ei\n\u00001\n2@Ean\n@\u0012;\nMs\n2\r0Asin\u0012@t\u0012= sin\u0012cos\u0012[2@s\u0012(\u0014+@s\u001e)\u0000\u0014\u001csin\u001e]\n+ sin2\u0012\u0002\n@ss\u001e+ 2\u001c@s\u0012cos\u001e\u0000\u001c2sin\u001ecos\u001e\u0003\n\u00001\n2@Ean\n@\u001e;\n(12)\nwhere Eanis the density of the anisotropy energy, see\n(9b).\nWe are mostly interested in the case of easy{tangential\nanisotropy, which is typical for the wires. In this case the\nanisotropy energy density has the form EET\nan, see (9b).\nFirst we discuss the limit case \u001c= 0 (ring wire instead\nof the helix). For any plane curve the energy functional\n(9) with easy{tangential or easy{normal anisotropy is\nminimized by the plane magnetization distribution, \u00120=\n\u0019=2. The energy minimization in respect to \u001eresults in\nthe pendulum equation\n{2@\u001f\u001f\u001e\u0000sin\u001ecos\u001e= 0;{=\u0014w (13)\nwith{being the reduced curvature.\nThe equilibrium magnetization state of a ring is a ho-\nmogeneous (in the curvilinear reference frame) vortex\nstate\u001evorin case of relatively small reduced curvature\n{<{0\u00190:657 and inhomogeneous onion solution \u001eon\nfor{>{011\n\u001evor= 0;\u0019; \u001eon=\u0019\n2\u0000am(x;k); x=2\u001f\n\u0019K(k):(14)\nHere am(x;k) is the Jacobi amplitude32and the modulus\nkis determined by condition\n2{kK(k) =\u0019 (15)\nwith K(k) being the complete elliptic integral of the \frst\nkind.32\n0 0.5 1 1.5 2π\n4π\n2\nReduced torsion σθt\nκ= 0.1 (am) /squaresolid(a)/squaredot(m)\nκ=κc(am)/diamondsolid(a)♦·(m)\nκ= 1 (am) •(a)⊙(m)\nboundary curve θt\nb(σ)Quasi-tangential\nstate\nOnion\nstateFIG. 2. (Color online) Equilibrium magnetization distri-\nbution in the quasi-tangential state of the helix wire with\nC= +1. Lines correspond to the analytics, see Eq. (17). Sym-\nbols correspond to simulations: (a) anisotropic Heisenberg\nmagnets [see (50)] ( Q= 2,w=`), (am) wires with account\nof dipolar interaction [see (49), (53)] ( Q= 2,we\u000b= 2`=p\n5),\nand (ms) isotropic wires with account of dipolar interaction\n[see (49),(53)] ( Q= 2,we\u000b= 2`). The boundary curve \u0012t\nb(\u001b)\ncorresponds to (23).\nA. Quasi-tangential state\nNow we consider the helix wire with a \fnite torsion,\n\u001c6= 0. Similar to the case of a ring wire, discussed above,\nwe look for the homogeneous (in the curvilinear reference\nframe) solution. Such kind of solutions is possible due\nto the constant curvature \u0014and the torsion \u001c. We can\neasily solve the static equations, see Eq. (12), using the\nsubstitution \u0012(s) =\u0012tand\u001e(s) =\u001et:\ntan 2\u0012t=\u00002C\u001b{\n1\u0000{2+\u001b2; \u001et= 0;\u0019; (16)\nwhere C= cos\u001et=\u00061, the quantity \u001b\u0011w\u001cis the\nreduced torsion. Explicitly for magnetization angles we\nget\n\u0012t=\u0019\n2\u0000arctan2C\u001b{\nV0; \u001et= 0;\u0019;\nV0= 1 +\u001b2\u0000{2+V1;\nV1=p\n(1\u0000{2+\u001b2)2+ 4{2\u001b2:(17)\nThe dependence \u0012t({;\u001b) is presented in Fig. 2.\nIn the limit case of very small curvature and torsion\n({;\u001b\u001c1), the magnetization distribution becomes al-\nmost tangential, see Fig. 3(a) with the asymptotic be-5\n(a) Quasi-tangential state: {=\u001b= 0:1\n(b) Quasi-tangential state: {=\u001b={0\u00190:657\n(c) Quasi-tangential state: {=\u001b= 1\n(d) Onion state: {= 1:5,\u001b= 1\nFIG. 3. (Color online) Magnetization distributions in the he-\nlix wire with C= +1 and easy-tangential anisotropy according\nto simulations data, see Sec. VI A.\nhavior\n\u0012t\u0019\u0019\n2\u0000C\u001b{; for{;\u001b\u001c1: (18)\nThat is why we refer to the state (17) as to the quasi{\ntangential state. Such a state is an analogue of the vortex\nstate for the case of the torsion presence.\nEven in the strong anisotropic case the magnetization\ndeviates from the tangential distribution: the inclination\nangle depends on the sign of C\u001b. One can interpret the\nsign of\u001bas the helix chirality (di\u000berent for right handed\nhelix when \u001b > 0 and left{handed one when \u001b < 0);\nthe quantity Ccan be interpreted as the magnetochiral-\nity, hence on can say about coupling between the two\nchiralities.\nThe energy density (9) of the quasi{tangential state(17) reads\nEt=\u00001\u0000{2\u0000\u001b2+V1\n2w2; (19)\nIt should be noted that the magnetization state in the\nhelix nanowire was recently studied:33in particular, the\nmagnon spectrum was shown to be a\u000bected by the curva-\nture, which acts mainly as e\u000bective anisotropy. However\nthe equilibrium state was forcedly supposed to be the\ntangential one in Ref. 33.\nB. Onion state\nLet us discuss the case of a large curvature and tor-\nsion. In analogy with the ring wire, we are looking for a\nsolution periodic with respect to \u001f, which is an analogue\nof the onion solution (14). Hence we look for solutions of\nthe following form\n\u0012on(s) =\u0019\n2+#(\u001f); \u001eon(s) =\u0000\u001f+'(\u001f) (20a)\nwith#(\u001f) and'(\u001f) being 2\u0019{periodic functions. Using\nan analogy with the ring case ( \u001b= 0) with exact onion\nsolution (14) we name (20a) an onion solution.\nNumerically we found onion solutions for {>{0\u0019\n0:657 in a wide range of \u001b, see Figs. 3(c), 4(a). The\nsymmetry of the static form of Eqs. (12) dictates the\nsymmetry of 2 \u0019{periodic functions #and', which has\nthe following Fourier expansion\n#(\u001f) =NX\nn=1#ncos(2n\u00001)\u001f; ' (\u001f) =NX\nn=1'nsin 2n\u001f;\n(20b)\nwhereN!1 . By substituting series (20b) into the\nstatic version of Eqs. (12), one get the set of nonlinear\nequations for amplitudes #nand'n, see (A5). Finally,\nthe energy of the onion state Eon(\u001b;{b), averaged over\nthe helix period, can be calculated numerically using am-\nplitudes#nand'n, see Appendix A for details.\nC. Phase diagram\nNow we summarize results on the equilibrium mag-\nnetization distribution. By comparing energies of di\u000ber-\nent states, we compute the energetically preferable states\nfor di\u000berent curvature and torsion values. The result-\ning phase diagram is presented in Fig. 4(a). There are\ntwo phases: (i) The quasi-tangential state is realized for\nrelatively small curvatures, when {<{b(\u001b); in such\na state the magnetization direction is close to the di-\nrection of easy{tangential anisotropy et, see Fig. 3(a,b)\nwith the limit vortex orientation in case of the ring wire\n(\u001c= 0). (ii) The onion state corresponds to the case,\nwhen{>{b(\u001b); the magnetization distribution is inho-\nmogeneous in accordance to (20), see Fig. 3(c).6\n0.5 1 1.5 200.511.52\nκ0\nReduced torsion σReduced curvature κ\nboundary curve κb(σ)\ninstability curve κc(σ)Quasi-tangential\nstateOnion\nstate\n0.2 0.4 0.6 0.8 100.20.40.60.81\nκ0\nReduced torsion σReduced curvature κboundary curve κb(σ)\ninstability curve κc(σ)\nNormal\nstateOnion state\n0 0.5 1 1.5 200.511.52\nReduced torsion σReduced curvature κboundary curve κb(σ)\ninstability curve κc(σ)\nQuasi-binormal\nstate\nOnion\nstate\n(a) Easy-tangential anisotropy (b) Easy-normal anisotropy (c) Easy-binormal anisotropy\nFIG. 4. (Color online) Phase diagram of equilibrium magnetization states for the helix wire with di\u000berent types of anisotropy.\nSymbols correspond to simulation data: green diamonds to homogeneous (in curvilinear reference frame) states and open circles\nto the onion ones. (a) Easy{tangential case, the curve {b(\u001b) (solid green line), calculated by (21) with N= 3, describes the\nboundary between the quasi-tangential and the onion states; the dashed-dot line corresponds to {b(\u001b) withN= 1. The curve\n{c(\u001b) (dashed red line) describes the boundary of linear instability of the quasi-tangential state, the dotted line is its \ftting\nby (33). In the region between lines {b(\u001b) and{c(\u001b) the quasi-tangential state is metastable. (b) and (c) correspond to\neasy-normal and easy-binormal anisotropy, respectively; all notations have the same sense as in (a). Note that Fig. (b) has\ndi\u000berent scale in order to show the normal state region in details.\nThe boundary between two phases {b={b(\u001b) can be\nderived using the condition\nEt(\u001b;{b) =Eon(\u001b;{b); (21)\nwhere Eonis energy density of the onion state averaged\nover the helix period 2 \u0019s0, see (A6). The onion solution\n(20) is energetically preferable when its energy is lower\nthan the energy of the quasi-tangential state (19). We\ncomputed the boundary curve numerically for N= 1 and\nN= 3, see dot-dashed and solid lines, respectively in the\nFig. 4(a). The obtained curves are very close, so the\napproximation N= 1 is reasonable. This is because the\nonion state of the helix wire is very close to an uniform\nmagnetization, see Fig. 3(c).\nFor the approximate description of the boundary de-\npendence we use the trial function\n{ET\nb=q\n{2\n0+ 2\u001b2; (22)\nwhich \fts the numerically calculated curve {b(\u001b) with an\naccuracy of about 5 \u000210\u00002.\nUsing the boundary dependence {b(\u001b), one can easily\ncompute domain of applicability of the quasi-tangential\nsolution (17):\n\u0012t2(\u0000\n\u0012t\nb;\u0019\n2\u0001\n;whenC\u001b>0;\u0000\u0019\n2;\u0012t\nb\u0001\n;whenC\u001b<0:(23a)\nHere\u0012t\nb=\u0012t\nb(\u001b) determines the boundary curve,\n\u0012t\nb(\u001b)\u0011\u0012t({b(\u001b);\u001b): (23b)IV. SPIN WAVE SPECTRUM IN A HELIX\nWIRE WITH EASY-TANGENTIAL\nANISOTROPY\nWe limit our consideration of spin waves by the case\nof the quasi-tangential magnetization state. First we lin-\nearize the Landau{Lifshitz equations (12) on the back-\nground of the quasi-tangential equilibrium state (17),\n\u0012(s;t) =\u0012t+#(s;t); \u001e (s;t) =\u001et+'(s;t)\nsin\u0012t:(24)\nThen for#and'we get the set of linear equations:\n@t0'=\u0000@\u0018\u0018#+V1#\u00002A@\u0018';\n\u0000@t0#=\u0000@\u0018\u0018'+V2'+ 2A@\u0018#;(25)\nwhere@t0is the derivative with respect to dimensionless\ntimet0= \n 0twith \n 0= 2K\r=Msand@\u0018is the derivative\nwith respect to dimensionless coordinate \u0018=s=w. Here\nV1is determined according to (17), the quantities V2and\nAhave the following form:\nV2=1 +{2+\u001b2+V1\n2;\nA=\u0000{cos\u0012t\u0000\u001bCsin\u0012t=\u0000\u001bCV2r\n2\nV1V0:(26)\nWhileV1andV2appear as scalar potentials, Aacts\nas a vector potential A=Aetof e\u000bective magnetic\n\feld. This becomes obvious if we combine the set of\nlinearized equations for #and'in a single equation for\nthe complex-valued function  =#+i',\n\u0000i@t0 =H +W \u0003; H = (\u0000i@\u0018\u0000A)2+U:(27a)7\nThis di\u000berential equation has a form of generalized\nScr odinger equation, originally proposed for the descrip-\ntion of spin waves on the magnetic vortex background.34\nThe \\potentials\" in Eq. (27a) read\nU=V1+V2\n2\u0000A2; W =V1\u0000V2\n2=\u00001 +w2Et\n2:\n(27b)\nAn e\u000bective magnetic \feld Ais originated from the cur-\nvature induced e\u000bective Dzyaloshinskii interaction, see\nEq. (7): the energy density ED\nex, harmonized using (24),\nreads35\nED\nex=\u00002\nw2Aj j2@\u0018arg : (28)\nNow we apply the traveling wave Ansatz for the spin-\nwave complex magnon amplitude\n (\u0018;t0) = uei\b+ ve\u0000i\b; \b =q\u0018\u0000\nt0+\u0011; (29)\nwithq=kwbeing the dimensionless wave number,\n\n =!=\n0the dimensionless frequency, \u0011is arbitrary\nphase, and u ;v2Rbeing constants. The corresponding\nwave vector is oriented along the wire, q=qet; its ori-\nentation with respect to the equilibrium magnetization\nis determined by Eq. (17). By substituting the Ansatz\n(29) into the generalized Scr odinger equation (27), one\ncan derive the spectrum of the spin waves:\n\n(q) = 2Aq+p\n(q2+V1) (q2+V2): (30)\nSimilar to the straight wire case with \n s(q) = 1 +q2,\nthe spectrum of spin waves in the helix wire has a gap,\ncaused, \frst of all, by the anisotropy (in dimensional\nunits the gap has an order of \n 0/K). However its\nvalue essentially depends on the curvature and the tor-\nsion. Moreover, the spectrum gap occurs at \fnite q=q0,\nsee Fig. 5. This means the asymmetry in the spectrum\nwith respect to the change q! \u0000q: spin waves have\ndi\u000berent velocities depending on the direction (along the\nhelix axis or in opposite direction). This asymmetry in\nthe dispersion law (30) occurs in the \frst term 2 Aq, which\nis originated from the e\u000bective Dzyaloshinskii interaction\nED\nex.\nIn this context it is instructive to mention that the\nspin wave spectrum in the presence of Dzyaloshinskii-\nMoriya interaction is known to be asymmetric with re-\nspect to wave vector inversion and has the minimum at\n\fnite wave vectors.25{27The curvature induced asymme-\ntry in the spin waves propagation in nanotubes and its\nanalogy with the Dzyaloshinskii-Moriya interaction was\ndiscussed recently in Ref. 36. The spin-wave spectrum\nfor the helix wire was calculated recently in Ref. 33, how-\never the deviations from the pure tangential state were\nno taken into account and the e\u000bective Dzyaloshinskii\nwas not considered.\nIn order to make analytical estimations, we consider\nnow the dispersion law in case of very small curvaturesand torsions:\n\n(q) = \n gap+ (q\u0000C\u001b)2+O\u0000\n{2;\u001b2;{\u001b\u0001\n;\n\ngap= 1\u0000{2\n2+O\u0000\n{2;\u001b2;{\u001b\u0001\n:(31)\nOne can see that the spin wave spectrum becomes asym-\nmetrical one with increasing the curvature and the tor-\nsion: the minimum of the frequency corresponds to\nq0=\u001bC(in dimensional units the corresponding wave\nnumberk0=\u001cC), its sign is determined by the product\nof the helix chirality and the magnetochirality.\nThe further increase of the curvature and torsion de-\ncrease the gap \n gap; there is a critical curve {c={c(\u001b),\nwhere the gap vanishes, \n( qc) = 0 and @q\n(qc) = 0.\nOne can easily \fnd that qc=Cp\nA2\u0000Uand the critical\ncurve{c={c(\u001b) can be found as a solution of algebraic\nequation\n4A2U=W2: (32)\nThe critical curve {c(\u001b), calculated numerically is plot-\nted in Fig. 4(a) (dashed red curve). For the approximate\ndescription of the critical dependence we use the trial\nfunction\n{trial\nc=p\n1 + 2\u001b2; (33)\nwhich \fts the numerical results with an accuracy of about\n5\u000210\u00003, see the dotted curve in Fig. 4(a). In the region\nbetween the boundary curve {b(\u001b) and the instability\ncurve{c(\u001b) [see Fig. 4] the quasi-tangential state be-\ncomes metastable.\nV. HELIX WITH OTHER ANISOTROPY\nORIENTATIONS\nLet us discuss other types of anisotropies: easy{normal\nand easy{binormal, see Eq. (9b) and Table I.\nA. Easy{normal anisotropy\nLet us start the analysis of the easy{normal anisotropy\nwith the limit case of the ring ( \u001c= 0). In this case,\nsimilarly to the easy{tangential anisotropy, the magneti-\nzation lies within the ring plane: \u0012=\u0019=2. The energy\nminimization with respect to \u001eresults in the pendulum\nequation:\n{2@\u001f\u001f\u001e+ sin\u001ecos\u001e= 0: (34)\nIn analogy with the easy{tangential anisotropy the equi-\nlibrium state is the exactly normal state \u001en=\u0006\u0019=2 in\ncase of relatively small reduced curvature {<{0and\ninhomogeneous onion solution \u001eon\nn(\u001f) =\u0019=2\u0000\u001eon(\u001f) for\n{>{0, where function \u001eon(\u001f) is de\fned by (14).8\n0 1 2 302468\nq0\nReduced wave numberr qReduced frequency Ωκ= 0,σ= 0\nκ= 0.5,σC= 0.5\nκ= 1.5,σC= 2\n0 0.5 1 1.501234\nReduced wave number qReduced frequency Ωκ= 0,σ= 0\nκ= 0.25,σC= 0.25\nκ= 0.45,σC= 0.45\n0 1 2 302468\nq0\nReduced wave number qReduced frequency Ωκ= 0,σ= 0\nκ= 0.5,σC= 0.5\nκ= 1.5,σC= 2\nReduced frequency Ω\nReduced wave number q0 1 2 302468\nκ= 0.5,σC= 0.5\n0 high\nReduced frequency Ω\nReduced wave number q0 0.5 1 1.501234\nκ= 0.25,σC= 0.25\n0 high\nReduced frequency Ω\nReduced wave number q0 1 2 302468\nκ= 0.5,σC= 0.5\n0 high(a) Easy–tangential anisotropy (b) Easy–normal anisotropy (c) Easy–binormal anisotropy\nFIG. 5. (Color online) Top row demonstrates dispersion laws for spin waves in the helix wire for di\u000berent anisotropies. The\nequilibrium states are homogeneous in the curvilinear reference frame. Symbols correspond to simulation data, see Sec. VI B,\nand lines to the analytics, see Eq. (30) and (39). Few examples of dispersion relation are shown at the bottom row in terms of\ndensity plots to demonstrate that (30) is a single frequency branch in the system.\nIn case of \fnite torsion there also exists exactly normal\nstate\n\u0012n=\u0019\n2; \u001en=C\u0019\n2; En=\u00001\u0000{2\u0000\u001b2\nw2;(35)\nwhere C=\u00061, see Fig. 6(a). Such a state is energet-\nically preferable for relatively small values of {and\u001b.\nThe magnetization in the normal state is directed ex-\nactly radially, which is well pronounced in experiments\nwith 3D microhelix coil strips.17\nIn case of large curvature, there is the periodic (in\ncurvilinear reference frame) onion solution, which has the\nform (20), see Fig. 6(b). Using the same numerical pro-\ncedure as in Sec. III B, we evaluate the onion solution\nand compute the phase diagram, see Fig. 4(b).\nFor the approximate description of the boundary\n{EN\nb(\u001b) between two phases we use the \ftting function\n{EN\nb={0s\n1\u0000\u0012\u001b\n\u001b0\u00132\n; \u001b 0\u00190:67; (36)\nwhich \fts the numerically calculated curve {EN\nb(\u001b) with\nan accuracy of about 3 \u000210\u00003.\n(a) Normal state: {=\u001b= 0:45\n(b) Onion state: {=\u001b= 1\nFIG. 6. (Color online) Magnetization distribution in the helix\nwire with C= +1 and easy-normal anisotropy according to\nsimulations data, see Sec. VI A.\nLet us discuss now the linear excitations on the back-\nground of the normal solution. Using the same approach9\nas in Sec. IV, we linearize Landau{Lifshitz equations (12)\non the background of the normal solution (35), \u0012=\u0012n+#,\n\u001e=\u001en+'. After linearization one gets a general-\nized Scr odinger{like equation for the complex variable\n =#+i',\n\u0000i@t0 = (\u0000@\u0018\u0018+Un) +Wn \u0003: (37a)\nHere the \\potentials\" read\nUn= 1\u0000{2+\u001b2\n2; Wn=1\n2(C\u001b\u0000i{)2:(37b)\nLet us compare this equations with the generalized\nScr odinger{like equation (27). First of all, there is no\ne\u000bective vector potential, since there is no asymme-\ntry by e\u000bective Dzyaloshinskii interaction like in easy{\ntangential case. The second di\u000berence is that the poten-\ntialWin (37b) is a complex{valued one, hence the scat-\ntering problem is similar to the two{channel scattering\nprocess. Similar to (29) we apply the following traveling\nwave Ansatz for the spin-wave complex magnon ampli-\ntude\n (\u0018;t0) = 1ei\b+ 2e\u0000i\b;\b =q\u0018\u0000\nt0+\u0011;  1;22C:\n(38)\nThe di\u000berence is that constants  1;2are complex ones.\nNow by substituting the Ansatz (38) into the generalized\nScr odinger equation (37), one can derive the spectrum of\nthe spin waves:\n\n(q) =p\n(1 +q2) (1 +q2\u0000{2\u0000\u001b2): (39)\nThis dispersion relation is reproduced by the numerical\nsimulations with a high accuracy, see Fig. 5(b). The crit-\nical dependence, where the gap of the spectrum vanishes,\nreads\n{c=p\n1\u0000\u001b2; (40)\nsee thick dashed curve in Fig. 5(b). In the region between\nsolid and dashed curves the normal state is metastable.\nThe dispersion law (39) is symmetric with respect to\nthe direction of the wave propagation: \n( q) = \n(\u0000q).\nUnlike the easy-tangential case there is no e\u000bective mag-\nnetic \feldA, because the curvature induced e\u000bective\nDzyaloshinskii interaction is absent in the harmonic ap-\nproximation, cf. (28). The reason is that the equilibrium\nstate is magnetized exactly in the normal direction en,\nwhich causes the degeneracy with respect to the sign of q.\nA similar behavior is known for thin \flms in the presence\nof Dzyaloshinskii{Moriya interaction, where the asymme-\ntry in the spin wave spectrum vanishes if the system is\nsaturated perpendicularly to the \flm plane.26\nB. Easy{binormal anisotropy\nIf the anisotropy axis is directed along eb, one has the\neasy{binormal anisotropy, EEB\nan, see (9b). The magneti-\nzation of the homogeneous (in the curvilinear reference\n(a) Quasi-binormal state: {=\u001b= 1:5\n(b) Onion state: {= 1; \u001b= 2\nFIG. 7. (Color online) Magnetization distribution in the helix\nwire with C= +1 and easy-binormal anisotropy according to\nsimulations data, see Sec. VI A.\nframe) state reads\ntan 2\u0012b=2C{\u001b\n1 +{2\u0000\u001b2; cos\u001eb=C=\u00061;(41)\nExplicitly\u0012breads\n\u0012b=\u0019\n2[1 + sgn ( C\u001b)]\u0000arctan2C\u001b{\nVb\n0;\nVb\n0= 1 +{2\u0000\u001b2+Vb\n1;\nVb\n1=p\n(1 +{2\u0000\u001b2)2+ 4{2\u001b2:(42)\nThe magnetization of this state is close to the direction\nof the helix axis, hence we name it quasi{binormal state,\nsee Fig. 7(a). It corresponds to the hollow{bar magneti-\nzation distribution in the helix microcoils.17For di\u000berent\nmagnetization distributions see also Table I.\nThe energy of the axial state reads\nEb=\u00001\u0000{2\u0000\u001b2+Vb\n1\n2w2: (43)\nLet us mention the formal analogy between the energy\nEb, the \\potentials\" Vb\n0,Vb\n1for the quasi-binormal state\nand the corresponding expressions Et[cf. (19)],Vt\n0,Vt\n1\n[cf. (17)] for the quasi-tangential state: the expressions\nfor the quasi-tangential state can be used for the quasi-\nbinormal one under the replacement {$\u001b.\nThe analogy between two states becomes deeper if we\nuse another parametrization for the magnetization m\nm= cos \u0002et\u0000sin \u0002 sin \ben+ sin \u0002 cos \b eb;(44)\nwhere \u0002 = \u0002( s) and \b = \b( s) are the angles in the\nFrenet{Serret frame of reference: the polar angle \u0002 de-\nscribes the deviation of magnetization from the tangen-\ntial curve direction, while the azimuthal angle \b corre-\nsponds to the deviation from the binormal. Similar to10\n(9), one can rewrite the energy terms as follows (cf. Ap-\npendix A from the Ref. 11 for details):\nEex= [\u00020\u0000\u0014sin \b]2+ [sin \u0002(\b0+\u001c)\u0000\u0014cos \u0002 cos \b]2;\nEEB\nan=\u0000sin2\u0002 cos2\b\nw2:\n(45)\nNow one can easily see that the energy functional of the\neasy{tangential magnet transforms to the energy func-\ntional of the easy{binormal magnet under the following\nconjugations: \u0012!\u0002,\u001e!\b, and {$\u001b.\nSimilarly to the easy{tangential case, there exist\ntwo equilibrium states: the homogeneous state (quasi-\nbinormal) and the periodic onion solution, see Fig. 7(b).\nThe phase diagram, which separates these two states, is\nplotted in the Fig. 4(c).\nNow we discuss the magnons for the easy{binormal\ncase. In analogy with the easy{tangential case, the\nlinearized equations can be reduced to the generalized\nScr odinger equation (27a) with the following \\poten-\ntials\":\nVb\n2=1 +{2+\u001b2+Vb\n1\n2;\nAb=\u0000{cos\u0012b\u0000\u001bCsin\u0012b=\u0000{CVb\n2s\n2\nVb\n1Vb\n0:(46)\nThe dispersion law has formally the form (30) with the\ncorresponding \\potentials\" described above. The disper-\nsion curve is plotted in the Fig. 5(c) for some typical\nparameters, it is con\frmed by the numerical simulations.\nThe critical curve {c(\u001b), where the gap of the spectrum\nvanishes, can be found numerically using condition (32).\nThe critical curve {c(\u001b), calculated numerically is plot-\nted in Fig. 4(c) (dashed red curve). For the approximate\ndescription of the critical dependence we use the trial\nfunction\n{trial\nc=r\n\u001b2\u00001\n2; (47)\nwhich \fts the numerical results of Fig. 4(c) with an accu-\nracy of about 2\u000210\u00002, see the dotted curve in Fig. 4(c).\nIn the region between solid and dashed curves the quasi-\nbinormal state is metastable.\nVI. SIMULATIONS\nIn order to verify our analytical results we numerically\nsimulate the magnetization dynamics of a helix-shaped\nchain of discrete magnetic moments miwithi=1;N.\nThe form of the chain is described by Eq. (11b). Mag-\nnetization dynamics of this system is determined by the\nset of Landau{Lifshitz equations\n1\n!0dmi\ndt=mi\u0002@E\n@mi+\u000bmi\u0002\u0014\nmi\u0002@E\n@mi\u0015\n;(48)where!0= 4\u0019\rMs,\u000bis the damping coe\u000ecient, Eis the\ndimensionless energy, normalized by 4 \u0019M2\ns\u0001s3with \u0001s\nbeing the sampling step of the natural parameter s. We\nconsider four contributions to the energy of the system:\nE=Eex+Ean+Ef+Ed: (49a)\nThe \frst term in Eq. (49a) is the exchange energy\nEex=\u00002`2\n\u0001s2N\u00001X\ni=1mi\u0001mi+1 (49b)\nwith`=p\nA=(4\u0019M2s) being the exchange length. The\nsecond term determines the uniaxial anisotropy contri-\nbution\nEan=\u0000Q\n2NX\ni=1(mi\u0001ean\ni)2; (49c)\nwhereean\niis the coordinate dependent unit vector along\nthe anisotropy axis and Qis the quality factor, see (4).\nThe third term determines interaction with the external\nmagnetic \feld b\nEf=\u0000NX\ni=1bi\u0001mi; (49d)\nwherebiis the dimensionless external \feld, normalized\nby 4\u0019Ms.\nThe last term in (49a) determines the dipolar interac-\ntion\nEd=(\u0001s)3\n8\u0019NX0\ni;j=1mi\u0001mj\njrijj3\u00003(mi\u0001rij) (mj\u0001rij)\njrijj5:(49e)\nwhererij\u0011\ri\u0000\rj.\nThe dynamical problem is considered as a set of 3 N\nordinary di\u000berential equations (48) with respect to 3 N\nunknown functions mx\ni(t); my\ni(t); mz\ni(t) withi=1;N.\nFor a given initial conditions the set (48) is integrated\nnumerically. During the integration process the condition\njmi(t)j= 1 is controlled.\nWe considered the helix wire with length L= 500\u0001s,\nthe exchange length `= 3\u0001sand the quality factor Q= 2\nare \fxed. The curvature \u0014and the torsion \u001cwere varied\nunder the restriction \u0014\u0001s=\u0019\u001c1.\nIn most of simulations we neglect magnetic dipolar in-\nteraction and consider the Heisenberg magnet with the\nenergy\nEH=Eex+Ean+Ef: (50)\nA. Equilibrium magnetization states\nWe start our simulations with easy{tangential mag-\nnets. In Sec. III A we found that the curvature and the11\ntorsion causes the deviation of the magnetization from\nthe anisotropy direction, which results in the magnetiza-\ntion distribution (17); such results are presented in Fig. 2\nby the curves for three di\u000berent value of the reduced cur-\nvature {= 0:1;{c;1 in the wide range of the torsion\n\u001b2(0; 2). In order to verify our theoretical predictions\nwe simulate numerically Landau{Lifshitz equations (48)\nin overdamped regime ( \u000b= 0:1) during a long time in-\nterval \u0001t\u001d(\u000b!0)\u00001.\nNumerically we model the anisotropic Heisenberg mag-\nnet with the energy (50) and Q= 2. Simulation data are\npresented in Fig. 2 by \flled symbols and labeled as (a);\none can see an excellent agreement between out theory\nand simulations. The typical magnetization distribution\nis shown in Figs. 3(a), (b) and (c) for the quasi-tangential\nstates and in Figs. 3(d) for the onion state.\nWe also perform simulations for other anisotropy\ntypes. The magnetization distribution for the helix wire\nwith easy-normal anisotropy is presented in Fig. 6(a) for\nthe normal state and Fig. 6(b) for the onion one. For\nthe case of easy-binormal anisotropy one has two possi-\nble states: the quasi-binormal one [see Fig. 7(a)] and the\nonion one [see Fig. 7(b)].\nThe second stage of our simulations is to \fnd the equi-\nlibrium magnetization state of a given helix wire. Numer-\nically we simulate Eqs. (48) as described above for \fve\ndi\u000berent initial states, namely the tangential, onion, nor-\nmal, binormal, and the random states. The \fnal static\nstate with the lowest energy is considered to be the equi-\nlibrium magnetization state. We obtain that for each\ntype of anisotropy the equilibrium state is either onion\none or anisotropy-aligned state (quasi-tangential, nor-\nmal and quasi-binormal state for easy-tangential, easy-\nnormal and easy-binormal anisotropy, respectively). We\npresent simulations data in Fig. 4 by symbols together\nwith theoretical results (plotted by lines). One can see a\nvery good agreement between simulations and analytics.\nB. Dispersion relations\nFor each anisotropy-aligned equilibrium state the\nmagnon dispersion relation is obtained numerically. It\nis carried out in two steps. In the \frst step the helix wire\nis relaxed in external spatially nonuniform weak magnetic\n\feld\nbj\ni=b0ed\nicossikj\nfor a range of wave-vectors kj=j=(300\u0001s) withj=\n0;300. Hereb0\u001c1 is the \feld amplitude, si= (i\u00001)\u0001s\nis position of the magnetic moment mi. The coordinate\ndependent unit vector ed\nidetermines the magnetic \feld\ndirection:ed\ni=enfor the quasi-tangential state and\ned\ni=etfor normal and quasi-binormal states.\nIn the second step we switch o\u000b the magnetic \feld and\nsimulate the magnetization dynamics with the damping\nvalue\u000b= 0:01 close to natural one. Then the space-timeFourier transform is performed for one of the magneti-\nzation components (we consider normal component for\nthe quasi-tangential state and tangential component for\nother two equilibrium states). The frequency \n which\ncorresponds to the maximum of the Fourier signal is\nmarked by a symbol for a given wave-vector qj=wkj,\nsee the top row of the Fig. 5. The absence of additional\npeaks in the spectrum is demonstrated by the dispersion\nmaps below, see bottom raw of Fig. 5.\nVII. DISCUSSION\nWe have performed a detailed study of statics and lin-\near dynamics of magnetization in the helix wire with dif-\nferent anisotropy. We have limited our study by hard\nmagnets, which can be well described by the model of\nanisotropic Heisenberg magnets. Our study was limited\nby the condition (5).\nLet us discuss how our model can be generalized\nwith account of the long range magnetostatics e\u000bects.\nThe non-local magnetostatic interaction for thin wires\nof circular and square cross-sections is known31to be\ncompletely reduced to a local e\u000bective easy{tangential\nanisotropy. It is important that such a conclusion sur-\nvives for the case of curved wires.31Thus the magneto-\nstatic interaction can be taken into account as additional\nanisotropy. In general, one has to consider the model\nof biaxial magnet. Here we limit ourselves by the helix\nwire with easy{tangential magnetocrystalline anisotropy.\nIn this case the magnetostatics e\u000bects can be taken into\naccount by a simple rede\fnition of the anisotropy con-\nstants, leading to a new magnetic length\nK!Ke\u000b=K+\u0019M2\ns;\nw!we\u000b=r\nA\nKe\u000b=2`p1 + 2Q:(51)\nThus our model (6) is also suitable for thin wires made\nof a magnetically softmaterial under the restriction\nh\u001cw; `;1\n\u0014;1\n\u001c: (52)\nIn order to check our predictions about e\u000bective\nanisotropy we perform numerical simulations with ac-\ncount of the nonlocal dipolar interaction as described in\nSec. VI. Numerically we integrate Eqs. (48) with the en-\nergy (49).\nFirst, we simulate the anisotropic wire with account\nthe dipolar interaction with the energy (49). In this case\nwe need to modify the magnetic length according to (51).\nThus we also need to rede\fne the reduced curvature and\ntorsion as follows\n{!{e\u000b=\u001cwe\u000b; \u001b!\u001be\u000b=\u001cwe\u000b: (53)\nFor the case Q= 2, one get `=wandwe\u000b= 2`=p\n5. One\ncan see that we have a very nice agreement between the12\nanalytical results (17) and simulations data, see yellow\nsymbols in Fig. 2; we label these data as (am).\nThe second kind of simulations with account of the\ndipolar interaction was aimed to verify the validity of\nour approach for softmagnets with Q= 0. For this pur-\npose we model the soft isotropic wire with account of the\ndipolar interaction. According to (51) we get we\u000b= 2`.\nSimulations data are presented in Fig. 2 by dotted sym-\nbols [labeled as (m)] for curvature and torsion rede\fned\naccording to (53). By comparing simulations data with\nanalytical results one can see the pretty good agreement\nin the wide range curvatures and torsions. Our simu-\nlations data for soft magnet di\u000ber from the theoretical\npredictions for hard magnets only for relatively high cur-\nvature in the vicinity of the boundary with the onion\nstate.\nThus we can conclude that our model of anisotropic\nHeisenberg magnet is physically sound also for thin wires\nmade of a magnetically soft material.\nIn conclusion, we have presented a detailed study of\nstatics and linear dynamics of magnetization in the he-\nlix wire. We have described equilibrium magnetization\nstates for three types of uniaxial anisotropy, according to\npossible curvilinear directions. All three cases have been\nrealized experimentally in rolled{up ferromagnetic micro-\nhelix coils.17We have calculated the phase diagram of\npossible states in case of easy-tangential anisotropy: the\nquasi-tangential con\fguration (17) is energetically prefer-\nable for the strong anisotropy case. In this case the de-\nviations from the strictly tangential direction (corkscrew\norientation17) are caused by the torsion, the direction\nof the deviation depends on both helix chirality and\nthe magnetochirality of the magnetization structure, see\nEq. (18). In case of high curvature there is the onion equi-\nlibrium state (20) in analogues to the onion state in mag-\nnetic ring wires37,38. The magnetization distribution (41)\nof the quasi-binormal state is directed almost along the\nbinormal (hollow{bar orientation17). In contrast to the\nquasi{tangential state and quasi{binormal one (which\nare realized for the easy{tangential and easy{binormal\nmagnets, respectively), the normal state for the easy{\nnormal magnets has several peculiarities: (i) it has the\nform of exact normal magnetization distribution along\nthe normal direction en, see (35); (ii) the normal state\nphase is realized for small curvatures and torsions only:\n{2={2\n0+\u001b2=\u001b2\n0<1, see Fig. 4(b); (iii) the spectrum of\nspin waves on the normal state background is symmetric\nwith respect to the direction of the wave propagation.\nThe torsion of the wire manifests itself in the magne-\ntization dynamics: an e\u000bective magnetic \feld, induced\nby the torsion breaks the mirror symmetry with the spin\nwave direction. The dispersion law of spin waves (30) is\nessentially a\u000bected by this \feld.\nThere is a connection between the helix geometry and\nthe tube one: when the helix pitch vanishes, we have a\nclose-coiled solenoid magnet, which properties are simi-\nlar to the thin shell nanotube. The spin-wave spectrum\nin the nanotube is known39to have a gap, caused by thecurvature. This conclusion is in agreement with the dis-\npersion law for the helix wire, see Fig. 5(a). One has to\nnote that the analogy between two systems is adequate\nunder the restriction of vanishing torsions ( \u001b!0); this\nexplains the absence of the linear shift in the dispersion\nlaw for the nanotube in comparison with (30). In gen-\neral the transition from 1D systems to 2D requires more\naccurate account of the dipolar interaction.\nWe considered the simplest example of the curved wire\nwith constant curvature and torsion. Our results can\nbe generalized for the case of variables parameters \u0014(s)\nand\u001c(s). To summarize we can formulate few general\nremarks about the curvature and torsion e\u000bects in the\nspin wave dynamics. The linear magnetization dynamics\ncan be described by the generalized Scr odinger equation\n(27). In case of the straight wire, one has the standard\nScr odinger equation for the complex magnon amplitude\n with the typical potential scattering. Th curvature\ninduces an additional e\u000bective potential, the `geometri-\ncal potential'.29This is described by the modi\fcation of\ne\u000bective potential Uin Eq. (27b). Besides, there is a\ncurvature induced coupling potential W: the problem be-\ncomes di\u000berent in principle from the usual set of coupled\nScr odinger equations, see the discussion in Ref.34. Due to\nthe torsion in\ruence there appears an e\u000bective magnetic\n\feld. The vector potential of this \feld is constant for\nthe helix wire, see (26), hence the e\u000bective magnetic \rux\ndensityB=r\u0002Avanishes. Nevertheless the presence\nof magnetic \feld with the vector potential Abreaks the\nmirror symmetry of the problem: the motion of magnetic\nexcitations in di\u000berent spatial direction is not identical.\nLet us mention the connection between the vector po-\ntential and the e\u000bective Dzyaloshinskii interaction: the\ntotal energy of the Dzyaloshinskii interaction ED\nex/R\ndsA\u0001jwith the current j=j j2rarg , see Eq. (28).\nUsing an explicit form of the integrand one can \fnd that\nED\nex/\u001bqC, which re\rects the relation between the topol-\nogy of the wire (namely, helix chirality) with the topology\nof the magnetic structure (namely, the magnetochiral-\nity). In this context it is instructive to note that there is\na deep analogy between the Dzyaloshinskii-Moriya inter-\naction and the Berry phase theory40.\nWe expect that our approach can be easily general-\nized for the arbitrary curved wires, where all potentials\nbecomes spatially dependent: U(s),W(s), andA(s). De-\npending on the curvature and the torsion these potentials\ncan repel or attract magnons. In latter case there can ap-\npear a well with possible bound states, i.e. local modes.\nACKNOWLEDGMENTS\nThe authors thank D. Makarov for stimulating dis-\ncussions and acknowledge the IFW Dresden, where part\nof this work was performed, for kind hospitality. The\npresent work was partially supported by the Program\nof Fundamental Research of the Department of Physics\nand Astronomy of the National Academy of Sciences of13\nUkraine (project No. 0112U000056). D.D.S. acknowl-\nedges the support from the Alexander von Humboldt\nFoundation.\nAppendix A: Onion-state solution\nWe start from the static form of Landau{Lifshitz equa-\ntions (12):\nF(\u0012;\u001e) = 0; G (\u0012;\u001e) = 0 (A1)\nwithFandGbeing the nonlinear operators,\nF(\u0012;\u001e) =\u0000@\u001f\u001f\u0012\u0000\u001bcos\u001e\u0000\n{cos 2\u0012\u00002@\u001f\u001esin2\u0012\u0001\n+ sin\u0012cos\u0012h\n({+@\u001f\u001e)2\u0000(1 +\u001b2)cos2\u001ei\n;\nG(\u0012;\u001e) = sin2\u0012\u0002\n\u0000@\u001f\u001f\u001e+ (1 +\u001b2)sin\u001ecos\u001e\n\u00002\u001b@\u001f\u0012cos\u001e] + sin\u0012cos\u0012[{\u001bsin\u001e\u00002@\u001f\u0012({+@\u001f\u001e)]:\n(A2)\nBy substituting here the expansion (20) in the form\n\u0012(\u001f) =\u0019\n2+\"NX\nn=1#ncos(2n\u00001)\u001f;\n\u001e(\u001f) =\u0000\u001f+\"NX\nn=1'nsin 2n\u001f;(A3)\nand expanding results into series over \"up to theN-th\norder, one get the Fourier expansion of operators FandGas follows\nF(\u0012;\u001e) =NX\nn=1Fn(#1;:::;#n;'1;:::;'n) cos(2n\u00001)\u001f;\nG(\u0012;\u001e) =NX\nn=1Gn(#1;:::;#n;'1;:::;'n) sin 2n\u001f:\n(A4)\nHereFnandGnare polynomials of the order Nwith re-\nspect to#kand'k. Then the Landau{Lifshitz equations\n(A1) results in the set of nonlinear polynomial equations\nFn(#1;:::;#n;'1;:::;'n) = 0\nGn(#1;:::;#n;'1;:::;'n) = 0;n=1;N; (A5)\nwhich can be solved numerically on #kand'kwith any\nprecision.\nIn order to calculate the energy of the onion state, we\nsubstitute the magnetization angles \u0012and\u001ein the form\n(A3) into the energy density (9), expand the results over\n\"up to the 2 N-th order and average the result over the\nhelix period,\nEon(\u001b;{) =1\n2\u00192\u0019Z\n0Ed\u001f;\nE=Eex+EET\nan=E(#1;:::;#n;'1;:::;'n):(A6)\n\u0003Corresponding author:sheka@univ.net.ua\nyvkravchuk@bitp.kiev.ua\nzyershov@bitp.kiev.ua\nxybg@bitp.kiev.ua\n1D. Makarov, C. Ortix, and L. Baraban, SPIN , 1302001\n(2013).\n2A. Saxena, R. Dandolo\u000b, and T. Lookman, Physica A:\nStatistical and Theoretical Physics 261, 13 (1998).\n3V. L. Carvalho-Santos and R. Dandolo\u000b, Brazilian Journal\nof Physics 43, 130 (2013).\n4V. L. Carvalho-Santos, A. R. Moura, W. A. Moura-Melo,\nand A. R. Pereira, Phys. Rev. B 77, 134450 (2008).\n5M.-W. Yoo and S.-K. Kim, Applied Physics Express 8,\n063003 (2015).\n6V. P. Kravchuk, D. D. Sheka, R. Streubel, D. Makarov,\nO. G. Schmidt, and Y. Gaididei, Phys. Rev. B 85, 144433\n(2012).\n7R. Streubel, V. P. Kravchuk, D. D. Sheka, D. Makarov,\nF. Kronast, O. G. Schmidt, and Y. Gaididei, Appl. Phys.\nLett. 101, 132419 (2012).\n8D. D. Sheka, V. P. Kravchuk, M. I. Sloika, and Y. Gaididei,\nSPIN 3, 1340003 (2013).\n9R. Streubel, D. J. Thurmer, D. Makarov, F. Kronast,\nT. Kosub, V. Kravchuk, D. D. Sheka, Y. Gaididei,\nR. Schaefer, and O. G. Schmidt, Nano Letters 12, 3961\n(2012), http://pubs.acs.org/doi/pdf/10.1021/nl301147h.\n10Y. Gaididei, V. P. Kravchuk, and D. D. Sheka, Phys. Rev.Lett. 112, 257203 (2014).\n11D. D. Sheka, V. P. Kravchuk, and Y. Gaididei, Journal\nof Physics A: Mathematical and Theoretical 48, 125202\n(2015).\n12V. L. Carvalho-Santos, R. G. Elias, J. M. Fonseca, and\nD. Altbir, Journal of Applied Physics 117, 17E518 (2015).\n13P. Landeros and \u0013Alvaro S. N\u0013 u~ nez, Journal of Applied\nPhysics 108, 033917 (2010).\n14J. Ot\u0013 alora, L.-L. J.A., P. Vargas, and P. Landeros, Applied\nPhysics Letters 100, 072407 (2012).\n15M. Yan, C. Andreas, A. Kakay, F. Garcia-Sanchez, and\nR. Hertel, Applied Physics Letters 100, 252401 (2012).\n16O. V. Pylypovskyi, V. P. Kravchuk, D. D. Sheka,\nD. Makarov, O. G. Schmidt, and Y. Gaididei, \\Magne-\ntochiral symmetry breaking in a M obius ring,\" ArXiv e-\nprints (2014), 1409.2049.\n17E. J. Smith, D. Makarov, S. Sanchez, V. M. Fomin, and\nO. G. Schmidt, Phys. Rev. Lett. 107, 097204 (2011).\n18E. J. Smith, D. Makarov, and O. G. Schmidt, Soft Matter\n7, 11309 (2011).\n19R. Streubel, J. Lee, D. Makarov, M.-Y. Im, D. Kar-\nnaushenko, L. Han, R. Sch afer, P. Fischer, S.-K. Kim, and\nO. G. Schmidt, Advanced Materials 26, 316 (2014).\n20I. M onch, D. Makarov, R. Koseva, L. Baraban, D. Kar-\nnaushenko, C. Kaiser, K.-F. Arndt, and O. G. Schmidt,\nACS Nano 5, 7436 (2011).14\n21F. Balhorn, S. Mansfeld, A. Krohn, J. Topp, W. Hansen,\nD. Heitmann, and S. Mendach, Physical Review Letters\n104, 037205 (2010).\n22F. Balhorn, C. Bausch, S. Jeni, W. Hansen, D. Heitmann,\nand S. Mendach, Phys. Rev. B 88, 054402 (2013).\n23A. A. Solovev, S. Sanchez, M. Pumera, Y. F. Mei, and\nO. G. Schmidt, Adv. Funct. Mater. 20, 2430 (2010).\n24K. E. Peyer, S. Tottori, F. Qiu, L. Zhang, and B. J. Nelson,\nChem. Eur. J. 19, 28 (2012).\n25K. Zakeri, Y. Zhang, J. Prokop, T.-H. Chuang, N. Sakr,\nW. X. Tang, and J. Kirschner, Phys. Rev. Lett. 104,\n137203 (2010).\n26D. Cortes-Ortuno and P. Landeros, Journal of Physics:\nCondensed Matter 25, 156001 (2013).\n27K. Zakeri, Physics Reports (2014),\n10.1016/j.physrep.2014.08.001.\n28A. Hubert and R. Sch afer, Magnetic domains: the analy-\nsis of magnetic microstructures (Springer{Verlag, Berlin,\n1998).\n29R. C. T. da Costa, Physical Review A 23, 1982 (1981).\n30V. S. Tkachenko, A. N. Kuchko, M. Dvornik, and V. V.Kruglyak, Applied Physics Letters 101, 152402 (2012).\n31V. V. Slastikov and C. Sonnenberg, IMA Journal of Ap-\nplied Mathematics 77, 220 (2012).\n32F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W.\nClark, eds., NIST Handbook of Mathematical Functions\n(Cambridge University Press, New York, NY, 2010).\n33V. S. Tkachenko, A. N. Kuchko, and V. V. Kruglyak, Low\nTemperature Physics 39, 163 (2013).\n34D. D. Sheka, I. A. Yastremsky, B. A. Ivanov, G. M. Wysin,\nand F. G. Mertens, Phys. Rev. B 69, 054429 (2004).\n35Up to a full derivative with respect to \u0018.\n36R. Hertel, SPIN 03, 1340009 (2013).\n37M. Kl aui, C. A. F. Vaz, L. Lopez-Diaz, and J. A. C. Bland,\nJournal of Physics: Condensed Matter 15, R985 (2003).\n38A. P. Guimar~ aes, Principles of Nanomagnetism ,\nNanoScience and Technology (Springer-Verlag Berlin\nHeidelberg, 2009).\n39A. Gonz\u0013 alez, P. Landeros, and \u0013Alvaro S. N\u0013 u~ nez, Journal\nof Magnetism and Magnetic Materials 322, 530 (2010).\n40F. Freimuth, S. Bl ugel, and Y. Mokrousov, J. Phys.: Con-\ndens. Matter 26, 104202 (2014)."    },    {        "title": "1606.09104v2.Dynamic_Elastic_Moduli_in_Magnetic_Gels__Normal_Modes_and_Linear_Response.pdf",        "content": "arXiv:1606.09104v2  [cond-mat.soft]  25 Aug 2016Dynamic Elastic Moduli in Magnetic Gels: Normal Modes and Li near Response\nGiorgio Pessot,∗Hartmut L¨ owen,†and Andreas M. Menzel‡\nInstitut f¨ ur Theoretische Physik II: Weiche Materie,\nHeinrich-Heine-Universit¨ at D¨ usseldorf, D-40225 D¨ uss eldorf, Germany\n(Dated: August 1, 2018)\nIn the perspective of developing smart hybrid materials wit h customized features, ferrogels and\nmagnetorheological elastomers allow a synergy of elastici ty and magnetism. The interplay between\nelastic and magnetic properties gives rise to a unique rever sible control of the material behavior by\napplying an external magnetic field. Albeit few works have be en performed on the time-dependent\nproperties so far, understanding the dynamic behavior is th e key to model many practical situations,\ne.g. applications as vibration absorbers. Here we present a way to calculate the frequency-dependent\nelastic moduli based on the decomposition of the linear resp onse to an external stress in normal\nmodes. We use a minimal three-dimensional dipole-spring mo del to theoretically describe the mag-\nnetic and elastic interactions on the mesoscopic level. Spe cifically, the magnetic particles carry\npermanent magnetic dipole moments and are spatially arrang ed in a prescribed way, before they\nare linked by elastic springs. An external magnetic field ali gns the magnetic moments. On the one\nhand, we study regular lattice-like particle arrangements to compare with previous results in the\nliterature. On the other hand, we calculate the dynamic elas tic moduli for irregular, more realistic\nparticle distributions. Our approach measures the tunabil ity of the linear dynamic response as a\nfunction of the particle arrangement, the system orientati on with respect to the external magnetic\nfield, as well as the magnitude of the magnetic interaction be tween the particles. The strength\nof the present approach is that it explicitly connects the re laxational modes of the system with\nthe rheological properties as well as with the internal rear rangement of the particles in the sample,\nproviding new insight into the dynamics of these remarkable materials.\nPACS numbers: 82.35.Np, 63.50.-x, 62.20.de, 75.80.+q\nI. INTRODUCTION\nThe class of smart hybrid materials encompassing fer-\nrogels and magnetorheological elastomers stands out for\ntheir unique capability of combining magnetic properties\nwith huge elastic deformability [1–4]. They typically con-\nsistofapermanentlycrosslinkedpolymermatrixinwhich\nmagnetic colloidal particles are embedded. The matrix\nis responsible for the elastic behavior typical of rubbers,\nwhile the particles magnetically interact with each other\nandwith externalmagneticfields. Thesematerialsdistin-\nguish themselves by the fascinating ability of reversible\non-demand tunability of shape and stiffness under the in-\nfluence of external magnetic fields [1, 2, 4–12] similarly\nto the tunability of viscosity in ferrofluids [13–23]. This\nmakes them ideal candidates for applications as soft ac-\ntuators [24], vibration absorbers [25, 26], magnetic field\ndetectors [27, 28], and even as model systems to study\naspects of hyperthermal cancer treatment [29, 30].\nThe core feature of these materials is their magneto-\nmechanicalcoupling[31–33], i.e. the waymagneticeffects\nsuch as the response to an external magnetic field couple\nto the overall mechanical properties (e.g. strain or elas-\ntic moduli) and vice versa. As was recently shown, such\ncoupling is responsible for surprising properties such as\n∗Electronic address: giorgpess@thphy.uni-duesseldorf.de\n†Electronic address: hlowen@thphy.uni-duesseldorf.de\n‡Electronic address: menzel@thphy.uni-duesseldorf.desuperelasticity [34], a characteristic buckling of chains of\nparticles under a perpendicular external magnetic field\n[35], qualitative reversal of the strain response [32], vol-\nume changes due to mesoscopic wrapping effects [36],\nor tunability of the electrical resistance [37]. There\nare several key factors that can influence the magneto-\nmechanicalcoupling: themagneticparticleconcentration\n[1, 38, 39], the stiffness of the gel [40], or whether the\nmagnetic moments of the particles can freely reorient or\nmust instead rotate synchronously with the whole parti-\ncle [2, 41]. The particles can be chemically bound to the\npolymernetwork[31,42,43]orbeconfinedinsidepockets\nof the matrix [44, 45]. Moreover, the magnetic material\nitselfcaneitherbe ferro-[43]or(super)paramagnetic[46].\nBecause of the variety of factors and parameters that\ncan characterize ferrogels and magnetic elastomers, it is\nno surprise that they are receiving increasing attention\nfrom the modeling side. In fact, gaining insight into\nthe mechanisms underlying the magneto-mechanical cou-\npling can be the key to devise smarter and more efficient\nmaterials. Macroscopic theories rely on a continuum-\nmechanical description of both the polymeric matrix and\nthe magneticcomponent [5,27,32, 47–50], whereasmeso-\nscopic approaches can take into account the granularity\nanddiscretenessofthe magneticparticles[34,51–53]. On\nthis mesoscopiclevel, simplified dipole-spring models rep-\nresent a convenient approach to address effects originat-\ning on the magnetic particle level. More precisely, in\nsuch models the particles carry a dipole magnetic mo-\nment and are linked with each other by a network of\nelasticsprings. Additionally,stericrepulsionandotheref-2\nfectslikeorientationalmemorytermscanbeincluded[54–\n59]. Finite-element descriptions are likewise employed to\naddress mesoscopic particle-based effects [10, 33, 60–63],\nand some works even resolve the individual polymers on\nthe microscopic scale [41, 64]. Moreover, in a coarse-\ngraining perspective, some routes have been outlined to\nconnect the different length scales listed above [65, 66].\nOften in material science, one aims at determining the\nmaterial parameters that characterize the system. Fun-\ndamental quantities to describe the time-dependent me-\nchanical behavior are the dynamic elastic moduli. They,\nfor instance, contain the information on the frequency-\ndependent stress response to imposed time-periodic de-\nformations. In the case of ferrogels and magnetic elas-\ntomers only few theoretical studies have so far addressed\nthe dynamic properties in special cases [56, 59, 67]. In\nthe present work we aim at calculating the dynamic (i.e.\nfrequency-dependent) elastic moduli of ferrogels. We use\na minimal three-dimensional (3D) dipole-spring model\nwith short-ranged steric repulsion between the magnetic\nparticles. Moreover, we consider the system around its\nequilibrium state of minimum total energy. Overdamped\nmotion of the particles is assumed, which is in general\na reasonable assumption for colloidal polymeric systems.\nWe focus on regular and more disordered particle ar-\nrangements of finite size with open boundary conditions\n(obc). In our particle-based approach this simply refers\nto a detached finite assembly of particles. This system\nis bounded in all three directions of space, in contrast\nto periodic boundary conditions ( pbc). We describe a\nsemi-analytical approach using a simple, direct connec-\ntion between the normal modes of the system and the\nlinear response to an oscillating external stress.\nThe paper is structured as follows. First, in section\nII we present our minimal dipole-spring model including\nsteric repulsion. To find the equilibrium configurations\nunder magnetic interactions, we use the methods as de-\nscribed in section III. Then, in section IV, we determine\nthe normal modes and in section V we connect them to\nthe static linear elastic response of the system. After\nthat, in section VI, we address the dynamic behavior of\nour system and show how to decompose it into the nor-\nmal modes. In section VII, we extend the elastic moduli\nexpressions obtained in section V to the dynamical case\nand show the corresponding numerical results in sections\nVIII, IX, and X before drawing our final conclusions in\nsectionXI. Appendix Aliststhespecificexpressionsused\nin modeling the steric repulsion, whereas appendices B\nand C list in detail the employed expressions for gradi-\nents and Hessian matrices. Appendix D describes in de-\ntail our procedure of obtaining a torque-free force field.\nIn appendix E we analytically estimate the Young mod-\nuli of regular lattices for comparison with our numerical\nresults. Last, in appendix F we present further data on\nthe loss components of the dynamic moduli, supporting\nour results in the main text.II. DIPOLE-SPRING MODEL\nFor simplicity we here work with a minimal 3D dipole-\nspring model. On the one hand, as a first approxima-\ntion, we represent the magnetic moments by permanent\npoint dipoles of constant magnitude. Possible magnetic\ncontributions due to the finite extension of the magnetic\nparticles are not considered. This is a valid approach for\ninterparticledistances largerthan the particle size (i.e. at\nlow densities) [68]. In a simplified manner, spatial varia-\ntions in dipole orientations and magnitudes due to their\nmutual feedback could be included in a subsequent step,\nsee Ref. [69]. On the other hand, the interaction between\nthe mesoscopic particles mediated by the polymeric ma-\ntrix is, in general, non-linear [66]. However, since we are\nmainlyinterestedinthelinearelasticmoduliforsmalldis-\nplacements around the equilibrium positions of the parti-\ncles, we confine ourselves to harmonic interactions in the\npresent study.\nOur system is made of Nidentical spherical magnetic\nparticles with positions Ri= (Rx\ni,Ry\ni,Rz\ni),i= 1...N.\nTo model the overdamped dynamics of the system, we\nconsider viscous drag forces −c˙Riduring particle dis-\nplacements, where the dot indicates the time derivative.\nEach particle carries an identical magnetic dipole mo-\nmentmof magnitude m=|m|. This situation re-\nflects, for instance, the case of ferromagnetic or super-\nparamagnetic particles under strong external magnetic\nfields. Neighboring particles iandjare coupled by har-\nmonic springs attached to the particle centers for sim-\nplicity. The unstrained spring length ℓ0\nijis set in the\ninitial ground state particle configuration in the absence\nof any magnetic interactions, while the spring constants\nare given by k/ℓ0\nij. Thus,kis related to the overall elas-\ntic modulus of the system and long springs are weakened\nwhen compared to short ones. We assume the polymeric\nmatrix—here represented by the network of springs—to\nhave vanishing magnetic susceptibility and therefore not\nto directly interact with magnetic fields. If magnetic par-\nticles come too close to each other, they interact steri-\ncally.\nThe total energy Uof the system is the sum of elastic\nUel, stericUs, and magnetic Umenergies [54, 55, 57, 58].\nElastic interactions are given by\nUel=1\n2/summationdisplay\ni/negationslash=jkij\n2/parenleftbig\nrij−ℓ0\nij/parenrightbig2, (1)\nwhere the sum runs over all particles iandj∝\\e}atio\\slash=i. More-\nover,kij=k/ℓ0\nijif particles iandjare connected by\na spring and vanishes otherwise. Furthermore, rij=\nRj−Riandrij=|rij|.\nWe model the steric interactions using a repulsive po-\ntential inspired by the Weeks-Chandler-Andersen form\n[70] but with different exponents. For instance, possibly\nabsorbed polymer chains on the surfaces of the particles\n[35]couldresultinasofterrepulsion. Ourstericpotential3\nreads\nUs=1\n2/summationdisplay\ni/negationslash=jvs(rij), (2)\nwhere\nvs(r) =εs/bracketleftBig/parenleftBigr\nσs/parenrightBig−4\n−/parenleftBigr\nσs/parenrightBig−2\n−/parenleftBigrc\nσs/parenrightBig−4\n+/parenleftBigrc\nσs/parenrightBig−2\n+cs(r−rc)2\n2/bracketrightBig\n(3)\nforr≤rcand zero otherwise. Here, εssets the strength\nofthe stericrepulsion, σscharacterizesthe rangeofsteric\nrepulsion, and rc=σs21/2is a cutoff distance. The\nparameter csis chosen such that altogether we have\nvs(rc) = 0,vs′(rc) = 0, and vs′′(rc) = 0 (see Appendix\nA).\nFinally, the magnetic energy is given by the dipole–\ndipole interaction\nUm=µ0m2\n4π1\n2/summationdisplay\ni/negationslash=jr2\nij−3(/hatwiderm·rij)2\nr5\nij,(4)\nwhere/hatwiderm=m/mandµ0is the magnetic permeability\nof vacuum. In the present work, we use reduced units as\nfollows: lengths are given in multiples of l0, energies in\nmultiples of kl0. The length l0is defined as l0=3/radicalbig\n1/ρ\nwhereρis the number density of the particles. Further-\nmore, we measure magnetic moments, velocities, and fre-\nquencies in multiples of m0=/radicalBig\n4πkl04/µ0,k/c, and\nk/cl0, respectively, with csetting the viscous friction co-\nefficient of each particle.1For our purposes, we assume\nσs= 0.2l0andεs=kl0.\nForreasonsthatwillbecomeclearinsectionV, itisuse-\nful to explicitly define and indicate the boundaries of our\nsystem. We here consider samples of cubelike shape with\nfaces perpendicular to /hatwidex,/hatwidey, and/hatwidez, the unit vectors defin-\ning our Cartesiancoordinatesystem. We can define “left”\nand “right”, “front” and “rear”, as well as “bottom” and\n“top” boundaries, namely the faces oriented by ∓/hatwidex,∓/hatwidey,\nand∓/hatwidez, respectively. Thecriteriatoidentify which parti-\nclesbelongtotheboundarieswillbedetailedlateraccord-\ning to the specific particle distribution. Subsequently, we\nindicate by Lx,Ly, andLzthe extension of the sample\nin thex-,y-, andz-direction, respectively. In the case\nof cubelike shape and uniform density, Lα(α=x,y,x)\nwill be proportional to N1/3l0. Otherwise, an additional\ngeometry-dependentprefactorcan be included. Then the\nscaling of cross-sectional areas (i.e. Sx=LyLz) and the\nvolumeV=LxLyLzfollow straightforwardly as N2/3l02\nandNl03, respectively.\n1There is a typo in the definition of m0in Ref. [55]: it should\nreadm0=/radicalBig\n4πkl05/µ0instead of m0=/radicalBig\n4πk2l05/µ0.III. EQUILIBRIUM STATE\nFirst, we need to find the equilibrium state of our\nsystem, i.e. the one that minimizes the total energy\nU=Uel+Us+Umwith respect to all degrees of free-\ndom. In our case the degrees of freedom are given by the\npositions Ri, which requires\n∂RiU=0,∀i= 1...N (5)\nin equilibrium. From Eqs. (1)–(4) it is straightforwardto\ncalculate the resulting gradients (see Appendix B). The\nsecond derivatives of the energy Uform the correspond-\ning Hessianmatrix, seebelow. Analytical expressionsare\nlisted in Appendices B and C.\nWe seek the minimum total energy Uof a sample com-\nposed of Nparticles arranged according to a prescribed\ndistribution, each carrying a prescribed magnetic dipole\nmoment m. Consequently, the equilibrium state is ob-\ntained as a function of m. To ease the convergenceof the\nminimization techniques, we gradually increase the mag-\nnitude of the magnetic moments from m= 0 (ground\nstate) to the required maximum value of mwhile min-\nimizing the total energy for each intermediate value of\nm. Because of the large number of degrees of freedom,\nthe only practical way to find the equilibrium state is\nto perform a numerical minimization of the energy. In\nthe present work we implemented a conjugated gradient\nalgorithm with guaranteed descent [71].\nWe wish to study the dynamic response of our systems\nfor different orientations while holding mfixed in space.\nHowever, once the orientation of the magnetic moments\nis fixed from outside, the system as a whole may start to\nrigidly rotate to minimize its overall energy. In real sam-\nples, such rotations are for instance suppressed by macro-\nscopic frictional and gravitational forces. Moreover, in\nour previous investigation, this macroscopic rotation was\nhindered by a “clamping” protocol of the boundaries [55].\nHere instead, we develop a new protocol to keep the sys-\ntem in the desired orientation. This is achieved by sub-\ntracting from the force field acting on the boundaries\nthose parts corresponding to rigid rotations (see below\nand Appendix D). This way, three constraints are ap-\nplied in the form of the suppressed rigid rotations and\nwe otherwise allow a complete internal relaxation of the\nsample.\nIV. NORMAL MODES\nNext, we describe a generic normal mode formalism\nand explain how it can be employed to characterize the\nlinear response of our systems to a small external pertur-\nbation. We do not assume regular, periodic particle dis-\ntributions. Instead,ourformalismcanlikewisebeapplied\nto irregularparticle arrangements, see, e.g., Refs. [72–74].\nIn the following, we indicate with a bra-ket notation\n|X∝a\\}bracketri}ht, theD-component vector containing all the Dde-\ngrees of freedom of the system. In our case, D= 3N4\nas we only consider translational degrees of freedom, but\nin principle |X∝a\\}bracketri}htcould also include, for instance, particle\nrotations.\nOnce we write down the total energy U(|X∝a\\}bracketri}ht), the equi-\nlibrium state |X∝a\\}bracketri}hteqis given by the condition\n∂XU(|X∝a\\}bracketri}hteq) =0. (6)\nIt is more convenient to discuss the problem in terms\nof displacement from equilibrium, |u∝a\\}bracketri}ht=|X∝a\\}bracketri}ht − |X∝a\\}bracketri}hteq.\nFurthermore, it is alwayspossible to shift the energy by a\nconstant so that U(|X∝a\\}bracketri}hteq) = 0. Around its minimum, we\ncan expand U(|X∝a\\}bracketri}ht) to lowest order in the displacement\n|u∝a\\}bracketri}ht:\nU(|u∝a\\}bracketri}ht)≃1\n2∝a\\}bracketle{tu|H|u∝a\\}bracketri}ht,withHij=∂ui∂ujU.(7)\nHere,His the Hessian matrix composed of the second\nderivatives of Uwith respect to |u∝a\\}bracketri}ht(see Appendices B\nand C). If U(|X∝a\\}bracketri}ht) has continuous second partial deriva-\ntives, then His symmetric. Moreover, being in a min-\nimum of U(|X∝a\\}bracketri}ht) implies that His positive-semidefinite.\nAll its eigenvalues are positive, except for the modes rep-\nresenting rigid translations and rotations, which cost no\nenergy and have vanishing eigenvalues.\nWe obtainthe linearizedgradientaroundthe minimum\nfrom Eq. (7) as\n∂uU(|u∝a\\}bracketri}ht)≃ H|u∝a\\}bracketri}ht. (8)\nWhen a small external force |f∝a\\}bracketri}htis applied, the system\nreacts to neutralize it and re-equilibrates:\n−∂uU(|u∝a\\}bracketri}ht)+|f∝a\\}bracketri}ht= 0⇒ H|u∝a\\}bracketri}ht ≃ |f∝a\\}bracketri}ht.(9)\nIn Eq. (9) we have used Eq. (8), which is justified for\nsmall|f∝a\\}bracketri}ht. Wediagonalize Handintroduceitseigenvalues\nλnand eigenvectors, i.e. the normal modes |vn∝a\\}bracketri}htwithn=\n1...DandDthenumberofdegreesoffreedom,suchthat\nH|vn∝a\\}bracketri}ht=λn|vn∝a\\}bracketri}ht,and∝a\\}bracketle{tvm|vn∝a\\}bracketri}ht=δmn,(10)\nwhereδmnis the Kronecker delta. Since the |vn∝a\\}bracketri}htform a\ncomplete basis, we can expand displacements and forces\nas\n|u∝a\\}bracketri}ht=D/summationdisplay\nn=1un|vn∝a\\}bracketri}htand|f∝a\\}bracketri}ht=D/summationdisplay\nn=1fn|vn∝a\\}bracketri}ht.(11)\nHere,un=∝a\\}bracketle{tu|vn∝a\\}bracketri}htandfn=∝a\\}bracketle{tf|vn∝a\\}bracketri}ht. Then, using these\nexpansions and the orthonormality of the eigenvectors,\nEq. (9) simply reduces to\nλnun=fn. (12)\nThis relation clearly shows that, under the influence of\nan external force |f∝a\\}bracketri}htexciting the n-th normal mode, the\namplitude unof the response is linearly related to the\nintensity fnof the force. In this perspective, the Hessianeigenvalue λnquantifies the magnitude of the static lin-\near response of the system within the nth mode to the\nexternal force. λnis therefore a sort of elastic constant.\nThus, the energy of the system around its minimum can\nbe written, using Eqs. (7), (11), and (12), as\nU=1\n2D/summationdisplay\nn=1λnun2=1\n2D/summationdisplay\nn=1fn2\nλn. (13)\nV. STATIC ELASTIC MODULI FROM\nNORMAL MODES\nIn numerical calculations there are two main ways to\nobtain elastic moduli in the zero-frequency limit, i.e., in\nthe static case. On the one hand, one can perform a fi-\nnite but small (linear-regime) strain of the whole system,\nboth forpbc[36, 75, 76] or obc[55]. The system is equi-\nlibrated under the prescribed amount of strain. In this\nway,the moduliaremeasuredfromtheslopeofthe result-\ning stress-strain curve or, equivalently, from the second\nderivatives of the free energy. On the other hand, when\nemploying pbcand working in thermodynamic equilib-\nrium, one can differentiate the free energy with respect\nto a macroscopic strain [75, 77, 78]. As a special case,\nand in the low-temperature limit, the elastic moduli of\napbcglassy system have recently been examined [79],\nwhereas the case of regular lattices was discussed under\nthe assumption of affinity in the deformation [51]. How-\never, it is important to remark that affinely mapping the\nmacroscopic strain down to all scales in the system does\nnot allow for internal relaxation [80] and can even lead\nto qualitatively incorrect results [55] in presence of non-\naffinity sources.\nIn the present work we consider the case of a finite\nsystem in the ground state neglecting thermal fluctua-\ntions of the mesoscopic particles. The semi-analytical\napproach that we use to calculate elastic moduli in the\nlinearregimedoesnot requirefinite macroscopicdisplace-\nments nor does it assume affinity of the deformation.\nThis method relies on the decomposition of the linear\nresponse over the eigenvectors of the Hessian matrix H.\nIt reduces the calculation to a problem of linear alge-\nbra and gives access to dynamic properties as well, see\nsections VI and VII. Physically, our procedure involves\nusing stress instead of strain as an independent variable.\nA. Macroscopic Stresses and Strains\nBelow we will focus on Young’s modulus Eand the\nshear modulus G. They can be defined via the stress-\nstrain relationships:\nσαα=Eααεαα, σαβ=Gαβεαβ,(14)\nwhereσαβ(α,β=x,y,z) denotes the force per area ap-\nplied in the β-direction acting on the boundary with the5\nsurface normal oriented in the α-direction. εαβindicates\nthe corresponding strain deformation, i.e. the total dis-\nplacement of the boundary in the βdirection divided by\nthe distance between the boundaries in the α-direction.\nHere, there is no summation over αandβ. In the first\nformula, αdefines the direction of imposed stretching or\ncompression, along which we evaluate Eαα. In the sec-\nond formula, the αβplane sets the shear plane within\nwhich we evaluate G, with the shear displacement on the\nboundaries introduced along the β-direction. Thus, only\nthe faces of the system perpendicular to the α-direction\nneed to be explicitly addressed to impose our boundary\nstresses, while the rest of the system is free to relax. This\nconfiguration conceptually reproduces an experimental\nsituation in which the sample would be enclosed between\nthe plates of a rheometer with the plates perpendicular\nto theα-direction [81].\nApplying during shear only forces oriented tangential\nto the surface planes typically induces rotations. In ex-\nperiments, these are hindered by the confining plates.\nAccordingly, we here suppress such global rotations by\nsubtracting them from the overall response of the system\n(see belowand appendix D). In this way, we maintainthe\ndefinition of σαβas above close to the experimental sit-\nuation and avoid symmetrization typically performed in\nthe context ofclassicalelasticity theory[82] (for a related\ndiscussion on anisotropic systems see also Ref. [65]).\nInthefollowingderivation,wefocusontheYoungmod-\nulusEααand drop the ααsubscripts. The calculation\nfor the shear modulus Gαβis analogous. Here, stresses\nand strains in Eq. (14) are interpreted as macroscopic\nquantities characterizing the overall deformation of the\nsystem. We measure them and accordingly define the\nelastic moduli of the system solely by the stresses on and\nthe displacements of the boundaries perpendicular to ˆα,\nrespectively. The stress is calculated from the ratio be-\ntween the external force and the surface over which it is\napplied. Similarly, the strain is obtained by measuring\nthe displacement of the boundaries and dividing by their\ninitial distance.\nTheenergyofastraindeformationis givenbythe work\nperformed by the stress in the whole volume, i.e., using\nEq. (14),\nU=V/integraldisplay\nσdε=VEε2\n2=Vσ2\n2E.(15)\nTherefore, the elastic modulus can be derived by differ-\nentiating the previous equation,\nE=1\nVd2U\ndε2=V/bracketleftbiggd2U\ndσ2/bracketrightbigg−1\n. (16)\nB. Mesoscopic Stress\nOur goal is to connect these macroscopic relations to\nthe mesoscopic level. On the mesoscopic scale, withinour linear response framework, it is impractical to use\nthe strain as a variable to impose an external perturba-\ntion of the system. Imposing a certain amount of strain\nby displacing the boundary particles in a prescribed way\ndoes not provide any information on the displacement of\nthe bulk particles because the internal relaxation of the\nsystem is not known a priori. Actually, the rearrange-\nment ofthe bulk particlesmainly determines the reaction\nof the system and contributes the most to the elastic re-\nsponse. In contrast to that, it is more convenient to use\nthe stress as a variable to impose the external perturba-\ntion when we connect the macroscopic to the mesoscopic\nlevel. As a matter of fact, we know that an externally\nimposed mechanicalstressleads to nonvanishingexternal\nforces on the boundary particles only.\nWe here describe the macroscopic mechanical stress σ\nin terms ofsets ofdiscretized forcesacting directly on the\nmesoscopic particles. We denote the number of particles\nonthe “left”and“right”boundaries(seesectionII) as Nl\nandNr, respectively. Ifweindicateby Sthecross-section\nover which a total external force Fis applied, then we\nhaveF=σS. The corresponding externally imposed\ndiscretized mesoscopic force field |f∝a\\}bracketri}htacting directly on\nthe particles can then be constructed using the following\nprotocol:\na)|f∝a\\}bracketri}htis non-vanishing only on the boundaries and\nhas components orientedin the stress-direction, see\nFig. 1 a).\nb) The total force Facting on one boundary must be\nequal in magnitude to the total force acting on the\nother boundary. First, we assume all individual\nforces acting on individual particles on the same\nboundarytobe equalinmagnitude. Weindicate by\nflandfrthose forces acting on a single individual\nparticle on the left or right boundary, respectively.\nThen the condition reads F=Nlfl=Nrfr, see\nFig. 1 b).\nc) The torque exerted by |f∝a\\}bracketri}hton the boundaries must\nvanish [see Fig. 1 c)]. This can be achieved using\nthe method described in Appendix D. The condi-\ntion is applied separately to each boundary.\nd) Finally, we must rescale all forces acting onto one\nboundary by a common factor so that the forces\nacting in the stress direction sum up to F=σS\n[see Fig. 1 d)]. Again, this condition is applied\nseparately to each boundary.\nThese steps serve as a protocol when generating the\ndiscretized boundary force field |f∝a\\}bracketri}htin numerical calcu-\nlations. In the following, we factor out Fand write\n|f∝a\\}bracketri}ht=σS|fu∝a\\}bracketri}ht, where|fu∝a\\}bracketri}htis a force field satisfying our\nrequirementsand representinga macroscopicforce ofuni-\ntary magnitude ( F= 1).6\na) b)\nc) d)xy\nFIG. 1: Protocol to connect a macroscopic stress ( σxx) act-\ningon thesystem boundaries toadiscretized mesoscopic for ce\nfield acting on the boundaryparticles. For simplicity, the c ase\nof an irregular two-dimensional (2D) system is shown here.\nParticles on the boundaries are colored in black and springs\nare represented by dotted lines. This figure is for illustrat ive\npurposes only, therefore lengths and vectors are scaled in a\nqualitative way. Our procedure is as follows: a) First, indi -\nvidual discrete forces of equal magnitude are introduced on\neach individual boundary particle, pointing into the stres s-\ndirection (here the x-direction). b) The forces are rescaled to\nbalance total forces on the left- and right-hand sides. c) An\nappropriate rotatory component is introduced to make the\ntorques vanish on each boundary (separately). d) All forces\non each boundary are rescaled by a common factor so that\ntheir sum in the stress-direction is normalized correctly.\nC. Calculation of Static Elastic Moduli\nWe now have all ingredients available to formulate the\nconnection between the macroscopic elastic modulus and\nour discretized mesoscopic normal modes. Following the\ndefinition of particle-resolved stress σS|fu∝a\\}bracketri}htthat we in-\ntroduced above, we write the energy in Eq. (13) as an\nexplicit function of σ,\nU=σ2S2\n2D/summationdisplay\nn=1fu\nn2\nλn,withfu\nn=∝a\\}bracketle{tfu|vn∝a\\}bracketri}ht.(17)\nCombining it with Eq. (16), we obtain\nE=L\nS/bracketleftBiggD/summationdisplay\nn=1fu\nn2\nλn/bracketrightBigg−1\n. (18)\nHere, again, Sis the surface area of the boundary on\nwhich the stress acts, while Lis the distance between the\ntwoboundariessothat LS=V.λnisthen-theigenvalue\noftheHessianmatrix, and fu\nnisgivenbyEq.(17). Ingen-\neral,SandLwill be proportional to N(d−1)/dl0d−1andN1/dl0, respectively, with dthe spatial dimensionality of\nthe system. Therefore, for 3D particle arrangements of\ncubelike shape we obtain L/S∼1/3√\nNl0. In other cases\na prefactor must be added, taking into account the shape\nof the sample or the unit cell structure in the case of reg-\nular lattices.\nIn the following numerical calculations we used the la-\npackdiagonalizationroutines[83]tofindeigenvaluesand\neigenvectors of H. Special care must be taken to avoid\nthe zero-energy modes when computing Eq. (18). We\nhere simply ignore contributions from the lowest 3 and\n6 eigenvalues when dealing with 2D and 3D systems, re-\nspectively. They correspond to rigid translations and ro-\ntations of the system.\nOverall, we have described a self-standing procedure\nto calculate elastic moduli in obcsystems. The system\nis required to be in a stable equilibrium state, where\nthe Hessian matrix of the total energy is positive semi-\ndefinite. Since the elastic moduli are properties of the\nground state, they can be directly obtained via the eigen-\nvalues and eigenvectors calculated in this configuration,\nsee Eq. (18), for a specified force field, see section VB.\nTherefore, it is not necessary to actually perform a fi-\nnite deformation and drive the sample out of equilibrium\nas e.g. in Refs. [36, 55, 76]. In the following section we\ncompare the results of our described method with those\nobtained by explicitly taking a system out of equilibrium\nvia actual boundary displacement.\nD. Comparison with 2D Calculations\nThe calculation we outlined in section VC has the ad-\nvantage of requiring knowledge of only the ground state\nto obtain all (linear) elastic moduli. Conversely, as we\njust mentioned, the previously taken path to determine\ntheelasticmoduliistodrivethesystemoutoftheground\nstate by prescribing a small amount of strain, determin-\ningitsdeformation, andtherebytrackingthe totalenergy\nvariations,see e.g.Ref. [34, 36, 55]. Totest the validity of\nthe present approach, we compare the method described\nabove with the numerical results obtained previously for\nthe 2D case via explicit boundary displacements [55].\nWe briefly sum up the technique applied in our for-\nmer work, see Ref. [55]. In that case, a 2D dipole-spring\nmodel, similar to the present one but without steric re-\npulsion, is considered. The left and right boundaries of\nthe system are set perpendicular to the x-direction and\nundergo a “clamping” protocol, i.e., all the particles in\nthe boundary are constrained to move along /hatwidexor/hatwideyin\na prescribed way and therefore the whole system under-\ngoes a determined amount of strain εxxorεxy. For ev-\nery prescribed position of the boundaries, the bulk of\nthe system is free to relax [see Fig. 2 (b), (d), and (f)].\nThen, the static Young’s modulus is obtained from the\nsecond derivative of the total energy with respect to a\nsmall strain in the linear elasticity regime.\nContrarily to the present case, in Ref. [55] we consid-7y [l0]LR |u〉, εxx (a) BD |u〉, εxx (b)y [l0]LR |u〉, εxy(c) BD |u〉, εxy(d)y [l0]\nx [l0]LR |u〉, εxy (e)\nx [l0]BD |u〉, εxy (f)\nFIG. 2: Non-affine displacement field |u/angbracketrightof exemplary square\nand triangular lattices composed of 100 particles (springs in-\ndicated by dashed lines) for m= 0 obtained with LRandBD\nmethods [panels (a), (c), (e) and (b), (d), (f), respectivel y]\nfor stretching/compression εxxand simple shear εxydeforma-\ntions [panels (a), (b) and (c), (d), (e), (f), respectively] . This\nsimple, exemplary case shows how the responses obtained\nfrom the two methods are both non-affine and similar, but\ncan yet present small differences (compare e.g. particles hi gh-\nlightedbyredsquares), explainingsmall deviations inthe elas-\ntic moduli resulting from the two methods, see Fig. 3. Panels\n(b), (d), (f) were obtained by imposing small (linear-elast icity\nregime) strains of εxx= 0.03 andεxy= 0.001, respectively.\nered springs of identical elastic constant, regardless of\nthe spring length. To allow a better comparison with our\nformer results we will—solely in this subsection—assign\nan equal elastic constant to all springs, i.e. kij=k∀i,j.\nMoreover, for the present 2D setup, the elastic moduli\nwill be measured in multiples of k. In the following,\nwe will address the previous calculations of Ref. [55] as\n“Boundary Displacement” ( BD) and those in the frame-\nwork of linear response theory of the present work as\n“Linear Response” ( LR).\nWe first consider the case of a 2D square spring lattice\nwith nonmagnetized ( m= 0) particles on the vertices.\nOn the one hand, and in the BDcase, we can apply aprescribed, small amount of strain εxxorεxyand, after\nfull internal energetic relaxation, observe the resulting\ndisplacement field BD|u∝a\\}bracketri}ht, see Fig. 2 (b), (d), and (f).\nOn the other hand, and in the present LRscheme, we\nstart from the small mesoscopic force field |f∝a\\}bracketri}htas con-\nstructed via the protocol described in section VB. The\ncorresponding coefficients fnare obtained from Eq. (11).\nThen, using the eigenvalues of the Hessian matrix λnas\nwell as Eq. (12), we obtain the response of the modes,\ni.e. the coefficients un. Finally, using the coefficients un,\nwe obtain via Eq. (11) the particle-resolved displacement\nLR|u∝a\\}bracketri}ht, which is the linear response of the system to the\nsmall applied force |f∝a\\}bracketri}ht, see Fig. 2 (a), (c), and (e).\nThe comparison between the resulting displacement\nfields is helpful to understand where small deviations be-\ntween the elastic moduli obtained via the two different\nmethods may arise from, see Fig. 3. Overall, the differ-\nences remain small, especially in the case of stretching\nand compression [see Fig. 2 (a) and (b)]. For shear defor-\nmations [see Fig. 2 (c) and (d)], such discrepancies are\nvisible and reflect small deviations in the resulting mod-\nuli. This effect seems to be stressed when the positions\nof boundary particles are not mirror symmetric with re-\nspect to the direction ofthe calculated modulus, as in the\ncase of the triangular lattice for Young’s modulus in x-\ndirection in Fig. 2 (e) and (f). In total, however, we may\nconcludethatourprotocoltoconstructtheforcefield, see\nsection VB, works well and reproduces the mesoscopic\ndisplacement fields previously obtained via BD.\nTo further test the performance of the present method,\nwe now consider magnetic particles ( m∝\\e}atio\\slash= 0). We com-\npare some of the elastic moduli obtained in Figs. 5, 6,\nand 7 of Ref. [55] as functions of mfor a few exemplary\ncases of regular lattice structures. As shown in Fig. 3,\nwe find the same behavior for E(m) depending on lattice\nstructure and neighbor orientation. Depending on the\nparticle arrangement, small discrepancies can appear, as\nexplained above. These deviations also seem to depend\non the specific shape of the boundaries and are more evi-\ndent for the case of the triangularlattice in Fig. 2 (e) and\n(f). From now on, we will turn back to the more general\n3D case.\nVI. DYNAMICS\nBecause of their often highly viscous character on the\nmesoscale, soft matter systems in motion typically un-\ndergo large dissipation and their dynamics is studied in\nthe overdamped regime [36, 56, 59, 84]. In the following\nwe describe the time-evolution of our systems, starting\nfrom the overdamped equation of motion. Then, a way\nto decouple the full equation of motion in the normal\nmodes is presented and the general solution for a single\nmode is shown.\nTo keep the derivation general, we here take up the\nnotationintroduced in section IVwith the difference that\nnow|u∝a\\}bracketri}ht(t) and|f∝a\\}bracketri}ht(t) depend on time. The full, coupled8\n 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6\n 0  0.02  0.04  0.06  0.08  0.1E(m) [k]\nm [m0]Rectangular, BD\nRectangular, LR\nSquare, BD\nSquare, LR\nTriangular, BD\nTriangular, LR\nFIG. 3: Young’s modulus Eas a function of mcalculated\nfor comparison with BDandLRtechniques. Three cases are\npresented (top to bottom): rectangular lattice of base-hei ght\nratiob/h= 2.5, square lattice ( b/h= 1), and triangular lat-\ntice [see panels (e) and (f) of Fig. 2] with moriented in the\nx-,z-, andy-direction, respectively. The number of particles\nin all of the three examples is N= 400. The triangular lattice\ncase shows a comparatively larger difference, which, howeve r,\ndoes not depend on m. We mostly attribute such deviations\nto the structure of the boundary, as detailed in Fig. 2 (e) and\n(f).\nequation of motion for the overdamped dynamics of the\nsystem can be written as\nC|˙u∝a\\}bracketri}ht(t)+H|u∝a\\}bracketri}ht(t) =|f∝a\\}bracketri}ht(t), (19)\nwhere the dot represents time differentiation, the ma-\ntrixCcontains the (viscous) friction coefficients, and we\nhave used the linearized version of the gradient H|u∝a\\}bracketri}ht\nas in Eq. (8). Here, for simplicity and as a first step,\nwe consider the case of mesoscopically isotropic building\nblocksundernegligiblelong-rangeddynamiccoupling,i.e.\nC=cI, withItheD×Didentity matrixand ctheviscous\nfriction coefficient for one isotropic particle.\nAs a consequence, the matrices CandHcommute and\ncan be simultaneously diagonalized, i.e. they have a com-\nmon base of eigenvalues, namely the |vn∝a\\}bracketri}htin Eq. (10).\nThen, using the normal modes, Eq. (19) of Dvariables\ncan be decoupled into Dindependent single-variable\nequations\nc˙un(t)+λnun(t) =fn(t), (20)\nwithn= 1...D. If the external force |f∝a\\}bracketri}ht(t) is periodic,\ni.e.|f∝a\\}bracketri}ht(t) =/vextendsingle/vextendsinglef0/angbracketrightbig\nexp(iωt), its projections onto the Hes-\nsian eigenvectors |vn∝a\\}bracketri}htwill be equally periodic,\nfn(t) =f0\nnexp(iωt), (21)\nwithf0\nn=/angbracketleftbig\nf0/vextendsingle/vextendsinglevn/angbracketrightbig\n. Thus, the solution un(t) of Eq. (20)\nafter all transients have decayed must be periodic as well,\ni.e.\nun(t) =u0\nnexp(iωt). (22)Substituting the last equations into Eq. (20), we obtain\nu0\nn=f0\nn/κn(ω) (23)\nwith\nκn(ω) =λn+icω (24)\n= eiδn(ω)λn/radicalbig\n1+τn2ω2,\nwhereδn(ω) = arctan( τnω).\nIn these expressions we introduced by τn=c/λnthe\nrelaxation time and by 1 /κn(ω) the dynamic linear re-\nsponse function of the n-th mode.\nAs described above, we focus on the overdamped dy-\nnamics and do not include inertial terms in Eq. (19). If\nan inertial term had been considered, it would have re-\nsulted in a ( λn−/tildewidemω2)2term inside the square root of\nEq. (24), with /tildewidemthe mass of one particle. Such a contri-\nbution would have showed up as a resonance frequency\n˜ωn=/radicalbig\nλn//tildewidemfor then-th mode. As a consequence,\nwhen the frequency of the driving force ωcoincides with\n˜ωn, large displacements can be induced by small exter-\nnal perturbations. Such an effect would result in a sig-\nnificant drop of the elastic moduli at frequencies close\nto the resonances of those modes that contribute most\nto the linear response. This behavior, however, is not\nobvious from experimental reports [85–88], thus support-\ning the overdamped approach. Eq. (24) implies that the\ndisplacement un(t), i.e. the response, chases the driving\nforcefn(t) with identical frequency. However, because of\nviscous friction, it follows with a phase lag δn(ω), which\nvanishes in the case of frictionless motion. Such a phase\nlagimplies an imaginarycomponent of κn(ω) correspond-\ning to a loss component of the elastic moduli, see below.\nVII. DYNAMIC ELASTIC MODULI\nWe aim at extending the normal modes treatment that\nwe carried out for Eq. (9) and transfer it to the dynamic\nsituation described by Eq. (19). The final goal will be\nto generalize Eq. (18) for the macroscopic overall elas-\ntic moduli to the case of periodically oscillating external\nstresses and thus obtain the dynamic elastic macroscopic\nmoduli. We here consider the case of a Young modulus\nE(ω) =Eαα(ω) for direction α∈ {x,y,z}. The discus-\nsion of a shear modulus Gαβ(ω) is entirely analogous,\nprovided that the protocol prescribed in section VB is\nfollowed.\nWe now start with a macroscopic, periodic, and single-\nfrequency stress\nσ(t) =σ0eiωt(25)\napplied to the sample, with σ0a real amplitude. The\nresulting macroscopic strain ε(t) varies with the same\nfrequency. Thus we write\nε(t) =ε0(ω)eiωt, (26)9\nwhereε0(ω) is, in general, a complex amplitude. Us-\ning these expressions in the single-frequency case, the\nfrequency-dependent dynamic modulus E(ω) follows via\nσ(t) =E(ω)ε(t)⇔E(ω) =σ0\nε0(ω).(27)\nThus,E(ω) =E′(ω) + iE′′(ω) has an imaginary part\nwhenever σ(t) andε(t) are not completely in phase and\ncan be divided into storage ( E′) and loss ( E′′) compo-\nnents.\nNow we take up againthe formalism of sections IV and\nV. On the mesoscopic level—see section VI—the time-\ndependent response/vextendsingle/vextendsingleu0/angbracketrightbig\nexp(iωt) of the system, after all\ntransientshavedecayed,isrelatedtoasmalldrivingforce/vextendsingle/vextendsinglef0/angbracketrightbig\nexp(iωt) by\n/vextendsingle/vextendsingleu0/angbracketrightbig\neiωt=D/summationdisplay\nn=1u0\nn|vn∝a\\}bracketri}hteiωt=D/summationdisplay\nn=1f0\nn\nκn(ω)|vn∝a\\}bracketri}hteiωt,(28)\nwhere, again, Dis the number of degrees of freedom,\nf0\nn=/angbracketleftbig\nf0/vextendsingle/vextendsinglevn/angbracketrightbig\n,u0\nn=/angbracketleftbig\nu0/vextendsingle/vextendsinglevn/angbracketrightbig\n, and we used Eq. (11).\nThe macroscopic dynamic stress is given by σ(t) =\nFexp(iωt)/S, withSthe boundary surface area and F\nthe macroscopic force acting on it. Moreover, the macro-\nscopic strain is ∆ /Lwith ∆ the change in separation of\nthe macroscopic sample boundaries and Lthe absolute\ndistance between them. The displacement ∆ is measured\nin the direction of the applied force inducing it. There-\nfore, and since |fu∝a\\}bracketri}htrepresents the mesoscopic direction\nof a force of magnitude unity ( F= 1, see section VB),\nwe define ∆ = ∝a\\}bracketle{tfu|u∝a\\}bracketri}htas a measure of the resulting dis-\nplacement. We recall here that/vextendsingle/vextendsinglef0/angbracketrightbig\nwas constructed to\napply only on the boundary, so ∝a\\}bracketle{tfu|u∝a\\}bracketri}htreally extracts the\ndisplacement of the boundaries. Consequently, we write\nEq. (27) on the mesoscopic level as\nFeiωt\nS=E(ω)/angbracketleftbig\nfu/vextendsingle/vextendsingleu0/angbracketrightbig\neiωt\nL. (29)\nUsing Eq. (28), as well as f0\nn=Ffu\nnandfu\nn=∝a\\}bracketle{tfu|vn∝a\\}bracketri}ht\n(see section VB), the dynamic modulus follows as\nE(ω) =L\nS/bracketleftBiggD/summationdisplay\nn=1fu\nn2\nκn(ω)/bracketrightBigg−1\n(30)\nwhich does not depend on the macroscopicforce intensity\nFand in the case ω= 0 recovers Eq. (18). Since κn(ω)\nis a complex number, E(ω) is complex as well and we\ncan separate it into storage and loss components E(ω) =\nE′(ω) + iE′′(ω). We remark that in the static case we\nalways find E′′(ω= 0) = 0 by definition [see Eq. (24)].\nOn the macroscopic level, Eq. (30) is connected to the\nKelvin-Voigt model, which correctly describes the prop-\nerties of permanently crosslinked polymers on long times\nscales, i.e., small ω. This is clear in a limit case when\na single mode, e.g. n= 1, has a relaxation time, e.g.\nτ1=c/λ1, much longer than the other modes. Then, 0 0.05 0.1 0.15 0.2 0.25\n 0 1 2 3 4 5 6 7 8 9 10g(ω) [cl0/k]\nω [k/cl0]Cubic lattice\nFcc lattice\nDisordered fcc\nFIG.4: Densityofstates g(ω)at vanishing mofacubiclattice\nwith springs up tosecond-nearest neighbors (see section VI II),\nanfcclattice with only nearest-neighbor springs taken into\naccount (see section IX), and a disordered lattice (see sect ion\nX) made of 4913, 6084, and 6084 particles, respectively. The\ndensity of states is shown from ω= 0 to the highest ωmax\nobtained from the Hamiltonian spectrum, which is usually\n/lessorsimilar10k/cl0. The standard deviation of the narrow Gaussians\nused to approximate the Dirac deltas appearing in Eq. (31) is\nchosen as 0 .005ωmax.\nthe long-frequency dynamics is dominated by this mode\nwhich gives, in fact, the largest contribution to the sum\nin Eq. (30). Eventually, in this case one would find\nE(ω)∝κ1(ω) =λ1+iωc, which is precisely the form of\nthe dynamic modulus in the Kelvin-Voigt model [89, 90].\nIn the following, we will apply the present approach to\ndifferent particle distributions, addressing the dynamic\nelastic moduli for varying ωandm. Although we will\ndisplay the behavior of the dynamic moduli up to rela-\ntively large values of ω, one should keep in mind our fo-\ncus on overdamped motion. At maximum our approach\nis meaningful up to a frequency ωmax=λmax/c, where\nλmaxis the largest eigenvalue of H.\nThe limit becomes visible from calculating the spec-\ntrum, i.e. the density of states g(ω) [91]. It is defined\nby\ng(ω) =1\nDD/summationdisplay\nn=1δ/parenleftbigg\nω−λn\nc/parenrightbigg\n, (31)\nwithδthe Dirac delta function. To determine it from\nour numerical calculations, we replace the Dirac delta\nfunction by a narrow normalized Gaussian. We chose\nthe Gaussians as narrow as possible to achieve a smooth\nrepresentation of the density of states.\nWe always find g(ω) to drop significantly beyond a\nmaximum value ωmax. The latter is of the order of a few\nk/cl0, see Fig. 4. Consequently, and because of our focus\non the overdamped regime, it is not sensible to take into\naccount the behavior for ω/greaterorsimilar10k/cl0.\nFirst, the exemplary case of a simple cubic lattice will\nbe studied. After that, we consider an fccparticle ar-10\nxyz\nFIG. 5: Illustration of the three principal shear geometrie s.\nmis rigidly oriented in the z-direction. Shear forces can\nbe applied to different boundaries and in different direction s,\ngivingorigin tothreemain geometries (toptobottom): (a)f or\nGxyforces are perpendicular to m, but the driven boundary\nplanes contain m; (b) for Gxzboth shear forces and driven\nboundary planes are parallel to the mdirection; (c) for Gzy\nthedrivenboundaryplanes andshearforces are perpendicul ar\ntom. We here define stresses directly via the forces acting\non the indicated planes along the desired directions.\nrangement, before we finally move on to the case of dis-\nordered and more realistic particle arrangements. For\nsimplicity, we will always keep the magnetic moment m\noriented in the z-direction. We measure the Young mod-\nuli in the perpendicular ( ExxandEyy) and parallel ( Ezz)\ndirections. Likewise, the shear moduli will be calculated\nin the three possible orientations depicted in Fig. 5: (a)\nshear corresponding to Gxydoes not directly modify dis-\ntances along the m-direction; (b) while Gxzis measured\nthe macroscopic shear displacements are oriented along\nm; and (c) the shear plane contains m, but the macro-\nscopic shear displacements are perpendicular to mwhen\nGzyis determined. Moreover, we here have Gyx=Gxy,\nGxz=Gyz, andGzx=Gzy.\nVIII. CUBIC LATTICE\nAs a first prototype, we consider the simple exemplary\ncase of a 3D cubic lattice with N= 3375 particles. Mag-\nnetic particles on the lattice are linked by springs up to\nsecond-nearest neighbors. Corresponding springs along\nthe diagonals of the faces of the unit cells are necessary\nto avoid unphysical soft-shear modes. The boundaries of\nthe system are simply identified as the outermost layers-2\n 0\n 2\n 4\n 6\n 8\n 10\n 12\n 14\n 16-2 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14\nz [l0]\nmInitial positions\nSprings\nFinal positions\nx [l0]\ny [l0]z [l0]\nFIG. 6: Deformation of an initially cubic lattice with sprin gs\nbetween up to second-nearest neighbors and N= 3375 when\na magnetic moment of m= 0.1m0/hatwidezis gradually switched on.\nFor illustrative purposes, only particles on the front, top , and\nright faces are depicted. Shrinking is observed along m, i.e.\nthez-direction, and dilation in the perpendicular directions.\nThe inset zooms in onto the deformation of the unit cell at\nthe bottom left corner of the sample.\nof particles in the respective directions. As explained in\nsection II, the lattice parameter and the typical interpar-\nticle distance l0follow from the number density ρ. In the\ncase of a simple cubic lattice structure, ρis given by one\nparticle per unit cell.\nUpon introducing a dipole magnetic moment in the\nparticles, the direct attraction between nearest neigh-\nbors causes the system to shrink in the m-direction\nand expand in the perpendicular directions (see Fig. 6).\nTechnically, in our numerical calculations, we gradually\nincreased the magnetic moment to the value under con-\nsideration, up to a maximum magnitude of m= 0.1m0.\nIn this regime, and despite the overall deformation, the\nlattice maintains a cuboidlike shape. The magnetic in-\nteractions are not as strong as to overcome the elastic\nsprings and the particles do not come into steric contact.\nA. Static moduli\nWe start by studying the static moduli EandG(i.e.\nthe storage components E′andG′of the dynamic mod-\nuli calculated for ω= 0) for increasing magnitude of the\nmagnetic moment m, see also Ref. [51]. Magnetic inter-\nactions between nearest neighbors are attractive in the\nz-directionand repulsivein the x- andy-direction. These\nattractive and repulsive magnetic interactions with corre-\nspondingly positive and negative second derivatives with11\n 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 2.35\n 0  0.02  0.04  0.06  0.08  0.1E(m) [k/l02]\nm [m0]Static Young's Modulus (a)\nExx\nEyy\nEzz\n 0.47 0.475 0.48 0.485 0.49 0.495 0.5 0.505 0.51 0.515\n 0  0.02  0.04  0.06  0.08  0.1G(m) [k/l02]\nm [m0]Static Shear Modulus (b)\nGxy\nGxz\nGzy\n 1e-05 0.0001 0.001 0.01 0.1 1\n 0.001  0.01  0.1|E(m)-E(m=0)| [k/l02]\nm [m0]Absolute Deviation | Eαα(m) - Eαα(m=0) | (c)\n∝m2Exx\nEyy\nEzz\n 1e-06 1e-05 0.0001 0.001 0.01 0.1\n 0.001  0.01  0.1|G(m)-G(m=0)| [k/l02]\nm [m0]Absolute Deviation | Gαβ(m) - Gαβ(m=0) | (d)\n∝m2Gxy\nGxz\nGzy\nFIG. 7: Static moduli (a) Eαα(m) =E′\nαα(ω= 0,m) and (b)\nGαβ(m) =G′\nαβ(ω= 0,m) (α,β=x,y,z) of a cubic lattice\nwithN= 3375 for increasing magnetic moment intensity m\n(moriented along the z-direction). (a) The Young moduli in\nthe directions perpendicular to mare increased by increasing\nmagnetic moments, whereas in the m-direction the modulus\nis decreased. Black dashed lines in panels (a) and (c) repre-\nsent the trends in Eq. (32) shifted vertically to compensate\nfor finite-size and boundary effects and to allow for a bet-\nter comparison of the m-dependence. (b) The shear modulus\nGxzobtained by shear displacements along the m-direction\ndecreases for increasing m, whereas GxyandGzyreveal an\nincreasing behavior. (c, d) All elastic moduli as functions of\nmshow quadratic behavior to lowest order, as required by the\nnecessary m→ −msymmetry.\nrespecttonearest-neighbordistancesinducedecreaseand\nincrease, respectively, of the Young moduli [55]. This\ntrend is observed in Fig. 7 (a). At vanishing magnetic\nmoment all Young moduli measured along the different\ndirections have the same value, as expected by the cu-\nbic lattice symmetry. Then, as mis slowly increased,\nthis symmetry is broken and Ezz(m) decreases, whereas\nExx(m) andEyy(m) increase identically, as expected by\nthe unbroken x↔ysymmetry. Moreover, all moduli\nshow to lowest order in ma quadratic behavior, as de-\nmanded by the necessary m→ −msymmetry [59], see\nFig. 7 (c).More explicitly, the trends of the static Young moduli\nin the simple cubic case can be explained by consider-\ning interactions between neighbors on a regular lattice,\nsee appendix E. When we focus on small magnetic inter-\nactions, i.e. m≪m0, the dipole–dipole forces are much\nweakerthan the restoring elastic ones and we can assume\nthey leave the particle positions unaltered.\nConsidering contributions up to neighbors as distant\nas 10l0, we obtain, see appendix E, the following trends\nfor the Young moduli\nExx(m)\nk/slashbig\nl02=Eyy(m)\nk/slashbig\nl02≈9+4√\n2\n7+15.61(m/m0)2,\nEzz(m)\nk/slashbig\nl02≈9+4√\n2\n7−31.21(m/m0)2.(32)\nThe trends provided by these expressions are in good\nagreement with our numerical results, see Fig. 7 (a).\nTheydescribe,respectively, increasingordecreasingmod-\nuliinthedirectionsperpendicularorparallelto m. More-\nover, Eq. (32) suggests a stronger dependence of Ezz\nonmcompared to ExxandEyy. This agrees with our\nnumerical results, see Fig. 7 (a) and (c). Furthermore,\nit confirms the major role played by the second deriva-\ntives of neighbor interactions in determining the trends\nforEαα(m) of regular distributions, as pointed out in\nRef. [55].\nIn our numerical calculations we obtain different be-\nhaviors for the different shear moduli as functions of m.\nHowever, at vanishing magnetic moment they all assume\nthe same value, as expected by lattice symmetry, see\nFig. 7 (b). Furthermore, as Young’s moduli, they are\nall, to lowest order, quadratic functions of m, as required\nby symmetry when mis flipped into −m, see Fig. 7 (d).\nThe shear modulus Gzy(m) shows an increasing behav-\nior for increasing m. It is, in fact, the only depicted\nshear deformation that breaks the spatial mutual align-\nment of the moments in the z-direction. This is hindered\nby increasing m, in agreement with an increasing modu-\nlusGzy(m). The shear deformation related to Gxz(m),\ninstead, induces the dipoles to move in parallel to their\nalignment direction. Nearest neighbors connected by l0/hatwidex\nlie on a maximum of the dipole–dipole interaction, see\nEq. (4). Therefore, increasing mfacilitates the displace-\nment induced by σxz, in agreement with a decreasing\nshear modulus Gxz(m), as found in Fig. 7 (b). Last,\nwe find an increasing trend for the Gxy(m) shear modu-\nlus, slightly weaker compared to the other two examined\nmoduli, as depicted in Fig. 7 (b) and (d).\nB. Dynamic moduli\nWe now focus on the dynamic properties, which are\nthe central aim of the present work. As a general trend,\nwe always find the storage moduli to tend to a finite\nvalue for large ω, see Fig. 8. Yet, as noted before, it12\n 0 5 10 15 20\n 0.01  0.1  1  10  100E(ω,m) [k/l02]\nω [k/c l0]Dynamic Young's Modulus E( ω, m=0.1m0) (a)\nE'αα(m=0)\nE'xx(m>0)\nE'yy(m>0)\nE'zz(m>0)\nE''αα(m=0)\nE''xx(m>0)\nE''yy(m>0)\nE''zz(m>0)\n 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5\n 0.01  0.1  1  10  100G(ω,m) [k/l02]\nω [k/c l0]Dynamic Shear Modulus G( ω, m=0.1m0) (b)\nG'αβ(m=0)\nG'xy(m>0)\nG'xz(m>0)\nG'zy(m>0)\nG''αβ(m=0)\nG''xy(m>0)\nG''xz(m>0)\nG''zy(m>0) 0.01\n 0.01\nFIG. 8: Dynamic elastic moduli (a) Eαα(ω) and (b) Gαβ(ω)\n(α,β=x,y,z) of a cubic lattice with N= 3375 for vanishing\nmagnetic moment (solid line, /circlecopyrt), andm= 0.1m0/hatwidez(dashed\nlines,/square,△,▽). Filled and unfilled markers correspond to\nstorage ( E′\nαα,G′\nαβ) and loss ( E′′\nαα,G′′\nαβ) components, respec-\ntively. Insetsinpanels(a)and(b)zoom ontothestorage par ts\n(a)E′\nαα(ω) and (b) G′\nαβ(ω) at small ωfor better resolution\n(see also Fig. 7).\nis not reasonable to consider the behavior for frequen-\ncies larger than 10 k/cl0. Conversely, the loss moduli\nas functions of ωshow a linear increase (see appendix\nF). This behavior we attribute to our model focusing on\noverdamped motion. In fact, under oscillatory motion,\nthe damping term in Eq. (19), which is the origin of the\nloss modulus, increases with frequency ω. This conforms\nwith a macroscopic Kelvin-Voigt model [89, 90] which\npredicts an imaginary component of the dynamic moduli\nlinearly increasing with frequency. Similarly, experimen-\ntal measurements of the loss moduli in polymeric mate-\nrials [81, 85, 87, 88] are compatible with a Kelvin-Voigt\nmodel [i.e. constant storage part and linearly increasing\nloss part of Eαα(ω) andGαβ(ω)] in the low-frequency\nregime. Furthermore, in the limit ω→0, we always\nfind vanishing loss moduli and the storage component\nto recover the corresponding static elastic modulus, see\nEqs. (18), (24), and (30).The storage Young moduli E′\nαα(ω) (α=x,y,z) in\nFig. 8 (a)—here calculated for m= 0.1m0/hatwidez—show at all\nfrequencies the trends as described in the static case, see\nFig.7. Theamountofvariationwithrespecttothe m= 0\nconfiguration, however, seems to be larger at larger fre-\nquencies. Furthermore, E′\nxx(ω) andE′\nyy(ω) show identi-\ncal behavior as functions of ω, as required by the sym-\nmetry of this geometry under switching x↔y. Likewise,\nat lowω, the loss moduli E′′\nzz(ω) andE′′\nxx,yy(ω) show a\ndecreasing and increasing trend, respectively, when the\nmagnetic moment is switched on and increased. Further-\nmore, for higher ω, all the loss components linearly in-\ncrease with ωwith identical coefficients, see also Fig. 17\n(b) in appendix F.\nThe storage shear moduli G′\nαβ(ω) at low frequencies\npresent the same trends of increase and decrease as in\nthe static case, see Figs. 7 (b) and 8 (b). We remark that\nat high frequencies (beyond 10 k/cl0), whileG′\nxy(ω) and\nG′\nxz(ω)showthesameandenhancedtrendasinthestatic\ncase,G′\nzy(ω)changesfromincreasetodecreasebyincreas-\ningm. This graphically results in a crossing between the\ncurves for G′\nαβ(ω,m= 0) and G′\nzy(ω,m= 0.1m0). The\nloss shear moduli G′′\nαβ(ω), instead, display the same in-\ncreasing or decreasing trends as the corresponding static\nGαβ(m= 0) both at low and high frequencies (see also\nappendix F).\nIX. FCC LATTICE\nWe now turn our focus onto the exemplary case of\na face-centered cubic ( fcc) lattice. Later in section X,\nwe will generate disordered samples by randomizing an\ninitiallyfccparticle arrangement. In this setup we in-\ntroduce springs connecting nearest neighbors only. This\nis enough to obtain a particle distribution stable under\nboth stretching and shearing. The boundaries of the sys-\ntem are chosen as the outermost layers of particles in a\ngiven direction. The typical interparticle distance l0fol-\nlows from the number density ρ, as explained in section\nII, which for the fcclattice is 4 particles per unit cell.\nWhen magnetic moments are introduced we here ob-\nserve an elongation of the system in the m-direction and\nshrinking in the perpendicular directions, see Fig. 9. The\nnearest neighborson the fcclattice are located along the\n/hatwidex+/hatwidey,/hatwidex+/hatwidez, and/hatwidey+/hatwidezdirections, i.e. at an angle of\nπ/4 with respect to the Cartesian axes. When the sys-\ntem elongates in the z-direction the angles between the\nnearest-neighbor directions and mreduce, thus lowering\nthe magnetic energy Um.\nA. Static moduli\nFirst, we present the behavior of the static moduli\nas functions of increasing magnetic moment, see also\nRef. [51]. We always find a monotonic, smooth behav-\nior for increasing m[see Fig. 10 (a) and (b)]. In fact,13\n-2\n 0\n 2\n 4\n 6\n 8\n 10\n 12\n 14\n 16-2 0 2 4 6 8 10 12 14 16-2 0 2 4 6 8 10 12 14 16\nz [l0]\nmInitial positions\nSprings\nFinal positions\nx [l0]\ny [l0]z [l0]\nFIG. 9: Deformation of an fcclattice with springs between\nnearest neighbors and N= 3430, when a magnetic moment\nofm= 0.1m0/hatwidezis switched on. For illustrative purposes, only\nthe first two particle layers on the front, top, and right face s\nare depicted. Elongation is observed in the m-direction and\ncontraction in the perpendicular ones. Inset zooms in onto\nthe deformation of the particles at the bottom left corner of\nthe sample.\nas shown in Fig. 10 (c) and (d), the elastic moduli as\nfunctions of mare to lowest order quadratic functions,\nin accord with the m→ −msymmetry. Again, and as\nrequired by lattice symmetry, at m= 0 all Young moduli\nand the shearmoduli in the examined directionscoincide,\nsee Fig. 10 (a) and (b).\nNext, we estimate the role played by the relative po-\nsitions of neighboring particles for the behavior of the\nYoung moduli. We consider the case of a regular fcc\nlattice and take into account contributions to the Young\nmoduli to lowest order in m, as explained in appendix\nE. Considering terms up to neighbors as far as 10 l0in\nEq. (E3), we obtain\nExx(m)\nk/slashbig\nl02=Eyy(m)\nk/slashbig\nl02≈27/6\n3−13.02(m/m0)2,\nEzz(m)\nk/slashbig\nl02≈27/6\n3+28.05(m/m0)2.(33)\nComparison with the behavior of the Young’s moduli\nresulting from our numerical calculations, see Fig. 10\n(a), leads to a good qualitative agreement. The modu-\nlus in the m-direction Ezz(m) increases with increasing\nm, whereas in the perpendicular directions Exx(m) and\nEyy(m) decrease with m. Thus, the fccarrangement\nshows a completely opposite behavior compared to the\nsimple cubic case, see section VIIIA. Moreover, Eq. (33)\nindicates the Ezz(m) modulus to have a stronger depen-\ndence on mcompared to Exx(m) andEyy(m), as also 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05\n 0  0.02  0.04  0.06  0.08  0.1E(m) [k/l02]\nm [m0]Static Young's Modulus (a)\nExx\nEyy\nEzz\n 0.29 0.295 0.3 0.305 0.31 0.315 0.32\n 0  0.02  0.04  0.06  0.08  0.1G(m) [k/l02]\nm [m0]Static Shear Modulus (b)\nGxy\nGxz\nGzy\n 1e-05 0.0001 0.001 0.01 0.1 1\n 0.001  0.01  0.1|E(m)-E(m=0)| [k/l02]\nm [m0]Absolute Deviation | Eαα(m) - Eαα(m=0) | (c)\n∝m2Exx\nEyy\nEzz\n 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1\n 0.001  0.01  0.1|G(m)-G(m=0)| [k/l02]\nm [m0]Absolute Deviation | Gαβ(m) - Gαβ(m=0) | (d)\n∝m2Gxy\nGxz\nGzy\nFIG. 10: Static moduli (a) Eαα(m) =E′\nαα(ω= 0,m) and (b)\nGαβ(m) =G′\nαβ(ω= 0,m) (α,β=x,y,z) of anfcclattice\nwithN= 3430 for increasing magnetic moment intensity m.\nmis oriented along the z-direction. The Young moduli for\nstretching perpendicular to mare reduced by increasing mag-\nnetic moments, whereas along the m-direction the modulus\nis increased. Black dashed lines in panels (a) and (c) repre-\nsent the trends in Eq. (33) shifted vertically to compensate\nfor finite-size and boundary effects and to allow for a better\ncomparison of the m-dependence. The shear modulus Gxy\nobtained by applying shear in the xyplane perpendicular to\nmincreases for increasing m, whereas GxzandGzyreveal a\ndecreasing behavior. (c, d) All elastic moduli as functions of\nmshow a quadratic behavior to lowest order, in accord with\nthem→ −msymmetry and as depicted by the log-log scale\nplots.\nfound in our numerical results and shown in Fig. 10 (a)\nand (c).\nSimilarly, the shear moduli are influenced by min dif-\nferent ways. Here we find the shear modulus Gxy(m) to\nincrease and Gxz(m) to decrease with increasing m, anal-\nogously to what we observed in the simple cubic case, see\nsection VIIIA. Contrarily to the simple cubic case, the\nshear modulus referring to displacements parallel to m,\nGzy(m), shows a decreasing trend when the magnetic\nmoments increase. Moreover, Gzy(m) displays a weaker\ndependence on mcompared to the remaining two shear14\n 0 2 4 6 8 10 12\n 0.01  0.1  1  10  100E(ω,m) [k/l02]\nω [k/c l0]Dynamic Young's Modulus E( ω, m=0.1m0) (a)\nE'αα(m=0)\nE'xx(m>0)\nE'yy(m>0)\nE'zz(m>0)\nE''αα(m=0)\nE''xx(m>0)\nE''yy(m>0)\nE''zz(m>0)\n 0 0.5 1 1.5 2 2.5\n 0.01  0.1  1  10  100G(ω,m) [k/l02]\nω [k/c l0]Dynamic Shear Modulus G( ω, m=0.1m0) (b)\nG'αβ(m=0)\nG'xy(m>0)\nG'xz(m>0)\nG'zy(m>0)\nG''αβ(m=0)\nG''xy(m>0)\nG''xz(m>0)\nG''zy(m>0) 0.01\n 0.01\nFIG. 11: Dynamic elastic moduli (a) Eαα(ω) and (b) Gαβ(ω)\n(α,β=x,y,z) of anfcclattice with N= 3430 for vanishing\nmagnetic moment (solid line, /circlecopyrt), andm= 0.1m0/hatwidez(dashed\nlines,/square,△,▽). Filled and unfilled markers correspond to\nstorage ( E′,G′) and loss ( E′′,G′′) components, respectively.\nInsets in panels (a) and (b) zoom onto the storage parts (a)\nE′\nαα(ω) and (b) G′\nαβ(ω) at small ωto better resolve the dif-\nferent curves (see also Fig. 10).\nmoduli, as depicted in Fig. 10 (d).\nB. Dynamic moduli\nFinally, we examine the behaviors of the dynamic elas-\ntic moduli for various frequencies ωand magnetic mo-\nment intensities m. The storage dynamic Young moduli\nE′\nαα(α=x,y,z) at all frequencies follow the same be-\nhavior as described in the static case (see Fig. 10). In the\ndirection parallel to m,E′\nzzincreases for increasing m,\nwhereas E′\nxxandE′\nyydecrease for the perpendicular di-\nrections, see Fig. 11 (a) and its inset for a zoom onto the\nlow-ωbehavior. As shown in appendix F, the loss compo-\nnentsE′′\nααpartially exhibit opposite trends compared to\ntheirstoragecounterparts(seeFig.18foradetailedplot).\nIn fact, at low frequencies, the loss modulus for the m\ndirection, E′′\nzz, decreases with increasing m, whereas forE′′\nxxandE′′\nyythe two perpendicular directions increase.\nAt higher frequencies, however, and as in the cubic lat-\ntice case, all the loss moduli E′′\nααrecover the behavior\nof their storage counterparts and show an identical de-\npendence on ω[see Figs. 11 (a) and Fig. 18 in appendix\nF].\nThe storage dynamic shear moduli G′\nαβ(α,β=x,y,z)\nare displayed in Fig. 11 (b). Here, at low- ωvalues the\nchanges in the shear moduli for the different geometries\nreproduce the trends shown in Fig. 10, see the inset of\nFig. 11 (b). However, when considering the behavior at\nhigherω,G′\nxyturnsfromincreasingtodecreasingwith m,\nwhileG′\nzyturns from decreasing to increasing when com-\npared with the shear modulus at m= 0. Although we\nalreadymentionedthatonlythebehaviorfor ω/lessorsimilar10k/cl0\nshould be interpreted, these data suggest the possibility\nthat some dynamic shear moduli could swap their ten-\ndency of increasing or decreasing with mto decreasing\nor increasing, respectively. Contrarily, the Young mod-\nuli consistently show a monotonic behavior as functions\nof bothωandm. Furthermore, at low ω, the loss shear\nmoduliG′′\nαβexhibit anoppositebehaviorwhencompared\nwith their storage complements. For shear deformations\nin the plane perpendicular to m,G′′\nxydecreases with in-\ncreasing magnetic moment, whereas the other two mod-\nuliG′′\nxzandG′′\nzyare increased by increasing m, see also\nappendix F, Fig. 18.\nX.3D DISORDERED SAMPLES\nA. Numerical generation\nWe start from a regular three-dimensional fcclattice.\nHaving a well defined density ρand neighbor structure,\nthis lattice allows us to define the average interparticle\ndistance l0as described in section IX. Then we introduce\ndisorder in the lattice by randomly displacing each parti-\ncle by 0.5l0in a stochastic direction. After that, we set\nthe elastic springs between nearest neighbors.\nIn the randomization step, we take care to generate an\ninitially stable disordered system so that magnetic inter-\nactions do not immediately overcome the elastic spring\ninteractionswhen the magnetic moments are switched on\n[54, 68]. In other words, the formation of collapsed clus-\nterswheretheparticlestoucheachotherinastuckconfig-\nuration shall be avoided for low strength of the magnetic\ninteractions. For this purpose, we impose that in the ran-\ndomized configuration for m= 0 no couples of particles\nare closer than 0 .5l0. Boundary particles are identified\nas the outermost layers of the initial fcclattice in each\ndirection. To help maintain an overall cubelike shape,\nwe move boundary particles by half the amount of other\nparticles. An example of the resulting initial distribution\nis given by the gray particles in Fig. 12.\nThus, we generate a disordered system of macroscopic\ncubelike shape with Nnon-overlapping magnetic parti-\ncles. In the following we set N= 1688. As described,15\nm=0.058 m0(a) m=0.060 m0(b)\nm=0.062 m0(c) m=0.064 m0(d)\nFIG. 12: Example deformation of a randomized particle dis-\ntribution (N=1688) of initially cubelike shape (gray parti cles)\nwhen a magnetic moment of m=m/hatwidezis switched on. Panels\n(a), (b), (c), and (d) show the equilibrated particle distri bu-\ntion (black) as the magnetic moment intensity is gradually\nincreased to m= 0.058m0,m= 0.06m0,m= 0.062m0, and\nm= 0.064m0, respectively. Panel (c) represents the onset of\nchain formation in the m-direction, see sections XB and XD.\nin the initial configuration, the springs are set before the\nmagnetic interactions are switched on. Then, we grad-\nually increase the magnitude of the magnetic moments\nand at each step find the minimum energy configuration,\nsee section III. When the equilibrium state for a given\nmis reached, we obtain the Young and shear moduli E\nandGas functions of both mandω, using the methods\ndescribed in sections VC and VII.\nAs the magnitude mof the magnetic moments in-\ncreases, we can principally distinguish between two\nregimes. On the one hand, the behavior for small m\nis controlled by magnetic Umand elastic Uelenergies,\nsee Fig. 13. The deformation is relatively small and the\nelastic moduli are, to lowest order, quadratic functions\nofm, as expected by the necessary m→ −msymme-\ntry. On the other hand, when attractive magnetic inter-\nactions become as strong as to overcome linear spring\nrepulsion, steric interactions come into play (see Fig. 13).\nThen, formation of chains is observed, as well as signif-\nicant changes in the system size (see Fig. 12). Further-\nmore, the close steric contact between particles generates\nextra stiffness, which is reflected by a significant change\nin the elastic moduli. This behaviorreflects a “hardening\ntransition” similar to the situation described in Ref. [54]\nfor one-dimensional systems.\nB. Static moduli\nFirst, we focus on the static elastic moduli of the ran-\ndomized system for increasing magnetic moment m. To\nextract a general trend we realized 80 different systems\nfollowing the protocol as described in section XA. Then-40000-20000 0 20000\n 0  0.02  0.04  0.06  0.08  0.1U [k l0]\nm [m0](a)UelUsUmU\n 1 20000\n 0  0.02  0.04  0.06  0.08  0.1|U| [k l0]\nm [m0](b)\nFIG. 13: (a) Equilibrium energies of the disordered fccsys-\ntemshown inFig. 12for increasing magnitude ofthemagnetic\nmoment m. (b) Two regimes are identified in a logarithmic\nplot. Up to m∼0.05m0the total energy Umostly comprises\nelasticUeland magnetic Umcontributions. For m/greaterorsimilar0.05m0\ninstead, the steric interaction energy Usbecomes higher than\nthe elastic energy Uel. This signals the subsequent formation\nof chains. The pronounced step at 0 .06m0/lessorsimilarm/lessorsimilar0.064m0is\nconnected to chain formation.\nwe obtain our results by averaging over the moduli for\nall different randomized realizations. Relative errors fol-\nlow from the standard deviations. The resulting static\nmoduli are depicted in Fig. 14. To lowest order in mand\nup to approximately m= 0.06m0, the Young moduli of\nthe system [see inset of Fig. 14 (a)] show a behavior sim-\nilar to the fcccase [compare with Fig. 10 (a)]: increas-\ning∝a\\}bracketle{tEzz∝a\\}bracketri}htfor imposed deformations in the mdirection\nand decreasing ∝a\\}bracketle{tExx∝a\\}bracketri}htand∝a\\}bracketle{tEyy∝a\\}bracketri}htfor the perpendicular\ncases. Moreover, in this regime the static Young mod-\nuli∝a\\}bracketle{tEαα(m)∝a\\}bracketri}ht(α=x,y,z) show a quadratic behavior as\nfunctions of min accord with the m→ −msymme-\ntry, see Fig. 14 (c). Similarly, the static shear moduli\n∝a\\}bracketle{tGαβ(m)∝a\\}bracketri}ht(α,β=x,y,z) in this regime show quadratic\nbehavior, see Fig. 14 (d), while the trends for ∝a\\}bracketle{tGαβ(m)∝a\\}bracketri}ht\nvary from those of the regular fcclattice [compare the\ninset of Fig. 14 (b) with Fig. 10 (b)].\nThis behavior changes dramatically for m/greaterorsimilar0.06m0,\nwhere magnetic interactions are as strong as to cause the\nparticles to come into steric contact and form chains in\nthem-direction. Hereweobserveasignificantincreasein\nall elastic moduli [see Fig. 14 (a) and (b)]. Still, Young’s\nmodulus for imposed deformations in the m-direction,\n∝a\\}bracketle{tEzz∝a\\}bracketri}ht, shows a much larger increase compared to ∝a\\}bracketle{tExx∝a\\}bracketri}ht\nand∝a\\}bracketle{tEyy∝a\\}bracketri}ht, in agreement with experimental observations\non anisotropic systems [38], see also the case of bi-axial\ntension [92]. ∝a\\}bracketle{tExx∝a\\}bracketri}htand∝a\\}bracketle{tEyy∝a\\}bracketri}htfeature an identical be-\nhavior within the errorbars, as expected by the largely\nunbroken isotropy of the systems within the xy-plane.\nLikewise, the shear moduli show an increase for all in-\nvestigated geometries. In a purely affine deformation of\nchains perfectly aligned along m, thezyshear geome-16\n 0 1 2 3 4 5 6 7 8 9 10 11\n 0  0.02  0.04  0.06  0.08  0.1〈E(m)〉 [k/l02]\nm [m0]Static Young's Modulus (a)\n〈Exx〉\n〈Eyy〉\n〈Ezz〉\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2\n 0  0.02  0.04  0.06  0.08  0.1〈G(m)〉 [k/l02]\nm [m0]Static Shear Modulus (b)\n〈Gxy〉\n〈Gxz〉\n〈Gzy〉 0 0.005 0.01 0.015 0.02 0.025 0.03\n 0 0.005 0.01 0.015 0.02 0.025 0.03\n 1e-05 0.0001 0.001 0.01 0.1 1 10\n 0.001  0.01  0.1|〈E(m)〉-〈E(m=0)〉| [k/l02]\nm [m0]Absolute Deviation | Eαα(m) - Eαα(m=0) | (c)\n∝m2〈Exx〉\n〈Eyy〉\n〈Ezz〉\n 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10\n 0.001  0.01  0.1|〈G(m)〉-〈G(m=0)〉| [k/l02]\nm [m0]Absolute Deviation | Gαβ(m) - Gαβ(m=0) | (d)\n∝m2〈Gxy〉\n〈Gxz〉\n〈Gzy〉\nFIG. 14: Static moduli (a) /angbracketleftEαα(m)/angbracketright=/angbracketleftE′\nαα(ω= 0,m)/angbracketrightand\n(b)/angbracketleftGαβ(m)/angbracketright=/angbracketleftG′\nαβ(ω= 0,m)/angbracketright(α,β=x,y,z) of a dis-\norderedfcclattice with N= 1688 for increasing m=|m|,\nwithmoriented in the z-direction. Statistics are collected\nover 80 differently randomized samples. Data points and bars\nrepresent the resulting averages and standard deviations, re-\nspectively. (c, d) All elastic moduli as functions of mshow\na quadratic behavior to lowest order for small m, in accord\nwith the m→ −msymmetry. For illustrative purposes\nwe have slightly shifted the bars for different data sets hor-\nizontally and reduced the number of points shown in pan-\nels (c) and (d) to better distinguish between the individual\nbars and data points. Dips in panels (c) and (d) occur when\n/angbracketleftEαα(m)/angbracketright ≈ /angbracketleftEαα(m= 0)/angbracketrightor/angbracketleftGαβ(m)/angbracketright ≈ /angbracketleftGαβ(m= 0)/angbracketright.\nThen, the logarithm of the absolute deviation from the value\nform= 0 diverges to −∞. The elastic moduli themselves,\nhowever, show smooth behavior, as displayed in panels (a)\nand (b) and respective insets.\ntry would be the only one displayed that leads to dis-\ntortions of the chains. Therefore it is conceivable that\n∝a\\}bracketle{tGzy∝a\\}bracketri}htgrows larger than ∝a\\}bracketle{tGxy∝a\\}bracketri}htand∝a\\}bracketle{tGxz∝a\\}bracketri}ht, although the\nsize of the standard deviations does not allow to draw a\nconclusive result.\nFinally, to avoid confusion, we stress that the dips in\nFig. 14 (c) and (d) simply mean that the elastic moduli\nform∝\\e}atio\\slash= 0 tend to the same values as those for m= 0.\nSince in Fig. 14 (c) and (d) the deviations of the elas-tic moduli from their values for m= 0 are plotted on\na logarithmic scale, the dips are not directly related to\na mechanical instability resulting from vanishing elastic\nmoduli. In fact, as shown in in Fig. 14 (a) and (b), for\na given value of mthe elastic moduli always remain pos-\nitive.\nC. Dynamic moduli, m/lessorsimilar0.06m0\nWe now move our attention to the dynamic proper-\nties of our disordered systems. Again, we have collected\nstatisticsover80different realizationsofourrandomizing\nprocess. The resulting averages and standard deviations\nare represented as data points and bars in the figures\nbelow.\nFirst we examine the dynamic moduli for the magni-\ntude of the magnetic moments below the onset of sig-\nnificant chain formation, i.e. m/lessorsimilar0.06m0. There, the\nstorage parts ∝a\\}bracketle{tE′\nαα(ω)∝a\\}bracketri}htof the dynamic Young moduli for\nincreasing mshow the same trends for the different ge-\nometries as the static moduli [see Fig. 15 (a) and com-\npare with the inset of Fig. 14 (a)]. Conversely, the loss\nparts∝a\\}bracketle{tE′′\nαα(ω)∝a\\}bracketri}htof the Young moduli feature a trend of\nincrease with increasing min all cases [see appendix F,\nFig. 19 (a)].\nSimilarly to the Young moduli, the storage parts\n∝a\\}bracketle{tG′\nαβ(ω)∝a\\}bracketri}htof the dynamic shear moduli approximately\nfollow their static counterparts at low ω[see the inset\nof Fig. 15 (b) and compare it to the inset of Fig. 14 (b)].\nHowever, with increasing frequencies ωand upon switch-\ningmfromm= 0 tom >0,∝a\\}bracketle{tG′\nzy(ω)∝a\\}bracketri}htswitches from\na slight decrease to a significant increase with respect to\nthevalueat m= 0, seeFig.15(b). Thisresultsinacross-\ning between the curvescorrespondingto ∝a\\}bracketle{tG′\nzy(ω,m= 0)∝a\\}bracketri}ht\nand∝a\\}bracketle{tG′\nzy(ω,m >0)∝a\\}bracketri}ht. Instead, the remaining two shear\nmoduli∝a\\}bracketle{tG′\nxy(ω)∝a\\}bracketri}htand∝a\\}bracketle{tG′\nxz(ω)∝a\\}bracketri}htalways show a decrease.\nAnalogously to ∝a\\}bracketle{tE′′\nαα(ω)∝a\\}bracketri}ht, the loss components ∝a\\}bracketle{tG′′\nαβ(ω)∝a\\}bracketri}ht\nare observed to increase at all frequencies when switch-\ning onm, independently of the chosen geometry [see ap-\npendix F, Fig. 19 (b)].\nD. Dynamic moduli, m/greaterorsimilar0.06m0\nIn the following, we consider the dynamic moduli of\nthe system at magnitudes mof the magnetic moment\nat the onset of chain formation [see Fig. 12 (c)]. Then\nsteric interactions play a major role in the total inter-\naction energy U(see Fig. 13). To better illustrate the\nbehavior of the storage dynamic moduli in this regime\nit is convenient to plot the deviation from the respective\nstatic value at m= 0, as shown in Fig. 16 (for brevity, al-\nthough deviations are plotted, the curves are still labeled\nby∝a\\}bracketle{tE′\nαα∝a\\}bracketri}htand∝a\\}bracketle{tG′\nαβ∝a\\}bracketri}ht). Thus the diminishing behavior of\nthe curves ∝a\\}bracketle{tE′∝a\\}bracketri}ht(m= 0) and ∝a\\}bracketle{tG′∝a\\}bracketri}ht(m= 0) for decreasing\nωrepresents a smooth convergence of the moduli to the17\n 0 1 2 3 4 5 6 7 8\n 0.001  0.01  0.1  1  10〈 E(ω,m) 〉 [k/l02]\nω [k/c l0]Dynamic Young's Modulus 〈 E(ω, m=0.056) 〉\n(a)〈E'〉(m=0)\n〈E'xx〉(m>0)\n〈E'yy〉(m>0)\n〈E'zz〉(m>0)\n〈E''〉(m=0)\n〈E''xx〉(m>0)\n〈E''yy〉(m>0)\n〈E''zz〉(m>0)\n 0 0.5 1 1.5 2\n 0.001  0.01  0.1  1  10〈 G(ω,m) 〉 [k/l02]\nω [k/c l0]Dynamic Shear Modulus 〈 G(ω, m=0.056) 〉\n(b)〈G'〉(m=0)\n〈G'xy〉(m>0)\n〈G'xz〉(m>0)\n〈G'zy〉(m>0)\n〈G''〉(m=0)\n〈G''xy〉(m>0)\n〈G''xz〉(m>0)\n〈G''zy〉(m>0) 0.001  0.01\n 0.001  0.01\nFIG. 15: Dynamic elastic moduli (a) /angbracketleftEαα(ω)/angbracketrightand (b)\n/angbracketleftGαβ(ω)/angbracketright(α,β=x,y,z) of randomized fcclattices with\nN= 1688 for vanishing magnetic moment (solid line, /circlecopyrt),\nandm= 0.056m0/hatwidez(dashed lines, /square,△,▽). Data points\nand standard deviations are obtained by averaging over 80\ndifferently randomized samples. Filled and unfilled markers\ncorrespond to storage ( E′,G′) and loss ( E′′,G′′) components,\nrespectively. Insets zoom onto the storage parts (a) /angbracketleftE′\nαα(ω)/angbracketright\nand (b) /angbracketleftG′\nαβ(ω)/angbracketrightat small ωto better resolve the different\ncurves. For illustrative purposes we have slightly shifted the\nbars for different data sets horizontally to better distingu ish\nthe individual bars.\nvalues for ω= 0, similarly to the results in Fig. 15. Ex-\nperimentally, deviations as small as 0 .01k/l2\n0−0.01k/l2\n0\nshould be accessible within rheometer sensitivities.\nThe main difference between the small- and large-\nmregimes is the qualitative change in ∝a\\}bracketle{tE′\nαα(ω)∝a\\}bracketri}htand\n∝a\\}bracketle{tG′\nαβ(ω)∝a\\}bracketri}ht(α,β=x,y,z) for increasing magnetic mo-\nment. For m/lessorsimilar0.06m0, and according to the different\ngeometries,weobservedincreaseordecreaseoftheelastic\nmoduli with increasing m. Conversely, for m/greaterorsimilar0.06m0\nwe observe all elastic moduli to increase with increasing\nmagnetic interaction for all frequencies and geometries.\nThe storage Young’s modulus for deformations in the\nm-direction ∝a\\}bracketle{tE′\nzz(ω)∝a\\}bracketri}htshows the most significant increase\nwhen compared to ∝a\\}bracketle{tE′\nxx(ω)∝a\\}bracketri}htand∝a\\}bracketle{tE′\nyy(ω)∝a\\}bracketri}ht. This trend\ncontinues at large ω[see inset of Fig. 16 (a)]. In a sim- 1e-05 0.0001 0.001 0.01 0.1 1 10\n 0.001  0.01  0.1  1  10| 〈 E'(ω,m) 〉 - 〈 E'(ω=0,m=0) 〉 | [k/l02]\nω [k/c l0]Absolute Deviation | 〈E'(ω, m=0.064) 〉 - 〈E'(ω=0, m=0) 〉 |\n(a)〈E'〉(m=0)\n〈E'xx〉(m>0)\n〈E'yy〉(m>0)\n〈E'zz〉(m>0)\n 1e-05 0.0001 0.001 0.01 0.1 1 10\n 0.001  0.01  0.1  1  10| 〈 G'(ω,m) 〉 - 〈 G'(ω=0,m=0) 〉 | [k/l02]\nω [k/c l0]Absolute Deviation | 〈G'(ω, m=0.064) 〉 - 〈G'(ω=0, m=0) 〉 |\n(b)〈G'〉(m=0)\n〈G'xy〉(m>0)\n〈G'xz〉(m>0)\n〈G'zy〉(m>0) 1  10\n 1  10\nFIG. 16: Storage dynamic elastic moduli (a) /angbracketleftE′\nαα(ω)/angbracketrightand\n(b)/angbracketleftG′\nαβ(ω)/angbracketright(α,β=x,y,z) of randomized fcclattices with\nN= 1688 for vanishing magnetic moment (solid line, /circlecopyrt), and\nm= 0.064m0/hatwidez(dashed lines, /square,△,▽). We plot on a double\nlogarithmic scale the absolute deviation from the respecti ve\naverage static modulus at m= 0. Data points and standard\ndeviations are obtained from statistics over 80 differently ran-\ndomized samples. For illustrative purposes we have slightl y\nshifted the bars for different data sets horizontally to bett er\ndistinguish the individual bars. Insets zoom onto the stora ge\nparts (a) /angbracketleftE′\nαα(ω)/angbracketrightand (b)/angbracketleftG′\nαβ(ω)/angbracketrightat large ωto better re-\nsolve the different curves. Small values of the curves for the\nm= 0 cases at low ωindicate smooth convergence to the\nrespective static moduli in Fig. 14.\nilar fashion, the large- ωbehavior of the storage modu-\nlus∝a\\}bracketle{tG′\nzy(ω)∝a\\}bracketri}htrelative to shear deformations of the chains\nalignedalong m[see inset ofFig. 16(b)] suggestsa larger\nincrease than for ∝a\\}bracketle{tG′\nxz(ω)∝a\\}bracketri}htand∝a\\}bracketle{tG′\nxy(ω)∝a\\}bracketri}ht. These overall\ntrends of the dynamic moduli are further enhanced and\nincreased for even larger m.\nThe loss components of the dynamic moduli, both\nYoung and shear, show again an increase with increas-\ningmover all frequencies and geometries. Furthermore,\nthe amount of increase follows approximately the same\ntrends as for the corresponding storage components (see\nappendix F, Fig. 20).18\nXI. CONCLUSIONS\nWe have described and applied a method to determine\nthe dynamic elastic moduli in discretized mesoscopic\nmodel systems representing magnetic elastic composite\nmaterials. More precisely, we have confined ourselves to\nparticle-based dipole-spring models [54–59] to character-\nize the behavior of magnetic gels and elastomers. The\nmagnitudesofYoungandshearmoduliwereevaluatedfor\ndifferent frequencies, particle distributions, magnitudes\nand orientations of the magnetic moments. We find the\nelasticmodulito lowestordertoincreaseordecreasewith\nthe magnitude of the magnetic moment according to the\nparticle distribution, the selected orientation, and the se-\nlected frequency.\nTo summarize our results, we find that increasing mag-\nnetic interactions tend to line up the particles in the\ndirection of the magnetic dipoles. This, in regular lat-\ntices, can result in different effects according to the con-\nsidered structure. In general, however, we find the Young\nmodulus in the directions of elongation to increase [51]\nand, vice versa, to decrease in the directions of shrinking.\nFor randomized particle arrangements we find a “hard-\nened” regime, where dipole–dipole attractions overcome\nthe elastic spring interactions and the elastic moduli sig-\nnificantly increase. Here, the increase of the storage part\nof the Young modulus in the direction parallel to the\nmagnetic moments is significantly largercompared to the\nperpendicular directions, in agreement with experiments\nreported in the literature [38, 92]. Furthermore, for all\ndistributions (except forthe randomizedarrangementsat\nhighm) we find the storage part of some of the investi-\ngated shear moduli to change tendency from increase to\ndecrease with m, or vice versa, for increasing frequency\nω. The loss component of the dynamic moduli follows\nan overall linear behavior for all cases at low and high ω\nwith a crossover regime in between. In conclusion, the\nbehavior of the dynamic elastic moduli with varying m\nandωstronglydepends on the spatialarrangementofthe\nmagnetic particles. The anglesbetween the magnetic mo-\nmentsandthe directionstofindthe nearestneighborsare\ncrucial to determine whether, for a selected direction, the\nsystem shrinks or elongates when switching on magnetic\ninteractions and whether the elastic moduli increase or\ndecrease.\nOur systems were of cubelike shape and finite size. On\ntwo opposing boundaries, we imposed prescribed force\nfields leading to an overall strain response of the whole\nsystem. The other boundaries remained unconstrained.\nSuch a geometry is characteristic for experimental inves-\ntigations using plate–plate rheometers. Assuming parti-\ncle sizesin the micrometerrange, our systemscorrespond\nto samples of several ten micrometers in thickness. Such\nexperimental samples can be analyzed using piezorheo-\nmetric devices [85, 93]. In fact, for anisotropic magnetic\ngels, corresponding piezorheometric measurements were\nperformed already more than a decade ago [81]. It will\nbe interesting to compare our approach in more detailwith such experimental investigations in the future.\nIt is important to model and understand the dynamic\nresponse of the materials at different frequencies in the\nview of many practical applications, from soft actuators\n[24] to vibration absorbers [25, 26]. Our method explic-\nitly connects the relaxational modes of the system on\nthe mesoscopic level [56] with the macroscopic dynamic\nresponse [47, 48, 50, 94]. Our approach allows to capture\nthe internal rearrangements of the system under an ex-\nternally applied stress or magnetic field and to link it to\nthe consequences for the overall system behavior. Fur-\nthermore, our technique can be applied to any particle\ndistribution, particularly also to those drawn from exper-\nimental analysis of real samples [55, 56].\nGeneralizations to systems composed of anisotropic\nparticles [95], as well as including rotational degrees\nof freedom [36, 54] and possibly induced-dipole effects\n[68, 69] could be added to the present framework in sub-\nsequentsteps. Apart fromthat, the mesoscopicallybased\ndynamicinvestigationscouldbeextendedtomorerefined\napproaches, where the elastic matrix between discretized\nparticles of finite volume is described in terms of contin-\nuum elasticity theory [34]. As indicated above, it will\nbe possible to use experimental data [44, 55, 56, 96] as\ninput for the initial particle positions and compare calcu-\nlated dynamic moduli with their measured counterparts,\nalso as a function of magnetic interaction strengths. In a\ncombinedeffort betweenexperiments and theory, suchan\napproach can serve to devise smarter and new materials\nwith optimized magnetic field dependence and adjusted\nbehavior at different frequencies.\nAcknowledgments\nThe authors thank the Deutsche Forschungsgemein-\nschaft for support of this work through the priority pro-\ngram SPP 1681.19\nAppendix A: Steric Repulsion Parameters\nThe relatively soft steric repulsion between two parti-\nclesiandjat positions RiandRjconnected by the\nvectorrij=Rj−Riis modeled by a generic potential\nvs(rij). Introducing the exponents pandq, the func-\ntional form of this potential is given by\nvs(r) =εs/bracketleftBigg/parenleftBigr\nσs/parenrightBig−p\n−/parenleftBigr\nσs/parenrightBig−q\n−/parenleftBigrc\nσs/parenrightBig−p\n+/parenleftBigrc\nσs/parenrightBig−q\n−cs(r−rc)2\n2/bracketrightBigg\n(A1)\nifr=|rij|< rcandvs(r) = 0 otherwise. The parameter\nrc=σs(p/q)1/(p−q)follows from the condition vs(rc)′=\n0, whereas csis chosen such that vs(rc)′′= 0. We find\ncs=p−2+q\np−q(p−q)q2+p\np−q\n(σs)2. (A2)\nAppendix B: Derivatives of Pair Interaction\nPotentials\nWe consider pair interactions between particles iand\nj, at positions RiandRj, respectively, and connected by\nrij=Rj−Ri. When the particles are linked by a har-\nmonic spring, their harmonic pair interaction potential\nis\nvel\nij=k\n2ℓ0\nij/parenleftbig\nrij−ℓ0\nij/parenrightbig2, (B1)\ncompare with Eq. (1). rij=|rij|andl0\nijis the un-\nstrained length of the spring. The gradient components\n(α=x,y,z) follow as (we here drop the ijsubscripts for\nsimplicity)\n∂vel\n∂rα=k\nℓ0/parenleftbig\nr−ℓ0/parenrightbigrα\nr. (B2)\nThe derivatives appearing below in Eq. (C4) are then\n∂2vel\n∂rβ∂rα=k\nℓ0/bracketleftbiggrαrβ\nr2+(r−ℓ0)δαβr2−rαrβ\nr3/bracketrightbigg\n.(B3)\nFurthermore, the steric repulsion pair potential vshas\nbeen addressed in detail in Appendix A. The gradient\ncomponents ( α=x,y,z) of the steric pair potential [see\nEq. (A1)] follow for r < rcas\n∂vs\n∂rα=−εsrα\nr/bracketleftbiggp\nr/parenleftBigr\nσs/parenrightBig−p\n−q\nr/parenleftBigr\nσs/parenrightBig−q\n+cs(r−rc)/bracketrightbigg\n(B4)andvanishfor r≥rc. Thederivativesbelowcontributing\nto Eq. (C4) are given by\n∂2vs\n∂rβ∂rα=−εs/braceleftbigg/parenleftbiggδαβ\nr2−2rαrβ\nr4/parenrightbigg/bracketleftbigg\np/parenleftBigr\nσs/parenrightBig−p\n−q/parenleftBigr\nσs/parenrightBig−q/bracketrightbigg\n−rαrβ\nr4/bracketleftbigg\np2/parenleftBigr\nσs/parenrightBig−p\n−q2/parenleftBigr\nσs/parenrightBig−q/bracketrightbigg\n+cs/bracketleftbiggrαrβ\nr2+(r−rc)δαβr2−rαrβ\nr3/bracketrightbigg/bracerightbigg\n(B5)\nforr < rcand vanish when r≥rc.\nFinally, the magnetic pair interaction potential vmas\nin Eq. (4) reads\nvm\nij=m2r2\nij−3(m·rij)2\nr5\nij(B6)\nin using reduced units, see also Eq. (4). The gradient\ncomponents ( α=x,y,z) of the previous expression read\n∂vm\n∂rα=−3\nr5/bracketleftBig\nm2rα+2mα(m·r)\n−5rα(m·r)2\nr2/bracketrightBig\n. (B7)\nThe derivatives appearing below in Eq. (C4) are given by\n∂2vm\n∂rβ∂rα=−3\nr5/bracketleftBigg\nm2δαβ−5m2rαrβr−2\n−10(m·r)r−2/parenleftbig\nmαrβ+mβrα/parenrightbig\n+2mαmβ\n−5(m·r)2r−2/parenleftbig\nδαβ−7rαrβr−2/parenrightbig/bracketrightBigg\n. (B8)\nAppendix C: Hessian Matrix for Pair Interaction\nPotentials\nHere we repeat in detail the derivation of the Hessian\nfor a system interacting solely via pair potentials. That\nis, any two particles iandjat positions RiandRjin-\nteract through a pair potential vdepending only on the\nconnecting vector rij=Rj−Ri. Then we can write\nU=1\n2N/summationdisplay\ni,j=1\ni/negationslash=jv(rij), (C1)\nwhereNisthetotalnumberofparticles. Again, Riisthe\nposition of the i-th particle ( i= 1...N),rij=Rj−Ri,\nand we denote by Rα\ni(α=x,y,z) theα-component of\nRi. For reasons of symmetry, v(rij) =v(rji). The sum\nin Eq. (C1) together with the prefactor1\n2then runs over\nall different pairs counting each of them only once. We\nabbreviate vij=v(rij). The gradient components ( α=20\nx,y,z) of the energy Ufollow as\n∂U\n∂Rα\nk=1\n2N/summationdisplay\ni,j=1\ni/negationslash=j∂vij\n∂Rα\nk(C2)\n=N/summationdisplay\nj=1\nj/negationslash=k∂vkj\n∂Rα\nk=−N/summationdisplay\nj=1\nj/negationslash=k∂vkj\n∂rα\nkj,\nsetting the force −∂U/∂Rkon the positional degrees of\nfreedom of the k-th particle.\nNext, we obtain the Hessian of the system as\n∂2U\n∂Rα\ni∂Rβ\nk=\n\n∂2vik\n∂Rα\ni∂Rβ\nk(i∝\\e}atio\\slash=k),\nN/summationdisplay\nj=1\nj/negationslash=i∂2vij\n∂Rα\ni∂Rβ\ni(i=k).(C3)\nThus, for pair interactions, the diagonal elements of the\nHessian contain the second derivatives of all pair interac-\ntions, whereas the off-diagonal elements are given by a\nsingle term. Since rij=Rj−Ri, the previous equation\ncan be expressed in terms of connecting vectors only:\n∂2U\n∂Rα\ni∂Rβ\nk=\n\n−∂2vik\n∂rα\nik∂rβ\nik(i∝\\e}atio\\slash=k),\nN/summationdisplay\nj=1\nj/negationslash=i∂2vij\n∂rα\nij∂rβ\nij(i=k).(C4)\nAppendix D: Torque-Free Force Fields\nOur scope is to describe the system behavior for pre-\nselected specified orientations. However, both during the\nsearch for the corresponding equilibrium state of the sys-\ntem (see section III) and the implementation of an ex-\nternal force (see section VB), the system may tend to\nperform a rigid rotation. We therefore must exclude such\nrigid rotations. Here we describe a simple method to re-\ndefine the generalized force field (or likewise the gradient\nof the total energy) so that the net overall torque on the\nsystem vanishes.\nWe consider the force field facting on the particles at\npositions Riwith components fi(i= 1,...N). The net\ntorqueτis given by\nτ=N/summationdisplay\ni=1qi×fi, (D1)\nwhereqi=Ri−Rcis the distance of the particle po-\nsitionsRifrom the center of mass Rc=1\nN/summationtext\niRi. To\nprevent, e.g., a global rotation of the system around thez-axis, the z-component of τ, i.e.τz, must vanish. We\ndefine a uniform, counter-clockwise rotational force field\naround the z-axisP(q) =cR(−qy,qx,0), withqa vector\nin thexy-plane and cRa constant. Next, we determine\ncRby imposing Pto have the same torque as given by\nf:\nN/summationdisplay\ni=1(qi×fi)z=τz=N/summationdisplay\ni=1[qi×P(qi)]z(D2)\n=cRN/summationdisplay\ni=1/bracketleftBig\n(qx\ni)2+(qy\ni)2/bracketrightBig\n.\nWe obtain the field Pby solving for the constant cR,\nleading to\ncR=τz\n/summationtextN\ni=1/bracketleftBig\n(qx\ni)2+(qy\ni)2/bracketrightBig. (D3)\nTherefore we can make f“torque-free” concerning the\nz-direction by subtracting P, i.e.fi→fi−P(qi) (i=\n1,...N). By repeating the procedure for the remaining\ndirections, we get rid of the rigid rotations induced by f\nand obtain a torque-free force field.\nAppendix E: Static Young Moduli of Regular\nLattices\nWe here present a simple energy argument to interpret\nthe behavior of the Young moduli of the regular lattices\npresented in sections VIII and IX. A regular lattice is\ngenerated by the basis vectors a1,a2, anda3. Therefore\na lattice point can be written as rijk=ia1+ja2+ka3,\nwithi,j,k∈Zintegers. If the particles interact by the\npair potential v, the total energy per particle in an in-\nfinitely extended lattice is given by\nUp=1\n2/summationdisplay\nn∈N0v(rn), (E1)\nwhere the sum runs over all lattice points (origin ex-\ncluded) labeled by the discrete index ncontained in the\nsetN0=Z3\\{(0,0,0)}.\nSince we consider the regular lattice to be the ground\nstate of the system, a small deformation that transforms\nrn→r′\nn(n∈ N0) has an energy-per-particle cost that\nto lowest order reads\n∆Up=1\n2/summationdisplay\nn∈N01\n2u⊺\nn·h(rn)·un (E2)\nwhereun=r′\nn−rn,⊺indicates transposition, and h(rn)\nis the Hessian matrix of the interaction v(rn) between\nthe particle fixed in the origin and the nth neighbor. Its\nelements are given by hµν(rn) =∂2v(rn)/∂rµ\nn∂rν\nn, with\nµ,ν=x,y,z.\nThe displacements un=D·rncorresponding to a\nuniform strain are given by the constant components of21\nthe displacement tensor D. The energy of the strain\ndeformation then follows as\n∆Up=1\n2/summationdisplay\nαβγδCαβγδ\n0DαβDγδ\nwithCαβγδ\n0=1\n2/summationdisplay\nn∈N0rα\nnhβγ(rn)rδ\nn,(E3)\nwhereα,β,γ,δ =x,y,z.\nIn the following we focus on compressive/dilative\nstrains and therefore consider diagonal Ddisplacement\ntensors. For an applied strain εααalong the α-direction\nDαα∝\\e}atio\\slash= 0 is imposed. The remaining components of D\nare relaxed to minimize the lattice energy\n∂∆Up\n∂Dµµ= 0,∀µ∝\\e}atio\\slash=α. (E4)\nThis leads to a system of linear equations the solution\nof which relates the components Dµµ(µ∝\\e}atio\\slash=α) to the im-\nposed deformation Dαα. As a result, we obtain Young’s\nmodulus Eαα[following the notation as in the main text,\nsee Eq. (16)] given by\nEαα=1\nVpd2∆Up\n(dDαα)2=1\nVp(Cαα\n0−Bα)\nwithBα=/summationdisplay\nβγCαβ\n0Cγγ\n0Cαβ\n0−Cαγ\n0Cβγ\n0\nCββ\n0Cγγ\n0−(Cβγ\n0)2(ǫαβγ)2,(E5)\nwhereVp= 1/ρ=V/Nis the volume per particle, we\nabbreviated Cαβ\n0=Cααββ\n0, andǫαβγis the Levi-Civita\nsymbol. The contributions −Bαto the elastic moduli\ntake into account relaxation along the remaining perpen-\ndicular axes and lower the moduli.For small values of the magnetic moment m, we write,\nto lowest order in m,h(rn) =h0(rn)+m2hm(rn), where\nthe elements of the matrix m2hm(rn) are as listed in\nEq. (B8). Thus, we can obtain both the static Young’s\nmodulus at m= 0 and the initial quadratic behavior for\nsmallm.\nAppendix F: Additional Information on the Loss\nPart of the Dynamic Elastic Moduli\nHere we show in more detail the various behaviors of\nthe loss part of the dynamic moduli as functions of fre-\nquencyωand magnitude of the magnetic moment mfor\nthe different considered geometries. As we have men-\ntioned before, we find as a general trend the loss parts to\nlinearly increase with ωat low and high frequencies. It\nresultsfromourviscousfriction term[seeEq.(19)]which,\nin the case of an oscillatory deformation as in Eq. (22),\nis proportional to ω. Moreover, it is consistent with the\npredicted loss component of the dynamic moduli in the\nKelvin-Voigt model [89, 90]. Therefore, and for better il-\nlustration, we plot the loss parts after division by ω. The\nagreement with linear behavior is confirmed in this way,\ni.e.E′′\nαα(ω)/ωandG′′\nαβ(ω)/ω(α,β=x,y,z) converge\nto a finite value in both the low- and high- ωlimit, see\nFigs. 17–20.\nOn the one hand, the regular lattices addressed in sec-\ntions VIII and IX show different trends for the loss parts\nas functions of mandω, as mentioned in the main text\nand illustrated in Figs. 17 and 18. On the other hand\nour randomized lattices generally show increasing loss\nparts with increasing mfor all frequencies, although the\namount of gain varies with the selected geometries, see\nFigs. 19 and 20.\n[1] G. Filipcsei, I. Csetneki, A. Szil´ agyi, and M. Zr´ ınyi, Adv.\nPolym. Sci. 206, 137 (2007).\n[2] A. M. Menzel, Phys. Rep. 554, 1 (2015).\n[3] M. Lopez-Lopez, J. D. Dur´ an, L. Y. Iskakova, and A. Y.\nZubarev, J. Nanofluids 5, 479 (2016).\n[4] S. Odenbach, Arch. Appl. Mech. 86, 269 (2016).\n[5] E. Jarkova, H. Pleiner, H.-W. M¨ uller, and H. R. Brand,\nPhys. Rev. E 68, 041706 (2003).\n[6] M. Zr´ ınyi, L. Barsi, and A. B¨ uki, J. Chem. Phys. 104,\n8750 (1996).\n[7] G. V. Stepanov, S. S. Abramchuk, D. A. Grishin, L. V.\nNikitin, E.Y.Kramarenko, andA.R.Khokhlov,Polymer\n48, 488 (2007).\n[8] X. Guan, X. Dong, and J. Ou, J. Magn. Magn. Mater.\n320, 158 (2008).\n[9] H. B¨ ose and R. R¨ oder, J. Phys.: Conf. Ser. 149, 012090\n(2009).\n[10] X. Gong, G. Liao, and S. Xuan, Appl. Phys. Lett. 100,\n211909 (2012).\n[11] B. A. Evans, B. L. Fiser, W. J. Prins, D. J. Rapp, A. R.\nShields, D. R. Glass, and R. Superfine, J. Magn. Magn.\nMater.324, 501 (2012).\n[12] D. Y. Borin, G. V. Stepanov, and S. Odenbach, J. Phys.:\nConf. Ser. 412, 012040 (2013).[13] R. E. Rosensweig, Ferrohydrodynamics (Cambridge Uni-\nversity Press, Cambridge, 1985).\n[14] S. Odenbach, Colloids Surf., A 217, 171 (2003).\n[15] S. Odenbach, Magnetoviscous effects in fer rofluids\n(Springer Berlin / Heidelberg, 2003).\n[16] S. Odenbach, J. Phys.: Condens. Matter 16, R1135\n(2004).\n[17] B. Huke and M. L¨ ucke, Rep. Prog. Phys. 67, 1731 (2004).\n[18] P. Ilg, M. Kr¨ oger, and S. Hess, J. Magn. Magn. Mater.\n289, 325 (2005).\n[19] C. Holm and J.-J. Weis, Curr. Opin. Colloid Interface Sc i.\n10, 133 (2005).\n[20] S. H. L. Klapp, J. Phys.: Condens. Matter 17, R525\n(2005).\n[21] S. M. Cattes, S. H. L. Klapp, and M. Schoen, Phys. Rev.\nE91, 052127 (2015).\n[22] S. D. Peroukidis and S. H. L. Klapp, Phys. Rev. E 92,\n010501 (2015).\n[23] S. H. L. Klapp, Curr. Opin. Colloid Interface Sci. 21, 76\n(2016).\n[24] K. Zimmermann, V. A. Naletova, I. Zeidis, V. B¨ ohm, and\nE. Kolev, J. Phys.: Condens. Matter 18, S2973 (2006).\n[25] H.-X. Deng, X.-L. Gong, and L.-H. Wang, Smart Mater.\nStruct.15, N111 (2006).22\n 22 22.2 22.4 22.6 22.8 23 23.2 23.4 23.6 23.8 24\n 0.001  0.01  0.1E''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Young's Modulus E''( ω, m=0.1)/ ω(a)\nE''αα/ω(m=0) \nE''xx/ω(m>0) \nE''yy/ω(m>0) \nE''zz/ω(m>0) \n 6 8 10 12 14 16 18 20 22 24\n 0.1  1  10E''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Young's Modulus E''( ω, m=0.1)/ ω(b)\nE''αα/ω(m=0) \nE''xx/ω(m>0) \nE''yy/ω(m>0) \nE''zz/ω(m>0) \n 7.5 7.55 7.6 7.65 7.7 7.75 7.8 7.85 7.9\n 10  100  1000E''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Young's Modulus E''( ω, m=0.1)/ ω(c)\nE''αα/ω(m=0)\nE''xx/ω(m>0) \nE''yy/ω(m>0) \nE''zz/ω(m>0) \n 6 6.5 7 7.5 8 8.5\n 0.001  0.01  0.1G''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Shear Modulus G''( ω, m=0.1)/ ω(d)\nG''αβ/ω(m=0) \nG''xy/ω(m>0) \nG''xz/ω(m>0) \nG''zy/ω(m>0) \n 1 2 3 4 5 6 7\n 0.1  1  10G''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Shear Modulus G''( ω, m=0.1)/ ω(e)\nG''αβ/ω(m=0) \nG''xy/ω(m>0) \nG''xz/ω(m>0) \nG''zy/ω(m>0) \n 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3\n 10  100  1000G''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Shear Modulus G''( ω, m=0.1)/ ω(f)\nG''αβ/ω(m=0) \nG''xy/ω(m>0) \nG''xz/ω(m>0) \nG''zy/ω(m>0) \nFIG. 17: Loss parts (a, b, c) E′′\nαα(ω) and (d, e, f) G′′\nαβ(ω)\n(α,β=x,y,z)forthedynamicelasticmoduliofasimplecubic\nlattice with N= 3375 for vanishing magnetic moment (solid\nline,/circlecopyrt), andm= 0.1m0/hatwidez(dashed lines, /square,△,▽). Because\nof the overall trend of a linear increase in frequency at low\nand high frequencies, we here present the moduli divided by\nω. Zoom-ins onto the low-, intermediate-, and high- ωregions\nare shown in panels (a, d), (b, e) and (c, f), respectively.\n[26] T. L. Sun, X. L. Gong, W. Q. Jiang, J. F. Li, Z. B. Xu,\nand W. Li, Polym. Test. 27, 520 (2008).\n[27] D. Szab´ o, G. Szeghy, and M. Zr´ ınyi, Macromolecules 31,\n6541 (1998).\n[28] R. V. Ramanujan and L. L. Lao, Smart Mater. Struct.\n15, 952 (2006).\n[29] M. Babincov´ a, D. Leszczynska, P. Sourivong, 19 20 21 22 23 24 25 26 27\n 0.001  0.01  0.1E''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Young's Modulus E''( ω, m=0.1)/ ω(a)\nE''αα/ω(m=0) \nE''xx/ω(m>0) \nE''yy/ω(m>0) \nE''zz/ω(m>0) \n 6 8 10 12 14 16 18 20 22\n 0.1  1  10E''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Young's Modulus E''( ω, m=0.1)/ ω(b)\nE''αα/ω(m=0) \nE''xx/ω(m>0) \nE''yy/ω(m>0) \nE''zz/ω(m>0) \n 6 6.05 6.1 6.15 6.2 6.25\n 10  100  1000E''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Young's Modulus E''( ω, m=0.1)/ ω(c)\nE''αα/ω(m=0)\nE''xx/ω(m>0) \nE''yy/ω(m>0) \nE''zz/ω(m>0) \n 4.5 5 5.5 6 6.5 7 7.5\n 0.001  0.01  0.1G''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Shear Modulus G''( ω, m=0.1)/ ω(d)\nG''αβ/ω(m=0) \nG''xy/ω(m>0) \nG''xz/ω(m>0) \nG''zy/ω(m>0) \n 1.5 2 2.5 3 3.5 4 4.5 5\n 0.1  1  10G''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Shear Modulus G''( ω, m=0.1)/ ω(e)\nG''αβ/ω(m=0) \nG''xy/ω(m>0) \nG''xz/ω(m>0) \nG''zy/ω(m>0) \n 1.58 1.6 1.62 1.64 1.66 1.68 1.7\n 10  100  1000G''(ω,m)/ω [c/l0]\nω [k/c l0]Loss Dynamic Shear Modulus G''( ω, m=0.1)/ ω(f)\nG''αβ/ω(m=0) \nG''xy/ω(m>0) \nG''xz/ω(m>0) \nG''zy/ω(m>0) \nFIG. 18: Loss parts (a, b, c) E′′\nαα(ω) and (d, e, f) G′′\nαβ(ω)\n(α,β=x,y,z) for the dynamic elastic moduli of an fcclattice\nwithN= 3430 for vanishing magnetic moment (solid line,\n/circlecopyrt), andm= 0.1m0/hatwidez(dashed lines, /square,△,▽). Since the\nloss moduli increase linearly with the frequency at low and\nhigh frequencies, we here show them divided by ω. Zoom-ins\nonto the low-, intermediate-, and high- ωregions are shown in\npanels (a, d), (b, e) and (c, f), respectively.\nP.ˇCiˇ cmanec, and P. Babinec, J. Magn. Magn. Mater.\n225, 109 (2001).\n[30] L. L. Lao and R. V. Ramanujan, J. Mater. Sci.: Mater.\nMed.15, 1061 (2004).\n[31] N. Frickel, R. Messing, and A. M. Schmidt, J. Mater.\nChem.21, 8466 (2011).\n[32] E. Allahyarov, A. M. Menzel, L. Zhu, and H. L¨ owen,23\n 1 10 100 1000\n 1e-08  1e-06  0.0001  0.01  1〈 E''(ω,m) 〉 /ω [c/l0]\nω [k/c l0]Loss Young's Modulus, 〈 E''(ω, m=0.056) 〉 /ω (a)\n〈E''〉/ω(m=0)\n〈E''xx〉/ω(m>0)\n〈E''yy〉/ω(m>0)\n〈E''zz〉/ω(m>0)\n 0.1 1 10 100 1000\n 1e-08  1e-06  0.0001  0.01  1〈 G''(ω,m) 〉 /ω [c/l0]\nω [k/c l0]Loss Shear Modulus, 〈 G''(ω, m=0.056) 〉 /ω (b)\n〈G''〉/ω(m=0)\n〈G''xy〉/ω(m>0)\n〈G''xz〉/ω(m>0)\n〈G''zy〉/ω(m>0) 1\n 1\nFIG. 19: Average loss parts (a, b) /angbracketleftE′′\nαα(ω)/angbracketrightand (c, d)\n/angbracketleftG′′\nαβ(ω)/angbracketright(α,β=x,y,z)forthedynamicelastic moduliofran-\ndomized fcclattices with N= 1688 for vanishing magnetic\nmoment (solid line, /circlecopyrt), andm= 0.056m0/hatwidez(dashed lines,\n/square,△,▽). Because of the overall trend of a linear increase in\nfrequency at low and high frequencies, we here present the\nmoduli divided by ω. Data points and standard deviations\nare obtained by averaging over 80 differently randomized sam -\nples. Because of the different randomizations, the initial s lope\nof the moduli in the ω→0 limit can vary significantly, thus\nleading to large bars in the small- ωregime and for the m >0\ncases, which are not shown here. Insets (a) and (b) zoom in\nonto the Young and shear loss moduli behavior, respectively ,\nat high frequencies for better resolving the individual cur ves.\nSmart Mater. Struct. 23, 115004 (2014).\n[33] M. K¨ astner, S. M¨ uller, J. Goldmann, C. Spieler, J. Bru m-\nmund, and V. Ulbricht, Int. J. Numer. Meth. Eng. 93,\n1403 (2013).\n[34] P. Cremer, H. L¨ owen, and A. M. Menzel, Appl. Phys.\nLett.107, 171903 (2015).\n[35] S. Huang, G. Pessot, P. Cremer, R. Weeber, C. Holm,\nJ. Nowak, S. Odenbach, A. M. Menzel, and G. K. Auern-\nhammer, Soft Matter 12, 228 (2016).\n[36] R. Weeber, S. Kantorovich, and C. Holm, J. Chem. Phys.\n143, 154901 (2015).\n[37] J. L. Mietta, P. I. Tamborenea, and R. Martin Negri, Soft\nMatter12, 6430 (2016).\n[38] Z. Varga, G. Filipcsei, and M. Zr´ ınyi, Polymer 47, 227\n(2006).\n[39] H. Haider, C. Yang, W. J. Zheng, J. Yang, M. X. Wang,\nM. Zr´ ınyi, S. Sang, Y. Osada, Z. Suo, Q. Zhang, et al., 1 10 100 1000 10000\n 1e-08  1e-06  0.0001  0.01  1〈 E''(ω,m) 〉 /ω [c/l0]\nω [k/c l0]Loss Young's Modulus, 〈 E''(ω, m=0.064) 〉 /ω (a)\n〈E''〉/ω(m=0)\n〈E''xx〉/ω(m>0)\n〈E''yy〉/ω(m>0)\n〈E''zz〉/ω(m>0)\n 0.1 1 10 100 1000\n 1e-08  1e-06  0.0001  0.01  1〈 G''(ω,m) 〉 /ω [c/l0]\nω [k/c l0]Loss Shear Modulus, 〈 G''(ω, m=0.064) 〉 /ω (b)\n〈G''〉/ω(m=0)\n〈G''xy〉/ω(m>0)\n〈G''xz〉/ω(m>0)\n〈G''zy〉/ω(m>0) 1\n 1\nFIG. 20: Average loss parts (a, b) /angbracketleftE′′\nαα(ω)/angbracketrightand (c, d)\n/angbracketleftG′′\nαβ(ω)/angbracketright(α,β=x,y,z) for the dynamic elastic moduli of\nrandomized fcclattices with N= 1688 for vanishing mag-\nnetic moment (solid line, /circlecopyrt), andm= 0.064m0/hatwidez(dashed\nlines,/square,△,▽). Since the loss moduli increase linearly with\nthe frequency at low and high frequencies, we here show them\ndivided by ω. Data points and standard deviations are ob-\ntained by averaging over 80 differently randomized samples.\nBecause of the different randomizations, the initial slope o f\nthe moduli in the ω→0 limit can vary significantly, thus\nleading to large bars in the small- ωregime and for the m >0\ncases, which are not shown here. Insets zoom in onto the (a)\nYoung and (b) shear loss moduli behavior at high frequencies\nfor better resolving the individual curves.\nSoft Matter 11(2015).\n[40] E. I. Wisotzki, M. Hennes, C. Schuldt, F. Engert,\nW. Knolle, U. Decker, J. A. Kas, M. Zink, and S. G.\nMayr, J. Mater. Chem. B 2, 4297 (2014).\n[41] R. Weeber, S. Kantorovich, and C. Holm, Soft Matter 8,\n9923 (2012).\n[42] L. Roeder, P. Bender, M. Kundt, A. Tsch¨ ope, and A. M.\nSchmidt, Phys. Chem. Chem. Phys. 17, 1290 (2015).\n[43] R. Messing, N. Frickel, L. Belkoura, R. Strey, H. Rahn,\nS. Odenbach, and A. M. Schmidt, Macromolecules 44,\n2990 (2011).\n[44] T. Gundermann and S. Odenbach, Smart Mater. Struct.\n23, 105013 (2014).\n[45] J. Landers, L. Roeder, S. Salamon, A. M. Schmidt, and\nH. Wende, J. Phys. Chem. 119, 20642 (2015).\n[46] C. Garcia, Y. Zhang, F. DiSalvo, and U. Wiesner, Angew.\nChem. Int. Edit. 42, 1526 (2003).24\n[47] H. R. Brand, A. Fink, and H. Pleiner, Eur. Phys. J. E\n38, 65 (2015).\n[48] H. R. Brand and H. Pleiner, Eur. Phys. J. E 37, 122\n(2014).\n[49] A. Y. Zubarev, Soft Matter 8, 3174 (2012).\n[50] S. Bohlius, H. R. Brand, and H. Pleiner, Phys. Rev. E\n70, 061411 (2004).\n[51] D. Ivaneyko, V. Toshchevikov, M. Saphiannikova, and\nG. Heinrich, Condens. Matter Phys. 15, 33601 (2012).\n[52] M. R.Dudek, B. Grabiec, andK. W. Wojciechowski, Rev.\nAdv. Mater. Sci. 14, 167 (2007).\n[53] D. S. Wood and P. J. Camp, Phys. Rev. E 83, 011402\n(2011).\n[54] M.A.Annunziata,A.M.Menzel, andH.L¨ owen, J.Chem.\nPhys.138, 204906 (2013).\n[55] G. Pessot, P. Cremer, D. Y. Borin, S. Odenbach,\nH. L¨ owen, and A. M. Menzel, J. Chem. Phys. 141,\n124904 (2014).\n[56] M. Tarama, P. Cremer, D. Y. Borin, S. Odenbach,\nH. L¨ owen, and A. M. Menzel, Phys. Rev. E 90, 042311\n(2014).\n[57] P. A. S´ anchez, J. J. Cerd` a, T. Sintes, and C. Holm, J.\nChem. Phys. 139, 044904 (2013).\n[58] J. J. Cerd` a, P. A. S´ anchez, C. Holm, and T. Sintes, Soft\nMatter9, 7185 (2013).\n[59] D. Ivaneyko, V. Toshchevikov, and M. Saphiannikova,\nSoft Matter 11, 7627 (2015).\n[60] C. Spieler, M. K¨ astner, J. Goldmann, J. Brummund, and\nV. Ulbricht, Acta Mechanica 224, 2453 (2013).\n[61] Y. L. Raikher, O. V. Stolbov, and G. V. Stepanov, J.\nPhys. D: Appl. Phys. 41, 152002 (2008).\n[62] O. V. Stolbov, Y. L. Raikher, and M. Balasoiu, Soft Mat-\nter7, 8484 (2011).\n[63] Y. Han, W. Hong, and L. E. Faidley, Int. J. Solids Struct.\n50, 2281 (2013).\n[64] R.Weeber, S. Kantorovich, and C. Holm, J. Magn. Magn.\nMater.383, 262 (2015).\n[65] A. M. Menzel, J. Chem. Phys. 141, 194907 (2014).\n[66] G. Pessot, R. Weeber, C. Holm, H. L¨ owen, and A. M.\nMenzel, J. Phys.: Condens. Matter 27, 325105 (2015).\n[67] I. A. Belyaeva, E. Kramarenko, G. V. Stepanov,\nV. Sorokin, D. Stadler, and M. Shamonin, Soft Matter\n12, 2901 (2016).\n[68] A. M. Biller, O. V. Stolbov, and Y. L. Raikher, J. Appl.\nPhys.116, 114904 (2014).\n[69] E.Allahyarov, H.L¨ owen, andL.Zhu, Phys.Chem. Chem.\nPhys.117, 034504 (2015).\n[70] J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem.\nPhys.54, 5237 (1971).\n[71] W. W. Hager and Z. Hongchao, SIAM J. Optimiz. 16,\n170 (2005).\n[72] H. Shintani and H. Tanaka, Nature Materials 7, 870\n(2008).[73] L. Rovigatti, W. Kob, and F. Sciortino, J. Chem. Phys.\n135, 104502 (2011).\n[74] E. Lerner, E. DeGiuli, G. D¨ uring, and M. Wyart, Soft\nMatter10, 5085 (2014).\n[75] D. Frenkel and B. Smit, Understanding Molecular Si mu-\nlation (Academic Press, San Diego, 2002), 2nd ed.\n[76] D. Frenkel and A. J. C. Ladd, Phys. Rev. Lett. 59, 1169\n(1987).\n[77] D. Squire, A. Holt, and W. Hoover, Physica 42, 388\n(1969).\n[78] M. Born, J. Chem. Phys. 7, 591 (1939).\n[79] I. Fuereder and P. Ilg, J. Chem. Phys. 142, (2015).\n[80] M. V. Jari´ c and U. Mohanty, Phys. Rev. B 37, 4441\n(1988).\n[81] D. Collin, G. K. Auernhammer, O. Gavat, P. Martinoty,\nand H. R. Brand, Macromol. Rapid Commun. 24, 737\n(2003).\n[82] L. D. Landau and E. M. Lifshitz, Elasticity theory (Perg-\namon Press, 1975).\n[83] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel ,\nJ. Dongarra, J. Du Croz, A. Greenbaum, S. Hammar-\nling, A. McKenney, et al., LAPACK Users’ Guide (So-\nciety for Industrial and Applied Mathematics, Philadel-\nphia, 1999), 3rd ed.\n[84] A. Ivlev, H. L¨ owen, G. Morfill, and C. P. Royall, Com-\nplex plas mas and colloi dal dispersions (World Scientific,\n2012).\n[85] J. J. Zanna, P. Stein, J. D. Marty, M. Mauzac, and\nP. Martinoty, Macromolecules 35, 5459 (2002).\n[86] M. Sedlacik, M. Mrlik, V. Babayan, and V. Pavlinek,\nComposite Structures 135, 199 (2016).\n[87] E. Roeben, L. Roeder, S. Teusch, M. Effertz, U. K. Deit-\ners, and A. M. Schmidt, Colloid Polym. Sci. 292, 2013\n(2014).\n[88] N. Hohlbein, A. Shaaban, and A. M. Schmidt, Polymer\n69, 301 (2015).\n[89] F. Mainardi and G. Spada, Eur. Phys. J., Spec. Top. 193,\n133 (2011).\n[90] L. B. Eldred, W. P. Baker, and A. N. Palazotto, AIAA\nJ.33, 547 (1995).\n[91] N. W. Ashcroft and N. D. Mermin, Solid State Physics\n(Saunders, 1976).\n[92] G. Schubert and P. Harrison, Smart Mater. Struct. 25,\n015015 (2016).\n[93] M. Roth, M. D’Acunzi, D. Vollmer, and G. K. Auern-\nhammer, J. Chem. Phys. 132, 124702 (2010).\n[94] A. M. Menzel, Phys. Rev. E 94, 023003 (2016).\n[95] C. Passow, B. ten Hagen, H. L¨ owen, and J. Wagner, J.\nChem. Phys. 143, 044903 (2015).\n[96] S. G¨ unther and S. Odenbach, Transp. Porous Med. 112,\n105 (2016)."    },    {        "title": "2003.08035v1.Thermally_induced_generation_and_annihilation_of_magnetic_chiral_skyrmion_bubbles_and_achiral_bubbles_in_Mn_Ni_Ga_Magnets.pdf",        "content": "1 \n Thermally induced generation and a nnihilation of magnetic chiral skyrmion \nbubbles and ac hiral bubbles in Mn -Ni-Ga Magnets  \nBei Ding1,2,3#, Junwei Zhang2,4#, Hang Li1,3, Senfu Zhang2, Enke Liu1,5, Guangheng \nWu1, Xixiang Zhang2* and Wenhong Wang1,5* \n \n1 Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, \nChinese Academy of Sciences, Beijing 100190, China  \n2 Physical Science and Engineering, King Abdullah University of Science and \nTechnology (KAUST), Thuwal 23955 -6900, Saudi Arabia  \n3 University of Chinese Academy of Sciences, Beijing 100049, China  \n4 Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education, \nLanzhou University, Lanzhou 730000, People’s Republic of China  \n5 Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China  \n \n \n \n# Contributions: Bei Ding  and Junwei Zhang  contributed equally to this work.  \n \n*Corresponding author.   E-mail: xixiang.zhang@kaust.edu.sa  \nwenhong.wang@iphy.ac.cn  \n  2 \n  \nABSTRACT:  \n \n Magnetic chiral skyrmion bubbles and achiral  bubbles are two independent magnetic \ndomain structures, in which the former with equivalent winding number to skyrmions \noffer s great promise as information carriers for further spintronic devices. Here, in this \nwork, we experimentally investigate the generation and annihilation  of magnetic chiral \nskyrmion bubbles  and achiral bubbles in the  Mn-Ni-Ga thin plate  by using the Lorentz \ntransmission electron microscopy (L -TEM).  The two independent magnetic domain \nstructures  can be directly controlled after th e field cooling  manipulation by varying the \ntitled angles  of external magnetic fields.  By imaging the magnetization reversal with \nincreasing temperature,  we foun d an extraordinary annihilation mode of  magnetic \nchiral skyrmion bubbles  and a non -linear frequ ency for the winding number reversal. \nQuantitative analysis of such dynamics was performed by using L -TEM to directly \ndetermine the barrier energy for the magnetization reversal of magnetic chiral skyrmion \nbubbles.  \n \n \n  3 \n Nanometric  magnetic textures  have long been the subject of intensive study, continuing \nto be fueled by the promise of spintronic  devices.  A vortex -like topological  spin texture, \ncalled skyrmion, is first theoretically predicted  1 and mostly  observed  2-4 in magnets \nwith non -centr osymmetric  cubic crystal structure.  Recently , the topological equiv alent \ntypes of magnetic chiral skyrmion bubbles  (SKBs)  are found in centrosymmetric  \nmagnets with dipole -dipole interaction and uniaxial magnetic anisotropy  5-9. Contrary \nto DMI -stabilized skyrmions with a fixed chirality, chiral  SKBs in centrosymmetric \nmagnets possess two degrees of freedom, i.e., chirality  and vorticity 2. The chirality  is \ndefined  by the in-plane magnetization , which  is rotated  direction along the perimeter , \nwhile  the vorticity is represented by a winding number  S, which c haracterizes the \ntopology of bubble s 10. The winding number  S for magnetic bubbles in the \ncentrosymmetric magnets  varies with the  defects of the  domain  wall,  thus resulting in \ndifferent magnetic  topological state such as chiral biskyrmions ( S = 2) 6, 11, 12, chiral \nSKBs ( S = 1) 5, 13 and achiral bubbles (ABs) or hard bubbles (S = 0) 14, 15. Since S = 1 \nSKBs  are considered as skyrmions, the dynamics of  metastable chiral SKBs including \ntheir generation and annihilation  behavior as  a function of external field  and \ntemperature , especially the relaxation time  𝜏 , is an important issue for  potential \nskyrmion -based applications.   \nThe magnetic field and temperature are the most critical controllable external \nparameter s for the skyrmion s’ generation and stability . For example, the dynamic s of \nmetastable skyrmion s with a changing external magne tic field was studied indicating a \nunique aggr egation and collapse between  skyrmions and the conical phase.16 Further \nexperiment s demonstrated that a hidden metastable skyrmion lattice c ould be obtained \nbehind non -topological magnetic order by applying  a sufficient high  field cooling rate.  \n17 In addition , the recent study  with the relaxation dynamics of zero field metastable \nskyrmion s via thermal manipulation in the chiral magnet FeGe  represented  three \ndistinct switching mode s from skyrmions to  stripe s, suggesting a non -Arrhenius law 4 \n behavior.  18 The creation of the skyrmion lattice with thermal  assist has been studied in \nPt/Co/Ta multilayer films . 19 However,  the existing work is  mostly restricted to  \nskyrmions in  chiral magnet s, the dynamic evolution of the formation and annih ilation  \nof chiral SKBs remain s elusive.  Here, in this letter, we report on the generation of \nmetastable chiral  SKBs and ABs in the Mn-Ni-Ga thin plate via field cooling (FC)  \nprocedure . By tuning the titled angles of external magnetic fields,  we investigated  the \ndynamic  annihilation  of the two states  with increasing  temperature as a control \nparameter for the stability. Importantly, t he chiral  SKBs annihilation dynamics is \ndirectly observed by in-situ L-TEM representing an extraordinary  evolution  into the \nparamagnetic state.  \nThe as -cast polycrystal line (Mn 1−xNix)65Ga35 (x = 0.45) is the same as  that used in \nour previous work.  6 We found that the formation of topological spin textures  \nsignificantly depended  on both the Mn-Ni-Ga crystal orientation and the applied \nmagnetic field.  20 Further neutron scattering  studies revealed the emergen ce of a non-\ncollinear canted magnetic  structure along the c -axis. 21 The Mn-Ni-Ga sample  for the \nL-TEM observation was prepared by the mechanic polishing , and the magnetic domain \nevolution behavior wi th thermal manipulation was observed by using FEI Titan G2  60–\n300 equipped with a heating hold er. The Curie temperature of the Mn-Ni-Ga thin plate  \nis about 325 K, and the specific FC process is similar to our previous studies . 11, 13  \nThe specific FC manipulation was summarized as follows: first, the sample was heated \nup to 360 K, which was higher than Curie temperature T C ∼ 325 K; Second, a small \nperpendicular magnetic field  was applied by increasing the objective lens current \ngradually in a very small increment ; Third, the temperature was cooled down gradually \nfrom 360  K to 300 K ; Finally, at 300  K, the perpendicular magnetic field was turned \noff. The experimental process was  recorded by the charge coupled device (CCD) \ncamera as shown in Supplementary Movie S1 and Movie S2 . The external magnetic \nfield was applied along the e -beam direction by controlling the current of the objective \nlens. To quantitatively analyze the domain s tructure , three images (under -, over - and 5 \n in-focus) were acquired with a CCD  camera and the in -plane magnetization distribution \nwas reconstructed  by the Qpt software based on the transport -of-intensity equation \n(TIE). \nWe first discuss the case at θ =0°; i.e., using the FC technique described, the magnetic \nfield is  applied normal to the (001) nano -sheet plane.  The results are summarized in the \nform of a magnetic phase diagram in Fig. 1(a). The spontaneous ground state of  stripe \ndomains  remain s unchanged when  the magnetic field  is lower than 300 Oe , shown in \nFig. 1(b). With the magnetic field increasing, the chiral  SKBs  appear . One can notice \nthat the density of chiral  SKBs  firstly increase and then decrease  with the increase  of \nthe magnetic field . The representative L -TEM image of chiral  SKBs after an optimized \n1000 Oe FC manipulation is shown in  Fig. 1(c). When a  higher magnetic field of 2000 \nOe is applied, the mixed stripe s and chiral  SKB s are shown up in Fig. 1(d). This is \nbecause the magnetic field is too strong  under this condition , the nucleation sites will \nbe forced to agglomerate, thereby forming  stripe domain again.  As a whole , the \nmagnetic field plays a critical role during the FC process, and an optimized magnetic \nfield exists to generate the chiral  SKB  phase with the highest density.  \nWe then investigate  the formation  of ABs  under different tilted angles of magnetic \nfields during  FC manipulation.  The schematic experimental configuration is shown in  \nFig. 2(a)  and (c), in which the inclined magnetic field H is realized by tilting the nano -\nsheet, while the magnetic field is fixed along the direction of electron beam  and the \nvalue  is fixed at H = 1 000 Oe. Figure 2(b) and (d) show the typical AB’s  arrangements \nat two typical tilted angles θ =  5°  and 10° .  We f ind that two kinds of bubbles c ould be \ntransferred by tilting the sample. When the angle is smaller than 3° , the chiral  SKB s \nform. With the increasing  of the angle,  most  chiral  SKBs evolve  into A Bs, shown in  \nFig. 2 (b). For θ > 10° , ABs are dominant  (Fig. 2 (d)). This effect is easily understood \nthat the in -plane magnetic field component compels  the magnetization orientated \nparallel to the in -plane magnetic field  direction  22, which results i n the transformation \nbetween chiral  SKBs and ABs . Most interestingly, the hexagonal lattice persists up to \nθ ～ 10°. The corresponding spin texture s are displayed in  Fig. 2(e). The white arrows 6 \n show the directions of the in  plane magnetic inductions, while the black regions \nrepresent the domains with out-of-plane magnetic inductions.  Chiral  SKBs  “1” and “2”  \nwith right - (C = +1 ) or left-handed chirality  (C = -1) show up as black  or white rings in \nthe under -focused images both resulting in winding nu mber S = 1, which is equal to \nthat of skyrmions . ABs “3” composed of a pair of open Bloch line s, is characteristic of \nthe domain structure commonly observed in ferromagnetic compounds with S = 0 (C = \n0). \n Based on these obtained chiral  SKBs  and ABs , the temperature dependent dynamic \nevolution of bubbles is investigate d by using the in-situ L-TEM technique . The \ntemperature is gradually increased from 301 K to 308 K, and the thermally activated \nexcitation of bubble s winding number reversal (S = 1 to S = 0) has been observed  (See \nSupplementary  Movie S 3). Figure  3 are snapshots of raw L -TEM image s for several \ntemperature  points with the exposure time 0.4  s per frame. As we increase the \ntemperature, the majority  of chiral  SKBs d ynamically and randomly reverse  their \nchirality and transfer into ABs, while mai ntaining the hexagonal lattice (see Fig. 3(a)-\n(g)). With further increasing the temperature to near T C, bubbles totally vanish into \nparamagnetic state.  This phenomenon is consistent with  previous study in the  Ba-Fe-\nSc-Mg-O 23. To further investigate the temperature de pendent dynamic evolution of  \nchiral  SKBs,  41 bubble s in the under -focus LTEM image s were indexed as mentioned \nabove. For each temperature, we calculated the averaged count of S = 1 and S = 0 \nbubbles based on the in-situ L-TEM video ( see Supplementary  Movie S 4). Figure 3(h) \nshows the temperature dependence of the  averaged statistic count of  S = 1 and S = 0 \nbubbles. As the temperature increase s, the count of chiral  SKBs  (S = 1)  gradually \ndecreases and sharply drops down to zero at 306 K. However, the ABs  (S = 0) represent \nopposite behavi or. For comparison, we perform  the same procedure on the ABs which \nare chosen as the initial state ( see Supplementary  Movie S 5), the corresponding thermal \nstability  shown in Fig. 3(i)-(l). Clearly , we can  see that the count of S = 1 and S = 0 \nbubbles mostly maintain  unchanged with the increas e of the temperature. These \nanalyses quantitatively de monstrate that the energy of  chiral  SKBs  (S = 1) is lower than 7 \n ABs (S = 0) at room temperature and thermally energy can active their transition. \nNoticeably , we f ind an extraordinary  dynamic annihilation mode of  chiral  SKBs under  \nthermally activation, that is, metastable chiral  SKBs firstly transfer into ABs with the \nincrease  of the temperature and then totally collapse into paramagnetic state near Tc. \nThis is very different from the coll apse dynamics of skyrmions in  chiral magnet FeGe18. \nTo clearly  understand  the dynamic behavior of chiral  SKBs and ABs, we  focus on  \nindividual  representative chiral  SKB. We found that a chiral  SKB permanently exhibits \nrepeated reversal of the winding number at a given T, as shown in Fig. 4(a). At a lower \ntemperature,  for example 300  K, the switching between the chiral  SKB and the AB is \ninfrequent and the chiral  SKB state is more  stable. In contrast, as the temperature \nincrease s up to 303 K , the winding number switches very quickly  between 0 and 1 . \nUpon increasing temperature to 3 05 K, t he switching frequency  slow s down and the \nAB state is favorable. The switching  process with the temperature further verified the \nannihilation mode of the metastable chiral  SKBs. In essence, the behavior shows the \nrelaxation dynamics. In Fig. 4(b), we show  the temperature dependence of the mean \nrelaxation time τn for the chiral  SKB and the AB , of which τn is calculated by  the \nexperiments shown in Fig. 4(a).  \nTo quantitatively evaluate the activation  energy of a single chiral SKB, we use  the \nArrhenius law,  𝜏=𝜏0exp(𝐸𝑠\n𝑘𝐵𝑇) , where 𝜏0  is the pre -exponential factor, 𝜅𝐵  is the \nBoltzman n constant and 𝐸𝑠  is the activation energy.  In estimating the order of \nmagnitude  of𝐸𝑠, we assume a simple temperature -dependent activation energy 𝐸𝑠/𝜅𝐵 \n= a (𝑇𝑐−𝑇) (See supplementary material  for detail s). 17 From the fitting result  (yellow  \nline), we can estimate the thermal activation energy 𝐸𝑠 (S = 1) to be ~9.9x10-20 J at \n301 K. This value is in the same magnitude with the previously reported switching \nenergy barrier skyrmions in Fe-Co-Si, 24 FeGe  16 and MnSi 17, but 2 order of magnitude \nlower than 10-17 J of skyrmions in La -Ba-Mn-O 25. As for the ABs, we f ind that the \ncurve i s poor fitted as a function of temperature using the Arrhenius equation, implying \nthat activation energy of the AB  represents a non -line variation with the temperature , 8 \n which represent s a similar behavior of  metastable skyrmions in  FeGe 13. (See \nsupplementary material for detail s). Based on the activation energy curve, the \nphenomenologic al free -energy landscapes of  chiral  SKBs and ABs can be sketched in \nFig. 4(c). At room temperature, the energetically lower state is the chiral  SKB. As the \ntemperature increased, the activation energy of chiral  SKB s appear linear decreasing \nwhile a subline decreases  of ABs, thus resulting in the same energy of two kinds of \nbubbles . When the temperature is near T c, ABs become more prominent with a lower \nenergy.  \nIn summary, the generation  and annihilation of chiral  SKBs and ABs with \ntemperature have  been clearly investigated in Mn -Ni-Ga by using  L-TEM. The two \nresidual magnetic states can be directly controlled after the FC manipulation by varying \nthe angle of the  magnetic fields. Systematic analysis of the in-situ L-TEM video  \ndemonstrate s an extraordinary  annihilation mode of chiral  SKBs and a non -linear \nfreque ncy for the winding number reversal with increasing temperature. The switching \nprocess are characterized by the temperature dependent activation energy showing the \nactivation energy ~ 9.9x10-20 J at 301 K. This work provides the basic information for \ncontr olling the topological magnetization texture and the stability.  \n \n \nSee supplementary material  for the calculation of  thermal activation energy  of chiral  \nSKBs and ABs . \nSee supplementary movie  for the evolution of magnetic domain structures as a \nfunction of temperature observed by using in -situ Lorentz -TEM.   \n \n \nThis work is supported partially by the National Key R&D Program of China (Grant \nNos. 2017YFA0303202  and 2017YFA0206303 ), the National N atural Science \nFoundation of China ( Grant No. 11974406  and 51801087 ), the Key Research Program \nof the Chinese Academy of Sciences, KJZD -SW-M01  and the King Abdullah \nUniversity of Science and Technology (KAUST) Office of Sponsored Research (OSR) 9 \n under Award  No. OSR -2016 -CRG5 -2977 . \n  10 \n Reference  \n1. A. N. Bogdanov and D. Yablonskii, Zh. Eksp. Teor. Fiz 95 (1), 178 (1989).  \n2. N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8 (12), 899 (2013).  \n3. S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii and P. \nBöni, Science 323 (5916), 915 -919 (2009).  \n4. X. Yu, Y. Onose, N. Kanazawa, J. Park, J. Han, Y. Matsui, N. Nagaosa and Y. Tokura, \nNature 465 (7300), 901 (2010).  \n5. X. Yu, M. Mostovoy, Y . Tokunaga, W. Zhang, K. Kimoto, Y. Matsui, Y. Kaneko, N. \nNagaosa and Y. Tokura, Proc Natl Acad Sci U S A 109 (23), 8856 -8860 (2012).  \n6. W. H. Wang, Y. Zhang, G. Z. Xu, L. C. Peng, B. Ding, Y. Wang, Z. P. Hou, X. M. Zhang, \nX. Y. Li, E. K . Liu, S. H. Wang, J. W. Cai, F. W. Wang, J. Q. Li, F. X. Hu, G. H. Wu, B. G. Shen \nand X. -X. Zhang, Adv. Mater. 28 (32), 6887 -6893 (2016).  \n7. Z. Hou, W. Ren, B. Ding, G. Xu, Y. Wang, B. Yang, Q. Zhang, Y. Zhang, E. Liu, F. Xu, W. \nWang, G. Wu, X. Zhang, B. Shen and Z. Zhang, Adv Mater 29 (29) (2017).  \n8. X. Xiao, L. Peng, X. Zhao, Y. Zhang, Y. Dai, J. Guo, M. Tong, J. Li, B. Li, W. Liu, J. Cai, \nB. Shen and Z. Zhang, Appl. Phys. Lett. 114 (14), 142404 (2019).  \n9. W. Zhang, B. Balasubramanian, A. Ullah, R. Pahar i, X. Li, L. Yue, S. R. Valloppilly, A. \nSokolov, R. Skomski and D. J. Sellmyer, Appl. Phys. Lett. 115 (17), 172404 (2019).  \n10. A. Rosch, Proc. Natl. Acad. Sci. U.S.A. 109 (23), 8793 -8794 (2012).  \n11. L. Peng, Y . Zhang, W. Wang, M. He, L. Li, B. Ding, J. Li,  Y. Sun, X. G. Zhang, J. Cai, S. \nWang, G. Wu and B. Shen,  Nano Lett. 17, 7075 (2017).  \n12. X. Z. Yu, Y. Tokunaga, Y. Kaneko, W. Z. Zhang, K. Kimoto, Y. Matsui, Y. Taguchi and Y. \nTokura, Nat. Commun. 5 (1), 3198 (2014).  \n13. B. Ding, J. Cui, G. Xu, Z. Hou, H.  Li, E. Liu, G. Wu, Y. Yao and W. Wang, Physical Review \nApplied 12 (5) (2019).  \n14. D. Morikawa, X. Z. Yu, Y. Kaneko, Y. Tokunaga, T. Nagai, K. Kimoto, T. Arima and Y. \nTokura, Appl. Phys. Lett. 107 (21), 212401 (2015).  \n15. P. J. Grundy and S. R. Herd, physi ca status solidi (a) 20 (1), 295 -307 (1973).  \n16. X. Yu, D. Morikawa, T. Yokouchi, K. Shibata, N. Kanazawa, F. Kagawa, T. -h. Arima and \nY. Tokura, Nat. Phys. 14 (8), 832 -836 (2018).  \n17. H. Oike, A. Kikkawa, N. Kanazawa, Y. Taguchi, M. Kawasaki, Y. Tokura and  F. Kagawa, \nNat. Phys. 12, 62 (2015).  \n18. L. Peng, Y. Zhang, L. Ke, T. H. Kim, Q. Zheng, J. Yan, X. G. Zhang, Y. Gao, S. Wang, J. \nCai, B. Shen, R. J. McQueeney, A. Kaminski, M. J. Kramer and L. Zhou, Nano Lett 18 (12), \n7777 -7783 (2018).  \n19. S. Zhang, J. Zh ang, Y. Wen, E. M. Chudnovsky and X. Zhang, Appl. Phys. Lett. 113 (19), \n192403 (2018).  \n20. B. Ding, H. Li, X. Li, Y. Wang, Z. Hou, G. Xu, E. Liu, G. Wu, F. Wang and W. Wang, APL \nMaterials 6 (7), 076101 (2018).  \n21. G. Xu, Y. You, J. Tang, H. Zhang, H. Li, X . Miao, Y. Gong, Z. Hou, Z. Cheng, J. Wang, A. \nJ. Studer, F. Xu and W. Wang, Phys. Rev. B 100 (5), 054416 (2019).  \n22. H. Nakajima, A. Kotani, K. Harada, Y. Ishii and S. Mori, Phys. Rev. B 94 (22), 224427 \n(2016).  \n23. X. Z. Yu, K. Shibata, W. Koshibae, Y. To kunaga, Y. Kaneko, T. Nagai, K. Kimoto, Y. 11 \n Taguchi, N. Nagaosa and Y. Tokura, Phys. Rev. B 93 (13) (2016).  \n24. J. Wild, T. N. G. Meier, S. Pöllath, M. Kronseder, A. Bauer, A. Chacon, M. Halder, M. \nSchowalter, A. Rosenauer, J. Zweck, J. Müller, A. Rosch, C.  Pfleiderer and C. H. Back,  3 (9), \ne1701704 (2017).  \n25. M. Nagao, Y. G. So, H. Yoshida, M. Isobe, T. Hara, K. Ishizuka and K. Kimoto, Nat \nNanotechnol 8 (5), 325-328 (2013).  \n \n  12 \n  \nFig.1. Magnetic phase diagram of chiral skyrmion  bubbles density  with FC \nprocedure  in the magnetic field versus the temperature plane.  The magnetic field \nis applied normal to the (001) nano -sheet plane . Experimentally measured position s are \nmarked by white  open  dots, and the representative L -TEM images of (b), (c) and (d) \nare labeled with star symbols.  (b), (c) and (d) Stripes, chiral  skyrmion bubbles and chiral  \nskyrmion bubbles + stripes after FC procedure with magnetic field at 100 Oe, 1 000 Oe \nand 2000  Oe. The scal e bar is 500 nm.  \n  \n13 \n  \n \nFig.2. Variation of topological magnetic state at room temperature under different \ntilted angles  after FC manipulation . (a) and (c)  The schematically experimental \nconfigurations. θ is the tilted angle.  (b) and (d) Under -focus L -TEM images of chiral  \nskyrmion bubbles and achiral bubbles with tilted angles θ = 5°and θ = 10°. (e) In-\nplane magnetization distribution map obtained by TIE analysis for magnetic texture \nindicated in ( b) and (d). Colors (the inset  of panel ( e) shows the color wheel) and white \narrows represent the direction of in -plane m agnetic induction, respectively. Panel 1 and \n2 display  the chiral  skyrmion bubbles with C= +1 and C = –1 chirality both resulting in \nS = 1 and panel 3 shows the achiral bubbles with C = 0  (S = 0).  \n14 \n  \n \nFig.3. Temperature dependence of zero field chiral skymrion bubbles and achiral \nbubbles.  (a) - (g) Chiral  skyrmion bubbles  are chosen as the initial state.  Series of \nunder -focus L -TEM image s observed at zero magnetic field  at 301  - 307 K. The chiral  \nskyrmion bubbles with S = 1 show up as dark  or white rings, while the achiral bubbles \nrepresent two arches ( S = 0) in the L -TEM contrast images.  Exposure time of each \nframe is 0.4  s. (h) Statistic al averaged count of zero field chiral  skyrmion bubbles  and \nachiral  bubbles with the increasing temperature.  The error bar was indicated by the gray \nline. (i)  - (k) Achiral  bubbles  are chosen as the initial state.  Snapshots of L -TEM video \nobserved at 301  K, 304  K and 306  K. (l) Statistical averaged count of  zero field chiral  \nskyrmion bubbles  and achiral  bubbles  with the increasing temperature .  \n  \n15 \n  \nFig.4. Analysis of dynamic transition of a single bubble.  (a) Switching between S = \n1 and S = 0 as a function of time in a specimen temperature range from 301  K to 305  \nK. (b) Temperature dependence of mean relaxation time τn for a single chiral  skyrmion \nbubble and achiral bubble. The colorful squares  represent  the experimental statistic  \nmean relaxation time τn. The yellow  line is fitted  by using Arrhenius’ law.  (c) The \nschematic free -energy landscape of the temperature -dependent potential for  chiral  \nskyrmion bubbles ( S = 1) and achiral bubbles ( S = 0). \n \n"    },    {        "title": "2106.06078v2.Conserving_Local_Magnetic_Helicity_in_Numerical_Simulations.pdf",        "content": "DRAFT VERSION APRIL 11, 2023\nTypeset using L ATEXtwocolumn style in AASTeX63\nConserving Local Magnetic Helicity in Numerical Simulations.\nYOSSEF ZENATI1,\u0003AND ETHAN T. V ISHNIAC1\n1Physics and Astronomy Department, Johns Hopkins University, Baltimore, MD 21218, USA\n(Dated: April 2021)\nABSTRACT\nMagnetic helicity is robustly conserved in systems with very large magnetic Reynolds numbers, including\nmost systems of astrophysical interest, and unlike kinetic and magnetic energy is not dissipated at small scales.\nThis plays a major role in suppressing the kinematic large-scale dynamo and may also be responsible for driving\nthe large-scale dynamo through the magnetic helicity flux. Numerical simulations of astrophysical systems\ntypically lack sufficient resolution to enforce global magnetic helicity over several dynamical times. In these\nsimulations, magnetic helicity is lost either through numerical errors or through the action of an unrealistically\nlarge resistivity. Errors in the internal distribution of magnetic helicity are equally important and typically larger.\nHere we propose an algorithm for enforcing strict local conservation of magnetic helicity in the Coulomb gauge\nin numerical simulations so that their evolution more closely approximates that of real systems.\n1.INTRODUCTION\nMagnetic helicity is a conserved quantity in ideal MHD\n(Woltjer 1958) and one of the primary ways to quantify the\ncomplexity of a magnetic field. In a weakly nonideal turbu-\nlent system, it is still almost conserved, in the sense that the\nturbulent cascade is ineffective at transferring it to resistive\nscales, even while the magnetic and kinetic energies are dis-\nsipated efficiently. Consequently, Taylor (1974) offered the\nconjecture that a laboratory system would first relax to the\nminimum energy state with the same total magnetic helicity\n(see also Taylor (1986)). For example, see Yamada (1999)\nfor a review of the laboratory evidence accumulated in the\nfollowing two decades. We note in particular Ji et al. (1996)\nwhich shows that the dynamical transport of magnetic helic-\nity is an important part of this balance. The conservation\nof magnetic helicity should be even more robust in astro-\nphysical systems where the microscopic resistivity is usually\nnegligible on large eddy scales. Minimizing energy for the\nsame global magnetic helicity drives the creation of large-\nscale magnetic fields. We can see this in simple terms by\nnoting that the magnetic helicity density has the dimensions\nof[length ]\u0002[energydensity ] =Wb2. For large-length scales,\nthe same magnetic helicity requires the least energy. This\nsame argument shows why magnetic helicity is so resistant\nto dissipation. It can’t participate in the turbulent cascade\nCorresponding author: Yossef Zenati\nyzenati1@jhu.edu\n\u0003CHE Israel Excellence Fellowshipsince the energy density corresponding to small-scale eddies\ndecreases with decreasing eddy size. The tendency of robust\nmagnetic helicity conservation to result in an inverse cascade\nmotivates the proposal that the large-scale dynamo (LSD) is\ndriven by magnetic helicity (Vishniac & Cho 2001) though\na combination of turbulent transport and the inverse cascade.\nOne of the main challenges in testing this proposal is to ex-\namine the flow of magnetic helicity spatially and between\nscales when numerical simulations tend to dissipate magnetic\nhelicity at significant rates.\nIn addition to its importance in determining the relaxation\nof complex systems, magnetic helicity may play a role in the\nlarge-scale dynamo process, a point that was first raised by\nPouquet et al. (1976). The evolution of the large scale mag-\nnetic field Bl, where the subscript denotes low pass filtering\non the scale l, is given by\n¶tBl=Ñ\u0002(vl\u0002Bl\u0000hJl)+Ñ\u0002el; (1)\nwhere\nel\u0011(v\u0002B)l\u0000vl\u0002Bl: (2)\nAnd Ohm’s law is\nE\u0011\u0000vl\u0002Bl+hJl: (3)\nhis the resistivity for a magnetic diffusivity. In a turbu-\nlent medium el, the electromotive force filtered on the scale\nl, is approximately the large-scale average of the cross prod-\nuct of the turbulent velocity and magnetic fields. Pouquet\net al. (1976) pointed out that elhas contributions from the\neddy scale contributions to the averaged kinetic and currentarXiv:2106.06078v2  [physics.plasm-ph]  8 Apr 20232\nhelicities multiplied by Bl, i.e. as a part of the aeffect in dy-\nnamo theory. Neither helicity is conserved in ideal MHD but\nin the Coulomb gauge the eddy scale current helicity, j\u0001b,\nis approximately k2times the eddy scale magnetic helicity,\na\u0001b, where k2is the mean square wavenumber associated\nwith the large scale eddies. While only the total magnetic he-\nlicity is conserved, the transfer of magnetic helicity between\nlarge and small scales depends on the parallel component of\nthe electromotive force. This connection was exploited by\nGruzinov & Diamond (1996) to show that a dynamo driven\nby the kinetic helicity saturates due to the local accumulation\nof magnetic helicity, and turns off long before the large scale\nfield reaches equipartition with the turbulent energy density\n(”asuppression”). Apparently, the conservation of magnetic\nhelicity can have profound implications for the large-scale\ndynamo.\nGiven the existence of large-scale magnetic fields, it seems\nsafe to assume that there is some way to avoid the negative\nconclusion offered by Gruzinov & Diamond (1996). Pos-\nsibly the kinematic dynamo operates, perhaps at a reduced\nefficiency, while the eddy scale magnetic helicity is ejected\nfrom the system (for a discussion of this idea see Branden-\nburg et al. (2009)). Alternatively, the kinetic helicity may\nbe irrelevant, and a magnetic helicity flux, induced by ro-\ntation, creates local accumulations of eddy scale magnetic\nhelicity which then drive a large scale dynamo as a side ef-\nfect of being transferred to large scale fields (Vishniac & Cho\n2001). Finally, it is possible that a large-scale dynamo (LSD)\ncould be produced through a combination of shear and fluc-\ntuating kinetic helicity (Vishniac & Brandenburg 1997; Mi-\ntra & Brandenburg 2012), although the relation between this\nmechanism and asuppression has never been fully explored.\nRegardless of which of these mechanisms dominate in real\nsystems, we can see that accurate simulations of dynamos\nrequire careful accounting of the distribution of magnetic he-\nlicity in a system. To model Taylor relaxation accurately one\nneeds only conserve the total magnetic helicity in a simula-\ntion box. To model dynamos, including asuppression and\nthe transport of magnetic helicity, one needs to extend this\ndown to scales not much larger than individual eddy scales.\nMoreover, since the magnetic helicity is a quadratic quan-\ntity, the average magnetic helicity on domain scales can arise\nfrom correlations on eddy scales. The dissipation of this con-\ntribution to the magnetic helicity proceeds at the ohmic dissi-\npation time at the large eddy scale (or a bit faster, see below).\nNumerical dissipation on this scale has to be slower than the\nlarge-scale dynamo growth rate for an accurate large-scale\ndynamo simulation. In practice, while this is a trivial thresh-\nold for real systems, it represents a substantial challenge for\nsimulations. This is a particularly difficult issue for simula-\ntions of magnetorotational instability, where the large-scale\neddy size is proportional to the strength of the magnetic field.Worse is that the MRI operates primarily in the ˆ rˆfplane, and\nthe vertical wave number in unstratified simulations tend to\nthe dissipation scale, which guarantees that magnetic helicity\nwill not be conserved for more than an eddy correlation time\nunless the magnetic Prandtl number is larger than one (see,\nBrandenburg & Subramanian 2005).\nThis raises the question of whether it might be useful to\nenforce magnetic helicity conservation in numerical simula-\ntions. In many cases, it is possible to enforce the conserva-\ntion of an important conserved quantity by choosing an ap-\npropriate numerical algorithm, but magnetic helicity is non-\nlocal and no local numerical algorithm conserves it exactly.\nEvaluating magnetic helicity in simulations of the MRI in a\nstratified accretion disk, for example, shows that in a single\norbit, the total magnetic helicity in the box can change by a\nnumber comparable to a density scale height times the mag-\nnetic energy density (see, Davis et al. 2010). What price do\nwe pay for adjusting the small-scale magnetic field in real-\ntime to compensate for numerical errors? As a general rule,\nrequiring strict conservation of any quantity will prioritize it\nabove other conserved quantities. In particular, adjusting the\nmagnetic field to conserve magnetic helicity will affect the\nstress-energy tensor, including the energy density. However,\nin a turbulent simulation, the fluctuating magnetic energy is\ncontinuously lost to the turbulent cascade, so energy conser-\nvation at small length scales is of secondary importance un-\nless the energy input (or loss) is comparable to the turbulent\nenergy dissipation rate. Any scheme that adjusts local mag-\nnetic field strengths will violate flux freezing, but this is al-\nready violated by turbulent transport in the turbulent cascade\n(Eyink et al. 2013a). Lastly, if there is no turbulence, then\nthere’s almost no numerical loss of magnetic helicity and any\nscheme for correcting the field will have a negligible effect.\nIn principle, there seems to be no fundamental obstacle to im-\nplementing a correction scheme to enforce magnetic helicity\nconservation. Whether or not a particular scheme improves\nthe accuracy of numerical simulations at an acceptable cost\ndepends on the details of the algorithm. In this paper, we will\nsuggest an approach to enforcing magnetic helicity conserva-\ntion and explore some of the constraints on its use.\nIn section 2 of this paper, we will review the definition\nof magnetic helicity, and other relevant quantities, and their\nevolution in a turbulent medium. We will review how fluctu-\nations in the magnetic helicity are generated and dissipated,\nincluding resistive losses. Finally, we will present an algo-\nrithm that allows us to impose strict conservation of mag-\nnetic helicity with minimal effect on other properties of the\nmagnetic field. In section 3 we will test the convergence of\nour algorithm. In section 4 we discuss the applications and\nlimitations of this tool.\n2.DEFINING AN ALGORITHM3\nIn a highly conducting fluid, the magnetic field evolves as\n¶tB=Ñ\u0002(v\u0002B\u0000hJ); (4)\nwhere the resistive term hJis small. Numerical effects may\nor may not mimic physical resistivity. We will assume here\nthat they do, but note places where the difference might be\nimportant. Consequently, the vector potential evolves as\n¶tA=v\u0002B\u0000hJ\u0000Ñf; (5)\nwhere the electrostatic potential, fsatisfies\nÑ2f=Ñ\u0001(v\u0002B\u0000hJ); (6)\nin the coulomb gauge. This gauge choice has the advantage\nof enforcing a tight correlation between the small-scale con-\ntribution to the total current helicity, J\u0001B, and the small-scale\ncontribution to the magnetic helicity. The former is gauge in-\nvariant and plays an important role in dynamo theory. We\nnote that other gauge choices have been used for compu-\ntational convenience or because the magnetic helicity has\nsome compelling interpretation in that gauge (for examples\nsee (Hubbard & Brandenburg 2011; Candelaresi et al. 2011;\nDel Sordo et al. 2013)).\nIn this gauge the magnetic helicity flux is\nJH\u0011A\u0002(v\u0002B\u0000hJ+Ñf); (7)\nand the magnetic helicity, H, evolves as\n¶tH=\u0000Ñ\u0001JH\u00002hJ\u0001B: (8)\nWhen the resistivity is small its contribution to JHwill be\nnegligible, but its role in breaking magnetic helicity conser-\nvation can still be important. We can define the small-scale\ncontribution to the magnetic helicity as hl\u0011Hl\u0000Al\u0001Blso\nthat the evolution equation is\n¶thl=\u0000Ñ\u0001JHl\u00002h([(J\u0001B)l\u0000Jl\u0001Bl]\u00002Bl\u0001el; (9)\nwhere the eddy scale magnetic helicity flux, JHl, is defined,\nas for the other eddy scale contributions, by subtracting from\nthe total flux the contribution from large scale fields, includ-\ningel. The last term on the RHS of equation (9) describes the\ntransfer of magnetic helicity between small and large scales\nand is critical for understanding how magnetic helicity is dis-\nsipated when the resistivity is not exactly zero.\nFor this purpose eq.(9) can be generalized by dividing the\nturbulent cascade into a series of shells in phase space, each\nwith a typical wave number amplitude k(Aluie 2017). Then\ninstead of large and small scale contributions to the total he-\nlicity on the scale lwe have contributions from each shell,\nhk. The transfer of magnetic helicity between large and small\nscales is still B\u0001el, but now elis the sum of multiple terms,each given by (v\u0002b)klthe contribution to elfrom the shell\nkand each term gives the transfer of averaged helicity, Hl,\nbetween large scales and the scale given by k\u00001see figure 1.\nThe rate at which magnetic helicity on some large scale l\nis dissipated is hhJ\u0001Bil, or in other words proportional to\nthe current helicity averaged over the same scale. The large-\nscale average of the current helicity is the sum of contribu-\ntions from smaller scales. We can find the expected distribu-\ntion of current helicity over scales by considering the transfer\nrate between scales, 2 Bl\u0001(v\u0002b)klwhere a positive sign rep-\nresents the transfer of magnetic helicity to large scale struc-\nture. In a stationary state, this goes to zero as the resistivity\ngoes to zero. The usual expansion of (v\u0002b)lto first order\nin the eddy correlation time ((Pouquet et al. 1976)) gives\nfour terms: a contribution proportional to large-scale aver-\nage magnetic helicity embedded in small-scale eddies, hkl, a\nsimilar contribution due to the kinetic helicity, but with the\nopposite sign, a term proportional to the gradient of the large\nscale field - usually written as a turbulent diffusivity, and a\ndrift term of the form Ve f f\u0002Bl. Obviously, the last term\nis orthogonal to Blis doe snot contribute to the transfer of\nmagnetic helicity between scales. The first term leads to the\ntransfer of hklto larger scales at a rate comparable to t\u00001\nk,\nthe small scale eddy turnover rate(Vishniac & Cho 2001)).\nThe second term depends on kinetic helicity, which is not a\nconserved quantity. Naively one would expect it to be em-\nbedded in small scales only to the extent that the symme-\ntry breaking of the large-scale environment biases the small-\nscale turbulence. However, the current helicity embedded on\nsmall scales induces a kinetic helicity, whose effect is to drive\nthe second term to cancel a fraction of the first term ((Vish-\nniac & Shapovalov 2014)). Finally, the third term contributes\nan effect which is approximate \u0000h(vivj+bibjileistBls¶jBlttk.\n(Here we have neglected a contribution due to pressure in\nthe force equation, which somewhat reduces the contribution\nfrom the correlated small-scale magnetic fields). This term\nis off order\u0000hv2iLtktimes the large-scale contribution to the\ncurrent helicity. This is the term that has to balance the first\ntwo terms. Since the small-scale contribution to the current\nhelicity is k2hkl, the ratio of the small-scale current helicity to\nthe large-scale current helicity is (kvktk)2, i.e. of order unity.\nThe current helicity, and consequently the ohmic dissipation\nof magnetic helicity, is distributed across the full range of the\ninertial cascade with a one-dimensional spectral index of \u00001.\nThis in turn implies that the dissipation of magnetic helicity\nwill be enhanced by some logarithmic factor above the ohmic\ndissipation rate on the large eddy scale. Simulations that use\nartificial forms of resistivity that depend on higher powers of\nthe wavenumber will damp magnetic helicity at slower rates\nbecause of the concentration of dissipation at the smallest\neddy scales and because those eddies will be smaller. How-\never, the reduced range of scales subject to dissipation will4\ntend to accumulate extra power, and presumably magnetic\nhelicity, due to the bottleneck effect which will work in the\nopposite direction.\nIn a turbulent cascade the energy dissipation rate is the en-\nergy density times the large-scale eddy turnover rate, i.e. the\ncharacteristic time scale is just L=vL. We have shown that the\ncorresponding time for the dissipation of magnetic helicity\nisL2=h(divided by some logarithmic measure of the dy-\nnamic range of the turbulent cascade). Dividing the former\nby the latter gives us a measure of how well magnetic helicity\nis conserved on dynamic times scales. This is inversely pro-\nportional to the magnetic Reynolds number VLL=h, which\nin most cases means that magnetic helicity is conserved to a\nlevel that cannot be adequately reproduced relying only on\nnumerical precision.\nThere is one final point worth stressing about the dis-\nsipation of magnetic helicity at different scales within the\nturbulent cascade. We can derive a maximum dissipation\nrate without making an explicit calculation of the transfer of\nmagnetic helicity between scales. On dimensional grounds,\nthe maximum current helicity at a given wave number will\nbe\u0018kb2\nk. However, highly magnetized turbulence will be\nhighly anisotropic and the current helicity depends on sym-\nmetry breaking in all three directions. Consequently, the ac-\ntual limit is kkb2\nk. If we multiply this by the Alfven speed we\nget the turbulent energy density times the Alfven frequency\nfor typical eddies with wave number k. Invoking the con-\ndition of critical balance, t\u00001\nnonlinear\u0018wAwhich is a generic\nfeature of MHD turbulence models, we see that the maximum\naverage current helicity embedded in eddies of wave number\nkis proportional to the energy cascade rate at those scales.\nFrom the conservation of energy and locality of nonlinear\ntransfer of energy in phase space, this shows that the maxi-\nmum dissipation rate of magnetic helicity is constant within\nthe turbulent cascade. In other words, the rate we derived\nin the previous paragraph, using a detailed but approximate\nanalysis of the transfer of magnetic helicity in phase space,\nis roughly the same as the maximum rate we could expect in\nany model of MHD turbulence.\nThe problem we wish to address in this paper is that the\ndistribution of magnetic helicity will diverge from the distri-\nbution given by evolving eqs. (7 and (8) with h=0, due to\nsmall numerical errors induced by discreteness effects, and\nby the necessity of using h>0 (or its equivalent) to promote\nnumerical stability. This leads to a magnetic field Bcalcwhich\ndiffers from the ideal result on small scales and whose asso-\nciated magnetic helicity is not conserved (Bodo et al. 2017).\nOf course, we can always redefine our goal as a simulation\nwith significant resistivity, but this fails to accurately model\nthe dynamics of realistic astrophysical systems. We can de-\nfine our error, i.e. the amount by which we have failed to\ntreat the magnetic helicity as a locally conserved quantity bycalculating Ñ\u0001JHat every time step and advancing Haccord-\ningly. Tracking magnetic helicity in a simulation requires\ncalculating Ñ\u0001JH, which implies a knowledge of Aandf\neverywhere. If the underlying code is pseudospectral or is\nat least based on a regular grid, then this is straightforward\n(Squire & Bhattacharjee 2016). Otherwise, the code has to\nbe supplemented with Green’s functions-based routines that\nwill calculate these quantities (Yousef et al. 2008; Singh &\nJingade 2015).\nHere we propose a rigorous procedure that will bring the\nmagnetic helicity back into alignment with its expected value\nwhile minimizing the size of the correction. We could write\nthis as a straightforward Gaussian minimization problem by\ndefining an action dependent on the mean square correction\nto the field, subject to constraints that ensure that the cor-\nrected field will have the correct magnetic helicity distribu-\ntion and will satisfy the usual constraints on the magnetic\nfield. This is\nS\u0011Z\n[(1=2)(dB)2+l(x)(H\u0000A\u0001B)+m(x)Ñ\u0001B]dV;(10)\nwhere His the magnetic helicity distribution that we expect\nandAandBare the corrected values of the vector potential\nand magnetic field. The last term in the integral enforces\nthe usual zero divergence condition on the magnetic field.\nWithout this term the corrected Bwould no longer satisfy\nMaxwell’s equations. It is not necessarily important to ob-\ntain the correct distribution of Hon all scales. Just correcting\nit on scales comparable to, and larger than, the large-scale\neddy size would improve the accuracy of dynamo simula-\ntions. Random fluctuations in the magnetic helicity flux pro-\nduce random fluctuations in the amplitude of magnetic he-\nlicity modes with scales comparable to the large-scale eddies\nor smaller. The large wavelength tail of these fluctuations\nmay be interesting, but small errors in the amplitude of eddy\nscale helicity fluctuations are unlikely to have any dynami-\ncal significance. Consequently, we are primarily concerned\nwith correcting the distribution of magnetic helicity at wave\nnumbers significantly smaller than the inverse of a large eddy\nscale.\nMinimizing the change to the magnetic field is not the only\nchoice. It’s more convenient and more physically reason-\nable to minimize the change to the vector potential. The\nsame change in the vector potential at small scales produces\na larger change in B, so the net effect is that this choice puts\nmore of the adjustment in the magnetic field at small scales.\n(This is not equivalent to correcting the magnetic helicity dis-\ntribution at small scales. The magnetic field on small scales\ncontributes to the large-scale distribution of magnetic helic-\nity.) This is a better choice since we’re trying to counteract\nthe effects of inaccuracies at small scales. In addition, an\nalgorithm that compensates for lost magnetic helicity by al-\ntering the large-scale magnetic field runs the risk of driving5\na large-scale dynamo by fiat. Conversely, fixing errors in the\nmagnetic helicity distribution by making correlated changes\non small scales implies that the bulk of the corrections will\noccur on scales within the inertial range of the turbulent cas-\ncade. On these scales magnetic helicity is passed between\nscales at eddy turnover rates and any excess energy is quickly\ndissipated. If we weight the algorithm by jdAj2then the ac-\ntion we choose to minimize is\nS\u0011Z\n[(1=2)(dA)2+l(x)(H\u0000A\u0001B)\u0000m(x)Ñ\u0001A]dV:(11)\nThe second Lagrangian constraint now fixes the gauge of A,\nsince the usual divergence-free condition for Bis automat-\nically satisfied. Integrating this by parts involves a surface\nterm\nDS=Z\n[lA\u0002dA\u0000mdA]\u0001dS: (12)\nIn a finite computational volume this gives us a pair of bound-\nary conditions, that landmvanish nearly everywhere on the\nboundary, the only exception being places where Ais nor-\nmal to the surface (for l). This guarantees that there is no\nnontrivial solution for lunless the calculated distribution of\nmagnetic helicity is nonzero somewhere. Here we will as-\nsume a periodic box and ignore this constraint. We obtain\ndA=2lB+Ñ(l)\u0002A\u0000Ñm; (13)\nand\nH=A\u0001B: (14)\nThe second equation is just a restatement of our goal, that\nthe field quantities should give the correct distribution of\nmagnetic helicity. The first equation is not as simple as it\nappears, since the fields on the RHS are the corrected fields,\nnot the calculated ones. The last term on the RHS arises from\nthe Coulomb gauge and leads to\nÑ2m=Ñl\u0001B=Ñ\u0001(lB): (15)\nIn order to make this problem more tractable, and to reduce\nthe numerical cost of implementing a correction scheme, we\nwill assume that lis small and that we can therefore ignore\nthe difference between the corrected and computed fields on\nthe RHS of the first equation. This should be reasonable if\nthe correction scheme is applied often enough. If we further\nassume that the correction to the magnetic helicity can be ap-\nproximated to the same order, i.e. linear in l, then we have\na second-order differential equation for l(x)in the computa-\ntional box. With some work, this can be shown to beDH\u0011H\u0000(Acalc\u0001Bcalc) (16)\n\u0019dA\u0001B+A\u0001(Ñ\u0002dA)\n=2lB2+B\u0001(Ñl\u0002A)\u0000B\u0001Ñm+2lJ\u0001A\n+2A\u0001(Ñl\u0002B)+AiAj¶i¶jl\u00001\n2Ñl\u0001Ñ(A2)\u0000A2Ñ2l\n=2l(B2+A\u0001J)\u0000Ñl\u0001(A\u0002Ñ\u0002A)\u00001\n2Ñl\u0001Ñ(A2)\n+(AiAj\u0000A2di j)¶i¶jl\n=2(B2+A\u0001J)l+¶i\u0000\n(AiAj\u0000A2di j)¶jl\u0001\n\u0000B\u0001Ñm\nwhere we employ implicit summation over repeated indices.\nThe last term can be written as\nB\u0001Ñm=B\u0001ÑÑ\u00002Ñ\u0001(lB): (17)\nThe operator ÑÑ\u00002Ñis just ˆkˆkin Fourier space. In real space,\nit is an integral operator acting on everything to its right and\ncan be written\nÑÑ\u00002Ñ\u0001(F) =F\n3(18)\n+Zd3r0\n4pr05\u0000\n3r0r0\u0001F(r+r0)\u0000r02F(r+r0)\u0001\n;\nwhere Fis any differentiable vector field. As before, we are\nassuming that boundary conditions can be ignored. To linear\norder, we obtain lfrom\nDH=2(B2+A\u0001J)l+¶i\u0000\n(AiAj\u0000A2di j)¶jl\u0001\n\u0000B\u0001ÑÑ\u00002Ñ(lB); (19)\nand use this in equation (13).\nWe need a general procedure for solving eq.(19). We can\ndo this through iteration. If we split two of the coefficients\non the RHS into varying and constant pieces defined by\nC0\u0011hB2+J\u0001Ai=2hB2i; (20)\nC1\u0011B2+J\u0001A\u0000C0; (21)\nD0i j\u0011hdi jA2\u0000AiAji; (22)\nand\nD1i j\u0011di jA2\u0000AiAj\u0000D0i j; (23)\nthen we can rewrite equation (19) as\nDH\u00002C1l+¶iD1i j¶jl+B\u0001Ñ\u0000\nÑ\u00002Ñ\u0001(lB)\u0001\n=\n2C0l\u0000¶iD0i j¶jl:(24)\nThe fourth term on the LHS of eq.(24) resists any simple\nreduction since the operator BÑ\u00002Balways depends on the6\nwavenumber of l(k). In Fourier space, the RHS of this equa-\ntion is a simple multiplication by the positive definite coeffi-\ncient which is always greater than 2 C0.\nln+1(k) =1\n2C0+kikjD0i jh\nDH(k)+\u0010\n\u00002C1ln+¶iD1i j¶jln+\nB\u0001Ñ\u0000\nÑ\u00002Ñ\u0001(lnB)\u0001\u0011\n(k)i\n: (25)\nThe choice of l0is not dictated by this method. One pos-\nsiblity is to usehDHi=2C0asl0so that\nl1=1\n2C0+kikjD0i j[DH(k)\u0000C1hDHi=C0]: (26)\nAlternatively, we could choose\nl0=DH\n2(C0+C1): (27)\nThe former captures the direct response to a helicity error\nat a wavenumber kand a small part of the indirect response\nthrough the interaction with the spatial variation in the coef-\nficient of l. The latter captures the nonlinear part a bit better\nbut misses the dependence on gradients of l.\nWe can get a sense of how important the various contri-\nbutions to lare by considering two simple cases. Note that\ndue to random fluctuations in the magnetic helicity flux, we\nexpect random fluctuations in the amplitude of modes with\nscales comparable to the large-scale eddies or smaller. Small\nerrors in the amplitude of these fluctuations are unlikely to\nhave any dynamic significance. Consequently, we are pri-\nmarily concerned with correcting the distribution of magnetic\nhelicity at wave numbers significantly smaller than the in-\nverse of a large eddy scale. For simplicity, we will assume\nthatDH(k)is nonzero only for these modes larger than the\nlarge-scale eddy modes. We expect the magnetic energy den-\nsity will be largest at the large eddy scale, and the rms vector\npotential will be either largest at that scale or the large scale\nmagnetic field domain scale. In the former case, we have\nC0\u0018B2\nT\u0018D0i jk2\nT, where kTis the wave number of the large-\nscale eddies. The fluctuating coefficients C1andD1i jwill be\nfunctions of scale, but at the large eddy scale, both are com-\nparable to their nonfluctuating pieces. We see from eq.(24)\nthat the contribution to lfrom the large eddy scale will be\ncomparable to that from the scales where DHis nonzero. On\nsmaller scales, l<<lTdominant term in C1will be A\u0001J,\nwhich will be of order ATAl=l2µl\u00002=3for the Goldreich-\nSridhar model of turbulence. Consequently the contribution\ntolfrom scales in the inertial cascade will decrease as l4=3\nThe magnetic domain scale vector potential can be larger\nthan ATeven in the early stages of the large-scale dynamo.\nIn this case, the previous argument goes through in a slightlydifferent form. Now the contribution to lfrom the turbulent\nscale is reduced by a factor of AT=A0relative to the scale\nwhere DHis nonzero. As before the contribution to lfalls\noff as l4=3on smaller scales.\nWe conclude that solutions for lneed to be calculated to\nscales somewhat smaller than the large eddy scale in order to\nrecover the desired distribution of magnetic helicity on mag-\nnetic domain scales.\nAfter solving for lto the required accuracy we correct the\nmagnetic potential:\ndA=Ñ\u0002(lA)+lB\u0000ÑÑ\u00002Ñ\u0001(lB): (28)\nIn phase space the last term does not have to be evaluated\nexplicitly, rather the sum of the first two terms can simply\nbe projected onto the plane perpendicular to k. If the initial\nerror in the magnetic helicity distribution is large then it may\nbe necessary to iterate to allow for the nonlinear term (Ñ\u0002\ndA)\u0001dA.\nIf we neglect the gradient of land consider the correction\nto magnetic helicity, we find that\nDH\u0019B\u0001(2lB)+A\u0001(Ñ\u00022lB) =2l(B2+A\u0001J):(29)\nThe reconstituted magnetic helicity is not deposited strictly\naccording to the energy density in the turbulent cascade, but\non average it works out to the same thing as a function of\nscale. This implies that the power spectrum of the deposited\nmagnetic helicity is much shallower than we would expect to\nfind in a turbulent cascade. This is a consequence of eq. (28).\nThere is some minimum scale, or maximum wavenumber,\nwhere the correction to the power spectrum is of order unity,\nkmaxl\u00181. In order to avoid highly unphysical effects on the\nsmall-scale turbulent spectrum we require that the dissipation\nscale wavenumber satisfy\n1>kdissl\u0018kdissDH\n4hB2i: (30)\nWe can use this to set a limit on the cadence of corrections\nsince we need\nDH<4hB2ildiss: (31)\n3.TESTING THE ALGORITHM\nIt is not immediately obvious that the procedure described\nin eq. (24) will converge efficiently, or at all. We have per-\nformed a simple test using a time slice from a simulation\nof MHD turbulence in the turbulence database of The Johns\nHopkins University ( (Li et al. 2008; Eyink et al. 2013b).\nIn this simulation, the turbulence is driven at wavenumbers\nof 2 and contains an inertial range extending roughly from\nwavenumbers of 101to 102. Rather than tracking the mag-\nnetic helicity over time and using our algorithm to correct\nto a predicted magnetic helicity distribution we deliberately7\nFigure 1. The current helicity HJspectrum compared to the ex-\npected k\u00001one dimensional spectrum.\ndistort the magnetic field and treat the new magnetic helic-\nity distribution as an error that the algorithm will correct. To\ndo this we take each Fourier mode in our randomly chosen\ntime slice and change the phase randomly, with an rms value\nof 0:5. For computational convenience, we degrade the res-\nolution by a factor of 2 to 5123. This eliminates about half\nof the strongly damped regime but leaves the inertial cascade\nuntouched. We can check the scale distribution of helicity\ndissipation by plotting the one-dimensional spectrum of the\ncurrent helicity or J(\u0000k)\u0001B(k)k2. We see a bump at small\nwavenumbers and rough agreement with our predicted slope\nfrom wave numbers jkj\u001820 to well into the damped regime.\nWe note that this implies that some of the dissipation is oc-\ncurring well within the damped regime where the magnetic\nspectrum is sharply declining.\nOne complication is that the gauge term given in eq. (17)\nis nonlocal and potentially expensive to evaluate. In Fourier\nspace, the central operator has a simple form, but calculating\nnonlinear terms in real space means that we need to perform\ntwo fast Fourier transforms every time we evaluate this ex-\npression. Instead, we use a local estimate of this term fol-\nlowing the method described by Wagner et al. (2017).\nApplying the algorithm to the altered map and using the\noriginal helicity distribution as the desired goal, we obtain\nconvergence to the limit of numerical accuracy for lat all\nscales in between 6 and 30-time steps. In fig. (2) we show\na comparison between the initial distribution of helicity and\nthe magnetic potential as a function of wavenumber and the\nfinal values after lconverged via the repeated application of\neq. (25). These plots show amplitudes, and not phase infor-\nmation, but since phase changes lead to amplitude changes\nin the helicity, this is an adequate proxy for the ability of\nthe algorithm to recover the magnetic helicity. From the sec-\nond panel, we see that we recover the original distributionof magnetic helicity without introducing significant errors in\nthe vector potential.\nFinally, we performed a test with a smaller shift in the\nphase of the modes (an rms phase shift of 0 :05). In this case,\nthe change in the magnetic helicity, and subsequent correc-\ntions, were small at higher wavenumbers, although still sub-\nstantial at large wavelengths. This is illustrated in fig. (3).\n4.DISCUSSION AND CONCLUSIONS\nMagnetic helicity is an important topological quantity that\nconstrains large-scale relaxation of magnetized systems and\nplays an important role in large-scale dynamos. Since the\nrate at which it is dissipated in a turbulent cascade is smaller\nthan the large-scale turnover rate (or the energy dissipation\nrate) requiring a simulation to treat it as a locally conserved\nquantity is roughly as important as requiring mass conser-\nvation. In this study, we have constructed an algorithm for\ncorrecting errors in the magnetic helicity distribution in nu-\nmerical simulations. For the purposes of this work, we de-\nfine an error as any deviation between the evolution of the\nsimulated system and one in which magnetic helicity is a lo-\ncally conserved quantity, as it typically is in the systems be-\ning modeled. Numerical simulations that include turbulence\nfrequently show a loss in global magnetic helicity. There is\nno published information on the ability of current algorithms\nto track magnetic helicity correctly on smaller scales but it\nis reasonable to suppose that they do significantly worse at\nthis more difficult task. While there is no unique way to re-\nstore lost information, the algorithm explored here has some\nattractive features, including the ability to embed the lost he-\nlicity to roughly the same scales where numerical errors elim-\ninated it. Adopting this algorithm has two associated costs.\nThe most obvious is the computational costs involved with\nthe algorithm itself. However, this method cannot be applied\nwithout tracking local magnetic helicity in a simulation, i.e.\nwithout the use of equations (7) and (8) with h=0. The lat-\nter implies a fractional increase in the computational costs of\nevery timestep. The former is hard to predict without know-\ning the necessary cadence of corrections but is probably less\nsignificant. We note also that in the event that the resistiv-\nity is large enough that its role in the system evolution is not\nentirely swamped by numerical noise that this algorithm will\nstill function properly as long the resistivity is included in\neqs. (7) and (8).\nWe have examined the cleaning algorithm for magnetic he-\nlicity in the JHU 10243MHD turbulence simulation. The al-\ngorithm restored the large-scale distribution of magnetic he-\nlicity after it was distorted by randomly altering the phases\nof Fourier components of the magnetic field. The iterative\nprocedure for constructing the solution converged, despite a\nlack of formal proof of convergence. As expected the algo-\nrithm does increase the amount of very small-scale power in8\nFigure 2. Left: Log scale of the power spectrum of the original magnetic helicity and final magnetic helicity after modification by eq.(19)\nas a function of modes. Right: Log scale of the power spectrum of the delta of original vector potential A and the final vector potential after\nmodified by eq.(28) as a function of modes\nFigure 3. The magnetic helicity H (solid line) from the simulation\ndata compared to the change produced by a smaller phase shift (rms\n0:05) of the Fourier modes. In this case the change in DH(dotted\npoint) calculated by eq.(24) is significantly smaller than H.\nthe turbulent cascade, a result which needs to be considered\nin setting the cadence of correction in a simulation. We also\nnote that while magnetic helicity is robustly conserved with\nany gauge choice our work is aimed specifically at enforcing\nconservation in the Coulomb gauge. We justify this choice\nby pointing to the close correlation between the magnetic he-\nlicity in this gauge and the current helicity, which appears in\nanalytic theories of the large-scale dynamo (and is not gauge\ndependent).\nOne complication is that we have not formally demon-\nstrated that our procedure for deriving lwill always con-\nverge. We have shown, as a practical matter, that it converges\nin our test case, a simulation of homogeneous isotropic mag-\nnetized turbulence. This is a sufficiently robust test that itis plausible that it will always converge. However, we note\nthat the fluctuating parts of the coefficients in the second-\norder equation for l, eq. (19), will be roughly as large as\nthe constant parts. Moreover, the gauge constraint term will\ntypically be of order lB2, i.e. the same order as the first RHS\nterm. We have already seen that in real space this term is the\nsum of lB2=3 plus a local quadrupole contribution which we\ncan reasonably expect to be smaller. We can compare it to\nthe first RHS term, whose average value is 4 B2l. We see\nthat the gauge constraint term will be smaller by an order of\nmagnitude than the other terms on the RHS.\nThe difficulty of enforcing strict conservation of magnetic\nhelicity in MHD simulations is a major obstacle in perform-\ning realistic simulations of large-scale dynamo processes in\nastrophysical objects. This is particularly true for magnetoro-\ntational instability, where the large-scale eddy size can vary\ndramatically over time. More generally, magnetic helicity\nconservation may play a key role in the shear-current effect\n(Squire & Bhattacharjee 2016). Moreover, for small-scale\nMHD fluctuations, this effect may define the off-diagonal\nturbulent diffusivity hxy(Singh & Jingade 2015; Squire &\nBhattacharjee 2016; Singh & Jingade 2015; Jingade & Singh\n2021; Dewar et al. 2020). As a result, this could be the cru-\ncial bound of for MHD instabilities. Our results show that a\nviable correction scheme that exists and can be used to pro-\nduce a realistic magnetic helicity distribution. In future work,\nwe will include this cleaning algorithm in a stratified peri-\nodic shearing box simulation (see, for example, Davis et al.\n(2010)) to follow the flow of magnetic helicity and its effect\non the magnetorotational instability-driven dynamo.\n5.ACKNOWLEDGEMENTS\nThis work makes use of the Johns Hopkins Turbulence\nDatabase. The authors wish to recognize and acknowledge9\nhelpful discussions with Amir Jafari and Greg Eyink. ETV\nthanks the AAS for supporting his research. YZ thanks Mor\nRozner for stimulating discussions. YZ thanks the CHE Is-\nraelis Excellence Fellowship for Postdoctoral for supportinghis research. Both authors wish to thank the referee, whose\nnumerous questions led us to add detailed explanations of\nwhy magnetic helicity conservation is an important goal. We\nbelieve this has led to a paper that is much more accessible\nto readers.\nREFERENCES\nAluie, H. 2017, New Journal of Physics, 19, 025008,\ndoi: 10.1088/1367-2630/aa5d2f\nBodo, G., Cattaneo, F., Mignone, A., & Rossi, P. 2017, ApJ, 843,\n86, doi: 10.3847/1538-4357/aa7680\nBrandenburg, A., Candelaresi, S., & Chatterjee, P. 2009, MNRAS,\n398, 1414, doi: 10.1111/j.1365-2966.2009.15188.x\nBrandenburg, A., & Subramanian, K. 2005, PhR, 417, 1,\ndoi: 10.1016/j.physrep.2005.06.005\nCandelaresi, S., Hubbard, A., Brandenburg, A., & Mitra, D. 2011,\nPhysics of Plasmas, 18, 012903, doi: 10.1063/1.3533656\nDavis, S. W., Stone, J. M., & Pessah, M. E. 2010, ApJ, 713, 52,\ndoi: 10.1088/0004-637X/713/1/52\nDel Sordo, F., Guerrero, G., & Brandenburg, A. 2013, MNRAS,\n429, 1686, doi: 10.1093/mnras/sts398\nDewar, R. L., Burby, J. W., Qu, Z. S., Sato, N., & Hole, M. J. 2020,\nPhysics of Plasmas, 27, 062504, doi: 10.1063/5.0005740\nEyink, G., Vishniac, E., Lalescu, C., et al. 2013a, Nature, 497, 466,\ndoi: 10.1038/nature12128\n—. 2013b, Nature, 497, 466, doi: 10.1038/nature12128\nGruzinov, A. V ., & Diamond, P. H. 1996, Physics of Plasmas, 3,\n1853, doi: 10.1063/1.871981\nHubbard, A., & Brandenburg, A. 2011, ApJ, 727, 11,\ndoi: 10.1088/0004-637X/727/1/11\nJi, H., Prager, S. C., Almagri, A. F., et al. 1996, Physics of\nPlasmas, 3, 1935, doi: 10.1063/1.871989\nJingade, N., & Singh, N. K. 2021, arXiv e-prints,\narXiv:2103.12599. https://arxiv.org/abs/2103.12599\nLi, Y ., Perlman, E., Wan, M., et al. 2008, Journal of Turbulence, 9,\nN31, doi: 10.1080/14685240802376389\nMitra, D., & Brandenburg, A. 2012, MNRAS, 420, 2170,\ndoi: 10.1111/j.1365-2966.2011.20190.xPouquet, A., Frisch, U., & Leorat, J. 1976, Journal of Fluid\nMechanics, 77, 321, doi: 10.1017/S0022112076002140\nSingh, N. K., & Jingade, N. 2015, ApJ, 806, 118,\ndoi: 10.1088/0004-637X/806/1/118\nSquire, J., & Bhattacharjee, A. 2016, Journal of Plasma Physics,\n82, 535820201, doi: 10.1017/S0022377816000258\nTaylor, J. B. 1974, PhRvL, 33, 1139,\ndoi: 10.1103/PhysRevLett.33.1139\n—. 1986, Reviews of Modern Physics, 58, 741,\ndoi: 10.1103/RevModPhys.58.741\nVishniac, E. T., & Brandenburg, A. 1997, ApJ, 475, 263,\ndoi: 10.1086/303504\nVishniac, E. T., & Cho, J. 2001, ApJ, 550, 752,\ndoi: 10.1086/319817\nVishniac, E. T., & Shapovalov, D. 2014, ApJ, 780, 144,\ndoi: 10.1088/0004-637X/780/2/144\nWagner, M., Dangel, F., Cartarius, H., Main, J., & Wunner, G.\n2017, Acta Polytecnica, 57, 470,\ndoi: https://doi.org/10.14311/AP.2017.57.0470\nWoltjer, L. 1958, Proceedings of the National Academy of Science,\n44, 489, doi: 10.1073/pnas.44.6.489\nYamada, M. 1999, Washington DC American Geophysical Union\nGeophysical Monograph Series, 111, 129,\ndoi: 10.1029/GM111p0129\nYousef, T. A., Heinemann, T., Rincon, F., et al. 2008,\nAstronomische Nachrichten, 329, 737,\ndoi: 10.1002/asna.200811018"    },    {        "title": "1910.03285v1.On_the_rigidity_of_Zoll_magnetic_systems_on_surfaces.pdf",        "content": "ON THE RIGIDITY OF ZOLL MAGNETIC SYSTEMS ON SURFACES\nLUCA ASSELLE AND CHRISTIAN LANGE\nAbstract. In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We char-\nacterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant\nmagnetic functions as the only magnetic systems such that the associated Hamiltonian \row is Zoll, i.e. every\norbit is closed, on every energy level. We also prove the persistence of possibly degenerate closed geodesics\nunder magnetic perturbations in di\u000berent instances.\n1.Introduction\nLet \u0006 be a closed oriented surface. A magnetic system on \u0006 is a pair ( g;f), wheregis a Riemannian\nmetric on \u0006 and f: \u0006!Ris a smooth function (the magnetic function ). Every magnetic system de\fnes a\n\row onS\u0006, the unit tangent bundle of \u0006, as we now brie\ry recall: a smooth arc-length parametrized curve\n\r:I!\u0006 is called a ( g;f)-geodesic , if it has geodesic curvature equal to f, that is, if it satis\fes\nr_\r_\r= (f\u000e\r)\u0001_\r?; (1.1)\nwhereris the Levi-Civita connection, and _ \r?is the unit tangent vector such that the angle between _ \rand\n_\r?is\u0019\n2(recall that gand the \fxed orientation yield a well-de\fned way of measuring angles in each tangent\nplane, as well as an area form \u0016gon \u0006). The \row on S\u0006 is given by\n\u001et\ng;f(q;v) = (\r(t);_\r(t));8t2R;\nwhere\ris the unique solution to (1.1) with \r(0) =q, _\r(0) =v. Such a \row is of physical interest since\nit models the motion of a charged particle in \u0006 under the e\u000bect of the magnetic \feld f\u0016g. The Legendre\ntransform provides a conjugacy between \bt\ng;fand the (restriction to the energy level1\n2of the) magnetic \row,\nthat is the Hamiltonian \row on T\u0003\u0006 de\fned by Hkin(q;p) =1\n2jpj2and the twisted symplectic form\n!g;f:= dp^dq\u0000\u0019\u0003(f\u0016g);\nwhere\u0019:T\u0003\u0006!\u0006 is the bundle projection. Moreover, for every \u0015>0 the reparametrization ~ \r(t) :=\r(t=\u0015)\nyields a correspondence between ( g;\u0015f )-geodesics and orbits of the magnetic \row contained in fHkin=1\n2\u00152g.\nHence, the magnetic \row can be seen as the collection f\bt\ng;\u0015fgof \rows associated with the family of magnetic\nsystemsf(g;\u0015f )j\u0015>0g; also we have a correspondence between high (low) energies and small (large) values\nof\u0015. After the pioneering work of Arnol'd [5] in 1960s, magnetic systems have received the attention of many\noutstanding mathematicians, such as Novikov, Ginzburg, and Contreras, among others. In particular, the\nproblem of \fnding periodic solutions to (1.1), which we will refer to as closed (g;f)-geodesics , turned out\nto be extremely di\u000ecult, and many questions in the topic still remain open or only partially answered. We\nrefer the reader to [7, 20, 23, 35] and references therein for an account of the main contributions to the closed\n(g;f)-geodesics problem, particularly for the case of surfaces. We shall recall that, in contrast with geodesic\n\rows, magnetic systems present very di\u000berent behaviors for di\u000berent values of \u0015; see e.g. [1, 19].\nIn this paper we will focus on the complementary problem, that is, in the study of systems ( g;f) whose\n\row (1.1) is orbit-equivalent to a free S1-action onS\u0006, hence in particular for which all orbits are closed.\nDe\fnition 1.1. A magnetic system ( g;f) is called Zoll if \bt\ng;fhas the same orbits as a free S1-action onS\u0006.\nRemark 1.2. Forf\u00110 we recover the notion of a Zoll metric, and in this case \u0006 must be the two-sphere (a\nthorough discussion of such metrics can be found e.g. in [15]). For \u0006 6=S2, Zoll magnetic systems can be\nequivalently de\fned as those pairs ( g;f) for which all ( g;f)-geodesics are closed (and contractible). Indeed,\nif all (g;f)-geodesics are closed, then \bt\ng;fhas by a theorem of Epstein [21] the same orbits as a \fxed-point\nDate : July, 2019.\n2000 Mathematics Subject Classi\fcation. 37J99, 58E10.\nKey words and phrases. Magnetic \rows, Zoll systems, waists.\n1arXiv:1910.03285v1  [math.DS]  8 Oct 20192 L. ASSELLE AND C. LANGE\nfreeS1-action onS\u0006, and hence de\fnes a Seifert \fbration of S\u0006. It follows from [28, Theorem 5.1] that this\nmust be the standard, regular S1-\fbration (given by S\u0006!\u0006) . For \u0006 = S2, the celebrated Katok's example\n[13, 29] yields magnetic systems ( g;f) for which all ( g;f)-geodesics are closed but whose \row on S\u0006 is only\norbit-equivalent to a semi-free S1-action. \u0004\nZoll magnetic systems exist for every closed oriented surface and, as established in [14], play a crucial role\nin local magnetic systolic inequalities: the minimal magnetic length of closed magnetic geodesics of magnetic\nsystems close to a Zoll one is bounded from above in terms of a quantity depending only on the g-volume,\nthe genus of \u0006, and the integral of fover \u0006, and the upper bound is attained precisely when the magnetic\nsystem is Zoll. Also, recently the \frst author and Benedetti [9] showed that integrable magnetic systems on\nthe two-torus admitting a global surface of section satisfy a sharp systolic inequality (see [2] for a similar\nresult for Riemannian spheres of revolution). For the applications of such systolic inequalities it is therefore\ncrucial to gain a better understanding of the space of Zoll magnetic systems.\nUntil last year, the only known examples of Zoll magnetic systems were pairs ( gcon;fcon), withgcona\nmetric of constant curvature Kconandfcon>0 any constant function such that1\nf2\ncon+Kcon>0:\nA breakthrough came only very recently with [9], in which explicit non-trivial 1-parameter families of rota-\ntionally symmetric Zoll magnetic systems on certain \rat tori are constructed. The result in [9] can be thought\nof as the \frst evidence of the \rexibility of magnetic \rows which are Zoll at a given energy. However, the trivial\nexamples are Zoll at every energy, that is, ( gcon;\u0015fcon) is Zoll for every \u0015>0 such that \u00152f2\ncon+Kcon>0.\nTherefore, since magnetic \rows strongly depend on the energy, it is natural to ask the following\nQuestion 1.3. Does the \rexibility in [9] turn into rigidity, if one requires the magnetic \row to be Zoll at\nmultiple energies?\nIn order to make the question more precise, we recall that if \u0006 has genus greater than or equal to 2, then\nthe two-form f\u0016gis weakly-exact, that is, its lift to the universal cover is exact. We \fx a primitive \u0012of the\nlift off\u0016gand de\fne the Ma~ n\u0013 e critical value of the universal cover as\nc(g;f) := inf\nu2C1(~\u0006)sup\nq2~\u00061\n2j\u0012q\u0000dquj2; (1.2)\nwherej\u0001jdenotes the dual norm on T\u0003~\u0006 induced by the lift of the metric g. It is well-known that c(g;f) is\nalways \fnite (see e.g. [6]), and vanishes if and only if f\u00110. Now we set\nh(g;f) :=1p\n2c(g;f)2(0;+1): (1.3)\nWe shall notice that c(g;f) is well de\fned also for the two-torus, and for any surface if f\u0016gis exact. However,\nas we will see, for our purposes we can always assume that f\u0016gis not exact. As it turns out, in this case c(g;f)\nis always in\fnite if \u0006 is a two-torus, and if \u0006 = S2we have that any primitive of f\u0016gjS2nfpgis unbounded.\nTherefore, if \u0006 is a two-sphere or a two-torus we set\nh(g;f) := 0:\nAlso, if \u0006 is a surface with genus at least two, then for every \u0015<h (g;f) there exists a closed ( g;\u0015f )-geodesic\nin every non-trivial free homotopy class; see [32]. In particular, ( g;\u0015f ) cannot be Zoll for \u0015<h (g;f).\nThe main goal of this paper is to provide a positive answer to Question 1.3 in the following form.\nTheorem 1.4. Let(g;f)be a magnetic system on a surface with genus greater than or equal to one such\nthat(g;\u0015nf)is Zoll for some bi-in\fnite sequence f\u0015ngn2Zwith\u0015n#h(g;f)forn!\u00001 and\u0015n\"+1for\nn!+1. Thenghas constant curvature and fis constant.\nWe now give an account on the strategy of the proof of Theorem 1.4 for the two-torus, which throughout\nthe paper will be identi\fed with the quotient T2ofR2by the lattice Z\u0002Z:\n1Takingfcon=Kcon= 0 for the two-torus yields the geodesic \row of the \rat metric, and in particular all closed ( gcon;fcon)-\ngeodesics are not contractible. A similar situation occurs for f2\ncon+Kcon<0 on higher genus surfaces, whereas for f2\ncon+Kcon= 0\nwe retrieve the celebrated horocycle \row of Hedlund (cf. [26]), and in particular no closed ( gcon;fcon)-geodesics at all.ON THE RIGIDITY OF ZOLL MAGNETIC SYSTEMS ON SURFACES 3\n\u000fThe decreasing sequence \u0015n#0 yields that gis \rat. This follows from the fact that closed non-\ncontractible geodesics which globally minimize the length in their homotopy class cannot all disappear\nunder magnetic perturbations, unless the metric is \rat. This fact will be proved in Section 2.\n\u000fOnce we know that the metric is \rat, we look at the magnetic systems ( g\rat;\u0015nf) forn!+1. In\nSection 3 we show that, if fis not constant, then for nsu\u000eciently large we \fnd both short and long\nclosed (g\rat;\u0015nf)-geodesics. The fact that ( g\rat;\u0015nf) is Zoll then contradicts the dichotomy between\nshort and long periodic orbits established in [12, Theorem 7.13].\nThe proof for higher genus surfaces is di\u000berent and hinges on the relation between the helicity and the\nMa~ n\u0013 e critical value c(g;f) proved in [33]; see the end of Section 3. Actually, in this case we only need the\nsequence\u0015n#h(g;f) to conclude rigidity; see Theroem 3.5. Therefore, we are prompted to ask the following\nQuestion 1.5. For \u0006 a two-torus, does Theorem 1.4 continue to hold if one only requires that ( g;\u0015nf) be Zoll\nfor some decreasing sequence \u0015n#0? More generally, can one detect the precise threshold between rigidity\nand \rexibility? Does this threshold depend on the topology of the surface?\nIn case of the two-sphere, a statement in the spirit of Theorem 1.4 turns out to be more di\u000ecult to\nprove, the main reason being that the space of Zoll metrics on the two-sphere is in\fnite dimensional [25].\nNevertheless, in Section 4 we prove some partial results in this direction: we show that the metric gmust\n\\generically\" be Zoll and that genericity can be dropped if one considers only rotationally invariant magnetic\nsystems. Further computations with rotationally invariant magnetic systems also support rigidity on S2\nwhich is why we make the following\nConjecture (Z). Let (g;f) be a magnetic system on S2such that ( g;\u0015f ) is Zoll for all \u0015 > 0. Theng\nhas constant curvature and fis constant.\nIn fact, after \fnishing this paper, we were able to con\frm Conjecture (Z) for rotationally invariant magnetic\nsystems of the form ( g;fcon). The proof will appear in a forthcoming paper.\nLet us also mention that Theorem 1.4 and Conjecture (Z) are related to two open problems about integrable\ndynamical systems on two-dimensional con\fguration spaces. The \frst one traces back to Birkho\u000b [18] and\naims at determining all metrics on the two-torus with an integrable geodesic \row. Despite several partial\nresults (see [16, 17, 31] and references therein), it is as of now not known whether there are metrics other than\nLiouville metrics which gives rise to integrable geodesic \rows. The second problem concerns exact magnetic\n\rows on the two-torus that admit a \frst integral on all energy levels: In [4] it is conjectured that such \rows\nmust be of a very particular type (cf. Example 1 in [4]), and the conjecture is con\frmed in the case of\nquadratic in momenta integrals.\nFinally, in Section 5 we prove a result of independent interest on the persistence of possibly degenerate\nclosed geodesics under magnetic perturbations which we can formulate roughly speaking as follows: Let gbe\na metric admitting a closed contractible geodesic which is a local minimizer of the length functional and is\nstable (namely, does not disappear after an arbitrarily small perturbation of the metric). Then such a closed\ngeodesic will persist also under magnetic perturbations. One major issue we have to overcome in the proof\nof such a statement is that the dynamics of a magnetic systems arising as perturbation of a geodesic systems\nis in general drastically di\u000berent from the geodesic dynamics, even if the perturbation is arbitrarily small.\nAlso, we have to deal with critical sets which may have complicated topology (such as e.g. a Cantor set).\nTheorem 1.6. Let\u0006be a closed orientable surface, and let (g;f)be a magnetic system on \u0006. Suppose that\ngpossesses a contractible stable waist, that is, a closed geodesic that locally minimizes the free-period action\nfunctional Ain(2.1) . Then there exists \u0003(g;f)>0such that for all 0< \u0015 < \u0003(g;f)there exists a closed\ncontractible (g;\u0015f )-geodesic which locally minimizes the free-period Lagrangian action functional A\u0015in(2.2) .\nMoreover, such closed (g;\u0015f )-geodesics can be chosen to lie in a small neighborhood of a waist for g.\nIn fact, in our proof of Theorem 1.6 the contractibilty assumption is only used in the case of the two-torus.\nAcknowledgments. We warmly thank Alberto Abbondandolo and Stefan Suhr for many fruitful discussions.\nWe are indebted to Gabriele Benedetti for suggesting us the reference [33]. L.A. is partially supported by the\nDFG-grant AS 546/1-1 \\Morse theoretical methods in Hamiltonian dynamics\". C.L. is partially supported\nby the DFG-grant SFB/TRR 191 \\Symplectic structures in Geometry, Algebra and Dynamics\".4 L. ASSELLE AND C. LANGE\n2.Zoll magnetic systems on T2for small values of the parameter \u0015\nIn this section we want to derive conditions on the metric gfor magnetic systems ( g;f) onT2such that\n(g;\u0015f ) is Zoll for \u0015>0 su\u000eciently small, or, equivalently, such that the corresponding magnetic \row is Zoll\nfor su\u000eciently large energies. More precisely, we want to show that being Zoll for small values of \u0015implies\nthat the metric is \rat. As it turns out, we don't need to require that ( g;\u0015f ) is Zoll for all\u0015>0 su\u000eciently\nsmall. Indeed, it is enough that ( g;\u0015nf) is Zoll for some sequence \u0015n#0.\nProposition 2.1. Let(g;f)be a magnetic system on T2such that (g;\u0015nf)is Zoll for some sequence \u0015n#0.\nThengis a \rat metric.\nRemark 2.2. The proof of Proposition 2.1 actually shows that there exists \u0015\u0000=\u0015\u0000(g;f)>0 such that if\n(g;\u00150f) is Zoll for some \u001502(0;\u0015\u0000) thengis a \rat metric. It would be interesting to see whether \u0015\u0000can be\nchosen independently of the magnetic systems in a neighborhood of ( g;f), and more generally if after some\nnormalization (such as e.g. Area( T2;g) = 1,kfk1\u00141, ...) the same assertion holds from some constant \u0015\u0000\nindependent of the magnetic system.\nProposition 2.1 is an immediate corollary of Proposition 2.3 below on the persistence of closed geodesics\nunder magnetic perturbations, whose statement requires the introduction of some notation (for the details we\nrefer e.g. to [1, 19]). We recall that closed arc-length parametrized geodesics on ( T2;g) one-to-one correspond\nto the critical points of the free-period Lagrangian action functional\nA:H1(T;T2)\u0002(0;+1)!R;A(\u0000;\u001c) :=1\n2\u001cZ1\n0j_\u0000(s)j2ds+\u001c\n2; (2.1)\nmeaning that \r:R=TZ!S2is an arc-length parametrized closed geodesic if and only if (\u0000 ;T) is a critical\npoint of A, where \u0000 is given by \u0000( s) :=\r(Ts). HereH1(T;T2) denotes the space of one-periodic loops in\nT2of Sobolev-class H1, and it is well-known that its connected components are in bijection with elements\n(actually conjugacy classes) of \u00191(T2). Hereafter we will identify a pair (\u0000 ;\u001c) with the corresponding curve \r,\nand write A(\r) instead of A(\u0000;\u001c) whenever more convenient. An analogous variational principle is available\nalso for exact magnetic systems (i.e. when f\u0016gis exact) and allows us to detect closed ( g;\u0015f )-geodesics as\ncritical points of a suitable action functional A\u0015, whose precise de\fnition will be recalled in (2.2).\nFor every homotopy class \u000b2\u00191(T2)nf0gwe denote by K\u000b6=;the compact set of global minimizers of\nAin the connected component of H1(T;T2)\u0002(0;+1) determined by \u000b, that is\nA(\r\u000b) = min\n\r2\u000bA(\r) i\u000b\r\u000b2K\u000b:\nWe would like to stress that in general the set K\u000bdoes not have more structure than a compact set (e.g. it\ncould be a Cantor set).\nProposition 2.3. Suppose that gis not a \rat metric on T2, and letf:T2!Rbe any smooth function. Then\nthere exist a homotopy class \u000b2\u00191(T2)nf0g, a bounded neighborhood U\u000bof the setK\u000b, and a constant\n\u0003 = \u0003(g;f;U\u000b)>0such that for every 0<\u0015< \u0003there exists a closed (g;\u0015f )-geodesic which is contained in\nU\u000band is a local minimizer of the functional A\u0015in(2.2) .\nProof of Proposition 2.1. Suppose that gis not \rat and set \u0015\u0000= \u0003(g;f;U\u000b), where\u000b2\u00191(T2)nf0gis given\nby Proposition 2.3. For n2Nsuch that\u0015n2(0;\u0015\u0000) we thus \fnd a closed non-contractible ( g;\u0015nf)-geodesic,\nin contradiction with the fact that all ( g;\u0015nf)-geodesics must be contractible, see Remark 1.2. \u0004\nBefore proceeding with the proof, we would like to make two comments on Proposition 2.3. First, to\nestablish if closed geodesics are stable under magnetic perturbations is a very natural question which has\nbeen already investigated in the past decades. Following [23] we see that, while on the one hand a non-\ndegenerate closed geodesic always \\survives\" when switching on a magnetic \feld, on the other hand the\nexample of a \rat torus with induced area form shows that we must impose some kind of condition on the\nclosed geodesic for it not to disappear. Therefore, Proposition 2.3 can be seen as a \frst step towards the\nstudy of the stability of degenerate closed geodesics under magnetic perturbations. Another instance of this\npersistence will be discussed in Section 4.\nSecond, we would like to stress that in Proposition 2.3 we do not require the magnetic function fto have\nvanishing integral over T2, or, equivalently, the two-form f\u0016gto be exact. Hence, in general, for the Ma~ n\u0013 eON THE RIGIDITY OF ZOLL MAGNETIC SYSTEMS ON SURFACES 5\ncritical value of the universal cover de\fned in (1.2) we have\nc(g;0) = 0; c(g;\u0015f ) = +1 8\u0015>0;\nor, equivalently, for the constant h(g;f) de\fned in (1.3)\nh(g;0) = +1; h(g;\u0015f ) = 08\u0015>0:\nThis can be rephrased by saying that the magnetic perturbation is notsmall even if \u0015is (arbitrary) small.\nProof of Proposition 2.3. We \frst assume that the image of K\u000bunder the evaluation map\nev :H1(T;T2)\u0002(0;+1)\u0002T!T2;ev(\r;s) :=\r(\u001cs);\nis a proper compact subset of T2for some\u000b2\u00191(T2)nf0g. Clearly, under this assumption we can \fnd a\nbounded neighborhood U\u000b\u0012H1(T;T2)\u0002(0;+1) ofK\u000bsuch that\nU:= ev(U\u000b\u0002T)\u001aev(U\u000b\u0002T)(T2:\nThus, we have f\u0016gjU= d\u0012for some bounded one-form \u00122\n1(U). According to [19] closed ( g;\u0015f )-geodesics\nwith image contained in Ucorrespond to critical points of the free-period Lagrangian action functional\nA\u0015(\u0000;\u001c) :=1\n2\u001cZ1\n0j_\u0000(s)j2ds\u0000\u0015Z1\n0\u0012\u0000(_\u0000) ds+\u001c\n2: (2.2)\nAt the same time, since Asatis\fes the Palais-Smale condition on the connected components of H1(T;T2)\u0002\n(0;+1), there exists \u000f>0 such that\ninf\n@U\u000bA>A(K\u000b) +\u000f;\nwhere A(K\u000b) denotes the action of any element in K\u000b(see e.g. [3, Lemma 3.1] for the proof). We now show\nthat, if\u0015>0 is su\u000eciently small, then\ninf\n@U\u000bA\u0015\u0015sup\n\r\u000b2K\u000bA\u0015(\r\u000b) +\u000f\n2: (2.3)\nFrom the fact that A\u0015satis\fes the Palais-Smale condition on bounded subsets of H1(T;S2)\u0002(0;+1) we\ntherefore deduce that for such values of \u0015there exists a closed ( g;\u0015f )-geodesic which is contained in U\u000band\nis a global minimizer of A\u0015inU\u000b, thus completing the proof.\nTo prove (2.3) we preliminarly compute using \u0012q(v) =hXq;viand the Cauchy-Schwarz inequality\n\f\f\f\fZ1\n0\u0012\u0000(_\u0000) ds\f\f\f\f\u0014Z1\n0j\u0012\u0000(_\u0000)jds=Z1\n0jhX\u0000;_\u0000ijds\u0014kX\u0000k2k_\u0000k2\u0014k\u0012k1k_\u0000k2\nand set\nr:= sup\n(\u0000;\u001c)2U\u000bk_\u0000k2;\u0003 = \u0003(g;f;U\u000b) :=\u000f\n4rk\u0012k1:\nFor all\u0015<\u0003, all\r= (\u0000;\u001c)2@U\u000b, and all\r\u000b= (\u0000\u000b;\u001c\u000b)2K\u000bwe thus have\nA\u0015(\r) =A(\r)\u0000\u0015Z1\n0\u0012\u0000(_\u0000) ds\n\u0015A(\r)\u0000\u0015k\u0012k1k_\u0000k2\n\u0015A(\r)\u0000\u000f\n4\n>A(\r\u000b) +3\n4\u000f\n\u0015A(\r\u000b) +\u0015k\u0012k1k_\u0000\u000bk2+\u000f\n2\n\u0015A\u0015(\r\u000b) +\u000f\n2;\nand (2.3) follows taking the in\fmum over \r2@U\u000band the supremum over \r\u000b2K\u000b.\nThus, we are left to consider the case in which each set K\u000bis mapped surjectively onto T2by the evaluation\nmap ev. In this case we will show that the metric gmust be \rat, in contradiction with the assumption.6 L. ASSELLE AND C. LANGE\nWe preliminarly observe that the same proof as above allows us to \fnd closed ( g;\u0015f )-geodesics close to\nany \\isolated\" \r\u000b2K\u000b. Here by isolated we mean that \r\u000bis a strict local (actually global) minimizer of A,\nthat is, there exists a neighborhood Uof the critical circle T\u0001\r\u000b:=f\r\u000b(\u0001+s)js2Tgsuch that\ninf\nUA=A(\r\u000b);andU\\A\u00001(A(\r\u000b)) =T\u0001\r\u000b:\nTherefore, we can further assume that all \r\u000b2K\u000bare non-isolated. We claim that, under this assumption,\nfor any\u000b2\u00191(T2)nf0gthe setK\u000byields a simple foliation of T2by closed geodesics. Observe that this\nimmediately implies that the metric is \rat by a theorem of Innami [27] (see also [11]).\nTo prove the last assertion we show that two elements \r\u000b;\u0017\u000b2K\u000bthat are not the same geometric curve\nmust have disjoint image. Thus, let us suppose that \u0017\u000b;\r\u000b2K\u000bintersect transversally, and denote with \u001a\u000b,\n\u001c\u000btheir periods. Then, we can \fnd lifts ~ \u0017\u000b;~\r\u000b:R!R2of\u0017\u000b,\r\u000btoR2respectively such that\n~\u0017\u000bj[0;\u001a\u000b](\u0001)\\~\r\u000bj[0;\u001c\u000b](\u0001)6=;:\nObserve that ~ \u0017\u000b, ~\r\u000bare embedded. Also, since \u0017\u000band\r\u000bbelong to the same homotopy class, up to shifting\nthe base point of ~ \u0017\u000bwe can suppose that there exist 0 \u0014s1<s2<\u001a\u000band 0\u0014t1<t2<\u001c\u000bsuch that\n~\u0017\u000b(si) = ~\r\u000b(ti);fori= 1;2:\nWe now de\fne the piecewise smooth curves\n~\u00111:= ~\u0017\u000bj[0;s1]# ~\r\u000bj[t1;t2]# ~\u0017\u000bj[s2;\u001a\u000b];\n~\u00112:= ~\r\u000bj[0;t1]# ~\u0017\u000bj[s1;s2]# ~\r\u000bj[t2;\u001c\u000b]:\nBy construction, ~ \u00111and ~\u00112project to closed curves \u00111and\u00112onT2in the homotopy class \u000bthat satisfy\nA(\u00111) +A(\u00112) =A(\u0017\u000b) +A(\r\u000b) = 2A(\r\u000b):\nIt follows that at least one of them, say \u00111, has action less or equal to A(\r\u000b). Therefore, \u00111is a closed geodesic\n(for it is a global minimizer of Ain the homotopy class \u000b), hence in particular smooth, a contradiction.\nUsing this we readily see that a non-isolated \r\u000bmust be:\n(1) embedded if \u000bis a primitive class in \u00191(T2)nf0g, or\n(2) them-th iterate of some \r\f2K\f, if\u000b=m\u0001\ffor somen2Nand some primitive class \f2\u00191(T2).\nIndeed, if\r\u000bhad a transversal self-intersection, then the image of elements in K\u000bsu\u000eciently close to \r\u000b\nwould not be disjoint from the image of \r\u000b. It is now straightforward to see that under our assumptions the\nsetK\u000byields a simple foliation of T2by geodesics if \u000bis a primitive class in \u00191(T2) and anm-fold foliation\nby geodesics for some m2Notherwise. This completes the proof. \u0004\n3.Zoll magnetic systems on flat tori for large values of \u0015\nLet (g;f) be a magnetic system on \u0006 = T2as in the statement of Theorem 1.4. In virtue of Proposition\n2.1 we can assume that g=g\ratis a \rat metric. In this section, by looking at large values of \u0015, we show\nthat the magnetic function fmust be constant as well, thus completing the proof of Theorem 1.4 in the case\n\u0006 =T2.\nThe theorem is an immediate corollary of Proposition 2.1 combined with the following\nProposition 3.1. Let(g\rat;f)be a magnetic sytem on T2such that (g;\u0015nf)is Zoll for some sequence \u0015n\"+1.\nThenfis constant.\nRemark 3.2. As for Proposition 2.1, we can improve the statement of Proposition 3.1 by saying that there\nexists\u0015+=\u0015+(g;f)>0 such that, if ( g\rat;\u00151f) is Zoll for some \u001512(\u0015+;+1), thenfmust be a constant\nfunction. Also here it would be very interesting to see if - after normalization - the constant \u0015+can be chosen\nindependently of the magnetic system.\nThe idea of the proof is that for \u0015 > 0 large enough we always \fnd short periodic ( g\rat;\u0015f)-geodesics,\nand iffis not constant also long ones (close to a regular level of f). The fact that ( g\rat;\u0015f) is Zoll then\ncontradicts the dichotomy between long and short periodic orbits established in [12, Theorem 7.13]. We shall\nnotice that [12, Theorem 7.13] deals only with magnetic systems on the two-sphere; however, the proof is\nbased on an argument of Bangert [10] which works for \rows converging to a free S1-action, and hence extends\ne.g. to all magnetic systems on closed surfaces.ON THE RIGIDITY OF ZOLL MAGNETIC SYSTEMS ON SURFACES 7\nTheorem. [12, Theorem 7.13] Let(g;f)be a magnetic system on \u0006, withf >0. For every \u000f>0andn2N\nthere is \u0003 = \u0003(\u000f;n)2(0;+1)such that for every \u00152(\u0003;+1)a periodic prime (g;\u0015f )-geodesic is either a\nsimple curve with length in (2\u0019\u0000\u000f\n\u0015maxf;2\u0019+\u000f\n\u0015minf), or has at least nself-intersections and length larger than1\n\u0015\u000f.\nIn order to apply this theorem we need some preliminary results on the properties of magnetic trajectories\ninR2for su\u000eciently small energies. Such properties are certainly well-known to the experts (see the discussion\nat the end of the section), however for the reader's convenience we include them here.\nLemma 3.3. Let(g;f)be a magnetic pair on \u0006. For any neighborhood Uof any compact subset K\u001a\u0006with\nfjU>0, anyT >0there exists some \u0003>0such that for every \u0015>\u0003every (g;\u0015f )-geodesic\rstarting inK\nstays completely in Uin forward and backward time up to time T, that is, satis\fes \rj[\u0000T;T]\u001aU.\nProof. The claim follows immediately from the fact that over the interior of supp( f) the magnetic \row\nC1\nloc-converges to rotations in the \fbers; see [30]. \u0004\nLemma 3.4. Let(g\rat;f)be a magnetic pair on R2and letq2R2be a regular point of f. Then for su\u000eciently\nlarge\u0015there is a (g\rat;\u0015f)-geodesic starting at qwhich has self-intersections.\nProof. Suppose that the level set ff=eg3qis tangential to the x-axis atqand that the gradient of fat\nqpoints in the positive y-direction. We look at ( g\rat;\u0015f)-geodesics starting at qin the negative y-direction.\nFor su\u000eciently large \u0015the trajectory q(t) schematically looks as the solid curve in Figure 1. For given \">0\nwe choose\u000e>0 such that\nkf(q+v)\u0000f(q)\u0000Dqf\u0001vk\u0014\"kvk\nfor allv2R2withkvk\u0014\u000e. Lett1andt2be the \frst and second time at which q(t) hits thex-axis. By\nthe proof of Lemma 3.3 there exists some c >1 such that for all su\u000eciently large \u0015the segment qj[0;t2]is\ncontained in a ball with radius r\u0015=c=(e\u0015) aroundq. We can suppose that this ball is contained in Bq(\u000e) by\nchoosing\u0015large enough. In order to prove that q(t) is not embedded, it su\u000eces to show that the \\drift\" \u0001 x\ndepicted in Figure 1 is positive for \u0015large enough.\nr1\nr0\nq(0) q(t2)x\n\u0001x\nr1\nr2\nq(t1)\nFigure 1. Magnetic trajectory in a magnetic \feld on R2.\nA curvature comparison shows that the segments qj[0;t1]andqj[t1;t2]ofq(t) lie completely below the dashed\nlines in Figure 1 which are speci\fed as follows: the dashed line above the x-axis is a circular arc with radius\nr2:=1\n\u0015e\u0000\"r\u0015\nstarting at q(t1) tangentially to q(t). The dashed line below the x-axis is tangential to q(t) atq(0) and\nconsists of three circular arcs of radii\nr1:=1\n\u0015e+\"r\u0015; r 0:=1\n\u0015e\u0000L\u0015\n2(\u0015e+\"r\u0015)+\"r\u0015;andr1\nrespectively, which meet tangentially on the horizontal line speci\fed by y= 2(\u0015e+\"r\u0015). Here we have set\nL=kDqfk. We compute8 L. ASSELLE AND C. LANGE\n\u0001x\u00152r1+ 2 cos(30\u000e)(r0\u0000r1)\u00002r2\n\u00152 cos(30\u000e)\n\u0015e\u0000L\u0015\n2(\u0015e+\"c=(e\u0015))+c\"\ne\u0015+2(1\u0000cos(30\u000e))\n\u0015e+c\"\ne\u0015\u00002\n\u0015e\u0000c\"\ne\u0015\n:= 2\u0001(\";\u0015)(3.1)\nfor some constant c>0. We need to show that there exists some \">0 and a su\u000eciently large \u0015such that\n\u0001(\";\u0015)>0. Sincej@\"\u00152\u0001(\";\u0015)jis bounded from above on ( \";\u0015)2(0;1]\u0002[1;1), it su\u000eces to show that\nlim inf\u0015!1\u00152\u0001(0;\u0015)>0. This is indeed the case, for\n\u00152\u0001(0;\u0015) =cos(30\u000e)\n\u0015e\u0000L\u0015\n2\u0015e+1\u0000cos(30\u000e)\n\u0015e\u00001\n\u0015e=L\u0015cos(30\u000e)\n2e2(\u0015e\u0000L\n2e): \u0004\nBefore proving Proposition 3.1 we shall recall that on T2the action functional A\u0015is well-de\fned over\nthe space of contractible loops even if f\u0016gis not exact. Indeed, one can replace the integral of \u0012along a\ncontractible loop \rby the integral of f\u0016gover a capping disk for \r, and the resulting functional will not\ndepend on the choice of the capping disk since T2is aspherical. Moreover, for any Zoll magnetic system\n(g;f) the value of the functional A\u0015is constant on the set of prime closed ( g;f)-geodesics; this latter fact\nfollows e.g. from the magnetic systolic-diastolic inequality in [14].\nProof of Proposition 3.1. We claim that f\u00150. Indeed, \frst by [14, Proposition 1.3] we can assume that the\nintegral offoverT2is positive. Now, if f\u00150 does not hold, then f\u0016gflatis oscillating in the sense of [8].\nIn particular, by [8, Theorem 4.1] for su\u000eciently large \u0015there exists a periodic ( g\rat;\u0015f)-geodesic\rwhich\nis a local minimizer of the action functional A\u0015. Another periodic magnetic geodesic \u0017starting at \r(0) with\ninitial velocity close to _ \r(0) has to intersect \ra second time. Now, since in the Zoll case the functional A\u0015\nis constant on the set of all prime magnetic geodesics, the same line of arguments as in the end of the proof\nof Proposition 2.3 applied to \rand\u0017and using the functional A\u0015instead of Ayields a contradiction.\nWe \frst consider the case f >0. We choose \u000f>0 so small that\n\u00142\u0019\u0000\u000f\nmaxf;2\u0019+\u000f\nminf\u0015\n\\\u00141\n\u000f;+1\u0013\n=;; (3.2)\nand set \u0003 := \u0003( \u000f;1)>0 as in [12, Theorem 7.13]. Following [22], up to enlarging \u0003 if necessary we \fnd for\nall\u0015>\u0003 one2embedded closed ( g\rat;\u0015f)-geodesic\r\u0015\nswhich is short, meaning that its length satis\fes\n2\u0019\u0000\u000f\n\u0015maxf< `(\r\u0015\ns)<2\u0019+\u000f\n\u0015minf;8\u0015>\u0003:\nOn the other hand, if fis not constant, then by Lemma 3.4 up to enlarging \u0003 further we also \fnd for\nall\u0015 > \u0003 a (g\rat;\u0015f)-geodesic\r\u0015\nlwhich has self-intersections. We set \u0015+:= \u0003 and take n2Nsuch that\n\u0015n> \u0015 +. Then, the ( g\rat;\u0015nf)-geodesic\r\u0015n\nlwith self-intersections is necessarily closed and hence by [12,\nTheorem 7.13] long, meaning that\n`(\r\u0015n\nl)>1\n\u0015n\u000f:\nWe denote by ( qs;vs), (ql;vl) the initial conditions of the closed ( g\rat;\u0015nf)-geodesics \r\u0015ns;\r\u0015n\nlrespectively,\nand consider a path r7!(q(r);v(r)) inST2connecting ( qs;vs) to (ql;vl). Since the period (and hence the\nlength) of closed ( g;\u0015nf)-geodesics varies smoothly, by the intermediate value theorem we \fnd r02(0;1)\nsuch that the length of the closed ( g;\u0015nf)-geodesic\r\u0015n\niwith initial conditions ( q(r0);v(r0)) satis\fes\n2\u0019+\u000f\n\u0015nminf< `(\r\u0015n\ni)<1\n\u0015n\u000f;\nthus contradicting [12, Theorem 7.13].\nThe casef\u00150 is more delicate since [12, Theorem 7.13] cannot be applied directly. Our strategy is\nto show that the problem can be reduced to the case f > 0 by suitably modifying the magnetic function.\nSuppose that f\u00150 is not constant and \fx three regular energy values 0 <e0<e1<e2<maxf. Without\n2Actually, at least two.ON THE RIGIDITY OF ZOLL MAGNETIC SYSTEMS ON SURFACES 9\nloss of generality we can suppose that the superlevel sets ff\u0015eigare connected, i= 0;1;2. Using a cut-o\u000b\nfunction we construct ~fsuch that\n~f\u0011fonff\u0015e1g;and ~f\u0011e0onff\u0014e0g:\nWe \fx\u000f>0 such that (3.2) holds with freplaced by ~fand choose\n\u0016`2\u00102\u0019+\u000f\ne0;1\n\u000f\u0011\n:\nWe also let \u0003 >0 be such that, for all \u0015>\u0003, closed (g\rat;\u0015~f)-geodesics are either embedded and short (in\nthe sense above) or long and have self-intersections. Up to enlarging \u0003 we \fnd for all \u0015 > \u0003 at least two\nshort embedded closed ( g\rat;\u0015~f)-geodesics, one of which is also a short ( g\rat;\u0015f)-geodesic (namely, the one\nthat is close to the S1-\fber over a maximum point of ~f). We denote such a short periodic orbit with \r\u0015\nsand\nrecall that\n`(\r\u0015\ns)2\u00102\u0019\u0000\u000f\n\u0015maxf;2\u0019+\u000f\n\u0015e0\u0011\n;8\u0015>\u0003:\nAt the same time, by Lemmas 3.3 and 3.4 applied to a neighborhood U\u001aff > e 1gof the compact set\nK=ff\u0015e2g, up to enlarging \u0003 further we can assume that for all \u0015 > \u0003 there is a ( g\rat;\u0015f)-geodesic\n\r\u0015\nlstarting at f\u00001(e2) which does not close up until time \u0016`=\u0015, and that all ( g\rat;\u0015f)-geodesics starting in\nKstay inUup to time \u0016`=\u0003. We set \u0015+:= \u0003 and take n2Nsuch that\u0015n> \u0015 +. In particular, the\n(g\rat;\u0015nf)-geodesic\r\u0015n\nlis closed and has length larger than \u0016`=\u0015n.\nLet now (qs;vs), (ql;vl) be the initial conditions of \r\u0015nsand\r\u0015n\nl, respectively, and let r7!(q(r);v(r)) be\na path inS\u0006 from (qs;vs) to (ql;vl) such that q(\u0001) is entirely contained in ff\u0015e2g. Since the period of\nclosed (g;\u0015nf)-geodesics varies smoothly, by the intermediate value theorem we \fnd r02(0;1) such that\nthe corresponding closed ( g;\u0015nf)-geodesic\r\u0015n\nihas length \u0016`=\u0015n, and in particular is completely contained in\nff\u0015e1g. This shows that \r\u0015n\niis also a closed ( g;\u0015n~f)-geodesic, thus contradicting [12, Theorem 7.13]. \u0004\nLemma 3.4 can also be deduced from Corollary 1.4 in [34], in which low energy magnetic dynamics in R2\nunder a non-constant magnetic \feld is described in terms of a fast rotating motion and a slow drift along\nthe level sets of the magnetic function. Let us mention that Proposition 3.1 extends with the same proof\nto magnetic systems ( g;f) wheregis any metric of constant curvature. For an arbitrary system ( g;f) an\nanalogue of Proposition 3.1, and therewith a step towards Conjecture (Z), would require to unscramble the\nin\ruences of an inhomogeneous magnetic \feld and an inhomogeneous metric on the drift motion. We intend\nto investigate this in future work.\nWe now prove Theorem 1.4 for higher genus surface. Actually, we will prove the following stronger\nTheorem 3.5. Let(g;f)be a magnetic system on a surface with genus at least two such that (g;\u0015n)is Zoll\nfor some sequence \u0015n#h(g;f). Thenghas constant curvature and fis a constant function.\nProof. Recall that the helicityH(g;\u0015f ) of (g;\u0015f ) vanishes if and only if \u0015=\u0015g;f, where\n\u00152\ng;f=\u00002\u0019\u001f(\u0006)A\n[f]2:\nHereAis the total area of (\u0006 ;g), and [f] :=R\n\u0006f\u0016gis the total integral of f\u0016g. For our purposes we don't\nneed to recall the de\fnition of helicity; all we need to know is that, by Lemma 2.12 in [14] the helicity\nH(g;\u0015f ) is related to the average magnetic curvature K\u0015f:=\u00152[f]2+ 2\u0019\u001f(\u0006)A\u00001by the formula\nH(g;\u0015f ) =A2\n2\u001f(\u0006)K\u0015f;\nhence in particular the function \u00157!H (g;\u0015f ) is strictly monotonically decreasing. Moreover, \u0015g;fandh(g;f)\nare related by the inequality h(g;f)\u0014\u0015g;f, with equality if and only if ghas constant curvature and fis a\nconstant function (for the details see [33] and references therein). Therefore, what we have to show is that\nour assumption implies that \u0015g;f=h(g;f).\nSince (g;\u0015nf) is Zoll, Corollary 2.13 in [14] implies that K\u0015nf>0. Thus,\nH(g;\u0015nf) =A2\n2\u001f(\u0006)K\u0015nf<0;8n2N:10 L. ASSELLE AND C. LANGE\nBy the monotonicity of \u00157!H (g;\u0015f ) we deduce that H(g;\u0015f )<0 for every \u0015 > h (g;f), and this implies\nthath(g;f)\u0015\u0015g;f, thus completing the proof. \u0004\n4.On Conjecture (Z) for \u0006 =S2\nThe goal of this section is to prove some partial results about Conjecture (Z) on \u0006 = S2. In particular,\nwe prove that if ( g;f) onS2is such that ( g;\u0015f ) is Zoll for \u0015su\u000eciently small then gmust \\generically\" be\nZoll, and that genericity can be dropped when restricting to rotationally invariant magnetic systems.\nProposition 4.1. Let(g;f)be a magnetic system on S2such that (g;\u0015nf)is Zoll for some sequence \u0015n#0.\nThengmust \\generically\" be Zoll.\nIn order to explain the sentence \\ gmust generically be Zoll\" we reformulate Proposition 4.1 as follows: if\nthe metricgpossesses a non-degenerate closed geodesic (which is well-known to be a generic property) then\nthere exists some \u0015\u0000=\u0015\u0000(g;f)>0 such that for all \u0015<\u0015\u0000the magnetic system ( g;\u0015f ) cannot be Zoll.\nThus, let (g;f) be a \fxed magnetic system such that ( g;\u0015nf) is Zoll for some sequence \u0015n#0. We set\n[f] :=1\nArea(S2;g)Z\nS2f\u0016g\nto be the average of f. Clearly, we can suppose [ f]\u00150. For every n2Nand everym2N0we de\fne\nEm\nn:=n\n(prime) closed ( g;\u0015nf)-geodesics with precisely mself-intersectionso\n;\nEn:=[\nm2N0Em\nn:\nIn particular,E0\nnis the set of embedded closed ( g;\u0015nf)-geodesics. Here self-intersections are counted with\nmultiplicity, meaning that, if there are more than two branches of the ( g;\u0015nf)-geodesic\rintersecting transver-\nsally at a point, then we slightly perturb \rso that the resulting curve has only simple transversal intersections\nand de\fne the multiplicity of the original intersection point as the minimal number of simple self-intersections\narising after perturbation. Notice that, unlike geodesics, ( g;\u0015nf)-geodesics might have self-tangencies; each\nself-tangency should count one in the total multiplicity.\nThe \frst step towards the proof of Proposition 4.1 is to show that it is possible to \fnd an upper bound\non the length of (closed) ( g;\u0015nf)-geodesics which depends only on the number of self-intersections. This\nproperty holds certainly true for general magnetic systems (i.e. independently of the fact that the system be\nZoll or not), but we don't need it here in its full generality.\nLemma 4.2. For everym2N0there exists c=c(m)>0such that for all n2Nand all\r2Em\nnwe have\n`g(\r)\u0014c:\nProof. By the systolic inequality for magnetic systems close to a Zoll one (see [14]) we have that, for every\nn2Nand for every (closed) ( g;\u0015nf)-geodesic\r,\n`g(\r) +Z\nD2C\u0003\n\r(\u0015nf\u0016g) =2\u0019\n\u0015n[f] +q\n\u00152n[f]2+2\u0019\nArea(S2;g)\n\u0014p\n2\u0019Area(S2;g);\nwhere`g(\r) denotes the length of \randC\r:D2!S2is any admissible capping disk for \r. Since for any\n\r2E0\nnthe admissible capping disk can be chosen to be embedded, we obtain for all n\u0015n0\n`g(\r)\u0014p\n2\u0019Area(S2;g)\u0000\f\f\f\fZ\nD2C\u0003\n\r(\u0015nf\u0016g)\f\f\f\f\n\u0014p\n2\u0019Area(S2;g) +\u0015nkfk1Area(S2;g)\n\u0014p\n2\u0019Area(S2;g) +\u00151kfk1Area(S2;g)\n=:c(0):\nForm2Nthe proof goes along the same lines, but employs a \fner estimate of the second integral. For\nthat we need to recall the de\fnition of admissible capping disks from [14]. For a Zoll magnetic system (~ g;~f)\nthe tangential lift ( \r;_\r) of a (closed) (~ g;~f)-geodesic is freely homotopic to the \fbre of the unit tangent bundle;ON THE RIGIDITY OF ZOLL MAGNETIC SYSTEMS ON SURFACES 11\ntherefore, we can \fnd a homotopy \u0000 between the \fbre and ( \r;_\r). Admissible capping disks arise now by\nprojecting such homotopies to S2under the bundle projection. It can be shown thatZ\nD2C\u0003\n\r(~f\u0016~g) (4.1)\ndoes not depend on the choice of the homotopy (hence, on the admissible capping disk).\nLet now\r2Em\nn, and let (\r;_\r) be its tangential lift. Up to an arbitrary small perturbation we can assume\nthat all self-intersections of \rare simple and that there are no self-tangencies (this perturbation will change\nthe value of the integral in (4.1) by a small constant, and hence it is for all estimates not relevant). Since\nthe tangential lift of \ris freely homotopic to the \fbre, we can \fnd t1>t0such that\nq:=\r(t0) =\r(t1);_\r(t0)6=\u0006_\r(t1);\nand\r0:=\rj[t0;t1]is an embedding (in particular, \r0has turning number \u00061 inS2nfpg, for somep2S2). We\n\fx\u000f>0 small and pick a homotopy hfrom (\r\u000f:=\rj[t0\u0000\u000f;t1+\u000f];_\r\u000f) to (\u0017\u000f;_\u0017\u000f) that \fxes qand (a neighborhood\nof) the endpoints of \r\u000f, such that \u0017\u000fis (again) an embedding outside qwith parallel tangent vectors at q.\nThis homotopy will clearly add some bounded quantity to the value of the integral in (4.1). Now we can\nhomotope the tangential lift of the part of \u0017\u000fthat starts from and ends in qwith the \fbre in SS2. This yields\na contribution to the integral in (4.1) which is given by the area of the disk bounded by the considered part\nof\u0017\u000fand hence is in absolute value smaller than \u0015nkfk1Area(S2;g) (see Figure 2).\nq\n\r\r1\n\r0\nFigure 2. Resolving one intersection point for \ryields a contribution to (4.1) smaller than \u0015nkfk1Area(S2;g).\nTherefore, up to the constants arising after perturbation we have that\f\f\f\fZ\nD2C\u0003\n\r(\u0015nf\u0016g)\f\f\f\f\u0014\u0015nkfk1Area(S2;g) +\f\f\f\fZ\nD2C\u0003\n\r1(\u0015nf\u0016g)\f\f\f\f;\nwhere\r1is the smooth loop given by the concatenation of \rj[0;1]n[t0\u0000\u000f;t1+\u000f]with the remaining portion of \u0017\u000f.\nBy construction \r1has one self-intersection less than \r. Repeating the same procedure recursively we obtain a\nhomotopy from ( \r;_\r) to somej-th iterate of the \fbre, j\u0014m, and then we still have to homotope the \\iterated\n\fbre\" to the simple \fbre. The \frst procedure will contribute (up to constants) to the integral with at most\nm-times\u0015nkfk1Area(S2;g), whereas the last homotopy yields at most another j-times\u0015nkfk1Area(S2;g).\nCombining all these estimates we obtain that\f\f\f\fZ\nD2C\u0003\n\r(\u0015nf\u0016g)\f\f\f\f\u0014(2m+ 1)\u0015nkfk1Area(S2;g): \u0004\nUsing the bound given by Lemma 4.2 we show now that we can generalize to the magnetic setting the\nfollowing fact for Riemannian metrics on S2all of whose geodesics are closed: the set of embedded closed\ngeodesics of a Riemannian metric on S2all of whose geodesics are closed is open and close. Notice that this\nis one main ingredient in the proof of Berger's conjecture for S2; see [24].\nLemma 4.3. For allm2N0there isn0=n0(m)2Nsuch that for all n\u0015n0the setEm\nnis open and closed\ninEn.\nProof. Fixm2N0and denote by gTS2the Sasaki metric on TS2. For allT > 0 there exists \u000e=\u000e(T)>0\nsuch that3\ndistgTS2\u0000\n(q;\u0000v);\u001et\ng(q;v)\u0001\n\u0015\u000e;8t2[0;T];8(q;v)2SS2;\n3Such an inequality might not be true if one does not \fx T, as there might be e.g. geodesics whose tangential lift is dense.12 L. ASSELLE AND C. LANGE\nwhere\u001et\ngdenotes the geodesic \row. By Lemma 4.2, we can \fnd c=c(m)>0 such that, for every n2N,\nevery\r2Em\nnhas length (hence period) less than c. Since (g;\u0015nf)-geodesics C1\nloc-converge to geodesics for\n\u0015!0, we \fndn0=n0(m)2Nsuch that for all n\u0015n0\ndistgTS2\u0000\n\u001et\ng(q;v);\u001et\ng;\u0015nf(q;v)\u0001\n<\u000e\n2;8t2[0;c];8(q;v)2SS2;\nwhere\u001et\ng;\u0015nfdenotes the \row on SS2induced by the magnetic system ( g;\u0015nf) via (1.1). Therefore, for all\nt2[0;c], all (q;v)2SS2, and alln\u0015n0, we obtain\ndistgTS2\u0000\n(q;\u0000v);\u001et\ng;\u0015nf(q;v)\u0001\n\u0015distgTS2\u0000\n(q;\u0000v);\u001et\ng(q;v)\u0001\n\u0000distgTS2\u0000\n\u001et\ng(q;v);\u001et\ng;\u0015nf(q;v)\u0001\n\u0015\u000e\n2;\nthat is, (g;\u0015nf)-geodesics cannot have self-tangencies before time c. Hence, in particular elements in Em\nn\ncannot have self-tangencies for all n\u0015n0. This implies that Em\nnis open and closed for every n\u0015n0. Indeed,\nopenness follows immediately from the fact that any sequence of ( g;\u0015nf)-geodesics which is C1\nloc-converging\nto\r2Em\nnis actuallyC1-converging to \r(the periods vary smoothly for a Zoll magnetic system) combined\nwith the fact that having mtransversal self-intersecions is a C1-open condition. On the other hand, the limit\n\rof a sequencef\r`g\u001aEm\nncannot have more self-intersections that any of the \r`'s, and since ( g;\u0015nf) is Zoll\nwe also have that \rcannot have less self-intersections than any of the \r`'s (since\rcannot have self-tangencies,\nthe number of self-intersections can only jump if there is a jump in the periods). \u0004\nCorollary 4.4. The following holds:\na) Letm2N0be \fxed. If for some n\u0015n0, wheren0is given by Lemma 4.3, we have that Em\nn6=;, then all\n(g;\u0015nf)-geodesics have precisely m(transversal) self-intersections.\nb) IfEmknk6=;for some sequence nk!+1and some bounded sequence mkthengis Zoll.\nProof. The \frst assertion is clear. Suppose now that there is a non-closed geodesic t7!\u001et\ng(q;v). Then from\ntheC1\nloc-convergence of ( g;\u0015nf)-geodesics to geodesics for n!+1we deduce\n`g(t7!\u001et\ng;\u0015nf(q;v))!+1;\nthus contradicting the assumption. Therefore, every geodesic in ( S2;g) is closed, and hence gis Zoll. \u0004\nProof of Proposition 4.1. Suppose that ( g;f) is such that ( g;\u0015nf) is Zoll for some sequence \u0015n#0, and that\nghas a non-degenerate closed geodesic \r(in particular gis not Zoll). Then, following [23] we \fnd for all n\nlarge enough a closed ( g;\u0015nf)-geodesic in a neighborhood of \r, which will therefore have the same number\nof self-intersections as \r. By Corollary 4.4, this implies that gis Zoll, a contradiction. \u0004\nWe believe that Proposition 4.1 holds also non generically. Indeed, Lemmas 4.2 and 4.3 do not require any\nnon-degeneracy assumption, and in order to apply Corollary 4.4 we only need the existence of a closed geodesic\nwhich persists under magnetic perturbations. However, to our best knowledge nothing like Ginzburg's result\n[23] is known for degenerate closed geodesics. Therefore, in order to extend Proposition 4.1 one has either\nto use other methods or to \frst better understand how degenerate closed geodesics behave under magnetic\nperturbations, which is in any case a problem of independent interest.\nWe \fnish this section showing that Proposition 4.1 can be improved in the rotationally symmetric setting\nby dropping the genericity assumption. In the statement of the next lemma we denote with ( \u0012;')2(0;\u0019)\u0002R\nspherical coordinates on S2.\nLemma 4.5. Let(g;f)be a rotationally symmetric magnetic system on S2such that (g;\u0015nf)is Zoll for some\nsequence\u0015n!0. Thengis a Zoll metric of revolution.\nProof. It is well-known (cf. [12, Section 5]) that rotationally invariant magnetic systems admit a \frst integral\nIwhose critical points correspond to latitudes , that is, periodic orbits with constant \u0012-coordinate. Since I\nhas at least two critical points, this yields the existence of at least two embedded periodic orbits; see [12,\nProposition 5.9]. The claim follows now from Corollary 4.4. \u0004ON THE RIGIDITY OF ZOLL MAGNETIC SYSTEMS ON SURFACES 13\n5.Persistence of stable waists.\nWe recall that a waist for a Riemannian metric gonS2is a closed geodesic which is a local minimizer of\nthe length, or, equivalently, of the free period action functional Agiven by (2.1). We call a waist \r= (\u0000;T)\nstable if there exists a bounded neighborhood U\u001aH1(T;S2)\u0002(0;+1) of the critical circle S1\u0001\rsuch that\ninf\nUA=A(\r);and inf\n@UA>A(\r): (5.1)\nTheorem 5.1. Letgbe a metric on S2possessing a stable waist. Then for every magnetic function fthere\nexists \u0003 = \u0003(g;f)>0such that for every \u00152(0;\u0003)the magnetic pair (g;\u0015f )has a closed magnetic geodesic\n\r\u0015which is a local minimizer of the free-period action functional A\u0015given by (2.2) .\nCombining Theorem 5.1 with either Corollary 4.4 or the argument in the \frst paragraph of the proof of\nProposition 3.1 we can con\frm the validity of Conjecture (Z) in case of metrics on S2admitting a stable\nwaist.\nCorollary 5.2. Let(g;f)be a magnetic system on S2, wheregis a metric possessing a stable waist. Then\nthere exists \u0015>0such that (g;\u0015f )is not Zoll. \u0004\nNotice that a non-stable waist might disappear after an arbitrarily small perturbation of the metric. Take\nfor instance a smooth sphere of revolution in R3with pro\fle function ': [\u00001;1]!(0;1] such that '\u00111\non (\u0000\u000f;\u000f) for some \u000f >0. The closed geodesics given by parallels fz=tg,t2(\u0000\u000f;\u000f), are indeed waists,\nbut we can \fnd arbitrarily small perturbations \u001fof'such that'+\u001fhas a unique maximum at 0 and no\nother critical points. The corresponding surface of revolution has no waists, and the closed geodesic given by\nfz= 0gis of mountain pass type. Therefore, an analogue of Theorem 5.1 for such waists is hopeless.\nIn case the stable waist in Theorem 5.1 is strict (isolated in the language of Section 2) the proof is identical\nto the proof of Proposition 2.3 and will be omitted. Therefore, hereafter we can assume that all stable waists\nare non-isolated. We choose such a non-isolated stable waist \rand \fx a neighborhood UofS1\u0001\rsatisfying\n(5.1). The main di\u000eculty that we have to face here is given by the fact that, unlike in Proposition 2.3, we\ncannot expect the compact set\nKU:=f\u00172UjA(\u0017) =A(\r)g\nto be mapped into a proper compact subset of S2by the evaluation map, for local minimizers which are not\nglobal need not have disjoint image. Observe that the compactness of KUfollows from (5.1) combined with\nthe fact that Asatis\fes the Palais-Smale condition.\nIn fact, the key part of the proof of Theorem 5.1 is to show that we can \fnd a possibly smaller neighborhood\nV\u001aU ofS1\u0001\rsuch that (5.1) still holds and such that the closure of ev( V) is a proper subset of S2. Once\nthis is done, the proof becomes identical to the one in the isolated case.\nFigure 3. The images of two waists need not be disjoint.\nAs a \frst step we show that \ris embedded, and that all \u00172KUwhich are su\u000eciently close to \rmust\nhave pairwise disjoint image. For the next lemma we actually do not even need that \ris stable.\nLemma 5.3. Let\rbe a non-strict local minimizer of A, and letUbe a neighborhood of S1\u0001\ras in (5.1) . Then\nthere exists a neighborhood W\r\u001aU ofS1\u0001\rsuch that all \r16=\r22KU\\W\rhave disjoint image unless\nthey belong to the same critical circle. In particular, every non-strict local minimizer must be embedded.14 L. ASSELLE AND C. LANGE\nProof. We \frst show that if f\rng\u001aKUis a sequence such that \rn!\rinH1(and hence in C1) then\neventually\rnand\rmust have disjoint images unless they belong to the same critical circle. Indeed, suppose\nby contradiction that (up to taking a subsequence) \rnand\rintersect transversally for every n2N. Since\nthe image of \rnis contained in an annular region around the image of \r, we have that \rnand\rmust intersect\nat least twice (even though the intersection point might be unique, as one easily sees by taking two \fgure\neight curves in the plane intersecting at zero). Thus we can \fnd tn\n0<tn\n1andsn\n0<sn\n1such that\n\r(tn\n0) =\rn(sn\n0); \r(tn\n1) =\rn(sn\n1)\nand de\fne the curves\n\u0011n;1:=\rj[tn\n0;tn\n1]#\rnj[sn\n1;sn\n0]; \u0011n;2:=\rnj[sn\n0;sn\n1]#\rj[tn\n1;tn\n0]:\nNow the same line of arguments as in the end of the proof of Proposition 2.3 yields a contradiction.\nNow suppose that we can \fnd sequences f\rngandf\r0\nnginKUthat converge to \rand such that \rnand\n\r0\nnintersect for every n2N. By the Jordan curve theorem \rnand\r0\nnmust intersect at least twice. Arguing\nas above, we see that \rnand\r0\nnare eventually the same geometric curve. \u0004\nIn particular, the evaluation map maps W\rto a proper subset of \u0006. However, Lemma 5.3 is not quite\nenough to complete the proof of Theorem 5.1 since it is a priori not clear that the in\fmum of Aover@W\r\nis strictly larger than A(\r). Recall indeed that all elements in KUare non-isolated; for the same reason, we\nsee that it does not make much sense to replace KUwith the connected component of \rinKU. We will\novercome this di\u000eculty proving an analogue of Lemma 5.3 for a suitable subset of KUwhich can be seen as\na generalized connected component of \rinKU, and which we now de\fne.\nLemma 5.3 implies that for all \u00172KUwe \fnd a neighborhood W\u0017such that every \r1;\r22W\u0017\\KU\neither have disjoint image or they are the same geometric curve. Since fW\u0017j\u00172KUgis an open covering of\nKUandKUis compact, we can \fnd \u00171;:::;\u0017`2KUsuch that\nKU\u001a`[\ni=1W\u0017i:\nBy Lebesgue's number lemma we see now that there exists \u001a > 0 such that, for all \u00172KU, every two\n\r1;\r22B\u001a(S1\u0001\u0017)\\KUeither have disjoint image or are the same geometric curve.\nDe\fnition 5.4. Fix\u000e<\u001a . Two elements \r0;\r12KUare said\u000e-connected if there exist M2Nand a family\nf\u0011igi=0;:::;M\u001aKUsuch that\r0=\u00110,\r1=\u0011M, and\u0011i+12B\u000e(S1\u0001\u0011i)nS1\u0001\u0011i;8i= 0;:::;M\u00001:\nLemma 5.5. For\u000e<\u001a small enough, every \u00172K\u000e(\r),\nK\u000e(\r) :=f\u00172KUj\u0017;\rare\u000e-connectedg;\ndoes not intersect \r, unless it belongs to S1\u0001\r.\nProof. We \frst observe that the C1-dependence of geodesics on the initial conditions, combined with the\nfact thatKUis compact, yields that there exists C > 0 such that the following holds for all \r1;\r22KU:\ndistgTS2((\r1(0);_\r1(0));((\r2(0);_\r2(0))<\u000f)distH1(\r1;\r2)<C\u000f: (5.2)\nChoose now \u000e<\u001a and let\u00172K\u000e(\r). Our aim is to show that \u0017and\rhave disjoint image if they are not\nthe same geometric curve. Thus, suppose that \u0017(\u0001)\\\r(\u0001)6=;, and letM2Nandf\u0011igM\ni=1be a sequence as in\nDe\fnition 5.4. By construction, \u0011iand\u0011i+1have disjoint image for every i= 0;:::;M\u00001. We now consider\na subfamilyf\u0011j;:::;\u0011j+`goff\u0011igwhich is minimal among the subfamilies (of the same form) that satisfy the\nfollowing properties:\ni)\u0011jand\u0011j+`intersect.\nii)\u0011j+rand\u0011j+shave disjoint image for all r<s withr6= 1 ands6=`.\nLet\u0011j(0) =\u0011j+`(t) be an intersection point. Since \u0011j+`and\u0011j+1have disjoint image, we deduce that\ndistgTS2((\u0011j(0);_\u0011j(0));((\u0011j+`(t);_\u0011j+`(t))<\u000f\nfor some\u000f>0 which, by compactness of KU, only depends on \u000eand goes to zero as \u000e#0. Then, by (5.2)\ndistH1(\u0011j;\u0011j+`)<C\u000f;ON THE RIGIDITY OF ZOLL MAGNETIC SYSTEMS ON SURFACES 15\nand hence\u0011jand\u0011j+`are the same geometric curve, provided we chose \u000e<\u001a so small that C\u000f<\u001a . Therefore,\nthe subfamilyf\u0011j;:::;\u0011j+`gcan be removed from f\u0011ig. Applying this procedure recursively we obtain that \r\nand\u0017are the same geometric curve. \u0004\nLemma 5.6. For\u000e>0as in Lemma 5.5 we can \fnd a bounded neighborhood V\u001aU ofS1\u0001\rsuch that\ninf\n@VA>A(\r);andKV:=KU\\V=K\u000e(\r):\nProof. Observe preliminarly that, if \u00172KUis not\u000e-connected to \r, then every \u00162B\u000b\u0017(\u0017)\\KUis not\n\u000e-connected to \r, where\n\u000e\u0014\u000b\u0017:= inf\nM2Ninf\n\u00110;:::;\u0011Mdist(\u0011M;\u0017);\nandf\u0011igi=0;:::;M is any sequence with \r=\u00110and\u0011i+12B\u000e(S1\u0001\u0011i)nS1\u0001\u0011ifor alli:For\u00172KUwe consider\nB\u000b\u0017(\u0017) if\u0017is not\u000e-connected with \r, andB\u000e(\u0017) otherwise. By compactness of KUwe \fnd\u00161;:::;\u0016`2KU\nwhich are not \u000e-connected with \rand\u0016`+1;:::;\u0016`+r2KUwhich are\u000e-connected with \rsuch that\nKU\u001a`[\ni=1B\u000b\u0016i(\u0016i)[r[\ni=`+1B\u000e(\u0016`+i):\nFor the sake of simplicity we assume that all balls are entirely contained in U(otherwise we can work with\nthe intersection of the balls with U). Notice that any \u0016which lies in\nK0\nU:=KU\\ `[\ni=1B\u000b\u0016i(\u0016i)n`[\ni=1B\u000b\u0016i(\u0016i)!\nis\u000e-connected to \r, that is, elements in KUwhich are on the boundary of some B\u000b\u0016i(\u0016i) but not in the\ninterior of some other B\u000b\u0016j(\u0016j) are necessarily \u000e-connected with \r.\nIfK0\nU=;then we take asVthe connected component of Un[`\ni=1B\u000b\u0016i(\u0016i) that contains \r. Otherwise,\nfor every\u00162K0\nUwe consider the open ball B\u000e=2(\u0016). Again by compactness we can \fnd \fnitely many\n\u00151;:::;\u0015s2K0\nUsuch that\nK0\nU\u0012s[\ni=1B\u000e=2(\u0015i):\nWe now observe that by construction\n@\u0010s[\nj=1B\u000e=2(\u0015j)\u0011\n\\\u0010`[\ni=1B\u000b\u0016i(\u0016i)\u0011\ncannot contain elements of KU. Therefore, for\nA:= `[\ni=1B\u000b\u0016i(\u0016i)!\nns[\ni=1B\u000e=2(\u0015i)\nwe have inf @AA>A(\r). The assertion follows now taking the component VofUnA that contains \r.\u0004\nProof of Theorem 5.1. LetVbe a neighborhood of the non-isolated stable waist \ras in Lemma 5.6. Then,\nsince the image of any \u00172KV=K\u000e(\r)nS1\u0001\ris entirely contained in one of the two disks in which S2is\ndivided by \r, and since all geodesics in KVhave the same length, Lemma 5.5 implies that KVis mapped by\nthe evaluation map into a subset of S2with proper closure, and hence up to shrinking Vfurther we also have\nthat the closure of the image of Vunder the evaluation map is a proper subset of S2. \u0004\nRepeating the proof above word by word we obtain the following result on the persistence of contractible\nstable waists on arbitrary closed surfaces.\nTheorem 5.7. Letgbe a metric on a closed surface \u0006possessing a contractible stable waist. Then for every\nmagnetic function fthere exists \u0003 = \u0003(g;f)>0such that for every \u00152(0;\u0003)the magnetic pair (g;\u0015f )has\na contractible closed (g;\u0015f )-geodesic\r\u0015which is a local minimizer of the functional A\u0015in(2.2) . \u0004\nIt would be very interesting to see whether analogues of Theorem 5.7 and Proposition 2.3 hold for higher\ndimensional manifolds, in particular for those whose fundamental group is ameanable, as in this case the\nMa~ n\u0013 e critical value c(g;f) is in\fnite whenever the magnetic function does not integrate to zero [32].16 L. ASSELLE AND C. LANGE\nReferences\n[1] A. Abbondandolo. Lectures on the free period Lagrangian action functional. J. Fixed Point Theory Appl. , 13(2):397{430,\n2013.\n[2] A. Abbondandolo, B. Bramham, U. Hryniewicz, and P. Salomao. Sharp systolic inequalities for Riemannian and Finsler\nspheres of revolution. arXiv:1808.06995, 2018.\n[3] A. Abbondandolo, L. Macarini, M. Mazzucchelli, and G. P. Paternain. In\fnitely many periodic orbits of exact magnetic\n\rows on surfaces for almost every subcritical energy level. J. Eur. Math. Soc. (JEMS) , 19(2):551{579, 2017.\n[4] S.V. Agapov, M.L. Bialy, and A.E. Mironov. Integrable magnetic geodesic \rows on 2-torus: New examples via quasi-linear\nsystem of PDEs. Comm. Math. Phys. , 351:993{1007, 2017.\n[5] V. I. Arnold. Some remarks on \rows of line elements and frames. Dokl. Akad. Nauk SSSR , 138:255{257, 1961.\n[6] L. Asselle and G. Benedetti. In\fnitely many periodic orbits in non-exact oscillating magnetic \felds on surfaces with genus\nat least two for almost every low energy level. Calc. Var. Partial Di\u000ber. Equ. , 54(2):1525{1545, 2015.\n[7] L. Asselle and G. Benedetti. The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent\nbundles. J. Topol. Anal. , 8(3):545{570, 2016.\n[8] L. Asselle and G. Benedetti. On the periodic motions of a charged particle in an oscillating magnetic \rows on the two-torus.\nMath. Z. , 286(3-4):843{859, 2017. online \frst.\n[9] L. Asselle and G. Benedetti. Integrable zoll magnetic systems on the two-torus. https://arxiv.org/abs/1909.13821, 2019.\n[10] V. Bangert. On the lengths of closed geodesics on almost round spheres. Math. Z. , 191(4):549{558, 1986.\n[11] V. Bangert. Busemann functions and monotone twist maps. Calc. Var. Partial Di\u000ber. Equ. , 2:49{63, 1994.\n[12] G. Benedetti. The contact property for magnetic \rows on surfaces . PhD thesis, University of Cambridge, 2014.\n[13] G. Benedetti. Magnetic Katok examples on the two-sphere. Bull. London Math. Soc. , 48(5):855{865, 2016.\n[14] G. Benedetti and J. Kang. On a systolic inequality for closed magnetic geodesics on surfaces. arXiv:1902.01262, 2018.\n[15] A. L. Besse. Manifolds all of whose geodesics are closed . Ergebnisse der Mathematik und ihrer Grenzgebiete, 1978.\n[16] M.L. Bialy and A.E. Mironov. Rich quasi-linear system for integrable geodesic \row on 2-torus. Discrete Contin. Dyn. Syst. ,\n29:81{90, 2011.\n[17] M.L. Bialy and A.E. Mironov. Integrable geodesic \rows on 2-torus: formal solutions and variational principle. J. Geom.\nPhys. , 81(1):39{47, 2015.\n[18] G. D. Birkho\u000b. Dynamical systems . With an addendum by Jurgen Moser. American Mathematical Society Colloquium\nPublications, Vol. IX. American Mathematical Society, Providence, R.I., 1966.\n[19] G. Contreras. The Palais-Smale condition on contact type energy levels for convex Lagrangian systems. Calc. Var. Partial\nDi\u000ber. Equ. , 27(3):321{395, 2006.\n[20] G. Contreras, L. Macarini, and G. P. Paternain. Periodic orbits for exact magnetic \rows on surfaces. Int. Math. Res. Not. ,\n(8):361{387, 2004.\n[21] D.B.A. Epstein. Periodic \rows on three-manifolds. Ann. of Math. (2) , 95:66{82, 1972.\n[22] V. L. Ginzburg. New generalizations of Poincar\u0013 e's geometric theorem. Funktsional. Anal. i Prilozhen. , 21(2):16{22, 96,\n1987.\n[23] V. L. Ginzburg. On closed trajectories of a charge in a magnetic \feld. An application of symplectic geometry. , volume 8,\npages 131{148. Cambridge Univ. Press, 1996.\n[24] D. Gromoll and K. Grove. On metric on s2all of whose geodesics are closed. Invent. Math. , 65:175{177, 1981.\n[25] V. Guillemin. The Radon transform on Zoll surfaces. Advances in Math. , 22(1):85{119, 1976.\n[26] G. A. Hedlund. Geodesics on a two-dimensional riemannian manifold with periodic coe\u000ecients. Ann. of Math. (2) ,\n33(4):719{739, 1932.\n[27] N. Innami. Families of closed geodesics which distinguish \rat tori. Math. J. Okayama Univ. , 28:207{217, 1986.\n[28] M. Jankins and W. D. Neumann. Lectures on Seifert manifolds . Brandeis Lecture Notes 2, Brandeis University, 1983.\n[29] A. B. Katok. Ergodic perturbations of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk SSSR Ser. Mat. ,\n37:539{576, 1973.\n[30] E. Kerman. Periodic orbits of Hamiltonian \rows near symplectic critical submanifolds. Int. Math. Res. Not. , (17), 1999.\n[31] V. V. Kozlov and D.V. Treschev. On the integrability of Hamiltonian systems with toral position space. Math. USSR\nSbornik , 63(1):121{139, 1989.\n[32] W. Merry. Closed orbits of a charge in a weakly exact magnetic \feld. Paci\fc J. Math. , 47(1):189{212, 2010.\n[33] G. P. Paternain. Helicity and the ma~ n\u0013 e critical value. Algebr. Geom. Topol. , 9:1413{1422, 2009.\n[34] N. Raymond and S. Vu Ngoc. Geometry and Spectrum in 2D mangetic wells. Ann. Inst. Fourier (Grenoble) , 65(1):137{169,\n2015.\n[35] I. A. Taimanov. Closed non-self-intersecting extremals of multivalued functionals. Sibirsk. Mat. Zh. , 33(4):155{162, 223,\n1992.\nJustus Liebig Universit at Giessen, Mathematisches Institut, Arndtstrasse 2,\nRaum 102, D-35392 Giessen, Germany\nE-mail address :luca.asselle@ruhr-uni-bochum.de\nMathematisches Institut der Universit at K oln, Weyertal 86-90, Raum -103, 50931, K oln, Germany\nE-mail address :clange@math.uni-koeln.de"    },    {        "title": "1706.04089v1.Dynamic_phase_transition_features_of_the_cylindrical_nanowire_driven_by_a_propagating_magnetic_field.pdf",        "content": "arXiv:1706.04089v1  [cond-mat.stat-mech]  13 Jun 2017Dynamic phase transition features of the cylindrical nanow ire driven by a\npropagating magnetic field\nErol Vatansever∗\nDepartment of Physics, Dokuz Eyl¨ ul University, Tr-35160 ˙Izmir, Turkey\nAbstract\nMagnetic response of the spin-1 /2 cylindrical nanowire to the propagating magnetic field wav e has been investigated\nby means of Monte Carlo simulation method based on Metropoli s algorithm. The obtained microscopic spin config-\nurations suggest that the studied system exhibits two types of dynamical phases depending on the considered values\nof system parameters: Coherent propagation of spin bands an d spin-frozen or pinned phases, as in the case of the\nconventional bulk systems under the influence of a propagati ng magnetic field. By benefiting from the temperature\ndependencies of variances of dynamic order parameter, inte rnal energy and the derivative of dynamic order parameter\nof the system, dynamic phase diagrams are also obtained in re lated planes for varying values of the wavelength of the\npropagating magnetic field. Our simulation results demonst rate that as the strength of the field amplitude is increased,\nthe phase transition points tend to shift to the relatively l ower temperature regions. Moreover, it has been observed\nthat dynamic phase boundary line shrinks inward when the val ue of wavelength of the external field decreases.\nKeywords: Cylindrical nanowire, Propagating magnetic field, Monte Ca rlo simulation.\n1. Introduction\nInteracting spin systems driven by a sinusoidally oscillat ing magnetic field can exhibit distinctive and fascinating\ndynamic behaviors, which can not occur for the correspondin g equilibrium spin systems. For the first time in Ref.\n[1], the authors applied their theoretical model to investi gate the physics underlying a simple ferromagnet being\nexposed to a time dependent magnetic field. From their analys is, the authors concluded that amplitude and period of\nthe oscillatory magnetic field play an important role on the d ynamic characters of the considered magnetic system.\nSince then, many theoretical [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and several experimental works [16, 17,\n18, 19, 20] have been carried to understand the origin of the d ynamic phase transitions. Based on the some of the\npreviously published studies, it is possible to say that the re is a good consensus between dynamic phase transitions\nand equilibrium phase transitions. For example, it has been reported that the critical exponents of the two dimensional\nkinetic Ising model subjected to a square-wave oscillatory magnetic field are consistent with the universality class\nof the corresponding equilibrium Ising model [5]. Moreover , it is recommended in these references [12, 15, 17, 19]\nthat bias field appears to be a conjugate field of the dynamic or der parameter, which is time averaged magnetization\nover a full cycle of the external field. In addition to the cons ensus in dynamic phase transitions and equilibrium phase\ntransitions, however, there are inconsistencies in the lit erature, in view of the universality class of the spin system s.\nFor instance, recent detailed Monte Carlo (MC) simulations studies show that there is a clear di fference between\ncritical dynamics of the magnetic system with surfaces and i ts equilibrium case [9]. From their finite-size scaling\nanalysis, the authors found that nonequilibrium surface ex ponents do not coincide with those of the equilibrium critic al\nsurface. Very recently, it is reported both experimentally and theoretically that there are metamagnetic fluctuations\nin the neighbourhood of dynamic phase transitions, which do not emerge in the thermodynamic behavior of typical\n∗Corresponding author. Tel.: +90 3019547; fax:+90 2324534188.\nEmail address: erol.vatansever@deu.edu.tr (Erol Vatansever)\nPreprint submitted to Turkish Journal of Physics November 7, 2018ferromagnets [20]. Keeping these facts mentioned above in m ind, it is possible to say that more work is required to\nunderstand the origin of dynamic phase transitions of magne tic systems driven by oscillatory magnetic field.\nIn addition to the works regarding the influences of an oscill ating magnetic field (uniform over space) on the dy-\nnamic characteristics of the magnetic systems, some e fforts were taken to investigate the dynamic features resulti ng\nfrom standing and/or propagating magnetic field waves [21, 22, 23, 24]. In Ref. [ 21], the authors applied their theo-\nretical model to elucidate the dynamical modes and nonequil ibrium phase transition of the spin-1 Blume-Capel model\nfor two dimensional square lattice, within the framework of MC simulation. It has been found that, the wavelength of\nthe external field plays an important role in the dynamic natu re of the considered system, in addition to the amplitude\nand period of the field. The system presents two dynamical pha ses: Propagating spin wave and spin frozen or pinned\nphases. In another interesting work, the physics behind a si mple ferromagnet under the influence of a propagating\nmagnetic field has been discussed in detail [22]. It is shown t hat dynamic phase boundary constructed in temperature\nversus applied field amplitude tends to shrink as the strengt h of the wavelength of the external field decreases. It is\nclear that particular attention in the works mentioned abov e has only been devoted to clarify the physics in the bulk\nmaterials. To the best of our knowledge, there is no any attem pt to address the same problem for nanoparticles with\nsurfaces, especially for magnetic nanowires, driven by a pr opagating or(and) standing magnetic field wave(s). Mag-\nnetic nanowires are important materials and also potential candidates for applications in advanced nanotechnology\nincluding magnetic memory devices [25, 26] and biomedical a pplications [27] due to their own distinctive magnetic\nproperties. For the sake of completeness, we would like to em phasize that nonequilibrium dynamics as well as phase\ntransition characteristics of various types of magnetic na nowire systems under the existence of an oscillating field\n(uniform over space) have been studied by employing e ffective-field theory [28, 29, 30, 31] and also MC simula-\ntion method [32, 33]. From this point of view, we intend to inv estigate the dynamic behaviors of cylindrical nanowire\nsystem being subjected to a propagating magnetic field, by ut ilizing MC simulation method. Our obtained results indi-\ncate that the present system exhibits unusual and interesti ng magnetic dynamics originating from the occurrence of the\npropagating magnetic field, which have not been observed and discussed so far, in view of the magnetic nanoparticles.\nThe outline of the remainder parts of the paper is as follows: In section 2, we present the model and simulation\ndetails. The results and discussion are given in section 3, a nd finally section 4 includes our conclusions.\n2. Model and Simulation Details\nThe Hamiltoanian of the spin-1 /2 cylindrical nanowire driven by a propagating magnetic fiel d can be written as\nfollows:\nˆH=−J/summationdisplay\n/angbracketlefti j/angbracketrightSz\ni(x,y,z,t)Sz\nj(x′,y′,z′,t)−/summationdisplay\nih(x,y,z,t)Si(x,y,z,t) (1)\nhere Sz\ni(x,y,z,t)=±1 is the Ising spin variable at any position x,yandzin the nanowire. J(>0) is the ferromagnetic\nspin-spin coupling term. The first summation in Eq. (1) is ove r the nearest-neighbour spin couplings while the second\none is over all lattice sites in the system. h(x,y,z,t) represents the propagating magnetic field, which has the fo llowing:\nh(x,y,z,t)=h0cos(ωt−kz). (2)\nFor the sake of simplicity, the magnetic field is propagating in the zdirection (long axis of the nanowire). Here,\nh0,ωandkrefer to the amplitude, angular frequency and wavevector of the external field, and tis time. Period and\nwavelength of the magnetic field are τ=2π/ω andλ=2π/k, respectively. It may be noted here that we have fixed\nthe period of the external field as τ=100 throughout the study.\nBy utilizing Metropolis algorithm [34, 35], we employ MC sim ulation to understand the magnetic nature of the\ncylindrical nanowire system driven by a propagating magnet ic field wave. The nanowire system is located on a simple\ncubic lattice, which has a length of Lz=200 and a radius r=10. For these selected values of the Lzandr, total\nnumber of the spins in the system is Nt=63400. In order to mimic the cylindrical nanowire, we follow the boundary\nconditions such that it is free in xyplane and periodic along the zdirection of the system. The simulation procedure we\nfollow in this study can be briefly summarized as follows. The simulation starts from a high temperature kBT/J=5.0\n(here, kBandTare Boltzmann constant and absolute temperature, respecti vely) using random initial configurations,\nwhich corresponds to paramagnetic phase. Next, it is slowly cooled down until the temperature reaches to the value of\nkBT/J=10−2with a relatively small temperature step kB∆T/J=25×10−2. In each temperature, 5 ×104Monte Carlo\n2steps per site (MCSS) are discarded for thermalization proc ess, and next 5×104MCSS are collected to determine\nthe thermal variations of physical quantities used in this s tudy. Numerical data were collected over 20 independent\nsamples.\nWe have calculated the instantaneous value of the magnetiza tion at time tas follows:\nM(t)=1\nNtNt/summationdisplay\ni=1Si(x,y,z,t). (3)\nUsing M(t), dynamic order parameter of the cylindrical nanowire syst em can be defined as follows:\nQ=1\nτ/contintegraldisplay\nM(t)dt. (4)\nIn order to detect the dynamic phase transition points of the present system, in addition to the derivative of dynamic\norder parameter dQ/dT, we check the variance of the Qand internal energy ( E), which are defined as follows:\nχQ=Nt/parenleftBig\n/angbracketleftQ2/angbracketright−/angbracketleftQ/angbracketright2/parenrightBig\n, (5)\nand\nχE=Nt/parenleftBig\n/angbracketleftE2/angbracketright−/angbracketleftE/angbracketright2/parenrightBig\n, (6)\nhere Eis the internal energy given in Eq. (1) over a full cycle of the propagating magnetic field, which is defined in\nthe following form:\nE=−1\nτNt/contintegraldisplay\nˆHdt. (7)\n3. Results and Discussions\n/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s97/s41/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s98/s41/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s99/s41\nFigure 1: (Color online) Coherent propagation of spin clust ers of up (green dots) and down (red dots) spins, swept by a pro pagating magnetic\nfield wave. The spin snapshots are taken for two di fferent values of the Monte Carlo simulation time (a) t=50100 MCSS, (b) t=50200 MCSS\nfor values of the reduced temperature kBT/J=4.0, amplitude h0/J=0.3, wavelengthλ=100 and periodτ=100 of the external magnetic\nfield. (c) corresponds to spin snapshot at time t=50100 MCSS, for the same system parameters with (a) and (b), e xcept from considered value of\ntemperature kBT/J=1.5.\nThe microscopic spin configurations of the present cylindri cal nanowire system show that the system demonstrate\ntwo different dynamical phases, which sensitively depend on the con sidered values of the system parameters. Two al-\nternate bands of spin values S=±1 are found, and they tend to propagate along the long axis of t he nanowire (namely\nalong zaxis of nanowire), in the high temperature regions. In Fig. 1 , we can see easily the coherent propagation\nof the spin bands. These spin snapshots are taken for two di fferent values of the MC simulation time (a) t=50100\nMCSS, (b) t=50200 MCSS for values of the reduced temperature kBT/J=4.0, amplitude h0/J=0.3 and wave-\nlengthλ=100 of the external magnetic field. The cylindrical nanowire includes two full waves for the considered\nvalue ofλ. Our MC simulation results also suggest that a reduction in t he value of temperature destroys the coherent\npropagating of spin bands of the cylindrical nanowire syste m, leading to a spin-frozen or pinned phase [21], as shown\n3in Fig. 1(c) which is displayed for value of kBT/J=1.5, with all other parameters of propagating magnetic field re -\nmain the same. At the lower temperature regions, ferromagne tic spin-spin coupling dominates against the propagating\nmagnetic field, hence, the most of the spins in the nanowire ar e frozen or pinned to any one value of S. It is clear that\nthese microscopic behaviors sensitively depend on the stud ied system parameters. We should note that these types of\ncoherent propagations of spin bands and also spin-frozen ph ases have been recently observed in bulk materials under\nthe existence of a propagating magnetic field [21, 22, 23].\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s49\n/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s51\n/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s53\n/s32/s32/s81\n/s107\n/s66/s84/s47/s74/s32/s61/s32/s49/s48/s48\n/s32/s61/s32/s49/s48/s48/s40/s97/s41\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s45/s49/s53/s45/s49/s50/s45/s57/s45/s54/s45/s51/s48\n/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s49\n/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s51\n/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s53\n/s32/s32/s100/s81/s47/s100/s84\n/s107\n/s66/s84/s47/s74/s32/s61/s32/s49/s48/s48\n/s32/s61/s32/s49/s48/s48 /s40/s98/s41\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/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/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s49\n/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s51\n/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s53\n/s32/s32/s81/s47/s78\n/s116\n/s107\n/s66/s84/s47/s74/s32/s61/s32/s49/s48/s48\n/s32/s61/s32/s49/s48/s48/s40/s99/s41\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s48/s46/s48/s48/s48/s48/s46/s48/s48/s49/s48/s46/s48/s48/s50/s48/s46/s48/s48/s51\n/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s49\n/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s51\n/s32/s32/s32/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s53\n/s32/s32/s69/s47/s78\n/s116\n/s107\n/s66/s84/s47/s74/s32/s61/s32/s49/s48/s48\n/s32/s61/s32/s49/s48/s48/s40/s100/s41\nFigure 2: (Color online) Thermal variations of the (a) dynam ic order parameter ( Q), (b) dQ/dT, (c) variance of Q(χQ/Nt) and (d) of E(χE/Nt) for\ndifferent values of reduced amplitude h0/Jof the external field. The curves are depicted for values of τ=100 andλ=100.\nIn Figs. 2(a-d), we focus our attention on the e ffects of the varying field amplitudes on the thermal variation s of\nthe dynamic order parameter ( Q), its derivative dQ/dT, variance of Q(χQ) and of E(χE). The curves are depicted\nfor three values of the applied field amplitudes, i.e., h0/J=0.1,0.3 and 0.5 withτ=100 andλ=100. As shown\nin Fig. 2(a), Qgradually decreases starting from its saturation value wit h increasing thermal energy, and it vanishes\ncontinuously at the critical temperature. Dynamic phase tr ansition point strongly depends on the component of the\npropagating magnetic field. Our MC findings underline that an increment in value of the applied field amplitude\nleads to a decrement in the location of phase transition poin t. At the higher temperature regions, Qis zero and\nthis corresponds to a phase where the coherent propagation o f spin bands exhibit. In the lower temperature regions,\nthe system exhibits dynamically ferromagnetic phase, wher eQis nonzero. As discussed earlier, the microscopic\nspin snapshots show that the system presents low temperatur e spin-frozen phase in these regions. Based on these\nspin configurations of the cylindrical nanowire system, it i s possible to say that the system undergoes a dynamical\nphase transition from a coherent spin propagation phase to a spin-frozen phase, when the temperature is decreased\nstarting from a relatively higher value. For a selected comb ination of the Hamiltonian parameters, the temperature\ndependencies of dQ/dTshow a sharp dip while χQandχEreveal a very sharp peak indicating the existence of a\nsecond order phase transition, as depicted in Figs. 2(b-d). These peaks are found to shift to the lower temperature\nvalues when the strength of h0/Jis increased.\nIn order to have a better understanding about the influences o f the varying values of wavelength of the propagating\nfield on the system, we give the thermal variations of Q,dQ/dT,χQand alsoχEin Figs. 3(a-d). The curves are given\n4/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s32/s32 /s32/s61/s32/s49/s48/s48\n/s32/s32/s32 /s32/s61/s32/s53/s48\n/s32/s32/s81\n/s107\n/s66/s84/s47/s74/s40/s97/s41\n/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s51\n/s32/s61/s32/s49/s48/s48\n/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53/s45/s49/s53/s45/s49/s50/s45/s57/s45/s54/s45/s51/s48\n/s32/s32/s32 /s32/s61/s32/s49/s48/s48\n/s32/s32/s32 /s32/s61/s32/s53/s48\n/s32/s32/s100/s81/s47/s100/s84\n/s107\n/s66/s84/s47/s74/s40/s98/s41/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s51\n/s32/s61/s32/s49/s48/s48\n/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53/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/s81/s47/s78\n/s116/s32/s32/s32 /s32/s61/s32/s49/s48/s48\n/s32/s32/s32 /s32/s61/s32/s53/s48\n/s32/s32\n/s107\n/s66/s84/s47/s74/s40/s99/s41\n/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s51\n/s32/s61/s32/s49/s48/s48\n/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53/s48/s46/s48/s48/s48/s48/s46/s48/s48/s49/s48/s46/s48/s48/s50/s48/s46/s48/s48/s51\n/s69/s47/s78\n/s116/s32/s32/s32 /s32/s61/s32/s49/s48/s48\n/s32/s32/s32 /s32/s61/s32/s53/s48\n/s32/s32\n/s107\n/s66/s84/s47/s74/s40/s100/s41\n/s104\n/s48/s47/s74/s32/s61/s32/s48/s46/s51\n/s32/s61/s32/s49/s48/s48\nFigure 3: (Color online) Temperature dependencies of the (a )Q, (b) dQ/dT, (c)χQ/Ntand (d)χE/Ntat various values of wavelength λof the\nexternal field. The curves are given for values of τ=100 and h0/J=0.3.\nfor two values of the wavelengths of the field, i.e., λ=50 and 100 with h0/J=0.3 andτ=100. It is found that\ndynamic phase transition points are observed to depend on th e selected value of λof the propagating magnetic field\nwave. As seen in these figures, thermal variations of Q,dQ/dT,χQandχEsupport that when the value of wavelength\nof the external field is increased, dynamically ferromagnet ic phase region gets wider. It may be noted here that such\ntypes of observations originating from the variation of the wavelength of the propagating magnetic field have been\nfound in Ref. [22], where kinetic Ising model on a 2 Dsquare lattice is exposed to a propagating magnetic field.\nIn order to elucidate the influences of the applied field ampli tude on the dynamic phase transition features of the\ncylindrical nanowire system, we obtain the phase diagrams i n (kBTc/J−h0/J) plane with two values of the wavelength\nof the propagating magnetic field with τ=100, in figure (4). We note that dynamic phase transition poin ts are\ndeduced from the peaks of the thermal variations of dQ/dT,χQandχEcurves. It is clear from the phase diagrams\nthat when the applied field amplitude is increased, the dynam ic phase transition points are shifted to the relatively\nlower temperature regions. The aforementioned behaviors s eem to be independent of the applied field wavelength\nλ. On the other hand, dynamic phase boundary line, which separ ates dynamically ordered phases from disordered\nphases, shrinks inward when the strength of the λof the external field decreases. Recently, magnetic respons e of the\nspin-1/2 Ising cylindrical nanowire system to an oscillating magne tic field (uniform over space) has been investigated\nby means of MC simulation with Metropolis algorithm [32] for the same system parameters with the present study.\nIn order to make a comparison between propagating and oscill ating magnetic fields, the results of the reference [32]\nare added to the Fig. 4 (green squares). It is obvious from the figure that there is no clear distinction between the\nmagnetic field sources for the small values of the applied fiel d amplitudes, in the sense of dynamic phase transition\npoint. However, as the strength of h0/Jis increased, dynamic phase boundaries begin to seperate fr om each other.\nFor considered values of the wavelengths of the propagating magnetic field in this study, our MC simulation findings\nindicate that phase transition points are always lower than those of the critical points obtained in reference [32],\nespecially at the higher applied field amplitude regions.\n5/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48\n/s32/s32/s82/s101/s102/s46/s32/s91/s51/s50/s93\n/s32/s32/s32 /s32/s61/s32/s49/s48/s48\n/s32/s32/s32 /s32/s61/s32/s53/s48\n/s32/s32/s32/s32/s100/s121 /s110/s97/s109/s105/s99/s97/s108/s108/s121 /s32\n/s32/s100/s105/s115/s111/s114/s100/s101/s114/s101/s100/s32/s112/s104/s97/s115/s101\n/s32/s32/s104\n/s48/s47/s74\n/s107\n/s66/s84\n/s99/s47/s74/s32/s32/s100/s121 /s110/s97/s109/s105/s99/s97/s108/s108/s121 /s32\n/s32/s111/s114/s100/s101/s114/s101/s100/s32/s112/s104/s97/s115/s101/s32/s61/s32/s49/s48/s48\nFigure 4: (Color online) Dynamic phase diagram in the (kBTc/J−h0/J)plane for the cylindrical nanowire system under the presenc e of a prop-\nagating magnetic field. Di fferent symbols correspond to the di fferent values of applied field wavelengths: λ=100 (•, red bullet) andλ=50 (/trianglesolid,\nblue triangle). Here, /squaresolid(green square) symbols correspond to the phase transition p oints of the spin-1/2 cylindrical nanowire system under a time\ndependent oscillating magnetic field, which is uniform over space [32], for the same system parameters used in this study . Dynamic phase diagrams\nare given for value of τ=100.\n4. Conclusions\nTo conclude, we study the nonequilibrium dynamics and phase transition features of the ferromagnetic spin-1 /2\ncylindrical nanowire under the existence of a propagating m agnetic field. For this investigation, we use Monte Carlo\nsimulation method with single-site update Metropolis algo rithm. Based on the obtained results in this study, it is\npossible to mention that there are two types of dynamical sta tes in the system, depending on the considered system\nparameters. The results can be summarized as follows: In the higher temperature regions, two alternate bands of\nspins are found, and they tend to coherently propagate along the long axis of the nanowire. However, a decrement\nin the value of temperature leads to the occurrence of spin-f rozen or spin pinned phase. This phase generally occurs\nat the relatively lower temperature regions, as in the case o f the conventional bulk systems under the influence of a\npropagating magnetic field [21, 22, 23].\nAdditionally, we give the dynamic phase diagram of the spin- 1/2 cylindrical nanowire in ( kBTc/J−h0/J) plane for two\nvalues of the wavelengths of the propagating magnetic field. Our MC simulation findings clearly indicate that as the\nstrength of the field amplitude is increased, the phase trans ition points tend to shift to the relatively lower temperatu re\nregions. In the sense of wavelength of the field, it is found th at dynamic phase boundary line shrinks inward in the\nrelated plane when the value of λof the external field decreases. We also observed that the dyn amic behavior of the\npresent system with a propagating magnetic field exhibits qu ite different characteristics in comparison with the same\nsystem, but only in the presence of a sinusoidally oscillati ng magnetic field (uniform over space) [32].\nVery recently, it has been shown that frequency dispersion o f the nanocubic core /shell particle driven by an oscil-\nlating field (uniform over space) can be categorized into thr ee groups, as in the case of the conventional bulk systems\n[11]. Keeping these in mind, it would be interesting to inves tigate the influences of the propagating magnetic field on\nthe frequency dispersion of dynamic loop area of the present system. From the theoretical perspective, it could also\nbe study the standing magnetic field e ffects on the present system and (or) on the quenched disordere d binary alloy\ncylindrical nanowire [36]. We believe that such types of stu dies will be beneficial to provide deeper understanding of\nphysics underlying of nanoscale materials driven by a time d ependent magnetic field.\n6Acknowledgements\nThe author is thankful to Muktish Acharyya from Presidency U niversity for valuable discussions and suggestions.\nThe numerical calculations reported in this paper were perf ormed at T ¨UB˙ITAK ULAKB ˙IM (Turkish agency), High\nPerformance and Grid Computing Center (TRUBA Resources).\nReferences\n[1] T. Tom´ e, M.J. de Oliveira, Phys. Rev. A 41 (1990) 4251.\n[2] W.S. Lo, R.A. Pelcovits, Phys. Rev. A 42 (1990) 7471.\n[3] S.W. Sides, P.A. Rikvold, M.A. Novotny, Phys. Rev. Lett. 81 (1998) 834.\n[4] G.M. Buend´ ıa, E. Machado, Phys. Rev. B 61 (2000) 14686.\n[5] G.M. Buend´ ıa, P.A. Rikvold, Phys. Rev. E 78 (2008) 05110 8.\n[6] B.K. Chakrabarti, M. Acharyya, Rev. Mod. Phys. 71 (1999) 847.\n[7] M. Keskin, O. Canko, U. Temizer, Phys. Rev. E 72 (2005) 036 125.\n[8] X. Shi, G. Wei, L. Li, Phys. Lett. A 372 (2008) 5922.\n[9] H. Park, M. Pleimling, Phys. Rev. Lett. 109 (2012) 175703 .\n[10] Y . Y¨ uksel, E. Vatansever, H. Polat, J. Phys.: Condens. Matter 24 (2012) 436004.\n[11] E. Vatansever, in Press. (Phys. Lett. A (2017) http: //dx.doi.org/10.1016/j.physleta.2017.03.012).\n[12] E. Vatansever, H. Polat, Thin Solid Films 589 (2015) 778 .\n[13] M. Acharyya, Phys. Rev. E 56 (1997) 1234.\n[14] M. Acharyya, Phys. Rev. E 56 (1997) 2407.\n[15] R.A. Gallardo, O. Idigoras, P. Landeros, A. Berger, Phy sica B 407 (2012) 1377.\n[16] Y .-L. He, G.-C. Wang, Phys. Rev. Lett. 70 (1993) 2336.\n[17] D.T. Robb, Y .H. Xu, O. Hellwig, J. McCord, A. Berger, M.A . Novotny, P.A. Rikvold, Phys. Rev. B 78 (2008) 134422.\n[18] J.-S. Suen, J.L. Erskine, Phys. Rev. Lett. 78 (1997) 356 7.\n[19] A. Berger, O. Idigoras, P. Vavassori, Phys. Rev. Lett. 1 11 (2013) 190602.\n[20] P. Riego, P. Vavassori, A. Berger, Phys. Rev. Lett. 118 ( 2017) 117202.\n[21] M. Acharyya and A. Halder, J. Magn. Magn. Mater. 426 (201 7) 53.\n[22] M. Acharyya, Acta Phys. Pol. B, 45 (2014) 1027.\n[23] M. Acharyya, J. Magn. Magn. Mater. 354 (2014) 349.\n[24] M. Acharyya, J. Magn. Magn. Mater. 382 (2015) 206.\n[25] M.I. Irshad, F. Ahmad, N.M. Mohamed, AIP Conf. Proc. 148 2 (2012) 625.\n[26] Y .P. Ivanov, A. Chuvilin, S. Lopatin, J. Kosel, ACS Nano 10 (2016) 5326.\n[27] Y .P. Ivanov, A. Alfadhel, M. Alnassar, J.E. Perez, M. Va zquez, A. Chuvilin, J. Kosel, Sci. Rep. 6 (2016) 24189.\n[28] B. Deviren, E. Kantar, M. Keskin, J. Magn. Magn. Mater. 3 24 (2012) 2163.\n[29] B. Deviren, M. Ertas ¸, M. Keskin, Phys. Scr. 85 (2012) 05 5001.\n[30] E. Kantar, B. Deviren, M. Keskin, Eur. Phys. J. B. 86 (201 3) 253.\n[31] M. Ertas ¸, Y . Kocakaplan, Phys. Lett. A 378 (2014) 845.\n[32] Y . Y¨ uksel, Phys. Rev. E, 91 (2015) 032149.\n[33] Y . Y¨ uksel, J. Magn. Magn. Mater. 389 (2015) 34.\n[34] K. Binder, Monte Carlo Methods in Statistical Physics ( Springer, Berlin, 1979).\n[35] M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Sta tistical Physics (Oxford University Press, New York, 1999) .\n[36] Z.D. Vatansever, E. Vatansever, J. Alloys Compd. 701 (2 017) 288.\n7"    },    {        "title": "0810.0090v1.Lattice_disorder_and_Ferromagnetism_in_La0_67Ca0_33MnO3_nanoparticle.pdf",        "content": "arXiv:0810.0090v1  [cond-mat.mtrl-sci]  1 Oct 2008Lattice disorder and Ferromagnetism in La 0.67Ca0.33MnO3\nnanoparticle\n1R.N. Bhowmik∗,2Asok Poddar,2R. Ranganathan, and2Chandan Majumdar\n1Department of Physics, Pondicherry University,\nR.V. Nagar, Kalapet, Pondicherry-605014, India\n2Experimental Condensed Matter Physics Division,\nSaha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkat a-700064, India\nWe study the ferromagnetism of La 0.67Ca0.33MnO3in bulk polycrystalline,\nnanocrystalline and amorphous phase. The structural chang e from crystalline phase\nto amorphous phase exhibited a systematic decrease of T C(paramagnetic to ferro-\nmagnetic transition temperature) and spontaneous magneti zation (M S). The ex-\nperimental results suggested few more features, e.g., appe arance of large magnetic\nirreversibility in the temperature dependence of magnetiz ation, lack of magnetic sat-\nuration at high magnetic field, blocking of magnetization be low TB, and enhance-\nmentofcoercivity. Inaddition, themagneticphasetransit ionneartoT Chaschanged\nfrom firstorder character in bulk sample to second order char acter in nanocrystalline\nand amorphous samples. We understand the observed magnetic features as the ef-\nfects of decreasing particle size and increasing magnetic ( spin-lattice) disorder. We\nnoted that magnetic dynamics of amorphous samples is distin ctly different from the\nnanocrystalline samples. The ferromagnetism of amorphous samples are comparable\nwith the properties of reported amorphous ferromagnetic na noparticles. We also\ndemonstrate the effect of disorder shell in controlling the dy namics of ferromagnetic\ncores.\n∗E-mail address for correspondence:\nrnbhowmik.phy@pondiuni.edu.in (RNB)\nr.ranganathan@saha.ac.in (RR)2\nI. INTRODUCTION\nLaMnO 3and CaMnO 3are two typical antiferromagnets of perovskite structure with\nTN∼140 K and 130 K, respectively [1]. The mixed compounds La 1−xCaxMnO3\n[(La3+\n1−xCa2+\nx)(Mn3+\n1−xMn4+\nx)O2−\n3] have shown a rich magnetic phase diagram both in hole\ndoped (x <0.5) and electron doped (x >0.5) regions [1, 2, 3]. The hole doped and electron\ndoped regions are dominated by Mn3+(3d4) and Mn4+(3d3) ions, respectively. The other im-\nportant aspect of these manganites is to understand the colossa l magnetoresistance (CMR)\nbelow room temperature [4]. Although CMR properties have been disc ussed in terms of\ndouble exchange (ferromagnetic) interactions between Mn3+-O-Mn4+ions, but double ex-\nchange (DE) interactions alone can not explain the complete mechan ism of CMR effect.\nRecently, many magnetic (spin-lattice) disorder effects, e.g., Jahn -Teller distortion (due to\nthe presence of Mn3+), and electron-phonon coupling (due to the lattice strain and defo r-\nmation of Mn3+-O-Mn4+bonds), were considered for CMR manganites [5, 6]. Most of the\nliterature works have attempted tostudy theroles ofmagnetic dis order by cation(La or Mn)\nsubstitution [6, 7, 8]. The dilution of magnetic cations (Mn) has shown the weakening of fer-\nromagnetic DE interactions. Finally, a competition between ferroma gnetic DE interactions\n(Mn3+-O-Mn4+) and antiferromagnetic (Mn3+-O-Mn3+and Mn4+-O-Mn4+) superexchange\ninteractions may result in the freezing of magnetic domains or phase separation phenomena\nin manganites [8, 9].\nThe present compound La 0.67Ca0.33MnO3(LCMN) is a ferromagnet with T Creported over a\nrange 250 K to 285 K [8, 9, 10, 11]. The understanding of magnetic ph ase transition near T C\nofLCMN, whether firstorderorsecondorder, isanissueofrecen t interest [12, 13, 14, 15, 16].\nThe study of ferromagnetic ordering of LCMN about T Ccould be relevant to get insight\nof the CMR behaviour in many magnetic oxides. La 0.67Ca0.33MnO3is a Mn3+(3d4) rich\ncompound, where crystal field splits the five-fold degenerate 3d4levels into t3\n2gand e1\nglevels.\nThe effective ferromagnetic coupling (J H) between e gelectrons (spin= 1/2) and t 2gelectrons\n(spin= 3/2) of on-site (Mn) is much more stronger than the hopping interaction (t0\nij) of the\negelectrons between holes of neighbouring Mn4+(3d3) at sites, i and j [17]. The Anderson-\nHasegawa relation [18] (t ij= t0\nijcos(θij/2)) suggests that the effective hopping interaction\n(tij) depends on the relative angle θijbetween the spins of neighbouring sites.\nIn our opinion, the effect of θijon the effective exchange interactions can be best studied3\nby reducing the particle size of manganites into nanometer scale, wh ere shell (surface) spins\nof the nanoparticle are expected to be more disordered in comparis on with the core spins.\nRecently, spin polarized tunneling between two grains (cores) via gr ain boundaries (shells)\n[19, 20] has been proposed to explain the large magnitude of low field m agnetoresistance\n(MR) in manganite nanomaterials. A proper knowledge of magnetic int eractions between\ncore-shell spins would be useful to realize the inter-grain tunneling of polarized spins, as well\nas the effect of disorder on double exchange ferromagnetism. A fe w reports are available to\ndemonstrate the underlying physics of CMR manganites by reducing the particle size in the\nnanometer scale [10, 21, 22, 23, 24]. A better understanding of ma gnetic disorder effects,\nrelated to the grain and grain boundaries, can be achieved by compa ring the properties\nof a CMR manganite in different structural (i.e., bulk polycrystalline, n anocrystalline and\namorphous) phase [25, 26]. It will be interesting to study, to what e xtent, the properties of\ndisordered magnets, such as spin glass, cluster glass, superpara magnetism and short-ranged\nmagnetic ordering above T C, are affected by the structural disorder in the amorphous phase\nof a ferromagnetic manganite. In the present work, we have highlig hted such magnetic as-\npects in La 0.67Ca0.33MnO3as a function of particle size, both in crystalline and amorphous\nphase.\nII. EXPERIMENTAL\nThe stoichiometric amounts of high purity La 2O3, CaCO 3, MnO 2oxides have been mixed\nto obtain the bulk La 0.67Ca0.33MnO3. The mixture has been ground for 2 hours and pel-\nletized. The pellets have been sintered at 11000C for 12 hours, at 12500C for 20 hours and\nat 13800C for 20 hours with several intermediate grindings by cooling the sam ple to room\ntemperature. Finally, the sample is cooled to room temperature at 2 -30C/min. The X ray\ndiffraction (XRD) spectrum at 300 K, using Philips PW1710 diffractome ter with Cu K α\nradiation, confirmed the formation of bulk La 0.67Ca0.33MnO3(LCMN) sample. The XRD\ndata have been recorded in the 2 θrange 10-900with step size 0.010. Fritsch Planetary\nMono Mill ”Pulverisette 6” has been used for mechanical milling of powd ered LCMN sam-\nple. The milling has been carried out in Argon atmosphere in an 80 ml aga te vial with 10\nmm agate balls. The ball to sample mass ratio was maintained to 7:1. The agate bowl and\nballs have been selected to avoid the magnetic contamination to the m aterial, if happens4\nat al during milling process. The samples with specific milling hours (X) ar e denoted as\nmhX. The structural phase evolution during milling process has been checked from the XRD\nspectrum of each sample. Crystal structure of the samples have been analyzed by standard\nfull profile fitting method using FULLPROF Program. Particle size of t he samples have\nbeen calculated using Transmission Electron Microscope (TEM) (mod el: Tecnai S-twin).\nThe surface morphology of samples (pellet form) have been studied by scanning electron\nmicroscope (SEM) (model: Hitachi S-3400N). Elemental compositio n of the samples have\nbeen determined from the energy dispersive analysis of X-ray (EDA X) spectrum (Thermo\nelectron corporation Instrument), attached with SEM.\nacsusceptibility anddcmagnetizationofthesampleshavebeenmeas uredinthetemperature\nrange 5 K to 340 K using SQUID magnetometer (MPMS-Quantum Desig n, USA). The real\n(χ′) and imaginary χ′′) part of ac susceptibility have been recorded in the frequency ran ge 1\nHz-1.5 kHz at ac field amplitude h rms≈1 Oe. The dc magnetization (M) measurement as a\nfunction of temperature (T) has been performed with convention al zero field cooled (ZFC)\nand field cooled (FC) modes. In ZFC mode, the sample has been zero fi eld cooled from 340\nK down to 5 K. Then, M(T) data are recorded in the presence of mea surement field while\ntemperature increases from 5 K. In FC mode, the sample has been c ooled from 340 K to\n5 K in the presence of magnetic field and the cooling field is maintained at the same value\nduring the M(T) measurement from 5 K to 340 K. The field dependenc e of magnetization\n(M(H)) has been carried out by cooling the sample at the measureme nt temperature under\nZFC mode.\nIII. RESULTS\nA. Structural phase\nFig. 1a shows the room temperature XRD spectra of selected samp les. The XRD spectra\nindicated that crystalline nature of the material decreases signific antly for the milling time\nmore than 61 hours and amorphous phase dominates in the spectru m for milling time more\nthan 98 hours. The milled samples upto 98 hours milling time do not show a ny additional\npeaks in comparison with the spectrum of bulk sample. This suggests that both bulk and\nmilled samples (upto mh98) are in similar crystallographic phase and fou nd to be matching5\nwith orthorhombic structure with Pnma space group. The lattice pa rameters are shown in\nTable I. The lattice parameters ( a,bandc) of bulk sample is within the range of reported\nvalues (a ∼5.4467˚A, b∼7.6914˚A, c∼5.4616˚A[27]; a∼5.4289˚A, b∼7.8194˚A, c∼5.4557˚A[28]).\nWe have seen the decrease of both aandcwith milling time, whereas bincreases. Over all,\ncell volume increases with the increase of milling time. The XRD line broad ening in milled\nsamples can be attributed to an effect of either increasing amorpho us phase or decreasing\nparticle size in the material. To get a quantitative estimation of retain ing crystalline phase\nin the milled samples, the intensity of (200) peak line of all the samples is divided by (200)\npeak intensity of bulk sample (Fig. 1b). The relative peak intensity (s hown in Table I) is\nsignificantly reduced for milling time more than 98 hours and approach ing to zero value at\nhigher milling time (e.g., ∼3% for mh200 sample). The structural change can be followed\nby denoting the bulk sample as polycrystalline (peak intensity 100%), the sample with\nrelative peak intensity between 90-10 % as nanocrystalline, the sam ple with relative peak\nintensity ≤5 % as amorphous, and the sample with relative peak intensity betwee n 9-6 %\nas nanocrystalline phase coexisting with amorphous phase. On the o ther hand, average size\nof the material (shown in Table I) from TEM micrographs is not monot onically decreasing\nwith the increase of milling time (size ∼65 nm, 12 nm and 90 nm for mh25, mh61, and\nmh200 samples, respectively). The results suggested the appear ance of graingrowth kinetics\nat milling time ≥98 hours, whereas the appearance of a large fraction of amorpho us phase\naftermillingtime ≥98hoursisevidenced fromthebroadhumpinXRDspectrum. TheSEM\npictures (Fig. 2a-c) suggest a relatively smooth surface of bulk sa mple in comparison with\nmilled samples. The change in surface morphology is understood from the breaking of µm\nsize (1µm to 10µm) particle size of bulk sample into the nanometer range for mechanic al\nmilled samples. This is confirmed from SEM pictures, as well as TEM pictu res. The\nestimated size of particles from SEM pictures is larger due to agglome ration of particles.\nThe elemental composition and impurity effect during mechanical milling have been checked\nfrom EDAX spectrum. Fig. 2d-f shows the spectrum of selected sa mples. The spectrum\nhas been recorded at 10 different points for each sample over a zon e of 125µm x 125µm.\nThe spectrum indicated the presence of La, Ca, Mn and O as main elem ents in the samples.\nThe spatial homogeneity of elements is also seen from the elemental mapping (data not\nshown). The elemental composition (La 0.67±0.01Ca0.33±0.01MnO3−δ) of the samples is close to\nthe expected compound La 0.67Ca0.33MnO3. The analysis indicated slight oxygen deficiency6\nin our samples with δ∼0.1 for mh200 sample. We do not find any significant amount of\nimpurity during mechanical milling for mh61 sample. However, a small am ount (less than\n3 atomic %) of Si impurity is detected in mh200 sample.\nB. ac susceptibility\nThe real ( χ′) and imaginary ( χ′′) parts at 10 Hz for selected samples are shown in Fig. 3.\nTheχ′of bulk sample increases sharply at about T m∼270 K, followed by a slow decrease\non lowering the temperature. The χ′′shows two peaks at T 1(= Tm)∼270 K and at T 2\n∼40 K, respectively. It may be noted that there is no significant chan ge inχ′near 40\nK. The sharp increase in χ′and associated peak in χ′′at about 270 K for bulk sample\nchanges into a broad maximum for milled samples. Both χ′andχ′′decreases below the\ntemperature of ac susceptibility maximum (T mforχ′and T 1forχ′′). Theχ′′of milled\nsamples does not show any low temperature peak, as seen in bulk sam ple at 40 K. The\nimportant change, we noted here, is the shift of both T m(position of χ′maximum) and T 1\n(position of χ′′maximum) to lower temperatures with milling time. The shift is much clear\nin T1as compared to T m. A gradual transformation in the magnetic dynamics of milled\nsamples is indicated from the fact that T 1(position of χ′′maximum) is higher than T m\n(position of χ′maximum) for milling time upto 98 hours (Fig. 3 (a-c)) and opposite is tr ue\n(i.e., T 1<Tm) for milling time more than 98 hours (Fig. 3 (d-e)). In order to under stand\nthe magnetic dynamics (spin ordering) for the samples with milling time le ss and more than\n98 hours, we have studied the ac susceptibility at different frequen cies (ν= 1 Hz to 1.5\nkHz) for mh61 (nanocrystalline) and mh146 (amorphous) samples ( Fig. 4). The magnitude\nofχ′andχ′′for mh61 sample (Fig. 4a) have shown certain changes with the freq uency\nvariation. There is a branching of ac susceptibility data in the temper ature range 40 K to\n200 K. The magnitude of χ′decreases with the increase of frequency, unlike the increase\nofχ′′magnitude. Such frequency dependence of ac susceptibility indicat es the presence of\nfinite magnetic disorder in the sample. The absence of a clear freque ncy shift either at T m∼\n165 K or at T 1∼200 K suggests that magnetic disorder in mh61 sample is not sufficient for\nexhibiting a typical spin glass freezing or superparamagnetic blockin g. On the other hand,\nac susceptibility ( χ′,χ′′) maximum of mh146 (amorphous) sample shows a clear shift with\nfrequency (Fig. 4b). The determination of T m(ν) shift from χ′may not be much accurate7\ndue to more broadening of maximum. Hence, we have analyzed the fr equency shift of χ′′\nmaximum at T 1(ν) (T1∼102 K, 108 K, 114 K, 119.5 K, 121.5 K and 123 K for 1 Hz,\n99 Hz, 575 Hz, 997 Hz and 1465 Hz, respectively). The temperatur e shift per decade of\nfrequency (∆T 1(ν)/T1(1 Hz)∆ln( ν)) is∼0.028. The fit of T 1(ν) according to Vogel-Fulcher\nlaw:ν=ν0exp(-E a/kB(T1-T0)) is shown in the inset of Fig. 4b. The fit parameters are\nν0(characteristic spin flip frequency) = 8.4 ×1014Hz, Ea(activation energy)= 0.23 ev and\nT0(interaction parameter) = 25 K. The values of fit parameters are n ot consistent with a\ntypical superparamagnetic blocking of magnetic particles (where T 0= 0 K,ν0∼1010-1012\nHz, temperatureshift per decadeoffrequency ∼0.1)abouttheacsusceptibility maximum of\nmh146 sample. The non-zero value of T 0indicates a sufficient amount of interactions among\nthe magnetic particles. The obtained value of temperature shift pe r decade of frequency is\nalso larger than the typical value 0.001 for classical spin-glasses like CuMn. However, the fit\nparameters of mh146 sample are comparable with other amorphous nanomagnets ( ν0∼108-\n1012Hz, T 0∼25 K-35 K) [29, 30], which have shown a collective freezing of interact ing\nmagnetic clusters at lower temperatures.\nC. dc magnetization\n1. Temperature dependence of dc magnetization\nThe temperature dependence of ZFC and FC magnetization at 100 O e are plotted in\nFig. 5. Some of the features of zero field cooled magnetization (MZF C) are similar to\nthe observations in the real part ( χ′) of ac susceptibility. For example, a sharp increase\nof magnetization about 275 K, followed by a slow decrease of MZFC fo r bulk sample, and\nthe appearance of a broad maximum in MZFC, followed by a sharp decr ease below 50 K\nfor milled samples were observed in ( χ′)(T) data (Fig. 3). There is no difference between\nMZFC and field cooled magnetization (MFC) above 270 K. MFC separat es out from MZFC\nbelow 270 K and the magnitude of MFC increases down to 10 K. The irre versibility tem-\nperature T irrdecreases with the increase of milling time (e.g., T irr∼265 K, 259 K, 245\nK, 225 K, 210 K, and 190 K for bulk, mh25, mh61, mh98, mh146 and mh 200 samples,\nrespectively). The irreversibility effect at 100 Oe is qualitatively demo nstrated (inset of Fig.\n5) from the temperature dependence of normalized thermoreman ent magnetization NTRM8\n(=[MFC(T)-MZFC(T)]/TRM(10K)). It is interesting to note that TR M (10 K) of nanocrys-\ntalline samples (4.87 emu/g, 3.55 emu/g, 2.31 emu/g for mh25, mh61, m h98, respectively)\nare larger than bulk sample (1.84 emu/g). On the other hand, TRM (1 0 K) of amorphous\nsamples (1.32 emu/g for mh146 and0.70 emu/g for mh200) not only low er than bulk sample,\nbut also decreases with the increase of amorphous character in th e material. The decrease\nof normalized TRM with the increase of temperature may be a genera l feature of magnetic\nmaterials, but a different kind of magnetic dynamics is marked in the ch ange of shape of\nNTRM(T) curves from down curvature for milling time ≥98 hours to up curvature for\nmilling time ≤61 hours.\nWe understand the nature of magnetic exchange interactions fro m the analysis of tempera-\nture dependence of inverse of dc susceptibility ( χ−1= H/M) curves at H = 100 Oe (Fig. 6).\nThe high temperature (T ≥300 K) data of bulk sample are fitted with simple Curie-Weiss\nlaw:\nχ=C/(T−θw) (1)\nIn the expression, C ( ∼0.0176) is Curie constant and θw(∼275 K) is paramagnetic Curie\ntemperature. The effective magnetic moment ( µ∼5.42µBper f.u.) is estimated from the\nCurie constant (C= N µ2/3kB, N is the number of formula unit (f.u.)). The estimated value\nofµis large compared to 4.56 µBper Mn atom for the spin contribution alone (assuming\n67% Mn3+with S =2 and 33% Mn4+with S = 3/2 and g S= 2, as suggested in Ref. [1]).\nAnother aspect is that the non-linear (slightly up curvature) varia tion ofχ−1(T) data above\n260 K for bulk sample can be fitted using\nχ=C/(T−θw)γ(2)\nwithγ= 1.20 and θw∼260 K. It is remarkable to note that χ−1(T) plots of milled samples\nshow different character in comparison with bulk sample. First, we fit the data (T >275 K)\nusing simple Curie-Weiss law. The fit parameters of Curie-Weiss law ( µ,θw) are shown in\nTable II. The hyperbolic shape of χ−1(T) (with down curvature) below 300 K suggests that\nmilled samples belong to the class of either ferrimagnet or double exch ange ferromagnet\n[18]. The χ−1(T) data in the temperature range 240 K-340 K are fitted with the f ollowing\nequation, generally applied for ferrimagnet [31].\n1/χ= (T−θ1)/Ceff−ξ/(T−θ2) (3)9\nIn our fitting method, we freely allow the parameters ( θ1, Ceff(µeff),ξandθ2) to take\ninitial values. As soon as the fitted curve comes close to the experim ental curve, we start\nto restrict the parameters one by one, except ξ, to obtain the best fit curve. The main\npanel of Fig. 6 suggests a good quality fit according to equation (3) . The fit parameters are\nshown in Table II. A comparative fits of equation (1) and (3) for mh6 1 sample suggests that\nequation (1) may be well valid at higher temperature, but equation ( 3) is more appropriate\nto describe the magnetic behaviour over a wide temperature range . The interesting point\nis that the change of effective moment with the milling time is identical, as obtained from\nsimple Curie-Weiss law (equation 1) and from equation (3), except th e magnitude of µeffis\nlarger than µ. In both cases, the effective moment increases with milling time upto 6 1 hours\n(nanocrystalline samples) and for milling time ≥98 hours the effective moment, again, de-\ncreases and attaining a constant magnitude for the amorphous (m h146 and mh200) samples.\nThe relatively large effective magnetic moment from equation (3) may be involved to the\naccuracy of fit parameters in the limited temperature range or the clustering of moments\nin the paramagnetic state [32, 33]. This limits the use of µefffor quantitative discussion.\nHowever, µcan be used for the qualitative understanding of paramagnetic mom ent with\nmilling time.\n2. Field dependence of dc magnetization\nThe field dependence of magnetization of the samples are studied at different tempera-\ntures. The magnetization is measured at 10 K with field range ±70 kOe, but M(H) data are\nshown within H= ±3 kOe for the clarity of Fig. 7a. We have noted some typical feature s of\ncoercive field H C(where magnetization reverses its sign), irreversible field H irr(where the\nloop open below this specified field) and remanent magnetization M R(which represents the\nresidual magnetism after making the field to zero value either from + 70 kOe or -70 kOe) and\nthe energy product (proportional to the hysteresis loss or loop a rea), defined by H CxMR.\nBoth H Cand H irr(in Fig. 7b) showed a rapid increase in the initial stage of milling time\n(nanocrystalline samples) and then, approaching to a constant va lue after 98 hours (amor-\nphous samples). The increase of H Cis related to the increasing anisotropy energy of the\nmaterial, whereas the increase of H irris related to the increasing magnetic disorder in the\nmaterial. The initial increase of M Rfor mh25 sampple is followed by a gradual decrease with10\nmilling time ≥61 hours. However, energy product (H CxMR) for the nanocrystalline samples\nis largecompared to bulk sample, but decreases forthe samples inam orphous phase. Similar\nvariation of energy product has been found in magnetic nanocompo sites, consisting of soft\n(core) andhard(shell) components [34]. Inspite of thesame orde r of energy product for both\nbulk and amorphous (mh200) samples, the experimental result indic ated that coercivity of\nsoft ferromagnet ( ∼30 Oe for bulk) can be enhanced ( ∼540 Oe for mh200) by making the\nmaterial in amorphous phase.\nFig. 8 represents the M(H) curve at different temperatures (10 K to 300 K) for selected\nsamples. A typical soft ferromagnetic character of bulk sample is c onfirmed from the rapid\nincrease of magnetization within 5 kOe field and a tendency to attain m agnetic saturation\nat higher fields. At the same time, a field induced magnetic behaviour is indicated from the\nsmall M-H loop (Fig. 8(b)) at about 30 kOe and 50 kOe for 10 K and 30 K , respectively,\nwhile the field was reversing back from +70 kOe. There is no field induce d transition at T\n≥50 K. A close look at the M(H) data suggested that magnetization of bulk sample is not\ncompletely saturated even at 70 kOe and the signature of magnetic non-saturation is more\nprominent at T ≥260 K. The non-saturation character of high field magnetization cle arly\ndominates in the material when the lattice structure changes from nanocrystalline phase to\namorphous phase (Fig. 8: (c) mh98, (d) mh200).\nIn order to determine the spontaneous magnetization (M S) and paramagnetic to ferromag-\nnetic transition temperature (T C), we have analyzed M-H data using Arrot plot (M2vs.\nH/M) [14]. The normalized Arrot plot (M(H)2/M(70 kOe)2vs. H/M ) at 10 K is shown in\nFig. 9a. The linear extrapolation of high field data intercepts on the p ositive M2axis for\nbulk, mh25 and mh61 samples, and the determination of M Sis easy. On the other hand,\na non-linear curve intercepts on the positive M2axis for the samples with higher milling\ntime. This is the signature of increasing disorder effect in the materia l [35]. We obtain the\nspontaneous magnetization for these samples from the polynomial fit of high field data. The\npolynomial fit is shown in Fig. 9a for mh98 and mh200 samples. The extr apolation of M S\n(T) curve to M S= 0 value determines T Cof the sample, whereas the extrapolation to T =\n0 K gives the spontaneous magnetization M S(0). The values of T Cand M S(0) are shown\nin Table II. Our experimental results show that both T Cand M S(0)decreases with milling\ntime, which indicates the loss of long range parameters in La 0.67Ca0.33MnO3ferromagnet.\nWe understand the order of magnetic phase transition near T Cof the samples from H/M vs.11\nM2isotherms, which has been recognised as an useful method [13, 14]. The extrapolated\nisotherms intercept the M2axis with positive slope (shown by dotted line in Fig. 9b) at T ≤\n260 K for bulk sample. In addition to the positive slope for high field isot herms, a negative\nslope also intercepts the M2axis at lower fields in the temperature range 270 K to 280 K. No\npositive slope intercepts the M2axis at 285 K, which is above the T C∼281 K of bulk sam-\nple. On the other hand, only a positive slope intercepts the M2axis up to T Cfor all milled\nsamples (Fig. 9c-d), irrespective of nanocrystalline or amorphous structure. It is established\n[13, 14, 36] that negative and positive slope gives the signature of fi rst order and second\norder character, respectively. This suggests a drastic change in the order of paramagnetic\nto ferromagnetic phase transition in milled samples. Fig. 10 showed th e lowering of reduced\nmagnetization curves(M S(T)/M S(0) vs. T/T C) for milled samples in comparison with bulk\nsample. The reduced curves qualitatively suggested more disorder in milled samples and\nindicated the increasing fluctuations about mean ferromagnetic ex change interactions [37],\nwhen lattice structure of the bulk ferromagnet changes into nano crystalline and amorphous\nphases. The Rhodes-Wohlfarth (P C/PSvs. TC) plot [38] plot is shown in the inset of Fig\n10, where P S= MS(0) per f.u. and effective magnetic moment per f.u. µ2\neff= PC(PC+2)\ngives P C. The plot shows that P C/PSfor all the samples is larger than the typical value\n1 for pure localized magnetism. The increase of P C/PSwith the decrease of T C(PC/PS=\n1.25, 46.29 and T C= 281 K, 212 K for the bulk and mh200 samples, respectively) sugges ts\nthe probability of increasing itinerant character in the ferromagne tic properties of milled\nsamples in comparison with localized magnetism of bulk sample [39].\nIV. DISCUSSION\nWe have employed mechanical milling to transform bulk La 0.67Sr0.33MnO3into different\nlattice structures. We preferred discontinuous milling of the mater ial. The milling proce-\ndure was stopped in every 6 hours interval for proper mixing of the milled powder and to\nminimize the agglomeration effect of particles. We are able to reduce t he particle size down\nto 12 nm for milling time 61 hours and grain growth kinetics appeared at higher milling\ntime. In contrast, there is a systematic change of lattice structu re with milling time from\nbulk polycrystalline to nanocrystalline and amorphous phase. It is int eresting to note that\nparticle size of mh25 (65 nm) and mh146 samples (60 nm) are nearly sa me order, but mh2512\nsample is more crystalline (87%) than mh146 sample (5%) and their mag netic properties are\ndrastically different from each other. This indicates that local heat ing during mechanical\nmilling upto 200 hours is not sufficient to exhibit the milling induced crysta llization effect\n[40]. A small amount of Si contamination (less than 3 atomic %) from th e agate bowl and\nballs has been observed at higher milling time (200 hours). Considering the non-magnetic\ncharacter of Si atoms, the effect of such small amount of impurity can be treated as some\nadditional disorder in the lattice structure whose magnetic effect is negligible. In fact no\nsignificant amount of contamination is found for mh61 sample, but ma gnetic properties are\nsignificantly changed with respect to bulk sample. Hence, we discuss the ferromagnetism of\nLa0.67Ca0.33MnO3in terms of particle size reduction and lattice disorder.\nFirst, we understand the magnetic ordering of bulk La 0.67Ca0.33MnO3. The T Cis, gener-\nally, determined from the ac susceptibility peak temperature or the temperature where dc\nmagnetization sharply increases or the temperature where spont aneous magnetization be-\ncomes zero and all these temperatures are nearly same for a long r anged ferromagnet. In the\npresent work, ac susceptibility ( χ′′) shows a sharp peak at ∼270 K, the dc magnetization\n(at 100 Oe) sharply increases at about 275 K, and M S(T) data indicated a non-zero value\nof spontaneous magnetization below 281 K. These results clearly su ggest that bulk sample\nis not in pure ferromagnetic phase below T C∼281 K (determined from M S(T) data). This\nis also confirmed from other experimental facts. For example, the fit exponent γ= 1.22\nfrom equation (2) is larger than the mean field theory predicted valu e (∼1) for second order\nmagnetic phase transition. However, the obtained value of γfalls in the range 1 to 1.36,\nwhich has been reported for first order magnetic transition [12, 13 ]. The next experimental\nevidence is the nature of magnetic phase transition, i.e., second ord er character at T ≤260\nK (positive slope in H/M vs M2plot) gradually changes into the first order character at T C\n∼281 K (negative slope in H/M vs M2plot). The mixture of first order and second order\ncharacter in between 270 K to 280 K represents a diffused magnetic state below T C, just like\nliquid-gas mixture below critical temperature of the thermodynamic phase diagram. Our\nobservations are consistent with literature reports [9, 15, 16, 41 , 42], which suggested that\ncoexisting clusters in the ferromagnetic ground state are playing a major role to show the\nfirst order character of magnetic phase transition. The existenc e of magnetic clusters in the\nferromagnetic matrix of our bulk sample is understood from the app earance of magnetic\nirreversibility between MZFC and MFC, and non-saturated charact er of magnetization even13\nat 70 kOe. The number of short ranged interacting clusters increa ses when the system ap-\nproaches to the paramagnetic state and reflected in a field induced magnetic state in the\ntemperature range 270 K-280 K, as reported earlier [32, 43, 44]. O ur magnetization data\nclearly indicated that most of these clusters are melting in the ferro magnetic matrix due to\nstrong ferromagnetic interactions [16, 43, 45] when the tempera ture decreases below 270 K.\nThe remaining fraction of clusters freezes into local magnetic stat es at lower temperature,\nwhich is evidenced from the χ′′peak at∼40 K. This is due to the high sensitivity of χ′′\n(T) to energy loss related to rotation/freezing of magnetic cluste rs and the fact is confirmed\nfrom the field induced magnetic behaviour at T ≤40 K. The absence of such anomaly in χ′\nsuggests a few numbers of freezing clusters in the bulk sample.\nNow, we demonstrate the magnetism of mechanical milled samples in na nocrystalline and\namorphous phase. We noted some significant features which indicat ed a gradual change in\nthe ferromagnetic behavour of milled samples. For example, the low t emperature χ′′peak\nat 40 K, as seen for bulk sample, is not appeared for milled samples, ex cept a shoulder\naround the same temperature for nanocrystalline samples. The eff ects of increasing mag-\nnetic disorder with milling time are realized from many observations: (1 ) increased magnetic\nirreversibility between ZFC and FC magnetization, (2) appearance o f a round shape max-\nimum in MZFC curves, (3) differenet character (down curvature) o fχ−1(T) data in the\nparamagnetic state, and (4) appearance of upturn in the Arrot p lot. Here, we would like\nto focus on the fit of χ−1(T) data, which seems to be more interesting for revealing the\ndynamics of milled samples. The paramagnetic Curie temperature ( θw), obtained from the\nfit of equation (1), is positive for all the milled samples, except the de creasing magnitude\nwith the increase of milling time (e.g., θw∼240 K and 30 K for mh25 and mh200 samples,\nrespectively). The positive value of θwsuggested that double exchange ferromagnetic inter-\nactions are, still, sfficiently strong [18] in the milled samples of La 0.67Ca0.33MnO3. On the\nother hand, equation (3) is applicable for ferrimagnetic materials, c onsisting of two magnetic\nsublattices of antiparallel directions [31, 35], where extrapolation o f high temperature χ−1\n(T) data is expected to intersect the temperature axis at negativ e value. Table II shows\nthat the fit parameter θ1is not negative for all the milled samples, but a systematic de-\ncrease is found as the milling time increases from 25 hours ( θ1∼150 K) to 200 hours ( θ1∼\n-70 K). The decrease of θ1suggests the reduction of ferromagnetic (FM) exchange interac -\ntions (or development of antiferromagnetic (AFM) exchange inter actions) in nanocrystalline14\nand amorphous samples. The spin glass like feature in amorphous (mh 146) sample clearly\nproves the development of antiferromagnetic exchange interact ions in the material, because\nspin glass like feature needs sufficient amount of both magnetic disor der and competition\nbetween FM/AFM interactions.\nThe validity of ferrimagnetic equation (3) in our milled samples can be ex amined by consid-\nering the core-shell spin structure of nanoparticles. The shell sp in structure is magnetically\ndisordered in comparison with the ferromagnetic ordered core spin s. The shell spins may\nnot be typical antiparallel with respect to core, but effective spin m oment of shell is ob-\nviously low in comparison with ferromagnetic core [46]. This allows us to c ompare the\nmagnetic contributions form shell and core of a nanoparticle with tw o unequal magnetic\nsublatticles of a ferrimagnet. Let us try to understand the possib le disorder effects in the\ncore-shell structure of nanoparticles. The increasing antiferro magnetic interactions in the\nmaterial can be associated with the spin canting and breaking of dou ble exchange ferro-\nmagnetic bonds (Mn3+-O-Mn4+) in the shell part of nanoparticle. The reduction of double\nexchange bonds may be affected due to slight oxygen deficiency in mille d samples, as seen\nfrom EDAX data. The oxygen deficiency has the effect to decrease Mn4+/Mn3+ratio, which\nresults in the possible increase of antiferromagnetic (Mn3+-O-Mn3+) superexchange interac-\ntions [47]. The experimental results of Zhao et al. [48] suggested n o alternation of T Cin\nLa0.5Ca0.5MnO3−δ(δ≤0.17) sample with the variation of oxygen deficiency ( δ). However, we\nobserve a drastic reduction of T C(∼281 K for bulk, 212 K for mh200) in La 0.67Ca0.33MnO3\nwith increasing milling time. This means slight oxygen deficiency may have a minor role\nin disturbing the double exchange ferromagnetic interactions in the present sample, but\nits effect is not enough to explain all the observed magnetic change in nanocrystalline and\namporphous samples. Another source of disorder in the material m ay be originated from\nthe mechanical strain induced effect [49], confined in the shell of nan oparticle. A typical\nvariation of coercivity in our samples (i.e., initial increase is followed by a lmost no change\nat higher milling time) suggests that strain induced disorder is satura ted at higher milling\ntime, whereas the magnetic behaviour of amorphous phase samples (milling time more than\n98 hours) is found to be markedly different from the nanocrystalline samples (milling time\nmore than 98 hours). Hence, mechanical strain induced disorder m ay not be the sufficient\ncause for increasing fraction of antiferromagnetic exchange inte ractions/disorder with the\nincrease of milling time that is suggested from the fit of equation (3). Here, we would like to15\nstate that coercivity of the material may be saturated at higher m illing time, but irreversible\nfield H irris continuously increasing with milling time. Cosidering irreversible field H irras\na good indicator of disorderness in magnetic nanoparticle, we sugge st that imparted me-\nchanical energy at higher milling time is being used to create mainly lattic e disorder in the\nmaterial. The fact is confirmed from the structural transformat ion of nanocrystalline phase\ninto amorphous phase for milling time more than 98 hours. On the othe r hand, similar\ntype variation of coercivity was observed in many materials [50], cons isting of ferromag-\nnetic particles surrounded by antiferromagnetic matrix. Coupling t he core-shell structure\nof nanopaticles [46] and increasing lattice disorder in the material, we assume that a large\nnumber offerromagneticclusters (cores) arecoexisting inthedis ordered matrix(contributed\nby shell spins and lattice disorder) of mechanical milled (nanocrysta lline and amorphous)\nsamples. This picture is different from the bulk sample where a few num ber of short-ranged\ninteracting clusters coexist inthelongrangedferromagneticmatr ix. Weproposeaschematic\ndiagram in Fig. 11 to represent the change of core-spin configurat ion as a consequence of\nincreasing disorder in the material. We have also compared the spin-la ttice configuration\nof polycrystalline sample (order in both spin and lattice) (Fig. 11a) wit h nanocrystalline\n(shell spin disorder and small lattice order) (Fig. 11b) and amorpho us (disorder in both spin\nand lattice) (Fig. 11c) samples. We suggest that ferromagnetic co herent length is confined\nwithin the cores and ferromagnetic order is disturbed in the shell pa rt. On the other hand,\nlattice disorder is spreading in both shell and core parts of the nano particles.\nThe above discussion suggests that the decrease of particle size in the nanometer range alone\nis not sufficient to controll the magnetic properties of La 0.67Ca0.33MnO3in nanocrystalline\nand amorphous phase, but increasing lattice disorder also playing an important role in de-\ntermining the magnetic parameters. The experimental results indic ated that ferromagnetic\n(spin-spin) interactions are sufficiently strong even in the amorpho us phase of the material.\nThe low temperature decrease of magnetization (both in ac and dc s usceptibility) occurs\ndue to the blocking/freezing of ferromagnetic cores in the disorde red matrix [47]. The is\nsignificant to note that blocking behaviour becomes prominent for h igher milling time as\nan effect of increasing lattice disorder, but not due to an effect of d ecreasing particle size.\nThe rapid increase of lattice disorder causes a sharp decrease of s pin-lattice interactions\nwith milling time and controlling a drastic reduction of T C(∼281 K for bulk to 212 K for\nmh200) and spontaneous magnetization at 0K ( ∼3.6µBfor bulk to 0.1 µBfor mh200). To16\nour knowledge it is not clear, why first order ferromagnetic phase t ransition of bulk LCMN\nsample transforms into second order character in nanomaterials. The existence of short-\nranged interacting clusters in the ferromagnetic matrix may be play ing a major role for first\norder phase transition in bulk LCMN sample. This is not true for milled sa mples, because\nfirst order character is not seen, although increasing number of m agnetic clusters (cores) is\nrealized from the magnetic measurements. The nature of magnetic phase transition may be\nvery much sensitive to the disturbance of spin-lattice interactions of bulk sample [11]. The\norigin of magnetic phase transformation in our milled samples is, most p robably, related to\nthe dilution of spin-lattice interactions and change in cluster(core) -matrix (shell) configu-\nration. The Rhodes-Wohlfarth (P C/PSvs. TC) plot suggested that itinerant character of\nLa0.67Ca0.33MnO3ferromagnet increases with milling time. Inspite of the non-monoton ic\nvariation of µ(hence P C) with milling time, the ratio of P C/PScontinuously increases. This\nsuggests that decrease of P S, mainly from shell spins, played a dominant role in determining\nPC/PSand increasing fraction of shell spins may be the probable source of itinerant spin\nmoments. The experimental evidence of itinerant ferromagnetic c haracter due to shell spins\nmay be realized from the large low field magnetoresistance (MR) in man ganite nanomate-\nrials, as reported in Ref. [19, 20]. The itinerant character of shell s pins is activated by the\nweakening of spin-lattice coupling [51] and excitation of spins to highe r energy states as an\neffect of mechanical strain induced anisotropy. The other magnet ic features of the amor-\nphous samples, i.e., lack of magnetic saturation and magnetic irrever sibility, are consistent\nwith reported amorphous ferromagnetic nanoparticles [25, 26].\nV. CONCLUSIONS\nLa0.67Ca0.33MnO3ferromagnet exhibited many interesting features in the nanocrys talline\nand amorphous phase, as an effect of increasing disorder in core-s hell spin morphology and\nlattice structure. The present work clearly showed the dominant r ole of lattice disorder\nplayed over the paricle size effect in determining the magnetic dynamic s of amorphous sam-\nples. We have also discussed some recent issues, such as: effects o f magnetic clustering in\nthe nature of magnetic phase transition, effect of spin-lattice inte ractions, unusual disorder\nmagnetic states that are different from the conventional spin glas s or superparamagnetism.\nThe distinction between nanoparticle magnetism and amorphous mag netism, and properties17\nrelated to structural and spin disorder, can be understood by ex tending similar experimental\nworkto othersystems. Theconcept offerrimagnetism isappliedto understand thecore-shell\nmagnetism in La 0.67Ca0.33MnO3ferromagnetic nanoparticles.\nAcknowledgement: We thank Pulak Roy for providing TEM data and CIF, Pondicherry\nUniversity for providing SEM and EDAX measurements.\n[1] E.O. Wollan and W.C. Koehler, Phys. Rev. 100, 545 (1955)\n[2] C. Martin, A. Maignan, M. Hervieu, and B. Raveau, Phys. Re v. B60, 12191 (1999)\n[3] A.P. Ramirez, S. W. Cheong, and P. Schiffer, J. Appl. Phys. 81, 5337 (1997)\n[4] Colossal Magnetoresistance, Charge Ordering and Relat ed Properties of Manganese Oxides,\nedited by B. Raveau and C. N. R. Rao (World Scientific, Singapo re, 1998).\n[5] J. Fontcuberta, B. Martinez, A. Seffar, S. Pinol, J.L. Garc ia-Munoz, and X. Obradors, Phys.\nRev. Lett. 76, 1122 (1996)\n[6] Y. Sun, W. Tong, X. Xu, and Y. Zhang, Phys. Rev. B 63, 174438 (2001)\n[7] X. Liu, X. Xu, and Y. Zhang, Phys. Rev. B 62, 15112 (2000)\n[8] S.M. Yusuf, M. Sahana, M.S. Hegde, K. D¨ orr and K.-H. M¨ ul ler, Phys. Rev. B 62, 1118 (2000)\n[9] E. Dagotto, S. Yunoki, A.L. Malvezzi, A. Moreo, J. Hu, S. C apponi, D. Poilblanc, and N.\nFurukawa, Phys. Rev. B 58, 6414 (1998)\n[10] J.W. Lynn, D.N. Argyriou, Y. Ren, Y. Chen, Y.M. Mukovski i, and D.A. Shulyatev, Phys.\nRev. B76, 014437 (2007)\n[11] C. Rettori, D. Rao, J. Singley, D. Kidwell, S.B. Oseroff, M .T. Causa, J. J. Neumeier, K.J.\nMcClellan, S-W. Cheong, and S. Schultz, Phys. Rev. B 55, 3083 (1997)\n[12] S. R¨ o βler, U. K. R¨ o βler, K. Nenkov, D. Eckert, S. M. Yusuf, K. D¨ orr, and K.-H. M¨ u ller, Phys.\nRev. B70, 104417 (2004)\n[13] D. Kim, B. Revaz, B.L. Zink, F. Hellman, J.J.Rhyne, and J .F. Mitchell, Phys. Rev. Lett. 89,\n227202 (2002)\n[14] J. Mira, J. Rivas, F. Rivadulla, C. Vazquez-Vazquez, an d M. A. Lpez-Quintela, Phys. Rev. B\n60, 2998 (1999)\n[15] R. H. Heffner, L. P. Le, M. F. Hundley, J. J. Neumeier, G. M. L uke, K. Kojima, B. Nachumi,\nY. J. Uemura, D. E. MacLaughlin, and S.-W. Cheong, Phys. Rev. Lett.77, 1869 (1996)18\n[16] P. Novak, M. Marysko, M.M. Savosta, and A.N. Ulyanov, Ph ys. Rev. B 60, 6655 (1999)\n[17] Y. Tokura and Y. Tomioka, J. Magn. Magn. Mater. 200, 1(1999)\n[18] P.W. Anderson and H. Hasegawa Phys. Rev. 100, 675 (1955)\n[19] H.Y. Hwang, S-W. Cheong, N.P. Ong, and B. Batlogg, Phys. Rev. Lett. 77, 2041 (1996)\n[20] S. Lee, H.Y. Hwang, B.I. Shraiman, W.D. Ratcliff,II, and S -W. Cheong, Phys. Rev. Lett. 82,\n4508 (1999)\n[21] M.A.L′opez-Quintela1, L.E. Hueso, J. Rivas and F. Rivadulla, Nano technology 14, 212(2003)\n[22] R. Rajagopal, J. Mona, S.N. Kale, T. Bala, R. Pasricha, P . Poddar, M. Sastry, B.L.V. Prasad,\nD.C. Kundaliya and S.B. Ogale, Appl. Phys. Lett. 89, 023107 (2006)\n[23] R.D. Sa′nchez, J. Rivas, C. Vazquez-, Vazquez, A. Lopez-Quintela, M .T. Causa, M. Tovar, S.\nOseroff, Appl. Phys. Lett. 68134(1996)\n[24] P. Dey and T.K. Nath, Phys. Rev. B 73, 214425 (2006)\n[25] E.De. Biasi, C.A. Ramos, R.D. Zysler and H. Romero, Phys . Rev. B 65, 144416 (2002)\n[26] R. Prozorov, Y. Yeshurun, T. Prozorov, and A. Gedanken, Phys. Rev. B 59, 6956 (1999)\n[27] P.R. Sagdeo, S. Anwar, and N.P. Lalla, Powder Diffraction 21, 40 (2006)\n[28] M.B. Liou, S. Islam, D.J. Fatemi, V.M. Browning, D.J. Gi llespie, and V.G. Harris, J. App.\nPhys.86, 1607 (1999)\n[29] M.D. Mukadam, S.M. Yusuf, P. Sharma, S.K. Kulshreshtha and G.K. Dey, Phys. Rev. B 72,\n174408 (2005)\n[30] J.A. De Toro, M.A. Lopez de la Torre, J.M. Riveiro, J. Bla nd, J.P. Goff, and M.F. Thomas,\nPhys. Rev. B 64, 224421 (2001)\n[31] J. S. Smart, Am. J. Phys. 23, 356 (1955).\n[32] Y.Q. Ma, W.H. Song, B.C. Zhao, R.L. Zhang, J. Yang, W.J. L u, J.J. Du, and Y.P. Sun,\nPhysica B 364, 117 (2005)\n[33] H. Terashita and J.J. Neumeier, Phys. Rev.B 71, 134402(2005)\n[34] R. Skomski and J.M. D. Coey, Phys. Rev. B 48, 15812 (1993)\n[35] R.N. Bhowmik, R. Ranganathan, and R. Nagarajan, Phys. R ev. B73, 144413(2006)\n[36] F. Rivadulla, J. Rivas, and J. B. Goodenough, Phys. Rev. B70, 172410 (2004)\n[37] S.J. Poon and J. Durand, Phys. Rev. B 16, 316 (1977)\n[38] V. Hardy, B. Raveau, R. Retoux, N. Barrier, and A. Maigan , Phys. Rev. B 73, 094418 (2006)\n[39] F. Rivadulla, J. Rivas, and J.B. Goodenough, Phys. Rev. B70, 172410 (2004)19\nTABLE I: The particle size of the milled samples are determin ed from the TEM data. % Intensity\nis the normalized XRD peak intensity of 200 line compared to b ulk sample. Structural notation:\nPCR = polycrystalline (bulk), NCR = Nanocrystalline, AMP= a morphous. Lattice parameters\nwithin±0.0003(a, b, c) are calculated from the XRD data.\nSample Milling hours Size (nm) % Intensity structural phase a (˚A) b (˚A) c (˚A)\nBulk 0 few µm 100 PCR 5.4615 7.7203 5.4634\nmh25 25 65 87 NCR 5.4569 7.7323 5.4607\nmh61 61 12 14 NCR 5.4479 7.7576 5.4549\nmh98 98 16 9 NCR + AMP 5.4444 7.7749 5.4493\nmh146 146 60 5 AMP - - -\nmh200 200 90 3 AMP - - -\n[40] Y.S. Kown, J.S. Kim, I.V. Povstugar, E.P. Yelsukov, and P.P. Choi, Phys. Rev. B 75, 144112\n(2007)\n[41] J. Burgy, M. Mayr, V. Martin-Mayor, A. Moreo, and E. Dago tto, Phys. Rev. Lett. 87, 277202\n(2001)\n[42] J. W. Lynn, R. W. Erwin, J. A. Borchers, A. Santoro, Q. Hua ng, J.-L. Peng, and R. L. Greene,\nJ. Appl. Phys. 81, 5488 (1997)\n[43] J. Mira, J. Rivas, A. Moreno-Gobbi, M.P. Macho, G.Paoli ni, and F. Rivadulla, Phys. Rev. B\n68, 092404 (2003)\n[44] F. Rivadulla, M. A. Lpez-Quintela, and J. Rivas, Phys. R ev. Lett. 93, 167206 (2004)\n[45] R.I. Zainullina, N.G. Bebenin, V.V. Ustinov, Ya.M. Muk ovskii and D.A. Shulyatev, Phys.\nRev. B76, 014408 (2007)\n[46] T. Zhang, T.F. Zhou, T. Quian, and X.G. Li, Phys. Rev. B 76, 174415 (2007)\n[47] K. D¨ orr, J. Phys. D: Appl. Phys. 39, R125 (2006)\n[48] Y.G. Zhao et al. Phys. Rev. B 65, 144406 (2002)\n[49] B.H. Liu, and H. Ding, Appl. Phys. Lett. 88, 042506 (2006)\n[50] J. Sort, J. Nogues, S. Surinach, J. S. Munoz, M. D. Baro, E . Chappel, F. Dupont, and G.\nChouteau, Appl. Phys. Lett. 79, 1142 (2001)\n[51] L. Sheng, D.Y. Xing, D.N. Sheng, and C.S. Ting, Phys. Rev . Lett.79,1710(1997)20\nTABLE II: The fit parameters are obtained from simple Curie-W eiss law (equation 1) and from\nequation (3) for different samples. The effective paramagnetic moments µandµeffare calculated\nfrom the Curie constant C and C eff. The Curie temperature (T C) and spontaneous magnetization\nat 0 K (M S(µB)) are calculated from the Arrot plot analysis.\nSampleµper f.u.(µB)θw(K)µeffper f.u. ( µB)θ1(K)θ2(K)ξ(arb. unit) T C(K) M S(0)(µB)\nBulk 5.42 275 – – – – 281 3.60\nmh25 5.75 240 7.20 150 252 91420 ±875 262 2.17\nmh61 6.9 164 9.11 -20 250 67200 ±395 250 0.87\nmh98 5.43 120 6.84 -54 248 90990 ±885 238 0.35\nmh146 5.33 55 6.20 -59 245 64700 ±1200 225 0.17\nmh200 5.33 30 6.11 -70 244 53840 ±730 212 0.10050 100 150 200 250 \n10 20 30 40 50 60 70 80 90 25 50 75 100 0300 600 900 1200 1500 1800 \n31 32 33 34 35 020 40 60 80 100 \n2θ (degree) Intensity (arb. unit) \n mh61 (12 nm) (nano crystalline) \n  \n  mh200 (90 nm) (amorphous) (a) (242) (224) (204) (220) (202) (110) \n(200)  \n mh0 (Bulk) (poly crystalline) (b) \nFig. 1 (Colour online) X ray diffraction spectra fo r selected samples in (a). \nRelative peak (200) intensity of all samples taking  100 for the bulk sample in (b). \n  \nmh200 (90 nm) mh146 (60 nm) mh98 (16 nm) mh61 (12 nm) mh49 (20 nm) mh25 (65 nm) mh0 (bulk) Relative peak intensity taking 100 for bulk \n2θ (degree)  \n \n \nFIG. 2: SEM pictures for bulk (a), mh61 (b) and \nmh200 (c) samples, and corresponding EDAX \nspectrum for bulk (d), mh61 (e) and mh200 (f) \nsamples.  \n \n \n \n \n (a) bulk \n(b) mh61 \n(c) mh200 (d) bulk \n(e) mh61 \n(f) mh200  \n \n 0 50 100 150 200 250 300 0246810 12 01234\n0510 15 0246\n0 50 100 150 200 250 300 01234\nT1Tm\nFig. 3 ac susceptibility data at 1 Oe and 10 Hz for  La 0.67 Ca 0.33 MnO 3 samples. T m and T 1 \nshows the peak positions for χ/ and χ// , respectively. T 2 represents the low temperature \nχ//  peak position for bulk sample. Note that χ//  data are multiplied by some factor.χ//  x 15 χ/ \n χac  (10 -3  emu/g/Oe) \nT (K) mh98 \n(16 nm) \n (c) T2T1Tm\n \n  \nχ//  x100 χ/χac  (10 -2  emu/g) Bulk \n(a) \nT1Tm\nχ// x2 χ/\n χac  (10 -3  emu/g/Oe) mh61 \n(12 nm) \n  (b) T1Tm\nχ// x10 χ/ \n χac  (10 -3  emu/g) mh146 \n(60 nm) \n  (d) \nT1Tm\nχ//  x 10 χ/\n  χac  (10 -3  emu/g/Oe) \nT (K) mh200 \n(90 nm) \n   (e) 0246810 12 14 16 0 50 100 150 200 250 300 \n0123456\n0 50 100 150 200 250 300 012345\n0.010 0.012 0246T1Tm\n1465 Hz 1 Hz \n1465 Hz \n1 Hz   mh61 \n(12 nm) \n   (a) \nFig. 4 (Colour online) ac susceptibility ( χ/, χ// ) data for mh61 (a) and mh146 (b) at ν = 1 Hz, 10 Hz, 99 Hz, \n575 Hz, 997 Hz and 1465 Hz. The arrows guide the sh ift of ( χ/, χ// ) maximum as ν increases from 1 Hz \nto 1465 Hz. Inset (b) shows the fit of T1(ν) data to Vogel-Fulcher law.χ// χ/ \n χ (10 -3  emu/g/Oe) \n1 Hz \n  mh146 \n (60 nm) \n    (b) \n χ/\n  χ/ (10 -3  emu/g/Oe) 1465 Hz \nT1 (1 Hz) \nχ//  (10 -4  emu/g/Oe) \nχ//  \n \nT (K) Fit data  \n ln ( ν Hz) \n1/(T 1-25) (K -1 )expt. data 0 50 100 150 200 250 300 0.0 0.2 0.4 0.6 0.8 1.0 \n0 50 100 150 200 250 300 350 0.01 0.1 1\nmh25 \n  \nmh200 bulk TRM (T)/TRM (10 K) \nT (K) mh98 \nmh146 mh61  \nFig. 5 (Colour online) Temperature dependence of ZF C and FC magnetization \nat 100 Oe for different samples. Inset shows the no rmalized thermoremanent \nmagnetization (NTRM= [MFC-MZFC]/TRM(10 K)) of the s amples. mh200 mh146 mh98 mh61 mh25 Bulk \n  M (emu/g) (log scale) \nT (K) 200 220 240 260 280 300 320 340 05000 10000 15000 20000 \n260 280 300 320 340 01000 2000 3000 4000 \nFit to eqn. (3) \nFit to eqn. (1) \nFig. 6. (Colour online) Inset shows the χ-1  vs. temperature data for bulk sample fit \nwith eqn. (1) (straight line) (left-bottom scale) a nd fit with eqn. (2) (right-top scale). \nMain panel shows the experimental data with non-lin ear fit to equn. (3). mh200 \nmh146 \nmh98 \nmh61 \nmh25 \nBulk \n  χdc -1  (Oe g/emu) \nT (K) ln[(T-260) (K)] \nln[ χ-1  (g Oe/emu)] χ-1  (g Oe/emu) \nT (K) 3.0 3.5 4.0 4.5 5.0 \n6.5 7.0 7.5 8.0 \n  0 50 100 150 200 0123456-3000 -2000 -1000 0 1000 2000 3000 -80 -60 -40 -20 020 40 60 80 \nMR (emu/g), H CxM R (kOe emu/g) Hirr HCx10  \n [H C, H irr ] (kOe) \nMilling time (hours) 0246810 12 14 16 \n(b) [H CxM R]x3 \nMR\nFig. 7 (a) (Colour online) M-H loop at 10 K for dif ferent samples and corresponding \nmagnetic parameters with milling time in (b), where  H C is coercive field, H irr  is \nirreversibility field, MR is remanent magnetization and HCxMR= energy product. mh200 \nmh98 \nmh61 \nmh25 \nBulk  \n M (emu/g) \nH (Oe) (a) M-H loop at 10 K 0510 15 20 \n0 20000 40000 60000 80000 0246810 12 14 \n20000 40000 60000 80000 96.0 96.5 97.0 0 20000 40000 60000 80000 020 40 60 80 100 \n (c) mh98 \n240 K 220 K 190 K 160 K 120 K 80 K 40 K 10 K \n M (emu/g) \nFig. 8 M(H) data at selected temperatures for (a) b ulk, (c) mh98, (d) mh200 samples. \nThe minor loops at higher field (20 kOe-80 kOe) and  temperature range 10 K-30 K are \nshown in (b). Solid and open symbols represent magn etization data during field increase \nand decrease, respectively.245 K 220 K 180 K 150 K 120 K 90 K 60 K 30 K 10 K (d) mh200 \n  M (emu/g) \n(b) bulk 50 K 30 K 10 K \n  M (emu/g) \nH (Oe) H (Oe) (a) Bulk 10 K \n90 K 140 K \n180 K \n210 K \n240 K \n260 K \n270 K \n275 K \n280 K \n285 K \n M (emu/g) 0 50 100 150 200 05000 10000 15000 20000 25000 \n0 2000 4000 6000 800010000 0200 400 600 800 1000 1200 0 200 400 600 800 10001200 02000 4000 6000 \n0 10002000300040005000 0.0 0.2 0.4 0.6 0.8 1.0 \nmh200 \n   (d) \n10 K 30 K 60 K 90 K 120 K 150 K 180 K 220 K 245 K \n  \nM2 (emu/g) 2H/M (Oe g/emu) H/M (Oe g/emu) \nbulk \n (b) \nFig. 9. (a) Normalized (M 2 vs. H/M) plot at 10 K for selected samples, along with line \nand polynomial fit to calculate spontaneous magneti zation. (b-d) H/M vs. M 2 plot for \nbulk, mh61 and mh200 samples. The dotted lines indi cate the slope at H = 0 limit.285 K \n280 K \n275 K \n270 K \n260 K \n240 K \n210 K \n140 K 10 K \n  H/M (Oe g/emu) \nM2 (emu/g) 2210 K mh61 \n (c) 270 K \n250 K \n230 K \n180 K 140 K \n100 K \n60 K 10 K \n  \nM2 (emu/g) 2(a) mh200 mh98 mh61 mh25 Bulk \n  [M(H)] 2/[M(70 kOe)] 2\nH/M (Oe g/emu) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 \n200 220 240 260 280 300 010 20 30 40 50 \nFig. 10 (Colour online) The reduced plot (M S(T)/M S(0) vs. T/T C) for selected samples. \nThe inset shows Rhodes-Wohlfarth plot (P C/P S vs. T C) for the samples. P C and P S are \nderived from the effective magnetic moment (P eff (µB)) and M S(0)( µB) per formula unit.\nThe notations: PCR, NCR and AMP are mentioned in Ta ble 1.  \n MS/M S(0) \nT/T C bulk (PCR) \n mh25 (NCR) \n mh61(NCR) \n mh98(NCR+AMP) \n mh200 (AMP) \nmh200 \nmh146 \nmh98 \nmh61 \nmh25 Bulk \n  PC/P S\nTC (K) Fig. 11 (a) \n(b) \n(c) "    },    {        "title": "1206.6929v2.Enhancement_of_critical_current_density_in_superconducting_magnetic_multi_layers_with_slow_magnetic_relaxation_dynamics_and_large_magnetic_susceptibility.pdf",        "content": "S. Z. Lin and L. N. Bulaevskii, Phys. Ref. B 86 , 064523 (2012).\nEnhancement of critical current density in superconducting /magnetic multi-layers with slow\nmagnetic relaxation dynamics and large magnetic susceptibility\nShi-Zeng Lin and Lev N. Bulaevskii\nTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n(Dated: August 3, 2021)\nWe propose to use superconductor-magnet multi-layer structure to achieve high critical current density by\ninvoking polaronic mechanism of pinning. The magnetic layers should have large magnetic susceptibility to\nenhance the coupling between vortices and magnetization in magnetic layers. The relaxation of the magnetiza-\ntion should be slow. When the velocity of vortices is low, they are dressed by nonuniform magnetization and\nmove as polarons. In this case, the viscosity of vortices proportional to the magnetic relaxation time is enhanced\nsignificantly. As velocity increases, the polarons dissociate and the viscosity drops to the usual Bardeen-Stephen\none, resulting in a jump in the I-V curve. Experimentally the jump shows up as a depinning transition and the\ncorresponding current at the jump is the depinning current. For Nb and proper magnet multi-layer structure, we\nestimate the critical current density Jc\u0018109A=m2at magnetic field B\u00191 T.\nPACS numbers: 74.25.Wx, 74.25.Sv, 74.25.Ha, 74.78.Fk\nI. INTRODUCTION\nOne fascinating property of superconductors is ability to\ncarry dissipationless current. With a transport current, vortices\nare induced inside the superconductor due to the magnetic\nfield generated by the current. These vortices are driven by the\nLorentz force exerted by the current and their motion causes\nvoltage and dissipation. In inhomogeneous superconductors,\nthe Lorentz force can be balanced by the pinning force due to\ndefects. The strength of pinning thus determines how much\ndissipationless current the superconductor can carry, which is\ndefined as the critical current. The random distributed point-\nlike defects where superconductivity are weakened, are im-\nportant for pinning of vortices.1One may also introduce arti-\nficially columnar pinning centers by heavy ion irradiation.2In\nthese cases, the pinning is caused by suppression of supercon-\nductivity.\nAn alternative approach to introduce pinning is to use mag-\nnetic moments, which interact strongly with vortex. Such op-\ntion may be present in magnetic superconductors.3,4The mag-\nnetic moments can also be introduced artificially in hybrid\nsystems consisting of superconducting and magnetic layers5.\nIt was proposed in Ref. 6 that the domain walls can provide\nstrong pinning with characteristic pinning energy \b0Md. Here\n\b0=hc=(2e) is the quantum flux and Mdis the magnetization\nat the wall. There are experimental attempts to enhance the\ncritical current by putting magnetic particles7, dots8,9or fer-\nromagnet with domain walls on top of superconductors.10\nReduction of dissipation can be also achieved by enhance-\nment of the vortex viscosity. At a given current J, the dis-\nsipation power for a superconductor without pinning due to\nquenched disorder is proportional to J2=\u0011where\u0011is the vor-\ntex viscosity. In nonmagnetic superconductors, \u0011is just the\nstandard Bardeen-Stephen (BS) drag coe \u000ecient accounting\nfor the dissipation in the normal vortex core. If one can in-\ncrease significantly the vortex viscosity, superconductors can\ncarry large current density with low dissipation, despite vor-\ntices are not pinned. It was shown that in magnetic supercon-\nductors, motion of vortex lattice excites magnons11. When the\nkinematic condition \n(G)=G\u0001vis satisfied, Cherenkov ra-diation of magnon occurs and the vortex viscosity is enhanced\ndue to transferring energy into the magnetic subsystem, where\nenergy is finally dissipated through magnetic damping. Here\nGis the lattice wave vector, vis the velocity of vortex lattice\nand\n(G) is the magnon spectrum. When the magnetic damp-\ning is weak, magnetic domain walls are created dynamically\ndue to the parametric instability and the viscosity is increased\nfurther.12\nRecently a polaronic mechanism of vortex pinning is pro-\nposed in Ref. 13 to explain the increase of critical current ob-\nserved in ErNi 2B2C below the incommensurate to commensu-\nrate spin density wave (SDW) transition at 2.3 K14. The tran-\nsition into the commensurate SDW phase leaves 1 /20 spins\nfree from molecular field15. These spins can be easily po-\nlarized by vortices. These spins are Ising spins and experi-\nence large crystal field splitting16, which results in slow re-\nlaxation dynamics17. When the velocity of vortex lattice is\nlow,a=v\u001d\u001c, the nonuniform component of free-spin magne-\ntization induced by vortex lattice follows the vortex motion,\nand the nonuniform magnetization and vortex form a polaron.\nHere ais the vortex lattice constant and \u001cis the relaxation time\nfor magnetizations. The e \u000bective viscosity of vortex lattice in-\ncreases with the relaxation time. For a large velocity, a=v\u001c\u001c,\nthe nonuniform magnetization cannot follow the motion of\nvortex lattice and they are decoupled from each other. The vis-\ncosity of the system recovers to the conventional BS one. The\ndecoupling or dissociation of polaron experimentally shows\nup as a depinning transition. The maximal critical current for\nErNi 2B2C is estimated as 1010A=m2at magnetic field B\u00190:1\nT. The polaronic mechanism is also at work in other borocar-\nbides, cuprate and iron-based superconductors with magnetic\nrear earth ions locating between superconducting layers.\nThe polaronic mechanism of pinning provides an additional\nroutine to achieve high critical current. To optimize such pin-\nning mechanism, we propose to use a multi-layer structure\nconsisting of superconducting (S) and magnetic (M) layers\nshown in Fig. 1, to achieve high critical current. For that\nthe magnetic layers should have high magnetic susceptibility\nat working magnetic field to ensure a strong coupling between\nmagnetic moments and vortices. Secondly, the relaxation timearXiv:1206.6929v2  [cond-mat.supr-con]  23 Aug 20122\nFIG. 1. (color online) Schematic view of multi-layer structure con-\nsisting of alternating magnetic (M) layers (green) with thickness dm\nand superconducting (S) layers (blue) with thickness ds. The distri-\nbution of the magnetic field is shown by red lines.\nof the magnetization should be long. Thirdly, the penetration\ndepth of the superconducting layers should be small.\nII. MODEL AND RESULTS\nUnder external magnetic fields, the vortex lattice is induced\ninside the S layers. With a transport current, vortex lattice\nmoves in response to the Lorentz force. In the quasistatic ap-\nproximation, the motion of vortex lattice is given by\n\u00152r\u0002r\u0002 B+B= \b 0X\ni\u000e[r\u0000ri(t)]ˆz; (1)\nwhere ri(t)=r0\u0000vtis the vortex coordinate, ˆzis the unit vec-\ntor along the zaxis and\u0015is the London penetration depth. In\nthe flux flow region, the quenched disorder is averaged out\nby vortex motion and the lattice ordering is improved18,19.\nThe magnetic field inside the M layers is determined by the\nMaxwell equations\nr\u0002(B\u00004\u0019M)=0;r\u0001B=0: (2)\nThe magnetization Mdepends on Band is determined by the\nmaterial properties. With a strong field and in static case, M\nis a nonlinear function of Band generally can be expressed as\nM(r)=R\ndr3f(r\u0000r0;B(r0)). The characteristic length of mag-\nnetic subsystem is much smaller than \u0015and we use a local ap-\nproximation f(r\u0000r0;B(r0))=\u000e(r\u0000r0)f(B(r0)).B(r) has com-\nponent uniform in space, B0, and the other component nonuni-\nform in space, ˜B(r), with B0\u001c¯B. Thus the spatially nonuni-\nform magnetization ˜M(r) is ˜M(r)\u0019@f(B0)=@B0˜B(r)\u0011\n\u001f0(B0)˜B(r). In the following we assume the magnetic subsys-\ntem is isotropic and is characterized by a susceptibility \u001f0(B0)\natB0in static case. The magnetic field inside the M layer isdetermined by the equation r2˜B=0. Since only the spatially\nnonuniform component ˜Mand ˜Bare responsible for pinning,\nwe will focus on the nonuniform components in the following\ncalculations. At the interface between the M and S layers, we\nuse the standard boundary condition for the field parallel to\nthez-axis Bzand field parallel to the interface Bjj\nBzjS=BzjM;BjjjS=(1\u00004\u0019\u001f0)BjjjM: (3)\nThen we can obtain the magnetic field inside the M layers\nBz\nm(G>0;z)=\u000bh\neGz0+e\u0000G(z0+dm)i\b0exp(\u0000iGxvxt)\n1+\u00152G2;(4)\nBjj\nm(G>0;z)=i\u000bh\neGz0\u0000e\u0000G(z0+dm)i\b0exp(\u0000iGxvxt)\n1+\u00152G2;(5)\n\u000b=\u0000edmG\u0010\n\u00001+edsks\u0011\n\u001f0\n(1\u0000\u001f0)(edsks\u0000eGdm)+(1+\u001f0)(1\u0000edmG+dsks);\nwith z0=z\u0000n(ds+dm),ks=p\n\u0015\u00002+G2, and\u001f0=(1\u0000\n4\u0019\u001f)\u00001ks=G. Here nis the layer index and the vortex motion\nis assumed to be along the xdirection. We consider square\nlattice G=(mx2\u0019=a;my2\u0019=a) with a=p\b0=B0the lattice\nconstant and mx,myintegers.\nWe assume a relaxational dynamics for the M layers,\nM(!)=\u001f(!)Bm(!), with a dynamic susceptibility\n\u001f(!)=\u001f0\n1+i!\u001c: (6)\nHere we have assumed that the relaxation dynamics is gov-\nerned by a single relaxation time. This assumption is not es-\nsential but just for convenience of calculations. In the steady\nstate, we have\nM(G;z;t)=Zt\n0exp[( t0\u0000t)=\u001c]\u001f0Bm(G;z;t0)\n\u001cdt0:(7)\nBecause of the relaxation, Mdepends on the history of vortex\nmotion. Due to slow relaxation of the magnetization, there is\nretardation between the time variation of induced nonuniform\nmagnetization and vortex motion. As a result, the magneti-\nzation exerts a drag force to the vortex which is opposite to\nthe driving force. The pinning force acting on a single vortex\ndue to the induced magnetization in one M layer is given by\nFp=@r0R\ndxdyR0\n\u0000dmdzM\u0001Bm, which yields\nFp=X\nG\u00021\u0000exp(\u00002Gdm)\u0003 2\u000b2\u001f0\u00001+\u00152G2\u00012a2Gv\u001c\b2\n0\n1+(Gv\u001c)2:(8)\nThe I-V curve is determined by the equation of motion for\nvortex ds\u0011BSv=dsFL\u0000Fpwith the electric field E=Bv=c\nand the Lorentz force FL=J\b0=c. Here\u0011BSis the BS vis-\ncosity\u0011BS= \b2\n0=(2\u0019\u00182c2\u001an) with\u001anthe resistivity just above\nTcand\u0018the coherence length. We consider a realistic case\nwhere a=(2\u0019)\u001cdm;ds. Taking into account only the domi-\nnant contribution Gx=2\u0019=aandGy=0 in the summation,\nwe obtain\nu=FL\u0000F pu\n1+u2; (9)3\n03 6 9 1 21 503691215r\netrapping(\nJr, Er)depinning(\nJc, Ec)2/s61552/s61556caE//s6151002\n/s61552J/s61556/s615100(dm+ds)/(/s61544ads) /s61510p=20 \n/s61510p=2\nFIG. 2. (color online) Calculated I-V curves for Fp=20 andFp=2.\nForFp=20 the system shows hysteresis in the I-V while for Fp=2\nno hysteresis is present. The green dotted line denotes the unstable\nsolution.\nFL=FL\n\u0011BSv0;Fp=2\u001c\n\u0011BSds 1\n2\u00004\u0019\u001f0!2\u001f0a\b2\n0\n\u00154(2\u0019)3; (10)\nwith u=v=v0andv0=a=(2\u0019\u001c).\nAt a small velocity u\u001c1, the velocity is given by u=\nFL=(1+Fp) which becomes inversely proportional to \u001cfor a\nlarge\u001c. For a large u\u001d1, we recover the conventional BS\nviscosity v=FL. The dependence of uonFLis shown in Fig.\n2. Hysteresis is developped when Fp\u00158. For typical param-\neters for Nb superconductor \u0018\u0019\u0015\u001940 nm,\u001an\u001910\u00006\n\u0001m\nanda=40 nm at B\u00191 T and\u001f0=0:05,Fp\u00158 requires\n\u001c > 1 ps. For the relaxation time of order \u001c\u00191\u0016s, the ef-\nfective viscosity is enhanced by a factor of 106compared to\nthe bare BS one at v<a=\u001c. Upon increasing the current,\nthe velocity increases and at a critical current (velocity), the\nsystem jumps at dFL=du=0 to the conventional BS branch\ndue to the dissociation of vortex polaron. The e \u000bective critical\ncurrent density for the whole system is given by\nJc\u0019 1\n2\u00004\u0019\u001f0!2\u001f0c\n(2\u0019)4\u00154\b0a2\nds+dm: (11)\nFords=dm=100 nm , we obtain Jc\u0019109A=m2. On the\nother hand, when one reduces the current from the conven-\ntional BS branch, the system jumps to the branches with high\nviscosity due to the formation of vortex polaron. We call this\nretrapping transition and the retrapping current Jris\nJr\u00191\n1\u00002\u0019\u001f0r\n\u001f0\u0011BSads\n\u0019\u001cac\n\u001524\u001921\nds+dm: (12)\nFor the parameters used before and \u001c=1\u0016s, we estimate\nJr\u00192\u0002106A=m2.III. DISCUSSIONS\nNow we discuss the optimal thickness of M and S layers.\nFordm\u001da, the magnetic induction and the magnetization\nis practically uniform in the lateral direction in the middle of\nthe M layer. This can be seen from Eqs. (4) and (5) that\nBm(G>0)\u0019exp(\u00002\u0019dm=a) when\u0000dm\u001cz0\u001c0. As a\nresult, the pinning force becomes practically dmindependent\nwhen dm\u001da. In other words, the pinning is e \u000bective only\nnear the boundaries between S and M layers in the area of\nthickness of the order a. On the other hand, the Lorentz force\nis proportional ds. Thus the e \u000bective critical current of the\nwhole system Jcis proportional to 1 =(ds+dm) as described by\nEq. (11). Therefore the thinner of both M and S layers, the\nhigher is the critical current of the system.\nLet us discuss the possible choice of S and M layers. The\ncritical current decreases as \u0015\u00004because the smaller \u0015, the\nmore nonuniform is the magnetic field distribution inside the\nM layers, hence stronger pinning. Thus superconductors with\nsmaller\u0015are preferred. The critical current does not depend\non\u001cfor su \u000eciently large \u001c, while the viscosity in the branch\nwith vortex polaron is proportional to \u001c. The slow magnetic\ndynamics can be realized in spin glasses. Their relaxation\nis described by a broad spectrum of time scale, with aver-\nage time of the order 0 :1\u0016s20,21. For CuMn 0:08,\u001f0\u00190:002\natB=1 T.22One may enhance \u001f0by tuning the concen-\ntration of magnetic metal in alloys.23One may use super-\nparamagnets with \u001cas large as 1 s and with huge \u001f0due to\nlarge magnetic moments in superparamagnets.24–26One may\nalso use the recently synthesized cobalt-based and rare-earth-\nbased single chain magnets with \u001f0\u00190:05 at B=1 T and\n10\u00006s<\u001c< 10\u00004s.27–30.\nNext we discuss the e \u000bect of quenched disorder. In the\npresence of quenched disorder, the vortex lines adjust them-\nselves to take the advantage of the pinning potential, which\ndestroys the long-range lattice order. Below a threshold cur-\nrent, vortices remain pinned (actually they creep between pin-\nning centers due to fluctuations). In this region, the polaronic\nmechanism does not play a role. When the current is high\nenough to depin the vortices from quenched disorder, vortices\nstart to move and the lattice ordering is enhanced. By for-\nmation of polaron with the nonuniformly induced magnetiza-\ntion, the vortex viscosity is enhanced. At a critical velocity\n(current), the polaron dissociates and the system jumps to the\nconventional BS branch. Pinning due to quenched disorder\nworks in the static region and polaronic pinning works in the\ndynamic region. The critical current of the whole system is\nthe sum of these two threshold currents. Note that magne-\ntostriction in combination with quenched disorder enhance the\npolaronic pinning mechanism.\nThe M /S multi-layer structure is naturally in some su-\nperconducting single crystals, such as RuSr 2GdCu 2O831and\n(RE)Ba 2Cu3O732,33, where RE is the rear earth magnetic\nions. In RuSr 2GdCu 2O8the magnetic moments order fer-\nromagnetically above Tcthus the dominant enhancement of\nvortex viscosity is due to the radiation of magnons11. For\n(RE)Ba 2Cu3O7, magnetic RE ions positioned between su-\nperconducting layers interact weakly with superconducting4\nelectrons and order at very low N ´eel temperatures of the\norder TN\u00181 K. The polaronic mechanism is important\nabove the magnetic ordering temperature, where spins are\nfree. The London penetration depth of cuprate supercon-\nductors is large \u0015\u0019200 nm, thus the critical current is re-\nduced significantly compared to that for Nb multi-layer struc-\nture, because Jcdrops as 1=\u00154. Another natural realization is\nthe recently discovered iron-based superconductors, such as\n(RE)FeAsO 1\u0000xFx, where RE ions ordered antiferromagneti-\ncally below TN\u00181K.34\nIV . CONCLUSION\nTo summarize, we have proposed superconductor-magnet\nmulti-layer structure to achieve high critical current densitybased on the polaronic pinning mechanism. The critical cur-\nrent is estimated to be 109A=m2atB\u00191 T for an optimal\nconfigurations of Nb and proper magnet multi-layer structure.\nIn the presence of quenched disorder, the polaronic pinning\nstarts to work when vortices depin from quenched potential.\nThus the total critical current of the system is the sum of de-\npinning current due to quenched disorder and depinning cur-\nrent due to the polaronic mechanism.\nACKNOWLEDGMENTS\nThe authors are indebted to Cristian D. Batista for helpful\ndiscussion. This publication was made possible by funding\nfrom the Los Alamos Laboratory Directed Research and De-\nvelopment Program, project number 20110138ER.\n1G. Blatter, M. V . Feigelman, V . B. Geshkenbein, A. I. Larkin, and\nV . M. Vinokur, Rev. Mod. Phys. 66, 1125 (1994).\n2L. Civale, A. D. Marwick, T. K. Worthington, M. A. Kirk,\nJ. R. Thompson, L. Krusin-Elbaum, Y . Sun, J. R. Clem, and\nF. Holtzberg, Phys. Rev. Lett. 67, 648 (1991).\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).\n6L. N. Bulaevskii, E. M. Chudnovsky, and M. P. Maley, Appl.\nPhys. Lett. 76, 2594 (2000).\n7Y . Otani, B. Pannetier, J. Nozires, and D. Givord, J. Magn. Magn.\nMater. 126, 622 (1993).\n8J. I. Mart ´ın, M. V `elez, J. Nogu ´es, and I. K. Schuller, Phys. Rev.\nLett. 79, 1929 (1997).\n9J. I. Mart ´ın, M. V `elez, A. Ho \u000bmann, I. K. Schuller, and J. L.\nVicent, Phys. Rev. Lett. 83, 1022 (1999).\n10V . Vlasko-Vlasov, U. Welp, G. Karapetrov, V . Novosad, D. Rosen-\nmann, M. Iavarone, A. Belkin, and W. K. Kwok, Phys. Rev. B 77,\n134518 (2008).\n11A. Shekhter, L. N. Bulaevskii, and C. D. Batista, Phys. Rev. Lett.\n106, 037001 (2011).\n12S. Z. Lin, L. N. Bulaevskii, and C. D. Batista, arXiv:1201.4195\n(2012).\n13L. N. Bulaevskii and S. Z. Lin, Phys. Rev. Lett. 109, 027001\n(2012).\n14P. L. Gammel, B. Barber, D. Lopez, A. P. Ramirez, D. J. Bishop,\nS. L. Bud’ko, and P. C. Canfield, Phys. Rev. Lett. 84, 2497 (2000).\n15P. Canfield, S. Bud’ko, and B. Cho, Physica C 262, 249 (1996).\n16U. Gasser, P. Allenspach, J. Mesot, and A. Furrer, Physica C 282,\n1327 (1997).\n17P. Bonville, J. A. Hodges, C. Vaast, E. Alleno, C. Godart, L. C.\nGupta, Z. Hossain, R. Nagarajan, G. Hilscher, and H. Michor, Z.\nPhys. B 101, 511 (1996).\n18A. E. Koshelev and V . M. Vinokur, Phys. Rev. Lett. 73, 3580\n(1994).19R. Besseling, N. Kokubo, and P. H. Kes, Phys. Rev. Lett. 91,\n177002 (2003).\n20Y . J. Uemura, T. Yamazaki, D. R. Harshman, M. Senba, and E. J.\nAnsaldo, Phys. Rev. B 31, 546 (1985).\n21K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986).\n22J. Prejean, M. Joliclerc, and P. Monod, J. Physique 41, 427\n(1980).\n23J. Kouvel, J. Phys. Chem. Solids 21, 57 (1961).\n24C. P. Bean and J. D. Livingston, J. Appl. Phys. 30, S120 (1959).\n25R. B. Goldfarb and C. E. Patton, Phys. Rev. B 24, 1360 (1981).\n26B. D. Cullity and C. D. Graham, Introduction to Magnetic Mate-\nrials, 2nd ed. (Wiley-IEEE Press, 2008).\n27A. Caneschi, D. Gatteschi, N. Lalioti, C. Sangregorio, R. Sessoli,\nG. Venturi, A. Vindigni, A. Rettori, M. G. Pini, and M. A. Novak,\nAngew. Chem. Int. Ed. 40, 1760(2001).\n28A. Caneschi, D. Gatteschi, N. Lalioti, C. Sangregorio, R. Sessoli,\nG. Venturi, A. Vindigni, A. Rettori, M. G. Pini, and M. A. Novak,\nEurophys. Lett. 58, 771 (2002).\n29L. Bogani, C. Sangregorio, R. Sessoli, and D. Gatteschi, Angew.\nChem. Int. Ed. 44, 5817 (2005).\n30K. Bernot, L. Bogani, A. Caneschi, D. Gatteschi, and R. Sessoli,\nJ. Am. Chem. Soc. 128, 7947 (2006).\n31A. C. McLaughlin, W. Zhou, J. P. Attfield, A. N. Fitch, and J. L.\nTallon, Phys. Rev. B 60, 7512 (1999).\n32P. H. Hor, R. L. Meng, Y . Q. Wang, L. Gao, Z. J. Huang, J. Bech-\ntold, K. Forster, and C. W. Chu, Phys. Rev. Lett. 58, 1891 (1987).\n33P. Allenspach, B. W. Lee, D. A. Gajewski, V . B. Barbeta, M. B.\nMaple, G. Nieva, S. I. Yoo, M. J. Kramer, R. W. McCallum, and\nL. Ben-Dor, Z. Phys. B 96, 455 (1995).\n34C. Wang, L. Li, S. Chi, Z. Zhu, Z. Ren, Y . Li, Y . Wang, X. Lin,\nY . Luo, S. Jiang, X. Xu, G. Cao, and Z. Xu, Europhys. Lett. 83,\n67006 (2008)."    },    {        "title": "2104.00688v1.The_Magnetic_Field_Across_the_Molecular_Warped_Disk_of_Centaurus_A.pdf",        "content": "The Magnetic Field Across the Molecular Warped Disk of\nCentaurus A\nEnrique Lopez-Rodriguez1;2\n1Kavli Institute for Particle Astrophysics & Cosmology (KIPAC), Stanford University, Stanford,\nCA 94305, USA\n2SOFIA Science Center, NASA Ames Research Center, Moffett Field, CA 94035, USA\nMagnetic fields are amplified as a consequence of galaxy formation and turbulence-driven\ndynamos. Galaxy mergers can potentially amplify the magnetic fields from their progeni-\ntors, making the magnetic fields dynamically important. However, the effect of mergers on\nmagnetic fields is still poorly understood. We use thermal polarized emission observations\nto trace the magnetic fields in the molecular disk of the nearest radio active galaxy, Centau-\nrus A, which is thought to be the remnant of a merger. Here, we detect that the magnetic\nfield orientations in the plane of the sky are tightly following the \u00183:0kpc-scale molecular\nwarped disk. Our simple regular large-scale axisymmetric spiral magnetic field model can\nexplain, to some extent, the averaged magnetic field orientations across the disk projected\non the sky. Our observations also suggest the presence of small-scale turbulent fields, whose\nrelative strength increases with velocity dispersion and column density. These results have\nstrong implications for understanding the generation and role of magnetic fields in the for-\nmation of galaxies across cosmic time.\n1arXiv:2104.00688v1  [astro-ph.GA]  1 Apr 20211 Introduction\nNearby galaxies are known to have regular large-scale magnetic fields (B-fields) with a spiral-\nlike pattern at kpc-scales1but they also appear to have an important small-scale (or turbulent, or\nrandom) component1, 2. These regular large-scale B-fields are thought to be generated by a mean-\nfield galactic dynamo, which relies on differential rotation of the galactic disk to amplify and order\na ‘seed’ B-field. This B-field is driven by supernova explosions at scales of l\u001850\u0000100pc3–5.\nThe turbulent or random B-fields are thought to be generated by small-scale dynamo, which rely on\nturbulent gas motions at scales smaller than the energy-carrying eddies4. The correlation length\nof the turbulent or random B-fields is compared to or smaller than the turbulent scale, l. Once\nthe B-fields are amplified and in close equipartition with thermal and turbulent forces, magnetic\nfields can influence galaxy evolution4, 6–10. Galaxies at redshifts up to z \u00182, which are thought\nto be the progenitors of present-day galaxies, have been observed to host magnetic fields11, 12.\nMagnetohydrodynamical simulations13suggest that during the violent feedback-dominated early\nphase in the galaxy formation history, weak seed B-fields can be first amplified by a small-scale\ndynamo. In the subsequent quiescent galaxy evolution phase, the turbulent or random B-fields can\nweaken or be maintained via large-scale dynamo action. These observations and simulations invite\nthe question of the origin and evolution of the B-fields in galaxy evolution.\nCentaurus A is thought to be the remnant of a merger about 1:6\u00003:2\u0002108yr ago14, 15.\nThe chaotic dust lane and shells within it form a warped disk of \u001812kpc along the east-west\ndirection with a median position angle of 122\u00064\u000e14, 16and inclination of \u001890\u000630\u000e17. The\n2observed parallelogram-shape structure in the mid-infrared (mid-IR) is caused by folds in a thin,\ndusty warped disk rich in molecular gas, which has the most active star formation present in the\ngalaxy18–21. The warped disk has a rapidly rotating gas of \u00183kpc in radius based on the velocity\nfield of several tracers22. A warped disk model was adequate to describe the mid-IR morphology\nand kinematics of the galaxy disk18, 21, 23–25. This model suggests that tidal forces during the merger\nmodified the original gas motions of the spiral galaxy, forming rings around the central elliptical\ngalaxy. In this scenario, the merger may enhance and amplify the B-fields by small-scale dynamo\naction when the B-fields are coupled to the gas component of the galaxy.\nPioneering work26using optical polarimetric observations with 24\u000069-inch telescopes\ndetected the polarization signature of dichroic absorption in the dust lane of Centaurus A. These\nauthors concluded that the most likely reason for this detection is that the B-fields have been\nconfined in the general orientation of motion of the gas in the galaxy disk. Although major efforts\nhave been undertaken using optical and IR polarimetric studies, only small regions around the\ncentral active nucleus and across the dust lane have been measured in the IR27–34. In general,\nthe position angle (PA) of polarization is measured to be in the range of 110\u0000117\u000e, which is\nparallel to the dust lane and shows small angle fluctuations of \u00189\u000e34. The degree of polarization\n(P) decreases from \u00186% in the optical to \u00182% in the IR, which is entirely consistent with\npolarization arising from dichroic absorption. These results indicate that the galaxy disk may have\nan ordered B-field, although the B-fields of the whole galaxy disk have not been traced yet using\nIR polarimetric techniques. Thus, a whole picture of the B-fields and how they relate to the gas\ndynamics in the molecular warped disk of Centaurus A is still missing.\n32 The Data\nWe observe Centaurus A using the High-resolution Airborne Wideband Camera-plus (HAWC+)\non the 2:7-m Stratospheric Observatory For Infrared Astronomy (SOFIA) telescope, with a beam\nsize (full width at half maximum) of 7:8000at89\u0016m. We performed observations using the on-\nthe-fly-map polarimetric mode (see Methods and Extended Data Figure 1). Figure 1 shows the\npolarization map of Centaurus A at 89\u0016m observed with HAWC+ overlaid on a composite image,\nas well as with the total and polarized intensities at 89\u0016m. The total flux image is consistent with\nthe70\u0000160\u0016mHerschel observations35. The near-constant PA of polarization from optical to\nfar-IR wavelengths indicates a single dominant polarization mechanism. We estimate that our 89\n\u0016m polarization measurements arise from dichroic emission of magnetically aligned dust grains in\nthe molecular disk (see Methods).\nThe most prominent polarization signature of Centaurus A is the measured B-fields with\norientations tightly following the \u00183:0kpc-scale molecular warped disk. The measurements of\nthe B-field orientations and warped disk morphology are those projected on the plane of the sky.\nWe measure that the B-field orientations have a dispersion of 8:6\u000015:5\u000eacross the observed \u0018\n181” (\u00183:0kpc) molecular warped disk (see Methods and Extended Data Figure 2 and Extended\nData Figure 3). However, the B-fields show a tightly orientation following the warped disk without\nsystematic dispersion. The polarized flux morphology is spatially coincident with the low-surface\nbrightness regions at the top and bottom of the edges of the parallelogram structure observed at\n8\u000015\u0016m with Spitzer21and70\u0000160\u0016m with Herschel35. These regions are the closest to our\n4line-of-sight (LOS) with low column density in the molecular disk35, where the dust is optically\nthin at far-IR wavelengths.\nWe report the measurement of a polarized radio-loud active nuclei of 1:5\u00060:2% and PA\nof151\u00064\u000e(B-field) within a 8” (128 pc) diameter at 89 \u0016m (See Extended Data Figure 4). We\nestimated that the polarization arises from magnetically aligned dust grains, where the B-field\norientation is found to be almost perpendicular to the radio jet axis, PA \u001851\u000e37.\nThere are striking morphological similarities with the B-fields in the disks of highly inclined\ngalaxies observed using radio polarimetric techniques36. However, radio polarimetric observations\nof the host galaxy of Centaurus A have not been performed. We note that Faraday rotation is not\na factor at far-IR wavelengths. The total gas column density is traced more effectively by our\nfar-IR observations than by the relativistic electrons producing the synchrotron emission at radio.\nNear-IR polarization is subject to scattering effects from the disk, and dichroic absorption is only\nsensitive to the outer layers of the dust lane. Our far-IR observations trace deeper regions of the\nmolecular disk than those at near-IR. In addition, far-IR emissive polarization observations reveal\nthe orientation of the ordered B-field, but not its direction. Hence we cannot distinguish between\nregular large-scale fields and anisotropic random fields with frequent reversals.\n3 The large-scale regular magnetic field\nSpiral arms have been reported in the central \u00183kpc of Centaurus A using12CO(2\u00001)obser-\nvations38, and no bar has been found using the HI 21 cm line15. We produce a three-dimensional\n5model of the regular large-scale B-field morphology using an axisymmetric spiral B-field config-\nuration39, 40, which is a mode of a galactic dynamo with a symmetric spiral pattern in the galactic\nmidplane and a helical component (see Methods for a full mathematical description of the B-field\nmodel). This model is used to investigate trends from regular large-scale B-fields and not intended\nto truly represent the B-fields of a warped disk or to account for turbulent small-scale B-fields.\nOur B-field morphological model (Figure 2) obtains an edge-on ( i= 89:5+0:7\n\u00000:8\u000e) galaxy with\na tilt of\u0012= 119:3+0:5\n\u00000:4\u000eeast from north in the counter clockwise direction. The axisymmetric\nspiral B-field on the plane of the galaxy has a pitch angle of \t0=\u000054:9+0:5\n\u00000:5\u000e, where the helical\ncomponent has a radial pitch angle of \u001f0= 74:8+10:5\n\u000016:5\u000eand vertical scale of z0= 1:7+0:3\n\u00000:4kpc.\u001f0\nis defined as the pitch angle at a radius z0along the vertical axis of the galaxy (see Methods and\nExtended Data Figure 5 and Extended Data Figure 6 for details of the fitting routine). Our model\nresults on a spiral B-field structure with a large pitch angle in the plane of the galaxy, potentially\ndue to the tidal effects of the galaxy interaction. However, the molecular disk of Centaurus A is\nlikely to be warped, where the inclinations, i, and tilted, \u0012, angles have been measured17to be in\nthe range of [60;120]\u000e, and [92;169]\u000e, respectively from 2 pc to 6500 pc. Specifically, the mean\ninclination and tilt angles are estimated to be 83\u00066\u000eand114\u000614\u000ein the range of [0:5\u00003]kpc\n17, respectively, in agreement with our inferred results. The complex dynamics along the disk may\nchange the pitch angle, \t0, as a function of the radius from the core.\nBoth our model and observations agree within 10\u000ealong the mid-plane of the dust lane\n(Figure 3). We estimate the median difference between the PA of the B-fields of HAWC+, PAH,\n6and our model, PAMto be \u0001PA=< PAH\u0000PAM>= 3:6\u000622:7\u000e. We find that the angular\ndispersions from our model and those from magnetically aligned dust grains at scales of 124:8pc\nresolution are greater than can be accounted for by errors from our observations ( \u001bPA\u00149:6\u000e)\nwithin the molecular disk (Figure 3). Thus, another B-field component (i.e. small-scale B-fields)\nis required to explain the angular dispersion between the regular axisymmetric B-field model and\nthe measured B-field.\nOur model also shows a vertical and twisted pattern in the central region of Centaurus A at\na PA\u001830\u000e, which is in close agreement with the radio jet at a PA \u001851\u000e(East of North) at the\ncore of the radio loud active nucleus. The central \u0018100pc of Centaurus A shows very complex\nstructures41that are expected to generate some level of misalignment between the radio jet axis\nand the kpc structures. It is also worth noting the change of the observed B-field orientation at\n(X;Y ) = (\u00000:6;+0:2)kpc in Figure 3, which may be explained by the change from the mid-\nplane spiral pattern to the helical pattern. However, on the other edge at (X;Y ) = (0:0;+0:3)\nkpc, the helical pattern seems to have a lower effect. We point out that we have excluded the\ncentral 0:8\u00020:8kpc2in our modeling because the polarization is affected by energetic processes\nassociated with the AGN. The exclusion zone is determined by the low polarized regions from\nour observations and the physical inner bubble42, which indicate that different mechanisms of\npolarization are taking place in the central 0:8\u00020:8kpc2. Thus, the aforementioned differences\nbetween our model and observations may be due to different physical mechanisms in play within\nthe central 800 pc. The largest differences between our observations and model appear in the NW\nand SE edges of the warped disk. The observed B-field tends to be parallel to the dust lane at scales\n7>3kpc, and another B-field configuration may be needed at these scales.\n4 The thermal polarization and the multi-phase interstellar medium\nThe B-field of the interstellar medium (ISM) in galaxies is turbulent, and its random fluctuations are\nnot necessarily isotropic (i.e. anisotropic random B-fields). Note the different nomenclature in the\nliterature: anisotropic random fields, tangled fields, striated, ordered random43, and our choice to\nfollow the nomenclature by1. The B-fields in the ISM are typically described using a combination\nof ordered and random components, where the relationship between the fractional polarization and\nintensity (I) provides a proxy to characterize the effect of the turbulent component. In general, the\nfractional polarization in the ISM has been found to decrease with increasing column density44, 45,\nwhich can be attributed to 1) variations in the alignment efficiency of dust grains with column\ndensity (NH+H2), 2) tangled B-fields along the LOS, and/or 3) turbulent fields. Note that the\nanisotropic random fields also contribute to the observed polarized dust emission. As dynamos\nconvert kinetic energy into magnetic energy, we use the measurements of the velocity dispersion\nas a proxy of the turbulence in the gas, where depolarization effects are expected if turbulence\nincreases with increasing column density.\nThe warped disk of Centaurus A is rich in molecular gas, where the molecular gas emis-\nsion generally originates in high-density regions of the ISM close to the spiral arms. We use the\n12CO(1\u00000)emission line to characterize the dynamics in the molecular disk of Centaurus A.\nThe measurements of the velocity dispersion of the12CO(2-1) emission line ( \u001bv;12CO(1\u00000), see Ex-\n8tended Data Figure 7) observed by the Atacama Large Millimeter/submillimeter Array (ALMA)\nare used as a proxy of the turbulence in the molecular gas. We use the polarized intensity vs. to-\ntal intensity ( PI\u0000I) plot to identify several physical regions in the galaxy disk of Centaurus A\n(Figure 4 and Extended Data Figure 8). We find three distinct regions: outer disk, molecular disk,\nand low polarized regions (see Methods for specific criteria). To quantify the effect of turbulent\nB-fields in these regions, we use the correlations between P,I,PI, temperature ( T),NH+H2, and\n\u001bv;12CO(1\u00000)(Figure 5, and Methods for details of how these maps have been computed). We show\nthe median values of these parameters for each region in Table 1.\nFor the whole galaxy, we find a relation between P,T,NH+H2, and\u001bv;12CO(1\u00000). We measure\nthatPdecreases with T, andNH+H2, while theIincreases with T, andNH+H2. These trends imply\nan increase of turbulent fields and/or tangle B-fields as NH+H2increases. This result is supported\nby the increase of Iand the decrease of Pwith increasing \u001bv;12CO(1\u00000). ThePIremains almost\nconstant with T, and\u001bv;12CO(1\u00000), with a marginal increase with NH+H2.\nFor the three regions, we find that each region has unique physical conditions (see Methods\nand Figure 5, Extended Data Figure 9). The outer disk of the galaxy, with a size of \u00186kpc\nin diameter, is characterized by low NH+H2,T,I, and\u001bv;12CO(1\u00000). We find that NH+H2and\n\u001bv;12CO(1\u00000)are correlated with high P. We conclude that the most plausible explanation is that an\noptically thin layer of diffuse ISM, with low velocity dispersion and ordered B-field, may be the\nmain physical component of the measured B-field.\nThe molecular disk, of a size of \u00183kpc in diameter, has a large \u001bv;12CO(1\u00000)= 18:4\u00069:2\n9km s\u00001(see Extended Data Figure 10) in comparison with the thermal velocity dispersions of \u00188\nkm s\u00001for molecular clouds in nearby galaxies46. The measured velocity dispersion may arise\nfrom multi-components along the LOS due to large velocity gradients in the molecular disk. This\nregion may be affected by 1) averaging out many B-field orientations along the LOS across the\nmolecular disk, and 2) high turbulence at the high density regions in the molecular disk.\nThe low polarized regions with high NH+H2andIare found to have large \u001bv;12CO(1\u00000)= 34\u0006\n4km s\u00001, which are spatially associated with the circumnuclear disk within the central 500pc and\nthe active nucleus. This region is spatially coincident with the inner bubble42, which is dominated\nby high energetic processes from the AGN. This region may be affected by 1) an increase of\nvelocity dispersion of the gas due to AGN activity, 2) a decrease of the dust grain alignment\nefficiency due to the large turbulence fields, and/or 3) competing mechanisms of polarization (i.e.\nB and/or K radiative torque alignment). At the location of the active nucleus, an ordered B-field\nin a dusty circumnuclear disk around the active nucleus may be the dominant physical structure\nproducing the measured far-IR polarization.\n5 The origin of the magnetic field in Centaurus A\nOur B-field model at radius >500pc reproduces the ordered B-fields parallel to the dust lane of the\ngalaxy from optical absorptive polarization observations26–28, 30, 32, 33. Our interpretation is that the\nouter layers of the dust lane may have a less turbulent B-field than at deeper regions of the galaxy,\nwhere the large velocity dispersion in the molecular gas may be enhancing the turbulent B-field\n10in the molecular warped disk. Our model also reproduces, to some extent, the orientations of the\nobserved B-fields along the central \u00183kpc of the galaxy from our far-IR observations. We find\nthat the morphology given by the thermal emission of the warped disk deviates from our regular\nlarge-scale B-field model. The inferred B-field orientations from our observations have higher\nspatial correspondence with the structure of the warped disk than with the regular large-scale B-\nfield model. It is difficult to combine the regular large-scale B-field model with warped ring models\n18, 21, 23–25to explain the warped disk. Furthermore, other physical mechanisms, i.e. small-scale\nturbulent fields, may play an important role in the B-fields of the warped disk of Centaurus A.\nWe show that our measured angular dispersions are larger than those purely arising from\nthe observational uncertainties. In addition, we find that the polarized emission is less affected by\ntheT,NH, and\u001bv;12CO(1\u00000)than the unpolarized emission. Our interpretation is that the isotropic\nturbulent field, traced by the unpolarized thermal emission, is more affected by these quantities\nthan the ordered field, where the latter consists of the regular large-scale and anisotropic small-\nscale components. A possible explanation is that the small-scale turbulent field is relatively more\nsignificant at higher velocity dispersions of the molecular gas and column densities than the large-\nscale axisymmetric field. Our results can be interpreted as a decreasing ratio of the large-to-small\nB-fields. Thus, a substantial amount of the observed B-field at far-IR wavelengths may arise from\nanisotropic small-scale turbulent (ordered) fields that also contribute to the polarized emission or\ntangled fields at scales below or beyond the 124:8pc from our observations.\nThe most likely scenario is that the observed B-fields have been generated by a dominant\n11small-scale dynamo across the fast rotating and turbulent gas and dust molecular warped disk. For\nthe outer disk, the turbulence may be driven by disk gravitational instabilities mixed with density\nwave or merge-driven streaming motions. In addition to the turbulent B-fields from the dense and\ncold ISM of spiral galaxies, galaxy interaction can enhance this turbulence, and produce disordered\ngas flows and shear. These gas flows can enhance and produce highly anisotropic turbulent fields\nand weak regular fields47, 48, and regenerate the B-fields at a faster rate than those observed in\nspiral galaxies49. Differential rotation by the high rotational velocity field, \u0018280km s\u0000118, 22, in\nthe molecular disk may also contribute to regenerate the B-fields at a fast rate3.\nThe B-field orientations within the central \u0018500pc diameter, excluding the active nuclei,\nfrom our 89\u0016m emissive polarization differs from the 1\u00002\u0016m absorptive polarization orientations\n31, 34. This difference can be explained by the far-IR emissive polarization tracing deep regions of\nthe galaxy disk. In addition, this region is also affected by the inner bubble42. Due to extinction\neffects, the near-IR absorptive polarization is only sensitive to the outer layers of the dust lane,\nwhich is not affected by the dynamics of the molecular warped disk or the inner \u0018500pc bubble.\nInterestingly, our B-field model within the central 500pc shows an ordered pattern that is more\ncompatible with the near-IR absorptive observations than the far-IR emissive observations.\nAt the location of the active nucleus, we measured a polarized point source with a B-field\norientation almost perpendicular to the direction of the jet. We interpret that the B-field may arise\nfrom a circumnuclear dusty structure around the obscured active nuclei. These physical conditions\nare similar to those observed in the radio-loud Cygnus A galaxy50.\n12We have demonstrated that far-IR polarization observations are a powerful tool to study the\nmagnetic field morphology in the cold ISM of galaxies, especially when radio polarimetric ob-\nservations are difficult due to contamination by radio jets or other radio sources. Due to the 180\u000e\nambiguity from our observations, we cannot study the B-field direction, so the regular large-scale\nB-field direction must be analyzed with future Faraday rotation measurements obtained from radio\npolarimetric observations. These observations are required to distinguish between the compressed,\namplified, highly anisotropic fields and unidirectional field generated by the dynamo process. In\naddition, further hydromagnetical simulations are required to refine our results to explain a regen-\nerated field by the merger of an elliptical and spiral galaxy. This work serves as a strong reminder\nof the potential importance of B-fields, usually completely overlooked, in the formation and evo-\nlution of galaxies.\n1. Beck, R., Chamandy, L., Elson, E., & Blackman, E. G. Synthesizing observations and theory to\nunderstand galactic magnetic fields: progress and challenges. Galaxies 8, 1-51 (2019).\n2. Beck, R. & Wielebinski, R. Magnetic fields in galaxies in Planets, Stars and Stellar Systems.\nVolume 5: Galactic Structure and Stellar Populations (ed. Oswalt, T. D. & Gilmore, G.) 641-\n723 (Springer, Berlin, 2013)\n3. Ruzmaikin, A., Sokolov, D. & Shukurov, A. Magnetism of spiral galaxies. Nature 336, 341-347\n(1988).\n4. Brandenburg, A. & Subramanian, K. Astrophysical magnetic fields and nonlinear dynamo the-\nory.Phys. Rep. ,417, 1-209 (2005).\n135. Haverkorn, M., Brown, J. C., Gaensler, B. M. & McClure-Griffiths, N. M. The outer scale\nof turbulence in the magnetoionized galactic interstellar medium. Astrophys. J. 680, 362-370\n(2008).\n6. Pakmor, R. & Springel, V . Simulations of magnetic fields in isolated disc galaxies. Mon. Not.\nR. Astron. Soc. 432, 176-193 (2013).\n7. Pakmor, R., Marinacci, F. & Springel, V . Magnetic fields in cosmological simulations of disk\ngalaxies. Astrophys. J. Lett. 783, 1-5 (2014).\n8. Marinacci, F., et al. First results from the IllustrisTNG simulations: radio haloes and magnetic\nfields. Mon. Not. R. Astron. Soc. 480, 5113-5139 (2018)\n9. Su, K.-Y . et al. Stellar feedback strongly alters the amplification and morphology of galactic\nmagnetic fields. Mon. Not. R. Astron. Soc. 473, 111-115 (2018)\n10. Ntormousi, E. Magnetic fields in massive spirals: The role of feedback and initial conditions.\nAstron. Astrophys. 619, 1-4 (2018)\n11. Bernet, M. L., Miniati, F., Lilly, S. J., Kronberg, P. P. & Dessauges-Zavadsky, M. Strong\nmagnetic fields in normal galaxies at high redshift. Nature 454, 302-304 (2008).\n12. Mao, S.A., et al. Detection of microgauss coherent magnetic fields in a galaxy five billion\nyears ago. Nat. Astron. 1, 621–626 (2017)\n1413. Rieder, M. & Teyssier, R. A small-scale dynamo in feedback-dominated galaxies as the origin\nof cosmic magnetic fields - I. The kinematic phase. Mon. Not. R. Astron. Soc. 457, 1722-1738\n(2016).\n14. Graham, J. A. The structure and evolution of NGC 5128. Astrophys. J. 232, 60-73 (1979).\n15. Struve, C., Oosterloo, T. A., Morganti, R. & Saripalli, L. Centaurus A: morphology and kine-\nmatics of the atomic hydrogen. Astron. Astrophys. 515, 1-12 (2010).\n16. Baade, W. & Minkowski, R. On the Identification of Radio Sources. Astrophys. J. 119, 215-\n231 (1954).\n17. Quillen, A. C., Neumayer, N., Oosterloo, T. & Espada, D. The warped disk of Centaurus A\nfrom a radius of 2 to 6500pc. Publ. Astron. Soc. Aust. 27, 396-401 (2010).\n18. Quillen, A. C., de Zeeuw, P. T., Phinney, E. S. & Phillips, T. G. The kinematics of the molecular\ngas in Centaurus A. Astrophys. J. 391, 121-136 (1992).\n19. Mirabel, I. F., et al. A barred spiral at the centre of the giant elliptical radio galaxy Centaurus\nA.Astron. Astrophys. 341, 667-674 (1999).\n20. Leeuw, L. L., Hawarden, T. G., Matthews, H. E., Robson, E. I. & Eckart, A. Deep submillime-\nter imaging of dust structures in Centaurus A. Astrophys. J. 565, 131-139 (2002).\n21. Quillen, A. C., et al. Spitzer observations of the dusty warped disk of Centaurus A. Astrophys.\nJ.645, 1092-1101 (2006).\n1522. van Gorkom, J. H., van der Hults, J. M., Haschick, A. D. & Tubbs, A. D. VLA HI observations\nof the radio galaxy Centaurus A. Astron. J. 99, 1781-1788 (1990).\n23. Nicholson, R. A., Bland-Hawthorn, J. & Taylor, K. The structure and dynamics of the gaseous\nand stellar components in Centaurus A. Astrophys. J. 387, 503-521 (1992).\n24. Quillen, A. C., Graham, J. R. & Frogel, J. A. The Warped Disk of Centaurus A in the Near-\nInfrared. Astrophys. J. 412, 550-567 (1993).\n25. Sparke, L. S. A dynamical model for the twisted gas disk in Centaurus A. Astrophys. J. 473,\n810-821 (1996).\n26. Elvius A. & Hall J. S. Polarimetric observations of NGC 5128 (Cen A) and other extragalactic\nobjects. Low. OB. 6, 123-134 (1964).\n27. Berry, D. S. Polarimetry of the peculiar elliptical galaxy, NGC 5128. Ph.D. Thesis, Univer-\nsity of Durham, England Available at Durham E-Theses Online: http://etheses.dur.ac.uk/7592/\n(1985).\n28. Bailey, J., Sparks, W. B., Hough, J. H. & Axon, D. J. Infrared polarimetry of the nucleus of\nCentaurus A: the nearest blazar? Nature 322, 150-151 (1986).\n29. Hough. J. H., Bailey, J. A., Rouse, M. F. & Whittet, D. C. B. Interstellar polarization in the\ndust lane of Centaurus A (NGC 5128). Mon. Not. R. Astron. Soc. 227, 1-5 (1987).\n30. Schreier, E. J., Capetti, A., Macchetto, F., Sparks, W. B. & Ford, H. J. Hubble Space Telescope\nimaging and polarimetry of NGC 5128 (Centaurus A). Astrophys. J. 459, 535-541 (1996).\n1631. Packham, C., et al. Near-infrared and millimetre polarimetry of Cen A. Mon. Not. R. Astron.\nSoc. 278, 406-416 (1996).\n32. Scarrott, S. M., Foley, N. B., Gledhill, T. M. & Wolstencroft, R. D. BVRI imaging polarimetric\nstudies of the galaxy NGC 5128. Mon. Not. R. Astron. Soc. 282, 252-262 (1996).\n33. Capetti, A., et al. Hubble Space Telescope Infrared imaging polarimetry of Centaurus A: im-\nplications for the unified scheme and the existence of a misdirected BL Lacertae nucleus. As-\ntrophys. J. 544, 269-276 (2000).\n34. Jones, T. J. The magnetic field geometry in M82 and Centaurus A. Astron. J. 120, 2920-2927\n(2000).\n35. Parkin, T. J., et al. The gas-to-dust mass ratio of Centaurus A as seen by Herschel. Mon. Not.\nR. Astron. Soc. 422, 2291-2301 (2012).\n36. Krause, M., et al. CHANG-ES. XXII. Coherent magnetic fields in the halos of spiral galaxies.\nAstron. Astrophys. 639, 1-39 (2020).\n37. Israel, F. P. Centaurus A - NGC 5128. Astron. Astrophys. Rev. 8, 237-278 (1998).\n38. Espada, D., et al. Disentangling the circumnuclear environs of Centaurus A: gaseous spiral\narms in a giant elliptical galaxy. Astrophys. J. Lett. 776, 1-5 (2012).\n39. Ruiz-Granados, B., Rubi ˜no-Mart ´ın, J. A. & Battaner, E. Constraining the regular Galactic\nmagnetic field with the 5-year WMAP polarization measurements at 22 GHz. Astron. Astrophys.\n522, 1-14 (2010).\n1740. Braun, R., Heald, G. & Beck, R. The Westerbork SINGS survey. III. Global magnetic field\ntopology. Astron. Astrophys. 514, 1-19 (2010).\n41. Espada, D., et al. Disentangling the circumnuclear environs of Centaurus A. I. High-resolution\nmolecular gas imaging. Astrophys. J. 695, 116-134 (2009).\n42. Quillen, A. C., et al. Discovery of a 500 parsec shell in the nucleus of Centaurus A. Astrophys.\nJ.641, 29-32 (2006).\n43. Jaffee, T. R. Practical modeling of large-scale galactic magnetic fields: status and prospects.\nGalaxies 7, 1-52 (2019).\n44. Jones, T. J., Infrared polarimetry and the interstellar magnetic field. Astrophys. J. 346, 728-734\n(1989).\n45. Hildebrand, R. H., Dotson, J. L., Dowell, C. D., Scheluning, D. A. & Vaillancourt, J. E. The\nfar-infrared polarization spectrum: first results and analysis. Astrophys. J. 516, 834-842 (1999).\n46. Cald ´u-Primo, A., et al. A High-dispersion molecular gas component in nearby galaxies. As-\ntron. J. 146, 1-14 (2013).\n47. Chy ˙zy, K. T. & Beck, R. Magnetic fields in merging spirals - the Antennae. Astron. Astrophys.\n417, 541-555 (2004).\n48. Drzazga, R. T., Chyz ˙y, K. T., Jurusik, W. & Wi ´orkiewicz, K. Magnetic field evolution in\ninteracting galaxies. Astron. Astrophys. 533, 1-17 (2011).\n1849. Rodenbeck, K. & Schleicher, D. R. G. Magnetic fields during galaxy mergers. Astron. Astro-\nphys. 593, 1-12 (2016).\n50. Lopez-Rodriguez, E., Antonucci, R., Chary, R.-R. & Kishimoto, M. The highly polarized\ndusty emission core of Cygnus A. Astrophys. J. 861, 1-7 (2018).\n51. Bland, J., Taylor, K. & Atherton, P. D. The structure and kinematics of the ionized gas within\nNGC 5128 (Cen A) - I. Taurus observations. Mon. Not. R. Astron. Soc. 228, 595-621 (1987).\n52. Eckart, A., et al. Observations of CO isotopic emission and the far-infrared continuum of\nCentaurus A. Astrophys. J. 363, 451-463 (1990).\n53. Phillips, T. G., et al. CO Emission from Centaurus A. Astron. J. Lett. 322, 73-77 (1987).\n54. Thomson, R. C. Galaxy shredding - I. Centaurus A, NGC 5237 and the Fourcade-Fegueroa\nshred. Mon. Not. R. Astron. Soc. 257, 689-698 (1992).\n55. Ferrarese, L., et al. The discovery of cepheids and a distance to NGC 5128. Astrophys. J. 654,\n186-218 (2007).\n56. Karachentsev, I. D., et al. New distances to galaxies in the Centaurus A group. Astron. Astro-\nphys. 385, 21-31 (2002).\n57. Rejkuba, M., Minniti, D., Silva, D. R. & Bedding, T. R. Long period variables in NGC 5128.\nII. Near-IR properties. Astron. Astrophys. 411, 351-360 (2003).\n58. Vaillancourt, J. E., et al. Far-infrared polarimetry from the Stratospheric Observatory for In-\nfrared Astronomy. Proc. Soc. Photo-Opt. Instrum. Eng. 6678 (2007).\n1959. Dowell, C. D., et al. HAWCPol: a first-generation far-infrared polarimeter for SOFIA. Proc.\nSoc. Photo-Opt. Instrum. Eng. 7735 (2010).\n60. Lopez-Rodriguez, E., et al. The emission and distribution of dust of the torus of NGC 1068.\nAstrophys. J. 859, 1-10 (2018).\n61. Kov ´acs, A. SHARC-2 350 micron observations of distant submillimeter-selected galaxies and\ntechniques for the optimal analysis and observing of weak signals. Dissertation (Ph.D.), Cali-\nfornia Institute of Technology. doi:10.7907/8ZM9-6671 (2006).\n62. Kov ´acs, A. CRUSH: fast and scalable data reduction for imaging arrays. Proc. Soc. Photo-Opt.\nInstrum. Eng. 7020 , doi:10.1117/12.790276 (2008).\n63. Harper, D. A., et al. HAWC+, the far-infrared camera and polarimeter for SOFIA. J. Astron.\nInstrum. 7, 1-41 (2018)\n64. Laing, R. A. Magnetic fields in extragalactic radio sources. Astrophys. J. 248, 87-104 (1981).\n65. Salvatier J., Wiecki T.V . & Fonnesbeck C. Probabilistic programming in Python using PyMC3.\nPeerJ Comput. Sci. 2:e55 (2016).\n66. Ramachandran, P. & Varoquaux, G., Mayavi: 3D Visualization of Scientific Data Comput. Sci.\nEng. 13, 40-51 (2011).\n67. Fujimoto, M. & Tosa, M. Spiral condensation of gas in disk galaxies by bisymmetric twisted\nmagnetic fields - Two-dimensional case. Publ. Astron. Soc. Jpn. 32, 567-580 (1980).\n2068. Weingartner, J. C. & Draine, B. T. Dust grain-size distributions and extinction in the Milky\nWay, Large Magellanic Cloud, and Small Magellanic Cloud. Astrophys. J. 548, 296-309 (2001).\n69. Aitken, D., K., Hough, J. H., Roche, P. F., Smith, C. H. & Wright, C. M. Mid-infrared po-\nlarimetry and magnetic fields: an observing strategy. Mon. Not. R. Astron. Soc. 348, 279-284\n(2004).\n70. Hildebrand, R. H. The determination of cloud masses and dust characteristics from submil-\nlimetre thermal emission. Q. J. R. Astron. Soc. 24, 267-282 (1983).\n71. Chuss, D. T., et al. HAWC+/SOFIA multiwavelength polarimetric observations of OMC-1.\nAstrophys. J. 872, 187-208 (2019).\n72. King, P. K., Chen, C.-Y ., Fissel, L., M. & Li, Z.-Y . Effects of grain alignment efficiency on\nsynthetic dust polarization observations of molecular clouds. Mon. Not. R. Astron. Soc. 490,\n2760-2778 (2019).\n73. Espada, D., et al. Star formation efficiencies at giant molecular cloud scales in the molecular\ndisk of the elliptical galaxy NGC 5128 (Centaurus A). Astrophys. J. 887, 1-14 (2019).\n74. Dobbs, C. L. & Price, D. J. Magnetic fields and the dynamics of spiral galaxies. Mon. Not. R.\nAstron. Soc. 383, 497-512 (2008).\nAcknowledgements We sincerely thank Kandaswamy Subramanian, Kostas Tassis, Ric Davies, B-G An-\ndersson, and Tom Osterloo for many useful discussions on theoretical approaches, hydromagnetic simu-\nlation, gas dynamics, and dust grain alignment theories. We are grateful to the four anonymous referees,\n21whose comments greatly helped to clarify and improve the manuscript. Based on observations made with\nthe NASA/DLR Stratospheric Observatory for Infrared Astronomy (SOFIA) under the 07 0032 Program.\nSOFIA is jointly operated by the Universities Space Research Association, Inc. (USRA), under NASA\ncontract NAS2-97001, and the Deutsches SOFIA Institut (DSI) under DLR contract 50 OK 0901 to the\nUniversity of Stuttgart.\nAuthor Contributions E.L.-R. is responsible for all aspects of this paper, i.e. lead the project, carried out\nobservations, developed the analysis methods and data reductions, interpreted results, and wrote the text.\nCompeting Interests The author declares that he have no competing financial interests.\nCorrespondence Correspondence and requests for materials should be addressed to E.L.-R. (email: en-\nloro@gmail.com).\nFigure 1: The measured B-field of Centaurus A using far-infrared polarimetric observations with\nHA WC+/SOFIA. a , Composite image with overlaid normalized B-field orientation (black) and total inten-\nsity contours (blue) using 89 \u0016m SOFIA/HAWC+ observations. The FOV of the image is 60000\u000240000\n(9:6\u00026:4kpc2). The composite image has 870 \u0016m observations from LABOCA/APEX (orange), X-ray\ndata from Chandra (blue), and V-band data from the Wide field Imager on the MPG/ESO 2.2-m telescope\nat La Silla, Chile showing the background stars and the dust lane. b, 89\u0016m total flux (color scale) with\noverlaid polarization measurements (white) with P=\u001bP>2:5and rotated by 90\u000eto show the measured\nB-field orientation within the central 25000\u000218000(4:0\u00022:9kpc2) region. Contours are shown in steps of\n2n\u001b, wheren= 2:0;2:5;3:0;::: and\u001b= 1:06mJy sqarcsec\u00001.c, Polarized flux (color scale) with total\nintensity contours and B-field orientation as image on b. A legend polarization of 10% (black) and beam\n22size of 7:8”(red circle) are shown.\nFigure 2: Three-dimensional representation of the best inferred B-field morphological model of Cen-\ntaurus A. a , The B-field morphology within 11\u000211kpc2from a face-on view, b, at the inclination and tilt\nangle as inferred from our model, c, and within the central 8\u00023kpc2are shown. The XYZ axes are shown\nfor each panel. A fake colormap has been added to visually distinguish the B-field lines as a function of the\ndistance to the core.\nFigure 3: B-field model vs. observations. a , Best inferred B-field model (black lines) at the same angular\nresolution ( 3:90”) as the HAWC+ observations at 89 \u0016m (red circle). The measured B-field (blue lines) by\nHAWC+ is shown. Low polarized emission and those within the central 0:5kpc have been masked to only\naccount for the magnetic field in the molecular disk. b, Angular difference between the B-field orientations\nfrom HAWC+, PA H, and model, PA M. Each pixel is at the Nyqvist sampling ( 3:90”) and central polarization\nmeasurements have been masked as described in Section 3. The red circle (bottom right) shows the beam\nsize of the HAWC+ observations. c, Histogram of the angular difference in bins of 5\u000e. The median (dashed\nvertical line) and 1\u001b(thin-dashed vertical lines) angular differences are shown.\nFigure 4: Physical regions of Centaurus A as a function of their polarization properties. a ,PI\u0000I\nplot vs. velocity dispersion of the12CO(2\u00001)emission line from ALMA observations (colorscale). The\nblue dotted vertical lines at I= 1000 and2700 MJy sr\u00001show the limits of the three physical regions\nfound in this analysis. The dashed diagonal black lines show the maximum expected polarization, P/I0\n= 15, 6.5, and 1.5% for each of these physical regions, respectively. The 1\u0000\u001buncertainty of the polarized\nfluxes are shown. b, The three physical regions, outer disk (black), molecular disk (blue), and low polarized\nregions (red), are shown. The measured B-field (white lines) is shown as in Figure 1.\n23Figure 5: Polarization measurements as a function of the multi-phase ISM. P,I, andPIvs.T,\nNH+H2, and\u001bv;12CO(1\u00000)for the outer disk (black circles), molecular disk (blue squares), and low polarized\nregions (red crosses) identified on Figure 4. For all figures, the 1\u0000\u001buncertainties are shown. A weighted\npower-law fit is shown for each region with the power-law indexes in Extended Data Figure 9.\nTable 1: Medians of the physical parameters of each region identified in Figure 4.\nMethods\nCentaurus A. Another alternative name is NGC 5128. The galaxy disk divides the elliptical\ngalaxy and obscures the nucleus and most of the structures at optical wavelengths within the central\n\u0018500 pc30. Estimations of the velocity field of several tracers are HI22H\u000b51,12CO(1-0)52, and\n12CO(2-1)53. The merger is estimated to be about 1:6\u00003:2\u0002108yr ago14, 15, 18, 23, 24, 54.\nDistance. We have taken a distance of 3:42\u00060:18(random)\u00060:25(systematic) Mpc by55. The\ndistance is estimated using Cepheid variables by the Hubble Space Telescope . Other authors quote\n3:63\u00060:07Mpc56based on median distance to the galaxy group formed by 17 galaxies (Centaurus\nA included); and 3:84\u00060:25Mpc57using two independent methods: Mira period luminosity and\nthe tip of the red-giant branch. In general, the range of distances is estimated to be 3:2\u00004:2Mpc\nbased on the review by37. We estimate that any potential uncertainties ( \u001821:8pc) due to the\nestimation of the distance to Centaurus A are smaller than a single detector pixel scale ( \u001870pc)\nfrom our observations.\n24Observations and Data Reduction. Centaurus A was observed (PI: Lopez-Rodriguez, E., ID:\n070032) at 89\u0016m using the High-resolution Airborne Wideband Camera-plus (HAWC+)58, 59, 63\non the 2:7-m Stratospheric Observatory For Infrared Astronomy (SOFIA) telescope. Observations\nwere performed on 20190717 during the SOFIA New Zealand deployment. HAWC+ polarimetric\nobservations simultaneously measure two orthogonal components of linear polarization arranged\nin two arrays of 32\u000240pixels each, with a pixel scale of 4:0200pixel\u00001, and beam size (full width\nat half maximum, FWHM) of 7:8000at89\u0016m. We performed observations using the on-the-fly-\nmap (OTFMAP) polarimetric mode. This technique is an experimental observing mode performed\nduring SOFIA Cycle 7 observations as part of engineering time to optimize the polarimetric ob-\nservations of HAWC+. We here focus on the scientific results of Centaurus A and describe the\nhigh-level steps of these observations.\nWe performed OTFMAP polarimetric observations in a sequence of four Lissajous scans,\nwhere each scan has a different halfwave plate (HWP) position angle (PA) in the following se-\nquence: 5\u000e,50\u000e,27:5\u000e, and 72:5\u000e. This sequence is called ‘set’ hereafter. In this new HAWC+\nobserving mode, the telescope is driven to follow a parametric curve with a nonrepeating period\nwhose shape is characterized by the relative phases and frequency of the motion. An example of\nthe OTFMAP for total intensity observations of NGC 1068 using HAWC+/SOFIA is shown by60.\nEach scan is characterized by the scan amplitude, scan rate, scan phase, and scan duration. A sum-\nmary of the observations is shown in Extended Data Figure 1. The scan amplitude is defined by the\nlength of the scan parallel (EL) and perpendicular (XEL) to the direction of the telescope elevation.\nWe performed rectangular scans to cover the central molecular warped disk (the ‘parallelogram’)\n25of\u00183:8kpc (\u001824000) in diameter along the diagonal of the scan.\nWe reduced the data using the Comprehensive Reduction Utility for SHARP II v.2.42-1\n(CRUSH61, 62) and the HAWC DRP V2.3.2 pipeline developed by the data reduction pipeline\ngroup at the SOFIA Science Center. Each scan was reduced by CRUSH, which estimates and\nremoves the correlated atmospheric and instrumental signals, solves for the relative detector gains,\nand determines the noise weighting of the time streams in an iterated pipeline scheme. Each\nreduced scan produces two images associated with each array. Both images are orthogonal com-\nponents of linear polarization at a given HWP PA. We estimated the Stokes IQU parameters using\nthe double difference method in the same manner as the standard chop-nod observations carried\nby HAWC+ described in Section 3.2 by63. The degree (P) and PA of polarization were corrected\nby instrumental polarization (IP) estimated using OTFMAP polarization observations of planets.\nWe estimated an IP of Q/I =\u00002:1% and U/I = 0:8% at 89\u0016m, with an estimated uncertainty\nof\u00180:8%. The IP using OTFMAP observations are in agreement within their uncertainties with\nthe estimated IP using chop-nod observations of Q/I =\u00001:6% and U/I = 0:8%. To ensure the\ncorrection of the PA of polarization of the instrument with respect to the sky, we took each set\nwith a fixed line-of-sight (LOS) of the telescope. For each set, we rotated the Stokes QU from\nthe instrument to the sky coordinates. The polarization fraction was debiased and corrected by\npolarization efficiency. The final Stokes IQU, P, PA, polarized intensity (PI), and their associated\nerrors were calculated and re-sampled to one-quarter of the beam size, 1:9500at89\u0016m.\nSeveral advantages and limitations are found with the OTFMAP polarization mode. The ad-\nvantages are the reduction of overheads and radiative offsets when compared with the chop-nod\n26technique. The overheads are improved by a factor of two in comparison with the chop-nod tech-\nnique. This improvement is due to the fact that the OTFMAP is constantly integrating with the\nsource on the FOV while covering off-source regions to estimate the background levels. For the\nOTFMAP method, the telescope is always on-axis, without chopping the secondary mirror as it\nis in the chop-nod technique. Thus, the radiative offset is not present and the sensitivity of the\nobservations was estimated to improve by a factor of 1.6. The limitation of this technique is the re-\ncovering of large-scale diffuse and faint emission from the astrophysical objects. This is a result of\nthe finite size of the array, variable atmosphere conditions, variable detector temperature, and the\napplied filters in the reduction steps to recover extended emission. In general, the noise increases\nas a function of the length, L, of the extended emission as \u0018L2. Although we can adjust CRUSH\nparameters to recover the extended emission in total intensity, the polarization is highly affected\nby the filter selection of the data reduction software. We applied several filters using CRUSH to\nrecover large-scale emission structures of Centaurus A without compromising the intrinsic polar-\nization pattern of the astrophysical object. We performed chop-nod and OTFMAP observations of\nwell-known objects, e.g. 30 Doradus and OMC-1, to test several filter options. We conclude that a\ncombination of the extended filter with 12 iterations using CRUSH can recover large-scale emission\nstructures up to 20000from our observations of Centarus A at 89\u0016m. Using Herschel images at 70\nand160\u0016m35, we estimate that the fluxes at the regions where we are not able to recover large-\nscale emission from our observations are \u00140:01Jy/sqarcsec at 89\u0016m. For the total on-source time\nof3200 s (Extended Data Figure 1) and assuming an expected degree of polarization of \u00185% from\nour observations at 89\u0016m, the signal-to-noise ratio (SNR) of the polarization is estimated to be\n27\u00141:5for the unrecovered regions (Sensitivities can be estimated using the SOFIA Instrument Time\nEstimator (SITE) at https://dcs.arc.nasa.gov/proposalDevelopment/SITE/index.jsp). Although we\nare missing some of the large-scale structures, our observations would have not been sensitive\nenough to provide statistically significant polarization measurements of this region. As the molec-\nular disk of Centaurus A has a size of \u001814000in diameter at 89\u0016m, we are able to achieve our\nscientific goals using our observations.\nPhysical regions as a function of the B-field orientation and its dispersion. We computed his-\ntograms of the degree of polarization and B-field orientations for all polarization measurements\nwithP=\u001bP\u00153:0. We identify three regions in the galaxy disk based on the overall orientation and\ndispersion of the magnetic field: 1) west side with a PA range of [90;120]\u000e, 2) east side with a PA\nrange of [120;175]\u000e, and 3) low-polarized regions with large angle dispersion at PA >175\u000e. For\neach region, we compute the mean of the degree of polarization and B-field orientation and their\ndispersions (Extended Data Figure 2). Extended Data Figure 3 shows a color code polarization\nmap of the three regions.\nThe west and east regions of the warped disk: We estimate a median PA of the B-field\norientation of 105\u000ein the west side of the galactic disk, and 147\u000ein the east side. Previous optical\nto near-IR polarimetric studies have measured a PA of polarization of 110\u0000117\u000efrom dichroic\nabsorption, which is roughly parallel to the dust lane27–34. These measurements were based on\nobservations of several patches across the dust lane and/or the central \u00181kpc.\nThe central kpc: The third region is mostly spatially coincident with low polarized areas\n28within the central\u00180:8kpc diameter around the AGN and at the edge of the northwest regions of\nthe galaxy disk. This region has the largest PA dispersion of 28:9\u000e. Extended Data Figure 4 shows\na zoom-in of the central 5000\u00025000(0:8\u00020:8kpc2), where the twist of the PA of polarization from\nthe dust lane to the nucleus can be seen. The central 0:8kpc region lacks molecular gas and dust\ndue to energetic processes near the AGN that have disturbed the inner regions of the warped disk\n21, 42. We conclude that the large PA dispersion, low polarization, and change of PA of polarization\nare due to the combination of the intrinsic polarization in the dust lane and the nucleus, both having\na different PA of polarization and arising from different physical structures.\nMagnetic field model. In this section, we describe the mathematical description of the magnetic\nfield model. The three components of the vectorial magnetic field in cylindrical coordinates ( r,\u001a,\nz) centered at the galactic center are described as\nBr=B0sin \t 0cos\u001fz (1)\nB\u001a=B0cos \t 0cos\u001fz (2)\nBz=B0sin\u001fz (3)\nwhereB0is the amplitude of the regular magnetic field strength, \t0is the pitch angle of the spiral\npattern, and \u001fzis the pitch angle vertical field given by\n\u001fz=\u001f0tanh(z\nz0) (4)\n29where\u001fzis interpreted as a helical angle with a radial pitch angle \u001f0, purely longitudinal with a\ngiven vertical scale at z064.\nThe magnetic field is projected on the plane of the sky, which adds two free parameters, the\ninclinationi, and tilt angle \u0012. For a face-on view i= 0\u000e, and an edge-on view i= 90\u000e. The tilt\nangle, also described as the position angle of the major axis of the projected galaxy plane, has a\nreference\u0012= 0\u000ealong the north-south direction and positively increases east from north. We use\nEuler rotations around the x-axis, Rx[i], and the z-axis, Rz[\u0012], to compute the final magnetic field\nBsky=Rx[i]Rz[\u0012]Bo, where Bo= (Bx,By,Bz) in cartesian coordinates.\nModel constraints. Thermal emissive polarization arising from magnetically aligned dust\ngrains is not directly sensitive to the magnetic field strength but rather to variations in dust grain\nalignment, and gradients in temperature and column density. Although we can include a variation\nof the magnetic field strength as a function of the radius in our model, this information is negligible\nfor interpreting the thermal emissive polarization. Radio polarimetric observations are sensitive to\nthe magnetic field strength, but these observations have not been acquired for the warped disk of\nCentaurus A. Thus, we do not have any information about the radial dependency of the magnetic\nfield strength in Centaurus A, where B0is an unknown variable with no constraints. If we still\ninclude a variable magnetic field strength as a function of the radius such as B0=r\u0000\u000band assume\nan isotropic strength variation on the XYZ axes, given that the projected PA of polarization on the\nplane of the sky from our model is PAB= arctan (By;sky=Bx;sky),PABdoes not depend on the\nvalue ofB0. A model using an isotropic variation of the magnetic field strength produces the same\nB-field orientation as our current model. If we consider the radial dependence of the B-field to be\n30different for each XYZ axes, then PA Bchanges as a function of the radius. This model provides a\ndifferent B-field orientation when compared with our model. As we do not have information about\nthe radial dependency of the magnetic field strength in Centaurus A, and we are only interested in\nthe orientation of the magnetic field, we assume a constant magnetic field strength, B0=const:\nThe polarization in the central 0:8\u00020:8kpc2is affected by energetic processes associated\nwith the AGN. Therefore, we have excluded this region in this analysis. The exclusion zone is\ndetermined by the low polarized regions (Extended Data Figure 3) and the physical inner bubble\nfound by42, which indicate a different mechanism of polarization than the one taking place in the\ngalactic disk.\nComputation of the magnetic field model. These definitions allow us to explore the param-\neter space of the five free parameters \t0,\u001f0,z0,i, and\u0012. We compute synthetic magnetic field\norientation maps, as well as images of the expected distribution of the magnetic field orientation\nin a full three-dimensional view of the magnetic field morphology of Centaurus A. We compute\nthe magnetic field morphology within a box of 231\u0002176\u000210pixels3with a physical pixel scale\nof64pc equal to the Nyqvist sampling of the 89\u0016m HAWC+ observations, which corresponds\nto a physical volume of 14:78\u000211:26\u00020:64kpc3. We perform a Markov Chain Monte Carlo\n(MCMC) approach using the dynamical evolution Metropolis sampling step in the PYTHON code\nPYMC 365. The prior distributions are set to flat within the ranges shown in Extended Data Figure\n5. We performed a blind exploration for \t0and\u001f0in the [\u000090;90]\u000erange and found that \ttends\nto be negative and \u001fpositive, so we constrained the prior distributions with the ranges shown in\nExtended Data Figure 5. We run 10 chains each with 5000 steps and an additional burning 2000\n31steps, which provides a total of 50000 steps for the full MCMC code useful for data analysis. Final\nposterior distributions, median values, and 1\u001buncertainties are shown in Extended Data Figure 5\nand Extended Data Figure 6.\nModel vs. Observations.38, 41measured a tilt angle, \u0012, of\u0018120\u000efor the\u00181kpc cen-\ntral region of the12CO(2\u00001)emission, in agreement with our inferred \u0012= 119:3+0:5\n\u00000:4\u000e. Using\n12CO(2\u00001)emission,38found a spiral arm from 0:2kpc to\u00181kpc, with a width of 0:5\u00060:2\nkpc and pitch angle of \u000020\u000e. These authors fitted by eye a logarithmic spiral to the deprojected\n12CO(2\u00001)emission using an inclination angle of 70\u000e. Our model uses a three-dimensional\naxisymmetric spiral pattern with a helical component of the magnetic field, which inferred an in-\nclination close to an edge-on view i= 89:5+0:7\n\u00000:8\u000e. The discrepancy of pitch angles may arise in the\ndifferences between model selection and projection effects. We tested our models with low incli-\nnations to compare with the results from38. In the range of [70\u000080]\u000e, this set of models provide\nreasonably good inferred magnetic field configurations only for the NW region of the molecular\ndisk. For these models, we measured an angular offset \u0001PA=PAH\u0000PAM>30\u000efor all\npolarization measurements in the SE region of the molecular disk. Thus, all these solutions are not\nconsidered in the final inferred magnetic field configuration presented above.\nIf we consider SNR PA=<PAH\u0000PAM>=\u001bPAHas a measurement of the signal-to-noise\nbetween the residuals and the observational uncertainties, we estimate that SNR PAis in the range\nof[2:4\u000010:4]\u001blevel, where 2:4\u001bmeans that the residual is 2:4times larger than the observational\nPA uncertainty. We estimate that \u001825:7% (78out of 303) of measurements are within [2:4\u00003:0]\u001b.\n32Three-dimensional B-field model. As we compute a full three-dimensinal magnetic field\nmodel, we have produced a representation of the magnetic field lines for the best inferred model\nin Figure 2 using the 3D data visualization PYTHON package M AYAVI (Mayavi can be found at\nhttps://docs.enthought.com/mayavi/mayavi/)66.\nAlternative magnetic field models. In this section, we describe the alternative B-field models.\nWe compare our magnetic field model with other alternative configurations. Bisymmetric magnetic\nfield configurations can also be used to describe the morphologies of magnetic field in galaxies.\nHowever, this morphology has only been argued for M81, which may probably be affected by\nFaraday depolarization1. We study this magnetic field configuration described as39\nBr=B0cos [\u001e\u0006\fln(r\nr0)] sin \t 0cos\u001fz (5)\nB\u001a=B0cos [\u001e\u0006\fln(r\nr0)] cos \t 0cos\u001fz (6)\nBz=B0sin\u001fz (7)\nwhere\u001eis the azimuthal angle, \f= 1=\t067, andr0is the radial scale of the pitch angle in the\nplane of the galaxy. For Milky Way studies, typically, r0is fixed at the Sun galactrocentric distance\nfor a given measurement of the magnetic field strength. However, we do not have this information\nfor Centaurus A and r0is an extra free parameter, which we assume to be in the range of [0;10]\nkpc in our model. We performed the same fitting methodology as described in ’Computation of\nthe magnetic field model’ but with the extra free parameter, r0. We found that for all models where\nr0<1:5kpc, the bisymmetric spiral model does not provide any magnetic field configuration\n33compatible with our observations. For r0>1:5kpc, the bisymmetric spiral model converges to\nan axisymmetric spiral morphology within the central 3kpc, and both models axisymmetric and\nbisymmetric are not distinguished. Because the bisymmetric and axisymmetric spiral field models\nprovide similar solutions for the magnetic field morphology within the central 3kpc of Centaurus\nA, and the axisymmetric spiral field model has less free parameters, we use the axisymmetric spiral\nmodel results for our analysis.\nThe thermal emission of the molecular disk of Centaurus A has been modeled using warped\ndisk configurations18, 21, 23, 24. This model consists of tilted concentric rings of material that predict\nthe structure of the warm and cold dust, as well as the velocity fields of the gas in the galaxy disk.\nAlthough warped disk models provide compatible solutions for total intensity and spectroscopic\nobservations of several tracers, the concentric and tilted rings do not provide a physical model\nfor magnetic field configurations. Thus, magnetic field configurations based on purely thermal\nemission or spectroscopic analysis are not considered.\nExpected emissive polarization from the ISM. In the following section, we show that our obser-\nvations trace the magnetic field morphology by means of thermal emission by magnetically aligned\ndust grains. We can estimate the expected emissive polarization based on previous measurements\nof the absorptive polarization. At 2.2 \u0016m, the typical degree of polarization in the galaxy disk\nis PK\u00182% with a visual extinction of AV= 7 mag. (\u001cK= 0:1AV= 0:7)31, 33, 34. Using the\ntypical extinction curve (The extinction curve used for the optical depth conversion can be found\nat https://www.astro.princeton.edu/ ˜draine/dust/dustmix.html) of the Milky Way for R V= 3:168,\nwe estimate the optical depth at 89\u0016m to be\u001c89= 1:18\u000210\u00002\u001cK= 8:26\u000210\u00003, which implies\n34that the dust lane is optically thin at 89 \u0016m. Under the optically thin condition, the emissive polar-\nization can be estimated as Pem\n89=\u0000Pabs\n89=\u001c89, where the negative sign indicates the change of 90\u000e\nfrom absorptive to emissive polarization69. We scaled the extinction curve, which is representative\nof the absorptive polarization, at 2\u0016m to be P K= 2% to estimate the expected absorptive polariza-\ntion at 89\u0016m, yield Pabs\n89= 0:024%. Finally, we estimate the expected emissive polarization at 89\n\u0016m to bePem\n89\u00182:9%. This result is in good agreement with our median P= 3:5\u00061:9% across\nthe whole galaxy disk and it shows that our far-IR polarization measurements at 89 \u0016m arise from\nthermal emission of magnetically aligned dust grains. These results also suggest that the ISM in\nthe dust lane of Centaurus A is similar to the ISM of our Galaxy.\nTemperature and column density maps. In this section, we show the details to compute the\ntemperature and column density maps. To compute the temperature and hydrogen column density\nmaps (Extended Data Figure 7), we registered and binned to a common 3:2”resolution the 70\u0000500\n\u0016mHerschel observations taken with PACS and SPIRE instruments. Then, for every pixel we fit\nan emissivity modified blackbody function with a constant dust emissivity index \f=\u00002:0735. We\nderived the molecular hydrogen optical depth as NH+H2=\u001c=(k\u0016mH), with the dust opacity k=\n0:1cm2g\u00001at250\u0016m70, and the mean molecular weight per hydrogen atom \u0016= 2:8. Temperature\nand column density values range from [20\u000030]K andlog(NH+H2[cm\u00002]) = [20:6\u000022:06], in\nagreement with35. Note that our maps cover the central 500pc, which was removed from the\nanalysis by35.\nThermal polarization vs. Intensity. In the following section, we discuss the analysis of the\nselected subregions across the galactic disk of Centaurus A. In the optically thin regime assuming\n35maximal dust alignment along the LOS and ordered B-field orientation, emissive polarization, P, is\nconstant with optical depth, \u001c(i.e.P/\u001c0). Thus, some level of variations in the B-field geometry\nalong the LOS will decrease the emissive polarization. For a completely random variation of the\nB-field orientation along the LOS with a well-defined optical depth scale length \u001c, thenP/\u001c\u00000:5\n44, where isotropic random variations of the B-fields do not give rise to polarization. However,\nobservations have found that the slope can be steeper than \u00000:5, e.g. molecular clouds such as\nOMC-171, and external galaxies such as NGC 106860. Hydromagnetic simulations have derived\nthat variations of dust grain alignment efficiency and turbulence with column density may explain\nsome of these trends in molecular clouds72, with a lower-limit of P/\u001c\u00001.\nWe have plotted (see Extended Data Figure 8) the debiased polarized flux against the total\nintensity at 89\u0016m becauseP\u0000Icontains selection effects due to the chosen quality cuts of the\ndegree of polarization and intensity. As the polarized flux is defined as PI=P\u0002I, the equivalence\nis such as a slope \u000binP/I\u000bbecome\f=\u000b+1inPI/I\f=I\u000b+1. ThePI\u0000Iplot (see Figure\n4 and Extended Data Figure 8) shows a complex structure. In the direction of increasing column\ndensity (also I),PIincreases up to log(NH+H2[cm\u00002])\u001821:40(I\u0018750MJy sr\u00001), and then\ndecreases with an inflection point at log(NH+H2[cm\u00002])\u001821:49(I\u00181000 MJy sr\u00001). After,\nPIincreases again up to log(NH+H2[cm\u00002])\u001821:72(I\u00182050 MJy sr\u00001), and then sharply\ndecreases with an inflection point at log(NH+H2[cm\u00002])\u001821:76(I\u00182700 MJy sr\u00001). The\nfinal trend of PIis an increase with increasing column density up to log(NH+H2[cm\u00002])\u001822:06\n(I\u00189150 MJy sr\u00001).\nOuter disk: Using the column density cut of log(NH+H2[cm\u00002])\u001421:49, we identify this\n36region as the outer disk of the galaxy (black in Figure 4) with a size of \u00186kpc in diameter. From\ntheP\u0000Iplot, the bulk of values shows a trend of P/\u001c\u00001. If only the outer layers of the disk have\nperfectly aligned dust grains, then the polarized emission will be diluted by additional unpolarized\nflux, and aP/\u001c\u00001is expected. If high velocity dispersion may be present, then the P\u0000Imay\nshow a steep decrease as in the molecular disk.\nMolecular disk: Using the range of column densities 21:49\u0014log(NH+H2[cm\u00002])\u001421:76,\nwe identify this region as the molecular disk (i.e. parallelogram) of a size of \u00183kpc in diameter\n(blue in Figure 4). In the PI\u0000Iplot, although the upper values follow the maximum P= 6:5%,\nthePIincreases with NH+H2with a steep drop at log(NH+H2[cm\u00002])\u001821:76and atT\u001830K.\nWe find that the velocity dispersion distribution with a mean of \u001810km s\u00001is spatially correlated\nwith a trend of P/\u001c\u00001. The velocity dispersion distribution peaking at \u001bv;12CO(1\u00000)\u001820km s\u00001\nis associated with the steep decrease in the PI\u0000Iplot, which creates a bump at the end of the\nP\u0000Iplot.\nCore and low polarized regions: This region contains several different physical structures.\nBecause we are focused on studying the magnetic fields in the galaxy disk, we only spatially iden-\ntify the polarized structures in this region. The values with high PI,I, andNH+H2are identified\nas the core of Centaurus A at the location of the AGN. The values with intermediate PI, and high\nIandNH+H2, are identified as star forming regions in the molecular disk. The values with low PI,\nand highIandNH+H2are identified as low polarized regions mostly located in the central 500pc\naround the AGN.\n37Thermal polarization vs.12CO(1-0) velocity dispersion. In this section, we describe the data\nanalysis using the12CO(1-0) observations. We use the12CO(1-0) emission line presented by73\nto estimate the velocity dispersions of the galaxy disk of Centaurus A. These observations pro-\nvide an angular resolution of 2:88\u00021:67”(46:1\u000226:9pc2), which will allow us to estimate\nthe kinematics at the turbulence scales of the galaxy disk. Observations were acquired from the\nALMA Archive using the Program ID 2013.1.083.3.S (12CO(1-0) ALMA data can be found at\nhttp://telbib.eso.org/?bibcode=2019ApJ...887...88E; PI: Espada, D.). We compute the moments\nusing IMMOMENTS task in CASA with a 1\u001bclip, where \u001b= 68:5mJy/beam. Moments were\nsmoothed to the beam size, 7:8”, of the HAWC+ observations. The integrated12CO(1-0) line\n(moment 0) and velocity dispersion (moment 2) are shown in Extended Data Figure 7. Using the\nseveral physical components distinguished in Figure 4, we measure the median velocity dispersion\n(see Extended Data Figure 9) across the molecular disk (parallelogram) to be \u001bMD\nv;CO= 18:4\u00069:2\nkm s\u00001, and outer disk to be \u001bOD\nv;CO= 6:4\u00066:0km s\u00001(Extended Data Figure 10).73estimated a ve-\nlocity dispersion of \u001815km s\u00001, and\u00185km s\u00001in the molecular disk and outer disk, respectively.\nThese two populations of velocity dispersion may be biasing the mean \u001bv;12CO(1\u00000)= 18:4\u00069:2\nkm s\u00001to larger values than those typically found, \u00188km s\u00001, in nearby galaxies.\nPower-law fits. We have fit a power-law, y/x\u000b, for each of the plots and physical regions shown\nin Figure 5. Extended Data Figure 9 shows the power-law indexes, \u000b, of the fits for each physical\nregion.\nAlternative scenario. Another possibility for the observed B-field could arise from field com-\npression and amplification due to density-shocks, where density-shocks are discontinuities in the\n38flow due to change of gas properties (density, temperature, velocity, etc.). These density-wave\nshocks bend the B-fields along the shocks, reducing the angular dispersion, which would make the\ngeneral B-field morphology indistinguishable from regular large-scale fields74. Therefore, small-\nscale anisotropic B-fields arising from compression and amplification due to density-shocks are\ninconsistent with our measured angular dispersions.\nData Availability The data that support the plots within this paper and other findings of this study are\navailable from https://galmagfields.com or from the corresponding author upon reasonable.\nCode Availability The code that support the algorithms within this paper and other findings of this study\nare available from https://github.com/galmagfields or from the corresponding author upon reasonable.\nExtended Data Figure 1: Summary of OTFMAP polarimetric observations.\nExtended Data Figure 2: Polarization measurements of the several regions of the galactic disk.\nExtended Data Figure 3: Physical regions based on B-field orientation and degree of polarization.\nHistograms of P ( a) and PA ( b) of polarization for measurements with P=\u001bP\u00153:0. Three distinct regions\nare found for the PA of polarization, which are identified with the west (orange), east (red) and low polarized\n(black) regions. The boundaries of each region are shown with vertical black dashed lines. c, The spatial\ncorrespondence of the three regions identified using the PA distributions are shown with the same colors as\nthe plots at b. The total intensity contours are shown as in Figure 1. A legend polarization of 10% (black)\nand beam size of 7:8”(red circle) are shown.\nExtended Data Figure 4: Magnetic field of the central 5000\u00025000(0:8\u00020:8kpc2) of Centaurus A. a ,\n39Total flux (colorscale) with overlaid B-field orientations (white lines). b, Polarized flux (colorscale) with\noverlaid B-field orientation (white lines). A legend polarization of 5% (black) and beam size of 7:8”(red\ncircle) are shown.\nExtended Data Figure 5: Parameters of the magnetic field morphological model.\nExtended Data Figure 6: Posterior distributions of the magnetic field morphological model. A refer-\nence of the parameter definitions, used priors, and median values is shown in Extended Data Figure 5.\nExtended Data Figure 7: Polarization map vs physical parameters. Temperature ( a) and column den-\nsity ( b) maps of Centaurus A with overlaid B-field orientation (while lines) with P=\u001bp\u00152:5andPI=\u001bPI\u0015\n2. Temperature contours start at 20K increasing in steps of 0:5K, and column density density contours start\natlog(NH+H2[cm\u00002]) = 20:6increasing in steps of 0.1.12CO(1\u00000)integrated line emission ( c) and\nvelocity dispersion ( d) of the warped disk of Centaurus A with overlaid B-field orientation (white lines)\nwithP=\u001bp\u00152:5andPI=\u001bPI\u00152.\nExtended Data Figure 8: Polarized flux vs. total intensity plots. P\u0000IandPI\u0000Iplots at 89\u0016m vs\ntemperature ( a,b) and column density ( c,d). The trend of the bulk of the P\u0000Iplot,P/\u001c\u00001, is shown\nas a black solid line in panels aandc. The uncertainties of the debiased polarized intensity in plots band\ndare shown. The blue dotted vertical lines at I= 1000 and2700 MJy sr\u00001show the limits of the three\nphysical regions found in this analysis. The black dotted lines in panels banddshow the maximum expected\npolarization, P/I0= 15, 6.5, and 1.5% for each of these physical regions, respectively.\nExtended Data Figure 9: Power-law index of plots from Figure 5.\n40Extended Data Figure 10: Velocity dispersion of the outer and molecular disk.12CO(1\u00002)veloc-\nity dispersion histograms of the outer disk (red) and molecular disk (blue) as identified in Figure 4. The\nmedian (solid line) and 1\u001b(dashed line) are shown for each physical structure. These values correspond to\n\u001bv;12CO(1\u00000)= 18:4\u00069:2(km s\u00001), and\u001bv;12CO(1\u00000)= 6:4\u00066:0(km s\u00001) for the molecular disk and\nouter disk, respectively.\nFigure 1:\n41Table 1: Medians of the physical parameters of each region identified in Figure 4\nParameter Outer Disk Molecular Disk Low Polarized\nT (K) 27:8\u00060:7 28:8\u00060:8 31:4\u00060:9\n\u001bv;12CO(1\u00000)(km/s) 6:4\u00066:0 18:4\u00069:2 34\u00064\nP (%) 9:5\u00063:3 3:9\u00062:4 1:2\u00060:6\nPI (MJy/sr) 55\u000613 63\u000618 71\u000626\nI (MJy/sr) 627\u0006264 1761\u0006554 6500\u00061876\nTable 2: Summary of OTFMAP polarimetric observations.\nWavelength Bandwidth Beam Size Scan Rate Scan Phase Scan Amplitude\n(\u0016m) ( \u0016m) (”) (”/s) (\u000e) (EL\u0002XEL; ”)\n89 17.0 7.80 100 0 180 \u0002120\nScan Duration #Sets t on\u0000source\n(s) (s)\n100 8 3200\n42Table 3: Polarization measurements of the several regions of the galactic disk. ?\u001bPand\n\u001bPAcorrespond to the dispersion of the measurements of the degree and PA of polariza-\ntion per region.\nRegions PA B\u001bPAB?P\u001bP?\n(\u000e) (\u000e) (%) (%)\nWest 105 8:6 3:5 1:6\nEast 147 15:5 3:6 2:0\nLow-P\u0000 28:9 1:5 1:7\nAll\u0000 \u0000 3:5 1:9\nTable 4: Parameters of the magnetic field morphological model\nParameter Symbol Priors Median Posteriors\nPitch angle (\u000e) \t0 [\u000090;0]\u000054:9+0:5\n\u00000:5\nRadial pitch angle (\u000e)\u001f0 [0;90] 74 :4+10:5\n\u000016:5\nVertical scale (kpc) z0 [0;10] 1 :7+0:3\n\u00000:4\nInclination (\u000e)i [0;90] 89 :5+0:7\n\u00000:8\nTilt angle (\u000e)\u0012 [0;180] 119 :3+0:5\n\u00000:4\n43Table 5: Power-law index of plots from Fig. 5\nParameters Regions T NH\u001bv;12CO(1\u00000)\nP Outer disk\u00006:89\u00060:47\u000016:49\u00061:24\u00000:18\u00060:04\nMolecular disk \u00005:35\u00060:49\u000031:91\u00061:21\u00000:13\u00060:03\nLow Polarized\u00005:00\u000610:12 15:39\u000612:06\u00000:02\u00060:05\nI Outer disk 5:27\u00060:47 31:07\u00065:99 0:12\u00060:02\nMolecular disk 5:33\u00060:17 53:87\u00062:78 0:18\u00060:01\nLow Polarized 9:73\u000610:33 63:79\u000652:19 0:21\u00060:01\nPI Outer disk\u00001:52\u00060:36 16:94\u00065:77\u00000:01\u00060:04\nMolecular disk 0:33\u00060:36 20:71\u00069:79 0:10\u00060:03\nLow Polarized 5:41\u000621:32 75:15\u000669:91 0:11\u00060:05\n44Figure 2:\n45Figure 3:\n46Figure 4:\n471.010.0P (%)\nOuter disk\nMolecular disk\nLow polarized regions\n100010000I (MJy/sr)\n25 26 27 28 29 30 31 32 33\nT [K]100PI (MJy/sr)\n21.0 21.2 21.4 21.6 21.8 22.0\nNH+H2 (cm2)\n25 5075\n12CO(10) (km/s)\nFigure 5:\n48Figure 6: Extended Data Figure 3\n49Figure 7: Extended Data Figure 4.\n50() = 54.90+0.50\n0.50\n506070800()\n0() = 74.40+10.50\n16.50\n1.21.51.82.1z0(kpc)\nz0(kpc) = 1.70+0.30\n0.40\n88899091i0()\ni0() = 89.50+0.70\n0.80\n56.0\n 55.2\n 54.4\n 53.6\n()\n118.2118.8119.4120.0120.6PA0()\n50 60 70 80\n0()\n1.2 1.5 1.8 2.1\nz0(kpc)\n88 89 90 91\ni0()\n118.2 118.8 119.4 120.0 120.6\nPA0()\nPA0() = 119.30+0.50\n0.40\nFigure 8: Extended Data Figure 6.\n51Figure 9: Extended Data Figure 7.\n52Figure 10: Extended Data Figure 8.\n530 10 20 30 40 50 60\nv,CO (km/s)\n020406080100Number of measurementsMolecular Disk: v,12CO(10)=18.4±9.2 (km/s)\nOuter Disk: v,12CO(10)=6.4±6.0 (km/s)\nFigure 11: Extended Data Figure 10.\n54"    },    {        "title": "0801.3668v1.Magnetic_properties_of_the_extended_periodic_Anderson_model.pdf",        "content": "arXiv:0801.3668v1  [cond-mat.str-el]  23 Jan 2008Typeset with jpsj2.cls <ver.1.2> Letter\nMagnetic properties of the extended periodic Anderson mode l\nAkihisaKoga,1,2NorioKawakami ,1RobertPeters2and Thomas Pruschke2\n1Department of Physics, Kyoto University, Kyoto 606-8502, J apan\n2Institut f¨ ur Theoretische Physik Universit¨ at G¨ ottinge n, G¨ ottingen D-37077, Germany\nWe study magnetic properties of the extended periodic Ander son model, which includes\nelectron correlations within and between itinerant and loc alized bands. By combining dynam-\nical mean-field theory with the numerical renormalization g roup we calculate the sublattice\nmagnetization and the staggered susceptibility to determi ne the phase diagram in the particle-\nhole symmetric case. We find that two kinds of magnetically or dered states compete with\nthe Kondo insulating state at zero temperature, which induc es non-monotonic behavior in\nthe temperature-dependent magnetization. It is furthermo re clarified that a novel magnetic\nmetallic state is stabilized at half filling by the competiti on between Hund’s coupling and the\nhybridization.\nKEYWORDS: periodic Anderson model, dynamical mean field the ory, numerical renormalization group\nStrongly correlated electron systems with degener-\nate orbitals have attracted great interest. One of the\npopular examples is the manganite (La ,Sr)MnO 3.1In\nthis system, itinerant electrons in the egband are cou-\npled to localized electrons in the t2gband through\nHund’s coupling, leading to a competition between an-\ntiferromagnetic (AF) correlations and the double ex-\nchange ferromagnetic correlations. This yields a com-\nplex phase diagram with various types of ordered ground\nstates. Other interesting examples are (Ca ,Sr)2RuO42\nand La n+1NinO3n+1.3,4In these compounds, the chem-\nical substitution or the change in temperature is sug-\ngested to trigger an orbital-selective Mott transition,5,6\nwhere some of the orbitals become localized by electron\ncorrelations, while the others still remain itinerant. It\nis also proposed that in these compounds, localized and\nitinerant electrons are hybridized with each other, induc-\ning heavy fermion or bad metal behavior at low temper-\natures.7–10\nAn important point in the above compounds is that\nthe localized and itinerant bands in the d-orbital and\ntheir correlations play a crucial role in stabilizing the\nmagnetically ordered state or heavy fermion state. Gen-\nerally speaking, in the systems with localized bands, the\nhybridization together with local electron correlations\nscreen spins, leading to heavy fermion behavior with the\nlarge density of states (DOS) around the Fermi level,\nthe so-called Kondo effect.11–13In contrast, Hund’s cou-\npling enforces parallel spins in different orbitals, enhanc-\ning magnetic correlations, as discussed for the mangan-\nites.14–17Therefore, an interesting question arises how\nrobust the nonmagnetic ground state is in systems with\nlocalized and itinerant bands. In a previous paper,9we\nhave investigated the extended periodic Anderson model\n(EPAM) to clarify how the Kondo and Mott insulat-\ning states compete with the metallic state in the para-\nmagnetic case. However, a magnetic instability has not\nbeen discussed so far, which may be important to un-\nderstand low-temperature properties in real materials\nsuch as some transition metal oxides and f-electron sys-tems. Furthermore, the competing interactions may lead\nto nontrivial behavior in the magnetically ordered state.\nTherefore, it is highly desirable to clarify the magnetic\npropertiesinthesystemwithlocalizedanditinerantelec-\ntrons.\nFor this purpose, we consider a correlatedelectron sys-\ntem which is described by the following Hamiltonian as,\nH=Ht+/summationdisplay\niH(i)\nloc, (1)\nHt=/summationdisplay\n/angbracketleftij/angbracketrightασ/bracketleftBig\nt(α)\nij−µδij/bracketrightBig\nc†\niασcjασ,\nH(i)\nloc=V/summationdisplay\nσ/parenleftBig\nc†\ni1σci2σ+c†\ni2σci1σ/parenrightBig\n+/summationdisplay\nαUαniα↑niα↓+/summationdisplay\nσσ′(U′−Jδσσ′)ni1σni2σ′\n−J/summationdisplay\nσc†\ni1σci1¯σc†\ni2¯σci2σ−J/summationdisplay\nαc†\niα↑c†\niα↓ci¯α↑ci¯α↓,\nwherec†\niασ(ciασ) creates (annihilates) an electron with\nspinσ(=↑,↓) and band index α(= 1,2) at the ith site,\nandniασ=c†\niασciασ. For the band α,t(α)\nijrepresents\nthe transferintegral, Vthe hybridizationbetween bands.\nThe intra-band and inter-band Coulomb interactions are\ndescribed by UαandU′, whileJdenotes Hund’s cou-\npling. Finally, µis the chemical potential.\nTo investigate the correlated electron system with one\nband localized and the other itinerant, we set the hop-\nping integral for the α= 2 band to t(2)\nij= 0, for sim-\nplicity. Then this model is regarded as the EPAM with\nnot only intraband interactions but also interband ones.\nHere, to discuss magnetic properties, we make use of\ndynamical mean-field theory (DMFT).18–21In DMFT,\na lattice model is mapped to an effective quantum im-\npurity, where local electronic correlations are taken into\naccount exactly. The requirement that the site-diagonal\nlattice Green function is equal to that of the effective\nquantum impurity then leads to a self-consistency con-\ndition for the parameters entering the impurity problem.\n12 J. Phys. Soc. Jpn. Letter Author Name\nThis treatment is formally exact in infinite spatial di-\nmensions and even for three dimensions reliable results\nare obtained if non-local correlations are allowed to be\nignored.\nWhen an AF instability is treated in the framework of\nDMFT,22–25the self-consistency equation for the sublat-\nticeγ[= (A,B)] is represented as,\n/bracketleftBig\nˆG−1\n0γ σ(z)/bracketrightBig\n11=z+µ−/parenleftbiggD\n2/parenrightbigg2/bracketleftBig\nˆGloc¯γ σ(z)/bracketrightBig\n11,(2)\nwhereˆG0is the non-interacting Green function for the\neffective impurity model and ˆGlocthe local Green func-\ntion. Here, we have used the Bethe lattice with infinite\ncoordination, where Dis the half bandwidth for the bare\nitinerant band. Note that the self-consistency equation is\nrepresented only by one component of the Green func-\ntion.9,26,27Therefore, we can introduce the effective im-\npurity model, where one of the impurity bands connects\nto the effective bath. The corresponding hybridization\nfunction is then defined by ∆ γσ=/parenleftbigD\n2/parenrightbig2/bracketleftBig\nˆGloc¯γ σ(z)/bracketrightBig\n11.\nTo solve the effective impurity model quantitatively, we\nuse the numerical renormalization group (NRG)28,29as\nan impurity solver.30,31This allows us to access low\nenergy properties, which are particularly important in\nthe Kondo insulating state. Details of the method can\nbe found in literature.32–34The hybridization function\n∆γσshould be determined self-consistently through the\nDMFT condition eq. (2).\nIn the half-filled system without frustration, the AF\ngroundstateisstabilizedifintersitecorrelationsarelarge\nenough. At each site, the ferromagnetic Hund’s coupling\ncompetes with the effective AF exchange coupling in-\nduced by the Coulomb interaction together with the\nhybridization.9Therefore, two possible spin configura-\ntions are naively expected for the AF states, which are\nschematically shown in Fig. 1 (a). A magnetization for\neach configuration may be given as m(I,II)\nAF=m1±m2\nwheremα=/summationtext\ni(−1)Pi(niα↑−niα↓)/(2N),Pi= 0(1)\nfori∈A(B), andNis the total number of sites. Note\nthat these magnetizations are not ordinary order param-\neters characterizing the AF (I) and (II) states since they\nshould be finite in both states. Nevertheless, we can dis-\ntinguish between these states: When |m(I)\nAF|−|m(II)\nAF|>\n0(<0), the AF (I) [(II)] is realized, and a first-order\nphase transition (crossover) between the two states oc-\ncurs atT= 0(T/negationslash= 0), where Tis the temperature. Fur-\nthermore, m(I)\nAFis an important quantity since it charac-\nterizesthemagnetizationofionsinrealmaterialsandcan\nbe observed in inelastic neutron scattering experiments.\nHere, to discuss how magnetic fluctuations develop at\nlow temperatures, we calculate the magnetization and\nthe staggered susceptibility χloc(=∂mAF/∂hAF), where\nhAFis the staggered magnetic field. The susceptibility is\nobtained as the slope of the sublattice magnetization for\na tiny field hAF/D∼0.002, which gives a good estimate\nexcept in the vicinity of the critical temperature.\nIn this paper, we focus on the half-filled EPAM to\ndiscuss magnetic properties. In particular, we fix the ra-\ntioλ=J/U= 0.1 andU=U′+2Jto clarify how thecompetition between Hund’s coupling and the hybridiza-\ntion affects the magnetic phase diagram at low temper-\natures. The effects of hole doping and/or the magnetic\nfield, which may yield a complex phase diagram includ-\ning several types of magnetically ordered states,35,36are\nalso interesting problems. However, these are out of the\nscope in this paper, and will be discussed elsewhere.\nIn Figs. 1 (b)-(d), we show the results obtained by the\nDMFT with the NRG. When U/D= 1.0andV/D= 0.3,\nConfiguration (I) \nConfiguration (II) 0 0.1 0.2 0123\nχloc  D \nT/D (b) U/D=1.0, V/D=0.3 \nχloc  D (I) (II) \nT/D (a) \n0 0.1 0.2 00.2 0.4 0.6 0.8 1\n01234\nmAF (II) (c) U/D=2.0, V/D=0.3 \nχloc  D χloc  D (I) \n(II) m1\nm2\n00.1 -0.5 00.5 \n0 0.1 0.2 00.2 0.4 0.6 0.8 1\n01234(d) U/D=2.0, V/D=0.2 \nmAF (I) χloc  D \nχloc  D (I) \n(II) \nT/D \nFig. 1. (a) Possible spin configurations. (b), (c) and (d) sho w the\nstaggered susceptibilities and the magnetizations as a fun ction of\nthe temperature T. Solid lines represent |m(I)\nAF|and dashed lines\nare guide to eyes. Solid and open symbols represent the resul ts\nfor the configuration (I) and (II).\nno singularity appears in both staggered susceptibilities,\nas shown in Fig. 1 (b). This suggests that the non-\nmagnetic Kondo singlet ground state is realized at low\ntemperatures.Wealsofindthatmagneticfluctuationsfor\nthe configuration (II) are enhanced at low temperatures\nand the system is closeto the AF (II) state. In fact, when\ntheparametersareslightlychanged,theAF(II) stateap-\npears at low temperatures. The staggered susceptibility\ndiverges at a critical temperature TNand a spontaneous\nmagnetization m(II)\nAFappears below TN, as shown in Fig.\n1 (c). It is also found that a shoulder structure appears\nin the temperature-dependent magnetization. This im-\nplies that magnetic correlationsforeach band arenot en-\nhancedat thesametemperature(seealsotheinset). This\ninteresting feature will be discussed later. On the other\nhand, when the Coulomb interaction and Hund’s cou-\npling are relatively large ( U/D= 2.0 andV/D= 0.2),\nχ(I)\nAFdiverges at the critical temperature and m(I)\nAFap-\npears at lower temperatures, as shown in Fig. 1 (d). The\nAF (I) state is then realized in the ground state.\nBy performing similar calculations for various model\nparameters, we end up with the low-temperature phase\ndiagram at half filling, shown in Fig. 2 (a). To discuss\nthe phase transitions between these states in detail, weJ. Phys. Soc. Jpn. Letter Author Name 3\nshow in Fig. 2 (b) the staggered magnetization for each\nband when V/D= 0.3. In the case of small U/D, the in-\n1.5 2 2.5 3-1 01\nU/D m1\nm2V/D=0.3 \n00.5 100.5 1V/D=0.05 (a) \n(b) 01234500.2 0.4 0.6 0.8 \nU/D V/D \nKondo Kondo \nAF(I) AF(I) AF(II) AF(II) \nFig. 2. (a) The phase diagram of the EPAM at T/D= 4.6×\n10−4. Solid squares, crosses and solid circles represent the Kon do\nsinglet, AF (I) and AF (II) phases, respectively. Open circl es\nindicate the metallic state realized in the shaded area, whi ch\nwill be discussed (Fig. 4). The phase boundaries are guide to\neyes. Dashed line is the phase boundary obtained fromthe str ong\ncoupling limit.(b)Themagnetization foreach band when V/D=\n0.3. Circles and triangles represent the results at T/D= 4.6×\n10−4and 0.015, respectively. The inset shows the results for\nV/D= 0.05.\nterband hybridization screens the local moments, and no\nmagnetizations appear in both bands. Therefore, in the\nregion(U/D)<(U/D)c1[∼1.5]theparamagneticKondo\nphase is realized, as shown in Fig. 2 (b). The increase\nin the interaction induces magnetic moments for both\nbands with opposite signs. This implies that a continu-\nous phase transition occurs to the AF (II) phase. Further\nincrease in the interaction leads to a jump singularity in\nthe temperature-dependent magnetization, at which the\nsublattice magnetizations become parallel. The first or-\nder phase transition then drives the system to the AF (I)\nstate at ( U/D)c2∼2.1. It is also found that the jump\nsingularity vanishes when the temperature is slightly in-\ncreased, as shown in Fig. 2 (b). Therefore, the phase\ntransition between two AF states is present at zero tem-\nperature only, while a crossover occurs at finite temper-\natures.\nThe competition between these phases may be ex-\nplained by considering the strong coupling limit ( U→\n∞). The system is then reduced to the Kondo neck-\nlace Heisenberg model as, H=Jinter/summationtext\nijSi1·Sj1+\nJintra/summationtext\niSi1·Si2, where Siα=/summationtext\nss′1\n2c†\niαsσss′ciαs′,\nJinteris the intersite effective exchange coupling, and\nthe intrasite one is represented by Jintra, instead of J.\nWe note that Jinter(= 4t2/U) is always positive, while\nJintradepends on the interactions and the hybridiza-\ntion. Its magnitude is given by the lowest singlet-triplet\ngap of the local Hamiltonian HlocasJintra= ∆E[=U(/radicalbig\nλ2+4(V/U)2−3λ)]. In the model, three distinct\nphases appear in the phase diagram, which is schemat-\nically shown in Fig. 3. When j(=zJintra/Jinter)≫1,\nj\nj=0 j=1/2 AF(I) AF(II) Kondo \nFig. 3. Ground-state phase diagram for the Kondo necklace\nmodel.\nthe Kondo singlet phase is stabilized, where zis the\ncoordination number. An increase in the intersite cou-\nplingJinterenhances AF correlations and a second-order\nphase transition, at last, occurs to the AF (II) phase\nat a critical value jc(= 1/2). The phase boundary ob-\ntained from the strong coupling limit, which is given\nby (V/D)c= (U/4D)/radicalbig\n(4λ+(D/U)2)(8λ+(D/U)2, is\nconsistentwith that in the EPAM, asshownin Fig. 2(a).\nOn the other hand, the AF (I) and (II) phases are sepa-\nrated by the condition j= 0[V/D=√\n2λ(U/D)], where\nthe localized spins Si2are completely decoupled and the\nphase transition never occurs. In contrast, two bands are\ncoupled through the hybridization Vin the EPAM, lead-\ning to a first-order phase transition. The corresponding\nphase boundary is in good agreement with the condition\nj= 0. When V/D= 0.05 andT/D= 4.6×10−4, we\ncould not find the AF (II) phase between AF (I) and\nKondo insulating phases, as shown in the inset of Fig.\n2 (b). This is consistent with the fact that in the weak\ncoupling region the energy scale of magnetic correlations\nis fairly small and the AF state is stable only at very low\ntemperatures. Therefore, we believe that in the ground-\nstate magnetic phase diagram, the AF (II) phase always\nappears between the AF (I) and Kondo phases.\nNext,wediscussthefinite-temperaturemagneticprop-\nertiessuchastheshoulderstructureinthemagnetization\nshownbefore.When oneconcentratesonthelocalHamil-\ntonianHloc, the low-lying singlet and triplet states can\nbe considered to be four-fold degenerate down to a cer-\ntaintemperature T∗=|∆E|.WhenT∗issufficientlylow,\nan intersite exchange stabilizes the magnetically ordered\nstate, where the itinerant band is almost fully polarized\nwhilenearly free localized spins appear in the other. This\nreveals that orbital-selective like features appear in the\nintermediate temperature T∗< T < T Nand magnetic\ncorrelations in the localized band are enhanced below\nT∗. Such behavior is clearly found in the vicinity of the\nphase boundary between AF (I) and (II) states, where\nT∗≪TN. In fact, when U/D= 2.0 andV/D= 0.3,\n(T∗,TN)∼(0.032D,0.075D) and the shoulder structure\nappears in the temperature-dependent magnetization, as\nshown in the inset of Fig. 1 (c). Furthermore, we find\nnontrivial behavior in the observable quantity |m(I)\nAF|.\nNamely, when decreasing the temperature, it once van-\nishes at a certain temperature below T∗, and is induced\nagain, shown as the solid line in Fig. 1 (c). This non-\nmonotonic behavior may be observed in the intensity of\nthe magnetic peak in the inelastic neutron scattering ex-\nperimentsforsometransition-metaloxideswith localized\nand itinerant bands such as Ca 2−xSrxRuO4(0<x<\n0.5), which then allows us to discuss the competition\nbetween Hund’s coupling and the hybridization at each\nsite.4 J. Phys. Soc. Jpn. Letter Author Name\nWe would like to mention another remarkable feature\nfor the metal-insulator (MI) transition at half filling. In\ninfinite dimensions, the Hubbard and the Kondo lattice\nmodel have a chance to realize the MI transition with-\nout magnetism when the system is strongly frustrated.\nHowever, in the system without frustration, the mag-\nnetically ordered state is more stable than the param-\nagnetic metallic state and the Mott insulating state at\nzero temperature, where the charge gap always opens\naround the Fermi level.24,25The EPAM treated here is\nnot frustrated, but has competing interactions between\nthe orbitals at each site, which may be referred to as or-\nbital frustration . Therefore, it is not trivial whether the\nground state is always insulating under orbital frustra-\ntion. To clarify this, we show the DOS in Fig. 4 (a).\nWhenU/D≤1.5, the Kondo insulating state is real-\n-3 -2 -1 012U/D=1.2 \nU/D=1.6 \nU/D=2.0 \nU/D=2.4 \nU/D=2.8 \nω/D (a) 0\n1\n2\n3ρ1(ω=0) ρ2(ω=0) U/D (b) \nFig. 4. (a) DOS in the system with the majority spin in the itin -\nerant band when V/D= 0.3 andU/D= 1.2,1.6,2.0,2.4,2.8 at\nT/D= 4.6×10−4. (b) DOS at the Fermi level as a function of\nU/D. Solid (dashed) lines represent the results for the itinera nt\n(localized) band.\nized, where the hybridization gap clearly appears in the\nlocalized band. In the case U/D≥2.8, the structures\nin the DOS in both bands appear far from the Fermi\nlevel, where the polarized ground state is realized with\nthe configuration (I). On the other hand, when the sys-\ntem approaches the intermediate region, spectral weight\nis built up around the Fermi level. In fact, Fig. 4 (b)\nclearly shows that a finite DOS at each band appears\nin the intermediate region, although the α= 2 band is\nalmost localized in the case (2 .4< U <2.8). This fact\nimplies that a metallic phase appears between the dif-\nferent insulating states. Since the DOS is continuously\nchanged at the metal-insulator transition points, a dras-\ntic change was not found in the magnetization shown in\nFig. 2 (b). In our phase diagram, this magnetic metal is\nstable in the weak coupling region shown as the shaded\narea in Fig. 2 (a). We recall that when V= 0(U= 0),\nthe system is reduced to the Kondo lattice (conventional\nperiodic Anderson) model, where the AF (Kondo) in-\nsulator is always realized except for the decoupled limit\nU=V= 0.Therefore,weclaim thatthe competition be-\ntweenHund’scouplingandthehybridizationintroducing\na kind of orbital frustration plays an important role in\nstabilizing the magnetic metal. This is in contrast to the\nother correlated systems such as the Hubbard model and\nthe Kondo lattice model, where the magnetic insulating\nstate is always realized, as mentioned above.\nIn summery, we have investigated the extended peri-\nodic Anderson model by combining the dynamical mean-field theory with the numerical renormalization group.\nWe have discussed the magnetic properties at low tem-\nperatures to clarify how the magnetically ordered states\ncompete with the Kondo insulating state. Furthermore,\nwe have for the first time found the magnetic metallic\nstate in the half-filled system without lattice frustration,\nwhich is instead stabilized by orbital frustration .\nThis work was partly supported by a Grant-in-Aid\nfrom the Ministry ofEducation,Science, Sports and Cul-\nture of Japan [19014013 (N.K.), 17740226 (A.K.)] and\nthe German Science Foundation (DFG) through the col-\nlaborative research grant SFB 602 (T.P., A.K.) and the\nproject PR 298/10-1(R.P.) A.K. would in particular like\nto thank the SFB 602for its support and hospitality dur-\ning his stays at the University of G¨ ottingen. Part of the\ncomputations were done at the Supercomputer Center at\nthe Institute for Solid State Physics, University of Tokyo\nand Yukawa Institute Computer Facility.\n1) A.Urushibara, et al., Phys.Rev.B 51,14103 (1995); Y.Morit-\nomo,et al., Nature 380, 141 (1996).\n2) S. Nakatsuji, et al., Phys. Rev. Lett. 90, 137202 (2003).\n3) K. Sreedhar et al., J. Solid State Chem. 110, 208 (1994); Z.\nZhanget al.,ibid.108, 402 (1994); 117, 236 (1995).\n4) Y. Kobayashi, et al., J. Phys. Soc. Jpn. 65, 3978 (1996)\n5) V. I. Anisimov et al., Eur. Phys. J. B 25, 191 (2002).\n6) A. Koga, et al., Phys. Rev. Lett. 92, 216402 (2004); Prog.\nTheor. Phys. Suppl. 160, 253 (2005).\n7) A. Koga, et al., Phys. Rev. B 72, 045128 (2005).\n8) L. de’ Medici, et al., Phys. Rev. B 72, 205124 (2005).\n9) A. Koga, et al., Phys. Rev. B 77, 045120 (2008).\n10) J. B¨ unemann, et al., J. Phys. Condens. Matter, 19, 436206\n(2007).\n11) N. Grewe and F. Steglich, in Handbook on the Physics and\nChemistry of Rare Earths , edited by K. A. Gschneidner, Jr.\nand L. Eyring (North-Holland, Amsterdam, 1991), vol. 14, p.\n343.\n12) N. Grewe, Solid State Commun. 50, 19 (1984); K. Yamada,\nand K. Yoshida, in Theory of Heavy Fermions and Valence\nFluctuations , edited by T. Kasuya and T. Saso (Springer,\nBerlin, 1985); Th. Pruschke, and N. Grewe, Z. Phys. B 74,\n439 (1989).\n13) A.C. Hewson, The Kondo Problem to Heavy Fermions , Cam-\nbridge University Press (Cambridge, 1993).\n14) C. Zener, Phys. Rev. 82, 4031 (1951).\n15) P.W.Anderson and H.Hasegawa, Phys.Rev. 100, 675 (1955).\n16) K. Kubo and N. Ohata, J. Phys. Soc. Jpn. 33, 21 (1975).\n17) N. Furukawa, J. Phys. Soc. Jpn. 64, 2734 (1995).\n18) W.Metzner and D.Vollhardt, Phys.Rev.Lett. 62, 324 (1989).\n19) E. M¨ uller-Hartmann, Z. Phys. B 74, 507 (1989).\n20) A. Georges, et al., Rev. Mod. Phys. 68, 13 (1996).\n21) T. Pruschke, et al., Adv. Phys. 42, 187 (1995).\n22) R. Chitra and G. Kotliar, Phys. Rev. Lett. 83, 2386 (1999).\n23) T. Momoi and K. Kubo, Phys. Rev. B 58, R567 (1998).\n24) R. Zitzler, et al., Eur. Phys. J. B 27, 473 (2002).\n25) R. Peters and Th. Pruschke, Phys. Rev. B 76, 245101 (2007).\n26) T. Schork and S. Blawid, Phys. Rev. B 56, 6559 (1997).\n27) R. Sato, et al., J. Phys. Soc. Jpn. 73, 1864 (2004).\n28) H. R. Krishna-murthy, et al., Phys. Rev. B 21, 1003 (1980).\n29) R. Bulla, et al., cond-mat/0701105 (2007).\n30) O.Sakai and Y.Kuramoto, Solid State Comm. 89, 307 (1994).\n31) R. Bulla, Phys. Rev. Lett. 83, 136 (1999).\n32) F.B. Anders, and A.Schiller, Phys. Rev. B 74, 245113 (2006).\n33) R. Peters, et al., Phys. Rev. B 74, 245114 (2006).\n34) A.Weichselbaum and J.von Delft, Phys.Rev.Lett. 99,076402\n(2007).\n35) T. Ohashi, et al., Phys. Rev. B 70, 245104 (2004).\n36) I.Milat, et al., Eur. Phys. J. B 38, 571 (2004)."    },    {        "title": "1412.8294v1.Magnetic_balltracking__Tracking_the_photospheric_magnetic_flux.pdf",        "content": "Astronomy &Astrophysics manuscript no. Magnetic_Balltracking c\rESO 2021\nOctober 22, 2021\nMagnetic balltracking: Tracking the photospheric magnetic flux\nR. Attie and D. E. Innes\nMax-Planck-Institut für Sonnensystemforschung, 37077 Göttingen, Germany\nOctober 22, 2021\nABSTRACT\nContext. One aspect of understanding the dynamics of the quiet Sun is to quantify the evolution of the flux within small-scale magnetic\nfeatures. These features are routinely observed in the quiet photosphere and were given various names, such as pores, knots, magnetic\npatches.\nAims. This work presents a new algorithm for tracking the evolution of the broad variety of small-scale magnetic features in the\nphotosphere, with a precision equal to the instrumental resolution.\nMethods. We have developed a new technique to track the evolution of the individual magnetic features from magnetograms, called\n\"magnetic balltracking\". It quantifies the flux of the tracked features, and it can track the footpoints of magnetic field lines inferred\nfrom magnetic field extrapolation. The algorithm can detect and quantify flux emergence, as well as flux cancellation.\nResults. The capabilities of magnetic balltracking are demonstrated with the detection and the tracking of two cases of magnetic flux\nemergence that lead to the brightening of X-ray loops. The maximum emerged flux ranges from 1018Mx to 1019Mx (unsigned flux)\nwhen the X-ray loops are observed.\nKey words. Sun: photosphere - Sun: magnetic fields\n1. Introduction\nDescribing the evolution of the magnetic flux on the surface\nof the Sun is a key component for understanding the complex\ncouplings involved in energetic events that release a consider-\nable amount of energy into the solar atmosphere and beyond.\nFor the quiet photosphere, tracking algorithms already exist that\ndescribe the motion and some physical properties of the mag-\nnetic features. Each of them have di \u000berent limitations. Because\nof that, the most complete description of the photospheric mag-\nnetic flux cannot be achieved by using only one of these algo-\nrithms. Instead, combining di \u000berent algorithms with di \u000berent\nlimitations, which are therefore complementary, gives a more\ncomplete description of the behaviour of the photospheric mag-\nnetic features.\nAmong the many algorithms that attempt to analyse the mag-\nnetic field, a consistent set of four algorithms have been devel-\noped and compared with each other in DeForest et al. (2007),\nwhere they are referred to as CURV (Hagenaar et al. 1999),\nMCAT (Parnell 2002), YAFTA (Welsch & Longcope 2003; De-\nForest et al. 2007; Stangalini 2014), and SWAMIS (DeForest\net al. 2007; Lamb et al. 2008). Other algorithms designed to track\nmagnetic features exist, such as those described in Démoulin &\nBerger (2003) and Welsch et al. (2004), and they focus on deriv-\ning averaged velocity flows in magnetised regions.\nThe present paper describes a new algorithm using a com-\npletely di \u000berent paradigm than the algorithms mentioned above.\nIt is also designed to track and label small magnetic features in\nthe quiet photosphere, as small as can be seen, and it measures\nthe magnetic flux they carry. This algorithm does not pretend to\nreplace these well-tested algorithms, but it aims at providing an-\nother alternative that simply broadens the spectrum of science\napplications that can benefit from feature tracking techniques in\nmagnetograms, more specifically at the smallest scales (a few\nMm or less).This technique uses the same paradigm as the one used in\na known algorithm called balltracking (Potts et al. 2004; Attie\net al. 2009), which is designed to track granules in continuum\nimages. In this technique, numerical spheres or \"balls\" are re-\nleased onto the image. Their position is known at any given time,\nand they settle in the local intensity minima that move with the\ngranules so we can track the group motions of the latter. How-\never, because of the very di \u000berent nature of the data provided by\nmagnetograms, we needed to write a variant of this algorithm,\nwhich we refer to as \"magnetic balltracking\".\nAfter a brief summary of the balltracking paradigm in\nSect. 2, the magnetic balltracking is explained in four main\nphases in Sect. 3. In Sects. 4 and 5 we discuss a few technical\naspects of the algorithm. Its practical application is detailed in\nSect. 6 and applied in Sect. 7 to a case study of flux emergence\nassociated with the brightening of X-ray loops.\n2. The balltracking paradigm\nThe full derivation of the equations of motion that animate the\nballs within balltracking can be found in Potts et al. (2004). Here\nwe schematise the balltracking paradigm in the context of track-\ning magnetic features. The general principle is sketched in Fig.1,\nwhere the forces and the integration of the position of one ball\nis represented across two consecutive data surfaces at the times\nt1andt2. These data surfaces are, in fact, pre-processed magne-\ntograms (more on this in Sect. 3.1).\nThe three forces governing the motion of each ball can sim-\nply be expressed with the vectorial equation (vectors noted in\nbold letters):\nm˙v=X\nifi(di)\u0000mgez\u0000\u000bv (1)\nArticle number, page 1 of 14arXiv:1412.8294v1  [astro-ph.SR]  29 Dec 2014A&A proofs: manuscript no. Magnetic_Balltracking\nez\nFlow directiong\nfi\ndi\nfk\ndkmgvt1\u0000\u000bvt1\nvt2X1\ninit(initial position) X1\nfinal(final position)Magnetogram 1\nMagnetogram 2Height (Rescaled flux density)\n\u0000\u000bvt2\nX2\ninit(initial position) X2\nfinal(final position)0\nmg\n0Height (Rescaled flux density)\nFig. 1. General concept of the balltracking algorithm for one ball. The blue squares represent the pixels drawn at a height equal to the rescaled\nvalue of the unsigned flux density so that maxima in flux density appear as local minima. Only a few pixels are represented.\nwhere vis the velocity of the ball and ˙vits time derivative. Here,P\nifi(di) is the sum over all the elementary buoyancy forces ex-\nerted by the data point at the ithpixel at the penetration depth di.\nThe user-defined gravity field gsets the maximum acceleration\nthat a ball can possibly reach. It is oriented downward, in a 3D\ncartesian frame of reference where ezis a unit vector pointing\nupward and \u000bis a damping coe \u000ecient that controls the stability\nof the tracking. For simplicity, the mass mof the balls can be\nset to unity. In Fig. 1, the ball is moving with a magnetic feature,\nwhere the intensity of the magnetic flux density is reversed and\nrescaled so it can be seen as a geometric height. In such a pic-\nture, the local maximum of unsigned flux density is, in fact, a\nlocal minimum. While floating on the data surface represented\nby the blue waves, the ball settles in the local minimum, while\nthe latter moves with the group motion. In this framework, we\nare only interested in using the final position of the balls at the\nend of the integration of Eq. 1 in each magnetogram.\n3. The magnetic balltracking algorithm\n3.1. Phase 1: preprocessing of the magnetograms\nThe first step of magnetic balltracking is to make the magnetic\nfeatures \"trackable\" by the balls. The main change from the orig-nal balltracking is to account for the signed values of the magne-\ntograms, and for the more contrasted maps, spanning typically\nover more than 2 orders of magnitude in the quiet Sun (from\na few G to hundreds of G). So we have to rescale the magne-\ntograms in order to rescale the magnetic features vertically into\n\"holes\" of reasonable depth, from either positively or negatively\nsigned magnetic flux density, and allow the balls to settle inside\nthem. This is the purpose of the following preprocessing (see\nalso Figs. 2 and 3).\nTo start with, let us consider the absolute value of the flux\ndensity in a single magnetogram as a scalar field jBz(x;y)j, like\nthe one in Fig. 2 (top). It is first reverse-scaled (non-linearly) into\nB\u0003\nz(x;y)=max(p\njBz(x;y)j)\u0000p\njBz(x;y)j\nNext, B\u0003\nzis o\u000bset by its mean value and normalised to its standard\ndeviation\u001bB\u0003ztaken across the whole frame:\nB\u0003\nzn(x;y)=B\u0003\nz(x;y)\u0000<B\u0003\nz(x;y)>\n\u001bB\u0003z\nwhere B\u0003\nznis displayed in Fig. 2 (bottom), with an intensity span-\nning over a few units, which is of the order of the horizontal size\nof the magnetic features that we want to track. This rescaled sig-\nnal can be seen as a geometrical height. In a 3D plot (Fig. 3), the\nmagnetic features look like holes into which the balls can settle\nArticle number, page 2 of 14R. Attie and D. E. Innes: Magnetic balltracking: Tracking the photospheric magnetic flux\nBz(G)\nY (px)X (px) 0255075100125\n−40−2002040\n0255075100125150\nB∗zn(σ-units)\nY (px)X (px) 0255075100125\n−8−6−4−202\n0255075100125150\nFig. 2. Preprocessing of the magnetic balltracking. Top: Initial mag-\nnetogram Bz, calibrated into Gauss units, scaled between \u000040 G and\n+40 G. Bottom: Bzn?obtained after rescaling Bz\neasily. This particular choice of rescaling (against for instance a\nlinear rescaling) is further justified in Sect. 4.\n3.2. Phase 2: Initialisation\nOnce the magnetograms are rescaled, the balls are initially posi-\ntioned at the pixels whose absolute intensity in the original (not\nrescaled) magnetogram is above a given threshold. This initial-\nisation is illustrated in Fig. 4 (top) where the balls’ centres are\nplotted on the original magnetograms.\nWith the magnetograms from the Narrowband Filter Im-\nager of the Solar Optical Telescope (NFI /SOT) onboard Hinode\n(Tsuneta et al. 2008), the threshold is usually set between 5 G\nand 20 G. Regardless, these values should be chosen so one does\nnot track random noise. In addition, this saves some computa-\ntional time by reducing the number of balls that is much smaller\nthan the total number of pixels. In the magnetic balltracking, we\ndo not make assumptions on the size of the magnetic features,\nand the minimum length between the balls’ centres, within each\nmagnetic feature, is 1 px at any time. Nonetheless, in a practical\ncase of instrumental data, the minimum ball spacing should ac-\ncount for the full-width-at-half-maximum (FWHM) of the point\nspread function (PSF) of the instrument. Should the pixel scale\nof the imager be smaller than the FWHM, the minimum spacing\nbetween the ball should be set to the nearest integer value. This\nwill optimise the total number of balls, and thus the computing\ntime.\nOnce the balls are positioned on the magnetic features, or\nmore precisely, within the \"magnetic holes\", the polarity of each\ntrue feature is retrieved from the signed intensity of the original\nmagnetogram, at the pixels mapped to the coordinates of each\nball’s centre. This polarity is stored, and is a constant associated\nB∗zn(σ-units)Z (σ-units)\nY (px)X (px) 0255075100125\n−8−6−4−202\n0255075100125150−82\nB∗zn(σ-units)Z(σ-units)Y( p x )X( p x )0255075100125−8−6−4−202\n0255075100125150−82Fig. 3. Top: Bzn\u0003in 3D, using the same data as in the bottom of Fig. 2,\nwith the intensity used as a geometrical height. The colormap is scaled\nexactly as the intensity. Bottom: same as the top figure, with balls that\nhave settled in the \"magnetic holes\" after a few integration steps. The\nposition of these balls on the original 2D magnetograms are shown in\nFig. 4.\nwith each ball. It is referred to as the initial \"ball polarity\".\nNext, a few integration steps, typically 10 to 20 depending on the\nsize of the features, are performed between the first and second\nframe, so the balls have time to converge down into the local\nminima. For instance, in Fig. 1, such a local minima in the first\nmagnetogram (Magnetogram 1) would be found at X1\nfinal.\nBecause a segmentation algorithm will be used on the\ntracked magnetic features (more on that later), it is not neces-\nsary to have several balls within the same feature. If several balls\nhave converged to the same local minima, only one ball is kept.\nAfter this stage, it is still possible to have one large magnetic\nfeature being tracked by several balls, if for instance the feature\nhas several local minima. This is illustrated in Fig. 4 (bottom).\nNote that this significantly reduces the number of balls between\nthe first (top) and the next frame (bottom), and consequently the\ncomputational time.\n3.3. Phase 3: main tracking phase\nAfter the initialisation, the next frames are loaded, and the balls\ntrack the local minima (the \"magnetic holes\") like they do within\nintergranular lanes with balltracking (Potts et al. 2004).\nIn the original magnetograms, the local minima of the rescaled\ndata correspond, respectively, to the local maxima and minima\nof the signed intensity (positive and negative, respectively) of the\nmagnetic flux density. At any time, the position of each ball is\nknown, and each ball is labelled with a unique number, referred\nto as the \"ball number\". The positions can be plotted on-the-fly,\nso that one can check by eye the quality of the tracking. An ex-\nample of the main tracking phase is visible in the snapshots of\nArticle number, page 3 of 14A&A proofs: manuscript no. Magnetic_Balltracking\nBz(G)Y (px)\nX (px)0 25 50 75 100 125−40−30−20−10010203040\n0255075100125150\nBz(G)Y( p x )\nX( p x )02 5 5 0 7 5 1 0 0 1 2 5−40−30−20−10010203040\n0255075100125150\nFig. 4. Initialisation of the magnetic balltracking. Top: The values in the\noriginal magnetogram Bzare used to dispatch the balls on pixels above\n20 G (red crosses). Bottom: New positions of the balls, after integrating\nthe equations of motion, on the same magnetogram.\nFig. 5, with three ball numbers identifying three distinct mag-\nnetic features that are being tracked.\nWhen a magnetic feature is moving too rapidly, the balls do\nnot have time to settle in the local minima. In this situation, at\nworst, the balls may be delayed by a few frames, and several in-\ntegration steps between each frame are necessary to make sure\nthat the balls do not get lost. This gives them more time to \"catch\nup\" on the fastest features. Typically, for magnetograms taken\nat cadences of up to 3 min, 10 to 20 intermediate integration\nsteps of the equation of motion are used between each frame.\nThis may or may not be the same number of intermediate steps\nused for the initialisation phase, these are indeed two indepen-\ndent \"tuning\" parameters that depend on the resolution of the\ndata and the time sampling rate of the instrument. Typically, the\nnumber of integration steps is proportional to the former, and\ninversely-proportional to the latter. Indeed, at higher resolution,\nthe features are relatively wider, and the balls must travel over a\nrelatively greater number of pixels. At increasing time sampling\nrates, the features evolve less rapidly between two consecutivemagnetograms, and so the balls need less time to catch up on the\nmotions of the local minima.\nFor large connected magnetic features, the shape of the mag-\nnetic features looks like an extended surface full of holes, where\neach hole can be filled with a ball. See for example the white\nmagnetic patch in Fig. 4 (bottom) around the coordinates (60,\n120) and the balls in the 3D view in Fig. 3 (bottom). This method\nis suitable for tracking clustered features that are made of several\nfragments, such as the ones in Fig. 5.\nAt each tracking step, the polarity under the current position\nof the ball’s centre is compared to the ball polarity. The tracking\nof a given ball ends as soon as the polarity of the current pixel is\nreversed with respect to the ball polarity.\nThis strategy has several advantages. Indeed, to \"see\" a reversed\npolarity, a ball needs either to keep tracking down to the noise\nlevel until the current pixel polarity reverses (which is known by\nlooking it up automatically in the original magnetograms), or it\nneeds to encounter a magnetic feature of opposite polarity. This\ndefines two conditions for the end of the tracking of a given ball.\nThey are described below separately.\nCondition 1: tracking down to the noise level\nTracking down to the noise level makes the algorithm use the\ntrue sensitivity of the instrument. Indeed, even if the initialisation\nstep uses a threshold, the features are tracked until they cannot\nbe detected by the instrument, i.e, to values below the thresh-\nold. Should we ever need to track the faintest magnetic features\nfrom the beginning, the threshold may simply be lowered down\nto the noise level, which has the only consequence of increasing\nthe computing time (more balls will be added). When the ball is\nfloating over random noise, the sign of the intensity in the orig-\ninal magnetograms eventually reverses. In this case the tracking\nof this ball ends. The stability of the ball within noisy data is set\nby the damping coe \u000ecient\u000b(Eq. 1). Typical values are between\n0.1 and 1, depending on the time sampling rate and spatial res-\nolution. In term of damping time ( defined as Td=1=\u000b), this is\nequivalent to values between 1 and 10, in units of time interval\nbetween frames. The damping force is necessary not only in the\npresence of noise, in which case the tracking is, in fact, more\nresilient with damping than without, but it is also necessary for\nthe general stability of the code, regardless of the noise. This is\nillustrated in more detail in Sect. 3.5.\nCondition 2: no crossing of opposite polarity\nAnother issue we had to solve is how to prevent the ball from\ncrossing a feature with positive flux to a neighbouring one with\nnegative flux. This would occur for example if the ball is in a\nlocal minimum, and if it has kept enough momentum to reach\nanother close-by local minimum of opposite polarity at the next\nintegration step. This can also occur, if the close-by local min-\nimum moves quickly towards the ball, like for example in the\ncase of the two footpoints of a loop being submerged.\nThis problem can be thought of as a \"numerical tunnel\ne\u000bect\", in the sense that at one time a ball is in a magnetic hole,\nand at the next, it has crossed a barrier and lies within the second\nmagnetic hole. The solution is as follows: because the true\npolarity of each pixel is known from the original magnetogram,\nthe local minima are always associated with a polarity, which\nis compared to the ball polarity (see § 3.2) at the end of each\nintegration step. Should a ball lie in a magnetic hole with a\npolarity opposite to the ball polarity, the tracking of this ball\nArticle number, page 4 of 14R. Attie and D. E. Innes: Magnetic balltracking: Tracking the photospheric magnetic flux\nBz(G)X (px)Frame 25 [15:49 UT]\nX (px)Y (px)Frame 20 [15:39 UT]Frame 15 [15:29 UT]Y (px)Frame 10 [15:19 UT]Frame 5 [15:09 UT]Y (px)Frame 1 [15:01 UT]\n5274\n275\n2485274\n275\n2485274\n275\n2485274\n275\n2485274\n275\n248275248\n−10 −5 0 5 100 25 50 75 100 125 0 25 50 75 100 1250255075100125\n02550751001250255075100125\n02550751001250255075100125\n0255075100125\nFig. 5. Magnetic balltracking of a small field of quiet Sun, at di \u000berent\ntime steps. The numbers bound to the crosses (red) are the labels that\nare unique to each ball (i.e. the ball number).\nends.\nConditions 1 and 2 are checked independently for each ball.\nIf either of these two conditions is fulfilled for a given ball, its\ntracking ends, so that the lifetime of a ball corresponds to the\nlifetime of the magnetic feature tracked so far. This does not end\nthe whole algorithm, which continues as long as other balls re-\nmain. Note also that in Phase 3, whenever there are overlapping\nballs, i.e, their centres are positioned on the same pixel, only the\noldest ball is kept, which allows the right estimation of the fea-\ntures lifetime (more details on this in Phase 4).\nNote also that the magnetic features may exist at very irreg-\nular places (see Fig. 4, top), and consequently, the balls end up\nscattered over a rather irregular grid. Therefore, contrary to the\noriginal balltracking algorithm, the conversion of the velocity,\ninitially calculated in a Lagrangian frame of reference, into a\nEulerian frame of reference cannot happen. In other words, this\nmethod tracks and follows the individual motions of the mag-\nnetic features. It is not designed to output a \"flow field\", with\nthe values of the velocity known at any given time at fixed posi-tions, on a regular grid, like in, e.g., Démoulin & Berger (2003)\nand Welsch et al. (2004). This is simply because there are parts\nof the magnetograms where the magnetic flux is not detected,\nand thus, there is no velocity defined there. At best, Eulerian\nflow fields can only be derived locally, in regions where there\nare enough magnetic features that provide a more \"reasonable\"\nsampling.\n3.4. Phase 4: detection of emerging flux\nNote that this phase is an optional module in the algorithm.\nWhen used, it runs simultaneously with Phase 3. During this\nphase, the algorithm permanently scans for new pixels that\nwould rise above a given \"detection threshold\", which must be\nat least greater than the noise level of the instrument. It may, or\nmay not be equal to the threshold defined in Phase 2. When new\npixels \"rise\" above this threshold, new balls are added on these\nareas so that emerging flux can be tracked. This phase needs an-\nother tunable parameter, referred to as the \"detection spacing\",\nwhich sets the minimum distance between the new balls and the\nones already present. Any pixel whose intensity rises above the\nthreshold must satisfy the condition of being at a distance larger\nthan this parameter, in which case a new ball is put there. Note\nthat this makes it also possible to follow large features whose\nsize varies significantly over time. Indeed, the large features are\nthe ones with many pixels above the detection threshold. If the\ndistance between these pixels and those that have balls already,\nis greater than the detection spacing, new balls will eventually\nbe put there too. Consequently, if a given feature, initially small\nand populated with one ball, grows over time (for instance, as a\nresult of flux emergence), it may be populated by more than one\nball. Note that the detection spacing also defines the resolution\nwith which emerging flux is detected. The subsequent tracking\nof the new balls is exactly the same as the other balls.\nAn example is given with the ball 5274 in Fig. 5 (starting\nfrom frame 5, near the upper right corner, seen as a small red\ncross). It is tracking emerging flux that, before frame 5, was be-\nlow the initialisation threshold of 5 G. When the magnetic fea-\nture emerges above 5 G, this new ball locks onto it and tracks\nit until the last frame. This is allowed because the nearest balls\ntracking other features are at a distance greater than the detec-\ntion spacing. So this phase makes the algorithm useful not only\nto track the flux visible from the start, but also to track the emerg-\ning flux, and little pieces of the largest features that may (or may\nnot) fragment over time.\nIn the first movie attached to the online version of this pa-\nper, one can compare the e \u000bects of di \u000berent values of the de-\ntection spacing (at a given detection threshold of 5 G), and vi-\nsualise Phases 3 and 4 of magnetic balltracking. A snapshot of\nthis movie is shown in Fig. 6. Three panels are shown. With\nthese data, the pixel size is 0 :2 arcsec and the resolution is about\n0:3 arcsec px\u00001(Chae et al. 2007). In the left-hand panel, the de-\ntection of the emerging is deactivated. In the middle panel, the\ndetection spacing is 15 px, which is still a bit \"loose\", and one\nmisses a few emerging features. Finally, in the right-hand panel,\nthe detection spacing is set to 5 px which is a rather \"aggres-\nsive\" detection. The latter may be harder to follow because the\nfiner the detection grid, the more balls are involved, and the more\n\"crowded\" these movies get. Nonetheless, it allows for more flux\nto be detected. A detection spacing of 10 px is typical. With a\npixel size of 0 :2 arcsec, which is about 6 to 7 seven times the size\nof the resolution element (0 :3 arcsec px\u00001). One can also see sev-\neral balls gathering from the boundaries toward the centre of the\nlargest features. This is due to new balls, added near the edges\nArticle number, page 5 of 14A&A proofs: manuscript no. Magnetic_Balltracking\nof the large features (Phase 4), converging toward the same local\nminima. Like in Phase 3, only the oldest ball is kept whenever,\nand wherever they overlap. This prevents \"young\" balls, i.e, the\nones added during the flux detection, from replacing the older\nones. Otherwise, we could not extract meaningful lifetimes dur-\ning Phase 3.\nNote that it is also possible to track emerging flux by simply\nreversing the timeline of the data series and recording submerg-\ning flux. In such a case, as far as the algorithm is concerned,\nflux removal is indistinguishable from flux emergence. Depend-\ning on the situation, this may be more suitable than using the\nmodule described in Phase 4.\n3.5. Notes on the damping force\nWithout damping, the balls have too much inertia and \"free-fall\"\ntoward the local minima. With too great a speed, they can move\npassed the local minimum. The damping coe \u000ecient prevents this\n\"overshoot\", and damps the subsequent oscillations of the posi-\ntions when the balls \"jiggle\" around the local minimum. This is\nillustrated in Fig. 7 using a synthetic, rescaled gaussian surface,\nused as a magnetic feature that has a maximum strength of 50 G,\nand a local minimum at x=11 px. The FWHM of the original\ngaussian curve (i.e, before rescaling) is \u00187 px. On the left col-\numn, three values of damping times are used ( Td=1, 5, and 10).\nOn the right column, we separated the damping times into a hor-\nizontal ( Tdh) and a vertical ( Tdz) damping time that are associ-\nated respectively with the horizontal and vertical components of\nthe damping force, whose usage are recommended in Potts et al.\n(2004) (although in the case of tracking granules). The balls are\nplotted at the integration step 1, 15, and 100. Note the overshoot\nseen with the red ball at Td=5 and Td=10, as well as at\n[Tdh=5;Tdz=5] and [ Tdh=5;Tdz=10]. The red ball has\nmoved passed the local minimum and falls back to it at a later\ntime (green). The overshoot is not seen with a shorter damping\ntime in both directions Td=1 and a shorter vertical damping\ntime [ Tdh=5;Tdz=0:5].\nThe x-coordinates of the ball in each case are plotted in\nFig. 8. The top and bottom panel (resp.) correspond to the dis-\nplacements the balls in the left and right column (resp.) in Fig. 7.\nThe overshoot corresponds to x>11. Note how the oscillations\nare reduced more e \u000eciently with a decreasing damping time. At\nTd=1, there is indeed no overshoot, and the ball settles in the\nlocal minimum after 30 steps. Yet it may be more e \u000ecient to\nuse [Tdh=5;Tdz=0:5], which induces a small overshoot, while\nwithin 25 steps the ball is less than 1 px from the local minimum.\n[Tdh=5;Tdz=10] is a limit case where the damping force on\nthe vertical axis is too small such that the tracking does not con-\nverge (the ball is not visible in Fig. 7 due to its z-coordinate that\nis o\u000bthe grid). If e \u000eciency is not an issue, the use of a unique\ndamping time Td=1 is su \u000ecient, as long as the convergence cri-\nterium is fulfilled (see later in Sect. 4). However, for an optimum\nstability on the vertical axis, and based on many experiences (not\nshown here), we recommend the use of Tdzwithin [0.5;1], and\nTdhwithin [1;10], which may vary according to the data charac-\nteristics: resolution, pixel size, cadence, noise level, and sizes of\nthe features.\n4. Motivations for the rescaling method and\nconvergence criterium\nThe choice of the rescaling method of the magnetograms is\ndriven by two objectives: (i) a successful tracking of the fea-\nX (px)Tdh= 5; Tdz= 10Tdh= 5; Tdz= 5Tdh= 5; Tdz= 0.5\nX (px)Height (px)Td= 10Height (px)Td= 5Height (px)Td= 1\n0 10 20 0 10 20−505−505−505Fig. 7. Tracking of the local minimum of a synthetic gaussian surface.\nThe balls are projected on the x-axis and z-axis at three di \u000berent num-\nbers of integration steps. Step 1 (yellow), 15 (red), 100 (green). The left\ncolumn uses the same damping time for the vertical and horizontal axis.\nThe right column shows the tracking with a horizontal damping time\n(Tdh) di\u000berent from the vertical damping time ( Tdz).\ntures, and (ii) a minimisation of the number of integration steps\nthat are necessary to achieve (i). In this section we demonstrate\nthe advantages of the choice of rescaling defined in Sect. 3.1.\n4.1. Formal definition of the rescaling methods\nOur \"generic\" rescaling method formalises as follows:\nB\u0003\nz(x;y)=max(jBz(x;y)j\r)\u0000jBz(x;y)j\r(2)\nB\u0003\nzn(x;y)=B\u0003\nz(x;y)\u0000<B\u0003\nz(x;y)>\n\f(3)\nin which we have introduced a normalisation factor \f, and\r:\nlinear rescaling is equivalent to \r=1 and non-linear rescaling\nequivalent to \r,0 and 1. Then we compare the following three\nscaling methods:\n1. [\r=1;\f=1]: no rescaling is actually performed.\n2. [\r=1;\f=\u001bB\u0003z]: linear rescaling.\n3. [\r=0:5;\f=\u001bB\u0003z]: non-linear rescaling.\nMethod 3 is the method defined in Sect. 3.1. \u001bB\u0003zwas originally\ndefined as the standard deviation of B\u0003\nz(x;y), it implicitly de-\npends on\r(Eq. 2). Note that subtracting by the mean value in\nEq. 3 only changes the zero-point of the data surface and does\nnot a\u000bect the results. It is used here to conveniently define the\nmean value of the data as the origin of the vertical axis.\nArticle number, page 6 of 14R. Attie and D. E. Innes: Magnetic balltracking: Tracking the photospheric magnetic flux\nBz (G)Detection spacing = 5px\nX (px)Detection spacing = 15px\nX (px)No emergence detectionY (px)\nX (px)Frame 50 [16:39 UT] Frame 50 [16:39 UT] Frame 50 [16:39 UT]\n−10 −5 0 5 100 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120020406080100120\nFig. 6. Snapshot comparing the e \u000bects of the detection spacing, at a given detection threshold of 5 G. The red crosses are plotted at the positions\nof the ball’s centre. Left: no flux detection. Middle: loose detection spacing at 15 px. Right: aggressive detection spacing at 5 px. A movie showing\nthe temporal evolution is available in the online edition.\n  \n  \nTdh= 5; Tdz= 10Tdh= 5; Tdz= 5Tdh= 5; Tdz= 0.5Distance (px)\nTime (# of frames)Td= 10Td= 5Td= 1Distance (px)\n0 50 100246810121416246810121416\nFig. 8. Coordinate on the x-axis of the ball’s centre during the tracking\nof Fig. 7. The top and bottom panel correspond to the motions of the\nball in the left and right column of Fig. 7.4.2. Convergence criterium\nThe positions of the balls at the end of 100 integration steps are\nshown in Fig. 9 for the three scaling methods 1, 2, 3, respectively,\nin blue dots, green and red crosses, respectively. The yellow con-\ntours are set at the initialisation threshold of 10 G. The initial po-\nsitions are the ones plotted in Fig. 4 (top). In the current example,\na successful tracking has the necessary (but not su \u000ecient) condi-\ntion that all the final positions are within the yellow contours in\nFig. 9. Yet the blue dots end up outside the contours (i.e, outside\nthe magnetic feature they are meant to track), whereas the others\nare all within the contours, which proves that the scaling method\n1 does not result in a successful tracking, and that Methods 2 and\n3 are more appropriate.\nIn Fig. 10 we have plotted three examples of the traveled dis-\ntance (in pixels) of three di \u000berent balls’ centres, using the lin-\near (green dotted line) and the non-linear (red continuous line)\nscaling. The chosen balls, respectively from top to bottom, cor-\nrespond to the ones pointed to by the cyan arrows in Fig. 9, re-\nspectively from left to right. The oscillations are due to the balls\novershooting to either side of the local minima before they can fi-\nnally settle. The oscillations are damped within one to two times\nthe horizontal damping time (here it is set to 4).\nThe time that the balls take to reach their final position de-\nfines a convergence factor represented in Fig. 11. The \"final posi-\ntion\" is set as the position reached by the ball after a su \u000eciently\nhigh number, here set to 100. The convergence factor is then de-\nfined as the ratio of the number of balls that have reached their\nfinal position at a given integration step, to the total number of\nballs. When the convergence factor reaches 1, the convergence\ncriterium is satisfied, and this sets the number of initialisation\nsteps. The scaling method 1 converges after about 95 integration\nsteps, the other two methods converge in less than 40 integration\nsteps, with the non-linear rescaling (red continuous line) con-\nverging more rapidly by a few integration steps. With the scal-\ning methods 2 and 3, 90% of the balls have converged within\nArticle number, page 7 of 14A&A proofs: manuscript no. Magnetic_Balltracking\nScaling 3 ( γ= 0.5)Scaling 2 ( γ= 1)Scaling 1 (no rescaling)Bz(G)Y (px)\nX (px)−20 −10 0 10 20\n0 25 50 75 100 1250255075100125150\nFig. 9. Final positions of the balls’ centre, after 100 integration steps,\nfor the three scaling methods. The original positions are the ones shown\nin Fig. 4 (top, red crosses).\n10 to 20 steps, and all the balls have converged within 40 steps\n(the convergence factor equals 1). 20 initial steps is typical but\na \"trial-run\" like the one here must be done in each case study\nto determine the optimum value for a given data set. Here, the\noptimum value would be 40.\n4.3. Comparisons between the rescaling methods 2 and 3\nAt some places, like in the yellow rectangle (top right) of Fig. 9,\nthe green crosses are not overlaid by the red ones, and instead,\na few of them ended up at di \u000berent positions within a few pix-\nels. This first suggests that whereas the linear rescaling (Method\n2) has converged according to Fig. 11, the balls, in fact, have\nnot converged in a local minimum, contrary to Method 3 where\nthe red crosses suggest that the balls have unambiguously settled\nin the same local minimum. Fig. 12 is a close-up in the yellow\nrectangle of Fig. 9 viewed from the top (top panels) and from\nthe side (along the left Y-axis) of the 3D rescaled data surface\n(bottom panels). The coloured circles are plotted at the final po-\nsitions of the balls’ centre. One sees that there is indeed only one\nlocal minimum in this area, but the balls in Method 2 (left) did\nnot settle in it. Some of them are standing still in the middle of\nthe steep slope (bottom left), which is not what one might expect\nin a real \"natural\" situation. This is a typical example of a linear\nrescaling being less appropriate than a non-linear rescaling with\n\r=0:5 (right). With a linear rescaling, the strongest magnetic\nfeatures (here above 100 G) still have a very steep slope (bottom\n# of integration stepsDistance (px) Distance (px)Scaling 2 ( γ= 0 .5)Scaling 2 ( γ= 1)Distance (px)\n0 20 40 60 80 100024680246801234Fig. 10. Distance from the initial position, for the three balls (resp. top\nto bottom) pointed to by the cyan arrows in Fig. 9 (resp. from left to\nright).\nScaling 3 ( γ= 0.5)Scaling 2 ( γ= 1)Scaling 1 (no rescaling)Convergence factor\n# of integration steps0 20 40 60 80 1000.70.80.91\nFig. 11. Convergence factor against the number of integration steps, for\nthe three scaling methods.\nleft), while the slope is more gradual with a non-linear rescaling\n(bottom right).\nThe reason for the very steep slope resulting in an incon-\nsistent tracking is related to the number of data points that are\nactually in contact with the balls, and therefore, contributing to\nthe total force fi(see also Eq. 1 and the blue squares in Fig. 1).\nIndeed, for maximum e \u000eciency, the 3D grid defined by the balls,\nand which samples the data surface, uses regularly spaced val-\nues with a grid size of 1 px. For a data point to be considered\nby the algorithm, it needs to be located within one ball radius\nfrom the ball’s centre. The data at these ball grid points are in-\nArticle number, page 8 of 14R. Attie and D. E. Innes: Magnetic balltracking: Tracking the photospheric magnetic flux\n  \n  \nY (px)Height\nY (px)X (px)Y (px)\nX (px)Scaling 3 (non-linear) Scaling 2 (linear)\n120130\n120130−10 −5 0\n115 120 125 130 135−20 −10 0\n115 120 125 130 135\n145 150 155 160\n145 150 155 160145150155160\n−8−6−4−20\n−20−15−10−50\nFig. 12. Close-up in the region of the yellow rectangle in Fig. 9. Left:\nlinear rescaling (Method 2). Right: non-linear rescaling (Method 3).\nThe small cyan dots are the initial positions. The coloured circles are\nthe final positions. The yellow lines are mapping the final position to\nthe corresponding initial positions. They are not the actual trajectories.\nBottom: 3D data surface viewed from the left Y-axis.\nterpolated. Note that nearest neighbour or linear interpolation\nmake no di \u000berence on the final positions (the former is pre-\nferred for e \u000eciency). At a given ball radius, the more of these\n\"contact points\", the better, as more contact points means that\nmore of the magnetic feature actual topology is accounted for,\nand thus the more coherent the tracking is. The number of con-\ntact points involved in the tracking in the three di \u000berent scaling\ncases is shown in Fig. 13. In the top panel, we use the number\nof contact points averaged over all the balls involved in Fig. 9.\nThe top panel again shows that in the case of no rescaling (blue\ndashed line), there is on average merely 1 contact point, which\nexplains why the scaling method 1 fails. Methods 2 and 3 di \u000ber\nonly slightly on average, by only 1 contact point. Locally how-\never (bottom panel), in the cases corresponding to the close-up\nin Fig. 12, this di \u000berence is more significant. Before the con-\nvergence criteria is reached (i.e, before the integration step 40),\nthe number of contact points varies from 1 to less than 6 with\na linear rescaling, whereas it varies from 1 to more than 8 with\na non-linear rescaling. Furthermore, with the latter method, this\nnumber increases more rapidly: it equals 8 while it is only 2 with\nthe linear rescaling, which can only result in an insu \u000ecient force\nto push the balls down the slope. Put simply, with too few contact\npoints, the balls can stall.\nThus, Method 3 has two advantages over Method 2: a better\nhandling of the steeper slope for the magnetic features with field\nstrength of the order of 102G, and a relatively higher e \u000eciency\n(a few integration steps less are needed). The tracking of faint\nmagnetic features is not a \u000bected by these e \u000bects. Visual inspec-\ntions show no di \u000berence in the final positions between Methods\n2 and 3 when tracking the smallest and weakest features close\nScaling 3 ( γ= 0 .5)Scaling 2 ( γ= 1)# of contact points\n# of integration stepsScaling 3 ( γ= 0.5)Scaling 2 ( γ= 1)Scaling 1 (no rescaling)# of contact points\n0 20 40 60 80 10002468100246810Fig. 13. Average number of contact points per ball, against the number\nof integration steps. Top: average over the FOV of Fig. 9 for the three\nscaling methods. Bottom: average over the balls in Fig. 12 for the linear\nand non-linear scaling methods.\nto the initialisation threshold. In addition, no case where Method\n3 is less accurate than Method 2 were found, and therefore, the\nnon-linear rescaling is the most appropriate choice among the\nthree tested methods. Note that other values of \rwithin ]0; 1],\nbut relatively close to 0.5, are possible, and they may be tuned\nto di\u000berent values for optimisation purposes with quiet Sun fea-\ntures, with negligible changes on the final positions.\n5. Effects of the balls radius\nThe appropriate choice of the ball radius relates to the size of\nthe magnetic features that are to be tracked. The sphere radius\nmust be greater than 1 px, regardless of how small the features\nare. At 1 px the number of contact points is too small to result in\na coherent tracking, which stays dominated by high-frequency\nnoise. Fig. 14 shows the final positions of a tracking using three\ndi\u000berent radii (Rs), represented by coloured crosses of increas-\ning sizes: Rs =2 px, Rs =3 px, and Rs =4 px. \"Nb\" is the number\nof di\u000berent local minima in which the balls have converged. It is\nhere equal to the number of crosses of a given colour that are vis-\nible in the figure. Nb is respectively equal to 124, 117, and 103.\nSo the number of di \u000berent local minima in which the balls man-\naged to settle decreases when the radius increases. The \"missing\"\nballs at Rs =3 px, and Rs =4 px have either \"fallen o \u000b\" the edges\nand/or passed \"through\" the smaller ones located near bigger lo-\ncal minima in which more of the bigger balls are pulled in more\ne\u000eciently: the yellow arrows show examples of local minima re-\nsolved by the use of a 2 px-radius, but that are not resolved by\nthe use of Rs =4 px and /or Rs =3 px. There is one place where\na ball with only the 4-px-radius converges (near the coordinates\n(100, 100)) and it is, in fact, a \"false positive\" (i.e, not an ac-\ntual local minimum): the lack of resolution made the bigger grid\nArticle number, page 9 of 14A&A proofs: manuscript no. Magnetic_Balltracking\nBz(G)\nNb =105Rs = 4 px\nNb =118Rs = 3 px\nNb =127Rs = 2 pxY (px)\nX (px)−15 −10 −5 0 5 10 15\n0 25 50 75 100 1250255075100125150\nFig. 14. Final positions using three di \u000berent ball radii: Rs=2 px (small\nred crosses), Rs=3 px (medium-sized green crosses), and Rs=4 px\n(large blue crosses). \"Nb\" is the number of balls visible in the figure for\neach ball radius. The yellow arrows point at local minima in which only\nthe balls with a 2-px-radius converged.\n(due to the greater radius) cover pixels on the neighbouring fea-\nture (black patch on the left), which averages the motion of the\ntracked white patch and of this nearby black patch. On the other\nhand, we do not find any \"false positive\" with the red crosses\n(Rs=2 px), which makes it, here, the best choice.\nNote that these test data are from NFI /SOT (Hinode). The\ndata were binned onboard to a pixel size of \u00180:2 arcsec while the\nspatial resolution of NFI magnetograms is 0 :3 arcsec px\u00001(Chae\net al. 2007), so a radius of 2 px or 3 px would be appropriate.\nNonetheless, for a given initial ball spacing (defined as the min-\nimum space between the balls at the initialisation phase), using\na greater radius reduces the resolution of the tracking, and also\nincreases the computing time and memory usage as a bigger ball\nencompasses more grid points.\nFig. 15 is a comparison between the horizontal distances\ntravelled by the balls at three locations chosen at random (from\ntop to bottom, respectively) pointed to by the cyan arrows in\nFig. 14 (from left to right, respectively). In each case, the three\nradii are tested. The distance has its origin at the initial position\nof the balls, and in each case, the three balls have the same initial\nposition. Fig. 15 shows there is less than one pixel of discrep-\nancy in the final position between Rs =2 px and Rs =3 px. This\ndiscrepancy is greater with Rs =4 px, about 2 px according to the\nbottom panel. Looking at other cases (not shown here) gives the\nsame conclusion that Rs =2 px and Rs =3 px have the least dis-\ncrepancy of less than one pixel, and that Rs =4 px is less accurate.\nNote that because the size of the features, in pixels, also depends\non the plate scale and on the resolution of the instrument, we\nRs = 4 pxRs = 3 pxRs = 2 px\n# of integration stepsDistance (px) Distance (px) Distance (px)\n0 20 40 60 80 100024680246801234Fig. 15. Travelled distances in three cases, with three di \u000berent ball radii:\nRs=2 px,Rs=3 px, and Rs=4 px. From top to bottom (resp.), the\nballs ending up at the location shown by the cyan arrows from left to\nright (resp.) in .\ncannot give ideal values that work in all situations. Such tests on\nsubsets of large data with di \u000berent radii are necessary to opti-\nmise this choice. In addition, one could combine the results of\ndi\u000berent radii for multi-scale analyses. The small radii would be\ndedicated to the smaller features, while the greater radii would\nbe tracking the broader patches.\nThe main output of magnetic balltracking is the series of fi-\nnal positions of the balls, for the purpose of tracking the time-\ndependent displacements of the local extrema of the flux. These\npositions are then used in a segmentation algorithm to integrate\nthe flux over the area of the tracked features. This is detailed in\nthe next section.\n6. Segmentation of the magnetic features\nAs mentioned earlier, the magnetic features can be very clus-\ntered in the quiet Sun. Then quantifying the evolution of each\nbit of each magnetic structure, which can be near the limit of the\ninstrumental resolution, is quite challenging. Magnetic balltrack-\ning facilitates this procedure. As explained in Sect. 3, this tech-\nnique tracks the time-dependent positions of the local extrema\nof individual magnetic features. Thus the next step in describing\ntheir evolution is to integrate the magnetic flux of these features.\nAn easy way to do this is by applying a \"region extraction\" algo-\nrithm. The technique is also known as \"region growing\", and has\nmany names and variations that depend on the scientific field in\nwhich this segmentation technique is applied. It is one of the ba-\nsic algorithms detailed in textbooks of digital image processing\n(see for example Gonzalez & Woods 2008, Chapter 10, § 10.4).\nThe \"region growing\" used here, consists in extracting the\nmagnetic features that have been tracked, from the rest of the\nmagnetograms, so we can easily integrate the intensity of the flux\nArticle number, page 10 of 14R. Attie and D. E. Innes: Magnetic balltracking: Tracking the photospheric magnetic flux\nover the extracted area. We use the tracked positions (the final\npositions of the balls in each frame) as so-called \"seed points\".\nFor each position (or each \"seed point\"), the di \u000berence between\nthe intensity of the neighboring pixels and the seed pixel is com-\npared to a given threshold. If the comparison is true (in the log-\nical sense), it is added to a list of pixels connected with each\nother, including the seed, which \"grows\" a region and thus seg-\nments the magnetic feature from the rest of the magnetogram.\nIf the comparison is false, the pixel is not added to the list. The\nregion stops growing when there are no more connected pixels.\nThe output of the region-growing algorithm is a binary mask: an\narray of logical values, co-spatial with the magnetograms, where\nthe pixels in the grown region are set to 1, and the other pixels\nare set to 0. These masks can directly be used to extract the fea-\ntures from the magnetogram in order to integrate the flux of the\ntracked features. Such extracted features are visible in Fig. 16,\nwhich used the tracked positions (i.e, the seeds) previously illus-\ntrated in Fig. 5 as the input of this segmentation.\nAs mentioned in the previous section, several balls may track\nthe same wide magnetic features, with as many balls as there are\nlocal extrema in it. As the position of these balls are used as\nseed points, they will ultimately extract the same connected pix-\nels, and output identical masks. Then we have to get rid of the\nduplicates, which we do by using a logical \"or\" (equivalent to\na logical \"union\") between all the extracted masks. If the same\nmasks of connected pixels are output for di \u000berent balls, the log-\nical \"or\" reduces them to one unique mask before integrating the\nflux. This makes sure that, when looping over the extracted re-\ngions to integrate their flux, we do not integrate it over the same\nregion more than once.\nFinally, each magnetogram is integrated over the extracted\nareas, which gives the flux carried by the tracked feature in each\nsingle frame. Repeating this for all the magnetograms provides\nthe time-dependent flux of all the tracked features, including\nthe emerging one if any has been detected during Phase 4. The\nball numbers are used as the labels of the magnetic features,\nwhich makes it easy to select them individually. This way we can\nchoose which one to extract with the region-growing algorithm,\nbut it is also possible to simply take all of the tracked features\nfor more global statistical analyses.\nOne limitation of this method is that it cannot grow a re-\ngion using too low thresholds, otherwise, close but probably\nnon-connected regions may be added to the list of \"good\" pix-\nels. This leads to a wrong segmentation, and ultimately a biased\nestimation of the flux. This could probably be solved using a\nmore sophisticated segmentation algorithm. For consistency, we\nuse thresholds close, but not necessarily equal to the thresholds\nused in the previous phases of the magnetic balltracking (defined\nin Phase 2). They are between, typically, 5 G and 15 G. We found\nthese values by \"trial and error\" and they turned out to be optimal\nin the very clustered flux of the quiet Sun. A direct unfortunate\nconsequence is that no flux is integrated below these thresholds.\nNonetheless, the actual value of the thresholds that we have used\nso far depend more on the instrumental detection limits rather\nthan being an intrinsic limitation of the algorithm.\nDifference with other algorithms\nTo identify the magnetic features, the four codes compared in\nDeForest et al. (2007) use a similarity metric in the (x,y,t) space\n(2 spatial dimensions and 1 time dimension) based on logical\ncomparisons between the shape and position of the segmented\nareas. This is done after the segmentation in the region of in-\nterest. The features are then identified and labeled as magnetic\nBz(G)X (px)Frame 25 [15:49 UT]\nX (px)Y (px)Frame 20 [15:39 UT]Frame 15 [15:29 UT]Y (px)Frame 10 [15:19 UT]Frame 5 [15:09 UT]Y (px)Frame 1 [15:01 UT]\n5274\n275\n2485274\n275\n2485274\n275\n2485274\n275\n2485274\n275\n248275248\n−10 −5 0 5 100 25 50 75 100 125 0 25 50 75 100 1250255075100125\n02550751001250255075100125\n02550751001250255075100125\n0255075100125Fig. 16. Region of the magnetograms extracted using the region-\ngrowing algorithm onto the balltracked seed-points of Fig. 5.\nentities at the end of this process. These are then the \"tracked\"\nmagnetic features. The motion of the features may or may not be\nderived using centre of gravity of the extracted area.\nIn magnetic balltracking, the magnetic features are already\nlabelled at the first phase, and in Phase 4 for the emerging ones,\nby the ball numbers that settle within their local maxima. Fur-\nthermore, the tracking is done before the discrimination of the\nmagnetic fragments, using the balltracking paradigm. The latter\noccurs as a post-processing of the magnetic balltracking. Here\nwe have used region growing as the post-processing method to\nbuild the masks that extract the magnetic fragment from the\nimages, which is similar to the so-called \"clumping\" used in\nMCAT. Regardless, the magnetic balltracking can be combined\nwith other methods of segmentation, as long as they can use\nthe balls as seed points. Within magnetic features large enough\nto have more than one local extrema, and thus more than one\nball tracking the whole feature, the duplicated masks associated\nwith each of the ball (used here as seed points) are unified into\none single mask. Further discrimination within these large fea-\ntures cannot be achieved with this method, and depending on\nthe science goals, one may revert to using a more discrimina-\nArticle number, page 11 of 14A&A proofs: manuscript no. Magnetic_Balltracking\ntive method like the so-called \"downhill\" method in YAFTA and\nSWAMIS.\nIn the next section, we demonstrate the capabilities of mag-\nnetic balltracking in a case study of flux emergence using\nMichelson Doppler Image (MDI) data and co-spatial X-ray im-\nages from the X-Ray Telescope (XRT) on Hinode.\n7. Magnetic balltracking on flux emergence\n7.1. Observation of flux emergence\nIn what follows, all times are given in Universal Time (UT). The\nobservations were made on September 26th, 2008, and consist\nof high-resolution, 1 min-cadence continuum images and mag-\nnetograms from MDI (Scherrer et al. 1995), and co-spatial im-\nages from XRT (Golub et al. 2007) in soft X-ray (pixel size\nof 1 arcsec), at\u001830 s-cadence. The di \u000berent time series last 4.5\nhours, between 15:00 until \u001819:30. The MDI magnetograms\nwere rigidly de-rotated using the local latitude at the centre of the\nfield-of-view (FOV), which is consistent with the preprocessing\nrecommendation in DeForest et al. (2007, § 5.1). The time se-\nries of XRT images were calibrated and registered using the rou-\ntines related to the XRT instrument in Solarsoft (xrt_prep.pro,\nxrt_jitter.pro). The co-alignment of the XRT images onto the\nMDI magnetograms was done using co-temporal XRT images\nand Extreme-ultraviolet Imaging Telescope (EIT) images (De-\nlaboudinière et al. 1995) whose frame-of-reference is put into\nalignment with the MDI frame-of-reference using the header in-\nformation. Accuracy of the latter was checked using limb fitting\nof the full solar disk. The XRT and EIT frame used for the co-\nalignment were taken near 19:00. We estimate the XRT and MDI\nframes to be co-spatial within less than 1 arcsec ( <1 Mm) .\nThe flux emergence is associated with the rise of X-ray loops\nobserved at the same location, and shown in Fig. 17 (pointed\nto by the orange arrows). The snapshots are taken in a FOV of\n60\u000260 Mm2. Balltracking was used to derive the flow fields, and\nthe associated supergranular network lanes are drawn as blue\ncontours. These are obtained with the algorithm of automatic\nrecognition of supergranular cells from Potts & Diver (2008).\nAt 15:12, the flux is barely visible in the internetwork, un-\ntil it emerges as a very fragmented, mixed-polarity flux after\n16:00 (left arrows). This occurred quite close to the supergran-\nular boundaries (blue lanes in Fig. 17, only 10 Mm away from\nit, which is consistent with previous observations of flux emer-\ngence (Wang 1988; Stangalini 2014). The clustered flux then\ndrifts away, and is finally observed with one clearly visible X-ray\nloop, at 17:55. At 18:12 another X-ray loop seems to connect the\nnegative-polarity footpoint from the left side of the network lane\nin the middle of the frame, to the positive one on the other side.\nA second, weaker emergence is seen near the bottom right part\nof the snapshots, starting at 16:42, and also gave rise to X-ray\nloops, visible at 19:05. The time scale of these emergences is a\nfew hours. The amount of flux in the emerged bipoles that are\nobserved at the footpoints of the X-ray loops is measured with\nthe magnetic balltracking. The results are presented in the next\nsection.\n7.2. Results of the segmentation\nMagnetic balltracking is performed on the FOV of the snapshots\nin Fig. 17 to track the features pointed to by the orange arrows.\nThe results are used by the region-growing algorithm: as ex-\nplained in Sect. 6, each ball can act as an identifier of a whole\nmagnetic feature, using the ball numbers as unique labels. This\n15:12\n 15:42\n 16:12\n16:42\n 17:00\n 17:25\n17:55\n 18:12\n 19:05\nFig. 17. Flux emergence observed at two di \u000berent places within the\ndisplayed field of view. Red /green contours are positive /negative flux\n(respectively). Thin unfilled contours are at 10 G, filled contours are at\n50 G. The orange arrows point at emerging flux regions that are fol-\nlowed by the rise of X-ray loops. The blue contours are the contours of\nthe supergranular boundaries.\n.\nmakes it easier to detect and isolate only the emerging flux. To\ndo so, we simply associate the number of the balls that were\ntracking these features to the time series of flux that is integrated\nby region-growing. Here, we selected only the flux that had in-\ncreased at the end of the time series by twice the standard devia-\ntion. This is su \u000ecient to isolate the emerging flux seen in Fig. 17.\nAt this stage, all the emerging flux is not associated with bright\nX-ray emission. Finally, we use the ball numbers to select pre-\ncisely the patches that are emerging underneath the X-ray loops.\nThis last selection is not an automated process as we need to\nknow where the X-ray loops are located, and to identify the ball\nnumbers by eye.\nOne can see the result of this selection more precisely in\nthe three snapshots shown in Fig. 18, and in the second movie\n(online supplemental). The left panel shows the magnetograms\nwith the positions of the balls (red crosses) that are tracking the\nemerging flux (see also Fig. 18). The small turquoise crosses\nshow the positions of the balls used throughout the magnetic\nballtracking, excluding the ones added during Phase 4. The big-\nger red crosses correspond to the balls selected for the region\nextraction of the emerging flux. The blue and green contours\noutline the boundaries of the extracted masks that are used to\nspatially integrate the flux density in the magnetograms. These\nmasks are shown as binary patches in the right panel (black for\nnegative flux, white for positive flux). They are the main output\nof the region extraction algorithm (§ 6). The detection thresh-\nold was set to 20 G, which is near the noise level of the magne-\nArticle number, page 12 of 14R. Attie and D. E. Innes: Magnetic balltracking: Tracking the photospheric magnetic flux\nMasks\nX (px)\nBz(G)Frame 213 18:32Y (px)\nX (px)Masks\nX (px)Frame 173 17:52Y (px)\nX (px)Masks\nX (px)Frame 60 15:59Y (px)\nX (px)\n25 50 75 100 125\n−40 −20 0 20 40255075 100 12525 50 75 100 125 25 50 75 100 125255075 100 125 25 50 75 100 125\n255075100125255075100125255075100125\nFig. 18. Snapshot of the region extraction of the emerging flux. The\nright column are the magnetograms. The left column are the masks of\nthe extracted regions. The contours are at \u000620 G. A movie showing the\ntemporal evolution is available in the online edition.\ntograms in MDI. The pixels with an intensity below this value are\nignored. It is possible to see the limitations of such a low thresh-\nold in the left panel of the movie, by considering the green con-\ntours on the left, near Frames 173-174 and again around Frame\n213. The extracted region suddenly includes small nearby fea-\ntures for 1 to 2 frames. This illustrates possible errors when us-\ning region growing with a detection threshold too close to the\nnoise level. Nonetheless, the short time scale (1 to 2 frames)\nof such extraction errors, compared to the lifetime of the cor-\nrectly extracted features (here, an order of magnitude greater)\nmakes it possible to filter them out, for instance with median\nfiltering over a few time steps, or using Fourier filtering in the\ntime-frequency domain, without significantly impairing the rest\nof the data. However, these techniques are not part of the mag-\nnetic balltracking and we will not discuss them further.\nThe integrated fluxes are plotted in Fig. 19. The flux in the\nfirst emergence (Fig. 17, left-side arrows, and Fig. 19, top) is bal-\nanced at the beginning (15:00), for \u001840 min with a positive and\nXRT intensityNegative fluxPositive flux\nXRT total intensity ( ×103DN s−1)MDI flux (×1018Mx)\nTime (UTC)15:00 15:30 16:00 16:30 17:00 17:30 18:00 18:301.01.52.02.53.03.5\n02468101214\nXRT intensityNegative fluxPositive flux\nXRT total intensity ( ×103DN s−1)MDI flux (×1018Mx)\nTime (UTC)15:00 15:30 16:00 16:30 17:00 17:30 18:00 18:300.70.80.91.01.11.21.31.41.5\n0.01.02.03.04.05.0Fig. 19. Top: evolution of the X-ray intensity (black line) and the mag-\nnetic flux in the region of emerging flux on the left part of the snapshots\nin Fig. 17. The colours of the curves of the flux are consistent with the\ncontours. Red is positive flux, green is negative flux (in absolute value).\nBottom: Same as the top panel, for the second region of flux emergence\nin the bottom right of the snapshots in Fig. 17. There is an X-ray data\ngap between 18:40 and 19:02 that has been filled in by the first available\nvalue after the gap.\na negative flux of a bit less than 2 \u00021018Mx. It is unbalanced\nfor\u00182:5 hr, until it is balanced again at \u001818:15 with an unsigned\nflux of\u00181019Mx. The X-ray emission increases \u00181 hr after the\nemergence is first detected, by \u001870% of the background intensity\nfrom\u00181200 DN s\u00001at 15:00 up to\u00183200 DN s\u000013after 18:30.\nIn the second case of emergence (Fig. 17, right-side arrows,\nand Fig. 19, bottom), the magnetic flux is about 50% weaker than\nin the first region, with a maximum positive and negative flux\n(respectively) between 3 :5\u00021018Mx and 4:5\u00021018Mx. The flux\nis unbalanced during about 2 hr, between \u001816:40 and\u001818:40,\nalthough the flux balance is not obvious afterwards. Like in the\nprevious case, this flux emergence is followed by the rise of an\nX-ray loop. The X-ray emission increases by about 50% from a\nbackground level of 1000 DN s\u00001at 15:00, up to\u00181500 DN s\u00001\nafter 19:00 when the X-ray loop is visible (Fig. 17, bottom right\npanel). Note that there is an X-ray data gap between 18:36 and\n19:02. The X-Ray gap is filled with the first value available after\nthe gap (19:02), which is only an arbitrary cosmetic correction.\n3DN s\u00001: Data Number per Second.\nArticle number, page 13 of 14A&A proofs: manuscript no. Magnetic_Balltracking\nComment on the results\nThe focus is given here on the current capability of the algo-\nrithm, regardless of the capability of the instrument. Therefore,\nfor a better assessment of future improvements of the algorithm,\nthis first example of a science application of magnetic balltrack-\ning is using data as close as possible to what is delivered by\nthe instrument. Therefore, the magnetograms are not free of P-\nmode oscillations that may contaminate the Zeeman-related sig-\nnals (DeForest et al. 2007). Here these oscillations directly a \u000bect\nthe region-extraction algorithm by changing the apparent flux\ndensity and area of the extracted feature. The less the signal-\nto-noise ratio in the extracted area, the more visible the oscilla-\ntions are. This is responsible for the relatively greater oscillations\nin the second case of the flux emergence in Fig. 19 (bottom),\nwhich we observed to be 50% weaker than in the first case (top).\nThis patently shows that for actual statistical analyses with mag-\nnetic balltracking, the preprocessing guidelines in DeForest et al.\n(2007), which are not, per se , part of the magnetic balltracking,\nshould be followed.\n8. Summary and prospects\nIn this paper, we have presented our implementation of an e \u000e-\ncient method called \"magnetic balltracking\" that tracks the mag-\nnetic features down to their finest scales. This new algorithm\nallows us to quantify the evolution of the magnetic flux. Applied\non MDI data , it allowed us to detect, track, and quantify the evo-\nlution of emerging flux between 1018Mx to 1019Mx on a very\nfine scale of a few Mm, and that are followed by the rise of soft\nX-ray loops within a few hours.\nAlthough we have presented here an application to estimate\nthe flux over the tracked features, magnetic balltracking has a\nmuch broader range of possible applications. It could be applied\nto the tracking of the footpoints of magnetic field lines when per-\nforming extrapolations, the study of MHD waves in flux tubes, or\nthe di \u000busion of internetwork elements whose emergence, trans-\nport and disappearance can now be tracked in detail. In addition,\nthe technique may be applied to the formation of Active Re-\ngions. Because magnetic balltracking can detect emerging flux\nand track the elements as they grow and move apart, it could\ncomplement other algorithms used in surveys of the solar cycle\nthat use di \u000berent algorithms to detect and track sunspots, e.g.,\nWatson et al. (2009, 2011) and Goel & Mathew (2014).\nAcknowledgements. This work was funded by the International Max Planck Re-\nsearch School in Göttingen (Germany), and by grant STFC /F002941 /1 from the\nUK’s Science and Technology Facilities Council, held at the School of Physics\nand Astronomy, University of Glasgow.\nReferences\nAttie, R., Innes, D. E., & Potts, H. E. 2009, A&A, 493, L13\nChae, J., Moon, Y ., Park, Y ., et al. 2007, PASJ, 59, 619\nDeForest, C. E., Hagenaar, H. J., Lamb, D. A., Parnell, C. E., & Welsch, B. T.\n2007, ApJ, 666, 576\nDelaboudinière, J., Artzner, G. E., Brunaud, J., et al. 1995, Sol. Phys., 162, 291\nDémoulin, P. & Berger, M. A. 2003, Sol. Phys., 215, 203\nGoel, S. & Mathew, S. K. 2014, Sol. Phys., 289, 1413\nGolub, L., Deluca, E., Austin, G., et al. 2007, Sol. Phys., 243, 63\nGonzalez, R. & Woods, R. 2008, Digital Image Processing, 3rd edn. (Pear-\nson/Prentice Hall)\nHagenaar, H. J., Schrijver, C. J., Title, A. M., & Shine, R. A. 1999, ApJ, 511,\n932\nLamb, D. A., DeForest, C. E., Hagenaar, H. J., Parnell, C. E., & Welsch, B. T.\n2008, ApJ, 674, 520\nParnell, C. E. 2002, MNRAS, 335, 389Potts, H. E., Barrett, R. K., & Diver, D. A. 2004, A&A, 424, 253\nPotts, H. E. & Diver, D. A. 2008, Sol. Phys., 248, 263\nScherrer, P. H., Bogart, R. S., Bush, R. I., et al. 1995, Sol. Phys., 162, 129\nStangalini, M. 2014, A&A, 561, L6\nTsuneta, S., Ichimoto, K., Katsukawa, Y ., et al. 2008, Sol. Phys., 249, 167\nWang, H. 1988, Sol. Phys., 116, 1\nWatson, F., Fletcher, L., Dalla, S., & Marshall, S. 2009, Sol. Phys., 260, 5\nWatson, F. T., Fletcher, L., & Marshall, S. 2011, A&A, 533, A14\nWelsch, B. T., Fisher, G. H., Abbett, W. P., & Regnier, S. 2004, ApJ, 610, 1148\nWelsch, B. T. & Longcope, D. W. 2003, ApJ, 588, 620\nArticle number, page 14 of 14"    },    {        "title": "2006.14127v1.Novel_dynamic_critical_phenomena_induced_by_superfluidity_and_the_chiral_magnetic_effect_in_Quantum_Chromodynamics.pdf",        "content": "Novel dynamic critical phenomena induced by\nsuper\ruidity and the chiral magnetic e\u000bect in\nQuantum Chromodynamics\nNoriyuki Sogabe\nJune 26, 2020arXiv:2006.14127v1  [hep-ph]  25 Jun 2020A Thesis for the Degree of Ph.D. in Science\nNovel dynamic critical phenomena induced by\nsuper\ruidity and the chiral magnetic e\u000bect in Quantum\nChromodynamics\nFebruary 2020\nGraduate School of Science and Technology\nKeio University\nNoriyuki SogabeAcknowledgements\nFirst and foremost, I would like to express the most profound appreciation to\nmy supervisor Prof. Naoki Yamamoto. He has taught me the joy of physics\nand led my life in an exciting direction. His broad knowledge and intuition\nhave strongly in\ruenced my understanding of physics. I am fortunate to have\nNaoki as my adviser and should thank him for his patience. It has been an\nhonor to be his \frst Ph.D. student.\nI am deeply grateful to my collaborator, Masaru Hongo, for my works\non the interplay between the CME and the dynamic critical phenomena. I\nwould also like to thank many people in the community, whom I met at\nconferences, workshops, seminars, and summer schools. I am particularly\ngrateful to Profs. Kenji Fukushima and Yoshimasa Hidaka. I would also like\nto express my gratitude to the referees of my Ph.D. defense, Profs. Youhei\nFujitani, Yasuhiro Nishimura, and Kohei Soga.\nI want to thank Aron Beekman, Gergely Fej} os, Tomoyuki Ishikawa, Ken-\ntaro Nishimura, Di-Lun Yang, who I was sharing the o\u000ece with for numerous\ndiscussions and relaxing chat. I am also very thankful to the other (includ-\ning previous) members of the theoretical physics group at Keio University,\nparticularly Daichi Kagamihara and Hiroaki Wakamura, for their friendship\nand encouragement.\nI appreciate the \fnancial support of research from the Japan Society for\nthe Promotion of Science. When I started writing this thesis during a journey\nto visit Germany, I was stuck at the Frankfurt airport a\u000bected by the typhoon\nHagibis. I want to thank my friend, Alexander Max Eller, and his mother,\nGalvi, to allow me to stay at their house.\nLast but not least, I would like to thank my family for all their love,\nsupport, encouragement, and patience. Without them, I could not have\ncompleted my Ph.D. thesis.\nNoriyuki Sogabe\n3Summary\nUnderstanding the phase structure of Quantum Chromodynamics (QCD) at\n\fnite temperature and \fnite baryon chemical potential is a long-standing\nproblem in the standard model of particle physics. So far, in addition to\nthe nuclear liquid-gas critical point, the possible existence of two critical\npoints is theoretically suggested in the QCD phase diagram: one is the high-\ntemperature critical point between the hadron phase and the quark-gluon\nplasma phase and the other is the high-density critical point between the\nnuclear and quark super\ruid phases. Since these critical points can be po-\ntentially tested in relativistic heavy-ion collision experiments, theoretical pre-\ndictions for critical phenomena near these critical points are important. On\nthe other hand, heavy-ion collision experiments have another goal to search\nfor the chiral transport phenomena related to the quantum anomaly. One\ntypical example is the chiral magnetic e\u000bect, which is the electric current\nalong the magnetic \feld. In particular, it is known that the chiral mag-\nnetic e\u000bect leads to the generation of a novel density wave called the chiral\nmagnetic wave.\nIn this thesis, we \frst construct the low-energy e\u000bective \feld theory near\nthe high-density QCD critical point and study its static and dynamic criti-\ncal phenomena. We \fnd that the critical slowing down of the speed of the\nsuper\ruid phonon near the critical point. Furthermore, we show that the dy-\nnamic universality class of the high-density critical point is not only di\u000berent\nfrom that of the high-temperature critical point, but also a new dynamic\nuniversality class beyond the conventional classi\fcation by Hohenberg and\nHalperin. Since this new universality class stems from the interplay speci\fc\nto QCD between the chiral order parameter and the super\ruid photon, the\nobservation of the dynamic critical phenomena in the vicinity of the high-\ndensity critical point would provide an indirect evidence of the super\ruidity\nin high-density QCD matter.\n56\nWe next consider the second-order chiral phase transition in massless\nQCD under an external magnetic \feld and study the interplay between the\ndynamic critical phenomena and the chiral magnetic e\u000bect. For this purpose,\nwe construct the nonlinear Langevin equations including the e\u000bects of the\nquantum anomaly and perform the dynamic renormalization group analysis.\nAs a result, we show that the presence of the chiral magnetic e\u000bect and the\nresulting chiral magnetic wave change the dynamic universality class of the\nsystem from the so-called model E into the model A within the conventional\nclassi\fcation. We also \fnd that the speed of the chiral magnetic wave tends\nto vanish when the phase transition is approached. This phenomenon is char-\nacterized by the same critical exponents as those for the critical attenuation\nof the sound wave near the critical points in liquid-gas phase transitions.Contents\nAcknowledgements 3\nSummary 5\n1 Introduction 11\n2 Overview of QCD 15\n2.1 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . 15\n2.1.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 15\n2.1.2 Asymptotic freedom . . . . . . . . . . . . . . . . . . . 16\n2.1.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 17\n2.2 Phase structure . . . . . . . . . . . . . . . . . . . . . . . . . . 20\n2.2.1 Order parameters . . . . . . . . . . . . . . . . . . . . . 20\n2.2.2 Formation of diquark condensate in high-density QCD 22\n2.2.3 Symmetry breaking patterns . . . . . . . . . . . . . . . 23\n2.3 Chiral magnetic e\u000bect . . . . . . . . . . . . . . . . . . . . . . 24\n2.3.1 Symmetry argument . . . . . . . . . . . . . . . . . . . 25\n2.3.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 26\n2.3.3 Chiral magnetic wave . . . . . . . . . . . . . . . . . . . 30\n3 Theory of dynamic critical phenomena 33\n3.1 Dynamic universality class . . . . . . . . . . . . . . . . . . . . 33\n3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35\n3.2.1 Ginzburg-Landau theory . . . . . . . . . . . . . . . . . 35\n3.2.2 Langevin theory . . . . . . . . . . . . . . . . . . . . . . 37\n3.2.3 Dynamic perturbation theory . . . . . . . . . . . . . . 39\n3.3 Renormalization group analysis . . . . . . . . . . . . . . . . . 40\n3.3.1 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . 41\n78 CONTENTS\n3.3.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 45\n4 Dynamic critical phenomena of the high-density QCD critical\npoint 47\n4.1 Hydrodynamic variables . . . . . . . . . . . . . . . . . . . . . 47\n4.2 Static critical phenomena . . . . . . . . . . . . . . . . . . . . . 48\n4.3 Dynamic critical phenomena . . . . . . . . . . . . . . . . . . . 50\n4.3.1 Langevin theory . . . . . . . . . . . . . . . . . . . . . . 50\n4.3.2 Hydrodynamic modes . . . . . . . . . . . . . . . . . . . 52\n4.3.3 Dynamic critical exponent . . . . . . . . . . . . . . . . 53\n4.4 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . 54\n5 Dynamic critical phenomena induced by the chiral magnetic\ne\u000bect in QCD 57\n5.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57\n5.1.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 57\n5.1.2 Hydrodynamic variables . . . . . . . . . . . . . . . . . 59\n5.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60\n5.2.1 Langevin theory . . . . . . . . . . . . . . . . . . . . . . 60\n5.2.2 Dynamic perturbation theory . . . . . . . . . . . . . . 61\n5.3 Renormalization-group analysis . . . . . . . . . . . . . . . . . 66\n5.3.1 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . 67\n5.3.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 69\n5.3.3 Physical consequences . . . . . . . . . . . . . . . . . . 78\n5.4 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . 79\n6 Summary and outlook 81\nA Static RG of the scalar \feld theory 83\nA.1 Perturbative RG equation . . . . . . . . . . . . . . . . . . . . 83\nA.2 Wilson-Fisher \fxed point . . . . . . . . . . . . . . . . . . . . . 88\nA.3 Useful intergrals . . . . . . . . . . . . . . . . . . . . . . . . . . 89\nB Coupling to energy-momentum densities 91\nB.1 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91\nB.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93\nB.2.1 Full Langevin equations . . . . . . . . . . . . . . . . . 93\nB.2.2 Decomposition of momentum density . . . . . . . . . . 95CONTENTS 9\nB.2.3 Hydrodynamic modes . . . . . . . . . . . . . . . . . . . 96\nB.2.4 Dynamic critical exponent . . . . . . . . . . . . . . . . 98\nC Calculation of the self-energies and the vertex function 101\nC.1 Self-energy \u0005 . . . . . . . . . . . . . . . . . . . . . . . . . . . 101\nC.2 Self-energy \u0006 . . . . . . . . . . . . . . . . . . . . . . . . . . . 103\nC.3 Vertex function V. . . . . . . . . . . . . . . . . . . . . . . . . 107\nD Linear-stability analysis on the \fxed points (iii) and (iv) 111Chapter 1\nIntroduction\nQuantum Chromodynamics (QCD) is the fundamental theory of the strong\ninteraction between quarks and gluons. One of the remarkable features of\nQCD is the asymptotic freedom: the coupling constant of the strong inter-\naction becomes small when the typical energy scale of the system is get-\nting large. When the energy scale is lower than the intrinsic scale of QCD,\n\u0003QCD\u0018200 MeV, the coupling constant becomes large, and the pertur-\nbation theory breaks down. In this low-energy regime, quarks and gluons\nare con\fned in color-neutral hadrons, such as baryons made of three quarks\nand mesons consisting of a quark and an antiquark. Another property of\nlow-energy QCD is the spontaneous breaking of chiral symmetry. The chiral\nsymmetry of QCD in the massless quark limit is broken in the vacuum. In\nthe real world with \fnite quark masses, the chiral symmetry is an approx-\nimate symmetry for the light quarks, especially up and down quarks whose\nmasses are su\u000eciently smaller than \u0003 QCD.\nWhat happens to matter when it is heated and/or squeezed up to the\norders of \u0003 QCD? Such matter under extreme conditions at high temperature\nT\u00181012kelvin and/or baryon chemical potential \u0016B(corresponding to mass\ndensity\u00181012kg=cm3), can also be described by QCD. However, it has\nbeen a long-standing problem to understand the phase structure of QCD at\n\fniteTand\u0016B[1]. Figure 1.1 shows a conjectured phase digram. At low\nTand low\u0016B, the chiral symmetry is spontaneously broken in the hadron\nphase, whereas it is restored at su\u000eciently high T, and the quark-gluon\nplasma is realized. On the other hand, at high \u0016B, quarks form Cooper\npairs induced by the attractive one-gluon exchange interaction. It follows\nthat the gluons acquire \fnite masses in an analogy of the gapped photons in\n1112 CHAPTER 1. INTRODUCTION\nHadronµB TQuark-Gluon Plasma \nNuclear SuperfluidityColor Superconductivity/Quark Superfluidity\nʙ200 MeV ʙ200 MeV \nFigure 1.1: Schematic phase diagram of QCD\nordinary superconductors. This state is called the \\color superconductivity.\"\nBesides, quark matter and nuclear matter can exhibit the super\ruidity as a\nconsequence of the spontaneous breaking of the symmetry associated with\nthe baryon number conservation.\nAs is also illustrated in Fig. 1.1, there are two possible QCD critical\npoints where the \frst-order chiral phase transition line (the doubled line in\nFig. 1.1) terminates. One is the high-temperature critical point between\nthe hadron phase and the quark-gluon plasma phase [2]; the other is the\nhigh-density critical point between the nuclear and quark super\ruid phases\n[3, 4, 5]. An important task is to understand the dynamic critical phenomena\nin QCD. In general, dynamic critical phenomena can be classi\fed based on\nthe symmetries and the low-energy gapless modes of systems near a second-\norder phase transition or a critical point. Such a classi\fcation of dynamic\ncritical phenomena is called the dynamic universality class [6]. In QCD, the\nhigh-temperature critical point belongs to the same dynamic universality\nclass as that of the nuclear liquid-gas critical point [7, 8, 9, 10], the so-called\nmodel H within the conventional classi\fcation by Hohenberg and Halperin\n[6]. On the other hand, the second-order chiral phase transition in massless\ntwo-\ravor QCD at \fnite Tand zero\u0016Bbelongs to the same class as that of\nO(4) antiferromagnets [11].\nRelativistic heavy-ion collision experiments can potentially test these13\nQCD critical points [12]. In particular, one of the big goals of the Beam\nEnergy Scan (BES) program at the Relativistic Heavy Ion Collider (RHIC)\nis to search for the high-temperature QCD critical point [13]. Moreover, the-\noretical predictions speci\fc to dense QCD matter would be crucial for the\nfuture low-energy heavy-ion collisions at Facility of Antiproton and Ion Re-\nsearch (FAIR), Nuclotron-based Ion Collider Facility (NICA), Japan Proton\nAccelerator Research Complex (J-PARC), and Heavy Ion Research Facility\n(HIRF).\nOn the other hand, the BES program at RHIC has another goal [13] to\nsearch for the anomalous chiral transport phenomena related to the quantum\nanomaly [14, 15]. One typical example is the chiral magnetic e\u000bect (CME),\nwhich is the generation of the electric current along the magnetic \feld [16, 17,\n18, 19]. In particular, a remarkable consequence of the CME is the creation of\nthe density-wave called the chiral magnetic wave (CMW) [20, 21]. Although\npossible signals consistent with the presence of the CME may have been\nobserved in RHIC [22, 23] and also in Large Hadron Collider (LHC) [24],\nit is pointed out that there are ambiguities in this interpretation due to\npossible background e\u000bects not related to the CME [25].1These ambiguities\nare currently tested by comparing collisions of the isobaric nuclei such as\n96\n44Ru and96\n40Zr, with the same background contributions but di\u000berent CME\nsignals due to the nuclear-charge di\u000berence [26].\nIn this thesis, we study the novel dynamic critical phenomena induced by\nthe super\ruidity and the CME in QCD, respectively . In each case, we use the\nlow-energy e\u000bective theory based on the symmetries and the hydrodynamic\nvariables of QCD.\nFor the \frst subject, we construct the low-energy e\u000bective theory of the\nsystem near the high-density QCD critical point and study its static and\ndynamic critical phenomena [28]. In particular, we \fnd the critical slowing\ndown of the speed of the super\ruid phonon. Moreover, we \fnd that the dy-\nnamic universality class of the high-density critical point is not only di\u000berent\nfrom that of the high-temperature critical but also is a new class beyond the\nHohenberg and Halperin's conventional classi\fcation [6]. This new universal-\nity class stems from the interplay speci\fc to QCD between the chiral order\nparameter and the super\ruid photon. Therefore, observation of the dynamic\n1On the other hand, the CME has already been observed in the table-top experiments\nof the Weyl/Dirac semimetals (see, e.g., Ref. [27]). In these systems, relativistic fermions\nappear as quasiparticles close to the band touching points.14 CHAPTER 1. INTRODUCTION\ncritical phenomenon in the vicinity of the high-density critical point would\nprovide indirect evidence of super\ruidity in the dense-QCD matter.\nFor the second subject, we study the dynamic critical phenomena of the\nsecond-order chiral phase transition in massless QCD under an external mag-\nnetic \feld [29], and clarify the interplay between the dynamic critical phe-\nnomena and the CME in QCD. For this purpose, we \frst construct the nonlin-\near Langevin equations incorporating the quantum-anomaly e\u000bects [30, 31],\nand study it by using the dynamic renormalization group [32, 33, 34]. As a\nresult, we show that the inclusion of the CME changes the dynamic univer-\nsality class from the model E into model A within the conventional classi\f-\ncation. We also \fnd that the speed of the CMW tends to zero near the phase\ntransition. We here observe the same critical exponents as those of the crit-\nical attenuation of the sound wave known in the system near the liquid-gas\ncritical point [35].\nThis thesis is organized as follows. In Chap. 2, we give a brief overview\nof QCD. In Chap. 3, we review the theory of dynamic critical phenomena.\nBased on the reviews above, in Chap. 4, we study the static and dynamic\ncritical phenomena of the high-density QCD critical point. In Chap. 5, we\nclarify the interplay between the dynamic critical phenomena and the CME\nin QCD. Finally, we conclude with Chap. 6. Among others, Chaps. 4 and 5\nare based on our original papers [28] and [29], respectively. The work [28] is\na collaboration with N. Yamamoto, and the work [29] is a collaboration with\nM. Hongo and N. Yamamoto.\nIn this thesis, we will work in natural units, where the speed of light, the\nreduced Plank constant, the elementary charge, and the Boltzmann constant\nare set equal to unity, c=~=e=kB= 1.Chapter 2\nOverview of QCD\nWe here give an overview of QCD. We start with its basic properties in\nSec. 2.1. Next, in Sec. 2.2, we review the phase structure of QCD. In Sec. 2.3,\nwe provide a brief explanation of the CME.\n2.1 Quantum Chromodynamics\nIn Sec. 2.1.1, we \frst introduce the Lagrangian of QCD. In section 2.1.2, we\nnext explain the asymptotic freedom. Then, in Sec. 2.1.3, we summarize the\ninternal symmetries of QCD.\n2.1.1 Lagrangian\nThe Lagrangian (density) of QCD is given by\nLQCD=Lquark+Lgluon; (2.1)\nwhere\nLquark = \u0016qi(i\r\u0016D\u0016\u0000mi)qi; (2.2)\nLgluon=\u00001\n2TrG\u0016\u0017G\u0016\u0017: (2.3)\nHere,Lquark is the kinetic term for the quark \feld q. The quark \feld q(x) has\nthe internal degrees of freedom in addition to the coordinates x\u0016= (t;x):\nthe \frst one is the \ravor labeled by i= 1;:::;N f. There are six \ravors of\nquarks in QCD: up (u), down (d), charm (c), strange (s), top (t), bottom\n1516 CHAPTER 2. OVERVIEW OF QCD\n(b). Quarks of di\u000berent \ravors have di\u000berent masses mi. U, c, and t quarks\nhave the electric charge + 2/3, while d, s, and b quarks have the electric\ncharge\u00001=3, in a unit of the elementary charge. The second one is the\nspinor component running 1 ;:::;4 with\r\u0016being a 4\u00024 matrix in the Dirac\nspace. The antiquark \feld is de\fned by \u0016 qi\u0011qy\ni\r0. The last one is the\ncolor labeled by 1 ;:::;N c. The quark \feld is coupled to the gluon \feld Aa\n\u0016\n(a= 1;:::;N2\nc\u00001) through the covariant derivative,\nD\u0016\u0011@\u0016\u0000igA\u0016; (2.4)\nwheregis the interaction strength; tais the generators of the color SU( Nc)\nspace with\n[ta;tb] = ifabctc;Tr(tatb) =\u000eab\n2: (2.5)\nHere,fabcis called the structure constants.\nLet us see thatLgluon includes the kinetic term for the gluon \felds and\ntheir self-interactions. In Eq. (2.3), Ga\n\u0016\u0017is the \feld strength de\fned in the\nfollowing form:\nG\u0016\u0017\u0011i\ng[D\u0016;D\u0017]: (2.6)\nBy using Eq. (2.4), one can write Eq. (2.6) into\nG\u0016\u0017=\u0000\n@\u0016Aa\n\u0017\u0000@\u0017Aa\n\u0016+gfabcAb\n\u0016Ac\n\u0017\u0001\nta: (2.7)\nFrom this expression, one can also rewrite Lgluon into\nLgluon=\u00001\n4\u0000\n@\u0016Aa\n\u0017\u0000@\u0017Aa\n\u0016\u00012\u0000gfabc(@\u0016A\u0017\na)Ab\n\u0016Ac\n\u0017\u0000g2\n4fabcfadeAb\n\u0016Ac\n\u0017A\u0016\ndA\u0017\ne:\n(2.8)\nHere, the \frst term is the kinetic term for the gluon \felds; the second and\nthe third terms are 3- and 4-points interaction terms among the gluons.\n2.1.2 Asymptotic freedom\nOne of the remarkable features of QCD is the asymptotic freedom [36, 37]:\nthe coupling constant gbecomes smaller when the typical energy scale is2.1. QUANTUM CHROMODYNAMICS 17\ngetting large. Generally, in \feld theories, when one takes into account per-\nturbative loop-corrections, there are some divergences in the calculations. To\nremove those divergences and obtain some physically meaningful results, the\nparameters in the Lagrangian should be no more constants but depend on\nthe typical energy scale at which such divergences are removed by renormal-\nizing the parameters. This is also the case for the interaction parameter g\nintroduced in Eq. (2.4) in QCD. It is known that the interaction strength g\nsatis\fes the following renormalization group equation to its leading order:\n\u0016@g\n@\u0016=\u0000b0\n(4\u0019)2g3+O(g5); (2.9)\nwith\u0016being the renormalization scale and b0\u001111\n3Nc\u00002\n3Nf. Since Eq. (2.9)\ncan be solved by the separation of variables method at this order, by de\fning\n\u000bs\u0011g2=(4\u0019), one \fnds the solution as\n\u000bs(\u0016) =2\u0019\nb0log(\u0016=\u0003QCD); (2.10)\nwhere \u0003 QCD\u0019200 MeV is the typical energy scale of QCD, and several ex-\nperiments determines its value [38]. This solution shows that the interaction\nstrength\u000bs(\u0016) runs towards a smaller value as /1=log(\u0016) when the typical\nenergy scale of the system \u0016increases.\n2.1.3 Symmetries\nThe internal symmetries of QCD can be summarized as\nG= SU(Nc)C\u0002SU(Nf)L\u0002SU(Nf)R\u0002U(1) B; (2.11)\nwhere each symmetry group is de\fned as follows. Note here that the U(1)\naxial symmetry is not included in Gfrom the reason given at the end of this\nsubsection.\nColor gauge symmetry\nThe color gauge symmetry SU( Nc)Cis the invariance under the following\ngauge transformation in color SU( Nc) space:\nq!VCq; A\u0016!VCA\u0016Vy\nC+i\ng(@\u0016VC)Vy\nC; V C\u0011e\u0000i\u0012a(x)ta: (2.12)18 CHAPTER 2. OVERVIEW OF QCD\nHere\u0012a(x) is a local transformation parameter that depends on the coordi-\nnates (for the de\fnition of ta, see Eq. (2.5)).\nTo see the invariance of QCD Lagrangian under Eq. (2.12), we \frst cal-\nculate the transformation laws for D\u0016qand the \feld strength G\u0016\u0017,\nD\u0016q!VC(D\u0016q); (2.13)\nG\u0016\u0017!VCG\u0016\u0017Vy\nC: (2.14)\nHere, one can derive the second transformation by using the \frst one. (See\nEqs. (2.4) and (2.6) for the de\fnitions of the covariant derivative D\u0016and\nG\u0016\u0017, respectively.) From Eqs. (2.13) and (2.14), one can con\frm that LQCD\nis invariant under the color gauge transformation (2.12).\nChiral symmetry\nThe chiral symmetry SU( Nf)L\u0002SU(Nf)Ris the invariance under the inde-\npendent rotation of right-handed and left-handed components of the quark\n\feld in the \ravor space. Here, the right-handed quark qRand the left-handed\nquarkqLare de\fned by\nqL\u00111\u0000\r5\n2q; q R\u00111 +\r5\n2q: (2.15)\nThese transformations under the chiral transformation are\nqL!VLqL; q L!VRqR; V L;R\u0011e\u0000i\u0012a\nL;R\u0015a: (2.16)\nHere,\u0015abeing the generator of SU( Nf)L;R. Note that the transformation\nparameters \u0012a\nL;Rdo not depend on x\u0016(generally such a transformation is\ncalled a global symmetry), unlike \u0012a(x) introduced in Eq. (2.12) for the color\ngauge symmetry.\nLet us see the chiral transformation (2.16) of the QCD Lagrangian (2.1).\nWe \frst decompose the quark \felds into\nq=qR+qL; (2.17)\nand rewrite the quark sector of the QCD Lagrangian (2.2) into\nLquark = \u0016qLi\r\u0016D\u0016qL+ \u0016qRi\r\u0016D\u0016qR\u0000\u0016qL^mqR\u0000\u0016qR^mqL: (2.18)2.1. QUANTUM CHROMODYNAMICS 19\nHere, ^mdenotes the matrix in the \ravor space, and its components are given\nbymi. In Eq. (2.18), the \frst two terms are invariant under the chiral\ntransformation, whereas the last two mass terms are not invariant except for\nthe particular vector -like choice of the transformation parameters \u0015a\nL=\u0015a\nR.1\nTherefore, the chiral symmetry is exact only when one sets the quark masses\nmito zero (this massless limit is called the chiral limit). Practically, the\nmass of u quark, mu\u00193 MeV and that of d quark, md\u00195 MeV are small\ncompared to \u0003 QCDintroduced in Sec. 2.1.2. In general, the chiral symmetry\nis an approximate symmetry of QCD, when miis small compared to any\ntypical energy scales of the system.\nBaryon number symmetry\nThe baryon number symmetry U(1) Bis the invariance under\nq!e\u0000i\u000bBq; (2.19)\nwhere\u000bBis the global transformation parameter. The QCD Lagrangian is\ninvariant under this transformation.\nU(1) axial symmetry and its quantum anomaly\nIn addition to the symmetry group G, the classical Lagrangian of QCD has\nthe approximate axial symmetry U(1) A, which is the invariance under\nq!e\u0000i\u000bA\r5q; (2.20)\nwhere\u000bAis the transformation parameter. This transformation is equivalent\nto\nqL!ei\u000bAqL; q R!e\u0000i\u000bAqR: (2.21)\nSimilarly to the chiral symmetry, U(1) Acan be regarded as an approximate\nsymmetry of QCD at the classical Lagrangian level in the presence of small\nquark masses mi. However, it is known that the quantum e\u000bects break the\nU(1) axial symmetry [39, 40],\nU(1) Aanomaly!Z(2Nf)A: (2.22)\n1Vector transformations are in-phase between left- and right-handed quarks. Mean-\nwhile, Axial transformations are opposite-phase rotation with di\u000berent chirality.20 CHAPTER 2. OVERVIEW OF QCD\nHere,Z(2Nf)Acorresponds to the symmetry under the parameter choices of\nthe parameter \u000bA=n\u0019=N f(n= 1;:::;2Nf).2Because of the presence of this\nquantum anomaly e\u000bect, we distinguish U(1) Afrom the symmetry group G.\nWe will discuss the residual part Z(2Nf)Ain the context of the continuity\nbetween the nuclear/quark super\ruid phases at the end of Sec. 2.2.3.\n2.2 Phase structure\nWe introduce the order parameters of QCD and discuss those symmetry\nbreaking patterns in Sec. 2.2.1. Section 2.2.2 is devoted to explaining why\nthe diquark condensation is favored at high-density QCD. In Sec. 2.2.3, we\nclassify the phases of Fig. 1.1 by using the order parameters.\n2.2.1 Order parameters\nThe symmetry group of QCD, Ggiven in Eq. (2.11) can be spontaneously\nbroken in the ground states. This spontaneous symmetry breaking can be\ncharacterized by the two representative order parameters in QCD de\fned as\nfollows.\nNote that in this subsection, we explicitly write the \ravor and the color\nindices:\ni;j;k;::: = u;d;s;::: (\ravor); (2.23)\nA;B;C;::: = r;g;b;::: (color); (2.24)\nwhere color indices range over red, green, and blue,... (r,g,b,...) in general.\nThe general transformation laws under all of the symmetries introduced in\nSec. 2.1.3,G\u0002U(1) Acan be written as follows:\n(qL)i\nA!ei\u000bAe\u0000i\u000bB(VL)ij(VC)AB(qL)j\nB; (2.25)\n(qR)i\nA!e\u0000i\u000bAe\u0000i\u000bB(VR)ij(VC)AB(qR)j\nB: (2.26)\n2Unlike the chiral anomaly in the background electromagnetic \feld in Eq. (2.66), the\ndivergence of the U(1) Acurrent in the presence of the gluons, discussed here, is related\nto the topological charge called the instanton number of the non-Abelian gauge \feld.\nThis topological nature leads to the invariance for some particular discrete choices of the\nparameters for U(1) Atransformation, Z(2Nf)A. See, e.g., Refs. [39, 41] for the details.2.2. PHASE STRUCTURE 21\nChiral order parameter\nOne of the important order parameters in QCD is the chiral order parameter\nde\fned by\n\bij\u0011\n(\u0016qR)j\nA(qL)i\nA\u000b\n; (2.27)\nwhich is the matrix in the \ravor space. By using Eqs. (2.25) and (2.26), one\n\fnds that this order parameter is transformed under G\u0002U(1) Aas\n\bij!e\u00002i\u000bA(VL)ik\bkl(Vy\nR)lj: (2.28)\nWe note that the determinant of \b characterizes the quantum anomaly [40].\nThe transformation law under (2.28):\ndet \b!e\u00002i\u000bAdet \b; (2.29)\nfollows that det \b preserves the chiral symmetry SU(N f)L\u0002SU(N f)Rbut\nbreaks U(1) A. Here, we use Eq. (2.28) and det VL= detVR= 1 to derive\nEq. (2.29).\nThe chiral order parameter breaks the chiral symmetry into the subgroup:\nSU(Nf)L\u0002SU(Nf)R!SU(3) L+R; (2.30)\nwhere SU(Nf)L+Ris the invariance under the vector -like choice of the param-\neters,VL=VR. Therefore, the chiral order parameter is an order parameter\ncharacterizing the spontaneous breaking of the chiral symmetry. Note here\nthat U(1) Ais not spontaneously broken, because the quantum e\u000bects have\nalready broken this symmetry as we explained in Sec. 2.1.3.\nDiquark condensate\nThe second order parameter in QCD is the diquark condensate. Each of the\nright-hand and left-handed quark \felds is de\fned by\n(dy\nL)Ai\u0011\"ABC\"ijk\n(qL)j\nBOC(qL)k\nC\u000b\n; (2.31)\n(dy\nR)Ai\u0011\"ABC\"ijk\n(qR)j\nBOC(qR)k\nC\u000b\n: (2.32)\nHere, two of the quark \felds in each of dy\nR;Lare chosen to be antisymmetric\nunder the exchanges of the color and \ravor, respectively (and also the spin22 CHAPTER 2. OVERVIEW OF QCD\nwhich is not explicit in these expressions). The labels of quarks in the right-\nhand sides will be con\frmed in Sec. 2.2.2. Besides, OCdenotes the charge\nconjugation operator\nOC\u0011i\r2\r0; (2.33)\nwhich is also necessary for the diquark condensate being a Lorentz scalar.\nBy using Eqs. (2.25) and (2.26), the diquark condensates are transformed by\nG\u0002U(1) Aas follows:\n(dL)Ai!e2i\u000bAe\u00002i\u000bB(VL)ij(VC)AB(dL)Bj; (2.34)\n(dR)Ai!e\u00002i\u000bAe\u00002i\u000bB(VR)ij(VC)AB(dL)Bj: (2.35)\nLet us consider the realistic colors and focus our attention on the light\nquarks, and set NC=Nf= 3. The diquark condensate breaks the color\ngauge symmetry and the chiral symmetry into their mixed subgroup:\nSU(3) C\u0002SU(3) L\u0002SU(3) R!SU(3) C+L+R: (2.36)\nHere, SU(3) C+L+R symmetry is the invariance under the particular combina-\ntion of the gauge transformation and the chiral transformation, Vy\nC=VL=\nVR. One can con\frm this invariance by using the transformation law of the\ndiquark condensate (2.34). The resulting phase is invariant under simulta-\nneous color and \ravor transformations. For this reason, it is called the color\n\ravor locked phase [42]. From Eq. (2.36), the diquark condensate is an order\nparameter characterizing the color superconductivity, which is the \\spon-\ntaneous breaking\" of the color gauge symmetry. In addition to this color\ngauge symmetry breaking, the diquark condensate also breaks the baryon\nsymmetry,\nU(1) B!Z(2)B; (2.37)\nwhereZ(2)Bis the discrete symmetry, which corresponds to the choices of the\nparameter, \u000bB= 0;\u0019in Eq. (2.19). Therefore, the diquark condensate is also\nan order parameter for the super\ruidity characterized by the spontaneously\nbreaking of the U(1) Bsymmetry.\n2.2.2 Formation of diquark condensate in high-density\nQCD\nIn the high-density QCD ( \u0016B\u001d\u0003QCD;T), quarks form a diquark condensate.\nIn this regime, the degenerate quarks near the Fermi surface can be described2.2. PHASE STRUCTURE 23\nby the weakly coupled QCD due to the asymptotic freedom. The interac-\ntion between two quarks is dominated by the one-gluon exchange which is\nproportional to a product of the SU(3) interaction vertices,\n(ta)AB(ta)CD=1\n6(\u000eAB\u000eCD+\u000eAC\u000eBD)\u00001\n3(\u000eAB\u000eCD\u0000\u000eAC\u000eBD): (2.38)\nHere, the \frst/last two terms represent the repulsive/attractive interaction\nbetween the quarks whose color labels are symmetric/antisymmetric under\nthe exchange of AandCorBandD, respectively. Degenerate fermionic\nsystems with a 2-body (4-point) attractive interaction have the instabilities\ntowards the formation of the Cooper pairs [43, 44]. According to this BCS\nmechanism, quarks form Cooper pairs in the attractive channels, and the\n\fnite diquark condensate is realized as the ground state of high-density QCD\n[45, 46, 47, 48].\nThe color, \ravor, and spin structure of the diquark condensate are de-\ntermined as follows. The color indices should be antisymmetric to have an\nattractive interaction. The spin indices should also be antisymmetric so that\nthe total spin of the diquark condensate is zero. This isotropic combination is\nenergetically favorable in general because it allows e\u000ecient use of the Fermi\nsurface. Finally, from the Pauli principle, the \ravor indices should also be\nantisymmetric.\n2.2.3 Symmetry breaking patterns\nHere, we will look at the phases classi\fed by the chiral order parameter \b ij\nand the diquark condensate dR;L[3, 4]. Let us consider NC=Nf= 3 and\nthe most symmetric ground state:\n\bij=\u001b\u000eij; (2.39)\n(dL)Ai=\u0000(dR)Ai=\u000eAi\u001e: (2.40)\nHere,\u001bis called the chiral condensate.\nIn principle, there are four possible phases characterized by \u001band\u001e,\nas summarized in Table 2.1. Here, the symmetry breaking patterns of the\nindividual phases are also listed ( Hdenotes the unbroken symmetry of G).\nThe chiral condensate \u001bclassi\fes the quark-gluon plasma phase and the\nhadron phase. It is approximately zero (of an order of the quark mass) in\nthe quark-gluon plasma phase and is \fnite in the hadron phase. On the other24 CHAPTER 2. OVERVIEW OF QCD\nPhaseUnbroken Symmetries Order ParametersQuark-Gluon PlasmaHadronColor Superconductivity/Quark SuperfluidityNuclear Superfluidity\u0000⇠0\n<latexit 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sha1_base64=\"jQ0H8KXzqH+mmbQ4zUq+9xGqPc4=\">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</latexit>\nTable 2.1: Symmetry breaking patterns of QCD phase diagram [3]\nhand, any symmetries cannot distinguish the two super\ruid phases, so that\nthese two phases can be continuously connected. This conjecture is called\nthe hadron-quark continuity [49].\nLet us see the crossover of the super\ruid phases on the aspects of the\nchiral symmetry and the U(1) Asymmetry following the argument in Ref. [3].\nWe consider the order parameter, ( dL)Ai(dy\nR)Aj=\u0000\u001e2\u000eij, which breaks these\nsymmetries,\n(dL)Ai(dy\nR)Aj!e4i\u000bA(VL)ik(dL)Ak(dy\nR)Al(VR)lj: (2.41)\nThe quark super\ruid phase with \fnite \u001e26= 0 breaks the chiral symmetry but\npreserves Z(4)A(\u000bA= 0;\u0019=2;\u0019;3\u0019=2) of U(1) Aas a subgroup. On the other\nhand, in the nuclear super\ruid phase, \u001b6= 0 and\u001e26= 0 break the chiral\nsymmetry but preserves Z(2)A(\u000bA= 0;\u0019). At a glance, the residual discrete\nU(1) Asymmetries look di\u000berent. However, the quantum anomaly has already\nbroken U(1) AintoZ(6)A, which includes Z(2)Abut not Z(4)A. The residual\nU(1) Asymmetry is the same Z(2)Ain both phases, in the presence of the\nquantum anomaly. Not only the chiral symmetry but also U(1) Asymmetry\ncannot classify the nuclear/quark super\ruid phases.\n2.3 Chiral magnetic e\u000bect\nSo far, we have discussed the static properties of QCD. In this section, we\ndiscuss its dynamics. Chiral transport phenomena are the novel macroscopic\ntransport phenomena induced by the chirality of quarks in QCD. As an exam-\nple, we here discuss the so-called chiral magnetic e\u000bect (CME). In Sec. 2.3.1,2.3. CHIRAL MAGNETIC EFFECT 25\nwe \frst demonstrate that the CME is a phenomenon speci\fc to the presence\nof the chirality imbalance from the symmetry arguments. Next in Sec. 2.3.2,\nwe derive the CME for noninteracting quarks under an external magnetic\n\feld. Finally, in Sec. 2.3.3, we discuss the relevance of the CME to the low-\nenergy hydrodynamic modes. In this section, we consider the chiral limit and\nsetmi= 0.\n2.3.1 Symmetry argument\nWe here discuss the symmetry aspects of the CME. Let us write the electric\ncurrentjby using the electric \feld Eand the magnetic \feld B. From the\nrotational symmetry of the system, we obtain\nj=\u001bEE+\u001bBB; (2.42)\nwhere\u001bEand\u001bBare some constants. The \frst term is nothing but the Ohmic\nlaw, which is allowed in the usual matter without chirality imbalance. On\nthe other hand, the second term is allowed when \u001bBis proportional to the\nchirality imbalance\n\u00165=\u0016R\u0000\u0016L\n2: (2.43)\nThe parity transformation leads to\nj!\u0000j;E!\u0000E;B!B; (2.44)\nand\n\u00165!\u0000\u00165: (2.45)\nOne can understand the last transformation by using the fact that the parity\ntransformation exchanges the chirality,\n\u0016R!\u0016L; \u0016 L!\u0016R: (2.46)\nAs we will see in Sec. 2.3.2 (in particular in Eq. (2.72)), the CME is the\ngeneration of such second term with [16, 17, 18, 19]\n\u001bB=\u00165\n2\u00192: (2.47)26 CHAPTER 2. OVERVIEW OF QCD\n2.3.2 Derivation\nWe here \frst derive the chiral anomaly in the presence of the background\nelectromagnetic \felds and the CME. In this subsection, we \frst neglect the\ngluons and discuss these e\u000bects later. The discussion here follows the original\nargument in Ref. [18] and also the recent lecture notes [50, 51].\nLandau Level\nLet us consider the Weyl equation for the left-handed quark qL,\ni\u0016\u001b\u0016D\u0016qL= 0; (2.48)\nwhere \u0016\u001b\u0016= (1;\u0000\u001bi),D\u0016=@\u0016\u0000iA\u0016, andA\u0016is the U(1) emgauge \feld. The\nWeyl equation can be written as\ni(@t\u0000\u001biDi)qL= 0: (2.49)\nWe set the magnetic \feld in the zdirection,B= (0;0;B), and take the\nparticular\n(Ax;Ay;Az) = (0;Bx; 0): (2.50)\nWe write the 2-component \feld qLas\nqL= e\u0000i(!t\u0000kyy\u0000kzz)\u0012q\u0000\nq+\u0013\n; (2.51)\nwith!andki(i=y;z) being the energy and the momentum of the quark,\nrespectively. Then, by substituting Eq. (2.51) into Eq. (2.49), we \fnd\n\u0012!+kz\u0000i(@x\u0000Bx+ky)\n\u0000i(@x+Bx\u0000ky)!\u0000kz\u0013\u0012q\u0000\nq+\u0013\n= 0: (2.52)\nWe can readily obtain the following solution of this equation:\n!=\u0000kz;q\u0000/exp\u0014\n\u0000(Bx\u0000ky)2\n2B\u0015\n; q += 0; (2.53)\nwhereas a similar form,\n!=kz;q+/exp\u0014(Bx\u0000ky)2\n2B\u0015\n; q\u0000= 0; (2.54)2.3. CHIRAL MAGNETIC EFFECT 27\nis not a solution because we cannnot normalize the eigenvector in a particular\nway atx!1 . On the other hand, the Weyl equation for the right-handed\nquark is\ni\u001b\u0016D\u0016qR= 0; (2.55)\nwith\u001b\u0016= (1;\u001bi) yields the solution for !=kzwith the normalized eigen-\nvector. Therefore, we obtain the zero eigenenergies of the Dirac fermion q\nincluding both of the Weyl fermions qLandqR,\n!=\u0006kz; (2.56)\nwhere the signs correspond to di\u000berent chirality of the quarks. These gapless\neigenmodes are called the lowest Landau level.\nIn general, the dispersion relation for Eq. (2.52) can be calculated as\n!2=k2\nz+B(2n+ 1\u0000s); (2.57)\nwheren= 0;1;2;:::ands=\u00061 is the eigenvalue of \u001bz. The lowest landau\nlevel corresponds to ( n;s) = (0;1). Note that Eq. (2.57) does not depend on\nky, so that any eigenvalues are degenerated by all of the degrees of freedom\nthatkycan take in our gauge choice. This is called the Landau degeneracy,\nand it is given by B=(2\u0019) per unit area.3\nChiral anomaly\nWe here derive the chiral anomaly. Figure. 2.1(a) shows the spectrum in a\nstatic magnetic \feld, i.e., the dispersion relation (2.57). Here, the branches\n3Let us calculate the Landau degeneracy. We consider the xy-plane bounded by 0 <\nx < Lx;0< y < Ly. The momentum in ydirection can take ky= 2\u0019ny=Ly, (ny2Z).\nHowever, the center of the wave function xc\u0011B=kymust be inside the range of x. This\nyields the upper limit of nysuch that\n0<B\nky<Lx,0<ny<LxLyB\n2\u0019: (2.58)\nAll of the modes included in this inequality contribute to the degeneracy. The number of\nsuch modes per unit area is:\nNy=B\n2\u0019: (2.59)28 CHAPTER 2. OVERVIEW OF QCD\nkz\n<latexit sha1_base64=\"e9Ht67rpTlg1N6IGEDCedjgaYuw=\">AAAGnXicjZTNbtNAEICnAUwJP23hgsSBiFAEl2hdkECcoqJIPSDUNqSp1FSR7W6L67+t7YQkVl4AiSs9cAKJA+It4MILcOgjII5F4sKB8dhtmljxYsvr2fF8M7Oz3tGFbQYhY0czhXPnLygXZy8VL1+5em1ufuH6RuB1fIM3DM/2/E1dC7hturwRmqHNN4XPNUe3eVO3nsXfm13uB6bnvgz7gm872p5r7pqGFqKqbrUH7fkyqzC6SllBTYVy9S7QteotFL5CC3bAAwM64AAHF0KUbdAgwHsLVGAgULcNEep8lEz6zmEIRWQ7aMXRQkOtheMezrZSrYvz2GdAtIFRbHx8JEuwyH6wz+yYfWdf2E/2d6qviHzEufTxrScsF+25Nzfrf6SUg+8QXo2o3JxD2IUnlKuJuQvSxKswEr47ODyuP11fjO6xj+wX5v+BHbFvuAK3+9v4tMbX3+fko2Mu0ysWYWSOK3TRSqR2LmpeU9UcWoeL+xTRbpnoYXhGFrRffZRCHEd741P2WS8tpCLifdSVcDbMidmi/EbWPNc6setSfrGsE+NLmV6G6UmZgwniQErs4J1ULt7ZHo4RafKpRoZpSBkrsx5LyogMI6TMcoZZ/g+Gn9Y7oSKi2mO7LK9/bcJHTUrsZ7LdlzID1I/HYafM9D/8pKInPUvDc5EfJ+5WZ6MkGp3OlixebG2TNafz51I3NcnffeK81DZKPTh4UofwgPwWsW+rk106K2wsVdSHlaW1R+XqStLAYRZuwR2MoMJjqMIKrOKfaWBHeQvv4FC5rdSU58qLxLQwkzI3YOxSmv8Ai8pjqQ==</latexit>!\n<latexit sha1_base64=\"YHNlSPY9AkQuUqTlI0TObLx+qC8=\">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</latexit>\n(a)E= 0\nkz\n<latexit sha1_base64=\"e9Ht67rpTlg1N6IGEDCedjgaYuw=\">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</latexit>!\n<latexit sha1_base64=\"YHNlSPY9AkQuUqTlI0TObLx+qC8=\">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</latexit> (b)E6= 0\nFigure 2.1: Occupied staes of Dirac fermions in a static magnetic in the\npresence/absence of the electric \feld E.\ncrossing the origin correspond to the lowest Landau level (2.56), and the\nother curves correspond to the higher Landau levels.\nNext, we apply an electric \feld in the z-direction,E= (0;0;E). Then,\nthe electric \feld accelerates the particles,\n@kz\n@t=E: (2.60)\nWhen we apply the electric \feld by time \u001c, the fermions acquire \fnite mo-\nmentum,\n\u0001kz=E\u001c: (2.61)\nWe illustrate the occupied state after applying the electric \feld Eas Fig. 2.1(b).\nWe can interpret here that some of the minus-energy particles are excited to\nget positive energy. In this accelerating process, the charges for the right-2.3. CHIRAL MAGNETIC EFFECT 29\nhanded and left-handed fermions in volume Vare varied by\n\u0001QR=BV\n2\u0019Z\u0001kz\n0dkz\n2\u0019=\u001cVEB\n4\u00192; (2.62)\n\u0001QL=\u0000BV\n2\u0019Z\u0001kz\n0dkz\n2\u0019=\u0000\u001cVEB\n4\u00192: (2.63)\nNote that the factor B=(2\u0019) corresponds to the Landau degeneracy (2.59).\nTherefore, the total charge does not change \u0001 Q=QR+QL= 0, whereas\nthe axial charge varies\n\u0001Q5= \u0001QR\u0000QL=\u001cVEB\n2\u00192: (2.64)\nWe can interpret this equation as an additional source of the classical con-\nservation law associated with the axial charge symmetry per unit time and\nvolume. Thus, we obtain\n@\u0016j\u0016\n5=EB\n2\u00192: (2.65)\nThis equation in static electric and magnetic \felds can be generalized into,\n@\u0016j\u0016\n5=CE\u0001B; (2.66)\nwhere\nC\u00111\n2\u00192: (2.67)\nEquation (2.66) is the same relation as that of the triangle anomalies in\nquantum \feld theories [14, 15].\nWe make some remarks on this triangle anomaly relation. It is exact at\nall orders of perturbation theory according to the Adler-Bardeen's theorem\n[52]. Moreover, this is an \\anomalous\" Ward-Takahashi identity, which does\nnot depend on any perturbative calculations [53]. Therefore, the presence of\ngluons does not a\u000bect Eq. (2.66).\nChiral magnetic e\u000bect\nLet us evaluate the currents in the presence of each chemical potential of left-\nor right-handed fermions, \u0016R;L. Using the Fermi-Dirac distribution function,30 CHAPTER 2. OVERVIEW OF QCD\nwe obtain\njz\nR=B\n2\u0019Z1\n0dkz\n2\u0019\u00121\n1 + ekz\u0000\u0016R\nT\u00001\n1 + ekz+\u0016R\nT\u0013\n=\u0016RB\n4\u00192; (2.68)\njz\nL=\u0000B\n2\u0019Z1\n0dkz\n2\u0019\u00121\n1 + ekz\u0000\u0016L\nT\u00001\n1 + ekz+\u0016L\nT\u0013\n=\u0000\u0016LB\n4\u00192: (2.69)\nHere the \frst and second terms of each equation correspond to the particle\nand anti-particle contributions, respectively. Note also that only the lowest\nLandau level contributes to the currents. By writing these in the vector form,\nwe get\njR=\u0016RB\n4\u00192; (2.70)\njL=\u0000\u0016LB\n4\u00192: (2.71)\nThus, the total and axial currents are\nj\u0011jR+jL=\u00165B\n2\u00192; (2.72)\nj5\u0011jR\u0000jL=\u0016B\n2\u00192; (2.73)\nwhere\u00165\u0011(\u0016R\u0000\u0016L)=2 and\u0016\u0011(\u0016R+\u0016L)=2. The creation of the vector\ncurrentjand the axial current j5are called the chiral magnetic e\u000bect (CME)\n[16, 17, 18, 19] and the chiral separation e\u000bect (CSE) [54, 55], respectively.\nAs we will see in Chap. 5.2.1, we can also derive the CME and CSE in the\nuse of the anomalous commutation relation (5.21), which is equivalent to the\ntriangle anomaly relation (2.66).\n2.3.3 Chiral magnetic wave\nWe here show the presence of the CME and CSE leads to the collective gapless\nmodes called the chiral magnetic wave (CMW) [20, 21]. In the absence of\nthe electric \feld, the conservation laws for the electric charge and the axial\ncharge are\n@n\n@t+r\u0001j= 0; (2.74)\n@n5\n@t+r\u0001j5= 0: (2.75)2.3. CHIRAL MAGNETIC EFFECT 31\nThe vector and the axial currents are generated by the CME and the CSE\nas\nj=\u00165B\n2\u00192=n5B\n2\u00192\u001f5; (2.76)\nj5=\u0016B\n2\u00192=nB\n2\u00192\u001f: (2.77)\nHere we write the conserved charge densities nandn5inton=\u001f\u0016and\nn5=\u001f5\u00165by using the the susceptibilities\n\u001f\u0011@n\n@\u0016; \u001f 5\u0011@n5\n@\u00165: (2.78)\nWe substitute Eqs. (2.76) and (2.77) into Eqs. (2.74) and (2.75), respectively.\nThen, we obtain\n@n\n@t+B\n2\u00192\u001f5\u0001rn5= 0; (2.79)\n@n5\n@t+B\n2\u00192\u001f\u0001rn= 0: (2.80)\nBy eliminating n5from these equations, we \fnd a wave function\n@2n\n@t2=\u0000B\n2\u00192\u001f5\u0001r@n5\n@t=B\n4\u00192\u001f\u001f5\u0001r(B\u0001rn) =B2\n4\u00192\u001f\u001f5r2\nsn; (2.81)\nwhere we de\fne the projection of the derivative into the direction of the\nmagnetic \feld,\nrs\u0011B\u0001r\njBj: (2.82)\nWe can interpret Eq. (2.81) as a propagating mode due to the \ructuation\nof charge density and axial charge density. This collective wave is called the\nCMW.Chapter 3\nTheory of dynamic critical\nphenomena\nIn this chapter, we give an overview of the theory of dynamic critical phe-\nnomena. In Sec. 3.1, we \frst review the dynamic universality class and\nexplain the idea of dynamic critical phenomena. In Sec. 3.2, we formulate\nthe static/dynamic low-energy e\u000bective theories of a general critical system.\nIn Sec. 3.3, we also illustrate the RG analysis of the e\u000bective theories.\n3.1 Dynamic universality class\nNear a second-order phase transition or a critical point, the large correlation\nlength of an order parameter, \u0018, leads to unusual hydrodynamics [6]. For\nexample, the typical time scale of di\u000busion becomes larger when the nuclear\nliquid-gas critical point is approached [56, 57, 58]. Remarkably, this critical\nslowing down is independent of the microscopic details of the system. In\nfact, several molecules, such as water, carbon dioxide, xenon, etc., exhibit\nexperimentally the same dynamic critical behavior [59]. Moreover, as we\nmentioned in the introduction, relativistic \ruid near the high-temperature\nQCD critical point also displays the same dynamic critical phenomenon as\nabove [7, 8, 9, 10]. This shows the universality between the quark-gluon\nsystem atT\u0018\u0003QCD\u0018200 MeV and the molecular systems at T\u0018300 K\n\u001830 meV.\nLet us look into the details of the universality of critical phenomena. We\nhere consider the general system near a second-order phase transition or a\n3334 CHAPTER 3. THEORY OF DYNAMIC CRITICAL PHENOMENA\nHydrodynamic variables:Same SymmetriesLow-energy effective theoryMicroscopic theory-Order parameters-Conserved charge densities-Nambu-Goldstone modes (broken phase)⇠\u0000⇤\u00001micro\n<latexit sha1_base64=\"6ryJ593iOTiOiFbMhwnflJBFjzc=\">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</latexit>Coarse graining \nFigure 3.1: Idea of the dynamic universality class\ncritical point. As schematically illustrated in Fig. 3.1, the correlation length\nof the order parameter, \u0018, is much larger than the typical microscopic length\nscale, \u0003\u00001\nmicro:1Therefore, to describe the dynamic critical phenomena char-\nacterized by \u0018, we can coarse grain or integrate out microscopic degrees of\nfreedom, such as quarks and gluons in QCD. Then, it is su\u000ecient to use the\nlow-energy e\u000bective theory of the hydrodynamic variables at large-length and\nlong-time scales of the system. Typical hydrodynamic variables are order pa-\nrameters (e.g., magnetization), conserved-charge densities (e.g., energy and\nmomentum densities), and Nambu-Goldstone modes associated with spon-\ntaneously symmetry breaking (e.g., super\ruid phonon). Symmetries of the\nlow-energy e\u000bective theory must be the same as those of the microscopic\ntheory before coarse graining or integrating out. We can classify dynamic\ncritical phenomena based on the hydrodynamic variables and the symmetries\nof the systems. The classi\fcation of dynamic critical phenomena is called the\ndynamic universality class [6].\nWe summarize the conventional classi\fcation of dynamic critical phenom-\nena studied by Hohenberg and Halperin [6] in Table 3.1. We here show, for\neach of the universality classes, internal symmetry, the property of order pa-\nrameter (whether it is conserved or not) and conserved charge (the symmetry\nrelated by the Noether's theorem), and a typical system.\n1Note here that we consider the system not directly at the critical point \u0018! 1 ,\nbut su\u000eciently near the critical point where \u0018\u001d\u0003\u00001\nmicro (\u0003micro6= 0). Otherwise, the\nhydrodynamic description breaks down. In other words, we extrapolate the calculation on\n\u0018\u001d\u0003\u00001\nmicro to the critical point perturbatively.3.2. FORMULATION 35\nClassSymmetryOrder ParameterConserved ChargeSystemANonconservedIsingmodelBConservedUniaxial FerromagnetsCNonconservedEnergyUniaxial AntiferromagnetsENonconservedEasy-plane magnetsFNonconservedSuperfluid 4He GNonconservedIsotropic AntiferromagnetsHConservedMomentumLiquid gasJConservedIsotropic FerromagnetsU(1)\n<latexit 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sha1_base64=\"npSCC2p1yMXXOsHM0eZlrj3nVAM=\">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</latexit>\nTable 3.1: Conventional dynamic universality class\n3.2 Formulation\nIn this section, we give more details about the low-energy e\u000bective theory\nof critical phenomena. In Secs. 3.2.1 and 3.2.2, we formulate the Ginzburg-\nLandau theory/Langevin theory to describe static/critical critical phenom-\nena, respectively. In Sec. 3.2.3, we demonstrate how to obtain the \feld theory\nequivalent to the Langevin theory derived in Sec. 3.2.2. This \feld theory will\nhelp us to apply the renormalization group approach in Sec. 3.3.2.\n3.2.1 Ginzburg-Landau theory\nThe static critical phenomena and the static universality class can be deter-\nmined only by the order parameters and the symmetries of the system. This\nnotion is called the Ginzburg-Landau theory, and its idea can be summarized\nas follows.\nSince a second-order phase transition or a critical point is/can be regarded\nas a continuous transition, we can consider the parameter region, e.g., tem-\nperature, with a small order parameter. Then, we can expand the free energy\nof the system with respect to the order parameter. Moreover, as we are in-\nterested in the long-range behavior of the system characterized by \u0018, we can\nalso expand this Ginzburg-Landau free energy with respect to the derivative.36 CHAPTER 3. THEORY OF DYNAMIC CRITICAL PHENOMENA\nHere we can neglect higher-order terms because such terms are suppressed\nbyO(r=\u0003). Here \u0003 is the momentum cuto\u000b, which gives the upper limit of\nthe applicability of the e\u000bective theory.2See, e.g., Refs. [60, 61, 62] for the\ngeneral construction of e\u000bective \feld theories.\nWe generalize the above expansion of the order parameter to all of the\nhydrodynamic variables. In the Ginzburg-Landau free energy, we write down\nall consistent terms with the symmetries of the system from the double ex-\npansion of the derivative and the hydrodynamic variables. Let us write one\nof the hydrodynamic variables of the system as  I(I;J;K = 1;:::;n ). Then,\nwe can formally write the free energy of the system in the following form:\nF[ ] =Z\ndr\u00141\n2 I\fIJ(r) J+1\n3!\fIJK(r) I J K\n+1\n4!\fIJKL(r) I J K L+\u0001\u0001\u0001\u0015\n:(3.1)\nHere\fIJ,\fIJK,\fIJKL are some functions of the spatial derivative. These\ncoe\u000ecients can also be expanded with respect to the derivative. Summation\nover repeated indices I;J;K , andLis implied.\nOnce we determine the functional of the free energy, we can calculate\nany correlation functions among the hydrodynamic variables by using the\nfollowing de\fnition of the expectation value:\nhO[ ]i=ZY\nID IO[ ]e\u0000\fF[ ]\nZY\nID Ie\u0000\fF[ ]: (3.2)\nHere,\f\u0011T\u00001is the inverse temperature, and O[ ] is some product of the\nhydrodynamic variables. We can also calculate thermodynamic quantities\nrelated to these static correlation functions. Static critical exponents char-\nacterize the critical phenomena of these thermodynamic quantities. Systems\nwith the same critical exponents belong to the same universality class. As we\nwill see later, we can uniquely determine static critical exponents by using the\n2Practically, we are constructing the low-energy e\u000bective theory by integrating out the\nmicroscopic degrees of freedom whose energy scale is larger than the typical energy scale\nof the critical phenomena, i.e., the inverse correlation length \u0018\u00001. It follows that \u0003 should\nbe chosen as to satisfy \u0018\u00001\u001c\u0003.3.2. FORMULATION 37\nrenormalization group. Thus, the static universality class is only determined\nby the order parameter and the symmetries, which constrains the possible\nterms of the Ginzburg-Landau free energy.3\n3.2.2 Langevin theory\nThe Langevin theory is the low-energy e\u000bective theory at large-length and\nlong-time scales, including macroscopic dissipation e\u000bects. The time evolu-\ntion of the hydrodynamic variable  Iis described by the following Langevin\nequation [63, 64, 65, 66]:\n@ I(t;r)\n@t=\u0000\rIJ(r)\u000eF[ ]\n\u000e J(t;r)\u0000Z\ndr0[ I(t;r); J(t;r0)]\u000eF[ ]\n\u000e J(t;r0)+\u0018I(t;r):\n(3.3)\nThese three terms in the right-hand side are called the dissipative term, the\nreversible term,4and the noise term, respectively.\nDissipative term\nThe \frst term of Eq. (3.3) describes the relaxation process of the hydrody-\nnamic variables to its equilibrium value. We consider the system slightly\napart from the equilibrium. It follows that we can interpret this \frst term\n3You may logically think that not only the order parameter but also conserved charge\ndensities can a\u000bect the static universality class. Such variables are normalized so that those\nmass terms in the Ginzburg-Landau free energy become unity (unlike the order parameter\n\feld) under the renormalization group. Due to this fact, the loop corrections on the\n\fxed point structure solely determined by the order parameters tend to be irrelevant. For\nexample, it is shown that the inclusion of the energy density does not change the static\nuniversality class governed by the Wilson-Fisher \fxed-point [67, 68].\n4Strictly speaking, there is another contribution in the reversible term at \fnite tem-\nperature,\nT\u000e\n\u000e J(t;r0)[ I(t;r); J(t;r0)]: (3.4)\nThis term is required so that the Fokker-Plank equation corresponding to Eq. (3.3) yields\nthe equilibrium distribution /e\u0000\fFas a steady-state solution (for more details, see, e.g.,\nRefs. [63, 65, 66]). Nevertheless, this term is found to be zero or unimportant in most\ncases.38 CHAPTER 3. THEORY OF DYNAMIC CRITICAL PHENOMENA\nas a leading term of the expansion with respect to the variation of the free\nenergy, \t I\u0011\u000eF=\u000e I,\nfI[\t] =fI[0] +\rIJ\tJ+\u0001\u0001\u0001; (3.5)\nwhere, \t I= 0 at the equilibrium. Note that fI[0] and \t Jare some functions\nof I, in general. The leading term fI[0] is the 0-th order of the perturbation\nwhich may be regarded as an equilibrium. On the other hand, the left-hand\nside of Eq. (3.3), @ I=@tvanish at this 0-th order. It follows that we can set\nfI[0] = 0. We also expand the coe\u000ecient \rIJwith respect to the derivative\nas we are interested in the long-range behavior of the system,\n\rij(r) =\r(0)\nij+\r(2)\nijr2+\u0001\u0001\u0001: (3.6)\nReversible term\nThe second term describes the reversible dynamics at a macroscopic scale.\nIn the expression of this second term, [ A(t;r);B(t;r0)] denotes the Pois-\nson bracket, which can be postulated from the symmetry algebra. One can\nobtain the expression of the Poisson bracket by computing the microscopic\ncommutation relation of the operators corresponding to the hydrodynamic\nvariables. We can interpret this term as the classical limit of the Heisenberg\nequation,\nZ\ndr0[ I(t;r); J(t;r0)]\u000eF[ ]\n\u000e J(t;r0)= [ I(t;r);F[ ]]: (3.7)\nOne may show this relation, order by order of the expansions in Eq. (3.1).\nOne distinct di\u000berence from the \frst dissipative term is that this second\nreversible term does not contribute to the time derivative of the free energy:\ndhF[ ]i\ndt=\u001c\u000eF[ ]\n\u000e I@ I\n@t\u001d\n=\u0000\rIJ\u001c\u000eF\n\u000e I\u000eF\n\u000e J\u001d\n\u0000Z\ndr0\u001c\n[ I; J]\u000eF\n\u000e I\u000eF\n\u000e J\u001d\n+\u001c\u000eF\n\u000e I\u0018\u001d\n=\u0000\rIJ\u001c\u000eF\n\u000e I\u000eF\n\u000e J\u001d\n: (3.8)\nTherefore, the second term describes the time evolution of the system without\nincreasing the entropy. Intuitively, the dissipative term breaks the time-\nreversal symmetry, whereas the reversible term does not.3.2. FORMULATION 39\nNoise term\nThe third term is the random driving force, which originates from the un-\nderlying microscopic degrees of freedom. Those statistical properties should\nbe dictated by the nature of equilibrium as follows:\nh\u0018I(t;r)i= 0; (3.9)\nh\u0018I(t;r)\u0018J(t0;r0)i= 2T\rIJ(r)\u000e(t\u0000t0)\u000ed(r\u0000r0): (3.10)\nEquation (3.9) stems from the requirement that h I(t;r)ishould not be\na\u000bected by the presence of \u0018I. The matrix \rIJin the random-force corre-\nlation (3.10) should be the same as that in the dissipative term (3.6). The\nrelation between the dissipation e\u000bects and the random force is called the\n(second) \ructuation-dissipation relation.5\n3.2.3 Dynamic perturbation theory\nWe here convert the Langevin theory derived in Sec. 3.2.2 into the path-\nintegral formulation of a \feld theory. This formulation is called the Martin-\nSiggia-Rose-Janssen-de Dominicis (MSRJD) formalism [32, 33, 34] and helps\nus to apply the renormalization group method systematically. In this sub-\nsection, we follow the derivation in Ref. [66].\nWe \frst formally write the Langevin equation (3.3) and the \ructuation-\ndissipation relation (3.10) (in the unit of T= 1) in the following form:\n@ I(r;t)\n@t=FI[ ] +\u0018I(r;t); (3.11)\nh\u0018I(r;t)\u0018J(r0;t0)i= 2\rIJ(r)\u000ed(r\u0000r0)\u000e(t\u0000t0): (3.12)\nHere,FImay involve all the hydrodynamic variables f Ig, and the noises\nf\u0018Igare assumed to obey the Gaussian white noise.\nWe consider the correlation functions for the hydrodynamic variables un-\nder various con\fgurations of the noise variables,\n\nO[\u0016 ]\u000b\n\u0018=NZ\nD\u0018O[\u0016 ] exp\u0014\n\u00001\n4Z\ndtZ\ndr\u0018I\r\u00001\nIJ\u0018J\u0015\n; (3.13)\n5In the context of the Brownian motion, this relation is due to the fact that both\n\ructuation and dissipation of the Brownian particles come from the same origin: random\nimpact of surrounding molecules.40 CHAPTER 3. THEORY OF DYNAMIC CRITICAL PHENOMENA\nwhere \u0016 Idenotes the formal solution of the Langevin equations, and Nis\na normalization factor. We can reproduce Eq. (3.12) by using the distribu-\ntion (3.13). To carry out the path integral of \u0018in Eq. (3.13) in the presence\nof the noise-dependent variable O[\u0016 ], we insert the following identity into\nthe right-hand side of Eq. (3.13):\n1 =Z\nD Y\nI\u000e( I\u0000\u0016 I) =Z\nD Y\nIY\nr;t\u000e\u0012@ I\n@t\u0000FI[ ]\u0000\u0018J\u0013\n:(3.14)\nWe here omit the Jacobian det ( @t\u0000\u000eF=\u000e ) in the right-hand side. We\ncan justify this by getting rid of some unnecessary graphs containing the\nso-called closed response loops in diagrammatic calculations (see Ref. [66] for\nmore details). Then, we can replace O[\u0016 ] byO[ ] and integrate over the\nnoise variables of Eq. (3.13). Finally, we get\nhO[ ]i=N0Z\niD~ Z\nD O[ ] exp\u0010\n\u0000S[~ ; ]\u0011\n: (3.15)\nHere,N0is a normalization factor, and we use the Fourier representation of\nthe delta function. We introduce the pure imaginary auxiliary \feld ~ Icalled\nthe response \feld for each hydrodynamic variable  I. The MSRJD e\u000bective\nactionS[~ ; ] is given by\nS[~ ; ] =Z\ndtZ\ndr\u0014\n~ I\u0012@ I\n@t\u0000FI[ ]\u0013\n\u0000~ J\rIJ(r)~ I\u0015\n: (3.16)\nBy the use of the path-integral technique, we can calculate any correlation\nfunctions (3.15) among the hydrodynamic variables  Iand the response \felds\n~ I.\n3.3 Renormalization group analysis\nThe renormalization group (RG) is a powerful method to solve the theory\nnear a second-order phase transition or a critical point. In Sec. 3.3.1, we \frst\nreview the static RG for the Ginzburg-Landau free energy following Ref. [63].\nNext, in Sec. 3.3.2, we move to the dynamic RG based on the MSRJD action\nand make remarks on its di\u000berence from the static one.3.3. RENORMALIZATION GROUP ANALYSIS 41\n3.3.1 Statics\nRG transformation\nWe perform static RG transformation on the path integral of the partition\nfunction:\nZ=Z\nD e\u0000\fF\u0003[ ]; (3.17)\nwithF\u0003[ ] being a Ginzburg-Landau free energy. The static RG consists of\nthe following steps:\n(i) Integrating over  (q) in the momentum shell:\n\u0003=b<q< \u0003; (3.18)\n(ii) Rescaling the momentum scale and all of the \felds:\nq!q0=bq; (3.19)\n (q)! 0(q0) = [\u0010(b)]\u00001 (q): (3.20)\nHere,q\u0011jqj, \u0003 is the momentum cuto\u000b, b>1 is the renormalization scale.\nWe will determine the explicit form of the scaling function \u0010(b) in a moment.\nAt the second-order phase transition or the critical point, the correlation\nlength diverges, and the typical length-scale disappears. Therefore, we expect\nthe RG invariance of the system. This emergent symmetry brings strong\nconstraints on the calculation.\nRG of the Ginzburg-Landau free energy\nTo apply the RG to F\u0003[ ], we decompose the \feld  (q) into the low momen-\ntum part <and the high momentum part  >:\n (q) = <(q) + >(q); (3.21)\nwhere\n <(q) =\u001a (q) (0<q< \u0003=b);\n0 (\u0003=b<q< \u0003);(3.22)\n >(q) =\u001a0 (0<q< \u0003=b);\n (q) (\u0003=b<q< \u0003):(3.23)42 CHAPTER 3. THEORY OF DYNAMIC CRITICAL PHENOMENA\nWe \frst integrate over the high-momentum \feld on the partition function:\nZ=Z\nD <(q)D >(q)e\u0000\fF\u0003[ <+ >]\n=Z\nD <(q)e\u0000\fF\u0003=b[ <]: (3.24)\nHere,\fF\u0003=b[ <(q)] is the e\u000bective action for  <, after integrating out  >:\ne\u0000\fF\u0003=b[ <]\u0011Z\nD >(q)e\u0000\fF\u0003[ <+ >]: (3.25)\nNext, we rescale the momentum q!q0=bqand the low momentum \feld\n <(q) into the original \feld  0(q) without the label \\ <\" or \\>;\"\n <(q)! 0(q) = [\u0010(b)]\u00001 <(q): (3.26)\nA series of these operations is the static RG transformation.\nScaling factor \u0010(b)\nLet us determine the explicit form of \u0010(b) from the RG invariance of the\nstatic correlation function.\nFrom the translational symmetry of the system, we can generally write\nthe static correlation function of the \felds as\nh (q1) (q2)i=C(q1)(2\u0019)d\u000e(q1+q2): (3.27)\nHere, the expectation value is de\fned in Eq. (3.2) and C(q) is some function\nofq. On the other hand, the correlation function after the RG transformation\nis\nh 0(q0\n1) 0(q0\n2)i=C(q0\n1)(2\u0019)d\u000e(q0\n1+q0\n2) =\u0010\u00002(b)h (q1) (q2)i: (3.28)\nWe plug Eq. (3.27) into Eq. (3.28) and \fnd\nC(q0\n1)(2\u0019)d\u000e(q0\n1+q0\n2) =\u0010\u00002(b)C(q1)(2\u0019)d\u000e(q1+q2);\nC(bq1)b\u0000d=\u0010\u00002(b)C(q1): (3.29)\nAt the critical point \u0018!1 , the anomalous dimension \u0011characterizes the\ncritical behavior of the correlation, C(q) =q2\u0000\u0011. We thus obtain the expres-\nsion for\u0010(b) as\n\u0010(b) =bd+2\u0000\u0011\n2: (3.30)\nIn particular, \u0011= 0 for the Gaussian distribution.3.3. RENORMALIZATION GROUP ANALYSIS 43\nHigher order terms\nWe here discuss the RG of the following general term:\n\fFp\n\u0003[ ] =Z\nddrup(r) p(r)\n=ZpY\ni=1ddqi\n(2\u0019)3up(q1;:::;qp) (q1)\u0001\u0001\u0001 (qp)\u000e(q1+\u0001\u0001\u0001+qp):(3.31)\nIn particular, the terms with p\u00142 are Gaussian terms; the terms with p>2\nare the non-Gaussian terms.\nWe can expand up(r) by using the rotational symmetry of the system as\nup(r) =up+u0\npr2+\u0001\u0001\u0001: (3.32)\nHere,upandu0\npare some constants. The parity symmetry under r!\n\u0000rprohibits the term proportional to r. The coe\u000ecient of the higher-\norder derivative acquires an additional scaling factor, b\u00002, after rescaling the\nmomentum and the \felds. Therefore, higher-order derivative terms with\n\fxedpare relatively suppressed under the RG procedure. In particular, we\nsetu0\np= 0 forp>2 from now on.\nBy applying the RG transformation to Eq. (3.31) we obtain\n\fFp\n\u0003[ ] =upb\u0000pd\u0010p(b)bdZpY\ni=1ddq0\ni\n(2\u0019)3 0(q0\n1)\u0001\u0001\u0001 0(q0\np)\u000e\u0000\nq0\n1+\u0001\u0001\u0001+q0\np\u0001\n:\n(3.33)\nThe new parameter u0\npafter the single RG step satis\fes\nu0\np=b\u0000pd\u0010p(b)bdup\u0011b\u0015pup: (3.34)\nHere,\u0015p\u0011p+d\u0000p(d+\u0011)=2. We can classify terms depending on the value\nof\u0015p,\n\u0015p=8\n<\n:>0 relevant ;\n= 0 marginal ;\n<0 irrelevant :(3.35)\nWhenupis a relevant parameter, it becomes larger under the RG transfor-\nmation; when upis an irrelevant parameter, it becomes smaller under the RG\ntransformation. Therefore, we can neglect all the irrelevant parameters from\nthe beginning. We can de\fne the above critical dimension dc(p) such that up\nis irrelevant when d>d c. In particular, we \fnd dc(3) = 6 and dc(4) = 4.44 CHAPTER 3. THEORY OF DYNAMIC CRITICAL PHENOMENA\nRG equation\nWe here show an outline of the RG of a scalar \feld theory (see Appendix A\nfor details). The Ginzburg-Landau free energy is given by\n\fF[ ] =Z\ndr\u0014r\n2 2+1\n2(r )2+u 4+h \u0015\n(3.36)\nwith being a scalar \feld. It is worth noting that Eq. (3.36) has Z(2)\nsymmetry under the following transformation:\n !\u0000 : (3.37)\nThis is the same symmetry as that of the Ising model, so that we can map\nthe scaler \feld  into the magnetization mand expect the same static critical\nphenomena.\nAccording to the discussion just under Eq. (3.35), relevant terms at d= 4\nare only the Gaussian terms. On the other hand, when dis slightly smaller\nthan 4,ugrows a little bit under the RG transformation. Therefore, by\nworking at\nd= 4\u0000\u000f (3.38)\nwith the small \u000f(0<\u000f\u001c1), we can construct the perturbation theory based\non the expansion with respect to u(or\u000f). Starting from this assumption, we\nwill eventually \fnd u=O(\u000f) and check the consistency of this perturbation\nscheme.\nWe can evaluate the parameters randuunder the RG transformation.\nWhen we write these values after the RG as r0andu0, we obtain\nr0=b2\u0014\nr+ 6uZ>\nq1\nr+q2+O(u2)\u0015\n; (3.39)\nu0=b\u000f\u0014\nu\u000036u2Z>\nq1\n(r+q2)2\u0015\n: (3.40)\nHere, the integralsR>represent the integration over the high-momentum\ndegrees of freedom  >(see Eq. (A.21) for the de\fnition of this integral);\noverall factors b2andb\u000fcome from the rescaling. See Appendix A for the\nderivation of Eqs. (3.39) and (3.40), which correspond to Eqs. (A.18) and\n(A.20), respectively.3.3. RENORMALIZATION GROUP ANALYSIS 45\nWe work on the thin momentum shell by setting b\u0011el'1 +lwith\nl\u001c1. Then, Eqs. (3.39) and (3.40) reduce to the following RG equations\n(for the derivation, see Eqs. (A.34) and (A.35)):\nd\u0016r\ndl= 2\u0016r+ 12\u0016u\u000012\u0016r\u0016u; (3.41)\nd\u0016u\ndl=\u000f\u0016u\u000036\u0016u2; (3.42)\nwhere\n\u0016r\u0011r\n\u00032;\u0016u\u0011u\n8\u00192\u0003\u000f; (3.43)\nare dimensionless parameters. These equations describe the parameters un-\nder the RG transformation at a general renormalization scale l= lnb. The\nstable \fxed-point solution is called the Wilson-Fisher \fxed point (its deriva-\ntion is given in Appendix A.2):\n\u0016r=\u0000\u000f\n6\u0016u=\u000f\n36: (3.44)\nWe can calculate the critical exponents from these \fxed point values.\n3.3.2 Dynamics\nWe can also apply the dynamic RG analysis to the MSRJD action S[ ; ]\ngiven in Eq. (3.16).We carry out the RG transformation of S[ ; ] on the\npath integral de\fned in the following partition function:\nZ=Z\nDi~ D e\u0000S[ ; ]: (3.45)\nThis is analogous to the static partition function (3.17). In this subsection,\nwe make some remark on the important di\u000berences from the static RG pro-\ncedure.\nDynamic critical exponent\nThe \frst di\u000berence is the presence of time tor frequency !. In addition to\nthe rescaling of the space coordinate in Eq. (3.19), frequency (or time) is\nregarded as a parameter which scales by\n!!!0=bz!; (3.46)\nwherezis called the dynamic critical exponent which characterizes the dy-\nnamic universality class.46 CHAPTER 3. THEORY OF DYNAMIC CRITICAL PHENOMENA\nResponse \felds\nThe second di\u000berence is the presence of the response \feld ~ . In the presence\nof this additional \feld, the Green function in the \feld theory becomes a\nmatrix form in (  ;~ ) space. In particular, the bilinear term of ~ in the\nMSRJD action yields the so-called noise vortex. We will look into these\npoints explicitly in our main analysis in Sec. 5.2.2.\nThere is also an extension of the RG procedure. In addition to the rescal-\ning for the hydrodynamic \feld  , Eq. (3.20), we also scale the response \feld\nas\n~ (r)!b~a~ (r): (3.47)\nHere, note that the scaling factor b~ais generally independent of that for the\nhydrodynamic variable, \u0010(b), in Eq. (3.30). We will see the rescaling of the\nresponse \felds in our main calculation, Eqs. (5.60){(5.62).\nFrequency-dependent internal loops\nThe third di\u000berence can be seen in the integrating out procedure of the\nhigher-momentum \felds. In addition to the same momentum integral as that\nof the static RG, Eq. (3.18), we carry out frequency integrals over \u00001<!<\n1in the dynamic formulation. Integrals in frequency space are performed\nby using contour integrals. These integral pick up hydrodynamic poles of the\nsystem.Chapter 4\nDynamic critical phenomena of\nthe high-density QCD critical\npoint\nIn this chapter, we study the static and dynamic critical phenomena near\nthe high-density QCD critical point. In Sec. 4.1, we summarize the hydrody-\nnamic variables of the system by following the strategy in Sec. 3.1. In Sec. 4.2,\nbased on the general discussion in Sec. 3.2.1, we study the static universality\nclass of the system. In Sec. 4.3, we apply the formulation in Sec. 3.2.2 and\nstudy its dynamic universality class. We conclude with Sec. 4.4 with some\ndiscussions.\n4.1 Hydrodynamic variables\nThe hydrodynamic variables of the system can be summarized as follows:\n(i) the chiral condensate, \u001b\u0011\u0016qq\u0000h\u0016qqi;\n(ii) the baryon number density, nB\u0011\u0016q\r0q\u0000h\u0016q\r0qi;\n(iii) the energy-momentum densities, \"\u0011T00\u0000hT00iand\u0019i\u0011hT0ii;\n(iv) the super\ruid phonon, \u0012.\nSome remarks are in order here: the chiral condensate \u001bis de\fned as the\nscaler channel of the general chiral order parameter (2.27), namely Eq. (2.39);\nthe baryon number density nBis the conserved charge density for the baryon\n4748 CHAPTER 4. HIGH-DEINSITY QCD CRITICAL POINT\nnumber symmetry under its transformation, Eq. (2.19); the super\ruid phonon\n\u0012is the Nambu-Goldstone mode associated with the spontaneous braking of\nthe baryon number symmetry, Eq. (2.37). The super\ruid phonon \u0012corre-\nsponds to the phase degrees of freedom of \u001ein Eq. (2.40). Among others, \u001b,\nnB, and\"are de\fned as the \ructuations around the equilibrium. Note that\nthe hydrodynamic variables in the system near the high-temperature QCD\ncritical point are the same as (i) \u0000(iii) of the high-density critical point.\nWe note that the following variables are not hydrodynamic variables. The\n\frst one is the amplitude \ructuation of the diquark condensate, or the \ructu-\nation ofj\u001ejin Eq. (2.40). This is because, the high-density QCD critical point\nis characterized by the massless chiral condensate \u001b, where the diquark con-\ndensate is nonvanishing. The second ones are the Nambu-Goldstone modes\nassociated with the chiral symmetry breaking in Eq. (2.36), e.g., the pions.\nAs we mentioned under Eq. (2.18), the chiral symmetry is explicitly broken\nby the \fnite quark mass. It follows that the pions acquire \fnite masses and\ncan be integrated out (or coarse grained) according to the general discussion\nin Sec. 3.1. The third ones are the gluons which are gapped due to the color\nMeissner e\u000bect in the color superconducting phase.\nAs we will give a discussion in Sec. 4.4, \"and\u0019ido not a\u000bect the static\nand dynamic critical phenomena of the system. Therefore, we \frst neglect\n\"and\u0019iand consider only \u001b,nB;and\u0012as hydrodynamic variables in the\nfollowing sections.\n4.2 Static critical phenomena\nWe \frst construct the Ginzburg Landau theory by following the argument\nin Sec. 3.2.1. The general Ginzburg-Landau free energy consistent with the\nsymmetries of the system, such as the chiral symmetry, the U(1) Bsymmetry,\nand the discrete CPT symmetries (charge conjugation, parity, and time-\nreversal symmetries, respectively) is given by\nF[\u001b;n B;\u0012] =Z\ndr\u0014a\n2(r\u001b)2+b0r\u001b\u0001rnB+c\n2(rnB)2+\u001a\n2(r\u0012)2+V(\u001b;n B)\u0015\n;\n(4.1)\nwhere\nV(\u001b;n B) =A\n2\u001b2+B\u001bn B+C0\n2n2\nB+\u0001\u0001\u0001: (4.2)4.2. STATIC CRITICAL PHENOMENA 49\nHerea;b0;c;\u001a;A;B;C0are the expansion parameters and \\ \u0001\u0001\u0001\" denotes higher\norder terms. The prime notation is used to distinguish from other characters\nintroduced so far. Note that b0andBcan be \fnite only when mq6= 0 and\n\u0016B6= 0, otherwise the term /\u001bnBis prohibited by the chiral symmetry and\nthe charge conjugate symmetry.\nAn important observation here is that the super\ruid phonon \u0012is decou-\npled from the other hydrodynamic variables \u001bandnBat the Gaussian level.\nThis is because the time-reversal symmetry prohibits the mixing between \u0012\nand\u001b(ornB). Beyond the mean-\feld level, one can \fnd that the nonlinear\ncouplings between \u001band\u0012, e.g.,\u001b2(r\u0012)2are irrelevant at the Wilson-Fisher\n\fxed point at d= 3.1Therefore, the static property of the system is only de-\ntermined by \u001bandnB, even when one takes into account higher-order terms.\nIt follows that the static universality class of the high-density critical point\nis the same as that of the high-temperature critical point. In particular, we\nobtain the following static correlation functions:\nh\u001b(r)\u001b(0)i=1\n4\u0019re\u0000r=\u0018; (4.3)\n\u001fB\u0011@nB\n@\u0016B=1\nVT\nn2\nB\u000b\nq!0=A\n\u0001; (4.4)\nwhereVandTbeing spatial volume and temperature of the system, respec-\ntively. We also de\fne\n\u0018\u0018\u0001\u00001\n2;\u0001\u0011AC0\u0000B2: (4.5)\nThe critical point is characterized by \u0001 !0, so that the correlation length\nof\u001b,\u0018, diverges. Because of the mixing between \u001bandnBmentioned under\nEq. (4.2),\u001fBand the susceptibility for the chiral condensate, \u001fmq, diverge\nas\n\u001fB\u0018\u00182\u0000\u0011; (4.6)\n\u001fmq\u0011@\u001b\n@mq\u0018\u00182\u0000\u0011: (4.7)\nHere, the anomalous dimension \u0011is determined by the static universality\nclass of the system ( \u0011'0:04 including non-Gaussian e\u000bects [63]).\n1Note that the couplings between \u0012and\u001bornBcontain the derivative due to the U(1) B\nsymmetry of the system, i.e., the invariance under \u0012!\u0012+\u000bB.50 CHAPTER 4. HIGH-DEINSITY QCD CRITICAL POINT\nWe here give the reason why the high-density QCD critical point belongs\nto the same static universality class as that of the Ising model. Since \u001band\nnBare mixed in the potential, it is useful to change the basis in the ( \u001b;n B)\nplane. We chose the basis such that the \rat direction, n0\nB/\u0000B\u001b+AnB,\nappears at the critical point \u0001 !0. After integrating out the gapped degrees\nof freedom orthogonal to n0\nB, we can obtain the free energy of the system as\nF[~n0\nB] =Z\ndr\u0014a\n2(rn0\nB)2+A0\n2n02\nB+vn03\nB+u0n04\nB\u0015\n; (4.8)\nwith some parameters A0,v, andu0. By taking the particular complete of\nsquare, we can map this functional into the same functional as that of the\nIsing model, namely, Eq. (3.36).\n4.3 Dynamic critical phenomena\nIn this section, we study the dynamic critical phenomena of the system in\nthe vicinity of the high-density QCD critical point. First, in Sec. 4.3.1, we\nconstruct the nonlinear-Langevin equations of the high-density QCD crit-\nical point. Next, in Sec. 4.3.2, we solve the linearized Langevin equation\nand obtain the hydrodynamic modes of the system. Then, in Sec. 4.3.3,\nwe calculate the dynamic critical exponent, which determines the dynamic\nuniversality class.\n4.3.1 Langevin theory\nWe can write down the Langevin equation of the system by following Eq. (3.3)\nas\n@\u001b\n@t=\u0000\u0000\u000eF\n\u000e\u001b+~\u0015r2\u000eF\n\u000enB+\u0018\u001b; (4.9)\n@nB\n@t=~\u0015r2\u000eF\n\u000e\u001b+\u0015r2\u000eF\n\u000enB\u0000Z\nV[nB;\u0012]\u000eF\n\u000e\u0012+\u0018nB; (4.10)\n@\u0012\n@t=\u0000Z\nV[\u0012;n B]\u000eF\n\u000enB\u0000\u0010\u000eF\n\u000e\u0012+\u0018\u0012: (4.11)\nHere, we use the simpli\fed notation of the integral,\nZ\nV[A;B]\u000eF\n\u000eB\u0011Z\ndr0[A(t;r);B(t;r0)]\u000eF\n\u000eB(t;r0): (4.12)4.3. DYNAMIC CRITICAL PHENOMENA 51\nThe kinetic coe\u000ecients \u0000, \u0015, and ~\u0015originate from the expansion (3.6) and\nare related to the noise terms \u0018\u001b,\u0018n, and\u0018\u0012by the \ructuation-dissipation\nrelations corresponding to Eq. (3.10),\nh\u0018\u001b(t;r)\u0018\u001b(t;r0)i= 2T\u0000\u000e(t\u0000t0)\u000ed(r\u0000r0); (4.13)\nh\u0018nB(t;r)\u0018nB(t;r0)i=\u00002T\u0015r2\u000e(t\u0000t0)\u000ed(r\u0000r0); (4.14)\nh\u0018\u001b(t;r)\u0018nB(t;r0)i=\u00002T~\u0015r2\u000e(t\u0000t0)\u000ed(r\u0000r0); (4.15)\nh\u0018\u0012(t;r)\u0018\u0012(t;r0)i= 2T\u0010\u000e(t\u0000t0)\u000ed(r\u0000r0): (4.16)\nHere,ddenotes the spatial dimension.\nWe postulate the following Poisson bracket,\n[\u0012(t;r);nB(t0;r0)] =\u000e(t\u0000t0)\u000ed(r\u0000r0); (4.17)\nso thatnBand\u0012satisfy the canonical conjugate relation. The reason why\nthese variables are canonical conjugate can be understood as follows. We\nstart from the QCD Lagrangian at \fnite baryon chemical potential,\nL0\nQCD=LQCD+\u0016B\u0016q\r0q: (4.18)\nWe regard \u0016Bas an auxiliary gauge \feld a\u0016= (\u0016B;0), and rewrite the\nLagrangian into\nL0\nQCD=LQCD+a\u0016\u0016q\r\u0016q; (4.19)\nwhich has the gauge invariance under the local U(1) Btransformation,\na\u0016!a\u0016\u0000@\u0016\u000bB; \u0012!\u0012+\u000bB: (4.20)\nNext, we consider the e\u000bective Lagrangian Le\u000bof the super\ruid phonon \u0012.\nSinceLe\u000bshould also possess the local gauge symmetry (4.20), the expression\nofLe\u000bis dictated by the covariant derivative\nD\u0016\u0012= (@t\u0012+a0;r\u0012): (4.21)\nThus, we \fnd [69, 70]\nLe\u000b=Le\u000b\u0012@\u0012\n@t+\u0016B;r\u0012\u0013\n; (4.22)52 CHAPTER 4. HIGH-DEINSITY QCD CRITICAL POINT\nwhich yields the canonical conjugation relation between nBand\u0012,\nnB\u0011\u000eLe\u000b\n\u000e\u0016B=\u000eLe\u000b\n\u000e_\u0012;_\u0012\u0011@\u0012\n@t: (4.23)\nSome remarks on the Langevin equations (4.9){(4.11) are in order here.\nThe derivative expansions for the \frst two terms of Eq. (4.10) start from r2\nso that the total baryon number is conserved _ nB!0, (q!0). The coe\u000e-\ncient of the second term of Eq. (4.9) should have the same kinetic coe\u000ecient\nas that of the \frst term in Eq. (5.13), according to the Onsager's principle\n(see Ref. [71] for the derivation of the Onsager's principle).\nBy substituting Eq. (4.17) and the variation of the free energy to the\nlinear level of hydrodynamic variables,\n\u000eF\n\u000e\u001b= (A\u0000ar2)\u001b+ (B\u0000b0r2)nB; (4.24)\n\u000eF\n\u000enB= (B\u0000b0r2)\u001b+ (C0\u0000cr2)nB; (4.25)\n\u000eF\n\u000e\u0012=\u0000\u001ar2\u0012; (4.26)\ninto Eqs. (4.9){(4.11), we obtain the Langevin equations to the order of O(q2)\nin frequency-momentum space ( !;q),\nM0\n@\u001b\nn\n\u00121\nA= 0; (4.27)\nwhere\nM\u00110\n@i!\u0000\u0000A\u0000(\u0000a+~\u0015B)q2\u0000\u0000B\u0000(\u0000b0+~\u0015C0)q20\n\u0000(~\u0015A+\u0015B)q2i!\u0000(~\u0015B+\u0015C0)q2\u001aq2\n\u0000B\u0000b0q2\u0000C0\u0000cq2i!\u0000\u0010\u001aq21\nA:\n(4.28)\nWe here omit the noise terms because these are not important in the following\nargument, where we only need the expressions of hydrodynamic modes.\n4.3.2 Hydrodynamic modes\nWe obtain the hydrodynamic modes of the system by solving the proper\nequation, detM= 0. This equation reduces to\n!3+ i(x1+x2q2)!2\u0000yq2!\u0000izq2= 0; (4.29)4.3. DYNAMIC CRITICAL PHENOMENA 53\nwhere\nx1\u0011\u0000A;\nx2\u0011\u0000a+ 2~\u0015B+\u0015C0+\u0010\u001a;\ny\u0011\u0000\u0015\u0001 +C0\u001a+ \u0000\u0010A\u001a;\nz\u0011\u0000\u001a\u0001: (4.30)\nIn Eq. (4.29), we ignore the higher-order terms of q. At this order, the\nleft-hand side of Eq. (4.29) can be factorized as\n\u0014\n!+ ix1+ i\u0012\nx2\u0000y\nx1+z\nx2\n1\u0013\nq2+O(q3)\u0015\n\u0002\u0014\n!\u0000rz\nx1jqj+i\n2\u0012y\nx1\u0000z\nx2\n1\u0013\nq2+O(q3)\u0015\n\u0002\u0014\n!+rz\nx1jqj+i\n2\u0012y\nx1\u0000z\nx2\n1\u0013\nq2+O(q3)\u0015\n= 0: (4.31)\nForm this factorized form, one can \fnd three hydrodynamic modes of the\nsystem near the high-density QCD critical point: the relaxation mode and\nthe pair of the super\ruid phonons with the dispersion relations,\n!1=\u0000i\u0000A+O(q2); (4.32)\n!2;3=\u0006csjqj+O(q2); (4.33)\nrespectively. Here,\ncs\u0011r\u001a\n\u001fB(4.34)\nis the speed of the super\ruid phonon, whose existence is dictated by the\nspontaneously symmetry breaking of the U(1) Bsymmetry, Eq. (2.37). We\nhere \fnd the critical slowing down of the speed of the super\ruid phonon,\ncs!0. Note that we have shown the divergence of \u001fBwhen the critical\npoint is approached \u0018!1 , in Eq. (4.6).\n4.3.3 Dynamic critical exponent\nWe can estimate the dynamic critical exponent zfrom!\u0018\u0018\u0000zby using\nEq. (4.33),\ncs\u0018\u00181\u0000z: (4.35)54 CHAPTER 4. HIGH-DEINSITY QCD CRITICAL POINT\nHere, we use the matching condition at \u0018q\u00181 between the critical regime\n(\u0018q\u001d1) and the hydrodynamic regime ( \u0018q\u001c1).\nTo determine the value of z, we use the following \u0018dependences of \u001aand\n\u001fBnear the high-density QCD critical point:\n\u001a\u0018\u00180; \u001f B\u0018\u00182\u0000\u0011: (4.36)\nHere,\u001ais the sti\u000bness parameter (or the \\decay constant\" for the super\ruid\nphonon). Note that it does not depend on \u0018near the high-density critical\npoint, which is generally away from the super\ruid phase transition charac-\nterized by the amplitude mode of the diquark condensate.\nFrom Eqs. (4.34), (4.35), and (4.36), we \fnd the dynamic critical exponent\nzas\nz= 2\u0000\u0011\n2: (4.37)\nThis dynamic critical exponent is di\u000berent from all the exponents reported in\nthe conventional classi\fcation by Hohenberg and Halperin [6]. In this sense,\nwe \fnd that the high-density QCD critical point belongs to a new dynamic\nuniversality class.\n4.4 Conclusion and discussion\nSo far, we have constructed the low-energy e\u000bective theory near the high-\ndensity QCD critical point and studied its static and dynamic critical phe-\nnomena. We have \frst shown that the static universality class of the system\nis the same as that of the high-temperature critical point, in Sec. 4.2. From\nthe speed of the super\ruid phonon, cs, obtained in Eq. (4.34), we found\nthe critical slowing down of csin the vicinity of the critical point. Further-\nmore, we have calculated the dynamic critical exponent of the system, z, in\nEq. (4.37), and found that the high-density QCD critical point belongs to\na new dynamic universality class beyond the conventional Hohenberg and\nHalperin's classi\fcation [6].\nLet us discuss nonlinear e\u000bects on the dynamic critical phenomena. In\ngeneral, there are nonlinear corrections to kinetic coe\u000ecients when a critical\npoint is approached. Nevertheless, we can argue that the expression for the\ndynamic critical exponent (4.37) is an exact relation, which does not receive4.4. CONCLUSION AND DISCUSSION 55\nany renormalization e\u000bects due to U(1) Bsymmetry of the system. In fact,\nwhen we rewrite (4.17) with some coupling constant g0\n[\u0012(r);n(r0)] =g0\u000e(r\u0000r0); (4.38)\nthe speed of the super\ruid phonon, Eq. (4.34) becomes\ncs\u0011s\ng2\n0\u001a\n\u001fB: (4.39)\nSince Eq. (4.38) is related to the U(1) Bsymmetry, g0is not a\u000bected by any\nrenormalization e\u000bects. Thus, the dynamic critical exponent obtained by\nusingcsis an exact relation. As is also shown in Appendix B, the speed of\nthe super\ruid phonon, Eq. (B.50), does not include any kinetic coe\u000ecients\nwhen the energy and momentum densities are taken into account.\nWhy the high-density QCD critical point belongs to a new dynamic uni-\nversality class beyond the conventional classi\fcation? First, the presence of\nthe super\ruid phonon in the high-density critical point leads to a di\u000berent\ndynamic universality class from that of the high-temperature QCD criti-\ncal point. We next compare with the super\ruid \u0015transition of4He, where\nthe super\ruid phonon also exists. Nevertheless, the interplay between the\nsuper\ruid phonon and the chiral order parameter cannot be found in this\ncondensed matter system. In fact, the second-order phase transition of the\n\u0015transition is characterized by the massless super\ruid gap.2\nOur \fndings suggest that the static quantities cannot distinguish two\npossible critical points in the QCD phase diagram, whereas the dynamic\ncritical phenomena can distinguish them. Moreover, since the uniqueness of\nthe dynamic critical phenomena of the high-density critical point is due to the\npresence of the super\ruid phonon, observation of the critical slowing down\nof the super\ruid phonon in the future heavy-ion collisions would provide\nindirect evidence of super\ruidity of high-density QCD matter.\nOther than the high-density QCD critical point, systems near critical\npoints associated with G1=Z2symmetry braking under G2= U(1) sponta-\nneously symmetry breaking can also belong to this new dynamic universality\n2Quantitatively, one can see a di\u000berence in the correlation length dependence of the\nsti\u000bness parameter \u001a, Eq. (4.36). Unlike the high-density QCD critical point, the phase\ntransition is characterized by the super\ruid gap in the helium system, so that the sti\u000bness\nparameter depends on \u0018as\u001a\u0018\u0018\u00001[6].56 CHAPTER 4. HIGH-DEINSITY QCD CRITICAL POINT\nclass with appropriate background \felds. The generalization of G1;2enables\nus to classify the dynamic critical phenomena involving a general interplay\nbetween order parameters and Nambu-Goldstone modes.Chapter 5\nDynamic critical phenomena\ninduced by the chiral magnetic\ne\u000bect in QCD\nIn this chapter, we study the interplay between the dynamic critical phenom-\nena and the CME in QCD. In Sec. 5.1, we \frst explain our setup and summa-\nrize the symmetries and the hydrodynamic modes of the system. In Sec. 5.2,\nwe construct the Ginzburg-Landau theory and the nonlinear Langevin equa-\ntion to describe the static and dynamic critical phenomena, respectively.\nWe also give the MSRJD action corresponding to the Langevin theory and\nsummarize its Feynman rules. In Sec. 5.3, we study the Ginzburg-Landau\nfree energy and the MSRJD action by using the static and dynamic RG,\nrespectively. We also show the results in Sec. 5.3. We conclude with Sec. 5.4.\n5.1 Setup\n5.1.1 Symmetries\nWe consider two-\ravor QCD with massless up and down quarks at \fnite\ntemperature Tand isospin chemical potential \u0016Iin an external magnetic\n\feldB. In this setup, there are two additional terms in the quark sector of\nmassless two-\ravor QCD Lagrangian. One is the \fnite isospin density term\n(the \frst term of Eq. (5.1)); the other is the coupling to the background\n5758CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\nelectromagnetic gauge \feld A\u0016(the second term of Eq. (5.1)):\nL0\nquark =Lquark+\u0016I\u0016q\r0\u001c3q+A\u0016\u0016q\r\u0016Qeq (5.1)\n=Lquark+\u0016I\n2(qy\nuqu\u0000qy\ndqd) +2\n3A\u0016\u0016qu\r\u0016qu\u00001\n3A\u0016\u0016qu\r\u0016qu: (5.2)\nHere,Lquarkis the same Lagrangian given in Eq. (2.2) except for mu=md= 0\nin the present case; nI= \u0016q\r0\u001c3qand\u0016Iare the isospin1conserved-charge\ndensity and the isospin chemical potential, respectively; the third component\nof the generators for the SU(2) algebra, \u001c3acts on the vector in the \ravor\nspace. In Eq. (5.2), we write the Dirac \felds for the up and down quarks as\nquandqd, by using the vector notation of the \ravor space:\nq=\u0012qu\nqd\u0013\n: (5.3)\nThe electric charge matrix Qein Eq. (5.1) are\nQe=0\nB@2\n30\n0\u00001\n31\nCA: (5.4)\nNote here that the up and down quarks possess the electric charge 2 =3 and\n\u00001=3 in the unit of the elementary charge.\nIn the presence of the electromagnetic gauge \feld (and/or the isospin\nchemical potential), the chiral symmetry Eq. (2.16) is explicitly broken into\nits subgroup [72]\nSU(2) L\u0002SU(2) R!U(1)\u001c3\nL\u0002U(1)\u001c3\nR: (5.5)\nHere we call U(1)\u001c3\nL\u0002U(1)\u001c3\nRthe partial chiral symmetry, which corresponds\nto the invariance under the following particular chiral transformation (corre-\nsponding to \u001c3),2\nqL!V\u001c3\nLqL; q L!V\u001c3\nRqR; V\u001c3\nL;R\u0011e\u0000i\u0012L;R\u001c3qL;R; (5.7)\n1The \ravor space at Nf= 2 is specially called the isospin space in an analogy to the\nspin 1=2.\n2This transformation can be also written as\nq!q0=e\u0000i\u000bV\u001c3e\u0000i\u000bA\u001c3\r5q; (5.6)\nwhere\u000bVand\u000bAdenote the phase-rotating angles associated with U(1)\u001c3\nVand U(1)\u001c3\nA\nsymmetries, respectively.5.1. SETUP 59\nwith\u0012L;Rbeing the independent parameters.\nSome remarks on our setup are in order here. First, we assume massless\nquarks so that possible quark-mass corrections to the CME can be ignored.\nSecond, we consider \fnite \u0016Iinstead of baryon chemical potential \u0016B, because\nbothnIandnI5are conserved in massless QCD, so that \u0016Iand\u0016I5are well-\nde\fned. On the other hand, the conservation of the axial charge is violated\nby the axial anomaly, even in massless QCD, Eq. (2.22).\n5.1.2 Hydrodynamic variables\nWe here order the hydrodynamic variables of the system. First one is the\ntwo-component order parameter \feld characterizing the symmetry breaking\nof the partial chiral symmetry U(1)\u001c3\nL\u0002U(1)\u001c3\nR,\u001e\u000b(\u000b= 1;2), i.e.,\n\u001e=\u0012\u001b\n\u00193\u0013\n; (5.8)\nwith\u001b= \u0016q\u001c0qand\u00193= \u0016qi\r5\u001c3qbeing the chiral condensate and the neutral\npion, respectively ( \u001c0= 1=2). We can show that the other variables in\nthe chiral order parameter \b in Eq. (2.27) are not hydrodynamic modes as\nfollows. At Nf= 2, the chiral order parameter \b can be decomposed\n\b =\u001b\u001c0+ i\u0011\u001c0+\u000ea\u001ca+ i\u0019a\u001ca; (5.9)\nwhere\u001b= \u0016q\u001c0q,\u0011= \u0016qi\r5\u001c0q,\u000ea= \u0016q\u001caq, and\u0019a= \u0016qi\r5\u001caq; we take the\nsum over repeated indices. In the absence of the magnetic \feld, \u001band\u0019a\nbecome massless near the second-order chiral phase transition, while \u0011and\n\u000eaacquire \fnite masses due to the U(1)Aanomaly [73].3When we switch on\nthe magnetic \feld, the charged pions \u00191;2also acquire a mass proportional to\n3Actually, in the presence of the U(1)Aanomaly, the Ginzburg-Landau potential for the\nchiral order parameter \b includes the so-called Kobayashi-Maskawa-'t Hooft interaction\n[39, 40],\n\u0000c\n2(det \b + det \by) =\u0000c\n2[\u001b2+ (\u0019a)2] +c\n2[\u00112+ (\u000ea)2]; (5.10)\nwithcbeing some positive parameter. By taking into account the mass terms\na\n2Tr\by\b =a\n2[\u001b2+ (\u0019a)2+\u00112+ (\u000ea)2]; (5.11)\nonly\u001band\u0019acan be massless at the phase transition point.60CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\np\njBjdue to the explicit chiral symmetry breaking (5.5). We can integrate\nout such massive degrees of freedom.\nThe second hydrodynamic variable is the conserved charge densities. We\nhere only take into account the conserved charge densities associated with\nthe symmetry U(1)\u001c3\nL\u0002U(1)\u001c3\nRwhich are coupled to \u001e\u000b, i.e.,nIand the\naxial isospin density nI5= \u0016q\r0\r5\u001c3q. Although the energy and momentum\ndensities can be also coupled to these hydrodynamic variables, we only focus\non\u001e\u000b,nI, andnI5.\nFrom now on, we omit the subscripts of nI,nI5,\u0016I, and\u0016I5and rewrite\nthese variables into n,n5, and\u0016for notational simplicity.\n5.2 Formulation\nIn this section, we give the formulation to study the static and the dynamic\nuniversality classes of the system. In Sec. 5.2.1, we \frst derive the nonlinear\nLangevin equations following general arguments in Sec. 3.2.2. We also con-\nstruct the Ginzburg-Landau theory in Sec. 5.2.1. In Sec. 5.2.2, we show the\nMSRJD \feld theory equivalent to the derived Langevin equations with the\nhelp of Sec. 3.2.3.\n5.2.1 Langevin theory\nApplying the general formation of the Langevin equation (3.3) to this system,\nwe obtain the nonlinear Langevin equations for the hydrodynamic variables\n\u001e\u000b,n, andn5as follows:\n@\u001e\u000b\n@t=\u0000\u0000\u000eF\n\u000e\u001e\u000b\u0000gZ\nV[\u001e\u000b;n5]\u000eF\n\u000en5+\u0018\u000b; (5.12)\n@n\n@t=\u0015r2\u000eF\n\u000en\u0000Z\nV[n;n 5]\u000eF\n\u000en5+\u0010; (5.13)\n@n5\n@t=\u00155r2\u000eF\n\u000en5\u0000gZ\nV[n5;\u001e\u000b]\u000eF\n\u000e\u001e\u000b\u0000Z\nV[n5;n]\u000eF\n\u000en+\u00105; (5.14)\nWe here use the simpli\fed notation of the integrals (4.12). The Ginzburg-\nLandau free energy Fis given by\nF=Z\ndr\u0014r\n2(\u001e\u000b)2+1\n2(r\u001e\u000b)2+u(\u001e\u000b)2(\u001e\f)2+1\n2\u001fn2+1\n2\u001f5n2\n5+\rn\u001e2\n\u000b\u0015\n;\n(5.15)5.2. FORMULATION 61\nwhere we take the summations over repeated indices; r;uand\rare some\nexpansion parameters; the isospin and axial isospin susceptibilities, \u001fand\n\u001f5, are de\fned as\n\u001f\u0011@n\n@\u0016; \u001f 5\u0011@n5\n@\u00165: (5.16)\nThe last term of Eq. (5.15), \rn\u001e2\n\u000b, is forbidden at \u0016= 0 by the charge\nconjugation symmetry, whereas it can appear at \u00166= 0. Returning to the\nLangevin equations (5.12){(5.14), \u0000 ;\u0015, and\u00155are the kinetic coe\u000ecients\nobtained by the derivative expansion (3.6), and gis the coupling constant\nbetween\u001e\u000bandn5. The noise terms \u0018\u000b; \u0010, and\u00105satisfy the \ructuation-\ndissipation relations:\nh\u0018\u000b(r;t)\u0018\f(r0;t0)i= 2\u0000\u000e\u000b\f\u000e(t\u0000t0)\u000ed(r\u0000r0); (5.17)\nh\u0010(r;t)\u0010(r0;t0)i=\u00002\u0015r2\u000e(t\u0000t0)\u000ed(r\u0000r0); (5.18)\nh\u00105(r;t)\u00105(r0;t0)i=\u00002\u00155r2\u000e(t\u0000t0)\u000ed(r\u0000r0); (5.19)\nandh\u0018\u000b\u0010i=h\u0018\u000b\u00105i=h\u0010\u00105i= 0. We here postulate the following Poisson\nbrackets from the symmetry algebra:\n[n5(r;t);\u001e\u000b(r0;t)] =\"\u000b\f\u001e\f\u000e(t\u0000t0)\u000ed(r\u0000r0); (5.20)\n[n(r;t);n5(r0;t)] =CB\u0001r\u000e(t\u0000t0)\u000ed(r\u0000r0): (5.21)\nHere,\"\u000b\fdenotes the anti-symmetric tensor in the order parameter space (5.8),\nandCis usually related to the anomaly coe\u000ecient (2.67) or the CME coef-\n\fcient. However, in our analysis near the second-order phase transition, we\nregardCas a free parameter, which will be determined by the RG equation of\nthe system. Since nonlinear \ructuations of massless \u001bcan potentially renor-\nmalizeC, it is a nontrivial question whether Cis exactly \fxed, related to the\nnonrenormalization away from the second-order phase transition mentioned\nunder Eq. (2.67). Nevertheless, in Sec. 5.3.2, we will show that this anomaly\ncoe\u000ecient or the CME coe\u000ecient does not receive the renormalization at the\none-loop level.\n5.2.2 Dynamic perturbation theory\nWe here summarize the MSRJD formulation of our Langevin theory. The\n\feld theoretical MSRJD action corresponding to the Langeinv theory in62CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\nSec. 5.2.1 is\nS=Z\ndtZ\ndr(L\u001e+Ln+L\u001en); (5.22)\nwhose general form is given by Eq. (3.16). We will explain each of the\nLagrangian in Eq. (5.22) and its brief derivation by assuming Eq. (3.16) (see\nSec. 3.1 for the derivation of Eq. (3.16) itself).\nFirst,L\u001erepresents the kinetic term of the order parameters \u001e\u000band the\n4-point interaction term:\nL\u001e=~\u001e\u000b\u0012@\n@t+ \u0000(r\u0000r2)\u0013\n\u001e\u000b\u0000\u0000~\u001e2\n\u000b+ 4\u0000u~\u001e\u000b\u001e\u000b\u001e2\n\f; (5.23)\nwhere ~\u001e\u000bdenotes the responsible \feld for \u001e\u000b. Almost all of the terms come\nfromF\u001e\u000b[ ] corresponding to the right-hand side of Eq. (5.12) (see Eq. (3.11)\nfor the de\fnition of FI[ ] with I=\u001e\u000b;n;n 5), but the bilinear of the response\n\feld,\u0000\u0000~\u001e2\n\u000b, comes from the last term of Eq. (3.16). The form of \rIJ(r) in\nEq. (3.16) can be obtained by comparing the original de\fnition of \rIJ(r),\nEq. (3.12) and Eq. (5.17).\nNext,Lnrepresents the bilinear part of the conserved charge densities n\nandn5, which are coupled to each other in the presence of the CME:\nLn=1\n2(~n;n; ~n5;n5)M0\nBB@~n\nn\n~n5\nn51\nCCA; (5.24)\nwhere\nM\u00110\nBBBBBBBBB@2\u0015r2@\n@t\u0000\u0015\n\u001fr20C\n\u001f5B\u0001r\n\u0000@\n@t\u0000\u0015\n\u001fr20\u0000C\n\u001fB\u0001r 0\n0C\n\u001fB\u0001r 2\u00155r2@\n@t\u0000\u00155\n\u001f5r2\n\u0000C\n\u001f5B\u0001r 0\u0000@\n@t\u0000\u00155\n\u001f5r201\nCCCCCCCCCA(5.25)\nand ~nand ~n5are the response \felds for nandn5, respectively. Almost\nall terms come from Fn[ ] andFn5[ ] corresponding to the linear terms of5.2. FORMULATION 63\nthe right-hand sides of Eqs. (5.13) and (5.14), respectively. Meanwhile, the\nbilinear terms of ~ nand ~n5originate from the last term of Eq. (3.16). Here,\nEqs. (5.18) and (5.19) give \rnn=\u0015r2and\rn5n5=\u00155r2, respectively.\nThe last term of Eq. (5.22), L\u001enrepresents the 3-point interaction between\nthe order parameters \u001e\u000band the conserved charge densities nandn5:\nL\u001en=\u0000g\"\u000b\f\n\u001f5\u0010\n~\u001e\u000b\u001e\fn5+\u001f5~n5(r2\u001e\u000b)\u001e\f\u0011\n+ 2\r\u0000~\u001e\u000b\u001e\u000bn\u0000\r\u0015~n(r2\u001e2\n\u000b) +\rC~n5B\u0001(r\u001e2\n\u000b):(5.26)\nWe obtain these terms from FI[ ] corresponding to the nonlinear terms\namong\u001e\u000b,n, andn5in the Langevin equations (5.12){(5.14).\nWe make some remarks on the interaction term L\u001en. There are two\ntypes of interactions: the \frst line of Eq. (5.26) ( /g) originates from the\nPoisson bracket (5.20), the second line of Eq. (5.26) ( /\r) originates from\nthe non-Gaussian term \rn\u001e2\n\u000bin the Ginzburg-Landau free energy (5.15). The\nformer (/g) exists even at \u0016= 0 and gives the couplings between di\u000berent\ncomponents of the order parameters \feld re\recting \"\u000b\fin the interaction\nvortex; the latter ( /\r) exists as long as \u00166= 0 and gives the couplings\nbetween the same order parameter components. Note also that the last\nterm of Eq. (5.26) may potentially generate the nonlinear corrections to the\nanomaly coe\u000ecient C. Nevertheless, we will show that this is not the case\nfrom the explicit computation.\nFeynman rules\nWe here summarize the Feynman rules for the action (5.22) (see Ref. [66]\nto obtain the Feynman rules from the MSRJD action. See also standard\ntextbooks of quantum \feld theory, e.g., Ref. [74] for getting the Feynman\nrules from a \feld theoretical action)\nThe bare propagator of the order parameter, G0\n\u000b\f, is obtained by calculat-\ning the 2-point correlation h\u001e\u000b~\u001e\fifrom the Gaussian part of L\u001ein momentum\nspace:\nG0\n\u000b\f(k;!) =G0(k;!)\u000e\u000b\f\u0011\u000e\u000b\f\n\u0000i!+ \u0000(r+k2); (5.27)\nwhich is diagonal with respect to \u000band\f. The bare propagator of the\nconserved \felds, D0\nij, is obtained by calculating the 2-point correlation hni~nji64CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\nfromLn, whereni=n;n 5. In particular, the inverse matrix of D0\nijhas the\nfollowing expression:\n[D0(k;!)]\u00001=0\nB@\u0000i!+\u0015\n\u001fk2iC\n\u001f5B\u0001k\niC\n\u001fB\u0001k\u0000i!+\u00155\n\u001f5k21\nCA; (5.28)\nwhich has o\u000b-diagonal components with respect to iandjbecause of the\nCME.\nAs we have brie\ry mentioned in Sec. 3.3.2, there is a 2-point noise vortex\ncorresponding to the bilinear of the response \felds in the action. In par-\nticular, from the bilinear term of the response \feld of the order parameter\n~ \u000binL\u001e, we obtain the bare noise vertex of the order parameter as 2\u0000 \u000e\u000b\f.\nSimilarly, we get the bare noise vertex of the conserved charge densities as\nL0(k) =\u00122\u0015k20\n0 2\u00155k2\u0013\n; (5.29)\nwhich is related to the bare correlation function of the conserved charge\ndensities,B0\nij, through the following equation:\nB0\nij(k;!) =D0\nil(k;!)L0\nlk(k)[D0(\u0000k;\u0000!)]T\nkj=D0\nil(k;!)L0\nlk(k)[D0(k;!)]y\nkj:\n(5.30)\nHere,B0\nijcan be calculated explicitly by the 2-point correlation hninjifrom\nLnas follows:\nB0\n11(k;!) =2\u0015k2(!2+\u00152\n5k4=\u001f2\n5) + 2\u00155k2(CB\u0001k=\u001f5)2\njdet[D0(k;!)]\u00001j2; (5.31)\nB0\n12(k;!) =B0\n21(k;!) =2 (\u0015=\u001f+\u00155=\u001f5)k2C(B\u0001k)!\njdet[D0(k;!)]\u00001j2; (5.32)\nB0\n22(k;!) =2\u00155k2(!2+\u00152k4=\u001f2) + 2\u0015k2(CB\u0001k=\u001f)2\njdet[D0(k;!)]\u00001j2: (5.33)\nThe 4-point interaction vertex is obtained from L\u001eas\nU0\n\u000b;\f\r\u000e=\u00004u\u0000\u000e\u000b\f\u000e\r\u000e; (5.34)5.2. FORMULATION 65\nwhere the indices \u000b;\f\r\u000eare the shorthand notation of the \felds ~\u001e\u000b\u001e\f\u001e\r\u001e\u000e.\nThe 3-point interaction vertices are obtained as\nV0\n\u000b;\fi=\u0012\u00002\r\u0000\u000e\u000b\f\ng\"\u000b\f=\u001f5\u0013\n; (5.35)\nV0\ni;\u000b\f(k;p) =\u0012\u00002\r\u0015k2\u000e\u000b\f\ng[(k\u0000p)2\u0000p2]\"\u000b\f\u00002i\rCB\u0001k\u000e\u000b\f\u0013\n; (5.36)\nwhere we use the vector notation for the label iwhich classi\fes ni=n;n 5.\nThe indices \u000b;\fiandi;\u000b\fare the shorthand notation of ~\u001e\u000b\u001e\fniand ~ni\u001e\u000b\u001e\f,\nrespectively. The vertex V0\ni;\u000b\f(k;p) is the function of the outgoing momentum\nkofniand the ingoing momentum pof\u001e\u000b(see also Fig. 5.2(c) for the\ncon\fguration of the external momentum).\nWe next summarize the diagrammatic representations of the above prop-\nagators, the noise- and interaction- vortices. First, we depict G0\n\u000b\fby the\nplane line, and D0\nijby the wavy line with the outgoing and ingoing compo-\nnentsiandj. SinceG0\n\u000b\fis diagonal with respect to \u000b;\fas one can con\frm\nin Eq. (5.27), we omit writing both of the outgoing and ingoing indices \u000b\nand\fin the diagram for G0\n\u000b\f. Instead, we shall write \u000balone at the center\nof plane lines. Each noise vertex of the order parameters and the conserved\ncharge densities can be understood as the diagrams with two outgoing lines\nas represented in Fig. 5.1. Note that the number of ingoing and outgoing\nlines in one of the diagrams corresponds to the power of the hydrodynamic\nvariable =\u001e\u000b;niand the response \feld ~ =~\u001e\u000b;~niin the action, respec-\ntively [66]. As is shown in Fig 5.2, each of the interaction vertices has one\noutgoing and three or two ingoing lines.\nOrdinary Feynman rules are applied to obtain full n-point correlations.\nAmong others, we obtain the full propagators G\u000b\fandDijby using the\nself-energies of the order parameter, \u0006 \u000b\f, and those of the conserved charge\ndensities, \u0005 ij, as\nG\u00001\n\u000b\f(k;!) = [G0\n\u000b\f(k;!)]\u00001\u0000\u0006\u000b\f(k;!); (5.37)\nD\u00001\nij(k;!) = [D0\nij(k;!)]\u00001\u0000\u0005ij(k;!): (5.38)\nWe also get the three-point vertex function V\u000b;\fi(k1;k2;!1;!2) by computing\none-particle irreducible diagrams with outgoing ~\u001e\u000band ingoing \u001e\f; ni. Here,\nk1and!1denote the ingoing momentum and frequency of \u001e\f, andk2and\n!2denote the ingoing momentum and frequency of ni. From the energy and66CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\nkω α\n−k−ω β\n(a) 2\u0000\u000e\u000b\f\nkω\n−k−ωi\nj (b)L0\nij\nFigure 5.1: Noise vertices\nαβ\nγ\nδ\n(a)U0\n\u000b;\f\r\u000e\niαβ (b)V0\n\u000b;\fi\nkp\nk−pα\nβi (c)V0\ni;\u000b\f\nFigure 5.2: Interaction vertices\nmomentum conservation laws, ~\u001e\fhas the outgoing momentum k3=k1+k2\nand the frequency !3=!1+!2. For later purposes, it is convenient to divide\nV\u000b;\fiinto its bare contribution V0\n\u000b;\fiand the correction term V\u000b;\fias follows:\nV\u000b;\fi(k1;k2;!1;!2) =V0\n\u000b;\fi+V\u000b;\fi(k1;k2;!1;!2): (5.39)\n5.3 Renormalization-group analysis\nIn this section, we apply the RG to the e\u000bective theories derived in Sec. 5.2. In\nSec. 5.3.1, we apply the static RG to the Ginzburg-Landau free energy (5.15)\nwithin the\u000fexpansion. In Sec. 5.3.2, we study the MSRJD action (5.22) by\nusing the dynamic RG with the help of Appendix C. We remark those results\nin Sec. 5.3.3.5.3. RENORMALIZATION-GROUP ANALYSIS 67\n5.3.1 Statics\nIn a single RG transformation, we integrate out the degrees of freedom in\nthe momentum shell \u0003 =b<q < \u0003 (with \u0003 being the momentum cuto\u000b and\nb>1,q\u0011jqj), and we rescale the coordinate and the \felds in the following\nway (For the details about the static RG, see Sec. 3.3):\nr!r0=b\u00001r; (5.40)\n\u001e\u000b(r)!\u001e0\n\u000b(r0) =ba\u001e\u000b(r); (5.41)\nn(r)!n0(r0) =bcn(r); (5.42)\nn5(r)!n0\n5(r0) =bc5n5(r): (5.43)\nHerea,c, andc5are some constants computed in the calculation below.\nHereafter, we work with the spatial dimension d\u00114\u0000\u000fwith small \u000fand\nperform the calculation to leading orders in the expansion of \u000fas we remarked\naround Eq. (3.38).\nLet us write the static parameters at the lth stage of the renormalization\nprocedure, rl; ul; \u001fland\rl. Then, these parameters satisfy the same recur-\nsion relation as that of the so-called model C for the system with a single\n(two-component) order parameter and the conserved charge densities [67].\nIn particular, we quote Eqs. (4.5){(4.8) in Ref. [67]. The recursion relations\nfor the static parameters in the leading-order of \u000fare given as\nrl+1=bd\u00002afrl+ 8\u0016ul[\u00032(1\u0000b\u00002)\u00002rllnb]g; (5.44)\n\u0016ul+1=bd\u00004a\u0016ul(1\u000040\u0016ullnb); (5.45)\n\u001f\u00001\nl+1=bd\u00002c\u001f\u00001\nl(1\u00004vllnb); (5.46)\n(\u001f5)\u00001\nl+1=bd\u00002c5(\u001f5)\u00001\nl; (5.47)\n\rl+1=b2\u0000c\rl[1\u0000(16\u0016ul+ 4vl) lnb]: (5.48)\nHere, the left-hand sides are the parameters at the ( l+ 1)-th step of the RG,\nand we introduced the following quantities,\nv\u0011\r2\u001f\u0003\u0000\u000f\n8\u00192;\u0016u\u0011u\u0003\u0000\u000f\n8\u00192: (5.49)\nNote that one may not \fnd the equation corresponding to Eq. (5.47) in\nRef. [67] which we quoted. Nevertheless, Eq. (5.47) can be readily introduced\nbecause there is only Gaussian term for n5in the Ginzburg-Landau free68CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\nenergy (5.15) and ( \u001f5)lis a\u000bected only by the trivial scale transformation\n(5.43). From the condition that ( \u001f5)lremains \fnite at the \fxed point, we\n\fnd the value of cfrom Eq. (5.47) as\nc5=d\n2: (5.50)\nIt is also worth writing that the recursion relations (5.45){(5.48) are the\nextension of Eqs. (3.39) and (3.40) to include the e\u000bects of the conserved\ncharge density nandn5. In particular, vcharacterizes the nonlinear coupling\nbetween the order parameter and the conserved-charge density n.\nLet us now compute aandc. Because the anomalous dimension \u0011is zero\nto the order of \u000f[63], the exponent ais solely determined by Eq. (3.30) as\na=d\u00002\n2: (5.51)\nTo determine c, we evaluate the interplay of the RG \row between vand \u0016u.\nCombining Eqs. (5.46) and (5.48), we obtain the recursion relation for vl,\nvl+1=b\u000fvl[1\u0000(32\u0016ul+ 4vl) lnb]: (5.52)\nThe RG equations corresponding to Eqs. (5.45) and (5.52) become4\nd\u0016ul\ndl= (d\u00004a\u000040\u0016ul)\u0016ul: (5.53)\ndvl\ndl= (\u000f\u000032\u0016ul\u00004vl)vl; (5.54)\nwhere Eq. (5.53) is the two-component order parameter version of Eq. (3.42),\nand Eq. (5.54) is the RG equation that evaluates the coupling between the\norder parameter \u001e\u000band the conserved charge density n.\n4The derived recursion relations typically have the following form:\nAl+1=bcAAl(1 +Bllnb):\nwithAlandBlbeing the variables and cAbeing some constant. One can derive the RG\nequation for Alby settingb=eland taking the limit l!0 as\ndAl\ndl= (cA+Bl)Al5.3. RENORMALIZATION-GROUP ANALYSIS 69\nEquations (5.53)-(5.54) tell us the \fxed-point values of \u0016 ulandvlas [67]\n\u0016u1=\u000f\n40; v1=\u000f\n20: (5.55)\nReturning to Eq. (5.46), we arrive at\nc=3d\u00002\n5: (5.56)\nStatic critical exponents\nThe physical parameters near the transition temperature Tcare solely given\nby the loop-correction terms proportional to ln bin the static recursion rela-\ntions [75]. In particular, we obtain the renormalized isospin charge suscepti-\nbility\u001f(T) by taking into account the correction calculated in Eq. (5.46) to\nthe bare value \u001f0at the cuto\u000b scale \u0003,\n\u001f(T) =\u001f0[1 + 4v1ln(\u0003\u0018)]\u0018\u0018\u000f=5; (5.57)\nwhere we use the relation 1+ xln \u0003\u0018+O(x2) = (\u0003\u0018)xforx\u001c1, by regarding\nx= 4v1as a small parameter when \u000f\u001c1. De\fning the critical exponents\n\u0017and\u000bin the usual manner,\n\u0018\u0018\u001c\u0000\u0017; \u001f\u0018\u001c\u0000\u000b; (5.58)\nwith\u001c\u0011(T\u0000Tc)=Tcbeing the reduced temperature, we obtain\n\u000b\n\u0017=\u000f\n5: (5.59)\n5.3.2 Dynamics\nAs we mentioned it around Eq. (3.47), we also need to rescale the response\n\felds ~ =~\u001e\u000b;~niin addition to those for the hydrodynamic variables  =\n\u001e\u000b;ni, (5.41){(5.43) in the dynamic RG:\n~\u001e\u000b(r)!~\u001e0\n\u000b(r0) =b~a~\u001e\u000b(r); (5.60)\n~n(r)!~n0(r0) =b~c~n(r); (5.61)\n~n5(r)!~n0\n5(r0) =b~c5~n5(r): (5.62)\nWe will determine ~ a, ~c, and ~c5in the following calculations.70CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\nWe compute the full inverse propagators at the ( l+ 1)th renormaliza-\ntion step. Then, we get the recursion relations of the dynamic parameters\n\u0000l; \u0015l; \u00155;andCl, in an analogy to those of the static quantities, (5.45){\n(5.48). In particular, we here demonstrate the derivation of the recursion\nrelation for \u0000 las an example. First, we calculate the full inverse propagator\nfor the order parameter as\n[G\u000b\u000b(k0;!0)]\u00001\nl+1= [G\u000b\u000b(k;!)]\u00001\nlb\u0000~a\u0000a\n=\u0014\n\u0000i!\u0012\n1\u0000i@\u0006\u000b\u000b(0;!)\n@!\f\f\f\f\n!!0\u0013\n+ \u0000lrl\u0000\u0006\u000b\u000b(0;0)\n+\u0012\n\u0000l\u00001\n2@2\u0006\u000b\u000b(k;0)\n@k2\f\f\f\f\nk!0\u0013\nk2+\u0001\u0001\u0001\u0015\nb\u0000~a\u0000a:(5.63)\nHere, we use Eq. (5.37) and expand \u0006 \u000b\u000bwith respect to the frequency !and\nwave number k. By regarding the term proportional to k2on the right-hand\nside as \u0000 l+1k02and including the overall factor bz+dcoming from rescaling\nthe measure of the action, we obtain the recursion relation for \u0000 l,\n\u0000l+1= \u0000l\u0012\n1\u00001\n2\u0000l@2\u0006\u000b\u000b(k;0)\n@k2\f\f\f\f\nk!0\u0013\nbd+z\u0000~a\u0000a\u00002: (5.64)\nFurthermore, we normalize the term proportional to \u0000i!including the overall\nfactor,\n1 =\u0012\n1\u0000i@\u0006\u000b\u000b(0;!)\n@!\f\f\f\f\n!!0\u0013\nbd\u0000~a\u0000a: (5.65)\nSimilarly, we can derive the following recursion relations by computing\nthe inverse propagator for conserved charge densities, [ Dij(k0;!0)]\u00001\nl+1, and\nthree-point vertex [ V\u000b;\fi(k1;k2;!1;!2)]l+1instead of [ G\u000b\u000b(k0;!0)]\u00001\nl+1as we5.3. RENORMALIZATION-GROUP ANALYSIS 71\nused in Eq. (5.63):\n\u0015l+1\n\u001fl+1=\u0015l\n\u001fl\u0012\n1\u0000\u001fl\n2\u0015l@2\u000511(k;0)\n@k2\f\f\f\f\nk!0\u0013\nbd+z\u0000~c\u0000c\u00002; (5.66)\n(\u00155)l+1\n(\u001f5)l+1=(\u00155)l\n(\u001f5)l\u0012\n1\u0000(\u001f5)l\n2(\u00155)l@2\u000522(k;0)\n@k2\f\f\f\f\nk!0\u0013\nbd+z\u0000~c5\u0000c5\u00002; (5.67)\nCl+1\n\u001fl+1B=\u0012Cl\n\u001flB+ i@\u000521(k;0)\n@k\f\f\f\f\nk!0\u0013\nbd+z\u0000~c5\u0000c\u00001; (5.68)\nCl+1\n(\u001f5)l+1B=\u0012Cl\n(\u001f5)lB+ i@\u000512(k;0)\n@k\f\f\f\f\nk!0\u0013\nbd+z\u0000~c\u0000c5\u00001; (5.69)\ngl+1\n(\u001f5)l+1\"\u000b\f=gl\n(\u001f5)l\u0012\n\"\u000b\f+(\u001f5)l\nglV\u000b;\f2(k1;k2;!1;!2)jk1;2!0; !1;2!0\u0013\nbd+z\u0000~a\u0000a\u0000c5;\n(5.70)\nwith the following constraints,\n1 =\u0012\n1\u0000i@\u000511(0;!)\n@!\f\f\f\f\n!!0\u0013\nbd\u0000~c\u0000c; (5.71)\n1 =\u0012\n1\u0000i@\u000522(0;!)\n@!\f\f\f\f\n!!0\u0013\nbd\u0000~c5\u0000c5: (5.72)\nTherefore, once we can evaluate the self-energies \u0006 \u000b\u000b(k;!), \u0005ij(k;!) and\nvertex function corrections V\u000b;\f2(k1;k2;!1;!2), we obtain the recursion rela-\ntions for the dynamic parameters in the MSRJD action. We put the details\nof the calculations on \u0006 \u000b\u000b, \u0005ij, andV\u000b;\f2in Appendix C.\nIn the calculations of Appendix C, the following parameters are intro-\nduced:\nf\u0011g2\u0003\u0000\u000f\n8\u00192\u00155\u0000; w\u0011\u0000\u001f\n\u0015; w 5\u0011\u0000\u001f5\n\u00155; h\u0011CBp\u0015\u00155\u0003; (5.73)\nX\u00112p\n(1 +w)(1 +w5) +h2+p\n(1 +w)(1 +w5)r\n1 +w\n1 +w5; X0\u00111 +w5\n1 +wX;\n(5.74)\nwhereB\u0011jBj. By using these new parameters and based on the detailed\nanalysis in appendix C, we can obtain the recursion relations for the dynamic72CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\nparameters \u0000 l; \u0015l; \u00155; Clandglat the one-loop level:\n\u0000l+1=bz\u00002\u0000l[1\u0000(4vlwlX0\nl\u0000flXl) lnb]; (5.75)\n\u0015l+1=bz\u0000d+2c\u00002\u0015l; (5.76)\n(\u00155)l+1=bz\u0000d+2c5\u00002(\u00155)l\u0012\n1 +fl\n2lnb\u0013\n; (5.77)\nCl+1=bz+c+c5\u0000d\u00001Cl; (5.78)\ngl+1=bz\u0000d+c5gl: (5.79)\nHere, Eqs. (5.75), (5.76){(5.78), (5.79) are derived in Appendix C.2, C.1, C.3,\nrespectively. Among these recursion relations, Eq. (5.78) shows that the CME\ncoe\u000ecientCis not renormalized by the critical \ructuations of the order pa-\nrameter in this order. This may be viewed as an extension of the nonrenor-\nmalization theorem for the CME coe\u000ecient at the second-order chiral phase\ntransition, where \u001bbecomes massless.\nTo obtain the \fxed point solutions of the recursion relations, it is useful\nto obtain the recursion relations for the parameters de\fned in Eq. (5.73). By\nusing the recursion relations for the dynamic parameters (5.75){(5.79) and\nthe static parameters (5.46){(5.48), we reach\nfl+1=b\u000ffl\u0014\n1 +\u0012\n4vlwlX0\nl\u0000flXl\u00001\n2fl\u0013\nlnb\u0015\n; (5.80)\n(w5)l+1= (w5)l\u0014\n1\u0000\u0012\n4vlwlX0\nl\u0000flXl+1\n2fl\u0013\nlnb\u0015\n; (5.81)\nwl+1=wl[1\u0000(4vlwlX0\nl\u0000flXl\u00004vl) lnb]; (5.82)\nhl+1=bhl\u0012\n1\u0000fl\n4lnb\u0013\n: (5.83)\nThe dynamic RG equations corresponding to Eqs. (5.80){(5.83) can be de-\nrived in a way similar to Eqs. (5.53)-(5.54) as (see also the footnote under5.3. RENORMALIZATION-GROUP ANALYSIS 73\nEq. (5.53) for the derivation of the RG equation from the recursion relation)\ndfl\ndl=\u0012\n\u000f+ 4vlwlX0\nl\u0000flXl\u00001\n2fl\u0013\nfl; (5.84)\nd(w5)l\ndl=\u0012\n\u00004vlwlX0\nl+flXl\u00001\n2fl\u0013\n(w5)l; (5.85)\ndwl\ndl= (\u00004vlwlX0\nl+flXl+ 4vl)wl; (5.86)\ndhl\ndl=\u0012\n1\u0000fl\n4\u0013\nhl; (5.87)\nfrom which we \fnd four possible nontrivial \fxed-point values of f; w 5; w; h :5\n(i)f1=\u000f;(w5)1= 1; w1=h1= 0; (5.88)\n(ii)f1=2\n3\u000f;(w5)1=w1=h1= 0; (5.89)\n(iii)f1=\u000f;(w5)1=3\n7; w1=h1=1; (5.90)\n(iv)f1= 2\u000f;(w5)1= 0; w1=h1=1: (5.91)\nSome remarks on the \fxed points above are in order here. Since the\nmagnetic \feld is external ( B6= 0), the \fxed points (i) and (ii) with h1= 0\nshould be interpreted as C= 0. We should note that the RG equations\n(5.84){(5.86) are nonuniform in the limits w!1 andh2!1 . If one takes\nw!1 \frst by \fxing h2to some particular value, the \fxed point (iii) is\nobtained; if one takes h2!1 \frst by \fxing wto some particular value, the\n\fxed point (iv) is obtained. In other words, the \fxed point (iii) corresponds\nto the case w1\u001dh2\n1\u001d1, and the \fxed point (iv) corresponds to the case\nh2\n1\u001dw1\u001d1. The competition between w!1 andh2!1 in Eq. (5.74)\nare characterized by the strength of the following parameter,\nh2\nw=C2B2\n\u00155\u0000\u001f\u00032: (5.92)\nFrom the expression of Eq. (5.92), we can see which parameters among C;\u0015\nand\u00155are dominant near the \fxed points, for a \fnite kinetic coe\u000ecient of the\n5Besides, there is a trivial \fxed point, f1= (w5)1=w1=h1= 0, which is stable\nonly for\u000f<0, and is not considered here.74CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\norder parameter, \u0000, and \fnite static susceptibilities, \u001fand\u001f5.6By looking\natf1, (w5)1, and the \fxed-point value of (5.92), one can see the \fxed point\n(iii) corresponds to C!0 and\u0015!0 with \fnite \u00155, where the CME can be\nneglected compared to the di\u000busion e\u000bect; the \fxed point (iv) corresponds\ntoC!1 ,\u0015!0, and\u00155!1 withC2=\u00155!1 , where the di\u000busion\ne\u000bect can be neglected compared to the CME. In short, we can regard the\ncompetition between the two limits w!1 andh2!1 as the competition\nbetween the CME and the di\u000busion of the axial isospin density n5.\nStability of \fxed points\nWe here summarize the stability analysis of the \fxed points (i){(iv). In order\nto investigate the in\ruence of the CME, we study the stability of the \fxed\npoints atC= 0, namely the \fxed points (i) and (ii). For this purpose, we\nconsider the linear perturbations around these \fxed points,\nfl=f1+\u000ef; (w5)l= (w5)1+\u000ew5; wl=\u000ew; hl=\u000eh: (5.93)\nSubstituting these expressions into Eqs. (5.84){(5.87) and setting vl=v1=\n\u000f=20 from Eq. (5.55),7the linearized equations with respect to \u000ef,\u000ew5,\u000ew5,\nand\u000ehread\nd\ndl0\nBB@\u000ef\n\u000ew5\n\u000ew\n\u000eh1\nCCA=M00\nBB@\u000ef\n\u000ew5\n\u000ew\n\u000eh1\nCCA; (5.94)\nwhere we de\fne\nM0\u00110\nBBBBBBB@\u0000f1\u0012\n\u00121+1\n2\u0013\nf2\n1\u00122\n1 4v1f1 0\n(w5)1\u0012\n\u00121\u00001\n2\u0013\nf1\u0014\n\u00121\u00001\n2\u0000(w5)1\u00122\n1\u0015\n\u00004v1(w5)1 0\n0 0 f1\u00121+ 4v1 0\n0 0 0 1 \u0000f1\n41\nCCCCCCCA\n(5.95)\n6One can con\frm the \fniteness of \u0000 ;\u001f;and\u001f5by putting back the \fxed-point values\nofv;f;w 5;w, andhto the recursion relations (5.46), (5.47), and (5.75) with the help of\nEqs. (5.50) and (5.56).\n7Here we can ignore the \ructuation of vl, because all of the \ructuation of vlwill be\nmultiplied by O(\u000ew) in Eqs. (5.84){(5.87) if we try to substitute vl=v1+\u000ev.5.3. RENORMALIZATION-GROUP ANALYSIS 75\nand\n\u00121\u00111\n1 + (w5)1=8\n<\n:1\n2for the case (i) ;\n1 for the case (ii) :(5.96)\nWe can test the stability of the \fxed points (i) and (ii) by substituting each\nof the \fxed point values into M0. Because of ( w5)1(\u00121\u00001=2) = 0 for both\ncases (i) and (ii), the matrix M0de\fned in Eq. (5.94) is reduced to an upper\ntriangular matrix. Thus, the eigenvalues of M0are just given by its diagonal\ncomponents for each \fxed point:\n(i)\u0012\n\u0000\u000f;\u0000\u000f\n4;7\n10\u000f;1\u0000\u000f\n4\u0013\nand (ii)\u0012\n\u0000\u000f;\u000f\n3;13\n15\u000f;1\u0000\u000f\n6\u0013\n:(5.97)\nFrom this result, we \fnd that the \fxed point (ii) is unstable in the w5di-\n(vl,wl,hl)=(v1,w1,h1)=(✏/20,0,0)\n<latexit sha1_base64=\"/L7eI1eY+byqeCViBcwi3DYHpMU=\">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</latexit><latexit sha1_base64=\"/L7eI1eY+byqeCViBcwi3DYHpMU=\">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</latexit><latexit sha1_base64=\"/L7eI1eY+byqeCViBcwi3DYHpMU=\">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</latexit><latexit sha1_base64=\"/L7eI1eY+byqeCViBcwi3DYHpMU=\">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</latexit>\n0.00.51.01.5-0.50.00.51.01.5\nfw5(i)(ii)\nFigure 5.3: RG \row of the parameters fandw5for \fxedwandh(\u000f= 1),\nwhich shows the \fxed points (i) and (ii).\nrection and that the RG \row runs to the \fxed point (i) (see also Fig. 5.3\nshowing the RG \rows in the ( f;w 5) plane atw=h= 0). We also \fnd that\nthe \fxed points (i) and (ii) are unstable in the directions of wandh, showing\nthat\u0015andCare relevant. It follows that small but nonzero values of wand\nhgrow around the \fxed point (i).76CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\nThere are two possibilities of the \fnal destination of the \row from the\n\fxed point (i): the \fxed point (iii) and the \fxed point (iv). Let us \frst\nqualitatively understand the conditions for obtaining each of the \fxed points\n(iii) and (iv) by using RG \row diagrams. For this purpose, we \frst forcibly\n\fxwandhto some \fnite values and investigate the RG \rows in the ( f;w 5)\nplane. As one can see in the RG \rows in Figs. 5.4(a) and 5.4(b), fandw5\n\row to the \fxed point (iii) when w\u001dh2\u001d1, while they \row to the \fxed\npoint (iv) when h2\u001dw\u001d1, related to the properties of the \fxed points\n(iii) and (iv) noted under Eq. (5.91). On the other hand, Fig. 5.4(c) shows\nthat (f;w 5) \row to the intermediate values between the \fxed-point values of\n(iii) and (iv) when w\u0018h2. Next, we vary wandharound those values of the\n\fxed point (i) with \fxed fandw5, and study which one is more relevant (iii)\nor (iv) around the \fxed points (i). As is shown in Fig. 5.5, the points in the\n(w;h) plane \row in the direction along the haxis unless w\u001dh. Therefore,\nthe system eventually \rows to the \fxed point (iv) for most of the parameter\nregion around the \fxed point (i).\nNext, we consider the RG \rows in all the parameter space ( f;w 5;w;h )\nwithout \fxing the parameters. Here, we \frst set the initial parameters near\nthe \fxed point (i) and consider the \row equations at a \fnite \row time.\nAs is shown within the linear-stability analysis in Appendix D, the initial\nparameter region that \rows to the \fxed point (iv) is much broader than\nthe region that \rows to the \fxed point (iii) in the ( w;h) plane. Therefore,\nin the almost whole region of the ( w;h) plane near the \fxed point (i), the\n\fxed point (iv) is favorable rather than the \fxed point (iii). We can also\nstudy the RG \row from the initial values near the \fxed point (iii). Expect\nfor the case of the RG evolution, starting from the parameters exactly at\nthe \fxed point (iii), all the parameters will eventually take the \fxed-point\nvalues of (iv). This is because hgrows much more rapidly than wdue to the\nadditional scaling factor bin the recursion relation (5.83) for h, compared\nto the relation (5.82) for w. From the above discussion, it follows that the\n\fxed point (iv) is stable in the almost whole region at \fnite wandh, while\ngenerally at a \fnite \row time there is a small parameter region that \rows to\nthe \fxed point (iii).5.3. RENORMALIZATION-GROUP ANALYSIS 77\n0.00.51.01.52.02.50.00.20.40.60.81.0\nfw5(iii)(iv)0.00.51.01.52.02.50.00.20.40.60.81.0\nfw5(iii)(iv)\n0.00.51.01.52.02.50.00.20.40.60.81.0\nfw5(iii)(iv)(a)\n<latexit 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sha1_base64=\"4/3VmZoTY0y4OFeaanHKwg4XbsM=\">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</latexit><latexit 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sha1_base64=\"8cadPy6Umc8aV4KE8NwQtcnKw9I=\">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</latexit>\nFigure 5.4: RG \rows of fandw5(\u000f= 1). We \fx the values of wandh2in\nseveral cases: (a) w\u001dh2\u001d1, (b)h2\u001dw\u001d1, and (c)w\u0018h2. These\n\fgures show the existence of the \fxed points (iii) and (iv), and the \row to\ntheir intermediate values.78CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\n02004006008001000-400-2000200400\nwh(i)(vl,fl,(w5)l)=(✏/20,✏,1)\n<latexit sha1_base64=\"t/5BiknVKA+2cP8RrHfnpe2X/SI=\">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</latexit>\nFigure 5.5: RG \row of wandhwith \fxing fandw5to those values at the\n\fxed point (i) ( \u000f= 1).\n5.3.3 Physical consequences\nDynamic universality class\nFrom the \fxed point values (i){(iv) obtained in Eqs. (5.88){(5.91), we can\nevaluate the dynamic critical exponent which characterizes the dynamic uni-\nversality class of the system. Practically, we substitute the \fxed point values\ninto the recursion relation (5.75) for \u0000 l, in each case of the \fxed-point values\n(i){(iv).\nThe \fxed points (i) and (iii) have the dynamic critical exponent of model\nE,z=d=2. As is also summarized in Tab. 3.1 the dynamic universality class\nof model E is generally determined only by two-component order parameter\n(mapped into U(1)) and oneconserved density that are coupled through the\nPoisson brackets. In our case, the order parameter \feld \u001e\u000band the axial\nisospin density n5are essential, whereas the isospin density ndoes not a\u000bect\nthe dynamic universality class.8\nOn the other hand, the \fxed point (iv) has the dynamic critical exponent\nz= 2. Up to O(\u000f), this exponent is the same as that of model A, which is the\n8The di\u000berence between nandn5can be seen in the Poisson brackets: there is a nonzero\nPoisson bracket among n5and\u001e\u000bin Eq. (5.20), whereas there are no nontrivial Poisson\nbrackets among nand\u001e\u000b.5.4. CONCLUSION AND DISCUSSION 79\nsimplest class determined only by nonconserved order parameters (see also\nTab. 3.1). In the system near the \fxed point (iv), the internal-momentum\nloop dominated by the CMW (the wavy lines in Fig. C.2) is suppressed, so\nthat not only nbut alson5do not a\u000bect the dynamic universality class. One\ncan con\frm that the factors XandwX0stemming from Fig. C.2 vanish.\nNow we remind the stability analysis in Sec. 5.3.2: the small \ructuation of\nthe \fxed point (i) in the absence of the CME can lead to the \fxed point (iv).\nWe \fnd that the inclusion of the CME can change the dynamic universality\nclass from model E into model A, corresponding to the stable \fxed point (i)\nand (iv), respectively. Strictly speaking, there is a small parameter region\nthat leads to model E, even B6=0andC6= 0 as we also noted in 5.3.2.\nNevertheless, such a region is small compared to the region that leads to\nmodel A.\nCritical attenuation of the CMW\nAs a result of the static critical behavior, Eq. (5.58), we \fnd the critical\nattenuation of the CMW: in the vicinity of the second-order chiral phase\ntransition, the speed of the CMW tends to zero as\nv2\nCMW\u0011C2B2\n\u001f\u001f5\u0018\u0018\u0000\u000b\n\u0017; (5.98)\nwherevCMW is the speed of the CMW [20] which have been already seen in its\nwave equation (2.81). We have already obtained the ratio \u000b=\u0017in Eq. (5.59).\nThis phenomenon is analogous to the critical attenuation of the speed of\nsound near the critical point associated with the liquid-gas phase transition\n[35].\n5.4 Conclusion and discussion\nIn this chapter, we have studied the critical dynamics near the second-order\nchiral phase transition in massless two-\ravor QCD under an external mag-\nnetic \feld and investigated the in\ruence of the CME on the dynamic critical\nphenomena in QCD. We found that the inclusion of the CME and the result-\ning CMW can change the dynamic universality class of the system from the\nmodel E into model A. We also found the critical attenuation of the CMW\nanalogous to that of the sound wave in the liquid-gas phase transition.80CHAPTER 5. DYNAMIC CRITICAL PHENOMENA INDUCED BY CME\nWe now discuss the analogy of the critical attenuation between the CMW\nand the sound wave of the compressive \ruids near the liquid-gas critical point.\nLet us \frst recall the critical attenuation of sounds near the critical point\nassociated with the liquid-gas phase transition, where the order parameter\n LGis a linear combination of the energy density \"and the mass density \u001am.\nIn this case, the speed of sound, cs, is attenuated with the correlation-length\ndependence [35],\nc2\ns\u0011\u0012@P\n@\u001am\u0013\nS=T\u0012@P\n@T\u00132\n\u001am\n\u001a2\nmCV\u0012\n1\u0000CV\nCP\u0013\u0018\u0018\u0000\u000b\n\u0017; (5.99)\nwherePandSare the pressure and total entropy per unit mass of the \ruids,\nrespectively. To obtain the last expression of Eq. (5.99), we use some thermo-\ndynamic relations and the fact that the speci\fc heat with constant volume\nCV\u0011T(@S=@T )\u001a, and that with constant pressure CP\u0011T(@S=@T )P, di-\nverge near the critical point as CV\u0018\u0018\u000b\n\u0017andCP\u0018\u0018\r\n\u0017, respectively. Here,\nthe critical exponents \u0017,\u000b, and\rde\fned by Eq. (5.58) and  LG\u0018\u001c\rare de-\ntermined by the static universality class of the 3D Ising model, \u000b\u00190:1; \u0017\u0019\n0:6; \r\u00191:2. We also use the approximation CV=CP\u001c1 near the criti-\ncal point. Remarkably, Eq. (5.99) takes exactly the same form as that of\nthe CMW which we obtained in Eq. (5.98), although the values of \u000band\u0017\nthemselves are di\u000berent due to the di\u000berence of the static universality classes.\nWhen the reduced temperature \u001cis su\u000eciently larger than \u0016 mq\u0011mq=Tc,\nquark mass e\u000bects on the critical phenomena can be negligible. Our analysis\nin the chiral limit would be relevant to such a parameter region around the\nhigh-temperature QCD critical point under an external magnetic \feld.Chapter 6\nSummary and outlook\nIn this thesis, we have studied the novel dynamic critical phenomena induced\nby super\ruidity and the CME in QCD.\nIn Sec. 4 we have \frst elucidated the in\ruence of the super\ruidity on the\ncritical phenomena near the high-density QCD critical point. In particular,\nwe have found that the static universality class is the same as that of the\nhigh-temperature critical point, independently of the existence of the super-\n\ruid phonon. On the other hand, we have found that the super\ruid phonon\nexhibits the critical slowing down when the critical point is approached. Fur-\nthermore, we have found that the dynamic universality class of the high-\ndensity critical point is not only di\u000berent from that of the high-temperature\ncritical point but also all of the conventional classes studied by Hohenberg\nand Halperin [6].\nExperimental signatures related to the dynamic critical phenomenon in\nthe heavy-ion collisions can distinguish the possible two QCD critical points.\nThough the vanishing speed of the super\ruid phonon characterizes the high-\ndensity critical point, little research has considered the super\ruid phonon\nitself in the context of the heavy-ion collisions. It would be essential to\ninvestigate the role of the super\ruid phonon on the evolution of the hot and\ndense medium created in the heavy-ion collisions.\nIn Sec. 5 we have studied the interplay between the dynamic critical phe-\nnomena and the CME in QCD. For this purpose, we considered the dynamic\ncritical phenomena of the second-order chiral phase transition under an ex-\nternal magnetic \feld. Then, we have found that the inclusion of the CME\nand the resulting CMW can change the dynamic universality class of the sys-\ntem from the model E into model A within the conventional classi\fcation.\n8182 CHAPTER 6. SUMMARY AND OUTLOOK\nWe have also found that the speed of the CMW tends to vanish near the\nphase transition point and that the speed can be characterized by the same\nstatic critical exponents as those of the sound wave in the vicinity of the\nliquid-gas critical point.\nWhile we have limited our analysis in Sec. 5 to the background mag-\nnetic \feld, dynamical electromagnetic \felds may a\u000bect the critical dynamics\nin QCD. In massless QCD coupled to dynamical electromagnetic \felds in\nan external magnetic \feld, there appears a nonrelativistic photon with a\nquadratic dispersion relation due to the quantum anomaly [77]. At \fnite\ntemperature, such a novel gapless mode will be crucial for dynamic critical\nphenomena near the second-order chiral phase transition. This study will be\nreported in detail elsewhere.Appendix A\nStatic RG of the scalar \feld\ntheory\nIn this appendix, we show the detailed RG analysis on the (one component)\nscaler \feld theory (3.36). In particular, we derive the RG equations (3.41)\nand (3.42), and the Wilson-Fisher \fxed-point solution (3.43).\nA.1 Perturbative RG equation\nTo derive the RG equations of the Ginzburg-Landau free energy (3.36), we\n\frst construct its perturbation theory. We start by decomposing the free\nenergy into the following form:\n\fF\u0003[ ] =\fF0\u0003[ ] +\fF0\n\u0003[ ] +Z\ndrh ; (A.1)\nwhere the free part and the perturbative part are given by\n\fF0\u0003[ ] =Z\ndr\u0014r\n2 2(r) +1\n2(r (r))2\u0015\n=Z\nq1\n2(r+q2) (q) (\u0000q); (A.2)\n\fF0\n\u0003[ ] =uZ\ndr 4(r)\n=uZ\nq1Z\nq2Z\nq3Z\nq4 (q1) (q2) (q3) (q4)\u000e(q1+q2+q3+q4);\n(A.3)\n8384 APPENDIX A. STATIC RG OF THE SCALAR FIELD THEORY\nrespectively. Here, we use the simple notation of the momentum integral,\nZ\nq\u0011Zddq\n(2\u0019)d: (A.4)\nUsing Eqs. (3.21) and (A.1), we can write the partition function as\nZ=Z\nD >D <e\u0000\fF\u0003[ ]=Z\nD <e\u0000\fF\u0003=b[ <]; (A.5)\nwith\fF\u0003=b[ <] being the e\u000bective free energy of the low-momentum \feld  <:\ne\u0000\fF\u0003=b[ <]\u0011Z\nD >e\u0000\fF0\u0003[ ]\u0000\fF0\n\u0003[ ]\u0000\fR\ndrh : (A.6)\nOne can write this e\u000bective free energy by using the generating functional\nZ[h],\nZ[h]\u0011D\ne\u0000\fR\ndrh >E\n; (A.7)\nwith the expectation value de\fned in the following form;\nhO[ <; >]i\u0011(Z>)\u00001Z\nD >e\u0000\fF0\u0003[ ]\u0000\fF0\n\u0003[ ]O[ <; >]; (A.8)\nZ>\u0011Z\nD >e\u0000\fF0\u0003[ ]\u0000\fF0\n\u0003[ ]: (A.9)\nIn fact, we rewrite Eq. (A.6) into\ne\u0000\fF\u0003=b[ <]+\fR\ndrh <=Z\nD >e\u0000\fF0\u0003[ ]\u0000\fF0\n\u0003[ ]\u0000\fR\ndrh >=Z[h]:(A.10)\nThen, we \fnd that \fF\u0003=b[ <] satis\fes\n\fF\u0003=b[ <] =\u0000lnZ[h] +\fZ\ndrh <: (A.11)\nThis relation means that the generating functional obtained from integrating\nover >gives the e\u000bective free energy for  <. Therefore, we can get theA.1. PERTURBATIVE RG EQUATION 85\n(a) 1-loop\n (b) 2-loop\nFigure A.1: \u0006\nFigure A.2: V\ne\u000bective free energy by calculating the sum of each vortex function,\n\fF\u0003=b[ <]\n=Z\nq\u000e(\fF\u0003=b[ <])\n\u000e <(q)\u000e <(\u0000q) <(q) <(\u0000q)\n+Z\nq1\u0001\u0001\u0001q4\u000e(\fF\u0003=b[ <])\n\u000e <(q1)\u000e <(q2)\u000e <(q3)\u000e <(q4) <(q1) <(q2) <(q3) <(q4)\n\u0002\u000e(q1+\u0001\u0001\u0001+q4) +\u0001\u0001\u0001; (A.12)\nwhere the 2- and 4- points vortex functions are given by\n\u000e(\fF\u0003=b[ <])\n\u000e <(q)\u000e <(\u0000q)=1\n2(r+q2) + \u0006(q); (A.13)\n\u000e(\fF\u0003=b[ <])\n\u000e <(q1)\u000e <(q2)\u000e <(q3)\u000e <(q4)=u+V(q); (A.14)\nrespectively. Here, \u0006 and Vare the 1-particle irreducible diagrams including\nthe internal momentum loops of  >. The lowest-loop contributions can be\nobtained as Figs. A.1 and A.2. The rules for calculating these diagrams are\nsummarized as follows:\n(i) Assign momentums for each of internal and external lines satisfying\nthe momentum conservation laws;\n(ii) Multiply the free propagator1\nr+q2for each of the internal lines;\n(iii) Multiply the free 4-point vortex function ufor each of the vertices\n(black dots);86 APPENDIX A. STATIC RG OF THE SCALAR FIELD THEORY\n(iv) Integrate over the momentum of the internal lines;\n(v) Multiply some numerical factor.1\nNext we carry out the rescaling of the vortex functions (A.13) and (A.14)\nand complete the RG transformation. Writing the vortex functions after one\nRG step as\n\u000e(\fF\u0003[ 0])\n\u000e 0(q0)\u000e 0(\u0000q0)=1\n2(r0+K0q02) +O(q04); (A.16)\n\u000e(\fF\u0003[ 0])\n\u000e 0(q0\n1)\u000e 0(q0\n2)\u000e 0(q0\n3)\u000e 0(q0\n4)=u0+O(q02); (A.17)\nwe obtain the parameters one step after the RG,\nr0=b2\u0014\nr+ 6uZ>\nq1\nr+q2+O(u2)\u0015\n; (A.18)\nK0= 1 +O(u2); (A.19)\nu0=b\u000f\u0014\nu\u000036u2Z>\nq1\n(r+q2)2\u0015\n: (A.20)\nHere, the overall factors b2andb\u000fcome from the rescaling. The short-hand\nnotation\nZ>\nq\u0011Z\u0003\n\u0003=bddq\n(2\u0019)d(A.21)\ndenotes the integration over the high momentum loop. We can carry out the\n1For the given diagram with Nvvertices, the numerical factor is given by\nNv!\nNsym\u0002NvY\ni=14!\nNi>!Ni<!; (A.15)\nwhereNi>andNi<are the number of the internal and the external lines connecting to\nthei-th vortex, and Nsymis the geometrical symmetry factor.A.1. PERTURBATIVE RG EQUATION 87\nintegrals in Eqs. (A.18) and (A.20) as\nZ>\nq1\nr+q2=Z>\nq1\nq21\n1 +r=q2\n'Z>\nq1\nq2\u0012\n1\u0000r\nq2\u0013\n=Z\u0003\n\u0003=bddq\n(2\u0019)d1\nq2\u0000rZ\u0003\n\u0003=bddq\n(2\u0019)d1\nq4\n=\u00032\u0000\u000f\n8\u00192\u0012\n1\u00001\nb2\u0013\n\u0000r\u0003\u0000\u000f\n8\u00192lnb; (A.22)\nZ>\nq1\n(r+q2)2'Z>\nq1\nq4\n=\u0003\u0000\u000f\n8\u00192lnb: (A.23)\nHere, we consider the system su\u000eciently near the phase transition point and\nsetr\u001c1. We also use the formulas (A.39) and (A.40). Then, we can write\nEqs. (A.18) and (A.20) into\nr0=b2\u0014\nr+ 6uKd\u00032\u0000\u000f\u0012\n1\u00001\nb2\u0000r\n\u00032lnb\u0013\u0015\n; (A.24)\nu0=b\u000f\u0000\nu\u000036u2Kd\u0003\u0000\u000flnb\u0001\n: (A.25)\nBy de\fning the dimension less parameters,\n\u0016r\u0011r\n\u00032;\u0016u\u0011Kdu\n\u0003\u000f; (A.26)\nwith\nKd\u00111\n8\u00192; (A.27)\nwe can rewrite these into\n\u0016r0=b2\u0014\n\u0016r+ 6\u0016u\u0012\n1\u00001\nb2\u0000\u0016rlnb\u0013\u0015\n; (A.28)\n\u0016u0=b\u000f\u0000\n\u0016u\u000036\u0016u2lnb\u0001\n: (A.29)88 APPENDIX A. STATIC RG OF THE SCALAR FIELD THEORY\nThese are the recursion relations which relate the parameters before and after\nthe RG procedure (at a single step).\nWe will next derive the di\u000berential equations equivalent to the recursion\nrelation. In order to do this, we consider the thin momentum shell with\n0<l\u001c1,\nb\u0011el'1 +l: (A.30)\nThen, Eqs. (A.28) and (A.29) reduce to\n\u0016r0= (1 + 2l) (\u0016r+ 24\u0016ul\u00006\u0016u\u0016rl) = \u0016r+ 24\u0016ul\u00006\u0016u\u0016rl+ 2\u0016rl+O(l2);(A.31)\n\u0016u0= (1 +\u000fl)\u0000\n\u0016u\u000036\u0016u2l\u0001\n= \u0016u\u000036\u0016u2l+\u000f\u0016ul: (A.32)\nBy writing\nd\u0016r\ndl\u0011lim\nl!0\u0016r0\u0000\u0016r\nl;d\u0016u\ndl\u0011lim\nl!0\u0016u0\u0000\u0016u\nl; (A.33)\nwe obtain the RG equations,\nd\u0016r\ndl= 2\u0016r+ 12\u0016u\u000012\u0016r\u0016u; (A.34)\nd\u0016u\ndl=\u000f\u0016u\u000036\u0016u2; (A.35)\nwhich give Eqs. (3.41) and (3.42).\nA.2 Wilson-Fisher \fxed point\nWe \frst assume randuare order of \u000fnear the critical point. Then we\ncan neglect the last term in the right-hand side of Eq. (A.34) and any other\nhigher-order terms coming from the perturbation expansion of u. Since the\ntypical length scale disappear at the critical point, the RG invariance emerges\nin the system. Therefore, we can set\nd\u0016r\ndl=d\u0016u\ndl= 0; (A.36)\nwhich yields two \fxed-point solutions: one is the Gaussian \fxed point:\n\u0016r= \u0016u= 0; (A.37)A.3. USEFUL INTERGRALS 89\nand the other is the so-called Wilson-Fisher \fxed point (we get Eq. (3.43)):\n\u0016r=\u0000\u000f\n6\u0016u=\u000f\n36: (A.38)\nThis solution is consistent with the assumption at the beginning of this sub-\nsection. From the liner stability analysis, the Gaussian \fxed point is unstable\nwith respect to \u0016 u, whereas the Wilson-Fisher \fxed point is stable [63].\nA.3 Useful intergrals\nZ\u0003\n\u0003=bddq\n(2\u0019)d1\nq2=1\n(2\u0019)dZ\nd\u0012Z\nd'1Z\nd'2:::Z\u0003\n\u0003=bdqqd\u00003sind\u00002\u0012sind\u00003'1:::\n=\u0003d\u00002\n(2\u0019)dZ\nd\u0012Z\nd'1Z\nd'2:::Z1\n1=bdxxd\u00003sind\u00002\u0012sind\u00003'1:::\n'\u00032\u0000\u000f\n(2\u0019)4Z\u0019\n0d\u0012sin2\u0012Z\u0019\n0d'1sin'1Z2\u0019\n0d'2Z1\n1=bdxx\n=\u00032\u0000\u000f\n8\u00192\u0012\n1\u00001\nb2\u0013\n; (A.39)\nZ\u0003\n\u0003=bddq\n(2\u0019)d1\nq4=1\n(2\u0019)dZ\nd\u0012Z\nd'1Z\nd'2:::Z\u0003\n\u0003=bdqqd\u00005sind\u00002\u0012sind\u00003'1:::\n=\u0003d\u00004\n(2\u0019)dZ\nd\u0012Z\nd'1Z\nd'2:::Z1\n1=bdxxd\u00003sind\u00002\u0012sind\u00003'1:::\n'\u0003\u0000\u000f\n(2\u0019)4Z\u0019\n0d\u0012sin2\u0012Z\u0019\n0d'1sin'1Z2\u0019\n0d'2Z1\n1=bdx1\nx\n=\u0003\u0000\u000f\n8\u00192lnb: (A.40)Appendix B\nCoupling to energy-momentum\ndensities\nIn Secs. 4.2 and 4.3, we have ignored the contributions of the energy and\nmomentum densities, \"and\u0019. In this appendix, we show that these con-\ntributions do not a\u000bect the static and dynamic universality classes of the\nhigh-density QCD critical point studied in Secs. 4.2 and 4.3. In particular,\nwe obtain the expression of the super\ruid phonon in the presence of \"and\n\u0019, Eq. (B.50), which will be mentioned in the discussion in Sec. 4.4.\nB.1 Statics\nLet us \frst consider the Ginzburg-Landau functional in terms of \u001b,nB,\",\n\u0019, and\u0012, up to the second-order expansion. Similarly to the argument\nin Sec. 4.2, the time reversal symmetry Tprohibits the mixing between\nxi\u0011\u001b;n B;\"and\u0019;\u0012:\nF[\u001b;n B;\";\u0019;\u0012] =F[\u001b;n B;\"] +F[\u0019;\u0012]: (B.1)\nThe presence of mqand\u0016Bgenerally allows the mixing among \u001b;n B;and\"\nin theT-even sector as\nF[\u001b;n B;\"] =1\n2Z\ndrxi\fij(r)xj; (B.2)\n\fij(r) =Vij\u0000vijr2: (B.3)\n9192APPENDIX B. COUPLING TO ENERGY-MOMENTUM DENSITIES\nHere, the subscripts i;jare the shorthand notations for xi;xj, respectively.\nTheT-odd sector is given by\nF[\u0019;\u0012] =1\n2Z\ndr\u0002\nV\u0019\u0019\u00192+ 2V\u0019\u0012\u0019\u0001r\u0012+V\u0012\u0012(r\u0012)2\u0003\n: (B.4)\nWe callVij,vij,V\u0019\u0019,V\u0019\u0012, andV\u0012\u0012the Ginzburg-Landau parameters.\nBy completing the square, Eq. (B.4) can be written as\nF[\u0019;\u0012] =1\n2Z\ndr\u0002\nV\u0019\u0019(\u00190)2+V0\n\u0012\u0012(r\u0012)2\u0003\n; (B.5)\nwhere\u00190\u0011\u0019+V\u0019\u0012r\u0012=V\u0019\u0019andV0\n\u0012\u0012\u0011V\u0012\u0012\u0000V2\n\u0019\u0012=V\u0019\u0019. We assume V0\n\u0012\u0012>0 and\nrede\fne\u00190andV0\n\u0012\u0012as\u0019andV\u0012\u0012for simplicity in the following caluculation.\nIn a similar manner to Eq. (4.3) when the \"and\u0019are absent, the corre-\nlation length \u0018is de\fned from the correlation function of \u001b. The form of the\ncorrelation function (4.5) can be generalized into\n\u0018\u0018(detV)\u00001\n2 (B.6)\nin the present case with Vbeing the 3\u00023 matrix in xi\u0011\u001b;n B;\"space. Thus,\nthe critical point is characterized by the condition, det V= 0, analogously to\nthe condition \u0001 = 0 in Eq. (4.5). At the critical point, only one of the linear\ncombinations of \u001b,nB, and\"becomes massless. This number of the gapless\nmode is the same as that in the absence of the \"and\u0019as we discussed around\nEq. (4.8), the static universality class remains the same as that of Sec. 4.2.\nWe de\fne the generalized susceptibilities,\n\u001fij\u0011\u000ehxiiXj\n\u000eXj\f\f\f\f\nXj=0; (B.7)\nwherehxiiXjis the the expectation value of xiin the same de\fnition of\nEq. (3.2) except for the replacement \fF!\fF+Z\ndrxjXj(where the\nsummation over the index jisnotimplied). Here, X\"\u0011\u0000\f,Xn\u0011\f\u0016B, and\nX\u001b\u0011\fmqwith\fdenotes the inverse temperature. Here and from now on,\nwe omit the subscript of nBwhennBis in the subscript of some character,\ne.g.,Xnrather than XnB. One can show the following relation between \u001fijB.2. DYNAMICS 93\nandVij\n\u001f\"\"=1\nVh\"2iq!0=T(V\u00001)\"\"; (B.8)\n\u001f\"n=1\nVh\"nBiq!0=T(V\u00001)\"n; (B.9)\n\u001fnn=1\nVhn2\nBiq!0=T(V\u00001)nn; (B.10)\nwhere (V\u00001)ijdenotes the ( i;j) component of the inverse matrix of V. Since\n(V\u00001)ij/(detV)\u00001, one \fnds\n\u001f\"\"\u0018\u001f\"n\u0018\u001fnn\u0018\u00182\u0000\u0011; (B.11)\nwhich can be regard as a generalization of Eq. (4.6).\nB.2 Dynamics\nB.2.1 Full Langevin equations\nWe next construct the Langevin equations of the full hydrodynamic variables\nxi=\u001b;n B;\"and\u0019;\u0012. The Langevin equations read\n@xi\n@t=\u0000\rij(r)\u000eF\n\u000exj\u0000Z\nV[~xi;\u0019]\u0001\u000eF\n\u000e\u0019\u0000Z\nV[~xi;\u0012]\u000eF\n\u000e\u0012+\u0018i; (B.12)\n@\u0019\n@t= \u0000\u0019\u0019rr\u0001\u000eF\n\u000e\u0019+ \u00000\n\u0019\u0019r2\u000eF\n\u000e\u0019\u0000\u0000\u0019\u0012r\u000eF\n\u000e\u0012\u0000Z\nV[\u0019;~xi]\u000eF\n\u000exi+\u0018\u0019;(B.13)\n@\u0012\n@t=\u0000\u0000\u0019\u0012r\u0001\u000eF\n\u000e\u0019\u0000\u0000\u0012\u0012\u000eF\n\u000e\u0012\u0000Z\nV[\u0012;~xi]\u000eF\n\u000exi+\u0018\u0012; (B.14)\nwhere we use the shorthand notation of the integrals, Eq. (4.12). Summation\nover repeated indices i;jis also understood. Here\n\rij(r)\u00110\n@\u0000\u001b\u001b\u0000\u0000\u001bnr2\u0000\u0000\u001b\"r2\n\u0000\u0000\u001bnr2\u0000\u0000nnr2\u0000\u0000n\"r2\n\u0000\u0000\u001b\"r2\u0000\u0000n\"r2\u0000\u0000\"\"r21\nA; (B.15)\n\u0000ab(a;b=xi;\u0019;\u0012), and \u00000\n\u0019\u0019are the kinetic coe\u000ecients. We also de\fne the\nnet variables ~ \u001b\u0011\u0016qq, ~nB\u0011\u0016q\r0q, and ~\"\u0011T00. On the other hand, we note\nagain that \u001b= ~\u001b\u0000\u001beq,nB= ~nB\u0000(nB)eq,\"\u0011~\"\u0000\"eqare the \ructuations94APPENDIX B. COUPLING TO ENERGY-MOMENTUM DENSITIES\naround the equilibrium values \u001beq, (nB)eq, and\"eq. The noise terms \u0018i;\u0018\u0019;\nand\u0018\u0012above are not important in the following calculation. In order to write\ndown the above Langevin equations, we use the momentum conservation law\nand the Onsager's principle, similarly to the derivation in Sec. 3.3.2.\nWe postulate the following Poisson brackets, in addition to Eqs. (4.17),\n[\u0019(r);~\u001b(r0)] = ~\u001b(r0)r\u000e(r\u0000r0); (B.16)\n[\u0019(r);~nB(r0)] = ~nB(r0)r\u000e(r\u0000r0); (B.17)\n[\u0019(r);~s(r0)] = ~s(r0)r\u000e(r\u0000r0); (B.18)\n[\u0012(r);~s(r0)] = 0; (B.19)\nwhere ~s(r) denotes the entropy density. The Poisson brackets concerning\n~\"(r) can be derived as\n[\u0019(r);~\"(r0)] = (T~s(r) +\u0016B~nB(r))r\u000e(r\u0000r0); (B.20)\n[\u0012(r);~\"(r0)] =\u0016B\u000e(r\u0000r0): (B.21)\nHere, we used the thermodynamic relation d \"=Tds+\u0016BdnBand the de\f-\nnition of the Poisson brackets [ \u0019(r);~yi(r0)]\u0011\u000e~yi(r0)=\u000eu(r) for ~yi\u0011~nB;~s;~\",\nwithu(r) being the in\fnitesimal translation of the coordinate, r!r+u(r).\n(See, e.g., Ref. [78] for the details of the Poisson brackets.)\nWe consider the small \ructuations of variables around the equilibrium\nvalues,\u001b(r) = ~\u001b(r)\u0000\u001beq,nB(r) = ~nB(r)\u0000(nB)eq, ands(r) = ~s(r)\u0000\nseq. Then, the linearized Langevin equations neglecting the higher order\n\ructuations become:\n@\u001b\n@t=\u0000\u0000\n\u0000\u001b\u001bV\u001bi\u0000\u0000\u001bjvjir2\u0001\nxi\u0000\u001beqV\u0019\u0019r\u0001\u0019; (B.22)\n@nB\n@t= \u0000njvjir2xi\u0000(nB)eqV\u0019\u0019r\u0001\u0019\u0000V\u0012\u0012r2\u0012; (B.23)\n@\"\n@t= \u0000\"jvjir2xi\u0000weqV\u0012\u0012r\u0001\u0019\u0000\u0016V\u0012\u0012r2\u0012; (B.24)\n@\u0019\n@t=\u0000[\u001beqV\u001bi+ (nB)eqVni+weqV\"i]rxi+ \u0000\u0019\u0019V\u0019\u0019rr\u0001\u0019+ \u00000\n\u0019\u0019V\u0019\u0019r2\u0019;\n(B.25)\n@\u0012\n@t=\u0000\u0002\n(Vni+\u0016BV\"i)\u0000(vn+\u0016Bv\"i)r2\u0003\nxi\u0000\u0000\u0019\u0012V\u0019\u0019r\u0001\u0019+ \u0000\u0012\u0012V\u0012\u0012r2\u0012;\n(B.26)\nwhereweq=Tseq+\u0016B(nB)eq.B.2. DYNAMICS 95\nB.2.2 Decomposition of momentum density\nIn this section, we decompose the momentum density \u0019iinto the longitudinal\nand transverse parts with respect to momentum qin (t;q) space,\n\u0019i=\u0019i\nL+\u0019i\nT; \u0019i\nL= (PL)ij\u0019j; \u0019i\nT= (PT)ij\u0019j; (B.27)\nwherePL;Tare the longitudinal and transverse projection operators de\fned\nby\n(PL)ij\u0011qiqj\njqj2;(PT)ij\u0011\u000eij\u0000qiqj\njqj2: (B.28)\nIn the following calculation, we show that the dynamics of \u0019i\nTis decoupled\nfrom that of the other hydrodynamic variables. We \frst write the linearized\nLangevin equations (B.22){(B.26) to the leading order of q, formally as\n@xk\n@t=Axk(q\u0001\u0019) +f(xk;\u0012); (B.29)\n@\u0019\n@t=A\u0019q(q\u0001\u0019) +B\u0019jqj2\u0019+qg(xk); (B.30)\n@\u0012\n@t=A\u0012(q\u0001\u0019) +h(xk;\u0012); (B.31)\nwheref(xk;\u0012),g(xk) andh(xk;\u0012) are some functions which may involve\nxkand\u0012, but not involve \u0019. The coe\u000ecients Axk,A\u0019,A\u0012, andB\u0019are\nthe parameters which depend on the Ginzburg-Landau parameters Vij; vij,\nkinetic coe\u000ecients, and equilibrium values of the thermodynamic quantities.\nThe explicit forms of these functions ( f,g, andh) and the coe\u000ecients ( Axk,\nA\u0019,A\u0012, andB\u0019) are not important for our purpose. By using \u0019L;T, we can\nwrite Eqs. (B.29){(B.31) into\n@xk\n@t=Axk(q\u0001\u0019L) +f(xk;\u0012); (B.32)\n@\u0019L\n@t= (A\u0019+B\u0019)jqj2\u0019L+qg(xk); (B.33)\n@\u0019T\n@t=B\u0019jqj2\u0019T; (B.34)\n@\u0012\n@t=A\u0012(q\u0001\u0019L) +h(xk;\u0012): (B.35)\nOne can con\frm that the dynamics of \u0019i\nLand\u0019i\nTare decoupled from each\nother at the mean-\feld level.96APPENDIX B. COUPLING TO ENERGY-MOMENTUM DENSITIES\nB.2.3 Hydrodynamic modes\nThanks to the decomposition in the previous section, we can readily obtain\nthe hydrodynamic mode for \u0019T,\n\u0000\ni!\u0000B\u0019q2\u0001\n\u0019T= 0; B\u0019= \u00000\n\u0019\u0019V\u0019\u0019; (B.36)\nwhich is a di\u000busion mode.\nThe other Langevin equations which involve \u0019L(but not\u0019T) can be\nexpressed in the form of the matrix equation,B.2. DYNAMICS 97\nMEM0\nBBBB@\u001b\nnB\n\"\n\u0019L\n\u00121\nCCCCA= 0; (B.37)\nwhere\nMEM\n\u00110\nBBBB@i!\u0000A\u001b\u001b\u0000a\u001b\u001bq2\u0000A\u001bn\u0000a\u001bnq2\u0000A\u001b\"\u0000a\u001b\"q2\u0000ia\u001b\u0019q 0\n\u0000an\u001bq2i!\u0000annq2\u0000an\"q2\u0000ian\u0019q\u0000an\u0012q2\n\u0000a\"\u001bq2\u0000a\"nq2i!\u0000a\"\"q2\u0000ia\"\u0019q\u0000a\"\u0012q2\n\u0000ia\u0019\u001bq\u0000ia\u0019nq\u0000ia\u0019\"qi!\u0000a\u0019\u0019q20\n\u0000A\u0012\u001b\u0000a\u0012\u001bq2\u0000A\u0012n\u0000a\u0012nq2\u0000A\u0012\"\u0000a\u0012\"q2\u0000ia\u0012\u0019qi!\u0000a\u0012\u0012q21\nCCCCA:\n(B.38)\nHereAabandaabare the parameters depending on Vij; vij, kinetic coe\u000ecients,\nand thermodynamic quantities. Note here that Aab; aabare not symmetric\nwith respect to aandb. The eigenfrequencies of Eq. (B.37) can be found\nfrom detM= 0, which yields\n!5+ i\u0000\nA\u001b\u001b+g4q2\u0001\n!4\u0000\u0002\ng3q2+O(q4)\u0003\n!3\u0000i\u0002\ng2q2+O(q4)\u0003\n!2\n+\u0002\ng1q4+O(q6)\u0003\n!+ i\u0002\ng0q4+O(q6)\u0003\n= 0;(B.39)\nwhereg0;:::;g 4are some functions of Aabandaab. Among others, we give\nthe explicit expressions only for A\u001b\u001b,g2, andg0which will be used in the\nfollowing arguments,\nA\u001b\u001b\u0011\u0000\u001b\u001b; (B.40)\ng2\u0011\u0000\u001b\u001b\u0002\n(nB)2\neqV\u0019\u0019+V\u0012\u0012\u0003\f\f\f\fV\u001b\u001bV\u001bn\nV\u001bnVnn\f\f\f\f\n+ 2\u0000\u001b\u001b[(nB)eqweqV\u0019\u0019+\u0016BV\u0012\u0012]\f\f\f\fV\u001b\u001bV\u001bn\nV\u001b\"Vn\"\f\f\f\f\n+ \u0000\u001b\u001b(w2\neqV\u0019\u0019+\u00162\nBV\u0012\u0012)\f\f\f\fV\u001b\u001bV\u001b\"\nV\u001b\"V\"\"\f\f\f\f; (B.41)\ng0\u0011\u0000\u001b\u001bV\u0019\u0019V\u0012\u0012T2s2\neqdetV: (B.42)98APPENDIX B. COUPLING TO ENERGY-MOMENTUM DENSITIES\nEquation (B.39) can be factorized as\n\u0014\n!+ iA\u001b\u001b+ i\u0012\ng4\u0000g3\nA\u001b\u001b+g2\nA2\n\u001b\u001b\u0013\nq2\u0015\n\u0002\u0014\n!\u0000s+jqj+ it+\n2q2\u0015\u0014\n!+s+jqj+ it+\n2q2\u0015\n\u0002\u0014\n!\u0000s\u0000jqj+ it\u0000\n2q2\u0015\u0014\n!+s\u0000jqj+ it\u0000\n2q2\u0015\n= 0; (B.43)\nwheres\u0006andt\u0006satisfy\ns2\n++s2\n\u0000=g2\nA\u001b\u001b; (B.44)\ns2\n+s2\n\u0000=g0\nA\u001b\u001b; (B.45)\nt++t\u0000=g3\nA\u001b\u001b\u0000g2\nA2\n\u001b\u001b; (B.46)\ns2\n+t\u0000+s2\n\u0000t+=g1\nA\u001b\u001b\u0000g0\nA2\n\u001b\u001b: (B.47)\nFrom Eq. (B.43), we \fnds the hydrodynamic modes of the system other\nthan the di\u000busion mode described by Eq. (B.36): one relaxation mode and\ntwo pairs of phonons. The speeds of phonons s\u0006can be obtained from the\nsolution of\ns4\u0000g2\nA\u001b\u001bs2+g0\nA\u001b\u001b= 0: (B.48)\nNote here that Eq. (B.42) shows that g0!0 when the critical point is\napproached det V!0. Therefore, near the critical point, we obtain\ns2\n+=g2\nA\u001b\u001b\u0000g0\ng2; s2\n\u0000=g0\ng2: (B.49)\nB.2.4 Dynamic critical exponent\nFrom the results above, we can show that one of the phonons with the speed\ncs\u0011s\u0000exhibits the critical slowing down. By using Eqs. (B.41) and (B.42),\ntogether with Eqs. (B.8){(B.10), we have\nc2\ns=V\u0019\u0019V\u0012\u0012T3s2\neq\n\u0014nn\u001fnn+ 2\u0014n\"\u001fn\"+\u0014\"\"\u001f\"\"; (B.50)B.2. DYNAMICS 99\nwhere\n\u0014nn\u0011(nB)2\neqV\u0019\u0019+V\u0012\u0012; (B.51)\n\u0014n\"\u0011(nB)eqweqV\u0019\u0019+\u0016BV\u0012\u0012; (B.52)\n\u0014\"\"\u0011w2\neqV\u0019\u0019+\u00162\nBV\u0012\u0012: (B.53)\nUsing Eq. (B.11), V\u0019\u0019\u0018\u00180, andV\u0012\u0012\u0018\u00180near the critical point, we obtain\nc2\ns\u0018\u0018\u00002+\u0011: (B.54)\nComparing Eq. (B.54) with Eq. (4.35), we obtain the same dynamic critical\nexponentzas that given by Eq. (4.37). Therefore, the dynamic universality\nclass remains the same as that without \"and\u0019in Sec. 3.3.2.Appendix C\nCalculation of the self-energies\nand the vertex function\nAs we mentioned under Eq. (5.72), in this appendix, we evaluate the self-\nenergies of the order parameter, \u0006 \u000b\f(k;!), and those of the conserved charge\ndensities, \u0005 ij(k;!), to the \frst order of \u000f, namely at the one-loop level.\nFrom these expressions, we can derive the recursion relations for the dynamic\nparameters, Eqs. (5.75){(5.78). In particular, we calculate \u0005 ijand derive\nEqs. (5.76){(5.78) in Sec. C.1; we calculate \u0006 \u000b\u000band derive Eq. (5.75) in\nSec. C.2. We also evaluate the three-point vertex function used in Eq. (5.70)\nat the one-loop level, and derive Eq. (5.79).\nC.1 Self-energy \u0005\nLet us begin with the self-energy for the conserved-charge densities, \u0005 ij(k;!),\nwhich corresponds to the diagrams with outgoing ~ niand ingoing nj. The\nleading diagram is given by Fig. C.1. According to the Feynman rules on the\nanalytic translation of the diagrams of the (plane and wavy) lines and the\nnoise vortices (Fig. 5.1) and the interaction vortices (Fig. 5.2), summarized\nin Sec. 5.2.2, we can calculate Fig. C.1 as\n\u0005ij(k;!) =Zd!0\n2\u0019Zddp\n(2\u0019)dV0\ni;\u000b\f(k;p)G0(k\u0000p;!\u0000!0)jG0(p;!0)j22\u0000V0\n\f;\u000bj\n=Zddp\n(2\u0019)dFij\nf\u0000i!+ \u0000[r+ (k\u0000p)2] + \u0000(r+p2)g(r+p2);(C.1)\n101102 APPENDIX C. CALCULATION ON \u0006,\u0005, ANDV\nkω\nkω−p−ω′\npω′k−p\nω−ω′\nβ\nαα ij\nFigure C.1: Diagram for \u0005 ijat one-loop level\nwhere we de\fned Fijas the components of the following matrix:\nF\u0011\u00128\r2\u0000\u0015k20\n8i\r2C\u0000B\u0001k\u00002g2[(k\u0000p)2\u0000p2]=\u001f5\u0013\n: (C.2)\nFrom this expression of the self energy \u0005 ij, we can calculate its derivatives\nwhich are required in the recursion relations (5.66){(5.69) and the normal-\nization conditions (5.71) and (5.72).\nIt is easy to calculate the latter conditions. Because \u0005 ij(0;!) = 0, the\nnormalization conditions (5.71) and (5.72) reduce to\n~c=d\u0000c;~c5=d\u0000c5: (C.3)\nLet us calculate the derivatives needed in the recursion relations (5.66){\n(5.69),\ni@\u000521(k;0)\n@k\f\f\f\f\nk!0=\u00004\r2CBZddp\n(2\u0019)d\u00121\np4+O(\u000f)\u0013\n=\u0000\r2CB\u0003\u0000\u000f\n2\u00192lnb;\n(C.4)\n1\n2@2\u000511(k;0)\n@k2\f\f\f\f\nk!0= 4\r2\u0015Zddp\n(2\u0019)d\u00121\np4+O(\u000f)\u0013\n=\r2\u0015\u0003\u0000\u000f\n2\u00192lnb; (C.5)\nwhere we carried out the integral over pin the shell \u0003 =b<jpj<\u0003 using the\nstandard formula in the dimensional regularization:\nZddp\n(2\u0019)d1\np4=\u0003\u0000\u000f\n8\u00192lnb: (C.6)C.2. SELF-ENERGY \u0006 103\nWe also compute\n1\n2@2\u000522(k;0)\n@k2\f\f\f\f\nk!0=g2\n\u001f5\u0000Zddp\n(2\u0019)d\u0012cos 2\u0012\np4+O(\u000f)\u0013\n=\u0000g2\u0003\u0000\u000f\n16\u00192\u001f5\u0000lnb;\n(C.7)\nwhere we parameterize pin the limit of d!4 by\n(p1;p2;p3;p4) =p(cos\u0012;sin\u0012cos\u001e;sin\u0012sin\u001ecos';sin\u0012sin\u001esin') (C.8)\nwithk= (1;0;0;0) and use\nZddp\n(2\u0019)dcos 2\u0012\np4=\u0000\u0003\u0000\u000f\n16\u00192lnb: (C.9)\nNow it ie ready to rewrite the recursion relations (5.66){(5.69). By using\nEqs. (C.3){(C.5), and (C.7), we obtain\n\u0015l+1\n\u001fl+1=bz\u00002\u0015l\n\u001fl(1\u00004vllnb); (C.10)\n(\u00155)l+1\n(\u001f5)l+1=bz\u00002(\u00155)l\n(\u001f5)l\u0012\n1 +fl\n2lnb\u0013\n; (C.11)\nCl+1\n\u001fl+1=bz+c5\u0000c\u00001Cl\n\u001fl(1\u00004vllnb); (C.12)\nCl+1\n(\u001f5)l+1=bz+c\u0000c5\u00001Cl\n(\u001f5)l: (C.13)\nThen, we can derive Eqs. (5.76){(5.78) by using Eqs. (5.46), (5.47), and\n(C.10){(C.12).\nC.2 Self-energy \u0006\nNext, let us evaluate the self-energy of the order parameter \feld \u001e\u000b, \u0006\u000b\f(k;!),\nwhose diagrams are shown in Fig. C.2,\n\u0006\u000b\f(k;!) = \u0006(a)\n\u000b\f(k;!) + \u0006(b)\n\u000b\f(k;!); (C.14)\nwhere \u0006(a)and \u0006(b)correspond to \fgures C.2(a) and C.2(b), respectively.\nSimilarly to the expression for Eq. (C.1), the diagonal ( \u000b\u000b) components104 APPENDIX C. CALCULATION ON \u0006,\u0005, ANDV\nkω β\nkω α−p−ω′\npω′k−p\nω−ω′\ni\nγγj\n(a)\nkω β\nkω α−p−ω′\npω′k−p\nω−ω′\ni\nlγj\nk (b)\nFigure C.2: Diagrams for \u0006 \u000b\fat one-loop level\nused in the recursion relation (5.64) and the normalization condition (5.65)\nare given by\n\u0006(a)\n\u000b\u000b(k;!)\n=Zd!0\n2\u0019Zddp\n(2\u0019)dV0\n\u000b;\riD0\nij(k\u0000p;!\u0000!0)jG0(p;!0)j22\u0000Vj;\r\u000b(k\u0000p;\u0000p)\n= 4\r2\u0000Zddp\n(2\u0019)d\u0015(k\u0000p)2[\u0000i!+ \u0000(r+p)2+\u00155(k\u0000p)2=\u001f5] +C2[B\u0001(k\u0000p)]2=\u001f5\n(r+p2) det[D0(k\u0000p;!+ i\u0000(r+p2))]\u00001\n+g2\n\u001f5Zddp\n(2\u0019)d(p2\u0000k2)[\u0000i!+ \u0000(r+p)2+\u0015(k\u0000p)2=\u001f]\n(r+p2) det[D0(k\u0000p;!+ i\u0000(r+p2))]\u00001; (C.15)\n\u0006(b)\n\u000b\u000b(k;!)\n=Zd!0\n2\u0019Zddp\n(2\u0019)dV0\n\u000b;\riG0(k\u0000p;!\u0000!0)B0\nij(p;!0)V\r;\u000bj\n= 8\r2\u00002Zd!0\n2\u0019Zddp\n(2\u0019)d\u0015p2(!02+\u00145p2) +\u00155(CB\u0001p)2=\u001f2\n5\njdet[D0(p;!0)]\u00001j2f\u0000i(!\u0000!0) + \u0000[r+ (k\u0000p)2]g\n\u00002g2\n\u001f2\n5Zd!0\n2\u0019Zddp\n(2\u0019)d\u00155p2(!02+\u0014p2) +\u0015(CB\u0001p)2=\u001f2\njdet[D0(p;!0)]\u00001j2f\u0000i(!\u0000!0) + \u0000[r+ (k\u0000p)2]g:\n(C.16)\nNote that we do not take a sum over \u000bhere.C.2. SELF-ENERGY \u0006 105\nThe integral over !0in Eq. (C.16) can be performed by closing contour\nbelow and picking up the poles on a lower half plane,\ndet[D0(p;!0)]\u00001=\u0002\n\u0000i!0+\u0014+p2+ i\n(p)\u0003\u0002\n\u0000i!0+\u0014+p2\u0000i\n(p)\u0003\n= 0;\n(C.17)\nwhere we de\fned\n\n(p)\u0011s\n(CB\u0001p)2\n\u001f\u001f5\u0000\u00142\n\u0000p4; \u0014\u0006\u00111\n2\u0012\u0015\n\u001f\u0006\u00155\n\u001f5\u0013\n: (C.18)\nThe result of the contour integral is given by\n\u0006(b)\n\u000b\u000b(k;!) =\u00002\r2\u00002Zddp\n(2\u0019)d\u0012\u0000\u0015\u0014\u0000p4+ (CB\u0001p)2=\u001f5\ni\n(p)K\u0000(p;k)\u0000\u0015p2K+(p;k)\u0013\n+g2\n2\u001f2\n5Zddp\n(2\u0019)d\u0012\u00155\u0014\u0000p4+ (CB\u0001p)2=\u001f\ni\n(p)K\u0000(p;k)\u0000\u00155p2K+(p;k)\u0013\n;\n(C.19)\nwhere we introduced\nK\u0006(p;k)\u00111\n[\u0014+p2+ i\n(p)][\u0000i!+ \u0001 +(p;k) + i\n(p)]\n\u00061\n[\u0014+p2\u0000i\n(p)][\u0000i!+ \u0001 +(p;k)\u0000i\n(p)]; (C.20)\n\u0001+(p;k)\u0011\u0000[r+ (k\u0000p)2] +\u0014+p2: (C.21)\nBy using an identity,\ndet[D0(p;!+ i\u0000[r+ (k\u0000p)2])]\u00001= [\u0000i!+ \u0001 +(p;k)]2+ \n(p)2;(C.22)\ntogether with some straightforward calculations, we obtain\n\u0006(b)\n\u000b\u000b(k;!) = 4\r2\u001f\u00002Zddp\n(2\u0019)d\u0000i!+\u00155p2=\u001f5+ \u0000[r+ (k\u0000p)2]\ndet[D0(p;!+ i\u0000[r+ (k\u0000p)2])]\u00001\n\u0000g2\n\u001f5Zddp\n(2\u0019)d\u0000i!+\u0015p2=\u001f+ \u0000[r+ (k\u0000p)2]\ndet[D0(p;!+ i\u0000[r+ (k\u0000p)2])]\u00001: (C.23)106 APPENDIX C. CALCULATION ON \u0006,\u0005, ANDV\nCombining Eqs. (C.15) and (C.23), we \fnd\n\u0006\u000b\u000b(k;!) = 4\r2\u001f\u0000Zddp\n(2\u0019)d\u0002(p;k;0)\u0002 5(p;k;!) + [CB\u0001(k\u0000p)]2=(\u001f\u001f5)\n(r+p2)[\u0002(p;k;!)\u00025(p;k;!) + (CB\u0001p)2=(\u001f\u001f5)]\n\u0000g2\n\u001f5Zddp\n(2\u0019)d(r+k2)\u0002(p;k;!)\n(r+p2)[\u0002(p;k;!)\u00025(p;k;!) + (CB\u0001p)2=(\u001f\u001f5)];\n(C.24)\nwhere \u0002(p;k;!) and \u0002 5(p;k;!) are de\fned by\n\u0002(p;k;!)\u0011\u0000i!+ \u0000(r+p2) +\u0015(k\u0000p)2=\u001f; (C.25)\n\u00025(p;k;!)\u0011\u0000i!+ \u0000(r+p2) +\u00155(k\u0000p)2=\u001f5: (C.26)\nNote that \u0006 11= \u0006 22and the \frst term on the right-hand side of Eq. (C.24)\nis independent of kwhen!= 0. From the expression (C.24), we obtain\n1\n2@2\u0006\u000b\u000b(k;0)\n@k2\f\f\f\f\nk!0\n=\u0000g2\n\u001f5Zddp\n(2\u0019)d\u0012\u0002(p;0;0)\n(r+p2)[\u0002(p;0;0)\u0002 5(p;0;0) + (CB\u0001p)2=(\u001f\u001f5)]+O(\u000f)\u0013\n;\n(C.27)\ni@\u0006\u000b\u000b(0;!)\n@!\f\f\f\f\n!!0\n=\u00004\r2\u001f\u0000Zddp\n(2\u0019)d\u00025(p;0;0)\n(r+p2)f\u0002(p;0;0)\u0002 5(p;0;0) + (CB\u0001p)2=(\u001f\u001f5)g\n+g2\n\u001f5Zddp\n(2\u0019)dr[\u00022(p;0;0)\u0000(CB\u0001p)2=(\u001f\u001f5)2]\n(r+p2)[\u0002(p;0;0)\u0002 5(p;0;0) + (CB\u0001p)2=(\u001f\u001f5)2]2:\n(C.28)\nTheO(\u000f) terms in Eq. (C.27) proportional to rare irrelevant for the following\ndiscussion. Let us carry out the integral over pin the shell \u0003 =b <jpj<\u0003.\nSettingB= (1;0;0;0) and using the parameterization (C.8), we obtain\n1\n2\u0000@2\u0006\u000b\u000b(k;0)\n@k2\f\f\f\f\nk!0\n=\u00002fp\n(1 +w)(1 +w5) +h2+p\n(1 +w)(1 +w5)r\n1 +w\n1 +w5lnb; (C.29)C.3. VERTEX FUNCTION V 107\nwherew; w 5; h; f are de\fned in Eq. (5.73). In order to obtain Eq. (C.29),\nwe carry out the integral over \u0012by using the following formula (with abeing\na real constant):\nZ\u0019\n0d\u0012sin2\u0012\na2+ cos2\u0012=\u0019\na(p\na2+ 1 +a2): (C.30)\nTo the leading order of \u000f, we can ignore the term proportional to ron the\nright-hand side of (C.28). Then, by comparing Eqs. (C.27) and (C.28), we\ncan rewrite Eq. (C.28) as\ni@\u0006\u000b\u000b(0;!)\n@!\f\f\f\f\n!!0\n=\u00008vwp\n(1 +w)(1 +w5) +h2+p\n(1 +w)(1 +w5)r\n1 +w5\n1 +wlnb; (C.31)\nwherevis de\fned in Eq. (5.49). Using Eqs. (C.29) and (C.31), Eqs. (5.64)\nand (5.65) become\n\u0000l+1= \u0000l(1 +flXllnb)bd+z\u0000~a\u0000a\u00002; (C.32)\n1 = (1 + 4vlwlX0\nllnb)bd\u0000~a\u0000a; (C.33)\nrespectively, where XandX0are de\fned in Eq. (5.74). By substituting\nEq. (C.33) into Eq. (C.32), we \fnally arrive at Eq. (5.75).\nC.3 Vertex function V\nThe lowest-order diagrams for V\u000b;\fiare depicted in Fig. C.3. We \fnd that\nthese diagrams with \u000b6=\fandi= 2 satisfy the following identity in the\nlimitk1;k2!0and!1;!2!0:\nV\u000b;\f2(0;0;0;0) =\u0000ig\"\u000b\f\n\u001f5@\u0006\r\r(0;!)\n@!\f\f\f\f\n!!0: (C.34)\nNote that the contribution from Fig. C.3(d) vanishes as long as \u000b6=\f.\nBy substituting Eqs. (C.34), (5.47), and (5.65) into Eq. (5.70), we obtain\nEq. (5.79). In particular, the loop correction of the interaction vortex,\nEq. (C.34), is canceled by the response-\feld scaling factor which includes\nthe loop correction originating from Eq. (5.65).108 APPENDIX C. CALCULATION ON \u0006,\u0005, ANDV\nk1ω1β\nk3ω3α\n−p−ω′pω′k3−p\nω3−ω′\nj\nγγk\nk2ω2iδ\n(a)\nk1ω1β\nk3ω3α−p−ω′\nk2+pk1−p\nω1−ω′\nj\nγ δk\nk2ω2\niδ\nω2+ω′ (b)\nk1ω1βk3ω3α−p−ω′pω′\nk3−p\nω3−ω′j\nγδm\nk2ω2ikl\n(c)\nk2ω2\nk3ω3α −p−ω′\npω′γi k1ω1β\nγ δ (d)\nFigure C.3: Diagrams for V\u000b;\fiat one-loop level\nInstead of showing Eq. (C.34) from the explicit calculation as we car-\nried out in Secs. C.1 and C.2, we show Eq. (C.34) as a consequence of the\nsymmetry algebra. Our derivation is similar to that of Ref. [79] (see its sec-\ntion III. A), where the Ward-Takahashi identity for O( N) symmetric systems\nis derived. We shall begin with the generating functional,\nZ[~|;j;~\u00165;\u00165]\u0011\u001c\nexpZ\ndtZ\ndr\u0010\n~|\u000b~\u001e\u000b+j\u000b\u001e\u000b+ ~\u00165~n5+\u00165n5\u0011\u001d\n:(C.35)\nHere, ~|\u000b; j\u000b;~\u00165;and\u00165are the external \felds of ~\u001e\u000b; \u001e\u000b;~n5andn5, respec-\ntively. We de\fne\n~\b\u000b\u0011\u000elnZ\n\u000e~|\u000b;\b\u000b\u0011\u000elnZ\n\u000ej\u000b;~N5\u0011\u000elnZ\n\u000e~\u00165; N 5\u0011\u000elnZ\n\u000e\u00165; (C.36)\nwhich reduce to the expectation values of ~\u001e\u000b; \u001e\u000b;~n5andn5, respectively,\nwhen the external \felds are set to zero. We also de\fne the e\u000bective actionsC.3. VERTEX FUNCTION V 109\nby the Legendre transformations of Eq. (C.36):\nW[~\b;\b;~\u00165;\u00165]\u0011\u0000lnZ[~|;j;~\u00165;\u00165] +Z\ndtZ\ndr\u0010\n~|\u000b~\b\u000b+j\u000b\b\u000b\u0011\n;(C.37)\n\u0000[~\b;\b;~N5;N5]\u0011W[~\b;\b;~\u00165;\u00165] +Z\ndtZ\ndr\u0010\n~\u00165~N5+\u00165N5\u0011\n: (C.38)\nOne can show the following identities which will be used later (see, e.g.,\nSec. 4. 4 of Ref. [66] for the derivations):\n\u000eW\n\u000e\b\u000b=\u000e\u0000\n\u000e\b\u000b=j\u000b;\u000e\u0000\n\u000eN5=\u00165; (C.39)\n\u000e2\u0000\n\u000e~\b\u000b(k;!)\u000e\b\f(k;!)=G\u00001\n\u000b\f(\u0000k;\u0000!); (C.40)\n\u000e2\u0000\n\u000e\b\u000b\u000e\b\f=\u000e2\u0000\n\u000e\b\u000b\u000eN5= 0; (C.41)\n\u000e3\u0000\n\u000eN5\u000e~\b\u000b\u000e\b\f=V\u000b;\f2: (C.42)\nHere, note that V\u000b;\f2in Eq. (C.42) satis\fes Eq. (5.39).\nWe use the condition that W[~\b;\b;~\u00165;\u00165] is invariant under the U(1)\u001c3\nA\ntransformation de\fned in Eq. (5.6). Note that W[~\b;\b;~\u00165;\u00165] is a functional\nof\u00165, and changing \u00165corresponds to the U(1)\u001c3\nAtransformation, because n5\nis its generator. Let us consider the variation of \u00165att= 0,\u000e\u00165=#\u00165with\n#being a small parameter. Then, the free energy (5.15) has an additional\ncontribution,\n\u000eF=Z\ndrn5\u000e\u00165: (C.43)\nBy using the reversible term of Eq. (5.12), an in\fnitesimal U(1)\u001c3\nAtransfor-\nmation can be written as\n\u000e\b\u000b=gZt\n0dt0\"\u000b\f\b\f\u000e\u00165=#g\"\u000b\f\b\f\u00165t: (C.44)\nBy applying Eq. (C.44) to W[~\b;\b;~\u00165;\u00165], we obtain its variation as\n\u000eW=#Z\ndtZ\ndr\u00165\u0014\u000eW\n\u000e\u00165+g\"\u000b\ft\u000eW\n\u000e\b\u000b\b\f\u0015\n(C.45)\n=#Z\ndtZ\ndr\u000e\u0000\n\u000eN5\u0014\nN5+g\"\u000b\ft\u000e\u0000\n\u000e\b\u000b\b\f\u0015\n; (C.46)110 APPENDIX C. CALCULATION ON \u0006,\u0005, ANDV\nwhere we use Eq. (C.39). Therefore, invariance of W[~\b;\b;~\u00165;\u00165] leads to\n\u000e\u0000\n\u000eN5\u0014\nN5+g\"\u000b\ft\u000e\u0000\n\u000e\b\u000b\b\f\u0015\n= 0: (C.47)\nBy taking a variation of this equation with respect to ~\b\u000band \b\f, we obtain\n\u000e3\u0000\n\u000eN5\u000e~\b\u000b\u000e\b\fN5=\u0000g\"\r\ft \n\u00165\u000e2\u0000\n\u000e~\b\u000b\u000e\b\r+j\r\u000e2\u0000\n\u000eN5\u000e~\b\u000b!\n; (C.48)\nwhere we use Eqs. (C.39) and (C.41). We take the functional derivative\nof Eq. (C.48) with respect to \u00165and set all the external \felds to zero. In\nfrequency space, we obtain,\n\u000e3\u0000\n\u000eN5\u000e~\b\u000b\u000e\b\f=\u0000ig\"\r\f\n\u001f5@\n@!\u000e2\u0000\n\u000e~\b\u000b\u000e\b\r\f\f\f\f\f\n!!0=g\"\r\f\n\u001f5\u0012\n1\u0000i@\u0006\u000b\r(0;!)\n@!\f\f\f\f\n!!0\u0013\n:\n(C.49)\nHere, we use \u001f5=\u000eN5=\u000e\u0016 5and Eqs. (C.40) and (5.37). By using Eqs. (C.42),\n(5.35) and (5.39), we \fnally arrive at Eq. (C.34) when \u000b6=\f. Note here that\none can write \"\r\f\u0006\u000b\r=\"\u000b\f\u000611(for\u000b6=\f) by using \u0006 11= \u0006 22, which can\nbe shown from Eq. (C.24).Appendix D\nLinear-stability analysis on the\n\fxed points (iii) and (iv)\nWe here show that the parameter region that leads to the \fxed point (iv)\nunder RG is much larger than (iii) within the linear-stability analysis around\nthe \fxed point (i). Let us look at the full propagator of the conserved \felds,\nDij=\u0016Dij(w;h) as a function of two relevant parameters wandh, which is\ntransformed under the rescaling as\n\u0016Dij(w;h)\u0018\u0016Dij(lyww;lyhh): (D.1)\nHere, we assume that wandhare scaled by the eigenvalues yw\u00117\u000f=10; yh\u0011\n1\u0000\u000f=4 obtained in Eq. (5.97), and we omit the overall factor. When the RG\n\row approaches l\u0018h\u00001=yh,\u0016Dijbecomes a function of wh\u0000'with'\u0011yw=yh\nbeing the crossover exponent. Then, \u0016Dijbehaves di\u000berently depending on\nwhetherwh\u0000'\u001d1 orwh\u0000'\u001c1: in the former region, we see the behavior of\nthe \fxed point (iii); in the latter region, we see the behavior of the \fxed point\n(iv). Since'=O(\u000f) andw\u001c1 near the \fxed point (i), the latter parameter\nregion is much larger than the former. Note also that these two regions\nshould be continuously connected and the crossover between the \fxed point\n(iii) and (iv) takes place. The right-hand side of the original RG equations\n(5.84){(5.87) are smooth functions of all the parameters, unless wandhare\nsimultaneously in\fnity.\n111Bibliography\n[1] K. Fukushima and T. Hatsuda, \\The phase diagram of dense QCD,\"\nRept. Prog. Phys. 74, 014001 (2011).\n[2] M. Asakawa and K. Yazaki, \\Chiral Restoration at Finite Density and\nTemperature,\" Nucl. Phys. A 504, 668 (1989).\n[3] T. Hatsuda, M. Tachibana, N. Yamamoto and G. Baym, \\New critical\npoint induced by the axial anomaly in dense QCD,\" Phys. Rev. Lett.\n97, 122001 (2006),\n[4] N. Yamamoto, M. Tachibana, T. Hatsuda and G. Baym, \\Phase struc-\nture, collective modes, and the axial anomaly in dense QCD,\" Phys.\nRev. D 76, 074001 (2007).\n[5] H. Abuki, G. Baym, T. Hatsuda and N. Yamamoto, \\The NJL model of\ndense three-\ravor matter with axial anomaly: the low temperature crit-\nical point and BEC-BCS diquark crossover,\" Phys. Rev. D 81, 125010\n(2010).\n[6] P. C. Hohenberg and B. I. Halperin, \\Theory of Dynamic Critical Phe-\nnomena,\" Rev. Mod. Phys. 49, 435 (1977).\n[7] H. Fujii, \\Scalar density \ructuation at critical end point in NJL model,\"\nPhys. Rev. D 67, 094018 (2003).\n[8] H. Fujii and M. Ohtani, \\Sigma and hydrodynamic modes along the\ncritical line,\" Phys. Rev. D 70, 014016 (2004).\n[9] D. T. Son and M. A. Stephanov, \\Dynamic universality class of the\nQCD critical point,\" Phys. Rev. D 70, 056001 (2004).\n113114 BIBLIOGRAPHY\n[10] Y. Minami, \\Dynamics near QCD critical point by dynamic renormal-\nization group,\" Phys. Rev. D 83, 094019 (2011).\n[11] K. Rajagopal and F. Wilczek, \\Static and dynamic critical phenomena\nat a second order QCD phase transition,\" Nucl. Phys. B 399, 395 (1993).\n[12] M. A. Stephanov, \\QCD phase diagram and the critical point,\" Prog.\nTheor. Phys. Suppl. 153, 139 (2004) [Int. J. Mod. Phys. A 20, 4387\n(2005)]\n[13] Brookhaven National Laboratory, \\Beam Energy Scan The-\nory (BEST) Collaboration,\" Retrieved November 4, 2019, from\nhttps://www.bnl.gov/physics/best/\n[14] S. L. Adler, \\Axial vector vertex in spinor electrodynamics,\" Phys. Rev.\n177, 2426 (1969).\n[15] J. S. Bell and R. Jackiw, \\A PCAC puzzle: \u00190!\r\rin the\u001bmodel,\"\nNuovo Cim. A 60, 47 (1969).\n[16] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, \\The e\u000bects of\ntopological charge change in heavy ion collisions: \\Event by event P\nand CP violation\",\" Nucl. Phys. A 803, 227 (2008).\n[17] K. Fukushima, D. E. Kharzeev and H. J. Warringa, \\The Chiral Mag-\nnetic E\u000bect,\" Phys. Rev. D 78, 074033 (2008).\n[18] H. B. Nielsen and M. Ninomiya, \\Adler-bell-jackiw Anomaly And Weyl\nFermions In Crystal,\" Phys. Lett. 130B , 389 (1983).\n[19] A. Vilenkin, \\Equilibrium Parity Violating Current In A Magnetic\nField,\" Phys. Rev. D 22, 3080 (1980).\n[20] D. E. Kharzeev and H. U. Yee, \\Chiral Magnetic Wave,\" Phys. Rev. D\n83, 085007 (2011).\n[21] G. M. Newman, \\Anomalous hydrodynamics,\" JHEP 0601 , 158 (2006).\n[22] B. I. Abelev et al. [STAR Collaboration], \\Azimuthal Charged-Particle\nCorrelations and Possible Local Strong Parity Violation,\" Phys. Rev.\nLett. 103, 251601 (2009).BIBLIOGRAPHY 115\n[23] B. I. Abelev et al. [STAR Collaboration], \\Observation of charge-\ndependent azimuthal correlations and possible local strong parity vi-\nolation in heavy ion collisions,\" Phys. Rev. C 81, 054908 (2010).\n[24] B. Abelev et al. [ALICE Collaboration], \\Charge separation relative to\nthe reaction plane in Pb-Pb collisions atpsNN= 2:76 TeV,\" Phys. Rev.\nLett. 110, no. 1, 012301 (2013).\n[25] S. A. Voloshin, \\Testing the Chiral Magnetic E\u000bect with Central U+U\ncollisions,\" Phys. Rev. Lett. 105, 172301 (2010).\n[26] W. T. Deng, X. G. Huang, G. L. Ma and G. Wang, \\Test the chiral\nmagnetic e\u000bect with isobaric collisions,\" Phys. Rev. C 94, 041901 (2016).\n[27] Q. Li et al. , \\Observation of the chiral magnetic e\u000bect in ZrTe5,\" Nature\nPhys. 12, 550 (2016).\n[28] N. Sogabe and N. Yamamoto, \\New dynamic critical phenomena in\nnuclear and quark super\ruids,\" Phys. Rev. D 95, no. 3, 034028 (2017).\n[29] M. Hongo, N. Sogabe and N. Yamamoto, \\Does the chiral magnetic\ne\u000bect change the dynamic universality class in QCD?,\" JHEP 1811 ,\n108 (2018).\n[30] R. Jackiw, \\Field theoretic investigations in current algebra,\" in Lectures\non Current Algebra and Its Applications, ed. S. B. Treiman, R. Jackiw\nand D. J. Gross (Princeton University Press, Princeton, 1972).\n[31] L. D. Faddeev, \\Operator Anomaly for the Gauss Law,\" Phys. Lett. B\n145, 81 (1984).\n[32] P. C. Martin, E. D. Siggia and H. A. Rose, \\Statistical Dynamics of\nClassical Systems,\" Phys. Rev. A 8, 423 (1973).\n[33] H-K. Janssen, \\On a Lagrangean for classical \feld dynamics and renor-\nmalization group calculations of dynamical critical properties,\" Z. Phys.\nB23, 377 (1976).\n[34] C. De Dominicis, \\Dynamics as a substitute for replicas in systems with\nquenched random impurities,\" Phys. Rev. B 18, 4913 (1978).116 BIBLIOGRAPHY\n[35] A. Onuki, \\Dynamic equations and bulk viscosity near the gas-liquid\ncritical point,\" Phys. Rev. E 55, 403 (1997).\n[36] H. D. Politzer, \\Reliable Perturbative Results for Strong Interactions?,\"\nPhys. Rev. Lett. 30, 1346 (1973).\n[37] D. J. Gross and F. Wilczek, \\Ultraviolet Behavior of Nonabelian Gauge\nTheories,\" Phys. Rev. Lett. 30, 1343 (1973).\n[38] M. Tanabashi et al. [Particle Data Group], \\REVIEW OF PARTICLE\nPHYSICS,\" Phys. Rev. D 98, no. 3, 030001 (2018).\n[39] G. 't Hooft, \\Symmetry Breaking Through Bell-Jackiw Anomalies,\"\nPhys. Rev. Lett. 37, 8 (1976).\n[40] M. Kobayashi and T. Maskawa, \\Chiral symmetry and eta-x mixing,\"\nProg. Theor. Phys. 44, 1422 (1970).\n[41] V. P. Nair, Quantum \feld theory: A modern perspective (Springer, New\nYork, 2005).\n[42] M. G. Alford, K. Rajagopal and F. Wilczek, \\Color-\ravor locking and\nchiral symmetry breaking in high density QCD,\" Nucl. Phys. B 537,\n443 (1999).\n[43] L. N. Cooper, \\Bound electron pairs in a degenerate Fermi gas,\" Phys.\nRev.104, 1189 (1956).\n[44] J. Bardeen, L. N. Cooper and J. R. Schrie\u000ber, \\Microscopic theory of\nsuperconductivity,\" Phys. Rev. 106, 162 (1957).\n[45] D. Bailin and A. Love, \\Super\ruidity and Superconductivity in Rela-\ntivistic Fermion Systems,\" Phys. Rept. 107, 325 (1984).\n[46] M. Iwasaki and T. Iwado, \\Superconductivity in the quark matter,\"\nPhys. Lett. B 350, 163 (1995).\n[47] M. G. Alford, K. Rajagopal and F. Wilczek, \\QCD at \fnite baryon\ndensity: Nucleon droplets and color superconductivity,\" Phys. Lett. B\n422, 247 (1998).BIBLIOGRAPHY 117\n[48] R. Rapp, T. Sch afer, E. V. Shuryak and M. Velkovsky, \\Diquark Bose\ncondensates in high density matter and instantons,\" Phys. Rev. Lett.\n81, 53 (1998).\n[49] T. Sch afer and F. Wilczek, \\Continuity of Quark and Hadron Matter,\"\nPhys. Rev. Lett. 82, 3956 (1999).\n[50] K. Landsteiner, \\Notes on Anomaly Induced Transport,\" Acta Phys.\nPolon. B 47, 2617 (2016).\n[51] Y. Hidaka, \\Field theory of equilibrium and nonequilibrium systems,\"\n(unpublished).\n[52] S. L. Adler and W. A. Bardeen, \\Absence of higher order corrections in\nthe anomalous axial vector divergence equation,\" Phys. Rev. 182, 1517\n(1969).\n[53] K. Fujikawa, \\Path Integral Measure for Gauge Invariant Fermion The-\nories,\" Phys. Rev. Lett. 42, 1195 (1979).\n[54] D. T. Son and A. R. Zhitnitsky, \\Quantum anomalies in dense matter,\"\nPhys. Rev. D 70, 074018 (2004).\n[55] M. A. Metlitski and A. R. Zhitnitsky, \\Anomalous axion interactions\nand topological currents in dense matter,\" Phys. Rev. D 72, 045011\n(2005).\n[56] K. Kawasaki, \\Kinetic equations and time correlation functions of crit-\nical \ructuations,\" Ann. Phys. (N.Y.) 61, 1 (1970).\n[57] B. I. Halperin, P. C. Hohenberg and E. D. Siggia, \\Renormalization-\nGroup Calculations of Divergent Transport Coe\u000ecients at Critical\nPoints,\" Phys. Rev. Lett. 32, 1289 (1974).\n[58] E. D. Siggia, B. I. Halperin and P. C. Hohenberg, \\Renormalization-\ngroup treatment of the critical dynamics of the binary-\ruid and gas-\nliquid transitions,\" Phys. Rev. B 13, 2110 (1976).\n[59] H. L. Swinney and D. L. Henry, \\Dynamics of Fluids near the Critical\nPoint: Decay Rate of Order-Parameter Fluctuations,\" Phys. Rev. A 8,\n2586 (1973).118 BIBLIOGRAPHY\n[60] J. Polchinski, \\E\u000bective \feld theory and the Fermi surface,\" In *Boul-\nder 1992, Proceedings, Recent directions in particle theory* 235-274,\nand Calif. Univ. Santa Barbara - NSF-ITP-92-132 (92,rec.Nov.) 39 p.\n(220633) Texas Univ. Austin - UTTG-92-20 (92,rec.Nov.) 39 p [hep-\nth/9210046].\n[61] D. B. Kaplan, \\E\u000bective \feld theories,\" arXiv:nucl-th/9506035.\n[62] D. B. Kaplan, \\Five lectures on e\u000bective \feld theory,\" arXiv:nucl-\nth/0510023.\n[63] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter\nPhysics (Cambridge University Press, Cambridge, 1995).\n[64] D. Forster, Hydrodynamic \ructuations, broken symmetry, and correla-\ntion functions (Perseus Books, New York, 1975).\n[65] G. F. Mazenko, Nonequilibrium Statistical Mechanics (WiLEY-VCH,\nWeinheim, 2006).\n[66] U. C. T auber, Critical dynamics: a \feld theory approach to equilib-\nrium and non-equilibrium scaling behavior (Cambridge University Press,\nCambridge, 2014).\n[67] B. I. Halperin, P. C. Hohenberg and Shang-keng Ma, \\Renormalization-\ngroup methods for critical dynamics: I. Recursion relations and e\u000bects\nof energy conservation,\" Phys. Rev. B 10, 139 (1974).\n[68] B. I. Halperin, P. C. Hohenberg and Shang-keng Ma, \\Renormalization-\ngroup methods for critical dynamics: II. Detailed analysis of the relax-\national models,\" Phys. Rev. B 13, 4119 (1976).\n[69] S. Weinberg, The quantum theory of \felds. Vol. 2: Modern applications\n(Cambridge University Press, Cambridge, 1996).\n[70] D. T. Son, \\Low-energy quantum e\u000bective action for relativistic super-\n\ruids,\" arXiv:hep-ph/0204199.\n[71] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (Pergamon\nPress, Oxford, 1980).BIBLIOGRAPHY 119\n[72] I. A. Shushpanov and A. V. Smilga, \\Quark condensate in a magnetic\n\feld,\" Phys. Lett. B 402, 351 (1997).\n[73] R. D. Pisarski and F. Wilczek, \\Remarks on the Chiral Phase Transition\nin Chromodynamics,\" Phys. Rev. D 29, 338 (1984).\n[74] M. E. Peskin and D. V. Schroeder, An Introduction to quantum \feld\ntheory (Addison-Wesley, Reading, 1995).\n[75] A. Onuki, Phase Transition Dynamics (Cambridge University Press,\nCambridge, 2007).\n[76] N. Goldenfeld, Lectures on phase transitions and the renormalization\ngroup (Addison-Wesley, Boston, 1992).\n[77] N. Sogabe and N. Yamamoto, \\Triangle Anomalies and Nonrelativis-\ntic Nambu-Goldstone Modes of Generalized Global Symmetries,\" Phys.\nRev. D 99, no. 12, 125003 (2019).\n[78] I. E. Dzyaloshinskii and G. E. Volovick, \\Poisson brackets in condensed\nmatter physics,\" Ann. Phys. (N.Y.) 125, 67 (1980).\n[79] U. C. T auber and Z. R\u0013 acz, \\Critical behavior of O(n)-symmetric systems\nwith reversible mode-coupling terms: Stability against detailed-balance\nviolation,\" Phys. Rev. E 55, 4120 (1997)."    },    {        "title": "2308.13408v1.Coexistence_of_non_trivial_van_der_Waals_magnetic_orders_enable_field_free_spin_orbit_torque_switching_at_room_temperature.pdf",        "content": " \n1 Coexistence of non-trivial van der Waals magnetic orders  \nenable field-free spin -orbit torque switching at room temperature  \nBing Zhao1, Lakhan Bainsla1,2, Roselle Ngaloy1, Peter Svedlindh3, Saroj P. Dash1,4* \n1Department  of Microtechnology and Nanoscience, Chalmers University of Technology,  \nSE-41296, Göteborg, Sweden.  \n2Department of Physics, Indian Institute of Technology Ropar, Roopnagar 140001, India . \n3Department of Materials Science and Engineering, Uppsala University,  Uppsala, SE -751 03 Sweden.  \n4Graphene Center, Chalmers University of Technology, Göteborg, SE -41296 Sweden.  \n \nAbstract  \nThe discovery of van der Waals (vdW) materials exhibiting  non-trivial and tunable magnetic interactions  \nat room temperature can give rise to exotic magnetic  states , which are not readily attainable with \nconventional materials . Such vdW magnets can provide a unique platform for studying new magnetic \nphenomena and realising magnetization dynamics for energy -efficient and non -volatile  spintronic memory \nand logic technologies.  Recent developments in vdW magnets have revealed their potential to enable \nspin-orbit torque (SOT) induced magnetization dynamics. However, the deterministic  and field-free SOT \nswitching o f vdW magnets  at room temperature  has been lacking, prohibiting their potential applications.  \nHere, we demonstrate magnetic field -free and deterministic SOT switching of a vdW magnet \n(Co 0.5Fe0.5)5GeTe 2 (CFGT) at room temperature, capitalizing on its non-trivial intrinsic magnetic ordering . \nWe discover a coexistence of ferromagnetic and antiferromagnetic orders  in CFGT at room temperature , \ninducing an intrinsic exchange bias and canted perpendicular magnetism. The resulting canted \nperpendicular magnetization of CFGT introduces  symmetry  breaking , facilitating  successful magnetic \nfield-free magnetization switching in the CFGT/Pt heterostructure device s. Furth ermore, t he SOT-induced \nmagnetization dynamic s and their efficiency are evaluated using 2nd harmonic Hall measurements. This \nadvancement opens new avenues for investigati ng tunable magnetic phenomena in vdW material  \nheterostructures and realizing field-free SOT-based spintronic technologies.  \n \n \n \nKeywo rds: (Co 0.5Fe0.5)5GeTe 2, 2D magnets, 2D materials, Room temperature, Spin-orbit torque, ferromagnet, anti -\nferromagnet, Exchange bias, canted magnetism, Field -free magnetization switching.  \nCorresponding author: saroj.dash@chalmers.se   \n2 Introduction  \nThe next generation of memory , logic, communication, and beyond von neumann computing architecture  \nare expected to benefit from the use of  energy -efficient and non -volatile spintronic technolog ies1,2. \nRecently, the spin -orbit torque (SOT)  phenomenon , which utilizes the spin -orbit interaction in materials \nand their hybrid structure s with magnets,  has emerged as an eff icient method to control the magnetization \nin spintronic devices3–5. The control of magnetization by SOT is an attractive alternative to spin -transfer \ntorque (STT) used in magnetic random access memory , spin-logic and spin nano -oscillator  technologies , \nas SOT -based devices are more energy efficient, ultrafast, and provide increased data density with high \nendurance5. However, for the commercial success of SOT -based technology, it is required to enhance the \nSOT efficiency, achieve magnetic field -free magnetization switching, and discover new  materials and \nmethods to control the device parameters2,6–9. \nRecently discovered t wo-dimensional van der Waals  (vdW) magnetic materials have significant  potential \nfor spintronic  devices because of their low dimensionality, tunable magnetic anisotropy, strong proximity -\ninduced interactions, and potential for voltage -controlled magnetism2,10–14. Recent advancements in vdW \nmagnets include the SOT-induced magnetization sw itching of Fe3GeTe 2 with perpendicular magnetic \nanisotropy (PMA) in heterostructures with different spin -orbit materials (SOMs) at cryogenic \ntemperatures6,15–18. Howe ver, devices including  conventional SOM s such as Pt require a ssistance from \nan external in -plane magnetic field  to break the symmetry19 or need materials with  low c rystal symmetry \nsuch as WTe 2 with an out -of-plane SOT component6,20 to deterministically switch the magnetization  of \nFe3GeTe 2 without an external field . Unfortuna tely, due to the low Curie temperature (T c) of Fe 3GeTe 2, the \noperation of these  SOT devices is limited to cryogenic temperature and the SOT eff icienc y of WTe 2 is \nmuch  smaller than  Pt6,20.  \nBeing able to control properties of vdW magnet s, such as the interlayer exchange coupling inherent to \nvdW magnets and the canted state of magnetization , could also help to ac hieve field -free SOT switching7–\n9,21.  To address th is, vdW magnets can be tailored to have unique non -trivial spin ordering due to \nexchange coupling depending on atomic position, stacking, and intra - and interlayer magnetic \ninteractions22–24. Specifica lly, the possibility of taking  advantage of coexisting ferromagnetic ( FM) and \nantiferromagnetic ( AFM ) phase s in a single vdW material to achieve  field-free SOT switching  remains \nunexplored . \nHere, we demonstrate deterministic SOT-induced magnetization switching of the vdW magnet \n(Co 0.5Fe0.5)5GeTe 2 (CFGT) at room temperature without using any external magnetic field. This has been \npossible due to the non -trivial spin ordering with the coexistence of FM and AFM  ordering in CFGT \naccompanied by a pronounced exchange bias effect and canted magneti zation  that persist above room  \n3 temperature , which i s not readily attainable with conventional materials . To gain deeper insights into the \nmagnetic properties of CFGT and their correlation with the SOT switching behavior in CFGT/Pt \nheterostructure Hall -bar devices, we employed the anomalous Hall effect, cur rent-induced magnetization \nswitching, and 2nd harmonic Hall measurements. Our findings reveal that the field -free SOT -induced \nmagnetization switching in CFGT arises from its canted magnetization , which can be induced by  the \nintrinsic exchange bias within t he CFGT magnet . This addresses critical aspects in the advancement of \nvdW-magnet -based spintronic technologies.  \n \nFigure 1. Coexistence of ferro - and antiferro - magnetism in ( Co0.5Fe0.5)5GeTe 2 above room temperature . a. \nSchematic of the crystal structure of CFGT with AA (AFM) and AA’ (FM) stacking order, respectively.  The arrows \nare the atomic  magnetic  moment in Fe(Co) -site. b. The out-of-plane  magnetic field H z dependence of the \nmagnetization  of bulk CFGT measured with SQUID. The inset is the zoom -in of the data at 300 K, the dashed curve  \ncorresponds to the magnetization measure d using an  in-plane  magnetic field  Hx. c. Temperature dependence of the \nmagnetization measured with a fixed in -plane (Hx) and out -of-plane (H z) magnetic field of 5T. d. Transversal Hall \nresistance (AHE) R xy=Vxy/I of a CFGT nanoflake as a function of the out -of-plane field (H z) at 300 K in the field range \nof +/-7 Tesla. The (blue/green) arrows show the AFM and FM magnetic  spin configurations. Inset shows R xy versus \nHx and  Hz in the field range +/ -4 Tesla. e. AHE signal at 300 K in a small field range after subtraction of a linear \nbackground.  \n \n \n \n4 Results and Discussion  \nCoexist ence of ferromagnetic and antiferromagnetic orders in CFGT at room temperature: \nExchange bias and canted magnetism  \n \nThe motivation behind using the vdW magnet CFGT  is it's beyond room temperature magnetic order  and \ntunable magnetic properties depending on atomic position s, stacking, and intra - and interlayer magnetic \ninteractions22–24. CFGT with the wurtzite crystal structure belongs to the P6 3mc space group and C 6v point \ngroup25. By substituting around 50% of the Fe atoms with Co, crystals of CFGT alloy were synthesized  \n(from Hq Graphene) , where AA’ (FM  intralayer  coupling ) and AA (AFM  interlayer coupling ) atomic crystal \nstructures possibly coexist  (Fig. 1a)23,25. Theoritical  calculations24 predi cted that it is the combination of \nhigh Co content and the concomitant change to primitive layer stacking that produces different non -trivial \nmagnetic  order s. The intralayer coupling is dominantly ferromagnetic, while the unique primitive unit cell \ncontain s the possibility of AFM interlayer coupling, which makes such a vdW magnet particularly \ninteresting and elusive. The coexistence of FM and AFM order provides an added advantage of an \nexchange bias effect in a single vdW magnetic material  and offers a unique platform for the possible \nrealization of field -free SOT magnetization switching.  \n \nOut-of-plane magnetization measurements  on bulk CFGT samples at different temperatures using a \nsuperconducting quantum interference device (SQUID)  show a characteristic magnetic field-induced spin-\nflop transition between antiferromagnetically and ferromagnetically aligned layers of CFGT  at large field  \n(Fig. 1b)26,27. Observing a hysteresis loop in the low field range at 300 K,  suggests the coexistence of both \nFM and AFM phases (Fig. 1b, inset). The temperature -dependent SQUID measurements with fixed in -\nplane and out -of-plane fields further confirm the magnetic behaviour of CFGT beyond 350 K, with an \nanomalous behavior observed at low temperature s, indicating a possible phase transition28 (Fig. 1c).  \n \nTo investiga te the magnetic properties of thin CFGT flakes (30-50 nm), magnetotransport measurements \nusing the anomalous Hall effect (AHE) were conducted on nanofabricated Hall -bar devices (see Methods \nsection Supplementary Fig. S1 for details). The AHE measurements ( Hall resistance R xy as a function of \nthe out -of-plane magnetic field H z) of CFGT thin flakes at 300 K  revealed the coexistence of FM and AFM \nbelow the spin -flop transition , as shown in Fig. 1d . Along with the observed spin -flop transitions at higher \nfields, a distinct hysteresis loop appeared in the low field range indicating the presence of a ferromagnetic \ncomponent of the magnetization . A further detailed measurement with both in -plane and ou t-of-plane fields \n(inset of Fig. 1d) and low field measurements (Fig. 1e) suggested an out-of-plane magnetism in CFGT \nwith finite remanence and coercivity at room temperature. Upon removal of the linear background, the  \n5 square loop -like AHE signal confirmed  the ferromagnetism and most importantly with an exchange bias \neffect29, which is manifested as a horizontal shift in a magnetic hysteresis loop due to the coexistence of \nintrinsic FM and AFM phases in CFGT.  \n \nFigure 2. Exchange bias and canted perpendicular magnetic anisotropy of CFGT. a. Transversal Hall \nresistance R xy with out -of-plane field H z at different temperatures in low field range . The dashed lines are the guide \nto the eye. b. AHE signals R xy with di fferent  out-of-plane angle s Φ of the  magnetic field at 300 K. c. The extracted \nmagnitude of the AHE signal ΔR xy and exchange bias field H EB as a function of the out -of-plane angle. The purple \ncurve in the bottom panel is a fit (see main text for details)  with uncertainty represented by dashed curves. The insets \nshow  the schematics of  the magnetization M with a canting angle ∆Φ and its relatio n with the external field H c.  \n \nFigures 2a illustrate the  evolution of the AHE signal of CFGT with temperature  in the range of 50-300 K, \nwhere  apparent  hyster esis is observed in a smaller  magnetic field  range . At lower temperatures, the \ncoercivity increases and the exchange bias effect become more prominent. The exchange bias effect is \nlarger around 120 K, consistent with a magnetic phase transition observed in SQUID measurements ( see \nSupplementary Fig.  S2). Due to the presence  of both FM and AFM properties in CFGT flakes, there is a \npossibility that its magnetic anisotropy  may be significantly impacted. To examine this, we conducted an \nangle -dependence study of the AHE on the CFGT/Pt Hall -bar device at room te mperature  (Fig. 2 b). The \nAHE signal with exchange bias effect can be observed at all angles, except for the in -plane case where \n \n6 the signal vanish es. The magnitude of the AHE signal remains constant up to higher angles, indicating a \nstrong magnetic anisotropy and minimal rotation of the magnetization towards the field direction (Fig. 2 c). \nThe extracted exchange bias field H EB=(H c++Hc-)/2 at different angles are presented in Fig. 2 c (bottom \npanel), whe re H c+/- are the switching fields in positive/negative directions. We consider Hc0+(-) to be  the \ncoercive field s required to switch -(+)M to +( -)M when the applied magnetic field is aligned with the easy \naxis of the magnetization. Further, assuming that th e rotation of the magnetization is negligible when \nchanging  the applied field by an angle Φ , implies that the projection of the applied field on the easy axis \nof the magnetization must be equal to Hc0+(-) to switch the magnetization; H c0+(-) = H c+(-) cos(Φ -∆Φ), where \n∆Φ defines the easy axis of magnetization. The formulate d relation between H EB and the field rotation \nangle Φ  is fitted to the angle -dependent results , indicat ing a magnetization canting  angle ∆Φ of ~10 \ndegrees  in CFGT  (see details in Supplementary Note 1).  \n \nRoom temperature field -free spin -orbit torque switc hing of CFGT/Pt heterostructure \ndevices  \n \nFigure 3. Field -free spin -orbit torque magnetization switching in CFGT/Pt heterostructure at room \ntemperature.  a. Schematics of CFGT/Pt van der Waals heterostructure used for spin -orbit torque (SOT) \nexperiments. The pulsed charge current I P applied to Pt generates a spin current J s due to the spin Hall effect, \ncreating a damping -like torque (τ DL) and field -like torque (τ FL) acting on the magnetization M of CFGT.  b. Left panel: \nOptical microscope image of the CFGT/Pt heterostructure Hall -bar device with measurement geometry for AHE a nd \nSOT experiments in Dev 1. The scale bar is 2 μm.  Right panel:  Schematic for SOT switching of CFGT with a canted \nmagnetization M. The dashed lines indicate the canted easy axis and hard axis of magnetization in CFGT. The \n \n7 arrows show the M switching direc tion. c. Pulsed write current I p induced transverse Hall signal R xy change due to \nSOT-induced magnetic switching without external magnetic field. d,e.  The measured SOT magnetization switching \nsignals with different in -plane fields H x.    \n \nSpin-orbit torqu e (SOT) experiments provide a valuable  platform for studying magnetization switching \ndynamic s, and realization field -free switching of vdW FMs will offer practical spintronic device applications. \nTo investigate the SOT-induced magnetization  switching and  dynamics in CFGT samples  with intrinsic \nexchange bias  and canted magnetization effects , CFGT/Pt heterostructure Hall bar devices were \nnanofabricated  as shown in Figure 3a  and 3b  (see details in the Methods section).  For an effective SOT \nmagnetization switching, it is essential to have a large  charge -spin conversion efficiency in Pt and high -\nquality CFGT/Pt interfaces for efficient transmission  of spin current . In the SOT-induced magnetization \nswitching experiment, a series of DC pulse currents applied along the x -direction through the spin -orbit \nmaterial (Pt), generates a spin current along the z -axis with spin polarization sy along the y -axis due to the \nspin Hall effe ct. The resulting spin current exerts SOTs in CFGT (field -like τFL and damping -like τDL \ntorques) , enabling the switching of magnetization M direction . Specifically, the field-like torque τFL ~ M×sy, \nprecesses M about the exchange field created by spin polarization, while the damping -like torque τDL ~ \nM×(M×sy) rotates the magnetization M towards the direction of spin polarization sy. Typically, damping -\nlike torque determines the switching of the magnetization30.  \n \nThe AHE signal of the CFGT/Pt Hall device shows an out-of-plane field magnetization and the  difference \nof the switching field for both field sweep directions suggests a strong exchange bias (H EB= 0.075 T) in \nCFGT  (Fig. 3c , the magnitude  of exchange bias  vary from device to device due to different flake thickness ). \nAs shown in Fig. 3d, w e observed the SOT -induced magnetization switching s using a  pulse d write current \nIp with pulse time 100 μs, followed by a small DC read current (I r~50 μA) to probe the magnetization state \nRxy=Vxy/Ir. As the signal R xy is proportional to the out -of-plane magnetization Mz41, the SOT signal shows  \ncurrent induced magnetization change between +M z and -Mz. The magnitude change of R xy in the SOT \nsignal is around 50% of the AHE signal with field sweep, which suggests around  half of the CFGT  \nmagnetiza tion is switched . Such partial switching  could be  due to  some domain structure in the Hall -bar \ndevice and /or also short spin diffusion length in CFGT  of ~ few nm , while the CFGT flakes are ~ 30 nm  \nthick. Interestingly, we observe a deterministic SOT switching of CFGT without applying any magnetic \nfield.  This is in contrast to the conventional requirement of an in -plane magnetic field H x to break the \ngeometrical symmetry for a deterministic switching of the PMA magnetization. To verify the origin, we \nperformed SOT measurements  with different in -plane magnetic fields H x (Fig. 3 e). We found that the H x \nfield does not aff ect the SOT switching and the magnetization switching direction (chirality) remains the  \n8 same in this field range of +/ -100 mT , suggesting that the external field does not play a significant  role due \nto exchange bias effects , instead  canted magnetization assist the deterministic switching  of CFGT21.  \n \nConventionally , chirality is determined by the external field, which favours opposite switching directions \nfor the positive and negative in -plane magnetic fields19. If no external field is applied, the magnetization \nwould be pulled to the in -plane  orientation  and broken into multip le domains, instead of the observed \ndeterministic switch ing. However, in the canted magnetization  scenario  of CFGT sample  (Fig. 3 b), the \nsymmetry is broken as the easy axis of the magnetization M is canted at an angle Φ with respect to the \nnormal directio n. The damping -like torque  τDL can rotate the M anticlockwise over the hard axis towards \nanother easy axis, while the other rotation direction is not possible. This also explains the unchanged \nchirality of SOT switching with  the external field direction , which is similar to that observed in other canted \nmagnets .21  Thermal fluctuations induced by pulse current  can randomize the magnetic domain of CFGT \nand make switching behavi or unstable16. This is not the case in our CFGT/Pt SOT device though the \nreducation of the SOT signal caused by the thermal fluctuations  can not be ruled out18. We can also rule \nout thermally assisted switching if the device reaches above T c, as this can result in unditerministic \nswitching d ue to formation of multiple domains31,32. Therefore, the observed deterministic and field -free \nswitching of CFGT can be attributed dominant SOT contribution and the canted magnetization of CFGT.  \n \n \nEstimation of spin -orbit torque magnetization dynamics using harmonic Hall measurement in  \nCFGT/Pt heterostructure  devices   \n9  \nFigure 4. Harmonic Hall signals  of the CFGT/Pt Hall -bar device at room temperature. a. 1st harmonic V xy1ω \nsignals as a function of the out -of-plane field H z and lon gitudinal in -plane field H x sweeps with +/ -Mz magnetic states \nof CFGT. The solid lines are the fitting results of the longitudinal V xy1ω signals. b. Longitudinal (transversal) in -plane \nfield H x(y) dependence of the 2nd harmonic signals V xy2ω. The pink and blue lines are linear fit s of the corresponding \nsignals. All the harmonic measurements were performed with ω=213.14 Hz and I ac=4.9 mA at 300 K in Dev 2. c. \nField dependence of the 1st harmonic V xy1ω signals with field directions indicated (H z and H x). The inset is the zoom -\nin around 0  T. d. In-plane angle dependence of the 2nd Harmonic signal at I ac=1.75  mA with in -plane field H ip. e, f. \nExtracted current dependence of the field -like effective field ∆HFL, Oersted  field HOe, damping -like effective field ∆HDL, \nand thermal contribution in Dev 3. g. Benchmark plot showing the state -of-the-art of the vdW magnets based SOT \ndevices , considering working temperature vs. in -plane assistant magnetic field H x. Benchmarking our work wit h state \nof the art results using vdW magnet based SOT devices of Fe3GeTe 2/WTe 26, Fe 3GeTe 2/WTe 218, Fe 3GeTe 2/Pt15, \n \n10 Fe3GeTe 2/Pt16, Cr 2Ge2Te6/Ta17, Cr 2Ge2Te6/Pt33, Cr 2Ge2Te6/(Bi 1-xSbx)2Te334 .Our work on CFGT/Pt device shows the \nfirst room -temperature SOT device operation without any assisted magnetic fields.  \n \nHarmonic Hall measurement is a powerful tool to study different SOT contributions  and effective spin-orbit \nfields  in magnetic heterostructures35. Application of an AC current I ac = I0sinωt with frequency ω modulates \nthe SOT amplitude and induces small oscillations of M about its equilibrium direction44. Such oscillations \ngenerate a 2nd harmonic contribution to the Hall voltage36 (Vxy=V0+𝑉𝑥𝑦1𝜔sinωt+𝑉𝑥𝑦2𝜔cos2 ωt), where  Vxy \ndepends on M z through the AHE and the product of Mx x My through the planar Hall effect (PHE). The 1st \nharmonic term 𝑉𝑥𝑦1𝜔 relates to the equilibrium direction of the magnetization and is independent of the \nmodulated SOT, which was measured as an in-phase  signal . The out -of-phase (Ø=90о) 2nd harmonic term \n𝑉𝑥𝑦2𝜔 measures the response of the magnetization to the current -induced spin-orbit fields. The current -\ninduced effective field ∆H can be decomposed into two components: the transverse field-like (∆H FL) and \nlongitudinal  damping -like (∆H DL) effective  spin-orbit fields, which can be extracted by the following \nequation35, \n∆HDL(FL)= −𝟐𝐁𝑳(𝑻)+𝟐𝝃𝐁𝑻(𝑳)\n𝟏−𝟒𝝃𝟐;          (Eq. 1) \nwhere BL(T)= −𝝏𝑽𝒙𝒚𝟐𝝎\n𝝏𝐇𝒙(𝒚)/𝝏𝟐𝑽𝒙𝒚𝟏𝝎\n𝝏𝑯𝒙(𝒚)𝟐 and 𝝃 = R PHE/RAHE ≈ 1 in CFGT/Pt heterostructure (see Supplementary Note 2 for \ndetails ). The 1st harmonic term 𝑉𝑥𝑦1𝜔 as a function of the external magnetic field H is shown in Fig. 4a. As a \ncomparison, the out -of-plane Hz component (AHE) is presented with the in-plane field (H x) dependent \ncomponents for +/- Mz. Further, the 2nd harmonic term 𝑉𝑥𝑦2𝜔 was measured vs. transverse (H y) and \nlongitudinal (H x) in-plane magnetic field (Fig. 4b). By fitting the 1st and 2nd harmonic terms with Eq.1 , we \ncalculate ∆HDL/Jac=1.9 mT per MA/cm2 and ∆H FL/Jac=1.8 mT per MA/cm2, where J ac is the current density \nin Pt. As the damping -like torque is attributed to the longitudinal effective spin–orbit field ∆HDL, the spin-\norbit torque efficiency can be calculated  using expression37,38, \nξDL=TiniθSH=2𝑒\nћμ0Ms𝑡𝐶𝐹𝐺𝑇𝑒𝑓𝑓∆HDL/Jac ;          (Eq. 2) \nwhere e is the electron charge, ℏ is the reduced Plank constant, μ 0Ms and 𝑡𝐶𝐹𝐺𝑇𝑒𝑓𝑓 are the saturation \nmagnetization  (net magnetization  at low field range)  and the effective thickness of CFGT, respectively. \nHere ξDL is determined by the spin Hall angle (θSH) of Pt and the interface transparency (T ini). By assuming \na fully transparent int erface, i.e.T ini=1, and spin Hall angle θ SH=0.12 for Pt15,16,38, we obtain the effective \nsaturation magnetization μ 0Ms ≈12.8 mT, which is smaller than the value of pure FM magnetic phase with \nthe AA’ crystal structure25. This suggests the AFM phase has no contribution to the effective saturation \nmagnetization  at lower field range , but coexists with the FM phase, which also agrees with the SQUID  \nmagnetization  and AHE results24.     \n11 To further confirm the SOT contributions, we also performed large field harmonic Hall measurements. In \nthe 1st harmonic Hall measurements with H z field sweep (Fig. 4c), we  observed the expected AFM phase \ntransition and FM hysteresis in high and low field ranges, respectively , whereas the H x field sweep \nmeasurement shows the FM magnetization component being pulled to the in -plane direction at around 2 \nT (i.e.  magnetic anisotropy field, H k), without any clear indication of the AFM transition could be detected \nup to 7 T. The 2nd harmonic Hall voltage signal as a function of a large in -plane field H ip can help to extract \nthe contributions from ∆ HDL, ∆HFL and the current -induced thermal effect  (Fig. 4d) . Furthermore , the \ncurrent dependence of the contributions from ∆ HDL, ∆HFL and the thermal effect are plo tted in Fig.  4e and \nFig. 4f. We extract ed ∆HDL/Jac=4.2 mT per MA/cm2  with an effective saturation magnetization μ 0Ms ≈ 5.9 \nmT (low field)  and ∆HDL/Jac=19.7 mT per MA/cm2. The current -induced Oersted  field is negligible  and \ncannot be the origin of the fi eld-free SOT -induced magnetization switching (see details in Supplementary \nNote 2), while the thermal effect contribution  to the harmonic Hall voltage signal  𝑉𝑥𝑦,𝑡ℎ𝑒𝑟𝑚𝑎𝑙 ∝𝐼2, \nincreasing quadratically with the current, is proportional to the Mx. Geometrically, 𝑉𝑥𝑦,𝑡ℎ𝑒𝑟𝑚𝑎𝑙  ~ Mx × ∇Tz, \nwhere ∇Tz is due to the different thermal conductivity of CFGT, Pt, SiO 2/Si substrate , and air  (see detailed \nanalysis in Supplement ary Note 2 and the summary of the key SOT device  parameters  in Supplement ary \nTable S1). Utilizing both SOT -induced magnetic switching and 2nd harmonic Hall measurements, we can \nconclude that the canted magnetization  in CFGT can be efficiently controlled with a switching current \ndensity of Jsw=2.4x107 A/cm2 at 300 K (see Supplementary Note 3), which is comparable to the reported \nvalues for  vdW magnets at low temperatures15. Moreover, the room temperature and field -free operation \nof CFGT/Pt heterostructure SOT devices without any external field using a vdW magnet with coexistence \nof FM and AFM phases are observed for the first time (Fig. 4g), as benchmarked against  the stat e-of-the-\nart. \nSummary  \nIn summary, we demonstrated the field -free and deterministic switching of the vdW magnet CFGT at room \ntemperature , exploiting the intrinsic coexistence of non -trivial magnetic orders. Remarkably, CFGT hosts \nboth FM and AFM properties  much above room temperature, with intrinsic exchange bias and canted \nperpendicular ferromagnetism  in a layered  vdW material, which is not possible to attend in conventional \nsystems . The detailed investigation of magnetic properties  of CFGT and magnetization dynamics using  \nSOT switching and 2nd harmonic  Hall measurements  of CFGT /Pt heterostructures show their relationship \nwith observed field-free SOT behaviour. By demonstrating the feasibility of deterministic SOT -induced \nmagnetizati on switching in a non-trivial vdW magnet at room temperature, our work lays the foundation \nfor the development of novel and efficient spintronic devices based on vdW magnets, which hold  \n12 tremendous promise for next -generation non-volatile memory, logic and neuromorphic computing  \napplications.  \nMethods  \nFabrication of devices and electrical measurements   \nDevice Fabrication:  Single crystals of (Fe 0.5Co0.5)5GeTe 2 (CFGT) were grown by chemical vapor transfer (CVT)  \nmethod by HqGraphene . The AHE measurements were measured in Hall -bar devices of CFGT (20 -50 nm) on \nSiO 2/Si substrate. The CFGT crystal was exfoliated onto the SiO 2/Si wafer, followed by electron -beam lithography \n(EBL) and Ar plasma etching to pattern in a Hall -bar geometry, a nd the Au/Ti contacts were deposited 2nd EBL step \nand lift -off technique. The CFGT/Pt heterostructures were prepared by exfoliating thin CFGT crystals (20 -30 nm) \ninside an N 2 glovebox onto pre -deposited Pt(10 nm)/Ti(2 nm) layers on a SiO 2/Si substrate. The  Pt layer was \nprepared by electron -beam evaporation onto a Si/SiO 2 substrate with a Ti seed layer. After exfoliation, the samples \nwere immediately transferred to an electron -beam evaporation system to deposit a 2 nm Al layer, forming an Al 2O3 \ncapping layer . The Hall bar and contacts were patterned using EBL and Ar ion beam etching.  \nMeasurements:  Low temperature and high magnetic field measurements of CFGT Hall bar devices were performed \nin the Quantum design DynaCool PPMs system with bias current in the ra nge of 50 -500 μA in the temperature range \nof 10 -300 K and magnetic field up to 10 Tesla (T).  \nSOT switching and Harmonic measurements  in low magnetic field range were carried out in a vacuum cryostat \nwith a magnetic field up to 0.7 T for CFGT/Pt devices. The electronic measurements were performed using a Keithley \n6221 current source, a nanovoltmeter 2182A. To monitor the longitudinal an d transverse Hall resistances, Keithley \n2182A nanovoltmeters were utilized. For SOT -induced magnetization  switching measurements, Keithley 2182A \nnanovoltmeters were used to monitor the response of the Hall resistances, while a Keithley 6221 AC source was \nutilized with a pulse current of 100 microseconds (μs) through the device, followed by a DC read current. The \nharmonic measurement was performed using Lockin SR830 to measure in -phase 1st and out -of-phase 2nd harmonic \nvoltages, respectively. The 2nd harmonic measurements in high magnetic field  range  were carried out in the PPMS \nsystem with an external electronic connection to Lockin SR830 to measure the 1st and 2nd harmonic voltages.  \nSQUID measurements: A Quantum Design superconducting quantum interfer ence device (SQUID) was used to \nmeasure the static magnetic properties of bulk CFGT crystals. We measured the magnetic hysteresis both in in -\nplane (H ∥xy) and out -of-plane (H ∥z) magnetic fields . The bulk CFGT crystal was glued on a Si substrate to properly \nalign the sample in the magnetic field during the measurements.  \nAcknowledgments  \nAuthors acknowledge support from European Union Graphene Flagship (Core 3, No. 881603), 2D TECH VINNOVA \ncompetence center (No. 2019 -00068), Swedish Research Council (VR) grant (No. 2021 –04821), FLAG -ERA project \n2DSOTECH (VR No. 2021 -05925), KAW – Wallenberg initiative on Materials Science for a Sustainable World (WISE), \nGraphene Center, AoA Nano  and AoA Materials progr am at Chalmers University of Technology. We acknowledge \nthe help of staff at the Quantum Device Physics and Nanofabrication laboratory at Chalmers.  \nData availability  \nThe data that support the findings of this study are available from the corresponding auth ors at a reasonable \nrequest.   \n13 Contributions  \nB.Z. and S.P.D.  conceived the idea and designed the experiments. B.Z. fabricated and characterized the devices  \nwith support from L.B. and R.N .. P.S. performed SQUID magnetization measurements. B.Z., S.P.D. analyze d and \ninterpreted the experimental data, compiled the figures, and wrote the manuscript  with feedback from all co -authors . \nS.P.D. supervised and managed the research project.  \nCorresponding author  \nCorrespondence to Saroj P. Dash, saroj.dash@chalmers.se  \nCom peting interests  \nThe authors declare no competing interests.  \nReferences  \n1. Lin, X., Yang, W., Wang, K. L. & Zhao, W. Two -dimensional spintronics for low -power electronics. \nNat. Electron.  2, 274 –283 (2019).  \n2. Yang, H. et al.  Two-dimensional materials prospects for non -volatile spintronic memories. Nature  \n606, 663 –673 (2022).  \n3. Liu, L. et al.  Spin-Torque Switching with the Giant Spin Hall Effect of Tantalum. Science (80 -. ). 336, \n555–558 (2012).  \n4. Miron, I. M. et al.  Perpendicular switching of a single ferromagnetic layer induced by in -plane current \ninjection. Nature  476, 189 –193 (2011).  \n5. Manchon, A. et al.  Current -induced spin -orbit torques in ferromagnetic and antiferromagnetic \nsystems. Rev. Mod. Phys.  91, 035004 (2019).  \n6. Kao, I. -H. et al.  Deterministic switching of a perpendicularly polarized magnet using unconventional \nspin–orbit torques in WTe2. Nat. Mater.  21, 1029 –1034 (2022).  \n7. Fukami, S., Zhang, C., DuttaGupta, S., Kurenkov, A. & Ohno, H. Magnetization switching by spin –\norbit torque in an antiferromagnet –ferromagnet bilayer system. Nat. Mater.  15, 535 –541 (2016).  \n8. Oh, Y. -W. et al.  Field -free switching of perpendicular magnetiz ation through spin –orbit torque in \nantiferromagnet/ferromagnet/oxide structures. Nat. Nanotechnol.  11, 878 –884 (2016).  \n9. Yu, G. et al.  Switching of perpendicular magnetization by spin –orbit torques in the absence of \nexternal magnetic fields. Nat. Nanotech nol. 9, 548 –554 (2014).  \n10. Kurebayashi, H., Garcia, J. H., Khan, S., Sinova, J. & Roche, S. Magnetism, symmetry and spin \ntransport in van der Waals layered systems. Nat. Rev. Phys. 2022 43  4, 150 –166 (2022).  \n11. Huang, B. et al.  Emergent phenomena and proximity effects in two -dimensional magnets and \nheterostructures. Nat. Mater. 2020 1912  19, 1276 –1289 (2020).  \n12. Zhao, B. et al.  A Room ‐Temperature Spin ‐Valve with van der Waals Ferromagnet Fe5GeTe \n2/Graphene Heterostructure. Adv. Mater.  35, 2209113 (2023).  \n13. Mak, K. F., Shan, J. & Ralph, D. C. Probing and controlling magnetic states in 2D layered magnetic \nmaterials. Nat. Rev. Phys. 2019 111  1, 646 –661 (2019).  \n14. Burch, K. S., Mandrus, D. & Park, J. -G. Magnetism in two -dimension al van der Waals materials. \nNature  563, 47–52 (2018).  \n15. Alghamdi, M. et al.  Highly Efficient Spin -Orbit Torque and Switching of Layered Ferromagnet \nFe3GeTe2. Nano Lett.  19, 4400 –4405 (2019).  \n16. Wang, X. et al.  Current -driven magnetization switching in a  van der Waals ferromagnet Fe3GeTe2. \nSci. Adv.  5, eaaw8904 (2019).  \n17. Ostwal, V., Shen, T. & Appenzeller, J. Efficient Spin ‐Orbit Torque Switching of the Semiconducting  \n14 Van Der Waals Ferromagnet Cr2Ge2Te6. Adv. Mater.  32, 1906021 (2020).  \n18. Shin, I. et al. Spin–Orbit Torque Switching in an All ‐Van der Waals Heterostructure. Adv. Mater.  34, \n2101730 (2022).  \n19. Liu, L., Lee, O. J., Gudmundsen, T. J., Ralph, D. C. & Buhrman, R. A. Current -Induced Switching of \nPerpendicularly Magnetized Magnetic Layers Using Spin Torque from the Spin Hall Effect. Phys. \nRev. Lett.  109, 096602 (2012).  \n20. MacNeill, D. et al.  Control of spin –orbit torques through crystal symmetry in WTe2/ferromagnet \nbilayers. Nat. Phys.  13, 300 –305 (2017).  \n21. Liu, L. et al.  Current -induced magnetization switching in all -oxide heterostructures. Nat. \nNanotechnol.  14, 939 –944 (2019).  \n22. Tian, C . et al.  Tunable magnetic properties in van der Waals crystals (Fe1−xCox)5GeTe2. Appl. \nPhys. Lett.  116, 202402 (2020).  \n23. Zhang, H. et al.  A room temperature polar magnetic metal. Phys. Rev. Mater.  6, 044403 (2022).  \n24. May, A. F., Du, M. -H., Cooper, V. R . & McGuire, M. A. Tuning magnetic order in the van der Waals \nmetal Fe5GeTe2 by cobalt substitution. Phys. Rev. Mater.  4, 074008 (2020).  \n25. Zhang, H. et al.  Room -temperature skyrmion lattice in a layered magnet (Fe0.5Co0.5)5GeTe2. Sci. \nAdv. 8, (2022).  \n26. Jiang, S., Shan, J. & Mak, K. F. Electric -field switching of two -dimensional van der Waals magnets. \nNat. Mater.  17, 406 –410 (2018).  \n27. Telford, E. J. et al.  Coupling between magnetic order and charge transport in a two -dimensional \nmagnetic semiconductor.  Nat. Mater.  21, 754 –760 (2022).  \n28. Telford, E. J. et al.  Layered Antiferromagnetism Induces Large Negative Magnetoresistance in the \nvan der Waals Semiconductor CrSBr. Adv. Mater.  32, 2003240 (2020).  \n29. Stamps, R. L. Mechanisms for exchange bias. J. Phys . D. Appl. Phys.  33, R247 –R268 (2000).  \n30. Zhang, S., Levy, P. M. & Fert, A. Mechanisms of Spin -Polarized Current -Driven Magnetization \nSwitching. Phys. Rev. Lett.  88, 236601 (2002).  \n31. Krishnaswamy, G. K. et al.  Time -Dependent Multistate Switching of Topological Antiferromagnetic \nOrder in Mn3Sn. Phys. Rev. Appl.  18, 024064 (2022).  \n32. Pal, B. et al.  Setting of the magnetic structure of chiral kagome antiferromagnets by a seeded spin -\norbit torque. Sci. Adv.  8, 593 0 (2022).  \n33. Gupta, V. et al.  Manipulation of the van der Waals Magnet Cr2Ge2Te6by Spin -Orbit Torques. Nano \nLett. 20, 7482 –7488 (2020).  \n34. Mogi, M. et al.  Current -induced switching of proximity -induced ferromagnetic surface states in a \ntopological insulator. Nat. Commun.  12, 1404 (2021).  \n35. Hayashi, M., Kim, J., Yamanouchi, M. & Ohno, H. Quantitative characterization of the spin -orbit \ntorque using harmonic  Hall voltage measurements. Phys. Rev. B  89, 144425 (2014).  \n36. Avci, C. O. et al.  Interplay of spin -orbit torque and thermoelectric effects in ferromagnet/normal -\nmetal bilayers. Phys. Rev. B  90, 224427 (2014).  \n37. Pai, C. -F., Ou, Y., Vilela -Leão, L. H., R alph, D. C. & Buhrman, R. A. Dependence of the efficiency \nof spin Hall torque on the transparency of Pt/ferromagnetic layer interfaces. Phys. Rev. B  92, 064426 \n(2015).  \n38. Nguyen, M. -H., Ralph, D. C. & Buhrman, R. A. Spin Torque Study of the Spin Hall Cond uctivity and \nSpin Diffusion Length in Platinum Thin Films with Varying Resistivity. Phys. Rev. Lett.  116, 126601 \n(2016).  \n "    },    {        "title": "2208.01282v1.Emergence_of_low_temperature_glassy_dynamics_in_Ru_substituted_non_magnetic_insulator_CaHfO3.pdf",        "content": "1 \n Emergence of  low–temperatu re glass y dynamics in Ru  substituted  non–\nmagnetic insulator CaHfO 3  \nGurpreet Kaur and K. Mukherjee  \nSchool of Basic Sciences, Indian Institute of Technology Mandi, Mandi 175005, Himachal \nPradesh, India  \n \nAbstract  \nNon–magnetic insulators/semiconductors with induced magnetism  introduced  via transition \nmetal substitution  are one of the promising materials in the field of  spintronic , \nmagnetoelectronics  and magneto –optical devices. In this context, h ere, we focus on magnetism \ninduced in a non–magnetic insulator CaHfO 3, by the substitution of  4d element  Ru, at Hf –site. \nStructural investigations indicate that substitution of Ru4+ (up to 50%) does not affect the \noriginal crystal structure of the parent compound. Magnetic studies divulge a crossover from a \ndiamagnetic to paramagnetic state with 20% Ru substitution. Further replacement of Hf results in \na glassy magnetic state in CaHf 1-xRuxO3 (0.3 ≤ x ≤ 0.5). The nature of the low temperature \nglassiness (below 20 K) in these compositions is confirmed through Vogel –Fulcher and Power \nlaw, along with, magnetic memory effect and relaxation dynamics. The observed glassiness is \nexplained through the  phenomenological “hierarchical model”.  Our studies indicate that  the \npresence  of competing short range  interactions among  random ly arranged Ru cations in non –\nmagnetic insulator  CaHfO 3 are responsible for the observed low temperature magnetic state in \nthis series with compositions  > 0.25.  \n \n \n \n \n \n 2 \n 1. Introduction  \n Transition metal –based perovskites  and their derivatives have been extensively studied \nover the last couple of decades due to the various novel properties like colossal \nmagnetoresistance, large magnetocaloric effect, magnetization reversal, multiferroicity, etc. \nexhibited by them [1 –5]. Dilute magnetic materials  (belonging to the above family)  in which \nferromagnetism  is induced by substitution of  magnetic impurities , has recently gained substantial \nattention bec ause of addition al spin functionality  [6–7]. Such  materials are desirable candid ates \nfor spintronic s, magne to–electronics, magneto –optics devices  etc., due to the interplay of charge \nand spin degree s of freedom. Dilute  magnetic alloys  formed by II–VI and III –V elements  are \nalso investigated for th ese phenomen a [8–10]. Generally, for all these cases, the magnetism has \nbeen induced by a low percentage of substitution. Further, the non –magnetic perovskites also \nprovide a vast playground to explore and manipulate the physical  properties of dilute magnetic \noxides (DMO). Fe/Cr/Co doped SrTiO 3, Co doped La 0.5Sr0.5TiO 3, Co/Mn  doped KTaO 3 and \nBaTiO 3 etc. are examples of DMOs  exhibiting room temperature ferromagnetism  [11–13]. \nSimilarly, enhance ment in thermoelectric p roperties  at high temperatures is  predicted in \nPr/Co/Nb doped SrTiO 3 [14,15].  BaSnO 3 is transparent to visible light and  exhibits room \ntemperature ferromagnetism . Ru doping  at the Sn –site makes it desirable candidate for the \nmagneto –optical devices [ 16]. The o bserved magnetism in these systems  has been mostly \nexplained by  Ruderman –Kittel –Kasuya –Yosida ( RKKY ) interactions among magnetic cations \nvia conduction electrons. Some of t hese materials , along with their insulating nature, also show \nglassy  magnetic  phases  at low temperatures [17–18]. Initially , spin glass  (SG) / cluster glass (CG)  \ndynamics ha d been anticipated in metallic alloys  due to RKKY interactions  [19]. But with time it \nwas observed that several  systems having semiconducting or insulating state show  a glassy \nmagnetic behaviour . This observed feature  was explained through short range super –exchange \ninteractions, and it was ascribed that these systems  belong to a different universality class  [20–\n21]. Additionally  glass y magnetic  state has also been induced in other  non–magnetic perovskite  \ninsulator s such as  Sr, Nb and Mn substituted LaCoO 3 [22–24].  \nIn this context, a non –magnetic insulator  CaHfO 3 can be interesting. This compound  has a \nwide bandgap of 6.4 eV and a high dielectric constant.  Insulating to half metallic transition has \nbeen theoretically predicted in this compound due to the presence of higher concentration of O -\ndefects [ 25]. Furthermore, it is optically transparent in  the range of  visible to deep ultraviolet 3 \n light, making it a suitable candidate for optoelectronic devices  [26–28]. Also due to its high \nmelting point, it can be used in electrochemical devices. This compound also acts as a most \npromising host for scintillators due to the high atomic number of Hf [ 29–30]. However, the \nevolution of the physical properties  of this compound  due to substitution of magnetic impurity \nhas not been explored . Another analogous perovskite compound, CaRuO 3, is metallic and \nisostructural to CaHfO 3. The magnetic state of CaRuO 3 is enigmatic because even after the \npresence of a magneti c atom, long–range ordering  is absent . It happens  due to the presence of  \nlattice distortion  [31–32]. The slow dynamic of spins at low temperature in this compound is \nconfirmed by studying the magnetic state of  the compounds via doping at Ru –site and by \napplying stress [ 33–39]. Since both  these  compounds have same structure, it is anticipated that \nCaHf 1-xRuxO3 display full  a solubility for whole x and presence of magnetic impurity will result \nin an evolution of novel magnetic properties in CaHfO 3. \nIn view  of the above, in this manuscript , we have investigated the structural, magnetic, \nand thermodynamic  properties of Ru doped CaHf 1-xRuxO3 (x = 0  – 0.5) series through X –ray \ndiffraction (XRD), dc and ac susceptibility, and heat ca pacity.  Studies are limited to x = 0.5 \ncomposition as for x ≥ 0.60, superstructure of calcium hafnate start s developing, in contrast to \nour expectations. Physical properties of CaRuO 3 compound are also added here for comparison . \nThe analysis of dc and ac suscepti bility  results indicate the absence of long –range magnetic \nordering in these  compound s, along with  the observation  of low temperature magnetic –glass \nstate in x = 0.3, 0.4, and  0.5 compounds . Nature of the glass –dynamics  investigated via different \nmeans indicat es CG behaviour in these  compounds. The magnetic properties below magnetic \nglass transition temperature have  also been studied via time evolution of isothermal remanent \nmagnetisation and memory effect as well. Our studies show  the evolution of magnetic CG state \ndue to competing short range magnetic interactions  among randomly distributed Ru cations  in \nthe respective compositions of the  insulating CaHf 1-xRuxO3 series .  \n2. Experimental details  \nThe polycrystalline compounds CaHf O3 (Ru_0.0), CaHf 0.8Ru0.2O3 (Ru_0.2), \nCaHf 0.75Ru0.25O3 (Ru_0.25), CaHf 0.7Ru0.3O3 (Ru_0.30), CaHf 0.6Ru0.4O3 (Ru_0.4), CaHf 0.5Ru0.5O3 \n(Ru_0.5) and CaRuO 3 (Ru_1.0) are prepared using standard solid –state reaction method. The \nstoichiometric  quantities of materials (purity > 99.9%) CaCO 3, HfO 2, and RuO 2 are mixed  and 4 \n grinded, followed by calcination for 12 hours at 1000° C. The calcinated mixture is regri nded to \nfine powder, pressed into pellets, and sintered at 1400° C for 48  – 60 hours. To investigate the \ncrystal structure, powder XRD pattern was collected at room temperature in the range 10  – 90° \nwith 0.02° step size, using Rigaku Smart Lab diffractometer with Cu –Kα source (λ = 1.54 Å). \nRietveld refinement  of the obtained XRD patterns i s performed using Fullprof Suite software. \nThe room temperature X–ray Photoelectron spectroscopy (XPS) was performed using NEXSA \nsurface analysis model by Thermo Fisher scientific. The dc and ac magnetization measurements \nare performed on a Magnetic Properties Measurement System (MPMS) from Quantum Design \nInc., USA. For heat capacity measurement, Physical Properties Measurement System (PPMS) \nfrom Quantum Design Inc., USA was used.  \n3. Results and discussions  \n3.1 Structural Analysis  \n3.1.1 X–Ray Diffraction  \nFig. 1 shows the Rietveld refined XRD pattern of CaHf 1-xRuxO3 series.  The Rietveld refinement \nhas been done by using Fullprof software. For this linear interpolation between the set point of \nthe background , and pseudo -voigt  function for peak fitting have been used.  All compounds \ncrystallize in a single –phase orthorhombic structure with space group Pnma . In Ru_0.4 and \nRu_0.5 compounds, high –intensity peaks of XRD patterns are accompanied by smaller peaks  as \ncan be seen in Fig. 1 (h). In this figure the peak at 2  ~ 23 of Ru_0.5 compound is accompanied \nwith the minor (extra) peak at higher side of 2  ~ 24. This minor peak is close to the peak \nobserved in Ru_1.0 (~ 23.12 ). Similar observations are noted f or other peaks of Ru_0.5 and \nRu_0.4 compounds. Thus, we have initially used the lattice parameters of Ru_1.0 for second \nphase which has same structure to Ru_0.0 to fit these peaks.  Except the lattice parameters, all \nthe other parameters remained same in both phases.  Lattice parameters, volume of the unit cell, \npercentage of both phases, along with goodness of fit parameters are listed in Table 1. The \npercentage of developed second phase is less than 5% (Table 1). The lattice parameters of the \nsecond phase  for Ru_0.4 and Ru_0.5 compounds are closer to the Ru_1.0  crystal structure. On \nexceeding Ru concentration above 50%, peaks of the superstructure Ca 6Hf19O44 have been \nobserved. Hence, we have restricted our study to compositions up to CaHf 0.5Ru0.5O3 (i.e.,  \nRu_0.5) . Fig. 1 (h) also shows the gradual shifting of peaks of doped compounds from  Ru_0.0 to 5 \n Ru_1.0. This observation implies that Ru ions (with ionic radius 0.62 Å) replace the Hf ions \n(with ionic radius 0.71 Å).  This observation is  also in accordance with the decrement in lattice \nparameters and unit cell volume  (shown in inset of Fig. 1 (g))  as we  move from Ru_0.0  to \nRu_1.0 .  \n3.1.2 X–Ray Photoelectron Spectroscopy  \nFor a proper analysis of the magnetic state of a compound, one should have  an idea about \nthe valence states of the constituent  ions. Therefore, we have performed the photoelectron \nspectroscopy  on selected compounds i.e., on the end members  and one of the substituted \ncompounds . As w e want ed to check the valence state of Ru and Hf  in each other’s surroundings , \nwe have chosen Ru_0.5 compound . Fig. 2 represents the core level XPS spectra of O 1 s, Hf 4 f, \nand Ru 3 p, for the se compound s. The inelastic background of the spectrum was subtracted using \nthe Touggard  method and then the peaks were fitted with the Voigt function. All compounds \nshow two major peaks around 529.98 and 531.53 eV in the photoelectron spectrum of O 1 s (Fig. \n2(a)–(c)). The 529.98 eV peak corresponds to O2- lattice oxygen while the latter resembles the \nsurface chemi sorbed oxygen or the oxygen –vacancy . On fitting XPS spectra of O 1 s, one extra \nweak peak around 532.8 eV is obtained which corresponds to physically adsorbed oxygen  \nspecies on the surface of compo unds. A similar peak was also reported in another perovskite \ncompound  LaFeO 3 [40–41]. Fig . 2 (d) and (e) show the XPS spectrum of Hf cation in Ru_0.0  \nand Ru_0.5 compounds respectively.  The Hf spectrum of Ru_0.0  shows two noticeable peaks at \n16.77  eV and 18.44 eV, which are allocated to spin –orbit splitting components Hf 4 f7/2 and 4 f5/2 \nrespectively . The spin –orbit splitting was calculated to 1.6 eV, which confirms the 4+ valence \nstate of the Hf cation. Another peak observed around 25 eV  corresponds to the loss part . The Hf \n4f7/2 and 4 f5/2 peaks have been shifted to 17.61 eV  and 19.37 eV respectively, in the presence of \nRu surrounding in Ru_0.5 compound.  Also, the asymmetr y observed around  16.29 eV  might \narise from the O -vacancies present in the system. Ru 3 p spectrum for Ru_0.5 and Ru_1.0 \ncompounds ha s been shown in Fig . 2 (f) and (g) respectively. Two peaks at 462.12 eV and \n484.26 eV corresponding to 3 p3/2 and 3 p1/2 spin–orbit splitting states  are noted  in the Ru 3 p \nspectrum of Ru_1.0 compound , and these peaks are shifted to 462.84 eV and 484.90 eV \nrespectively in Ru_0.5 compound . These peaks are accompanied by two satellite peaks, seen as \nhumps on higher binding energy sides. It confirms the  predo minate existence of  4+ state of Ru \ncation [ 41]. Furthermore, the  shifting of  core–level spectrum of Hf 4 f and Ru 3 p to higher 6 \n binding  energy  in the presence of each other’s surroundings in the Ru_0.5 compound has been \nassociated to the disordered arrangement of these elements in B site of perovskite , CaHfO 3. \n3.2 DC magnetization study  \nThe temperature dependent dc susceptibility  (χ (T)) for all the compounds of the series,  \nmeasured under the zero –field (ZFC) and field–cooled (FC) protocols in presence of  magnetic \nfield of 100 Oe is shown in the Fig . 3 (a) – (f). Ru_0.0  shows the crossover from diamagnetic to \nparamagnetic behaviour  around 100 K on decreasing the temperature (Fig. 3 (a)).  Here , it is to be \nnoted that t he minor thermomagnetic irreversibility  observed  between the  ZFC and FC curves is \nan artefact of the measurement . Like HfO 2 [42], this magnetic behaviour of Ru_0. 0 is not  an \nintrinsic property of th e compound but arises from presence of oxygen vacancies , confirmed via \nXPS analysis (a s mentioned in the previous section ). Thus, local magnetic moment induced by \noxygen defects is responsible for the low temperature paramagnetic state of no n-magnetic \nRu_0.0 compound. However,  when  20% of Ru is substitut ed, the χ (T) curve shows a positive \nvalue  in the temperature range 2 – 300 K (Fig. 3 (b)). The magnetisation grows as the \ntemperature is reduced to 2 K, with no bifurcation between  the ZFC and FC curves.  This implies \nthe absence of any magnetic ordering/spin freezing  in this compound.  Also , the observed  \nmagnetic susceptibility of Ru_0.2 is couple of  orders larger than Ru_0.0 compound. Thus,  \ncontribution  of oxygen defects can be neglected  in this case . The presence  of Ru cations  in \nRu_0.2 is sufficient to produce a paramagnetic state  and suppresses the diamagnetic behaviour  \nobserved in the parent compound.  As noted from Fig. 3 (d), o n increasing the Ru  concentration \nto 30%, a maxima at ~ 17 K  (Tf) is observed in the ZFC curve, while  FC curve continues to \nincrease as the temperature is decreased.  The bifurcation between ZFC and FC curves starts \naround the irreversible temperature ( Tirr) ~ 53 K (>>  Tf). A similar kind of behaviour  in χ (T) \ncurves  is observed for compounds with Ru_0. 4 and Ru_0. 5 (Fig. 3 (e) and (f)) . Notably, the \nvalues of Tf and Tirr, as well as the difference between them lessen  as Ru concentration  rises. \nThese observed features indicate the presence of magnetic ordering, yet no feature  or anomaly is \nnoticed in heat capacity data ( as discussed in supplementary  data). This suggests that  these \ncompositions  lack any kind of long–range  ordering . This type  of behaviour is not unusual , and it \nhas also been observed in  other  insulating as well as semiconducting materials  having magnetic \nglass state  or in superparamagnetic system  [22–24, 43–45]. As mentioned earlier , a paramagnetic \nstate is present in Ru_0.2 , while Ru_0 .3 exhibits a magnetic anomaly  at low temperature . Hence, 7 \n to find the percolation threshold at which short range magnetic  interactions starts dominating , we \nhave prepared an intermediate  compound  Ru_0.25 . The ZFC and FC curves of this compound as \na function of temperature are shown in Fig . 3 (c). A clear bifurcation between ZFC and FC \ncurves is noted at Tirr ~ 66 K  while a weak anomaly in the ZFC curve  is noted around  ~ 15 K. \nSuch features might arise due to the local ordering of magnetic moments at higher temperatures \nas compared to the macroscopic ordering temperature. Additionally, t he bifurcation between the \nZFC and FC curves in the other end member, Ru_1.0  (Fig. S2 (a) ), occurs at 85 K, which is \nconsistent with previous studies [ 46].  \nInset of Fig. 3 (b) represents modified Curie Weiss law fitting of χ-1(T) using equation χ = \nχ0+C/(T-θp) (where symbols having their standard meanings), in the paramagnetic region for all \ncompounds (except diamagnet Ru_0.0 ). All the parameters obt ained from the modified Curie –\nWeiss law are given in Table 2. The negative value of the Curie –Weiss tempera ture (θp) \nindicat es the dominance of antiferromagnetic interactions in  Ru_0.2  to Ru_0.5  compounds , and \nin Ru_1.0 (shown in  section  S2 of supplementary data ). In all compositions, a S–type feature is \nnoted in the  χ-1 vs T curve  below paramagnetic range . It indicates the presence of short –range  \nferromagnetic interactions as also reported in CaRu 1-xTixO3 series  [46]. The temperature \nindependent susceptibility ( χ0) is found  to be in order of  10-5 emu/mol –Oe. Also, the \nexperimental  value  of magnetic moment for Ru_1.0  (~ 2.96 μB) is close to the moment of Ru4+ \nion in its low spin ( S = 1) state. The parameters obtained  (shown in section S2 of supplementary \ndata)  for Ru_1.0  matched  well with the literature [ 46]. On incorporating Ru atom at the Hf site of \na non –magnetic matrix, the slope of  χ-1 (T) curves i.e. , Curie –Weiss constant  (C) varies  linearly \nwith the Ru concentration . However,  P does not vary  linearly with x, implying the strength of \nmagnetic interactions among Ru cations  does not follow any pattern . This is not in accordance \nwith the disordered Heisenberg antiferromagnet , where P and C values  scale linearly with \nconcentration of magnetic impurity in the system  [47]. Also,  the high value of P in comparison \nwith Tf indicate s to the presence of frustration  caused due to the presence of  disorder at B–site of \nthe perovskite , which might be  responsible for observed magnetic glass state  [48]. The \nexperimental value of magnetic moment matches  with calculated value of magnetic moment  \n√(xTheor(Ru4+))2), implying  that the valency  of Ru cation is matched  well with the results of \nXPS analysis .  8 \n To further investigate the low –temperature magnetic state, we have measured dc χ (T) of \nRu_0.3, Ru_0.4 and Ru_0.5  compounds at different magnetic fields as shown in the respective \ninsets of Fig . 3 (d) – (f). Both the bifurcation temperature between ZFC  – FC curves (Tirr) and \npeak temperature ( Tf) of ZFC curve  reduce s on increasing the  applied  magnetic field. Such kind \nof features ha s been reported  in unconventional SG systems [ 18, 49–50]. For a conventional SG, \nit is noted,  Tirr ≤ Tf, and below this temperature, FC curve becomes independent of temperature \n[51]. However, some studies report  a temperature dependent FC curve with minima after Tirr or \nTf, in conventional SG s [52, 53]. The observation of  Tirr ≥ Tf has also been noted in insulating SG \nsystems [ 44, 49]. Hence, to determine the nature of the low temperature complex magnetic state \nin our system, isothermal magnetisation and ac susceptibility measurement are carried out .  \nIsothermal magnetization as a function of magnetic field ( M (H)) at 2 K and 300 K for all \nthe prepared compounds are represented in the Fig . 4 (a) and (b). Within ±10 kOe, the M (H) \ncurve for Ru_0.0  (right  panel  of Fig . 4 (a))  show  a linear dependence , but beyond this region, it \nshows the diamagnetic behaviour. The paramagnetic type of behaviour in Ru_0.0 compound  at \nlow temperature and low magnetic field observed in χ (T) and M (H) curves might arise due to \nthe presence of oxygen vacancies.  In Ru_1.0 compound , the linear behaviour of M (H) curve and \nnegative value of P indicates strong  antiferromagnetic interactions  (Fig. S2 of supplementary \ndata). While M (H) curves for Ru_0.2 and Ru_0.25 compounds at 2 K form S –like shape with no \nhysteresis at low field, or  saturation at high field . On further dilut ing the Hf –site ( 0.30 ≤ x ≤ 0.5), \nwe have noticed  the opening of  magnetic hysteresis in the low field region. However, magnetic \nsaturation is not achieved even at 70 kOe.  Such kind of M (H) hysteresis without saturation even \nat 2 K is observed in systems having competing antiferromagnetic and ferromagnetic \ninteractions. The se competing  magnetic interactions are responsible for glassy magnetic state or \nsuperparamagnetic behaviour or  canted antiferromagnetism  [22, 49–52]. The variation of \ncoercive field and remanent  magnetisation  at 2 K  with Ru concentration ha s been displayed in \nFig. 4 (c). Increment in both parameters with Ru concentration  (except Ru_1.0  due to its \nparamagnetic nature)  have been observed, implying magnetic interactions in the system solely \ndepend on  the magnetic entity Ru4+ cations . To get a better idea about the magnetic state, \ntemperature response of the coercivity is noted. For this  purpose,  we have measured isothermal \nM (H) curves at different temperatures below Tirr for Ru_0.3, Ru_0.4 and Ru_0.5  compositions  \n(Fig. 4 (d) – (f)). The substantial magnetic hysteresis has been noticed below the Tf which 9 \n reduces with temperature. The coercive field (Hc) for single domain particle i.e., \nsuperparam agnet varies as  T-1/2, while it follows parabolic curve  for canted antiferromagnet s [5, \n51, 54]. In magnetic glasses, t he value of Hc drops  exponentially  with temperature below Tirr, \nwhich is fitted with the following function  [55]: \n                                                                                                   (1)                                                     \nwhere α is the temperature exponent. The  respective  inset s of Fig . 4 (d) – (f) show the variation \nof Hc with T and fit using  equation  (1). The obtained parameters for all the three compounds  are \nlogged in Table 3. The exponential  decrement  of Hc on increasing temperature ha s been  reported  \nfor systems having slow dynamics of spins or group of spins [ 55–57]. This implies that Ru_0.3, \nRu_0.4 and Ru_0.5  compo unds  have  complex  spin dynamics at low temperature.   \n3.3 AC susceptibility  \nAC susceptibility is a powerful tool to identify the glassiness in the system and its nature \nbecause the relaxation of spins/ cluster of spins  slows down in the  glassy  magnetic  state on \ncooling the system from a paramagnetic region . The maximum relaxation τ = 1/f (f is the \nfrequency of applied Hac) is obtained at freezing temperature Tf, thus, its value shifts to higher \ntemperature on increasing the frequency. Therefore, we have measured temperature dependent ac \nsusceptibility of compounds showing magnetic relaxation behaviour  in the above context. \nCompound s Ru_0.2, Ru_0.25 and Ru_1.0 show no anomaly in ac susceptibility data  and the \nvalue  of ac susceptibility  decreases with increasing temperature irrespective of frequency of \napplied ac magnetic field Hac, indicating the absence of any type of magnetic ordering  in th ese \ncompounds  (shown  in Fig. S3 of supplementary data ). Fig . 5 (a) – (c) presents the temperature \ndependent in –phase ( χ′ (T)) and out –phase ( χ″ (T)) components (in their respective insets) at \ndifferent frequencies (13  – 931 Hz) of the ac magnetic field ( Hac) with amplitude of 1 Oe for \nRu_0.3, Ru_0.4, and Ru_0.5 compounds  respectively . A clear frequency dependent broad peak \naround Tf is detected in the ac susceptibility  (both in –phase and out –phase part) in  terms of both \npeak position and peak intensity, indicative of a glass iness /slow spin dynamics  in the  system. \nAlso, peaks with different frequencies tend to converge below Tf. A SG state appears when \ndisorder or mixed exchange interactions give rise to an atomistic glassy phase with frozen spins \nbelow a well -defined fr eezing temperature. In contrary , in cluster glass ( CG), the clusters having \nregularly arranged spins are frozen below  the freezing temperature. Thus , this observed 10 \n glassiness might arise from the interacting magnetic entities which are larger than atomic spins \nthat constitute conventional SG system , as implied by the convergence of ac susceptibility curves \nat different f requencies below Tf  [58-60].  \nThe ac susceptibility has also been measured at different applied dc fields ( Hdc) with \nfixed Hac (1 Oe) of a single frequency  (331 Hz)  for Ru_0.3 compound, and Ru_0.4 and Ru_0.5 \ncompounds  Fig. 5 (d) – (f) shows the in –phase χ′ (T) component of these compounds. Here the \nstrong dependence of peak intensity on superimposed Hdc is observed. On applying 100 Oe, the \npeak shifts towards the lower temperature side with a decrease in intensity. On further increasing \nHdc, the peak changes to a broad hump and gets flattened. A similar feature has been observed in \ndc χ (T) on increasing the  magnetic field. The occurrence of frequency and Hdc dependent peak \nin ac susceptibility confirms the magnetic glass state in the system  [54].  \nThe variation of ac susceptibility with Hdc could also be analysed  by non –mean field \ntheory (shown in insets of Fig . 5 (d) – (f)). So, we have fitted the field variation of inflection \ntemperature calculated from d χ′ (T)/dT (not shown here) with the following equation  [61] \n                                                                                                                                 (2) \nwhere A is the anisotr opic strength parameter , Tf (0) is the value of Tf in absence of magnetic \nfield and Φ represents the crossover exponent. This Φ exponent distinguishes between weak \nanisotropy regime and strong anisotropy regime. In strong anisotropic (strong irreversibility) \nregime, the T–H phase tra nsition line follows the Almeida –Thouless (AT) line with a value of Φ \n~ 3.  The AT line distinguishes the non-ergodic  (i.e. SG) phase from the  ergodic (i.e.  \nparamagnetic ) phase . While in a weak anisotropic regime, Gabay –Toulouse (GT) line is \nfollowed in the T–H phase diagram, with a value of Φ ~ 1. For Ru_0.3 compound, obtained \nvalue s of Φ ~ 2.059  and Tf (0) = 19.0  ± 0.2 K. In the case of Ru_0.4 and Ru_0.5 compounds, \nvalues  of Φ are increased to 4.55 and 4.097 , and Tf (0) = 20.0 ± 0.1 K and 17.2  ± 0.2 K  \nrespectively. The obtained values of Tf (0) are very close to freezing temperature, as observed in \nfrequency dependent data  of χ′ (T). From values of Φ, it can be concluded that observed glassy \nbehaviour  in Ru  substituted compounds follow the no n–mean field model  and belong to different \nuniversality class . Such kind of behaviour  is also  seen in other magnetic glasses  like Er 5Pd2, Fe2-11 \n xMn xCrAl , Cr0.5Fe0.5Ga, etc [62–64]. To conclude, it can be said that a transformation from weak \nanisotropy to strong anisotropy is observed  in this series  on increasing the Ru concentration.  \nTo further explore the spin /clusters  dynamics, Mydosh parameter δTf = ΔT/TfΔlnf, \nindicating the shift of peak temperature in response of applying Hac of varying frequencies, is \ndetermined from Tf. The shifting of peak (δTf) depends upon the interactions among the magnetic \nentities (spins/magnetic  clusters) . Thus, stronger the interaction among magnetic particles, \nweaker is  the sensitivity of Tf to the frequency  of ac field. The values of δTf for canonical SG’s \nlike AuMn and CuMn is ~  0.005, while for superparamag nets δTf  ~ 0.5 , due to very weak  \ninteractions among particles  [51]. The values δTf for our compounds are given in Table 4. It is \nnoted that the value s lie in–between that of  SG and superparamagnets. However, the  δTf value is \ncomparable to that reported  CG’s in metallic as well as in the insulating system (0.02 –0.06) [ 58–\n66].  \nTo further probe the spin dynamics of the system, the frequency dependent peak is \nanalysed  with the Vo gel–Fulcher (V –F) law : \n          \n                                                                       (3) \nwhere T0 is the V –F temperature (which tells about the strength of interaction between \nspins/clusters), kB is the Boltzmann constant, Ea is the average thermal activation energy, and τ0 \nis the characteristic relaxation time. The scaling according to equation  (3) is represented in Fig . 6 \n(a) – (c) for Ru_0.3, Ru_0.4 and Ru_0.5  compounds. The obtained values are presented in Table \n4. The non –zero value of T0 and agreement with V –F law indicates the presence of finite \ninteractions, in contrast to superparamagnets  or non –interacting spin dynamics [ 48, 51]. The \npresence of finite interactions has also been confirmed from comparable values of Ea/kB and T0. \nIt is to be noted that  in weak interactions regime, the ratio of Ea/kB to T0 is substantially lower \nthan 1 , whereas it is reverse for the  strong interactions  regime . Also, the obtained value of τ0 ~ \n10-6 to 10-8 s, far greater than for conventional SG systems (10-13 s), has also been reported in \nother insulating CG system [ 57]. The temperature dependent relaxation time is also analysed  by \nPower  law described as:  \n       \n        \n                                                           (4) 12 \n where z is the dynamic exponent which describes the slowing down of relaxation, ν is the \ncorrelation length exponent, τ* describes the flipping time for relaxing entities, and Tg is the true \nSG temperature. Fig . 6 (d) – (f) shows the scaling of τ with reduced temperature ε = (Tf/Tg-1). \nAll the parameters are given in Table 4. For SG, zν falls in the range of ~ 4  – 10, and τ* falls in \nthe range  of 10-10 – 10-12 s. For CGs, it is reported that the value of  τ* lies in the range of 10-7 – \n10-10 s [63]. In our case, even though the value of zν lies in the range of SG’s, the value of τ* fall \nin the range reported for CGs. Thus,  presence of Ru at B –site of non –magnetic insulator CaHfO 3 \nleads to formation of insulating CGs at low temperature.  The origin of magnetic  glassiness in \ninsulator s has been explained via the presence of competing magnetic super –exchange \ninteractions  [27] and the geometrical frustration  in the system  [55, 56]. Hence, it can be said that \nin Ru_0.3, Ru_0.4 and Ru_0.5  compounds  the presence of competing magnetic interactions  \nmight be responsible for the CG formation.  Similar kind of behaviour has also been reported in \nother disordered perovskites [2 3–24, 48, 5 7].  \nTo understand the nature of magnetism in this series, the T – x phase diagram , based on \nthe results of dc and ac susceptibility have been plotted in Fig. 7. As mentioned before, the \nnature of magnetism in Ru_0.0 is diamagnetic.  For x ≥ 0.25, p aramagnetic phase noted above \n100 K , whereas,  Ru_0.2 is paramagnet to the lowest measured temperature of 2 K . The \nparamagnetic phase is presented by a grey region in the phase diagram  (Phase I) . On further \ndecreasing the temperature below 100 K,  the irreversibility  between ZFC and FC curves noted, \nindicating toward the local ordering of magnetic moments in contrast  to long ranged  ordering. \nThe irreversibility decreases  on moving from Ru_0.25 to Ru_0.5 . This phase between \nparamagnetic state and frozen state is represented  by violet region  (Phase II)  in the phase \ndiagram.  The CG state has been observed at low temperatures for Ru_0. 3 – 0.5 compound s \n(shown as green coloured region (Phase III) ). The magnetic interactions among randomly \ndistributed Ru cations in no n-magnetic insulator Ru_0.0  are mainly antiferromagnetic, as implied \nfrom the negative value of p. In an  insulator, generally the magnetic interactions are of super -\nexchange in nature . In Ru_0.25 compound, these interactions arise but are not strong enough to \ngive rise to an ordered or frozen state. For this composition, the interconnected network  of \nnearest neighbouring magnetic atoms at the atomic sites is not formed. However, this network  \nbecomes strong  at higher concentration  and a  magnetic glassy state is noted . The observed  \nglassiness could be explained through the  random distribut ion of  Ru cations and competing 13 \n super -exchange interactions between nearest and next nearest neighbouring magnetic cations \nRu4+. The origin magnetic glass state from competing super -exchange interactions is not unusual. \nSuch behaviour has also been observed in Ca(Fe 1/2Nb1/2)O3 [48].  \n3.4 Aging effect and iso thermal rem anent magnetization  \nSince aging effect and isothermal rem anent magnetisation are the features of magnetic \nglass system, we have carried out these measurements on Ru_0.3, Ru_0.4 and Ru_0.5  \ncompo unds  and for comparison, also on Ru_0.25 compound . For the aging effect phenomenon , \nfollowing protocol is carried out. The sample is cooled in ZFC mode from paramagnetic region \nto measuring temperature (2 K ) and then the system is allowed to age for different  waiting time \n(tw) of 10 , 100 and 10 ,000 s. After that, a magnetic field of 500 Oe  is applied and the time \nresponse of ZFC magnetization  is noted  (Fig. 8 (a) – (d)). The value of magnetization and its \ngrowth (except for x = 0.25), gets reduced on increasing tw. This indicates the influence of aging \non the size of magnetic entities  and frozen energy barriers associated with the  non-ergodic phase. \nThis is an obvious feature for a glassy magnetic  system , where spins/magnetic entities fall into a \nmore stable and deeper energy valley on increasing the waiting time tw [65–66]. This age \ndependent phenomenon also confirms the presence of metastable states below Tf in 30 –50% Ru \nsubstituted compounds. On the other hand,  Ru_0.25 compound  also shows the aging effect but \nthe difference between MZFC(t) curves for tw = 10 s and 1000 s  is very small . For the  other  \nmembers of this series  which do not show any frequency dependent feature in ac susceptibility , \nthis measurement was not carried out .  \n To further explore the low –temperature magnetic state of these compounds, isothermal \nremanent magnetic relaxation ( MIRM (t)) was performed at temperatures primarily below  Tf. In \nthis protocol , the system is again cooled in ZFC mode from paramagnetic state to the desired \ntemperature, then a magnetic field greater than coercive field  (Hc) is applied for 20 minutes. \nAfter switching off the field, the MIRM(t) is recorded as a function of time for 1800 s. Such \nprotocol for MIRM(t) is recorded at dif ferent measuring temperatures 5 , 10, and 20 K. In Fig . 8 (e) \n– (h), MIRM(t)/MIRM(0) curves at different temperatures are shown for Ru_0.25 to Ru_0.5 \ncompounds.  Pattern followed by MIRM  (t)/MIRM (0) curves for Ru_0.25 compound  is dif ferent \nfrom the other compounds , implying  the absence of a significant glassy nature  at low \ntemperature . In Ru_0.3, Ru_0.4  and Ru_ 0.5 compounds,  the slowing down of decay rate of the 14 \n remanent magnetization  on decreasing temperature  is clearly observed. The magnetic relaxations \nat these temperatures are well fitted  with the stretched exponential equation:  \n                               \n   \n                                    (5) \nHere M0(H) and M∞(H) are magnetization values at t → 0 and t → ∞ respectively whereas  is \nthe characteristic relaxation time and  is the stretching exponent . The value of  depends upon \nthe energy barrier s involved for relaxation process.   = 0 indicate the absence of any relaxation \nin the system while  = 1 indicate relaxation with single time constant. The intermediate value of \n talks  about the distribution of relaxation times due to the presence of multiple degenerate \nlevels  in the frozen state.  All the parameters  obtained from equation ( 5) are presented in Table 5. \nThe value of M0 (H)/M∞ (H) approaches 1 on decreasing temperature. Also, the relaxation time  \nincreases on lowering the temperature  (shown in inset of Fig. 8 (h)). The variation of \nM0(H)/M∞(H) and  with temperature supports the slowing down of magnetic relaxation on \ndecreasing temperatures and the presence of metastability below  Tf. The value  < 1 indicates the \npresence of magnetic anisotropy in the system. Inset of Fig. 8 (h) shows very weak variation of β \nwith temperature. The value of β is independent of Ru concentration. Also, t he obtained values \nof β ~ 0.5, is close to the values reported in  other  CG compounds like Zn3V3O8, Nd5Ge3, \nCr0.5Ga0.5Ga, and insulating magnetic glass compounds SrTi0.5Mn 0.5O3, Sr 2-xLaxCoNbO 6 [18, 64-\n66]. The non–zero value of  at 20 K (above Tf) suggest s the presence of short –range \ninteractions among magnetic entities even above the Tf. Thus,  the results of aging effect and IRM \nmagnetization also support  the presence CG dynamics in Ru_0.3, Ru_0.4  and Ru_ 0.5 \ncompounds.  \n3.5 Memory effect  \nThe memory effect is also a characteristic feature of glassy system  due to degenerate \nground state  and it  provides insight about  the spin dynamics of such system s. As a result, we \ninvestigated the memory effect and rejuvenation effect on Ru_0.3, Ru_0.4 and Ru_0.5  \ncompounds using ZFC and FC protocols. The results  of this measurement  are shown in Fig . 9. \nFor FC memory effect (Fig . 9 (a) – (c)), the protocol reported in R ef [68] is followed. \nTemperature dependent magnetization has been recorded from 300 – 2 K in the presence of 100 15 \n Oe, with temporary halts at TH = 25, 15, 8 , and 5 K.  At each halt temperature TH, the magnetic \nfield is turned  off and waited for 2 hours. After each stop at waiting t emperature , the field of 100 \nOe is applied again, and the magnetization measurement under field cooled cooling (FCC)  \nprotocol is resumed.  The obtained M (T) curve is presented as MFCCstop\n. At the halt temperatures, \nmagnetization of above –mentioned  compounds decays . It rises again when the magnetic field is \nresumed, resulting in a step –like behaviour  in MFCCstop curve . When the system reaches  2 K, the \nM (T) is measured  immediately  under  FC warming  protocol  of the compounds  without any stop s. \nThe obtained M (T) curve is presented as Mmem\nFCW which also shows the smeared step –like \nanomalies at the halted temperatures  except at 25 K . This behaviour  of Mmem\nFCW  indicates  that \nthe compounds  (for Ru_0.3 to Ru_0.5) in their glassy state remember their earlier  thermal history \nof magnetization.  As TH = 25 K is much beyond  the glass transition temperature Tf, no effect  on \nMmem\nFCW  has been observed.  The compound s are cooled down again in presence of 100 Oe  \nmagnetic field without any pauses  and the FC warming M (T) is recorded , presented as Mref\nFCW. \nThe Mref\nFCW curve does not show  any anomaly at  the previous halted temperatures, indicating the \nthermal history – memory effect of the se compounds  has an intrinsic origin. The FC memory \neffect has been reported  for both  superparamagnet ic and magnetic glass systems [ 69–70]. Hence, \nto further scrutinize the nature of the low –temperature magnetic phase of the system, ZFC \nmemory effect is performed on the  above –mentioned  compounds  (shown in Fig . 9 (d) – (f)). The \nZFC memory effect , which  is a distinctive  signature of SG’s  emerging  from cooperative spin –\nspin interactions , it is absent  in superparamgnetic  systems [ 70–71]. In this pro cedure , the \ncompounds  are cooled to 2 K in absence of field with a 2–hours  halt at temperatures of 25, 15 \nand 8 K. During aging , at the halt temperatures below Tf  (i.e. in non -ergodic state), the growth of \nmagnetic entities  and frozen energy barriers increases simultaneously. At temperature  2 K, \nmagnetic field (100 Oe) is switched on  and Mstop\nZFC(T) was noted  during warming the system to \n300 K . For the reference curve, the system is again cooled down to 2 K in the absence of \nmagnetic field  without any halts  and then Mref\nZFC curve is measured on warming  the system  in \npresence of 100 Oe field.  At the halt temperature of 15 K, the deviation of Mstop\nZFC curve from \nMref\nZFC curve has been clearly observed  due to aging at this temperature. This deviation vanishes  \non increasing the temperature in between the 15 K and Tf. Thus,  the system rejuvenates above 15 \nK even in the non -ergodic phase.  Insets of Fig . 9 (d) – (f) present  the difference between these \ntwo curves  Mref\nZFC - Mstop\nZFC as a function of temperature of respected compounds  with Ru_0.3, 16 \n Ru_0.4 and Ru_0.5. The curve  Mref\nZFC - Mstop\nZFC for all compounds clearly  display s the dips at \nthe halt temperatures of 8 and 15 K , implying the presence  of interaction among the magnetic \nentities instead of independent relaxation s of them giving rise to superparamagnetic behaviour . \nThus, memory and rejuvenation effect are obtained under ZFC protocol. Again,  there is no ZFC \nmemory effect at the  halted temperature of 25 K  because it  is above  Tf. Such behaviour  of \nmemory effect under ZFC and FC procedures supports non–equilibrium characteristics of the \nmagnetic glass dynamics at low temperature  in the Ru substituted compounds from 30 –50% \nconcentration . \nTo further explore the memory effect, cooling  and heating  temperature  cycling has been \nperformed on magnetic relaxation of Ru_0.3, Ru_0.4 and Ru_0.5 compounds. The protocol \ndescribed in Refs. [68, 71] was followed  for this relaxation memory measurement .  \nFor cooling T cycle,  the magnetic relaxation under ZFC and FC protocols are measured.  \nTherefore, the system is cooled down to 8 K (below Tf) in absence of field, then the logarithmic \nincrease of magnetization with time ( M (t)) is recorded in the presence of 50 Oe  for ZFC method \nfor time t1 = 1800 s, respectively. There  after the compound  is quenched to a lower temperature \nof 5 K in the presence and again M (t) is recorded for second time t2 = 1800 s. After that, the \ncompound  is heated  back to 8 K temperature in the presence, then M (t) is recorded for  third time  \nt3 = 1800 s in presence of field. The upper panels of Fig. 10 (a) – (c) represent s the ZFC \nrelaxation  during the cooling  T cycle  for Ru_0.3, Ru_0.4 and  Ru_0.5 compounds respectively . \nNow for FC mode, the system is field cooled (in 50 Oe) down to the temperature below Tf.  On \nreaching the measuring temperature 8 K, magnetic field is switched  off and t hen logarithmic \ndecay of magnetisation as function of time is recorded  for 1800 s at  8, 5 and 8 K respectively . \nThe lower panels of Fig. 10 (a) – (c) shows the magnetic relaxation under FC mode of the  \ncooling cycle.   From the figures it is noted that : (i) relaxation of t3 starts where relaxation of t1 \nends. This clearly indicates that the se compounds  recall  their previous history i.e., before \ntemporarily cooling down in both ZFC and FC cooling cycles and (ii) observation of weak \nrelaxation at t2 during immediate quenching to 5 K, implies the system does not get ample time \nto get relaxed at this temperature.   \nSimilarly,  to see the effect of memory during  heating T cycle , the above protocol s for \nZFC and FC mode  are followed,  and magnetic relaxation of the compound s has been noted for 17 \n the sequence of temperatures 8, 11 and 8 K.  The outcome of the measurements is represented in  \nFig. 10 (e) – (f) for Ru_0.3, Ru_ 0.4 and Ru_0.5 compounds  (in FC mode and ZFC mode). It is \nobserved that t he heating T cycle  appears to erase the pervious memory effect and re –initialise \nthe magnetic relaxation during temporary heating in both ZFC and FC situations . Thus, the \nsystem rejuvenates the magnetic relaxation during positive cycle.  \n The memory effect observed in magnet ic relaxation of glassy state is generally described \nby either Droplet model [ 68, 72] or Hierarchical  model [ 73–74]. In droplet model, the overlap \nlength  lT is introduced which describes the critical length scale at which correlation of spins at \ntwo different temperatures (below magnetic glass transition temperature) is same. Thus,  one \nexpects the symmetric al response of magnetic relaxation irrespective T cycles. However  \nasymmetric response of magnetic relaxation is expected  for Hierarchical  model.  In Hierarchical  \nmodel, the system has multi –valley energy landscape , which corresponds to metastable  \nconfiguration s that existed at temperatures below Tf, and these valleys split into sub valleys on \ndecreasing temperature . The magnetic relaxation  of Ru_0.3, Ru_0.4 and Ru_0.5 compounds  \nduring cooling and heating T cycle follows the Hierarchical  model  i.e., when  the temperature of \nthese compounds is reduced from 8 K to 5 K, the system becomes locked in sub –valleys or \nmetastable states. If the temperature changes significantly, the energy barriers between primary \nvalleys become  too high , and the system does not have enough time to overcome these barriers  \nand get relaxed within the sub valley.  As a result, during the cooling T cycle, weak relaxation is \nobtained at a temporarily halted  temperature of 5 K. On returning to  8 K these sub –valleys merge \nto form  the original  valleys  and the system restores its previous memory. However, in the \nheating T cycle, when the  system returns  to temperature 8 K from temporary halt at  11 K, the  \nvalue of magnetic relaxation has been re –initialised.  This is because valleys at 8 K merge to form \nsuper –valley or the energy barriers among these valleys reduced on increasing temperature to  11 \nK. When  the temperature  is dropped once again  to 8 K , the super  – valley  at 11 K  restored to the \noriginal valleys  at 8 K  but the relative occupancy of each valley has been changed.  Thus,  \nobservation of memory effect in cooling  temperature  cycle and re –initialisation in heating  \ntemperature  cycle in our compounds supports the hierarchical  arrangement of metastable states \nin glassy phase. This observation  in magnetic relaxation as well as analysis of dynamic \nsusceptibility confirms  the presence of magnetic interactions among the clusters of spins rather \nthan independent behaviour of individual spins.  18 \n 4 Conclusions  \nIn conclusion, we  have investigated the phys ical properties of a series of insulating \nperovskite CaHf 1-xRuxO3 (0 ≤ x ≤ 0.50). Our studies  reveal that  long range magnetic interaction  \nis absent across the series. For the 0.30 ≤ x ≤ 0.50 compounds,  magnetic glass dynamics at low \ntemperature is confirmed through both static and dynamic susceptibility. Further investigations \nreveal the  cluster glass nature of the glassy phase . The asymmetric  response  of magneti c \nrelaxation during heating and cooling T cycles  implies that the energy  landscape picture of this \nglassy phase is described in terms of hierarchical model . Our study indicates that the competing \nshort –range  interactions among the inhomogeneous distributed magnetic Ru4+ cations in non –\nmagnetic matrix CaHfO 3 is responsible for the low tem perature cluster glass dynamics in this \ninsulating series . \nAcknowledgements  \nThe authors acknowledge IIT Mandi for the experimental facilities and financial support.  \nData availability statement  \nAll data that support the findings of this study are included within the article.  \n \n \n \n \n \n \n \n \n 19 \n References  \n[1] Phan M H , and Yu S C 2007 J. Magn. Mag . Mat. 308, page 325 –340 \n[2] Mao J, Sui Y, Zhang  X, Su Y, Wang  X, Liu Z, Wang  Y, Zhu R, Wang  Y, Liu W, \nand Tang  J 2011 Appl . Phys . Lett. 98,192510  \n[3] Sharma M K, Singh K, and Mukherjee K 2016 J. Magn . Mag. Mater. 414, 116–121 \n[4] Catalan G, and Scott J F 2009 Adv. Mater . 21, 2463 –2485  \n[5] Sharma M K, Basu T, Mukherjee K, and Sampathkumaran E V 2016 J. Phys.: Condens . \nMatter  28 426003  \n[6] Furdyna J K 1988 J. Appl . Phys . 64, R29  \n[7] Li X, Qi S, Jiang F, Quan Z, and Xu X.  2013  Sci. China Phys . Mech . Astron . 56, 111–123  \n[8] Ohno H, Shen A, Matsukura F, Oiwa A, Endo A, Katsumoto S, Iye Y 1996  Appl . Phys . \nLett. 69, 363  \n[9] Fukumura T, Zhengwu Jin, Ohtomo A, Koinuma H, Kawasaki M 1999 Appl . Phys . Lett. \n75, 3366  \n[10] Lee H J, Jeong S Y, Cho C R, Park C H 2002 Appl . Phys . Lett. 81, 4020  \n[11] Malo S, and Maignan A 2004, Inorg . Chem  43, 8169  \n[12] Zhao Y G, Shinde S R, Ogale S B, Higgins R, Choudhary R J, Kulkarni V N, Lofland S \nE, Lanci C, BuBan J P, Browning N D, Sharma S D and Millis A 2003  Appl . Phys . Lett. 83, 11 \n[13] Lee J S, Khim  Z G, Park Y D, Norton D P, Theodoropoulou N A, Hebard A F, Budai J \nD, Boatner L A, Pearton S J, and Wilson R G 2003 Solid–State Electronics  47, 2225  \n[14] Kahaly M U, and Schwingenschogl U 2014 J. Mater . Chem. A 2, 10379  \n[15] Kahaly M U, Ozdogan K, and Schwingenschogl U 2013 J. Mater . Chem. A 1, 8406  \n[16] Zhang Y, Cui Z, Zhu L, Zhibo Z, Liu H, Wu Q, Wang J, Huang H, Fu Z and Lu Y 2020 \nAppl . Phys . Lett. 117, 052406  \n[17] Paul M S and Tholence J L 1986 Solid State Comm . 60, 621 –624 \n[18] Sharma S, Yadav P, Sau T, Yanda  P K, Baker P J, Silva I da, Sunderasan A, and Lalla N \nP 2019 J. Magn . Magn . Mater . 492, 165671  \n[19] Mydosh J A, Spin glasses – Recent experiments and systems 1978 J. Magn . Magn . Mater . \n7, 237   \n[20] Saito H, Zaets W, Yamagata S, Suzuki Y , and Ando K 2002 J. Appl . Phys . 91, 8085  \n[21] Goya G F, and Sagredo V 2001 Phys . Rev. B 64, 235208  20 \n [22] Wu J, and Leighton C 2003 Phys . Rev. B 67, 174408  \n[23] Shukla R, and Dhaka R S 2018 Phys. Rev. B 97, 024430  \n[24] Bull C L, Playford H Y, Knight K S, Stenning G B G, and Tucker M G 2016 Phys. Rev. \nB 94, 014102  \n[25] Alay -e-Abbas S M, Nazir S, Wong K M, Shaukat A, and Schwingenschlogl U 2014  EPL \n106, 27003  \n[26] Noh M, Choi S, Lee D, Cho M, Jeon C, and Lee Y 2013 New Phys.: Sae Mulli  63 (8), \n939–944  \n[27] Rauwel E, Galeckas A, Rauwel P, Fjellvåg H 2012 Adv. Funct. Mater.  22, 1174 –1179.  \n[28] Shibuya K, Ohnishi T, Lippmaa M, Kawasaki M, and Koinuma H 2004 Appl. Phys. Lett.  \n84, 2142 –2144.  \n[29] Jia W, Jia D, Rodriguez T, Wang Y, Jiang H, Li K, 2007 J. Lumin.  122–123, 55  \n[30] Grezer A, Zych E,  and Kępiński L, 2010 Radiat. Meas.  45, 386 \n[31] Cao G, McCall S, Bolivar J, Shepard M, Freibert F, Henning P, Crow J E, and Yuen T \n1996 Phys . Rev. B 54, 15144  \n[32] Cao G, McCall M, Shepard, Crow J E, and Guertin R P 1997 Phys . Rev. B 56,321  \n[33] Bradarić I M, Matić V M, Savić I, Rakočević Z, Popović M, Destraj D , and Keller H \n2018 Phys . Rev. B 98, 134436  \n[34] Tripathi S, Rana R, Kumar S, Pandey P, Singh R S, and Rana D S 2014  Sci. Rep. 4, 3877  \n[35] Mazin I I, and Singh D J 1997 Phys . Rev. B 56, 2556  \n[36] Yoshimura K, Imai T, Kiyama T, Thurber K R, Hunt A W, and Kosuge  K 1999 Phys . \nRev. Lett. 83, 4397  \n[37] Felner I, Nowik I, Bradaric I, and Gospodinov M 2000 Phys . Rev. B 62,11332  \n[38] Felner I, Asaf U, Nowik I, and Bradaric I 2000  Phys . Rev. B 66, 054418  \n[39] Hardy V, Raveau B, Retoux R, Barrier N, and Maigan A 2006  Phys . Rev. B 73, 094418  \n[40] Moulder J F, Sticle W F, Sobol P E, and Bomben K D 1992 Handbook of X –ray \nphotoelectron spectroscopy  Perkin –Elmer Corp.,  Eden Prairie, MN  \n[41] Li X, Zhang H, Liu X, Li S, and Zhao M 1994 Mater. Chem. Phys.  38, 355 –362 \n[42] Coey J M D, Venkatesan M, Stamenov  P, Fitzgerald C B, and Dorneles L S 2005 Phys . \nRev. B 72, 024450  \n[43] Beauvillain P, Dupas C, Renard J P, and Veillet P  1984 Phys . Rev. B 29, 4086.  21 \n [44] Westerholt K, and Bach H 1985 Phys . Rev. B 31, 7151.  \n[45] Borgermann F J, Maletta H, and Zinn W 1987 Phys . Rev. B 35, 8455.  \n[46] He T, and Cava R J 2001 Phys . Rev. B 63,172403  \n[47] Foya G F, and Sagredo V  2001  Phys . Rev. B 64, 235208  \n[48] Kumar A, Senyshyn A, and Pandey D 2019 Phys . Rev. B 99, 214425  \n[49] Maletta H, and Felsch W 1979 Phys . Rev. B 20, 1245  \n[50]  Sharma M K, and Mukherjee  K 2018 J. Magn . Magn . Mater . 466, 317 \n[51] Mydosh J A, Spin Glasses: An Experimental Introduction  (Taylor & Francis, London \n1993)  \n[52] Bouchiat H and Mailly D 1985 J. Appl. Phys.  57, 3453  \n[53] Pal S, Kumar K, Banerjee A, Roy S B, and Nigam A K 2020 J. Phys.: Condens. \nMatter  33, 025801  \n[54] Sharma  M K , Basu  T, Mukherjee  K, and Sampathkumaran E V 2017 J. Phys.: Condens . \nMatter 29, 085801  \n[55] Dong Q Y, She B J, Chen J, Shen J, and Sun J R  2011 Solid State Commun . 151, 112  \n[56] Dho J, Kim W S, and Hur N H 2002 Phys . Rev. Lett. 89, 027202  \n[57] Kumar A, and Pandeya D  2020 J. Magn . Magn . Mater . 511, 166964  \n[58] Kumar P A, Nag A, Mathieu R, Das R, Ray S, Nordblad P, Hossain A, Cherian D, \nVenero D A, Schmitt L D B, Karis O, and Sharma D D 2020 Phys . Rev. Res. 2, 043344  \n[59] Maji  B, Suresh K G, and Nigam A K 2011 J. Phys .: Condens . Matter  23, 115601  \n[60] Sharma M K, Kaur G, and Mukherjee K 2019, J. Alloys Compd . 782, 10–16 \n[61] Malozemo A P, Barnes S E, and Barbara B,  1983 Phys . Rev. Lett. 51, 1704.  \n[62] Sharma M K, Yadav K, and Mukherjee  K 2018 J. Phys.: Condens . Matter 30, 215803  \n[63] Yadav K, Sharma M K, and Mukherjee K 2019 Sci. Rep. 9, 15888  \n[64] Bagh  P, Baral P R, and Nath R  2018 Phys. Rev. B 98, 144436  \n[65] Chakrabarty T, Mahajan A V, and Kundu S J. Phys .: Condens . Matter 26, 405601  \n[66] Kumar A, Schwarz B, Ehrenberg H, and Dhaka R S 2020 Phys . Rev. B 102, 184414  \n[67] Binder K, and Young A P 1986 Rev. Mod. Phys . 58, 801  \n[68] Sun Y, Salamon M B, Garnier K, and Averback R S 2003  Phys . Rev. Lett. 91, 167206  22 \n [69] De D, Karmakar A, Bhunia M K, Bhaumik A, Majumdar S, and Giri S 2012 J. Appl . \nPhys . 11, 033919; Markovich V, Fita I, Wisniewski A, Jung G, Mogilyansky D, Puzniak R, \nTitleman L, and Gorodetsky G 2010 Phys . Rev. B 81, 134440  \n[70] Sasaki M, Jönsson P E, and Takayama H 2005 Phys . Rev. B 71, 104405  \n[71] Mukherjee K, and Banerjee A 2008 Phys. Rev. B 77, 024430  \n[72] Lefloch F, Hammann J, Ocio M, and E Vincent 1992 Eur. Phys . Lett. 18, 647  \n[73] Fisher D S, and Huse D A 1988 Phys. Rev. B  38, 373; Fisher D S, and Huse D A 1988 \nPhys . Rev. B 38, 386  \n[74] Mimiya H , and Nimori  S 2012  J. Appl . Phys . 111, 07E147  23 \n Table 1 Structural  parameters  obtained from Rietveld refinement of XRD pattern using Fullprof \nsuit Software   \nCompounds  a (Å) b (Å) c (Å) V (Å3) Phase % χ2 \nRu_0.0  5.733 (1) 7.980  (0) 5.570  (1) 254.83  (1) 100 3.08 \nRu_0.2  5.692 (1) 7.935 (1) 5.546  (2) 250.48  (1) 100 2.21 \nRu_0.25  5.681 (0) 7.923 (1) 5.540 (0) 249.34  (2) 100 2.69 \nRu_0.3  5.686 (1) 7.921 (1) 5.541( 1) 249.55  (4) 100 2.58 \nRu_0.4  Phase 1 5.684 (1) 7.918 (1) 5.536  (1) 249.14  (3) 98.29  \n2.33 \nPhase2  5.542  (0) 7.670  (4) 5.408  (6) 229.9 0 (4) 1.71 \nRu_0.5  Phase1  5.672 (0) 7.909 (1) 5.536 (0) 248.34  (3) 95.24  \n1.95 \nPhase2  5.561  (1) 7.717  (3) 5.358  (4) 230.0 0 (2) 4.76 \nRu_1.0  5.528  (1) 7.663 (1) 5.359 (1) 227.0 2 (1) 100 1.97 \nTable 2 Parameters obtained from Curie –Weiss fit on χ-1 vs T at an applied field of 100 Oe \nCompounds  Tf (K) χ0 10-5 (emu/mol/Oe)  θP (K) μeff (μB) μtheor (μB) \nRu_0.2  - 3.32 (3) -84.9(2) 1.25 1.26 \nRu_0.25  23.58  6.45 (1) -194.2(1) 1.46 1.41 \nRu_0.3  17.00 3.53 (1)  -111.4(1) 1.58 1.54 \nRu_0.4  17.80  3.29 (3) -35.8(1) 1.51 1.78 \nRu_0.5  14.57  3.52 (2) -65.2(1) 1.98 2.00 \nTable 3 Parameters obtained from equation (1) fitted on coercive field Hc vs T \nCompounds  H0 (kOe)  α (K-1) \nRu_0.3  3.52 (22) 0.21 (2) \nRu_0.4  8.96 (13) 0.33 (1) \nRu_0.5  3.43 (22) 0.23 (2) \n 24 \n Table 4 Mydosh parameters and p arameters obtained from Vogel –Fulcher and power law fitted \non Tf obtained from ac susceptibility  \nCompounds  Tf (K) δTf Vogel –Fulcher law  Power law  \nT0 (K)   \n   (K) τ0 (s) 10-7 Tg (K) zν τ*\n (s) 10-8 \nRu_0.3  17.89  0.036  16.0 21 (1) 10.00 (0) 16.9  5.6 (2) 1.10 (1)  \nRu_0.4  19.29  0.028 17.7  18 (1) 12.50 (1)  18.6  4.9 (3) 1.00 (1) \nRu_0.5  16.50 0.030  14.0 39 (1) 0.11 (2)  15.5  6.4 (1) 0.14 (4) \nTable 5 Parameters obtained from stretched exponential formula using equation ( 5) on MIRM vs t \nCompounds  T (K) M∞/M0   (s) \nRu_0.3  5 0.931  0.465 (1) 540 (2) \n10 0.830 0.477  (1) 493 (2) \n20 0.810  0.480  (1) 403 (2) \nRu_0.4  5 0.925 0.498  (1) 529 (3) \n10 0.833 0.490  (1) 524 (3) \n20 0.778  0.482  (1) 420 (2) \nRu_0.5  5 0.927  0.484  (1) 619 (6) \n10 0.845  0.478  (1) 487 (3) \n20 0.807 0.476  (1) 398 (2) \n \n 25 \n  Figures : \n \nFigure 1 Rietveld refined X –ray diffraction pattern of CaHf 1-xRuxO3: (a) Ru_0. 0 (b) Ru_0.2 (c) \nRu_0.25 (d) Ru_0.3 (e) Ru_0.4 ( f) Ru_0.5 ( g) Ru_1.0 and (h) shows the shifting of peaks with \nincreasing Ru doping . Inset of ( g) shows the variation of unit cell v olume  of these compounds.  \n26 \n   \nFigure 2 XPS spectra of (a) – (c) O 1s; (d) and (e) Hf 4 f; (f) and (g) Ru 3 p; of Ru_0.0, Ru_0.5 \nand Ru_1.0  compounds  respectively . The black circles present the experimental data, and the \nresultant fit from individual deconvoluted peaks  (represented in blue, pink, green and cyan \ncolours) ) is presented by solid red line.  \n27 \n   \nFigure 3 Temperature dependent dc χ under ZFC and FC protocols  at 100 Oe for (a) Ru_0.0 (b) \nRu_0.2 ( c) Ru_0.25 ( d) Ru_0.3 ( e) Ru_0.4 (f) Ru_0.5 compounds . Inset of ( b) shows  χ-1 vs T \nwith Curie -Weiss fit (solid red line). Insets  of (d), (e) and ( f) show  illustrated view of  ZFC – FC \ncurves under different magnetic field s (0.1 – 10.0 kOe) . \n28 \n   \nFigure 4 Isothermal M (H) curves at (a) 2 K and (b) 300 K for CaHf 1-xRuxO3 compounds  (c) \nvariation of coerciv e field  (Hc) and remanent magnetization  (Mr) at 2 K  with increasing  Ru \nconc entration.  Isothermal M (H) curves at different T are shown for (d) Ru_0.3 (e) Ru_0.4 and  \n(f) Ru_0.5 compounds . Respective in sets show  the variation of Hc with temperature  and solid red \nline represents the  fit using equation (1).  \n29 \n   \nFigure 5 Temperature dependent i n–phase (χ′ (T)) component s of ac susceptibility  is shown  for \n(a) Ru_0.3 (b) Ru_0.4 (c) Ru_0.5 and their respective insets  show the out–phase components ( χ″ \n(T)). Fig (d) – (f) presents the effect of Hdc on χ′ (T) and their respective insets present the H–T \nphase diagram with fit according to equation ( 2). \n30 \n  \nFigure 6 (a) – (c) Dynamic scaling on frequency dependent Tf by Vogel –Fulcher law fit of \nrelaxation time ( ) as a function of reduced temperature 1/( T0-Tf) using equation ( 3) and (d) – (f) \nCritical slowing down of relaxation time as function of reduced temperature log ( Tf/Tg-1) using \nPower law equation (4) in (d) – (f) for Ru_0.30, Ru_0.4 , and Ru_  0.5 compounds.  \n31 \n  \nFigure 7 T-x phase diagram of CaHf 1-xRuxO3 series.  \n \n32 \n  \nFigure 8 (a) – (d) shows the aging effect on MZFC(t) curves at 2 K for different waiting times ( tw \n= 100 s, 1000 s and 10000 s)  (e) shows the isothermal remanent magnetization of Ru_0.25  (f) – \n(g) shows the isothermal remanent  magnetization with fitting  (solid red line)  using  equation ( 5) \nat different temperatures for CaHf 1-xRuxO3 (Ru_0. 3 to Ru_0.5 ) compounds.  Inset of ( f) shows \ntemperature variation of  and  of these compounds.  \n  \n33 \n  \nFigure 9 Memory effect as a function of temperature ( a) – (c) under FC protocol  (d) –(f) under \nZFC protocol  for Ru_0.3, Ru_0.4 and Ru_0.5 compounds. Respective  insets  of (d) – (f) show the \ntemperature response of Mref\nZFC-Mstop\nZFC. \n34 \n  \nFigure 10 Magnetic relaxations under ZFC and FC protocols during (a) – (c) cooling  T cycle  (d) \n– (f) heating  T cycle for Ru_0.3, Ru_0.4 and Ru_0.5 compounds  respectively . \n \n \n35 \n Supplementary data  \nEmergence of low –temperature glassy dynamics in Ru substituted  non–\nmagnetic insulator CaHfO 3  \nGurpreet Kaur and K. Mukherjee  \nSchool of Basic Sciences, Indian Institute of Technology Mandi, Mandi 175005, Himachal \nPradesh, India  \n \nS1. Raman Spectroscopy  \n \nRaman spectroscopy is used to understand the local structure of Ru_0.0, Ru_0.5, and  Ru_1.0 \ncompounds.  Hence to obtain  Raman spectrum, Horiba HR spectrometer is used with 532 nm \nlaser as an excitation source with back scattering geometry.   Generally, orthorhombic perovskite \ncompound has 24 Raman active modes (7A g + 5B 1g + 7B 2g + 5B 3g). Due to low intensity or \noverlapping of modes,  we have observed only 10 and 8 Raman active modes in Ru_0.0 and \nRu_1.0 compound respectively, from 100 –600 cm-1 (shown in Fig. S1), which is consistent with \nliterature  [1, 2]. The Raman modes observed in Ru_0.5 are a combination of modes obtained  in \nRu_0.0  and Ru_1.0. This confirms that the crystal structure of the doped compound resembles its \nend members. The sharpness of peaks reduces when both Ru4+ and Hf4+ ions are present at the B \nsite. This might be due to the presence of disorder in the doped compound. The nomenclature of \nRaman modes for Ru_1.0 is  obtained using Ref [2]. The peak at 262 cm-1 in the Ru_0.0  spectrum \ncorresponds to A g(3) peak   (~ 268 cm-1), as observed in Ru_1.0 . This peak shifts to 253.5 cm-1 \nin Ru_0.5  compound. Thi s implies that the distortion in BO 6 octahedra is increased due to the \npresence of both cations, Ru4+ and Hf4+, at B –site of  this perovskite.  36 \n         \nFigure S1 Raman spectrum for Ru_0.0 , Ru_0.5 and Ru_1.0 compounds at room temperature.  \n \n \nS2. DC and AC  Magnetisation of CaRuO 3 \nFig. S2 shows temperature and magnetic field dependent dc magnetisation of CaRuO 3 \n(Ru_1.0). The bifurcation between the ZFC and FC curves (Fig. S2 (a)) in this compound occurs \nat 85 K, which is consistent with previous studies [3, 4 ]. The linear behaviour of M (H) curves at \n300 and 2 K (Fig. S2 (b)), and the negative value of  (= -168.47±0.61) obtained from  modified \nCurie –Weiss law (inset of Fig. S 2 (a)) indicates to the presence of dominant antiferromagnetic \ninteractions. The value of magnetic moment obtained from Curie –Weiss constant (2.96 B) is \nclose to the moment of Ru4+ ion in its low spin ( S = 1) state . To further explore the low \ntemperature magnetic state ac susceptibility as function of temperature at different frequencies \nhas been performed (shown in Fig. S3 (c)). No frequency or temperature dependent anomaly has \nbeen obtained indicating the absence magn etic ordering in this compound. This has been further \nconfirmed by the heat capacity data (Fig . S4).  \n37 \n   \nFigure S2 ( a) Temperature and ( b) magnetic field dependent dc magnetisation of Ru_1.0 \ncompound. Inset shows the modified Curie –Weiss fit on χ–1 (T). \n \nFigure S3 Temperature dependent in –phase  component  (χ′ (T)) of ac susceptibility at different \nfrequencies for ( a) Ru_0.2, ( b) Ru_0.25, and ( c) Ru_1.0 compounds.  \n38 \n S3. Heat capacity  \nTo further investigate the physical  properties in th is series , temperature dependent heat capacity \n(C (T)) for selected compounds (i.e. for Ru_0.0, Ru_0.25, Ru_0.4, and Ru_1.0) is measured from \n2–200 K in O Oe (shown in Fig. S4 (a)). Inset Fig. S4 (a) shows the C/T vs T of these \ncompounds in 2 –15 K range. The nonexistence of any anomaly in temperature dependent C and \nC/T in compounds Ru_0.0 and Ru_0.25 indicate the absence of magnetic ordering in these \ncompounds.  The Ru_0.4 compound shows a sharp peak at 7.31 K far below Tf and this anomaly \nis field independent (not shown here).  Also, no kind of long –range magnetic ordering was \nobserved in static as well as dynamic magnetic susceptibility, which abolishes its magnetic \norigin. Thus, this anomaly might arise from structural phase transition. The Ru_1.0 compound \nshows an upturn belo w 7 K in C/T vs T and the value of C/T approaches 81  mJ/mol –K2 as T \napproaches  to zero, consistent with Ref [3]. It has also been noted that the value of heat capacity \nat low temperatures increases with increasing Ru  content in the system. This might happen due \nto the magnetic contribution of Ru4+cations.   \nTo find the non –magnetic contribution, we have fitted the zero –field heat capacity ( C(T)) \nfrom 100 –200 K by the following equation.  \n               \n    \n        \n           \n \n               \n   \n    \n \n    \n                   (1) \nwhere      \n    \nThe first term accounts for the electronic contribution while the second and third terms \nfor lattice contribution to the heat capacity at high temperature. The parameters γ, θD and θE are \nthe Sommerfeld coefficient, Debye temperature, and Einstein temperat ure respectively. The \nvalues of m and (1 –m) give information about the contribution of the Debye and Einstein model \nto the lattice part of heat capacity, respectively. The value of Sommerfeld coefficient ( γ = 18  \nmJ/mol –K2) and Debye temperature ( θD = 431.6 8 K) for Ru _1.0 are close to the values repo rted \nin ref [37,  39], with Einstein temperature, θE = 897.73 K. The values of γ, θD and θE for other \ncompounds are tabulated in Table S1. As we move from insulating Ru_0.0  to metallic Ru_1.0 , \nthe electronic contribution to heat capacity increases thus γ increases from zero to 18 mJ/mol –39 \n K2. Also, the θD and θE decreases while replacing Ru by Hf. To find the magnetic contribution at \nlow temperature, we have subtracted the fitted non –magnetic part by equation ( 1) from the total \nheat capacity as shown in Fig . S4 (b). No anomaly around Tf is observed in C vs T but a broad \npeak at ~ 50 K in Cm vs T, have been reported in well –known magnetic glass system. Maximum \nentropy change calculated for Ru_0.25, Ru_0. 4 and Ru_1.0 compositions using        \n    \nis much smaller from the expected value Rln(2S+1), unlike the case of long range ordered \nsystems [5]. Such significant reduction in entropy change indicates the presence of multiple \ndegenerate s tates at low temperature [6,7].  It has also been noted that the magnetic contribution \nincreases with the concentration of magnetic ion Ru4+ in the system.    \n \nFigure S4 (a) Temperature dependent heat capacity at 0 Oe for selected compounds Ru_0 .0, \nRu_0.25, Ru_0.4 and Ru_1 .0, with red solid line fit by equation (5), inset shows zoomed view of \nlow temperature heat capacity. ( b) shows Cm vs T for Ru_0.25, Ru_0.4 and Ru_ 1.00 compounds.  \n \nTable S1 Parameters obtained from equation (1) fitted on temperature dependent heat capacity at \nzero field.  \nCompounds  γ (mJ/mol –K2) θD (K) θE (K) m \nRu_0.0  0 328.4  (3) 671.3  (5) 0.537  \nRu_0.25  1 339.6  (2) 649.7  (3) 0.553  \nRu_0.4  2.6 389.5  (2) 759.0 (0)  0.640  \nRu_1.0  18 431.7  (3) 897.0 (0)  0.650 \n \n \n40 \n References  \n[1] Noh M, Choi  S, Lee  D, Cho  M, Jeon  J, and Lee  Y 2013 New Phys.: Sae Mulli  63 (8), \n939–944 \n[2] Kolev  N, Chen  C L, Gospodinov  M, Bontchev  R P, Popov  V N , Litvinchuk  A P, \nAbrashev  M V, Hadjiev  V G,  and Iliev  M N 2002 Phys. Rev. B 66, 014101  \n[3] Baran A, Zorkovskà A, Kajńakovà M, Śebek J, Śantavà E, Bradaric I, and Feher A 2012 \nPhys. Status Solidi B 249, 1607  \n[4] He T, and Cava  R J 2001 Phys. Rev. B 63,172403  \n[5] Werner J, Koo C , and Klingeler R 2016 Phys. Rev. B 94, 104408.  \n[6] Chakrabarty T, Mahajan A V, and Kundu S J. Phys.: Condens. Mat. 26, 405601  \n[7] Kundu S, Dey T, Mahajan A V, and Büttgen 2020 J. Phys.: Condens. Matter 32 \n115601  \n \n \n \n "    },    {        "title": "2111.12342v1.Thermal_generation_of_droplet_soliton_in_chiral_magnet.pdf",        "content": "Thermal generation of droplet soliton in chiral magnet\nVladyslav M. Kuchkin,1, 2,\u0003Pavel F. Bessarab,3, 4and Nikolai S. Kiselev1\n1Peter Grünberg Institute and Institute for Advanced Simulation,\nForschungszentrum Jülich and JARA, 52425 Jülich, Germany\n2Department of Physics, RWTH Aachen University, 52056 Aachen, Germany\n3Science Institute of the University of Iceland, 107 Reykjavík, Iceland\n4ITMO University, 197101 St. Petersburg, Russia\n(Dated: November 25, 2021)\nControlled creation of localized magnetic textures beyond conventional \u0019-skyrmions is an impor-\ntant problem in the field of magnetism. Here by means of spin dynamics simulations, Monte Carlo\nsimulations and harmonic transition state theory we demonstrate that an elementary chiral mag-\nnetic soliton with zero topological charge – the chiral droplet – can be reliably created by thermal\nfluctuations in the presence of the tilted magnetic field. The proposed protocol relies on an unusual\nkinetics combining the effects of the entropic stabilization and low energy barrier for the nucleation\nof a topologically-trivial state. Following this protocol by varying temperature and the tilt of the\nexternal magnetic field one can selectively generate chiral droplets or \u0019-skyrmions in a single sys-\ntem. The coexistence of two distinct magnetic solitons establishes a basis for a rich magnetization\ndynamics and opens up the possibility for the construction of more complex magnetic textures such\nas skyrmion bags and skyrmions with chiral kinks.\nThe model of chiral magnets allows surprisingly many\nspatially localized statically stable solutions. Together\nwith originally reported solutions, also known as k\u0019-\nskyrmions (Sks) [1] a large diversity of non-axially sym-\nmetric solitons with an arbitrary topological index has\nrecently been discovered in the two-dimensional (2D)\nmodel of chiral magnet. The latter include skyrmion\nbags [2, 3] and Sks with chiral kinks (CKs) [4–7]. Re-\ncently, the direct observation of skyrmion bags by means\nof Lorentz transmission electron microscopy and their\ncurrent-induced motion have been reported in Ref. [8].\nThe experimental evidence for CKs has been provided\nin Ref. [9]. Co-existence of various types of solitons in\na single system is fundamentally interesting and techno-\nlogically appealing. However, since the localized states\nbeyond conventional Sks are typically metastable states,\ntheir controllable nucleation is challenging.\nHere we suggest a reliable protocol for generating an\nelementary magnetic soliton containing a single CK –\nthe chiral droplet (CD), also referred to as a chimera\nskyrmion [10] – by means of thermal fluctuations and\noblique magnetic field. We refer to CD as an elementary\nchiral soliton because it is the most compact non-axially\nsymmetric soliton containing only one CK. The CD tex-\nture has previously been reported as a statically stable\nsolution [5, 10] and a transient state during the asym-\nmetric Sk collapse [11, 12]. In contrast to k\u0019-skyrmions,\nthe interparticle interaction potentials for CD with other\nsolitons are a strongly asymmetric due to the presence\nof the CK [4]. As a consequence, CDs may attract or\nrepel other solitons depending on their mutual orienta-\ntion. This provides a basis for the skyrmion fusion, and,\nthereby, creation of more complex magnetic textures.\nAlthough CDs represent excitations in the ferromag-\nnetic (FM) background, their large entropy enables en-tropic stabilization, similar to what was reported for con-\nventional\u0019-skyrmions [13–15]. On the other hand, CDs\nbelong to a class of topologically trivial solitons [5]. Be-\ncause of that one may expect lower energy barriers for\ntheir nucleation compared to that for topologically non-\ntrivial textures. As a result, there are prerequisites for\nan effective thermal generation of CDs. Indeed, we found\nthat under tilted magnetic field and moderate thermal\nfluctuations, the spontaneous nucleation of CDs domi-\nnates the\u0019-Sk nucleation by several orders of magni-\ntude. Noteworthy, by varying the temperature and the\ntilt angle of the external field one can selectively nucleate\neither\u0019-Sks or CDs. These findings are supported by the\nconsistency of stochastic Landau-Lifshitz-Gilbert (LLG)\nsimulations, Monte Carlo simulations, and analysis based\non the transition state theory.\nWe consider a classical spin Hamiltonian on a square\nlattice:\nE=\u0000JX\nhi;jini\u0001nj\u0000X\nhi;jiDij\u0001[ni\u0002nj]\u0000\u0016sBX\nini;(1)\nwereniis the normalized magnetization vector at lat-\ntice sitei,JandD=D^rijare the Heisenberg exchange\nconstantandDzyaloshinskii-Moriya(DM)vector, respec-\ntively, ^rijistheunitvectorbetweensites iandj,\u0016sisthe\nmagnitude of the magnetic moment at each site, and B\nis the external magnetic field. The symbol hi;jidenotes\nsummation over unique nearest neighbor pairs. The ratio\nbetweenJandDdefines the equilibrium period of helical\nspin spirals, LD= 2\u0019Ja=D, withabeing the lattice con-\nstant, and characteristic magnetic field, BD=D2=(J\u0016s).\nWe consider the case when magnetic filed is tilted\nwith respect to the plane normal, h=B=BD=\nh(sin#cos';sin#sin';cos#)and parametrized by the\npolar angle #and azimuthal angle '. For the parame-\nters ofJandDused in our simulations and providing aarXiv:2111.12342v1  [cond-mat.mes-hall]  24 Nov 20212\nBext Bext(b) (a) T=0.18 T=0.3 T=0.4 T=0.18 T=0.3 T=0.4\nFIG. 1. (a)Illustrates the case of perpendicular magnetic field, h= 0:643,#= 0and(b)corresponds to the tilted magnetic\nfield,h= 0:645,#= 0:4. The simulations were performed on a square domain, Lx=Ly= 8LD, with periodic boundary\nconditions in the xy-plane, and n(r)jjBextin the initial state. The top row of images represents the snapshots of the system at\ndifferent temperatures taken at thermal equilibrium after \u0018106LLG iterations. Each image in the bottom raw corresponds to\nthe top image after setting T= 0and energy relaxation.\nrelatively large LD= 64a, the Hamiltonian (1) becomes\nnearly isotropic in the xy-plane. In this case, the choice\nof angle'does not affect the results, but for definiteness\nwe fix'=\u0000\u0019=4.\nWe simulate spin dynamics at finite temperature using\nthe stochastic LLG equation:\n@ni\n@t=\u0000ni\u0002\u0000\nBi\ne\u000b+Bi\n\ruc\u0001\n+\u000bni\u0002@ni\n@t;(2)\nwheretis a dimensionless time scaled by J\r\u0016\u00001\ns, with\n\rbeing the gyromagnetic ratio, \u000bis the Gilbert damp-\ning parameter, Bi\ne\u000b=\u00001\nJ@E\n@niis a dimensionless effec-\ntive field and Bi\n\rucis the the fluctuating field represent-\ning uncorrelated Gaussian white noise with correlation\ncoefficient proportional to temperature, T. For the nu-\nmerical integration of Eq. (2), we use the semi-implicit\nmethod provided in Ref. [16] assuming \u000b= 0:3and time\nstep \u0001t= 0:01. For the chosen coupling parameters\nwe estimate the critical temperature, Tc'0:7J=kB(see\nRef. [18]). For the results presented below, the tempera-\nture is always T <T cand given in units of J=kB.\nThe top row of images in Fig. 1 provides representative\nsnapshots of the LLG simulations showing the system at\nvarious temperatures and different tilt angles of the ex-\nternal magnetic field, #= 0inaand#= 0:4inb. To\nmake the presence of the localized magnetic textures in\nthe system more evident, in the bottom row of images we\nprovide corresponding snapshots of the system after cool-\ning by setting T= 0in (2). In the case of perpendicular\nmagnetic field, #= 0we observe spontaneous nucleation\nof\u0019-Sks only, while at tilted magnetic field, #= 0:4, we\nobserve the nucleation of CDs. Noticeably, the temper-\nature required for the nucleation of CD is significantly\nlower than that for the \u0019-Sk nucleation, compare the\nsnapshots in aandbatT= 0:18J=kB. Thus, apply-ing the tilted field and varying the temperature one can\nselectively nucleate either CDs or \u0019-Sks. At high tem-\nperature, however, in both cases, we observe emergence\nof\u0019-Sks which in appropriate range of fields tend to form\na regular lattice. There is a critical tilt angle, #c\u001950\u000e,\nabove which the \u0019-Sk lattice becomes unstable [17].\nThe CD orientation in 2D space can be defined by the\nin-plane component of the net magnetization of the CD,\nm=P\nini\u0000n0, where n0is the magnetization far from\nthe soliton n0=n(r), forr!1. In the presence of a\ntilted magnetic field, vector mof CD is always parallel\nto the in-plane projection of the external field, h.\nFigure 2 illustrates the details of the CD nucleation\nprocess, as obtained from the LLG simulations. The\nblack curve in Fig.2 ashows an averaged out-of-plane\ncomponent of magnetization, N=P\nini;z=Nwherei\nruns over all Nspins. With time, Nconverges to its\nequilibrium value, which for chosen parameters equals\n0:74. The Monte Carlo (MC) simulations [18] are fully\nconsistent with the results of LLG simulations and show\nsimilar behavior of Nin reaching thermal equilibrium.\n\u0019-Sks are characterized by topological charge Q=\u00001,\nhence the event of their nucleation can be identified by\ncalculation of Q. In contrast, the CDs have zero topo-\nlogical charge, which makes this approach not applica-\nble. To identify the presence of CDs in the system we\nemploy an alternative method based on time tracing of\nthe net magnetization, as described in the following. We\nsplit the whole simulated domain into overlapping sub-\ndomains \njof fixedLD\u0002LD-size containing N0=L2\nD=a2\nspins. Taking into account the periodic boundary condi-\ntions, the total number of such sub-domains equals the\nnumber of spins in the system, N. At each time step,\nwe calculate the averaged out-of-plane magnetization for\neach sub-domain, Nj=P\nini;z=N0,i2\nj. Then, we\nidentify the sub-domain \nmwith minimalNjdenoted3\n(a)\nAfter cooling\n6×106\n2×1064×106\nLDy\nx\n2 4 0\nLLG iterations×10-66 8 100.2\n00.60.8\n0.41.0Instant magnetization\ny\nx(b) (c)\nthreshold valuem\nFIG. 2. (a)shows the evolution of the average magnetization, N, (black) and its maximal value, Nm, (red) at temperature\nT= 0:18, as obtained in the LLG simulations. The dashed blue line corresponds to the threshold value of 0:3forNm. The\nposition of sub-domain \nmcorresponding to Nmis given in (b), the colors encode the time [see the color bar in (a)]. Blue\ndots in (a)mark instants of time for which magnetic textures are shown in (c). The white squares of size LD\u0002LDcorrespond\nto the sub-domains \nm. The right bottom image in (c)is obtained after relaxation at T= 0.\nNm. The red curve in Fig.2 ashows representative de-\npendency ofNmon the LLG simulation step. The event\nof soliton nucleation is signaled by a drop of Nmbelow\nan empirically estimated threshold value, and then con-\nfirmed by abrupt cooling, see corresponding images in\nFig. 2 c. The expected position of the soliton is given by\nthe coordinates of the \nm, as depicted in Fig.2 b. The\nCD nucleation is well reproduced in MC simulations [18].\nFigure 3 aillustrates the stability range of CD in\ntermsofabsolutevalueoftheappliedmagneticfieldfield,\nh=jhj, and the tilt angle, #, forT= 0. The CD stabil-\nity domain is confined between the critical lines defined\nby the collapse field hc(#), and by the stretching (or el-\nliptical) instability field hs(#):hs(#)< h < h c(#). The\nrange ofhwhere CD is stable shrinks with increasing\n#, and at#'1:1the CD becomes unstable. Remark-\nably, the critical field hscoincides with the phase transi-\ntion line between skyrmion lattice and spin spiral states.\nThereby, the CD always represents a metastable solu-\ntion while the lowest energy state, in that range of fields,\nis the skyrmion lattice. Nevertheless, at moderate tem-\nperatures, the nucleation of the topologically trivial CDs\ndominates over the nucleation of topologically protected\nand energetically more favorable \u0019-Sks. At elevated tem-\nperatures, we observe the transition into skyrmion lattice\nwithin an accessible simulation time irrespective of the\nfield tilt angle, see Fig. 1.\nFurther understanding of thermal nucleation of the\nCD states can be obtained using the harmonic transition\nstate theory (HTST) [19, 20]. Within the HTST, the rate\nof transition between states XandYis described by theArrhenius law,\nkX!Y=\u0017X!Yexp\u0012\n\u0000\u0001EX!Y\nkBT\u0013\n;(3)\nwhere the energy barrier \u0001EX!Ycan be identified from\ntheMEPconnecting XandYastheenergydifferencebe-\ntween the highest point along the MEP – the first-order\nsaddle point (SP) on the energy surface of the system\n– and the minimum at X. The pre-exponential factor\n\u0017X!Yincorporating the dynamical and entropic contri-\nbutions to the transition rate is defined by the curvature\nof the energy surface at the minimum and at the SP [15].\nThe MEP calculations using the geodesic nudged elas-\ntic band method [21, 22] show that direct nucleation of\nthe Sk state from the FM background is only possible for\n#.0:1. For larger tilts of the external field, the MEP\nfor the Sk nucleation and annihilation passes through\nan intermediate energy minimum corresponding to the\nCD state [see Fig. 3(b)]. Therefore, the system initially\nprepared in the FM state undergoes a transition to the\nCD state before it may reach the Sk state, which is also\nobserved in the spin dynamics and Monte Carlo simu-\nlations [18]. As seen from Fig. 3 b, the energy barrier\nbetween the FM state and CD state gradually decreases\nwith#. This explains enhancement in the CD genera-\ntion as the tilt of the field increases. On the other hand\nthe energy of the saddle point between the CD and \u0019-Sk\nstates depends weakly on the tilt angle. It is mostly de-\nfined by the discretization of the system. Approaching\nthe micromagnetic regime with increasing LD, the energy\nbarrier \u0001EFM!Skincreases [23], which makes the \u0019-Sk\nnucleation less probable.\nThe CD energy minimum appears to be quite shallow\n[Fig. 3 b], suggesting a quick collapse of the CD state.4\n0 20 4000.050.10.150.20.25\n0 20 40 60 80 100 120 140 160 180\nReaction coordinate, rad\n(a)\nExternal magnetic filed, h\n ϑ Field tilt angle, 0.610.630.650.670.690.71\n0 0.2 0.4 0.6\nSpin spiral\n0.8\nChiral droplet\nField saturated state(b) (c)\n−4−20246Energy, (E-EFM)/J\n0.0\n0.00.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.1\n0.2\n0.3\n0.40.5\n0.60.7    [rad] ϑ\nSk\nFMCD\nReaction coordinate, radhc\nhs\nEnergy, (E-ECD)/J\n(ϑ)(ϑ)\nFIG. 3. Stability diagram of CD in terms of the magnetic field, h, and its tilt, #. Forh<h s(blue region), the CD becomes\nunstable with respect to stretching. For h > h c(red region), the CD collapses to FM. The red line corresponds to the\nintermediate values of the stability range in terms of h; magnetic textures of CDs for some points on this curve are shown in\ninsets. (b)shows the energy variation along the MEPs connecting the Sk, CD and FM states for different parameters h;#\nalong the red line in ( a). Red circles denote the energy minima corresponding to CD, red triangles mark the saddle points\nbetween CD and FM states. ( c) shows the zoom of ( b), with the zero of energy defined at the CD state.\nHowever, the HTST calculations, in agreement with the\nspin dynamics simulations, predict the opposite. The\nrates of transitions involving the CD state (FM \u001dCD,\nCD\u001dSk) are characterized by very different values of the\nenergy barrier and pre-exponential factor, as can be seen\nfrom the Arrhenius plots shown in Fig. 4. In particu-\nlar, the pre-exponential factor \u0017FM!CDfor the FM!CD\ntransition – the nucleation of CDs – is much larger than\nthatforthebackwardtransition. Despitetheverylowen-\nergybarrierfortheCD !FMtransitionaboveacrossover\ntemperature, the nucleation of CDs becomes more inten-\nsive than their annihilation into the FM state. For small\ntilts of the external field, the CD quickly transits into\nthe Sk state. However, with increasing #due to increas-\ning energy barrier \u0001ECD!Skthe CD!Sk transition is\nprogressively suppressed, see Fig. 3 b. The temperature\nrange where the CD nucleation dominates the annihi-\nlation by several orders of magnitude increases with #,\nsee gray color domain in Fig. 4. The HTST calculations\ntherefore provide a consistent interpretation of thermally\ninduced creation of CDs observed in our spin dynamics\nand MC simulations.\nIn conclusion, we proposed a robust protocol for the\ncreation of long-lived CDs by means of thermal fluctu-\nations in a 2D chiral magnet under the tilted magnetic\nfield. The protocol takes advantage of the entropic sta-\nbilization and relatively low energy barrier for the nu-\ncleation of a topologically-trivial magnetic soliton. By\nvarying the temperature and the tilt of the applied field,\nCDs and Sks can be generated selectively in a single sys-\ntem. Co-existing CDs and Sks can further be used as\nbuilding blocks for creating more complex magnetic soli-\ntons in chiral systems.\nThe authors would like to thank T. Sigurjónsdót-\nTransition rate, k\n10-810-610-4\n10-21102104\nSk → CD\nCD→ Sk\nCD→ FM\nFM → CD\nInverse temperature, 1/T0 5 10 0 5 10 0 5 10 ϑ=0.2  ϑ=0.3  ϑ=0.4FIG. 4. Rates of various magnetic transitions (as indicated\nin the legend) as functions of the inverse thermal energy for\nvarious tilts, #, of the magnetic field. The amplitude of h\ncorresponds to the middle line of the CD stability range in\nFig. 3 a. The grey regions mark the temperature range with\nthe maximal intensity of the FM !CD transition. Vertical\ndashed line corresponds to the Curie temperature, 1=Tc.\ntir for helpful discussions. This work was funded by\nDeutsche Forschungsgemeinschaft (DFG) through SPP\n2137 \"Skyrmionics\" Grant No. KI 2078/1-1, the Russian\nScience Foundation (Grant No. 19-72-10138), the Ice-\nlandic Research Fund (Grant Nos. 184949 and 217750),\nand the University of Iceland Research Fund.\n\u0003v.kuchkin@fz-juelich.de\n[1] A. Bogdanov and A. Hubert, J. Mag. Mag. Mat. 195,\n182 (1999).\n[2] F. N. Rybakov and N. S. Kiselev, Phys. Rev. B 99,\n064437 (2019).5\n[3] D. Foster, C. Kind, P. J. Ackerman, J.-S. B. Tai,\nM. R. Dennis and I. I. Smalyukh, Nat. Phys. 15, 655\n(2019).\n[4] V. M. Kuchkin and N. S. Kiselev, Phys. Rev. B 101,\n064408 (2020).\n[5] V. M. Kuchkin, B Barton-Singer, F. N. Rybakov, S.\nBlügel, B. J. Schroers and N. S. Kiselev, Phys. Rev. B\n102, 144422 (2020).\n[6] V. M. Kuchkin, K. Chichay, B. Barton-Singer, F. N. Ry-\nbakov, S. Blügel, B. J. Schroers and N. S. Kiselev, Phys.\nRev. B 104, 165116 (2021).\n[7] R. Cheng, M. Li, A. Sapkota, A. Rai, A. Pokhrel, T.\nMewes, C.Mewes, D.Xiao, M.DeGraef, andV.Sokalski,\nPhys. Rev. B 99, 184412 (2019).\n[8] J. Tang, Y. Wu, W. Wang, L. Kong, B. Lv, W. Wei,\nJ. Zang, M. Tian, H. Du, Nat. Nanotechnol. 16, 1086\n(2021).\n[9] M. Li, A. Sapkota, A. Rai, A. Pokhrel, T. Mewes, C.\nMewes, D.Xiao, M.DeGraef, V.Sokalski, J.Appl.Phys.\n130, 153903 (2021).\n[10] L. Rózsa, K. Palotás, A. Deák, E. Simon, R. Yanes,\nL. Udvardi, L. Szunyogh, U. Nowak, Phys. Rev. B 95,\n094423 (2017).\n[11] F. Muckel, S. von Malottki, C. Holl, B. Pestka,\nM. Pratzer, P.F. Bessarab, S. Heinze, M. Morgenstern,\nNat. Phys. 17, 395 (2021).[12] S. Meyer, M. Perini, S. von Malottki, A. Kubetzka,\nR. Wiesendanger, K. von Bergmann, S. Heinze, Nat.\nCommun. 10, 3823 (2019).\n[13] L.Desplat, D.Suess, J.-V.Kim, R.L.Stamps, Phys.Rev.\nB98, 134407 (2018).\n[14] S. von Malottki, P.F. Bessarab, S. Haldar, A. Delin,\nS. Heinze, Phys. Rev. B 99, 060409(R) (2019).\n[15] A.S. Varentcova, S. von Malottki, M.N. Potkina,\nG. Kwiatkowski, S. Heinze, and P.F. Bessarab, NPJ\nComput. Mater. 6, 193 (2020).\n[16] J. H. Mentink, M. V. Tretyakov, A. Fasolino, M. I. Kat-\nsnelson and Th, Rasing, J. Phys.: Condens. Matter 22,\n176001 (2010).\n[17] A. O. Leonov and I. Kézsmárki, Phys. Rev. B 96, 214413\n(2017).\n[18] See Supplemental Material at http://link.aps.org/ sup-\nplemental/ for details of Monte Carlo simulations, and\nestimation for the Curie temperature.\n[19] P.F. Bessarab, V.M. Uzdin, and H. Jónsson, Phys. Rev.\nB85, 184409 (2012).\n[20] P.F. Bessarab, V.M. Uzdin, and H. Jónsson, Z. Phys.\nChem. 227, 1543 (2013).\n[21] P.F. Bessarab, V.M. Uzdin, and H. Jónsson, Comput.\nPhys. Commun. 196, 335 (2015).\n[22] P.F. Bessarab, Phys. Rev. B 95, 136401 (2017).\n[23] B. Heil, A. Rosch, J. Masell, Phys. Rev. B 100, 134424\n(2019)."    },    {        "title": "2305.10111v1.Material_Parameters_for_Faster_Ballistic_Switching_of_an_In_plane_Magnetized_Nanomagnet.pdf",        "content": "arXiv:2305.10111v1  [cond-mat.mes-hall]  17 May 2023Journal of the Physical Society of Japan FULL PAPERS\nMaterial Parameters for Faster Ballistic Switching of an In -plane Magnetized\nNanomagnet\nToshiki Yamaji1*and Hiroshi Imamura1 †\n1National Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan\nHigh-speed magnetization switching of a nanomagnet is nece ssary for faster information processing. The ballistic\nswitching by a pulsed magnetic filed is a promising candidate for the high-speed switching. It is known that the switch-\ning speed of the ballistic switching can be increased by incr easing the magnitude of the pulsed magnetic field. However\nit is difficult to generate a strong and short magnetic field pulse in a sm all device. Here we explore another direction\nto achieve the high-speed ballistic switching by designing material parameters such as anisotropy constant, saturati on\nmagnetization, and the Gilbert damping constant. We perfor m the macrospin simulations for the ballistic switching of\nin-plane magnetized nano magnets with varying material par ameters. The results are analyzed based on the switching\ndynamics on the energy density contour. We show that the puls e width required for the ballistic switching can be re-\nduced by increasing the magnetic anisotropy constant or by d ecreasing the saturation magnetization. We also show that\nthere exists an optimal value of the Gilbert damping constan t that minimizes the pulse width required for the ballistic\nswitching.\n1. Introduction\nIn modern information technologies huge amount of data\nare represented as the direction of the magnetization in a sm all\nmagnet such as magnetic grains in magnetic tapes or hard\ndisk drives. To write information on the conventional mag-\nnetic recording media an external magnetic field is applied i n\nthe opposite direction of the magnetization to switch the di -\nrection of the magnetization. During the switching the mag-\nnetization undergoes multiple precessions around the loca l ef-\nfective field consisting of the external field, anisotropy fie ld,\nand demagnetizing field. The typical switching time or write\ntime is of the order of nanoseconds.\nTo meet the growing demand for fast information process-\ning it is important to develop a faster switching scheme. The\nballistic switching is a promising candidate for high-spee d\nswitching, and much e ffort has been devoted to developing\nthe ballistic switching both theoretically1–8)and experimen-\ntally.9–16)In ballistic switching a pulsed magnetic field is ap-\nplied perpendicular to the easy axis to induce the large-ang le\nprecession around the external magnetic field axis. The dura -\ntion of the pulse is set to a half of the precession period. Aft er\nthe pulse the magnetization relaxes to the equilibrium dire c-\ntion opposite to the initial direction. The switching speed of\nthe ballistic switching can be increased by increasing the m ag-\nnitude of the pulsed field. However, it is di fficult to generate a\nstrong and short field pulse in a small device. It is desired to\nfind a way to speed up the ballistic switching without increas -\ning magnetic field.\nThe magnetization dynamics of the ballistic switching is\ndetermined by the torques due to the external magnetic field,\nthe uniaxial anisotropy field, the demagnetizing field, and t he\nGilbert damping. The torques other than the external mag-\nnetic field torque are determined by the material parameters\nsuch as the anisotropy constant, the saturation magnetizat ion,\nand the Gilbert damping constant. There is room to speed up\n*toshiki-yamaji@aist.go.jp\n†h-imamura@aist.go.jpthe ballistic switching by designing the appropriate mater ial\nparameters.\nIn the early 2000s the several groups each independently\nreported the optical microscope measurements of the ballis -\ntic switching by picosecond pulse magnetic field.9–13)Then\nthe mechanism of a ballistic switching was analyzed in terms\nof the nonlinear dynamics concepts such as a fixed point, at-\ntractors, and saddle point.2, 3, 6)Especially the minimal field\nrequired for a ballistic switching was investigated by comp ar-\ning the so-called Stoner-Wohlfarth (SW) type.2, 3)The damp-\ning constant dependence of the minimal switching field was\nalso studied.2)The characteristics of the parameters of a pulse\nmagnetic field, i.e., magnitude, direction, and rise /fall time on\nthe mechanism of a ballistic switching had been also studied\nby the simulations and experiments.6, 7, 14, 15)\nAs described above, in 2000s and 2010s a ballistic switch-\ning technique had received much attention for the fast magne -\ntization reversal with ringing suppression by fine-tuning t he\nmagnetic pulse parameters. Due to the recent advance of an\nultra-fast measurement17)the studies of a ballistic switching\nhave attracted much attention again. Last year the in-plane\nmagnetization switching dynamics as functions of the pulse\nmagnetic field duration and amplitude was calculated and\nanalyzed by using the conventional Landau-Lifshitz-Gilbe rt\n(LLG) equation and its inertial form, the so-called iLLG\nequation.16)The solutions of both equations were compared\nin terms of the switching characteristics, speed and energy\ndensity analysis. Both equations return qualitatively sim ilar\nswitching dynamics. However the extensive material param-\neter dependences of a ballistic switching region have not\nyet been sufficiently explored. Therefore it is worth clearing\nthe extensive material parameter dependences of the ballis tic\nswitching of an in-plane magnetized nanomagnet.\nIn this paper, we study the ballistic switching of an in-\nplane magnetized nanomagnet with systematically varying\nthe material parameters by using the macrospin simulations .\nThe results show that the pulse width required for the bal-\nlistic switching can be reduced by increasing the magnetic\n1J. Phys. Soc. Jpn. FULL PAPERS\nHp\nmz\nyx(a)\n(c) my at t = 10 ns (b) \n(d) 0 200 400-1 01\nt [ps]my\ntp [ps]0 1 2 3 4 5tSW  [ps] \n110 10 210 3\ntl tutSW \n0 1 -1 \n0 1 2 3 4 502.55.010.0\n7.5\ntp [ps]Hp [T] \nFig. 1. (a) Schematic illustration of the in-plane magnetized nano magnet.\nThe pulse field, Hp, is applied along the x-direction. The initial direction of\nthe magnetization is in the positive y-direction. (b) Gray scale map of myat\nt=10 ns as a function of the pulse field width, tp, and Hp. The black and\nwhite regions represent the success and failure of switchin g. The parameters\nareµ0Ms=0.92 T,µ0HK=0.1 T, andα=0.023. (c) Typical example of\nthe time evolution of mywhen the magnetization switches ( Hp=5 T and tp\n=0.4 ps). The switching time, tSW, is defined as the time when mychanges\nthe sign. (d) tpdependence of tSWalong the dashed horizontal line at Hp=5\nT shown in Fig. 1(b). tlandtuare 3.15 ps and 3.93 ps, respectively. tSWat\ntl≤tp≤tuis 1.7 ps.\nanisotropy constant or by decreasing the saturation magnet i-\nzation. There exists an optimal value of the Gilbert damping\nconstant that minimizes the pulse width required for ballis -\ntic switching. The simulation results are intuitively expl ained\nby analyzing the switching trajectory on the energy density\ncontour.\n2. Model and Method\nIn this section we show the theoretical model, the numer-\nical simulation method, and the analysis using the trajecto ry\nin the limit ofα→0. The macrospin model of the in-plane\nmagnetized noanomagnet and the equations we solve to simu-\nlate the magnetization dynamics are given in Sec. 2.1. In Sec .\n2.2 we show that the switching conditions can be analyzed by\nusing the trajectory on the energy density contour in the lim it\nofα→0 if theα≪1.\n2.1 Macrospin Model Simulation\nFigure 1(a) shows the schematic illustration of the in-\nplane magnetized nanomagnet. The pulsed magnetic field,\nHp, is applied along the x-direction. The unit vector m=\n(mx,my,mz) indicates the direction of the magnetization. The\nsize of the nanomagnet is assumed to be so small that the dy-\nnamics of mcan be described by the macrospin LLG equation\ndm\ndt=−γm×/parenleftBigg\nHeff−α\nγdm\ndt/parenrightBigg\n, (1)\nwhere tis time,γis the gyromagnetic ratio, αis the Gilbert\ndamping constant. The e ffective field, Heff=Hp+HK+Hd,\ncomprises the pulse field, Hp, the anisotropy field, HK, andthe demagnetizing field, Hd. The anisotropy field and the de-\nmagnetizing field are defined as\nHK=/bracketleftbig2K/(µ0Ms)/bracketrightbigmyey, (2)\nand\nHd=µ0Msmzez, (3)\nrespectively, where Kis the uniaxial anisotropy constant, µ0\nis the magnetic permeability of vacuum, Msis the saturation\nmagnetization, and ejis the unit vector along the j-axis ( j=\nx,y,z).\nThe switching dynamics are calculated by numerically\nsolving the LLG equation. The initial ( t=0) direction is set\nasmy=1. The rectangular shaped pulse magnetic field with\nduration of tpis applied at t=0. The time evolution of magne-\ntization dynamics are calculated for 10 ns. Success or failu re\nof switching is determined by whether my<−0.5 att=10\nns.\nFigure 1(b) shows the gray scale plot of myatt=10 ns\non the tp-Hpplane. Following Ref. 16 the parameters are as-\nsumed to beµ0Ms=0.92 T, K=2.3 kJ/m3, i.e.µ0HK=\n0.1 T, andα=0.023. The black and white regions represent\nthe success and failure of switching, respectively. The wid e\nblack region at upper right of Fig. 1(b) represents the balli stic\nswitching region (BSR). A typical example of the time evolu-\ntion of mywhen the magnetization switches is shown in Fig.\n1(c). The switching time, tSW, is defined as the time when my\nchanges the sign. Figure 1(d) shows the tpdependence of tSW\nalong the horizontal line shown in Fig. 1(b), i.e. at Hp=5\nT. The BSR indicated by shade appears between tl=3.15\nps and tu=3.93 ps, where tSW=1.7 ps independent of tp.\nThe lower and upper boundary of the BSR are represented by\ntlandtu, respectively. We investigate the material parameter\ndependence of tlandtuwith keeping Hp=5 T.\n2.2 Analysis of the Switching Conditions for α≪1\nIf the Gilbert damping constant is much smaller than unity\nthe approximate value of tlandtucan be obtained without\nperforming macrospin simulations. In the limit of α→0, the\ntrajectory is represented by the energy contour because the en-\nergy is conserved during the motion of m. The energy density,\nE, of the nanomagnet is defined as18)\nE=1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ), (4)\nwhereθandφare the polar and azimuthal angles of the mag-\nnetization, respectively. The color plot of the energy dens ity\ncontour is shown in Fig. 2. The separatrix representing the\nenergy contour with E=Kis indicated by the white curve,\nwhich is expresses as\n1\n2µ0M2\nscos2θ−Ksin2θsin2φ=0. (5)\nThe green dot indicates the initial direction of matt=0. The\nblack curve represents the trajectory of munder the pulse field\nofHpin the limit ofα→0. Under the pulse field the energy\ndensity is given by\nE=1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ)\n−µ0MsHpsinθcosφ. (6)\n2J. Phys. Soc. Jpn. FULL PAPERS\n01 5 4 3 26E/K\ntltu\nθ\nφ\nFig. 2. (Color online) Color plot of the energy density contour give n by\nEq. (4).θandφare the polar and azimuthal angles of the magnetization, re-\nspectively. The material parameters, MsandKare same as in Fig. 1. The\nseparatrix given by Eq. (5) is indicated by the white curve. T he initial direc-\ntion of mis indicated by the green dot at ( θ,φ)=(π/2,π/2). The black curve\nrepresents the trajectory of the magnetization under the fie ld of Hp=5 T in\nthe limit ofα→0, which is given by Eq. (7). The yellow stars indicate the\nintersection points of the separatrix and the trajectory, w hich correspond to tp\n=tlandtu. If the pulse is turned o ffattl≤t≤tu, the magnetization switches\nballistically. The yellow triangle indicates the turning p oint of the trajectory\nof the magnetization near mz=1, at whichφ=0.\nSince the energy density of the initial direction, θ=φ=π/2,\nisE=0, the trajectory under the pulse field is expressed as\n1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ)\n−µ0MsHpsinθcosφ=0. (7)\nThe yellow stars indicate the points where the trajectory\ncrosses the separatrix surrounding the equilibrium point a t\nφ=−π/2. The upper and lower points indicates the direc-\ntion of mat the end of the pulse with tp=tuandtl, re-\nspectively. The corresponding angles ( θl,φl) and (θu,φu) can\nbe obtained by solving Eqs. (5) and (7) simultaneously. If\ntl≤tp≤tu, the magnetization relaxes to the equilibrium di-\nrection at (θ,φ)=(π/2,−π/2) after the pulse to complete the\nswitching. We can obtain the approximate expressions of tl\nandtuas follows. Assuming that the pulse field is much larger\nthan the other fields, the angular velocity of the precession ,ω,\nis approximated as γHp/(1+α2), and tlandtuare analytically\nobtained as\ntl=π−2θturn\nω−1\n2∆θ\nω, (8)\nand\ntu=π−2θturn\nω+1\n2∆θ\nω, (9)\nwhere∆θ=θu−θl, andθturnis the polar angle at the turning\npoint (φ=0) indicated by the yellow triangle.3. Results and Discussion\nIn this section we discuss the dependence of the BSR on\nthe material parameters by analyzing the numerical simula-\ntion results and Eqs. (8) and (9). The results for the variati on\nof the magnetic anisotropy constant, K, saturation magnetiza-\ntion, Ms, and the Gilbert damping constant, α, will be given\nin Secs. 3.1, 3.2, and 3.3, respectively.\n3.1 Anisotropy Constant Dependence of the BSR\nFigure 3(a) shows the anisotropy constant, K, dependence\nof the BSR. The parameters are Hp=5 T,µ0Ms=0.92 T, and\nα=0.023. The simulation results of tlandtuare indicated\nby the orange and blue dots, respectively. The analytical ap -\nproximations of tlandtuobtained by solving Eqs. (5),(7),(8),\nand (9) are represented by the orange and blue curves, respec -\ntively. The simulation and analytical results agree well wi th\neach other because the Gilbert damping constant is as small a s\n0.023. As shown in Fig. 3(a), tlis a monotonically decreasing\nfunction of Kwhile tuis a monotonically increasing function\nofK. As a result the width of the BSR, tu-tl, is a monoton-\nically increasing function of Kas shown in the inset of Fig.\n3(a).\nIn the left panel of Fig. 3(b) the separatrix and the trajecto ry\nwithα=0 for K=2.3 kJ/m3are shown by the blue and\nblack curves, respectively. The same plot for K=9.3 kJ/m3\nis shown in the right panel. As shown in these panels, the\nincrease of Kdoes not change the trajectory much. However,\nthe increase of Kchanges the separatrix significantly through\nthe second term of Eq. (5). Assuming that the angular velocit y\nof the precession is almost constant, the spread of the area\nsurrounded by the separatrix results in the spread of the tim e\ndifference between tlandtu. As a result the BSR is spread by\nthe increase of Kas shown in Fig. 3(a)\n3.2 Saturation Magnetization Dependence of the BSR\nFigure 4(a) shows the saturation magnetization dependence\nof the BSR obtained by the numerical simulation and the ana-\nlytical approximations. The horizontal axis represents th e sat-\nuration magnetization in unit of T, i.e µ0Ms. The parameters\nareHp=5 T,K=2.3 kJ/m3, andα=0.023. The symbols are\nthe same as in Fig. 3(a). The lower boundary of the BSR, tl,\nincreases as theµ0Msincreases while the upper boundary of\nthe BSR, tu, decreases with increase of µ0Ms. Therefore, the\nfaster switching is available for smaller Ms. Theµ0Msdepen-\ndence of the BSR ( tu-tl) is also shown in the inset of Fig. 4(a).\nThe BSR decreases with increase of µ0Ms. In other words, the\nwider BSR is obtained for smaller Ms.\nIn the right panel of Fig. 4(b) the separatrix and the trajec-\ntory withα=0 forµ0Ms=0.35 T are shown by the blue and\nblack curves, respectively. The same plot for µ0Ms=0.92 T is\nshown in the left panel. As shown in these panels, the increas e\nofMsdoes not change the trajectory much but decrease the\nseparatrix significantly through the first term of Eq. (5). As -\nsuming that the angular velocity of the precession is almost\nconstant, the reduction of the area surrounded by the separa -\ntrix results in the reduction of the time di fference between tl\nandtu. As a result the BSR decreases with increase of Msas\nshown in Fig. 4(a)\n3J. Phys. Soc. Jpn. FULL PAPERStl, t u [ps] \n0 10 20 30 40 2.03.04.05.0\nK [kJ/m3]ballistic switching region (a)\ntltu\ntu - t l [ps] \nK [kJ/m 3]0 10 20 30 40 0.0 0.5 1.0 1.5 2.0 2.5 \n0π\nπ/2 \n0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ\nφ φˑˑ\nˑˑ(b) K = 2.3 kJ/m3K = 9.3 kJ/m3\ntltltutu\nFig. 3. (Color online) (a) Anisotropy constant, K, dependence of the BSR\n(orange shade). Simulation results of tlandtuare plotted by the orange and\nblue dots, respectively. The analytical results are indica ted by the solid curves\nwith the same color. The parameters are Hp=5 T,µ0Ms=0.92 T, andα=\n0.023. In the inset the simulation and analytical results of the width of the\nBSR, tu-tl, are plotted by the dots and the solid curve, respectively. ( b)\nTypical examples of the trajectory of the magnetization (bl ack curve) and the\nseparatrix (blue curve). The left and right panels show the r esults for K=2.3\nkJ/m3andK=9.3 kJ/m3, respectively. The orange and blue stars indicate\nthe direction at t=tlandtu, respectively. The green dots indicate the initial\ndirection of m.\n3.3 Gilbert Damping Constant Dependence of the BSR\nFigure 5(a) shows the simulation results of the Gilbert\ndamping constant, α, dependence of the BSR. The width of\nthe BSR is shown in the inset. The symbols are the same as\nin Fig. 3(a). The approximate values obtained by Eqs. (8) and\n(9) are not shown because the αis not limited toα≪1. The\nparameters are Hp=5 T, K=2.3 kJ/m3, andµ0Ms=0.92\nT. There exists an optimal value of αthat minimizes tl. The\noptimum value in Fig. 5 (a) is αopt=0.35.\nTo understand the mechanism for minimization of tlat a\ncertain value ofαone need to consider two di fferent effects of\nαon the magnetization dynamics. One e ffect is the decrease\nof the precession angular velocity with increase of α. The pre-\ncession angular velocity around the e ffective field of Heffis\ngiven by (γHeff)/(1+α2), which decreases with increase of α.\nThis effect causes the increase of tlandtu.\nThe other effect is the increase of the energy dissipation rate\nwith increase ofα. Let us consider the trajectory in the cases\nof small damping ( α=0.023) and large damping ( α=αopt).\nIn Fig. 5 (b) the typical examples of the trajectory for the\nsmall damping are shown by the yellow and green curves\nand dots on the energy density contour. The pulse widths are\ntp=tl(=3.15 ps) and 3.14 ps. The trajectories during the\npulse are represented by the solid curves and the trajectori es\nafter the pulse are represented by the dots. The white curve\nshows the separatrix and the black dot indicates the initial di-\ntl, t u [ps] \n2.03.04.05.0\n0.0 0.3 0.6 0.9 1.2\nμ0Ms [T]ballistic switching region \ntltu(a)\n(b) μ0Ms = 0.92 T μ0Ms = 0.35 T\n0π\nπ/2 \n0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ\nφ φˑˑ\nˑˑ\ntl tltu tu1.5tu - t l [ps] \n0123\n0.0 0.3 0.6 0.9 1.2 \nμ0Ms [T] 1.5 \nFig. 4. (Color online) (a) Saturation magnetization dependence of the\nBSR. The horizontal axis represents the saturation magneti zation in unit of T,\ni.eµ0Ms. The parameters are Hp=5 T, K=2.3 kJ/m3, andα=0.023. The\nsymbols are the same as in Fig. 3 (a). (b) Typical examples of t he trajectory\nof the magnetization (black curve) and the separatrix (blue curve). The right\nand left panels show the results for µ0Ms=0.35 T and 0.92 T, respectively.\nThe symbols are the same as in Fig. 3 (b).\nrection. The yellow and green stars indicate the points wher e\nthe trajectories cross the separatrix surrounding the targ et and\ninitial states, respectively. The arrows indicate the dire ction\nof the movement of the magnetization. For the small damp-\ning, even very close to the separatrix around the target stat e at\nthe end of the pulse, the magnetization flows to the sepatrari x\naround the initial state and relax to the initial state after many\nprecessions with the slow energy dissipation.\nFigure 5 (c) shows the tpdependence of tSWat the large\ndamping (α=αopt). All parameters except αare the same\nas in Fig. 1 (d). t′\nl,tl, and tuare 0.82 ps, 1.98 ps, and 4.54\nps, respectively. t′\nlis the time when for the large damping the\nmagnetization goes across the e ffective separatrix around the\ninitial state during the pulse duration. In Fig. 5 (d) the typ ical\nexamples of the trajectory for the large damping are shown\nby the yellow ( tp=0.9 ps), green ( tp=tl=1.98 ps), and\npurple ( tp=4.55 ps) curves and dots on the energy density\ncontour. The symbols are the same as in Fig. 5 (b). In the\nregion 1 ( tp<t′\nl) of Fig. 5 (c) the magnetization is located on\nthe inside of the effective separatrix at the end of the pulse and\nreturn to the initial state. The trajectory is not shown in Fi g. 5\n(d). In the region 2 ( t′\nl≤tp<tl) of Fig. 5 (c) after the pulse is\nremoved the magnetization move toward the target state unde r\nthe effective field of Heffand goes across the separatrix. Then\nthe magnetization finishes the switching. The typical examp le\nof the trajectory is shown by the yellow curve and dots in Fig.\n5 (d). In the region 3 ( tl≤tp≤tu) after the pulse is removed\nthe magnetization ballistically switches. The typical exa mple\n4J. Phys. Soc. Jpn. FULL PAPERS\ntl, t u [ps] \n2.04.06.08.0\n0.0 0.2 0.4 0.6 0.8\nαballistic switching region \ntltu(a)\n0\ntu - t l [ps] \n0.00123\n0.2 0.4 0.6 0.8 \nα45\nαopt \nαopt\n01 5 4 3 26E/K \nα = 0.023 \ntp = t l = 3.15 ps \ntp = 3.14 ps (b)\n0 -π/2 π/2 π -π0π/2 π\nφθ\n< <<\n< <\nˑ\nˑ\n(c)\ntp [ps] 0 1 2 3 4 5tSW  [ps] \n110 10 210 3\ntl tu1 2 4 3\ntl’\n01 5 4 3 26E/K \nα = 0.35 \ntp = 0.9 ps \ntp = t l = 1.98 ps \ntp = 4.55 ps (d)\n0 -π/2 π/2 π -π0π/2 π\nφθ\n<\n<\n<ˑˑ <\n< <\nˑ\nFig. 5. (Color online) (a) Simulation results of the tlandtuas a function\nofα. The parameters are Hp=5.0 T, K=2.3 kJ/m3, andµ0Ms=0.92\nT. The symbols are the same as in Fig. 3 (a). (b) The trajectori es atα=\n0.023 with tp=3.15 ps (yellow) and 3.14 ps (green) on the energy density\ncontour. The trajectory during the pulse is represented by t he solid curve. The\ntrajectory after the pulse is represented by the dots. The wh ite curve shows\nthe separatrix.The direction of the trajectory is indicate d by the arrow. The\nstar indicates the intersection point of the trajectory and the separatrix. The\ninitial direction is indicated by the black dot. (c) tpdependence of tSWat\nα=αopt. All parameters except αare the same as in Fig. 1 (d). t′\nl,tl, and tu\nare 0.82 ps, 1.98 ps, and 4.54 ps, respectively. tSWattl≤tp≤tuis 1.98 ps.\n(d) The trajectories at α=αoptwith tp=0.9 ps (yellow), 1.98 ps (green), and\n4.55 ps (purple) on the energy density contour. The symbols a re the same as\nin Fig. 5 (b).\nof the trajectory is shown by the green curve and dots in Fig.5 (d). As explained in Sec. 2.1, in this study the tSWis defined\nas the time when mychanges the sign, in other words, the\nmagnetization reaches the turning point ( φ=0) on the energy\ndensity contour. Therefore the tlfor the large damping is equal\nto the ballistic tSW, i.e. the time when under the pulse field the\nmagnetization reaches the turning point. The ballistic tSWat\nthe region 3 is 1.98 ps.\nAs described above, for the large damping the magnetiza-\ntion can relax to the target state even at greater distances f rom\nthe separatrix by moving under Heffwith the fast energy dis-\nsipation. The effect can be regarded as the e ffective spread\nof the separatrix and results in the decrease of tland the in-\ncrease of tuwith increase ofα. Therefore, there exists the\noptimal value ofαthat minimizes tlwhile tumonotonically\nincreases with increase of αas shown in Fig. 5(a). In the re-\ngion 4 ( tp>tu) after the pulse is removed the magnetization\nmoves toward the separatrix around the initial state under Heff\nand relaxes to the initial state. We find that the BSR for the\nlarge damping can be explained by the anisotropic spread of\nthe effective separatrix with increasing α, which is fundamen-\ntally due to the breaking of the spatial inversion symmetry o f\nthe spin dynamics. The broken symmetry of the spatial inver-\nsion of the spin dynamics for the large damping can be easily\nconfirmed by comparing Fig. 5 (c) with Fig. 1 (d).\n4. Summary\nIn summary, we study the material parameter dependence\nof the ballistic switching region of the in-plane magnetize d\nnanomagnets based on the macrospin model. The results show\nthat the pulse width required for the ballistic switching ca n be\nreduced by increasing the magnetic anisotropy constant or b y\ndecreasing the saturation magnetization. The results also re-\nvealed that there exists an optimal value of the Gilbert damp -\ning constant that minimizes the pulse width required for the\nballistic switching. The simulation results are explained by\nanalyzing the trajectories on the energy contour. The resul ts\nare useful for further development of the high-speed inform a-\ntion processing using the ballistic switching of magnetiza tion.\nThis work is partially supported by JSPS KAKENHI Grant\nNumber JP20K05313.\n1) L. He and W. D. Doyle: J. Appl. Phys. 79(1996) 6489.\n2) Z. Z. Sun and X. R. Wang: Phys. Rev. B 71(2005) 174430.\n3) D. Xiao, M. Tsoi, and Q. Niu: Journal of Applied Physics 99(2006)\n013903.\n4) Y . Nozaki and K. Matsuyama: Jpn. J. Appl. Phys. 45(2006) L758.\n5) Y . Nozaki and K. Matsuyama: Journal of Applied Physics 100(2006)\n053911.\n6) Q. F. Xiao, B. C. Choi, J. Rudge, Y . K. Hong, and G. Donohoe: J ournal\nof Applied Physics 101(2007) 024306.\n7) P. P. Horley, V . R. Vieira, P. M. Gorley, V . K. Dugaev, J. Ber akdar, and\nJ. Barna´ s: Journal of Magnetism and Magnetic Materials 322(2010)\n1373.\n8) Y . B. Bazaliy: Journal of Applied Physics 110(2011) 063920.\n9) T. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. B¨ ar, and T . Rasing:\nNature 418(2002) 509.\n10) I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann , J. St¨ ohr,\nG. Ju, B. Lu, and D. Weller: Nature 428(2004) 831.\n11) H. W. Schumacher, C. Chappert, R. C. Sousa, P. P. Freitas, and J. Miltat:\nPhys. Rev. Lett. 90(2003) 017204.\n12) W. K. Hiebert, L. Lagae, J. Das, J. Bekaert, R. Wirix-Spee tjens, and\nJ. De Boeck: Journal of Applied Physics 93(2003) 6906.\n13) W. K. Hiebert, L. Lagae, and J. De Boeck: Phys. Rev. B 68(2003)\n5J. Phys. Soc. Jpn. FULL PAPERS\n020402.\n14) H. W. Schumacher: Appl. Phys. Lett. 87(2005) 042504.\n15) N. Kikuchi, Y . Suyama, S. Okamoto, O. Kitakami, and T. Shi matsu:\nAppl. Phys. Lett. 104(2014) 112409.\n16) K. Neeraj, M. Pancaldi, V . Scalera, S. Perna, M. d’Aquino , C. Serpico,\nand S. Bonetti: Phys. Rev. B 105(2022) 054415.17) K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Zhou Hagstr ¨ om, S. S.\nP. K. Arekapudi, A. Semisalova, K. Lenz, B. Green, J.-C. Dein ert,\nI. Ilyakov, M. Chen, M. Bawatna, V . Scalera, M. d’Aquino, C. S erpico,\nO. Hellwig, J.-E. Wegrowe, M. Gensch, and S. Bonetti: Nat. Ph ys.17\n(2021) 245.\n18) W. F. Brown: Phys. Rev. 130(1963) 1677.\n6"    },    {        "title": "0705.0406v1.Planar_spin_transfer_device_with_a_dynamic_polarizer.pdf",        "content": "arXiv:0705.0406v1  [cond-mat.mtrl-sci]  3 May 2007Planar spin-transfer device with a dynamic polarizer.\nYa. B. Bazaliy,1D. Olaosebikan,2and B. A Jones1\n1IBM Almaden Research Center, 650 Harry Road, San Jose, CA 951 20\n2Department of Physics, Cornell University, Ithaca, NY 1485 3\n(Dated: July, 2006)\nIn planar nano-magnetic devices magnetization direction i s kept close to a given plane by the large\neasy-plane magnetic anisotropy, for example by the shape an isotropy in a thin film. In this case\nmagnetization shows effectively in-plane dynamics with onl y one angle required for its description.\nMoreover, the motion can become overdamped even for small va lues of Gilbert damping. We\nderive the equations of effective in-plane dynamics in the pr esence of spin-transfer torques. The\nsimplifications achieved in the overdamped regime allow to s tudy systems with several dynamic\nmagnetic pieces (“free layers”). A transition from a spin-t ransfer device with a static polarizer to\na device with two equivalent magnets is observed. When the si ze difference between the magnets\nis less than critical, the device does not exhibit switching , but goes directly into the “windmill”\nprecession state.\nPACS numbers: 72.25.Pn, 72.25.Mk, 85.75.-d\nI. INTRODUCTION\nThe prediction1,2and first experimental\nobservations3,4,5,6,7,8of spin-transfer torques opened a\nnew field in magnetism which studies non-equilibrium\nmagnetic interactions induced by electric current. Since\nsuch interactions are relatively significant only in very\nsmall structures, the topic is a part of nano-magnetism.\nThe current-induced switching of magnetic devices\nachieved through spin-transfer torques is a candidate\nfor being used as a writing process in magnetic random\naccess memory (MRAM) devices. The MRAM memory\ncell is a typical example of a spintronic device in which\nthe electron spin is used to achieve useful logic, memory\nor other operations normally performed by electronic\ncircuits.\nTo produce the spin-transfer torques, electric currents\nhave to flow through the spatially non-uniform mag-\nnetic configurations in which the variation of magneti-\nzation can be either continuous or abrupt. The first case\nis usually experimentally realized in magnetic domain\nwalls.3,9,10,11Here we will be focusing on the second case\nrealized in the artificially grown nano-structures. Such\nspin-transfer devices contain severalmagnetic pieces sep-\narated by non-magnetic metal spacers allowing for arbi-\ntraryanglesbetweenthemagneticmomentsofthepieces.\nMagnetizationmyvarywithineachpieceaswell, butthat\nvariation is usually much smaller and vanishes as the size\nof piece is reduced, or for larger values of spin-stiffness\nof magnetic material. The typical examples of a system\nwith discrete variation of magnetization are the “nano-\npillar” devices8(Fig.1A). Their behavior can be reason-\nablywellapproximatedbyassumingthatmagneticpieces\nare mono-domain, each described by a single magnetiza-\ntion vector /vectorM(t) =Ms/vector n(t) where /vector nis the unit vector\nandMsis the saturation magnetization. The evolution\nof/vector n(t) is governedby the Landau-Lifshitz-Gilbert (LLG)\nequation with spin-transfer terms.2,12\nIt is often the case that magnetic pieces in a spin-transfer device have a strong easy-plane anisotropy. For\nexample, in nano-pillars both the polarizer and the free\nmagnetic layer are disks with the diameter much larger\nthan the thickness. Consequently, the shape anisotropy\nmakes the plane of the disk an easy magnetic plane. In\nthe planar devices13built from thin film layers (Fig. 1B)\nthe shape anisotropy produces the same effect. When\nthe easy-plane anisotropy energy is much larger then all\notherenergies,thedeviationsof /vector n(t)fromthein-planedi-\nrection are very small. An approximation based on such\nsmallness is possible and providesan effective description\nof the magnetic dynamics in terms of the direction of the\nprojection of /vector n(t) on the easy plane, i.e. in terms of one\nazimuthalangle. Inthispaperwederivetheequationsfor\neffective in-plane motion in the presence of spin-transfer\neffect and discuss their use by considering several exam-\nples.\nIn the absence of spin-transfer effects the large easy-\nplane anisotropy creates a regime of overdamped mo-\ntion even for the small values of Gilbert damping con-\nstantα≪1.14In that regime the equations simplify\nfurther. Here the overdamped regime is discussed in\nthe presence of electric current. The reduction of the\nnumber of equations allows for a simple consideration\nof a spin-transfer device with two dynamic magnetic\npieces. We show how an asymmetry in the sizes of these\npieces createsa transition between the polarizer-analyzer\n(“fixed layer - free layer”) operation regime2,8,12,15and\nthe regime of nearly identical pieces where current leads\nns\n\rjj j\nsnA B\nFIG. 1: Planar spin-transfer devices2\nnot to switching, but directly to the Slonczewski “wind-\nmill” dynamic state.2Finally, we point out the limita-\ntions of the overdamped approximation in the presence\nof the spin-transfer torques.\nII. DYNAMIC EQUATIONS IN THE LIMIT OF\nA LARGE EASY-PLANE ANISOTROPY\nMagnetizationdynamicsin thepresenceofelectriccur-\nrent is governed by the LLG equation with the spin-\ntransfer term.2,12For each of the magnets in the device\nshown on Fig. 1A\n˙/vector n=γ\nMs/bracketleftbigg\n−δE\nδ/vector n×/vector n/bracketrightbigg\n+u[/vector n×[/vector s×/vector n]]+α[/vector n×˙/vector n] (1)\nwhere/vector s(t) is the unit vector along the instantaneous\nmagnetization of the other magnet and the spin-transfer\nmagnitude\nu=g(P)γ(¯h/2)\nVMsI\ne(2)\nis proportional to the electric current I. Hereeis the\n(negative) electroncharge, so uis positivewhen electrons\nflow into the magnet. Due to the inverse proportionality\nto the volume V, the larger magnets become less sensi-\ntive to the current and can serve as spin-polarizers with\na fixed magnetization direction. As for the other pa-\nrameters, γis the gyromagnetic ratio, g(P,(/vector n·/vector s)) is the\nSlonczewski spin polarization factor2which depends on\nmanysystemparameters,16,17andαis the Gilbert damp-\ning which also depends on /vector nand/vector swhen spin pumping18\nis taken into account. We will restrict our treatment to\nthe constant gandαto focus on the effects specific to\nthe strong easy plane anisotropy.\nIn terms of the polar angles ( θ,φ) the LLG equation\n(1) has the form\n˙θ+α˙φsinθ=−γ\nMsinθ∂E\n∂φ+u(/vector s·/vector eθ)\n˙φsinθ−α˙θ=γ\nM∂E\n∂θ+u(/vector s·/vector eφ) (3)\nwhere the tangent unit vectors /vector eθand/vector eφare defined in\nAppendix A.\nWe will consider a model for which the energy of a\nmagnet is given by\nE=K⊥cos2θ\n2+Er(φ) (4)\nwithK⊥being the easy-plane constant, Erbeing the\n“residual”in-plane anisotropy energy and z-axis directed\nperpendicular to the easy plane. The limit of a strong\neasy-planeanisotropyisachievedwhenthe maximalvari-\nation of the residual energy is small compared to the\neasy-plane energy, ∆ Er≪K⊥. In this case θ=π/2+δθ\nwithδθ≪1.To estimate δθ, consider the motion of magnetization\ninitially lying in-plane offthe minimum of Erand neglect\nfor the moment the spin-transfer terms in Eq.(3). Mag-\nnetization starts movingand a certain deviation from the\neasy plane is developed. For the estimate, assume that\nthe energy is conserved during this motion (the presence\nof damping will only decrease δθ). Then\n|δθ| ∼/radicalbigg\n∆Er\nK⊥≪1 (5)\nWecannowlinearizetherighthandsidesofequations(3)\nin smallδθ. On top of that, some terms on the left hand\nsides of (3) turn out to be small and can be discarded.\nIndeed, taking into account the smallness of αone gets\nthe estimates\n˙θ∼ −γ\nMs∂Er\n∂φ∼ −γ\nMs∆Er\n˙φ∼γ\nMsK⊥δθ∼γ\nMs/radicalbig\nK⊥∆Er\nConsequently ˙θ∼˙φ/radicalbig\n∆Er/K⊥≪˙φand˙φ≫α˙θ, there-\nfore the second term on the left hand side of the second\nequation of the system (3) can be discarded. No simpli-\nfication happens on the left hand side of the first equa-\ntion, where ˙θandα˙φcan be of the same order when\nα<∼/radicalbig\n∆Er/K⊥.\nPutting the spin-transfer terms back we get the form\nof equations in the limit of large easy-plane anisotropy:\n˙δθ+α˙φ=−γ\nMs∂E\n∂φ+u(/vector s·/vector eθ)\n˙φ=γK⊥\nMsδθ+u(/vector s·/vector eφ) (6)\nExpressions for the scalar products in (6) in terms of\npolar angles are given in Appendix A.\nThe second equation shows that δθcan be expressed\nthrough ( φ,˙φ). Small out-of-plane deviation becomes a\n“slave” of the in-plane motion.14We get\nMs\nγK⊥/parenleftbigg\n¨φ−ud(/vector s·/vector eφ)\ndt/parenrightbigg\n+αi˙φ=−γ\nMs∂Er\n∂φ+u(/vector s·/vector eθ) (7)\nThe term with the second time derivative ¨φdecreases\nwith increasing K⊥. As pointed out in Ref. 14, in the\nabsence of spin-transfer this term can be neglected when\nK⊥>∆Er/α2. Mathematically this corresponds to a\ntransition from an underdamped to an overdamped be-\nhavior of an oscillator as the oscillator mass decreases.\nWith spin-transfer terms the overdamped approxima-\ntion gives an equation\nα˙φ−ξd\ndt(/vector s·/vector eφ) =−γ\nMs∂Er\n∂φ+u(/vector s·/vector eθ) (8)\nwhereξ=uMs/(γK⊥). The range of this equation’s\nvalidity will be discussed in Sec. IV. The scalar products\nin Eq. (8) have to be expressed through the polar angles3\n(θs(t),φs(t)) ofvector /vector s, and linearizedwith respectto δθ\n(see Appendix, Eq. A4), which is then substituted from\nEq. (6). Finally, the equation is linearized with respect\nto small spin-transfer magnitude u. We get:\nα˙φ−ξ/parenleftbiggd\ndt/bracketleftbig\nsinθssin(φs−φ)/bracketrightbig\n−sinθscos(φs−φ)˙φ/parenrightbigg\n=−γ\nMs∂Er\n∂φ−ucosθs, (9)\ndescribing the in-plane overdamped motion of an ana-\nlyzer with a polarizer pointed in the arbitrary direction.\nNext, we show how some known results on spin-transfer\nsystems are recovered in the approximation (9).\nConsider the device shown on Fig. 1A and assume that\nthe first magnet is very large. As explained above, this\nmagnetisnotaffectedbythecurrentandservesasafixed\nsource of spin-polarized electrons for the second magnet\ncalled the analyzer, or the “free” layer. The magneti-\nzation dynamics of the analyzer is described by Eq. (3).\nThe case of static polarizer is extensively studied in the\nliterature.\nFirst, consider the case of collinear switching , exper-\nimentally realized in a nano-pillar device with the ana-\nlyzer’s and polarizer’s easy axes along the ˆ xdirection:\nEr= (1/2)K||sin2φ,/vector s= (1,0,0).7Using Eq. (9) with\nθs=π/2,φs= 0 we get\n(α+2ξcosφ)˙φ=−γK||\n2Mssin2φ (10)\nWithout the current, there are four possible equilibria\nof the analyzer. Two stable equilibria are the parallel\n(φ= 0) and anti-parallel ( φ=π) states. Two perpen-\ndicular equilibria ( φ=±π/2) are unstable. Lineariz-\ning Eq. (10) near equilibria one finds solutions the form\nδφ(t)∼exp(ωt) with eigenfrequencies\nω=−γK||\nMs(α+2ξ),(φ≈0)\nω=−γK||\nMs(α−2ξ),(φ≈π)\nω=γK||\nMsα,(φ≈ ±π/2)\nThe equilibria are stable for ω <0 and unstable other-\nwise. Thus the parallel state is stable for ξ >−α/2, the\nantiparallel state is stable for ξ < α/2, and the perpen-\ndicular states cannot be stabilized by the current. These\nconclusions agree with the results of Refs. 2,7,12. The\nstability regions are shown in Fig. 2A.\nNote how Eq. (10) emphasizes the fact that spin-\ntransfer torque destabilizes the equilibria by making the\neffective damping constant αeff=α+2ξcosφnegative,\nwhiletheequilibriumpointsremainaminimumoftheen-\nergyEr. Any appreciable influence of the current on the\nposition and nature (minimum or maximum) of the equi-\nlibrium can only be observed at the current magnitudes\n1/αtimes larger than the actual switching current.12ξ\nα/2−α/2(A) static polarizer\n−α/[2(1−ε)](B) dynamic polarizer\nα/2 −α/(2ε)\"windmill\"\nprecession\"windmill\"\nprecession ξ\nFIG. 2: Stability regions for systems with static (A) and\ndynamic (B) polarizers as a function of applied current,\nξ=g(P)(¯h/2VK⊥)I/e∝I.\nSecond, consider the case of magnetic fan .19Here the\neasy axis of the polarizer is again directed along ˆ x, but\nthe polarizer is perpendicular to the easy plane: /vector s=\n(0,0,1),θs= 0. This arrangement is known to produce\na constant precession of vector /vector n. Eq. (9) gets a form:\nα˙φ=−γK||\n2Mssin2φ−u (11)\nfor|u|< γK ||/(2Ms) the current deflects the analyzer\ndirection from the easy axis direction. For larger values\nofuthere is no time-independent solution. The angles\nφgrows with time which corresponds to /vector nmaking full\nrotations. At |u| ≫γK||/(2Ms) the rotation frequency\nof the magnetic fan is given by ω∼u/α.\nIII. DEVICE WITH TWO DYNAMIC\nMAGNETS (TWO “FREE LAYERS”)\nNo let us assume that both magnets in Fig. 1A have\nfinitesize. Eachmagnetservesasapolarizerfortheother\none. Without approximations, the evolution of two sets\nof polar angles ( θi,φi),i= 1,2 is described by two LLG\nsystems of equations\n˙θ(i)+αi˙φ(i)sinθ(i)=−γ\nMsisinθ(i)∂E(i)\n∂φ(i)+\n+uji(/vector n(j)·/vector e(i)\nθ) (12)\n˙φ(i)sinθ(i)−αi˙θ(i)=γ\nMsi∂E(i)\n∂θ(i)+uji(/vector n(j)·/vector e(i)\nφ)\nwherejmeanstheindexnotequalto iandnosummation\nis implied.\nWe now apply the overdamped, large easy-plane\nanisotropyapproximationtobothmagnets. Equation(9)4\nfor each magnet is further simplified since for the magnet\nithe angle θs=θj=π/2 +δθj,δθj≪1. Expanding\n(9) in small δθjand using the slave condition (6) for δθj\nwith (/vector s·/vector eφ) = (/vector n(j)·/vector e(i)\nφ) expanded in both small angles\n(see Eq. (A5)) we get the system:\n(αi+2ξjicos(φj−φi))˙φi− (13)\n−ξji(cos(φj−φi)+1)˙φj=−∂E(i)\n∂φi,\nwithξji=ujiMsi/(γK⊥). It was assumed that K⊥is\nthe same for both magnets.\nThe spin-transfer torque parameters u21andu12have\nopposite signs and their absolute values are different due\nto different volumes of the magnets, accordingto Eq. (2).\nWe assume V1≥V2and denote u12=u,u21=−ǫu.\nThe larger magnet experiences a relatively smaller spin\ntransfer effect, and the asymmetry parameter satisfies\n0≤ǫ≤1. In general, material parameters α1,2,Ms1,2\nand magnetic anisotropy energies E(1,2)of the two mag-\nnets are also different, but here we focus solely on the\nasymmetry in spin-transfer parameters. Both E(1)and\nE(2)are assumed to be given by formula (4) with the\nsame direction of in-plane easy axis. The situation can\nbe viewed as a collinear switching setup with dynamic\npolarizer. Equations (13) specialize to\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleα−2ǫξC ǫξ(C+1)\n−ξ(C+1)α+2ξC/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg˙φ1\n˙φ2/bracketrightbigg\n=−ω0\n2/bracketleftbigg\nsin2φ1\nsin2φ2/bracketrightbigg\nC= cos(φ1−φ2), ω0=γK||\nMs(14)\nNext, we study the stability of all equilibrium configu-\nrations ( φ1,φ2) of two magnets. There are four equilib-\nrium states that are stable without the current: two par-\nallel states along the easy axis (0 ,0) and (π,π), two an-\ntiparallelstatesalongthe easyaxis(0 ,π)and(π,0). Four\nmore equilibrium states have magnetization perpendicu-\nlar to the easy axis and are unstable without the current:\n(±π/2,±π/2). Once again, since spin-transfer does not\ndepend on the relative direction of current and magneti-\nzation, the configurations which can be transformed into\neach other by a rotation of the magnetic space as a whole\nbehaveidentically. Thusitisenoughtoconsiderfourcon-\nfigurations: (0 ,0), (0,π), (π/2,π/2), and ( π/2,−π/2).\nWe linearize equations (14) near each equilibrium and\nsearch for the solution in the form δφi∼exp(ωt). The\neigenfrequencies are found to be:\n(0,0) :ω1=−ω0\nα, ω2=−−ω0\nα+2ξ(1−ǫ)\n(0,π) :ω1=−ω0\nα+2ǫξ, ω2=−ω0\nα−2ξ\n(π\n2,π\n2) :ω1=ω0\nα, ω2=ω0\nα−2ξ(1−ǫ)\n(π\n2,−π\n2) :ω1=ω0\nα+2ǫξ, ω2=ω0\nα−2ξThe state is stable when both eigenfrequencies are\nnegative. We conclude that initially unstable states\n(π/2,±π/2) are never stabilized by the current, while\nthe (0,0) and (0 ,π) state remain stable for\n(0,0) :ξ >−α\n2(1−ǫ)\n(0,π) :−α\n2ǫ< ξ <α\n2\nThese regions of stability are shown schematically in\nFig. 2B in comparison with the case of static magnetic\npolarizer (Fig. 2A) which is recovered at ǫ→0.\nAs the size of the polarizer is reduced, the asymme-\ntry parameter ǫgrows. The stability region of the an-\ntiparallel state acquires a lower boundary ξ=−α/(2ǫ).\nUp toǫ= 1/2, this boundary is still below the lower\nboundary of the parallel configuration stability region.\nConsequently, the parallel configuration is switched to\nthe antiparallel at a negative current ξ=−α/(2(1−ǫ)).\nThe system then remains in the antiparallel state down\ntoξ=−α/(2ǫ). Below that threshold no stable configu-\nrations exist, and the system goes into some type of pre-\ncession state. This dynamic state is related to the “wind-\nmill” state predicted in Ref. 2 for two identical magnets\nin the absence of anisotropies. Obviously, here it is mod-\nified by the strong easy-plane anisotropy.\nTheǫ= 1/2 value represents a transition point in the\nbehavior of the system. For 1 /2< ǫ <1, the stability\nregion of the parallel configuration completely covers the\none of the antiparallel state. A transition without hys-\nteresis now happens at ξ=−α/(2(1−ǫ)) between the\nparallel state and the precession state. If the system is\ninitially in the antiparallel state, it switches to the par-\nallel state either at a negative current ξ=−α/(2(1−ǫ))\nor at a positive current ξ=α/2, and never returns to\nthe antiparallel state after that.\nIV. CONCLUDING REMARKS\nWe studied thebehaviorofplanarspin-transferdevices\nwith magnetic energy dominated by the large easy-plane\nanisotropy. The overdamped approximation in the pres-\nence of current-induced torque was derived and checked\nagainst the cases already discussed in the literature. In\nthe new “dynamic polarizer” case, we found a transition\nbetween two regimes with different switching sequences.\nThe large asymmetry regime is similar to the case of\nstatic polarizer and shows hysteretic switching between\nthe parallel and antiparallel configurations, while in the\nsmall asymmetry regime the magnets do not switch, but\ngo directly into the “windmill” precession state.\nWe saw that the current-induced switching occurs\nwhen the effective damping constant vanishes near a par-\nticularequilibrium. Thismakestheoverdampedapproxi-\nmationinapplicableinthe immediatevicinityofthetran-\nsition and renders Eqs. (14) ill-defined at some points.\nHowever, the overall conclusions about the switching5\nevents will remain the same as long as the interval of\ninapplicability is small enough.\nWe also find that the overdamped planar approxima-\ntion does not work well when a saddle point of magnetic\nenergy is stabilized by spin-transfer torque, e.g. during\nthe operation of a spin-flip transistor.20Description of\nsuch cases in terms of effective planar equations requires\nadditional investigations.\nV. ACKNOWLEDGEMENTS\nWe wish to thank Tom Silva, Oleg Tchernyshyov,Oleg\nTretiakov, and G. E. W. Bauer for illuminating discus-\nsions. This work was supported in part by DMEA con-\ntract No. H94003-04-2-0404, Ya. B. is grateful to KITP\nSanta Barbara for hospitality and support under NSF\ngrant No. PHY99-07949. D. O. was supported in part\nby the IBM undergraduate student internship program.\nAPPENDIX A: VECTOR DEFINITIONS\nz\neφ\nθen\nφθ\r\nxFIG. 3: Definitions of the tangent vectors and polar angles.\nWe use the standard definitions of polar coordinates\nand tangent vectors (see Fig. 3):\n/vector n= (sinθcosφ,sinθsinφ,cosθ)\n/vector eθ= (cosθcosφ,cosθsinφ,−sinθ) (A1)\n/vector eφ= (−sinφ,cosφ,0)\nWhenθ=π/2+δθa linearization in δθgives\n/vector n≈(cosφ,sinφ,−δθ)\n/vector eθ≈ −(δθcosφ,δθsinφ,1) (A2)\n/vector eφ≈(−sinφ,cosφ,0)\nFor two unit vectors /vector n(i),i= 1,2 with polar angles\n(θi,φi) the scalar product expressions are\n(/vector n(j)·/vector e(i)\nθ) = sin θjcosθicos(φj−φi)−cosθjsinθi\n(/vector n(j)·/vector e(i)\nφ) = sin θjsin(φj−φi) (A3)\nLinearizing (A3) with respect to small δθifor arbitrary\nvalues of θjone gets:\n(/vector n(j)·/vector e(i)\nθ)≈ −sinθjδθicos(φj−φi)−cosθj\n(/vector n(j)·/vector e(i)\nφ)≈sinθjsin(φj−φi) (A4)\nLinearization of (A3) with respect to both δθiandδθj\ngives\n(/vector n(j)·/vector e(i)\nθ)≈ −δθicos(φj−φi)+δθj\n(/vector n(j)·/vector e(i)\nφ)≈sin(φj−φi) (A5)\n1L. Berger, J. Appl. Phys., 49, 2160 (1978); Phys. Rev. B\n33, 1572 (1986); J.Appl.Phys. 63, 1663 (1988).\n2J.Slonczewski, J.Magn.Magn.Mater. 159, L1 (1996).\n3L. Berger, J. Appl. Phys., 71, 2721 (1992)\n4M. Tsoi, A.G.M. Jansen, J. Bass, W.-C. Chiang, M. Seck,\nV. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 (1998).\n5J. Z. Sun, J. Magn. Magn. Mater., 202, 157, (1999).\n6J.E. Wegrowe, D. Kelly, Y. Jaccard, Ph. Guittienne, and\nJ.-Ph. Ansermet, Europhys.Lett. 45, 626 (1999).\n7E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, R.A.\nBuhrman, Science 285, 867 (1999).\n8J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and\nD.C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n9E. Saitoh, H. Miyajima, T. Yamaoka, and G. Tatara, Na-\nture,432, 203 (2004).\n10M. Klaui, P.-O. Jubert, R. Allenspach, A. Bischof, J. A.\nC. Bland, G. Faini, U. Rudiger, C. A. F. Vaz, L. Vila, and\nC. Vouille, Phys. Rev. Lett., 95, 026601 (2005).\n11M. Hayashi, L. Thomas, Ya. B. Bazaliy, C. Rettner, R.\nMoriya, X. Jiang, and S. S. P. Parkin, Phys. Rev. Lett.,\n96, 197207 (2006).12Ya. B. Bazaliy, B. A. Jones, and Shou-Cheng Zhang, Phys.\nRev. B,69, 094421 (2004).\n13A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep.,\n427, 157 (2006).\n14C. J. Garica-Cervera, Weinan E, J. Appl. Phys., 90, 370\n(2001).\n15S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph ,\nNature,425, 380 (2003).\n16X. Waintal, E. B. Myers, P. W. Brouwer, and D. C. Ralph,\nPhys. Rev. B, 62, 12317 (2000).\n17Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n18Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, Phys. Rev.\nLett.88, 117601 (2002).\n19X. Wang, G. E. W. Bauer, and A. Hoffmann, Phys. Rev.\nB,73, 054436 (2006), and references therein.\n20A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000); X. Wang, G. E. W. Bauer,\nand T. Ono, Japan. J. Appl. Phys., 45, 3863 (2006)."    },    {        "title": "2005.05742v1.Analysis_in_k_space_of_Magnetization_Dynamics_Driven_by_Strong_Terahertz_Fields.pdf",        "content": "1\nAnalysis in k-space of Magnetization Dynamics Driven by Strong\nTerahertz Fields\nV . Scalera1, M. Hudl2, K. Neeraj2, S. Perna1, M. d’Aquino3, S. Bonetti2, C. Serpico1\n1Department of Electrical Engineering and ICT, University of Naples Federico II, 80125 Naples, Italy\n2Department of Physics, Stockholm University, 106 91 Stockholm, Sweden\n3Department of Engineering, University of Naples ”Parthenope”, 80143 Naples, Italy\nAbstract —Demagnetization in a thin film due to a terahertz\npulse of magnetic field is investigated. Linearized LLG equation\nin the Fourier space to describe the magnetization dynamics is\nderived, and spin waves time evolution is studied. Finally, the\ndemagnetization due to spin waves dynamics and recent exper-\nimental observations on similar magnetic system are compared.\nAs a result of it, the marginal role of spin waves dynamics in\nloss of magnetization is established.\nIndex Terms —ultrafast magnetization dynamics, demagnetiza-\ntion, spin waves analysis\nI. I NTRODUCTION\nThe mode of operation of magnetic storage technologies\nstrongly relies on the control of fast magnetization reorienta-\ntion in a small region of a ferromagnetic media. Thus, in order\nto increase the efficiency and the speed of these technologies,\nit is crucial to develop techniques to obtain increasingly faster\nmagnetic dynamics. In this respect, large research efforts are\ncurrently carried out to achieve fast magnetic reorientation\ndynamics by using intense electromagnetic pulses [1]. In\nthe early pioneering work in this area, femtosecond optical\npulses were used to induce magnetization dynamics indirectly\nvia electronic excitation [2], [3]. More recently, it has been\nshown that intense terahertz (THz) pulses can be used to\nachieve ultrafast magnetization dynamics by direct Zeeman\ncoupling of magnetization with the magnetic field component\nof the pulse [4]. It turns out that this technique enables to\napproach the fastest possible magnetization reversal [6]. A\nsurprising result of these experiments is the reduction of the\nmagnetization module [4], even when the THz pulses have\nenergies too small to heat the medium. Such demagnetization\nprocess has been explained in terms of ultrafast scattering of\nspin polarized currents [5].\nIn this work, the role of spin waves dynamics in the demag-\nnetization process is investigated. In particular, the occurrence\nof inhomogeneities in the magnetization pattern due to spin\nwaves excitation for a thin film excited by THz pulses, similar\nto the one considered in ref. [4], is evaluated. The linear\nregime is considered and magnetization dynamics in terms of\nplane waves [7], [8], [9] in the Fourier transform domain ( k-\nspace) is described. The dispersion relations are derived and\nthe demagnetization effect due to the spin waves excitation\nis numerically computed and compared with experimental\nresults. It is found that the role of spin waves induced\ninhomogeneities is several order of magnitude smaller than\nthat measured in ref. [4], highlighting the importance of spin-\ntransport phenomena in ultrafast magnetization dynamics.II. M AGNETIZATION DYNAMICS\nThe system considered is a thin film where the magne-\ntization dynamics is assumed to be described by the LLG\nequation, expressed by the following equation:\n@M\n@t=\rM\u0002\u0012\nHeff\u0000\u000b\n\rMS@M\n@t\u0013\n; (1)\nwhere\ris the gyromagnetic ratio, \u000bis the damping constant,\nM(r;t)is the magnetization, MSis the saturation magnetiza-\ntion,Heff(r;t)is the effective magnetic field.\nThe effective magnetic field is given by the sum of several\ncontributions according to the following equation:\nHeff=Ha(r;t) +HM[M] +`2\nEXr2M; (2)\nwhere Hais the applied field, `EXis the exchange length,\nand the demagnetizing field HMis given by\nHM(r;t) =\u00001\n4\u0019rr\u0001Z\nR3M(s)\njr\u0000sjd3s: (3)\nLet the applied magnetic field be\nHa(r;t) =HDC+HTHz(r;t); (4)\nwhere HDCis constant in time and space, and HTHz(r;t)is\na pulse with sub-picosecond time duration. In the following\nit is considered the case where the film thickness dsatisfies\nthe relation d\u0018`EX. This produces that the magnetization\ndoes not change significantly along the film thickness and then\nmakes reasonable the following assumption:\nM(r;t) =M(x;y;t ) =M(\u001a;t); (5)\nwhere\u001a2R2is used to denote points of the xy-plane.\nIn this case only the average value of magnetic field (over the\nfilm thickness) affects the magnetization dynamics, namely\nhHeffi(\u001a;t) =1\ndZ+d=2\n\u0000d=2Heff(x;y;z;t ) dz : (6)\nSimilarly, we define hHTHziandhHMias the average over\nthe film thickness of HTHzandHMrespectively.\nAt this point, it is useful to express the magnetization vector\nfield through its Fourier transform:\nM(\u001a;t) =1\n4\u00192Z\nR2M(k;t) exp (ik\u0001\u001a) d2k; (7)\nwhich expresses the fact that the magnetization is seen as a\ncontinuous superposition of spin waves. When we apply the\ndemagnetizing field and the exchange field operator to thearXiv:2005.05742v1  [cond-mat.mes-hall]  12 May 2020single spin wave, we obtain an other spin wave with the same\nwave vector of the magnetization. In fact we have:\nhHMi\u0002\nM(k;t)ei\u001a\u0001k\u0003\n=\u0000D(k)M(k;t)ei\u001a\u0001k;(8)\nand\n`2\nEXr2\u0000\nM(k)ei\u001a\u0001k;t\u0001\n=\u0000k2`2\nEX\u0000\nM(k;t)ei\u001a\u0001k\u0001\n;(9)\nwherek=jjkjjandD(k)is a symmetric 3\u00023matrix (see\nappendix A).\nAs we will see in the following, equations (8) and (9) imply the\nwell known fact that the linear dynamics of spin waves with\ndifferent wave vectors are independent. When HTHz(t) =0,\nthe equilibrium condition of (1) is satisfied by uniform mag-\nnetization and is expressed by the following Brown’s equation\nM0\u0002hHeff;0i=0: (10)\nwhich in scalar form reads as\nHDCsin(\u0012H\u0000\u0012M)\u0000MScos\u0012Msin\u0012M= 0; (11)\nwhere\u0012Mand\u0012Hare the angles between the unit vector ez\nand the vectors M0andHDCrespectively.\nIntroducing a Cartesian coordinate system fem;e\u0012;e'g, given\nby\nem=M0\nMS;e'=ez\u0002em\njez\u0002emj;e\u0012=e'\u0002em\nje'\u0002emj;(12)\nthe magnetization is written as\nM(\u001a;t) =MSem+M\u0012(\u001a;t)e\u0012+M'(\u001a;t)e';(13)\nwhereM\u0012(\u001a;t)andM'(\u001a;t)are the first order variations.\nBy linearizing (1) and projecting it on the plane orthogonal to\nM0, it eventually yields\n@M\u0012\n@t\u0000\u000b@M'\n@t=\rH0M'\u0000\rMSH'; (14)\n\u0000\u000b@M\u0012\n@t\u0000@M'\n@t=\rH0M\u0012\u0000\rMSH\u0012; (15)\nwhere\nH'=hHeffi\u0001e'; H\u0012=hHeffi\u0001e\u0012; H 0=jjhHeff;0ijj:\nIn this framework the field hHTHzi(\u001a;t)is treated as a first\norder perturbation.\nReplacing expression (2) of the effective field (along with (9)\nand (8) ) into (14) and (15) and taking the Fourier transform\nin space, we arrive to the following system of equations:\n@\n@t\u0014\nM\u0012\nM'\u0015\n=A(k)\u0014\nM\u0012\nM'\u0015\n\u0000\rMS\u0014\nH\u0012\nH'\u0015\n;(16)\nwhereH\u0012andH'are the components along e\u0012ande'of\nthe Fourier transform in space of the applied THz field, the\ndynamical matrix is\nA=\u00141\u0000\u000b\n\u0000\u000b\u00001\u0015\u00001\u0014\rMSD\u0012'(k)\r^H\u0012(k)\n\r^H'(k)\rMSD'\u0012(k)\u0015\n;\n(17)\nand\n^Hi(k) =H0+MSDii(k) +MSk2`2\nEX: (18)For every k, equation (16) describes a 2\u00022linear system\nwithout interactions between spin waves with different k.\nDynamics can be simulated separately for every wave vector,\nwhereas the magnetization in real space is obtained through\n(7).\nAt this point, it is useful to derive the dispersion relations for\nspin waves dynamics, which will be of help in explaining the\nnumerical results of the next section. Dispersion relations are\nobtained by imposing det(A\u0000i!I) = 0 with\u000b= 0.\nTwo cases are considered: first, when the exchange field in\nnegligible compared with the demagnetizing field and second,\nwhen the demagnetizing field is negligible compared to the\nexchange field.\nLet us start with the first case. This situation occurs when\nthe demagnetizing coefficients Dij(k)are much greater than\n`2\nEXk2, sincejDijj\u00141and then the condition required is\n`2\nEXk2\u001c1.\nThis type of waves are called magnetostatic waves.\nLet us focus on the case with the magnetization out of plane\n(\u0012M= 0 ). This occurs when the applied field is out of\nplane andHa> MS. By replacing H0=Ha\u0000MSand\nthe demagnetizing factors with their expressions in (41), it\neventually yields\n!=\u0000\rp\n(Ha\u0000MS)(Ha\u0000MSSk); (19)\nwhereSkis defined in (40). The dispersion relation (19) has a\npositive group velocity and it is called forward magnetostatic\nwave [9].\nNext we consider the magnetization is in plane ( \u0012M=\u0019=2),\nwhich occurs when the applied field is in plane. Replacing\nH0=Haand the expressions of the demagnetizing factors\nfrom (41) yields\n!=\u0000\rq\n(Ha+MSSk)(Ha+MS(1\u0000Sk) sin2\u001ek);(20)\nwhere\u001ekis the angle between M0andk. When\u001ek= 0,\n(20) has a negative group velocity and it called backward\nmagnetostatic wave [9].\nThe plot of (19) and (20) are in Figure 1.\nConsider now the case `2\nEXk2\u001d1, i.e. the magnetostatic\nfield is negligible. In this case the dispersion relation does not\ndepend on the orientation k. We have\n!=\u0000\r(H0+MS`2\nEXk2): (21)\nFor spatially uniform, or nearly uniform, terahertz pulse\ndistribution (kd\u001c1)the resonance frequencies for spin\nwaves excitation are in the order of \u001810GHz . Therefore,\nit is expected that the nonlinear spin waves dynamics regime\nis not reached, even for high power magnetic field pulse used\nfor the experimental investigations of ref. [4].\nIII. S IMULATION\nWe consider a system similar to the one used in [4]: The\nspecimen is a thin film with thickness d= 5 nm, the material\nparameters are \u00160MS= 1:84T,\r=\u0000176 rad=(T\u0001ns),\n\u000b=\u00000:007 and`EX= 4:4nm, and the applied magnetic\nfield out of plane component \u00160H?= 0:448 T whereas the\nin plane component is \u00160Hk= 0:056T. The terahertz field isFig. 1. Dispersion relations of magnetostatic waves ( `EX = 0 ). The\nsaturation magnetization is \u00160MS= 1 T and the applied field at the\nequilibrium is \u00160H0= 0:5T, i.e. the applied field is \u00160Ha= 0:5T for\nthe magnetization in plane and \u00160Ha= 1:5T for the magnetization out of\nplane. The blue curve is represents (19), the red and yellow curves represent\n(20) for\u001ek= 0 and\u001ek=\u0019=2respectively.\napplied in plane (orthogonally to the constant field) and the\nintensity is\nHTHz(\u001a;t) =HMax\n\u001b2\nt(t2\u0000\u001b2\nt) exp\u0012\n\u0000t2\n2\u001b2\nt\u0000\u001a2\n2\u001b2x\u0013\n;(22)\nwhere\u00160HMax= 0:06T,\u001bx= 500\u0016m and\u001bt= 0:5ps.\nThe time evolution of HTHz is displayed in the left panel of\nFigure 2.\nThe system (16) is simulated within a range of wave number\nFig. 2. Time profile of the applied terahertz pulse.\nup to 10\u00005nm\u00001, higher wave numbers are not excited by the\napplied field. The time evolution of M(k;t)is shown Figure\n5 for several values of k.\nIn the range of wave vector excited the dynamics does not\nchange appreciably, hence the space profile of either M\u0012andM'are almost the same as HTHz(\u001a;0)rescaled. The space\ndependence ofM\u0012andM'are in Figure 3.\nFig. 3. Space dependence of the applied terahertz field HTHz and the\nperturbation of the magnetization M\u0012andM'. The red solid line is M\u0012,\nthe dashed blue line is M'and the dashed purple line is HTHz. The plot are\nrescaled so that the in \u001a= 0 the three curves have the same value.\nIn order to evaluate the demagnetization observed the mag-\nnetization is normalized, since the linearized model does not\npreserve the magnetization modulus. We have\nM=MSM0+M\u0012e\u0012+M'e'\njjM0+M\u0012e\u0012+M'e'jj=MSM0+\u000eM\njjM0+\u000eMjj;\n(23)\nwhere\u000eM=M\u0012e\u0012+M'e'.\nThe measured magnetization is given by\nMmeas=1\nj\nMj\r\r\r\rZ\n\nMMdS\r\r\r\r(24)\nwhere \nMis the area hit by the probe, i.e. a circular area with\nradius 125\u0016m, andj\nMjis the measure of the area.\nDeveloping the Taylor series up to the second order in\n\u000eM=MSand using M0?\u000eM, the integral is approximated\nby\nZ\n\nMMdS\u0019Z\n\nM \nM0\u0000M0\n2\r\r\r\r\u000eM\nMS\r\r\r\r2\n+\u000eM!\ndS :\n(25)\nBy replacing (25) in (23), neglecting the terms of order greater\nthan two in \u000eM=MS, we eventually obtain the demagnetiza-\ntion\nMmeas=q\nM2\nS\u0000I2+I1; (26)\nwhere\nI1=1\nj\nMj2\u0012Z\n\nM\u000eMdS\u00132\n; (27)\nand\nI2=1\nj\nMjZ\n\nM\u000eM2dS : (28)\nIt is noteworthy that I2\u0015I1because of Cauchy-Schwartz\ninequality, hence the measured magnetization can only be\nsmaller than MS.\nThe plot of the relative reduction of the measured magnetiza-\ntion is displayed in Figure 4.Fig. 4. Relative reduction of the measured magnetization.\nRemarkably, the reduction of the observed magnetization\ndoes not grow over 2:5\u000110\u000010, whereas the demagnetization\nobserved in [4] is several order of magnitude higher (roughly\n2\u000110\u00003).\nIV. C ONCLUSION\nIn conclusion, the demagnetization effect induced in a\nthin film excited by a strong terahertz pulse is investigated.\nThe magnetization dynamics is described in terms of linear\nspin waves dynamics governed by the linearized LLG\nprojected into the Fourier’s k-space. In this micromagnetic\nframework, we tried to reproduce the experiment described\nin [4] and determine whether the spin waves play a role\nin the experimental observed demagnetization explained in\nterms of ultrafast scattering of spin polarized currents [5].\nIt is found, that the reduction of the magnetization module\ncomputed in the simulation is several orders of magnitudes\nsmaller than the demagnetization observed in the experiments.\nMoreover, despite the field intensity is high, the pulse is so\nshort that the magnetization barely moves from equilibrium\nand spin waves do not grow enough to cause relevant\nnonlinear effects [7], [8]. Then, the magnetization reduction\ndue to non-uniformities is negligible and the role of ultrafast\nspin-transport phenomenon is prevalent.\nAPPENDIX A\nDEMAGNETIZING COEFFICIENTS COMPUTATION\nThis section derives the analytical expressions for the de-\nmagnetizing factor D(k)defined in equation (8). Its elements\nexpress the linear relation between the averaged demagnetizing\nfield and the magnetization\n2\n4hHMxi\nhHMyi\nhHMzi3\n5=\u00002\n4DxxDxyDxz\nDxyDyyDyz\nDzxDzyDzz3\n52\n4Mx\nMy\nMz3\n5:(29)TheDij(k)can be obtained from (3), which is the general\nsolution of the magnetostatic problem\nr\u0002HM= 0;r\u0001HM=\u0000r\u0001M: (30)\nAlternatively, the system (30) can be restated in terms of a\nscalar potential  , that is\nHM=\u0000r ;r2 =r\u0001M: (31)\nLet us focus on the demagnetizing field in the thin film with\nmagnetization given by\nM(r) =(\nM0exp(ik\u0001\u001a) ifjzj\u0014d=2\n0 ifjzj>d=2: (32)\nThe particular solution of (31) is\n p(r) =8\n<\n:k\u0001M0\nik2exp(ik\u0001\u001a) ifjzj\u0014d=2\n0 ifjzj>d=2:(33)\nThe potential (33) does not satisfy the interface conditions on\nthe thin film surfaces, namely the continuity of the normal\ncomponent of the magnetic density flux\n(Hm\nM(r;t) +M(r;t))\u0001ez=Ha\nM(r;t)\u0001ez; (34)\nand the continuity of the tangent component of the magnetic\nfield\nHm\nM(r;t)\u0002ez=Ha\nM(r;t)\u0002ez; (35)\nwhere the superscript ‘ m’ and ‘ a’ denote the fields in the\nmagnetic medium and in the air respectively, and equations\n(34) and (35) are imposed at z=\u0006d=2.\nThe solution of (31) is obtained by summing to (33) a linear\ncombination of harmonic functions\n \u0006(r) = exp(ik\u0001\u001a\u0006kz): (36)\nThe homogeneous solution is\n 0(r) =8\n><\n>:C1 \u0000 ifz > d= 2\nC2 \u0000+C3 + ifjzj\u0014d=2\nC4 + ifz <\u0000d=2;(37)\nwhereC1,C2,C3andC4are constant to be determined.\nBy using (34) and (35), we obtain a set of linear equations in\nC1,C2,C3andC4, which yields\nC1=\u0000sinh(kd=2)\nk[(M0\u0001ek) + (M0\u0001ez)];\nC2=exp(\u0000kd=2)\n2k[(M0\u0001ek) + (M0\u0001ez)];\nC3=exp(\u0000kd=2)\n2k[(M0\u0001ek)\u0000(M0\u0001ez)];\nC4=\u0000sinh(kd=2)\nk[(M0\u0001ek)\u0000(M0\u0001ez)];(38)\nwhereek= (1=k)k.\nThe derivation of the averaged demagnetizing field is straight-\nforward. We eventually have\nhHMi=\u0000[(M0\u0001ez)Sk+(M0\u0001ek)(1\u0000Sk)] exp(ik\u0001\u001a);(39)\nwhere\nSk= [1\u0000exp(\u0000kd)]=kd : (40)Fig. 5. Amplitude of the spin waves of the system described in the simulation section. Lager curves correspond to spin waves with smaller wave number,\nthe orientation of khas almost no impact on the dynamic.\nFig. 6.Skfunction defined in (40)\nThe function Skis shown in Figure 6. By using the rotated\nreference frame defined in (12), we obtain the demagnetizing\nfactors used in (17) and (18). They are\nD\u0012\u0012= (1\u0000Sk) cos2\u0012Mcos2\u001ek+Sksin2\u0012M;\nD\u0012'=D'\u0012=1\n2(1\u0000Sk) cos\u0012Msin(2\u001ek);\nD''= (1\u0000Sk) sin2\u001ek;(41)\n[2] E. Beaurepaire, J. Merle, A. Daunois, J. Bigot, “Ultrafast Spin Dynamics\nin Ferromagnetic Nickel”, Physical Review Letters , vol. 76, pp. 4250-4253\n(1996).where\u0012Mis the angle between ezandem, and\u001ekis angle\nbetweenekand the projection of either e\u0012oremin thexy-\nplane.\nREFERENCES\n[1] J. Walowski and M. Mnzenberg,, “Perspective: Ultrafast magnetism and\nTHz spintronics”, Journal of Applied Physics , vol. 120, 140901 (2016).\n[3] A. Kirilyuk, A. V . Kimel, T. Rasing, “Ultrafast optical manipulation of\nmagnetic order”, Reviews of Modern Physics , vol. 82, pp. 2731-2784,\n(2016).\n[4] M. Hudl, M. dAquino, C. Serpico, M. Pancaldi, S.-H. Yang, M.G.\nSamant, S.S.P. Parkin, H.A. Durr, M.C. Hoffmann, S. Bonetti, “Nonlinear\nmagnetization dynamics driven by strong terahertz fields”, Physical Review\nLetters , vol. 123 , issue 19, pp 197-204 (2019)\n[5] S. Bonetti, M. C. Hoffmann, M. J. Sher, Z. Chen, S. H. Yang, M. G.\nSamant, S. S. P. Parkin, and H. A. Durr, “THz-Driven Ultrafast Spin-\nLattice Scattering in Amorphous Metallic Ferromagnets”, Physical Review\nLetters , vol. 117, 087205, (2016).\n[6] S. J. Gamble, M. H. Burkhardt, A. Kashuba, R. Allenspach, S. S. P.\nParkin, H. C. Siegmann, and J. Sthr, “Electric Field Induced Magnetic\nAnisotropy in a Ferromagnet”, Physical Review Letters , vol. 102, 217201\n(2009).\n[7] H. Suhl, “The theory of ferromagnetic resonance at high signal powers”,\nJournal of Physics and Chemistry of Solids , vol. 1, issue 4, pp. 209-227\n(1957)\n[8] G. Bertotti, I.D. Mayergoyz, C. Serpico, “Spin-Wave Instabilities in\nLarge-Scale Nonlinear Magnetization Dynamics”, Physical Review Letters ,\nvol. 87, issue 21, pp. 217203 (2001)\n[9] D.D. Stancil, A Prabhakar, “Spin Waves: Theory and Applications”,\nSpringer (2009)"    },    {        "title": "2211.15091v1.Superfluid_like_spin_transport_in_the_dynamic_states_of_easy_axis_magnets.pdf",        "content": "Super\ruid-like spin transport in the dynamic states of easy-axis magnets\nJongpil Yun and Se Kwon Kim\nDepartment of Physics, Korea Advaced Institute of Science and Technology, Daejeon 34141, Republic of Korea\n(Dated: November 29, 2022)\nThe existing proposals for super\ruid-like spin transport have been based on easy-plane magnets\nwhere the U(1) spin-rotational symmetry is spontaneously broken in equilibrium, and this has been\nlimiting material choices for realizing super\ruid-like spin transport to restricted class of magnets. In\nthis work, we lift this limitation by showing that super\ruid-like spin transport can also be realized\nbased on easy-axis magnets, where the U(1) spin-rotational symmetry is intact in equilibrium but\ncan be broken in non-equilibrium. Speci\fcally, we \fnd the condition to engender a non-equilibrium\neasy-cone state by applying a spin torque to easy-axis magnets, which dynamically induces the\nspontaneous breaking of the U(1) spin-rotational symmetry and thereby can support super\ruid-like\nspin transport. By exploiting this dynamic easy-cone state, we show theoretically that super\ruid-\nlike spin transport can be achieved in easy-axis magnets under suitable conditions and con\frmed the\nprediction by micromagnetic simulations. We envision that our work broadens material library for\nrealizing super\ruid-like spin transport, showing the potential utility of dynamic states of magnets\nas venue to look for spin-transport phenomena that do not occur in static magnetic backgrounds.\nI. INTRODUCTION\nSpintronics is a \feld that harnesses spin degree of free-\ndom of electrons to store, transport, and process infor-\nmation in order to go beyond the conventional electronics\nwhere only charge degree of freedom has been used [1{3].\nSince information is encoded in the form of spin, it is im-\nportant to achieve e\u000ecient spin transport in spintronics,\nwhich demands us to identify and employ low-dissipation\nmagnetic materials and spin transport therein. In this\nregard, magnetic insulators have emerged as energy-wise\ne\u000ecient material platforms for spintronics since they have\nno Joule heating associated with an electric current and\ntherefore can enable low-power spin-based information\ntransport and processing. In a magnetic insulator, spin is\ntransported by its collective excitations, i.e., spin waves,\nwhose quanta are called magnons [4{7]. Since magnon-\nbased information transport and processing can be real-\nized without the Joule heating in principle, generating\nand controlling magnons have been actively investigated\nto realize magnonic devices [8{10], which includes the ex-\nperimental demonstration of long-distance spin transport\nin certain magnets [11{13]. Most of the previously stud-\nied magnonic spin transport have been based on di\u000busion\nof magnons, which has a critical problem in that the spin\ncurrent exponentially decays away from the spin-current\nsource [11]. To circumvent this problem of rapidly decay-\ning spin current of di\u000busive magnons, a novel type of spin\ntransport referred to as super\ruid-like spin transport has\nemerged as an alternative for e\u000ecient spin transport [14].\nSuper\ruid-like spin transport is a spin-analogue of\nmass super\ruidity. The mass super\ruidity can occur\nwhen the wave function de\fned by  =pnei\u0012, where\nnis a particle density and \u0012is an arbitrary phase, breaks\nthe U(1) phase symmetry spontaneously [15]. Like-\nwise, super\ruid-like spin transport can occur in mag-\nnetically ordered systems when the magnetic order pa-\nrameter breaks the U(1) spin-rotational symmetry spon-\ntaneously [14, 16]. In contrast to the exponential de-\n<latexit sha1_base64=\"HA1GJXdAB5Hew9Hgsu3zSpKlO1E=\">AAACDHicbVDLSgNBEJyNrxhfUY9eBoPgxbArQT1GvHiMYKKQBOmddHTI7Owy0ysJSz7Ai7/ixYMiXv0Ab/6NszEHXwUDRVV1D11hoqQl3//wCjOzc/MLxcXS0vLK6lp5faNl49QIbIpYxeYyBItKamySJIWXiUGIQoUX4eAk9y9u0VgZ63MaJdiN4FrLvhRATroqVzqEQ8oQ7GgPhtLy3EeSgkttUwUUm7FL+VV/Av6XBFNSYVM0rsrvnV4s0gg1CQXWtgM/oW4Gxu1VOC51UosJiAFcY9tRDRHabjY5Zsx3nNLj/di4p4lP1O8TGUTWjqLQJSOgG/vby8X/vHZK/aNuJnWSEmrx9VE/VZxinjfDe9KgIDVyBISReQfiBgwIcv2VXAnB75P/ktZ+NTio1s5qlfrxtI4i22LbbJcF7JDV2SlrsCYT7I49sCf27N17j96L9/oVLXjTmU32A97bJ+CvnCc=</latexit>easy-axis magnetic insulator\n<latexit 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Schematics of the experimental setup for realizing\nsuper\ruid-like spin transport in a magnetic insulator having\nthezaxis as an easy axis subjected to a spin torque, denoted\nby a spin-torque oscillator. In the magnetic insulator, the\nblue arrows depict the spatially varying magnetization and\nthe dashed cones represent the precession trajectories (form-\ning cones with tilting angle \u0012from the easy axis) of the local\nspins driven by the spin torque. The black arrow of the left\nmetal represents the charge current, which injects the spin\ncurrent (JL) to the left end of the ferromagnet via the spin\nHall e\u000bect. The red arrows of the left and the right metals are\nthe direction of spin accumulations, at the interfaces between\nthe metals and the ferromagnet. The spin accumulation \u0016R\nat the right metal generated by the nonlocal spin transport\nfrom the left metal can be detected via the inverse spin Hall\ne\u000bect.\ncaying of di\u000busive spin transport, super\ruid-like spin\ntransport has the characteristic of algebraically decay-\ning spin current, which enables long-distance spin trans-\nport in certain magnets [17{26]. The existing research\nof super\ruid-like spin transport has been focused only\non easy-plane magnets in which the system breaks the\nU(1) spin-rotational symmetry spontaneously in equilib-\nrium [27{30].\nIn this work, for a potential material platform for\nsuper\ruid-like spin transport, we consider easy-axis mag-\nnets, where the magnetization aligns with easy-axis and\nthus does not break U(1) spin-rotational symmetry spon-\ntaneously in equilibrium. Instead of using the equilib-\nrium U(1) symmetry breaking as done for previous pro-arXiv:2211.15091v1  [cond-mat.mes-hall]  28 Nov 20222\nposals based on easy-plane magnets, we turn to the dy-\nnamic breaking of the U(1) spin-rotational symmetry.\nMore speci\fcally, to break U(1) spin-rotational symme-\ntry spontaneously in easy-axis magnets, we apply a spin\ntorque and engender a dynamic easy-cone state to sup-\nport super\ruid-like spin transport [31{37]. The system is\nschematically illustrated in Fig. 1. The easy-axis magnet\nsubjected to a suitable spin torque forms a spin-torque\noscillator, in which the local magnetization (shown as\nthe blue arrows) precesses within cones (depicted by the\ndashed black lines with the cone angle \u0012) by breaking the\nU(1) spin-rotational symmetry dynamically. Applying a\ncharge current in the left metal injects a spin current JL\nfrom the left metal to the left end of the magnet via the\nspin Hall e\u000bect [34, 38{40]. The injected spin current is\ntransported through the magnet by super\ruid-like spin\ntransport in the form of the spatially varying order pa-\nrameter. The spin transport generates a spin accumula-\ntion\u0016Rat the interface between the ferromagnet and the\nright metal, which can be probed either by spin pump-\ning [41, 42] or by the inverse spin Hall e\u000bect. In our\nsystem, non-local spin transport refers to the generation\nof the spin accumulation \u0016Rat the right end of the fer-\nromagnet by the spin-current injection from the left end.\nBy the theoretical analysis based on the Landau-Lifshitz-\nGilbert (LLG) equation [43, 44] and the micromagnetic\nsimulations, we show that the spin accumulation \u0016Rde-\ncays algebraically as the ferromagnet length increases,\nexhibiting super\ruid-like spin transport.\nThe paper is organized as follows. In Sec. II, we de-\nscribe the model system, namely the easy-axis ferromag-\nnet subjected to a spin torque, and identify the condi-\ntion with which super\ruid-like spin transport can be re-\nalized by theoretical analysis and micromagnetic simula-\ntions. In Sec. III, we theoretically and numerically show\nthat our system can indeed support algebraically decay-\ning spin current, i.e., super\ruid-like spin transport. In\nSec. IV, we summarize our results.\nII. MODEL\nOur model system which is illustrated in Fig. 1 is a\nquasi-one-dimensional ferromagnetic wire whose energy\nis given by\nU=Z\ndV\u0014Am02\u0000Ke\u000bm2\nz+K2m4\nz\n2\u0000H\u0001m\u0015\n;(1)\nwhere mis the three-dimensional unit vector in the di-\nrection of the magnetization, the prime (0) is the gradient\nwith respect to the z-coordinate along the wire, Ais the\nexchange coe\u000ecient, Ke\u000b>0 is the \frst-order e\u000bective\nanisotropy which combines the shape anisotropy and the\n\frst-order easy-axis crystalline anisotropy, K2>0 is the\nsecond-order anisotropy [45{50], and His the external\nmagnetic \feld. We assume that the system is quasi-one-\ndimensional so that the magnetization varies only along\nthezdirection: m(z;t). The external magnetic \feld isapplied along the easy-axis direction: H=H^z. We con-\nsider the cases where the ground state is given by the\nuniform magnetization in the zdirection m(z;t)\u0011^z,\nwhich is satis\fed when K2<(Ke\u000b+H)=2. Note that\nthe energy Uis invariant under uniform rotations of the\nmagnetization about the z-axis, i.e., m7!Rzmwith an\narbitrary rotation matrix Rzabout thezaxis, indicating\nthat the system possess the U(1) spin-rotational symme-\ntry about the zaxis. Since the ground state m(z;t)\u0011^z\nis invariant under the spin rotations, it does not break\nthe U(1) spin-rotational symmetry.\nThe equation of motion for the dynamics of the magne-\ntization msubjected to a spin torque is given by the LLG\nequation [43, 44] augmented by the spin-torque term :\ns_m\u0000\u000bsm\u0002_m=\u0000m\u0002he\u000b+\u001cST; (2)\nwheresis the saturated (scalar) spin density, the dot\n( _ ) denotes di\u000berentiation with respect to time, \u000b > 0\nis the dimensionless Gilbert damping parameter, he\u000b=\n\u0000\u000eU=\u000emis the e\u000bective \feld, and \u001cST=\u001cSTm\u0002(m\u0002^z)\nis a externally-applied spin torque polarized along the\nzdirection, which can be realized either by the spin-\ntransfer torque [51] or by spin-orbit torque [52]. We as-\nsume that this spin torque \u001cSTis exerted uniformly on\nthe ferromagnet.\nTo endow the ferromagnet with the capability to sup-\nport super\ruid-like spin transport, it is necessary to in-\nduce the ferromagnet to break the U(1) spin-rotational\nsymmetry dynamically, which can be done by driving it\ninto a self-oscillatory mode with the su\u000eciently strong\nspin torque. The detailed condition for this oscillating\nphase can be obtained as follows. The LLG equation in\nterms of the polar angle ( \u0012) and the azimutal angle ( \u001e)\nwithm= sin\u0012cos\u001e^x+ sin\u0012sin\u001e^y+ cos\u0012^zis given by\ns\u0010\n_\u0012sin\u0012+\u000b_\u001esin2\u0012\u0011\n=A\u0000\n\u001e0sin2\u0012\u00010+\u001cSTsin2\u0012;(3)\ns\u0010\n_\u001esin\u0012\u0000\u000b_\u0012\u0011\n=A\u0000\n\u001e02sin\u0012cos\u0012\u0000\u001200\u0001\n+Hsin\u0012+Ke\u000bsin\u0012cos\u0012\n\u00002K2sin\u0012cos3\u0012:(4)\nEquation (3) has a clear physical meaning: It is the\nspin continuity equation: The \frst term and the sec-\nond term on the left-hand side are the time evolution\nof thezcomponent of spin density and the damping\nterm, respectively. The \frst term on the right-hand\nside of Eq. (3) is the divergence of spin current density:\njs=\u0000Asin2\u0012@x\u001eand the second term is the spin cur-\nrent coming from the bulk spin torque.\nNow, we look for a condition under which the spin\ntorque induces a dynamic easy-cone state and thus the\nspontaneous breaking of the U(1) spin-rotational sym-\nmetry. A steady-state solution of Eqs. (3) and (4) with\nconstant polar angle with _\u0012= 0 satis\fes\n\u001cST=\u000b\u0000\nKe\u000bcos\u0012+H\u00002K2cos3\u0012\u0001\n: (5)3\nThen, the condition that the system is in a dynamic easy-\ncone state with 0 <\u0012<\u0019 is given by\n\u000b(Ke\u000b+H\u00002K2)<\u001cST<\u000b \n2\n3r\nKe\u000b\n6K2Ke\u000b+H!\n;\n(6)\nKe\u000b\n6<K2<Ke\u000b+H\n2; (7)\nwhere\u001cc;1=\u000b(Ke\u000b+H\u00002K2) is the lower critical torque\nthat is given by the value of the right-hand side of Eq. (5)\nat\u0012= 0 and\u001cc;2=\u000b\u0010\n2\n3q\nKeff\n6K2Ke\u000b+H\u0011\nis the upper\ncritical torque. The lower critical torque \u001cc;1is the mini-\nmum torque that is required to drive the ferromagnet into\nthe dynamic easy-cone state, i..e, self-oscillatory state.\nWhen we apply the torque lower than the lower critical\ntorque, the magnetization is kept along the zaxis with-\nout breaking the U(1) spin-rotational symmetry. The\nphysical meaning of the upper critical \feld is as follows.\nWhen we apply the spin torque larger than the upper\ncritical torque \u001cc;2, the magnetization reverses into the\nnegativezdirection completely, i.e, m=\u0000^z, which does\nnot show the dynamic oscillation of the magnetization.\nEquation (7) states the condition for the system param-\neters,Ke\u000b;K2, andH. WhenK2is smaller than Ke\u000b=6,\nthe system does not possess a dynamic easy-cone state,\nbut is saturated along either the positive zdirection (for\nweak spin torques) or the negative zdirection (for strong\nspin torques). The condition K2<(Ke\u000b+H)=2 is to\nensure that the unperturbed ground state of the ferro-\nmagnet is given by the zdirection, as discussed above.\nWhen the spin torque and the system parameters sat-\nisfy Eqs. (6) and (7), the system is driven into a dynamic\neasy-cone state, where the magnetization precesses about\nthezaxis with arbitrary initial values for the azimuthal\nangle\u001eand thereby spontaneously break the U(1) spin-\nrotational symmetry.\nTo con\frm the spontaneous U(1) spin-rotatinal sym-\nmetry breaking under conditions [Eqs. (6) and (7)], we\nrun the micromagnetic simulation using MuMax3[53]\nwith the following material parameters of NdCo 5:Ke\u000b=\n2:4\u0002106J/m3,H= 1T; Ms= 1:1\u0002106A/m; A =\n1:05\u000210\u000011J/m and\u000b= 0:1 [54{57]. The demagne-\ntization e\u000bects are captured by the \frst-order e\u000bective\nanisotropyKe\u000bthrough the shape anisotropy in our sim-\nulations, as done analytically in Refs. [46, 49, 50, 58],\nwhich is expected to be a good approximation when\nthe system is quasi-one-dimensional cylindrical wire [59{\n61]. We run the simulation with three di\u000berent val-\nues of second-order anisotropy: K2= 0:4\u0002106J/m3,\n1\u0002106J/m3and 1:6\u0002106J/m3. For the polar angle in\nthe presence of a spin torque, the analytical result (which\ncan be obtained by solving Eq. (5) for \u0012) and the simu-\nlation results are shown in Fig. 2(a) as the lines and the\nsymbols, respectively, which agree with each other. In\nFig. 2(a), the blue and the red symbols correspond to the\ncases withK2= 1:6\u0002106J/m3andK2= 1\u0002106J/m3,\n<latexit sha1_base64=\"6o24vuTY54UwtoL+zl+HXursiqM=\">AAAB6nicbVBNS8NAEJ34WetX1aOXxSLUS0mkqMeCF48V7Qe0oWy2m3bpZhN2J0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekEhh0HW/nbX1jc2t7cJOcXdv/+CwdHTcMnGqGW+yWMa6E1DDpVC8iQIl7ySa0yiQvB2Mb2d++4lrI2L1iJOE+xEdKhEKRtFKD5Xgol8qu1V3DrJKvJyUIUejX/rqDWKWRlwhk9SYrucm6GdUo2CST4u91PCEsjEd8q6likbc+Nn81Ck5t8qAhLG2pZDM1d8TGY2MmUSB7YwojsyyNxP/87ophjd+JlSSIldssShMJcGYzP4mA6E5QzmxhDIt7K2EjaimDG06RRuCt/zyKmldVr2rau2+Vq7X8zgKcApnUAEPrqEOd9CAJjAYwjO8wpsjnRfn3flYtK45+cwJ/IHz+QOMhI1S</latexit>(b)\n<latexit 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(a) The polar angle ( \u0012) of the magnetization as a func-\ntion of the bulk spin torque ( \u001cST). The lines show the theoret-\nical results [Eq. (5)] and the symbols represent the simulation\nresults. The circles, the triangles, and the squares correspond\ntoK2= 2Ke\u000b=3,K2= 5Ke\u000b=12, andK2=Ke\u000b=6, respec-\ntively. (b) The spin accumulation ( j\u0001\u0016Rj) at the right bound-\nary of the ferromagnet induced by a spin-current injection JL\nfrom the left boundary as a function of the ferromagnet length\n(L). The lines show the theoretical result [Eq. (13)] and the\nsymbols correspond to the simulations results. The circles,\nthe triangles, and the squares correspond to input spin cur-\nrentJL= 2\u000210\u00006J/m2, 6\u000210\u00006J/m2and 10\u000210\u00006J/m2,\nrespectively.\nrespectively. For these cases, the ferromagnet is in a\nsteady state with nontrivial polar angle \u00126= 0;\u0019, i.e.,\nin a dynamic easy-cone state under suitable spin-torque\nvalues. The black symbols correspond to the cases with\nK2=Ke\u000b=6, where the polar angle is either 0 or \u0019re-\ngardless of the spin-torque values as discussed above and\nthe dynamic easy-cone state is not available.\nThe obtained dynamic easy-cone state can be inter-\npreted in the framework of the Gross-Pitaevskii equa-\ntion [62, 63] as follows. The LLG equation (2) for\nthe magnetization mcan be recast into the equation\nfor the complex order parameter de\fned by  (x;t) =\nmx(x;t)\u0000imy(x;t) =p\u001aei\u001e, where\u001a= sin\u0012and\u001eare\nanalogous to the density and the phase of the conden-\nsate [23, 64]. The LLG equation in terms of the complex\norder parameter  is given by\nis@ \n@t=\u0012\n(Ke\u000b\u00002K2+H) +\u0012\n\u00001\n2Ke\u000b+ 3K2\u0013\nj j2\u0013\n \n+s\u000b\u0012\n1\u00001\n2j j2\u0013@ \n@t+i\u001cST\u0012\n1\u00001\n2j j2\u0013\n ;\n(8)4\nup to the third order in  . The \frst term on the right-\nhand side of the equation originates from the potential\nenergy of our system [Eq. (1)], in which ( Ke\u000b\u00002K2+\nH) can be interpreted as the single particle potential\nand (\u0000Ke\u000b=2 + 3K2) can be regarded as the interaction\nstrength. The condition to possess a vacuum ground\nstate and the condition to have a stable condensate under\npumping (i.e., the repulsive interaction) are respectively\ngiven byK2<(Ke\u000b+H)=2 andKe\u000b=6<K 2, which are\nidentical to the conditions in Eq. (7) that were obtained\ndirectly from the LLG equation.\nThere are two additional terms on the right-hand side\nin the equation that break the time-reversal symmetry:\nThe second term is the damping term and the third term\nrepresents the particle pumping by the spin torque. To\nhave a \fnite condensate  6= 0 in a steady state, the\npumping/\u001cSTshould be su\u000eciently strong to dominate\nthe damping term. This condition is given by \u001cST>\n\u000b(Ke\u000b+H\u00002K2), which is identical to the lower critical\ntorque that we obtained above. The upper critical torque\nis not available in Eq. (8), since it is truncated to the third\norder in the order parameter and thus cannot capture the\ndynamics of the dense condensate.\nIII. SUPERFLUID-LIKE SPIN TRANSPORT\nNow, let us investigate the nonlocal spin transport be-\nhavior of an obtained dynamic easy-cone state of the\neasy-axis ferromagnet by assuming that our system sat-\nis\fes Eqs. (6) and (7) so that it breaks the U(1) spin-\nrotational symmetry dynamically. The situation that we\nconsider is depicted in Fig. 1. To inject a spin current JL\nto the ferromagnet through the left end, one heavy metal\nwith a \fnite charge current is attached to the left end of\nthe ferromagnet. To detect a spin accumulation \u0016Rat\nthe right end of the ferromagnet, the other heavy metal\nwith no external current is attached to the right end of\nthe ferromagnet. When we attach the metals to the left\nand right boundaries of the ferromagnet, there arises two\ne\u000bects: the spin-current injection from the metal with a\n\fnite current and spin pumping from the ferromagnet to\nthe metals [24], which determines the boundary condi-\ntions for the spin current at the left end x= 0 and the\nright endx=L:\njs(0) =JLsin2\u0012\u0000\rsin2\u0012_\u001e(0); (9)\njs(L) =\rsin2\u0012_\u001e(L); (10)\nwherejs=\u0000Asin2\u0012@x\u001eis the spin current of the ferro-\nmagnet,Lis the length of the ferromagnet, \u0012is the polar\nangle of the magnetization, \r=~g\"#=4\u0019, andg\"#is the\ne\u000bective interfacial spin-mixing conductance between the\nferromagnet and the normal metal [39, 65]. The \frst term\n/JLon the right-hand side of Eq. (9) is the spin current\ninjected from the left metal to the ferromagnet by the\nspin Hall e\u000bect, where JLis proportional to the product\nof the charge current \rowing in the left metal and thee\u000bective spin Hall angle of the interface between the fer-\nromagnet and the left metal. The second term /\ron\nthe right-hand side is the spin current ejected from the\nferromagnet to the left metal by the spin pumping. The\nright-hand side of Eq. (10) is the spin current ejected from\nthe ferromagnet to the right metal by the spin pumping.\nBy solving the bulk LLG Eq. (3) with the boundary\nconditions [Eqs. (9) and (10)] for a steady state, we ob-\ntain the following spin current density and the preces-\nsional velocity of the azimuthal angle:\njs(x;t) = (JL\u0000(\r+\u000bsx)!+\u001cSTx) sin2\u0012; (11)\n_\u001e(x;t)\u0011!=JL+\u001cSTL\n2\r+\u000bsL; (12)\nwhere the value of the polar angle \u0012changes from the\nvalue obtained from Eq. (5) due to the additional input\nspin current from the left boundary [66]. The precession\nof the magnetization induces a \fnite spin accumulation\ngiven by\u0016R=\u0000~^z\u0001m\u0002_m=\u0000~sin2\u0012_\u001ewith ~the\nreduced Planck constant, which can be measured exper-\nimentally [20, 41, 42].\nTo investigate the non-local spin transport from the\nleft endx= 0 to the right end x=Lthrough the dy-\nnamic ferromagnet, we employ the spin accumulation \u0016R\nat the interface between the ferromagnet and the right\nheavy metal and extract the component that is induced\nby the spin-current injection JLfrom the left metal. In\nother words, we use the di\u000berence of the spin accumu-\nlation\u0016Rbetween the two cases: with spin-current in-\njection from the left end ( JL6= 0) and without the\nspin-current injection ( JL= 0). Using _\u001eof Eq. (12),\n\u0001\u0016R=\u0016R(JL6= 0)\u0000\u0016R(JL= 0) is given by\n\u0001\u0016R=\u0000\u0012JL\n2\r+\u000bsLsin2\u0012\u0013\n~\n\u0000\u001cSTL\n2\r+\u000bsL\u0000\nsin2\u0012\u0000sin2\u00120\u0001\n~;(13)\nwhere\u00120is the polar angle obtained from Eq. (5) in the\nabsence of an input spin current JL= 0. The \frst term\non the right-hand side is the spin accumulation at the\nright end induced by injecting a spin current JLat the\nleft end. It decays algebraically \u00181=Lfor su\u000eciently\nlong samples as a function of the ferromagnet length\nL, which is the characteristic super\ruid-like spin trans-\nport. The second term can be interpreted as the e\u000bect\nof the polar-angle change (from \u00120to\u0012) induced by the\nspin-current injection JLfrom the left end. The alge-\nbraic decaying behavior of the second term is not evident\nfrom the analytical expression above, but we can show\nthat, by linearizing Eq. (4) with respect to the injected\nspin current JL, the second term is approximately given\nby [2 ~scos\u00120=(6K2cos2\u00120\u0000Ke\u000b)]JL\u001cSTL=(2\r+\u000bsL)2,\nwhich decays as 1 =Lfor su\u000eciently long samples. There-\nfore, the spin accumulation \u0001 \u0016Rat the right end induced\nby the spin-current injection from the left end decays as\n1=Las the ferromagnet length Lincreases, exhibiting\nsuper\ruid-like spin transport.5\nTo con\frm our theoretical prediction of super\ruid-\nlike spin transport in a dynamic cone state of ferro-\nmagnets, we perform the micromagnetic simulations and\ncompare the simulation results against the theoretical re-\nsults [Eq. (13)]. In simulations, we use the same mate-\nrial parameters that were mentioned above and the \fxed\nsecond-order anisotropy K2= 1:6\u0002106J/m3. For sim-\nplicity, we assumed that the spin pumping e\u000bect is neg-\nligible by setting \r= 0J/m2. Figure 2(b) plots the spin\naccumulation di\u000berence \u0001 \u0016RbetweenJL6= 0 andJL= 0\nas a function of the ferromagnet length Lfor several dif-\nferent values of the input spin current JL. The non-local\nspin transport \u0001 \u0016Rdecays algebraically, not exponen-\ntially, as the ferromagnet length Lincreases. Our ana-\nlytical [Eq. (13)] and simulation results [Fig. 2(b)] show\nthat we can realize super\ruid-like spin transport using\ndynamic states of easy-axis ferromagnets. These are our\nmain results.\nIV. SUMMARY\nTo go beyond the previous works on super\ruid-like spin\ntransport that have been restricted to easy-plane mag-\nnets, we have investigated the possibility of super\ruid-\nlike spin transport in an easy-axis ferromagnet driven to\na spin-torque oscillating regime. We have identi\fed the\ncondition for the spin torque with which the system can\nbe stabilized to a dynamic easy-cone state that breaks the\nU(1) spin-rotational symmetry spontaneously. By com-bining the theoretical analysis and the micromagnetic\nsimulations, we have shown that the spin current injected\nfrom one end of the ferromagnet decays algebraically,\nrather than exponentially, as the system length increases,\nwhereby demonstrating that super\ruid-like spin trans-\nport can be achieved in an easy-axis ferromagnet under\nsuitable dynamic biases. We hope that our work stimu-\nlates further investigations of super\ruid-like spin trans-\nport and other unconventional spin transport in various\ntypes of magnets, departing from simple easy-plane or\neasy-axis magnets.\nACKNOWLEDGMENTS\nWe acknowledge the discussion with Yaroslav\nTserkovnyak. This work was supported by Brain Pool\nPlus Program through the National Research Founda-\ntion of Korea funded by the Ministry of Science and ICT\n(NRF-2020H1D3A2A03099291), by the National Re-\nsearch Foundation of Korea(NRF) grant funded by the\nKorea government(MSIT) (NRF-2021R1C1C1006273),\nby the National Research Foundation of Korea funded\nby the Korea Government via the SRC Center for\nQuantum Coherence in Condensed Matter (NRF-\n2016R1A5A1008184), and by Basic Science Research\nProgram through the KAIST Basic Science 4.0 Priority\nResearch Center funded by the Ministry of Science and\nICT.\n[1] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Spintronics: Fun-\ndamentals and applications, Rev. Mod. Phys. 76, 323\n(2004).\n[2] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Moln\u0013 ar, M. L. Roukes, A. Y.\nChtchelkanova, and D. M. Treger, Spintronics: A spin-\nbased electronics vision for the future, Science 294, 1488\n(2001).\n[3] L. Guo, X. Gu, X. Zhu, and X. Sun, Recent advances\nin molecular spintronics: Multifunctional spintronic de-\nvices, Adv. Mater. 31, 1805355 (2019).\n[4] C. Kittel, P. McEuen, and P. McEuen, Introduction to\nsolid state physics , Vol. 8 (Wiley New York, 1996).\n[5] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando,\nK. Takanashi, S. Maekawa, and E. Saitoh, Transmis-\nsion of electrical signals by spin-wave interconversion in\na magnetic insulator, Nature 464, 262 (2010).\n[6] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi,\nJ. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai,\nG. E. W. Bauer, S. Maekawa, and E. Saitoh, Spin seebeck\ninsulator, Nat. Mater. 9, 894 (2010).\n[7] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. Hillebrands, Magnon spintronics, Nat. Phys. 11, 453\n(2015).[8] A. Navabi, Y. Liu, P. Upadhyaya, K. Murata,\nF. Ebrahimi, G. Yu, B. Ma, Y. Rao, M. Yazdani, M. Mon-\ntazeri, L. Pan, I. N. Krivorotov, I. Barsukov, Q. Yang,\nP. Khalili Amiri, Y. Tserkovnyak, and K. L. Wang, Con-\ntrol of spin-wave damping in yig using spin currents from\ntopological insulators, Phys. Rev. Applied 11, 034046\n(2019).\n[9] I. Barsukov, H. K. Lee, A. A. Jara, Y.-J. Chen, A. M.\nGon\u0018 calves, C. Sha, J. A. Katine, R. E. Arias, B. A.\nIvanov, and I. N. Krivorotov, Giant nonlinear damping\nin nanoscale ferromagnets, Sci. Adv. 5, eaav6943 (2019).\n[10] A. Etesamirad, R. Rodriguez, J. Bocanegra, R. Verba,\nJ. Katine, I. N. Krivorotov, V. Tyberkevych, B. Ivanov,\nand I. Barsukov, Controlling magnon interaction by a\nnanoscale switch, ACS Appl. Mater. Inter. 13, 20288\n(2021).\n[11] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and\nB. J. van Wees, Long-distance transport of magnon spin\ninformation in a magnetic insulator at room temperature,\nNat. Phys. 11, 1022 (2015).\n[12] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A.\nDuine, and B. J. van Wees, Magnon spin transport driven\nby the magnon chemical potential in a magnetic insula-\ntor, Phys. Rev. B 94, 014412 (2016).\n[13] D. Wesenberg, T. Liu, D. Balzar, M. Wu, and B. L. Zink,\nLong-distance spin transport in a disordered magnetic6\ninsulator, Nat. Phys. 13, 987 (2017).\n[14] E. Sonin, Spin currents and spin super\ruidity, Adv. Phys.\n59, 181 (2010).\n[15] J. F. Annett et al. ,Superconductivity, super\ruids and\ncondensates , Vol. 5 (Oxford University Press, 2004).\n[16] E. B. Sonin, Spin super\ruidity, coherent spin precession,\nand magnon bec, J. Low Temp. Phys. 171, 757 (2013).\n[17] J. K onig, M. C. B\u001cnsager, and A. H. MacDonald, Dissi-\npationless spin transport in thin \flm ferromagnets, Phys.\nRev. Lett. 87, 187202 (2001).\n[18] H. Chen, A. D. Kent, A. H. MacDonald, and I. Sode-\nmann, Nonlocal transport mediated by spin supercur-\nrents, Phys. Rev. B 90, 220401 (2014).\n[19] W. Chen and M. Sigrist, Spin super\ruidity in coplanar\nmultiferroics, Phys. Rev. B 89, 024511 (2014).\n[20] H. Skarsv\u0017 ag, C. Holmqvist, and A. Brataas, Spin super-\n\ruidity and long-range transport in thin-\flm ferromag-\nnets, Phys. Rev. Lett. 115, 237201 (2015).\n[21] W. Chen and M. Sigrist, Dissipationless multiferroic\nmagnonics, Phys. Rev. Lett. 114, 157203 (2015).\n[22] J. Linder and J. W. A. Robinson, Superconducting spin-\ntronics, Nat. Phys. 11, 307 (2015).\n[23] B. Flebus, S. A. Bender, Y. Tserkovnyak, and R. A.\nDuine, Two-\ruid theory for spin super\ruidity in mag-\nnetic insulators, Phys. Rev. Lett. 116, 117201 (2016).\n[24] S. K. Kim and Y. Tserkovnyak, Interaction between a\ndomain wall and spin supercurrent in easy-cone magnets,\nPhys. Rev. B 94, 220404 (2016).\n[25] R. Cheng, D. Xiao, and A. Brataas, Terahertz antiferro-\nmagnetic spin hall nano-oscillator, Phys. Rev. Lett. 116,\n207603 (2016).\n[26] A. Qaiumzadeh, H. Skarsv\u0017 ag, C. Holmqvist, and\nA. Brataas, Spin super\ruidity in biaxial antiferromag-\nnetic insulators, Phys. Rev. Lett. 118, 137201 (2017).\n[27] S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak,\nSuper\ruid spin transport through antiferromagnetic in-\nsulators, Phys. Rev. B 90, 094408 (2014).\n[28] S. Takei and Y. Tserkovnyak, Super\ruid spin transport\nthrough easy-plane ferromagnetic insulators, Phys. Rev.\nLett.112, 227201 (2014).\n[29] S. Takei and Y. Tserkovnyak, Nonlocal magnetoresis-\ntance mediated by spin super\ruidity, Phys. Rev. Lett.\n115, 156604 (2015).\n[30] E. B. Sonin, Super\ruid spin transport in ferro- and anti-\nferromagnets, Phys. Rev. B 99, 104423 (2019).\n[31] J. Slonczewski, Current-driven excitation of magnetic\nmultilayers, J. Magn. Magn. Mater 159, L1 (1996).\n[32] D. Ralph and M. Stiles, Spin transfer torques, J. Magn.\nMagn. Mater 320, 1190 (2008).\n[33] Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Theory\nof current-driven magnetization dynamics in inhomoge-\nneous ferromagnets, J. Magn. Magn. Mater 320, 1282\n(2008).\n[34] X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Spin transfer\ntorque on magnetic insulators, Eur. Phys. Lett 96, 17005\n(2011).\n[35] Z. Duan, A. Smith, L. Yang, B. Youngblood, J. Lindner,\nV. E. Demidov, S. O. Demokritov, and I. N. Krivoro-\ntov, Nanowire spin torque oscillator driven by spin orbit\ntorques, Nat. Commun. 5, 5616 (2014).\n[36] Y. Tserkovnyak, Perspective: (beyond) spin transport in\ninsulators, J. Appl. Phys 124, 190901 (2018).\n[37] B. Flebus, R. A. Duine, and H. M. Hurst, Non-hermitian\ntopology of one-dimensional spin-torque oscillator arrays,Phys. Rev. B 102, 180408 (2020).\n[38] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Observation of the spin hall e\u000bect in semi-\nconductors, Science 306, 1910 (2004).\n[39] Y. Tserkovnyak and S. A. Bender, Spin hall phenomenol-\nogy of magnetic dynamics, Phys. Rev. B 90, 014428\n(2014).\n[40] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Spin hall e\u000bects, Rev. Mod. Phys. 87,\n1213 (2015).\n[41] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, En-\nhanced gilbert damping in thin ferromagnetic \flms, Phys.\nRev. Lett. 88, 117601 (2002).\n[42] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Nonlocal magnetization dynamics in ferromag-\nnetic heterostructures, Rev. Mod. Phys. 77, 1375 (2005).\n[43] L. D. Landau and E. M. Lifshitz, Quantum Mechanics\n3rd ed. (Butterworth-Heinemann, Oxford, 1976).\n[44] T. Gilbert, A phenomenological theory of damping in\nferromagnetic materials, IEEE Trans. Magn. 40, 3443\n(2004).\n[45] R. Matsumoto, H. Arai, S. Yuasa, and H. Imamura, Spin-\ntransfer-torque switching in a spin-valve nanopillar with\na conically magnetized free layer, Appl. Phys. Exp. 8,\n063007 (2015).\n[46] J. M. Shaw, H. T. Nembach, M. Weiler, T. J. Silva,\nM. Schoen, J. Z. Sun, and D. C. Worledge, Perpen-\ndicular magnetic anisotropy and easy cone state in\nTa/Co60Fe20B20/MgO, IEEE Magn. Lett 6, 1 (2015).\n[47] P.-H. Jang, S.-W. Lee, and K.-J. Lee, Spin-transfer-\ntorque-induced zero-\feld microwave oscillator using a\nmagnetic easy cone state, Curr. Appl. Phys. 16, 1550\n(2016).\n[48] P.-H. Jang, S.-H. Oh, S. K. Kim, and K.-J. Lee, Domain\nwall dynamics in easy-cone magnets, Phys. Rev. B 99,\n024424 (2019).\n[49] H. K. Gweon, H.-J. Park, K.-W. Kim, K.-J. Lee, and\nS. H. Lim, Intrinsic origin of interfacial second-order\nmagnetic anisotropy in ferromagnet/normal metal het-\nerostructures, NPG. Asia. Mater. 12, 23 (2020).\n[50] H. K. Gweon and S. H. Lim, Evolution of strong second-\norder magnetic anisotropy in pt/co/mgo trilayers by\npost-annealing, Appl. Phys. Lett. 117, 082403 (2020).\n[51] D. Ralph and M. Stiles, Spin transfer torques, J. Magn.\nMagn. Mater 320, 1190 (2008).\n[52] A. Manchon, J. \u0014Zelezn\u0013 y, I. M. Miron, T. Jungwirth,\nJ. Sinova, A. Thiaville, K. Garello, and P. Gambardella,\nCurrent-induced spin-orbit torques in ferromagnetic and\nantiferromagnetic systems, Rev. Mod. Phys. 91, 035004\n(2019).\n[53] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez, and B. Van Waeyenberge, The design\nand veri\fcation of mumax3, AIP Adv. 4, 107133 (2014).\n[54] M. Ohkoshi, H. Kobayshi, T. Katayama, M. Hirano,\nT. Katayama, M. Hirano, and T. Tsushima, Spin reori-\nentation in NdCo5 single crystals, AIP Conf. Proc. 29,\n616 (1976).\n[55] M. Seifert, L. Schultz, R. Sch afer, V. Neu, S. Hankemeier,\nS. R ossler, R. Fr omter, and H. P. Oepen, Domain evolu-\ntion during the spin-reorientation transition in epitaxial\nNdCo5thin \flms, New J. Phys 15, 013019 (2013).\n[56] M. Seifert, L. Schultz, R. Sch afer, S. Hankemeier,\nR. Fr omter, H. P. Oepen, and V. Neu, Micromag-\nnetic investigation of domain and domain wall evolution7\nthrough the spin-reorientation transition of an epitaxial\nNdCo5\flm, New J. Phys 19, 033002 (2017).\n[57] S. Kumar, C. E. Patrick, R. S. Edwards, G. Balakrish-\nnan, M. R. Lees, and J. B. Staunton, Torque magne-\ntometry study of the spin reorientation transition and\ntemperature-dependent magnetocrystalline anisotropy in\nNdCo5, J. Phys.: Condens. Matter 32, 255802 (2020).\n[58] A. A. Timopheev, R. Sousa, M. Chshiev, H. T. Nguyen,\nand B. Dieny, Second order anisotropy contribution in\nperpendicular magnetic tunnel junctions, Sci. Rep. 6\n(2016).\n[59] J. M. D. Coey, Magnetism and Magnetic Materials (Cam-\nbridge University Press, Cambridge, England, 2009).\n[60] R. Wieser, E. Y. Vedmedenko, and R. Wiesendanger, Do-\nmain wall motion damped by the emission of spin waves,\nPhys. Rev. B 81, 024405 (2010).[61] P. Yan and G. E. W. Bauer, Magnonic domain wall heat\nconductance in ferromagnetic wires, Phys. Rev. Lett.\n109, 087202 (2012).\n[62] E. P. Gross, Structure of a quantized vortex in boson\nsystems, Il Nuovo Cimento 20, 454 (1961).\n[63] L. P. Pitaevskii, Vortex lines in an imperfect bose gas,\nSov. Phys. JETP. 13, 451 (1961).\n[64] S. A. Bender, R. A. Duine, A. Brataas, and\nY. Tserkovnyak, Dynamic phase diagram of dc-pumped\nmagnon condensates, Phys. Rev. B 90, 094409 (2014).\n[65] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Finite-\nelement theory of transport in ferromagnet{normal metal\nsystems, Phys. Rev. Lett. 84, 2481 (2000).\n[66] The modi\fed value of the polar angle \u0012can be calculated\nby solving Eq. (4) and Eq. (12) self-consistently for \u0012."    },    {        "title": "2206.08624v1.Theory_of_ultrafast_magnetization_of_non_magnetic_semiconductors_with_localized_conduction_bands.pdf",        "content": "Theory of ultrafast magnetization of non-magnetic semiconductors with localized\nconduction bands\nGiovanni Marini1,\u0003and Matteo Calandra2, 3, 1,y\n1Graphene Labs, Fondazione Istituto Italiano di Tecnologia, Via Morego, I-16163 Genova, Italy\n2Department of Physics, University of Trento, Via Sommarive 14, 38123 Povo, Italy\n3Sorbonne Université, CNRS, Institut des Nanosciences de Paris, UMR7588, F-75252 Paris, France\nThe magnetization of a non-magnetic semiconductor by femtosecond light pulses is crucial to\nachieveanall-opticalcontrolofthespindynamicsinmaterialsandtodevelopfastermemorydevices.\nHowever, the conditions for its detection are largely unknown. In this work we identify the criteria\nfor the observation of ultrafast magnetization and critically discuss the difficulties hindering its\nexperimental detection. We show that ultrafast magnetization of a non magnetic semiconductor\ncan be observed in compounds with very localized conduction band states and more delocalized\nvalence bands, such as in the case of a p-d charge transfer gap. By using constrained and time\ndependent density functional theory simulations, we demonstrate that a transient ferrimagnetic\nstate can be induced in diamagnetic semiconductor V 2O5via ultrafast pulses at realistic fluences.\nThe ferrimagnetic state has opposite magnetic moments on vanadium (conduction) and oxygen\n(valence) states. Our methodology outruns the case of V 2O5as it identifies the key requirements\nfor a computational screening of ultrafast magnetism in non-magnetic semiconductors.\nThe ultrafast all-optical control of the spin dynam-\nics in materials via femtosecond light pulses is an ex-\ntremely appealing perspective as it could lead to a new\ngeneration of memory devices much faster than the\nwidespread magneto-optical storage media or random ac-\ncess memories[1–3]. The investigation of magnetic phe-\nnomena at the femtosecond scale [4] includes ultrafast\ndemagnetization [5], namely the melting of the magnetic\norder of a material by femtosecond pulses, magnetization\nreversal[2, 5], i.e. the use of ultrafast laser pulses to in-\nvertthedirectionofthemagnetizationandthepossibility\nto induce a phase transition from an antiferromagnetic\nphase to a ferromagnetic one[6]. Furthermore, the en-\nhancement of a pre-existent ferromagnetic order using ul-\ntrafast irradiation has been demonstrated in GaMnAs[7]\nandmorerecentlyinthelayeredcompoundFe 3GeTe 2. In\nFe3GeTe 2the Curie temperature has been enhanced and\ntuned by ultrafast laser pulses by manipulating the mag-\nnetic anisotropy energy and, consequently, the itinerant\nmagnetic order of the material[8].\nIn all these cases, the material is assumed to be mag-\nnetic in its ground state and the magnetic state is altered\nvia excitation. A different (yet related) phenomenon, of\nwhich there is still little evidence, is the ultrafast mag-\nnetization of a non-magnetic material via femtosecond\nlight pulses. Ultrafast magnetism has been induced in\nnonmagnetic materials using circularly polarized light\nthrough inverse Faraday effect[9, 10], although the re-\nsulting magnetism is short lived and the non-magnetic\nnature of the irradiated material is not altered. The\naforementioned experimental results demonstrate that in\nsome conditions light can modify the magnetic properties\nof the material. It is then natural to wonder whether\nthe ultrafast stabilization of a ordered magnetic state\nin a non-magnetic material via femtosecond pulses is a\nrealistic possibility. Namely, by promoting electrons inthe conduction band of a non-magnetic material and the\nconsequent stabilization of a thermalized electron-hole\nplasma, is it possible to stabilize a magnetic transient\nstate at the ultrashort timescale before electron-hole re-\ncombination takes place? Finally, under what conditions\ncan this happen and what are the criteria to identify ma-\nterials possessing this remarkable property?\nWhile at present there are still many open questions\nregarding the possibility to induce a magnetic state in a\nnon-magnetic material via ultrafast light, it is clear that\nthe ability to do so would open new perspectives in tai-\nloring magnetic properties of materials. The goal of this\nwork is to address these fundamental questions from a\ntheoretical point of view. We give general criteria for the\noccurrence of ultrafast magnetization in non-magnetic\nmaterials and we target the layered diamagnetic semi-\nconductor V 2O5as a system that can be made magnetic\nvia ultrafast light pulses.\nFor the sake of simplicity let us consider a non-\nmagnetic semiconducting or insulating compound with\na gap of the order of the eV or larger. This assumption\nis needed to ensure that the electron-hole recombination\nrate is long enough, namely of the order of nanoseconds.\nBy using ultrafast laser pulses a substantial number of\nelectrons [11] can be promoted in the optically active\nconduction band and an electron-hole plasma (hole in\nthe valence band and electrons in the conduction band)\nis induced in the system. At large enough photocarrier\nconcentrationstheexcitonsarescreenedandtheelectron-\nhole plasma thermalizes fairly quickly, below the picosec-\nond scale, so that the system effectively feels an unbal-\nanced population of carriers that can be described by\ntwo Fermi distributions, one for the holes and one for\nthe electrons[12–14]. The question we want to answer is\nunder what circumstances this transient state occurring\nbefore recombination is magnetic.arXiv:2206.08624v1  [cond-mat.mtrl-sci]  17 Jun 20222\nUltrafast magnetization can happen via several mate-\nrial specific mechanisms (e.g. enhancement of the mag-\nnetic anisotropy energy in a 2D material), however a gen-\neral one is the following. If the conduction band states\nhave very different localization properties than the va-\nlence band states, as it happens in the case of a p-d gap\n(p-statesinvalenceandd-statesintheconductionbands)\norinthecaseofaweaklydispersiveconductionband(flat\nband) and a strongly dispersive valence band, then the\nelectron-electron interaction felt by the excited electrons\nis very different from the one in the ground state. As a\nconsequence a magnetic instability can in principle occur\nin the transient state.\nWe give a practical demonstration of our idea for the\nvan-der-Waals compound vanadium pentoxide (V 2O5).\nThis compound is mainly studied for its performances as\ncathode material in the realization of batteries[15]. V 2O5\nhas a rich structural phase diagram with several compet-\ning phases[16]. The most commonly stabilized structure\nisthe\u000b-structure(spacegroup59, Pmmn)andischarac-\nterized by one-dimensional ladders along the y-direction,\ndepictedinFig.1. Electronically, thematerialbehavesas\nan insulator, due to the presence of an indirect band gap\nbetween the valence and the conduction band. The opti-\ncal band-gap is\u00192.35 eV[17], however different values in\nthe 2.2-2.8 eV range have also been reported[18–20]. Ex-\nperimentally V 2O5is non magnetic, however, antiferro-\nmagnetismcanbeinducedbyNadoping[21]andNaV 2O5\nhas a Néel temperature of 320\u000650K, in agreement\nwith theory[22]. Due to its layered nature, it has been\npossible to synthesize few layer V 2O5via liquid exfoli-\nation, as well as supported V 2O5/TiO 2monolayers[23].\nSome experimental ultrafast investigations on V 2O5are\nalready present in literature, including a study of carrier\nrelaxation in V 2O5nanowires[24] and the possibility to\nsynthesize VO 2from V 2O5via femtosecond pulsed laser\ndeposition[25].\nWe simulate the ground state and excited states prop-\nerties within density functional theory (DFT) by using\nthe Quantum ESPRESSO (QE) package[26, 27]. The\nexcited electron-hole plasma is modeled by using con-\nstraineddensityfunctionalperturbationtheory(cDFPT)\nthat we recently developed [14] extending previous theo-\nreticalresults[13]. Thisamountstosuitablyconstrainthe\noccupations of the Kohn-Sham conduction eigenstates in\norder to mimic the thermalized photocarrier population.\nIn more details, electrons are removed from the oxygen\nvalence states (leaving behind a thermal distribution of\nholes) and are added to the conduction bands. All struc-\ntural and electronic properties are then calculated by\nconstraining two thermal distributions of electrons and\nholes in the conduction and valence band, respectively.\nWe include correlation effects employing the GGA +U\napproach[28] (including the Hubbard parameters on the\nvanadium d states). In order to maintain a completely\nab-initioapproach, the Uvalue is determined self consis-tently from first principles, using the method introduced\nby Timrov et al:[28, 29] that we modified to include the\npresence of an electron-hole plasma. Thus, we calculate\nthe on-site repulsion Uparameter both in presence and\nabsence of the electron-hole plasma, finding in both cases\nvery similar values of U\u00194:4eV. Further technical de-\ntails are given in the Supplemental Material[30], which\nincludes Refs.[14, 16, 22, 26–29, 31–45].\nIn Fig. 1, panels a) and b) we report the structure\nof\u000b-V2O5. Each vanadium atom is surrounded by five\noxygen atoms: one oxygen is exclusively bound to the\nvanadium, three of them are bound to three vanadium\natoms in the same ladder, while one oxygen behaves as a\nspacer between ladders and is shared between two vana-\ndium atoms belonging to different ladders. The ground\nstateelectronicstructureisshowninFig.1c). Itdisplays\nanindirectbandgapof2.47eV,ingoodqualitativeagree-\nment with the experimental value of 2.35 eV[17]. The\nprojection of the Kohn-Sham states onto atomic vana-\ndium and oxygen orbitals is also shown and the size of\nthe dots is proportional to the atomic character. Pro-\njections over atomic orbitals demonstrate that valence\nbands are mainly composed by oxygen( p)-derived states,\nwhile vanadium( d) states mainly constitute the conduc-\ntion band. Furthermore, the narrow bandwidth of the\nconduction band should also favour the formation of a\ncorrelated state.\nWethenperformconstrainedDFTcalculationsforsev-\neral photocarrier concentrations (PCs) in order to inves-\ntigate how the presence of an electron-hole plasma af-\nfects the electronic structure of this compound. We find\nthat ultrafast magnetization can occur in this material,\nas shown in Fig. 1 where panel d) schematically repre-\nsents the magnetic ordering observed after irradiation.\nMagnetism is characterized by the vanadium and oxygen\nsub-lattices presenting opposite magnetic moments. The\nmagnetic state can be well understood by looking at the\nspin up minus spin down charge plot in Fig. 1 e). Here,\nyellow surfaces represent a spin up excess charge while\nthebluesurfacesrepresentspindownexcesscharges. The\nalternating magnetic moments of the vanadium and oxy-\ngen sub-lattices leads to a transient ferrimagnetic order\nafter irradiation, resulting in a very weak (close to zero)\ntotal magnetic moment.\nThe effect of the electron-hole plasma on the electronic\nstructure is shown in Fig. 1 f), where the Kohn-Sham\neigenvalues for \u000b-V2O5are plotted for a PC value ne=\n0:25e\u0000=f:u:. The excitations are of the charge trans-\nfer type from the O-valence band to the V-conduction\nband. The ferrimagnetic state is signalled by the non-\ndegeneracy of the up and down bands. The observed fer-\nrimagnetic configuration can be understood thinking to\nthe simultaneous effect of hole and electron doping. This\nis confirmed by performing calculations in a rigid dop-\ning approach (see Supplemental Material) demonstrat-\ning that hole and electron doping produce different mag-3\nFIG. 1. Light induced magnetism in V 2O5. Panel a,b): V 2O5in the\u000bstructure before irradiation (top and lateral views).\nPanel c): ground state electronic structure of V 2O5including the projection onto atomic vanadium (green) and oxygen (red)\natomic orbitals. Panel d): ferrimagnetic order stabilized by ultrafast radiation. Panel e): three-dimensional plot of spin-\npolarized charge density in the photoexcited state. Yellow (cyan) regions represent spin up (down) charge excess. Panel f):\nElectronic structure of \u000b-V2O5forne= 0:25e\u0000=f:u:, projected onto spin-resolved vanadium (green) and oxygen (red) orbitals.\nThe imaginary part of the dielectric function resolved in the three spatial direction for the ground state (panel a) and in\nthe presence of an electron-hole plasma (panel d) is plotted on the right hand side of the corresponding electronic structures\n(panels c and f).\nnetization and magnetic moments residing on vanadium\n(oxygen) atoms for electron (hole) doping.\nIn panels c) and f), we also show the imaginary part of\nthe dielectric function corresponding to the the ground\nstateandphotoexcitedelectronicstructures,respectively.\nThe dielectric function is resolved into the three spa-\ntial directions. For the ground state, we find that\nour calculations are in qualitative agreement with opti-\ncal measurements[36] (see Supplemental Material), thus\njustifying our procedure. Laser irradiation completely\nchanges the optical properties of the system. In the low-\nenergy region (0-1.5 eV) a metallic-like contribution ap-\npears due to intraband scattering and a reduction of the\nbandgap caused by irradiation occurs. As a consequence,\nthe prominent peak observed at \u00193 eV in panel c) for\nthe incident field parallel to the adirection (cyan curve)is shifted towards lower energies in panel f)( \u00192:3eV).\nSince the reduction of the gap only happens if the sys-\ntem becomes magnetic after irradiation (see Supplemen-\ntal Material where we simulate the optical properties of\nphotoexcited V 2O5neglecting magnetic ordering), opti-\ncal measurements can be seen as a fingerprint to directly\nprobe the irradiation-induced magnetization in V 2O5.\nThe discussion presented here pertains to a quasi-\nequilibriumsituationwherebothelectronsandholeshave\nreached an intraband thermalization via carrier-carrier\nscattering. Nevertheless, constrained DFT simulations\ndonotallowtodescribetheevolutiontowardsamagnetic\nstate occurring in the first femtoseconds after irradiation.\nIn order to answer to this question, we performed time-\ndependent density functional theory (TDDFT) simula-\ntions. InFig.2, weplottheabsolutemagneticmomentas4\nFIG. 2. Light induced magnetization dynamics in TDDFT:\ngrey curve represents the absolute magnetic moment per unit\ncell (a unit cell includes two formula units), red and blue\ncurves represent the separated contribution of oxygen and\nvanadium atoms to the absolute magnetic moment, respec-\ntively.\na function of time for photoexcited V 2O5, as obtained in\nTDDFT: an ever increasing absolute magnetic moment is\nobserved in the first 100 fs, reaching a value greater than\n0.1\u0016B/unit cell. The TDDFT results, extensively dis-\ncussed in the Supplemental Material, demonstrate that\na finite absolute magnetic moment develops in the com-\npound following the laser irradiation and that a funda-\nmental role is played by the spin-orbit interaction. We\nconclude that no dynamical barrier between the ground\nstate and the quasi-equilibrium magnetic state predicted\nusing cDFPT is present, and thus that the photoexcited\nmagnetic state can be experimentally realized.\nThe Raman frequencies of the ground state structure\nand at a photocarrier concentration of ne= 0:25e\u0000=f:u:\nare shown in Fig. 3 a) (blue lines for the ground\nstate). For the ground state, we find a good agreement\nwith previously published theoretical and experimental\nresults[16, 46]. The grey filled curve are experimental\ndata from Ref.[46]. Dashed red lines label Raman fre-\nquencies after photoexcitation. We observe a general\nsoftening of the high frequency modes (above 400 cm\u00001),\ntogether with a general hardening of the modes in the\nrange 0-400 cm\u00001. This prediction and the optical fea-\ntures discussed before are the fingerprints of the photoex-\ncited ferrimagnetic state.\nWhile the physical model presented here is conceptu-\nally simple, some caveats are needed. First of all, the\nfluence of the ultrafast pulse must be low enough so that\nthe energy transferred to the sample in the region illu-\nminated by the laser is smaller than kBTc, wherekBis\nthe Boltzmann constant and Tcis the Curie tempera-\nture (i.e. the magnetic temperature below which mag-\nnetism can occur). Moreover, the different localization\nproperties of the valence and conduction bands often im-ply that the excited electrons feel very different forces\nfrom the lattice at scale larger than the picoseconds. As\na consequence, a structural phase transitions could in\nprinciple occur with unpredictable consequences on the\nmagnetic properties[47]. If, however, these two difficul-\nties are avoided, then transient ultrafast magnetization\ncan occur.\nWe discuss these two points for the case of V 2O5. The\noccurrence of a ferrimagnetic state could be hindered by\na low Curie temperature so that the energy transferred\nto the sample by the laser overrules kBTceven at low\nfluences. Despite the fact that it is difficult to obtain a\nreliable estimation of the Curie temperature from first\nprinciples, especially in the case of a complex system like\n\u000b-V2O5, we can try to compare our situation with the\noneofelectrondopedV 2O5. InV 2O5electrondopingcan\nbe achieved by creating oxygen vacancies[48] that leaves\nuncompensated electrons in the vanadium d xystates, an\neffect well described by first principles calculations[49].\nThe presence of oxygen vacancies would cause the ap-\npearance of a ferromagnetic state in V 2O5\u0000xfor x<\n0.13 or 0.19 <x<0.45[49]. Experimentally, the total\nmagnetic moment is found to be unrelated to the type of\nvacancies (there are several non equivalent oxygen sites)\nand the ferromagnetic Curie temperature is fairly above\nroomtemperature[48]. WeinferthatsimilarvaluesforT c\ncan be expected in our system , as the photoexcited mag-\nnetic state reported here and the magnetic state originat-\ning from electron doping possess a common origin and a\ntotal magnetic moment of the same order is calculated\nfor both cases (see Supplemental Material).\nWe now evaluate the energy transferred to the crys-\ntal by the laser light as a function of the photocarrier\nconcentration (i.e. as a function of the fluence of the\nincoming laser) to understand if magnetism is possible.\nIn Fig. 3 b) we calculate the maximum plasma electron-\nhole density generated by a 3.1 eV laser having a fluence\nof 5 mJ=cm2with in-plane polarized light (E ?c) (see\nthe Supplemental Material for more details). Averaging\nbetween the two planar polarization directions, we calcu-\nlate that an electron-hole density as high as 0:22e\u0000=f:u:\ncan be generated in the first layer of the sample. The\nplot in Fig. 3 c) demonstrates that magnetization sets in\nmuch earlier, namely at 0:015e\u0000=f:u:, corresponding to\n1:35\u00021020e\u0000=cm3. Here, the red and green curves\nrepresent the absolute magnetic moment calculated with\nand without performing structural optimization with re-\nspect to the unit cell parameters in the cDFT calcula-\ntion, respectively. The absolute magnetic moment grows\nlinearly for ne\u00140:25e\u0000=f:u:. For higher electron densi-\nties, weobserveasub-linearincreaseoftheabsolutemag-\nnetic moment in both cases. We find that a photocarrier\nconcentration of 0:22e\u0000=f:u:corresponds already to an\nabsolute magnetic moment in excess of 1 \u0016B=f:u:. The\ncomparison between the two curves shows that structural\nchanges contribute to enhance the photoinduced magne-5\nFIG.3. CharacterizationofphotoexcitedV 2O5. Panela): CalculatedRamanfrequenciesfor \u000b-V2O5(bluelines)comparedwith\ntheexperimentalRamanspectrum[46](greyshadedregion)andcalculatedphotoexcitedRamanfrequenciesat ne= 0:25e\u0000=f:u:\n(red dashed lines). Panel b): density of states (red and green indicate filling indicate valence and conduction, respectively)\nand corresponding energy-resolved photocarrier concentration (light blue and yellow filling) induced by a 3.1 eV laser having\na fluence of 5 mJ/cm2. Panel c): absolute magnetic moment vs photocarrier concentration with (red) and without (green)\nperforming structural optimization with respect to the unit cell parameters.\ntization.\nThe 3.1 eV energy absorbed by each photocarrier cor-\nresponds to an excess energy of 0.63 eV with respect\nto the gap. This translates to an excess energy \u0001E=\n0:14eV=f:uatne= 0:22e\u0000=f:u:, which is mostly trans-\nferred to the lattice via intraband thermalization (here\nwe neglect the energy transfer to the spin subsystem).\nCombining these findings with experimental heat capac-\nity data[50], we now estimate the quasi-equilibrium tem-\nperature for the system (before carrier recombination\ntakes place). Assuming an initial temperature T= 50K\nand neglecting energy dissipation, we estimate a quasi-\nequilibrium temperature T f= 168:5K at 0:22e\u0000=f:u:(\nTf= 250K if we consider ne= 0:5e\u0000=f:u:). Both\nthese temperatures are lower than room temperature,\nsupporting the possibility to experimentally observe ul-\ntrafast magnetization in V 2O5.\nWe now discuss the structural stability of the pho-\ntoexcited phase. As we stressed before, injection of a\nlarge number of photocarriers can trigger reversible or\nirreversible phase transitions due to the possible popu-\nlation of antibonding states resulting in strong forces on\nthe ions. For this reason it is important to calculate\nthe structural and vibrational properties of the photoex-\ncited transient phase. We find that the transient pho-\ntoexcited ferrimagnetic state is dynamically stable, i:e:\nno imaginary phonon frequencies are observed (see Sup-\nplemental Material). Interestingly, another crystal struc-\nture structure, commonly referred to as \f-V2O5(P21m,\nspace group 11), becomes the lowest energy structure un-\nder photoexcitation. However, as imaginary phonons are\nabsent from Pmmnphonon spectrum, an energy barrier\nexists between the two structures. We calculate a very\nlarge energy barrier of 1:25eV/atom by interpolating be-\ntween atomic coordinates of the two structures. Thus,\nwe do not expect any structural transition to occur un-\nder photoexcitation.\nIn conclusion, we identify the general criteria for ul-trafast magnetization of non-magnetic semiconductors to\noccur. The key requirement is that the conduction band\nshould host very localized states or flat bands while the\nvalence more delocalized ones, like in the case of a p-d\ncharge transfer gap. Moreover, we discuss the possible\ndifficulties hindering the experimental detection of this\neffect. By using constrained density functional perturba-\ntion theory calculations, we demonstrate that femtosec-\nond light pulses can induce a magnetic state in an oth-\nerwise non-magnetic material, namely the diamagnetic\nsemiconductor V 2O5. Our calculations show that a tran-\nsition from the ground state non-magnetic state to a fer-\nrimagnetic state develops upon laser irradiation, even for\nvery low plasma densities. Thus, light can shape mag-\nnetic properties, notably not just suppressing an existing\nmagnetic phase, but also inducing a new magnetic phase.\nOur methodology outruns the case of V 2O5and paves\nthe way to a computational screening of semiconducting\nmaterials displaying ultrafast magnetization.\nFinally, we point out that, while our conditions for the\noccurrence of ultrafast magnetization are very general,\nin specific materials, such as 2D materials, other effects\ncould cooperate in inducing the magnetic order. For ex-\nample, as the Mermin and Wagner theorem forbids the\nbreaking of a continous symmetry in 2D, any mechanism\nenhancing the magnetic anisotropic energy and making\nthe photoexcited magnetic system more Ising like could\nlead to an enhancement of magnetic order in a 2D or a\nquasi 2D material. This is the mechanism claimed to be\nat the origin of the enhancement of magnetic order in Fe 3\nGeTe 2by ultrafast light pulses.\nThe authors acknowledge support from the European\nUnion’s Horizon 2020 research and innovation program\nGraphene Flagship under grant agreement No 881603.\nWe acknowledge the CINECA award under the ISCRA\ninitiative and PRACE, for the availability of high perfor-\nmance computing resources and support.6\n\u0003giovanni.marini@iit.it\nym.calandrabuonaura@unitn.it\n[1] A. V. Kimel and M. Li, Nature Reviews Materials 4, 189\n(2019).\n[2] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett.\n99, 047601 (2007).\n[3] C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad,\nY. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski,\nK. Hono, Y. Fainman, M. Aeschlimann, and E. E. Fuller-\nton, Science 345, 1337 (2014).\n[4] M. Maiuri, M. Garavelli, and G. Cerullo, Journal of\nthe American Chemical Society 142, 3 (2020), pMID:\n31800225, https://doi.org/10.1021/jacs.9b10533.\n[5] E.Beaurepaire, J.-C.Merle, A.Daunois,andJ.-Y.Bigot,\nPhys. Rev. Lett. 76, 4250 (1996).\n[6] G. Ju, J. Hohlfeld, B. Bergman, R. J. M. van de Veer-\ndonk, O. N. Mryasov, J.-Y. Kim, X. Wu, D. Weller, and\nB. Koopmans, Phys. Rev. Lett. 93, 197403 (2004).\n[7] J. Wang, I. Cotoros, K. M. Dani, X. Liu, J. K. Furdyna,\nand D. S. Chemla, Phys. Rev. Lett. 98, 217401 (2007).\n[8] B. Liu, S. Liu, L. Yang, Z. Chen, E. Zhang, Z. Li, J. Wu,\nX. Ruan, F. Xiu, W. Liu, L. He, R. Zhang, and Y. Xu,\nPhys. Rev. Lett. 125, 267205 (2020).\n[9] O. H.-C. Cheng, D. H. Son, and M. Sheldon, Nature Pho-\ntonics 14, 365 (2020).\n[10] R. V. Mikhaylovskiy, E. Hendry, and V. V. Kruglyak,\nPhys. Rev. B 86, 100405 (2012).\n[11] Modern ultrafast laser sources allow for concentrations\nas large as 1022=cm3photoexcited carriers.\n[12] P.TangneyandS.Fahy,Phys.Rev.Lett. 82,4340(1999).\n[13] P. Tangney and S. Fahy, Phys. Rev. B 65, 054302 (2002).\n[14] G. Marini and M. Calandra, Phys. Rev. B 104, 144103\n(2021).\n[15] J. Yao, Y. Li, R. C. Massé, E. Uchaker, and G. Cao,\nEnergy Storage Materials 11, 205 (2018).\n[16] M. B. Smirnov, E. M. Roginskii, K. S. Smirnov,\nR. Baddour-Hadjean, and J.-P. Pereira-Ramos, Inor-\nganic Chemistry 57, 9190 (2018), pMID: 30044091,\nhttps://doi.org/10.1021/acs.inorgchem.8b01212.\n[17] N. V. Hieu and D. Lichtman, Journal of Vacuum Science\nand Technology 18, 49 (1981).\n[18] S. Shin, S. Suga, M. Taniguchi, M. Fujisawa, H. Kan-\nzaki, A. Fujimori, H. Daimon, Y. Ueda, K. Kosuge, and\nS. Kachi, Phys. Rev. B 41, 4993 (1990).\n[19] E. E. Chain, Appl. Opt. 30, 2782 (1991).\n[20] J. Meyer, K. Zilberberg, T. Riedl, and A. Kahn,\nJournal of Applied Physics 110, 033710 (2011),\nhttps://doi.org/10.1063/1.3611392.\n[21] A. Carpy, A. Casalot, M. Pouchard, J. Galy, and P. Ha-\ngenmuller, Journal of Solid State Chemistry 5, 229\n(1972).\n[22] C. Bhandari and W. R. L. Lambrecht, Phys. Rev. B 92,\n125133 (2015).\n[23] T. Machej, J. Haber, A. M. Turek, and I. E. Wachs, Ap-\nplied Catalysis 70, 115 (1991).\n[24] A. Othonos, C. Christofides, and M. Zervos,\nApplied Physics Letters 103, 133112 (2013),\nhttps://doi.org/10.1063/1.4823506.\n[25] E. Kumi-Barimah, D. E. Anagnostou, and\nG. Jose, AIP Advances 10, 065225 (2020),https://doi.org/10.1063/5.0010157.\n[26] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,\nC. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ-\ncioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris,\nG. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis,\nA. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari,\nF. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello,\nL. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P.\nSeitsonen, A. Smogunov, P. Umari, and R. M. Wentzcov-\nitch, Journal of Physics: Condensed Matter 21, 395502\n(2009).\n[27] P. Giannozzi, O. Baseggio, P. Bonfà, D. Brunato, R. Car,\nI. Carnimeo, C. Cavazzoni, S. de Gironcoli, P. Delugas,\nF. Ferrari Ruffino, A. Ferretti, N. Marzari, I. Timrov,\nA. Urru, and S. Baroni, The Journal of Chemical Physics\n152, 154105 (2020), https://doi.org/10.1063/5.0005082.\n[28] M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71,\n035105 (2005).\n[29] I. Timrov, N. Marzari, and M. Cococcioni, Phys. Rev. B\n98, 085127 (2018).\n[30] See Supplemental Material at [URL] for methodological\ndetails, comparison with the rigid doping case, structural\nstability and optical characterization of the photoexcited\nmagnetic phase.\n[31] P. Scherpelz, M. Govoni, I. Hamada, and\nG. Galli, Journal of Chemical Theory and Com-\nputation 12, 3523 (2016), pMID: 27331614,\nhttps://doi.org/10.1021/acs.jctc.6b00114.\n[32] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Re-\nview Letters 77, 3865 (1996).\n[33] H. J. Monkhorst and J. D. Pack, Physical Review B 13,\n5188 (1976).\n[34] S. Grimme, Journal of Computa-\ntional Chemistry 27, 1787 (2006),\nhttps://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.20495.\n[35] G. Grosso and G. P. Parravicini, in Solid State Physics ,\nedited by G. Grosso and G. P. Parravicini (Academic\nPress, London, 2000) pp. 425–472.\n[36] C. Lamsal and N. M. Ravindra, Journal of Materials Sci-\nence 48, 6341 (2013).\n[37] P. Elliott, M. Stamenova, J. Simoni, S. Sharma, S. San-\nvito, and E. K. U. Gross, Time-dependent density func-\ntional theory for spin dynamics, in Handbook of Materi-\nals Modeling : Methods: Theory and Modeling , edited by\nW. Andreoni and S. Yip (Springer International Publish-\ning, Cham, 2018) pp. 1–26.\n[38] E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997\n(1984).\n[39] N. Tancogne-Dejean, M. J. T. Oliveira, X. Andrade,\nH. Appel, C. H. Borca, G. Le Breton, F. Buch-\nholz, A. Castro, S. Corni, A. A. Correa, U. De Gio-\nvannini, A. Delgado, F. G. Eich, J. Flick, G. Gil,\nA. Gomez, N. Helbig, H. Hübener, R. Jestädt, J. Jornet-\nSomoza, A. H. Larsen, I. V. Lebedeva, M. Lüders,\nM. A. L. Marques, S. T. Ohlmann, S. Pipolo,\nM. Rampp, C. A. Rozzi, D. A. Strubbe, S. A. Sato,\nC. Schäfer, I. Theophilou, A. Welden, and A. Rubio,\nThe Journal of Chemical Physics 152, 124119 (2020),\nhttps://doi.org/10.1063/1.5142502.\n[40] C. Hartwigsen, S. Goedecker, and J. Hutter, Phys. Rev.\nB58, 3641 (1998).\n[41] X. Andrade, A. Castro, D. Zueco, J. L. Alonso,\nP.Echenique, F.Falceto,andA.Rubio,JournalofChem-\nical Theory and Computation 5, 728 (2009), pMID:7\n26609578, https://doi.org/10.1021/ct800518j.\n[42] E. P. Silaeva, E. Bevillon, R. Stoian, and J. P. Colombier,\nPhys. Rev. B 98, 094306 (2018).\n[43] N. A. Modine and R. M. Hatcher, The Jour-\nnal of Chemical Physics 142, 204111 (2015),\nhttps://doi.org/10.1063/1.4921690.\n[44] Y. Miyamoto, Scientific Reports 11, 14626 (2021).\n[45] K. Krieger, J. K. Dewhurst, P. Elliott, S. Sharma,\nand E. K. U. Gross, Journal of Chemical\nTheory and Computation 11, 4870 (2015),\nhttps://doi.org/10.1021/acs.jctc.5b00621.\n[46] P. Shvets, O. Dikaya, K. Maksimova, and A. Goikhman,\nJournal of Raman Spectroscopy 50, 1226 (2019).[47] F. Vidal, Y. Zheng, L. Lounis, L. Coelho, C. Laulhé,\nC. Spezzani, A. Ciavardini, H. Popescu, E. Ferrari, E. Al-\nlaria, J. Ma, H. Wang, J. Zhao, M. Chollet, M. Seaberg,\nR. Alonso-Mori, J. M. Glownia, M. Eddrief, and M. Sac-\nchi, Phys. Rev. Lett. 122, 145702 (2019).\n[48] A. B. Cezar, I. L. Graff, J. Varalda, W. H. Schreiner, and\nD. H. Mosca, Journal of Applied Physics 116, 163904\n(2014).\n[49] Z. R. Xiao and G. Y. Guo, The Journal of Chemical\nPhysics 130, 214704 (2009).\n[50] C. T. Anderson, Journal of the Amer-\nican Chemical Society 58, 564 (1936),\nhttps://doi.org/10.1021/ja01295a006."    },    {        "title": "2005.03493v1.Ultrafast_All_Optical_Magnetization_Control_for_Broadband_Terahertz_Spin_Wave_Generation.pdf",        "content": "Ultrafast All Optical Magnetization Control for Broadband Terahertz Spin Wave Generation  \n \nSaeedeh Mokarian Zanjani1, Mehmet C. Onbaşlı1,2,* \n1Graduate School of Materials Science and Engineering, Koç University, Sarıyer, 34450 Istanbul, Turkey.  \n2 Department of Electrical and Electronics Engineering, Koç University, Sarıyer, 34450 Istanbul, Turkey.  \n* Corresponding Author: monbasli@ku.edu.tr   \n \nTerahertz spin waves could be generated on -demand via all -optical manipulation of \nmagnetization by femtosecond laser pulse. Here, we present an energy balance model, which \nexplains the energy transfer rates from laser pulse to electron bath coupled with phonon, spin, and \nmagnetization of five different ma gnetic metallic thin films: Iron, Cobalt, Nickel, Gadolinium and \nNi2MnSn Heusler alloy. Two types of transient magnetization dynamics emerge in metallic \nmagnetic thin films based on their Curie temperatures (T C): type I (Fe, Co, and Ni with T C > room \ntemperature, RT) and type II films (Gd and Ni 2MnSn with T C ≈ RT). We study the effect of laser \nfluence and pulse width for single Gaussian laser pulses and the effect of metal film thickness on \nmagnetization dynamics. Spectral dynamics show that br oadband spin waves up to 24 THz could be \ngenerated by all -optical manipulation of magnetization in these nanofilms.\n \nI. INTRODUCTION  \nTerahertz (THz) spintronic systems have been \nemerging due to their new fundamental physics on the \ninteractions of magnons, phonons and photons [1-9] as \nwell as their applications in sensing, spectroscopy [10], \nand on -chip communications [11,12]. THz band \ncontains many molecule -specific absorption  peaks \nwhich could be used for highly -specific sensor designs. \nWhile electromagnet ic sensing in the THz region using \nTHz time -domain spectroscopy (TDS) has been \ndemonstrated to have high specificity [13-18], \nspintronic THz spectroscopic systems could enable \nmuch higher sensitivities  since THz spin waves are \nhighly localized (sub -10 nm) compared to \nelectromagnetic THz modes (240 μm) [10]. Thus, \nspintronic THz -TDS systems could be used for label -\nfree single molecule detection.   \nIn the design of such THz spintronic sensor \nsystems, broad band THz spin waves could be \ngenerated by applying femtosecond (fs) laser pulses on \nmetallic magnetic thin films [8, 9] . A detailed \nunderstanding of how the key material and laser \nparameters determine the generated THz spin wave \nspectra is essential for bo th sensing applications and \nfor understanding phonon, magnon and photon \ninteractions. In this study, we developed an energy \nbalance model, which describes the coupled interaction \nbetween electron, phonon, and spin temperatures with \nnormalized magn etization  (extended M3TM) after \nillumination with fs Gaussian single laser pulse for five   \ndifferent metallic magnetic thin films: Iron (Fe), \nCobalt (Co), Nickel (Ni), Gadolinium (Gd), and \nNi2MnSn Heusler alloy. We investigated the effect of \nthe material propertie s of these metals and fs laser \npulse characteristics (laser fluence and pulse width) on \nmagnetization dynamics. Magnetization dynamics \nindicate that b roadband THz spin wave s could be \ngenerated inside the films due to exchange coupling. \nWe provide the guide lines for obtaining a desired spin \nwave spectrum using the appropriate metallic film \ntype, thickness, laser pulse width and fluence. By \nidentifying the regimes in which T C, phonon scattering , \nthermal mass and spin -flip scattering dominates, we \nexplain the mechanisms that determine the se spectra.  \nThe interaction of fs laser pulse with Nickel (Ni) \nmetallic magnetic materials was first explained using \nthree temperature model (3TM) [19]. Since \nmagnetization dynamics were not considered in this \nmodel, Landau -Liftshitz -Bloch (LLB) equation [20] \nwas developed to model the laser -matter interaction. \nThe LLB model explains  the thermal effects of laser \npulse on magnetization dynamics. Nevertheless, this \nmodel does not capture the thermal interaction of laser \npulse w ith electron, phonon, and spin temperatures  and \ntheir interactions . This model also depends on \ntemperature dependent damping parameter, which was \nnot measured for most materials. Koopmans  et al.  [21] \ndeveloped a model considering the coupling of \nmagnetizat ion dynamics with electron and phonon \ntemperatures  and could extract the spin wave emission spectra ; however, the y neglected the coupling of spin \ntemperature with that of phonon and electron in their \nmodel. Our extended M3TM model describes each of \nthese e ffects for different pulse fluences, widths , film \nthicknesses and metal types.  By capturing the heat \nexchange between electron, phonon and spin baths , \ntheir heat exchanges, the magnetization dynamics and \ntheir spectral characteristics, our model is the fir st to \nidentify the conditions needed for  broadband  and \ntunable  THz spin wave emission .  \nII. ENERGY BALANCE MODEL AND \nMAGNETIZATION DYNAMICS  \nWe developed an extended three -temperature \nmodel coupled with transient magnetization dynamics \nto capture the energy  transfer rates between spin, \nphonon and electron thermal baths. These thermal \nbaths are assumed to behave in the classical or semi -\nbulk regime (i.e. films are sufficiently thick with no \nquantum confinement effects , t > 20 nm ). The energy \nbalance model des cribes that the fs l aser pulse injects \nenergy into the coupled baths and the transient \nelectron, phonon, and spin temperatures (T e, T p, T s) \nand magnetization are described with the rate \nequations (1) to (4) [19, 21]. \nCedTe\ndt=−Gep(Te−Tp)−Ges(Te−Ts)+P(t)    (1) \nCpdTp\ndt=−Gep(Tp−Te)−Gps(Tp−Ts)       (2) \nCsdTs\ndt=−Ges(Ts−Te)−Gps(Ts−Tp)          (3) \ndm\ndt=RmTp\nTC (1−mcoth (mTC\nTe))            (4) \nIn these equations, T e, Tp, Ts, and T C are electron, \nphonon, spin, and Curie temperatures, respectively. R  \nis the  spin-flip ratio [21] and determines the kinetics of \ntransient changes of T e, Tp, Ts, and magnetization. The \nelectron -phonon, electron -spin, and phonon -spin \ncoupling constants G ep, G es, G ps, follow a similar \nformalism described in ref. [21] and [22]. The heat \ncapacities of pho non and spin are C p and C s. The \nelectron heat capacity is a temperature -dependent \nparameter and is defined as C e=γT e, where γ is a \nparameter that depends on free electron density and \nFermi energy level [23] as used in ref. [21] as Cp=5 \nγTC). Finally, P( t) represents the laser pulse power \ninjected per unit volume and m is normalized \nmagnetization defined as |M z|/M s. The material \nparameters and constants used in the numerical solutions to the equations (1) to (4) for Ni, Co, Fe, Gd, \nand Ni 2MnSn are shown i n Supplementary Table 1 . \nIII. RESULTS AND DISCUSSION  \nFig.1 (a) shows a schematic of THz spin wave \ngeneration inside the metallic magnetic thin film. As a \nfs laser pulse hits the metallic film, the film’s magnetic \nmoment undergoes precession and damped \noscillations. These oscillations trigger THz spin waves \nover their neighbors due to exchange coupling.  \n \nFIG. 1. Effect of fs laser pulse on metallic magnetic thin \nfilms.  (a) THz spin wave generation in metallic magnetic \nthin films after illum ination with a  fs laser pulse. (b) \nTransient magnetization dynamics of 50 nm thick Fe, Co, \nNi, Gd, and Ni 2MnSn Heusler alloy films illuminated with \n100 fs laser pulse with 70 J·m-2 fluence . (c) Exchange -\ncoupled s pin wave spectra generated after laser pulse \n(Fourier  transform of magnetization in b). \nFig. 1(b) shows the calculated time -dependent \nrelaxation of the magnetic moments of 50 nm thick Fe, \nNi, Co, Gd, and Ni 2MnSn upon each receiving a 100 \nfs-long pulse with a moderate (70 J·m-2) fluence. \nAccording to Fig.1 (b ), magnetizations in Fe, Ni, Co \n(type I) reach the dip in less than 200 fs and reach \nsteady state moments in 6, 3, and 0.5 ps, respectively.  \nWe group Fe, Co, and Ni as type I since their Curie \ntemperatures greatly exceed room temperature (Fe: \n1043 K, Co: 1388 K, Ni: 627 K). Thus, these films \nnever undergo thermal demagnetization after \ninteracting with fs laser pulse. On the other hand, Gd \nand Ni 2MnSn (type II) with Curie temperatures near \nroom temperature (Gd: 297 K, Ni 2MnSn: 319 K) \nundergo thermal demagne tization upon interacting \nwith the fs laser pulse and reach zero magnetization \nwithin 15 and 6 ps, respectively. The laser -induced loss \nof spin angular momentum could be due to spin -flip \nscattering with phonons and hot electrons (electron -\nphonon/electron -electron interaction), known as \nElliot -Yafet scattering [24, 25]. The spin wave spectra \nfor these thin films are shown in Fig. 1(c) by \ncalculating the fast Fourier transform (FFT) of \nmagnetization in Fig. 1(b). Broadband spin waves \ncould be generated using Fe, Ni, and Co magnetic thin \nfilms up to 24 THz. The intensity of the THz signal is \nhigher for Fe and Co due to their higher Curie temperatures than that for Ni. In addition, the \nrecovered fraction of magnetization after a few \npicoseconds is higher in Fe a nd Co (M/M s > 97%) \ncompared to Ni (M/M s ~50%). The recovered fraction \nof magnetization increases with increasing the film \nthickness.   \nIn the rest of this section, the e ffect of film \nthickness, fs laser pulse fluence and pulse width on the \nmagnetization dynamics and the spin wave spectra in \nboth type I and II films are investigated.  \n1. Thin films with type I dynamics (Fe, Co, Ni)  \nType I dynamics is defined as ultrafast \nswitching/decrease of magnetization in fs timescales, \nfollowed by slow recovery of magnetization in \npicosecond timescales. Hot electrons excite  transient \nmagnetization loss and a subsequent excitation of spin \nwaves  (magnons)  due to exchange coupling . Magnons \nundergo inelastic scattering with phonons and \nelectronic charges  in the lattice . The lattice ser ves as a \nreservoir for absorbed, dissipated  and scattered energy  \nand causes evanescent decay of the generated spin \nwaves. Magnetization recovery time is slightly longer \nthan electron -lattice relaxation time  due to the different \ncoupling strengths  among lattice, phonon and electron \ntemperatures  (i.e. strength of Hamiltonian terms for \neach quasiparticle) . In our model, we do not consider \nthe quantum nature of these baths or their \nHamiltonians  and lump their couplings into G ep, G es, \nGps parameters . These parameters are known for the \nmetallic magnetic thin films in our study [21].  \nFig. 2 (a) show s the magnetization dynamics o f (i) \n20 nm and (ii) 100 nm thick Fe films  under \nillumination with a 100 fs Gaussian laser pulse.   \nFIG. 2. Effect of film thickness, fs laser fluence and pulse width on magnetization dynamics and lattice temperature \nof Fe thin film.  (a) Transient electron (T e), phonon (T p), spin (T s) temperatures  and magnetization (m=|M z|/M s) of (i) 20 nm \nand (ii) 100 nm thick Fe, illuminated with 100 fs  laser pulse with 70 J·m-2 fluence. (b) Fluence dependence of magnetization \nin 20 nm thick Fe film illuminated with (i) 100 fs and (ii) 500 fs laser pulse.  \nFollowing type I dynamics, magnetization of Fe \nstarts switching around 300 fs for 20 nm film and \nslightly shorter than that for 100 nm Fe film. The \nperpendicu lar magnetization starts recovering and \neventually reaches ~78% of saturation moment after 12 \nps in 20 nm Fe film. The recovered magnetization \nfraction reaches 99% for 100 nm thick film after 8 ps. \nDue to the interaction of spins with the fs laser pulse, \nthe electron, spin and lattice temperatures increase. \nAlthough the electron temperature undergoes a sudden \nincrease to above Curie temperature (T C) of Fe, due to \nelectron’s  lower heat capacity (C e), it drops quickly \nand the equilibrium lattice temperature ( Tp) stays below T C. Thus, both 20 and 100 nm Fe films retain \ntheir magnetization.  \nIn Fig. 3 (a), the magnetization dynamics of (i) 20 \nand (ii) 100 nm thick Co, are shown after illumination \nwith Gaussian single pulse with 100 fs duration and 70 \nJ·m-2 fluence . Co magnetization (type I) decreases \nwithin 250 fs in 20 nm film and recovers back to 75% \nof its saturation moment in around 10 ps. Magnetic \nmoment in 100 nm Co starts switching in ~160 fs and \nalmost completely recovers (~99.5%) its saturation \nstate after  800 fs. In this case, T p /TC <1 so the film does \nnot demagnetize completely.  \n \nFIG. 3. Effect of film thickness, fs laser fluence and pulse width on magnetization dynamics and lattice temperature \nof Co thin film.  (a) Transient electron (T e), phonon (T p), spin (T s) temperatures, and normalized magnetization (m=|M z|/M s) \nof (i) 20 nm and (ii) 100 nm thick Co illuminated with 100 fs laser pulse with 70 J·m-2 fluence . (b) Fluence  dependence of \nnormalized magnetization of 20 nm thick Co film illuminated with (i) 100 fs and (ii) 500 fs laser pulse.  \nIn Fig. 4, the magnetization dynamics of (i) 20 and \n(ii) 100 nm thick Ni thin films upon illumination with \n100 fs Gau ssian pulses are shown. The 20 nm Ni film \ndemagnetizes completely after 130 fs, since the lattice \ntemperature (T p) greatly exceeds nickel’s Curie \ntemperature. Nickel’s T C (627 K) is lower than those \nof Fe (1043 K) and Co (1388 K); so 20 nm thick Ni \nfilm ca nnot retain its magnetization after interacting \nwith the fs laser pulse. However, the thicker Ni film \n(100 nm) exhibits type I dynamic similar to Fe and Co \nfilms  due to its larger heat capacity . The film’s \nmagnetic moment decreases in ~170 fs and recovers \n~83% of its saturation moment within 1.2 ps. \nIncreasing the film thickness in Ni decreases both \nswitching and recovery times, but it increases the \nrecovered fraction of magnetization. The fluen ce and pulse width dependence of magnetizations for 20 nm \nFe (Fig. 2b), Co (Fig. 3b), Ni (Fig. 4b) films are shown. \nThe magnetization dynamics have been calculated for \nfluences from 20 to 150 J·m-2. Illuminated with both \n100 fs (Fig. 2.b (i), Fig. 3. b (i) , Fig. 4.b (i)) and 500 fs \n(Fig. 2.b (ii), Fig. 3. b (ii), Fig. 4.b (ii)), both \nmagnetization switching and recovery times of 20 nm \nfilms increase  with increasing laser fluence. The \nrecovered magnetization fraction, however, decreases \nwith increasing laser  fluence. In addition, there is a \nthreshold fluence above which the films are thermally \ndemagnetized completely. This threshold fluence \ndepends on the film type, thickness and pulse width. \nIncreasing the pulse width increases the response time \nof the films , which is attributed to the larger energy \ninjected and dissipated with the films . The recovered \nfraction of magnetization is higher in case of thicker \nfilms, since the higher thermal mass prevents the \nexcess heat accumulation and magnetization \ndisturbance  in these films. The fluence dependence of magnetiz ation dynamics of 100 nm films are presented \nin the Supplemental Material.  \n \n \nFIG. 4. Effect of film thickness, fs laser fluence and pulse width on magnetization dynamics and lattice temperature \nof Ni thin film.  a(i) Transient electron (T e), phonon (T p), spin (T s) temperatures, and normalized magnetization (m=|M z|/M s) \nof 20 nm and (ii) 100 nm thick Ni illuminated with 100 fs laser pulse with 70 J·m-2 fluence . b Fluence dependence of \nnormalized magnetization of (i) 20 nm thick Ni film illuminated with 100 fs and (ii) 500 fs  laser pulse.  \n2. THz spin wave spectra generated in type I films  \nFig. 5 shows the fast Fourier transform (FFT) \nresults of the fluence dependence of temporal \nmagnetization change for Fe, Co and Ni. As shown in \nFig. 5, by illuminating the Fe, Co, and Ni films with \nsingle 100 fs Gaussian laser pulses each, spin waves up \nto 10 THz could be generated. The THz spin wave \nintensity i s higher when fs laser fluence is lower. \nIncreasing film thickness to 100 nm increases the THz spin wave intensity and the allowed threshold fluence \nfor THz spin wave generation. Laser pulses with sub -\npicosecond widths have minimal effect on the \nbandwidth and the intensity of the THz spin wave \nsignal  (see the spectra of the films under illumination \nof 500 fs laser pulse  in Supplemental Material ). From \nthe spectra of Fe and Co, one can infer that the THz \nspin wave intensity is slightly higher compared to Ni,  \nsince these two metals have higher T C and recovered \nmagnetization fractions (> 95%).  \n \n \nFIG. 5. THz spin wave generation in type I metallic magnetic thin films. Effect of laser fluence on the bandwidth and \nintensity  of THz spin wave generated in a(i) 20nm thick, a(ii) 100 nm thick Fe, b(i) 20nm thick, a (ii) 100 nm thick Co, and  \na(i) 20nm thick, a(ii) 100 nm thick Ni thin films under illumination of 100 fs laser pulse.  \n3. Thin films with type II d ynamics (Gd, Ni 2MnSn)  \nType II magnetization dynamics arise when the \nfilms undergo thermal demagnetization (magnetic \nmoment decays to zero in tens of picoseconds). Fig. 6 and 7 show the magnetization dynamics of type II \nfilms, Gd and Ni 2MnSn. Fig. 6a (i) a nd (ii) show the \ntemporal change of T e, Tp, Ts, and the magnetization of \n20 nm and 100 nm Gd thin film s after interacting with \na 100 fs Gaussian single laser pulse of 70 J·m-2 fluence. \nThe spin -flip ratio of Gd is small (R= 0.092×1012 s-1) \ncompared to the type I metallic magnetic thin films; so \nthe magnetization vanishes slowly after 2.5 ps in 20 nm \nfilm and 50 ps in 100 nm Gd film. Since the Curie \ntemperature of Gd is near room temperature (297 K), \nthe equilibrium electron, spin and the l attice \ntemperatures exceed the Curie temperature and the \nfilm is thermally demagnetized completely with the \nlaser pulse. The equilibrium lattice temperature of 20 \nnm thick Gd film (Fig. 6.a (i)) is higher compared to \nthe 100 nm thick film (Fig. 6.a (ii)) d ue to the lower \nthermal mass of the thinner film. Similarly, Ni 2MnSn \nalso undergoes thermal demagnetization and its \nmagnetization decays faster than Gd due to the alloy’s \nlarger spin -flip ratio (R =0.1×1012s-1). Demagnetization \ntimes of 20 nm and 100 nm thi ck Ni 2MnSn film s are 1 ps and 35 ps, respectively, showing that a larger \nthermal mass delays thermal equilibration. Fig. 6(b) \nand 7(b) show the dependence of magnetization \ndynamics to the laser pulse fluence and duration.  \n Similar to type I dynamics, incre asing the laser \nfluence and pulse duration, increases the response time \nin both type II films. On the other hand, excitation of \nthe films with smaller laser fluences keeps the films \nmagnetized for longer times. Unlike type I films, the \nresponse time increa ses in type II by increasing the \nfilm thickness. According to Fig. 6(b) and Fig. 7(b), \nincreasing the incoming laser pulse width increases the \nresponse time.  \n \n \nFIG. 6. Effect of film thickness, fs laser fluence and pulse width on magnetization dynamics and lattice temperature \nof Gd magnetic thin film.  (a) transient electron (T e), phonon (T p), spin (T s) temperatures, and normalized magnetization \n(m=|M z|/M s) of (i) 20 nm and (ii) 100 nm thick Gd illuminated with 100 fs laser pulse with 70 J·m-2 fluence . (b) Fluence \ndependence of normalized magnetization of 20 nm thick Gd film illuminated with (i) 100 fs and (ii) 500 fs Gaussian laser \npulse . \n \nFIG. 7. Effect of film thickness, fs laser fluence and pulse width on magnetization dynamics and lattice temperature \nof Ni 2MnSn magnetic thin film.  a(i) Transient electron (T e), phonon (T p), spin (T s) temperatures, and normalized \nmagnetization (m=|M z|/M s) of 20 nm and a(ii) 100 nm thick Ni 2MnSn illuminated with 100 fs laser pulse with 70 J·m-2 \nfluence. b(i) Fluence dependence of magnetization of 20 nm thick Ni 2MnSn film illuminated with 100 fs and b(ii) 500 fs \nlaser pulse.  \n4. THz spin wave spectra generated in type II films   \nComparison of the spin wave spectra in 20 nm and \n100 nm Gd and Ni 2MnSn thin films show that THz spin \nwaves up to 8 THz could be generated in such thin \nfilms by manipulation of magnetization with 100 fs \nGaussian laser pulse. One can tune the intensity and the \nbandwidth of the generated THz signal by changing the \nfilm thickness and laser pulse fluence. Laser pulses \nwith sub -picosecond widths have minimal effect on the \nbandwidth and the intensity of the THz spin wave \nsignal  (see the spectra corr esponding to 500 fs pulse \nwidth s in Supplemental Material ). According to Fig. 8, the bandwidth and the intensity of the THz signal is \nhigher in both 100 nm thick Gd and Ni 2MnSn thin \nfilms compared to 20 nm films since thicker films stay \nmagnetized for long er. Hence, decreasing the laser \nfluence helps generate broadband THz spin waves with \nhigher intensity.  \nComparison of Fig. 5 and Fig. 8 shows that using \nmagnetic metallic thin films which follow the type I \ndynamics results in generation of higher intensity and \nbroader band THz spin waves, since type I metals \npreserve their magnetization for longer after \ninteracting with the pulse.  \n \nFIG. 8. THz spin wave generation in type II metallic magnetic thin films. Effect of laser fluence on the bandwidth and \nintensity of THz spin wave generated in (a) (i) 20nm and  (ii) 100 nm thick Gd and (b) (i) 20nm and  (ii) 100 nm thick Ni 2MnSn \nfilms under illumination of 100 fs laser pulse.  \nIV. CONCLUSION  \nWe developed a model which explains the energy \ntransfer from fs laser pulse to coupled electron, \nphonon, spin, and normalized magnetization in Fe, Co, \nNi (type I films), Gd and Ni 2MnSn (type II films). Our \nmodel and numerical solutions show that type I \nmetallic thin films with Curie temperatures much \nhigher than room temperature recover their \nmagnetization. The films were first illuminated with \nGaussian single laser pulse. Then the magnetization \nvector decreases briefly and reverts to its initial \norientatio n within the next 130 -200 fs. In the following \n1-2 ps, the magnetization settles to its steady -state \norientation. Increasing the laser pulse width and \nfluence increases the film response time and decreases \nits recovered fraction of magnetization. Type II f ilms \nundergo thermal demagnetization gradually in about \n35-50 ps after interaction with Gaussian laser pulse. Although higher laser fluences and pulse widths result \nin higher response times in these films, similar to type \nI films, increasing the thickness does not shorten the \nthermal demagnetization time. In thicker type I films \n(i.e. 100 nm), unlike type II films, the response time of \nmagnetization is shorter. FFT of normalized \nmagnetization dynamics show the broadband THz \nexchange -mediated spin wave gener ation up to 24 THz \ndepending on laser pulse width, fluence and film \nthickness.  \nOur study suggests the need for optimizing the \nexperimental parameters, such as laser pulse width and \nenergy, film type and thickness for more energy \nefficient and faster respon se. Broadband THz spin \nwaves generated with ultrafast all optical manipulation \nof magnetization could be utilized in many \napplications such as communications, imaging, \nsecurity and sensing using THz spintronic time -\ndomain spectroscopy (THz -sTDS).  \nACKNOWLED GMENTS  \nM.C.O. acknowledges BAGEP 2017 Award, \nTÜBA -GEBİP Award by Turkish Academy of \nSciences and TUBITAK Grant No. 117F416.  \n \nREFERENCES  \n[1] B. Lenk, H. Ulrichs, F. Garbs, and M. Münzenberg, \nThe building blocks of magnonics,  Phys. Rep., \n507, 107 -136 (2011).  \n[2] S. Mizukami, A. Sugihara, S. Iihama, Y. Sasaki, K. \nSuzuki, and T. Miyazaki, Laser -induced THz \nmagnetization precession for a tetragonal Heusler -\nlike nearly compensated fe rrimagnet,  Appl. Phys. \nLett., 108, 012404 (2016).  \n[3] M. Hoefer, M. J. Ablowitz, B. Ilan, M. R. Pufall, \nand T. J. Silva, Theory of magnetodynamics \ninduced by spin torque in perpend icularly \nmagnetized thin films,  Phys. Rev. Lett., 95, \n267206 (2005).  \n[4] T. Kampfrath, M. Battiato, P. Maldonado, G. \nEilers, J. Nötzold, S. Mährlein, et al. , Terahertz \nspin current pulses controlle d by magnetic \nheterostructures,  Nat. Nanotechnol., 8, 256 (2013).  \n[5] G. Torosyan, S. Keller, L. Scheuer, R. Beigang, and \nE. T. Pa paioannou, Optimized spintronic terahertz \nemitters based on epitaxia l grown Fe/Pt laye r \nstructures,  Sci. Rep., 8, 1311 (2018).  \n[6] Y. Wu, M. Elyasi, X. Qiu, M. Chen, Y. Liu, L. Ke, \nH.Yang , High‐Performance THz Emitters Based \non Ferromagnetic/Nonmagnetic  Heterostructures,  \nAdv. Mater., 29, 1603031 (2017).  \n[7] E. T. Papaioannou, G. Torosyan, S. Keller, L. \nScheuer, M. Battiato, V. K. Mag -Usara, J. \nL’huillier, M. Tani, R. Beigang , Efficient terahertz \ngeneration using Fe/Pt spintronic emitters pumped \nat diff erent wavelengths,  IEEE Trans. Magn., 54, \n1 (2018).  \n[8] H. Qiu, K. Kato, K. Hirota, N. Sarukura, M. \nYoshimura, and M. Nakajima, Layer thickness \ndependence of the terahertz emission based on spin \ncurrent in ferromagnetic heterostructures,  Opt. \nExpress, 26, 15247  (2018).  \n[9] D. Yang, J. Liang, C. Zhou, L. Sun, R. Zheng, S. \nLuo, Y. Wu, J. Qi , Powerful and tunable THz \nemitters based on the Fe/Pt magnetic \nheterostructure,  Adv. Opt. Mater., 4, 1944 (2016).  [10] Y. -J. Lin, S. Cakmakyapan, N. Wang, D. Lee, M. \nSpearrin, and M. Jarrahi, Plasmonic heterodyne \nspectrometry  for resolving the spectral signatures \nof ammonia over a 1 -4.5 THz frequency range, \nOpt. Express, 27, 36838 (2019).  \n[11] X. Chen, X. Wu, S. Shan, F. Guo, D. Kong, C. \nWang, et al. , Generation and manipulation of \nchiral broadband terahertz waves from cascade  \nspintronic terahertz emitters,  Appl. Phys. Lett., \n115, 221104 (2019).  \n[12] Y. Yang, Y. Yamagami, X. Yu, P. Pitchappa, J. \nWebber, B. Zhang, M. Fujita, T. Nagatsuma & R. \nSingh, Terahertz topological photonics for on -chip \ncommunication, Nat. Photon. (2020).  \nhttps://doi.org/10.1038/s41566 -020-0618 -9 \n[13] M. Walther, B. M. Fischer, A. Ortner, A. B itzer, \nA. Thoman, and H. Helm, Chemical sensing and \nimaging w ith pulsed terahertz radiation,  Anal. \nBioanal. Chem., 397, 1009 -1017 (2010).  \n[14] Z.-P. Zheng, W. -H. Fan, and H. Yan, Terahertz \nabsorption spectra of benzene -1, 2-diol, benzene -\n1, 3-diol and benzene -1, 4-diol, Chem. Phys. \nLett., 525, 140 -143 (2012).  \n[15]  P. Upadhya, Y. Shen, A. Davies, and E. Linfield, \nTerahertz time -domain spectrosc opy of glucose \nand uric acid, J. Biol. Phys., 29, 117 -121 (2003).  \n[16] K. M. Mølster, T. Sørgård, H. Laurell, C. \nCanalias, V. Pasiskevicius, F. Laurell, U. \nÖsterberg , Time -domain spectroscopy of \nKTiOPO 4 in the frequency range 0.6 –7.0 THz, \nOSA Continuum, 2, 3521 -3538 (2019).  \n[17] B. Fischer, M. Walther, and P. U. Jepsen, Far -\ninfrared vibrational modes of DNA components \nstudied by terahertz time -domain spectroscopy, \nPhys. Med. Biol., 47, 3807 (2002).  \n[18]  M. Leahy -Hoppa, M. Fitch, X. Zheng, L. Hayden, \nand R.  Osiander, Wideband terahertz \nspectroscopy of explosives, Chem. Phys. Lett., \n434, 227 (2007).  \n[19]  E. Beaurepaire, J. -C. Merle, A. Daunois, and J. -\nY. Bigot, Ultrafast spin dynamics in \nferromagnetic nickel, Phys. Rev. Lett., 76, 4250 \n(1996).  \n[20] U. Atxitia, D. Hinzke, and U. Nowak, \nFundamentals and applications of the Landau –\nLifshitz –Bloch equation, J. Phys. D: Appl. Phys., \n50, 033003 (2016).  \n[21]  B. Koopmans, G. Malinowski, F. Dalla Longa, D. \nSteiauf, M. Fähnle, T. Roth, M. Cinchetti, M. \nAeschli mann, Explaining the paradoxical diversity of ultrafast laser -induced \ndemagnetization, Nat. Mater., 9, 259  (2010).  \n[22] J. Kimling, J. Kimling, R. Wilson, B. Hebler, M. \nAlbrecht, and D. G. Cahill, Ultrafast \ndemagnetization of FePt: Cu thin films and the \nrole of magnetic heat capacity,  Phys. Rev. B, 90, \n224408 (2014).  \n[23] Z. Lin, L. V. Zhigilei, and V. Celli, Electron -\nphonon coupling and electron heat capacity of \nmetals under conditions of strong electron -\nphonon nonequilibrium, Phys. Rev. B, 77, \n075133 (200 8). [24] K. Bühlmann, R. Gort, G. Salvatella, S. Däster, A. \nFognini, T. Bähler, Ch. Dornes, C. A. F. Vaz, A. \nVaterlaus, Y. Acremann,  Ultrafast \ndemagnetization in iron: Separating effects by \ntheir nonlinearity, Struct. Dynam., 5, 044502 \n(2018).  \n[25] B. Y. M ueller, T. Roth, M. Cinchetti, M. \nAeschlimann, and B. Rethfeld, Driving force of \nultrafast magnetization dynamics, New J. Phys., \n13, 123010 (2011).  \n Supplementary Information for “ Ultrafast All Optical Magnetization Control for Broadband \nTerahertz Spin Wave Generation ” \nSaeedeh Mokarian Zanjani1, Mehmet Cengiz Onbaşlı1,2,* \n1Graduate School of Materials Science and Engineering, Koç University, Sarıyer, 34450 Istanbul, Turkey.  \n2 Department of Electrical and Electronics Engineering, Koç University, Sarıyer, 34450 Istanbul, Turkey.  \n* Corresponding Author: monbasli@ku.edu.tr  \n \nSUPPLEMENTAL MATERIAL  \n \nSUPPLEMENTAL NOTE 1  \nIn the main manu script , we solved four coupled \ndifferential equations , which describe  the energy \ntransfer between electron, phonon (lattice), spin baths \nand the normalized magnetization, after interaction \nwith femtosecond (fs) Gaussian single laser pulse. \nCoefficients and material parameters used in solving \nthe equations (1) -(4) in the main manuscript are shown \nin the Supplementary Table I . We have shown the \nmagnetization dynamics and transient change of th ree temperature baths (Te: electron, T p: phonon/lat tice, T s: \nspin) for five different metallic thin films: Iron (Fe), \nCobalt (Co), Nickel (Ni), Gadolinium (Gd), and \nNi2MnSn Heusler alloy. In addition, fast Fourier \ntransform (FFT) results of the transient magnetization \nare plotted as the frequency -domain ch ange of \nmagnetization to illustrate the broadband THz spin \nwave generation. We applied our model to different \nthicknesses of the films illuminated wi th different laser \npulse widths.\n \nSUPP . TABLE I. Material parameters used in the extended M3TM model. C p, Cs are in J·m-3·K-1; G ep, G es, G ps are in W·m-\n3·K-1, TC is the Curie temperature (Kelvin), R: spin -flip ratio (s-1), γ in J·m-3·K-2 [1-4]. \nFilm  Cp ×106 Cs ×106 Gep ×1018  Ges ×1018 Gps ×1018 TC R×1012 γ =C p/5T C \nNi 2.33 0.2 4.05 0.6 0.03 627 17.2 743.22  \nCo 2.07 0.2 4.05 0.6 0.03 1388  25.3 298.28  \nFe 3.46 0.17 0.7 0.06 0.03 1043  1.86 663.47  \nGd 1.78 0.2 0.21 0.6 0.03 297 0.092  1200  \nNi2MnSn  0.615  0.2 0.24 0.6 0.03 319 0.1 385.58  \nSUPPLEMENTAL NOTE 2  \nThis supplementa l part is organized to present the \ncalculated plots of sensitivity of the transient \nmagnetization to t he incoming laser pulse fluence  and \nfilm thickness after illumination with 500 fs pulse \nwidth . Supp. Fig S1 and S2 show  the sensitivity of the \nmagnetization dyn amics to the laser fluence and pulse \nwidth for 100 nm thick  type I and type II thin films , \nrespectively.  Supp. Fig. S3 and S4 show  the sensitivity \nof the spectral change of magnetization to the laser \nfluence and film thickness for 500 fs laser pulse width \nfor type I and type II thin films, respectively . \n \n SUPPLEMENTAL NOTE 3  \nThe fluence dependence of magnetization \ndynamics of 20 nm films are presented in t he main \nmanuscript. The sensitivity of magnetization to the \nfluence and pulse width for 100 nm thick Fe, Co, and \nNi films are shown  in Supp. Fig. S1(a, b, c), \nrespectively . The magnetization dynamics have been \ncalculated for fluences from 20 to 150 J·m-2. \nIlluminated with both 100 fs ( Supp. Fig. S1. a(i), b(i), \nc(i)) and 500 fs (Supp. Fig. S1. a(ii), b(ii), c(ii)) , both \nmagnetization switching and recovery times of 100 \nnm-thick  films increase with increasing laser fluence. \nThe recovered magnetization fracti on, however, \ndecreases with increasing laser fluence. There is a threshold fluence above which the films are thermally \ndemagnetized. This threshold fluence depends on the film type, thickness and pulse width. Increasing the \npulse width increases the respon se time of the films. \n  \nSUPP. FIG . S1. Effect of fs laser fluence  and pulse width  on magnetization dynamics and lattice temperature of \ntype I magnetic thin films.  Sensitivity of transient magnetization (m=|M z|/M s) of 100 nm thick ( a) Fe, ( b) Co, and ( c) \nNi under illumination of (i) 100 fs and (ii) 500 fs laser pulse widths . \n \n \nSUPP. FIG. S2. Effect of fs laser fluence and pulse width on magnetization dynamics and lattice temperature of \nmagnetic type II thin films.  Sensitivity of transient magnetization (m=|M z|/M s) of 100 nm thick ( a) Gd, ( b) Ni 2MnSn under \nillumination of (i) 100 fs and (ii) 500 fs laser pulse widths.  \nSupp.  Fig. S2(a) and S2(b) show the dependence \nof magnetization dynamics to the pulse fluence and \npulse width  for 100 nm thick Gd and Ni 2MnSn films, \nrespectively . Type II films, unlike type I, undergo total \nthermal demagnetization and their moments cannot be \nrecovered. Similar to type I dynamics, increasing the \nlaser fluence and pulse durat ion increases the response \ntime in both type II films. On the other hand, excitation \nof the films with smaller laser fluences keeps the films \nmagnetized for longer times. Unlike type I films, the \nresponse time increases in type II films by increasing \nthe f ilm thickness. Besides, increasing the incoming \nlaser pulse width increases the response time  in both \nfilms, as shown in Supp. Fig. S2(a(ii)) and (b(ii)) . \n \n SUPPLEMENTAL NOTE 4  \nSupp.  Fig. S3 shows the FFT of the fluence \ndependence of temporal magnetization dynamics for \nFe, Co and Ni. The spectra for films under 100 fs laser \npulse illumination are presented in the main \nmanuscript. Supp. Fig. S3 shows  the spectra for Fe, Co, \nand Ni films  each i lluminated with single 5 00 fs \nGaussian laser pulses . Spin waves up to 10 THz could \nbe generated. The spin wave intensity is higher when \nfs laser fluence is lower. Increasing film thickness to \n100 nm increases the THz spin wave intensity slightly, \nand the a llowed threshold fluence for THz spin wave \ngeneration. From the spectra of Fe and Co, one can \ninfer that the THz spin wave intensity is slightly higher \ncompared to Ni, since these two metals have higher T C \nand recovered magnetization fractions (> 95%).  \n \n \nSUPP. FIG. S3. THz spin wave generation in t ype I metallic magnetic thin films. Effect of laser fluence on the bandwidth \nand intensity of THz spin wave s generated in  a(i) 20nm and a(ii) 100 nm thick Fe,  b(i) 20nm  and a(ii) 100 nm thick Co, and  \nc(i) 20nm  and c(ii) 100 nm thick Ni films each illuminated under 500 fs single laser pulse.  \n \nSUPP. FIG. S4. THz spin wave generation in t ype II metallic magnetic thin films. Effect of laser fluence on the bandwidth \nand intensity  of THz spin wave generated in a(i) 20nm and  a(ii) 100 nm thick Gd , b(i) 20nm and  b(ii) 100 nm thick Ni 2MnSn \nfilms under illumination of 500 fs single laser pulse.  \nComparison of the spin wave spectra in 20 nm and \n100 nm Gd  and Ni 2MnSn thin films show that THz spin \nwaves up to 8 THz could  be generated in such thin \nfilms by mani pulation of magnetization with 5 00 fs \nGaussian laser pulse. One can tune the intensity and the \nbandwidth of the generated THz spin waves by \nchanging t he film thickness and laser pulse fluence. \nLaser pulse s with sub -picosecond  width s have minimal \neffect on the bandwidth and the intensity of the THz \nspin wave signal . According to Supp. Fig. S4, the \nbandwidth and the intensity of the THz spin waves  is \nhigher in both 100 nm Gd and Ni 2MnSn thin films \ncompared to 20 nm films since thicker films stay \nmagnetized longer. Hence, decreasing the laser fluence \nhelps generate broadband THz spin waves with higher \nintensity.  Comparison of Supp. Fig. S4 and S3 shows \nthat using type I films results in generation of higher \nintensity and broader band THz spin waves, since type I metals preserve their magnetization for longer after \ninteracting with the pulse.  \nREFERENCES  \n[1] B. Koopmans, G. Malinowski, F. Dalla Longa, D. \nSteiauf, M. Fähnle, T. Roth, M. Cinchetti, M. \nAeschlimann,  Explaining the paradoxical diversity of \nultrafast  laser -induced demagnetization,  Nat. Mater., 9, \n259-265 (2010).  \n[2] J. Kimling, J. Kimling, R. Wilson, B. Hebler, M. \nAlbrecht, and D. G. Cahill, Ultrafast demagnetization of \nFePt: Cu thin films and the role of magnetic heat \ncapacity,  Phys . Rev. B  90, 224408 ( 2014 ). \n[3] Z. Lin,  L. V. Zhigilei, and V. Celli, Electron -phonon \ncoupling and electron heat capacity of metals under \nconditions of strong electron -phonon nonequilibrium,  \nPhys . Rev.  B 77, 075133 ( 2008 ). \n[4] E. Beaurepaire, J. -C. Merle, A. Daunois, and J. -Y. Bigot, \nUltrafast spin dynamics in  ferromagnetic nickel,  Phys.  \nRev. Lett. 76, 4250  (1996 ). \n"    },    {        "title": "1804.05391v1.Re_orientation_of_easy_axis_in___varphi_0__junction.pdf",        "content": "arXiv:1804.05391v1  [cond-mat.supr-con]  15 Apr 2018Re-orientation of easy axis in ϕ0junction\nYu. M. Shukrinov1,2, A. Mazanik1,3, I. R. Rahmonov1,4, A. E. Botha5, A. Buzdin6\n1BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n2Dubna State University, Dubna, Russian Federation\n3Moscow Institute of Physics and Technology, Institutsky la ne 9,\nDolgoprudny, Moscow region, 141700, Russian Federation\n4Umarov Physical Technical Institute, TAS, Dushanbe, 73406 3, Tajikistan\n5Department of Physics, University of South Africa, Private Bag X6, Florida 1710, South Africa\n6University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence C edex, France\n(Dated: November 9, 2018)\nWe study theoretically a dynamics of ϕ0junction with direct coupling between magnetic moment\nandJosephson currentwhichshows features close toKapitza pendulum. Wehavefoundthatstarting\nwith oscillations along z-axis, the character of magnetization dynamics changes cru cially and stable\nposition of magnetic moment /vector mis realized between z−andy-axes depending on parameters of the\nsystem. Changes in critical current and spin-orbit interac tion lead to the different stability regions\nfor magnetization. An excellent agreement between analyti cal and numerical results is obtained for\nlow values of the Josephson to magnetic energy ratio.\nA ferromagnet makes a strong influence on Josephson\njunction (JJ), particularly, superconductor-ferromagnet-\nsuperconductor (SFS) JJ oscillates between 0 −and\nπ−junctions as thickness of ferromagnet increases [1–\n3]. In a closed superconducting loop with π-junction a\nspontaneous circulating current appears without applied\nmagnetic flux [4]. Very special situation is possible when\nthe weak link is a non-centrosymmetric magnetic metal\nwith broken inversion symmetry like MnSiorFeGe. In\nthis case, Rashba type spin-orbit coupling leads to a gen-\neral current-phase relation j=jcsin(ϕ−ϕ0), where ϕ0\nis proportional to the strength of the broken inversion\nsymmetry magnetic interaction [5–7]. An important is-\nsue is that the ϕ0-junction provide a mechanism of a\ndirect coupling between suppercurrent (superconducting\nphase) and magnetic moment.\nThe mechanical analog with a simple pendulum with\napplied torque occurred to be very useful to get a bet-\nter insight into a physics of Josephson devices [8]. Here\nwe introduce an another analog between magnetic ϕ0-\njunction and a pendulum with oscillating point of sus-\npension. A particle moving simultaneously in the per-\nmanent field and in the field oscillating with a high fre-\nquency, demonstratesa veryinteresting feature: new sta-\nbility points appear at some parameters of the particle\nand fields [9, 10]. The system, where such feature is\nrealized, usually is called as Kapitza pendulum. Partic-\nularly, in pendulum with vibrating point of suspension\n(pivot point), the external sinusoidal force can invert the\nstability position of the pendulum [9]. In this case the\nbottom equilibrium position is nolongerstable. Any tiny\ndeviation from the vertical increases in amplitude with\ntime. In the presence of a periodic drive, the unstable\nfixed point can become dynamically stable. Kapitza pro-\nvided an analytical insight into the reasons of stability\nby splitting the motion into “fast” and “slow” variables\nand by introducing an effective potential. This resultwasfirst obtained by Kapitza in the high frequency limit.\nBy averaging the classical equations of motion over the\nfast oscillations of the drive, Kapitza found that the “up-\nper” extremum becomes stable for large enough driving\namplitudes. This pioneering work initiated the field of\nvibrational mechanics, and the Kapitza method is used\nfor description of periodic processes in different physical\nsystems such as atomic physics, plasma physics, and cy-\nberneticalphysics(see[11, 12] andreferencestherein). In\nnonlinearcontroltheorythe Kapitzapendulum isused as\nan example of a parametric oscillator that demonstrates\nthe concept of “dynamic stabilization”.\nIn this paper we study dynamics of ϕ0junction with\ndirect coupling between magnetic moment and Joseph-\nson current and found that applying the external volt-\nage may lead to the different stability regions for mag-\nnetization. We have found the manifestation of Kapitza\npendulum features in the ϕ0-junction [13, 14]. Devel-\noping our previous approach [15], we have investigated\nthe effect of superconducting current on the dynamics of\nmagnetic momentum. We show that starting with oscil-\nlationsalong z-axis,the characterof /vector mdynamicschanges\ncrucially and stable position of /vector mbecomes between z−\nandy-axes depending on parameters of the system.\nThe ac Josephson effect provides an ideal tool to study\nmagnetic dynamics in a ϕ0–junction. To realize the ac\nJosephson effect, we apply the constant voltage Vto the\nconsidered ϕ0–junction. In such a case, the supercon-\nducting phase varies with time like ϕ(t) =ωJt, where\nωJ= 2eV//planckover2pi1is Josephson frequency [14, 16]. Dynamics\nofconsidered system is described by the Landau-Lifshitz-\nGilbert equation\ndM\ndt=γHeff×M+α\nM0/parenleftbigg\nM×dM\ndt/parenrightbigg\n(1)2\nwith effective magnetic field Heffin the form [14]\nHeff=K\nM0/bracketleftbigg\nΓsin/parenleftbigg\nωt−ϕ0/parenrightbigg\n/hatwidey+Mz\nM0/hatwidez/bracketrightbigg\n(2)\nwhereγ– gyromagnetic ratio, α– phenomenological\ndamping constant, M0=/bardblM/bardbl,Miare components of\nM, and we will use normalized units below mi=Mi\nM0.\nHereϕ0=rMy\nM0,r=lυso/υF,l= 4hL//planckover2pi1υF,L–length\nofFlayer,h–exchange field of the Flayer, Γ = Gr,\nG=EJ/(KV),EJ= Φ0Ic/2πis the Josephson energy.\nHere Φ 0is the flux quantum, Icis the critical current,\nυFis Fermi velocity, the parameter υso/υFcharacter-\nizes a relative strength of spin-orbit interaction, Kis the\nanisotropic constant, and Vis the volume of the Flayer.\nTo investigate the dynamics of considered system numer-\nically, wewritetheequation(1)inthedimensionlessform\n(see system of equations (1) in the Supplement ). In that\nsystem time is normalized to the inverse ferromagnetic\nresonance frequency ωF=γK/M 0: (t→tωF), soωis\nnormalized to the ωF.\nFirstwepresentresultsofnumericalsimulationsofsys-\ntem according to equation (1). Figure 1 shows dynam-\nics of magnetization components mzandmyat different\nparameter Gdemonstrating the re-orientation of the os-\ncillations around zaxis to the oscillations around yaxis.\nWith increase in G, the component mygoes from zeroth\ntomy= 1. Figure 1(a) demonstrates the time depen-\ndence of mzatG= 5π. The character of oscillations\nin the beginning and in the middle of time interval is\nshown in the insets. We see that the average value of mz\ndeviates from one. Figure 1(b) shows the corresponding\noscillation of my. Figure 1(c) demonstrates a stabiliza-\ntion ofmyoscillations around of some average values\nofmybetween z- andy- directions with increase in G,\nshows the re-orientation of oscillations at three values of\nG= 10π,20π,50π. With an increase in G, time of re-\norientation from z-direction to y-direction is decreased\nessentially. At enough large value of G(seeG= 400π\nin Fig.1(d)), the average value of myis getting close to\n1. The oscillations show periodically splashing related to\nthe Josephson frequency. Inset demonstrates the jumps\nofmyfrom the value my= 1 with Josephson frequency.\nThe amplitude of jumps decreases in time and it is get-\nting smaller and in a shorter time with an increase in G.\nWe note that the effect of rleads to the similar features\nas the changes in G. So, 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 [9]. Varia-\ntion of average myas a function of frequency ωandG\nis shown in Fig.2. We see that an increase in Gmakes\norientation of myalong y-axis stable, but frequency de-\npendence differs from characteristic Kapitza pendulumn\nbehavior, as we can see below from the results presented\nin Fig.4. The system of equations (1) describing dynam-\ntmy\n024681000.20.40.60.81\n10 π20 π50 π(c)\ntmy\n010 20 3000.20.40.60.81(d)\ntmy\n0 1 200.51\nFigure 1. (a) Dynamics of mzcomponent at G= 5π,r=\n0.1; (b) The same for my; (c) Effect of G at r= 0.5. The\nnumbers indicate the nvalue in G=nπ. (d) Dynamics of\nmycomponent at G= 400π,r= 0.1. In all figures α= 0.1,\nω= 5.\nGω\n  \n05010015020020406080100\n0.20.40.60.8\nmy\nFigure 2. The ω−G-diagram for averaged myshown bycolor.\nics of magnetic moment in angular variables mz= cosθ,\nmx= sinθcosφ,my= sinθsinφcan be written as\n\n\n˙φ=cosθ\n1+α2−\nGr\n1+α21\nsinθ[cosθsinφ−αcosφ]sin[ωτ−rsinθsinφ],\n˙θ=−αsin2θ\n2(1+α2)+\nGr\n1+α2[αcosθsinφ+cosφ]sin[ωτ−rsinθsinφ].\n(3)\nIn the case of absence of coupling between the magnetic\nmoment and the Josephson junction at G= 0\n/braceleftBigg˙φ=cosθ\n1+α2,\n˙θ=−αsin2θ\n2(1+α2).(4)\nthe solution has a form\nθ(τ) = ArcTan/bracketleftbigg\nexp/braceleftbigg\nlogtanθ0−ατ\n1+α2/bracerightbigg/bracketrightbigg\n,(5)3\nwhere tan θ0is considered to be positive for simplicity.\nSo, the characteristic scale in time to order magnetic\nmoment along the easy axes is τ⋆=1+α2\nα. To real-\nize the Kapitza pendulum we should investigate the case\nω≫2π\nτ⋆=2πα\n1+α2, whereωis a frequency of the external\nfast-varying field.\nLet us consider that\nθ→Θ+ξ,\nφ→Φ+η.(6)\nHere Θ and Φ describe slow movement, while ξandηare\ncoordinates for fast-varying movement. Conditions for\nξandηare discussed in the Supplement. We consider\nθ= Θ,φ= Φ, where averaging is taken over the period\nof the fast-varying force T=2π\nω.\nThe system of equations for slow movement has a form\n(see Supplement)\n\n\n˙Φ =cosΘ\n1+α2−(Gr)2rα\n2ω(1+α2)2·1\nsinΘ[cosΘsinΦ\n−αcosΦ]/braceleftbig\n1−sin2Θsin2Φ/bracerightbig\n,\n˙Θ =−αsin2Θ\n2(1+α2)+(Gr)2rα\n2ω(1+α2)2[αcosΘsinΦ\n+cosΦ]/braceleftbig\n1−sin2Θsin2Φ/bracerightbig(7)\nTherearesomeequilibriumpointsofthesystemfollowing\nfrom˙Θ = 0,˙Φ = 0.\nFigure 3. Demonstration of equilibrium points of equation\n(18). Arrows show stable points, other two are unstable.\nThe first pair is located on the equator\nΘ0=π/2,Φ0=π/2 and Θ 0=π/2,Φ0= 3π/2.(8)\nLinearization of (18) after substitution Φ = Φ 0+δφ,Θ =\nΘ0+δθgives\n/braceleftBigg˙δφ=1\n1+α2δθ,\n˙δθ=α\n1+α2δθ.(9)\nIt’s clear that δθ∼eατ\n1+α2, so these points are unstable.\nUsing (18) we find that the equilibrium points at Φ 0=\nπ\n2are described by equation\nsinΘ0=−1+/radicalbig\n1+4β2\n2β, (10)whereβ=(Gr)2rα\n2ω(1+α2), which can be approximated as\nsinΘ0=βat small β. It has two solutions in the in-\nterval 0 ≤Θ0≤π. Note that at Φ 0=3π\n2we have\nsinΘ0=1−√\n1+4β2\n2βwhich leads to the negative sinΘ 0,\nbut Θ is always positive. So, there are no any stable\npoints at Φ 0=3π\n2.\nTo find out if these points are stable we have to make\nlinearization of equation (18) near these points and test\neigenvaluesoflinearizedsystem. Thestraightforwardcal-\nculation shows, that the real parts of eigenvalues of the\ncorresponding system are always negative. It means that\nthe second pair of points are always stable (see Supple-\nment).\nResults of numerical calculations of the averaged my\nas a function Gat different frequencies are presented in\nFig. 4. We see that system reminiscent the Kapitza pen-\ndulum behavior: averaged mycomponent characterizing\nchanges of stability direction is growing with G. Char-\nacter of behavior depends essentially on the frequency of\nthe fast-varying field getting very sharp at small ω. Fig-\nure 4 also compares the my(G) dependence obtained by\nanalytical and numerical calculations. We see results at\nthree frequencies: ω= 0.5, 20 and 70. Analytical de-\npendence calculated according to the formula (19). Two\nmethods are used for numerics: numerics-1 is based on\nstandard program “Mathematica” (shown by triangles,\ncalculated for ω= 70) and numerics-2 presents results of\ndirect solution of system (1) based on the Runge-Kutta\nfourth order. Both numerics gives practically the same\nresults. An excellent agreement between analytical and\nnumerical results is obtained at low G, depending on fre-\nquency of the fast-varying field. For high frequencies the\ncoincidence is still good at rather large G.\nGmy\n50 100 15000.20.40.60.81\nThin lines - theoretical \nDiamonds - numerics 1, ω=70\nCircles - numerics 2, ω=70\nGradients - numerics 2, ω=20\nSquare - numerics 2, ω=0.50.5\n7020\nFigure 4. Comparison the data of theoretical and numerical\ncalculations for G-dependence of the averaged myat different\nfrequencies and r= 0.5 andα= 1. Thin lines show theoreti-\ncal G-dependence of myaccording to formula (19). Symbols\nshow numerical results, thick lines guide the eyes.\nTable shows the values of the averaged myobtained by4\nTable I. myfrom the theory and the numerical results\nG Θ0analytic mynumerics-1 numerics-2\n15.7 1.462 0 .109 0 .108 0 .109\n31.4 0.387 0 .369 0 .376 0 .378\n47.1 0.663 0 .577 0 .611 0 .616\n62.8 0.859 0 .689 0 .748 0 .756\n157 1 .272 0 .817 0 .945 0 .956\nanalytic and numerical calculations at different G. The\ndifference in values of the averaged myobtained by all\nthree used methods is rather small. There is an essential\ndifference between the original Kapitza pendulum and\nour system. In Kapitza pendulum the stability of new\nequilibrium point is determined by amplitude and fre-\nquency of external force. In our case, two new points are\nalways stable and their positions on the sphere is deter-\nmined by parameters of the system.\nIn our case, twonew points arealwaysstable and there\npositions on the sphere is determined by parameters of\nthe system. Figure 5 shows vector fields in the plane\n0≤Φ≤2π, 0<Θ< πaccording to the equation (18)\nat two different values of G: 5πand 50π. It demonstrates\nthat with an increase in Gtwo equilibrium positions are\napproaching the third unstable one Φ =π\n2, Θ = 0. This\nresult is in agreement with (19), which shows that an\nincrease in Gleads to the increase in β, and to the ap-\nproaching of sinΘ 0to one.\nFigure 5. Phase planes (18) at G= 10π(left) and G=\n50π(right). Here r= 0.5,α= 1,ω= 70. Red points\nare stable equilibrium points which are calculated from (19 ),\nblack points are unstable (8). With increase in Gstable red\npoints are approaching unstable ones Φ 0=π/2, Θ0=π/2.\nThere is another interesting phenomena which is real-\nized in our case. With changing parameters of the sys-\ntem, the stableequilibrium positionsof“slow”movement\nare approaching unstable one Φ =π\n2, Θ =π/2. Whenthe distance between them is getting comparable with an\namplitude of “fast” one, all three special points of “slow”\nsystem effectively are merged.\nIn conclusion, we have demonstrated that the coupling\nbetween magnetic moment and Josephson phase differ-\nence inϕ0junction may effectively lead to the strong\nre-orientation of magnetic easy axis under the applied\nvoltage. This serves a manifestation of the Kapitza-like\npendulum behavior and open a new way to the magneti-\nzation control by a superconducting current.\nThe authors thank K. Sengupta, K. Kulikov, K. Sen-\ngupta, I. Bobkova, A. Bobkov for useful discussions.\nThe study was partially funded by the RFBR (research\nproject 18-02-00318), and the SA-JINR collaborations.\nYu. M. S. gratefully acknowledges support from the Uni-\nversity of South Africa’s visiting researcher program.\n[1] V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov, A. V.\nVeretennikov, A. A. Golubov, and J. Aarts, Phys. Rev.\nLett.86, 2427 (2001).\n[2] V. A. Oboznov, V. V. Bol’ginov, A. K. Feofanov, V. V.\nRyazanov, A. I. Buzdin, Phys. Rev. Lett. 96, 197003\n(2006).\n[3] J. W. A. Robinson, S. Piano, G. Burnell, C. Bell, M. G.\nBlamire, Phys. Rev. Lett. 97, 177003 (2006).\n[4] L. N. Bulaevskii, V. V. Kuzii, A. A. Sobyanin, JETP\nLetters25, 314 (1977).\n[5] A. Buzdin, Phys. Rev. Lett. 101, 107005 (2008).\n[6] I. V. Krive, A. M. Kadigrobov, R. I. Shekhter, and M.\nJonson, Phys. Rev. B 71, 214516 (2005).\n[7] A. A. Reynoso, G. Usaj, C. A. Balseiro, D. Feinberg, and\nM. Avignon, Phys. Rev. Lett. 101, 107001 (2008).\n[8] Antonio Barone and Gianfranco Paterno, Physics and\nApplications of the Josephson Effect (John Wiley and\nSons Inc. 1982).\n[9] P. L. Kapitza, Soviet Phys. JETP 21, 588–592 (1951); P.\nL. Kapitza, Usp. Fiz. Nauk, 44, 7–15 (1951).\n[10] L. D. Landau, E. M. Lifshitz, Mechanics. Vol. 1 (1st ed.)\n(Pergamon Press. ASINB0006AWV88 1960).\n[11] Roberta Citro, Emanuele G. Dalla Torre, Luca\nD’Alessiod, Anatoli Polkovnikov, Mehrtash Babadi,\nTakashi Oka, Eugene Demler, Annals of Physics, 360,\n694–710 (2015).\n[12] Erez Boukobza, Michael G. Moore, Doron Cohen, and\nAmichay Vardi Phys. Rev. Lett. 104, 240402 (2010).\n[13] A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n[14] F. Konschelle, A. Buzdin, Phys. Rev. Lett. 102, 017001\n(2009).\n[15] Yu. M. Shukrinov, I. R. Rahmonov, K. Sengupta, and A.\nBuzdin, Appl. Phys. Lett. 110, 182407 (2017).\n[16] B. D. Josephson, Superconductivity (in two volumes),\nVol. 1, Chap. 9. (R. D. Parks, Marcel Dekker, New York,\n1968),5\nSupplement for paper Re-orientation of easy axis in ϕ0junction\nYu. M. Shukrinov1,2, A. Mazanik1,3, I. R. Rahmonov1,4, A. E. Botha5, A. Buzdin6\n1BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n2Dubna State University, Dubna, 141980, Russia\n3Moscow Institute of Physics and Technology, Institutsky lane 9, D olgoprudny, Moscow region, 141700, Russia\n4Umarov Physical Technical Institute, TAS, Dushanbe, 734063, T ajikistan\n5Department of Physics, University of South Africa, Private Bag X6 , Florida 1710, South Africa\n6University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence Cede x, France\nMETHOD\nFull system of equations for magnetization:\n\n\n˙mx=−mzmy+Γmzsin[ωτ−ϕ0]+α[my˙mz−mz˙my],\n˙my=mzmx+α[mz˙mx−mx˙mz],\n˙mz=−Γmxsin[ωτ−ϕ0]+α[mx˙my−my˙mx].(11)\nThemx,y,z=Mx,y,z/M0satisfytheconstraint/summationtext\nα=x,y,zm2\nα(t) = 1. Inthissystemofequationstimeisnormalizedto\nthe inverse ferromagnetic resonancefrequency ωF=γK/M 0: (t→tωF),γis the gyromagneticratio, and M0=/bardblM/bardbl.\nSystem of equations of magnetic moment motion in angular variables mz= cosθ,mx= sinθcosϕ,my= sinθsinϕ\nhas a form:\n/braceleftBigg˙φ=cosθ\n1+α2−Gr\n1+α21\nsinθ[cosθsinφ−αcosφ]sin[ωτ−rsinθsinφ],\n˙θ=−αsin2θ\n2(1+α2)+Gr\n1+α2[αcosθsinφ+cosφ]sin[ωτ−rsinθsinφ].(12)\nWe consider that\nθ→Θ+ξ,\nϕ→Φ+η.(13)\nHere Θ and Φ describe slow movement, while ξandηare coordinates for fast-varying movement. We consider θ= Θ,\nφ=φ, where averaging is taken over the period of the fast-varying for ceT=2π\nω.\nUsing (13) in the system of equations (equ. (3) in main text), and ex panding on the ξandηtill members of first\norder, we get\n\n\n˙Φ+ ˙η≈cosΘ\n1+α2−ξsinΘ\n1+α2+Fφ(Θ,Φ,τ)+\n∂Fφ(Θ,Φ,τ)\n∂Θξ+∂Fφ(Θ,Φ,τ)\n∂Φη,\n˙Θ+˙ξ=αsin2Θ\n2(1+α2)+ξαcos2Θ\n(1+α2)+Fθ(Θ,Φ,τ)+\n∂Fθ(Θ,Φ,τ)\n∂Θξ+∂Fθ(Θ,Φ,τ)\n∂Φη.(14)\nwhere\n/braceleftBigg\nFφ(Θ,Φ,τ) =−Gr\n1+α21\nsinΘ[cosΘsinΦ −αcosΦ]sin[ ωτ−rsinΘsinΦ] ,\nFθ(Θ,Φ,τ) =Gr\n1+α2[αcosΘsinΦ+cosΦ]sin[ ωτ−rsinΘsinΦ] .(15)\nFor the fast movement we have\n/braceleftBigg\n˙η=−Gr\n1+α21\nsinΘ[cosΘsinΦ −αcosΦ]sin[ ωτ−rsinΘsinΦ] ,\n˙ξ=Gr\n1+α2[αcosΘsinΦ+cosΦ]sin[ ωτ−rsinΘsinΦ] ,(16)\nbecause other terms are proportional to ξorη, while ˙η∼ωη≫η,˙ξ∼ωξ≫ξ.\nBy direct integration we see that\n/braceleftBigg\nη=Gr\nω1\n1+α21\nsinΘ[cosΘsinΦ −αcosΦ]cos[ ωτ−rsinΘsinΦ] ,\nξ=−Gr\nω1\n1+α2[αcosΘsinΦ+cosΦ]cos[ ωτ−rsinΘsinΦ] .(17)6\nUsing these ξ,ηin equation 14 and do averaging over T=2π\nω, we are obtaining the system of equations for slow\nmovement\n\n\n˙Φ =cosΘ\n1+α2−(Gr)2rα\n2ω(1+α2)2·1\nsinΘ[cosΘsinΦ\n−αcosΦ]/braceleftbig\n1−sin2Θsin2Φ/bracerightbig\n,\n˙Θ =−αsin2Θ\n2(1+α2)+(Gr)2rα\n2ω(1+α2)2[αcosΘsinΦ\n+cosΦ]/braceleftbig\n1−sin2Θsin2Φ/bracerightbig(18)\nSTABLE POINTS\nLinearization of (18) after substitution Φ = Φ 0+δφ,Θ = Θ 0+δθat the equilibrium points\nΦ0=π\n2,\nΘ0= ArcSin−1+/radicalbig\n1+4β2\n2βor Θ0=π−ArcSin−1+/radicalbig\n1+4β2\n2β(19)\ngives\n\n\n˙δφ=−α\n1+α2δφ+/braceleftbigg√\n1+4β2\nβ2(1+α2)/bracerightbigg\nδθ,\n˙δθ=√\n1+4β2−1\n2β(1+α2)δφ+/braceleftbigg\nα(−1−4β2+√\n1+4β2)\n2β2(1+α2)/bracerightbigg\nδθ.(20)\nEigenvalues of this system are\nλ1=−/radicalBig\n4α2β4+8α2β2−4α2/radicalbig\n4β2+1β2−2α2/radicalbig\n4β2+1+2α2−32β4+8/radicalbig\n4β2+1β2−8β2\n4(α2+1)β2+\n+−6αβ2+α/radicalbig\n4β2+1−α\n4(α2+1)β2,\nλ2=/radicalBig\n4α2β4+8α2β2−4α2/radicalbig\n4β2+1β2−2α2/radicalbig\n4β2+1+2α2−32β4+8/radicalbig\n4β2+1β2−8β2\n4(α2+1)β2+\n+−6αβ2+α/radicalbig\n4β2+1−α\n4(α2+1)β2.(21)\nλ1andλ2are negative. So, these points are stable. Let’s test it. The denom inator is always positive. α >0,β >0.\nA=−α(1+6β2−/radicalbig\n4β2+1),\nB=/radicalBig\n4α2β4+8α2β2−4α2β2/radicalbig\n4β2+1−2α2/radicalbig\n4β2+1+2α2−32β4+8β2/radicalbig\n4β2+1−8β2(22)\nWe see, that imaginary part may occur only when Bis imaginary. If Bis imaginary, the points are stable because\nAis always negative. Let Bbe the real, in this case we can find instability ( λ1orλ2are positive) if it does exist.\nA=−α(1+6β2−/radicalbig\n4β2+1)<0 always. We have\n|A|2−|B|2= (α2+1)8β2/bracketleftBig\n1+4β2−/radicalbig\n1+4β2/bracketrightBig\n≥0. (23)\nIt means that\n|A| ≥ |B|,\n−|A|+|B| ≤0.(24)\nSo, these points are stable equilibrium point.7\nCONDITIONS FOR THE THEORY\nThere are two ways to formulate conditions when our theory works . In the first way we imply that adding to angles\nΘ and Φ, arriving from the fast movement are small in comparison with 1. So, we need to estimate ξandηon the\ntrajectory of slow movement. To do it, we need Θ( τ) and Φ(τ). From the phase portrait (see Fig. 5 in the main text)\nwe see that starting from an arbitrary point on the sphere, the ma gnetic moment tends to line up towards to one of\nthe stable points, so the trajectory is attracted by these points . We may evaluate ξandηnear the equilibrium points\nand make them to be small in comparison to 1.\n\n\nη=Gr\nω1\n1+α21\nsinΘ[cosΘsinΦ −αcosΦ]cos[ ωτ−rsinΘsinΦ]/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΘ=Θ0, Φ=Φ 0∼Gr\nω1\n1+α2cosΘ0\nsinΘ0≪1,\nξ=−Gr\nω1\n1+α2[αcosΘsinΦ+cosΦ]cos[ ωτ−rsinΘsinΦ]/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΘ=Θ0, Φ=Φ 0∼Gr\nωαcosΘ0\n1+α2≪1.(25)\nThe second approach is to calculate the fast movement changes mx(τ),my(τ),mz(τ) near equilibrium points and\nmake terms arriving from the fast movement slightly deflect the mom ent from the equilibrium points. Let’s consider\nmy(τ) near stable my0, Φ0=π/2, sinΘ 0=−1+√\n1+4β2\n2β,ξandηwritten in (17). So, we have\nmy= sin(Θ 0+ξ)sin(Φ 0+η) =\n= [sinΘ 0cosξ+cosΘ 0sinξ][sinΦ 0cosη+cosΦ 0sinη] =\n= sinΘ 0cosξcosη+cosΘ 0sinξcosη.(26)\nWe are interested in the situation when the fast movement slightly ch anges the slow movement. It means that we\nshould imply cosξcosη≈1 andsinξcosη≈0 when the averaging is taken over the period 2 π/ω. It may be read as\ncosξcosη=1\n2π/ω/integraldisplay2π/ω\n0dτcos/bracketleftbigg\n−Gr\nωαcosΘ0\n1+α2cos[ωτ−rsinΘ0]/bracketrightbigg\n×\n×cos/bracketleftbiggGr\nω1\n1+α2cosΘ0\nsinΘ0cos[ωτ−rsinθsinφ]/bracketrightbigg\n=\n=1\n2π/integraldisplay2π\n0dycos/bracketleftbiggGr\nωαcosΘ0\n1+α2cosy/bracketrightbigg\ncos/bracketleftbiggGr\nω1\n1+α2cosΘ0\nsinΘ0cosy/bracketrightbigg\n=\n=1\n2π/integraldisplay2π\n0dycos[Acosy]cos[Bcosy] =1\n2[J0(A−B)+J0(A+B)] =\n=1\n2/bracketleftbigg\nJ0/parenleftbiggGr\nωαcosΘ0\n1+α2−Gr\nω1\n1+α2cosΘ0\nsinΘ0/parenrightbigg\n+J0/parenleftbiggGr\nωαcosΘ0\n1+α2+Gr\nω1\n1+α2cosΘ0\nsinΘ0/parenrightbigg/bracketrightbigg\n≈1.(27)\nHereA=Gr\nωαcosΘ0\n1+α2,B=Gr\nω1\n1+α2cosΘ0\nsinΘ0. In the same way one may notice that cosξsinη=sinξcosη= 0 (ξ(τ) and\nη(τ) are 2π/ω-periodical functions in τ).\nAlso, for zdirection we have to calculate new condition\nmz= cos(Θ 0+ξ) = cosΘ 0cosξ+sinΘ 0sinξ (28)\nSo, we have to imply cosξ≈1,sinξ≈0.\ncosξ=1\n2π/ω/integraldisplay2π/ω\n0dτcos/bracketleftbigg\n−Gr\nωαcosΘ0\n1+α2cos[ωτ−rsinΘ0]/bracketrightbigg\n=\n=J0/parenleftbiggGr\nωαcosΘ0\n1+α2/parenrightbigg\n≈1.(29)\nsinξ=1\n2π/integraldisplay2π\n0dysin(Acosy) = 0. (30)\nForxdirection\nmx= sin(Θ 0+ξ)cos(Φ 0+η) =\n= sinΘ 0cosξsinη+cosΘ 0sinξsinη.(31)8\nSo,\n/vextendsingle/vextendsinglesinξsinη/vextendsingle/vextendsingle=\n=1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingleJ0/parenleftbiggGr\nωαcosΘ0\n1+α2−Gr\nω1\n1+α2cosΘ0\nsinΘ0/parenrightbigg\n−J0/parenleftbiggGr\nωαcosΘ0\n1+α2+Gr\nω1\n1+α2cosΘ0\nsinΘ0/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1.(32)\nSo, new conditions are\n\n\n1\n2[J0(A−B)+J0(A+B)]≈1,\nJ0(A)≈1,\n1\n2|J0(A−B)−J0(A+B)| ≪1.(33)\nOne should notice that conditions (33) are fulfilled, when AandBare small. It implies\n\n\nA=Gr\nωαcosΘ0\n1+α2≪1,\nB=Gr\nω1\n1+α2cosΘ0\nsinΘ0≪1.(34)\nWe see that these conditions are coincide with (25)."    },    {        "title": "2306.00174v1.Magnetization_dynamics_in_a_three_dimensional_interconnected_nanowire_array.pdf",        "content": "Magnetization dynamics in a three-dimensional interconnected nanowire array\nRajgowrav Cheenikundil,1Massimiliano d’Aquino,2and Riccardo Hertel1\n1Universit´ e de Strasbourg, CNRS, Institut de Physique et Chimie des Mat´ eriaux de Strasbourg, F-67000 Strasbourg, France∗\n2Department of Electrical Engineering and ICT, University of Naples Federico II, Naples, Italy\n(Dated: June 2, 2023)\nThree-dimensional magnetic nanostructures have recently emerged as artificial magnetic material\ntypes with unique properties bearing potential for applications, including magnonic devices. Inter-\nconnected magnetic nanowires are a sub-category within this class of materials that is attracting\nparticular interest. We investigate the high-frequency magnetization dynamics in a cubic array of\ncylindrical magnetic nanowires through micromagnetic simulations based on a frequency-domain\nformulation of the linearized Landau-Lifshitz-Gilbert equation. The small-angle high-frequency\nmagnetization dynamics excited by an external oscillatory field displays clear resonances at distinct\nfrequencies. These resonances are identified as oscillations connected to specific geometric features\nand micromagnetic configurations. The geometry- and configuration-dependence of the nanowire\narray’s absorption spectrum demonstrates the potential of such magnetic systems for tuneable and\nreprogrammable magnonic applications.\nINTRODUCTION\nOver many years, cylindrical soft-magnetic nanowires\nhave been the subject of intense research in micro- and\nnanomagnetism [1]. Their magnetic properties have been\nanalyzed regarding the structure and motion of domain\nwalls [2–5], which, depending on the nanowire thickness,\ncan display remarkable stability and high propagation ve-\nlocities [6, 7]. Magnetic nanowires and nanotubes have\nalso attracted interest within the magnonics community\n[8] and in the nascent field of curvilinear micromagnetism\n[9], as it was found that, due to their rounded cylindrical\nouter shape, these geometries can exhibit non-reciprocal\nspin-wave propagation [10]. In addition to studies on\nindividual nanowires, research has also been conducted\non the collective behavior of magnetostatically interact-\ning nanowire arrays [11, 12]. Recently, structurally even\nmore complex types of ensembles have been investigated\nin the form of arrays of interconnected nanowires [13].\nIn these systems, the intersection sites of crossing wires\nintroduce additional geometric and micromagnetic fea-\ntures, which provide further potential for functionali-\nties in future magnetic nanodevices [14–16]. Such in-\nterconnected nanowires arrays can be fabricated, e.g., by\noblique ion irradiation of a polymer matrix followed by\na filling of the generated pores with magnetic materi-\nals through electrodeposition [17], which results in ex-\ntended, irregular networks with statistically distributed\nspacing between nanowires. Alternatively, by means of\nlayer-by-layer nanofabrication techniques such as FEBID\n(focussed electron-beam induced deposition), it is possi-\nble to generate smaller but highly regular arrays of con-\nnected magnetic nanowires [18–20].\nExtending the research from studies on single\nnanowires to interconnected arrays not only expands the\nknowledge of the magnetic properties of nanowires but\nalso contributes to furthering the more general topic\nof three-dimensional (3D) nanomagnetism—an emergingfield of research [21, 22] which aims at identifying the im-\npact of nanoscale 3D geometric features on the magneti-\nzation. The ability to fabricate arbitrarily shaped nano-\nmagnets and nanoarchitectures [19, 23, 24] may open up\na plethora of possibilities, which could ultimately enable\nmagnetic materials with tailored properties governed by\ntheir geometric features. Potential applications of 3D\nnanomagnetic materials are foreseen, e.g., in the field of\nmagnonics [25], high-density data storage, and neuromor-\nphic computing [26].\nFIG. 1: Perspective view on the studied sample geome-\ntry. The array consists of three orthogonal sets of equidistant\ncylindrical magnetic nanowires. The intersection points con-\nstitute a regular cubic lattice with a constant of 70 nm. The\nstructure’s total size is 420 nm ×350 nm ×280 nm.\nWithin this type of artificial magnetic material, regu-\nlar arrays of interconnected nanowires represent an inter-\nesting subcategory athat combines the unique micromag-\nnetic features of cylindrical nanowires with the magnonic\nproperties of 3D nanoarchitectures. In addition, since thearXiv:2306.00174v1  [cond-mat.mes-hall]  31 May 20232\ndominant magnetostatic effect tends to orient the mag-\nnetization along the nanowire axis, the arrays are akin to\nassemblies of Ising-type nanomagnets, which imparts to\nthese systems the characteristics of 3D artificial spin ices\n[27]. These qualities render arrays of magnetic nanowires\na complex class of systems whose study could pave the\nway toward the design of three-dimensional magnetic\nmetamaterials [28]. Understanding the high-frequency\noscillatory magnetization dynamics and its dependence\non the magnetic structure and the geometry could help\nidentify the potential of such material in the field of three-\ndimensional magnonics [25].\nMODEL SYSTEM\nAs an example of this type of artificial magnetic ma-\nterial, we consider here a regular array of intercon-\nnected nanowires, as illustrated in Fig. 1. We investi-\ngate, through advanced micromagnetic simulations, the\nfrequency-dependent small-angle oscillatory magnetiza-\ntion dynamics excited by a low-amplitude time-harmonic\nexternal field, and how this dynamic response depends on\nthe static magnetic configuration of the nanowire array.\nOur sample consists of 47 magnetic nanowires, arranged\nin a regular 5 ×4×3 grid. Twelve of them are aligned\nalong the xdirection, 15 are parallel to the yaxis, and\n20 are oriented in the zdirection. The intersection sites\nare located on a regular cubic grid, with a lattice con-\nstant of 70 nm. Each cylindrical nanowire has a diame-\nter of 12 nm and their length varies between 280 nm and\n420 nm, depending on the wire orientation. The geomet-\nric parameters of this array are compatible with what is\ncurrently achievable using modern FEBID nanofabrica-\ntion technologies. Accordingly, we use typical material\nparameters of FEBID-deposited amorphous cobalt, i.e.,\nan exchange constant of A= 15 pJ m−1, a spontaneous\nmagnetization of µ0Ms= 1.2 T, and zero magnetocrys-\ntalline anisotropy [29] ( µ0is the vacuum permeability).\nAs a model for the simulation, we use an unstructured\ntetrahedral finite-element mesh consisting of more than\n170 000 cells, and approximately 57 000 nodes. The edge\nlengths of the tetrahedra are smaller than 4 nm, which is\nbelow the material’s exchange length ls=p\n2A/µ 0M2s≃\n5.1 nm and yields sufficiently small elements to obtain a\ngood geometric approximation of the rounded nanowire\nshape.\nSIMULATION METHOD\nTo analyze the static and the oscillatory high-\nfrequency micromagnetic properties of the system, we\nemploy a two-step approach in which the two aspects\nare treated separately. In the first step, we use our\ngeneral-purpose finite-element micromagnetic simulationsoftware tetmag to determine the zero-field equilibrium\nstructure of the magnetization. The equilibrium state at\nzero field is not unique and depends on the sample’s mag-\nnetic history. Analogous to the typical situation in artifi-\ncial spin-ice lattices, a large number of quasi-degenerate\nconfigurations are possible, which can be characterized\nby the magnetic structures developing at the vertices, as\nwill be discussed in section .\nThe static equilibrium is simulated by numerically in-\ntegrating the Landau-Lifshitz-Gilbert (LLG) equation\n[30]\ndM\ndt=−1\n1 +α2\u0014\nγ(M×Heff) +α\nMs[M×(M×Heff)]\u0015\n(1)\nuntil convergence is reached, where αis the Gilbert\ndamping constant, γis the absolute value of the gyromag-\nnetic ratio, and Heffis the micromagnetic effective field\n[31] obtained from the variational derivative of the micro-\nmagnetic free energy functional, which contains contribu-\ntions from the ferromagnetic exchange, the magnetostatic\n(dipolar) interaction, and, if applicable, any static exter-\nnal field. In practice, the time integration of the LLG\nequation is continued until the value of the local torque\n|M×Heff|drops below a user-defined threshold at each\ndiscretization point. This first part of the calculation\nis a standard micromagnetic simulation task, which we\nperform reliably using numerical methods that we have\ndeveloped over many years and used in numerous previ-\nous works.\nIn a second step, once the static equilibrium con-\nfiguration is computed, we simulate the stationary\nand frequency-dependent response of the system to a\nweak externally applied sinusoidal field, δhext(t) =\nδˆhextexp(iωt). As a convention, we refer to vector\nfields by lower-case letters when they are represented\nin reduced units, i.e., hext=Hext/Ms,m=M/Ms,\netc. Moreover, we use variables prefixed by a δto de-\nnote oscillating, small-amplitude quantities, a zero sub-\nscript for static quantities, and the circumflex diacritic\nto denote complex oscillation amplitudes associated with\nsmall-amplitude fields. The applied time-harmonic field\nδhextexcites small-angle oscillations of the magnetization\naround the equilibrium state m0, i.e., m=m0+δm,\nwhich are accompanied by fluctuations of the effective\nfieldheff=heff\n0+δheff+δhext(we explicitly separate\nthe external applied field δhextfrom the remaining terms\nδheffinheff). Inserting this perturbative approach into\nthe LLG equation, assuming |δm| ≪ |m0|, results in\n−iω(αδˆm+m0×δˆm) =P ·\u0010\nh0δˆm−δˆheff−δˆhext\u0011\n(2)\nwhere h0=m0·heff\n0andP=I −m0⊗m0.\nIn discretized form, eq. (2) yields a linear system that\ncan be solved for δˆmfor any given frequency ωand exter-\nnal field amplitude δˆhext. In this linearized dynamics ap-3\nproach, the micromagnetic constraint of constant magni-\ntude,|m|= 1, translates into the condition δm·m0= 0.\nTherefore, the variations δmat each discretization point\ncontain only two degrees of freedom (not three), as only\nvariations perpendicular to m0are admissible. Corre-\nspondingly, assuming that the numerical model contains\nNdiscretization points, the system of equations to cal-\nculate ˆmneeds to be solved for 2 Nunknowns. The re-\nduction of variables from 3 Nto 2Nis achieved by apply-\ning suitable coordinate transforms through local rotation\noperators [32]. Note that the numerical solution of the\ndiscrete counterpart of eq. (2) would normally involve\na dense matrix with O(N2) dimensions, which becomes\nsoon unpractical even for a moderate number of compu-\ntational cells. However, we achieve a large-scale matrix-\nfree solution of eq. (2) by using an operator-based formal-\nism [32, 33], which allows to preserve the almost linear\nO(N) complexity of the effective-field calculation [34].\nThe solution of the linear system provides the\nfrequency-dependent dynamic oscillation profile of the\nmagnetization,\nδm(x, t) =δˆm(x)·exp(iωt), (3)\nwhich is obtained directly in the frequency domain, i.e.,\nwithout performing a time-consuming integration of the\nLLG equation. To solve problems of this type, we\nhave recently developed a powerful software package, the\nmatrix-free micromagnetic linear-response solver (MF-\nµMLS) [33].\nFor this part of the simulation in the frequency domain,\nwe assume a Gilbert damping of α= 0.01 and a weak field\namplitude\f\f\fδˆhext\f\f\fµ0Ms= 0.5 mT. In our simulations,\nthe frequency ωis varied in steps of 50 MHz in a range\nfrom 1 GHz to 30 GHz, and the field amplitude δˆhextis\noriented along the xaxis.\nSTATIC ZERO-FIELD CONFIGURATIONS\nThe array of interconnected nanowires shown in\nFig. (1) can be regarded as an assembly of 227 nanowire\nsegments connected to the intersection points (the ver-\ntices). Each segment behaves like an Ising-type magnet,\ni.e., it can be magnetized in one of two possible axial di-\nrections. Accordingly, a large number of magnetization\nstates can be attained—a situation that is well-known\nfrom artificial spin-ice systems [35, 36]. To reduce the\ncomplexity related to the numerous possible magnetic\nconfigurations, we consider only three qualitatively dif-\nferent states, the analysis of which will help us identify\nthe general properties of the system.\nThe first configuration, which we call the reference\nstate, is one in which the magnetization in each wire\nis uniform, such that all wire segments belonging to the\nthree orthogonal sets are magnetized in the same way,\nFIG. 2: Four qualitatively different vertex configurations:\n(a) uncharged state of type I, (b) uncharged state of type\nII, (c) double-charge configuration, and (d) quadruple-charge\nconfiguration. The color of the arrows displays the local mag-\nnetostatic volume charge density ρ=−∇ ·M, with red rep-\nresenting positive and blue negative values.\ni.e., along the positive x,y, and zaxis, respectively. As\na second configuration, which we call the charged state,\nwe select a remanent state that develops at zero field\nafter saturating the sample in the xdirection. In this\nconfiguration, all nanowire segments oriented parallel to\nthexaxis remain magnetized in the positive xdirection.\nThe magnetization in the other two sets of wires, in con-\ntrast, is not necessarily uniform, and the corresponding\nnanowire segments can be either magnetized along the\npositive or negative yandzdirections, respectively. Fi-\nnally, as a third state, we consider a zero-field magnetic\nconfiguration that develops after numerically relaxing the\nsystem to equilibrium when starting from a fully random-\nized initial magnetic structure. In this random state,\nthe magnetization is a priori different in each nanowire\nsegment, pointing towards the negative or positive x,y,\nandzdirections. In practice, however, not all combina-\ntions are possible as certain local magnetic configurations\nare energetically unstable, in particular the “hedgehog”-\ntype structure where the magnetization in all six adjacent\nwires points towards or away from a vertex.\nDifferent magnetic configurations can develop at the\nvertices throughout the array. As the magnetization in\nadjacent nanowire segments points towards or away from\na vertex, it carries positive or negative magnetic flux to-\nward the intersection site. An imbalance in the number\nof nanowire segments magnetized toward and away from\nthe vertex generates an effective magnetic charge, which\nmakes it possible to distinguish between charged and un-\ncharged vertex sites. The charge can be visualized by\nplotting the divergence of the magnetization field, which,\nexcept for a sign convention, is equivalent to the defini-\ntion of the magnetostatic volume charge density [37].4\nFigure 2 shows the four basic magnetic vertex config-\nurations found in the array. Frames (a) and (b) display\ntwo different zero-charge structures (0 q), with a “three-\nin/three-out” magnetic configuration in the adjacent wire\nsegments. In configuration (a), the magnetization pre-\nserves its direction along each wire after traversing the\nvertex, whereas configuration (b) contains a head-to-head\ndomain wall [38] along one wire and a tail-to-tail wall\nalong another. Configuration (c) is a (double) charged\nstate of the type “two-in/four-out” ( −2q), and configu-\nration (d) contains a quadruple charge structure (“one-\nin/five-out”, −4q). Several equivalent permutations and\nvariations of these configurations are possible, which can\nbe mapped onto each other through rotations, mirror\noperations, and time-inversion operations M→ −M.\nFor example, the negatively charged “one-in/five-out”\nstructure shown in panel (d) is equivalent to a positively\ncharged configuration of the type “five-in/one-out” (not\nshown).\nIn the reference state, all vertices are in the zero-charge\nmagnetic configuration of type I, while the charged state\nalso contains zero-charge configurations of type II along-\nside double-charge vertex structures. Quadruple-charge\nvertex structures of the type shown in Fig. 2d develop\nonly in the random state. These differences in the mag-\nnetic configurations result in distinct changes in the high-\nfrequency magnetization dynamics.\nHIGH-FREQUENCY MAGNETIZATION\nDYNAMICS\nUsing the numerical methods described in section , we\nsimulate situations in which a low-amplitude harmonic\nexternal field is applied to probe the frequency-dependent\nsystem’s response. At specific frequencies, this field ex-\ncitation generates resonant oscillations in the magnetic\nnanowire array. Distinct absorption peaks can be seen\nin the results displayed in Fig. 3, which compare the\npower spectrum of the three zero-field states described\nabove. The spectra display the frequency-dependence of\nthe power that the system absorbs from the applied rf\nfield [33].\nIn the case of the reference state, represented in\nthe frame on the top of Fig. 3, we observe four well-\ndefined sharp resonances in the absorption spectrum,\nnamely mode #1 at 10 .65 GHz, #2 at 13 .25 GHz, #3\nat 17 .0 GHz, and #4 at 18 .85 GHz. A weak fifth absorp-\ntion peak, whose amplitude is too small to be discerned\nin the figure, develops at 28 .35 GHz. Each of these reso-\nnances can be ascribed to different modes, which develop\nat specific locations within the array and, in some cases,\nare characteristic of certain magnetic structures. The\nprofiles of these five modes are shown in Fig. 4. The\nlow-frequency mode #1 concerns magnetic oscillations\nat the free ends of the nanowires, i.e., at the outer sur-\nFIG. 3: Absorption spectra of the nanowire array for the three\nmagnetic states discussed in the main text. The reference\nstate in the top frame shows sharp resonances at well-defined\nfrequencies. With increasing magnetic disorder, new absorp-\ntion peaks appear at low frequencies, highlighted in the mid-\ndle and bottom frame, while the high-frequency peaks near\n17 GHz become broader.\nface of the array, as shown in Fig. 4a. Peak #2 at the\nnext-higher frequency is due to the resonant excitation\nof the type-I zero-charge vertices, whose configuration is\nshown in Fig. 2. The profile of this vertex mode is shown\nin Fig. 4b. In the ordered magnetization state, all ver-\ntex structures oscillate at essentially the same frequency,\nwith only minor variations within the array. Upon close\ninspection, one can identify that the resonance frequency\nof vertices at specific positions, e.g., at the edges, near\nthe surface, or at the corners, is slightly different from\nthat of the vertices in the bulk of the array, which we at-\ntribute to changes in the local magnetostatic field. The\nremaining modes numbered 3, 4, and 5, develop at higher\nfrequencies and result from standing-wave type oscilla-\ntions within the nanowires. At these frequencies, the\nmagnetization in the ensemble of nanowire segments os-\ncillates with a specific profile, as shown in Fig. 4c. The\nmode profiles suggest that the ends of the wires, i.e., the\nvertex positions, neither act as fixed nor as free bound-\naries, indicating a frequency dependence of the effective\nboundary condition [39].5\nFIG. 4: Spatial distribution of the oscillation amplitude for the reference state at the frequencies labeled 1-5 in the top frame\nof Fig. 3. The mode profile (a) shows a localization of the oscillations at the dangling, free wire ends at the array surfaces, while\nframe (b) displays the resonance at the intersection points (vertices). The profiles in panel (c) show different standing-wave\nmodes in individual nanowire array segments. These segments are embedded in the array and are graphically extracted (c)\nto improve the visualization of the mode profile. The blue lines on top of modes 3, 4, and 5 are numerically determined\noscillation amplitude profiles. The color scale indicates the local oscillation strength and ranges from purple (minimum) to\nyellow (maximum). In each mode, the scale has been adapted to the maximum and minimum amplitude values developing at\nthe given frequency.\nThe resonances discussed so far, with the absorption\nspectrum shown in Fig. 3a and the mode profiles dis-\nplayed in Fig. 4, refer to the magnetically ordered ref-\nerence state. The spatial distribution of these modes is\nprimarily determined by the geometry, as they appear\nat specific positions of the three-dimensional structure—\nthe surface, the intersection points, and the nanowire\nsegments. As shown in Fig. 3b, the spectrum of the\ncharged magnetic state obtained after saturating the\nsample along the xdirection contains further absorption\npeaks, labeled 6 and 7, in the lower frequency range.\nThese additional peaks are signatures of specific mag-\nnetic vertex configurations.\nThe first peak (mode 6) refers to the uncharged ver-\ntex configuration of type II, shown in Fig. 5b. Vertices\nwith this magnetic configuration, which combines a head-\nto-head type transition along one direction and a tail-\nto-tail one along another, become active at 8 .6 GHz, as\ndisplayed in Fig. 5a. The configuration considered here\ncontains six vertices of this type. At a slightly higher fre-\nquency of 9 .3 GHz, the charged vertex configurations of\nthe type “four-in/two-out” and “two-in/four-out”, shown\nin Figs. 2b, become resonant. In our example, this ver-\ntex configuration occurs in a total of 14 vertices, equally\ndistributed in terms of positive and negative charges.\nFig. 5b displays the localized magnetic oscillations of\nthese vertices, constituting mode 7 in the spectrum of\nFig. 3b. The previously discussed modes of the reference\nstates also occur in the charged state, such as, e.g., the\nsurface mode of the free ends at 10 .65 GHz and the mode\nof the uncharged type-I vertices at 13 .2 GHz. Fig. 5c\nshows the spatial distribution of the oscillations of these\nvertices in the charged configuration. The results show\nthat specific absorption peaks can be seen as “finger-prints” of different types of vertex configurations, mak-\ning it possible to deduce the presence of specific mag-\nnetic structures by inspecting the absorption spectrum\n[40]. To a certain extent, the relative peak heights can\nmoreover serve as an indication for the density of spe-\ncific configurations in the array, as the intensity of the\nvertex-related peaks depends on the number of vertices\nparticipating in the resonant oscillation. For example,\npeak 2 of the charged state is diminished compared to\nthe reference state as the latter contains fewer vertices\nwith type-I uncharged configuration.\nThe spectrum of the third configuration, the “random\nstate” obtained after starting from a randomized initial\nmagnetic strate, is shown in Fig. 3c. It displays an addi-\ntional peak labeled 8 at the lower frequency range. This\nmode is due to the oscillation of vertices with quadruple\ncharge, as shown in Fig. 2d. Our version of the random\nstate contains two such vertices, one with a positive and\nthe other with a negative charge. The mode profile re-\nlated to the oscillation of these vertices, which become\nactive at 4 .0 GHz, is shown in Fig. 5d.\nIn the charged state, and even more so in the ran-\ndom state, the nanowire modes 3 and 4 broaden and\nmerge, evolving into a nearly continuous absorption re-\ngion between 16 and 19 GHz. This behavior can be ex-\nplained by magnetostatic fields arising from the charged\nvertices. Since the vertex charges are sources of magne-\ntostatic fields, they modify the effective field acting along\nthe nanowire segments, which, depending on the charge\ndistribution in the adjacent vertices, may be stronger or\nweaker than in the ordered state. These variations in\nthe effective field strength modify the nanowire segments’\nresonance frequency and, thus, lead to a broadening of\nthe absorption peaks.6\nFIG. 5: Magnetic modes localized at the vertices. Modes (a),\n(b) and (c) refer to the charged configuration. Each vertex\ntype becomes resonant at a different frequency. The mode\ndisplayed in panel (a) refers to vertices with a configuration\nas shown in Fig. 2a, while the localized mode of the vertices\nwith double-charge configuration (cf. Fig. b) is displayed in\npanel (b). Panel (c) shows the oscillation of the remaining\nvertices, which are magnetized in the uncharged configura-\ntion of type I and oscillate at the same frequency as in the\nordered state, previously labeled as mode 2. Panel (d) refers\nto the disordered (“random”) configuration, which contains\ntwo vertices with quadruple charge (cf. Fig. d) that become\nresonant at a particularly low frequency.\nCONCLUSION\nWe studied the static magnetization and the high-\nfrequency modes developing in a three-dimensional mag-\nnetic nanoarchitecture consisting of a regular array of\ninterconnected nanowires through frequency-domain mi-\ncromagnetic simulations. The system displays distinct\nabsorption peaks at specific frequencies when exposed\nto a low-amplitude harmonic external magnetic field in\nthe GHz frequency range. The simulations reveal that\nthe magnetic oscillations of these resonances are local-\nized at different geometric constituents of the array: the\nnanowire segments, the intersection points, and the sur-\nfaces. In addition to these modes determined by geo-\nmetric parameters, we find modes that depend on the\nmicromagnetic configuration developing at the vertex\npoints. Such appearance of configuration-dependent ver-\ntex modes is similar to analogous effects known from two-\ndimensional artificial spin-ice systems [41]. The corre-\nsponding characteristic absorption lines can thus serve as\na means to indirectly detect specific micromagnetic struc-\ntures [40], with the peak height indicating their density\nwithin the array. By combining three-dimensional as-\npects of magnonic crystals and artificial spin-ice systems,regular arrays of magnetic nanowires of the type studied\nin this article provide a variety of micromagnetic proper-\nties, particularly regarding the high-frequency magneti-\nzation dynamics, which could render these artificial ma-\nterial types interesting for reprogrammable magnonic ap-\nplications [42].\nThis work was funded by the French National Research\nAgency (ANR) through the Programme d’Investissement\nd’Avenir under contract ANR-11-LABX-0058 NIE and\nANR-17-EURE-0024 within the Investissement d’Avenir\nprogram ANR-10-IDEX-0002-02. The authors acknowl-\nedge the High Performance Computing center of the Uni-\nversity of Strasbourg for supporting this work by provid-\ning access to computing resources.\n∗Electronic address: riccardo.hertel@ipcms.unistra.fr\n[1] M. V´ azquez, Magnetic Nano- and Microwires: Design,\nSynthesis, Properties and Applications (Woodhead Pub-\nlishing, 2015), ISBN 978-0-08-100181-3.\n[2] H. Forster, T. Schrefl, D. Suess, W. Scholz, V. Tsiantos,\nR. Dittrich, and J. Fidler, Journal of Applied Physics\n91, 6914 (2002), ISSN 00218979, URL http://link.\naip.org/link/JAPIAU/v91/i10/p6914/s1&Agg=doi .\n[3] R. Hertel and J. Kirschner, Physica B: Condensed Matter\n343, 206 (2004), number: 1-4.\n[4] M. Yan, A. K´ akay, S. Gliga, and R. Hertel,\nPhysical Review Letters 104, 057201 (2010), num-\nber: 5, URL http://link.aps.org/doi/10.1103/\nPhysRevLett.104.057201 .\n[5] R. Wieser, U. Nowak, and K. D. Usadel, Physical Review\nB69, 064401 (2004), number: 6, URL http://link.\naps.org/doi/10.1103/PhysRevB.69.064401 .\n[6] M. Yan, C. Andreas, A. K´ akay, F. Garc´ ıa-S´ anchez,\nand R. Hertel, Applied Physics Letters 99, 122505\n(2011), ISSN 0003-6951, 1077-3118, number: 12, URL\nhttp://scitation.aip.org/content/aip/journal/\napl/99/12/10.1063/1.3643037 .\n[7] R. Hertel, Journal of Physics: Condensed Matter 28,\n483002 (2016), ISSN 0953-8984, number: 48, URL http:\n//stacks.iop.org/0953-8984/28/i=48/a=483002 .\n[8] H. Wang, L. Flacke, W. Wei, S. Liu, H. Jia, J. Chen,\nL. Sheng, J. Zhang, M. Zhao, C. Guo, et al., Applied\nPhysics Letters 119, 152402 (2021), ISSN 0003-6951,\npublisher: American Institute of Physics, URL http:\n//aip.scitation.org/doi/full/10.1063/5.0064134 .\n[9] D. Makarov and D. D. Sheka, eds., Curvilinear Micro-\nmagnetism: From Fundamentals to Applications , vol.\n146 of Topics in Applied Physics (Springer Interna-\ntional Publishing, Cham, 2022), ISBN 978-3-031-09085-1\n978-3-031-09086-8, URL https://link.springer.com/\n10.1007/978-3-031-09086-8 .\n[10] J. A. Ot´ alora, M. Yan, H. Schultheiss, R. Hertel, and\nA. K´ akay, Physical Review Letters 117, 227203 (2016),\nnumber: 22, URL http://link.aps.org/doi/10.1103/\nPhysRevLett.117.227203 .\n[11] R. Hertel, Journal of Applied Physics 90, 5752\n(2001), ISSN 00218979, URL http://link.aip.org/\nlink/JAPIAU/v90/i11/p5752/s1&Agg=doi .7\n[12] L. Clime, P. Ciureanu, and A. Yelon, Journal of\nMagnetism and Magnetic Materials 297, 60 (2006),\nISSN 0304-8853, URL https://www.sciencedirect.\ncom/science/article/pii/S030488530500301X .\n[13] E. C. Burks, D. A. Gilbert, P. D. Murray, C. Flores,\nT. E. Felter, S. Charnvanichborikarn, S. O. Kucheyev,\nJ. D. Colvin, G. Yin, and K. Liu, Nano Letters 21, 716\n(2021), ISSN 1530-6984, number: 1 Publisher: American\nChemical Society, URL https://doi.org/10.1021/acs.\nnanolett.0c04366 .\n[14] A. Frotanpour, J. Woods, B. Farmer, A. P. Kaphle, and\nL. E. De Long, Applied Physics Letters 118, 042410\n(2021), ISSN 0003-6951, publisher: American Institute\nof Physics, URL http://aip.scitation.org/doi/full/\n10.1063/5.0035195 .\n[15] E. Saavedra and J. Escrig, Journal of Magnetism and\nMagnetic Materials 513, 167084 (2020), ISSN 0304-\n8853, URL https://www.sciencedirect.com/science/\narticle/pii/S030488532030487X .\n[16] R. Cheenikundil, J. Bauer, M. Goharyan, M. d’Aquino,\nand R. Hertel, APL Materials 10, 081106 (2022), pub-\nlisher: American Institute of Physics, URL http://aip.\nscitation.org/doi/10.1063/5.0097695 .\n[17] T. da Cˆ amara Santa Clara Gomes, F. Abreu Araujo, and\nL. Piraux, Science Advances 5, eaav2782 (2019), pub-\nlisher: American Association for the Advancement of\nScience, URL https://www.science.org/doi/10.1126/\nsciadv.aav2782 .\n[18] J. M. D. Teresa, A. Fern´ andez-Pacheco, R. C´ ordoba,\nL. Serrano-Ram´ on, S. Sangiao, and M. R. Ibarra,\nJournal of Physics D: Applied Physics 49, 243003\n(2016), ISSN 0022-3727, URL https://doi.org/10.\n1088/0022-3727/49/24/243003 .\n[19] L. Keller, M. K. I. Al Mamoori, J. Pieper, C. Gspan,\nI. Stockem, C. Schr¨ oder, S. Barth, R. Winkler, H. Plank,\nM. Pohlit, et al., Scientific Reports 8, 6160 (2018),\nISSN 2045-2322, number: 1 Publisher: Nature Publish-\ning Group, URL https://www.nature.com/articles/\ns41598-018-24431-x .\n[20] L. Skoric, D. Sanz-Hern´ andez, F. Meng, C. Donnelly,\nS. Merino-Aceituno, and A. Fern´ andez-Pacheco, Nano\nLetters 20, 184 (2020), ISSN 1530-6984, number: 1, URL\nhttps://doi.org/10.1021/acs.nanolett.9b03565 .\n[21] A. Fern´ andez-Pacheco, R. Streubel, O. Fruchart, R. Her-\ntel, P. Fischer, and R. P. Cowburn, Nature Communi-\ncations 8, ncomms15756 (2017), ISSN 2041-1723, URL\nhttps://www.nature.com/articles/ncomms15756 .\n[22] P. Fischer, D. Sanz-Hern´ andez, R. Streubel, and\nA. Fern´ andez-Pacheco, APL Materials 8, 010701 (2020),\npublisher: American Institute of Physics, URL http:\n//aip.scitation.org/doi/full/10.1063/1.5134474 .\n[23] A. Fern´ andez-Pacheco, L. Skoric, J. M. De Teresa,\nJ. Pablo-Navarro, M. Huth, and O. V. Dobrovolskiy,\nMaterials 13, 3774 (2020), number: 17 Publisher: Mul-\ntidisciplinary Digital Publishing Institute, URL https:\n//www.mdpi.com/1996-1944/13/17/3774 .\n[24] S. Gliga, G. Seniutinas, A. Weber, and C. David,\nMaterials Today 26, 100 (2019), ISSN 1369-7021, URL\nhttp://www.sciencedirect.com/science/article/\npii/S1369702119304961 .\n[25] G. Gubbiotti, ed., Three-Dimensional Magnonics: Lay-\nered, Micro- and Nanostructures (Jenny Stanford Pub-\nlishing, New York, 2019), ISBN 978-0-429-29915-5.[26] J. Grollier, D. Querlioz, K. Y. Camsari, K. Everschor-\nSitte, S. Fukami, and M. D. Stiles, Nature Electronics 3,\n360 (2020), ISSN 2520-1131, number: 7 Publisher: Na-\nture Publishing Group, URL https://www.nature.com/\narticles/s41928-019-0360-9 .\n[27] S. Sahoo, A. May, A. van Den Berg, A. K. Mondal,\nS. Ladak, and A. Barman, Nano Letters 21, 4629 (2021),\nISSN 1530-6984, publisher: American Chemical So-\nciety, URL https://doi.org/10.1021/acs.nanolett.\n1c00650 .\n[28] R. Hertel and R. Cheenikundil, Defect-sensitive\nHigh-frequency Modes in a Three-Dimensional Arti-\nficial Magnetic Crystal (2022), URL https://www.\nresearchsquare.com .\n[29] A. Fernandez-Pacheco, Private Communication (2020).\n[30] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004), ISSN 0018-9464, number: 6.\n[31] W. F. Brown, Micromagnetics (Interscience Publishers,\n1963).\n[32] M. d’Aquino, C. Serpico, G. Miano, and C. Forestiere,\nJournal of Computational Physics 228, 6130\n(2009), ISSN 0021-9991, number: 17, URL\nhttp://www.sciencedirect.com/science/article/\npii/S0021999109002642 .\n[33] M. d’Aquino and R. Hertel, Journal of Applied Physics\n133, 033902 (2023), ISSN 0021-8979, publisher: Ameri-\ncan Institute of Physics, URL https://aip.scitation.\norg/doi/full/10.1063/5.0131922 .\n[34] R. Hertel, S. Christophersen, and S. B¨ orm, Journal of\nMagnetism and Magnetic Materials 477, 118 (2019),\nISSN 0304-8853, URL https://www.sciencedirect.\ncom/science/article/pii/S0304885318327689 .\n[35] C. Nisoli, R. Wang, J. Li, W. F. McConville, P. E. Lam-\nmert, P. Schiffer, and V. H. Crespi, Physical Review Let-\nters98, 217203 (2007), URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.98.217203 .\n[36] Y. Perrin, B. Canals, and N. Rougemaille, Nature\n540, 410 (2016), ISSN 1476-4687, URL http://\nwww-nature-com/articles/nature20155 .\n[37] W. F. Brown, Magnetostatic Principles in Ferromag-\nnetism (North-Holland Pub. Co., 1962).\n[38] R. Hertel and A. K´ akay, Journal of Magnetism\nand Magnetic Materials 379, 45 (2015), ISSN 0304-\n8853, URL http://www.sciencedirect.com/science/\narticle/pii/S0304885314011925 .\n[39] K. Y. Guslienko, S. O. Demokritov, B. Hillebrands, and\nA. N. Slavin, Physical Review B 66, 132402 (2002),\nnumber: 13, URL http://link.aps.org/doi/10.1103/\nPhysRevB.66.132402 .\n[40] A. Vanstone, J. C. Gartside, K. D. Stenning, T. Dion,\nD. M. Arroo, and W. R. Branford, New Journal of\nPhysics 24, 043017 (2022), ISSN 1367-2630, publisher:\nIOP Publishing, URL https://dx.doi.org/10.1088/\n1367-2630/ac608b .\n[41] S. Gliga, A. K´ akay, R. Hertel, and O. G. Heinonen,\nPhysical Review Letters 110, 117205 (2013), num-\nber: 11, URL http://link.aps.org/doi/10.1103/\nPhysRevLett.110.117205 .\n[42] S. Gliga, E. Iacocca, and O. G. Heinonen, APL Mate-\nrials8, 040911 (2020), number: 4 Publisher: American\nInstitute of Physics, URL https://aip.scitation.org/\ndoi/10.1063/1.5142705 ."    },    {        "title": "1904.00564v1.Mechanism_of_Electron_Spin_Relaxation_in_Spiral_Magnetic_Structures.pdf",        "content": "arXiv:1904.00564v1  [cond-mat.mes-hall]  1 Apr 2019Mechanism of Electron Spin Relaxation in Spiral Magnetic St ructures\nI.I. Lyapilin∗\nInstitute of Metal Physics of Ural Branch of Russian Academy of Sciences, 620990 Yekaterinburg, Russia\nDepartment of Theoretical Physics and Applied Mathematics ,\nUral Federal University, Mira St. 19, 620002 Yekaterinburg , Russia\n(Dated: April 2, 2019)\nSpin dynamics in spiral magnetic structures has been invest igated. It has been shown that the\ninternal spatially dependent magnetic field in such structu res produces a new mechanism of spin\nrelaxation.\nI. INTRODUCTION\nOne of the central problems facing spintronics, such as\nthe development of methods of injection, generation, and\ndetection of spin-polarized charge carriers, is to analyze\nand study the mechanisms of spin relaxation. Spin-orbit\ninteraction(SOI)iswell-knowntohaveasignificantinflu-\nence on the mechanisms of electron spin relaxation. SOI\ncan impact on spin degrees of freedom through transla-\ntionalones1. SOIitselfdoesnotcausespinrelaxationbut\nin combination with the scattering of the electron mo-\nmentum leadstoits relaxation. Elliottexamined the pro-\ncess of electron spin relaxation through the momentum\nscattering at impurity centers, conditional upon induc-\ning the spin-orbit interaction by lattice ions2. Phonons\ncan also participate in spin relaxation. The nature of\nspin relaxation involving phonons is to modulate time-\ndependent spin-orbit relaxation by lattice oscillations3.\nAs a result, spin relaxation occurs. The Elliott-Jafet\nmechanism offers for the spin relaxation frequency ωs\nto be proportional to the electron momentum relaxation\nfrequency ωp. In systems without a center of inversion,\na fundamentally different mechanism of spin relaxation\ncan be realized. In such systems, the spin states ↑and\n↓are not degenerate Ek↑/ne}ationslash=Ek↓, but the condition is\nsatisfied Ek↑=E−k↓. The lack of a center of symmetry\ncontributes to splitting the states subjected to SOI. The\nsplitting can be described by entering an intrinsic mag-\nnetic field Bi(k) around which the electron spins precess\nat a Larmor frequency. The precession of the electron\nspin in the effective magnetic field together with the elec-\ntron momentum scattering leads to spin relaxation, and\nωs∼ω−1\np4. Below we consider the spin relaxation mech-\nanisminmagneticallyorderedstructures, whicharechar-\nacterized by a spiral arrangement of magnetic moments\nrelative to some crystal axes5. The simplest case of such\nstructures is an antiferromagnetic spiral or a helicoid;\nthey can be found in rare earth metals ( Eu, Tb, Dy ) and\nsomeoxidecompounds. The structureofthis type canbe\nrepresented as a sequence of atomic planes perpendicular\nto the axis of a helicoid. In this case, atoms in each of\nthe planes have ferromagnetically ordered magnetic mo-\nments. However, the magnetic moments in neighboring\nplanes turn at some angle θdepending on the ratio of ex-\nchange interactions. This is because of the coexistence of\npositive exchange interaction between the nearest atomicneighbors and negative exchange interaction between the\nneighbors following the closest ones. The components of\nthe magnetic moments of atoms oscillate in the plane\nof the magnetic layer Sx=S0xsinkz, S y=S0ycoskz.\nIf, in this case, Sz/ne}ationslash= 0, we have a ferromagnetic spiral\nwith a resulting moment. If also oscillates Szby a har-\nmonic law, a complex magnetic structure emerges. In\nsuch structures, the exchange interaction between zone\nchargecarriersand localizedmoments is described by the\nwell-known expression Hex=−J/summationtext\nis·Swherejis the\nexchangeconstant. Inthe mean-fieldapproximation,this\ninteraction can be represented as Hex=−m(r)Hef(r),\nwherem(r) =gµBS(r),g– is the spectroscopic split-\nting factor, µBis the Bohr magneton, and Hef(r) is the\ninternal effective magnetic field.\nIt is obvious that the spatial variation of the internal\nfield in spiral magnetic structures should affect the spin\ndynamics of conduction electrons. Indeed, the spin of an\nelectron in a state with quasi-momentum ( k) precesses in\nthe effective magnetic field Hef(r) only for a time of the\norder of the elastic scattering time τp. After scattering,\nthe electron goes into a state ( k′) where the effective\nmagnetic field has a different direction. Consequently,\nthe evolution of the spin dynamics under such conditions\nturns out to be associated with the electron momentum\nrelaxation.\nLet us look into a simple spiral when the effec-\ntive magnetic field rotates in a plane ( xy), where\nHef(H0cos(Qz), H0sin(Qz),0) ,Q= 2π/Λ, Λ is the\nperiod of the spiral structure. To gain insight into the\nspin dynamics, we estimate the spin relaxation mecha-\nnism realized under above conditions. The system at\nhand is assumed to expose to an electric field directed\nalong the axis z ( E= (0,0E0) . The Hamiltonian of the\nsystem can be written in the form H=Hk+Hs+HeE+\nHv+Hev,, whereHk, Hsare the operators of the kinetic\nand spin energy of electrons interacting with the lattice\nHevand the external electric field HeE=−e/summationtext\niEriand\nHvis the Hamiltonian of the lattice.\nHk=/summationdisplay\njp2\ni/2m,\nHs=−gµB/summationdisplay\njsiHef=−¯hωsf/summationdisplay\nj(s+eiQzj+s−\nje−iQzj)\n,(1)\nwhereωsf=geH0/2mocis the electron precession fre-2\nquency in the effective field, s±=sx±isy. The spin\ndynamics is determined by macroscopic equations of mo-\ntionforthespinsubsystemofelectrons. Themacroscopic\nequations of motion can be deduced by averaging micro-\nscopic equations of motion over the non-equilibrium sta-\ntistical operator ρ(t) . To begin with, we write down\nthe microscopic equations of motion for the longitudinal\ncomponents of the spin and momentum of electrons. We\nhave\n˙sz=iωsf{s+eiQz−s−e−iQz}+ ˙sz\nsv,(2)\n˙pz=−eEN+iQ¯hωsf{s+eiQz−s−e−iQz}+ ˙pz\nev(r),.\n(3)\n˙A= (i¯h)−1[A, H],˙Aij= (i¯h)−1[A, Hij].Averaging\nthe microscopic equations of motion over the operator\nρ(t), we arrive at < Ai>t=Sp{Aiρ(t)}. The explicit\nform of the operator ρ(t) can be found within the NSO\nmethod6. Suppose that we have carried out this proce-\ndure. As a result, we come to the following system of\nmacroscopic equations.\n<˙sz>t=iωsf{< s+eiQz>t−< s−e−iQz>t}+<˙sz\nev>t,\n(4)\nand\n<˙pz>=iQ¯hωsf{< s+eiQz>t−< s−e−iQz>t}−\n−eEn0+<˙pz\nev>t. (5)\nwhere< N >t=nis the electron concentration. The\ncollisional summands in the balance equations (4) and\n(5) can be represented as follows6,7:\n<˙sz\nev>t∼(sz, sz)0ωs, <˙pz\nev>t∼(pz, pz)0ωp,(6)\nwhere\n(A, B)0=1/integraldisplay\n0dτ Sp{Aρτ\n0∆Bρ1−τ\n0},(7)\nwhere ∆ A=A−< A > 0, <···>0=Sp{···ρ0},ρ0is\nthe equilibrium Gibbs distribution. ωsis the relaxationfrequency of the longitudinal spin components, and ωp\nis the momentum relaxation frequency; both latter are\ncalculated in the Born approximation over the scatter-\nelectron interaction.\nωs=gµB\nχ0/integraldisplay\n−∞dteǫt(˙sz\nev,˙sz\nev(t))0,\nωp=1\nmnT0/integraldisplay\n−∞dteǫt(˙pz\nev,˙pz\nev(t))0, ǫ→+0.(8)\nHereχis the paramagnetic susceptibility of an electron\ngas\nχ=(gµB)2\n2T·(s+, s−)0=gµB\nH0·< sz>0,(9)\nT≡kBTis the temperature in energy units. Taking\na stationary case into account, we finally obtain the ex-\npression for the spin relaxation frequency in a helicoidal\nmagnet:\nωs∼mnT\n¯hQ(sz, sz)0·ωp (10)\nThus, in spiral magnets as well as in the Elliott-Jafet\nmechanism, thespiralrelaxationisrelatedtothe electron\nmomentum relaxation and ωs∼ωp. However, against\nthe Elliott-Jafet mechanism, the spin relaxation mech-\nanism in spiral magnetic structures is due to the pres-\nence of the internal spatially-dependent magnetic field in\nthem. The evolution of the spin dynamics is determined\nby both the period of the magnetic structure and the\nmomentum relaxation frequency.\nACKNOWLEDGMENTS The research was carried\nout within the state assignment of Minobrnauki of Rus-\nsia (theme ”Spin No AAAA-A18-118020290104-2), sup-\nported in part by RFBR (project No. 19-02-00038/19).\n∗ligor47@mail.ru\n1E.I. Rashba, Electron spin operation by electric elds: spin\ndynamics and spin injection, Physica E 20, 189 (2004).\n2R. J. Elliott, Theory of the effect of spin-orbit coupling on\nmagnetic rresonance in some semiconductors, Phys. Rev.,\n96266 (1954).\n3Y. Y a f e t, Sol. g factors and spin-lattice relaxation of\nconduction electrons, in Solid State Physics, F. Seitz and\nD. Turnbull, Eds. New York: Academic, 141 (1963).\n4M. I. Dyakonov, V. I. Perel, Spin relaxation of conductionelectrons in noncentrosymetric semiconductors, Sov. Phys .\nSolid State, 13, 3023 (1972).\n5Yu. A. Izyumov, Modulated, or long-periodic, magnetic\nstructures of crystals” Sov. Phys. Usp 27, 845 (1984).\n6I. I. Lyapilin, M. S. Okorokov, V.V. Ustinov, Spin effects in-\nduced by thermal perturbation in a normal metal/magnetic\ninsulator system, Phys.Rev B. 91, 195309-7 (2015).\n7H.M. Bikkin, I. I. Lyapilin, Non equlibrium thermodynam-\nics and physical kinetics, Walter de Gruyter GmbH, Berlin\n2014, p. 359."    },    {        "title": "1912.02500v1.Steering_magnonic_dynamics_and_permeability_at_exceptional_points_in_a_parity_time_symmetric_waveguide.pdf",        "content": "Steering magnonic dynamics and permeability\nat exceptional points in a parity-time\nsymmetric waveguide\nXi-guang Wang,y,zGuang-hua Guo,yand Jamal Berakdar\u0003,z\nySchool of Physics and Electronics, Central South University, Changsha 410083, China\nzInstitut für Physik, Martin-Luther Universität Halle-Wittenberg, D-06120 Halle/Saale,\nGermany\nE-mail: jamal.berakdar@physik.uni-halle.de\nAbstract\nTuning the low-energy magnetic dynamics is a key element in designing novel mag-\nnetic metamaterials, spintronic devices and magnonic logic circuits. This study uncov-\ners a new, highly effective way of controlling the magnetic permeability via shaping\nthe magnonic properties in coupled magnetic waveguides separated by current carrying\nspacer with strong spin-orbit coupling. The spin-orbit torques exerted on the waveg-\nuidesleadstoanexternallytunableenhancementofmagneticdampinginonewaveguide\nandadecreaseddampingintheother, constitutingsoamagneticparity-time(PT)sym-\nmetric system with emergent magnetic properties at the verge of the exceptional point\nwhere magnetic gains/losses are balanced. In addition to controlling the magnetic per-\nmeability, phenomena inherent to PT-symmetric systems are identified, including the\ncontrolonmagnonpoweroscillations, nonreciprocalmagnonpropagation, magnontrap-\nping and enhancement as well as the increased sensitivity to magnetic perturbation and\nabrupt spin reversal. These predictions are demonstrated analytically and confirmed\n1arXiv:1912.02500v1  [cond-mat.mes-hall]  5 Dec 2019by full numerical simulations under experimentally feasible conditions. The position of\nthe exceptional points and the strength of the spontaneous PT symmetry breaking can\nbe tuned by external electric and/or magnetic fields. The roles of the intrinsic magnetic\ndamping, and the possibility of an electric control via Dzyaloshinskii-Moriya interac-\ntion are exposed and utilized for mode dispersion shaping and magnon amplification\nand trapping. The results point to a new route to designing optomagnonic waveguides,\ntraps, sensors, and circuits.\nKeywords\nMagnonic circuits, PT-symmetry breaking, spin orbit torque, non-Hermitian dynamics, Op-\ntomagnonics, magnetic switching\nIntroduction\nNanomagnetism is the backbone of spin-based memories, data processing and sensorics. In a\ngeneric magnet, the permeability, meaning the magnetic response to a weak external pertur-\nbation is governed by the behavior of the spin waves which are collective transverse oscilla-\ntions (with their quantum termed magnon) around the ground state. Miniaturized magnonic\nlogiccircuits1–6andwaveguidesoperatedatlowenergycostwithnegligibleOhmiclosseswere\ndemonstrated. Furthermore, geometric confinements, nanostructuring, and material design\nallow a precise spectral shaping and guiding of magnons, which is reflected respectively in a\nmodified magnetic response. Here we point out an approach based on a magnonic gain-loss\nmechanism in two waveguides with a normal-metal spacer. The two magnetic waveguides\nare coupled via the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Driving charge\ncurrent in a spacer layer with a strong spin-orbit coupling (cf. 1(a)), spin orbit torques\n(SOTs) are exerted on the magnetizations of the waveguides. In effect, SOT adds to the\nintrinsic magnetic damping, as evident from Eq. (1). It is thus possible to achieve a case\n2where SOT-induced magnetic losses in one waveguide are balanced by antidamping in the\nother waveguide. This is a typical case of a PT-symmetric system as realized for instance\nin optical systems.7–12A hallmark of PT-symmetric systems is that, even if the underly-\ning Hamiltonian is non-Hermitian, the eigenvalues may be real,13–15and turn complex when\ncrossing the \"exceptional point\" and entering the PT-symmetry broken phase upon varying a\nparametric dependence in the Hamiltonian. Examples were demonstrated in optics and pho-\ntonics,8–12,16–21optomechanics,22,23acoustics24,25and electronics.26–29Also, PT-symmetric\ncavity magnon-polaritons were discussed involving phonon dissipation or electromagnetic\nradiation as well as parametric driving or SOT effects.30–35\nOur main goal is the design and demonstration of PT-symmetric magnonic waveguides\nwhich are controllable by feasible external means that serve as a knob to tune the system\nacross the exceptional point. In addition to the documented advantages of magnons, this\nwould bring about new functionalities that can be integrated in optomagnonic, spintronic,\nand magnonic circuits.\nSpin-torque driven PT-symmetric waveguides\nOur magnon signal propagates along two magnetic waveguides (which define the ~ xdirection)\ncoupled via RKKY exchange interaction (cf. Fig, 1(a)). A charge current flowing in a spacer\nwith a large spin Hall angle (such as Pt) applies a SOT ~T1k~ yon first waveguide enhancing\nthe effective damping, and a SOT ~T2k\u0000~ yon second waveguide weakening the effective\ndamping. The polarization directions of the spin Hall effect induced transverse spin currents\n~T1=~ z\u0002~jPtin WG1 and ~T2= (\u0000~ z)\u0002~jPtin WG2, are related to the charge current density\n~jPt. In a generic ferromagnet and for the long wavelength spin excitations of interest here,\nto describe the magnetic dynamics it is sufficient to adopt a classical continuous approach\nand solve for the equations of motion of the magnetization vector fields ~Mp(~ r;t)(p= 1;2\n3a \nFigure 1: (a) Two magnetic waveguides (labeled as WG1 and WG2, as an example we\nuse YIG in the numerical simulations) are coupled via RKKY interaction with metallic\nspacer that has a large spin Hall angle (here Pt). Driving a charge current ~jPtalong the\nspace (xdirection) results in spin-Hall torques acting on the magnetic waveguides. The\ntorques damp or antidamp the magnetic dynamics in WG1 and WG2 resulting so in PT-\nsymmetric structure with a new (PT) symmetry behavior of the magnetic permeability.\nMagnon wave packets are launched locally at one end of WG1 or WG2 (left side in the figure)\nand the propagation characteristics of the magnonic signal is steered, amplified or suppressed\nby external fields that drive the waveguides from the PT-symmetric to the PT-symmetry\nbroken phase through the exceptional point where magnetic losses in WG1 balance magnetic\nantidamping in WG2. This can be achieved for instance by changing the ratio between\nthe intrinsic coupling strength between the waveguides \u0014and the strength of the spin-Hall\ntorques!J. (b) Real and (c) imaginary parts of two eigenmode frequency f=!=(2\u0019)as\nwe scan!J=\u0014at the wave vector kx= 0:1nm\u00001. (d-f) Spatial profiles of propagating spin\nwave amplitude for different loss/gain balance (different !J), when the spin waves are locally\nexcited either in WG1 or WG2. The color change from blue to red corresponds to a linear\namplitude change ranging from 0 to the maximum of input signal. The local microwave field\nexcites spin waves at the left side of the waveguide and has a frequency of 20 GHz. The\nlength (along xaxis) of waveguides in (d-f) is 580 nm. 4enumerates the two waveguides), which amounts to propagating the Landau-Lifshitz-Gilbert\n(LLG) equation,36–39\n@~Mp\n@t=\u0000\r~Mp\u0002~He\u000b;p+~Mp\nMs\u0002\"\n\u000b@~Mp\n@t\u0000\rcJ~Tp\u0002~Mp#\n: (1)\nThe waveguides are located at z= +z0andz=\u0000z0. We are interested in small transversal\nexcitations and hence it is useful to use the unit vector field ~ mp=~Mp=MswhereMsis the\nsaturation magnetization and \ris the gyromagnetic ratio. \u000bis the conventional Gilbert\ndamping inherent to magnetic loses in each of the waveguides. The effective field ~He\u000b;p=\n2Aex\n\u00160Msr2~ mp+JRKKY\n2\u00160Mstp~ mp0+H0~ yconsists of the internal exchange field, the interlayer RKKY\ncoupling field, and the external magnetic field applied along the yaxis, where p;p0= 1;2,\nandp06=p.Aexis the exchange constant, JRKKYis the interlayer RKKY exchange coupling\nstrength,tpis the thickness of the pth layer, and \u00160is the vacuum permeability. Of key\nimportance to this study is the strength cJ=T\u0012SH~Je\n2\u00160etpMsof SOT which is proportional\nto charge-current density Jeand the spin Hall angle \u0012SHin the spacer layer, for instance,\nat the exceptional point defined in the following study, cJ= 1\u0002105A/m corresponds to a\ncharge current density of Je= 9\u0002108A/cm2in Pt.40Tis the transparency at the interface,\nandeis the electron charge. Our proposal applies to a variety of settings, in particular\nsynthetic antiferromagnets41offer a good range of tunability. To be specific, we present\nhere numerical simulations for Pt interfaced with a Yttrium-Iron-Garnet (YIG) waveguides\nas experimentally realized for instance in Ref. [ 40] corresponding to the following values\nMs= 1:4\u0002105A/m,Aex= 3\u000210\u000012J/m (technical details of the numerical realization are\nin the supplementary materials). For the Gilbert damping we use \u000b= 0:004but note that\ndepending on the quality of the waveguides \u000bcan be two order of magnitude smaller. The\ninterlayer exchange constant JRKKY = 9\u000210\u00005J/m2, which is in the typical range.42For\nthe waveguide thickness we used t1;2= 4nm. A large enough magnetic field H0= 2\u0002105\nA/m is applied along +ydirection to bring the WGs to a remnant state.\n5Magnonic coupled wave-guide equations with spin-orbit\ntorque\nFor a deeper understanding of the full-fledge numerical simulations presented below, it\nis instructive to formulate an analytical model by considering small deviations of ~ ms;p=\n(\u000emx;p;0;\u000emz;p)away from the initial equilibrium ~ m0;p=~ y. Introducing  p=\u000emx;p+i\u000emz;p\nwe deduce from linearizing Eq. (1) the coupled waveguide equations\ni@ 1\n@t\u0000[(!0\u0000\u000b!J)\u0000i(!J+\u000b!0)] 1+q 2= 0;\ni@ 2\n@t\u0000[(!0+\u000b!J) +i(!J\u0000\u000b!0)] 2+q 1= 0:(2)\nFor convenience, we introduce in addition to the coupling strength q=\rJRKKY\n(1+i\u000b)\u00160Mstp, the SOT\ncoupling at zero intrinsic damping \u0014=\rJRKKY=(2\u00160Mstp) =qj\u000b!0. The intrinsic frequency\nof the waveguides is given by !0=\r\n1+\u000b2(H0+2Aex\n\u00160Msk2\nx+JRKKY\n2\u00160Mstp)which is for the material\nstudied here is in the GHz. Essential for the behavior akin to PT-symmetric systems is\nthe SOT-driven gain-loss term !J=\rcJ\n1+\u000b2. The wavevector along xdirection is kx. Eq. (2)\nadmits a clear interpretation: The magnonic guided modes in the first waveguide ( WG1) are\nsubject to the confining complex potential V(z) =VR(z)+iVi(z)withVR(z0) =!0\u0000\u000b!Jand\nVi(z0) =\u0000!J\u0000\u000b!0. In WG2 the potential is VR(\u0000z0) =!0+\u000b!JandVi(\u0000z0) =!J\u0000\u000b!0.\nThe mode coupling is mediated by qwhich determines the periodic magnon power exchange\nbetween WG1 and WG2 in absence of SOT.\nFor a PT symmetric system the condition VR(z0) =VR(\u0000z0)andVi(z0) =\u0000Vi(\u0000z0)must\napply, which is obviously fulfilled if the intrinsic damping is very small ( \u000b!0). Comparing\nthecurrentandthephotoniccase, inthelattercasethesignoftheimaginarypartoftheWGs\nrefractive index is tuned. Here we control with SOT the imaginary part of the permeability\nwhich we explicitly prove by deriving and analyzing of the magnetic susceptibility (cf. Supp.\nMaterials). ThisfindingpointstoanewroutefordesigningPT-symmetricmagneto-photonic\nstructures via permeability engineering. We note, for a finite magnetic damping \u000ba PT-\n6behavior is still viable as confirmed by the full numerical simulations that we discuss below.\nMagnon dynamics across the spontaneous PT-symmetry\nbreaking transition\nThe dispersion !(kx)of the modes governed by Eq. (2) reads\n/s45/s50/s120/s49/s48/s56\n/s45/s49/s120/s49/s48/s56\n/s48\n/s49/s120/s49/s48/s56\n/s50/s120/s49/s48/s56/s48/s50/s48/s52/s48/s54/s48\n/s32/s32/s102/s32/s32/s40/s71/s72/s122/s41\n/s107\n/s120/s32/s40/s109 /s41\nFigure 2: Merging of the acoustic ( !J= 0, solid squares) and optical magnon ( !J= 0,\nsolid dots) modes dispersion Re[!](kx)when approaching the loss/gain-balanced exceptional\npoint!J=\u0014(open dots).\n!= (1\u0000i\u000b)!0\u0006q\nq2\u0000!2\nJ+ 2i\u000b!2\nJ+\u000b2!2\nJ (3)\nwhich describes both the acoustic and optical magnon modes43and depends parametrically\non!Jandq. For\u000b!0(in which case q\u0011\u0014) the eigenvalues are always real in the\nPT-symmetric regime below the gain/loss-balanced threshold !J=\u0014< 1. At the exceptional\npoint!J=\u0014= 1, the two eigenvalues and eigenmodes become identical. For !J=\u0014 > 1(by\nincreasing the current density for instance) we enter the PT-symmetry broken phase, and the\neigenvalues turn complex, as typical for PT-symmetric systems.21,44The splitting between\nthe two imaginary parts is determined by 2\u0014[(!J=\u0014)2\u00001]1=2and is tunable by external\nfields. This fact is useful when exploiting the enhanced waveguides sensitivity to magnetic\nperturbations round the exceptional point. Allowing for a small damping \u000bdoes not alter\nthe modes behavior, as demonstrated by the full numerical results in a Fig. 1(b-c). The\n7full magnon dispersions ( Re[!]versuskxcurves ) for !J=\u0014< 1and!J=\u0014= 1are shown in\nFig. 2. The symmetry of our waveguide brings in a special behavior of the magnon signal\ntransmission, meaning the propagation of a superposition of eigenmodes: Without charge\ncurrent in the spacer ( !J= 0), a signal injected at one end in one waveguide oscillates\nbetween WG1 and WG2 (due to the coupling \u0014) in a manner that is well-established in\ncoupled wave guide theory (cf. Fig. 1(d) ). Switching on the charge current, !J=\u0014becomes\nfinite and the beating of the magnon power between WG1 and WG2 increases (cf. Fig. 1(e)),\nas deducible from Eqs. (2), and also encountered in optical wave guides.44Eqs. (2) also\nindicatethatneartheexceptionalpoint, amagnonicwavepacketinjectedinonewaveguideno\nlonger oscillates between the two waveguides but travels simultaneously in both waveguides,\nas confirmed in Fig. 1(f) by full numerical simulations. This behavior resembles the optics\ncase.10We note that in our waveguides, this limit is simply achieved by tuning the external\nelectric and magnetic fields that then change the ratio !J=\u0014. We also found in line with\nRef. [ 10] a non-reciprocal propagation below the exceptional point. Passing the exceptional\npoint (!J=\u0014> 1) the magnonic signal always propagates in the guide with gain and is quickly\ndamped in the guide with loss.\nEnhanced sensing at PT-symmetry breaking transition\nTo assess the susceptibility of our setup to external magnetic perturbations we apply an\nexternal microwave field ~hpwhich adds to effective field in the LLG equation. In frequency\nspace we deduced that e p=P\np0\u001fpp0\rehm;p0(tilde stands for Fourier transform), with hm;p=\nhx;p+ihz;p, and\u001fpp0is the dynamic magnetic susceptibility which has the matrix form\n\u001f=1\n(!k\u0000i\u000b!\u0000!)2+!2\nc\u0000\u001420\n@(!k\u0000i\u000b!) + (i!c\u0000!) \u0014\n\u0014 (!k\u0000i\u000b!)\u0000(i!c\u0000!)1\nA;(4)\nwith!c=\rcJand!k=\r(H0+2Aexk2\nx\n\u00160Ms+JRKKY\n2\u00160Mstp).\n8/s49/s53\n/s51/s48/s48/s46/s48/s49/s46/s48/s120/s49/s48/s45/s56\n/s48/s49\n/s49/s53\n/s51/s48/s48/s46/s48/s51/s46/s48/s120/s49/s48/s45/s56\n/s48/s49\n/s48 /s53/s48/s48/s48/s55\n/s48 /s53/s48/s48/s48/s49/s48\n/s48 /s52/s48/s48/s48/s48/s56/s48\n/s48 /s52/s48/s48/s48/s48/s54/s48\n/s50/s48 /s52/s48/s45/s50/s46/s48/s120/s49/s48/s52/s48/s46/s48/s50/s46/s48/s120/s49/s48/s52\n/s48 /s49/s48/s48/s45/s49/s46/s52/s120/s49/s48/s53/s48/s46/s48/s49/s46/s52/s120/s49/s48/s53\n/s73/s109\n/s91\n/s93\n/s74/s32/s47/s32\n/s102/s32/s40/s71/s72/s122/s41/s32/s74/s32/s47/s32/s73/s109\n/s91\n/s93\n/s102\n/s32/s40/s71/s72/s122/s41/s98/s65/s109/s112/s32/s40/s65/s47/s109/s41\n/s32/s32/s65/s109/s112/s32/s40/s65/s47/s109/s41\n/s120 /s32/s40/s110/s109/s41/s97\n/s99\n/s74\n/s65/s109/s112/s32/s40/s65/s47/s109/s41\n/s32/s32/s65/s109/s112/s32/s40/s65/s47/s109/s41\n/s120 /s32/s40/s110/s109/s41/s100\n/s74\n/s74\n/s32/s32/s32\n/s120 /s32/s40/s110/s109/s41/s101/s74\n/s32/s32/s32\n/s120 /s32/s40/s110/s109/s41/s102/s32/s32/s77\n/s120/s32/s40/s65/s47/s109/s41\n/s116/s32/s40/s110/s115/s41/s103\n/s32/s32/s77\n/s121/s32/s40/s65/s47/s109/s41\n/s116/s32/s40/s110/s115/s41/s104Figure 3: Magnetic susceptibility Im[\u001f11](a) and Im[\u001f12](b) as functions of fand!J.\nPeaks in Im[\u001f11]andIm[\u001f12]are found at the exceptional point !J=\u0014. (c-f) Exciting spin\nwaves of frequency 20 GHz at x= 0in WG1, spatial profiles of spin wave amplitude for\ndifferent!J. Black solid line and red dashed line represents the amplitudes in WG1 and\nWG2, respectively. Near !=\u0014, a slight variation in !Jcauses marked changes in the spin\nwave amplitudes (e-f), while the change is negligible near != 0:7\u0014(c-d). (g-h) At !=\u0014,\nincreasing the external magnetic field ( H0= 2\u0002105A/m) in WG1 by 100 A/m, the time\ndependence of Mx(g) andMy(h) atx= 2000nm in WG2.\n9Near the exceptional point the system becomes strongly sensitive, for instance to changes\nin the charge current term !c, as testified by the behavior of the susceptibility which is\ndemonstrated for the imaginary parts of \u001f11and\u001f12in Fig. 3. The high sensitivity of the\nexcited spin waves on !cnear the exception point (see Fig. 3(c-f)) is exploitable to detect\nslight changes in charge current density cJ.\nFurthermore, near the exceptional point, our setup is strongly sensitive to changes in\nthe magnetic environment. As an example, at !J=\u0014, if the magnetic field H0(or local\nmagnetization) is increased by 100 A/m in WG1, large amplitude spin-wave oscillations\nare generated in WG2, as evidenced by the time dependence of Mx(x= 2000nm) in WG2\n(Fig. 3(g)). The spin wave amplification leads eventually to a reversal of Myin WG2 (Fig.\n3(h)). Away from the PT-breaking transition, e.g. for !J= 0:7\u0014, whenH0is reduced by\nthe same amount in WG1, virtually no changes in propagating spin waves are observed (not\nshown). Obviously, this magnon amplification may serve as a tunable sensor for the magnetic\nenvironment.\nCurrent-induced switching in magnetic PT-symmetric junc-\ntions\nA special feature of magnetic systems is the possibility of current-induced switching (de-\nscribed by Eq. (1) but not by Eqs. (2)).45In fact, for large current densities we are well\nabove the exceptional point. In this case the magnetic system becomes unstable towards\nswitching. We find with further increasing the charge current density (enhancing !J), the\nlocal magnetization in guide 2 is indeed switched to \u0000y. Magnon dynamics above the excep-\ntional point is still possible however by tuning the spacer material properties or its thickness\nto obtain a smaller \u0014, for instance with JRKKY = 9\u000210\u00007J/m2and\u000b= 0:01. In this case\nthe condition !J\u001d\u000b!0is not satisfied anymore, and the influence of intrinsic magnetic\nlosses (\u000b) in both wave guides is important. Nonetheless, even without reaching the strict\n10a\nc\ne\nf\nb\nd\nωJ = 2κ ωJ = 2κ\nωJ = 3κ\nωJ = 3κFigure 4: (a-b) Real and imaginary parts of the eigenmodes !as varying the loss/gain\nbalance by scanning !J(meaning, the SOT strength). The wave vector is kx= 0:03nm\u00001\nand the intrinsic coupling between WG1 and WG2 \u0014is lowered, as compared to Fig. 1 (by\nchoosingJRKKY = 9\u000210\u00007J/m2). (c-d) Spatial profiles of magnon wave amplitudes (as\nnormalized to their maxima) for !J= 2\u0014, and ( Re[!] = 2\u0019\u000220GHz). Black dashed lines\ncorresponds to WG1 and red solid line to WG2. (e-f) Time dependence (at the location\nx= 2000nm) and the spatial profiles (at t= 40ns) of thexcomponent of the magnetization\nMxfor!J= 3\u0014. The color variation from blue to red corresponds to a Mxchange from the\nnegative maximum to the positive maximum.\n11PT-symmetric condition, we still observe that the real parts of the two eigenvalues merge at\nthe same point !J=\u0014, and the two imaginary parts become different when !J>\u0014, as shown\nby Fig. 4(a-b). When !J= 2\u0014, the two imaginary parts are both negative, meaning that\nboth modes are evanescent. The propagation of magnonic signal launched in one waveguide\nend is shown in Fig. 4 (c-d) evidencing that the spin waves in the two waveguides decay\ndifferently. An input signal in the waveguide with enhanced damping leads to an evanescent\nspin wave in WG1. Injecting the signal in WG2, the attenuation of spin wave is weaker, and\nits amplitude is always larger. When !J= 3\u0014andIm[!]of the optical magnon mode turns\npositive, we observe that SOT induces spin wave amplification with time (Fig. 4(e-f)). This\nfinding is interesting for cavity optomagnonics.46\nFor input signal in WG1 or WG2, the spin wave amplitude is always larger in WG2 with a\nnegative effective damping. Also, the excited spin wave amplitude is much larger when the\ninput is in the WG2. Thus, no matter from which waveguide we start, the output signal is\nalways distributed at the end of WG2, a fact that can be employed for constructing magnonic\nlogic gates.\nDzyaloshinskii-Moriya interaction in electrically controlled\nPT-symmetric waveguides\nInmagneticlayersandattheirinterfacesanantisymmetricexchange,alsocalledDzyaloshinskii-\nMoriya (DM), interaction47,48may exist. In our context it is particularly interesting that\nthe DM interaction may allow for a coupling to an external electric field ~Eand voltage\ngates. The contribution to the system free energy density in the presence of DM and ~E\nisEelec=\u0000~E\u0001~P, with the spin-driven polarization ~P=cE[(~ m\u0001r)~ m\u0000~ m(r\u0001~ m)].49,50\nThis alters the magnon dynamics through the additional term ~Helec=\u00001\n\u00160Ms\u000eEelec\n\u000e~ min the\neffective field ~He\u000b. To uncover the role of DM interaction on the magnon dynamics in PT\nsymmetric waveguides we consider three cases: (i) The two waveguides experience the same\n12-0.15 0.00 0.15030 60 \n-0.15 0.00 0.15030 60 \n0 1 2-5 05\n0 1 2912 kx (nm-1 ) kx (nm-1 )f  (GHz) f  (GHz) \nf  (GHz) f  (GHz) Ez = 2 MV/cm Ez = −2 MV/cm\nRe[ω/2 π]Im[ω/2 π]\nWG1 WG1 \nWG2 WG2 ωJ / κ ωJ / κaa bb\nddccFigure 5: Control of coupled magnonic waveguide characteristics by an external electric\nfield in presence of Dzyaloshinskii-Moriya interaction. (a-b) Applying a static electric field\n~E1;2= (0;0;Ez)withEz=\u00062MV/cm to both waveguides modify the magnon dispersion\nRe[!](kx)curves for!J= 0(solid dots) and !J=\u0014(open dots). (c) Real and imaginary\nparts of two eigenmodes !as functions of !Jand in the presence of two static electric\nfields (or voltages) applied with opposite polarity to the two waveguides ( ~E1= (0;0;Ez)\nand~E2= (0;0;\u0000Ez), andEz= 2MV/cm) at kx= 0:1nm\u00001. (d) Spatial profiles of the\npropagating spinwave amplitudes when applying electric field in WG1 ( ~E1= (0;0;Ez)with\nEz= 2MV/cm, and ~E2= (0;0;0)). Color scale from blue to red corresponds to amplitude\nchange from 0 to its maximum.\n13static electric field ~E1;2= (0;0;Ez).\n(ii) The electric fields in the two waveguides are opposite to each other, i.e. ~E1= (0;0;Ez)\nand~E2= (0;0;\u0000Ez).\n(iii) The electric field is applied only to waveguide 1. These situations can be achieved by\nelectric gating.\nFor the case (i) with ~E1;2= (0;0;Ez),!0=\r\n1+\u000b2(H0\u00002cEEzkx\n\u00160Ms+2Aex\n\u00160Msk2\nx+JRKKY\n2\u00160Mstp)in Eq. 2,\nand the condition for PT-symmetry still holds. Applying an electric field along the zaxis\ncauses an asymmetry in the magnon dispersion. As shown by Fig. 5, the positive Ezshifts\nthe dispersion towards positive kxwhile a negative Ezshifts it in the opposite direction.\nWith increasing !J, the changes of Re[ !] and Im[!] (not shown) are similar to these in Fig.\n1(b-c).\nAs for the case ~E1= (0;0;Ez)and~E2= (0;0;\u0000Ez), in the two equations (2) !0is\ndifferent. Explicitly: !0=\r\n1+\u000b2(H0\u00072cEEzkx\n\u00160Ms+2Aex\n\u00160Msk2\nx+JRKKY\n2\u00160Mstp)where the\u0000sign applies\nfor WG1 and the +sign corresponds to WG2. Hence, under an asymmetric electric field\nthe potential VRis not even ( VR(z0)6=VR(\u0000z0)), and the PT-symmetry condition can not\nbe satisfied. The !Jdependence of Re[ !] and Im[!] are shown in Fig 5, and no exceptional\npoint can be strictly identified in this case.\nFor case (iii), we set ~E1= (0;0;Ez)and~E2= (0;0;0). The PT-symmetry condition is\nnot satisfied. When the electric field is applied only to a single guide, it shifts selectively the\nmagnon dispersion relation in this guide. Therefore, the magnon wave in the lower frequency\nrange propagates solely in the guide with the electric field. As shown in Fig. 5(d), we excite\nthe magnonic wavepacket with a frequency in the WG1 or WG2, the magnonic wave always\npropagates in the waveguide 1 which amounts to a magnon channeled by the electric field,\nwhile the propagation in the other guide is suppressed. This example illustrates yet another\nhandle to steer magnonic waves swiftly and at low energy consumption by pulsed electric\ngating.\n14Conclusions\nMagnonic waveguides based on magnetic junctions that exhibit a transition from a PT-\nsymmetric to a PT-symmetry broken phase may act near the transition (exceptional) point\nas effective sensor for changes in external fields and in the magnetic environments and also\nserve as magnonic amplifier or magnetic switch. The particular behavior of the waveguides\nmagnetic susceptibility is also reflected in the permeability (cf. supplementary materials)\npointing to a new route to PT-symmetric magneto-photonics. Magnonic propagation is\nhighly controllable by external electric and magnetic fields that can derive the system across\nthe exceptional point and lead to controlled power distribution in the waveguides as well as\nnon-reciprocal or amplified magnon waves. DM interaction allows for dispersion engineering\nvia external electric fields, and for PT-symmetry based large-amplitude spin excitations.\nThese observations underline the potential of PT-symmetric magnonics as the basis for\nadditional functionalities of magnetophotonic, spintronics and cavity magnonic devices that\nare highly controllable by external parameters.\nAcknowledgement\nThis research is financially supported by the DFG through SFB 762 and SFB TRR227,\nNational Natural Science Foundation of China (No. 11704415, 11674400, 11374373), and\nthe Natural Science Foundation of Hunan Province of China (No. 2018JJ3629).\nAuthor contributions\nXGW performed all numerical simulations and analytical modeling. JB conceived and su-\npervised the project. XGW and JB wrote the paper . XGW , JB, and GHG discussed,\ninterpret and agreed on the content.\nCompeting interests:\nThe authors declare no competing\n15financial and non-financial interests.\nFull data availability statements: All technical details for producing the figures are\nenclosed in the supplementary materials. Data are available from the authors upon request.\nSupporting Information Available\nThe following files are available free of charge.\n\u000fpt-symmetry-supp.pdf: Numerical simulation details, dynamic magnetic permeability\nin separated waveguides, and the influence of intrinsic damping.\nReferences\n(1) Chumak, A. V.; Vasyuchka, V. I.; Serga, A. A.; Hillebrands, B. Magnon spintronics.\nNat. Phys. 2015,11, 453–461.\n(2) Wang, Q.; Pirro, P.; Verba, R.; Slavin, A.; Hillebrands, B.; Chumak, A. V. Reconfig-\nurable nanoscale spin-wave directional coupler. Sci. Adv. 2018,4, e1701517.\n(3) Kruglyak, V. V.; Demokritov, S. O.; Grundler, D. Magnonics. J. Phys. D: Appl. Phys.\n2010,43, 264001.\n(4) Vogt, K.; Fradin, F. Y.; Pearson, J. E.; Sebastian, T.; Bader, S. D.; Hillebrands, B.;\nHoffmann, A.; Schultheiss, H. Realization of a spin-wave multiplexer. Nat. Commun.\n2014,5, 3727.\n(5) Chumak, A. V.; Serga, A. A.; Hillebrands, B. Magnon transistor for all-magnon data\nprocessing. Nat. Commun. 2014,5, 4700.\n(6) Sadovnikov, A. V.; Beginin, E. N.; Sheshukova, S. E.; Romanenko, D. V.;\nSharaevskii, Y. P.; Nikitov, S. A. Directional multimode coupler for planar magnonics:\nSide-coupled magnetic stripes. Appl. Phys. Lett. 2015,107, 202405.\n16(7) Ruschhaupt, A.; Delgado, F.; Muga, J. G. Physical realization of PT-symmetric po-\ntential scattering in a planar slab waveguide. J. Phys. A 2005,38, L171–L176.\n(8) Feng,L.; Wong,Z.J.; Ma,R.-M.; Wang,Y.; Zhang,X.Single-modelaserbyparity-time\nsymmetry breaking. Science 2014,346, 972–975.\n(9) Makris, K. G.; El-Ganainy, R.; Christodoulides, D. N.; Musslimani, Z. H. Beam Dy-\nnamics inPTSymmetric Optical Lattices. Phys. Rev. Lett. 2008,100, 103904.\n(10) Rüter, C. E.; Makris, K. G.; El-Ganainy, R.; Christodoulides, D. N.; Segev, M.; Kip, D.\nObservation of parity-time symmetry in optics. Nat. Phys. 2010,6, 192–195.\n(11) Regensburger, A.; Bersch, C.; Miri, M.-A.; Onishchukov, G.; Christodoulides, D. N.;\nPeschel, U. Parity-time synthetic photonic lattices. Nature 2012,488, 167–171.\n(12) Zhang, X.-L.; Wang, S.; Hou, B.; Chan, C. T. Dynamically Encircling Exceptional\nPoints: In situ Control of Encircling Loops and the Role of the Starting Point. Phys.\nRev. X 2018,8, 021066.\n(13) Bender, C. M.; Boettcher, S. Real Spectra in Non-Hermitian Hamiltonians Having PT\nSymmetry. Phys. Rev. Lett. 1998,80, 5243–5246.\n(14) Bender, C. M.; Brody, D. C.; Jones, H. F. Complex Extension of Quantum Mechanics.\nPhys. Rev. Lett. 2002,89, 270401.\n(15) Bender, C. M. Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 2007,\n70, 947–1018.\n(16) Peng, B.; Özdemir, S. K.; Lei, F.; Monifi, F.; Gianfreda, M.; Long, G. L.; Fan, S.;\nNori, F.; Bender, C. M.; Yang, L. Parity-time-symmetric whispering-gallery microcav-\nities. Nat. Phys. 2014,10, 394–398.\n17(17) Lin, Z.; Ramezani, H.; Eichelkraut, T.; Kottos, T.; Cao, H.; Christodoulides, D. N.\nUnidirectional Invisibility Induced by PT-Symmetric Periodic Structures. Phys. Rev.\nLett.2011,106, 213901.\n(18) Nazari, F.; Nazari, M.; Moravvej-Farshi, M. K. A 2 \u00022 spatial optical switch based on\nPT-symmetry. Opt. Lett. 2011,36, 4368–4370.\n(19) Kartashov, Y. V.; Szameit, A.; Vysloukh, V. A.; Torner, L. Light tunneling inhibition\nand anisotropic diffraction engineering in two-dimensional waveguide arrays. Opt. Lett.\n2009,34, 2906–2908.\n(20) Miri,M.-A.; Regensburger,A.; Peschel,U.; Christodoulides,D.N.Opticalmeshlattices\nwithPTsymmetry. Phys. Rev. A 2012,86, 023807.\n(21) Guo, A.; Salamo, G. J.; Duchesne, D.; Morandotti, R.; Volatier-Ravat, M.; Aimez, V.;\nSiviloglou, G. A.; Christodoulides, D. N. Observation of PT-Symmetry Breaking in\nComplex Optical Potentials. Phys. Rev. Lett. 2009,103, 093902.\n(22) Lü, X.-Y.; Jing, H.; Ma, J.-Y.; Wu, Y. PT-Symmetry-Breaking Chaos in Optomechan-\nics.Phys. Rev. Lett. 2015,114, 253601.\n(23) Xu, H.; Mason, D.; Jiang, L.; Harris, J. G. E. Topological energy transfer in an op-\ntomechanical system with exceptional points. Nature 2016,537, 80–83.\n(24) Zhu, X.; Ramezani, H.; Shi, C.; Zhu, J.; Zhang, X. PT-Symmetric Acoustics. Phys.\nRev. X 2014,4, 031042.\n(25) Fleury, R.; Sounas, D.; Alù, A. An invisible acoustic sensor based on parity-time sym-\nmetry. Nat. Commun. 2015,6, 5905.\n(26) Schindler, J.; Li, A.; Zheng, M. C.; Ellis, F. M.; Kottos, T. Experimental study of active\nLRC circuits with PTsymmetries. Phys. Rev. A 2011,84, 040101.\n18(27) Schindler, J.; Lin, Z.; Lee, J. M.; Ramezani, H.; Ellis, F. M.; Kottos, T. PT-symmetric\nelectronics. J. Phys. A 2012,45, 444029.\n(28) Assawaworrarit, S.; Yu, X.; Fan, S. Robust wireless power transfer using a nonlinear\nparity-time-symmetric circuit. Nature 2017,546, 387–390.\n(29) Chen, P.-Y.; Sakhdari, M.; Hajizadegan, M.; Cui, Q.; Cheng, M. M.-C.; El-Ganainy, R.;\nAlù,A.Generalizedparity-timesymmetryconditionforenhancedsensortelemetry. Nat.\nElectron. 2018,1, 297–304.\n(30) Lee, J. M.; Kottos, T.; Shapiro, B. Macroscopic magnetic structures with balanced gain\nand loss. Phys. Rev. B 2015,91, 094416.\n(31) Galda, A.; Vinokur, V. M. Parity-time symmetry breaking in magnetic systems. Phys.\nRev. B 2016,94, 020408.\n(32) Galda, A.; Vinokur, V. M. Parity-time symmetry breaking in spin chains. Phys. Rev.\nB2018,97, 201411.\n(33) Zhang,D.; Luo,X.-Q.; Wang,Y.-P.; Li,T.-F.; You,J.Q.Observationoftheexceptional\npoint in cavity magnon-polaritons. Nat. Commun. 2017,8, 1368.\n(34) Cao, Y.; Yan, P. Exceptional magnetic sensitivity of PT-symmetric cavity magnon\npolaritons. Phys. Rev. B 2019,99, 214415.\n(35) Yang, H.; Wang, C.; Yu, T.; Cao, Y.; Yan, P. Antiferromagnetism Emerging in a\nFerromagnet with Gain. Phys. Rev. Lett. 2018,121, 197201.\n(36) Krivorotov, I. N.; Emley, N. C.; Sankey, J. C.; Kiselev, S. I.; Ralph, D. C.;\nBuhrman, R. A. Time-Domain Measurements of Nanomagnet Dynamics Driven by\nSpin-Transfer Torques. Science 2005,307, 228–231.\n19(37) Liu, L.; Lee, O. J.; Gudmundsen, T. J.; Ralph, D. C.; Buhrman, R. A. Current-Induced\nSwitching of Perpendicularly Magnetized Magnetic Layers Using Spin Torque from the\nSpin Hall Effect. Phys. Rev. Lett. 2012,109, 096602.\n(38) Garello, K.; Miron, I. M.; Avci, C. O.; Freimuth, F.; Mokrousov, Y.; Blügel, S.; Auf-\nfret,S.; Boulle,O.; Gaudin,G.; Gambardella,P.Symmetryandmagnitudeofspin-orbit\ntorques in ferromagnetic heterostructures. Nat. Nanotechnol. 2013,8, 587–593.\n(39) Hoffmann, A. Spin Hall Effects in Metals. IEEE Trans. Magn. 2013,49, 5172–5193.\n(40) Collet, M.; de Milly, X.; d’Allivy Kelly, O.; Naletov, V. V.; Bernard, R.; Bortolotti, P.;\nBen Youssef, J.; Demidov, V. E.; Demokritov, S. O.; Prieto, J. L.; Muñoz, M.; Cros, V.;\nAnane, A.; de Loubens, G.; Klein, O. Generation of coherent spin-wave modes in yt-\ntrium iron garnet microdiscs by spin-orbit torque. Nat. Commun. 2016,7, 10377.\n(41) Duine, R. A.; Lee, K.-J.; Parkin, S. S. P.; Stiles, M. D. Synthetic antiferromagnetic\nspintronics. Nat. Phys. 2018,14, 217–219.\n(42) Heinrich, B.; Burrowes, C.; Montoya, E.; Kardasz, B.; Girt, E.; Song, Y.-Y.; Sun, Y.;\nWu, M. Spin Pumping at the Magnetic Insulator (YIG)/Normal Metal (Au) Interfaces.\nPhys. Rev. Lett. 2011,107, 066604.\n(43) Konovalenko, A.; Lindgren, E.; Cherepov, S. S.; Korenivski, V.; Worledge, D. C. Spin\ndynamicsoftwo-couplednanomagnetsinspin-floptunneljunctions. Phys. Rev. B 2009,\n80, 144425.\n(44) Klaiman, S.; Günther, U.; Moiseyev, N. Visualization of Branch Points in PT-\nSymmetric Waveguides. Phys. Rev. Lett. 2008,101, 080402.\n(45) Baumgartner,M.; Garello,K.; Mendil,J.; Avci,C.O.; Grimaldi,E.; Murer,C.; Feng,J.;\nGabureac, M.; Stamm, C.; Acremann, Y.; Finizio, S.; Wintz, S.; Raabe, J.; Gam-\n20bardella, P. Spatially and time-resolved magnetization dynamics driven by spin-orbit\ntorques. Nat. Nanotechnol. 2017,12, 980–986.\n(46) Osada, A.; Hisatomi, R.; Noguchi, A.; Tabuchi, Y.; Yamazaki, R.; Usami, K.; Sad-\ngrove, M.; Yalla, R.; Nomura, M.; Nakamura, Y. Cavity Optomagnonics with Spin-\nOrbit Coupled Photons. Phys. Rev. Lett. 2016,116, 223601.\n(47) Zakeri, K.; Zhang, Y.; Prokop, J.; Chuang, T.-H.; Sakr, N.; Tang, W. X.; Kirschner, J.\nAsymmetric Spin-Wave Dispersion on Fe(110): Direct Evidence of the Dzyaloshinskii-\nMoriya Interaction. Phys. Rev. Lett. 2010,104, 137203.\n(48) Moon,J.-H.; Seo,S.-M.; Lee,K.-J.; Kim,K.-W.; Ryu,J.; Lee,H.-W.; McMichael,R.D.;\nStiles, M. D. Spin-wave propagation in the presence of interfacial Dzyaloshinskii-Moriya\ninteraction. Phys. Rev. B 2013,88, 184404.\n(49) Wang, X.-g.; Chotorlishvili, L.; Guo, G.-h.; Berakdar, J. High-Fidelity Magnonic Gates\nfor Surface Spin Waves. Phys. Rev. Applied 2019,12, 034015.\n(50) Zhang, X.; Liu, T.; Flatté, M. E.; Tang, H. X. Electric-Field Coupling to Spin Waves\nin a Centrosymmetric Ferrite. Phys. Rev. Lett. 2014,113, 037202.\n21"    },    {        "title": "2305.14753v1.Time_reversal_Invariance_Violation_and_Quantum_Chaos_Induced_by_Magnetization_in_Ferrite_Loaded_Resonators.pdf",        "content": "Time-reversal Invariance Violation and Quantum Chaos Induced by Magnetization in\nFerrite-Loaded Resonators\nWeihua Zhang,1, 2Xiaodong Zhang,1and Barbara Dietz1, 2,∗\n1Lanzhou Center for Theoretical Physics and the Gansu Provincial Key Laboratory of Theoretical Physics,\nLanzhou University, Lanzhou, Gansu 730000, China\n2Center for Theoretical Physics of Complex Systems,\nInstitute for Basic Science (IBS), Daejeon 34126, Korea\n(Dated: May 25, 2023)\nWe investigate the fluctuation properties in the eigenfrequency spectra of flat cylindrical mi-\ncrowave cavities that are homogeneously filled with magnetized ferrite. These studies are motivated\nby experiments in which only small pieces of ferrite were embedded in the cavity and magnetized\nwith an external static magnetic field to induce partial time-reversal ( T) invariance violation. We\nuse two different shapes of the cavity, one exhibiting an integrable wave dynamics, the other one a\nchaotic one. We demonstrate that in the frequency region where only transverse-magnetic modes\nexist, the magnetization of the ferrites has no effect on the wave dynamics and does not induce\nT-invariance violation whereas it is fully violated above the cutoff frequency of the first transverse-\nelectric mode. Above all, independently of the shape of the resonator, it induces a chaotic wave\ndynamics in that frequency range in the sense that for both resonator geometries the spectral prop-\nerties coincide with those of quantum systems with a chaotic classical dynamics and same invariance\nproperties under application of the generalized Toperator associated with the resonator geometry.\nI. INTRODUCTION\nFlat, cylindrical microwave resonators are used since three decades to investigate in the context of Quantum\nChaos [1–9] the properties of the eigenfrequencies and wave functions of quantum systems with a chaotic dynamics\nin the classical limit [10–15]. Here, the analogy to quantum billiards of corresponding shape is used, which holds\nbelow the frequency of the first transverse-magnetic mode. Namely, in the frequency range, where the electric field\nstrength is parallel to the cylinder axis, the associated Helmholtz equation is identical to the Schr¨ odinger equation\nof the quantum billiard of corresponding shape. The classical counterpart of a quantum billiard consists of a two-\ndimensional bounded domain, in which a point-like particle moves freely and is reflected specularly on impact with\nthe boundary [16–19]. The dynamics of the billiard depends only on its shape. Therefore, such systems provide a\nsuitable model to investigate signatures of the classical dynamics in properties of the associated quantum system.\nExperiments have also been performed with three-dimensional microwave resonators [13, 20–24], where the analogy\nto the quantum billiard of corresponding shape is lost, because of the vectorial nature of the Helmholtz equations.\nTheir objective was the study of wave-dynamical chaos.\nIn the above mentioned experiments properties of quantum systems with a chaotic classical counterpart and pre-\nserved time-reversal ( T) invariance were investigated. In order to induce T-invariance violation in a quantum system\na magnetic field is introduced. Quantum billiards with partially violated Tinvariance were modeled experimentally\nwith flat, cylindrical microwave resonators containing one or more pieces of ferrite that were magnetized with an\nexternal magnetic field [25–32]. Time-reversal invariance is violated through the coupling of the spins in the ferrite,\nwhich precess with the Larmor frequency about the external magnetic field, to the magnetic-field component of the\nresonator modes, which depends on the rotational direction of polarization of the latter. The effect, that leads to\nT-invariance violation is especially pronounced in the vicinity of the ferromagnetic resonance and its harmonics or\nat the resonance frequencies of the ferrite piece that lead to trapped modes inside in it [30]. The motivation of the\npresent study is the understanding of the electrodynamical processes that take place inside a magnetized cylindrical\nferrite, especially of its wave-dynamical properties and dependence on its shape.\nIn this work we compute with COMSOL multiphysics the eigenfrequencies and electric field distributions of flat,\ncylindrical metallic resonators, that are homogeneously filled with fully magnetized ferrite material and investigate\nthe fluctuation properties in the eigenfrequency spectra in the realm of Quantum Chaos. We choose two different\nshapes, one which has the shape of a billiard with integrable dynamics, the other one with chaotic dynamics. The\nspectral properties are studied in the frequency range where only transverse-electric modes exist and the Helmholtz\n∗bdietzp@gmail.comarXiv:2305.14753v1  [nlin.CD]  24 May 20232\nequation is scalar, and in the region where it is vectorial. In Sec. II we review properties of ferrite and the associated\nwave equations. They reduce to that of the quantum billiard of corresponding shape with no magnetic field in the\ntwo-dimensional case. In Sec. III we present the models that are investigated and in Sec. IV the results for the\nspectral properties. Interestingly, in the region where also transverse-magnetic modes exist, even for the resonator\nwith the shape corresponding to a dielectric-loaded cavity with integrable wave dynamics, they coincide with those\nof a quantum system with chaotic classical dynamics. Our findings are discussed in Sec. V.\nII. REVIEW OF THE WAVE EQUATIONS FOR A METALLIC RESONATOR HOMOGENEOUSLY\nFILLED WITH MAGNETIZED FERRITE\nA ferrite is a non-conductive ceramic with a ferrimagnetic crystal structure. Similar to antiferromagnets, it consists\nof different sublattices whose magnetic moments are opposed and differ in magnitudes. When applying a static\nexternal magnetic field, these magnetic moments become aligned. This process can be effectively described as a\nmacroscopic magnetic moment. We consider a flat cylindrical resonator made of ferrite [33, 34] which is magnetized\nby a static magnetic field perpendicular to the resonator plane, and enclosed by a perfect electric conductor (PEC).\nThe macroscopic Maxwell equations for an electromagnetic field with harmonic time variation [35],\n⃗E(⃗ x, t) =⃗E(⃗ x)e−iωt,⃗B(⃗ x, t) =⃗B(⃗ x)e−iωt(1)\npropagating through the resonator read\n⃗∇ × ⃗E =iω⃗B (2)\n⃗∇ ×(ˆµ−1⃗B) =−iϵr(⃗ x)\nc2ω⃗E (3)\n⃗∇ ·⃗B = 0 (4)\n⃗∇ ·⃗E = 0, (5)\nwhere c=1√ϵ0µ0is the speed of light in vacuum, ϵ0andµ0are the permittivity and permeability in vacuum and\nϵris the relative permittivity, which due to the assumed homogeneity of the material does not depend on ⃗ xin\nthe bulk of the resonator. Furthermore, ˆ µrdenotes the permeability tensor, which may be expressed in terms\nof the susceptibility tensor ˆ χas ˆµr= 1+ ˆχ, and results from the magnetization ⃗M(⃗ x) = ˆχ⃗H(⃗ x) of the ferrite,\n⃗B(⃗ x) =µ0ˆµr⃗H(⃗ x) =µ0\u0010\n⃗H(⃗ x) +⃗M(⃗ x)\u0011\n.\nCombination of the first two equations yields wave equations for ⃗E(⃗ x) and ⃗B(⃗ x) [14, 25],\n⃗∇ ×h\nˆµ−1\nr∇ × ⃗E(⃗ x)i\n−ϵr(⃗ x)k2\n0⃗E(⃗ x) = 0 , (6)\n⃗∇ ×n\n⃗∇ ×h\nˆµ−1\nr⃗B(⃗ x)io\n−ϵr(⃗ x)k2\n0⃗B(⃗ x) = 0 , (7)\nwith k0=ω\ncdenoting the wavenumber in vacuum. The magnetization follows the equation of motion\nd⃗M\ndt=|γ|⃗M×⃗H, (8)\nresulting from the torque exerted by the magnetic field ⃗H(⃗ x) on ⃗M(⃗ x). Here, γ=−2.21·105mA−1s−1denotes the\ngyromagnetic ratio of the electron. The magnetic field ⃗H(⃗ x) is composed of the static magnetic field ⃗H0=H0⃗ ezapplied\nin the direction of the cylinder axis of the resonator, which is chosen parallel to the zaxis, and the magnetic-field\ncomponent of the electromagnetic field,\n⃗H(⃗ x) =H0⃗ ez+⃗h(⃗ x)eiωt. (9)\nWe assume that the strength H0is sufficiently large to ensure that the magnetization attains its saturation value\nMs=χ0H0, with χ0denoting the static susceptibility. Similarly, ⃗M(⃗ x) is given by a superposition of the static\nmagnetization Ms⃗ ezand the magnetization resulting from the electromagnetic field,\n⃗M=Ms⃗ ez+⃗ meiωt. (10)3\nThe magnetization ˆMis obtained by inserting Eqs. (9) and (10) into Eq. (8). We may assume that the contributions\noriginating from the electromagnetic field in Eqs. (9) and (10) are sufficiently small compared to that of the static\nparts, so that only terms linear in ⃗hand⃗ mneed to be taken into account, yielding ⃗M(⃗ x) = ˆχ⃗H(⃗ x) with\nˆχ=\nχ−iκ0\niκ χ 0\n0 0 χ0\n, (11)\nandχ(ω) =ωLωM\nω2\nL−ω2, κ(ω) =ωωM\nω2\nL−ω2. Here, ωM=|γ|MandωL=|γ|H0denote the precession frequency about the\nsaturation magnetization ⃗Msand the Larmor frequency, with which the magnetization ⃗Mpresesses about ⃗H0, respec-\ntively. The quantities χ(ω) and κ(ω) exhibit a pronounced resonance behavior around the ferromagnetic resonance\nω=ωL. With these notations the inverse of ˆ µris given by\nˆµ−1\nr= (1 + ˆχ)−1=\n1+χ\nδiκ\nδ0\n−iκ\nδ1+χ\nδ0\n0 01\n1+χ0\n, δ= (1 + χ)2−κ2. (12)\nThe resonators under consideration have a cylindrical shape with a non-circular cross section and the external magnetic\nfield is constant and perpendicular to the resonator plane. Furthermore, the ferrite material is homogeneous, that\nis, in the bulk the entries of ˆ µrandϵrare spatially constant and only depend on the angular frequency ωof the\nelectromagnetic field. This is distinct from the experiments presented in Refs. [25–32], where Tinvariance violation\nwas induced by inserting cylindrical ferrites with circular cross section into an evacuated metallic resonator. There,\nthe origin of the Tinvariance violation is the coupling of the spins of the ferrite to the magnetic field components of\nthe electromagnetic field excited in the resonator, which depends on their rotational direction. It is strongest in the\nvicinity of the ferromagnetic resonances and at resonance frequencies of the ferrite, were modes are trapped in it [30].\nIn these microwave resonators, ˆ µrandϵrare spatially dependent, since they experience a jump at the surface of the\nferrite.\nFor a ferrite enclosed by a PEC the boundary conditions are given by\n⃗ n(⃗ xS)×⃗E=⃗0, ⃗ n(⃗ xS)·⃗B= 0, (13)\nwith ⃗ n(⃗ xS) denoting the normal to the ferrite surface at ⃗ xSpointing away from the resonator. Here, the surface\ncharge density and surface current density may be neglected due to the high resistivity of the ferrite. Defining\n⃗E=Ex⃗ ex+Ey⃗ ey+Ez⃗ ez, this yields at the bottom and top planes of a flat, cylindrical resonator of height h, where\n−⃗ n(x, y, z = 0) = ⃗ n(x, y, z =h) =⃗ ez, the boundary conditions\nEx(x, y, z = 0) = Ex(x, y, z =h) =Ey(x, y, z = 0) = Ey(x, y, z =h) = 0 . (14)\nAlong the side wall they read\nnxEy=nyEx, Ez= 0, ⃗ nt= (nx(s), ny(s),0) (15)\nwith sparametrizing the contour of the resonator in a plane parallel to the ( x, y) plane. This condition leads to a\ncoupling of ExandEy.\nAccordingly, we may separate the electromagnetic field into modes propagating in the resonator plane, denoted by\nan index tand modes perpendicular to it, i.e., in zdirection.\n⃗E=⃗Et+Ez⃗ ez,⃗B=⃗Bt+Bz⃗ ez,⃗∇=⃗∇t+⃗ ez∂\n∂z(16)\nand\nˆµ−1\nr⃗∇= ˆm⃗∇t+m0⃗ ez∂\n∂z,ˆµ−1\nr⃗B= ˆm⃗Bt+m0Bz⃗ ez (17)\nwith\nˆm=\u00121+χ\nδiκ\nδ\n−iκ\nδ1+χ\nδ\u0013\n, m0=1\n1 +χ0. (18)\nFurthermore, due to the cylindrical shape we may assume that\n⃗E(x, y, z ) =⃗E(x, y)e−ikzz,⃗B(x, y, z ) =⃗B(x, y)e−ikzz. (19)4\nThe electromagnetic waves are reflected at the PECs terminating the resonator at the top and bottom, implying that\nkz=qπ\nh, q= 0,1, . . . , (20)\nthat is, k2=k2\nt+k2\nz=k2\nt+q2\u0000π\nh\u00012fork≥qπ\nh.\nThe Maxwell equations become\niω⃗Bt=h\nikz⃗Et+⃗∇tEzi\n×⃗ ez, iωB z=h\n⃗∇t×⃗Eti\n·⃗ ez (21)\n−iϵr(⃗ x)\nc2ω⃗Et=h\nm0⃗∇tBz+ikzˆm⃗Bti\n×⃗ ez, −iϵr(⃗ x)\nc2ωEz=h\n⃗∇t×\u0010\nˆm⃗Bt\u0011i\n·⃗ ez (22)\n⃗∇t·⃗Et=ikzEz, ⃗∇t·⃗Bt=ikzBz. (23)\nThe in-plane modes can be expressed in terms of the modes perpendicular to the plane. For this we insert the first\nequation of Eq. (21) into the first one of Eq. (22) and vice versa yielding\n−i\u0002\nϵrk2\n01−k2\nzˆm\u0003⃗Et=ωm0⃗∇tBz×⃗ ez−kzˆm⃗∇tEz (24)\ni\u0002\nϵrk2\n01−k2\nzˆm\u0003⃗Bt=kzm0⃗∇tBz+ϵr(⃗ x)\nc2ω⃗∇tEz×⃗ ez. (25)\nThe wave equation Eq. (6) can also be separated into in-plane modes and modes perpendicular to the resonator plane.\nNamely,\n⃗∇ ×h\nˆµ−1\nr⃗∇ × ⃗E(⃗ x)i\n= ˆµ−1\nr⃗∇\u0010\n⃗∇ ·⃗E\u0011\n+\u0010\nˆµ−1\nr⃗∇\u0011← −∇ ·⃗E−⃗∇ ·\u0010\nˆµ−1\nr⃗∇\u0011\n⃗E (26)\nwhere the gradient← −∇is applied to the term to its left. According to Eq. (5) the first term on the right hand side\nvanishes. Inserting this equation into Eq. (6) and separating into modes in the resonator plane and perpendicular to\nit yields\nh\n⃗∇t·\u0010\n1+χ\nδ⃗∇t\u0011\n−ikz∂\n∂zn\n1\n1+χ0o\n+i⃗∇t·\u0010\n⃗ ez×κ\nδ⃗∇t\u0011i\n⃗E (27)\n−\n\u0010\n⃗∇\b1+χ\nδ\t∂\n∂x−⃗∇\biκ\nδ\t∂\n∂y\u0011\n·⃗E\u0010\n⃗∇\b1+χ\nδ\t∂\n∂y+⃗∇\biκ\nδ\t∂\n∂x\u0011\n·⃗E\n−ikz⃗∇n\n1\n1+χ0o\n·⃗E\n=\u0010\n1\n1+χ0k2\nz−ϵrk2\n0\u0011\n⃗E, (28)\nwhere curly brackets mean that ⃗∇is only applied to the terms framed by them. For q= 0 in Eq. (20), i.e., kz= 0\nthe electric field is perpendicular to the resonator plane, ⃗E(r) =E(x, y)⃗ ezand Eq. (27) becomes\n\u0014\n⃗∇t·\u00121 +χ\nδ⃗∇t\u0013\n+i⃗∇t·\u0010\n⃗ ez×κ\nδ⃗∇t\u0011\u0015\nE(x, y) =−ϵrk2\n0E(x, y), E(x, y)\f\f\n∂Ω= 0 (29)\nwith Dirichlet boundary conditions along the boundary ∂Ω. For the case considered here, i.e., for spatially constant\nˆµr, the wave equation reduces to the scalar Helmholtz equation\n∆tE(x, y) =−ϵrδ\n1 +χk2\n0E(x, y) =−k2E(x, y), E(x, y)\f\f\n∂Ω= 0 (30)\nyielding the dispersion relation\nk=s\nϵrδ\n1 +χk0=s\nϵr(ωL+ωM)2−ω2\nωL(ωL+ωM)−ω2k0. (31)\nEquations (29)- (31) hold up to\nkcrit=π\nh(32)5\nor, equivalently, with ¯ ω2=(ωL+ωM)2+\u0010\ncπ\nh√ϵr\u00112\n2\nωcrit=vuut¯ω2±s\n¯ω4−ωL(ωM+ωL)\u0012cπ\nh√ϵr\u00132\n, (33)\nwhere for vanishing external field, i.e., for ωL=ωM= 0 the plus sign has to be taken in the radicand, yielding\nωcrit\nr=cπ\nh√ϵr. (34)\nFor nonzero ωLandωMthe minus sign applies. Thus, below the cutoff circular frequency of the first transverse-\nelectric mode, referred to as critical in the following, ωcritthe wave equation coincides with that of a quantum\nbilliard [16, 17, 19] in a dispersive medium [36–39]. This correspondence between quantum billiards and the scalar\nHelmholtz equation of flat, cylindrical microwave cavities has been used in numerous experiments to determine their\neigenvalues and eigenfunctions [10–12, 14, 15]. The corresponding classical billiard consists of a point particle which\nmoves freely inside a bounded two-dimensional domain and is reflected specularly at the wall.\nIII. NUMERICAL ANALYSIS\nWe investigated the spectral properties of ferrite-loaded metallic resonators with the shapes shown in Fig. 1, using\nCOMSOL multiphysics. We set the properties of the ferrite material to those of 18G 3ferrite from the Y-Ga-In\nseries, which has a low loss, of which the relative permittivity and saturation magnetization are εr= 14 .5 and\nMs= 1.47·105A/m, respectively. The other parameters are given in Tab. I.\nShape h AreaA ωL ωM ωcrit\nr ωcrit\nSector 20 mm 0.3355 m287.96 GHz 32.49 GHz 24.71 GHz 10.56 GHz\nAfrica 10 mm 0.0377 m243.98 GHz 32.49 GHz 12.35 GHz 18.34 GHz\nTABLE I. Parameters for the two resonator realizations.\nThe sector has a radius of 800 mm. The wave dynamics of microwave resonators with this shape is integrable [13, 20–\n23]. The boundary of the Africa shape [ x(r, φ), y(r, φ)] is defined in the complex plane w(r, φ) =x(r, φ) +iy(r, φ)\nby\nw(r, φ) =r0\u0010\nz+ 0.2z2+ 0.2z3eiπ/3\u0011\n, (35)\nwith z=reiφandr0= 100 mm. Below ωcritthe wave equation Eq. (27) reduces to the Schr¨ odinger equation for\nFIG. 1. Sketch of the ferrite-loaded cavities, which have the shape of a circle-sector billiard (left) with inner angle 60◦and of a\nAfrica billiard (right). The cavity is made of a PEC (brown lines), and the filling consists of magnetized ferrite (gray domain).\nthe quantum billiard of corresponding shape; see Eq. (30). For the sector quantum billiard the solutions of Eq. (30),6\nnamely, the eigenvalues kp,νand eigenfunctions Ψ p,ν(r, φ) are known,\nΨp,ν(r, φ) = sin\u00123\n2pφ\u0013\nJ3\n2p(kp,νr), J 3\n2p(kp,νr0) = 0 , (36)\nwhereas for the Africa billiard the dynamics is fully chaotic [40] so that they need to be computed numerically, e.g.,\nwith the boundary integral method [41]. We computed the eigenstates with COMSOL multiphysics which employes\na finite element method using the parameters listed in Tab. I. Note, that beyond the critical frequency ω≳ωcrit,\nwhere the Helmholtz equation becomes three-dimensional, the analogy to the three-dimensional quantum billiard of\ncorresponding shape is lost.\nIV. RESULTS\nA. Electric-Field Distributions\nBelow the critical frequency fcrit=ωcrit\n2πcorresponding to kz= 0, the electric field is perpendicular to the resonator\nplane ⃗E(x, y)eikzz=E(x, y)⃗ ez. Figures 2 and 3 present examples for the electric-field distributions E(x, y) of the\nsector and Africa resonators for four eigenfrequencies f=ω\n2π. As expected, the electric-field components in the\nFIG. 2. Electric field distributions for the sector-shaped resonator for ω < ωcritin the z= 10 mm plane for, from top left to\nbottom right, f=0.6598 GHz, 0.6625 GHz, 0.6649 GHz, 0.6668 GHz.\nresonator plane, Ex(r) and Ey(r), are identical to zero, and Ez(r) is constant in z-direction. This is no longer the\ncase for frequencies beyond the critical frequency, ω≳ωcritwhere for all components the zdependence is given\naccording to Eq. (20) by e−iqπ\nhzwith q≥1 and ⃗E(x, y) is governed by the wave equation Eq. (27) together with\nthe boundary conditions Eq. (14) and Eq. (15). In Fig. 4 and Fig. 5 we show examples for the case q= 1 for the\nsector- and Africa-shaped resonators, respectively. In the top and bottom plane the electric field has opposite signs\nfor given values ( x, y) as illustrated in the first row of both figures. Furthermore, in the\u0000\nz=h\n2\u0001\n-plane it vanishes,\nthus confirming that the zdependence is given by sin qπ\nhz. The second row of both figures shows ⃗E(x, y),Ex(x, y)\nandEy(x, y) for z=h\n2. We also confirmed that they fulfill the boundary condition Eq. (14), that is, vanish in the\ntop and bottom plane.7\nFIG. 3. Electric field distributions for the Africa-shaped resonator for ω < ωcritin the z= 5 mm plane for, from top left to\nbottom right, f= 0.9303 GHz, 0.9395 GHz, 0.9540 GHz, 0.9854 GHz.\nFIG. 4. Electric field distribution for the sector-shaped resonator in the ( x, y) plane at the eigenfrequency f= 2.0142 GHz,\nfor which q= 1. The top left and right figures show Ez(x, y) in the bottom ( z= 0) and top ( z=h) planes, respectively. The\nbottom left and right figures show Ex(x, y) and Ey(x, y) in the z=h\n2plane.\nB. Spectral Properties\nA central prediction within the field of Quantum Chaos is the Bohigas-Gianonni-Schmit (BGS) conjecture [42–\n44], which states that for typical quantum systems, whose corresponding classical dynamics is chaotic, the universal\nfluctuation properties in the eigenvalue spectra coincide with those of random matrices from the Gaussian orthogonal\nensemble (GOE) if Tinvariance is preserved, and from the Gaussian unitary ensemble (GUE) if it is violated [9, 14,\n19, 45]. On the other hand, if the classical dynamics is integrable they are well described by uncorrelated random8\nFIG. 5. Electric field distribution for the Africa-shaped resonator in the ( x, y) plane at the eigenfrequency f= 2.9652 GHz,\nfor which q= 1. The top left and right figures show Ez(x, y) in the bottom ( z= 0) and top ( z=h) planes, respectively. The\nbottom left and right figures show Ex(x, y) and Ey(x, y) in the z=h\n2plane.\nnumbers drawn from a Poisson process according to the Berry-Tabor (BT) conjecture [46]. To obtain information on\nuniversal fluctuation properties in the eigenfrequency spectra of the ferrite resonators, system-specific properties need\nto be extracted, that is, the eigenfrequencies have to be unfolded to a uniform average spectral density, respectively,\nto average spacing unity. Below ωcritthe integrated spectral density is well described by Weyl’s formula [47], as\nlong as the frequency interval is chosen such that the frequency dependence of the dispersion factor in Eq. (31) can\nbe neglected. Then, according to Weyl’s formula, the smooth part of the integrated spectral density is given by\nNWeyl(k) =A\n4πk2+L\n4πk+C0withAandLdenoting the area and perimeter of the resonator shape. Unfolding is\nachieved by replacing the eigenwavenumbers kpof Eq. (30) by the Weyl term ϵp=NWeyl(kp) [9].\nAbove the critical frequency, the Helmholtz equation becomes vectorial. For a three-dimensional metallic cavity\nwith a non-dispersive medium the smooth part of the integrated spectral density is given by a polynomial of third\norder in k[48], where the quadratic term vanishes. For a dispersive medium, like the cavities filled with magnetized\nferrite, it still provides a good description of the smooth part of the integrated spectral density, if the frequency\nrange is chosen such that the variation of the dispersion term with frequency is small. In Fig. 6 we show as red\nsolid lines the fluctuating part of the integrated spectral density, Nfluc(k) =N(k)−NWeyl(k) for the sector- and\nAfrica-shaped resonators. The wave dynamics of a three-dimensional sector-shaped PEC cavity is integrable, whereas\nthe Africa-shaped one comprises non-chaotic bouncing-ball orbits corresponding to microwaves that bounce back and\nforth between the top and bottom plate [22, 24, 49]. These occur in both resonators for ω≳ωcrit, that is, kz≥π\nh.\nThey are non-universal, since they depend on the height of the cavity and lead to deviations from BGS predictions\nfor cavities with otherwise chaotic dynamics, which are similar to those induced by the bouncing-ball orbits in the\ntwo- and three-dimensional stadium billiard [24, 49–51]. The slow oscillations Nosc(kp) in the fluctuating part of the\nintegrated spectral density, depicted as dashed black curves in Fig. 6, originate from these bouncing-ball orbits. We\nremoved them by unfolding the eigenvalues with ϵp=NWeyl(kp) +Nosc(kp) [24, 49] which, in addition to the smooth\nvariation, takes into account these oscillations. The resulting ˜Nfluc(kp) =Nfluc(kp)−Nosc(kp) is shown as thin\ndashed turquoise line.\nIn Fig. 7 we show length spectra, that is, the modulus of the Fourier transform of the fluctuating part of the spectral\ndensity from wavenumber to length. They are named length spectra because they exhibit peaks at the lengths of\nperiodic orbits of the corresponding classical system, as may be deduced from the semiclassical approximation for the\nfluctuating part of the spectral density [52, 53]. Shown are the length spectra for the sector-shaped (left) and Africa-\nshaped (right) resonators (turquoise solid lines) below the critical frequency compared to those computed from the\neigenvalues of the quantum billiard of corresponding geometry taking into account a similar number of eigenvalues\n(red solid lines). To match the lengths of the periodic orbits we employed the dispersion relation Eq. (31) which9\nFIG. 6. Fluctuating part of the integrated spectral density (red curves) and the slow oscillations (dashed black curve) resulting\nfrom bouncing-ball orbits for ω≳ωcritfor the sector-shaped (left) and Africa-shaped (right) resonators. The cyan curves show\nthe fluctuations after removal of the contributions from bouncing-ball orbits.\nprovides the relation between the eigenwave numbers of the empty metallic cavity, i.e., the quantum billiard, and\nferrite-loaded one. The black diamonds mark the lengths of classical periodic orbits. The agreement between the\nlength spectra is very good, as may be deduced from the fact that below the critical frequency the underlying wave\nequations are mathematically equivalent. Above the critical frequency this analogy is lost, because of the different\nstructures of the wave equation for an empty metallic cavity [35] and Eq. (27) for a cavity filled with magnetized\nferrite and the implicated dispersion relation, which also becomes vectorial.\nFIG. 7. Comparison of the length spectra of the sector-shaped and Africa-shaped resonators below the critical frequency\n(turquoise curves) with those obtained from the lowest 300 eigenvalues of the quantum billiard of corresponding shape (red\nlines). The black diamonds mark the lengths of classical periodic orbits.\nTo study the spectral properties of the ferrite-loaded resonators we analyzed the nearest-neighbor spacing distribu-\ntionP(s), the integrated nearest-neighbor spacing distribution I(s), the number variance Σ2(L) and the Dyson-Mehta\nstatistic ∆ 3(L), which is a measure for the rigidity of a spectrum [45, 54]. Furthermore, we computed distributions of\nthe ratios [55, 56] of consecutive spacings between nearest neighbors, rj=ϵj+1−ϵj\nϵj−ϵj−1. These are dimensionless implying\nthat unfolding is not required [55–57]. For the sector-shaped resonator, the spectral properties below (red histograms\nand dashed lines) and above (cyan histograms, circles and dots) the critical frequency fcrit= 1.68 GHz are shown\nin Fig. 8. They agree well with those of Poissonian random numbers, and thus with those of the corresponding\nquantum billiard below fcritand exhibit GOE statistics in the other case. Here, we used 500 eigenfrequencies in the\nfrequency range f∈[0.0854,1.5010] GHz and 321 eigenfrequencies in the range f∈[1.7104,1.9314] GHz, respectively.\nFor the Africa-shaped resonator, the spectral properties below (red histograms and dashed lines) and above (cyan10\nFIG. 8. Spectral properties of the eigenfrequencies of the sector-shaped resonator below (red histogram and dashed lines) and\nabove (cyan histograms, circles and dots) the critical frequency. They are compared to the spectral properties of Poissonian\nrandom numbers (dashed-dotted black lines), GOE (black solid lines), and the GUE (dashed black lines).\nhistograms, circles and dots) fcrit= 2.92 GHz are shown in Fig. 9, and agree with those of the corresponding quantum\nbilliard, that is with GOE below fcrit. Above fcritthey are well described by the GUE. We used 229 eigenfrequencies\nin the frequency range f∈[0.2115,2.6333] GHz and 309 eigenfrequencies in the range f∈[3.1655,3.9094] GHz.\nV. DISCUSSION AND CONCLUSIONS\nFor both realizations of a PEC resonator loaded with magnetized ferrite, the spectral properties agree with those of\nthe corresponding quantum billiard for f≲fcrit. Above the critical frequency, the spectral properties of the sector-\nshaped resonator coincide with those of random matrices from the GOE, implying that there the wave dynamics is\nchaotic, even though the shape corresponds to that of a three-dimensional billiard with integrable classical dynamics.\nAbove all, the spectral properties of a sector-shaped PEC resonator filled with a homogeneous dielectric exhibit\nPoissonian statistics [36, 58], that is, their wave dynamics is integrable. Thus we may conclude that the GOE\nbehavior of the sector-shaped resonator and the GUE behavior of the Africa-shaped one have their origin in the\nmagnetization of the ferrite, as may also be concluded from the structure of the wave equation Eq. (27). It comprises\npurely complex parts containing derivatives of the entries of ˆ µr, which are spatially independent in the bulk of the11\nFIG. 9. Same as Fig. 8 for the eigenfrequencies of the Africa-shaped resonator.\nferrite but experience jumps at the ferrite surface where it is terminated with a PEC. Thereby, the electric-field\ncomponents of ⃗E(x, y) are coupled for non-vanishing static external magnetic field H0, thus leading to the complexity\nof the dynamics. For H0= 0, that is for a dielectric medium, ˆ µrequals the identity matrix, so that such a coupling is\nabsent. The spectral properties of the sector-shaped resonator do not exhibit GUE behavior, but are well described\nby GOE statistics. This is attributed to the mirror symmetry, which implies a generalized Tinvariance [9]. In the\nexperiments presented in Refs. [25–30] cylindrical ferrites were introduced in a flat, metallic microwave resonator and\nmagnetized with an external magnetic field to induce T-invariance violation. Based on our findings we expect that,\nwhen choosing a circular shape of the resonator and inserting the ferrite at the circle center, it acts like a potential\nwhich induces wave-dynamical chaos above its critical frequency. XDZ is currently performing such experiments, and\npreliminary results confirm this assumption.\nDECLARATIONS\n•Funding: This work was supported by the NSF of China under Grant Nos. 11775100, 12047501, and\n11961131009. WZ acknowledges financial support from the China Scholarship Council (No. CSC-202106180044).\nBD and WZ acknowledge financial support from the Institute for Basic Science in Korea through the project12\nIBS-R024-D1. XDZ thanks the PCS IBS for hospitality and financial support during his visit of the group of\nSergej Flach.\n•Conflict of Interest: The authors declare no conflict of interest.\n•Data Avalaibility Statement: All data were generated with COMSOL multiphysics under license number\n9409425. The parameters to reproduce the data are provided in Tab. I and Sec. III. All data needed to\nsupport our findings are included in the figures of this article.\n•Authors’ Contributions: WZ and XZ contributed equally.\n[1] T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Random-matrix physics: spectrum and\nstrength fluctuations, Rev. Mod. Phys. 53, 385 (1981).\n[2] T. Zimmermann, H. K¨ oppel, L. S. Cederbaum, G. Persch, and W. Demtr¨ oder, Confirmation of random-matrix fluctuations\nin molecular spectra, Phys. Rev. Lett. 61, 3 (1988).\n[3] T. Guhr and H. A. Weidenm¨ uller, Coexistence of collectivity and chaos in nuclei, Ann. Phys. 193, 472 (1989).\n[4] T. Guhr, A. M¨ uller-Groeling, and H. A. Weidenm¨ uller, Random-matrix theories in quantum physics: common concepts,\nPhys. Rep. 299, 189 (1998).\n[5] H. Weidenm¨ uller and G. Mitchell, Random matrices and chaos in nuclear physics: Nuclear structure, Rev. Mod. Phys. 81,\n539 (2009).\n[6] J. G´ omez, K. Kar, V. Kota, R. Molina, A. Rela˜ no, and J. Retamosa, Many-body quantum chaos: Recent developments\nand applications to nuclei, Phys. Rep. 499, 103 (2011).\n[7] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn, C. Makrides, A. Petrov, and S. Kotochigova, Quantum chaos in\nultracold collisions of gas-phase erbium atoms, Nature 507, 474 (2014).\n[8] J. Mur-Petit and R. A. Molina, Spectral statistics of molecular resonances in Erbium isotopes: How chaotic are they?,\nPhys. Rev. E 92, 042906 (2015).\n[9] F. Haake, S. Gnutzmann, and M. Ku´ s, Quantum Signatures of Chaos (Springer-Verlag, Heidelberg, 2018).\n[10] S. Sridhar, Experimental observation of scarred eigenfunctions of chaotic microwave cavities, Phys. Rev. Lett. 67, 785\n(1991).\n[11] J. Stein and H.-J. St¨ ockmann, Experimental determination of billiard wave functions, Phys. Rev. Lett. 68, 2867 (1992).\n[12] H.-D. Gr¨ af, H. L. Harney, H. Lengeler, C. H. Lewenkopf, C. Rangacharyulu, A. Richter, P. Schardt, and H. A. Weidenm¨ uller,\nDistribution of eigenmodes in a superconducting stadium billiard with chaotic dynamics, Phys. Rev. Lett. 69, 1296 (1992).\n[13] S. Deus, P. M. Koch, and L. Sirko, Statistical properties of the eigenfrequency distribution of three-dimensional microwave\ncavities, Phys. Rev. E 52, 1146 (1995).\n[14] H.-J. St¨ ockmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 2000).\n[15] B. Dietz, T. Klaus, M. Miski-Oglu, and A. Richter, Spectral properties of superconducting microwave photonic crystals\nmodeling dirac billiards, Phys. Rev. B 91, 035411 (2015).\n[16] Y. G. Sinai, Dynamical systems with elastic reflections, Russ. Math. Surv. 25, 137 (1970).\n[17] L. A. Bunimovich, On the ergodic properties of nowhere dispersing billiards, Commun. Math. Phys. 65, 295 (1979).\n[18] M. V. Berry, Regularity and chaos in classical mechanics, illustrated by three deformations of a circular ’billiard’, Eur. J.\nPhys. 2, 91 (1981).\n[19] M. Giannoni, A. Voros, and J. Zinn-Justin, eds., Chaos and Quantum Physics (Elsevier, Amsterdam, 1989).\n[20] R. L. Weaver, Spectral statistics in elastodynamics, J. Acoust. Soc. Am. 85, 1005 (1989).\n[21] C. Ellegaard, T. Guhr, K. Lindemann, H. Lorensen, J. Nyg˚ ard, and M. Oxborrow, Spectral statistics of acoustic resonances\nin aluminum blocks, Phys. Rev. Lett. 75, 1546 (1995).\n[22] H. Alt, H.-D. Gr¨ af, R. Hofferbert, C. Rangacharyulu, H. Rehfeld, A. Richter, P. Schardt, and A. Wirzba, Chaotic dynamics\nin a three-dimensional superconducting microwave billiard, Phys. Rev. E 54, 2303 (1996).\n[23] H. Alt, C. Dembowski, H.-D. Gr¨ af, R. Hofferbert, H. Rehfeld, A. Richter, R. Schuhmann, and T. Weiland, Wave dynamical\nchaos in a superconducting three-dimensional sinai billiard, Phys. Rev. Lett. 79, 1026 (1997).\n[24] C. Dembowski, B. Dietz, H.-D. Gr¨ af, A. Heine, T. Papenbrock, A. Richter, and C. Richter, Experimental test of a trace\nformula for a chaotic three-dimensional microwave cavity, Phys. Rev. Lett. 89, 064101 (2002).\n[25] P. So, S. M. Anlage, E. Ott, and R. Oerter, Wave chaos experiments with and without time reversal symmetry: Gue and\ngoe statistics, Phys. Rev. Lett. 74, 2662 (1995).\n[26] D. H. Wu, J. S. A. Bridgewater, A. Gokirmak, and S. M. Anlage, Probability amplitude fluctuations in experimental wave\nchaotic eigenmodes with and without time-reversal symmetry, Phys. Rev. Lett. 81, 2890 (1998).\n[27] H. Schanze, E. R. P. Alves, C. H. Lewenkopf, and H.-J. St¨ ockmann, Transmission fluctuations in chaotic microwave billiards\nwith and without time-reversal symmetry, Phys. Rev. E 64, 065201 (2001).\n[28] B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A. Richter, F. Sch¨ afer, J. Verbaarschot, and H. A. Weidenm¨ uller,\nInduced violation of time-reversal invariance in the regime of weakly overlapping resonances, Phys. Rev. Lett. 103, 064101\n(2009).13\n[29] B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A. Richter, F. Sch¨ afer, and H. A. Weidenm¨ uller, Quantum chaotic\nscattering in microwave resonators, Phys. Rev. E 81, 036205 (2010).\n[30] B. Dietz, T. Klaus, M. Miski-Oglu, A. Richter, and M. Wunderle, Partial time-reversal invariance violation in a flat,\nsuperconducting microwave cavity with the shape of a chaotic africa billiard, Phys. Rev. Lett. 123, 174101 (2019).\n[31] M. Bia lous, B. Dietz, and L. Sirko, How time-reversal-invariance violation leads to enhanced backscattering with increasing\nopenness of a wave-chaotic system, Phys. Rev. E 102, 042206 (2020).\n[32] W. Zhang, X. Zhang, J. Che, J. Lu, M. Miski-Oglu, and B. Dietz, Experimental study of closed and open microwave\nwaveguide graphs with preserved and partially violated time-reversal invariance, Phys. Rev. E 106, 044209 (2022).\n[33] B. Lax and K. J. Button, Microwave ferrites and ferrimagnetics (McGraw-Hill, New York, 1962).\n[34] R. F. Soohoo, Theory and application of ferrites (Englewood Cliffs, Prentice-Hall, 1960).\n[35] J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1999).\n[36] M. Lebental, N. Djellali, C. Arnaud, J.-S. Lauret, J. Zyss, R. Dubertrand, C. Schmit, and E. Bogomolny, Inferring periodic\norbits from spectra of simply shaped microlasers, Phys. Rev. A 76, 023830 (2007).\n[37] E. Bogomolny, R. Dubertrand, and C. Schmit, Trace formula for dielectric cavities: General properties, Phys. Rev. E 78,\n056202 (2008).\n[38] S. Bittner, B. Dietz, R. Dubertrand, J. Isensee, M. Miski-Oglu, and A. Richter, Trace formula for chaotic dielectric\nresonators tested with microwave experiments, Phys. Rev. E 85, 056203 (2012).\n[39] S. Bittner, E. Bogomolny, B. Dietz, M. Miski-Oglu, and A. Richter, Application of a trace formula to the spectra of flat\nthree-dimensional dielectric resonators, Phys. Rev. E 85, 026203 (2012).\n[40] M. V. Berry and M. Robnik, Statistics of energy levels without time-reversal symmetry: Aharonov-Bohm chaotic billiards,\nJ. Phys. A 19, 649 (1986).\n[41] A. B¨ acker, Numerical Aspects of Eigenvalue and Eigenfunction Computations for Chaotic Quantum Systems, in The\nMathematical Aspects of Quantum Maps , edited by M. D. Esposti and S. Graffi (Springer Berlin Heidelberg, Berlin,\nHeidelberg, 2003) pp. 91–144.\n[42] M. Berry, Structural stability in physics (Pergamon Press, Berlin, 1979).\n[43] G. Casati, F. Valz-Gris, and I. Guarnieri, On the connection between quantization of nonintegrable systems and statistical\ntheory of spectra, Lett. Nuovo Cimento 28, 279 (1980).\n[44] O. Bohigas, M. J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation\nlaws, Phys. Rev. Lett. 52, 1 (1984).\n[45] M. L. Mehta, Random Matrices (Elsevier, Amsterdam, 2004).\n[46] M. V. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. R. Soc. London A 356, 375 (1977).\n[47] H. Weyl, ¨Uber die Abh¨ angigkeit der Eigenschwingungen einer Membran und deren Begrenzung, J. Reine Angew. Math.\n141, 1 (1912).\n[48] R. Balian and C. Bloch, Distribution of eigenfrequencies for the wave equation in a finite domain: I. Three-dimensional\nproblem with smooth boundary surface, Annals of Physics 60, 401 (1970).\n[49] B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A. Richter, F. Sch¨ afer, and H. A. Weidenm¨ uller, Chaotic scattering\nin the regime of weakly overlapping resonances, Phys. Rev. E 78, 055204 (2008).\n[50] M. V. Berry, Semiclassical theory of spectral rigidity, Proc. R. Soc. London A 400, 229 (1985).\n[51] M. Sieber, U. Smilansky, S. C. Creagh, and R. G. Littlejohn, Non-generic spectral statistics in the quantized stadium\nbilliard, J. Phys. A 26, 6217 (1993).\n[52] M. V. Berry and M. Tabor, Closed orbits and the regular bound spectrum, Proc. R. Soc. London A 349, 101 (1976).\n[53] M. C. Gutzwiller, Periodic orbits and classical quantization conditions, J. Math. Phys. 12, 343 (1971).\n[54] O. Bohigas and M. J. Giannoni, Level density fluctuations and random matrix theory, Ann. Phys. 89, 393 (1974).\n[55] V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).\n[56] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the ratio of consecutive level spacings in random\nmatrix ensembles, Phys. Rev. Lett. 110, 084101 (2013).\n[57] Y. Atas, E. Bogomolny, O. Giraud, P. Vivo, and E. Vivo, Joint probability densities of level spacing ratios in random\nmatrices, J. Phys. A 46, 355204 (2013).\n[58] H. G. L. Schwefel, A. D. Stone, and H. E. Tureci, Polarization properties and dispersion relations for spiral resonances of\na dielectric rod, J. Opt. Soc. B 22, 2295 (2005)."    },    {        "title": "2202.07488v1.The_effect_of_inter_particle_hydrodynamic_and_magnetic_interactions_in_a_magnetorheological_fluid.pdf",        "content": "arXiv:2202.07488v1  [physics.flu-dyn]  15 Feb 20221\nThe effect of inter-particle hydrodynamic\nand magnetic interactions in a\nmagnetorheological fluid\nV. Kumaran\nDepartment of Chemical Engineering, Indian Institute of Sc ience, Bangalore 560 012, India.\n(Received )\nA magnetorheological fluid, which consists of magnetic particles sus pended in a viscous\nfluid, flows freely with well-dispersed particles in a the absence of a ma gnetic field, but\nparticle aggregation results in flow cessation when a field is applied. Th e mechanism of\ndynamical arrest is examined by analysing interactions between mag netic particles in a\nmagnetic field subject to a shear flow. An isolated spherical magnet ic particle undergoes\na transition between a rotatingstate at lowmagnetic field and astat ic orientationat high\nmagnetic field. The effect of interactions for spherical dipolar and p olarisable particles\nwith static orientation is examined for a dilute viscous suspension. Th ere are magnetic\ninteractions due to the magnetic field disturbance at one particle ca used by the dipole\nmomentofanother,hydrodynamicinteractionsduetotheantisym metricforcemomentof\na non-rotatingparticle in a shearflow, and a modification ofthe magn etic field due to the\nparticle magnetic moment density. When there is a concentration va riation, the torque\nbalance condition results in a disturbance to the orientation of the p article magnetic\nmoment. The net force and the drift velocity due to these disturba nces is calculated, and\nthe collective motion generated is equivalent to an anisotropic diffusio n process. When\nthe magnetic field is in the flow plane, the diffusion coefficients in the two directions\nperpendicular to the field directionarenegative,implying that conce ntrationfluctuations\nare unstable in these directions. This instability could initiate field-indu ced dynamical\narrest in a magnetorheological fluid.\n1. Introduction\nA magnetorheological fluid is a suspension of magnetic particles of siz e about 1 −10µ\nm in a viscous fluid (de Vicente et al.(2011); Morillas & de Vicente (2020)). Brownian\nmotion is not important in these suspensions, because the particle s ize exceeds 1 µm, and\ntheviscosityofthecarrierfluidcouldbe2-3ordersofmagnitudelar gerthanthatofwater.\nThe volume fraction of the particles could be as low as 10% or lower (An upamaet al.\n(2018)). The salientfeatureofthesefluids is the rapidreversiblet ransitionbetweenalow-\nviscosity state in the absence of a magnetic field, where the particle s are well dispersed,\nandahighviscositystateunderamagneticfieldwheretheparticlesf ormsample-spanning\nclusters which arrest flow in the conduit (Sherman et al.(2015)). This transition takes\nplace reversibly and rapidly within time periods of tens to hundreds of milliseconds.\nDue to this rapid switching, magnetorheological fluids are used in app lications such as\ndampers and shock absorbers (Klingenberg (2001)).\nMagnetorheological fluids are characterised by measuring their ‘yie ld stress’ as a func-\ntion of the magnetic field (Sherman et al.(2015)). In the field of rheology,the yield stress2 V. Kumaran\nfor a Bingham plastic fluid delineates solid-like and fluid-like behaviour — t he material\nbehaves as an elastic solid when the stress is less than the yield stres s, and flows like\na viscous liquid when the stress exceeds the yield stress (Barnes et al.(1989)). In the\nBingham model, the stress is the sum of a yield stress τyand a contribution which is\nlinear in the strain rate, and the slope of the stress-strain rate cu rve is called the plastic\nviscosity. For characterisation,the magnetorheologicalfluid is pla ced in a rheometer, and\na magnetic field is applied across the sample. The strain rate is then se t at progressively\nincreasing values, and the stress is measured. The stress-strain rate curve is fitted to the\nBingham plastic model to determine the yield stress and the plastic vis cosity. The stress-\nstrain rate curves for magnetorheological fluids typically increase continuously, and they\ndo not exhibit a discontinuous change in slope at yield. In order to det ermine the yield\nstress and plastic viscosity, their high strain rate behaviour of the se fluids are extrap-\nolated linearly to zero strain rate. There are two dimensionless numb ers that are used\nto characterise magnetorheological fluids. The first is the Mason n umber, the ratio of\nthe shear stress and a reference magnetic stress, the latter is t he ratio of the magnetic\ndipole-dipole interaction force between pairs of particles and the sq uare of the particle\ndiameter (Sherman et al.(2015)). The second is the Bingham number which is the ratio\nof the yield stress and the fluid stress. The yield stress does increa se as the applied mag-\nnetic field is increased, and it has been proposed that the Bingham nu mber (ratio of yield\nstress and viscous stress) is inversely proportional to the Mason number (Sherman et al.\n(2015)).\nMore sophisticated constitutive relations for magnetorheological fluids have been ex-\nplored in simulations and experiments. Empirical fitting functions of t he type\nη\nη0= 1+/parenleftbiggMn∗\nMn/parenrightbiggα\n(1.1)\nhave been used for the effective viscosity η, which is the ratio of the stress and strain\nrate (Vagberg & Tighe (2017)). Here, η0is the viscosity in the absence of a magnetic\nfield, Mn is the Mason number which is proportional to the strain rate , and Mn∗and\nαare fitting parameters. Equation 1.1 reduces to the Bingham equat ion forα= 1. For\nα <1, there is a transition from a power law form for the constitutive re lation at low\nMn to a Newtonian form for high Mn. While some simulation and experimen tal studies\n(Sherman et al.(2015); Marshall et al.(1989); Bonnecaze & Brady (1992)) on dipolar\nparticles in an external field have found that αis close to 1, others (Melrose (1992);\nFeltet al.(1996); Martin et al.(1994)) report that the exponent is less than 1. In the\nsimulations of Vagberg & Tighe (2017), the exponent αis found to depend on the details\nof the interactions between particles. More sophisticated rheolog ical models, such as the\nCasson model in Ruiz-L´ opez et al.(2017), have also been employed for the rheology of\nmagnetorheological fluids.\nThe above characterisation procedure examines the ‘unjamming’ o r dynamical release\ntransition, where the particle structures formed by the magnetic field are disrupted due\nto shear. However, rapid flow cessation involves the opposite dyna mical arrest, where\ninitially dispersed particles cluster and block the flow in the conduit whe n a magnetic\nfield is applied. The mechanism for dynamical arrest is different from t hat probed in\ncharacterisation experiments. The clustering has to be initiated by interactions between\nwell-dispersed particles upon application of a magnetic field in the pres ence of flow. Here,\nsome insight is obtained into the initiation of the dynamical arrest pro cess by considering\nthe effect of the hydrodynamic and magnetic interactions between dispersed particles in\nthe sheared state in the presence of a magnetic field.\nFerrofluids(Schumacher et al.(2008);Moskowitz & Rosensweig(1967);Zaitsev & ShliomisParticle interactions in a magnetorheological fluid 3\n(1969); Chaves et al.(2008)) form another class of suspensions of magnetic particles. In\nthis case, the particles are of nanometer size, Brownian diffusion is s ignificant, and the\neffect of interactions on the fluctuating motion of the particles may be less important.\nSuspensions of conducting particles in shear flow also experience a t orque in a magnetic\nfield (Moffat (1990); Kumaran (2019, 2020 b)). This is because an eddy current is induced\nwhen a particle rotates in a magnetic field, and this induces a magnetic dipole which in-\nteracts with the field. Suspensions of conducting particles are not considered here, but a\nsimilar calculation procedure can be used to predict the effect of inte ractions.\nRather than fitting the measured or simulated rheology to a specific constitutive rela-\ntion, the approach here is to examine the effect of particle interact ions in a suspension\nwith well-dispersed particles. This is similar to the effect of particle inte ractions on the\nviscosity of a non-Brownian particle suspension (Batchelor (1970) ; Hinch (1977)). The\ndynamics of a spheroidal particle subjected to a shear flow and a ma gnetic field has been\nstudied (Almog & Frankel (1995); Sobecki et al.(2018); Kumaran (2020 a, 2021a,b)). In\nthe absence of a magnetic field, the particle axis rotates in closed ‘Je ffery orbits’ (Jeffery\n(1923); Hinch & Leal (1979)) on a unit sphere. When a magnetic field is applied, the\nmagnetic torque tends to align the particle along the field direction. W hen the mag-\nnetic field is below a threshold, the particle rotates with frequency lo wer than the Jeffery\nfrequency. When the magnetic field exceeds the threshold, the pa rticle has a steady\norientation which progressively inclines towards the field direction as the field strength\nis increased. The transition between steady and rotating states d epends on the particle\nshape factorand the magnetisationmodel, and therecould alsobe m ultiple steady states.\nHere, the effect of interactions is studied for a suspension of sphe rical particles. Though\nthis configuration is sufficiently simple that the particle orientation ca n be determined\nanalytically, it does provide physical insight into the mechanisms that drive the collective\nbehaviour of the particles.\nConsider a suspension of spherical magnetic particles sheared bet ween two plates, as\nshown in figure 1. The red arrows show the magnetic moment of the p articles, and the\nblue arrows indicate the direction of rotation of the particles due to the fluid shear.\nWhen there is no magnetic field, there is no magnetic torque, and the hydrodynamic\ntorque is zero in the viscous limit. The particles rotate with angular ve locity equal to the\nlocal fluid rotation rate, which is one half of the vorticity. When a sma ll magnetic field\nis applied, there is a torque on the particle which depends on the part icle orientation.\nThe particles do rotate, but with average angular velocity smaller th an the fluid rotation\nrate, as shown in figure 1 (a). When the magnetic field is increased be yond a threshold,\nthe particles do not rotate; the static orientation is determined by a balance between\nthe magnetic and hydrodynamic torques, as shown in figure 1 (b). T here is a transition\nbetween rotating and steady states when the dimensionless param eter Σ, defined later\nin equation 2.43, exceeds1\n2. In the limit of large magnetic field, the particle magnetic\nmoments align closer to the field direction, as shown in figure 1 (c).\nThe formation of sample-spanning clusters for dynamical arrest in magnetorheological\nfluids, shown in figure 1 (d), requires an additional mechanism not pr esent in the single-\nparticle dynamics. While alignment ofparticle dipoles is expected for hig h magnetic field,\nit is not clear how the non-Brownian particles approach and cluster in the stream-wise\ndirection in a highly viscous fluid where particle contact is prevented b y lubrication. For-\nmation of isotropic clusters is not sufficient to explain the dynamical a rrest phenomenon;\nsuch clusters would be rotated and stretched by the shear flow (V argaet al.(2019)).\nHere, we show that anisotropic clustering is initiated by interactions between dispersed\nparticles. The interactionsareoftwo types, magnetic and hydrod ynamic.It is shownthat\nhydrodynamic interactions between particles amplify concentratio n variations along the4 V. Kumaran\nHV\nxy\nz\n(a)V\nxyH\nz\n(b)\nV\nxyH\nz\n(c)xyH\nz\n(d)\nFigure 1. A suspension of dipolar spherical particles in a magnetic fie ld; (a) for low field\nintensity, the rotation rate is smaller then the fluid rotati on rate; (b) above a threshold, the\nparticles align in the direction determined byabalance bet ween the hydrodynamicand magnetic\ntorque; (c) at high magnetic field, the particles align close to the field direction; (d) dynamical\narrest occurs due to sample-spanning aggregation of partic les.\nflow direction, while magnetic interactions dampen concentration va riations along the\ncross-stream direction. The amplification of density of waves in the stream-wise direc-\ntion, combined with the rapid equalisation of concentration in the cro ss-streamdirection,\ncould result in the formation of sample-spanning aggregates that a rrest the flow.\nThe effect of magnetic interactions between particles on magnetop horesis has been\ncalculated (Morozov (1993, 1996)), and this has been included in mo dels for suspensions\nof magnetic particles in the presence of magnetic fields (Pshenichnik ovet al.(2011);\nPshenichnikov & Ivanov (2012)). Magnetic interactions enhance t he diffusion coefficient,\ndampen concentration and fluctuations and stabilise the suspensio n. This is contrary to\nthe aggregationrequired for dynamical arrest in magnetorheolog icalfluids. The following\nsimple calculation illustrates the diffusion enhancement due to magnet ic interactions.\nThe force on a particle in a magnetic field is assumed of the form F=µ0∇(M·H),\nwhereµ0is the magnetic permeability which is considered a constant, Mis the particle\nmoment and His the magnetic field. The net force is zero when the magnetic field is\nuniform in the case of non-interacting particles. The magnetic field d isturbance H′(x)\non a particle at the location xdue to the presence of another particle at the location x′\nwith magnetic moment Mis,\nH′(x) =1\n4π/parenleftbigg3(x−x′)(x−x′)\n|x−x′|5−I\n|x−x′|3/parenrightbigg\n·M. (1.2)\nIn a uniform suspension where the number density is a constant, th e integral of the\nright side of 1.2 over all space is zero by symmetry. In a suspension o f particles with a\nspatially varying number density n(x), the magnetic field at the particle location xdue\nto interaction with other particles is,\nH′(x) =1\n4π/integraldisplay\ndx′/parenleftbigg3(x−x′)(x−x′)\n|x−x′|5−I\n|x−x′|3/parenrightbigg\n·(Mn(x′)).(1.3)Particle interactions in a magnetorheological fluid 5\nThese disturbances are expressed in Fourier space using the tran sform,\nˆ⋆k=/integraldisplay\ndxexp(ık·x)⋆′(x), (1.4)\nwhere the field variable ⋆could be the magnetic field, velocity, vorticity, concentration\nor the orientation vector. The Fourier transform of the disturba nce to the magnetic field\nis,\nˆHk=−kk·Mˆnk\nk2, (1.5)\nwhere ˆnkistheFouriertransformofthenumberdensityvariations.TheFou riertransform\nof the force F=µ0∇(M·H) acting on the particle is,\nˆFk=−ıkµ0(M·ˆHk) =ık/parenleftbiggµ0(M·k)(M·k)ˆnk\nk2/parenrightbigg\n. (1.6)\nThe particle velocity drift due to the magnetic interaction force is th e ratio of the force\nand the friction coefficient,\nˆuk=Fk\n3πηd=ık/parenleftbiggµ0(M·k)(M·k)ˆnk\n3πηdk2/parenrightbigg\n, (1.7)\nwhere the Stokes expression for the friction coefficient for a sphe rical particle, 3 πηd, has\nbeen used, ηis the fluid viscosity and dis the particle diameter. The variation in number\ndensity due to the drift velocity is determined from the conservatio n equation,\n∂ˆnk\n∂t−ık·(¯nˆuk) = 0. (1.8)\nHere ¯nis the averagenumber density ofthe particles, and it is assumed tha t the variation\nin number density ˆ nkis much smaller than ¯ n. This results in a diffusion-type equation,\n∂ˆnk\n∂t+kk:(Dˆnk) = 0, (1.9)\nwhere the diffusion tensor Dis,\nD=µ0MM¯n\n3πηd. (1.10)\nThe diffusion tensor depends on the magnetic moment of the particle s, and not the\nmagnetic field, because it is caused by the interaction force betwee n the particles, as\nnoted in Sherman et al.(2015) and Klingenberg et al.(2007). This effect is anisotropic,\nand it operates only in the direction of the particle moment. Importa ntly, the effect of\ninteractions dampens fluctuations in the direction of the magnetic m oment since the\ndiffusivity is positive, and it has no effect in the direction perpendicular to the particle\nmagnetic moment.\nThe clustering of particles can be predicted if the variation of the ma gnetic field due\nto the magnetic moments of the particles is included in an effective med ium approach.\nThe force on the particle is F=µ0∇[M·(H+nM)],wherenMis the magnetisation\nor the magnetic moment per unit volume. The force on the particles is , instead of 1.6,\nˆFk=−ıkµ0(M·ˆHk+ ˆnkM·M) =ık/parenleftbiggµ0(M·k)(M·k)ˆnk\nk2−M2ˆnk/parenrightbigg\n.(1.11)\nThe last term in the bracket on the right arises from the modification of the variation\nin the magnetic moment per unit volume due to the variation in the part icle number6 V. Kumaran\ndensity. With this inclusion, the diffusion tensor is,\nD=µ0(MM−IM2)¯n\n3πηd, (1.12)\nwhereIis the identity tensor. The above diffusion tensor is diagonal in a co-o rdinate\nsystem where one the co-ordinate directions is along the particle ma gnetic moment. The\ndiffusion coefficient along the magnetic moment is identically zero, while t hat in the two\ndirections perpendicular to the magnetic moment is negative. This pr edicts spontaneous\namplification of concentrationfluctuations in the direction perpend icular to the magnetic\nmoment of the particles, and no amplification or damping along the par ticle magnetic\nmoment. This is consistent with the formation of particle chains along the magnetic field\ndirection if the magnetic moment is aligned along the magnetic field. How ever, this is\nstill an effective medium or a mean-field approach for the initial growt h of perturbations.\nThecontrastintheelectricalpermittivity ormagneticpermeabilityo nthestabilityofa\nsuspension was considered by von Pfeil et al.(2003) for a sheared particle suspension. In\nthe dilute limit, the effective magnetic permeability is expressed as µeff=µ0(1+µ′φ),\nwhereφis the volume fraction, µ′= 3(µR−1)/(µR+ 2), where µ0is the magnetic\npermeability of the fluid in the absence of particles and µRis the ratio of the magnetic\npermeabilities of the particles and fluid. For µR>1 where the magnetic permeability of\nthe particles is higher than that for the fluid, the magnetic flux lines p referentially pass\nthrough regions of higher particle concentration. In the present analysis, the change in\nthe magnetic field due to the particles is the magnetisation or the mag netic moment per\nunit volume, and latter is the product of the particle moment and the number density.\nWhen there is fluid flow, there is one other effect, which is the variatio n in the effective\nviscosity of the suspension with the particle concentration. For a d ilute suspension of\nsphericalparticles subject to shearflow, the effective viscosityis η= (1+η′φ), whereη′=\n5\n2andφis the volume fraction. The viscosity variation with volume fraction is in cluded\nin the analysis in sections 2.1 and 3.1, where an effective diffusion coeffic ient is derived\nin the equation for the concentration fluctuation. This effective diff usion coefficient is\nproportional to η′.\nThe objective of the present analysis is to analyse the stability of a fl owing suspen-\nsion incorporating magnetic and hydrodynamic interactions, and th e effect of particle\nconcentration on the viscosity and magnetic field. When there is a diff erence between\nthe particle and fluid rotation rates, there is a torque exerted by t he particle on the\nfluid. This torque results in an antisymmetric dipole force moment at t he particle center,\nwhich generates a velocity disturbance. The resulting torque at th e center of a test par-\nticle due to interactions with other particles is zero if the number den sity is uniform (see\nAppendix D1 in Kumaran (2019)). However, when there is a perturb ation in the number\ndensity, there is a net torque exerted due to hydrodynamic intera ctions. In the presence\nof spatial concentration variations, there is also a net torque exe rted at the test particle\ndue to magnetic interactions with other particles. There is a small sp atial variation in\nthe particle orientation vector in order to satisfy the zero torque condition. This results\nin a small correction to the magnetic moment of the particle M, and consequently an\nadditional contribution to the force. The two additional effects of hydrodynamic inter-\nactions and the zero torque condition are incorporated in the forc e due to interactions,\nand it is shown that this force amplifies concentration fluctuations a nd destabilises the\nsuspension.\nThe expression for the force, F=∇(M·B), results from the Ampere model for a\nmagnetic dipole, where the dipole is considered as an infinitesimal curr ent loop (Jackson\n(1975); Griffiths (2013)). Here, B=µ0(H+nM) is the magnetic flux density. AnParticle interactions in a magnetorheological fluid 7\nalternate expression is based on the Gilbert model, F=M· ∇B, where the dipole is\nconsidered the superposition of two magnetic ‘charges’ of opposit e signs separated by an\ninfinitesimal distance. Though the two models are equivalent in most c ases, there are\ndifferences in special situations, such as the magnetic field of a prot on which fits the\nAmpere model. The present analysis is another situation where ther e are differences in\nthe results of the two models. For small perturbations M′(x) andB′(x) about the base\nstate where the particle moment ¯Mand magnetic field ¯Bare spatially uniform, the force\nin the Gilbert model linearised in the perturbations is ¯M· ∇B′, whereas that in the\nAmpere model is ∇(M′(x)·¯B+¯M·B′(x)). Variations in the magnetic permeability\nwith position due to a contrast in the permeabilities of the particles an d fluid are also\nincorporated in the Ampere model. The force in the Gilbert model not depend on the\nperturbations to the particle magnetic moment, while that in the Amp ere model does.\nAnother distinction is that the force in the Ampere model can be writ ten as−∇UM,\nwhereUM=−M·Bis the magnetic energy,whereasthere is no equivalent relationto an\nenergy for the Gilbert model. Since the Ampere model is also known to fit experimental\nresults, such as the magnetic field of a proton measured in hyperfin e splitting (Griffiths\n(1982)), the Ampere model is used here for the force on the part icle.\nThe analysis is carried out for two magnetisation models, the perman ent dipole in\nsection 2, where the magnitude of the magnetic moment is independe nt of the field, and\nthe induced dipole model in section 3, where the magnitude of the mom ent is propor-\ntional to the component of the magnetic field along the orientation a xis. These models\nhave been used earlier to study the single-particle dynamics of a sph eroid in a magnetic\nfield(Almog & Frankel (1995); Sobecki et al.(2018); Kumaran (2020 a, 2021a,b)). In the\ninduced dipole model, it is assumed that the magnetic moment of the pa rticle is propor-\ntional to the component of the magnetic field along that axis. This mo del is applicable to\nsuperparamagnetic particles, which are usually nanometer sized sin gle-domain particles\npolarisable along one axis. Superparamagnetic particles have very lit tle hysteresis, and\nso the constant susceptibility assumption is valid for low magnetic field . This also applies\nto micron sized multi-domain ferromagnetic particles, or soft magne ts. These are magne-\ntised along an ‘easy axis’ Rikken et al.(2014) for several reasons. Though the domains\nwithin the particles are aligned along the magnetic field for soft magne ts, those on the\nsurface are aligned tangential to the surface, resulting in a magne tisation along the par-\nticle axis for non-spherical particles. In addition, strain anisotrop ies could also result in a\nhigher susceptibility along the easy axis. At low magnetic field, this res ults in a constant\nanisotropic susceptibility tensor. It was shown Kumaran (2021 a) that for axisymmetric\nparticles, if the susceptibility is axisymmetric about the easy axis, th e magnetic moment\ncan be modeled as a vector directed along the easy axis whose magnit ude is proportional\nto the component of the magnetic field along the easy axis. The induc ed dipole model\nused here applies to this case, provided the magnetic moment is small compared to the\nsaturation moment.\nThe permanent dipole model applies to particles in ferrofluids, which a re suspensions\nof nanometer sized single-domain magnetic particles stabilised using s urfactants. Here,\nthe orientations of the particles are randomised by Brownian motion in the absence of\na magnetic field, but the particles align when a field is applied. The analys is here also\napplies to superparamagnetic or ferromagnetic particles at high ma gnetic field when\nthe magnetic moment reaches it saturation value. In contrast to p ermanent magnets,\nthe magnetic moments align along the component of the field along the easy axis, and\nmagnetic moment reverses when the direction of the field is reverse d. Therefore, the\norientation dynamics for rotating states is different from that for a permanent magnet\n(Kumaran (2020 a, 2021a,b)). However, for states with a steady orientation that are8 V. Kumaran\nstudied here, the effect ofinteractionsis identical to those forpe rmanent dipolarparticles\nprovided the orientation disturbance is small and it does not revers e the orientation of\nthe magnetic moment.\nFor the permanent and induced dipole models, the dynamics of an isola ted spheroid\nsubjected to a shearflowand a magneticfield hasbeen studied (Almo g & Frankel(1995);\nSobeckiet al.(2018); Kumaran (2020 a, 2021a,b)). The results for a spherical particle\nfrom these earlier studies are used here. For a permanent dipole, t he single-particle\ndynamics depends on the parameter Σ defined in equation 2.43, which is the ratio of\nthe magnetic and hydrodynamic torques. The effective diffusion coe fficient is calculated\nin section 2.1, and the rotating and steady states in the absence of particle interactions\nare summarised in section 2.2. The effect of particle interactions for a parallel magnetic\nfield, where the field is in the flow plane, is discussed in 2.3, and this is gen eralised to\nan oblique magnetic field in section 2.4. The analysis for an induced dipole is provided\nin section 3, where the single particle dynamics depends on the param eter Σ idefined in\nequation 3.19. The diffusion coefficient is calculated in section 3.1, and t he orientation of\nan isolated particle is outlined in section 3.2. The diffusion coefficients fo r a parallel and\noblique magnetic field are provided in sections 3.3 and 3.4 respectively. In section 4, the\nmajor conclusions are summarised, and numerical estimates for th e diffusion coefficients\nare provided.\n2. Permanent dipole\n2.1.Diffusion due to interactions\nThe magnetic force and torque on a particle with magnetic moment Min a magnetic\nfieldHin a suspension of particles with number density nare,\nFm=∇(M·µ0(H+nM)), (2.1)\nTm=µ0M×H, (2.2)\nwhereµ0is the magnetic permeability. In the expression for the force, 2.1, t he magnetic\nflux density is expressed as B=µ0(H+nM), where nMis the magnetic moment per\nunit volume. This term is an effective-medium approximation for the va riations in the\nmagnetic field due to variations in the number density of particles. Th e contribution to\nthe magnetic moment per volume in the expression for the torque 2.2 is zero, since it\ncontains the cross product of the particle moment at a location with itself.\nThe hydrodynamic force and torque are,\nFh= 3πηd(v−u), (2.3)\nTh=πηd3(1\n2ω−Ω), (2.4)\nwheredis the particle diameter, vandωare the fluid velocity and vorticity at the\nparticle center, uandΩare the particle linear and angular velocities. The imposed\nmagnetic field is denoted ¯H, and the vorticity due to the imposed shear flow at the\nparticle center is ¯ω. The particle magnetic moment is expressed as M=Mo, whereM\nis the magnitude of the magnetic moment and ois the orientation vector. Linear models\nare used to incorporate the effect of the variations in the volume fr action on the viscosity\nand the magnetic permeability,\nη=η0(1+η′φ′), (2.5)\nwhereφ′=φ−¯φ,φis the local volume fraction of the particles, ¯φis the average volume\nfraction in the uniform suspension, and η0is the viscosity for the uniform suspension.Particle interactions in a magnetorheological fluid 9\nFor a viscous suspension of spherical particles in the limit of low volume fraction, the\nresultη′=5\n2is obtained for the Einstein correction to the viscosity.\nTheorientationvectorofthe particle oisdeterminedfromthetorquebalanceequation,\nTm+Th= 0,\nπηd3(1\n2ω−Ω)+µ0M(o×H) = 0. (2.6)\nThe magnetic force on the particle 2.1 is zero in a uniform suspension. The force balance\nrequires that the hydrodynamic force is also zero, that is, the par ticle moves with the\nsame velocity as the fluid, ¯u=¯v.\nFor a stable stationary state where the particle orientation is stea dy, the component\nof the angular velocity Ωperpendicular to the orientation vector is zero. However, the\nparticle does spin around the orientation vector, and the compone nts of the particle\nangular velocity and fluid rotation rate (one half of the vorticity) alo ng the orientation\nvectorareequal.This is becausethe torqueexerted bythe magne tic field is perpendicular\nto thedirection ofthe magneticmomentand themagnetic field,and c onsequentlythereis\nno magnetic torque along the orientation vector. Therefore, the particle angular velocity\nis necessarily along the orientation vector, Ω= Ωo. It is necessary to consider the\ntorque balance equations parallel and perpendicular to the orienta tion vector in order\nto determine the particle orientation. The torque balance equation along the orientation\nvector is obtained by taking the dot product of equation 2.6 with o, and this is solved to\nobtain,\nΩ =1\n2ω·o. (2.7)\nThe above solution 2.7 for the angular velocity is substituted into the torque balance\nequation 2.6 to obtain,\n1\n2πηd3(I−oo)·ω+µ0M(o×H) = 0, (2.8)\nwhereIis the identity tensor, and ( I−oo)·is the transverse projection operator which\nprojects a vector on to the plane perpendicular to o. One relation between the vorticity,\nmagnetic field and orientation vector is obtained by taking the dot pr oduct of equation\n2.8 with H,\nH·ω−(o·H)(o·ω) = 0. (2.9)\nThe torque balance for the base state in the absence of interactio ns is obtained by sub-\nstituting ¯o,¯ωand¯Hforo,ωandHin equation 2.8.\nThere is a disturbance to the magnetic field H′and the fluid vorticity ω′due to a\nnumber density fluctuation n′(x) imposed on a uniform suspension with number density\n¯n. The orientation vector of the particle is expressed as o=¯o+o′(x), where o′(x)\nis the disturbance to the orientation vector due to hydrodynamic a nd magnetic inter-\nactions resulting from concentration inhomogeneities; the latter is calculated from the\ntorquebalanceequation.Inthelinearapproximationwheretermsq uadraticintheprimed\nquantities are neglected, the mean and disturbance to the orienta tion vector satisfy\n|¯o|= 1,¯o·o′= 0. (2.10)\nThe reason for the above conditions is that both ¯oand (¯o+o′) are unit vectors, that\nis,¯o·¯o= 1 and ( ¯o+o′)·(¯o+o′) = 1.In the linear approximation, this implies that\n(¯o·¯o+2¯o·o′) = 1. Therefore, ¯o·o′= 0.\nThe Fourier transform ˆokof the disturbance to the orientation vector is defined using\nequation 1.4. The Fourier transform of the force on a particle due t o interactions is10 V. Kumaran\ndetermined from equation 2.1,\nˆFk=−ıkµ0M[¯o·ˆHk+¯H·ˆok+Mˆnk], (2.11)\nwhereMis the magnitude of the magnetic moment. The Fourier transform of expression\n2.5has been substituted forthe magneticpermeability, and linearise din the perturbation\nto the volume fraction field, to derive the third term in the square br ackets on the right\nin equation 2.11. The last term in the square brackets on the right in e quation 2.11 is\nthe force due to the variation in the number density in the expressio n 2.1.\nThe conservation equation for the particle number density is,\n∂n\n∂t+∇·(un) =DB∇2n, (2.12)\nwhereuis the particle velocity and DBis the Brownian diffusion coefficient. The particle\nvelocity is expressed as the sum of the fluid velocity vand the particle velocity relative to\nthe fluid, ( u−v)+v. Since there is no net force on the particles in a uniform suspension,\n¯u=¯v. In the linear approximation, the particle conservation equation is e xpressed as,\n∂n′\n∂t+∇·(¯vn′+v′¯n)+∇·((u′−v′)¯n) =DB∇2n, (2.13)\nThe Fourier transform of the above equation is,\n∂ˆnk\n∂t−ık·(¯vˆnk)−ık·(¯nˆvk)−ık¯n(ˆuk−ˆvk))+DBk2ˆnk= 0.(2.14)\nThe first two termson the left of2.14are the rateof changeandth e convectiondue to the\nbase flow of concentration fluctuations. The third term on the left is the rate of change\nof concentration disturbances due to fluid velocity perturbation. The perturbation to the\nfluid velocity due to the rotation of other particles, which is calculate d later in 2.22, is\nperpendicular to k. Consequently, the third term on the left in 2.14 is zero. The fourth\nterm on the left is the relative motion between the particles and the fl uid due to the\nforces on the particle.\nIn a dilute suspension, the difference between the particle and fluid v elocity distur-\nbances,ˆukandˆvk, is given by Stokes law,\nˆuk−ˆvk=ˆFk\n3πη0d. (2.15)\nThe drift velocity does not depend on the coefficient η′in 2.5 in the linear approximation,\nbecause the force on a particle is zero in a uniform suspension.\nThe expression 2.15 for the drift velocity is substituted into the linea rised conservation\n2.14,\n∂ˆnk\n∂t−ık·(ˆnk¯v)+DBk2ˆnk\n−k2¯nµ0M(¯o·ˆHk+¯H·ˆok+Mˆnk)\n3πη0d= 0. (2.16)\nThedisturbance ˆHkduetomagneticinteractionsisexpressedasafunctionof ˆ nk,andthe\norientation disturbance ˆok, currently unknown, is determined from the first correction\nto the torque balance. These are substituted into equation 2.16 to obtain an equation of\nthe form,\n∂ˆnk\n∂t−ık·(ˆnk¯u)+k·D·kˆnk+DBk2ˆnk= 0, (2.17)Particle interactions in a magnetorheological fluid 11\nwhereDis the magnetophoretic diffusion tensor due to hydrodynamic and ma gnetic\ninteractions. In the following calculation, the disturbance to the ma gnetic field and the\nvorticity due to particle interactions are first discussed, and then the disturbance to the\norientation vector is calculated using torque balance. This is inserte d into equation 2.16\nto extract the diffusion tensor D.\nThe change in the magnetic field at the location xdue to interactions with other\nparticles is a generalisation of equation 1.3 which incorporates the dis turbance to the\norientation vector,\nH′(x) =1\n4π/integraldisplay\ndx′/parenleftbigg3(x−x′)(x−x′)\n|x−x′|5−I\n|x−x′|3/parenrightbigg\n·[M(¯on′(x′)+ ¯no′(x′))],(2.18)\nwheredx′is the volume element. It is easily verified that the contribution to H′due\nto the product of the background constant particle density and m agnetic field, M¯n¯o, is\nzero. Equation 1.4 is used to determine the disturbance to magnetic field in reciprocal\nspace,\nˆHk=−kk·[M(¯oˆnk+ ¯nˆok)]\nk2=ˆH′\nk+ˆH′′\nk·ˆok, (2.19)\nwhere\nˆH′\nk=−Mˆnkkk·¯o\nk2,ˆH′′\nk=−M¯nkk\nk2. (2.20)\nNote that ˆH′\nkis a vector, and ˆH′′\nkis a second order tensor.\nFor a particle located at x′in a fluid with vorticity ¯ωin the absence of the particle,\nthe fluid velocity disturbance v′at the location xis,\nv′(x) =/integraldisplay\ndx′n(x′)(Ω(x′)−1\n2ω(x′))×d3(x−x′)\n8|x−x′|3\n=−/integraldisplay\ndx′n(x′)[I−o(x′)o(x′)]·ω(x′)×d3(x−x′)\n16|x−x′|3\n=−/integraldisplay\ndx′[n′(x′)(I−¯o¯o)·¯ω+ ¯n(I−¯o¯o)·ω′(x′)\n−¯n(¯o¯ω+I¯ω·¯o)·o′(x′)]×d3(x−x′)\n16|x−x′|3. (2.21)\nThe expression 2.7 for the particle angular velocity has been used to simplify in the\nsecond step in equation 2.21. The linearisation approximation has bee n used in the third\nstep in equation 2.21, resulting in an equation that is linear in the distur bances due to\nparticle interactions. The Fourier transform of the velocity field is,\nˆvk=−[ˆnk(I−¯o¯o)·¯ω+ ¯n(I−¯o¯o)·ˆωk−¯nˆok·(¯ω¯o+I¯ω·¯o)]×πd3ık\n4k2.(2.22)\nThe Fourier transform of the disturbance to the fluid vorticity is,\nˆωk=−ık׈vk\n=πd3(kk−Ik2)·[ˆnk(I−¯o¯o)·¯ω+ ¯n(I−¯o¯o)·ˆωk−¯n(¯o¯ω+I¯ω·¯o)·ˆok]\n4k2.(2.23)\nThe above implicit equation is solved for ˆωk,\n/parenleftbigg\nI−πd3¯n(kk−Ik2)·(I−¯o¯o)\n4k2/parenrightbigg\n·ˆωk=ˆω′+ˆω′′·ˆok, (2.24)12 V. Kumaran\nwhere\nˆω′=πˆnkd3(kk−Ik2)·(I−¯o¯o)·¯ω\n4k2, (2.25)\nˆω′′=−π¯nd3(kk−Ik2)·(¯o¯ω+I¯ω·¯o)\n4k2. (2.26)\nNote that ˆω′\nkis a vector, and ˆω′′\nkis a second order tensor. The latter is proportional to\n(d3¯n) or the volume fraction. In the limit of small volume fraction, this ter m is small\ncompared to 1.\nThe velocity due to the force exerted by particles is higher order in g radients or higher\npower in k, and so it will result in a higher gradient of the concentration field. Th e\ncontribution to the velocity due to the magnetic field perturbation, in Fourier space, is\n(|ˆFk|/3πηd)∼(kµ0MˆHk/3πηd)∼(kµ0M2ˆnk/3πηd), where kis the magnitude of the\nwave number. Here, 2.11 is used for the force and 2.19, 2.20 are use d for the magnetic\nfield perturbation. The magnitude of the fluid velocity due to particle rotation, from\n2.22, is|ˆvk| ∼π|¯ω|ˆnkd3/k. The ratio of the velocities due to the force and torque\nis (k2µ0M2/3πηd4|¯ω|)∼(kd)2M∗Σ, where the dimensionless groups Σ and M∗are\ndefined later in 2.43 and 2.44. The continuum approximation is valid only f or (kd)≪1,\nthat is, the length scale for the gradients in the concentration field is much larger than\nthe particle diameter. In this limit, the velocity disturbance due to th e particle force is\nneglected compared to that due to the torque exerted on the fluid .\nThe disturbance to the magnetic field and the vorticity at the partic le locationcauses a\ndisturbance to the particle orientation o, which is determined from the correction to the\ntorque balance equation in Fourier space. The torque balance equa tion, 2.8, is divided\nbyµ0M, and linearised in the primed quantities to obtain,\nπd3η0\n2µ0M[(I−¯o¯o)·ˆωk−¯o(ˆok·¯ω)−ˆok(¯o·¯ω)+η′ˆφk(I−¯o¯o)·¯ω]\n+(¯o׈Hk+ˆokׯH) = 0,(2.27)\nwhereˆφk= ˆnk(πd3/6) is the Fourier transform of the fluctuation in the volume fraction .\nThe expressions 2.19 and 2.24 for ˆHkandˆωkare substituted into equation 2.27,\nπd3η0\n2µ0M{(I−¯o¯o)·ˆω′\nk+[(I−¯o¯o)·ˆω′′\nk−¯o¯ω−(¯o·¯ω)I]·ˆok\n+η′ˆφk(I−¯o¯o)·¯ω}+[¯o׈H′\nk+ˆokׯH+¯o×(ˆH′′\nk·ˆok)] = 0.(2.28)\nThe first term on the left side of equation 2.28 is the correction to th e hydrodynamic\ntorque due to interactions and due to the concentration depende nce of the viscosity and\nmagnetic permeability. In this, the contribution proportional to η′ˆφkis the correction to\nthe hydrodynamic torque due to the concentration dependence o f the viscosity. In the\nfirst term on the left in 2.28, within the flower brackets, the prefac tor ofˆokcontains\none contribution proportional to ˆω′′, and the second which is proportional to |¯ω|. From\nequation 2.26, the formerisproportionalto ¯ nd3|¯ω|, whichis small comparedtothe latter,\nbecause the volume fraction is small. Therefore, the term ( I−¯o¯o)·ˆω′′\nkis neglected in\nthe first term on the left in equation 2.28. The second term on the lef t in equation 2.28\nis the contribution due to the magnetic torque. There are two term s containing ˆokin the\nmagnetic torque, the first which is proportional to |¯H|and the second proportional to\n|ˆH′′\nk|. From equation 2.20, the latter scales as ¯ nM, whereMis the magnetic moment of\na particle. From dimensional analysis, ¯ nMis the disturbance to the magnetic field at a\ndistance ¯ n−1/3from a particle, comparable to the inter-particle distance in the sus pen-Particle interactions in a magnetorheological fluid 13\nsion. The weak interaction limit is considered here, where the magnet ic field disturbance\ndue to particle interactions is small compared to the applied field, ¯ nM≪ |¯H|. Therefore,\nthe term containing ¯o×(ˆH′′\nk·ˆok) is neglected in comparison to ˆokׯHin equation\n2.28.\nEquation 2.28 is solved to obtain the unknown orientation disturbanc eˆok. From equa-\ntion 2.10, ˆokis perpendicular to ¯o, and it is expressed as,\nˆok= ˆo†\nk(¯H−(¯o·¯H)¯o)+ ˆo‡\nk(¯oׯH), (2.29)\nwhere ˆo†\nkand ˆo‡\nkare scalar functions of the wavenumber. The specific form 2.29is ch osen\nbecause it satisfies the requirement ˆok·¯o= 0. The expression 2.29 is inserted into the\ntorque balance equation, 2.28,\nπd3η0\n2µ0M{(I−¯o¯o)·ˆω′\nk−¯o[ˆo†\nk(¯H−(¯o·¯H)¯o)·¯ω+ ˆo‡\nk(¯oׯH)·¯ω]\n−(¯o·¯ω)[ˆo†\nk(¯H−(¯o·¯H)¯o)+ ˆo‡\nk(¯oׯH)]+η′ˆφk(I−¯o¯o)·¯ω}\n+[¯o׈H′\nk−ˆo†\nk(¯o·¯H)(¯oׯH)−ˆo‡\nk(¯o|¯H|2−¯H(¯o·¯H))] = 0.(2.30)\nFromequation2.9,theunderlinedtermintheaboveequationiszero. Thescalarfunctions\nˆo†\nkand ˆo‡\nkare evaluated by taking the dot product of equation 2.30 with ¯oׯHand¯o\nrespectively,\nπd3η0\n2µ0M[ˆω′\nk·(¯oׯH)−ˆo‡\nk(¯o·¯ω)(|¯H|2−(¯o·¯H)2)+η′ˆφk(¯oׯH)·¯ω]\n+[¯H·ˆH′\nk−(¯o·¯H)(ˆH′\nk·¯o)−ˆo†\nk(¯o·¯H)(|¯H|2−(¯o·¯H)2] = 0,(2.31)\n−πd3η0\n2µ0Mˆo‡\nk(¯oׯH)·¯ω−ˆo‡\nk(|¯H|2−(¯o·¯H)2)) = 0.(2.32)\nThese equations are solved to obtain,\nˆo†\nk=πη0d3ˆω′\nk·(¯oׯH)\n2µ0M(¯o·¯H)(|¯H|2−(¯o·¯H)2)+¯H·ˆH′\nk−(¯o·¯H)(¯o·ˆH′\nk)\n(¯o·¯H)(|¯H|2−(¯o·¯H)2)\n+πd3η0η′ˆφk(¯oׯH)·¯ω\n2µ0M(¯o·¯H)(|¯H|2−(¯o·¯H)2), (2.33)\nˆo‡\nk= 0. (2.34)\nThe magnetophoretic diffusion term in equation 2.16 is simplified by negle cting the\ntermˆH′′\nk·ˆokin equation 2.19, since it is much smaller than ˆH′\nk,\nk·D·k=−¯nµ0Mk2(¯o·ˆHk+ˆok·¯H+Mˆnk\n3πη0dˆnk\n=−¯nk2\n3πη0dˆnk\nµ0M1○/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n¯o·ˆHk+πd3η0ˆωk·(¯oׯH)\n2(¯o·¯H)+πd3η0η′ˆφk¯ω·(¯oׯH)\n2(¯o·¯H)\n+µ0M(¯H·ˆHk−2○/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\n(¯o·¯H)(ˆHk·¯o))\n(¯o·¯H)+µ0M2ˆnk\n,(2.35)14 V. Kumaran\nHere, the right sides of equations 2.29 and 2.33-2.34 have been subs tituted for ˆokin\nthe second step. In equation 2.35, the terms 1○and2○cancel. The term 1○is the\ncontribution due to the interaction between the undisturbed part icle magnetic moment\nandthedisturbancetothemagneticfieldduetoparticleinteraction s.Thisexactlycancels\nwith one of the terms resulting from the disturbance to the particle orientation; the\nremaining terms are entirely due to the interaction between the und isturbed magnetic\nfield and the disturbance to the particle orientation due to interact ions,\nk·D·k=−¯nk2d2ˆωk·(¯oׯH)\n6(¯o·¯H)ˆnk−π¯nk2d5η′¯ω·(¯oׯH)\n36(¯o·¯H)\n−¯nk2µ0M¯H·ˆHk\n3πηd(¯o·¯H)ˆnk−k2¯nµ0M2\n3πη0d. (2.36)\nHere, the substitution ˆφk= ˆnk(πd3/6) has been used to relate the volume fraction and\nnumber density in the third term on the right. In equation 2.36, the fi rst term on the\nright is due to the disturbance to the orientation vector caused by the hydrodynamic\ninteractions, and the third term results from the disturbance to t he orientation vector\ncaused by magnetic interactions. The expressions 2.20 and 2.25 are substituted for the\ndisturbance to the magnetic field and vorticity due to particle intera ctions to obtain,\nk·D·k=−π¯nd5{[k·(¯ω−¯o(¯o·¯ω))][k·(¯oׯH)]−k2[¯ω−¯o(¯o·¯ω)]·[¯oׯH]}\n24(¯o·¯H)\n+¯nµ0M2(k·¯o)(k·¯H)\n3πη0d(¯o·¯H)−π¯nk2d5η′¯ω·(¯oׯH)\n36(¯o·¯H)\n−k2¯nµ0M2\n3πη0d. (2.37)\nThe cross product ¯oׯHin equation 2.37 is expressed in terms of the vorticity using\nthe torque balance equation 2.8 for the base state,\nk·D·k=π¯φd5η0{[k·(¯ω−¯o(¯o·¯ω))][k·(¯ω−¯o(¯o·¯ω))]−k2[|¯ω|2−(¯ω·¯o)2]}\n8µ0M(¯o·¯H)\n+2¯φµ0M2(k·¯o)(k·¯H)\nπ2d4η0(¯o·¯H)+πk2d5¯φη0η′(|¯ω|2−(¯ω·¯o)2)\n12µ0(¯o·¯H)\n−2k2¯φµ0M2\nπ2η0d4. (2.38)\nHere, the substitution ¯ n=¯φ(πd3/6)−1has been made, where ¯φis the volume fraction\nin the base state. The diffusion coefficient Dis extracted from equation 2.38. This is\nwritten in scaled form as the sum of a ‘hydrodynamic’ contribution du e to the vortic-\nity disturbance, ‘magnetic’ contribution due to the magnetic field dis turbance, and an\nisotropic contribution due to the variation in viscosity and magnetic p ermeability with\nparticle volume fraction,\nD=¯φd2|ω|(Dh+Dm+D′\niI), (2.39)Particle interactions in a magnetorheological fluid 15\nwhere the scaled diffusivities are,\nDh={(e¯ω−¯o(¯o·e¯ω))(e¯ω−¯o(¯o·e¯ω))−I[1−(e¯ω·¯o)2]}\n8Σ(¯o·e¯H),(2.40)\nDm=ΣM∗\nπ/parenleftbigg(¯oe¯H+e¯H¯o)\n(¯o·e¯H)−2I/parenrightbigg\n, (2.41)\nD′\ni=η′(1−(e¯ω·¯o)2)\n12Σ(¯o·e¯H). (2.42)\nHere,e¯ωande¯Hare the unit vectors along the vorticity and magnetic field, and the\nsubstitutions ¯ω=|¯ω|e¯ωand¯H=|¯H|e¯Hhave been used. There are two dimension-\nless numbers in equations 2.40-2.41, the scaled ratio of the hydrody namic and magnetic\ntorques Σ and the scaled magnetic moment M∗,\nΣ =µ0M|¯H|\nπη0d3|¯ω|, (2.43)\nM∗=M\nd3|¯H|. (2.44)\nIn the diffusion tensor 2.39, the contribution Dmdue to the magnetic torqueis propor-\ntional to ¯φ|¯ω|d2Σ∼(¯φµ0M|¯H|/dη0) is independent of the vorticity, but it does depend\non the viscosity, the particle magnetic moment and the magnetic field . The contribution\nDhduetothehydrodynamictorqueisproportionalto ¯φ|¯ω|d2Σ−1∼(¯φd5η0|¯ω|2/µ0M|¯H|)\nis proportional to the square of the angular velocity and inversely p roportional to the\nmagnetic field. The isotropic component DiIis due to the variation of the viscosity and\nthe magnetic permeability with the particle volume fraction. This is pos itive, and it has\na damping effect on concentration fluctuations.\n2.2.Steady solution\nThe rotating and steady solutions for the orientation vector for a n isolated dipolar\nspheroid in a magnetic field were derived in Kumaran (2020 a). Here, the results for\na spherical particle are briefly summarised. The orientation of the p article depends on\nthe dimensionless parameter Σ in equation 2.43. The projections oft he orientation vector\nalong the directions of the magnetic field and the vorticity are shown as a function of the\nparameter Σ in figure 2. The stable solutions are shown by the blue line s, and unstable\nsolutions by the red lines and the neutral solutions by the brown lines .\nIn the limit Σ → ∞, the orientation vector is parallel to the magnetic field; this\ncorrespondsto ¯o·e¯H→1 for the stable solution. Thereis alsoan unstable solution where\nthe orientation vector is anti-parallel to the magnetic field, in which c ase¯o·e¯H→ −1.\nAs Σ decreases, there is a gradual decrease in ¯o·e¯H, while¯o·e¯ωincreases and tends to\n1 for the stable solution in this limit. This implies that the particle is aligned along the\nvorticity direction in the limit Σ →0.\nThe parallel magnetic field, e¯ω·e¯H= 0, is a special case. In figure 2, it is observed\nthat there is a transition with a slope discontinuity in this case for Σ =1\n2. The stable\nand unstable branches merge and bifurcate into two center nodes , around which there\nare periodic closed orbits. There are no steady solutions in this case , and the distribution\nof orientations in the different orbits depends on the initial distribut ion. The effect of\ninteractions for a parallel magnetic field is examined in the following sec tion 2.3 for the\nparameter regime Σ >1\n2where there are steady solutions. The same study for an oblique\nmagnetic field is presented in section 2.4 for Σ >0, since there is a steady solution for\nall values of Σ.16 V. Kumaran\n10-11 101-1-0.500.51\nPSfrag replacements\nΣ¯o·e¯H\n(a)10-11 101-1-0.500.51\nPSfrag replacements\nΣ¯o·e¯ω\n(b)\nFigure 2. The variation of ¯o·e¯H(a) and ( ¯o·e¯ω) (b) with Σ for a particle with a permanent\ndipole. The solid lines are the results for a parallel magnet ic field (e¯ω·e¯H) = 0 and the dashed\nlines are the results for an oblique magnetic field with e¯ω·e¯H= 0.1 (◦),1\n2(△),1√\n2(∇) and\n√\n3\n2(⋄). The blue lines are the stable stationary nodes, the red lin es are the unstable stationary\nnodes and the brown lines are the neutral stationary nodes ar ound which there are periodic\norbits.\n2.3.Parallel magnetic field\nHere, we consider the special case where the imposed magnetic field is in the flow plane\nand perpendicular to the vorticity, ¯H·¯ω= 0. From equation 2.9, ¯o·¯ω= 0 for steady\nsolutions, that is, the orientation vector and the vorticity are also orthogonal. The con-\nfiguration and co-ordinate system are shown in figure 3 (a). A linear shear flow, shown\nby grey arrows, is applied in the fluid far from the particle. The mean v orticity is in\nthe direction perpendicular to the plane of shear, and the magnetic field vector is in\nthe plane of shear. The orthogonal unit vectors e¯ω= (¯ω/|¯ω|) ande¯H= (¯H/|¯H|) are\ndefined along the vorticity and magnetic field directions, and the thir d orthogonal vector\ne⊥= (e¯ω×e¯H) is perpendicular to the ( e¯ω−e¯H) plane.\nThe solution for the orientation vector for the stable stationary n ode is,\n¯o·e¯H=√\n4Σ2−1\n2Σ, (2.45)\n¯o·e⊥=1\n2Σ. (2.46)\nIn the equation 2.39 for the diffusion coefficient, the orientation vec tor is expressed as\n¯o= ˆoHe¯H+(¯o·e⊥)e⊥, and equations 2.45-2.46 are substituted to obtain the diffusion\ntensor due to hydrodynamic and magnetic interactions,\nDh/m=/parenleftbige¯ωe¯He⊥/parenrightbig\nDh/m\n¯ω¯ω0 0\n0Dh/m\n¯H¯HDh/m\n¯H⊥\n0Dh/m\n¯H⊥Dh/m\n⊥⊥\n\ne¯ω\ne¯H\ne⊥\n,(2.47)Particle interactions in a magnetorheological fluid 17\nωHo\neωeH\ne⊥\n(a)¯ω¯o\ne¯ωe/bardble⊥\ne¯H\n(b)\nFigure 3. The configuration and co-ordinate system for analysing the e ffect of interactions for\na parallel magnetic field (a) and an oblique magnetic field (b) . The vorticity vector, shown in\nblue, is perpendicular to the plane of flow. The applied magne tic field shown in brown and the\norientation vector shown in red are in the plane of flow in (a) a nd at an angle to the plane of\nflow in (b). The unit vector e/bardblis perpendicular to ¯ωin the¯ω−¯Hplane in (b). The unit vector\ne⊥is perpendicular to the ¯ω−¯Hplane.\nwhere\nDh\n¯ω¯ω= 0, Dm\n¯ω¯ω=−2ΣM∗\nπ(2.48)\nDh\n¯H¯H=−1\n4√\n4Σ2−1, Dm\n¯H¯H= 0, (2.49)\nDh\n¯H⊥= 0, Dm\n¯H⊥=M∗Σ\nπ√\n4Σ2−1, (2.50)\nDh\n⊥⊥=−1\n4√\n4Σ2−1, Dm\n⊥⊥=−2ΣM∗\nπ. (2.51)\nThe isotropic contributions to the diffusion tensor due to the conce ntration dependence\nof the viscosity and the magnetic permeability are,\nD′\ni=η′\n6√\n4Σ2−1. (2.52)\nThe values of the coefficients in the limit Σ ≫1 and Σ −1\n2≪1 are listed in table\n1. The eigenvalues and eigenvectors of the diffusion matrix are also p rovided. Since the\ndiffusion matrix is symmetric, the eigenvalues are real, the eigenvect ors are orthogonal,\nand the eigenvectors are the principal directions of amplification (a long the directions\nwith negative eigenvalues) or damping (along the directions with posit ive eigenvalues).\nThe isotropic part of the diffusion tensor Didue to the dependence of viscosity on\nconcentration is not included in the calculation of the eigenvalues in ta ble 1. When\nthe isotropic part is included, the eigenvectors remain unchanged a nd the eigenvalues\ntransform as λ→λ+Di.\nEquation 2.47 shows that one of the principal directions is always the vorticity direc-\ntion, and the diffusion coefficient in this direction is negative, resulting in concentration\namplification in this direction for all values of Σ. The magnitude of this d iffusion coeffi-\ncient increases proportional to Σ M∗for Σ≫1. This diffusion is due to the modification\nof the magnetic field by fluctuations in the particle number density, w hich is the last\nterm in the square brackets on the right in 2.11.18 V. Kumaran\nThe diffusion in the flow plane is anisotropic, and the diffusion coefficient s contain\ncontributions due to hydrodynamic and magnetic interactions. The principal directions\nfor diffusion in the plane, which are the eigenvectors of the 2 ×2 square sub-matrix in\n2.47, are presented in table 1 for high magnetic field in the limit Σ ≫1, and close to the\ntransition between static and rotating states (Σ −1\n2)≪1. For Σ ≫1, the two principal\naxes are along the e¯Hande⊥directions. The eigenvalue is negative along the e⊥with\nmagnitude (2Σ M∗/π), which is the same as that in the e¯ωdirection. The eigenvalue in\nthee¯Hdirection is positive for M∗> π, and negative otherwise, and the magnitude\nis proportional to Σ−1in this limit. This implies a weak damping of fluctuations for\nM∗> π, and a weak amplification for M∗< π. Thus, there is anisotropic clustering with\na strong amplification of fluctuations in the two directions perpendic ular to the magnetic\nfield, and weak damping or amplification along the magnetic field for Σ ≫1.\nFor Σ−1\n2≪1, the diffusion coefficient in the vorticity direction tends to a finite va lue,\n−(M∗/π). The diffusion coefficients in the flow plane divergeproportionalto ( Σ−1\n2)−1/2.\nThe eigenvectors in the flow plane are along the directions rotated b y an angle −π/4\nand +π/4 respectively from the magnetic field direction. One of the eigenvalu es is always\nnegative,whiletheotherispositivefor M∗>(π/2)andnegativeotherwise.Inbothcases,\nthe eigenvalues diverge proportional to (Σ −1\n2)−1/2. Thus, there is strong concentration\namplification along one principal direction in the flow plane and strong d amping in\nthe perpendicular direction for M∗>(π/2), and strong amplification in both principal\ndirections for M∗<(π/2).\nIt should be noted that the divergence proportional to (4Σ2−1)−1/2is the result of a\nmean-field calculation; a more complex renormalisation group calculat ion is required to\ninclude the effect of fluctuations.\n2.4.Oblique magnetic field\nAnalytical solutions for the steady orientation have been derived f or a spherical particle\nin an obliquemagneticfield, wherethe vorticityandmagneticfield aren ot perpendicular.\nThe orthogonal co-ordinate system shown in figure 3 (b) is used, w here the unit vector\ne¯ωis along the vorticity direction, e/bardblis perpendicular to e¯ωin thee¯ω−e¯Hplane, and\ne⊥is perpendicular to the e¯ω−e¯Hplane. The following notations are used for the dot\nproducts,\nˆωH=e¯ω·e¯H,& ˆoH=¯o·e¯H. (2.53)\nIt should be noted that ˆ ωHis specified, since it depends on the relative orientation of\nthe vorticity and magnetic field, whereas ˆ oHis determined from the solution for the\norientation vector. From equation 2.9, the dot product ¯o·e¯ω= (ˆωH/ˆoH). The unit\nvectorse/bardblande⊥are,\ne/bardbl=e¯H−ˆωHe¯ω/radicalbig\n1−ˆω2\nH, (2.54)\ne⊥=e¯ω×e¯H/radicalbig\n1−ˆω2\nH. (2.55)Particle interactions in a magnetorheological fluid 19\nPermanent dipole Induced dipole\nΣ≫1(Σ−1\n2)≪1Σi≫1(Σi−1)≪1\nD¯ω¯ω−2ΣM∗\nπ−M∗\nπ−4Σiχ∗\nπ−χ∗\n√\n2π√\nΣi−1\nD¯H¯H−1\n8Σ−1\n8√\nΣ−1\n2−1\n4Σi−1\n4√\n2√\nΣi−1\nD¯H⊥M∗\n2πM∗\n4π√\nΣ−1\n2χ∗\nπχ∗\n2√\n2π√\nΣi−1\nD⊥⊥−4ΣM∗\nπ−1\n8√\nΣ−1\n2−4Σiχ∗\nπ−4χ∗+π\n4√\n2π√\nΣi−1\nD′\niη′\n12Ση′\n12√\nΣ−1\n2η′\n6Σiη′\n6√\n2√\nΣi−1\nλ1(M∗−π)\n8πΣ(2M∗−π)\n8π√\nΣ−1\n2χ∗−π\n4πΣi2(√\n2−1)χ∗−π\n4√\n2π√\nΣi−1\ne1e¯He¯H+e⊥√\n2e¯H(1+√\n2)e¯H+e⊥\n23/4√√\n2+1\nλ2−2M∗Σ\nπ−(2M∗+π)\n8π√\nΣ−1\n2−4χ∗Σi\nπ−2(√\n2+1)χ∗+π\n4√\n2π√\nΣi−1\ne2e⊥e¯H−e⊥√\n2e⊥−(√\n2−1)e¯H+e⊥\n23/4√√\n2−1\nλ3−4M∗Σ\nπ−M∗\nπ−4χ∗Σi\nπ−χ∗\n√\n2π√\nΣi−1\ne3e¯ω e¯ω e¯ω e¯ω\nTable 1. The asymptotic behaviour of the diffusion coefficients for Σ ,Σi≫1 and close to the\ntransition from steady to rotating states for a parallel mag netic field for permanent and induced\ndipoles.\nThe the orientation vector for the steady solution is specified by th e relation,\n¯o·e¯ω=/radicaligg\n−4Σ2−1\n2+/radicalbig\n(4Σ2−1)2+16Σ2ˆω2\nH\n2, (2.56)\nˆoH=1\n2Σ/radicaligg\n4Σ2−1\n2+/radicalbig\n(4Σ2−1)2+16Σ2ˆω2\nH\n2. (2.57)20 V. Kumaran\nThe projection of the orientation vector on the unit vector e/bardblis,\n¯o·e/bardbl=ˆωH(1−(¯o·e¯ω)2)\n(¯o·e¯ω)/radicalbig\n1−ˆω2\nH. (2.58)\nThe dot product ¯o·e⊥is determined from the torque balance equation,\n¯o·e⊥=¯o·(e¯ω×e¯H)/radicalbig\n1−(e¯ω·e¯H)2=−e¯ω·(¯o×e¯H)/radicalbig\n1−(e¯ω·e¯H)2\n=e¯ω·(I−¯o¯o)·e¯ω\n2Σ/radicalbig\n1−(e¯ω·e¯H)2=1−(¯o·e¯ω)2\n2Σ/radicalbig\n1−ˆω2\nH. (2.59)\nThe torque balance equation 2.8 and the definition 2.43 of Σ has been u sed in the third\nstep in the above equation. Equations 2.57- 2.59 are substituted int o equation 2.39,\nto obtain equation 2.47 for the diffusion coefficient, where DhandDmare symmetric\ntensors with the form,\nDh/m=/parenleftbig\ne¯ωe/bardble⊥/parenrightbig\nDh/m\n¯ω¯ωDh/m\n¯ω/bardblDh/m\n¯ω⊥\nDh/m\n¯ω/bardblDh/m\n/bardbl/bardblDh/m\n/bardbl⊥\nDh/m\n¯ω⊥Dh/m\n/bardbl⊥Dh/m\n⊥⊥\n\ne¯ω\ne/bardbl\ne⊥\n.(2.60)\nThe solutions for the elements of the tensors DhandDm, and the asymptotic expansion\nof these elements for small and large Σ, are provided in table 2. The e lements of Dhand\nDmdo depend on the solution 2.57 for ˆ oH, and the asymptotic expansions employ the\nsmall and large Σ approximations for the ˆ oHsolution,\nˆoH= ˆωH+2Σ2ˆωH(1−ˆω2\nH) for Σ≪1, (2.61)\nˆoH= 1−1−ˆω2\nH\n4Σ2for Σ≫1. (2.62)\nIt is easily verified that the elements of DhandDmreduce to equation 2.39 for a parallel\nmagnetic field with ˆ ωH= 0.\nFor an oblique magnetic field, all the elements of DhandDmare non-zero. This is in\ncontrast to the diffusion matrix 2.39 for a parallel magnetic field, whe re there are only\nfour independent non-zero components. For a parallel magnetic fi eld, there is a transition\nfrom a static to a rotating state for the orientation vector for a p arallel magnetic field\nat Σ =1\n2. In contrast, there is no transition for an oblique magnetic field. Th erefore, the\ndiffusioncoefficientsforaparallelmagneticfieldaredefinedonlyinthe range1\n2<Σ<∞,\nwhereas those for an oblique magnetic field are defined for 0 <Σ<∞in figure 8.\nTable 2 shows that all components of the hydrodynamic contributio n to the diffu-\nsion tensor Dhare negative, with the exception of the component Dh\n/bardbl⊥. In contrast,\nall components of the magnetic contribution to the diffusion tensor Dmare positive.\nThis implies that the magnetic interactions tend to dampen concentr ation fluctuations\nand stabilise the uniform state, whereas hydrodynamic interaction s tend to destabilise\nthe concentration fluctuations. The diffusion in the direction perpe ndicular to the vor-\nticity and magnetic field is entirely due to hydrodynamic interactions a nd the diffusion\ncoefficient D⊥⊥is negative. Therefore, concentration fluctuations are amplified in the\ndirection perpendicular to the vorticity and magnetic field. In other directions, the sta-\nbility is determined by a balance between the contributions due to hyd rodynamic and\nmagnetic interactions.\nThe components of Dare shown as a function of Σ for different values of ˆ ωHin figure\n8. Also shown by solid lines in figures 8 (a), (d), (e) and (f) are the re sults for a parallelParticle interactions in a magnetorheological fluid 21\nΣ≪1 Σ≫1\nDh\n¯ω¯ω −(ˆo2\nH−ˆω2\nH)ˆω2\nH\n8Σˆo5\nH−Σ(1−ˆω2\nH)\n2ˆωH−ˆω2\nH(1−ˆω2\nH)\n8Σ\nDm\n¯ω¯ω −2M∗Σ(ˆo2\nH−ˆω2\nH)\nπˆo2\nH−8M∗Σ2(1−ˆω2\nH)\nπ−2M∗Σ(1−ˆω2\nH)\nπ\nDh\n¯ω/bardbl −ˆωH(ˆo2\nH−ˆω2\nH)2\n8Σˆo5\nH√\n1−ˆω2\nH−2Σ3(1−ˆω2\nH)3/2−ˆωH(1−ˆω2\nH)3/2\n8Σ\nDm\n¯ω/bardblM∗ΣˆωH(1+ˆo2\nH−2ˆω2\nH)\nπˆo2\nH√\n1−ˆω2\nHM∗Σ√\n1−ˆω2\nH\nπˆωH2M∗ΣˆωH√\n1−ˆω2\nH\nπ\nDh\n¯ω⊥ −ˆωH(ˆo2\nH−ˆω2\nH)2\n16Σ2ˆo6\nH√\n1−ˆω2\nH−Σ2(1−ˆω2\nH)3/2\nˆωH−ˆωH(1−ˆω2\nH)3/2\n16Σ2\nDm\n¯ω⊥M∗ˆωH(ˆo2\nH−ˆω2\nH)\n2πˆo3\nH√\n1−ˆω2\nH2M∗Σ2√\n1−ˆω2\nH\nπM∗ˆωH√\n1−ˆω2\nH\n2π\nDh\n/bardbl/bardbl(ˆo2\nH−ˆω2\nH)(2ˆo2\nHˆω2\nH−ˆo2\nH−ˆω4\nH)\n8Σˆo5\nH(1−ˆω2\nH)−Σ(1−ˆω2\nH)\n2ˆωH−(1−ˆω2\nH)2\n8Σ\nDm\n/bardbl/bardbl −2M∗Σˆω2\nH\nπˆo2\nH−2M∗Σ\nπ−2M∗Σˆω2\nH\nπ\nDh\n/bardbl⊥ˆω2\nH(ˆo2\nH−ˆω2\nH)2\n16Σ2ˆo6\nH(1−ˆω2\nH)Σ2(1−ˆω2\nH)ˆω2\nH(1−ˆω2\nH)\n16Σ2\nDm\n/bardbl⊥M∗(ˆo2\nH−ˆω2\nH)\n2πˆo3\nH2M∗Σ2(1−ˆω2\nH)\nπˆωHM∗(1−ˆω2\nH)\n2π\nDh\n⊥⊥−ˆo2\nH−ˆω2\nH\n8Σˆo3\nH+ˆω2\nH(ˆo2\nH−ˆω2\nH)2\n32Σ3ˆo7\nH(1−ˆω2\nH)−2Σ3(1−ˆω2\nH)\nˆωH−1−ˆω2\nH\n8Σ\nDm\n⊥⊥ −2ΣM∗\nπ−2ΣM∗\nπ−2ΣM∗\nπ\nD′\niη′(ˆo2\nH−ˆω2\nH)\n12ˆo3\nHΣΣη′(1−ˆω2\nH)\n3ˆωHη′(1−ˆω2\nH)\n6Σ\nλ1Σ(1−ˆωH)(2M∗−π(1+ˆωH))\n2πˆωH(M∗−π)(1−ˆω2\nH)\n8πΣ\ne1√\n1+ˆωHe¯ω+√\n1−ˆωHe/bardbl√\n2ˆωHe¯ω+/radicalbig\n1−ˆω2\nHe/bardbl\nλ2 −Σ(1+ˆωH)(2M∗+π(1−ˆωH))\n2πˆωH−2ΣM∗\nπ\ne2−√\n1−ˆωHe¯ω+√\n1+ˆωHe/bardbl√\n2−/radicalbig\n1−ˆω2\nHe¯ω+ ˆωHe/bardbl\nλ3 −2ΣM∗\nπ−2ΣM∗\nπ\ne3 e⊥ e⊥\nTable 2. The asymptotic behaviour of the diffusion coefficients for Σ ≫1 and Σ ≪1 for an\noblique magnetic field for particles with permanent dipoles .22 V. Kumaran\n10-11 10110-210-11\nPSfrag replacements\nΣD′\ni/η′\nFigure 4. The scaled isotropic part of the diffusion tensor, D′\ni/η′as a function of the parameter\nΣ for particles with a permanent dipole. The orientation of t he vorticity and magnetic field are\ne¯ω·e¯H= 0.1 (◦),1\n2(△),1√\n2(∇) and√\n3\n2(⋄). The solid line is the result for a parallel magnetic\nfield.\nmagnetic field. The components of the diffusion tensor for a parallel magnetic field do\nnot extend to Σ <1\n2, due to the transition to a rotating state; the divergence in the\ndiffusion coefficients Dh¯H¯H,Dm¯H⊥andDh\n⊥⊥predicted by equations 2.49-2.51 is apparent\nin figure 8. The results for ˆ ωH= 0.1 are in close agreement with those for a parallel\nmagnetic field for Σ /greaterorsimilar1\n2, but the divergence in the coefficients Dh¯H¯H,Dm¯H⊥andDh\n⊥⊥is\ncut-off and the diffusion coefficients are finite Σ <1\n2.\nThe coefficients Dh\n¯ω¯ω,Dh\n¯ω/bardbl,Dm\n¯ω/bardbl,Dh\n¯ω⊥,Dm\n¯ω⊥,Dm\n/bardbl/bardbl,Dh\n/bardbl⊥are all zero for a parallel mag-\nneticfieldwith ˆ ωH= 0.Figure8showsthatthesecoefficientsdodecreaseas ˆ ωHdecreases\nfor Σ>1\n2, in accordance with the Σ ≫1 expressions table 2. For Σ <1\n2, these do not as\nˆωHdecreases. The latter regime is not accessible for a parallel magnet ic field, since there\nis no steady orientation. Therefore, the diffusion due to interactio ns for a nearly parallel\nmagnetic field is qualitatively different from that in a parallel magnetic fi eld, due to the\ntransition to rotating states in the latter.\nThe magnetic contributions to the diffusion tensor are larger than t he hydrodynamic\ncontributions for Σ ≫1. For Σ ≫1, the largest contributions to the diffusion tensor\nareDm\n¯ω¯ω,Dm\n¯ω/bardbl,Dm\n/bardbl/bardblandDm\n⊥⊥which diverge proportional to Σ. The coefficient Dm\n⊥⊥=\n−(2M∗Σ/π) is negative for all Σ, indicating that concentration fluctuations ar e amplified\nin thee⊥direction. The diffusion in the e¯ω−e/bardblplane is along two principal directions.\nThe eigenvalue −(2M∗Σ/π) alonge2in table 2 also increases proportional to Σ for\nΣ≫1, indicating strong amplification of concentration fluctuations in th is direction.\nThe eigenvalue along the e1direction decreases proportional to Σ−1for Σ≫1, and the\nconcentration fluctuations are dampened/amplified for M∗≷π. Thus, the behaviour\nof concentration fluctuations for Σ ≫1 is very similar to that for a parallel magnetic\nfield, with strong concentration amplifications in two directions e⊥ande2, and weak\namplification/damping in the third perpendicular direction.\nIn the limit Σ ≪1, the diffusion coefficient along the e⊥direction perpendicular to\nthe vorticity and magnetic field is Dm\n⊥⊥=−(2M∗Σ/π). TheO(Σ) contributions to the\ndiffusion tensor are Dh\n¯ω¯ω,Dm\n¯ω¯ω,Dh\n/bardbl/bardbl,Dm\n¯ω/bardblandDm\n⊥⊥. For diffusion in the e¯ω−e/bardblplane;\nconcentration fluctuations are unstable along the e2direction where the eigenvalue λ2isParticle interactions in a magnetorheological fluid 23\n10-11 10110-510-410-310-210-11101\nPSfrag replacements\nΣ−Dh\n¯ω¯ω,−Dm\n¯ω¯ω/M∗\n(a)10-11 10110-510-410-310-210-11101\nPSfrag replacements\nΣ−Dh\n¯ω/bardbl,Dm\n¯ω/bardbl/M∗\n(b)\n10-11 10110-510-410-310-210-11\nPSfrag replacements\nΣ−Dh\n¯ω⊥,Dm\n¯ω⊥/M∗\n(c)10-11 10110-510-410-310-210-11101\nPSfrag replacements\nΣ−Dh\n/bardbl/bardbl,−Dm\n/bardbl/bardbl/M∗\n(d)\n10-11 10110-510-410-310-210-11\nPSfrag replacements\nΣDh\n/bardbl⊥,Dm\n/bardbl⊥/M∗\n(e)10-11 10110-510-410-310-210-11101\nPSfrag replacements\nΣ−Dh\n⊥⊥,−Dm\n⊥⊥/M∗\n(f)\nFigure 5. The components of the diffusion tensor −Dh\n¯ω¯ω&−Dm\n¯ω¯ω/M∗(a),−Dh\n¯ω/bardbl&Dm\n¯ω/bardbl/M∗(b),\n−Dh\n¯ω⊥&Dm\n¯ω⊥/M∗(c),−Dh\n/bardbl/bardbl&−Dm\n/bardbl/bardbl/M∗(d),Dh\n/bardbl⊥&Dm\n/bardbl⊥/M∗(e) and−Dh\n⊥⊥&−Dm\n⊥⊥/M∗(f)\ndue to hydrodynamic interactions (blue lines) and magnetic interactions (red lines) as a function\nof the parameter Σ for particles with a permanent dipole. The orientation of the vorticity and\nmagnetic field are e¯ω·e¯H= 0.1 (◦),1\n2(△),1√\n2(∇) and√\n3\n2(⋄). The solid red and blue lines\nare the non-zero results for a parallel magnetic field.24 V. Kumaran\nnegative, while they are stable in the perpendicular direction e1for 2M∗> π(1 + ˆωH)\nand unstable otherwise.\n3. Induced dipole\n3.1.Diffusion due to interactions\nFor an induced dipole, the particle magnetic moment is modeled as,\nM=χo[o·(H+nM)], (3.1)\nthatis,themagneticmomentisproportionaltothecomponentoft hemagneticfieldalong\ntheparticleorientation.Here, χisthemagneticsusceptibilityoftheparticle.Substituting\nM=Moin 3.1, the magnitude of the particle magnetic moment is,\nM=χo·H\n1−χn. (3.2)\nThe torque balance equation in the absence of interactions in the dir ection perpendic-\nular to the orientation vector, analogous to equation 2.8, is\n1\n2πηd3(I−oo)·ω+µ0χ(o·H)(o×H)\n1−χn= 0. (3.3)\nThe substitution M=χ(o·H)ois made in equation 2.1 to obtain the expression for\nthe force,\nF=∇/bracketleftbiggχ(o·H)\n1−χno·/parenleftbigg\nH+χno(o·H)\n1−χn/parenrightbigg/bracketrightbigg\n=∇/bracketleftbiggχ(o·H)2\n(1−χn)2/bracketrightbigg\n. (3.4)\nWhentheaboveexpressionislinearisedintheperturbationsandtra nsformedintoFourier\nspace, the equivalent of equation 2.11 for the interaction force is,\nˆFk=−2ıkµ0χ(¯o·¯H)/bracketleftigg\n(¯o·ˆHk+¯H·ˆok)\n(1−χ¯n)2+χˆnk(¯o·¯H)\n(1−χ¯n)3/bracketrightigg\n. (3.5)\nSince we are consideringthe limit where the disturbance is small compa red to the applied\nmagnetic field, the approximation ¯ nχ≪1 is made in 3.5. With this approximation, the\nequivalent of equation 2.16 for the particle concentration field is,\n∂ˆnk\n∂t+ık·(¯vˆnk)+DBk2ˆnk\n−/parenleftigg\n2k2¯nµ0χ(¯o·¯H)(¯o·ˆHk+¯H·ˆok+χˆnk(¯o·¯H))\n3πη0d/parenrightigg\n= 0. (3.6)\nComparing the expression 3.6 with the expression 2.17, the magneto phoretic diffusion\nterm in the particle number density equation is,\nk·D·k=−2k2¯nµ0χ(¯o·¯H)(¯o·ˆHk+ˆok·¯H+χˆnk¯o·¯H)\n3πη0dˆnk. (3.7)\nThe equivalent of equation 2.19 for the disturbance to the magnetic field due to inter-Particle interactions in a magnetorheological fluid 25\nactions is,\nˆHk=−kk\nk2·/bracketleftbiggˆnkχ¯o(¯o·¯H)\n(1−χ¯n)2+¯nˆok·(I(¯o·¯H)+¯H¯o)\n1−χ¯n/bracketrightbigg\n. (3.8)\nThe second term in the square brackets on the right proportional toˆok, equivalent of the\nterm proportionalto ˆH′′in equation 2.19, is neglected. The reasonfor this is discussed in\nthe paragraph following equation 2.28. In the denominator of the fir st term in the square\nbrackets on the right of 3.8, χ¯nis neglected in comparison to 1, because the disturbance\nto the magnetic field is small compared to the applied magnetic field. Th e disturbance\nto the vorticity field is given in equation 2.24. In this expression, the t erm proportional\nto ¯nd3on the left side is neglected because the volume fraction is small, and t he term\nproportional to ˆω′′on the right is neglected for the reason discussed after equation 2 .28.\nWith these approximations, the torque balance equation, equivalen t of equation 2.28, is\nπd3η0\n2µ0χ[(I−¯o¯o)·ˆωk−¯o(ˆok·¯ω)−ˆok(¯o·¯ω)+η′ˆφk(I−¯o¯o)·¯ω)]\n+{(¯o·¯H)(¯o׈Hk)+(¯o·ˆHk)(¯oׯH)+(¯o·¯H)(ˆokׯH)\n+(ˆok·¯H)(¯oׯH)+χˆnk(¯o·¯H)(¯oׯH)}= 0,(3.9)\nThe expression 2.29, is substituted into equation 3.9 to obtain,\nπd3η0\n2µ0χ{(I−¯o¯o)·ˆωk−¯oˆo‡\nk(¯oׯH)·¯ω−(¯o·¯ω)[ˆo†\nk(¯H−(¯o·¯H)¯o)\n+ ˆo‡\nk(¯oׯH)]+η′ˆφk(I−¯o¯o)·¯ω)}+{(¯o·¯H)(¯o׈Hk)\n+(¯o·ˆHk)(¯oׯH)+ ˆo†\nk[|¯H|2−2(¯o·¯H)2](¯oׯH)\n+ ˆo‡\nk(¯o·¯H)[¯H(¯o·¯H)−¯o|¯H|2]+χˆnk(¯o·¯H)(¯oׯH)}= 0.(3.10)\nThe functions ˆ o†\nkand ˆo‡\nkare determined by taking the dot product of equation 3.10 with\n¯oׯHand¯oand respectively,\nπd3η0\n2µ0χ[ˆωk·(¯oׯH)−ˆo‡\nk(¯o·¯ω)(|¯H|2−(¯o·¯H)2)+η′ˆφk¯ω·(¯oׯH)]\n+{(¯o·¯H)[¯H·ˆHk−2(¯o·¯H)(ˆHk·¯o)]+(¯o·ˆHk)|¯H|2\n+[|¯H|2−(¯o·¯H)2][ˆo†\nk(|¯H|2−2(¯o·¯H)2)+χˆnk(¯o·¯H)]}= 0,(3.11)\n−πd3η0\n2µ0χ[ˆo‡\nk(¯oׯH)·¯ω]−{ˆo‡\nk(¯o·¯H)[|¯H|2−(¯o·¯H)2]}= 0.(3.12)\nThese are solved to obtain,\nˆo†\nk=πη0d3\n2µ0χ[ˆωk·(¯oׯH)+η′ˆφk¯ω·(¯oׯH)]\n[2(¯o·¯H)2−|¯H|2][|¯H|2−(¯o·¯H)2]+χˆnk¯o·¯H\n2(¯o·¯H)2−|¯H|2\n+{(¯o·ˆHk)[|¯H|2−2(¯o·¯H)2]+(¯H·ˆHk)(¯o·¯H)}\n[2(¯o·¯H)2−|¯H|2][|¯H|2−(¯o·¯H)2]. (3.13)\nˆo‡\nk= 0. (3.14)26 V. Kumaran\nThe expression for the magnetophoretic diffusion in equation 3.7 is,\nk·D·k=−2k2¯nµ0χ(¯o·¯H)\n3πη0dˆnk/bracketleftig\n¯o·ˆHk+ ˆo†\nk[|¯H|2−(¯o·¯H)2]+χˆnk(¯o·¯H)/bracketrightig\n=−2k2¯nµ0χ(¯o·¯H)\n3πη0dˆnk/bracketleftigg\nπd3η0[ˆωk·(¯oׯH)+η′ˆφk¯ω·(¯oׯH)]\n2µ0χ[2(¯o·¯H)2−|¯H|2]\n+(¯H·ˆHk)(¯o·¯H)\n[2(¯o·¯H)2−|¯H|2]+χˆnk¯o·¯H[|¯H|2−(¯o·¯H)]\n2(¯o·¯H)2−|¯H|2+χˆnk(¯o·¯H)/bracketrightigg\n.\n=−2k2¯nµ0χ(¯o·¯H)\n3πη0dˆnk/bracketleftigg\nπd3η0[ˆωk·(¯oׯH)+η′ˆφk¯ω·(¯oׯH)]\n2µ0χ[2(¯o·¯H)2−|¯H|2]\n+(¯H·ˆHk)(¯o·¯H)\n[2(¯o·¯H)2−|¯H|2]+χˆnk(¯o·¯H)3\n[2(¯o·¯H)2−|¯H|2]/bracketrightigg\n.(3.15)\nSubstituting equations 3.8 and 2.24 for ˆHkandˆωk, the final expression for the diffusion\ncoefficient is,\nk·D·k=−d2¯φ(¯o·¯H)[(k·¯ω−(k·¯o)(¯o·¯ω))(k·(¯oׯH))−k2(¯oׯH)·(¯ω−(¯ω·¯o)¯o)]\n2(2(¯o·¯H)2−|¯H|2)\n−k2¯nd2η′(¯o·¯H)ˆφk¯ω·(¯oׯH)\n6ˆnk(2(¯o·¯H)2−|¯H|2)+4µ0χ2¯φ(k·¯H)(k·¯o)(¯o·¯H)3\nπ2d4η0[2(¯o·¯H)2−|¯H|2]\n−4k2¯φµ0χ2(¯o·¯H)4\nπ2η0d4[2(¯o·¯H)2−|¯H|2]. (3.16)\nHere,thenumberdensityisexpressedintermsofthevolumefract ionusingtheexpression\n¯n=¯φ/(πd3/6). Equation 3.3 is used to express ( ¯oׯH) in the first term on the right in\nequation 3.16,\nk·D·k=π¯φη0d5[(k·¯ω−(k·¯o)(¯o·¯ω))(k·¯ω−(k·¯o)(¯o·¯ω))−k2(|¯ω|2−(¯ω·¯o))2]\n4µ0χ(2(¯o·¯H)2−|¯H|2)\n+πk2¯φη0d5η′(|¯ω|2−(¯ω·¯o)2)\n6µ0χ(2(¯o·¯H)2−|¯H|2)+4µ0χ2φ(k·¯H)(k·¯o)(¯o·¯H)3\nπ2d4η0(2(¯o·¯H)2−|¯H|2)\n−4k2¯φµ0χ2(¯o·¯H)4\nπ2η0d4[2(¯o·¯H)2−|¯H|2]. (3.17)\nThe diffusion coefficient extracted from the above equation is of the form 2.39, where\nDhandDmare,\nDh=[(e¯ω−¯o(¯o·e¯ω))(e¯ω−¯o(¯o·e¯ω))−I(1−(e¯ω·¯o))2]\n4Σi[2(¯o·e¯H)2−1],\nDm=2Σiχ∗(¯o·e¯H)3(e¯H¯o+¯oe¯H)\nπ[2(¯o·e¯H)2−1]−4Σiχ∗(¯o·e¯H)4I\nπ[2(¯o·e¯H)2−1],\nD′\ni=η′(1−(e¯ω·¯o)2)\n6Σi[2(¯o·e¯H)2−1]. (3.18)\nHere, the dimensionless ratio of the magnetic and hydrodynamic tor ques is,\nΣi=µ0χ|¯H|2\nπη0d3|¯ω|, (3.19)Particle interactions in a magnetorheological fluid 27\nand the scaled susceptibility per unit volume is,\nχ∗=χ\nd3. (3.20)\nIn going from 3.5 to 3.6, we had made the approximation ¯ nχ≪1. If this approximation\nis not made, there is only one change in the diffusion coefficients, χ∗(1+ ¯nχ) should be\nsubstituted for χ∗in the first term on the right in the expression for Dm.\n3.2.Steady state\nThe orthogonal basis vectors ( e¯ω,e/bardbl,e⊥) in figure 3 (b) are used for an oblique magnetic\nfield, where e¯ωis along the vorticity, e/bardblis perpendicular to the vorticity in the ¯ω−¯H\nplane, and e⊥is perpendicular to the ¯ω−¯Hplane. The unit vector e/bardblande⊥are defined\nin equations 2.54-2.55. The solution ˆ oH(equation 2.53) has to be determined numerically\nin this case; this is in contrast to the permanent dipole in an oblique mag netic field where\nit is possible to obtain an analytical solution, 2.57. The solutions have b een derived in\nKumaran (2021 a,b) for a spheroid. The solution ˆ o2\nHfor a spherical particle satisfies the\ncubic equation,\n4Σ2\niˆo4\nH[1−ˆo2\nH]+[ˆω2\nH−ˆo2\nH] = 0, (3.21)\nwhere ˆωHand ˆoHare defined in equation 2.53.\nIn the limit Σ i≫1, the particle aligns along the magnetic field and ˆ oH→1. As Σ iis\ndecreased, there are steady stable solutions for the orientation vector for the parameter\nregimes\nΣ2\ni>1+18ˆω2\nH−27ˆω4\nH−/radicalbig\n(1−ˆω2\nH)(1−9ˆω2\nH)3\n32ˆω2\nHfor 0≤ˆω2\nH≤1\n9,(3.22)\n>9(1−3ˆω2\nH)\n8for1\n9≤ˆω2\nH≤1\n3, (3.23)\n>0 for ˆω2\nH≥1\n3. (3.24)\nWhen the conditions 3.22-3.24 are not satisfied, there are stable limit cycles and possibly\nan unstable steady solution. The boundary between the steady an d rotating solutions in\nthe Σi−ˆωHparameter space is shown by the blue line in figure 6.\nThe solutions of equation 3.21 for ¯o·e¯Hand the corresponding solutions of ¯o·e¯ω\nare shown as a function of Σ in figure 7. These are qualitatively differe nt from those for\nparticles with a permanent dipole shown in figure 2 because ¯o·e¯His necessarily positive\nfor an induced dipole; the parameter space ¯o·e¯H<0 does not exist in this case. The\nsolutions for the orientation of a particle acted upon by a shear flow and a magnetic\nfield are derived in orientation space consisting of the azimuthal and meridional angles\nof the particle orientation Kumaran (2021 a,b). In this orientation space, the red lines in\nfigure 7 are the unstable steady solutions, the blue lines are the sta ble steady solutions\nand the brown lines are saddle points. The evolution of the fixed point s for 0<ˆω2\nH<1\n9\nis illustrated by the curve for ˆ ωH= 0.1 in figure 7. There is one stable fixed point for\nΣi≫1. As Σ iis decreased, there appears one unstable fixed point and one sadd le point\nin the phase diagram. When there is a further decrease in Σ i, the saddle and stable fixed\npoint merge, and there remains one unstable fixed point and one sta ble limit cycle. For\n1\n9<ˆω2\nH<1\n3, the phase plot contains an stable fixed point and an unstable limit cyc le\nfor Σi≫1, and there is an exchange of stability to an unstable fixed point and a stable\nlimit cycle for Σ i≪1, as shown by the curve for ˆ ωH= 0.5 in figure 7. For1\n3<ˆω2\nH<1,28 V. Kumaran\n0 0.2 0.4 0.6 0.8 100.51\nPSfrag replacements\nˆωHΣi\nFigure 6. The boundary between the stable stationary solutions (abov e) and rotating states\n(below) for the single-particle dynamics, equations 3.24 i s shown by the blue line, and the\nboundary for the dynamical transition at Σ i=/radicalbig\n1−2ˆω2\nHis shown by the red line in the\nˆωH−Σiplane.\nthere is a stable fixed point which is alined with the magnetic field for Σ i≫1, and with\nthe vorticity for Σ i≪1.\n3.3.Parallel magnetic field\nEquation 3.21 has analytical solutions for when the magnetic field is pa rallel to the flow\nplane, ˆωH= 0. The stable solution for the orientation vector exists only for Σ i>1, and\nthe orientation vector is given by,\nˆoH=/radicaligg\n1\n2+/radicalbig\nΣ2\ni−1\n2Σi,¯o·e⊥=/radicaligg\n1\n2−/radicalbig\nΣ2\ni−1\n2Σi. (3.25)\n2ˆo2\nH−1 =/radicalbig\nΣ2\ni−1\nΣi,ˆoH(¯o·e⊥) =1\n2Σi. (3.26)\nThe orthogonal basis unit vectors ( e¯ω,e¯H,e⊥) shown in figure 3 (a) are used, where\ne¯ωis along the vorticity direction, e¯His the along the direction of the magnetic field and\ne⊥=e¯ω×e¯H. The diffusion coefficient due to hydrodynamic and magnetic interact ions,\nequation 3.18, is reduced to the form in equation 2.47, where\nDh\n¯ω¯ω= 0, Dm\n¯ω¯ω=−χ∗(Σi+/radicalbig\nΣ2\ni−1)2\nπ/radicalbig\nΣ2\ni−1, (3.27)\nDh\n¯H¯H=−1\n4/radicalbig\nΣ2\ni−1, Dm\n¯H¯H= 0, (3.28)\nDh¯H⊥= 0, Dm¯H⊥=χ∗(Σi+/radicalbig\nΣ2\ni−1)\n2π/radicalbig\nΣ2\ni−1, (3.29)\nDh\n⊥⊥=−1\n4/radicalbig\nΣ2\ni−1, Dm\n⊥⊥=−χ∗(Σi+/radicalbig\nΣ2\ni−1)2\nπ/radicalbig\nΣ2\ni−1. (3.30)Particle interactions in a magnetorheological fluid 29\n10-11 10100.20.40.60.81\nPSfrag replacements\nΣi¯o·e¯H\n(a)10-11 10100.20.40.60.81\nPSfrag replacements\nΣi¯o·e¯ω\n(b)\nFigure 7. The variation of ¯o·e¯H(a) and ( ¯o·e¯ω) (b) with Σ ifor a particle with an induced\ndipole. The solid lines are the results for a parallel magnet ic field (e¯ω·e¯H) = 0 and the dashed\nlines are the results for an oblique magnetic field with e¯ω·e¯H= 0.1 (◦),1\n2(△),1√\n2(∇) and\n√\n3\n2(⋄). The blue lines are the stable stationary nodes, the red lin es are the unstable stationary\nnodes and the brown lines are the saddle nodes. The value of ( e¯ω·e¯H) for parallel a magnetic\nfield is not shown in sub-figure (b) because it is zero.\nThe isotropic part of the diffusion tensor due to the concentration dependence of the\nviscosity and magnetic permeability is,\nD′\ni=η′\n6/radicalbig\nΣ2\ni−1. (3.31)\nThe asymptotic values of the diffusion matrix for Σ i≫1 and Σ i−1≪1 are reported\nin table 1. The qualitative characteristics of the diffusion matrix are v ery similar to that\nfor permanent dipoles in a parallel magnetic field discussed at the end of section 2.3. An\nimportant difference is that Dm\n¯ω¯ωandDm\n⊥⊥diverge for Σ i→1 where a transition occurs\nbetween static and rotating states.\nTheeigenvaluesandeigenvectorsofthe diffusionmatrixareprovide din table1.Forthe\nsymmetric diffusion tensor, the eigenvalues are real, and the ortho gonal eigenvectors are\nthe principalaxesofextension/compression.Thereisextensiona longdirectionswith pos-\nitiveeigenvalues,indicatingthatconcentrationfluctuationsareda mped, andcompression\nalongdirectionswith negativeeigenvaluesresultingin amplification ofc oncentrationfluc-\ntuations. The eigenvalues in 1 are calculated without including the isot ropic component\nDiIin the diffusion tensor; these do not include the effect of variations o f viscosity with\nconcentration. The eigenvectors are unchanged when the isotro pic diffusion tensor with\nthe viscosity correction is included, and the eigenvalues are transf ormed as λ→λ+Di.\nThe component Dm\n¯ω¯ωof the diffusion tensor is always negative, and it diverges as\n(−4χ∗Σi/π) for Σ i≫1. Therefore, perturbations are always unstable in the vorticity\ndirection perpendicular to the flow, and there is clustering in this dire ction for a parallel\nmagnetic field. For Σ i≫1, the principal eigenvalues are aligned along the e¯ω,e/bardbl,e⊥\ndirections. The principal eigenvalue along the e⊥direction diverges as ( −4χ∗Σi/π). The\nprincipal eigenvalue along the e¯Hdirection is positive/negative for χ∗≷π, and its30 V. Kumaran\nmagnitude decreases proportional Σ−1\niin this limit. Thus, depending on the value of χ∗,\nthere is weakamplification/damping of fluctuations in the direction of the magnetic field,\nand strong amplification of fluctuations in the other two directions. When the magnetic\nfield is perpendicular to the fluid velocity, this would result in the forma tion of long and\nnarrow clusters aligned along the magnetic field.\nFor Σ i−1≪1 near the transition between rotating and steady states, all thr ee\neigenvalues diverge proportional to (Σ i−1)−1/2. Two of the eigenvalues, λ3in thee⊥\ndirection and λ2in thee¯ω−e¯Hplane, are negative, and therefore strong amplification\nof perturbations is predicted in these two directions. The other eig envalueλ1in thee1\ndirection is positive/negative for 2(√\n2−1)χ∗≷π, indicating damping of fluctuations if\nχ∗exceeds a threshold. It should be noted that the divergence expo nent−1\n2is a mean\nfield exponent, which could be altered if fluctuations are included.\n3.4.Oblique magnetic field\nThe orthogonal co-ordinate system shown in figure 3 (b) is used fo r an oblique magnetic\nfield, where e¯ωis along the direction of the vorticity perpendicular to the flow plane,\nthe unit vector e/bardblis perpendicular to e¯ωin the¯ω−¯Hplane, and e⊥is perpendicular\nto the¯ω−¯Hplane. After the solution of equation 3.21 for ˆ oHis determined, ( ¯o·e¯ω)\nis calculated using equation 2.9. This completely specifies the orientat ion vector ¯o. It is\neasily verified that for a parallel magnetic field with e¯ω·e¯H= 0, the non-trivial solution\nreduces to equation 3.25. The product ¯o·e/bardblis given by equation 2.58, and the product\n¯o·e⊥is,\n¯o·e⊥=¯o·(e¯ω×e¯H)/radicalbig\n1−(e¯ω·e¯H)2=−e¯ω·(¯o×e¯H)/radicalbig\n1−(e¯ω·e¯H)2\n=e¯ω·(I−¯o¯o)·e¯ω\n2Σi(¯o·e¯H)/radicalbig\n1−ˆω2\nH=[ˆo2\nH−ˆω2\nH]\n2Σiˆo3\nH/radicalbig\n1−ˆω2\nH. (3.32)\nHere, the torque balance equation 3.3 has been used to substitute for (¯o×e¯H), and\nequation 2.9 is used to substitute ˆ oH= ˆωH/(¯o·e¯ω). Equations 2.58 and 3.32 are used\nto substitute for ¯oin equation 3.18, and the diffusion tensor is of the form 2.60, and the\nelements of the matrix are listed in table 3.\nAlso presented in table 3 are the expansions for the components of DhandDmfor\nthe limits of small and large Σ irespectively. These are determined using the expansion\nfor the solutions of equation 3.21,\nˆoH= ˆωH+2Σ2\niˆω3\nH(1−ˆω2\nH) for Σ i≪1, (3.33)\nˆoH= 1−1−ˆω2\nH\n8Σ2\nifor Σi≫1. (3.34)\nThe denominators of the diffusion coefficients in the third column in tab le 3 decrease\nto zero for Σ i≪1 and ˆωH=1√\n2. In this case, it is necessary to first substitute the\nexpression 3.33 into the expressions in the second column in table 3, a nd then substitute\nˆωH=1√\n2and take the limit Σ i≪1. The results obtained in this manner are presented\nin the fourth column in table 3.\nThe striking feature of the diffusion tensor elements in table 3, is the singularity at\nˆo2\nH=1\n2. From equation 3.21, this corresponds to Σ i=/radicalbig\n1−2ˆω2\nH, shown by the red\nline in figure 6, and this singularity exists only for ˆ ωH<1√\n2. This boundary is different\nfrom the blue boundary between stationary and rotating states f or the single-particle\ndynamics, and this represents a dynamical transition due to inter- particle interactions.Particle interactions in a magnetorheological fluid 31\nΣi≪1,ˆωH/negationslash=1√\n2Σi≪1,ˆωH=1√\n2Σi≫1\nDh\n¯ω¯ω−(ˆo2\nH−ˆω2\nH)ˆω2\nH\n4Σiˆo4\nH[2ˆo2\nH−1]−Σiˆω2\nH(1−ˆω2\nH)\n(2ˆω2\nH−1)−1\n4Σi−ˆω2\nH(1−ˆω2\nH)\n4Σi\nDm\n¯ω¯ω−4χ∗Σiˆo2\nH(ˆo2\nH−ˆω2\nH)\nπ(2ˆo2\nH−1)−16χ∗Σ3\niˆω6\nH(1−ˆω2\nH)\nπ(2ˆω2\nH−1)−χ∗Σi\nπ−4χ∗Σi(1−ˆω2\nH)\nπ\nDh\n¯ω/bardbl−ˆωH(ˆo2\nH−ˆω2\nH)2\n4Σiˆo4\nH[2ˆo2\nH−1]√\n1−ˆω2\nH−4Σ3\niˆω5\nH(1−ˆω2\nH)3/2\n(2ˆω2\nH−1)−Σi\n4−ˆωH(1−ˆω2\nH)3/2\n4Σi\nDm\n¯ω/bardbl2χ∗Σiˆo2\nHˆωH(1+ˆo2\nH−2ˆω2\nH)\nπ[2ˆo2\nH−1]√\n1−ˆω2\nH2χ∗Σiˆω3\nH√\n1−ˆω2\nH\nπ(2ˆω2\nH−1)χ∗\n4Σi4χ∗ΣiˆωH√\n1−ˆω2\nH\nπ\nDh\n¯ω⊥−ˆωH(ˆo2\nH−ˆω2\nH)2\n8Σ2\niˆo6\nH[2ˆo2\nH−1]√\n1−ˆω2\nH−2Σ2\niˆω3\nH(1−ˆω2\nH)3/2\n(2ˆω2\nH−1)−1\n4−ˆωH(1−ˆω2\nH)3/2\n8Σ2\ni\nDm\n¯ω⊥χ∗ˆωH(ˆo2\nH−ˆω2\nH)\nπ[2ˆo2\nH−1]√\n1−ˆω2\nH4χ∗Σ2\niˆω5\nH√\n1−ˆω2\nH\nπ(2ˆω2\nH−1)χ∗\n2πχ∗ˆωH√\n1−ˆω2\nH\nπ\nDh\n/bardbl/bardbl(ˆo2\nH−ˆω2\nH)(2ˆo2\nHˆω2\nH−ˆo2\nH−ˆω4\nH)\n4Σiˆo4\nH[2ˆo2\nH−1](1−ˆω2\nH)−Σiˆω2\nH(1−ˆω2\nH)\n(2ˆω2\nH−1)−1\n4Σi−(1−ˆω2\nH)2\n4Σi\nDm\n/bardbl/bardbl−4χ∗Σiˆo2\nHˆω2\nH\nπ(2ˆo2\nH−1)−4χ∗Σiˆω4\nH\nπ(2ˆω2\nH−1)−χ∗\nπΣi−4χ∗Σiˆω2\nH\nπ\nDh\n/bardbl⊥ˆω2\nH(ˆo2\nH−ˆω2\nH)2\n8Σ2\niˆo6\nH[2ˆo2\nH−1](1−ˆω2\nH)2Σ2\niˆω4\nH(1−ˆω2\nH)\n(2ˆω2\nH−1)1\n4ˆω2\nH(1−ˆω2\nH)\n8Σ2\ni\nDm\n/bardbl⊥−4χ∗Σiˆo4\nH\nπ(2ˆo2\nH−1)4χ∗Σ2\niˆω4\nH(1−ˆω2\nH)\nπ(2ˆω2\nH−1)χ∗\n2πχ∗(1−ˆω2\nH)\nπ\nDh\n⊥⊥ −(ˆo2\nH−ˆω2\nH)\n4Σiˆo2\nH[2ˆo2\nH−1]+\nˆω2\nH(ˆo2\nH−ˆω2\nH)2\n16Σ3\niˆo8\nH[2ˆo2\nH−1](1−ˆω2\nH)−4Σ3\niˆω4\nH(1−ˆω2\nH)\n(2ˆω2\nH−1)−Σi\n2−(1−ˆω2\nH)\n4Σi\nDm\n⊥⊥−4χ∗Σiˆo4\nH\nπ(2ˆo2\nH−1)−4χ∗Σiˆω4\nH\nπ(2ˆω2\nH−1)−χ∗\nπΣi−4χ∗Σi\nπ\nD′\niη′(ˆo2\nH−ˆω2\nH)\n6Σiˆo2\nH(2ˆo2\nH−1)2η′Σiˆω2\nH(1−ˆω2\nH)\n3(2ˆω2\nH−1)η′\n6Σiη′(1−ˆω2\nH)\n6Σi\nλ1Σiˆω2\nH(1−ˆωH)(2χ∗ˆωH−π(1+ˆωH))\nπ(2ˆω2\nH−1)2χ∗(√\n2−1)−π\n4πΣi(χ∗−π)(1−ˆω2\nH)\n4πΣi\ne1√\n1+ˆωHe¯ω+√\n1−ˆωHe/bardbl√\n2e¯ω+(√\n2−1)e/bardbl\n23/4√√\n2−1ˆωHe¯ω+/radicalbig\n1−ˆω2\nHe/bardbl\nλ2 −Σiˆω2\nH(1+ˆωH)(2χ∗ˆωH+π(1−ˆωH))\nπ(2ˆω2\nH−1)−2χ∗(√\n2+1)+π\n4πΣi−4χ∗Σi\nπ\ne2−√\n1−ˆωHe¯ω+√\n1+ˆωHe/bardbl√\n2−e¯ω+(√\n2+1)e/bardbl\n23/4√√\n2+1−/radicalbig\n1−ˆω2\nHe¯ω+ ˆωHe/bardbl\nλ3 −4χ∗Σiˆω4\nH\nπ(2ˆω2\nH−1)−χ∗\nπΣi−4χ∗Σi\nπ\ne3 e⊥ e⊥ e⊥\nTable 3. The asymptotic behaviour of the diffusion coefficients for Σ i≫1 and Σ i≪1 for an\noblique magnetic field for particles with induced dipoles.32 V. Kumaran\n10-11 10110-210-11\nPSfrag replacements\nΣiD′\ni/η′\nFigure 8. The scaled isotropic part of the diffusion tensor, D′\ni/η′as a function of the parameter\nΣifor particles with a permanent dipole. The orientation of th e vorticity and magnetic field are\ne¯ω·e¯H= 0.1 (◦),1\n2(△),1√\n2(∇) and√\n3\n2(⋄). The solid line is the result for a parallel magnetic\nfield.\nAt this boundary, the ‘susceptibility’ multiplying ˆ o†\nkin equation 3.11 decreases to zero,\nand therefore the perturbation to the orientation vector ˆ o†\nkin equation 3.13 diverges.\nThe divergence at ˆ o2\nH=1\n2is observed in all components of the diffusion tensor in figure\n9 with the exception of Dm\n⊥⊥. Of course, the analysis is not accurate at this point because\nit is carried out assuming ˆ o†\nk≪1, and it is necessary to include non-linear fluctuation\neffects in the vicinity of this point to extract the nature of the diver gence. This is a\nsubject for future study.\nFor Σi≫1, the characteristics of the diffusion tensor are similar to those fo r dipolar\nparticles in figure 8. The largest contributions to the diffusion tenso r areDm\n¯ω¯ω,Dm\n¯ω/bardbl,Dm\n/bardbl/bardbl\nandDm\n/bardbl⊥; all of these diverge proportional to Σ i. The eigenvalue λ3corresponding to the\ne⊥direction is negative, and diverges proportional to Σ i, and therefore concentration\nfluctuations are amplified in the direction perpendicular to the plane c ontaining the\nvorticity and magnetic field. In the e¯ω−e/bardblplane, one of the eigenvalues λ2diverges\nproportional to −(4χΣi/π), indicating strong amplification of concentration fluctuations\nin thee2direction. The third eigenvalue λ1decreases proportional to Σ−1\ni, and this is\npositive/negativefor χ∗≷π, indicating weakamplification/dampingoffluctuations. The\ntwo directions e1ande2align with e¯Hande¯ωfor a parallel magnetic field ˆ ωH= 0.\nThe characteristics of the diffusion tensor are similar to those for d ipolar particles for\nΣi≪1 and ˆωH>1√\n2. The diffusion coefficients Dh\n¯ω¯ω,Dm\n¯ω/bardbl,Dh\n/bardbl/bardbl,Dm\n/bardbl/bardblandDm\n⊥⊥, and all\nthree eigenvalues, decrease proportional to Σ i. The eigenvalue of the diffusion tensor in\nthee⊥direction is negative for ˆ ωH>1√\n2,λ3=−(4χ∗Σiˆω2\nH/π(2ˆω2\nH−1)) and this is an\nunstabledirection.Oneoftheeigenvaluesinthe e2directionisalsoalwaysnegative,while\nthe third eigenvalue is positive/negative for 2 χ∗ˆωH≷π(1 + ˆωH). Thus, concentration\nfluctuations are unstable in the e⊥ande2directions, and they could be stable/unstable\nin thee1direction depending on the value of χ∗.\nFor ˆωH>1√\n2, there is a dynamical transition at ˆ oH=1\n2along the red line in figure\n6. This is a transition in the collective dynamics of the particles, and is d istinct from the\nsingle-particle transition between stationary and rotating states along the blue line in\nfigure 6. All the coefficients of the diffusion matrix diverge proportio nal to (ˆoH−1\n2)−1/2Particle interactions in a magnetorheological fluid 33\n10-11 10110-410-310-210-11101102\nPSfrag replacements\nΣi−Dh\n¯ω¯ω,−Dm\n¯ω¯ω/χ∗\n(a)10-11 10110-410-310-210-11101102\nPSfrag replacements\nΣi−Dh\n¯ω/bardbl,Dm\n¯ω/bardbl/χ∗\n(b)\n10-210-11 10110-410-310-210-11101102\nPSfrag replacements\nΣi−Dh\n¯ω⊥,Dm\n¯ω⊥/χ∗\n(c)10-11 10110-410-310-210-11101102\nPSfrag replacements\nΣi−Dh\n/bardbl/bardbl,−Dm\n/bardbl/bardbl/χ∗\n(d)\n10-11 10110-410-310-210-11101102\nPSfrag replacements\nΣiDh\n/bardbl⊥,Dm\n/bardbl⊥/χ∗\n(e)10-11 10110-410-310-210-11101102\nPSfrag replacements\nΣi−Dh\n⊥⊥,−Dm\n⊥⊥/χ∗\n(f)\nFigure 9. The components of the diffusion tensor −Dh\n¯ω¯ω&±Dm\n¯ω¯ω/χ∗(a),−Dh\n¯ω/bardbl&Dm\n¯ω/bardbl/χ∗(b),\n−Dh\n¯ω⊥&Dm\n¯ω⊥/χ∗(c),−Dh\n/bardbl/bardbl&±Dm\n/bardbl/bardbl/χ∗(d),Dh\n/bardbl⊥&Dm\n/bardbl⊥/χ∗(e) and−Dh\n⊥⊥&−Dm\n⊥⊥/χ∗(f) due\nto hydrodynamic interactions (blue lines) and magnetic int eractions (red lines) as a function of\nthe parameter Σ ifor particles with an induced dipole moment. The orientatio n of the vorticity\nand magnetic field are e¯ω·e¯H= 0.1 (◦),1\n2(△),1√\n2(∇) and√\n3\n2(⋄). The solid blue and red\nlines are the non-zero results for a parallel magnetic field.34 V. Kumaran\nat this transition. For ˆ ωH=1√\n2, the elements of the diffusion tensor have a different\nscaling than that for ˆ ωH>1√\n2. The elements Dh\n¯ω¯ω,Dm\n¯ω/bardbl,Dh\n/bardbl/bardbl,Dm\n/bardbl/bardblandDm\n⊥⊥actually\nincreaseproportionalto Σ−1\niin this limit. The eigenvaluesofthe diffusion matrixincrease\nproportional to Σ−1\ni. Two of these is negative, and therefore concentration fluctuat ions\nare amplified in one direction in the e¯ω−e/bardblplane and in the e⊥direction. The third is\npositive for 2 χ∗(√\n2−1)> π, and negative otherwise.\n4. Conclusions\nThe principal result of the present calculation is that the hydrodyn amic and magnetic\ninter-particle interactions manifest as an anisotropic diffusion tens or in the equation for\nthe particle concentrationfield for spherical particles with a perma nent or induced dipole\nmoment. The magnetic dipole due to neighbouring particles results in t he disturbance to\nthe magnetic field at a test particle. When a neighbouring particle is st ationaryin a shear\nflow, there is a hydrodynamic disturbance in the form of an antisymm etric force moment,\nwhich causes a disturbance to the vorticity at the location of a test particle. In addition,\nthere is the modification of the applied magnetic field due to the partic le magnetisation,\nwhichdependsontheparticleconcentration.Theneteffectofthe sedisturbancesiszeroin\na spatially uniform suspension. When there are variations in the part icle concentration\nfield, the net effect of these interactions in the torque balance equ ation results in a\ndisturbance to the orientation vector. There is a force due to the gradient in the dot\nproduct of the magnetic moment and the magnetic field, which cause s a drift velocity\nof the particles relative to the fluid. There is a contribution to the dr ift velocity due to\nthe variation in the magnetic moment per unit volume when there is a co ncentration\nvariation. The divergence of the drift velocity in the concentration equation has the\nform of an anisotropic diffusion term, and the elements of the diffusio n tensor have been\ncalculated for both permanent and induced dipoles.\nThe dimensionless parameters are the ratio of the magnetic and hyd rodynamic torque\non a particle, Σ for permanent dipoles (equation 2.43) and Σ ifor induced dipoles (equa-\ntion 3.19), and the ratio of the magnetic moment per unit volume and t he magnetic field,\nM∗(equation 2.44) for permanent dipoles and χ∗(equation 3.20) for induced dipoles.\nThe diffusion tensor consists of two distinct contributions, one due to magnetic interac-\ntionswhich isproportionalto M∗&χ∗,andthe secondduetohydrodynamicinteractions\nwhich does not depend on M∗&χ∗. The product Σ M∗= (µ0M2/πη0d6|¯ω|), which is\nindependent of the magnetic field, represents the effects of the in teraction between par-\nticles and the modification of the magnetic field due to the particle mag netic moment.\nThis is the inverse of the Mason number (Sherman et al.(2015)), for simple shear flows\nwhere the magnitudes of the strain rate and vorticity are equal. Fo r particles with an\ninduced dipole moment, the product Σ iχ∗= (µ0χ2|¯H|2/πη0d6|¯ω|) is the inverse of the\nMason number for a simple shear flows.\nThe components of the diffusion tensor are listed in table 1 for the pa rticular case\nˆωH= 0 where the magnetic field is in the flow plane. For high magnetic field, Σ ,Σi≫1,\ntwo principal directions of the diffusion tensor are along the magnet ic field and along\nthe vorticity direction (perpendicular to the flow plane), while the th ird component is\northogonal to the first two. For particles with a permanent and ind uced dipoles, the\neigenvalues of the diffusion tensor in the two directions perpendicula r to the magnetic\nfield are negative and diverge proportional to Σ ,Σi; this would lead to strong amplifi-\ncation of concentration fluctuations in these two directions. The e igenvalue of the dif-Particle interactions in a magnetorheological fluid 35\nfusion tensor along the magnetic field is positive when M∗,χ∗exceed a threshold and\nnegative otherwise, and it decreases proportional to Σ−1; this would lead to weak am-\nplification/damping of fluctuations along the direction of the magnet ic field. When the\nmagnetic field is perpendicular to the velocity, this explains the exper imental observation\nof the formation of particle chains in the direction of the field directio n when a magnetic\nfield is applied.\nThe diffusion matrix exhibits interesting behaviour close to the trans ition between ro-\ntating and stationary solutions of the orientation vector, Σ −1\n2≪1 and Σ i−1≪1. The\ncomponent of the diffusion coefficient perpendicular to the plane of fl ow is negative, in-\ndicating amplification of concentration fluctuations perpendicular t o the flow plane. The\ncomponents of the diffusion tensor in the flow plane diverge proport ional to (Σ −1\n2)−1/2\nand (Σ i−1)−1/2for permanent and induced dipoles. The eigenvalue of the diffusion\nmatrix in one of the principal directions in the flow plane is negative, ind icating strong\nclustering, while that in the other principal direction is positive when M∗orχ∗exceed\na threshold. This implies a strong anisotropic clustering tendency in t he flow plane as\nthe magnetic field is reduced near the transition to rotating states . This intriguing phe-\nnomenon should be observable in experiments similar to those perfor med in the field of\ndynamical critical phenomena (Hohenberg & Halperin (1977)). The exponents −1\n2cal-\nculated here is a mean field exponent; renormalisation group calculat ions are required to\ndetermine how these exponents change when fluctuations are inco rporated.\nFor a permanent dipole, there is no transition between rotating and stationary states\nfor an oblique magnetic field ( ˆ ωH/ne}ationslash= 0). The elements of the diffusion tensor, and their\nasymptotic behaviour for Σ ≪1 and Σ ≫1 are shown in table 2. In all cases, one of\nthe principal directions e⊥, which is orthogonal to the vorticity and the magnetic field,\nand the eigenvalues of the diffusion matrix in this direction are all nega tive, indicating\nthat there is strong amplification of concentration fluctuations in t his direction when a\nmagnetic field is applied. For particles with permanent dipoles in the limit Σ ≫1, one\neigenvalue of the diffusion tensor is negative and its magnitude increa ses proportional to\nΣ in one principal direction in the ¯ω−¯Hplane, but the eigenvalue perpendicular to the\n¯ω−¯Hplane is positive when M∗exceeds a threshold and it decreases proportional to\nΣ−1. For Σ≪1, the eigenvalues in the ¯ω−¯Hplane increase proportional to Σ. One of\nthe eigenvalues is negative, and the second could be positive or nega tive depending on\nthe value of M∗.\nForasuspensionofparticleswithinduceddipoles,thereisadynamica ltransitionatthe\nredlineinfigure6,whichisdifferentfromthebluelinewherethereisatr ansitionbetween\nstatic and rotatingstates in the single-particledynamics. The dyna mical transitionis due\nto inter-particle interactions. Of course, the linearisation approx imation is not applicable\nclose to the transition where the disturbance to the orientation ve ctor diverges, and more\nanalysis is required to examine how the divergence in the diffusion coeffi cients is cut-off\ndue to non-linear effects.\nThe eigenvalues of the diffusion matrix for particles with induced dipole s are qualita-\ntively similar to those with permanent dipoles. For Σ i≫1, there is strong amplification\nof fluctuations perpendicular to the ¯ω−¯Hplane, and in one principal direction in the\n¯ω−¯Hplane, and the magnitude of the eigenvalue increases proportional to Σiin this\nlimit. There is amplification or damping in the third direction depending on the value\nofχ∗, and the magnitude of the eigenvalue is proportional to Σ−1\ni. For Σ i≪1, the\nmagnitudes of the eigenvalues decrease proportional to Σ i. The eigenvalue in the direc-\ntion perpendicular to the ¯ω−¯Hplane is negative, and concentration fluctuations are\namplified in this direction. The eigenvalue in one principal directions in th e¯ω−¯Hplane36 V. Kumaran\nis also negative, and the second is positive or negative depending on t he value of χ∗and\nthe angle ˆ ωHbetween the magnetic field and the vorticity direction.\nThe effect of viscosity variations due to variations in the particle con centration has\nalso been analysed, considering a linear model for the dependence o f the viscosity on the\nconcentration. This results in a positive contribution to the isotrop ic part of the diffusion\ntensor, which dampens concentration fluctuations. For particles with permanent dipoles,\nthis contribution decreases proportional to Σ for Σ ≪1, and proportional to Σ−1for\nΣ≫1. For particles with induced dipoles, the diffusion coefficient decreas es proportional\nto Σ−1\nifor Σi≫1, and it decreases proportional to Σ i≪1 for values of ˆ ωHwhere there\nis no dynamical transition.\nThe estimates forthe hydrodynamicand magnetic contributions to the diffusion tensor\nare as follows. The hydrodynamic contribution operates perpendic ular to the magnetic\nfield for a parallel magnetic field (equations 2.51, 3.30), and this scale s as\nDh∼φd2|¯ω|. (4.1)\nThe characteristic diffusion time, the time for a particle to diffuse a dis tance comparable\nto its diameter, is τh= (φ|¯ω|)−1, is independent of particle diameter. The magnetic\ncontribution is proportional to Dm∼φd2|¯ω|M∗Σ∼(φµ0M2/πηd4) (equation 2.49).\nThis depends onlyonthe magneticmomentoftheparticles,and noto nthe magneticfield\nor the particle angular velocity. The magnetic moment per particle is t he product of the\nmagnetic moment per unit volume Mvand the particle volume, M=Mv(πd3/6)A m2.\nBased on this, the estimate for the diffusion coefficient is\nDm∼(φd2µ0M2\nv/η). (4.2)\nThe characteristicdiffusion time, which isthe time taken todiffuse a dis tancecomparable\nto the particle diameter, is τm= (η/φµ0M2\nv). Thus, both DhandDmare proportional\ntod2φ, and the characteristic diffusion times are independent of diameter , ifDmis ex-\npressed in terms of magnetic moment per unit volume. The vorticity in magnetorheolog-\nical applications, which is of the same magnitude as the strain rate, c ould vary between\n1−104s−1. Therefore, the minimum hydrodynamic diffusion time is τh∼(10−4/φ) s.\nFor particles with a permanent magnetic dipole, the dipole moment is us ually expressed\nas the magnetic moment per unit mass, emu/gm, and the dipole momen ts are in the\nrange 1-100 emu/gm. The magnetic moment per unit volume Mvis the product of the\nmagnetic moment per unit mass and the mass density, which is in the ra ngeMv∼1−103\nemu/cm3∼103−106A/m, if we assume the material has a mass density of 1-10 gm/cm3.\nTherefore, the minimum magnetic diffusion time is (10−6/φ)s. Thus, the time required\nfor a particle to diffuse across a distance comparable to its diameter is in the ms −µs\nrange if the strain rate is sufficiently high. This provides a plausible mec hanism for the\nformation of sample-spanning clusters when a magnetic field is applied .\nThe analysis of the diffusion due to interactions has been comprehen sive, coveringpar-\nticle suspensions with permanent and induced dipoles, and different r elative orientations\nof the flow and gradient directions and the applied magnetic field. The diffusion tensors\ndetermined here can be incorporated in continuum equations for ma gnetorheological flu-\nids in order to capture the effect of interactions on the particle dyn amics and orientation.\nThey provide an opportunity for a more granular design of magneto rheological devices,\nwhere the dimensionless parameters and the relative orientation of the flow and magnetic\nfield couldbedesignedfordispersion/clusteringofdesiredmagnitud esalongspecificaxes.\nFrom a fundamental perspective, the present analysis reveals a r ich dynamical landscape\ninviting detailed inspection of several interesting phenomena, such as the divergence of\nthe diffusivities at the transition between steady and rotating orien tation states for aParticle interactions in a magnetorheological fluid 37\nsuspension of particles with a permanent dipole, and the dynamical t ransition where the\ndiffusivities diverge due to collective effects for particles with an induc ed dipole. There is\nalso scope for incorporating features such as spheroidal particle s, where the shape factor\ncould lead to additional interesting phenomena not present for sph erical particles.\nThe author would like to thank the Department of Science and Techn ology, Govern-\nment of India for financial support.\nThe author reports no conflict of interest.\nREFERENCES\nAlmog, Y. & Frankel, I. 1995 The motion of axisymmetric dipolar particles in ahomog eneous\nshear flow. J. Fluid Mech. 289, 243–261.\nAnupama, A. V., Kumaran, V. & Sahoo, B. 2018 Magnetorheological fluids containing rod-\nshaped lithium-zinc ferrite particles: the steady-state s hear response. Soft Matter 14, 5407–\n5419.\nBarnes, H. A., Hutton, J. F. & Walters, K. 1989An Introduction to Rheology . Amsterdam,\nThe Netherlands: Elsevier Science B. V.\nBatchelor, G. K. 1970 The stress in a suspension of force-free particles. J. Fluid Mech. 41,\n545–570.\nBonnecaze, R. T. & Brady, J. F. 1992 Dynamic simulation of an electrorheological fluid.\nThe Journal of Chemical Physics 96(3), 2183–2202.\nChaves, A., Zahn, M. & Rinaldi, C. 2008 Spin-up flow of ferrofluids: Asymptotoic theory\nand experimental measurements. Phys. Fluids 20, 053102.\nFelt, D. W., Hagenbuchle, M., Liu, J. & Richard, J. 1996 Rheology of a magnetorheo-\nlogical fluid. Journal of Intelligent Material Systems and Structures 7, 589–593.\nGriffiths, D. J. 1982 Hyperfine splitting in the ground state of hydrogen. Am. J. Phys. 50,\n698–703.\nGriffiths, D. J. 2013Introduction to electrodynamics; 4th ed. . Boston, MA: Pearson.\nHinch, E. J. 1977 An averaged equation approach to particle interaction s in a fluid suspension.\nJ. Fluid Mech. 83, 695–720.\nHinch, E. J. & Leal, L. G. 1979 Rotation of small non-axisymmetric particles in a simp le\nshear flow. J. Fluid Mech. 92, 591–608.\nHohenberg, P. C. & Halperin, B. I. 1977 Theory of dynamic critical phenomena. Rev. Mod.\nPhys.49, 435–479.\nJackson, J. D. 1975Classical electrodynamics; 2nd ed. . New York, NY: Wiley.\nJeffery, G. B. 1923 The motion of ellipsoidal particles immersed in a visco us fluid. Proc. Roy.\nSoc. A123, 161–179.\nKlingenberg, D. J. 2001 Magnetorheology: Applications and challenges. AIChE Journal 47,\n246–249.\nKlingenberg, D. J., C., Ulicny. J. & Golden, M. A. 2007 Mason numbers for magnetorhe-\nology.J. Rheol. 51, 883–893.38 V. Kumaran\nKumaran, V. 2019 Rheology of a suspension of conducting particles in a ma gnetic field. J.\nFluid Mech. 871, 139–185.\nKumaran, V. 2020aBifurcations in the dynamics of a dipolar spheroid in ashear flow subjected\nto an external field. Phys. Rev. Fluids 5, 033701.\nKumaran, V. 2020bA suspension of conducting particles in a magnetic field - the maxwell\nstress.J. Fluid Mech. 901, A36.\nKumaran, V. 2021aDynamics of polarizable spheroid in a shear flow subjected to a parallel\nmagnetic field. Phys. Rev. Fluids 6, 043702.\nKumaran, V. 2021bSteadyandrotatingstatesofapolarizable spheroidsubjec tedtoamagnetic\nfield and a shear flow. Phys. Rev. Fluids 6, 063701.\nMarshall, L., Zukoski, C. F. & Goodwin, J. W. 1989Effects ofelectric fieldsontherheology\nof non-aqueous concentrated suspensions. J. Chem. Soc., Faraday Trans. 1 pp. 2785–2795.\nMartin, J. E., Odinek, J. & Halsey, T. C. 1994 Structure of an electrorheological fluid in\nsteady shear. Phys. Rev. E 50, 3263–3266.\nMelrose, J. R. 1992 Brownian dynamics simulation of dipole suspensions un der shear: the\nphase diagram. Molecular Physics 76, 635–660.\nMoffat, H. K. 1990 On the behaviour of a suspension of conducting particle s subjected to a\ntime-periodic magnetic field. J. Fluid Mech. 218, 509–529.\nMorillas, J. R. & de Vicente, J. 2020 Magnetorheology: a review. Soft Matter 16, 9614–\n9642.\nMorozov, K. I. 1993 The translational and rotational diffusion of colloida l ferroparticles. Jour-\nnal of Magnetism and Magnetic Materials 122, 98–101.\nMorozov, K. I. 1996 Gradient diffusion in concentrated ferrocolloids unde r the influence of a\nmagnetic field. Phys. Rev. E 53, 3841–3846.\nMoskowitz, R. & Rosensweig, R. E. 1967 Nonmechanical torque-driven flow of a ferromag-\nnetic fluid by an electromagnetic field. Appl. Phys. Lett. 11, 301–303.\nvon Pfeil, K., Graham, D., Klingenberg, D. J. & Morris, J. F. 2003 Structure evolution\nin electrorheological and magnetorheological suspension s from a continuum perspective. J.\nApp. Phys. 93, 5769–5779.\nPshenichnikov, A. F., Elfimova, E. A. & Ivanov, A. O. 2011 Magnetophoresis, sedimen-\ntation, and diffusion of particles in concentrated magnetic fluids.J. Chem. Phys. 134,\n184508.\nPshenichnikov, A. F. & Ivanov, A. S. 2012 Magnetophoresis of particles and aggregates in\nconcentrated magnetic fluids. Phys. Rev. E 86, 051401.\nRikken, R. S. M., Nolte, R. J. M., Maan, J. C., van Hest, J. C. M. , Wilson, D. A.\n& Christianen, P. C. M. 2014 Manipulation of micro- and nanostructure motion with\nmagnetic fields. Soft Matter 10, 1295–1308.\nRuiz-L´opez, J. A., Hidalgo-Alvarez, R. & de Vicente, J. 2017 Towards a universal master\ncurve in magnetorheology. Smart Materials and Structures 26, 054001.Particle interactions in a magnetorheological fluid 39\nSchumacher, K. R., Riley, J. J. & Finlayson, B. A. 2008 Homogeneous turbulence in\nferrofluids with a steady magnetic field. J. Fluid Mech. 599, 1–28.\nSherman, S. G., Becnel, A. C. & Wereley, N. M. 2015 Relating mason number to bingham\nnumber in magnetorheological fluids. J. Magn. Magn. Mater. 380, 98–104.\nSobecki, C. A., Zhang, J., Zhang, Y. & Wang, C. 2018 Dynamics of paramagnetic and\nferromagnetic ellipsoidal particles in shear flow under a un iform magnetic field. Phys. Rev.\nFluids3, 084201.\nVagberg, D. & Tighe, B. P. 2017 On the apparent yield stress in non-brownian magnetorh e-\nological fluids. Soft Matter 13, 7207–7221.\nVarga, Z., Grenard, V., Pecorario, S., Taberlet, N., Doliqu e, V., Manneville, S.,\nDivoux, T., McKinley, G. H. & Swan, J. W. 2019Hydrodynamicscontrolshear-induced\npattern formation in attractive suspensions. Proc. Nat. Acad. Sci. 116, 12193–12198.\nde Vicente, J., Klingenberg, D. J. & Hidalgo-Alvarez, R. 2011Magnetorheological fluids:\na review. Soft Matter 7, 3701–3710.\nZaitsev, V. M. & Shliomis, M. I. 1969 Entrainment of ferromagnetic suspension by a rotating\nfield.J. Appl. Mech. Tech. Phys. 10, 696–700."    },    {        "title": "1509.00788v3.Dynamo_model_for_the_inverse_transfer_of_magnetic_energy_in_a_nonhelical_decaying_magnetohydrodynamic_turbulence.pdf",        "content": "arXiv:1509.00788v3  [physics.plasm-ph]  15 Feb 2016Dynamo model for the inverse transfer of magnetic energy in a nonhelical\ndecaying magnetohydrodynamic turbulence\nKiwan Park\nDepartment of Physics, UNIST, Ulsan, 689798, Korea; pkiwan@u nist.ac.kr\nABSTRACT\nThe inverse cascade of magnetic energy occurs when helicity or rotational instability\nexists in the magnetohydrodynamic (MHD) system. This well k nown phenomenon has\nbeen considered as a basis for the large scale magnetic field i n universe. However\nnonhelical magnetic energy in a decaying MHD system also mig rates toward the large\nscale, which holds vital clues to the origin of large scale ma gnetic field in a quiescent\nastrophysical system. Zeldovich’s rope dynamo model is con sidered as a basic and\nsymbolistic model of magnetic field amplification. However, the rope model assuming\nspecific external forces like buoyancy or Coriolis force is n ot appropriate for a decaying\nturbulent system without any external force. So we suggest a new dynamo model based\non magnetic induction equation excluding a forcing source. This model shows the\nexpansion and growth of magnetic field (flux) is basically the redistribution of energy in\nthe system. Thetransfer of magnetic energy is in fact a succe ssive induction of magnetic\nfield resulted from the interaction between the fluid motion a nd seed magnetic field. We\nalso discuss about an analytic theorem based on the scaling i nvariant MHD equation.\nSubject headings: galaxies: magnetic fields, decaying turbulence, dynamo\nIntroduction\nMagnetic field is a ubiquitous phenomenon in universe which i s full of conducting fluids (plas-\nmas). The interaction between magnetic field and plasma is kn own to play a crucial role in the\nevolution of celestial objects like pulsars, jets, galaxy, or GRBs etc. So better comprehension of\nthe rudimentary interaction causing the MHD phenomena will more clearly explain the past and\npresent universe, and how the universe will evolve. However since the interaction between the\nmagnetic field and conducting fluid is basically a nonlinear p rocess, the intuitive understanding\nof MHD phenomena without its exact solution is not easy nor re liable. At the moment the gen-\neral solution of MHD equation is not yet known. Only a few appr oximate (stochastic) solutions\n(Kraichnan and Nagarajan 1967; Kazantsev 1968; Pouquet et a l 1976; Krause and Radler 1980)\nare available in a limited way, which means our comprehensio n is also restricted as much. So com-\nputational simulation is widely used as an alternative meth od. However the exact interpretation\nof simulation, which is the numerical calculation of MHD equ ation, requires precise understanding– 2 –\nof the analytic theories on MHD. So the investigation of a the ory-based intuitive model with the\nnumerical simulation is a practical and helpful way to get th e point of MHD phenomena.\nWe briefly introduce couple of models on the origin of magneti c field in the early universe. We show\nsimulation results for the amplification of (nonhelical) ma gnetic field in a decaying MHD system.\nThen we discuss about a theory derived from the scaling invar iant MHD equation (Olesen 1997).\nFinally we will introduce an intuitive dynamo model for the g eneral mechanism of magnetic energy\ntransfer in plasma. They give us clues to the origin of magnet ic field in the quiescent astrophysical\nsystem.\nFor the origin of magnetic field roughly two hypotheses are ac cepted: primordial (top-down) and\nastrophysical (bottom-up) model. Primordial hypothesis s upposes magnetic field could be gener-\nated as the conformal invariance of electromagnetic field wa s broken during the inflationary period\nof early universe (Turner and Widrow 1988). After the expans ion, magnetic field could be succes-\nsively generated through the cosmic phase transition such a s electroweak phase transition (EWPT)\nor quantum chromodynamic transition (QCD) from quark to har dron (Grasso and Rubinstein\n2001; Subramanian 2015). The magnitude of generated magnet ic field are thought to have been\nB0∼10−62G on a 1 Mpc comoving scale (during inflation), ∼10−29G (EWPT), and ∼10−20G\n(QCD) on a 10 Mpc scale (Sigl and Olinto 1997). The correlatio n lengths of these seed fields were\nlimited by the scale of particle horizon: ∼1cm (EWPT) and ∼104cm (QCD). On the other\nhand astrophysical hypothesis, posterior to the primordia l inflation, suggests that the seed field\nwas generated by the plasma effect in the primeval astrophysic al objects like galaxies or clusters.\nThe strengths of magnetic seed fields in these models are infe rred to be in the range between\nB0∼10−21G (Biermann battery effect) and ∼10−19G (Harrison effect).\nHowever, whether the seed magnetic fields were originated fr om primordial or astrophysical model,\nthe inferred strengths are too weak for the currently observ ed magnetic fields ( ∼µG). Also the\ninferred correlation length, which should be limited to the particle horizon at that time, is too tiny\ncompared with the that of present magnetic field in space. Defi nitely the initial seed fields must\nhave been amplified (Cho 2014). This process, called dynamo, is essentially the redistribution of\nenergy in the MHD system. If the magnetic energy is cascaded t oward large scale with its inten-\nsity increasing, this is called large scale dynamo (LSD, Bra ndenburg (2001), Blackman and Field\n(2002)). In contrast if the energy is cascaded toward small s cale, the process is called small scale\ndynamo (SSD, Kulsrud and Anderson (1992)). Also if the growt h rate of magnetic field depends\non the magnetic resistivity, the process is called slow dyna mo; otherwise, it is called fast dynamo.\nThe migration and increase of magnetic field are influenced by many factors whose critical condi-\ntions are still in debate. But, we will not discuss about the c riteria at this time. We focus on the\nqualitative mechanism of energy transfer in the MHD system.– 3 –\nAs mentioned dynamo is essentially energy redistribution. In the turbulent system the direction of\nenergy transfer is related to the conservation or minimizat ion of system variables. For example in\n(quasi) two dimensional ideal hydrodynamics the kinetic en ergyu2and enstrophy ω2= (∇×u)2\nare conserved. Then usingd\ndtu2∼d\ndt/integraltext∞\n0EV(k,t)dk= 0 andd\ndtω2∼d\ndt/integraltext∞\n0k2EV(k,t)dk= 0 we\ncan trace the migration of kinetic energy like below (Davids on 2004):\nd\ndt/angb∇acketleftk/angb∇acket∇ight=d\ndt/parenleftbigg/integraltext∞\n0kEV(k,t)dk/integraltext∞\n0EV(k,t)dk/parenrightbigg\n=d\ndt/integraltext∞\n0kEV(k,t)dk/integraltext∞\n0EV(k,t)dk. (1)\nWith the definition of initial centroid of /angb∇acketleftEV/angb∇acket∇ightbykc=/integraltext\nkE(k)dt//integraltext\nE(k)dt, this result can be\nrewritten like\nd\ndt/angb∇acketleftk/angb∇acket∇ight=−1\nkcd\ndt/integraltext∞\n0(k−kc)2EV(k,t)dk/integraltext∞\n0EV(k,t)dk. (2)\nSince the naturally spreading energy spectrum EV(k,t) makes/integraltext∞\n0(k−kc)2EV(k,t)dkincrease,\nthe peak of EVretreats toward the large scale with time; i.e., the inverse cascade ofEV. In three\ndimensional case where the enstrophy is not conserved, EVmigrates toward the smaller scale which\nhas larger damping effect. But with the kinetic helicity HV(=/angb∇acketleftu·ω/angb∇acket∇ight, ω=∇×u∼u), which is\na conserved variable in the ideal hydrodynamic system, the e nergy can be inversely cascaded with\nthe triad interaction among eddies (Biferale et al 2012).\nIn the MHD system without helicity, the kinetic and magnetic energy cascade toward small\nscale. But if the MHD system has a nontrivial kinetic helicit yHV, or magnetic helicity HM(=\n/angb∇acketleftA·B/angb∇acket∇ight,B=∇×A∼A), magnetic energy is cascaded inversely (Brandenburg and S ubramanian\n2005). This phenomenon can be derived using the conservatio n of energy and helicity, which\nis also related to the minimization of energy in the system. S ince the MHD system converges\ninto the absolute equilibrium state, the system variables c an be generally described by Gibbs\nfunctional composed of ideal invariants: /angb∇acketleftu2+B2/angb∇acket∇ight,/angb∇acketleftA·B/angb∇acket∇ight, and/angb∇acketleftu·B/angb∇acket∇ight. The ensemble average of\na N-dimensional system variable is represented by the parti tion function ‘ Z’ and the generalized\ncoordinate ‘ p’, ‘q’:/angb∇acketleftA/angb∇acket∇ight=/integraltext\nρA(p, q)dp3Ndq3N,ρ=Z−1exp[−W],Z=/integraltext\nexp[−W]dp3Ndq3N,\nW=λ1E+λ2A·B+λ3u·B. However, the system invariant itself does not always guara ntee the\ninverse cascade. What determine the direction of cascade ar e the structures of ideal invariants and\nLagrangian multipliers: λ1,λ2, andλ3(Frisch et al. 1975). From a fluid point of view, in addition\nto the statistical concept, we can explain the inverse casca de ofEMusing the mean field theory\n(Blackman and Field 2002; Park and Blackman 2012a,b). The pr ofiles of large scale magnetic\nenergyEMand helicity /angb∇acketleftA·B/angb∇acket∇ightin Fourier space ( k= 1) are represented like below (Park 2014):\n∂\n∂tEM=α/angb∇acketleftA·B/angb∇acket∇ight−2(β+η)EM, (3)\n∂\n∂t/angb∇acketleftA·B/angb∇acket∇ight= 4αEM−2(β+η)/angb∇acketleftA·B/angb∇acket∇ight, (4)\nwhere\nα=1\n3/integraldisplayt/parenleftbig\n/angb∇acketleftj·b/angb∇acket∇ight−/angb∇acketleftu·ω/angb∇acket∇ight/parenrightbig\ndτ, β=1\n3/integraldisplayt\n/angb∇acketleftu2/angb∇acket∇ightdτ. (5)– 4 –\nHere,/angb∇acketleftj·b/angb∇acket∇ight,/angb∇acketleftu·ω/angb∇acket∇ight, and/angb∇acketleftu2/angb∇acket∇ightare respectively small scale current helicity, kinetic hel icity, and kinetic\nenergy. These equations indicate the energy in small scale i s transferred to the large scale through\nthe small scale helical field ( α∼ /angb∇acketleftj·b/angb∇acket∇ight−/angb∇acketleftu·ω/angb∇acket∇ight), and the dissipation of large scale magnetic field\nis related to magnetic resistivity and small scale kinetic e nergy (β∼ /angb∇acketleftu2/angb∇acket∇ight). This equation is valid\nfor both a driven and decaying helical MHD system.\nAlso the cross field diffusive flow (Ryu and Yu 1998) or Bohm diffusi on effect (Lee and Ryu 2007)\nwas forced to increase the flux of magnetic field. Their concep ts are somewhat different from the\ntypical dynamo theory, but include the fundamental questio n on the dynamo.\nIn a driven MHD system the saturated field profile is decided by the injection scale, external driving\nforce, and the intrinsic properties like viscosity & magnet ic resistivity, not by the initial conditions\nwhose effect disappears within a few simulation time steps (Pa rk 2013, 2014). In contrast in a\nfree decaying MHD system the profile of decaying field is deter mined by the initial conditions and\nselective decaying speed besides other intrinsic properti es. Especially the selective decaying speed\ndue to the different spatial derivative order in the ‘system in variants’ arouses academic interest\nbesides many applications: the life time of star-forming cl ouds, measurement of galactic magnetic\nfield, or fast magnetic reconnection (Biskamp 2008). Howeve r, we should note the free decay\ndoes not mean the interaction between magnetic field and plas ma is forbidden. While the total\nenergy decays with time, the magnetic field still interacts w ith the fluid to migrate among eddies.\nWhether or not the MHD system is driven, the helical EMis inversely cascaded as long as ‘ α’ in\nEq.(3) is predominant over the dissipation effect. Moreover t he magnetic helicity in large scale, the\nstatistical correlation between different components of mag netic fields, is resilient to the turbulent\ndiffusion so that the decay speed slows down (Blackman and Subr amanian 2013). On the other\nhand the inverse transfer of EMin a decaying system without helicity or shear is contradict ory to\nthe typical MHD dynamo theory (Olesen 1997; Shiromizu 1998; Zrake 2014; Brandenburg et al\n2014). This phenomenon implies the energy released from the past events like supernovae could be\na source of large scale magnetic field structure in a quiescen t astrophysical object which does not\nhave a significant driving source.\nUsually the amplification of magnetic field (flux) due to the flu id motion is explained by Zeldovich’s\nrope model ‘stretch, twist, fold, and merge’ (Zeldovich et a l 1983). However, since the model as-\nsumes the influence of external force, it is not appropriate t o the growth of large scale magnetic\nfield in a decaying turbulence.– 5 –\n1. Simulation and method\nWe used PENCIL CODE (Brandenburg 2001) for the weakly compre ssible fluid in a periodic\nbox (8π3). MHD equations are basically coupled partial differential e quations composed of density\n‘ρ’, velocity ‘ u’, and vector potential ‘ A’ (or magnetic field B=∇×A).\nDρ\nDt=−ρ∇·u, (6)\nDu\nDt=−∇lnρ+1\nρ(∇×B)×B+ν/parenleftbig\n∇2u+1\n3∇∇·u/parenrightbig\n, (7)\n∂A\n∂t=u×B−η∇×B+f. (8)\nHereD/Dt(=∂/∂t+u·∇) is Lagrangian time derivative to be calculated following t he trajectory\nof fluid motion. In simulation we used 0.015 and 2 ×10−5for the kinematic viscosity ‘ ν’ and\nmagnetic diffusivity ‘ η’ respectively. This setting is to realize the large magneti c prandtl number\nPrM(=ν/η) in the early universe as similar as we can. The function f(x,t) generates the random\nnonhelical force with the dimension of electromotive force (EMF,u×B). The nonhelical magnetic\nenergy maximizes Lorentz force J×Bso that the fluid motion in plasma is efficiently excited.\nTurning on and off the function (0 < t <1, simulation time unit), we imitate a celestial system\nwhich was left decaying after it had been driven magneticall y by an event in the past.\n2. Results\nFig.1(a) shows the evolution of kinetic and magnetic energy spectrum in a decaying MHD\nsystem after the initial nonhelical magnetic forcing (0 <t<1). Black dotted line indicates kinetic\nenergy spectrum EV, and red solid one indicates magnetic energy spectrum EM. The increasing\nthickness of line means the lapse of time from t= 5 (thinnest) to t= 1342 (thickest). The overall\nEVandEMdecrease after the forcing stops, but each spectrum does not monotonically decay. For\n5<t<60EVin the small scale ( k >∼25) grows before it turns into the decay mode at t∼60,\nand the lagging decrease of EVin the large scale gives some hints of the influence of eddy tur nover\ntime ‘τ’. The profile of EMshows the similar, but somewhat different pattern. EMin small scale\ngrows faster than that of large scale, surpasses its initial magnitude, and then begins to decrease\natt∼60. In contrast, EMin large scale keeps growing far longer before it eventually decays. The\noverall evolution of magnetic field lags behind that of veloc ity field, which is a typical feature of a\nlargePrMMHD system\nFig.1(b) in fact has the same information as that of Fig.1(a) . But it shows the important features\nof a decaying nonhelical field more clearly. After the event s tops att∼1, bothEVandEMgrow\ntemporarily before they decay. The onsets of large scale EMandEVlag behind those of small\nscale energy but keep growing longer, which implies the influ ence ofτ,PrM, and energy supply on– 6 –\nthe evolution of large scale field. The advection term ‘ −u·∇u’ or pressure ‘ −∇p’ transfersEVto\nmake the system homogeneous and isotropic without the help o fEM. But for the transfer of EM,\nEVis a prerequisite as the source term ‘ ∇ × /angb∇acketleftu×b/angb∇acket∇ight’ in the magnetic induction equation shows.\nThe energy in the kinetic eddy is non locally transferred to t he magnetic eddy through B·∇u, and\nthe magnetic energy is locally transferred through - u·∇B. Therefore the long lasting EVin large\nscale can induce the migration of EMtoward the large scale, which also keeps longer due to the\nproportionally increasing eddy turnover time ( ∼1/k) and decreasing magnetic diffusivity ( ∼k2).\nThe inverse transfer of EMwithout helicity or shear deviates from the principle of typ ical dynamo\ntheory. However the seemingly inconsistent phenomenon is t rue and implies the fundamental prin-\nciple of energy migration in the magnetized plasma system. O lesen (1997) suggested the (inverse)\ntransferof energy bean essential phenomenonin thescaling invariant (self-similar) MHD equations,\nand the direction of energy transfer should be decided by the initial distribution of given energy\n‘E(k,0)∼kq’. The self similarity theorem shows that energy is inversel y transferred if q <−3;\notherwise, the energy cascades forward. The inverse transf er ofEMin a decaying MHD system im-\nplies that an event, which emitted electromagnetic energy i n the past, can be an origin of large and\nsmall scale magnetic field observed in the present universe. As an application of this plot, an ob-\nserver at ‘t∼1000’ can find large and small scale EV&EMwithout any driving source nor helicity.\nFig.2(a), 2(b) show the decaying EVand (helical) EMin Fourier space. Fig.2(c) includes the\nevolving profiles of total EM(B2/2, solid red line), helical EM(/angb∇acketleftkA·B/angb∇acket∇ight/2, dotted red line), and\nEV(dashedblack line) inreal space. Fig.2(d) showstheevolvi ng helicity ratio ‘ kHM/2EM’atk=1,\n5, and 8. The conditions are the same as those of nonhelical ca se except that the system is initially\ndriven by the helical magnetic energy. The helicity makes so me distinct features discriminated\nfrom the nonhelical decaying turbulence. The migration of ‘ EMpeak’ appears clearly, the strength\nof kinetic energy and overall decay rate of energy are lower t han those of the nonhelical energy\nsystem. Like the nonhelical MHD system the magnetic energy i s transferred to the kinetic eddy\nthrough Lorentz force. In principle the fully helical magne tic energy cannot be transferred to the\nkineticeddies. However theunstableturbulentmotionlack inginmemoryeffectgeneratesnonhelical\ncomponent so that kinetic eddies receive energy from magnet ic eddies. So although the strength\nofEVis weaker, there is limited contribution of EVto the induction of nonhelical magnetic field.\nCareful comparison of Fig.1(b), 2(c), 2(d) shows that the ki netic energy is transferred toward both\ndirections; consequently, EVin large and small scale leads to the forward and backward mig ration\nofEM. In the early time regime (1 < t <∼50), the helicity ratio of magnetic energy keeps\nrelatively constant in spite of the elevation of EM. When the helicity ratio in large scale EMbegins\nto accelerate ( t >∼50, Fig.2(d)), that of small scale EMstarts falling. We can infer the transfer\nof totalEMdue toEVprecedes that of helical EMdue to the small scale helical magnetic field\n(‘αeffect). And then ‘ αcoefficient’ interacts with large scale magnetic field direct ly leading to the\ncascade of small scale helical magnetic energy inversely. T he role of pressure and advection term\nseems to be independent of the helicity ratio in the system.– 7 –\n3. Theory: Models of energy transfer\n3.1. Inverse cascade of helical magnetic energy\nAs Fig.1(a)-2(d) show, the magnetic energy in a decaying MHD system can be transferred\ntoward the larger scale whether the field is helical or not. In case of helical field, toroidal mag-\nnetic field interacts with the helical fulid motion to genera te (amplify) poloidal magnetic field, and\nthe poloidal component generates (amplifies) the helical co mponent through the interaction with\nplasma. The effect of helicity in small scale is represented by ‘α’, which can be considered as\nan independent coefficient in a homogeneous and isotropic (wi thout reflection symmetry) system.\nAccording to the relative contribution to the field amplifica tion, the helical dynamo is divided into\n‘α2dynamo’, ‘αΩ dynamo’, ‘ α2Ω dynamo’ (Here ‘Ω’ indicates the differential rotation effect) .\nThe large scale magnetic energy and helicity in a decaying sy stem are represented by the solutions\nof coupled Eq.(3), (4) (Park 2014):\n2HM(t) = (HM(0)+2EM(0))e2/integraltextt\n0(α−β−η)dτ+(HM(0)−2EM(0))e−2/integraltextt\n0(α+β+η)dτ,(9)\n4EM(t) = (HM(0)+2EM(0))e2/integraltextt\n0(α−β−η)dτ−(HM(0)−2EM(0))e−2/integraltextt\n0(α+β+η)dτ.(10)\n(HereEM(0) andHM(0) are the initial large scale magnetic energy and helicity .)\nWith the positive initial magnetic helicity ( α >0), the first terms on the right hand side (RHS)\nare dominant. However, since there is no driving source, ‘ α(∼ /angb∇acketleftj·b/angb∇acket∇ight−/angb∇acketleftu·ω/angb∇acket∇ight)’ and ‘β(∼ /angb∇acketleftu2/angb∇acket∇ight)’\neventually decay and converge to ‘ zero’. But the combined index ‘ α−β−η’ decreases to become\nnegative with the finite diffusivity ‘ η’. Consequently the strengths of (large scale) magnetic ene rgy\nandhelicity increase first, reach thepeak, decay andconver ge tozeroeventually, whichis consistent\nwith the field evolution shown in Fig.2(c). If the initial mag netic helicity in small scale is negative,\ni.e.,α<0 the second terms in RHS decide the profiles of EM(t) andHM(t) with the negative sign.\n3.2. Inverse transfer of nonhelical magnetic energy\nThe inverse transfer of decaying nonhelical magnetic energ y cannot be explained by Eq.(9), (10)\nbecause of the negligible ‘ α’. Instead there have been trials to explain the phenomenon q ualita-\ntively using the scaling invariant Navier Stokes and magnet ic induction equation (Olesen 1997;\nDitlevsen et al. 2004) with numerical simulation (Brandenb urg et al 2014). The incompressible\nfree decaying MHD equations\n∂u\n∂t=−u·∇u+b·∇b−∇P/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nj×b−∇p+ν∇2u, (11)\n∂b\n∂t=b·∇u−u·∇b/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n∇×/angb∇acketleftu×b/angb∇acket∇ight+η∇2b. (12)– 8 –\nare invariant under the scaling transformation: r→lr,t→l1−ht,u→lhu,ν→l1+hν,b→lhb,\nη→l1+hη,P→l2hP, where ‘l’ and ‘h’ are arbitrary parameters. Then the scaled kinetic energy i s\nEV(k/l,l1−ht,Ll,K/l ) =l42πk2\n(2π)3/integraldisplayL\n2π/Kd3xd3y eik·(x−y)/angb∇acketleftu(lx,l1−ht)u(ly,l1−ht)/angb∇acket∇ight\n=l4+2hEV(k,t,L,K ). (13)\nSimilarlythescaledmagneticenergyisrepresentedlike‘ EM(k/l,l1−ht,Ll,K/l ) =l4+2hEM(k,t,L,K )’.\nSo the energy density spectrum is ‘ EV,M(k/l,l1−ht,lL,K/l ) =l1+2hEV,M(k,t,L,K )’, which can\nbe defined as ‘ EV,M(k, t) =k−1−2hψV,M(k, t)’ with an arbitrary function ‘ ψ(k, t)’ (Olesen (1997),\nDitlevsen et al. (2004), references therein). These scaled energy density relations lead to\nψV,M(k/l, l1−ht) =ψV,M(k, t). (14)\nIf we differentiate Eq.(14) with respect to ‘ l’ and then set ‘ l= 1’, we derive a differential equation:\n−k∂ψV,M\n∂k+(1−h)t∂ψV,M\n∂t= 0. (15)\nThegeneralsolutionofthisequationimplies‘ ψV,M(k, t)’isthefunctionof‘ k1−ht’, whichimpliesthe\ninverse transfer of energy with increasing time ‘ k∼t1/(h−1)’. For example if the primordial energy\nis ‘EV,M(k,0) =k−1−2h(q≡ −1−2h)’, the energy at ‘ t’ is ‘EV,M(k, t) =k−1−2hψV,M(k1−ht) =\nkqψV,M(k(3+q)/2t)’. More clearly, when h<1 (i.e.,q>−3), the decaying energy can be transferred\ninversely. Briefly the energy migration is the result of scal ed energy density relation, which is the\nintrinsic property of MHD equations. However this criterio n is not valid in the whole range. The\nintegration of EV,M∼kqψV,M(k(3+q)/2t) to get the total energy is known to yield an inconsistent\nresult(Ditlevsen et al. 2004). Thesaturated energy spectr umofEVandEMin a‘largePrMMHD\nsystem’ driven by the nonhelical kinetic energy has the rela tion of ‘E2\nM=k2EV’ in the subviscous\nscale (Park 2015). When the background magnetic field ‘ bext’ is strong, turbulence due to the\nnonlinear effect becomes relatively weak compared with that o f the guiding background magnetic\nfield. This leads to the energy spectrum EV∼k−4andEM∼k−1(Lazarian et al. 2004). If the\nforcing stops, EVshould be cascaded toward small scale but EMshould be transferred toward large\nscale. However EVandEMcannot migrate oppositely. With the strong bextthe assumption of\nself-similarity in the MHD equations is not valid. Moreover it is not yet clear if the scaled energy\ndensity relation can be applied to the helical field MHD syste m. More study is necessary.\n3.3. New model of energy transfer based on MHD equation\nWe suggest a new dynamo model for the transfer of EM. This model is not limited to the\nfree decaying turbulence; rather, it covers the general mec hanism of converting kinetic into mag-\nnetic energy and the local magnetic energy transfer. As ment ioned dynamo is in fact the result\nof interaction between the velocity field ‘ u’ and magnetic field ‘ b’, i.e., EMF /angb∇acketleftu×b/angb∇acket∇ightand cross– 9 –\nhelicity/angb∇acketleftu·b/angb∇acket∇ight. While EMF plays a role of explicit source of the magnetic fiel d, the cross helicity\nconstrains or suppresses the field profile implicitly (Yokoi 2013). In this paper we do not consider\nthe effect of cross helicity; instead, we show how the curl of EM F∇ × /angb∇acketleftu×b/angb∇acket∇ightand its decom-\nposed terms ‘ b·∇u’, ‘−u·∇b’ induce the magnetic field. The MHD system is filled with vario us\nkinds (scale, magnitude) of magnetic and velocity fields. If both velocity and magnetic field are\ncomposed of a single field vector, ‘ ∇’ or ‘∇×’ is meaningless and no dynamo occurs. The fields\nshould be at least locally inhomogeneous and anisotropic al though the macroscopic system may\nbe homogeneous and isotropic. So we consider two simplest ca ses that show the physical process\nof dynamoclearly: ‘ u’ (orb) isasinglefieldvector, but‘ b’ (oru) iscomposedof pluralfieldvectors.\nIn Fig.3(a) we assume the magnetic field heads for the gradien t of velocity field. That is, we set\nthe magnetic field: ‘ b= (0,0, b)’ and velocity field: ‘ ui= (0, ui(z),0)’. The strengths of velocity\nfields ‘ui(i= 1,2)’ and ‘ U’ are assumed to be the function of ‘ z’ in order of u2< u1< U. Also\nthe velocity field is not too strong compared with ‘ b’. Then we can make clear the meaning of first\nterm ‘b·∇u’ in Eq.(12). Mathematically it is the contraction of second order tensor into the first\norder one, i.e., vector:\nb·∇u→bˆz·/parenleftbigg\nˆx∂\n∂x+ ˆy∂\n∂y+ ˆz∂\n∂z/parenrightbigg\nui(z)ˆy=b∂ui(z)\n∂zˆy. (16)\nSince strength of ‘ ui(z)’ increases toward ‘ˆ z’, the induced magnetic field ‘ bind’ is parallel to ‘ ui’.\nAnd ‘bind’ is merged with ‘ b/ba∇dbl’ which is partially dragged ‘ b’ by the fluid motion ‘ ui’. If we assume\n‘b/ba∇dbl’ is thestrongest at ‘ U’ and‘∂ui/∂z’ is uniform, thetransferredmagnetic field‘ bind+b/ba∇dbl’ will have\nthe largest value at ‘ U’. However since ‘ b·∇u’ is the part of ∇×(u×b), some physical information\nis missing. So we need to derive the induction of magnetic fiel d through the curl of EMF directly.\nAs mentioned ‘ b’ interacts with the fluid ‘ ui’ to generate EMF varying from the smallest /angb∇acketleftu2×b/angb∇acket∇ight\nto the largest /angb∇acketleftU×b/angb∇acket∇ight. These differential magnitudes in EMF yield the clockwise rot ational effect\n∇×/angb∇acketleftui×b/angb∇acket∇ight>0, which induces the magnetic field bind∼/integraltext\ndτ∇×/angb∇acketleftui×b/angb∇acket∇ight. Therefore, bindand\nb/ba∇dbl, iare in the complementary relation to cause the growth of net m agnetic field. If the strength of\n‘b/ba∇dbl’ is assumed to be uniform, bind+b/ba∇dbl, iwill be the strongest at u2. However if ‘ b/ba∇dbl’ is proportional\nto the strength of ‘ ui’, the combined magnetic field will have the largest value bet ween ‘ui’ and\n‘U’. This can be one of the reasons why the peak of EMin a non helically driven MHD system\n(small scale dynamo) is located between the injection and vi scous scale. Of course the influence of\neddy turnover time ‘ τi’ and dissipation effect ‘ ∼k2bi’ need to be considered in order to explain the\nreason more exactly. On the other hand if the direction of bis reversed ( b· ∇u<0),bindis in\nthe opposite direction of b/ba∇dblresulting in the dissipation of net magnetic field. This is th e process\nof nonlocal energy transfer from EVtoEM.\nIn the same way, we can explain the local energy transfer in ma gnetic eddies as shown in Fig.3(b).\nAt this time the fluid ‘ u’ is heading for the decreasing equi-magnetic field line ( u·∇b<0). Each\nvelocity and magnetic field can be represented like ‘ u= (0,0, u)’ and ‘bi= (0, bi(z),0)’. The– 10 –\nstrength of magnetic field is the function of ‘ z’ in the order of b2< b1< Band∂bi(z)/∂z <0.\nThen the second term ‘ −u·∇bi’ in Eq.(12) is\n−uˆz·/parenleftbigg\nˆx∂\n∂x+ ˆy∂\n∂y+ ˆz∂\n∂z/parenrightbigg\nbi(z) ˆy=−u∂bi(z)\n∂zˆy→/vextendsingle/vextendsingle/vextendsingle/vextendsingleu∂bi(z)\n∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆy. (17)\nSo the direction of induced magnetic field ‘ bind’ is ˆy, which is parallel to ‘ bi’. Also their sum-\nmation is the largest at B. However, this cannot explain local energy transfer. So we n eed to\nderive the induced magnetic field using the mathematical defi nition of curl operator again. The\ninteraction between ‘ u’ and ‘b’ generates EMF varying from the smallest /angb∇acketleftu×b2/angb∇acket∇ightto the largest\n/angb∇acketleftu×B/angb∇acket∇ight. Their differential strengths create the rotational effect, wh ich induces magnetic field\nbind∼/integraltext\ndτ∇×/angb∇acketleftu×bi/angb∇acket∇ight. This induced ‘ bind’ is parallel to ‘ bi’ and largest at ‘ b2’ indicating the\nenergy migration from the strongest magnetic field to the wea kest one, which is independent of an\neddy scale. This explains the inverse transfer of EMin a decaying MHD system. Like the nonlocal\nenergy transfer the direction of local EMtransfer is determined by the location of velocity field,\nrelative strength of energy among eddies, eddy turnover tim e, dissipation effect. Also we can see\nthat ‘-bind’ generated by the oppositely directed ‘ u’ annihilates ‘ bi’.\nTheinducedmagnetic fieldalso constrains thefluidmotion. T hecurrentdensity ‘ J’ in Lorentzforce\n(Eq.11) is derived from the flow of electric charge ‘ qu’ carried by the fluids. In view of macroscopic\nfluid model, ‘ J’ is replaced by ∇×B, meaning the induced magnetic field is also able to constrain\nthe fluid motion. In Fig.3(a), where the magnetic field ‘ bˆz’ crosses the velocity fields, the directions\nofbindand∇ ×bindare ‘ˆy’ and ‘ˆx’ respectively, the direction of ( ∇ ×bind)×bindis ‘ˆz’. This\nappears to perturbthe fluids‘ ui’ perpendicular. However, ( ∇×bind)×bindcan bedecomposed into\nthe magnetic pressure‘ −∇b2\nind/2’ and magnetic tension ‘ bind·∇bind’. Magnetic tension cancels the\nmagnetic pressure parallel to the field so that ‘ jind×bind’ seems to press the fluid perpendicular.\nHowever, this tension is actually ˆbindˆbind·∇b2\nind/2+b2\nindˆbind·∇ˆbind=ˆbind∇/ba∇dblb2\nind/2+b2\nindˆκwhere\n‘ˆκ’ is a measure of the curvature of bind. So the exact Lorentz force is −∇⊥b2\nind/2+b2\nindˆκ. This\ncurvature related force gives energy to the fluid motion. The rmal and magnetic pressure suppress\nthe fluid motion, but magnetic tension boosts it. On the other hand in Fig.3(b), ( ∇×bind)×bind\nperturbs the fluid motion.\nThese two cases explain the amplification, more exactly migr ation ofEMand its constraint on\nEV. Real distribution of EVandEMis more complicated, but basically they can be replaced by\nthe combination of these two structures. What we have neglec ted is the effect of cross helicity\n/angb∇acketleftu·b/angb∇acket∇ight. When magnetic field ‘ b’ is (anti) parallel to the fluid motion ‘ u’, EMF iszeroso that\n‘bind’ is not generated. But more detailed stochastic analysis sh ows the effect of cross helicity is\ngenerated when the fourth order moment is decomposed into th e combination of second order one\nto close the MHD equations: /angb∇acketleftuubbind/angb∇acket∇ight ≃ /angb∇acketleftuu/angb∇acket∇ight/angb∇acketleftbbind/angb∇acket∇ight+/angb∇acketleftub/angb∇acket∇ight/angb∇acketleftubind/angb∇acket∇ight(Quasi Normal approximation,\nKraichnan and Nagarajan (1967), Park (2015)), which qualit atively matches the cross helicity\nterm ‘/angb∇acketleftui·bind/angb∇acket∇ight’ showninFig.3(a) or3(b). For detailed investigation of cr oss helicity andanisotropy– 11 –\nin the magnetized plasmasystem, more elaborate method is re quired. But we will not discussabout\nthose topics here.\n4. Summary\nWe have shown the inverse transfer of EMin a free decaying MHD turbulence. In the typical\n2D hydrodynamic and 3D MHD turbulence system, the inverse tr ansfer of energy is related to\nthe conservation of physical invariants such as energy, ens trophy, or magnetic helicity. The overall\nmechanism of inverse cascade of helical EMis well understood. However it is still tentative if the\nmagnetic helicity is prerequisite to LSD. Moreover since th e helicity is negligible or completely\nabsent from some celestial objects that are filled with vario us scales of magnetic fields, the exact\ncomprehension of (inverse) transfer of general EMis important to understand the origin and mech-\nanism of magnetic field evolution in the present universe.\nAs we have seen, the inverse transfer of nonhelical EMis the result of intrinsic property of scaling\ninvariant MHD equations. The evolving energy spectrum at ‘ t’ is represented by the initial energy\ndistribution EV,M(k,0)∼kqandψ(k(3+q)/2t). However this relation, limited to the local range,\nmay induce some inconsistent inference for the direction of energy transfer.\nWe introduced a dynamo model based on the magnetic induction equation∂b/∂t∼ ∇×(u×b)∼\nb·∇u−u·∇b. This model is not limited to the inverse transfer of decayin gEM, but it explains\nthe general mechanism of EMtransfer. The migration of EMis in fact the continuous induction\nof magnetic field guided by EV. The energy in the kinetic eddies is non locally transferred to the\nmagnetic ones through ‘ b· ∇u’; and, the energy in the magnetic eddies is locally transfer red to\ntheir adjacent ones through ‘ −u·∇b’. As the model shows, what decides the direction of magnetic\nenergy transfer is not the eddy scale size, but the guide of ve locity field and relative energy differ-\nence between eddies. In a mechanically driven system, the pr essure and advection term transfer\nEVchiefly toward the smaller scale whose eddy turnover time dec reases by ∼1/kbut dissipation\neffect elevates by ∼k2. ThenEVcascaded to the small scale induces EMcontinuously forming\nthe peak of EMin small scale. But strictly speaking energy migrates towar d both directions. If\nthe forcing in the system stops, EVin small scale fades away more quickly than that of large\nscale. Consequently EVin large scale generates EMin large scale and decides the profile of energy\nspectrum in the system. Again magnetic back reaction due to L orentz force generated by this EM\nconstrains (suppresses or boosts) the fluid motion, which is consistent with other numerical results.\nIn addition to this current model and numerical simulation, a more detailed analytic theory and\nvarious numerical simulations with arbitrary helicity rat io are required. Since we need to trace\nthe evolution of EVandEMtogether, Eddy Damped Quasi Normalized Markovianized (EDQ NM)– 12 –\napproximation is a more suitable method than other stochast ic models (Kraichnan and Nagarajan\n1967; Pouquet et al 1976; Park 2015). According to the numeri cal test of EDQNM for a unit PrM\nMHDsystem(Son 1999), thespectrumof low kis notmodified, butthepeaksof EVandEMclearly\nmigrates toward large scale. We think the approximation of E DQNM with large PrMcan show\nclearer transfer of EMtoward the large scale and give us the hints about the exact re presentation\nofψV,M(k, t) beyond the inertial range. We leave these topics for the fut ure work.\n5. acknowledgement\nKP acknowledges support from the National Research Foundat ion of Korea through grant\n2007-0093860.\nREFERENCES\nBiferale L., Musacchio S., & Toschi F., 2012, Phys. Rev. Lett ., 108, 164501\nBiskamp, D., Magnetohydrodynamic Turbulence, 2008, Cambr idge University Press, UK\nBlackman E. G., Field G. B., 2002, Phys. Rev. Lett., 89, 26500 7\nBlackman E. G., Subramanian K., 2013, MNRAS, 429, 1398\nBovino, S., Schleicher, D., R., G., & Schober, 2013, J., New J . Phys., 15, 013055\nBrandenburg, A., 2001, ApJ, 550, 824\nBrandenburg A., Subramanian K., 2005, Phys. Rep., 417, 1\nBrandenburg, A., Kahniashvili, T., & Tevzadze, A., G., 2014 , Phys. Rev. L, 114, 075001\nCho, J., 2014, ApJ, 797, 133\nCornwall, J., M., 1997, Phys. Rev. D, 56, 6146\nDavidson, P., A., An introduction for scientists and engine ers, 2004, Oxford University Press\nDitlevsen, P., D., Jensen, M., H., and Olesen, P., 2004, Phys . A, 342, 471\nFrisch, U., Pouquet, A., L´ eorat, 1975, J., Mazure, A. J. Flu id Mech. 68, 769-778\nGrasso, D. & Rubinstein, H. R., 2001, Phys. Rep., 348, 163\nKazantsev, A., P., 1968, JETP, 26, 1031\nKraichnan R. H. & Nagarajan, S., 1967, Physics of Fluids, 10, 859– 13 –\nKrause, F. & R¨ adler, K. H., 1980, Mean-field magnetohydrody namics and dynamo theory\nKulsrud, R. M. & Anderson, 1992, S. W., ApJ, 396, 606\nLazarian, A., Vishniac, E. T., & Cho, J., ApJ, 2004, 603, 180\nLee, Tae-Yeon & Ryu, Chang-Mo, 2007, J. Phys. D: Appl. Phys., 40, 5912\nOleson, P., 1997, PhLB., 398, 321O\nPark, K., & Blackman, E. G., 2012, MNRAS, 419, 913\nPark, K., & Blackman, E. G., 2012, MNRAS, 423, 2120\nPark, K., 2013, MNRAS, 434, 2020\nPark, K., 2014, MNRAS, 444, 3837\nPark, K. & Park, Dongho, Eq.13-A10, Eq.29-30, 2015, arXiv:1 504.02940\nPouquet, A., Frisch, U., & Leorat, J., 1976, J. Fluid Mech., 7 7, 321\nRyu, Chang-Mo & Yu, M. Y., 1998, PHYS. SCRIPTA, 57, 601\nShiromizu, T., 1998, Phys. Rev. D., 58, 107301\nSigl, G. & Olinto, A. V. 1997, Phys. Rev. D., 55, 4582\nSon, D., 1999, Phys. Rev. D., 59, 063008\nSubramanian, K., 2015, arXiv:1504.02311\nTurner, M. S. & Widrow, L. M., 1988, Phys. Rev. D., 37, 2743\nYokoi, N., 2013, Geophys. Astro. Fluid, 107, 114\nZeldovich, I., B., Ruzmaikin, A., A., Sokolov, D., D, 1983, M agnetic Fields in Astrophysics, Gordon\nand Breach Science Publishers\nZrake, J., 2014, ApJL, 794, 26\nThis preprint was prepared with the AAS L ATEX macros v5.2.– 14 –\n(a) (b)\nFig. 1.— (a) Free decaying EVandEMspectrum after nonhelical forcing. Vertical axis means\nthe energy, and horizontal axis means the wave number. (b) Th e time evolution of EVandEM.\nVertical axis is the energy, and horizontal one is time. The l ines ofk=1, 5, and 8 represent the\nlarge, injection, and small scale energy spectrum respecti vely. The plot shows the increasing energy\nin large and small scale is in fact the transferred one from th e injection scale at k= 5.– 15 –\n(a) (b)\n(c) (d)\nFig. 2.— (a) Spectrum of decaying kinetic and magnetic energ y. (b) Spectrum of decaying total\nmagnetic energy /angb∇acketleftB2/angb∇acket∇ight/2 and helical magnetic energy /angb∇acketleftkA·B/angb∇acket∇ight/2. Statistically helicity is the cor-\nrelation between different components of fields ∼Σi/negationslash=j\ni,j/angb∇acketleftBiBj/angb∇acket∇ight. Magnetic energy includes helical\nand nonhelical component. (c) Inverse cascade of total magn etic energy (solid red line), helical\nmagnetic energy (dotted red line), and kinetic energy (dash ed black line) of k=1, 5, 8. (d) Helicity\nratiokHM/2EM.– 16 –\n(a)\n(b)\nFig. 3.— (a) The energy in kinetic eddies is cascaded toward m agnetic eddies in a nonlocal way.\n(b) The energy in a magnetic eddy is transferred toward its ad jacent eddy."    },    {        "title": "1010.4771v1.Dipole_Quadrupole_dynamics_during_magnetic_field_reversals.pdf",        "content": "arXiv:1010.4771v1  [physics.flu-dyn]  22 Oct 2010Dipole-Quadrupole dynamics during magnetic field reversal s\nChristophe Gissinger1\n1Department of Astrophysical Sciences, Princeton Universi ty, Princeton, NJ 08544.\nThe shape and the dynamics of reversals of the magnetic field i n a turbulent dynamo experiment\nare investigated. We report the evolution of the dipolar and the quadrupolar parts of the magnetic\nfield in the VKS experiment, and show that the experimental re sults are in good agreement with\nthe predictions of a recent model of reversals: when the dipo le reverses, part of the magnetic energy\nis transferred to the quadrupole, reversals begin with a slo w decay of the dipole and are followed\nby a fast recovery, together with an overshoot of the dipole. Random reversals are observed at the\nborderline between stationary and oscillatory dynamos.\nPACS numbers: 47.65.-d, 52.65.Kj, 91.25.Cw\nDespite a large variability of the internal structure of\nplanets and stars, most of the observed astrophysical\nbodies possess a coherent large scale magnetic field. It\nis widely accepted that these natural magnetic fields are\nself-sustained by dynamo action [1]. Although reversals\nofthemagneticfieldinplanetaryandstellardynamosare\nnow consideredto be a common feature, they still remain\npoorly understood. Whereas the Sun shows periodic os-\ncillations of its magnetic field, the polarity of the Earth’s\ndipole field reverses randomly. During the last decades,\nseveral mechanisms have been proposed for geomagnetic\nreversals,amongwhich wecanmention, the analogywith\na bistable oscillator [2], a mean-field dynamo model [3],\nor interaction between dipolar and higher axisymmetric\ncomponents of the magnetic field [4, 5]. The comprehen-\nsion of dynamo reversals have also benefited from direct\nnumerical simulations of the MHD equations, which have\ndisplayed several possible mechanisms for reversals [6],\n[7], [8].\nReversals of a dipolar magnetic field have also been\nreported in the VKS (Von Karman Sodium) dynamo\nexperiment [9]. In this experiment, periodic or chaotic\nflips of polarity can be observed depending on the mag-\nnetic Reynolds number. Based on these results, a model\nfor reversals has been recently proposed by P´ etr´ elis and\nFauve [11]. It relies on the interaction between the dipo-\nlar and the quadrupolar magnetic components, and de-\nscribe transitions to periodic oscillations or randomly re-\nversing dynamos. It has been claimed that such a mech-\nanism could apply to the reversals of the Earth magnetic\nfield [12], and temptatively be connected to the periodic\noscillations of the Solar dynamo. Unfortunately, as for\nmany other models, the lack of observations of the mag-\nnetic field during a reversal limits a direct comparison\nwith the actual geomagnetic reversals. From this point\nof view, the VKS experiment is a unique opportunity to\ntest the validity of different models of turbulent revers-\ning dynamos. In particular, the model [11] makes predic-\ntions about the dynamics that are easily confrontable to\nexperimental results. We propose a simple way to ana-\nlyze data from the VKS experiment in order to test this\nmodel. To wit, we extract from the data the dipolar and\nFIG. 1: (Color online) Sketch of the VKS experiment\nthe quadrupolar components of the magnetic field. We\nshow that the characteristics of the reversals in the VKS\nexperiment are in very good agreement with the predic-\ntions, and that the dynamics of the magnetic field in this\nturbulent dynamo mainly result from an interaction be-\ntween dipolar and quadrupolar modes.\nIn the VKS experiment, a turbulent von Karman flow\nof liquid sodium is generated inside a cylinder by two\ncounter-rotating impellers, with independent rotation\nfrequencies F1andF2(see figure 1, and [13] for the de-\nscription of the set-up). When F1=F2, the system is\ninvariant about a rotation of an angle πaround any axis\nlocated in the mid-plane. On the contrary, if the im-\npellers rotate at different rates, this symmetry, hereafter\nreferredto as the Rπsymmetry, is broken. The dynamics\nof the magnetic field observed in the experiment strongly\ndepend on this symmetry. When F1=F2, a statistically\nstationary magnetic field with either polarity is gener-\nated, with a dominant axial dipolar component. Dynam-\nical regimes, including periodic oscillations and chaotic\nreversals of the magnetic field, are observed only when\ntheRπsymmetry is broken ( F1/negationslash=F2).\nPrevious studies have suggested that the evolution of\nthe magnetic field in the VKS experiment results from\nlow dimensional dynamics, involving only a few modes in\ninteraction [10]. This can be ascribed to the proximity of2\nthebifurcationthresholdandtothesmallnessofthemag-\nnetic Prandtl number in liquid metals ( Pm <10−5). In-\ndeed, inthelow- Pmregime,themagneticfieldisstrongly\ndamped compared to the velocity field. Hence, the dy-\nnamics are governed only by a small number of magnetic\nmodes. Based on this observation, P´ etr´ elis and Fauve\n[11] have proposed that close to the dynamo threshold,\nthe magnetic field can be decomposed into two compo-\nnents:\nB=D(t)d(r)+Q(t)q(r) (1)\nwhereD(respectively Q) represents the amplitude of\ndipolard(r) (resp. quadrupolar q(r)) component of the\nfield. As emphasized in [11], these components do not\nonly involve a dipole or a quadrupole, but also all the\nhigher components with the same symmetry in the trans-\nformation Rπ. In other words, d(r) (resp. q(r)) is the\nantisymmetric (resp. symmetric) part of the magnetic\nfield.\nThe evolution equations for D(t) andQ(t) can then\nbe obtained by symmetry arguments (see [11] for a de-\ntailed description of the model). Since d→ −dand\nq→qin the transformation Rπ, dipolar and quadrupo-\nlar modes cannot be linearly coupled when F1=F2.\nBreaking the Rπsymmetry by rotating the impellers at\ndifferent speeds allows a linear coupling between dipo-\nlar and quadrupolar modes. For a sufficiently strong\nsymmetry-breaking, this coupling can generate a limit\ncycle that involves an energy transfer between dipolar\nand quadrupolar modes. This mechanism has been re-\ncently validated on a numerical model of the VKS exper-\niment [15] and also in the case of a mean-field α2dynamo\nmodel [17].\nTwo scenarios of transition from a stationary dy-\nnamo to a periodically reversing magnetic field can be\ndescribed in the framework of this dipole-quadrupole\nmodel. When the coupling is such that the system is\nclose to both a stationary and a Hopf bifurcation, i.e.\nin the vicinity of a codimension-two bifurcation point,\none can have bistability between a stationary and a\ntime periodic reversing dynamo [13]. We thus get a\nsubcritical transition from a stationary dynamo to a\nperiodic one with a finite frequency at onset. Turbulent\nfluctuations can generate random transitions between\nthese two regimes [14]. Far from this codimension-two\npoint, a reversing magnetic field can be generated\nthrough an Andronov bifurcation when the stationary\nstate disappears through a saddle-node bifurcation [11].\nThen, the frequency of the limit cycle vanishes at onset.\nIn the vicinity of this transition, turbulent fluctuations\ndrive random reversals of the magnetic field. As a\nconsequence, random reversals always occur at the\nborderline between stationary and oscillatory dynamos.\nThis simple mechanism also yields several predictions\nabout the shape of the reversals. First, when the dipole\nDvanishes, part of the magnetic energy is transferred to8 10 12 14 1618 20\nF1 [Hz]182022242628F2[Hz]\nStationnary dynamo\noscillations or reversals\nNo dynamoF1/F2=0.6\nFIG. 2: (Color online) Parameter space: (cross) no\ndynamo, (circle) stationary dynamos, (star) oscillatory\nor random reversing dynamos.\nthe quadrupole Q. An overshoot of the dipolar ampli-\ntude is expected after each reversal. Random reversals\nare asymmetric. During a first phase, fluctuations push\nthe system from the stable solution to the unstable one,\nthus acting against the deterministic dynamics. This\nphase is slow compared to the one beyond the unstable\nfixed point, where the system is driven to the opposite\npolarity under the action of the deterministic dynamics.\nInthispaper, weusedataoftheVKSexperimentinor-\nder to reconstruct the dipolar and the quadrupolar parts\nofthe magnetic field and study their behavior in the time\ndependent regimes. Time-dependent regimes only occur\nfor specific values of F1andF2, inside three delimited\nregions of the parameter space [14]. We will focus on the\nregimes observed when following the line F1/F2= 0.6\nin the parameter space. Figure 2 shows that when the\nrotation rates are increased along this line, one first bi-\nfurcates to a stationary dynamo, then to time-dependent\nregimes. Figure 3(top) shows the time-recordings of the\nthreecomponentsofthemagneticfieldclosetothefastest\ndisk, displaying the bifurcation from stationary to time-\ndependent dynamo when the frequencies of the two disks\nare increased from 14 .4/24 Hz (F1+F2= 38.4 Hz) to\n15/25 Hz (F1+F2= 40 Hz). After a short transient\nstate, thethreecomponentsofthe magneticfieldundergo\na transition to nearly periodic oscillations.\nHowever, it is hard to test the pertinence of the\nmodel [11] from the time recording of the magnetic\nfield at a single point. Using measurements obtained\nfrom two probes 1 and 2, symmetric with respect to\nthe mid-plane, we compute the dipolar part, D(t)di=\n(Bi(1,t)+Bi(2,t))/2 and the quadrupolarpart, Q(t)qi=\n(Bi(1,t)−Bi(2,t))/2. In order to obtain observables3\n160 200 240\nTime [s]-150-100-50050100150Magnetic field [G]BR\nBθ\nBZ\n150 200 250 300\nTime [s]-505Magnetic field [arb. units]Dipole\nQuadrupole\nFIG. 3: (Color online) Top: time recordings of the three\ncomponents of the magnetic field. The rotation\nfrequencies are increased from 14 .4/24 Hz to 15 /25 Hz,\nleading to a transition from a stationary low amplitude\ndynamo regime to a limit cycle. Bottom: Behavior of\nthe dipolar and quadrupolar parts of the magnetic field.\nNote the transfer between the two components during\nreversals.\nwhich are independent of the spatial component i, each\nof these vectors is projected on its value at a given time\nt0. We thus extract D(t) andQ(t) up to a multiplicative\nconstant. In the measurements displayed here, the dipo-\nlar and quadrupolar components are projected on their\nstationary values obtained at F1+F2= 38.4 Hz. Note\nhowever that different methods could be used to recon-\nstruct these amplitudes. In particular, plotting the sum\nand the difference of a given component does not change\nthequalitativebehavior[19]. Figure3(bottom)showsthe\nevolution of D(t) andQ(t) during periodic oscillations of\nthe magnetic field. We observethat when the dipole van-\nishes, the amplitude of the quadrupole reaches its max-\nimum. This shows that the field reversals observed in\nthe VKS experiment do not correspond to a vanishing\nmagnetic field, but rather to a change of shape from a\ndominant dipolar field to a quadrupolar one. Immedi-\nately after each reversal, one can also note that during\nitsrecovery,thedipolaramplitudestronglyovershootsitsmeanvalue. Therefore, inagreementwith themodel [11],\nreversals in the VKS experiment involve a strong com-\npetition between dipolar and quadrupolar components of\nthe magnetic field.\nThe decomposition between dipolar and quadrupolar\ncomponents is not only relevant to study these oscilla-\ntions but is alsouseful to follow the bifurcations observed\nalong the line F1/F2= 0.6 in figure 2. We now investi-\ngate the evolution of the dynamics in the phase space\n(D,Q) displayed in figure 4 as F1+F2is modified. The\nlimit cycle described in figure 3 bifurcates from a low\namplitude stationary dynamo when F1+F2is increased\nfrom 38.4 to 40 Hz. This limit cycle is shown in green in\nfigure 4a. When F1+F2is decreased again to 38 .4 Hz, a\nsmaller amplitude limit cycle is obtained (orange curve,\ncircles) instead of a fixed point. We need to decrease the\nrotation frequencies further to recover the low amplitude\nstationary dynamo (black dot). Therefore, this transi-\ntion is hysteretic and within some frequency range we\nhave bistability involving stationary and time-periodic\ndynamos. This oscillation appears at finite amplitude\nand finite period [14]. The oscillation of figure 3 displays\na slowing down in the vicinity of two symmetric fixed\npoints, as expected for a system close to the saddle-node\nbifurcation of Andronov type. Note however that the on-\nset of the cycle when F1+F2is increased, does not corre-\nspond to such a saddle-node bifurcation, since these two\nstagnationpointsaredistinctfromthelowamplitudesta-\ntionary dynamo regime obtained at F1+F2 = 38.4. In\nfact, thistransitionfromalowamplitudestationarymag-\nnetic field to an oscillatory regime at finite period, rather\ncorresponds to the model taken close to its codimension-\ntwo bifurcation point [13].\nWhenF1+F2is increased further, the amplitude of\nthelimitcyclecontinuouslyincreases(figure4b). Inaddi-\ntion, the system slows down in the vicinity of two points\n(±Ds,±Qs) (figure 4c). Thus, the period of the limit\ncycle significantly increases. For F1+F2= 44 Hz, the\nsystems stops on one of these two fixed points and we get\na stationary dynamo (although we cannot rule out the\noccurrence of other reversals with a longer experiment).\nAs explained in the framework of the model [11], this\nsecond transition corresponds to a saddle-node bifurca-\ntion or more precisely an Andronov bifurcation: the sta-\nble fixed points ( ±Ds,±Qs) collide with unstable fixed\npoints (±Du,±Qu) whenF1+F2is decreased and dis-\nappear. A limit cycle is thus created, and its period is\nexpected to diverge in the vicinity of the saddle-node bi-\nfurcation. Turbulent fluctuations of course saturate this\ndivergence by kicking the system away from the points\n(±Ds,±Qs) where it slows down. They also strongly\nmodify the dynamics on the other side of the bifurca-\ntion. Indeed, when the stable and unstable fixed points\nareverycloseoneto the other, turbulent fluctuations can\nrandomlydrivethe systemfrom astable fixedpoint to its\nneighboring unstable one, and thus trigger a reversal of4\n(a)-5 0 5\nDipole [arb. units]-6-4-20246Quadrupole [arb. units]F1+F2=37.6\nF1+F2=38.4\nF1+F2=40\n(b)-5 0 5\nDipole [arb. units]-10-50510 Quadrupole [arb. units]F1+F2=41.8\nF1+F2=40.8\nF1+F2=40\n(c)-5 0 510\nDipole [arb. units]-50510 Quadrupole [arb. units]F1+F2=42.4\nF1+F2=44\nFIG. 4: (Color online) (a): Evolution of the magnetic field in the phase space (D,Q) at low frequencies: there exists\na range of bistability in which a stationary dynamo (black dot) and a pe riodic limit cycle (orange circles) are both\nmetastable. (b): Evolution of the limit cycle when F1+F2is increased from 40 to 41 .8 Hz. (c): Chaotic reversals\nobtained for large values of F1+F2in the vicinity of a saddle-node bifurcation.\n200 300 400 500600700\nTime [s]-10-50510 Magnetic field [arb. units]Dipole\nQuadrupole\n220225230235240\nFIG. 5: (Color online) Time evolution of the magnetic\nfield during a regime of chaotic reversals, for\nF1+F2= 42.4 Hz. Inset: zoom on the reversal occuring\natt= 231s.\nthe magnetic field. Therefore, random reversals are ex-\npected in the vicinity of the saddle-node bifurcation [11].\nThis is what is observed here as shown below.\nFigure 5 displays the time-recordings of the dipolar\nand quadrupolar components for F1+F2= 42.4 Hz.\nWe observe that both components fluctuate around\nconstant values as if they have reached a stable fixed\npoint. The time spent in both polarities is random but\nmuch longer than the magnetic diffusion time scale (of\norder 1 s). One also clearly observes that the amplitude\nof the dipole slowly decreases before rapidly changing\nsign. In the phase space ( D,Q) displayed in figure 4c,\nthis slow decay corresponds to random motion in the\nregions in the form of elongated spots located along the\nlimit cycle in the vicinity of the fixed points. Indeed, the\nmotion from each stable fixed point to the neighboring\nunstable one, occurs under the influence of fluctuationsacting against the deterministic dynamics. It is thus a\nslow random drift compared to the fast reversal phase\ndriven by the deterministic dynamics once the system\nhas been pushed beyond the unstable fixed point.\nFigure 4c also show that the spots become more and\nmore elongated when F1+F2is increased. This tells\nthat the distance between each stable fixed point and its\nunstable neighbor increases. Correspondingly, reversals\nare less frequent. For F1+F2= 44 Hz (red (light\ngrey) cycle in figure 4c), fluctuations can hardly drive\nreversals. As for periodic oscillations obtained above the\nAndronov bifurcation, the modal decomposition ( D,Q)\nunderlines the short transfer from an axial dipole to\na quadrupolar magnetic field during random reversals\nobtained below the bifurcation threshold (see inset of\nfigure 5). The dipolar amplitude displays the expected\nbehavior, characterized by a slow decay followed by a\nrapid recovery, and showing a typical overshoot after\neach reversal. Evolution in the phase space ( D,Q)\nalso illustrate how the transfer between dipolar and\nquadrupolar components yields very robust cycles,\nsystematically avoiding the origin B= 0.\nIn conclusion, wehaveused asimple method to extract\nthe dipolar (antisymmetric) and quadrupolar (symmet-\nric) components of the magnetic field in the VKS exper-\niment. We have shown that this decomposition allows\nto investigate the morphology of the magnetic field dur-\ning reversals, and to compare experimental results to the\npredictions of a recent model proposed in [11]. We have\nshown that the results of the VKS experiment are in very\ngood agreement with these predictions:\n- reversalsare characterized by a strong transfer to the\nquadrupole when the dipole vanishes,\n-thedipolarmodesystematicallydisplaysanovershoot\nafter each reversal,\n- random reversals are asymmetric, i.e. involve two5\nphases: a slow one triggered by turbulent fluctuations\nfollowed by a fast one mostly governed by the determin-\nistic dynamics.\nThis agreement between the VKS experiment and the\nmodel has significant consequences. It first shows that\na fluid dynamo, even generated by a strongly turbu-\nlent flow, can exhibit low dimensional dynamics, involv-\ning mostly dipolar and quadrupolar modes. Further-\nmore, because such a model is based on symmetry ar-\nguments, the mechanisms described here are expected\nto apply beyond the VKS experiment. For instance, al-\nthough 3-dimensional simulations do not involve a sim-\nilar level of turbulence, a transfer between dipole and\nquadrupole during reversals has been observed in sev-\neral numerical studies of the geodynamo [6], [7]. This\nis consistent with indirect evidences from paleomagnetic\nmeasurements, suggesting a dipole-quadrupole interac-\ntion [4] and asymmetric reversals [18]. Observations of\nthe Sun’s magnetic field also suggest a transfer between\ndipolar and higher components [16]. In numerical simu-\nlations based on the VKS experiment [8], a good agree-\nment with the three predictions reported here has been\nobtained, but only when the magnetic Prandtl number\nis sufficiently small. In this context, our simple method\ncould be used to investigate data from numerical simula-\ntions of the geodynamo at low magnetic Prantl number.\nThis opens new perspectives to understand the dynamics\nof planetaryand stellarmagnetic fields with a simple and\nlow dimensional description.\nI aknowledge my colleagues of the VKS team with\nwhom the experimental data used here have been ob-tained [14] and ANR-08-BLAN-0039-02 for support. I\nalsothankanonymousrefereesforveryhelpfulcomments.\n[1] Dormy E., Soward A.M. (Eds), Mathematical Aspects of\nNatural dynamos, CRC-press 2007.\n[2] P. Hoyng,M. A. J. H. Ossendrijver and D.Schmitt, Geo-\nphys. Astrophys. Fluid Dyn. 94, 263 (2001).\n[3] F. Stefani and G. Gerbeth, Phys. Rev. Lett. 94, 184506\n(2005).\n[4] P. L. McFadden et al., J. Geophys. Research 96, 3923\n(1991).\n[5] B. M. Clement, Nature (London) 428, 637 (2004)\n[6] Glatzmaier G. A. and Roberts P. H., Nature, 377(1995).\n[7] Coe, R.S. and Glatzmaier G. A. , Geophys. Rev. Lett.\n33, L21311 (2006)\n[8] C. Gissinger, E. Dormy and S. Fauve, Europhys. Lett.\n90, 49001 (2010).\n[9] M. Berhanu et al., Europhys. Lett. 77, 59001 (2007).\n[10] F. Ravelet et al., Phys. Rev. Lett. 101, 074502 (2008)\n[11] F. Petrelis and S. Fauve, J. Phys. Condens. Matter 20,\n494203 (2008).\n[12] F. Petrelis et al., Phys. Rev. Lett. 102, 144503 (2009).\n[13] M. Berhanu et al., J. Fluid Mech. 641, 217 (2009).\n[14] M. Berhanu et al, to be published in EPJB (2010).\n[15] C.J.P. Gissinger, Europhys. Lett. 87, 39002 (2009)\n[16] R. Knaack and J. O. Stenflo, Astron. and Astro., 438\n(2005)\n[17] B.Gallet andF.Petrelis, Phys.Rev.E 80, 035302(2009).\n[18] J.P. Valet et al ,Nature, 435802-805 (2006)\n[19] C. Gissinger, PhD Thesis (2010)."    },    {        "title": "1912.10513v2.Observation_of_magnetic_solitons_in_two_component_Bose_Einstein_condensates.pdf",        "content": "Observation of magnetic solitons in two-component Bose-Einstein condensates\nA. Farol\f, D. Trypogeorgos, C. Mordini, G. Lamporesi,\u0003and G. Ferrari\nINO-CNR BEC Center and Dipartimento di Fisica, Universit\u0012 a di Trento, 38123 Povo, Italy and\nTrento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy\n(Dated: July 10, 2020)\nWe experimentally investigate the dynamics of spin solitary waves (magnetic solitons) in a har-\nmonically trapped, binary super\ruid mixture. We measure the in-situ density of each pseudospin\ncomponent and their relative local phase via an interferometric technique we developed, and as such,\nfully characterise the magnetic solitons while they undergo oscillatory motion in the trap. Magnetic\nsolitons exhibit non-dispersive, dissipationless long-time dynamics. By imprinting multiple magnetic\nsolitons in our ultracold gas sample, we engineer binary collisions between solitons of either same\nor opposite magnetisation and map out their trajectories.\nWaves have the natural tendency to spread while prop-\nagating. In nonlinear media, this tendency can be coun-\nterbalanced through a self-focusing mechanism creating\nlocalized and long-lived solitary waves, a.k.a. solitons.\nTheir dissipationless nature makes them invaluable tools\nfor technological applications and information transport\n[1, 2]. They play a fundamental role across science, clas-\nsical and quantum alike, and have been observed in dif-\nferent physical systems, such as classical \ruids, liquid\nHe, plasmas, optical waveguides, polaritons, and ultra-\ncold atomic gases [3{8]. The latter can be widely ma-\nnipulated to explore soliton behaviour, by altering the\nshape of the gas, the characteristic interactions among\nparticles, and their energy dispersion [9{18].\nTwo-component mixtures display an even richer ex-\ncitation spectrum, showing new types of solitons. These\nsolitons were long-sought in the liquid He community, but\nwere never observed due to the absence of an experimen-\ntal realisation of interpenetrable super\ruids. However,\nmixtures of ultracold atomic gases can be used instead\n[19{23]. A mixture can be perturbed from its ground\nstate by creating either excitations in the total density,\nwith an in-phase response of the two components, or ex-\ncitations in the population imbalance (magnetisation),\nwith an out-of-phase response. This implies the exis-\ntence of both unmagnetised solitons, similar to those in\na single component super\ruid, and magnetised ones [21].\nAmong the latter, magnetic solitons (MS) are denoted\nby a localised population imbalance in an otherwise bal-\nanced and symmetrically interacting mixture [24].\nAtomic mixtures (superpositions) of23Na lowest-\nhyper\fne-state atoms in the jF;m Fi=j1;\u00061iare fully\nmiscible and not subject to buoyancy [25]. The two\nground-state components experience the same trapping\npotential, show the same spatial pro\fle, and occupy the\nsame volume [20, 22]. These are prerequisite conditions\nfor the excitation and characterisation of MSs [24], which\nare ful\flled in our system [20]; however this is not the case\nin other atomic species, such as87Rb [21].\nHere, we create MSs via spin-sensitive phase imprint-\ning. We characterize them in-situ using a fully to-\nmographic method with quasi-concurrent density andrelative-phase measurements, that show a characteris-\ntic\u0019jump. The MSs perform oscillatory dynamics in\na harmonically con\fned BEC that show only minimal\ndispersion and dissipation for times as long as 1 sec. In\naddition, we engineer collisions between MSs with same\n\"\"and opposite\"#magnetisation and monitor their be-\nhaviour close to the collision point.\nFIG. 1. (a) A spin-selective optical potential generates a pair\nof MSs that travel in opposite directions along yin our two-\ncomponent elongated BEC. The overall imparted phase of 2 \u0019\nis dealt symmetrically on the two spin components. (b) \u0003-\ncoupling scheme showing all the hyper\fne transitions that are\nused for preparation of the mixture and inducing an e\u000bective\nquadratic shift. (c) Full tomography of a pair of MSs 15 ms\nafter their creation. Left column: Optical densities (OD) of\nj1;\u00001i(red) andj1;+1i(blue), and relative phase (purple).\nRight column: The measured apparent magnetisation (top)\nis of the order of 0.5 and the expected relative-phase pro\fle\n(bottom) shows two \u0019jumps at the soliton positions.arXiv:1912.10513v2  [cond-mat.quant-gas]  9 Jul 20202\nExperiment. All experiments described here begin with\na thermal cloud of23Na atoms in a hybrid trap [26{29]\nin thej1;\u00001istate, which we then transfer into an elon-\ngated crossed optical trap (Fig. 1a). Further evaporative\ncooling leads to a BEC of typically N'2\u0002106atoms\nwith negligible thermal component, T'250 nK. The \f-\nnal trap frequencies are ffy; f?g=f8:7(1:2);585(2)gHz\ngiving axial and transverse Thomas-Fermi radii Ry'\n250\u0016m andR?'3:7\u0016m, respectively. A uniform mag-\nnetic \feld is applied along the z-axis with a Larmor fre-\nquency of 182.3(1) kHz. Atoms are transferred into a\nmixture ofj1;\u00061ivia a two-photon microwave radiation\nwith an e\u000bective Rabi frequency of 268(2) Hz using an\nadiabatic rapid passage technique; an initially large de-\ntuning of\u00194 kHz is gradually reduced to zero in 60 ms\n(see Fig. 1b). After the two-photon coupling is switched\no\u000b, a dressing radiation with Rabi frequency 2.27(5) kHz\nis turned on, 20 kHz blue-detuned from j1;0i!j 2;0i,\nthat creates an e\u000bective quadratic shift and stabilizes the\nmixture against spin-relaxation [20, 30, 31].\nWe produce MSs by applying a step-like, purely vecto-\nrial, optical dipole potential to the right half of the BEC\n(Fig. 1a) [32]. The light is circularly polarized in order\nto maximize the vector term of the light shift and, since\nthe atomic states have opposite angular momentum, the\nphase imprinted on the j1;\u00061istates is opposite by con-\nstruction,\u001e+=\u0000\u001e\u0000=\u001e=2 (Fig. 1a). Using a pulse-\ntime of\u001c= 70\u0016s we imprint a phase of \u0006\u0019onto the\nj1;\u00061istates. The amount of imprinted phase is inde-\npendently calibrated (see Supplemental Materials). The\nphase imprint pulse does not introduce additional spin or\ndensity excitations since \u001c <h=ng\u001ch=n\u000eg , wherenis\nthe total atomic density, gis the intracomponent interac-\ntion constant, \u000egthe di\u000berence between intra- and inter-\ncomponent interactions, and his Planck's constant. The\nlight beam is elliptical with an aspect ratio of 5:1 and its\nmasked intensity goes from 10% to 90% over 2 \u0016m'3\u0018s,\nwhere\u0018s=~=p2mn\u000eg is the spin healing length and m\nthe atomic mass. This produces two MSs since the to-\ntal magnetisation of the system is conserved. They move\nin opposite directions, have opposite magnetisation, and\nare robust against transverse instabilities.\nThe stability of a soliton depends on the ratio between\nits transverse extension and its thickness [33, 34]. If the\nlatter is much smaller than the former, the soliton re-\nsembles a fragile thin membrane that decays into vortical\nstructures through snaking instability [11, 16, 18, 35, 36].\nAssuming the same stability criterion valid for density\nsolitons with the relevant quantities replaced by their\nspin equivalents, we expect stability for R?.6\u0018s. In\nour system R?'5\u0018sand the MSs show stable one-\ndimensional dynamics.\nTime-of-\right measurements do not reveal any phase\ndislocations [16, 35, 37{39]| associated with the forma-\ntion of vortices or vortex rings | in either component\neven after 1 sec from the creation of MSs (see Supple-\n0 200 400 600 800 1000\nt(ms)150\n100\n50\n050100150y(m)\n0.0 0.2 0.4 0.6 0.8 1.0\nv/cs0.00.51.0L/Ry\n(a)\n(b)FIG. 2. (a) Two MSs of opposite magnetisation oscillate out-\nof-phase in the trap with a period of 550 ms. The BEC is\ncentered at y= 0. Finite temperature e\u000bects lead to the\nslight oscillation amplitude increase and decay of magnetisa-\ntion (see Supplemental Material). Since the magnetisation of\none of the MSs is slightly smaller, it goes below our detec-\ntion threshold sooner. (b) The oscillation amplitude L=R yof\na single MS increases monotonically with its initial velocity\nv, in agreement with the theoretical prediction with no free\nparameters [24].\nmental Materials). Density dynamics of the BEC are\nstill three-dimensional, with phase coherence along the\nwhole sample since \u0016=~!?= 9:3, where\u0016is the chemi-\ncal potential of the BEC.\nDynamics. We detect the density pro\fles of the two\ncomponents and their relative phase throughout the sam-\nple (Fig. 1c). We measure density in-situ by separately\ntransferring 14% of the atoms from j1;+1iandj1;\u00001i\nto thej2;0istate, and then imaging them using the\nF= 2!F0= 3 transition. Where one spin com-\nponent shows a local density dip, the other one has a\npeak and vice versa. Together they comprise MSs with\npositive or negative magnetisation, while the total den-\nsity is unperturbed. The two transfers happen 600 \u0016s\napart in which time the soliton has traversed a distance\nof\u0019600 nm which is smaller than our optical resolution\nof roughly 2 \u0016m. After 2.5 ms we acquire an image of the\nrelative phase of the two components. Figure 1c shows a\nfull tomographic snapshot of two MSs 16 ms after their\ncreation at the center of the trap. They have an opposite\napparent magnetisation jm0j=jn+\u0000n\u0000j=n= 0:5 and\nhave travelled to \u000620\u0016m with a velocity v'1.2 mm/s;\nthe relative phase shows two discontinuities at these po-3\n(a)\n(b) (c)\nFIG. 3. (a) The input state of the interferometer is an equal\nsuperposition of the two components (red-blue). All phases\nare applied to one arm without loss of generality. The two\ncomponents are projected onto the readout state (purple) us-\ning a bichromatic microwave pulse. (b) Typical data showing\nthe OD in the readout state (top), which is a direct mea-\nsurement of the phase di\u000berence of the two components. The\njump in\u001e(y) is proportional to the ratio of the average counts\non either side of a MS (shaded regions). (c) Average counts in\nthe left versus right regions across each of the two MSs. The\nphase di\u000berence of the left- and right-projected populations is\nindependent of \u001eLO. The anticorrelated behaviour is consis-\ntent with a \u0019phase jump for both MSs within our statistical\nuncertainty.\nsitions.\nFigure 2a shows the MS long-time dynamics in the har-\nmonic trap for up to 1 second since their formation. They\nundergo multiple oscillations with no discernible disper-\nsion. The oscillation amplitude is \u00190:4Ryand the period\n4:7(1)Ty, whereTythe axial harmonic oscillator period.\nThroughout the oscillatory dynamics we observe a slight\noscillation amplitude increase and decay of magnetisa-\ntion, a dissipation likely induced by the residual thermal\nfraction, while our limited imaging resolution does not\nallow to appreciate an evolution of the width of the MS.\nVarying the BEC atom number and the intensity gra-\ndient of the phase imprint light we launch MSs with dif-\nferentv=cs, wherecs=p\nn\u000eg= 2mis the spin sound ve-\nlocity. Figure 2b shows the dependence of the amplitude\nof oscillation L=R yon the peak velocity v=csand is in\ngood agreement with theory [24].\nInterferometer. We measure the relative phase of\nthe two components using a generalised Ramsey tech-\nnique. The input state of the interferometer j1;\u00001i+\nexp [i(\u001e(y) +\u001eLO)]j1;+1iis an equal superposition ofthe ground hyper\fne levels, where, without loss of gen-\nerality, thej1;+1istate carries all the relative phase; we\nignore any global phase associated with the unitary evo-\nlution of the Hamiltonian under Larmor precession. The\nterm\u001e(y) corresponds to the phase-imprint pulse which\nacts locally on the BEC and initially imprints a bipar-\ntite left/right phase which is later redistributed along the\nBEC due to the relative motion of the MSs. The phase\nof the local oscillator \u001eLOis the phase di\u000berence of the\ntwo microwave \felds that form the interferometer and\nact globally on the BEC (see Fig. 3a). A bichromatic\nmicrowave pulse, resonant with the j1;\u00001i!j 2;0iand\nj1;+1i!j 2;0itransitions with respective Rabi frequen-\ncies \n\u0000= \n += \n (Fig. 3a), acts as a beamsplitter that\nprojects the ground-excited-state superposition onto the\npopulation ofj2;0i. We then image the full spatial dis-\ntribution of Pj2;0i= [1 + cos(\u001e(y) +\u001eLO)] sin2\nt, where\nthe populations of j1;\u00061iare normalised to unity; the\nrelative local phase of the two ground-state components\nis mapped onto the population of j2;0i. The duration of\nthe microwave pulse is 20 \u0016s and the two arms of the in-\nterferometer are balanced with \n =2\u0019= 1:12(1) kHz (see\nSupplemental Material).\nWe are interested in how \u001e(y) changes across a MS.\nFor any random \u001eLO, a\u0019jump in\u001e(y) results in the\nanti-correlated output of the interferometer when com-\nparing the counts on the two sides of each MS. However\ne\u000bects such as the residual spin-dependent curvature of\nthe potential and the excitation of long-wavelength spin\nwaves [40] [41], adversely a\u000bects the output of the inter-\nferometer. We circumvent these by restricting the phase\nmeasurement to the regions adjacent to the position of\nthe MSs (see shaded regions in Fig. 3b). Figure 3c shows\nthe anti-correlation in the interferometer output chan-\nnels which appears as a distribution with \u00001 slope when\nplotted against each other.\nCollisions. We engineer\"\"collisions by phase imprint-\ning two pairs of MSs in our BEC, \u000650\u0016m from the center,\nusing a di\u000berent optical mask. We ignore the two that\ntravel towards the edges of the BEC and focus on the \"\"\npair that collides at the center 45 ms later (Fig. 4b).\nSince the size of the MSs is smaller than our optical res-\nolution, we infer their magnetisation from their velocity,\ninstead. The relative velocity of the two \"\"MSs increases\nwith respect to its pre-collision value from 1 :83(8) mm/s\nto 2:8(1) mm/s (Fig. 4d), implying a change in m0from\n0.86 to 0.62. Such a dissipative behaviour is in direct\ncontrast with the expected solitonic interactions, that are\nnormally dissipationless. Our observations can be naively\nexplained by considering that for low-magnetisation \"\"\ncollisions, the total magnetisation at the collision is ap-\nproximately the linear sum of the two MSs. However\nforjm0j>0:5 this would lead to magnetisation larger\nthan 1, that is unphysical. The system responds to this\napparent impasse by introducing dissipation.\nCollisions of MSs with opposite magnetisation \"#hap-4\n100\n50\n050100y(m)\n200 300 400100\n0100y (m)\n30 50 70\n(a) (b)\n(c) (d)\ntime from phase imprint (ms)\nFIG. 4. Collisions of MSs with opposite (left column) or same\n(right column) magnetisations. (a-b) Time evolution of the\nMSs in the reference frame of the center-of-mass of the soli-\ntons. (c-d) Relative position of the MSs close to the collision\npoint at 300 ms for \"#collisions (c) and 45 ms for \"\"(d). For\nthese\"\"collisions (m0= 0:86), the MSs dissipate energy dur-\ning the collision as evident by the di\u000berent velocities (slopes)\nbefore and after. The missing points in (d) are due to not\nbeing able to distinguish same magnetisation MSs when they\nare very close together. The variance of the data is larger\nin the left column than in the right one since the collision\nhappens at much later times.\npen naturally in our system when the two MSs reach\nthe centre of the trap half an oscillation later (Fig.2a).\nFigure 4a shows their trajectories in the reference frame\nwhere the center-of-mass of the solitons is \fxed. In \"#\ncollisions MSs go through each other, but our signal-to-\nnoise ratio does not allow us to conclude whether there is\na change in their relative velocity (Fig. 4c). The noise in\n\"#collisions is larger than the one in \"\"collisions since\nthe former happen 300 ms instead of 45 ms after the MSs\ncreation.\nConclusions. We produced and characterised various\naspects of MSs using the tomographic techniques de-\nscribed above. Our MSs are stable, nondispersive, rela-\ntively long-lived, and their dynamic behaviour is in good\nagreement with theory [24]. Collisions of \"\"solitons of\nlarge magnetisation show violation of the solitonic prop-\nerty of nondissipative interaction. This dissipative be-\nhaviour is reminiscent of light bullets: non-linear, stable\nstructures that appear in dispersive optical media and\nlose energy when they collide [42]. Our observations may\ntrigger further studies on soliton interaction mechanisms,\nincluding the role of \fnite temperature e\u000bects.\nWith a suitable low-magnetic-\feld-noise environ-ment [29, 43], our techniques can be readily extended\ntowards investigating the physics of MSs in the presence\nof coherent coupling between the two components [44{\n46]. Note that this work does not correspond to the limit\nof zero coupling between the two components. The soli-\ntons supported by the coupled and uncoupled systems\nhave a distinct topological character [44, 47], since the\nphase across a coherently-coupled MS cannot be contin-\nuously unwound to \u0006\u0019.\nWhile \fnalising this manuscript we became aware of\nsimilar work on magnetic excitations [48].\nThe authors are grateful to S. Stringari, F. Dal-\nfovo, L. Pitaevskii, A. Recati, C. Qu, A. Gallem\u0013 \u0010, M.\nLewenstein and L. Tarruell for stimulating discussions.\nWe acknowledge funding from the project NAQUAS of\nQuantERA ERA-NET Cofund in Quantum Technolo-\ngies (Grant Agreement N. 731473) implemented within\nthe European Union's Horizon 2020 Programme, from\nProvincia Autonoma di Trento and from INFN-TIFPA\nunder the project FISh.\n\u0003giacomo.lamporesi@ino.it; http://bec.science.unitn.it\n[1] H. A. Haus and W. S. Wong, Rev. Mod. Phys. 68, 423\n(1996).\n[2] M. Nakazawa, H. Kubota, K. Suzuki, E. Yamada, and\nA. Sahara, IEEE Journal of Selected Topics in Quantum\nElectronics 6, 363 (2000).\n[3] F. Ancilotto, D. Levy, J. Pimentel, and J. Eloranta,\nPhys. Rev. Lett. 120, 035302 (2018).\n[4] K. E. Lonngren, Plasma Physics 25, 943 (1983).\n[5] T. T. Dauxois and . Peyrard, M. (Michel), Physics of\nsolitons (Cambridge, UK ; New York : Cambridge Uni-\nversity Press, 2006) formerly CIP.\n[6] A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet,\nI. Carusotto, F. Pisanello, G. Lem\u0013 enager, R. Houdr\u0013 e,\nE. Giacobino, C. Ciuti, and A. Bramati, Science 332,\n1167 (2011).\n[7] L. D. Carr and J. Brand, \\Multidimensional soli-\ntons: Theory,\" in Emergent Nonlinear Phenomena\nin Bose-Einstein Condensates: Theory and Experi-\nment , edited by P. G. Kevrekidis, D. J. Frantzeskakis,\nand R. Carretero-Gonz\u0013 alez (Springer Berlin Heidelberg,\nBerlin, Heidelberg, 2008) pp. 133{156.\n[8] B. P. Anderson, \\Dark solitons in becs: The \frst\nexperiments,\" in Emergent Nonlinear Phenomena in\nBose-Einstein Condensates: Theory and Experiment ,\nedited by P. G. Kevrekidis, D. J. Frantzeskakis,\nand R. Carretero-Gonz\u0013 alez (Springer Berlin Heidelberg,\nBerlin, Heidelberg, 2008) pp. 85{96.\n[9] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Seng-\nstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewen-\nstein, Phys. Rev. Lett. 83, 5198 (1999).\n[10] J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark,\nL. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley,\nK. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I.\nSchneider, and W. D. Phillips, Science 287, 97 (2000).\n[11] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder,\nL. A. Collins, C. W. Clark, and E. A. Cornell, Phys.5\nRev. Lett. 86, 2926 (2001).\n[12] K. E. Strecker, G. B. Partridge, A. G. Truscott, and\nR. G. Hulet, Nature 417, 150 (2002).\n[13] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cu-\nbizolles, L. D. Carr, Y. Castin, and C. Salomon, Science\n296, 1290 (2002).\n[14] B. Eiermann, T. Anker, M. Albiez, M. Taglieber,\nP. Treutlein, K.-P. Marzlin, and M. K. Oberthaler, Phys.\nRev. Lett. 92, 230401 (2004).\n[15] C. Becker, S. Stellmer, P. Soltan-Panahi, S. D orscher,\nM. Baumert, E.-M. Richter, J. Kronj ager, K. Bongs, and\nK. Sengstock, Nature Physics 4, 496 (2008).\n[16] I. Shomroni, E. Lahoud, S. Levy, and J. Steinhauer,\nNature Physics 5, 193 (2009).\n[17] C. Hamner, J. J. Chang, P. Engels, and M. A. Hoefer,\nPhys. Rev. Lett. 106, 065302 (2011).\n[18] M. J. H. Ku, B. Mukherjee, T. Yefsah, and M. W. Zwier-\nlein, Phys. Rev. Lett. 116, 045304 (2016).\n[19] I. Ferrier-Barbut, M. Delehaye, S. Laurent, A. T. Grier,\nM. Pierce, B. S. Rem, F. Chevy, and C. Salomon, Science\n345, 1035 (2014).\n[20] T. Bienaim\u0013 e, E. Fava, G. Colzi, C. Mordini, S. Sera\fni,\nC. Qu, S. Stringari, G. Lamporesi, and G. Ferrari, Phys.\nRev. A 94, 063652 (2016).\n[21] I. Danaila, M. A. Khamehchi, V. Gokhroo, P. Engels,\nand P. G. Kevrekidis, Phys. Rev. A 94, 053617 (2016).\n[22] E. Fava, T. Bienaim\u0013 e, C. Mordini, G. Colzi, C. Qu,\nS. Stringari, G. Lamporesi, and G. Ferrari, Phys. Rev.\nLett.120, 170401 (2018).\n[23] J. H. Kim, D. Hong, and Y. il Shin, \\Observation of\ntwo sound modes in a binary super\ruid gas,\" (2019),\narXiv:1907.10289 [cond-mat.quant-gas].\n[24] C. Qu, L. P. Pitaevskii, and S. Stringari, Phys. Rev.\nLett.116, 160402 (2016).\n[25] The intra- and intercomponent scattering lengths for\n23Na area= 54:54a0anda\u0006= 50:78a0respectively [49],\nand correspond to an interaction di\u000berence \u000ea=a\u0000a\u0006=\n3:76a0= 0:07a.\n[26] G. Lamporesi, S. Donadello, S. Sera\fni, and G. Ferrari,\nRev. Sci. Instrum. 84, 063102 (2013), 10.1063/1.4808375.\n[27] G. Colzi, G. Durastante, E. Fava, S. Sera\fni, G. Lam-\nporesi, and G. Ferrari, Phys. Rev. A 93, 023421 (2016).\n[28] G. Colzi, E. Fava, M. Barbiero, C. Mordini, G. Lam-\nporesi, and G. Ferrari, Phys. Rev. A 97, 053625 (2018).\n[29] A. Farol\f, D. Trypogeorgos, G. Colzi, E. Fava, G. Lam-\nporesi, and G. Ferrari, Review of Scienti\fc Instruments\n90, 115114 (2019).\n[30] F. Gerbier, A. Widera, S. F olling, O. Mandel, and\nI. Bloch, Phys. Rev. A 73, 041602 (2006).\n[31] K. Jim\u0013 enez-Garc\u0013 \u0010a, A. Invernizzi, B. Evrard, C. Frapolli,\nJ. Dalibard, and F. Gerbier, Nature Communications\n10, 1422 (2019).\n[32] We set the scalar part of the potential equal to zero by an\nappropriate choice of the laser frequency \u0015= 589:557 nm,\nwhich is detuned to the blue of the 32S1=2!32P1=2\n23Na transition and to the red of the 32S1=2!32P3=2\ntransition [50].\n[33] J. Brand and W. P. Reinhardt, Phys. Rev. A 65, 043612\n(2002).\n[34] S. Komineas and N. Papanicolaou, Phys. Rev. A 68,\n043617 (2003).\n[35] C. Becker, K. Sengstock, P. Schmelcher, P. G. Kevrekidis,\nand R. Carretero-Gonz\u0013 alez, New Journal of Physics 15,113028 (2013).\n[36] A. Mu~ noz Mateo and J. Brand, Phys. Rev. Lett. 113,\n255302 (2014).\n[37] M. J. H. Ku, W. Ji, B. Mukherjee, E. Guardado-Sanchez,\nL. W. Cheuk, T. Yefsah, and M. W. Zwierlein, Phys.\nRev. Lett. 113, 065301 (2014).\n[38] S. Donadello, S. Sera\fni, M. Tylutki, L. P. Pitaevskii,\nF. Dalfovo, G. Lamporesi, and G. Ferrari, Phys. Rev.\nLett.113, 065302 (2014).\n[39] M. Tylutki, S. Donadello, S. Sera\fni, L. Pitaevskii,\nF. Dalfovo, G. Lamporesi, and G. Ferrari, Eur. Phys.\nJ. Special Topics 224, 577 (2016).\n[40] C. Hamner, Y. Zhang, J. J. Chang, C. Zhang, and P. En-\ngels, Phys. Rev. Lett. 111, 264101 (2013).\n[41] Note for instance the di\u000berence between Fig. 1c lower\npanel and Fig. 3b top panel which, despite being taken\nin similar experimental settings, show a qualitative di\u000ber-\nent pattern. From single magnetisation images of atomic\nclouds with a much larger aspect ratio 100:1, we estimate\nthe energy curvature to be 450 Hz/mm2, or 28 Hz for our\nRy= 250\u0016m BEC.\n[42] C. Mili\u0013 an, Y. V. Kartashov, and L. Torner, Phys. Rev.\nLett.123, 133902 (2019).\n[43] D. Trypogeorgos, A. Vald\u0013 es-Curiel, N. Lundblad, and\nI. B. Spielman, Physical Review A 97, 013407 (2018).\n[44] C. Qu, M. Tylutki, S. Stringari, and L. P. Pitaevskii,\nPhys. Rev. A 95, 033614 (2017).\n[45] S.-W. Su, S.-C. Gou, A. Bradley, O. Fialko, and\nJ. Brand, Phys. Rev. Lett. 110, 215302 (2013).\n[46] S. S. Shamailov and J. Brand, SciPost Phys. 4, 18 (2018).\n[47] J. Sanz, A. Frlian, C. S. Chisholm, C. R. Cabrera, and\nL. Tarruell, \\Interaction control and bright solitons in\ncoherently-coupled bose-einstein condensates,\" (2019),\narXiv:1912.06041 [cond-mat.quant-gas].\n[48] X. Chai, D. Lao, K. Fujimoto, R. Hamazaki, M. Ueda,\nand C. Raman, \\Magnetic solitons in a spin-1 bose-\neinstein condensate,\" (2019), arXiv:1912.06672 [cond-\nmat.quant-gas].\n[49] S. Knoop, T. Schuster, R. Scelle, A. Trautmann, J. App-\nmeier, M. K. Oberthaler, E. Tiesinga, and E. Tiemann,\nPhys. Rev. A 83, 042704 (2011).\n[50] R. Grimm, M. Weidem uller, and Y. B. Ovchinnikov\n(Academic Press, 2000) pp. 95 { 170.\n[51] L. Salasnich and B. A. Malomed, Phys. Rev. A 74,\n053610 (2006).1\nSupplemental Materials\nMS DETECTION\nWe identify the position of MSs in the sample by taking\nan in-situ image of the density distribution for each state\nof the mixture. The image is integrated along the xaxis\nto compute two 1D density distributions (Fig. S1a).\nThe inset shows the density pro\fle of the two compo-\nnents of a MS moving with a velocity v=cs= 0:6. This\ncorresponds to a MS with a width of \u0019\u0018s\u00190:8\u0016m\n(dashed line) and a true magnetisation m0= 0:8. Due\nto our \fnite resolution \u00192\u0016m, the pro\fle is broad-\nened and the height is reduced (solid line). We infer\nthe true magnetisation from velocity measurements since\nm0=p\n1\u0000v2=c2s.\nThe two 1D density distributions are used to compute\nthe apparent magnetisation m0. Due to the presence\nof long-wavelength spin excitations superimposed to the\npro\fle of the solitons, we apply a high-pass \flter to the\nmagnetization pro\fle (Fig. S1b) before locating the posi-\ntion of the MSs. To avoid false positives, only the peaks\nwith magnitude exceeding a given threshold (gray dashed\nlines) are considered as solitons (red and blue dots).\n102030integrated OD\n0 50 100 150 200 250 300\ny (m)\n0.2\n0.00.2m0\n10\n 0 101\n1(a)\n(b)\nFIG. S1. (a) Pro\fle of the integrated OD of the two com-\nponents. The inset shows how the computed MS pro\fle with\nm0= 0:8 (dashed) is broadened taking the \fnite optical reso-\nlution into account (solid). (b) Apparent magnetisation of the\nsystem along y. The points indicate the detected location of\nthe MSs. Dashed lines show the threshold for peak detection.\nDECAY OF THE MAGNETIC SOLITONS\nThe stability of a soliton is given by the ratio between\nits transverse extension and its thickness R?=\u0018[33, 34].In the case of MSs the existence of dynamically stable so-\nlutions is expected for R?=\u0018.6 [24], while for larger val-\nues of this parameter the soliton resembles a fragile thin\nmembrane that decays into vortical structures through\nthe snaking instability [10, 11, 18, 36]. Solitonic vortices\nand vortex rings are the most probable decay products,\nbeing the least energetic excitations supported by the\nsystem.\n(a)\n(b)\n(c)(d)\n200 \ny\nFIG. S2. (a) MSs persist for about 1 sec in our system\nbefore their peak magnetisation becomes comparable to our\nsignal-to-noise ratio. (b) The apparent width of the MSs is\nconstant throughout the lifetime of the MS. (c) Even after\n700 ms of in-trap dynamics, no phase distortions, character-\nistic of solitonic vortices, appear in images after a ballistic\nexpansions of the gas for 30 ms. (d) Pro\fle of the magneti-\nsation in the transverse xdirection, shown consecutively in\ntime. The transverse shape of MSs seems to remain stable\nand shows no evidence of snaking instability.\nSince the transverse size of our BEC is of the order of\ncritical value expected to allow relaxation of the MSs via\nsnaking instability, it is important to verify the dynami-\ncal evolution of the MSs. The lifetime of the MS is about\n1 sec, past which its magnetisation becomes comparable\nto the imaging noise in our system (Fig. S2a). The appar-\nent width (FWHM) of the MS remains constant to 5 \u0016m.\nIts real width its of the order of \u0018swhich is smaller than\nour resolution.2\nWe look for snaking-instability-like decay of the soli-\nton in the transverse pro\fle of the MS magnetisation.\nCurving (snaking) of the MS along the long axis of the\nBEC would manifest as an amplitude reduction along its\nprojection on the transverse axis. In-situ imaging, which\nhas a resolution of the order of the radial Thomas-Fermi\nradius does not provide any evidence of relaxation of the\nsoliton into a vortex line or ring. In order to increase\nour sensitivity to vortical excitations we turn to imaging\none single component after ballistic expansion but also\nin this case we never observe signatures possibly related\nto the presence of circulation in the sample such as local-\nized dislocations along the depleted line of the soliton, or\nrotation of the soliton plane (see Fig. S2d).\nThe question about the fate of the MS when the binary\nmixture is not properly balanced arises naturally, as well\nas what happens in the \fnite-temperature regime. We\ndo not systematically investigate unbalanced or \fnite-\ntemperature mixtures since they are beyond the scope of\nthe present work. On the other hand in the early stage of\nour measurements we notice that population imbalance\noften results in the formation of a single MS steadily cen-\ntered at the center of the BEC and exhibiting a peak in\nthe population of the most abundant component. With\nregard to the e\u000bect of temperature on the MS dynam-\nics, we remark that we observe reliable MS oscillations of\nmore than one period only after minimizing the temper-\nature in the binary mixture prior to the phase imprint.\nCALIBRATION OF THE PHASE IMPRINT\nWe calibrate the phase imprint pulse by measuring the\nrelative phase imprinted on a balanced j1;\u00061imixture.\nWe vary the length of the pulse and measure the inter-\nferometer output in the illuminated and dark halves of\nthe BEC. By scanning the local oscillator phase \u001eLOfor\na given pulse time, we measure the relative phase im-\nprinted. Figure S3a shows the outputs of the illuminated\n(top) and dark (bottom) half of the BEC for two di\u000berent\nphase imprint pulses (red and blue).\nA pre-requisite for this is for the ground-state popu-\nlation to be equally transferred to j2;0iso the Rabi fre-\nquencies of the two interferometer arms need to be the\nsame and on resonance. In order to balance the interfer-\nometer, we use a reduced Rabi frequency (1.12(1) kHz)\nto avoid spectral crowding from saturation of the rf elec-\ntronics. Figure S3b shows the populations of all three\nparticipating states. The system undergoes symmetric\noscillations between j2;0iandj1;\u00061i. For this, we pre-\npared the system in the j2;0istate and induced unitary\ndynamics with a resonant bichromatic microwave pulse.\nThe interferometric measurement allows us to readily ac-\ncess the imprinted relative phase \u001eand set its value to\nthe desired amount 2 \u0019.\n0.00.51.0\n0 50 100 150 200 250 300 350\nLO (degrees)\n0.00.51.0\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6\nt (ms)0.00.51.0populations\n(a)\n(b)\npopulationsFIG. S3. (a) Absolute calibration of the interferometer us-\ning the phase of the local oscillator to convert atom counts\nto radians. The measured population in j2;0iis plotted as\na function of the set \u001eLO. The two bipartite splits have a\nrelative phase of 2.6 \u0019for a 90\u0016s phase-imprint pulse (top),\nand 2\u0019for a 70\u0016s (bottom). (b) Symmetric Rabi oscillations\nfromj2;0itoj1;\u00061ishow that the interferometer is balanced,\nwith a Rabi frequency \n =2\u0019= 1:12 kHz\nSPIN SOUND EXCITATIONS\nTwo-component systems can sustain spin sound wave\npackets, travelling with a phase jump which is not locked\nto\u0019but is rather dependent on their group velocity. Due\nto the nonlinearity in the phononic dispersion relation,\nsuch excitations are expected to show a weak disper-\nsive behaviour, increasing their width and losing contrast\nwhile propagating through the system. There is the pos-\nsibility that these magnetic objects are excited together\nwith MSs by the phase imprint procedure if the total\nphase di\u000berence imprinted by the optical potential is dif-\nferent from 2 \u0019.\nIn order to gain insight on this spurious e\u000bect we sim-\nulate the generation of magnetic excitations after an im-\nprint of a relative phase corresponding either to \u0019or 2\u0019.\nA two-component non-polynomial Schr odinger equation\n(NPSE) [51] is suited to simulate the one-dimensional\ndynamics in our elongated system while taking into ac-\ncount the full three-dimensional scattering properties of\nthe condensate in this density regime. The e\u000bective wave\nequation in the high-density regime ( aj 1;2j2\u001d1) reads\ni~@t 1=\u0014\n\u0000~2\n2m@2\ny+1\n2m!2\nyy2\u0015\n 1+\n\u00143\n2\u0015~!yp\n2aj 1j2+ 2a\u0006j 2j2\u0015\n 1;(S1)\nfor the \frst component, and symmetrically (1 $2) for3\n20\n020y(a) (b) tA\n0.00.4\nm0tB\n01\nA/\n01020304050t20\n020y\n5\n05yy0\n0.00.4\nm0\n5\n05yy0\n01\nA/\nFIG. S4. (a) NPSE simulation of the dynamics of two pairs of di\u000berent kinds of magnetic excitations. In the upper (lower) row\nthe initial phase imprint is \u0019(2\u0019). The time and space coordinates are given in units of the harmonic oscillator frequency and\nlength. (b) Magnetisation and jump in the relative phase across the magnetic excitation at time tAandtB.\nthe second component. Here aanda\u0006are respectively\nthe intra- and intercomponent scattering lengths, !y=\n2\u0019fy,\u0015=f?=fyis the aspect ratio of the system, and the\nequation is valid within the approximation ( a\u0000a\u0006)=a=\n\u000eg=g\u001c1.\nFigure S4 shows that in both cases two excitations\nwith opposite magnetisations are created, but with dif-\nferent properties. For a total imprinted phase of \u0019, spin\nsound wave packets are produced. They are dispersive\nand spread their width while they oscillate in the trap\nbecause of nonlinearity of the medium. We show theirshape at two di\u000berent times, at tA, shortly after their\ncreation, and at tB, nearly one oscillation period later.\nClearly the sound excitation (Fig. S4a top) spreads over a\nlarger region and changes its shape. When, instead, the\ntotal imprinted phase is 2 \u0019(Fig. S4a bottom), a mag-\nnetic soliton is created, as clearly visible from its ability\nto preserve the initial shape and magnetisation in time.\nThe latter is much closer to what we observe experimen-\ntally, where the magnetization (Fig. S2a) after one full\noscillation is reduced to 75% of the initial one."    },    {        "title": "2303.16820v1.General_relativistic_simulations_of_the_formation_of_a_magnetized_hybrid_star.pdf",        "content": "Draft version March 30, 2023\nTypeset using L ATEXtwocolumn style in AASTeX631\nGeneral-relativistic simulations of the formation of a magnetized hybrid star\nAnson Ka Long Yip\n ,1Patrick Chi-Kit Cheong ( 張志杰)\n,2, 3and Tjonnie Guang Feng Li\n1, 4, 5\n1Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong\n2Department of Physics & Astronomy, University of New Hampshire, 9 Library Way, Durham NH 03824, USA\n3Department of Physics, University of California, Berkeley, Berkeley, CA 94720, USA\n4Institute for Theoretical Physics, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium\n5Department of Electrical Engineering (ESAT), KU Leuven, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium\nABSTRACT\nStrongly magnetized neutron stars are popular candidates for producing detectable electromagnetic\nand gravitational-wave signals. A rapid density increase in a neutron star core could also trigger the\nphase transition from hadrons to deconfined quarks and form a hybrid star. This formation process\ncould release a considerable amount of energy in the form of gravitational waves and neutrinos. Hence,\nthe formation of a magnetized hybrid star is an interesting scenario for detecting all these signals.\nThese detections may provide essential probes for the magnetic field and composition of such stars.\nThus far, a dynamical study of the formation of a magnetized hybrid star has yet to be realized.\nHere, we investigate the formation dynamics and the properties of a magnetized hybrid star through\ndynamical simulations. We find that the maximum values of rest-mass density and magnetic field\nstrength increase slightly and these two quantities are coupled in phase during the formation. We\nthen demonstrate that all microscopic and macroscopic quantities of the resulting hybrid star vary\ndramatically when the maximum magnetic field strength goes beyond a threshold of \u00185\u00021017G but\nthey are insensitive to the magnetic field below this threshold. Specifically, the magnetic deformation\nmakes the rest-mass density drop significantly, suppressing the matter fraction in the mixed phase.\nTherefore, this work provides a solid support for the magnetic effects on a hybrid star, so it is possible\nto link observational signals from the star to its magnetic field configuration.\n1.INTRODUCTION\nNeutron stars are natural laboratories for studying\nphysics under extreme conditions, which terrestrial ex-\nperiments cannot reproduce. On the one hand, the den-\nsity of a neutron star reaches above the nuclear satu-\nration density \u001a0\u00182:8\u00021014g cm\u00003, at which the\ncanonical atomic structure of matter is disrupted. The\ndetailed microphysics and the concerning equation of\nstate at supra-nuclear densities still remain elusive. Ex-\notic matter, such as deconfined quark matter and hy-\nperons, could exist in this ultradense regime (See e.g.\nRezzolla et al. 2018 for a review).\nSeveral studies have long proposed compact stars that\nare partly or wholly composed of deconfined quark mat-\nter (Itoh 1970; Bodmer 1971; Witten 1984). These\nstars are typically interpreted as the products of the\nphase transition of the hadrons in the original neutron\nklyip@phy.cuhk.edu.hkstars. In particular, when the density inside a neutron\nstar reaches a threshold, a phase transition converting\nhadrons into deconfined quarks could happen. If this\nphase transition only occurs in the stellar core, the re-\nsulting star is usually called a ‘hybrid star’. A hybrid\nstar generally has a smaller radius and higher compact-\nness than the progenitor neutron star. Therefore, gravi-\ntational potential energy of the order of \u00181052erg is ex-\npected to be released when a hybrid star is formed. Sig-\nnificant portions of the released energy could give rise to\nthe emission of neutrinos and gravitational waves. De-\ntecting these phase transition signals provides evidence\nof deconfined quark matter. Newly born neutron stars\nin supernovae and accreting neutron stars in binary sys-\ntems are possible hosts for such a phase transition (See\ne.g. Weber 1999; Abdikamalov et al. 2009 for reviews).\nOn the other hand, neutron stars have the strongest\nmagnetic field found in the Universe. Dipole spin-down\nmodels allow for the estimation of the surface magnetic\nfield strength of neutron stars. With the surface field\nstrengthBs, we can classify neutron stars into millisec-arXiv:2303.16820v1  [astro-ph.HE]  29 Mar 20232 Yip et al.\nond pulsars with Bs\u0018108\u00009G, classical pulsars with\nBs\u00181011\u000013G, and magnetars with Bs\u00181014\u000015\nG. Although there is still no direct observation of the\ninterior magnetic field of neutron stars, virial theorem\nsuggests that it could reach 1018\u000020G (see e.g. Ferrer\net al. 2010, Lai & Shapiro 1991, Fushiki et al. 1989, and\nCardall et al. 2001). Furthermore, binary neutron star\nsimulations have demonstrated that the local maximum\nmagnetic field can be amplified up to \u00181017G during\nthemerger(Price&Rosswog2006;Kiuchietal.2015b,a;\nAguilera-Miret et al. 2020).\nHighly magnetized neutron stars are promising can-\ndidates for explaining some puzzling astronomical phe-\nnomena, including soft gamma-ray repeaters and X-ray\npulsars (Kouveliotou et al. 1998; Hurley et al. 1999;\nMereghetti & Stella 1995; Mereghetti et al. 2000; van\nParadijs et al. 1995). Moreover, neutron stars can be\ndeformed by the magnetic field, depending on the geom-\netryofthemagneticfield. Apurelytoroidalfieldinduces\nprolateness (Kiuchi & Yoshida 2008; Kiuchi et al. 2009;\nFrieben & Rezzolla 2012), while a purely poloidal field\ncauses oblateness to neutron stars (Bocquet et al. 1995;\nKonno 2001; Yazadjiev 2012). These magnetic-field-\ninduced distortions make rotating neutron stars possible\nsources for the emission of detectable continuous gravi-\ntational waves (Bonazzola & Gourgoulhon 1996). How-\never, the actual field geometry inside neutron stars is\nstill unknown. Stability analyses of magnetized stars\nsuggest that simple geometries are subjected to insta-\nbilities (Tayler 1957, 1973; Markey & Tayler 1973, 1974;\nWright 1973). Magnetohydrodynamics simulations pro-\npose a mixed configuration of toroidal and poloidal\nfields as the most favored configuration (Braithwaite &\nNordlund 2006; Braithwaite & Spruit 2006; Braithwaite\n2009). This configuration is usually referred to as the\n‘twisted torus’.\nDeconfined quarks and strong magnetic fields are ex-\npected to be present inside neutron stars, so studying\nmagnetized hybrid stars is necessary to probe the com-\nbined effects of these two features. Previous studies\nhave investigated the properties of magnetized hybrid\nstars by constructing equilibrium models (e.g. Rabhi\net al. 2009; Dexheimer et al. 2012; Isayev 2015; Chat-\nterjee et al. 2015; Franzon et al. 2016b,a; Mariani et al.\n2022). In particular, Chatterjee et al. (2015) and Fran-\nzon et al. (2016a) have demonstrated that the pure field\ncontribution to the energy–momentum tensor primarily\ncontributes to the macroscopic properties of magnetized\nhybrid stars. In contrast, the magnetic effects in the\nequation of state and the field-matter interactions have\nnegligible effects on these properties. Moreover, a mag-\nnetic field reduces the central density and prevents theappearance of quark matter. Dynamics of hybrid star\nhave been studied by numerical simulations (Lin et al.\n2006; Abdikamalov et al. 2009; Herzog & Röpke 2011;\nPrasad & Mallick 2018, 2020). Nonetheless, these stud-\nies did not take the magnetic field into account. Since\nthe dynamical stability and the possible observational\nsignal of a magnetized hybrid star could not be thor-\noughly examined through equilibrium modeling, a dy-\nnamical study of this star is still indispensable.\nIn this work, we numerically study the formation of a\nmagnetizedhybridstarthroughgeneralrelativisticmag-\nnetohydrodynamics simulations. Specifically, we first\nconstruct magnetized neutron star equilibrium models\nby the open-sourced code XNS(Bucciantini & Del Zanna\n2011; Pili et al. 2014, 2015, 2017; Soldateschi et al. 2020)\nand then dynamically evolve these models using the new\ngeneral relativistic magnetohydrodynamics code Gmunu\n(Cheong et al. 2020, 2021, 2022). The details of the\ninitial neutron star models, hybrid star mdoels and evo-\nlutions are described in Section 2. Next, the results of\nthe formation process and the properties of the result-\ning star are presented in Sections 3 and 4 respectively.\nFinally, we provide the conclusions in Section 5.\n2.NUMERICAL METHODS\n2.1. Initial neutron star models\nWe construct the non-rotating magnetized neutron\nstar equilibrium models in axisymmetry by the open-\nsourced code XNS(Bucciantini & Del Zanna 2011; Pili\net al. 2014, 2015, 2017; Soldateschi et al. 2020). These\nequilibrium models serve as initial data for our simula-\ntions.\nThe initial neutron star models are constructed with\na polytropic equation of state,\nP=K\u001a\r; (1)\nwherePis the pressure, \u001ais the rest-mass density and\nwe choose a polytropic constant K= 1:6\u0002105cm5g\u00001\ns\u00002(whichequalsto110intheunitof c=G=M\f= 1)\nand a polytropic index \r= 2.\nWe specify the specific internal energy \u000fon the initial\ntime-slice by\n\u000f=K\n\r\u00001\u001a\r\u00001: (2)\nWe adopt a magnetic polytropic law for the toroidal\nfields\nB\u001e=\u000b\u00001Km(\u001ah$2)m(3)\nwhere\u000bis the laspe function, Kmis the toroidal mag-\nnetization constant, his the specific enthalpy, $2=\n\u000b2 4r2sin2\u0012, is the conformal factor, (r;\u0012)are theGeneral-relativistic simulations of the formation of a magnetized hybrid star 3\nTable 1. Properties of the 9 initial neutron star mod-\nels constructed by the XNScode. All numerical values are\nrounded off to two decimal places. \u001acis the central rest-mass\ndensity,Mgis the gravitational mass, reis the equatorial ra-\ndius, andBmaxis the maximum toroidal field strength inside\nthe neutron star. All the models have a fixed baryonic mass\nM0= 1:68M\fand the 8 magnetized models also have the\nsame toroidal magnetization index m= 1.\nModel \u001acMgreBmax\n(1014g cm\u00003) (M\f) (km) ( 1017G)\nREF 8.56 1.55 11.85 0.00\nT1K1 8.56 1.55 11.85 3:45\u000210\u00002\nT1K2 8.56 1.55 11.85 6:89\u000210\u00002\nT1K3 8.57 1.55 11.85 3:44\u000210\u00001\nT1K4 8.63 1.55 11.92 1.36\nT1K5 8.81 1.56 12.15 2.63\nT1K6 9.10 1.58 14.43 5.52\nT1K7 8.81 1.59 16.21 6.01\nT1K8 8.27 1.60 18.64 6.14\nradial and angular coordinates in 2D spherical coordi-\nnates, andm\u00151is the toroidal magnetization index.\nIn total, 9 models are constructed, where ‘REF’ is\nthe non-magnetized reference model and the remaining\n8 neutron star models are magnetized. They are part of\nthe models used in Leung et al. (2022). Because we do\nnot intend to perform a comprehensive study of neutron\nstars with different masses in this work, all models have\na fixed baryonic mass M0= 1:68M\f, which is within\nthe typical range of neutron star mass. Also, the 8 mag-\nnetized models have the same toroidal magnetization in-\ndexm= 1but different values of toroidal magnetization\nconstantKm. They are arranged in the order of increas-\ning maximum magnetic field strength Bmax, where the\nmodel ‘T1K1’ has the lowest strength, and ‘T1K2’ has\nthe second-lowest strength, so on and so forth. (‘T1’\nrepresents the toroidal magnetization index m= 1and\n‘K’ indicates the toroidal magnetization constant Km).\nThe configuration of these models allows a phase tran-\nsition that occurs inside the stellar core and facilitates\ncomparison with Leung et al. (2022). Table 1 summa-\nrizes the detailed properties of all 9 models.\n2.2. Hybrid star models and evolution\nThe MIT bag model equation of state introduced by\nJohnson et al. (1975) has been widely used to model\nquark matter inside compact stars (see e.g. Weber 1999;\nGlendenning 2012 for a review). The MIT bag model\nequationofstateformasslessandnon-interactingquarks\nis given by\nPq=1\n3(e\u00004B); (4)wherePqis the pressure of quark matter, eis the total\nenergy density and Bis the bag constant.\nFor the normal hadronic matter, we adopt an ideal\ngas type of equation of state for the evolution\nPh= (\r\u00001)\u001a\u000f (5)\nwherePhis the pressure of hadronic matter and \ris\nkept to be 2.\nEither two or three parts constitute the hybrid star\nformed after the phase transition: (i) a hadronic matter\nregion with a rest-mass density below the lower thresh-\nold density \u001ahm, (ii) a mixed phase of the deconfined\nquark matter and hadronic matter for the region with\na rest-mass density in between the lower threshold den-\nsity\u001ahmand the upper threshold density \u001aqm, and (iii) a\nregion of pure quark matter phase with a rest-mass den-\nsity beyond \u001aqm(this might or might not be present in\npractice, depending on the maximum density reached).\nWith this picture, the equation of state for hybrid stars\nis given by\nP=8\n>><\n>>:Ph for\u001a<\u001a hm;\n\u000bqPq+ (1\u0000\u000bq)Phfor\u001ahm\u0014\u001a\u0014\u001aqm;\nPq for\u001aqm<\u001a;(6)\nwhere\n\u000bq= 1\u0000\u0012\u001aqm\u0000\u001a\n\u001aqm\u0000\u001ahm\u0013\u000e\n(7)\nis a scale factor to quantify the relative contribution due\nto hadronic and quark matters to the total pressure in\nthe mixed phase. The exponent \u000eadjusts the pressure\ncontribution due to quark matter. We set 3 values of \u000e2\nf1;2;3gto investigate the dynamical effects of varying\nquark matter contributions. We choose \u001ahm= 6:97\u0002\n1014g cm\u00003,\u001aqm= 24:3\u00021014g cm\u00003andB1=4= 170\nMeV. This treatment of phase transition is similar to\nthat of Abdikamalov et al. (2009).\nWeemploythenewgeneralrelativisticmagnetohydro-\ndynamicscode Gmunu(Cheongetal.2020,2021,2022)to\nevolve the stellar models in dynamical spacetime. Gmunu\nsolves the Einstein equations in the conformally flatcon-\ndition approximation based on the multigrid method.\nWe perform 2D ideal general-relativistic magnetohy-\ndrodynamics simulations in axisymmetry with respect\nto thez-axis and equatorial symmetry using cylindrical\ncoordinates (R;z). The computational domain covers\n[0,100] for both Randz, with the base grid resolution\nNR\u0002Nz= 32\u000232and allowing 6 AMR levels (effec-\ntive resolution = 1024\u00021024). The refinement criteria\nof AMR is the same as that in Cheong et al. (2021);4 Yip et al.\nLeung et al. (2022). Our simulations adopt TVDLF ap-\nproximate Riemann solver (Tóth & Odstrčil 1996), 3rd-\norder reconstruction method PPM (Colella & Wood-\nward 1984) and 3rd-order accurate SSPRK3 time in-\ntegrator (Shu & Osher 1988). The region outside the\nstar is filled with an artificial low-density ‘atmosphere’\nwith rest-mass density \u001aatm\u001810\u000010\u001ac. Since we are\nrestricted to purely toroidal field models and axisym-\nmetry for the simulations, we do not use any divergence\ncleaning method.\n3.FORMATION DYNAMICS\nFor each of the 9 equilibrium models, we perform sim-\nulations for three times, once for each value of the expo-\nnent\u000e2f1;2;3g. Consequently, 9\u00023 = 27simulations\nare performed in total.\nSince all simulations exhibit the same behavior, we\ntake one of them as an example to describe the features\nof the formation dynamics. Here, we choose the simu-\nlation with an initial maximum magnetic field strength\nBmax= 5:52\u00021017G (i.e. Initial model T1K6) and an\nexponent\u000e= 3. The exponent \u000e= 3corresponds to\na more substantial phase transition effect, which favors\nthe demonstration of the formation dynamics. As illus-\ntrated by the radial profiles of the rest-mass density \u001a(r)\n(top panel) and the magnetic field strength B(r)(bot-\ntom panel) at t=0 ms (grey solid lines), 10 ms (yellow\ndash-dotted lines), and 20 ms (red dotted lines) in Fig.\n1, the resulting hybrid star has a slightly higher central\nrest-mass density and maximum magnetic field strength\nafter phase transition. In addition, the magnetic field\ninside the star becomes more concentrated towards the\ncore with a tiny shift of the maximum magnetic field\nstrength position rB(dashed lines) to smaller values.\nFurthermore, new configurations of \u001a(r)andB(r)are\nobtained at t= 10ms and remain until at least t= 20\nms.\nFig. 2 shows the time evolution of the maximum val-\nues of the rest-mass density \u001amax(t)(brown solid line)\nandthemagneticfieldstrength Bmax(t)(darkcyandash-\ndotted line) relative to their initial values \u001amax(0)and\nBmax(0). The equilibrium values obtained at t=20\nms are plotted with dashed lines. We observe similar\ndamped oscillatory behaviors for both quantities and\nand the star is relaxed into a new equilibrium config-\nuration after the phase transition. As discussed in Ab-\ndikamalov et al. (2009), this damping is mainly due to\nnumerical dissipation and shock heating. Importantly,\nthese two quantities are coupled in phase during the for-\nmation process. Moreover, after reaching their peak val-\nues att\u00180:5ms, the oscillation amplitudes decrease by\na factor ofe\u00001att\u00186ms. This damping explains the\n10−1100101ρ[1014g cm−3]\nMixed phase Hadronic phaset= 0 ms t= 10 ms t= 20 ms\n0 2 4 6 8 10 12 14\nr[km]0246B[1017G]rBFigure 1. The radial profile of the rest-mass density \u001a(r)\n(top panel) and the magnetic field strength B(r)(bottom\npanel) in the equatorial plane for the simulation with the\ninitial model T1K6 and the exponent \u000e= 3att= 0ms\n(grey solid lines), 10 ms (yellow dash-dotted lines), and 20\nms (red dotted lines). Model T1K6 has an initial maximum\nmagnetic field strength Bmax = 5:52\u00021017G and\u000eis an\nexponent describing the pressure contribution due to quark\nmatter in the mixed phase. The dashed lines in the lower\npanel represent the maximum magnetic field strength po-\nsitionrB. Cadet blue region represents the portion of the\nmatter in the mixed phase while light blue region denotes\nthe portion of matter in hadronic phase. After phase transi-\ntion, the resulting hybrid star obtains a slightly higher cen-\ntral rest-mass density and maximum magnetic field strength.\nAlso, the magnetic field inside the star becomes more con-\ncentrated towards the core with a tiny shift of rBto smaller\nvalues. Moreover, these new configurations of \u001a(r)andB(r)\nare obtained at t= 10ms and remain the same until at least\nt= 20ms.\nminor discrepancy between the radial profiles at t= 10\nms andt= 20ms as demonstrated in Fig. 1.\n4.PROPERTIES OF THE RESULTING\nMAGNETIZED HYBRID STARS\nTo better examine the properties of the resulting mag-\nnetized hybrid stars, we plot in Fig. 3 different mi-\ncroscopic and macroscopic quantities against the max-\nimum magnetic field strength Bmaxof the stars. The\ndata points are arranged into 3 sequences with 3 val-\nues of\u000e2f1;2;3g, where\u000eis an exponent quantifying\nthe pressure contribution due to quark matter in the\nmixed phase. Here, we define the equatorial radius and\npolar radius of the resulting hybrid stars as the radial\npositions where the rest-mass density \u001ais less than or\nequal to 10\u00002of the lower threshold density \u001ahm(i.e.\n\u001a\u00146:97\u00021012g cm\u00003).General-relativistic simulations of the formation of a magnetized hybrid star 5\n0 2 4 6 8 10 12 14 16 18 20\nt[ms]0.91.01.11.21.31.4\nρmax(t)/ρmax(0)\nBmax(t)/Bmax(0)\nFigure 2. The time evolution of the maximum values\nof the rest-mass density \u001amax(t)(brown solid line) and the\nmagnetic field strength Bmax(t)(dark cyan dash-dotted line)\nrelative to their initial values \u001amax(0)andBmax(0)the simu-\nlation with the initial model T1K6 and the exponent \u000e= 3.\nModel T1K6 has an initial maximum magnetic field strength\nBmax = 5:52\u00021017G and\u000eis an exponent describing\nthe pressure contribution due to quark matter in the mixed\nphase. Dashed lines are the equilibrium values of the two\nquantities obtained at t= 20ms. Similar damped oscilla-\ntorybehaviorsareobservedforbothquantitiesandthestaris\nrelaxed into a new equilibrium configuration after the phase\ntransition. Importantly, these two quantities are coupled in\nphase during the formation process. Moreover, after reach-\ning the peak values at t\u00180:5ms, the oscillation amplitudes\nare reduced by a factor of e\u00001att\u00186ms.\nWe find that forBmax&5\u00021017, all microscopic and\nmacroscopic quantities vary strongly with Bmax, irre-\nspective of \u000e. WhenBmax.3\u00021017G, all quantities\nvary slightly with Bmax. This means that it may be\npossible to link observational signals from a magnetized\nhybrid star to the magnetic field of the star.\nSpecifically, the central rest-mass density \u001ac(top left\npanel) and the baryonic mass fraction of the matter in\nthe mixed phase Mmp=M0(bottom left panel) decrease\nwithBmax. These decreasing behaviors could be under-\nstood in terms of magnetic pressure. As the magnetic\npressure becomes more dominant due to the increasing\nBmax, matter is pushed off-center to a greater extent.\nAs a result, the rest-mass density \u001ain the stellar core\nreduces, giving a smaller \u001ac. Moreover, as described in\nEq. (6), reducing \u001ain the core contributes to a smaller\nfraction of matter that undergoes the phase transition\nand thus gives a smaller Mmp=M0.\nMoreover, the equatorial radius re(top middle panel),\nthe polar radius to equatorial radius ratio rp=re(bottom\nmiddle panel) and the gravitational mass Mg(top right\npanel) all increase with Bmax. The increase in Mgis\ndue to the increasing contribution of the magnetic fieldtoMg(corresponds to the increasing B2term of Eq.\n(B1) in Pili et al. 2014 for example). The other two\nincreasing trends could be interpreted in terms of mag-\nnetic deformation. As the matter is pushed off-center\nby the increasing magnetic pressure, both rpandrein-\ncrease and the star then deviates from spherical symme-\ntry. Previousstudiesofmagnetizedneutronstarequilib-\nrium models (e.g. Kiuchi & Yoshida 2008; Kiuchi et al.\n2009; Frieben & Rezzolla 2012) indicate that a purely\ntoroidal field deforms the stars to prolate shape, corre-\nsponding to rp=re>1. Thus, increasing Bmaxof the\ntoroidal field in our models causes the increase in rp=re.\nWe also examine the effect of pressure contribution\ndue to quark matter \u000eon different quantities of the hy-\nbrid stars. \u000ehas a negligible effect on re,rp=reandMg\nfor all values ofBmax. On the contrary, \u001acandMmp=M0\nincrease substantially with \u000eforBmax.3\u00021017G\nbut they become less sensitive to \u000eforBmax&5\u00021017\nG. These increasing trends could be interpreted in re-\nlation to pressure reduction. With the increasing value\nof\u000e, the contribution due to quark matter to the total\npressure becomes more important and the pressure re-\nduction is enlarged. As a result, this enlarged pressure\nreductionmakesthestarcollapsetoaconfigurationwith\na higher\u001acandMmp=M0.\nWe compare our resulting hybrid stars with Franzon\net al. (2016a). The magnetized hybrid star models con-\nsidered in this study are also in axisymmetry, but the\nmagnetic field is purely poloidal. A poloidal field would\nmake the stars oblate instead of prolate. Also, these\nmodels have a different baryonic mass M0= 2:2M\f.\nThese equilibrium models are constructed by solving\nthe coupled Maxwell–Einstein equations. They also em-\nployed a more realistic equation of state with both mag-\nnetic and thermal effects taken into account.\nWe plot the normalized gravitational mass Mg=M\u0003\ng\nagainstBmax(bottom right panel) to compare with\nthe models computed in Franzon et al. (2016a) (Red\nstars), where M\u0003\ngis the gravitational mass of the non-\nmagnetized reference models. We observe that Mg=M\u0003\ng\nincreases withBmaxsimilarly for the models in our sim-\nulations and Franzon et al. (2016a). Besides, this pre-\nvious study also found that the magnetic field reduces\nthe central baryon number density and hinders the ap-\npearance of matter in quark and mixed phases. These\nalso agree with the trends of \u001acandMmp=M0for our\nmodels. Accordingly, despite the disparity in field ge-\nometry, baryonic mass, and construction method, our\nmodels agree qualitatively with the models in the pre-\nvious study. This similarity provides additional support\nthat the properties of hybrid the properties of the mag-\nnetised hybrid stars presented here are robust.6 Yip et al.\n8.59.09.510.010.511.011.5ρc[1014g cm−3]\nδ= 1 δ= 2 δ= 3\n12131415re[km]\n1.551.561.571.581.591.60Mg[M⊙]\n0123456\nBmax[1017G]51015Mmp/M0[%]\n0123456\nBmax[1017G]1.01.11.21.3rp/re\n0123456789\nBmax[1017G]1.001.011.021.031.041.05Mg/M∗\ng\nFranzon et al. (2016)\nFigure 3. Plots of different microscopic and macroscopic quantities against the maximum magnetic field strength Bmaxof\nour resulting hybrid star models. In particular, we plot the central rest-mass density \u001ac(top left panel), the baryonic mass\nfraction of the matter in the mixed phase Mmp=M0(bottom left panel), the equatorial radius re(top middle panel), the ratio\nbetween polar and the equatorial radii rp=re(bottom middle panel), and the gravitational mass MgagainstBmax(top right\npanel). The data points are arranged into 3 sequences with 3 values of \u000e2f1;2;3g, where\u000eis an exponent quantifying the\npressure contribution due to quark matter in the mixed phase. The macroscopic and microscopic quantities of all hybrid star\nmodels are not sensitive to the magnetic field for Bmax.3\u00021017G. However, these quantities noticeably vary with Bmaxfor\nBmax&5\u00021017G. In addition, \u000ehas a negligible effect on re,rp=reandMgfor all values ofBmax. In contrast, \u001acandMmp=M0\nincrease substantially with \u000eforBmax.3\u00021017G but they become less sensitive to \u000eforBmax&5\u00021017G. Furthermore, we\nplot the normalized gravitational mass Mg=M\u0003\ngagainstBmax(bottom right panel) as a comparison with the models computed\nin Franzon et al. (2016a) (Red stars), where M\u0003\ngis the gravitational mass of the non-magnetized reference models. We find that\nMg=M\u0003\ngincreases withBmaxsimilarly for the models in our simulations and Franzon et al. (2016a). Hence, we find agreement\namongst vastly different methods and thus provide a solid support for the magnetic effects on a hybrid star.\n5.CONCLUSIONS\nIn this paper, we studied the formation of a mag-\nnetized hybrid star by performing 2D axisymmetric\ngeneral-relativistic magnetohydrodynamics simulations.\nWe first found that the maximum values of rest-mass\ndensity and magnetic field strength in the stars rise\nslightly after a phase transition. The magnetic field\nalso becomes more concentrated towards the center. In\naddition, the magnetic field and the rest-mass density\nare coupled during the process. We then investigated\nthe properties of the resulting magnetized hybrid stars.\nBoth macroscopic and microscopic quantities of the hy-\nbrid stars are not sensitive to the magnetic field until\nBmax&5\u00021017G, where all quantities change signifi-\ncantly. Specifically, the magnetic deformation decreases\nthe rest-mass density dramatically, leading to a substan-tial reduction in the matter fraction in the mixed phase.\nSimilar trends for these quantities are found compared\nwith Franzon et al. (2016a).\nThis work takes the first step to dynamically study-\ning magnetized hybrid stars. Several natural extensions\nshould be considered to model them more realistically.\nFirst, a more realistic equation of state, which includes\nthermal and magnetic effects, should be adopted. In ad-\ndition, since magnetized stars with purely toroidal fields\nare expected to be unstable, the suppression of insta-\nbility in this work is mainly due to the restriction to\n2D axisymmetry. The effects of purely poloidal fields\nand the twisted torus configurations should also be in-\nvestigated. Since these field geometries extend to the\nouter region of neutron stars, a force-free/resistive mag-\nnetohydrodynamics solver is necessary for more realisticGeneral-relativistic simulations of the formation of a magnetized hybrid star 7\nmodeling. Also, 3D simulations without axisymmetry\nshould be conducted to include the instability of mag-\nnetic fields. Finally, as most observations suggested that\nneutron stars rotate, rotations should also be included\nin future studies.WeacknowledgethesupportoftheCUHKCentralHigh-\nPerformance Computing Cluster, on which the simu-\nlations in this work have been performed. This work\nwas partially supported by grants from the Research\nGrants Council of Hong Kong (Project No. CUHK\n14306419), the Croucher Innovation Award from the\nCroucher Foundation Hong Kong, and the Direct Grant\nfor Research from the Research Committee of The Chi-\nnese University of Hong Kong. P.C.-K.C. acknowl-\nedges support from NSF Grant PHY-2020275 (Network\nfor Neutrinos, Nuclear Astrophysics, and Symmetries\n(N3AS)).1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\nSoftware: XNS(Bucciantini & Del Zanna 2011; Pili\net al. 2014, 2015, 2017; Soldateschi et al. 2020), Gmunu\n(Cheong et al. 2020, 2021, 2022)\nREFERENCES\nAbdikamalov, E. B., Dimmelmeier, H., Rezzolla, L., &\nMiller, J. C. 2009, MNRAS, 392, 52,\ndoi: 10.1111/j.1365-2966.2008.14056.x\nAguilera-Miret, R., Viganò, D., Carrasco, F., Miñano, B., &\nPalenzuela, C. 2020, PhRvD, 102, 103006,\ndoi: 10.1103/PhysRevD.102.103006\nBocquet, M., Bonazzola, S., Gourgoulhon, E., & Novak, J.\n1995, A&A, 301, 757, doi: 10.48550/arXiv.gr-qc/9503044\nBodmer, A. R. 1971, PhRvD, 4, 1601,\ndoi: 10.1103/PhysRevD.4.1601\nBonazzola, S., & Gourgoulhon, E. 1996, A&A, 312, 675,\ndoi: 10.48550/arXiv.astro-ph/9602107\nBraithwaite, J. 2009, MNRAS, 397, 763,\ndoi: 10.1111/j.1365-2966.2008.14034.x\nBraithwaite, J., & Nordlund, Å. 2006, A&A, 450, 1077,\ndoi: 10.1051/0004-6361:20041980\nBraithwaite, J., & Spruit, H. C. 2006, A&A, 450, 1097,\ndoi: 10.1051/0004-6361:20041981\nBucciantini, N., & Del Zanna, L. 2011, A&A, 528, A101,\ndoi: 10.1051/0004-6361/201015945\nCardall, C. Y., Prakash, M., & Lattimer, J. M. 2001, ApJ,\n554, 322,\ndoi: 10.1086/32137010.48550/arXiv.astro-ph/0011148\nChatterjee, D., Elghozi, T., Novak, J., & Oertel, M. 2015,\nMNRAS, 447, 3785, doi: 10.1093/mnras/stu2706\nCheong, P. C.-K., Lam, A. T.-L., Ng, H. H.-Y., & Li, T.\nG. F. 2021, MNRAS, 508, 2279,\ndoi: 10.1093/mnras/stab2606\nCheong, P. C.-K., Lin, L.-M., & Li, T. G. F. 2020, Classical\nand Quantum Gravity, 37, 145015,\ndoi: 10.1088/1361-6382/ab8e9cCheong, P. C.-K., Pong, D. Y. T., Yip, A. K. L., & Li, T.\nG. F. 2022, ApJS, 261, 22, doi: 10.3847/1538-4365/ac6cec\nColella, P., & Woodward, P. R. 1984, Journal of\nComputational Physics, 54, 174,\ndoi: 10.1016/0021-9991(84)90143-8\nDexheimer, V., Negreiros, R., & Schramm, S. 2012,\nEuropean Physical Journal A, 48, 189,\ndoi: 10.1140/epja/i2012-12189-y\nFerrer, E. J., de La Incera, V., Keith, J. P., Portillo, I., &\nSpringsteen, P. L. 2010, PhRvC, 82, 065802, doi: 10.\n1103/PhysRevC.82.06580210.48550/arXiv.1009.3521\nFranzon, B., Dexheimer, V., & Schramm, S. 2016a,\nMNRAS, 456, 2937, doi: 10.1093/mnras/stv2606\nFranzon, B., Gomes, R. O., & Schramm, S. 2016b,\nMNRAS, 463, 571, doi: 10.1093/mnras/stw1967\nFrieben, J., & Rezzolla, L. 2012, MNRAS, 427, 3406,\ndoi: 10.1111/j.1365-2966.2012.22027.x10.48550/arXiv.\n1207.4035\nFushiki, I., Gudmundsson, E. H., & Pethick, C. J. 1989,\nApJ, 342, 958, doi: 10.1086/167653\nGlendenning, N. K. 2012, Compact stars: Nuclear physics,\nparticle physics and general relativity (Springer Science\n& Business Media)\nHerzog, M., & Röpke, F. K. 2011, PhRvD, 84, 083002,\ndoi: 10.1103/PhysRevD.84.083002\nHurley, K., Li, P., Kouveliotou, C., et al. 1999, ApJL, 510,\nL111, doi: 10.1086/311820\nIsayev, A. A. 2015, in Journal of Physics Conference Series,\nVol. 607, Journal of Physics Conference Series, 012013,\ndoi: 10.1088/1742-6596/607/1/0120138 Yip et al.\nItoh, N. 1970, Progress of Theoretical Physics, 44, 291,\ndoi: 10.1143/PTP.44.291\nJohnson, K., et al. 1975, Acta Phys. Pol. B, 6, 8\nKiuchi, K., Cerdá-Durán, P., Kyutoku, K., Sekiguchi, Y., &\nShibata, M. 2015a, PhRvD, 92, 124034,\ndoi: 10.1103/PhysRevD.92.124034\nKiuchi, K., Kotake, K., & Yoshida, S. 2009, ApJ, 698, 541,\ndoi: 10.1088/0004-637X/698/1/54110.48550/arXiv.0904.\n2044\nKiuchi, K., Sekiguchi, Y., Kyutoku, K., et al. 2015b,\nPhRvD, 92, 064034, doi: 10.1103/PhysRevD.92.064034\nKiuchi, K., & Yoshida, S. 2008, PhRvD, 78, 044045, doi: 10.\n1103/PhysRevD.78.04404510.48550/arXiv.0802.2983\nKonno, K. 2001, A&A, 372, 594, doi: 10.1051/0004-6361:\n2001055610.48550/arXiv.gr-qc/0105015\nKouveliotou, C., Dieters, S., Strohmayer, T., et al. 1998,\nNature, 393, 235, doi: 10.1038/30410\nLai, D., & Shapiro, S. L. 1991, ApJ, 383, 745,\ndoi: 10.1086/170831\nLeung, M. Y., Yip, A. K. L., Cheong, P. C.-K., & Li, T.\nG. F. 2022, Communications Physics, 5, 334,\ndoi: 10.1038/s42005-022-01112-w\nLin, L. M., Cheng, K. S., Chu, M. C., & Suen, W. M. 2006,\nApJ, 639, 382, doi: 10.1086/499202\nMariani, M., Tonetto, L., Rodríguez, M. C., et al. 2022,\nMNRAS, 512, 517, doi: 10.1093/mnras/stac546\nMarkey, P., & Tayler, R. J. 1973, MNRAS, 163, 77,\ndoi: 10.1093/mnras/163.1.77\n—. 1974, MNRAS, 168, 505, doi: 10.1093/mnras/168.3.505\nMereghetti, S., Cremonesi, D., Feroci, M., & Tavani, M.\n2000, A&A, 361, 240\nMereghetti, S., & Stella, L. 1995, ApJL, 442, L17,\ndoi: 10.1086/187805\nPili, A. G., Bucciantini, N., & Del Zanna, L. 2014,\nMNRAS, 439, 3541, doi: 10.1093/mnras/stu215—. 2015, MNRAS, 447, 2821, doi: 10.1093/mnras/stu2628\n—. 2017, MNRAS, 470, 2469, doi: 10.1093/mnras/stx1176\nPrasad, R., & Mallick, R. 2018, ApJ, 859, 57,\ndoi: 10.3847/1538-4357/aabf3b\n—. 2020, ApJ, 893, 151, doi: 10.3847/1538-4357/ab7f2b\nPrice, D. J., & Rosswog, S. 2006, Science, 312, 719,\ndoi: 10.1126/science.1125201\nRabhi, A., Pais, H., Panda, P. K., & Providência, C. 2009,\nJournal of Physics G Nuclear Physics, 36, 115204,\ndoi: 10.1088/0954-3899/36/11/115204\nRezzolla, L., Pizzochero, P., Jones, D. I., et al. 2018,\nAstrophysics and Space Science Library, Vol. 457, The\nPhysics and Astrophysics of Neutron Stars (Springer),\ndoi: 10.1007/978-3-319-97616-7\nShu, C.-W., & Osher, S. 1988, Journal of Computational\nPhysics, 77, 439, doi: 10.1016/0021-9991(88)90177-5\nSoldateschi, J., Bucciantini, N., & Del Zanna, L. 2020,\nA&A, 640, A44, doi: 10.1051/0004-6361/202037918\nTayler, R. J. 1957, Proceedings of the Physical Society B,\n70, 31, doi: 10.1088/0370-1301/70/1/306\n—. 1973, MNRAS, 161, 365, doi: 10.1093/mnras/161.4.365\nTóth, G., & Odstrčil, D. 1996, Journal of Computational\nPhysics, 128, 82, doi: 10.1006/jcph.1996.0197\nvan Paradijs, J., Taam, R. E., & van den Heuvel, E. P. J.\n1995, A&A, 299, L41\nWeber, F. 1999, Journal of Physics G Nuclear Physics, 25,\nR195\nWeber, F. 1999, Journal of Physics G: Nuclear and Particle\nPhysics, 25, R195\nWitten, E. 1984, PhRvD, 30, 272,\ndoi: 10.1103/PhysRevD.30.272\nWright, G. A. E. 1973, MNRAS, 162, 339,\ndoi: 10.1093/mnras/162.4.339\nYazadjiev, S. S. 2012, PhRvD, 85, 044030, doi: 10.1103/\nPhysRevD.85.04403010.48550/arXiv.1111.3536"    },    {        "title": "2204.05387v2.Deformation_of_a_magnetic_liquid_drop_and_unsteady_flow_inside_and_outside_it_in_a_non_stationary_magnetic_field.pdf",        "content": "arXiv:2204.05387v2  [physics.flu-dyn]  17 Jun 2022Deformation of a magnetic liquid drop and unsteady flow insid e and outside it in a\nnon-stationary magnetic field\nAlexander N. Tyatyushkin\nInstitute of Mechanics, Moscow State University, Michurin skiy Ave., 1, Moscow 119192, Russia\nSmall steady-state deformational oscillations of a drop of a viscous magnetic liquid in a non-\nstationary uniform magnetic field are theoretically invest igated for low Reynolds numbers. The drop\nis suspended in another viscous magnetic liquid immiscible with the former. The non-stationary\nmagnetic field causes flow inside and outside the drop. The ine rtia is taken intoaccount byregarding\ntheflowas essentially unsteady. The variation of the magnet ic field is so slow thatthe approximation\nof quasi-stationary magnetic field may be used.\nINTRODUCTION.\nA non-stationary magnetic field applied to a drop of a\nmagnetic liquid suspended in an ordinary liquid or to a\ndrop ofan ordinaryliquid suspended in a magnetic liquid\ncausesamotionoftheliquidsinsideandoutsidethedrop.\nInvestigation of this phenomenon is interesting from the\npoint of view of application to the actuation of motion\nof liquids in various devices including microfluidic ones.\nBesides, the study ofthis phenomenon is interesting from\nthe point of view of basic science.\nResultsofexperimentalandtheoreticalinvestigationof\nthe behavior of magnetic liquid drops in non-stationary\nmagneticfieldsarepresentedin works[1–10]. Thebehav-\nior of an ordinary liquid drop suspended in a magnetic\nliquid was also studied [11].\nIn works [2–4], [6], and [8] for the theoretical descrip-\ntion of the behavior of a magnetic liquid drop, models\nbased on rather strong assumptions about the shape of\nthe drop and about the flow inside and outside it were\nused. In works [9] and [10] the tree-dimensional bound-\nary element method algorithm was used for the numeri-\ncal calculations. In works [1], [5], [7], and [9] asymptotic\nmethods were used. In the present work the problem\nabout the shapeofthe dropandflowinside and outsideit\nisalsosolvedwithasymptoticmethodswithouttheuseof\nany assumptions about them. Within this approach, the\nshape of the drop is determined by equations obtained\nwith the help of asymptotic methods from the system of\nequations and boundary conditions that determine the\nmagnetic field and the flow. In order to provide the ap-\nplicability of the asymptotic methods, the scope of the\npresent theoretical investigation is restricted by the case\nof weakly deformed drops. The unsteady flow inside and\noutside a drop of a viscous magnetic liquid suspended in\nanother viscous magnetic liquid in non-stationary mag-\nnetic field is investigated with taking into account the\ninfluence of form of the drop on the field and flow. This\napproachwasused alsoin works[12–14], in which the be-\nhavior of a magnetic liquid drop in non-stationary fields\nfor low Reynolds numbers was investigated within the\nquasi-steady flow approximation, and in [15], in which\nthe Reynolds numbers were regarded as high and the vis-\ncosity was neglected. Thus, the inertia was completely\nneglected in [12–14]. In the present work, the inertia istaken into account simultaneously with the viscosity by\nrefusing the quasi-steady flow approximation.\nThe Reynolds number in the problem about defor-\nmational oscillations of a drop under action of non-\nstationary fields tends to zero as the amplitude of the\noscillation tends to zero. Indeed, the velocity in that\ncase has the order of magnitude ω∆a, whereωis the an-\ngularfrequency of the oscillationsofthe applied field, ∆ a\nis the amplitude of the deformational oscillations of the\ndrop (maximal variation of the distance from the center\nof the drop to its surface). Then the Reynolds number\ntends to zero as the amplitude of the oscillations tends\nto zero. Thus, in the case of small oscillations of the\ndrop, the influence of the convection term in the momen-\ntum equation can be neglected, and the influence of the\ninertia can be described by the unsteady therm. This is\ndirectly confirmed by the fact that, in the problem about\ndeformational oscillations of a drop in the case of when\nthe influence of the viscosity is completely neglected (see\n[15]), the influence of the convection term in the correc-\ntion of the first order over the parameter the smallness of\nwhich provides the smallness of the deformations is ab-\nsent, althoughitistakenintoaccountintheinputsystem\nof equations. It is also followed from this fact that the\nsolution of the problem about small oscillations of the\ndrop with taking into account both the viscosity and in-\nertia should tend to the solution in [15] as the maximal\nviscosity of the liquids tends to zero with exception of\nthe case when the frequency of the applied field is close\nto the resonance frequency.\nIn order to use the asymptotic methods for the in-\nvestigation of small oscillations of drops under action of\nnon-stationary fields developed in the works [12–15] with\nthe use of unsteady term in the momentum equation, it\nis necessary to obtain the general solution to the sys-\ntem of hydrodynamic equations that contains this term.\nThis solution was obtained and is written down below.\nThe problem stated above presents a good opportunity\nto use this general solution since the problems solved in\nthe works [12–15] allow checking the results for the limit\ncases when the influence of either the inertia or the vis-\ncosity is neglected.2\nI. SETTING OF THE PROBLEM.\nA. Object of investigation.\nConsider a drop of incompressible magnetic liquid in\nan applied non-stationary uniform magnetic field with\nintensity/vectorHa=/vectorHa(t), wheretis the time. The density,\nviscosity, and magnetic permeability of the liquid inside\nthe drop are ρi,ηi, andµi. The radius of the drop in\nthe undeformed state is a. The drop is suspended in\nan incompressible magnetic liquid with the density ρe,\nviscosityηe, and magnetic permeability µe. The surface\ntension of the interfacebetween the liquids is σs. The liq-\nuids are regarded as sufficiently viscous so that the low\nReynolds number approximation is valid. The influence\noftheinertiaoftheliquidsistakenintoaccountbymeans\nof regarding the flow as essentially unsteady. The varia-\ntionofthe magneticfieldissoslowandtheconductivities\nof the liquids are so small that the ferrohydrodynamics\napproximation is valid (see [16]).\nB. System of equations.\nIn order to find the magnetic field intensity, /vectorH, veloc-\nity,/vector v, and pressure, p, as functions of the radius-vector,\n/vector r, and time, t, as well as the shape of the drop under\nthe above made assumptions, the system of equations of\nferrohydrodynamicswritten downin the quasi-stationary\nmagnetic field and low Reynolds number approximations\nis used. This system consists of the continuity equation\nfor an incompressible liquid\n∇·/vector v= 0, (1)\nthe motion equation in the low Reynolds number approx-\nimation for arbitrary Strouhal numbers\nρ∂/vector v\n∂t=∇·/parenleftBig\n−pˆI+ ˆσm+ ˆσv/parenrightBig\n, (2)\nthe Maxwell’s equations in the ferrohydrodynamics and\nquasi-stationary field approximations\n∇·/vectorB= 0,∇×/vectorH= 0, (3)\nand the constitutive relation\n/vectorB=µ/vectorH. (4)\nHere, the formulas are written down for Gaussian system\nof units, ·and×denote the scalar and vector products,\n/vectorBis the magnetic induction, ρ=ρi,η=ηi, andµ=µi\ninside the drop, ρ=ρe,η=ηe, andµ=µeoutside\nit,∇is the nabla operator, ˆIis the identity tensor, ˆ σm\nand ˆσvare the tensors of magnetic and viscous stresses\nexpressed as follows\nˆσm=1\n4π/vectorB/vectorH−1\n8π/vectorB·/vectorHˆI, (5)\nˆσv= 2η(∇/vector v)S, (6)where/vectorA/vectorBdenotes the dyadic product of the vectors /vectorA\nand/vectorB,∇/vectorfdenotes the dyadic product of the nabla op-\nerator and the vector field /vectorf=/vectorf(/vector r),ˆTSdenotes the\nsymmetric part of the tensor ˆT.\nUsing the continuity equation Eq. (1) and Maxwell’s\nequations Eq. (3) and taking into account that the\nmagnetic permeability is uniform, the motion equation\nEq. (2) can be rewritten in the form of the Navier–Stokes\nequation in the low Reynolds number approximation for\narbitrary Strouhal numbers\nρ∂/vector v\n∂t=−∇p+η∆/vector v, (7)\nwhere ∆ is the Laplacian.\nC. Boundary conditions.\nThe boundary conditions on the interface between the\nliquids include the impenetrability condition\n/vector v|e·/vector n=/vector v|i·/vector n=vsn, (8)\nthe no-slip condition\n[/vector v]s×/vector n= 0, (9)\nthe conditions for the jumps of the normal and tan-\ngential components of the stress vector /vector σn=/vector n·/parenleftBig\n−pˆI+ ˆσm+ ˆσv/parenrightBig\n/vector n·/bracketleftBig\n−pˆI+ ˆσm+ ˆσv/bracketrightBig\ns·/vector n=−σsH,(10)\n/vector n·[ˆσm+ ˆσv]s×/vector n= 0, (11)\nthe continuity conditions for the tangential component of\nthe magnetic field intensity\n/bracketleftBig\n/vectorH/bracketrightBig\ns×/vector n= 0 (12)\nand for the normal component of the magnetic field in-\nduction\n/bracketleftBig\n/vectorB/bracketrightBig\ns·/vector n= 0. (13)\nHere,A|iandA|edenote the values of the quantity Aon\nthe interface between the liquids approached from inside\nand outside the drop, respectively, [ A]s=A|e−A|ide-\nnotes the jump of the quantity Aat the interface when\nmovingfromtheinsidetotheoutside, ∇sdenotesthesur-\nface nabla operator, /vector nis the external normal unit vector\nat a given point of the interface, vsnis the normal com-\nponent of the velocity of the surface of the drop at a\ngiven point, His the mean curvature at a given point of\nthe surface of the drop. Note that definition of the mean\ncurvature used in the present work is such that it takes\nnegative values on the surface of a convex domain.3\nThe continuity conditions for the tangential compo-\nnents of the velocity and electric field intensity allow de-\ntermining the following vector fields defined on the sur-\nface of the drop\n/vector vsτ=/vector n×(/vector v|i×/vector n) =/vector n×(/vector v|e×/vector n),(14)\n/vectorHs=/vector n×/parenleftBig\n/vectorH/vextendsingle/vextendsingle/vextendsingle\ni×/vector n/parenrightBig\n=/vector n×/parenleftBig\n/vectorH/vextendsingle/vextendsingle/vextendsingle\ne×/vector n/parenrightBig\n.(15)\nThe boundary conditions at infinity have the form\n/vector v→0 as r→ ∞,(16)\np→p∞ asr→ ∞,(17)\n/vectorH→/vectorHa asr→ ∞,(18)\nwherep∞is the pressure at infinity.\nBesides,/vector v(/vector r,t),p(/vector r,t), and/vectorH(/vector r,t) should be bounded\nfor all the bounded values of /vector r.\nII. SOLUTION.\nA. Deformation of the drop.\nLet the surface of the drop be given by the following\nequation\nr=|/vector r|=a+h/parenleftbigg/vector r\nr,t/parenrightbigg\n. (19)\nIn order the deformations of the drop to be small, the\nfollowing condition should be fulfilled\nh=h/parenleftbigg/vector r\nr,t/parenrightbigg\n≪a. (20)\nThe function h(/vector r/r,t) can be represented in the form\nh=h/parenleftbigg/vector r\nr,t/parenrightbigg\n=∞/summationdisplay\nn=2ˆhnn·/vector rn\nrn. (21)\nHere,ˆhn=ˆhn(t) (n>1) are some tensors each of which\nis an arbitrary irreducible tensor of nth order, i.e., a ten-\nsor ofnth order symmetric with respect to any pair of\nindices and such that its contraction with the identity\ntensor over any pair of indices is equal to zero (see [17]).\nHere and in what follows, /vectorbndenotesnth dyadic degree\nofthe vector /vectorb(i.e., the (n−1)-multipledyadicproductof\nthe vector/vectorbby itself),n·denotesn-multiple contraction\nof tensors over the adjacent indices.\nThe impenetrability condition (8) yields the following\nequation for the function h(/vector r/r,t)\n/vector n·/vector r\nr∂h\n∂t=vsn. (22)B. Magnetic field.\nThe intensity of the magnetic field is sought for in the\nform\n/vectorH=∇ψ, (23)\nwhere\nψ=\n\n/vectorHa·/vector r−∞/summationdisplay\nn=1ˆMenn·/vector rn\nr2n+1, r>a+h,\n/vectorHa·/vector r−∞/summationdisplay\nn=1ˆMinn·/vector rn\na2n+1, r/lessorequalslanta+h.(24)\nHere,ˆMen=ˆMen(t) andˆMin=ˆMin(t) are tensorswhich\nare either some vectors for n= 1 or, for n>1, some ir-\nreducible tensors of nth order. For a field given in this\nform, Maxwell’s equations Eq. (3), the constitutive rela-\ntion Eq. (4), the condition at infinity Eq. (18), and the\ncondition of boundedness of /vectorHare automatically satis-\nfied.\nC. Flow.\nThe velocity and pressure in the flow are sought for in\nthe form\n/vector v=/vector v(/vector r,t) =1\n2π∞/integraldisplay\n−∞e−iωt/vector v1ωdω\n+1\n2π∞/integraldisplay\n−∞e−iωt/vector v2ωdω,(25)\np=p(/vector r,t) =1\n2π∞/integraldisplay\n−∞e−iωtpωdω\n+/braceleftBigg\np∞, r>a +h,\np0, r<a +h,(26)\n/vector v1ω=/braceleftBigg\n/vector v1eω, r>a +h,\n/vector v1iω, r/lessorequalslanta+h,(27)\n/vector v1eω=∇×\n\n\n∞/summationdisplay\nn=1\nQ−n−1/parenleftBig\nκer\na/parenrightBig\nκ2eˆBen+ˆCen\n\nn−1·/vector rn−1an+2\nr2n+1\n\n×/vector r\n,(28)4\n/vector v1iω=∇×\n\n\n∞/summationdisplay\nn=1\nQn/parenleftBig\nκir\na/parenrightBig\nκ2\niˆBin+ˆCin\n\nn−1·/vector rn−1\nan−1\n\n×/vector r\n,(29)\n/vector v2ω=/braceleftBigg\n/vector v2eω, r>a +h,\n/vector v2iω, r/lessorequalslanta+h,(30)\n/vector v2eω=∞/summationdisplay\nn=1n/bracketleftbigg\nκ2n+1\neG−n−1/parenleftBig\nκer\na/parenrightBig\nˆΩen\nn−1·/vector rn−1\nan−1/bracketrightbigg\n×/vector r,(31)\n/vector v2iω=∞/summationdisplay\nn=1n/bracketleftbigg\nGn/parenleftBig\nκir\na/parenrightBig\nˆΩinn−1·/vector rn−1\nan−1/bracketrightbigg\n×/vector r,(32)\npω=/braceleftBigg\npeω, r>a +h,\npiω, r<a +h,(33)\npeω=ηe\na/bracketleftBig\n2(2n−1)ˆBen+κ2\neˆCen/bracketrightBign·/vector rnan+1\nr2n+1,(34)\npiω=ηi\na/bracketleftbigg2(n+1)(2n+3)\nnˆBin−κ2\ni(n+1)\nnˆCin/bracketrightbigg\nn·/vector rn\nan,\n(35)\nκe= (1−i)/radicalBigg\nρeωa2\n2ηe,κi= (1−i)/radicalBigg\nρiωa2\n2ηi,(36)\nGn+1(z) =2n+3\nzG′\nn(z), G0(z) =sinhz\nz,(37)\nG−n−2(z) =−1\n(2n+1)zG′\n−n−1(z), G−1(z) =e−z\nz,\n(38)\nQn(z) = 2(2n+3)[Gn(z)−1], n/greaterorequalslant0,(39)\nQ−n−1(z) =−2(2n−1)/bracketleftbig\nz2n+1G−n−1(z)−1/bracketrightbig\n, n/greaterorequalslant0.\n(40)\nHere,ˆΩen=ˆΩen,ω,ˆBen=ˆBen,ω,ˆCen=ˆCen,ω,ˆΩin=\nˆΩin,ω,ˆBin=ˆBin,ω,ˆCin=ˆCin,ωare some tensors de-\npending on ωeach of which is either an arbitrary vector\nforn= 1 or an arbitrary irreducible tensor for n >1,\np0=p0(t) is the pressureat the center of the drop, whichis to be found. For a flow given in this form, the conti-\nnuity equation Eq. (1), the motion equation Eq. (7), the\nconditions at infinity Eqs. (16)–(17), and the condition\nof boundedness of /vector vandpare automatically satisfied.\nThe general solution Eqs. (25)–(40) of the equations\nEq. (1)–(7) was obtained as the generalization of the\ngeneral Lamb’s solution for the case of unsteady flows\nwith arbitrary Strouhal numbers. It turns into the gen-\neral Lamb’s solution (cf. [18], art. 336 and [12]) for the\nquasi-steady flow approximation.\nD. Asymptotic expansion.\nThus, in order to solve the problem set above, it is\nnecessary to find the unknown vector and tensor func-\ntionsˆMen=ˆMen(t),ˆΩen=ˆΩen,ω,ˆBen=ˆBen,ω,\nˆCen=ˆCen,ω,ˆMin=ˆMin(t),ˆΩin=ˆΩin,ω,ˆBin=ˆBin,ω,\nandˆCin=ˆCin,ω(n= 1,2,...) as well as the tensor\nfunctions ˆhn=ˆhn(t) (n= 2,...) and the scalar func-\ntionp0=p0(t) using the remaining boundary conditions\nEqs. (8)–(13), and the equation Eq. (22). The functions\nare sought for in the form of the following asymptotic\nexpansions over the parameter\nα=9aµeH2\nam\n32πσs, (41)\nˆMen∼∞/summationdisplay\nj=0αjˆMen,j,ˆΩen∼∞/summationdisplay\nj=1αjˆΩen,j,\nˆBen∼∞/summationdisplay\nj=1αjˆBen,j,ˆCen∼∞/summationdisplay\nj=1αjˆCen,j,\nˆMin∼∞/summationdisplay\nj=0αjˆMin,j,ˆΩin∼∞/summationdisplay\nj=1αjˆΩin,j,\nˆBin∼∞/summationdisplay\nj=1αjˆBin,j,ˆCin∼∞/summationdisplay\nj=1αjˆCin,j,\np0∼∞/summationdisplay\nj=0αjp0j,ˆhn∼∞/summationdisplay\nj=1αjˆhn,jasα→0.(42)\nHere,Hamis the maximal absolute value of the intensity\nvector of the applied magnetic field. The parameter αis\nthe magnetic Bond number for the present problem.\nE. Steady-state oscillations.\nFor the forced steady-state oscillations, the found\nnonzero terms of the asymptotic expansions are as fol-\nlows\nˆMe1,0=/vectorMe1,0=ˆMi1,0=/vectorMi1,0=µi−µe\nµi+2µea3/vectorHa,(43)5\np0,0=p∞+2σs\na+3µe\n8πµe−µi\nµi+2µeH2\na,(44)\nˆBe2,1=−iωτ\n2ˆh2,1,ω/bracketleftbigg\n14ηi\nηeQ′\n2(κi)\nκi+5ηi\nηeQ′′\n2(κi)\n+16Q′\n2(κi)\nκi/bracketrightbigg\n/parenleftbigg\n3/braceleftbiggηi\nηeQ′\n−3(κe)\nκe/bracketleftbigg\n4Q′\n2(κi)\nκi+Q′′\n2(κi)/bracketrightbigg\n+Q′\n2(κi)\nκi/bracketleftbigg\n6Q′\n−3(κe)\nκe−Q′′\n−3(κe)/bracketrightbigg/bracerightbigg/parenrightbigg−1\n,(45)\nˆCe2,1=iωτ\n2ˆh2,1,ω/braceleftbiggηi\nηeQ′\n2(κi)\nκi/bracketleftbigg\n14Q−3(κe)\nκ2e\n−4Q′\n−3(κe)\nκe/bracketrightbigg\n+ηi\nηeQ′′\n2(κi)/bracketleftbigg\n5Q−3(κe)\nκ2e−Q′\n−3(κe)\nκe/bracketrightbigg\n+Q′\n2(κi)\nκi/bracketleftbigg\n16Q−3(κe)\nκ2e\n−6Q′\n−3(κe)\nκe+Q′′\n−3(κe)/bracketrightbigg/bracerightbigg\n/parenleftbigg\n3/braceleftbiggηi\nηeQ′\n−3(κe)\nκe/bracketleftbigg\n4Q′\n2(κi)\nκi+Q′′\n2(κi)/bracketrightbigg\n+Q′\n2(κi)\nκi/bracketleftbigg\n6Q′\n−3(κe)\nκe−Q′′\n−3(κe)/bracketrightbigg/bracerightbigg/parenrightbigg−1\n,(46)\nˆBi2,1=iωτ\n2ˆh2,1,ω/bracketleftbigg\n6ηi\nηeQ′\n−3(κe)\nκe\n+14Q′\n−3(κe)\nκe−5Q′′\n−3(κe)/bracketrightbigg\n/parenleftbigg\n3/braceleftbiggηi\nηeQ′\n−3(κe)\nκe/bracketleftbigg\n4Q′\n2(κi)\nκi+Q′′\n2(κi)/bracketrightbigg\n+Q′\n2(κi)\nκi/bracketleftbigg\n6Q′\n−3(κe)\nκe−Q′′\n−3(κe)/bracketrightbigg/bracerightbigg/parenrightbigg−1\n,(47)\nˆCi2,1=−iωτ\n2ˆh2,1,ω/braceleftbiggηi\nηeQ′\n−3(κe)\nκe/bracketleftbigg\n6Q2(κi)\nκ2\ni\n+4Q′\n2(κi)\nκi+Q′′\n2(κi)/bracketrightbigg\n+Q′\n−3(κe)\nκe/bracketleftbigg\n14Q2(κi)\nκ2\ni+6Q′\n2(κi)\nκi/bracketrightbigg\n−Q′′\n−3(κe)/bracketleftbigg\n5Q2(κi)\nκ2\ni+Q′\n2(κi)\nκi/bracketrightbigg/bracerightbigg\n/parenleftbigg\n3/braceleftbiggηi\nηeQ′\n−3(κe)\nκe/bracketleftbigg\n4Q′\n2(κi)\nκi+Q′′\n2(κi)/bracketrightbigg\n+Q′\n2(κi)\nκi/bracketleftbigg\n6Q′\n−3(κe)\nκe−Q′′\n−3(κe)/bracketrightbigg/bracerightbigg/parenrightbigg−1\n,(48)/vectorMe1,1=2\n5(µe−µi)13µe+11µi\n(µi+2µe)2a2/vectorHa·ˆh2,1,(49)\n/vectorMi1,1=4\n5(µe−µi)7µi+5µe\n(µi+2µe)2a2/vectorHa·ˆh2,1,(50)\nˆMe3,1=µi−µe\nµi+2µe5µi−8µe\n3µi+4µea4/bracketleftbigg/parenleftBig\n/vectorHaˆh2,1/parenrightBigS/bracketrightbiggD\n,(51)\nˆMi3,1=−4µi−µe\nµi+2µeµi−µe\n3µi+4µea4/bracketleftbigg/parenleftBig\n/vectorHaˆh2,1/parenrightBigS/bracketrightbiggD\n,(52)\nwhere\nτ=4ηea\nσs, (53)\nˆh2,1=ˆh2,1(t) =1\n2π∞/integraldisplay\n−∞e−iωtˆh2,1,ωdω,(54)\n/bracketleftbigg/parenleftBig\n/vectorHaˆh2,1/parenrightBigS/bracketrightbiggD\n=/parenleftBig\n/vectorHaˆh2,1/parenrightBigS\n−3\n5/parenleftBig\n/vectorHa·ˆh2,1ˆI/parenrightBigS\n,(55)\nandˆh2,1,ωis the solution of the following equation\n1\n2π∞/integraldisplay\n−∞e−iωt/bracketleftbigg\n1−iωτ\n48F(κe,κi)/bracketrightbigg\nˆh2,1,ωdω\n=(µi−µe)2a\n(2µe+µi)2H2am/parenleftbigg\n/vectorH2\na−1\n3ˆIH2\na/parenrightbigg\n,(56)\nF(κe,κi)\n=PNs(κe,κ2\ni)sinhκi+PNc(κe,κ2\ni)κicoshκi\nPDs(κe,κ2\ni)sinhκi+PDcn(κe,κ2\ni)κicoshκi,(57)\nPDs(κe,κ2\ni) = (κ4\ni+15κ2\ni+30)(κe+1)ηi\nηe\n−3(2κ2\ni+5)(κ2\ne+5κe+5),(58)\nPDc(κe,κ2\ni) =−5(κ2\ni+6)(κe+1)ηi\nηe\n+(κ2\ni+15)(κ2\ne+5κe+5),(59)\nPNs(κe,κ2\ni) = 3(κ6\ni+25κ4\ni+252κ2\ni+480)(κe+1)η2\ni\nη2e\n+(2κ3\neκ4\ni+9κ2\neκ4\ni+30κ3\neκ2\ni−45κeκ4\ni\n+540κ2\neκ2\ni−45κ4\ni+60κ3\ne+198κeκ2\ni\n+1260κ2\ne+198κ2\ni+720κe+720)ηi\nηe\n−6(2κ2\ni+5)(κ4\ne+5κ3\ne+45κ2\ne+72κe+72),(60)6\na1=a2a3/vectorHa(t) =Ham/vectorkcos(ωt)\n/vectorj/vectork\n/vectori\nFIG. 1. Drop in an applied uniform harmonically oscillating\nmagnetic field.\nPNc(κe,κ2\ni) =−3(5κ4\ni+92κ2\ni+480)(κe+1)η2\ni\nη2e\n−(−3κ2\neκ4\ni+10κ3\neκ2\ni−15κeκ4\ni+120κ2\neκ2\ni\n−15κ4\ni+60κ3\ne−42κeκ2\ni+1260κ2\ne\n−42κ2\ni+720κe+720)ηi\nηe\n+2(κ2\ni+15)(κ4\ne+5κ3\ne+45κ2\ne+72κe+72).(61)\nHere, i is the imaginary unit, sinh and cosh are the hy-\nperbolic sine and cosine.\nThe obtained relations for the terms of the asymptotic\nexpansions allow one also to find the expressions for the\ncorresponding unknown scalar, vector, and tensor func-\ntions for the free proper oscillations of the drop. They\nhave, in general, infinite number of modes determined by\nthe initial disturbances of the spherical shape and decay\ndue to the viscosity. These expressions are too cumber-\nsome and are not written down here.\nWith accuracy up to the terms of the first order, in an\noscillating applied magnetic field with the intensity\n/vectorHa=Hamcos(ωt)/vectork, (62)\nthe dropisaprolatespheroidwith theaxisdirectedalong\n/vectork(see Fig. 1) and with the semiaxes\na1=a2\n=a/braceleftBigg\n1−α\n6/parenleftbiggµi−µe\nµi+2µe/parenrightbigg2\n[1+|ζ|cos(2ωt−2φ)]/bracerightBigg\n,\n(63)\na3=a/braceleftBigg\n1+α\n3/parenleftbiggµi−µe\nµi+2µe/parenrightbigg2\n[1+|ζ|cos(2ωt−2φ)]/bracerightBigg\n,\n(64)\nwhere\nζ=/bracketleftbigg\n1−iωτ\n96F(κe,κi)/bracketrightbigg−1\n, (65)\nφ=1\n2arctan/parenleftbigg\n−Imζ\nReζ/parenrightbigg\n. (66)a3a2/vectorHa(t) =Ham/bracketleftBig\n/vectoricos(ωt)+/vectorjsin(ωt)/bracketrightBig\n/vectork/vectorj\n/vectoriφ\nFIG. 2. Drop in an applied uniform rotating magnetic field.\nHere, Re and Im denote the real and imaginary parts of\na complex quantity, arctan is the arc tangent. Thus, the\ndrop performs deformational oscillations with the angu-\nlar frequency 2 ωand the phase lag 2 φ.\nWith accuracy up to the terms of the first order, in a\nrotating applied magnetic field with the intensity\n/vectorHa=Ham/bracketleftBig\ncos(ωt)/vectori+sin(ωt)/vectorj/bracketrightBig\n,(67)\nthedroptakestheshapeofatri-axialellipsoid(seeFig.2)\nwith the semiaxes\na1=a/bracketleftBigg\n1−α\n3/parenleftbiggµi−µe\nµi+2µe/parenrightbigg2/bracketrightBigg\n, (68)\na2=a/bracketleftBigg\n1+α\n6/parenleftbiggµi−µe\nµi+2µe/parenrightbigg2\n(1−3|ζ|)/bracketrightBigg\n,(69)\na3=a/bracketleftBigg\n1+α\n6/parenleftbiggµi−µe\nµi+2µe/parenrightbigg2\n(1+3|ζ|)/bracketrightBigg\n.(70)\nTheellipsoidrotatesarounditsminoraxis,directedalong\n/vectork=/vectori×/vectorj, with the angular speed ωso that its major axis\nlags from/vectorHaby the angle φ. Here,/vectori,/vectorj, and/vectorkform a\nright triple of orthonormal vectors.\nThe dependencies of |ζ|andφuponωfor a drop of\nmagnetic liquid on the basis of magnetite in kerosene\nsuspended in water at various values of the radius of the\ndrop,a, are presented in Fig. 3.\nF. Limit cases.\nFor the limit case ρeσsa/(2η2\ne)→0,\n|ζ| →1/radicalBig\n1+4τ2\n2,1ω2, φ→1\n2arctan(2τ2,1ω)\nasρeσsa\n2η2e→0,(71)\nτ2,1=(16ηe+19ηi)(3ηe+2ηi)a\n40(ηe+ηi)σs.(72)7\nτω\n0,10,20,30,40,5 0|ζ|\n123\n1\n23\n4\nτω\n0,10,20,30,40,5 0φ\nπ\n4π\n23π\n4π1\n2 34\nFIG. 3. The dependencies of |ζ|andφupon the angular frequency ωforηi= 20 mPa s, ρi= 1.4 kg/m,ηe= 0.89 mPa s,\nρe= 1 kg/m,σs= 20 mN /m (1:a= 1 mm, 2: a= 100 µm, 3:a= 10µm, 4:a= 1µm).\nτω\n0.1 0.2 0.3 0.4 0.5 0|ζ|\n1231\n2\n3\n4\nFIG. 4. The dependencies of |ζ|upon the angular frequency ωforηi= 20 mPa s, ρi= 1.4 kg/m,ηe= 0.89 mPa s, ρe= 1 kg/m,\nσs= 20 mN /m (1:a= 1 mm, 2: a= 100 µm, 3:a= 10 µm, 4:a= 1µm) as compared with those for the limit cases: the\ndotted line when the inertia is neglected and the dashed line s when the viscosity is neglected.\nThis case corresponds to the case when the inertia of the\nliquids can be neglected, considered in [12–14].\nFor the limit case ρeσsa/(2η2\ne)→ ∞,\n|ζ| →ω2\n2,1/vextendsingle/vextendsingleω2\n2,1−4ω2/vextendsingle/vextendsingle, φ→/braceleftBigg\n0,2ω<ω2,1\nπ,2ω>ω2,1\nasρeσsa\n2η2e→ ∞,(73)\nω2,1=/radicalbigg24ρe\n2ρe+3ρiσs\nρea3. (74)\nThis case corresponds to the case when the viscosity of\nthe liquids can be neglected, considered in [15]. The\nlimit resonance circular frequency ω2,1coincides with the\nfrequency of the proper free deformational oscillations\n(for the mode with n= 2) found by Rayleigh (see [18]\nart. 275).\nIn Fig. 4, the dependencies of |ζ|upon the angular fre-\nquencyωare presented together with those for the limit\ncases. As can be seen from the figure, the inertia may be\nneglected forsmall oscillationsofa dropofa typicalmag-\nnetic liquid suspended in water if the radius of the drop\ndoes not exceed 10 microns. And, for sufficiently large-\nscale drops, the formulas in which the influence of the\nviscosity is neglected may be used only if the frequencyof the oscillations of the applied field is sufficiently far\nfrom the resonance frequency.\nIII. CONCLUSION\nThe behavior of a drop of a viscous magnetic liquid\nsuspended in another viscous magnetic liquid immisci-\nble with the former in an applied non-stationary uniform\nmagnetic field is theoretically investigated within the ap-\nproximation of low Reynolds numbers with taking into\naccount that the flow of the liquids is unsteady. The\ngeneral solution of the Navier–Stokes equation in the low\nReynolds number approximation is found for unsteady\nflows of an incompressible liquid. With the use of this\ngeneral solution in the first order approximation over the\nsmall parameter smallness of which provides the small-\nness of the deformationof the drop, the equations are ob-\ntained that describe the variation of the form of the drop\nin an arbitrarily varying uniform magnetic field. The\nsolutions of these equations are found for forced steady-\nstate oscillations of the drop in harmonically oscillating\nand rotating applied magnetic fields.\nAs the solution of the given problem shows, rather\ncumbersome expressions even for the corrections of the\nfirst order appear in taking into account the unsteady\nterm. So more careful study of the general solution with8\nthe goal to find more simple asymptotic expressions for\nthe case when the viscosity tends to zero acquires actual-\nity. Those asymptotic expressions should allow describ-\ning the process of the oscillation decay and finiteness of\nthe oscillation amplitude for the resonance.ACKNOWLEDGMENTS\nThe partial support by RFBR grant 19-01-00056is ac-\nknowledged.\n[1]G. I. Subkhankulov . Dynamics of small deformable\ndrops of magnetic liquid. Magnetohydrodynamics , vol. 20\n(1984), no. 4, pp. 370–375.\n[2]J.-C. Bacri, A. Cebers, and R. Perzynski . Beha-\nvor of Magnetic Fluid Microdrop in a Rotating Magnetic\nField.Phys. Rev. Lett. , vol. 72 (1994), pp. 2705–2708.\n[3]A. V. Lebedev and K. I. Morozov . Dynamics of a\ndrop of magnetic liquid in a rotating magnetic field.\nJournal of Experimental and Theoretical Physics Letters ,\nvol. 65 (1997), pp. 160–165.\n[4]K. I. Morozov . Rotation of a droplet in a viscous fluid.\nJournal of Experimental and Theoretical Physics , vol. 85\n(1997), pp. 728–733.\n[5]Y. K. Bratukhin, A.V. Lebedev, and A. F.\nPshenichnikov . Motion of a deformable droplet of mag-\nnetic fluid in a rotating magnetic field. Fluid Dynamics ,\nvol. 35 (2000), no. 1, pp. 17–23.\n[6]K. I. Morozov and A. V. Lebedev . Bifurcations of the\nshape of a magnetic fluid droplet in a rotating magnetic\nfield.Journal of Experimental and Theoretical Physics ,\nvol. 91 (2000), pp. 1029–1032.\n[7]Yu. K. Bratukhin and A. V. Lebedev . Forced oscilla-\ntions of a magnetic liquid drop. Journal of Experimental\nand Theoretical Physics , vol. 94 (2002), pp. 1114–1120.\n[8]A. V. Lebedev, A. Engel, K. I. Morozov, and H.\nBauke. Ferrofluid drops in rotating magnetic fields. New\nJ. Phys., vol. 5 (2003), pp. 57.1–57.20.\n[9]A. P. Stikuts, R. Perzynski and A. Cebers . Small\ndeformation theory for a magnetic droplet in a rotat-\ning field. Phys. Fluids , vol. 34 (2022), pp. 052010.1–052010.14.\n[10]A. Langins, A. P. Stikuts and A. Cebers . A three-\ndimensional boundaryelementmethodalgorithm forsim-\nulations of magnetic fluid droplet dynamics. Phys. Flu-\nids, vol. 34 (2022), pp. 062105.1–062105.18.\n[11]Yu. I. Dikansky and A. R. Zakinyan . Dynamics of\na nonmagnetic drop suspended in a magnetic fluid in a\nrotating magnetic field. Technical Physics , vol. 55(2010),\npp. 1082–1086.\n[12]A. N. Tyatyushkin . Deformation of a magnetic liquid\ndrop in anapplied non-stationary uniform magnetic field.\nEPJ Web of Conferences , vol. 185 (2018), pp. 09006-1–\n09006-4.\n[13]A. N. Tyatyushkin . Drop of viscous magnetic liquid in\na non-stationary magnetic field. Magnetohydrodynamics ,\nvol. 54 (2018), nos. 1–2, pp. 151–154.\n[14]A. N. Tyatyushkin . Magnetization and Deformation\nof a Drop of a Magnetic Fluid in an Alternating Mag-\nnetic Field. Bulletin of the Russian Academy of Sciences:\nPhysics, vol. 83 (2019), pp. 804–805.\n[15]A. N. Tyatyushkin . Deformation of an Inviscid Magne-\ntizable Liquid Drop in a Non-Stationary Magnetic Field.\nFluid Dynamics , vol. 56 (2021), no. 5, pp. 732–744.\n[16]R. E. Rosensweig .Ferrohydrodynamics (Cambridge\nUniversity Press, New York, 1985).\n[17]L. D. Landau and E. M. Lifshits .The classical theory\nof fields (Pergamon Press, 1975).\n[18]H. Lamb .Hydrodynamics (Cambridge University Press,\n1932)."    },    {        "title": "1804.04022v1.Continuous_Nucleation_Dynamics_of_Magnetic_Skyrmions_in_T_shaped_Helimagnetic_Nanojunction.pdf",        "content": "Continuous Nucleation Dynamics of Magnetic Skyrmions in\nT-shaped Helimagnetic Nanojunction\nYa-qing Zheng1and Yong Wang1,\u0003\n1School of Physics, Nankai University, Tianjin 300071, China.\nAbstract\nMagnetic skyrmions are topologically-protected spin textures existing in helimagentic materials,\nwhich can be utilized as information carriers for non-volatile memories and logic circuits in spin-\ntronics. Searching simple and controllable way to create isolated magnetic skyrmions is desirable\nfor further technology developments and industrial designs. Based on micromagnetic simulations,\nwe show that the temporal dissipative structure can be developed in the T-shaped helimagnetic\nnanojunction when it is driven to the far-from-equilibrium regime by a constant spin-polarized\ncurrent. Then the magnetic skyrmions can be continuously nucleated during the periodic magne-\ntization dynamics of the nanojunction. We have systematically investigated the e\u000bects of current\ndensity, Dzyaloshinskii-Moriya interaction, external magnetic \feld, and thermal \ructuation on the\nnucleation dynamics of the magnetic skyrmions. Our results here suggest a novel and promis-\ning mechanism to continuously create magnetic skyrmions for the development of skyrmion-based\nspintronics devices.\n1arXiv:1804.04022v1  [cond-mat.mes-hall]  11 Apr 2018I. INTRODUCTION\nMagnetic skyrmions are nano-sized noncollinear magnetization con\fgurations character-\nized by the topological skyrmion number,1{3which can exist in the noncentrosymmetric\nhelimagnet with \fnite Dzyaloshinskii-Moriya interaction (DMI).4,5Due to their topologi-\ncal stability and \rexible controllablity by spin-polarized current, magnetic skyrmions show\ngreat advantages as information carriers to develop spintronics devices,6such as memories,7\nlogic gates,8transistors,9etc. For the successful operations of these skyrmion-based devices,\nthe \frst step is to create isolated magnetic skyrmions in a simple and controllable way.\nPhysically, the generation of an isolated mgnetic skyrmion implies the topological transition\nof magnetization con\fgurations, which is accompanied by overcoming a potential barrier.\nUp to date, several methods to create isolated magnetic skyrmions from the uniform fer-\nromagnetic(FM) states have been proposed and realized, including pulsed magnetic \feld,10\nspin wave,11local heating,12, laser pulse,13electric \feld,14and spin-polarized current.15{22\nTo avoid a brute-force destruction of the FM state, the e\u000bect of geometry con\fnement on\nthe nucleation dynamics of magnetic skyrmions can be utilized.17Furthermore, it has been\nshown that the isolated magnetic skyrmion can be converted from either a domain-wall(DW)\npair23or a chiral stripe domain24by spin torque in a junction geometry.\nIn this paper, we report a novel mechanism to continuously generate magnetic skyrmions\nin the T-shaped helimagnetic nanojunction based on micromagnetic simulations. When the\nnanojunction is driven far from the equilibrium by applying a spin-polarized current, the\nperiodic magnetization dynamics can arise for certain simulation parameters, which is known\nas \\temporal dissipative structure\".25Since one magnetic skyrmion can be formed during\neach cycle of the dynamics, the magnetic skyrmions will be continuously generated by the\ntemporal dissipative structure realized here. Unlike the previous mechanisms to generate\nmagnetic skyrmions in the junction structure,23,24where a DW pair or a stripe domain\nshould be prepared in priori , the ingredients in our mechanism here only include the special\ngeometry shape of the nanojunction and the suitable spin-polarized current.\n2xy\nz\nx CNarrow partWide part\nJFIG. 1. Schematic diagram of the T-shaped helimagnetic nanojunction from top view. The junction\nconsists of a narrow part and a wide part, and an adjunctive layer (not shown explicitly) carrying\na spin-polarized current Jalong thex-direction.L(l) : the length of the wide (narrow) part; W(w)\n: the width of the wide (narrow) part. C: the point at the lower-right corner of the narrow part.\nII. STRUCTURE AND METHOD\nThe T-shaped helimagnetic nanojunction considered here is schematiclaly shown in Fig. 1,\nwhich consists of a narrow part and a wide part. The length and width of the wide(narrow)\nregion of the junction are denoted by L(l) andW(w) respectively, and the layer thickness\nis denoted by d. The average energy density of the helimagnet includes the Heisenberg ex-\nchange interaction (HEI), the DMI, the perpendicular magnetic anisotropy, and the Zeeman\nenergy, which is explicitly de\fned as6\nE[M] =A(rm)2+EDM\u0000Ku(ez\u0001m)2\u0000MsB\u0001m: (1)\nHere,Msis the saturation magnetization and m(r) =M(r)=Msis the normalized magneti-\nzation distribution, Ais the exchange sti\u000bness, Kuis the perpendicular magnetic anisotropy\ncoe\u000ecient, ezis the anisotropic axis, and Bis the external magnetic \feld. Depending on the\nmicroscopic origin, the form of EDMcan be either bulk DMI with EDM=Dm\u0001[r\u0002m] or\ninterfacial DMI with EDM=D[mzr\u0001m\u0000(m\u0001r)mz],6whereDis the strength of DMI. Due\nto the competition between DMI and HEI, the helimagnetic materials can host Bloch-type\nskyrmion for bulk DMI and N\u0013 eel-type skyrmion for interfacial DMI.6\n3TABLE I. Geometry size and material parameters for the T-shaped helimagnetic nanojunction in\nthe micromagnetic simulations. L(l): length of the wide (narrow) part of the junction; W(w):\nwidth of the wide (narrow) part of the junction; d: thickness of the junction; Ms: saturation\nmagnetization; A: exchange sti\u000bness; Ku: perpendicular magnetic anisotropy coe\u000ecient; \u000b:\nGibert damping co\u000ecient.\nL(nm)l(nm)W(nm)w(nm)d(nm)Ms(kA/m)A(pJ/m)Ku(MJ/m3)\u000b\n100 100 100 20 0.4 580 15 0.8 0.3\nIn our strategy, an adjunctive layer that can carry a spin-polarized current is placed on\nthe helimagnetic layer. When the spin-polarized current is injected along the x-direction, the\nspin torque e\u000bect will be exerted on the helimagnet to excite the magnetization dynamics,\nwhich is described by the Landau-Lifshitz-Gilbert(LLG) equation16\ndm\ndt=\u0000\rm\u0002Heff+\u000b(m\u0002dm\ndt) +u\nd(m\u0002mp\u0002m)\u0000\u0018u\ndm\u0002mp: (2)\nHere,\ris the gyromagnetic ratio of electron, Heff=\u0000@E[M]\n@Mis the e\u000bective magnetic \feld\ndetermined by the energy density functional E[M],\u000bis the Gilbert damping coe\u000ecient. The\nlast two terms in Eq. (2) describe the damping-like and \feld-like torque.16The spin-polarized\ncurrent is characterized by the current density J, the spin polarization degree P, and the\nspin polarization direction mp. The spin torque coe\u000ecient uis given by26u=j\r~JP\n2eMsj, where\n~denotes the reduced Planck constant and eis the elementary charge. The dimensionless\nparameter\u0018is adjusted to set the ratio between the \feld-like torque and the damping-like\ntorque.16\nThe restriction on spin polarization direction mpto develop the temporal dissipative\nstructure depends on the type of DMI. Here, we consider the continuous generation of\nN\u0013 eel-type magnetic skyrmions, which can be realized by the spin-polarized current with\nmp=\u0000ex. For Bloch-type magnetic skyrmions, a spin-polarized current with mp=eyis\nrequired, and the similar phenomena can also be observed (see supporting information for\ndetails). The micromagnetic simulations for the magnetization dynamics are performed with\nthe \fnite di\u000berence OOMMF code27. The material parameters for Co/Pt \flms have been\nexploited,16which is listed in Table I together with the geometry size of the nanojunction. To\n4take into account the e\u000bect of \feld-like torque, which acts as an e\u000bective magnetic \feld along\nthempdirection, we set the ratio \u0018= 0:75 during all the simulations. The imperfection of\nspin polarization is also considered by setting P= 0:4. More details about the calculations\ncan be found in the supporting information.\nIII. RESULTS AND DISCUSSION\nA. Nucleation Dynamics of Single Magnetic Skyrmion\n0ps 48ps\n150ps 246ps(a)mx\nmy\nmz(b)\n(c)\nFIG. 2. (a) Snapshots of the nucleation dynamics of a magnetic skyrmion driven by spin-polarized\ncurrent in the T-shaped helimagnetic nanojunction at di\u000berent time. (b) The oscillatory dynamics\nof the normalized magnetic moment mat corner point C. (c) The time-evolution of the topological\ncharge in the whole region of the junction.\nWe \frst demonstrate the nucleation dynamics of one magnetic skyrmion in the T-shaped\nnanojunction in Fig. 2(a). The DMI strength is set as D= 3:8 mJ/m2and the current\ndensity at the narrow region is set as J= 2:0\u00021012A/m2. The external magnetic \feld is\nnot applied at the moment. Before the spin-polarized current is applied, the magnetization\ndirection is uniformly aligned along the ez-direction due to the perpendicular magnetic\nanisotropy energy( t= 0 ps). When the current is applied, the magnetization at the narrow\nregion of the junction will \frst be driven out of equilibrium, most of which will align along\nthe\u0000exdirection due to the spin torque e\u000bect. This e\u000bect is however signi\fcantly weak\nin the wide region, where the current density is only J=5 due to the larger cross-sectional\n5area. Nevertheless, the magnetization near the lower-right corner of the narrow region will\nrotate from ezto\u0000ezdirection, which can be regarded as a seed for the magnetic skyrmion.\nBecause of the spin torque e\u000bect, the size of the \\skyrmion seed\" will gradually expand in\nthe wide region of the junction( t= 48 ps). Later on, the \\skyrmion seed\" will be detached\nfrom the narrow region when the magnetization near its lower-right corner rotates back to\nez(t= 150 ps). The isolated skyrmion seed then will continue to move in the wide region\nof the junction under the spin torque e\u000bect, during which its shape will gradually evolve\nand shrink down to a N\u0013 eel-type magnetic skyrmion( t= 246 ps). The motion of the formed\nmagnetic skyrmion will obey the Thiele equation6until it is annihilated at the boundary of\nthe junction. The same dynamical process will happen repeatly, which then results in the\ncontinuous nucleation of the magnetic skyrmions in the T-shaped nanojunction.\nFor a better understanding of the dynamical process described above, we investigate the\nmagnetization dynamics at the corner point Cof the narrow region. As shown in Fig. 2\n(b), the magnetic moment at point Cwill oscillate with a period T0= 112 ps, which is\naccompanied by the repeating nucleation dynamics of magnetic skyrmions. In fact, it is\nthe collective dynamics of all the oscillatory magnetic moments in the nanojunction that\ngives rise to the temporal dissipative structure. Furthermore, we numerically calculate the\ntime-evolution of the topological skyrmion number Q=\u00001\n4\u0019R\nm\u0001(@m\n@x\u0002@m\n@y)d2r,1,6which is\nintegrated over the whole area of the junction. For the simulated magnetization dynamics\nhere, the skyrmion number will \frst increase from 0 to 4, and then oscillate between the\nvalues 3 and 4 :7 ( Fig. 2(c)), which corresponds to the creation and annihilation of magnetic\nskyrmions. Due to the \fnite size of the junction, the maximal number of coexisting magnetic\nskyrmions in the temporal dissipative structure here is 4, and the extra 0 :7 comes from the\npartially-formed skyrmion seed.\nBesides, we found that the evolution dynamics of the skyrmion seeds can be complicated\nby the skyrmion-skyrmion interaction or skyrmion-edge interaction in the wide region of the\njunction. For example, the evolution from a skyrmion seed to a magnetic skyrmion requires\nenough space. When the skyrmion seed gets close to the bottom boundary of the junction,\nit may quickly evolve to a magnetic skyrmion due to the skyrmion-edge interaction. On\nthe other hand, the distance between the magnetic skyrmions and their sizes can vary with\ntime due to skyrmion-skyrmion interaction. If two magnetic skyrmions get close, both of\nthem will shrink down due to the repulsive interaction; then their distance will get longer\n6and their sizes will become large again. Besides, the motion of magnetic skyrmions will\nget slower near the edge, such that the following magnetic skyrmion will catch up with the\nformer one and compress it through the skyrmion-skyrmion interaction.\nB. Dzyaloshinskii-Moriya interaction and Current Density\nNow we investigate the e\u000bect of DMI strength Dand current density Jon the nucleation\ndynamics of the magnetic skyrmions in the junction. The same geometry size and the mate-\nrial parameters have been set as above. By sweeping the DMI strength D2[3:4;4:9] mJ/m2\nand the current density J2[17;26] A/m2, we \fnd that the magnetization dynamics in the\njunction can be classi\fed as the following cases (see supporting information for more details):\n(I) no skyrmion seed can be formed in the the junction for small current density;\n(II) all the skyrmion seeds will move along the left boundary and fail to form magnetic\nskyrmions;\n(III) some but not all the skyrmion seeds can evolve into the magnetic skyrmions;\n(IV) all the skyrmion seeds will evolve into the magnetic skyrmions continuously, which\nis the case we are interested in;\n(V) all skyrmion seeds will move and stick to the bottom boundary of the wide region,\nand no magnetic skyrmion can be formed.\nThe parameter space for case (IV) is displayed in the D-Jphase diagram in Fig. 3(a).\nFor each value of DMI strength D, there exist a lower critical current density Jc1which\ntriggers the temporal dissipative structure to continuously create magnetic skyrmions, and\nan upper critical current density Jc2beyond which the dynamics will vanish. Therefore, the\ncontinuous formation of magnetic skyrmions from the skyrmion seeds can only happen with\na moderate current density. From the D-Jphase diagram, we see that Jc1slightly depends\non the DMI D, since the energy barrier to form a magnetic skyrmion mainly depends on the\nHeisenberg exchange interaction. In contrast, Jc2becomes larger along with the increasement\nofD, since the formed magnetic skyrmion will be more stable with stronger DMI. In the\nconventional current-perpendicular-to-plane(CPP) con\fguration,17where the spin-polarized\ncurrent is vertically injected to the helimagnetic layer to create magnetic skyrmions, the spin\ntorques take the same forms as Eq. (2). However, we expect that higher current density\nis required to overcome the energy barrier to create an isolated magnetic skyrmion, since\n7(a)\n(c) (d)3.4 3.6 3.8 4.0 4.2 4.4-505101520253035\n \n  \nD(mJ/m2) J(u1012A/m2)\n 10nm\n 20nm\n 30nm\n T- junction\n1.8 1.9 2.0 2.1 2.2100105110115120125130 \n \n  T0(ps)\nJ(u1012A/m2) 3.6mJ/m2\n 3.8mJ/m2\n 4.0mJ/m2\n 4.2mJ/m2\n3.4 3.6 3.8 4.0 4.2 4.4100105110115120125130\n 1.85u\u0014\u0013\u0014\u0015A/m2\n 2.00u\u0014\u0013\u0014\u0015A/m2\n 2.15u\u0014\u0013\u0014\u0015A/m2 \n T0(ps)\nD(mJ/m2)3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.81.82.02.22.42.6\n J(u1012A/m2)\nD(mJ/m2)(b)FIG. 3. (a) The D-Jphase diagram. The yellow region gives the parameter space for which\nthe continuous nucleation dynamics of magnetic skyrmions can happen. (b) The critical current\ndensities to generate magnetic skyrmions in the current-perpendicular-to-plane (CPP) con\fguraton\nand the T-shaped nanojunction (T-junction). Three di\u000berent diameters 10 ;20;30 nm have been\nset for the contact region in CPP con\fguration. (c) The dependence of nucleation period T0on the\ninjected current density Jfor di\u000berent DMI strength D. (d) The dependence of nucleation period\nT0on the DMI strength Dfor di\u000berent injected current density J.\nthe geometry con\fnement is not utilized in this con\fguration. In Fig. 3(b), we show the\ncritical current density Jcto create a magnetic skyrmion in the wide region with the CPP\nmethod. We see that Jcwill gradually decrease when the DMI strength Dis increased for\n\fxed contact area. While for given D, the critical density Jcwill signi\fcantly decrease when\nthe diameter of the contact region is increased from 10 nm to 30 nm. For comparison, we\nalso show the critical current density Jc1to continuously create magnetic skrymions in the T-\nshaped junction, which is lower than the critical current densities in CPP con\fguration here.\nTherefore, the mechanism proposed here has the advantage to create magnetic skyrmions\nwith lower current density.\nBased on the D-Jphase diagram, we further investigate the e\u000bect of current density\n8Jand the DMI strength Don the period T0of the continuous nucleation dynamics in\ncase (IV), which is retrieved from the oscillation of magnetic moment at the corner point\nCas in Fig. 2(b). As shown in Fig. 3(c), when the current density Jis increased from\n18:5 A/m2to 22 A/m2, the range of nucleation period T0will become narrower for D=\n3:6;3:8;4:0;4:2 mJ/m2. Thus, the DMI will become less importance on the nucleation period\nwhen the current density Japproaches the upper critical value Jc2. This feature becomes\nmore transparent in the T0-Drelations for di\u000berent current density J, as shown in Fig. 3(d).\nHere, the slope of T0-Dcurve will decrease when the current density Jincreases. Another\nfeature in Fig. 3(c) is that the nucleation period T0is almostly independent on the current\ndensityJwhenD\u00193:8 mJ/m2, which results in a cross region near D\u00193:8 mJ/m2;T\u0019\n110 ps for the T0-Dcurves in Fig. 3(d). Therefore, when the current density is increased,\nthe nucleation period T0will decrease if D > 3:8 mJ/m2but increase if D < 3:8 mJ/m2,\nas given in Fig. 3(c). The results here imply that the nucleation dynamics of the magnetic\nskyrmions in the T-shaped nanojunction can be tuned by the injected current density.\nC. External Magnetic Field\nWe further investigate the response of the continuous nucleation dynamics of magnetic\nskyrmions in the presence of external magnetic \feld. When the amplitude of magnetic \feld\nis very large and the Zeeman term is dominant over the other contributions in Eq. (1),\nwe expect that the nucleation dynamics will be suppressed or even eliminated. For several\ntypical parameter sets fD;JgwithD= 3:7;3:8;3:9 mJ/m2and current density J= 1:9\u0002\n1012;2:0\u00021012;2:1\u00021012A/m2in theD-Jphase diagram, we sweep the magnetic \feld\nin the ex,ey, and ezdirection respectivley, and retrieve the period T0of the nucleation\ndynamics. We \fnd that the continuous nucleation dynamics can robustly exist in a rather\nwide range of external magnetic \feld, and the dependence of nucleation period T0on the\nmagnetic \feld Bis shown in Fig. 4. Firstly, when the external magnetic \feld B= (Bx;0;0)\nis applied, the continuous nucleation dynamics will exist if Bx2[\u0000200;50] mT for all the\nparameter sets. The nucleation period T0can increase exponentially by tens of picoseconds\nwhen the magnetic \feld reaches Bx=\u0000200 mT. Besides, the amplitude of increasement\nis larger for higher current density J, namely, the nucleation dynamics driven by higher\ncurrent density is more sensitive to the magnetic \feld along ex-direction. Secondly, when\n9105110115120125130 1.9u\u0014\u0013\u0014\u0015A/m2\n 2.0u\u0014\u0013\u0014\u0015A/m2\n 2.1u\u0014\u0013\u0014\u0015A/m2 T0(ps)\n  \n111114117120123126  1.9u\u0014\u0013\u0014\u0015A/m2\n 2.0u\u0014\u0013\u0014\u0015A/m2\n 2.1u\u0014\u0013\u0014\u0015A/m2T0(ps)\n  \n-200 -150 -100 -50 0 50111114117120123126\n 1.9u\u0014\u0013\u0014\u0015A/m2\n 2.0u\u0014\u0013\u0014\u0015A/m2\n 2.1u\u0014\u0013\u0014\u0015A/m2T0(ps)\nBx(mT)\n  939699102105108111  1.9u\u0014\u0013\u0014\u0015A/m2\n 2.0u\u0014\u0013\u0014\u0015A/m2\n 2.1u\u0014\u0013\u0014\u0015A/m2T0(ps)\n  \n99102105108111114\n 1.9u\u0014\u0013\u0014\u0015A/m2\n 2.0u\u0014\u0013\u0014\u0015A/m2\n 2.1u\u0014\u0013\u0014\u0015A/m2\n  \n T0(ps)\n0 100 200 300 400102105108111114117\n 1.9u\u0014\u0013\u0014\u0015A/m2\n 2.0u\u0014\u0013\u0014\u0015A/m2\n 2.1u\u0014\u0013\u0014\u0015A/m2T0(ps)\nBy(mT)\n  9095100105110115  1.9u\u0014\u0013\u0014\u0015A/m2\n 2.0u\u0014\u0013\u0014\u0015A/m2\n 2.1u\u0014\u0013\u0014\u0015A/m2 T0(ps)859095100105110  1.9u\u0014\u0013\u0014\u0015A/m2\n 2.0u\u0014\u0013\u0014\u0015A/m2\n 2.1u\u0014\u0013\u0014\u0015A/m2T0(ps)\n  \n0 20 40 60 809095100105110115  1.9u\u0014\u0013\u0014\u0015A/m2\n 2.0u\u0014\u0013\u0014\u0015A/m2\n 2.1u\u0014\u0013\u0014\u0015A/m2T0(ps)\nBz(mT)\n  (b) (c)\nD=3.7mJ/m2\nD=3.8mJ/m2\nD=3.9mJ/m2D=3.7mJ/m2\nD=3.8mJ/m2\nD=3.9mJ/m2\nD=3.9mJ/m2D=3.8mJ/m2D=3.7mJ/m2(a)FIG. 4. The response of nucleation period T0to the external magnetic \feld Bfor di\u000berent DMI\nstrengthDand injected current density J. The direction of Bis set as : (a) x; (b)y; (c)z\nrespectively.\nthe external magnetic \feld is applied as B= (0;By;0), its range to keep the continuous\nnucleation dynamics for all the parameter sets is By2[0;300] mT. The nucleation period\nT0shows a pattern of oscillatory decay when Byis increased, and the variation of T0is less\nthan 20 ps. In contrast to the magnetic \feld along exdirection, the nucleation dynamics\nis more sensitive to the magnetic \feld along eydirection when the current density is lower.\nLastly, the condition to keep the continuous nucleation dynamics for the magnetic \feld\nB= (0;0;Bz) isBz2[0;70] mT for all the parameter sets. Here, the nucleation period\nT0will decay linearly when the Bzis increased, and the variation of T0is less than 30 ns.\nBesides, the slope of T0\u0000Bzcurve shows less dependence on the current density. Therefore,\nthe continuous nucleation dynamics will response in di\u000berent way when the external magnetic\n\feld is applied in di\u000berent direction, which can also be used to tune the nucleation dynamics\nof magnetic skyrmions.\n10/s45/s50/s48 /s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48 /s49/s54/s48 /s49/s56/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48\n/s45/s50/s48 /s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s32/s112/s114/s111/s98/s97/s98/s105/s108/s105/s116/s121\n/s84/s40/s75/s41/s32/s32 /s32/s49/s46/s57 /s65 /s47/s109/s50\n/s32/s32/s32/s32/s32/s32 /s32/s32 /s32/s50/s46/s48 /s65 /s47/s109/s50\n/s32/s32/s32/s32/s32/s32 /s32/s32 /s32/s50/s46/s49 /s65 /s47/s109/s50\n/s32/s32/s78/s117/s109/s98/s101/s114\n/s84/s40/s75/s41FIG. 5. The dependence of number of skyrmion seed Nseed]and magnetic skyrmion Nskyron the\ntemperature Tfor three di\u000berent injected current density. Here, the DMI strength D= 3:8 mJ/m2.\nInset: the probability p=Nskyr=Nseedto successfully generate magnetic skyrmions at di\u000berent\ntemperature.\nD. Thermal Stability\nFinally, we investigate the thermal stability of the continuous nucleation dynamics of\nmagnetic skyrmions at \fnite temperatures. When the thermal \ructuation is present, some\nof the skyrmion seeds will not evolve into magnetic skyrmions, but will be destroyed and\nthen disappear at the bottom boundary of the wide region in the junction (see supporting\ninformation for more details). Fig. 5 shows the numbers of skyrmion seeds Nseedand success-\nfully formed magnetic skyrmions Nskrduring the nucleation dynamics for 10 ns at di\u000berent\ntemperature, where the DMI strength is \fxed as D= 3:8 mJ/m2and three di\u000berent current\ndensitiesJ= 19;20;21 A/m2have been chosen. We see that Nseeddoesn't obviously depend\non the temperature, but Nskrquickly decays when the temperature is higher. The depen-\ndence of ratio p=Nskr=Nseedon the temperature is further shown as inset in Fig. 5. At low\ntemperatures, most of the skyrmion seeds will evolve into magnetic skyrmions and pis close\n11to 1. For higher temperatures, the probability to form magntic skyrmions will decrease and\n\fnally become zero at a critical temperature. For \fxed temperature, the skyrmion seeds\nare more easily destroyed and pgets smaller if the current density is larger, which implies a\nsmaller critical temperature above which no magnetic skyrmion can be generated. Besides,\nthe critical temperatures are far below the Curie temperature, which is about 220 K as given\nin the supporting information.\nIV. CONCLUSION\nIn conclusion, we have revealed that the magnetic skyrmions can be continuously created\nby spin torque e\u000bect in the T-shaped helimagnetic nanojunction. Here, the magnetization\ndynamics of the nanojunction is driven to the far-from-equilibrium regime by spin-polarized\ncurrent, and the temporal dissipative structures can be developed. In certain parameter\nspace, one magnetic skyrmion can be generated during per cycle of the periodic magnetiza-\ntion dynamics. The nucleation period of the magnetic skyrmions can be further tuned by\ncurrent density and external magnetic \feld, and the nucleation dynamics can be destroyed\nby thermal \ructuations. The discovery here o\u000bers a simple and controllable mechanism to\ncontinuously create magnetic skyrmions, which can have potential applications to develop\nskyrmion-based spintronics devices at nanoscale. Furthermore, it is also valuable to under-\nstand how the topological defects can be continuously generated in the far-from-equilibrium\ndynamics in general.\nACKNOWLEDGEMENTS\nThis work is supported by NSFC Project No. 61674083 and No. 11604162.\n\u0003yongwang@nankai.edu.cn\n1N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899 (2013).\n2R. Wiesendanger, Nat. Rev. Mater. 1, 16044 (2016).\n3A. Fert, N. Reyren, and V. Cros, Nat. Rev. Mater. 2, 17031 (2017).\n4I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958).\n125T. Moriya, Phys. Rev. 120, 91 (1960).\n6W. Kang, Y. Huang, X. Zhang, Y. Zhou, and W. Zhao, Proc. IEEE 44, 2040 (2016).\n7N. S. Kiselev, A. N. Bogdanov, R. Sch afer, and U. K. R a \fler, J. Phys. D: Appl. Phys. 44,\n392001 (2011).\n8X. Zhang, M. Ezawa, and Y. Zhou, Sci. Rep. 5, 9400 (2015).\n9X. Zhang, Y. Zhou, M. Ezawa, G. P. Zhao, and W. Zhao, Sci. Rep. 5, 11369 (2015).\n10O. Dzyapko, I. Lisenkov, P. Nowik-Boltyk, V. E. Demidov, S. O. Demokritov, B. Koene, A. Kir-\nilyuk, T. Rasing, V. Tiberkevich, and A. Slavin, Phys. Rev. B 96, 064438(R) (2017).\n11Y. Liu, G. Yin, J. Zang, J. Shi, and R. K. Lake, Appl. Phys. Lett 107, 152411 (2015).\n12W. Koshibae and N. Nagaosa, Nat. Commun. 5, 5148 (2014).\n13M. Finazzi, M. Savoini, A. R. Khorsand, A. Tsukamoto, A. Itoh, L. Du\u0012 o, A. Kirilyuk, T. Rasing,\nand M. Ezawa, Phys. Rev. Lett. 110, 177205 (2013).\n14P.-J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von Bergmann, and R. Wiesendanger, Nat.\nNanotechnol. 12, 123 (2017).\n15N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka,\nand R. Wiesendanger, Science 341, 636 (2013).\n16J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat. Nanotechnol. 8, 839 (2013).\n17J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nanotechnol. 8, 742 (2013).\n18Y. Tchoe and J. H. Han, Phys. Rev. B 85, 174416 (2012).\n19S. Lin, C. Reichhardt, and A. Saxena, Appl. Phys. Lett. 102, 222405 (2013).\n20G. Yin, Y. Li, L. Kong, R. K. Lake, C. L. Chien, and J. Zang, Phys. Rev. B 93, 174403 (2016).\n21H. Y. Yuan and X. R. Wang, Sci. Rep. 6, 34898 (2016).\n22W. Legrand, D. Maccariello, N. Reyren, K. Garcia, C. Mouta\fs, C. Moreau-Luchaire, S. Collin,\nK. Bouze-houane, V. Cros, and A. Fert, Nano. Lett. 17, 2703 (2017).\n23Y. Zhou and M. Ezawa, Nat. Commun. 5, 4652 (2014).\n24W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jung\reisch, F. Y. Fradin, J. E. Pearson,\nY. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Ho\u000bmann, Science\n349, 283 (2015).\n25D. Kondepudi and I. Prigogine, Modern Thermodynamics:From Heat Engines to Dissipative\nStructures , 2nd ed. (Wiley, Chichester, United Kingdom, 2015).\n1326A. V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin, M. Krounbi, K. A. Zvezdin, A. Anane,\nJ. Grollier, and A. Fert, Phys. Rev. B 87, 020402 (R) (2013).\n27M. Donahue and D. Porter, OOMMF User's Guide, Version 1.0 Interagency Report NISTIR\n6376, (National Institute of Standards and Technology, Gaithersburg, MD, USA, 1999).\n14"    },    {        "title": "1103.1044v1.Dynamical_TAP_equations_for_non_equilibrium_Ising_spin_glasses.pdf",        "content": "arXiv:1103.1044v1  [cond-mat.dis-nn]  5 Mar 2011Dynamical TAP equations for non-equilibrium Ising\nspin glasses\nYasser Roudi\nNordita, 106 91 Stockholm, Sweden\nKavli Institute for Systems Neuroscience, NTNU, 7010 Trondheim , Norway\nE-mail:yasser@nordita.org\nJohn Hertz\nNordita, 106 91 Stockholm, Sweden\nThe Niels Bohr Institute, 2100 Copenhagen, Denmark\nE-mail:hertz@nordita.org\nAbstract. We derive and study dynamical TAP equations for Ising spin glasses\nobeying both synchronous and asynchronous dynamics using a gen erating functional\napproach. The system can have an asymmetric coupling matrix, and the external\nfields can be time-dependent. In the synchronouslyupdated mode l, the TAP equations\ntake the form of self consistent equations for magnetizations at t imet+1, given the\nmagnetizations at time t. In the asynchronously updated model, the TAP equations\ndetermine the time derivatives of the magnetizations at each time, a gain via self\nconsistent equations, given the current values of the magnetizat ions. Numerical\nsimulations suggest that the TAP equations become exact for large systems.\nPACS numbers: 64.60.De,75.10.NDynamical TAP equations for non-equilibrium Ising spin gla sses 2\n1. Introduction\nWithinthemeanfieldapproximation, spinmodelswithquencheddisorde rcanbestudied\nby analyzing their quenched averaged behavior or, alternatively, f or a specific realization\nofthequenched disorder [1]. Intheequilibrium case, theformer typ eofanalysisincludes\nthe replica method, while the latter one is usually formulated as naive m ean field, TAP\nequations [2], or, more generally, a Plefka expansion [3]. These equat ions can be derived\nbycalculating thefreeenergy inahightemperature(weak coupling) expansion, withthe\nfirst order calculation giving the mean field free energy, the second order the TAP free\nenergy and so on. For non-equilibrium and kinetic spin glass models, so ft spin systems\nwere the first ones to be analyzed, using the Martin-Siggie-Rose ge nerating functional\nformalism[4]. Spinglassmodelswithhardspinswerefirsttreatedin[5,6 ,7]. Apowerful\ngenerating functional approach was then developed by Coolen and collaborators [8, 9],\nand it forms the basis of our analysis here; see also [10, 11] for revie ws of the techniques\nused in both soft and hard spin models. However, dynamical TAP equ ations to describe\nthe kinetics of order parameters for a specific realization of the dis order have been only\nderived for the spherical p-spin model [12] and the stationary state of the Ising spin\nmodel with asynchronous update dynamics [13].\nIn the same way that studying the quenched averaged kinetics of h ard spin\nmodels usually involves a different approach compared to soft spin mo dels, deriving\nthe dynamical TAP equations for hard spin models is somewhat differe nt from doing so\nfor their soft spin counterparts. The aim of this paper is to develop a dynamical mean\nfield theory that relates the dynamics of mean magnetizations, pot entially time varying\nexternal fields, and the quenched couplings for two kinetic version s of the Sherrington-\nKirkpatrick model, one with synchronous update, the other with as ynchronous update.\nUsing a generating functional approach, we derive the dynamical n aive mean-field and\nTAP equations as first and second orders of a high temperature ex pansion, similar to\nthe equilibrium case for these two kinetic models.\nIn addition to the technical issues, the recent use of hard spin mod els with discrete\nstates, e.g. Potts and Ising models [14, 15, 16, 17], in reconstruct ing the connectivity\nof biological networks encourages the study of the dynamics of th ese models in more\ndetail. Once the forward dynamics is described, it is possible to use th e results to\nconstruct approximations at the corresponding levels for the inve rse problem: finding\nthe couplings, given the magnetizations and correlation functions. In this way, one\ncan develop effective approximate reconstruction techniques tha t exploit the temporal\nstructure of data and significantly improve the quality of network r econstruction in\nbiological systems. In fact, the results of this paper have been re cently used in two\nother recent papers on the inverse problem [18, 19].\nThe paper is organized as follows. After defining the dynamical mode ls in the\nfollowing section, we derive dynamical naive mean-field equations usin g the generating\nfunctional for the synchronous updated model. We report the TA P equations, for which\nthe details of the derivations are reported in the Appendices. We th en numericallyDynamical TAP equations for non-equilibrium Ising spin gla sses 3\ncalculate the errors for these kinetic equations as a function of th e strength of the\ncouplings for the synchronous dynamics.\n2. Dynamical Model\nWe consider a system of NIsing spins si=±1,i= 1,···,Nand assume that its state\nat timet,s(t) ={s1(t),...,sN(t)}, follows one of the following dynamics:\n(i)Synchronous dynamics. In this case time is discretized and the probability of\nbeing in state sat time step t,pt(s), is given by\npt(s) =/summationdisplay\ns′Wt[s;s′]pt−1(s′) (1a)\nWt[s;s′] =/productdisplay\niexp(siθi(t−1))\n2cosh(θi(t−1))(1b)\nθi(t) =hi(t)+/summationdisplay\njJijs′\nj(t), (1c)\nThis is, in other words, a Markov chain with transition probability Wt.\n(ii)Asynchronous dynamics. In this case time is continuous and, pt(s) satisfies the\nfollowing equation\nd\ndtpt(s) =/summationdisplay\ni[pt(Fis)wi(Fis;t)−pt(s)wi(s;t)] (2 a)\nwi(s;t) =1\n2[1−sitanh[θi(s;t)]], (2b)\nwhere the operator Fiacting on sflips itsith spin.\nFor each of these processes one can define a generating function al. For the synchronous\ncase it takes the form of\nZ[ψ,h] =/angbracketleftBigg\nexp/bracketleftBig/summationdisplay\ni,tψi(t)si(t)/bracketrightBig/angbracketrightBigg\n, (3)\nwhere for any quantity Adefined as a function of a path ( s(T),...,s(0)),/an}bracketle{t···/an}bracketri}htindicates\naveraging over the paths taken by s(t) under the stochastic dynamics of Eqs. (1 a) –\n(1c), i.e.\n/an}bracketle{tA/an}bracketri}ht= TrWT−1[s(T);s(T−1)]···W0[s(1);s(0)]p0(s(0))A(s(T),...,s(0)), (4)\nand\nTr≡/summationdisplay\ns(T)/summationdisplay\ns(T−1)···/summationdisplay\ns(0). (5)\nTheasynchronous caseissimilar expect thatthesumover tinEq. (3)shouldbereplaced\nby an integration; see Appendix B.\nItisuseful torewritethegeneratingfunctional byconsidering θi(t)foreachspinand\neach time step as a free parameter, integrate over it, and make su re that the definitionDynamical TAP equations for non-equilibrium Ising spin gla sses 4\nEq. (1c) is satisfied by inserting an appropriate delta function. This yields\nZ[ψ,h] =/integraldisplay\nDθ/angbracketleftBigg\nexp/bracketleftBig/summationdisplay\ni,tψi(t)si(t)/bracketrightBig/angbracketrightBigg/productdisplay\ni,tδ/parenleftBig\nθi(t)−hi(t)−/summationdisplay\njJijsj(t)/parenrightBig\n=/integraldisplay\nDθˆθ/angbracketleftBigg\nexp/bracketleftBig\ni/summationdisplay\ni,tˆθi(t){θi(t)−hi(t)−/summationdisplay\njJijsj(t)}+/summationdisplay\ni,tψi(t)si(t)/bracketrightBig/angbracketrightBigg\n(6)\nwhereDθ=/producttext\ni,tdθi(t) andDθˆθ=/producttext\ni,t1\n2πdθi(t)dˆθi(t). Using Eq. (4) in Eq. (6), we get\nZα[ψ,h] =/integraldisplay\nDθˆθTrexp[Lα] (7a)\nLα=/summationdisplay\ni,t{iˆθi(t)[θi(t)−α/summationdisplay\njJijsj(t)]+si(t+1)θj(t)−logcoshθi(t)\n−ihi(t)ˆθi(t)+ψi(t)si(t)}, (7b)\nwhere the parameter αis introduced to control the magnitude of the couplings, as will\nbecome clear later.\nThe generating functional has the property that its derivatives w ith respect to ψ\nandhgive the averages of the correlators involving the spins and auxiliary fields. In\nparticular, defining\n/an}bracketle{tA/an}bracketri}htα=/integraltextDθˆθTrAexp[Lα]\n/integraltextDθˆθTrexp[Lα](8)\nand using Eqs. (7 a) and (7b), we can define mi(t) and ˆmi(t) as\n−iˆmi(t)≡∂logZ\n∂hi(t)=−i/an}bracketle{tˆθi(t)/an}bracketri}htα (9a)\nmi(t)≡∂logZ\n∂ψi(t)=/an}bracketle{tsi(t)/an}bracketri}htα. (9b)\nFrom Eq. (6), we can see that the ψ→0 limit of ˆmi(t) is the expected value of the\nauxiliary field ˆθi(t) under the measure inside the integral of Eq. (6). It is easy to sho w\nthat this average, similar to the soft spin case, is zero. The same limit formi(t) gives\nus the mean magnetizations. We therefore have\n/an}bracketle{tˆθi(t)/an}bracketri}ht= lim\nψ→0ˆmi(t) = 0 (10)\n/an}bracketle{tsi(t)/an}bracketri}ht= lim\nψ→0mi(t). (11)\nForadetaileddiscussion abouttheseandotherdynamical process es onIsingspinmodels\nsee [11].\nTo derive the dynamical mean-field and TAP equations, one first calc ulates the\nLegendre transform of the logarithm of the generating functiona l of the process defined\nby Eqs. (1 a) – (1c). In this dynamical case, the logarithm of the generating function al\nplays the role of the Helmholtz free energy in the equilibrium statistica l mechanics while\nits Legendre transform corresponds to the Gibbs free energy. O ne then expands this\ndynamical Gibbsfreeenergyaroundthezerocouplingslimit, similarlyt otheequilibrium\ncase [3] and the soft spin model [12]. In the following, we do this for Is ing spins upDynamical TAP equations for non-equilibrium Ising spin gla sses 5\nto linear order in the couplings for the synchronous update and use it to derive the\ndynamical mean-field equations. The details of how to proceed to th e TAP for the\nsynchronous and asynchronous dynamics are provided in the Appe ndices.\n3. Outline of the derivation of the dynamical equations\nThe Legendre transform of the logarithm of the generating funct ional with respect to\nthe real fields, hi, and the auxiliary fields, ψireads\nΓ[ˆm,m]≡logZ[ψ[ˆm,m],h[ˆm,m]]−/summationdisplay\ni,tψi[ˆm,m](t)mi(t)+i/summationdisplay\ni,thi[ˆm,m](t)ˆmi(t),(12)\nwhereψandhare now treated as functions of ˆ mandmthrough the following equalities\n∂Γ\n∂mi(t)=−ψi[ˆm,m](t) (13a)\n∂Γ\n∂ˆmi(t)=ihi[ˆm,m](t) (13b)\nEqs. (13a) and (13 b) together with the definition of Γ αin Eq. (12) imply Eqs. (9 a)\nand (9b). Using Eq. (3) in Eq. (7 b), Γαcan also be written as\nΓα[ˆm,m] = log/integraldisplay\nDθˆθTreΩα(14a)\nΩα=/summationdisplay\ni,t/braceleftBig\niˆθi(t)[θi(t)−α/summationdisplay\njJijsj(t)]+si(t+1)θi(t)−logcosh(θi(t)) (14 b)\n−ihi(t)[ˆθi(t)−ˆmi(t)]+ψi(t)[si(t)−mi(s)]/bracerightBig\nThe idea now is that for α= 0 the generating functional and its Legendre transform\ncan be easily calculated, as the spins will be independent of each othe r. For the\ngenerating functional we have\nZ0[ψ,h] =/productdisplay\niT/productdisplay\nt=12cosh[hi(t−1)+ψi(t)]\n2cosh(hi(t−1)), (15)\nand for the Legendre transform of log Z0we have\nΓ0[ˆm,m] =/summationdisplay\ni,t[log(2cosh( h0\ni(t)+ψ0\ni(t+1))−log2cosh(h0\ni(t)) (16)\n−ψ0\ni(t)mi(t)+ih0\ni(t)ˆmi(t)],\nwhereh0andψ0aretherealandauxiliaryfields forwhich Eqs. (9 a)and(9b)aresatisfied\nfor givenmand ˆmat zero coupling ( α= 0), i.e.\nmi(t) = tanh[h0\ni(t−1)+ψ0\ni(t)] (17 a)\n−iˆmi(t) = tanh[h0\ni(t)+ψ0\ni(t+1)]−tanh[h0\ni(t)]. (17b)\nThis can be used to express h0andψ0in terms of mand ˆmas\nh0\ni(t) = tanh−1Mi(t) (18a)\nψ0\ni(t) = tanh−1mi(t)−tanh−1Mi(t−1), (18b)\nwhereMi(t) =mi(t+1)+iˆmi(t).Dynamical TAP equations for non-equilibrium Ising spin gla sses 6\nTo calculate the integral on the right hand side of Eq. (14 a) forα= 1, we can\nexpand Γ αaroundα= 0 and eventually set α= 1. Using the fact that\n/integraltextDθˆθTrAexp[Ωα]\n/integraltextDθˆθTrexp[Ω α]=/integraltextDθˆθTrAexp[Lα]\n/integraltextDθˆθTrexp[Lα]=/an}bracketle{tA/an}bracketri}htα, (19)\nfor the first derivative of Γ αwith respect to αwe have\n∂Γα\n∂α=/angbracketleftBigg∂Ω\n∂α/angbracketrightBigg\nα, (20)\nyielding\n∂Γα\n∂α=−i/summationdisplay\nij,tJij/an}bracketle{tˆθi(t)sj(t)/an}bracketri}htα (21)\n−i/summationdisplay\ni,t∂hα\ni(t)\n∂α/an}bracketle{t[ˆθi(t)−ˆmi(t)]/an}bracketri}htα+/summationdisplay\ni,t∂ψα\ni(t)\n∂α/an}bracketle{t[si(t)−mi(t)]/an}bracketri}htα.\nThe last two terms in Eq. (21) are zero because of Eqs. (9 a) and (9b); hence\n∂Γα\n∂α=−i/summationdisplay\nijtJij/an}bracketle{tˆθi(t)sj(t)/an}bracketri}htα. (22)\nThe correlation function /an}bracketle{tˆθi(t)sj(t)/an}bracketri}htαcan also be easily calculated at α= 0 yielding\n−i/an}bracketle{tˆθi(t)sj(t)/an}bracketri}ht0=1\nZ0∂2Z0\n∂hi(t)∂ψj(t)\n= (tanh[h0\ni(t)+ψ0\ni(t+1)]−tanh(h0\ni(t)])tanh[h0\nj(t−1)+ψ0\nj(t)],\n=−iˆmi(t)mj(t) (23)\nwhere the last equality follows from Eqs. (17 a)–(17b). Consequently, to first order in α,\nwe have\nΓα[ˆm,m] =/summationdisplay\ni,t[log2cosh(h0\ni(t−1)+ψ0\ni(t))−log2cosh(h0\ni(t))]\n−/summationdisplay\ni,tψ0\ni(t)mi(t)+/summationdisplay\ni,tih0\ni(t)ˆmi(t)−iα/summationdisplay\ni,j,tJijˆmi(t)mj(t) (24)\nwhich combined with Eqs. (18 a) – (18b) gives\nΓα[ˆm,m] =−1\n2/summationdisplay\ni,t/braceleftBig\nlog/bracketleftBig1+mi(t)\n2/bracketrightBig\n+/braceleftBig\nlog/bracketleftBig1−mi(t)\n2/bracketrightBig/bracerightBig\n(25)\n+1\n2/summationdisplay\ni,t/braceleftBig\nlog/bracketleftBig1+Mi(t)\n2/bracketrightBig\n+log/bracketleftBig1−Mi(t)\n2/bracketrightBig/bracerightBig\n−/summationdisplay\ni,tmi(t) tanh−1[mi(t)]+/summationdisplay\ni,tMi(t) tanh−1[Mi(t)]−iα/summationdisplay\ni,j,tJijˆmi(t)mj(t)\n+O(α2)\nUsing Eq. (13 b) yields\ntanh−1Mi(t) =hi(t−1)+/summationdisplay\njJijmj(t−1). (26)Dynamical TAP equations for non-equilibrium Ising spin gla sses 7\nIn the limit ψ→0 for which Eq. (11) is satisfied, we have\nmi(t+1) = tanh\nhi(t)+/summationdisplay\njJijmj(t)\n. (27)\nThis is the dynamical (naive) mean-field equation for the evolution of the mean\nmagnetization. The TAP equations can be derived in a similar way by exp anding Γ αto\nsecond order in α, as shown in Appendix A. This yields the dynamical TAP equations\nmi(t+1) = tanh\nhi(t)+/summationdisplay\njJijmj(t)−mi(t+1)/summationdisplay\njJ2\nij[1−mj(t)2]\n. (28)\nTo find the time evolving magnetizations for given external field and c oupling\nwithin the TAP approximation, the above equation should be solved se lf consistently\nformi(t+1) at each time step. Note the form of the Onsager correction (t he last term\nin Eq. (28)). The (1 −m2\nj) term is evaluated at time step t, butmiis evaluated at time\nstept+1. Thus (28) is a set of equations to be solved for mi(t+1), not just a simple\nexpression for mi(t+1) in terms of the mj(t), as in naive mean field theory.\nThe derivations of dynamical naive mean-field and TAP equations for the case of\nasynchronous dynamics defined in Eqs. (2 a) and (2b) are given in Appendix B. As\nshown there, these equations read\nmi(t)+dmi(t)\ndt= tanh\nhi(t)+/summationdisplay\njJijmj(t)\n (29)\nmi(t)+dmi(t)\ndt= tanh\nhi(t)+/summationdisplay\njJijmj(t)−/parenleftBig\nmi(t)+dmi(t)\ndt/parenrightBig/summationdisplay\njJ2\nij(1−m2\nj(t))\n(30)\n4. Numerical results\nTo test the dynamical naive mean field (hereafter: nMF) and TAP eq uations (27) and\n(28), we ransimulations inwhich we simulated the process define by (1 a)-(1c) forLtime\nsteps, for couplings drawn from a zero mean Gaussian distribution w ith variance g2/N\n(Jijis drawn independent of Jji) and subjected to two alternative types of external field.\nOne was a temporally constant field with a magnitude drawn independe ntly for each\nspinfromazero mean, unitvarianceGaussiandistribution. Theothe r wasasinusoidally\nvarying external field. For each sample of the Js and the fields, we generated data from\nthesystem for rrepeats, calculated mi(t)fromtheserepeats, anduseditin(27)and(28)\nto predictmi(t+1). Finally, we calculated the mean squared errors of these predic ted\nvalues\nMSEnMF/TAP=1\nLNN/summationdisplay\ni=1L/summationdisplay\nt=1[mnMF/TAP\ni(t+1)−mi(t)]2. (31)\nThe results for the two external fields used are shown below.Dynamical TAP equations for non-equilibrium Ising spin gla sses 8\n4.1. Uniform field\nFig. 1A shows the dependence of the error for predicting the magn etizations at time\nt+1, given the measured magnetizations at t. Both TAP and nMF errors increase as\ngincreases, but the error of nMF is always larger than that of TAP. F urthermore, how\nclose to the true ( r→ ∞) values the measured magnetizations are systematically affects\nthe nMF and TAP predictions: increasing rdecreases the errors for all g. This can also\nbe seen in Fig. 1B, where the errors at g= 0.3 are shown as functions of r, also for two\ndifferent values of N.\ng  \nnMF r=1E3\nTAP r=1E3\nnMF r=1E4\nTAP r=1E4\nnMF r=5E4\nTAP r=5e4\n10 010 -1 MSE \n10 -5 10 -4 10 -3 10 -2 10 -1 (A)\nMSE \n10 3\nr10 410 510 -6 10 -5 10 -4 10 -3 (B)\nnMF r=1E3\nTAP r=1E3\nnMF r=1E4\nTAP r=1E4\nnMF r=5E4\nTAP r=5e4MSE \ng10 010 -1 10 -5 10 -4 10 -3 10 -2 10 -1 (C)\n10 3\nr10 410 5MSE \n10 -6 10 -5 10 -4 10 -3 (D)\nFigure 1. (A) The effect of magnitude of the couplings, g, on the the error of TAP\nand nMF in predicting the magnetization at t+1 given the measured magnetizations\nattusingrruns. (B) The effect of the number of runs, r, on the error of TAP (red)\nand nMF (blue), for N= 10 (full curve) and N= 50 (dashed), and g= 0.3. All\nerrors are averages over 25 samples of the system. The errors b ars show the standard\ndeviation of these samples. (C) and (D) the same as (A) and (B) but for a sinusoidal\nfield.\n4.2. Sinusoidal field\nFigs. 1 C and D show the same thing as Fig. 1 A and B, but now the syste m is subjected\nto a sinusoidal external field with a peak amplitude of 0 .1 and a period of 20 time steps.Dynamical TAP equations for non-equilibrium Ising spin gla sses 9\nTheresultsarequalitatively thesame. Forthiscase, wealsolookatt hetimedependence\nof the errors in TAP and nMF equations.\nFig. 2 shows the time dependent error (i.e. the right hand side of Eq. (31) without\naveraging over time) versus time. For weak coupling, the error of b oth nMF and TAP\nare very small. At intermediate values of g, the error of nMF is still comparable to TAP.\nbut fluctuating. At yet stronger couplings, the nMF prediction ver y rapidly becomes\ndifferent from the actual measured values of the magnetizations.\n0123x10-5 MSE \n10 20 40 30 50 t(A) \n10 20 40 30 50 t0369x10-5 MSE (B) \n10 20 40 30 50 t10 -6 10 -4 10 -2 10 0MSE    (C)\n10 20 40 30 50 t-1 01 m\n10 20 40 30 50 t-1 01 m\n10 20 40 30 50 t-1 01 m\nFigure 2. (A) Timedependence ofthenMF (blue) andTAPerrors(red) toget herwith\nan example of the measured magnetization (lower panel) from 50000 repeats (black),\nnMF prediction (blue) and TAP (red) for one spin and for g= 0.1 andN= 40. (B)\nand (C) show the same thing but for g= 0.28 andg= 1.5 respectively. Note the\ndifference in the scale of the y axes.\n5. Discussion\nThe TAP approach, formulated as a high temperature Taylor series expansion of the\nequilibrium Gibbs free energy [3], is a powerful method for studying e quilibrium spin\nglass models. Similarly, dynamical TAP equations allow analyzing the dyn amics of a\nsingle sample of a disordered system away from equilibrium. In this pap er, we derived\nthese equations for Ising spin glasses with both synchronous and a synchronous updates.\nThe main idea behind the derivation is similar to the one used by Biroli [12] for the\nsoftp-spin model obeying a Langevin equation, with the difference that ins tead of a\nMSR formalism, we had to use the generating functional approach o f Coolen. For the p-\nspin model the spherical condition results in the appearance of the the autocorrelation,\n/an}bracketle{tsi(t)si(t′)/an}bracketri}ht, and response functions as order parameters in dynamical TAP. F or the hard\nspin Ising model, this is not the case. The response function can, of course, be directly\ncalculated from its definition and the TAP equations, but calculating c orrelations,Dynamical TAP equations for non-equilibrium Ising spin gla sses 10\n/an}bracketle{tsi(t)sj(t′)/an}bracketri}ht, function requires a different approach.\nThederivationdoesnotrelyonthesymmetryofthecouplingsandca n, therefore, be\napplied tosystems without detailed balance. Forthestationarycas e, theTAP equations\nare identical to those derived for the equilibrium model with symmetr ic connections.\nThis has been previously shown by Kappen and Spanjer [13] using an in formation\ngeometric derivation for the stationary state of the asynchrono usly updated model.\nNumerical simulations with both a constant external field and a rapid ly evolving one\nshow that the TAP equations predict the dynamics of the individual s ite magnetizations\nvery well. This may not be surprising given the fact that the model we studied here was\na kinetic variant of the SK model for which the equilibrium TAP equation s provide the\nexact picture.\nIt is intriguing that the Onsager term in Eqs. (28) and (30) does not get the form\nJijJji(1−m2\nj), as would be expected from a simple reaction argument. This obser vation\nhas also been made earlier by Kappen and Spanjer [13]. A naive argume nt showing that\nthe true correction to the mean-field equations is of the type J2\nij(1−m2\nj) is as follows.\nStarting from the exact equation\nmi(t+1) =/an}bracketle{ttanh[hi(t)+/summationdisplay\njJijsj(t)]/an}bracketri}ht, (32)\nwe expand tanh around bi(t) =hi(t) +/summationtext\njJijmj(t) to quadratic order in/summationtext\njJijδsj(t)\nwhereδsj(t) =si(t)−mi(t). The linear term vanishes, and using /an}bracketle{t[δsj(t)]2/an}bracketri}ht= 1−m2\nj(t)\nwe have\nmi(t+1) = tanh[ bi(t)]−(1−tanh2[bi(t)])tanh[bi(t)]/an}bracketle{t[/summationdisplay\njJijδsj(t)]2/an}bracketri}ht\n= tanh[bi(t)]−(1−tanh2[bi(t)])mi(t+1)/summationdisplay\njJ2\nij(1−m2\nj(t))\n= tanh[hi(t)+/summationdisplay\njJijmj(t)−mi(t+1)/summationdisplay\njJ2\nij(1−m2\nj(t))]\nwhere in the second line we have used the mean field equation mi(t+1)≈tanh(bi).\nAn important issue that we have left out in this paper is the expected number\nof solutions to the TAP equations for arbitrary couplings. It has be en known for a\nlong time that, at low temperature, the expected number of solutio ns of the TAP\nequations for the SK model with symmetric couplings is exponential in N[20]. It\nis also possible to calculate the number of metastable states for cou plings with an\nantisymmetric component at zero temperature [7]. The TAP equatio ns presented here\nallow extending the calculation in [20] to the type of couplings consider ed in [7] for\nnon-zero temperatures. This calculation will be presented elsewhe re.\nThe equilibrium TAP equations, derived for spin glass models with symme tric\ncouplings, can be used in deriving efficient approximations for solving t he inverse\nproblem of reconstructing a spin glass model from samples of its sta tes [21, 22]. As\nhas been recently shown [18, 19], the dynamical equations derived here can be employed\nfor taking the reconstruction to a more powerful level, allowing for the reconstruction\nof systems outside equilibrium.Dynamical TAP equations for non-equilibrium Ising spin gla sses 11\nAcknowledgment\nThe authors thank Erik Aurell and Bert Kappen for discussions at v arious stages of this\nwork. The use of computing resources at Gatsby Computational N euroscience Unit is\ngratefully acknowledged.\nAppendix A. TAP equations for synchronous update\nFor deriving the TAP equations, we note that\n∂2Γα\n∂α2=/angbracketleftBigg∂2Ω\n∂α2/angbracketrightBigg\nα+/angbracketleftBigg/bracketleftBigg∂Ω\n∂α/bracketrightBigg2/angbracketrightBigg\nα−/bracketleftBigg/angbracketleftBigg∂Ω\n∂α/angbracketrightBigg\nα/bracketrightBigg2\n(A.1)\nThe first term on the right hand side of the above equation is equal t o zero. To calculate\nthe next two terms, we use the Maxwell equations\ni∂hα\ni(t)\n∂α=∂\n∂ˆmi(t)∂Γα\n∂α=−i/summationdisplay\njJijmj(t) (1.2 a)\n∂ψα\ni(t)\n∂α=−∂\n∂mi(t)∂Γα\n∂α=i/summationdisplay\njˆmj(t)Jji (1.2b)\nto write\n∂Ω\n∂α=−i/summationdisplay\nijtJijˆθi(t)sj(t)+i/summationdisplay\nijtJij[ˆθi(t)−ˆmi(t)]mj(t)+i/summationdisplay\nijtJji[si(t)−mi(t)]ˆmj(t).(1.3)\nWe are therefore interested in calculating\n∂2Γα\n∂α2=/angbracketleftBigg/bracketleftBigg\nδ/parenleftBigg∂Ω\n∂α/parenrightBigg/bracketrightBigg2/angbracketrightBigg\nα=/angbracketleftBigg/parenleftBigg∂Ω\n∂α−/angbracketleftBigg∂Ω\n∂α/angbracketrightBigg\nα/parenrightBigg2/angbracketrightBigg\nα(1.4)\nwhere\nδ/parenleftBigg∂Ω\n∂α/parenrightBigg\n=−i/summationdisplay\nijtˆθi(t)Jijsj(t)+i/summationdisplay\nijtδˆθi(t)Jijmj(t)\n+i/summationdisplay\nijtˆmi(t)Jijδsj(t)+i/summationdisplay\nijtˆmi(t)Jijmj(t). (1.5)\nDefiningδsj(t) =sj(t)−mj(t) andδˆθi(t) =ˆθi(t)−ˆmi(t), this can be rearranged into\nthe following form:\nδ/parenleftBigg∂Ω\n∂α/parenrightBigg\n=−i/summationdisplay\nijtδˆθi(t)Jijδsj(t). (1.6)\nNow it is simple to evaluate Eq. (1.4)\n/angbracketleftBigg/bracketleftBigg\nδ/parenleftBigg∂Ω\n∂α/parenrightBigg/bracketrightBigg2/angbracketrightBigg\nα=−/summationdisplay\nijti′j′t′/an}bracketle{tδˆθi(t)Jijδsj(t)δˆθi′(t′)Ji′j′δsj′(t′)/an}bracketri}htα (1.7)\nThe factors have to be paired and for the pair averages we use\n/an}bracketle{t(−iδˆθi(t))2/an}bracketri}htα=∂logZ0\n∂hi(t)2=−ˆm2\ni(t)+2iˆmi(t)mi(t+1) (1.8 a)Dynamical TAP equations for non-equilibrium Ising spin gla sses 12\n/an}bracketle{t−iδˆθi(t−1)δsi(t))/an}bracketri}htα=∂logZ0\n∂hi(t−1)∂ψi(t)= 1−m2\ni(t) (1.8 b)\n/an}bracketle{t(δsi(t))2/an}bracketri}htα=∂logZ0\n∂ψi(t)2= 1−m2\ni(t). (1.8c)\nThe terms containing products of two averages of the form /an}bracketle{tδˆθδs/an}bracketri}htvanish, because one\npair factor has to have t′=t−1 and the other has to have t=t′−1, which cannot be\nsatisfied simultaneously. This leaves\n/angbracketleftBigg/bracketleftBigg\nδ/parenleftBigg∂Ω\n∂α/parenrightBigg/bracketrightBigg2/angbracketrightBigg\n0=−/summationdisplay\nijj′t/an}bracketle{t(δˆθi(t))2/an}bracketri}ht0JijJij′/an}bracketle{tδsj(t)δsj′(t)/an}bracketri}ht0 (1.9)\n=/summationdisplay\nijt[−ˆm2\ni(t)+2iˆmi(t)mi(t+1)]J2\nij[1−m2\nj(t)]\nUsing this to calculate Γ αto the quadratic order in α, differentiating with respect\nto ˆmj(t), and setting ˆ mj= 0 yields the dynamical TAP equations (28).\nAppendix B. Asynchronous Dynamics\nIn the asynchronous case the generating functional takes the f orm\nZα[ψ,h] =/integraldisplay\nDθˆθ/productdisplay\ni/angbracketleftBigg\nexp/bracketleftBig\ni/integraldisplay\ndtˆθi(t)[θi(t)−hi(t)−α/summationdisplay\njJijsj(t)]+/integraldisplay\ndtψi(t)si(t)/bracketrightBig/angbracketrightBigg\n(2.1)\nand/an}bracketle{t···/an}bracketri}htnowindicates averaging withrespect tothedistribution defined byt hesolution\nto the differential equation Eq. (2 a). This solution can be written as\npt(s) =/productdisplay\ni/bracketleftBigg1+µi(t)\n2δsi(t),1+1+µi(t)\n2δsi(t),−1/bracketrightBigg\n(2.2a)\ndµi\ndt=−µi+tanhθi(t), µi(0) =si(0). (2.2b)\nThe solution to Eq. (2.2 b) can be written as\nµi(t) =/integraldisplayt\n0dt′et′−ttanh(θi(t′))+e−tµi(0). (2.3)\nThe dynamical Gibbs free energy (i.e. the Legendre transform of t he log generating\nfunctional) is then\nΓα[ˆm,m] = log/integraldisplay\nDθˆθ/productdisplay\ni/angbracketleftBigg\nexp/bracketleftBig\ni/integraldisplay\ndtˆθi(t)[θi(t)−hi(t)−α/summationdisplay\njJijsj(t)]+/integraldisplay\ndtψi(t)si(t)/bracketrightBig/angbracketrightBigg\n+i/summationdisplay\ni/integraldisplay\ndt hi(t)ˆmi(t)−/summationdisplay\ni/integraldisplay\ndt ψi(t)mi(t) (2.4)\nAppendix B.1. nMF for asynchronous update\nAs we did for the synchronous case, we first calculate the non-inte racting (α= 0)\ngenerating functional\nlogZ0=/summationdisplay\ni/integraldisplay\ndtlog[cosh(ψi(t))+µ0\ni(t)sinh(ψi(t))] (2.5 a)\nµ0\ni(t) =/integraldisplayt\n0dt′et′−ttanh(h0\ni(t′))+e−tµi(0) (2.5 b)Dynamical TAP equations for non-equilibrium Ising spin gla sses 13\nand\nΓ0[ˆm,m] =/summationdisplay\ni/integraldisplay\ndt/braceleftBig\nlog[cosh(ψ0\ni(t))+µ0\ni(t)sinh(ψ0\ni(t))]−ψ0\ni(t)mi(t)+ih0\ni(t)ˆmi(t)/bracerightBig\n(2.6)\nwhere now ψ0andh0are functions of mand ˆmfrom the following equations\nδlogZ0\nδψ0\ni(t)=sinh[ψ0\ni(t)]+µ0\ni(t)cosh[ψi(t)]\ncosh[ψi(t)]+µ0\ni(t)sinh[ψi(t)]=mi(t) (2.7 a)\nδlogZ0\nδhi(t)=/integraldisplay\ndt′χ0\ni(t′,t)sinh[ψ0\ni(t′)]\ncosh[ψ0\ni(t′)]+µ0\nj(t′)sinh[ψ0\ni(t′)]=−iˆmi(t) (2.7b)\nand\nχ0\ni(t′,t) =δµ0\ni(t′)\nδhi(t)= Θ(t′−t)et−t′(1−tanh2[h0\ni(t)]). (2.8)\nFor nMF, we need to calculate the linear term in α. This is\n∂Γα\n∂α=−i/summationdisplay\nijJij/integraldisplay\ndt/an}bracketle{thi(t)sj(t)/an}bracketri}htα=−i/summationdisplay\nijJij/integraldisplay\ndtˆmi(t)mj(t) (2.9)\nwhere the last equality follows from\n/an}bracketle{tθi(t)sj(t)/an}bracketri}htα=i\nZ0δ2Z0\nδhi(t)δψjt)= ˆmi(t)mj(t), i/ne}ationslash=j. (2.10)\nConsequently, up to the linear term in α, we have\nΓα[ˆm,m] = Γ0[ˆm,m]−iα/summationdisplay\nijJij/integraldisplay\ndtˆmi(t)mj(t) (2.11)\nUsing the fact that ∂Γ0/∂ˆmi(t) =ih0\ni(t), we find that\nihi(t) =ih0\ni(t)−i/summationdisplay\njJijmj(t) (2.12)\nTogether with the fact that for ψ0= 0, we have mi(t) =µ0\ni(t), the mean-field equation\nis\ndmi(t)\ndt+mi(t) = tanh[hi(t)+/summationdisplay\njJijmj(t)] (2.13)\nAppendix B.2. TAP equations for asynchronous update\nTo derive the TAP equations, we need to calculate the second deriva tive of Γ with\nrespect toα. Similar to the synchronous update case, we have\n∂2Γα\n∂α2=−/summationdisplay\niji′j′JijJi′j′/integraldisplay\ndtdt′/an}bracketle{tδˆθi(t)δsj(t)δˆθi′(t′)δsj′(t′)/an}bracketri}htα (2.14)\nand the non-zero contributions come from pairing the terms inside t he averages.\nNon-zero contributions come from /an}bracketle{tδˆθi(t)2/an}bracketri}htα. A correlation function of the form\n/an}bracketle{tδˆθi(t)δsj′(t′)/an}bracketri}htαis nonzero for t′< tbut since it always appears multiplied byDynamical TAP equations for non-equilibrium Ising spin gla sses 14\n/an}bracketle{tδsj(t)δˆθi′(t′)/an}bracketri}htα, which is zero for t′< t, it does not contribute to the final results.\nWe therefore have\n∂2Γα\n∂α2/vextendsingle/vextendsingle/vextendsingle\nα=0= (2.15)\n−/summationdisplay\nijJ2\nij/integraldisplay\ndt/an}bracketle{t[δˆθi(t)]2/an}bracketri}ht0/an}bracketle{t[δsj(t)]2/an}bracketri}ht0\n−/summationdisplay\nijJijJji/integraldisplay\ndt/an}bracketle{tδˆθi(t)δsi(t)/an}bracketri}ht0/an}bracketle{tδˆθj(t)δsj(t)/an}bracketri}ht0 (2.16)\nTo evaluate the above expression we first note that\n/an}bracketle{t[δsj(t)]2/an}bracketri}ht0=δ2logZ0\nδψj(t)2= 1−m2\nj(t) (2.17a)\n/an}bracketle{t[δˆθi(t)]2/an}bracketri}ht0=−δ2logZ0\nδhi(t)2=−/integraldisplay\ndt′δχ0\ni(t′,t)\nδhi(t)γi(t′)+/integraldisplay\ndt′[χi(t′,t)γi(t′)]2(2.17b)\n/an}bracketle{tδˆθj(t)δsj(t)/an}bracketri}ht0=iδ2logZ0\nδhj(t)δψj(t)= 0. (2.17c)\nwhere the last equality follows from δµ0\nj(t)/δhj(t) = 0 and\nγ0\ni(t)≡sinh[ψ0\ni(t)]\ncosh[ψ0\ni(t)]+µ0\ni(t)sinh[ψ0\ni(t)](2.18a)\nδχ0\ni(t′,t)\nδhi(t)=δ2µ0\ni(t′)\nδh2\ni(t)=−2tanh[h0\ni(t)][1−tanh2[h0\ni(t)]]et′−tΘ(t′−t)\n=−2tanh[h0\ni(t)]χ0\ni(t′,t) (2.18b)\nUsing Eq. (2.18 b) in Eq. (2.17 b) gives\n/an}bracketle{t[δˆθi(t)]2/an}bracketri}ht0=−2itanh[h0\ni(t)]ˆmi(t)+/integraldisplay\ndt′[χ0\ni(t′,t)γ0\ni(t′)]2(2.19)\nThe dynamical Gibbs free energy can then be written as\nΓα[ˆm,m] = Γ0[ˆm,m]−iα/summationdisplay\nijJij/integraldisplay\nˆmi(t)mj(t)−1\n2α2/summationdisplay\nijJ2\nij/integraldisplay\ndt/an}bracketle{t[δˆθi(t)]2/an}bracketri}ht0(1−m2\nj(t))(2.20)\nwhere in the last sum /an}bracketle{t[δˆθi(t)]2/an}bracketri}ht0should be considered as a function of mand ˆm.\nAppendix B.2.1. Stationary case For the stationary case we have h0\ni(t) =hs0\niand we\nhave to take t→ ∞. This gives\nµ0\nj= tanh(hs0\nj) (2.21a)\nmi=µ0\ni+tanh[ψi]\n1+µ0\njtanh[ψi](2.21b)\n−iˆmi= (1−[µ0\ni]2)tanh[ψi]\n1+µ0\nitanh[ψi](2.21c)\n/integraldisplay\ndt′[χ0\ni(t′,t)γ0\ni(t′)]2= (−iˆmi)2(2.21d)\nUsing Eqs. (2.21 a), (2.21b), (2.21c) yields\nmi+iˆmi= tanh(hs0\ni) (2.22)Dynamical TAP equations for non-equilibrium Ising spin gla sses 15\nConsequently, for the stationary case, Eq. (2.19) can be written as\n/an}bracketle{t[δˆθi(t)]2/an}bracketri}ht0=−2imiˆmi+O(ˆm2). (2.23)\nand therefore\n∂2Γα\n∂α2/vextendsingle/vextendsingle/vextendsingle\nα=0= 2i/summationdisplay\nijJ2\nijˆmimi(1−m2\nj)+O(ˆm2) (2.24)\nUsing this the TAP equations in the stationary case would be\ntanh−1mi=hi+/summationdisplay\njJijmj−mi/summationdisplay\njJ2\nij(1−m2\nj) (2.25)\nwhich is identical to the result of [13]\nAppendix B.2.2. General case Under general conditions we cannot express handψ\nexplicitly in terms of mand ˆm. However, we can still calculate ∂Γ/∂ˆmi(t)iat ˆm= 0,\nwhich is what we need for deriving the TAP equations.\nFirst note that the second term on the right-hand side of Eq. (2.19 ) is of quadratic\norder inψin the limit ψ→0 (from Eq. (2.18 a)). But ˆmis linear inψ(from Eq. (2.7 b)),\nso this term is of second order in ˆ mand its derivative with respect to ˆ mvanishes as\nˆm→0. Thus we can discard it in finding the TAP equations.\nWe are now interested in the following quantity\nδ\nδˆmi(t)/integraldisplay\ndt′/an}bracketle{t[δˆθj(t′)]2/an}bracketri}ht0(1−m2\nk(t′)) =\n−2ixj(t)(1−m2\nk(t))δij−2i/integraldisplay\ndt′δxi(t′)\nδˆmi(t)ˆmj(t′)(1−m2\nk(t′)) (2.26 a)\nwherexi(t) = tanh(h0\ni(t)). For ˆmj(t)→0, The only term that will be nonzero on the\nright hand side of Eq. (2.26 a) is the first, as long δxi(t′)/δˆmi(t) does not diverge as fast\nas or faster than 1 /ˆmas ˆm→0. Whether δxi(t′)/δˆmi(t) is regular in the limit ˆ m→0\nor not depends on whether the functional matrix δ(m,ˆm)/δ(h,ψ) is regular in this limit.\nThe latter is not singular when the generating functional is regular u nless the system is\nat a phase transition. Assuming that this is not the case, we can igno re the last term\nin Eq. (2.26 a).\nNow we can proceed the way we did in the naive mean-field case, but ev aluating\nΓαto second order in α. The functional derivative of Γ αwith respect to ˆ m, evaluated\natα= 1, givesih:\nihi(t) =ih0\ni(t)−i/summationdisplay\njJijmj(t)+itanh[h0\ni(t)]/summationdisplay\njJ2\nij[1−m2\nj(t)],(2.27)\ntanhh0\ni(t) can be related to µ0\ni(t) through\ndµ0\ni\ndt=−µ0\ni+tanhh0\ni(t), (2.28)\nandµ0\ni→miwhenψand ˆm→0, yielding the TAP equations\ndmi(t)\ndt+mi(t) = (2.29)Dynamical TAP equations for non-equilibrium Ising spin gla sses 16\ntanh\nhi(t)+/summationdisplay\njJijmj(t)−/parenleftBiggdmi(t)\ndt+mi(t)/parenrightBigg/summationdisplay\njJ2\nij[1−m2\ni(t)]\n(2.30)\nNote that these are of the same form as those for the synchrono us-update model with\nmi(t+1) replaced by mi+dmi/dt.\nReferences\n[1] M. Mezard, G. Parisi, and M.A. Virasoro. Spin Glass Theory and Beyond . World Scientific,\nSingapore, 1987.\n[2] D. J. Thouless, P. W. Anderson, and R. G. Palmer. Solution of ’solv able model of a spin glass’.\nPhilosophical Magazine , 35:593–601, 1977.\n[3] T. Plefka. Convergencecondition of the tap equation for the infi nite-ranged ising spin glass model.\nJ. Phys. A: Math. Gen. , 15:1971–78, 1981.\n[4] H. Sompolinsky and A. Zippelius. Dynamic theory of the spin-glass p hase.Phys. Rev. Lett. ,\n47:359–62, 1981.\n[5] H. J. Sommers. Path-integralapproachto ising spin-glassdyna mics.Phys. Rev. Lett. , 58:1268–71,\n1987.\n[6] H. Rieger, M. Schreckenberg, and J. Zittartz. Glauber dynamic s of neural network models. J.\nPhys. A: Math. Gen. , 21:L263, 1988.\n[7] A. Crisanti and H. Sompolinsky. Dynamics of spin systems with ran domly asymmetric bonds:\nIsing spins and glauber dynamics. Phys. Rev. A , 37:4865–4874, 1988.\n[8] A. C. C. Coolen and D. Sherrington. Dynamics of fully connected a ttractor neural networks near\nsaturation. Phys. Rev. Lett. , 71:3886–89, 1993.\n[9] A.C.C. CoolenandD. Sherrington. Order-parameterflowin the skspinglass.i.replicasymmetry.\nJ. Phys. A: Math. Gen. , 27:7687–7707, 1994.\n[10] K.H. Fischer and John A. Hertz. Spin Glasses . Cambridge University Press, 1991.\n[11] A. C. C. Coolen. Statistical mechanics of recurrent neural ne tworks ii. dynamics. arXiv:cond-mat ,\n0006011, 2000.\n[12] G. Biroli. Dynamial tap approach to mean field glassy systems. J. Phys. A: Math. and Gen. ,\n32:8365–8388, 1999.\n[13] H. J. Kappen and J.J. Spanjers. Mean field theory for asymmet ric neural networks. Phys.l Rev.\nE, 61:5658–63, 2000.\n[14] T.R.Lezon,J.R.Banavar,M.Cieplak,A.Maritan,andN.Fedo roff. Usingtheprincipleofentropy\nmaximization to infer genetic interaction networks from gene expre ssion patterns. Proc. Natl.\nAcad. Sci. USA , 103, 2006.\n[15] Martin Weigt, Robert A. White, Hendrik Szurmant, James A. Hoc h, and Terence Hwa.\nIdentificationofdirectresiduecontactsinprotein-proteinintera ctionbymessagepassing. PNAS,\n106:67–72, 2009.\n[16] S. Cocco, S. Leibler, and R. Monasson. Neuronal couplings bet ween retinal ganglion cells inferred\nby efficient inverse statistical physics methods. Proc Natl Acad Sci U S A , 106:14058–62, 2009.\n[17] Y. Roudi, J. Tyrcha, and J. Hertz. Ising model for neural dat a: Model quality and approximate\nmethods for extracting functional connectivity. Phys. Rev. E , 79(5):051915, 2009.\n[18] Y. Roudi and J. Hertz. Mean field theory for nonequilibrium netw ork reconstruction. Phys. Rev.\nLett., 106:048702, 2011.\n[19] H.-L. Zeng, M. Alava, H. Mahmoudi, and E. Aurell. Network infere nce using asynchronously\nupdated kinetic ising model. arXiv:1011.6216v1 , 2010.\n[20] A. J. Bray and M. A. Moore. Metastable states in spin glasses. J. Phys. C: Solid St. Phys. ,\n13:L469–76, 1980.\n[21] H. J. Kappen and F. B. Rodriguez. Efficient learning in boltzmann m achines using linear response\ntheory.Neur. Comp. , 10:1137–1156, 1998.Dynamical TAP equations for non-equilibrium Ising spin gla sses 17\n[22] T. Tanaka. Mean-field theory of boltzmann machine learning. Phys. Rev. E , 58:2302–2310, 1998."    },    {        "title": "2103.07244v1.Dynamically_induced_magnetism_in_KTaO__3_.pdf",        "content": "Dynamically induced magnetism in KTaO 3\nR. Matthias Geilhufe1, Vladimir Juri\u0014 ci\u0013 c1;2, Stefano Bonetti3;4, Jian-Xin Zhu5, and Alexander V. Balatsky1;6\n1Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden\n2Departamento de F\u0013 \u0010sica, Universidad T\u0013 ecnica Federico Santa Mar\u0013 \u0010a, Casilla 110, Valpara\u0013 \u0010so, Chile\n3Department of Physics, Stockholm University, 10691 Stockholm, Sweden\n4Department of Molecular Sciences and Nanosystems,\nCa' Foscari University of Venice, 30172 Venice, Italy\n5Theoretical Division and Center for Integrated Nanotechnologies,\nLos Alamos National Laboratory, Los Alamos, New Mexico 87545,\nUSA6Department of Physics, University of Connecticut, Storrs, CT 06269, USA\n(Dated: March 15, 2021)\nDynamical multiferroicity features entangled dynamic orders: \ructuating electric dipoles induce\nmagnetization. Hence, the material with paraelectric \ructuations can develop magnetic signatures\nif dynamically driven. We identify the paraelectric KTaO 3(KTO) as a prime candidate for the\nobservation of the dynamical multiferroicity. We show that when a KTO sample is exposed to a\ncircularly polarized laser pulse, the dynamically induced ionic magnetic moments are of the order\nof 5% of the nuclear magneton per unit cell. We determine the phonon spectrum using ab initio\nmethods and identify T 1uas relevant soft phonon modes that couple to the external \feld and induce\nmagnetic polarization. We also predict a corresponding electron e\u000bect for the dynamically induced\nmagnetic moment which is enhanced by several orders of magnitude due to the signi\fcant mass\ndi\u000berence between electron and ionic nucleus.\nIntroduction. Dynamical multiferroicity [1], the phe-\nnomenon where the \ructuating electrical dipoles induce\nmagnetization, represents the dynamical counterpart of\nthe Dzyaloshinskii-Moriya mechanism [2]. The origin of\nthis e\u000bect lies in the duality between the electric and\nmagnetic properties [3]. Quite generally, the e\u000bect fea-\ntures entangled quantum orders. Most notably, dis-\nplacive paraelectrics (PE) exhibiting a ferroelectric (FE)\nphase transition [4{11] can display an elevated magnetic\nresponse induced by either quantum [12] or thermal \ruc-\ntuations [13] close to the critical point. On the other\nhand, the dynamical magnetization can be induced by ex-\nternally driving the material, e.g. by applying the light or\na lattice strain [1]. Dynamic multiferroicity is an example\nof the nonlinear phononics phenomenology [14], where a\ntwo phonon process induces magnetization. From the\nperspective of the materials where dynamical multifer-\nrroicity can be realized, the prime candidate to search\nfor the e\u000bect is SrTiO 3(STO), the paradigmatic quan-\ntum critical paraelectric where ferroelectricity is induced\nby displacive \ructuations. It has been recently predicted\nthat the dynamically induced magnetization both by ex-\nternal means and intrinsically, close to the FE QCP in\nthis material, may be in a measurable range [12, 13].\nIn contrast to STO, KTaO 3(KTO) is a quantum dis-\nordered paraelectric at low temperatures with a signi\f-\ncantly gapped transverse optical mode [5]. At zero stress,\nKTO retains its cubic structure down to helium temper-\natures [15]. The transition into a ferroelectric phase in\nKTO can be induced as well, e.g., by impurities [16, 17]\nor strain [18]. It is assumed that the ground state of\nKTO is a quantum disordered phase and signi\fcantly\naway from quantum critical \ructuations. Since KTO be-\nhaves as a regular quantum paraelectric quantum criticalmodes are gapped. Furthermore, on the paraelectric side\nof the quantum critical point, the \ructuations of the po-\nlarization are expected to be stronger and might give rise\nto a more dominant signal of a dynamically induced mag-\nnetic moment. So far, no prediction regarding the e\u000bect\nof a dynamically induced magnetization has been made\nfor KTO, and this is precisely the aim of the current\npaper.\nFollowing the formalism of dynamical multiferroicity\n[1, 12, 13], we investigate the induction of magnetic mo-\nments by applying circularly polarized terahertz radia-\ntion resonant with the phonon frequency that yield \ruc-\ntuating local electric dipoles, according to\nM=\u000bP\u0002@\n@tP=\ru\u0002m@\n@tu: (1)\nHere, Mdenotes the local magnetic moment, Pthe elec-\ntric polarization, uthe atomic displacement (associated\nwith the relevant phonon mode in our analysis), mthe\nparticle mass, while \u000band the gyromagnetic ratio \rare\nthe respective coupling constants. By performing an ab\ninitio analysis of the phonon spectrum (see Fig. 1), we\nsingle out T 1usoft phonon modes as relevant for the dy-\nnamical multiferroicity. As we show, using both single\nmode approximation and the full dynamical matrix ap-\nproach, when the system is subjected to a resonant cir-\ncularly polarized laser pulse (Fig. 2), one obtains a mea-\nsurable magnetic signal. Taking a realistic value of the\ndamping for the mode, we \fnd that the induced magnetic\nmoment per unit cell can reach the values of \u00180:05\u0016N,\nwhere\u0016Nis the nuclear magneton. We also predict an\nenhancement of the e\u000bect due to the coupling of the ion\ndynamics with the electronic one, which should be de-\ntectable experimentally.arXiv:2103.07244v1  [cond-mat.mtrl-sci]  12 Mar 20212\n(a)\nbc\naKTaO 3(b)\nR Γ M X0102030ω(THz)K\nTa\nO\nFIG. 1. (a) Unit cell of KTaO 3. (b) Calculated phonon spec-\ntrum and phonon density of states.\nPhonon spectrum: First-principles calculation. KTO\ncrystallizes in a cubic lattice with space group Pm 3m\n(Fig 1(a)). We chose the experimental lattice constants\nas determined by Zhurova et al. [19], with a unit cell\nvolume of 63.44 \u0017A3. The phonon spectrum was calcu-\nlated using Phonopy [20]. The related force matrix was\nobtained from a 2 \u00022\u00022 supercell with automatically gen-\nerated displacements, where forces were calculated using\nthe Vienna ab initio simulation package VASP [21]. The\nexchange correlation functional was approximated by the\nPBE functional [22]. We chose 8 \u00028\u00028 points for the\nBrillouin zone integration which corresponds to a k-mesh\ndensity of\u00191050k-points=\u0017A\u00003. We used a cut-o\u000b en-\nergy of 700 eV. Additionally, we calculated the Hessian\nmatrix for the energy landscape using density functional\nperturbation theory. This approach also provides a force\nmatrix and phonon frequencies at the \u0000 point, which we\nused to estimate the dynamically induced magnetization,\nas explained below.\nThe KTO unit cell contains 5 inequivalent sites, re-\nsulting in 15 phonon modes. We studied the symmetry\nof the phonon modes using GTPack [23, 24]. Construct-\ning a \fve-dimensional permutation representation \u0000 pfor\nthe point group O hand the 5 unit cell sites and com-\nputing the direct product with the vector representation\n\u0000v=T1uwe obtain \u0000 p\n\u0000v'4T1u\bT2ucorrespond-\ning to the expected modes at the \u0000-point in the Brillouin\nzone [24]. Using Phonopy we verify 4 T 1umodes at fre-\nquencies 0.0 THz, 3.02 THz, 6.16 THz, and 16.38 THz,\nas well as one T 2umode at 7.94 THz. The former modes\nbeing soft but \fnite-frequency modes are instrumental\nfor the dynamical multiferroicity, as shown below. The\nfull phonon spectrum showing 3 acoustical and 12 op-\ntical modes is plotted in Fig. 1(b). The values are in\ngood agreement with previous experiments on KTO [25].\nThese frequencies slightly change when calculated by us-\ning the density functional perturbation theory, giving 0.0\nTHz, 3.17 THz, 6.19 THz, 8.05 THz, and 16.53 THz. We\nnotice that, in contrast to STO, KTO does not give rise\nto negative energy modes in the phonon spectrum for\nthe cubic phase, indicating the absence of a structural\nphase transition at low temperatures. After identifyingthe T 1usoft phonon modes, we analyze the magnetic sig-\nnal resulting from the exposure of the KTO system to an\nexternally applied circularly polarized laser pulse.\nDynamical Multiferroicity. The polarization contains\nan ionic and an electronic contribution and can be writ-\nten as\nPi\u000b=Z\u0003\ni\u000b\fui\f+\u000f0(\u000f\u000b\f\u0000\u000e\u000b\f)E\f: (2)\nHereui\u000bdenotes a displacement of atom ialong the\nCartesian coordinate \u000b. The Born e\u000bective charge Z\u0003\ni\u000b\f\ndescribes the response of the macroscopic polarization\nper unit cell to the displacement of atom i,Z\u0003\ni\u000b\f=\n\n@P\f\n@ui\u000b\f\f\f\nE=0, with \n the unit cell volume [26]. The calcu-\nlated Born e\u000bective charges for KTO are given in Tab.\nII. The electronic response of the polarization to the elec-\ntric \feld is approximated in terms of the static dielectric\ntensor\u000fij. Due to the cubic symmetry of the unit cell\nthe dielectric tensor is diagonal and we obtain\n\u000fxx=\u000fyy=\u000fzz= 5:4: (3)\nThis value is sensitive to the chosen computational pa-\nrameters, but consistent with other references [27]. \u000f0\u0019\n5:52 e2keV\u00001\u0017A\u00001is the vacuum permittivity. We calcu-\nlate atomic displacements uiat the siteiusing classical\nequations of motion,\nui\u000b(t) +\u0011_ui\u000b(t) +X\nj\fKi\u000bj\fuj\f(t) =Zi\u000b\nmiE\u0003\n\u000b(t):(4)\nHere,Zi\u000b=Z0\ni+P\n\fZ\u0003\ni\u000b\fui\f, withZ0\nibeing the bare\ncharge of the ion (see Tab. I). mithe mass of atom i,\n\u0011is a damping factor, and K is the dynamical matrix.\nThe electric \feld within the medium E\u0003is related to the\nvacuum electric \feld Eby\nE\u0003=\u000f\u00001E: (5)\nIn experiments an additional loss in the \feld strength\nhas to be taken into account due the polarization pro-\ncess. In our approach, the electric \feld induces a col-\nlective displacement of the ionic positions by coupling to\nthe charge. Note that we do not include higher order\ncorrections to the dielectric screening [28].\nWe continue by discussing the size of the dynamically\ninduced magnetic moment using a simpli\fed analytical\nmodel. The full set of coupled di\u000berential equations\nis solved numerically afterwards. We start by solving\nK Ta O\ncharge [e \u00191;602\u000210\u000019C] 0.867 4.954 -1.940\nmass [u \u00191;66\u000210\u000027kg] 39.1 180,95 16.0\nTABLE I. Site parameters. Charge values according to DFT\ncalculations performed in this study.3\nZ\u0003\nxxZ\u0003\nyyZ\u0003\nzz\nK 1.13 1.13 1.13\nO -6.58 -1.69 -1.69\nO -1.69 -6.58 -1.69\nO -1.69 -1.69 -6.58\nTa 8.83 8.83 8.83\nTABLE II. Calculated Born e\u000bective charges in units of the\nelementary charge e.\nEq. (4) within a single-mode approximation, by consid-\nering one relevant mode !i= 2\u0019fi, corresponding to one\nrelevant site,\nu\u000b(t) +\u0011_u\u000b(t) +!2\niu\u000b(t) =q\nmE\u0003\n\u000b(t): (6)\nWe choose circularly polarized light, i.e., E\u0003(t) =\nE\u0003\n0(sin(!t);cos(!t);0). In a coarse approximation, from\n(6), we notice that the displacement scales linearly with\nthe applied \feld, u\u0019qE\u0003\nm!2. For a harmonic displace-\nment, we can estimate the corresponding time derivative\nas_u\u0019!u. Using equation (1) and replacing the gyro-\nmagnetic ratio by \r=q\n2m, we can estimate the asymp-\ntotic behavior for the dynamically induced magnetic mo-\nment by\nMz\u0018q3E\u00032\nm2!3: (7)\nHence, the e\u000bect increases quadratically in the \feld\nstrength, but decreases with !\u00003in the driving frequency.\nThe corresponding values for the charge qand the mass m\nfor KTO are given in Tab. I. The charges calculated using\nDFT are close to the chemistry picture of an ionic crystal,\nwith integer oxydation states O\u00002, K+1, and Ta+5.\nEquation (6) can be solved exactly. As we are solely\ninterested in the contribution to the atomic displace-\nment emerging due to exposure to an external laser \feld,\nwe only keep the inhomogeneous part of the solution of\nEq. (4) that can be written as\nu(t) =1\n\u00014!+ 4\u00112!2 \n\u00012\n!\u00002\u0011!\n2\u0011! \u00012\n!!\nq\nmE\u0003;(8)\nwith \u00012\n!=!2\ni\u0000!2. Evaluating the polarization as P=\nq\nVu, the!-dependent part of Eq. (8) can be interpreted\nas the susceptibility \u001f, by transforming it into the well-\nknown expression P=\u001f\u000f0E. Hence, we obtain for the\nmagnetization\nMz=q3!E\u00032\n2m2(\u00112!2+ \u00014!): (9)\nIn the limit !\u001d!i, we obtain \u00014\n!\u0019!4. Neglecting the\ndamping term \u00112!2\u001c!4gives a similar expression to\nEq. (7).\n0 5 10 150.0000.0200.0400.060\ntime (ps)Mz(µN)\n0 5 10 15η= 0.05 (THz)\nη= 0.10 (THz)\nη= 0.15 (THz)\ntime (ps)f= 3.17 THz f= 6.19 THz\n024-101\ntime (ps)Ex(MV/cm)FIG. 2. Dynamically induced total moment per unit cell for a\nlaser pulse with driving frequencies 3.17 THz (left panel) and\n6.19 THz (right panel).\nSystem driven with a terahertz pulse. Next, we con-\nsider a more realistic terahertz pulse and solve Eq. (4)\nnumerically. Such terahertz pulses are nowadays avail-\nable [29] and allow for large peak electric \feld to drive\nphonons, but with an average deposited energy which is\nnot enough to melt the sample. We set u\u000b(0) = _u\u000b(0) =\n0. The pulse is modeled by a Gaussian embedding as\nfollows,\nE(t) =E0e\u0000(t\u0000t0)2\n2\u001b0\nB@sin(!t)\ncos(!t)\n01\nCA: (10)\nThe considered driving frequencies are 3.17 THz and 6.19\nTHz, being resonant with the phonon modes. We choose\na total width of 2 ps with a peak at 2 ps and obtain the so-\nlution for a window up to 16 ps. The dynamically induced\nmagnetic moments are shown in Fig. 2 for various val-\nues of the damping parameter ( \u0011= 0.05 THz, 0.10 THz,\n0.15 THz). Depending on the damping factor we observe\na slow decay of the dynamically induced magnetic mo-\nment. The maximal total dynamically induced magnetic\nmoment is\u00190:7\u0016Nfor small damping of \u0011 <0:1 THz.\nThe dynamically induced magnetic moment decreases by\nabout one order of magnitude for a driving frequency in\nresonance with the T1umode at 6.19 THz. Due to the\nopposite local charges of the ions, the induced moments\nhave opposite strength for O, compared to Ta and K.\nThe site resolved dynamically induced moments due to\nlocal displacements are shown in Fig. 3. We observe that\nthe main contributions to the total induced magnetiza-\ntion per unit cell come from Ta and O, being of the order\nof 0:2\u0016Nand\u00000:1\u0016Nfor a small value of the damping\nparameter, \u0011= 0:05 THz.\nConclusion and Outlook. We showed that KTO is a\nprominent candidate for the observation of the dynami-\ncal multiferroicity. By performing an ab initio analysis,\nwe \frst \fnd that the T 1usoft phonon modes may be rel-\nevant for the observation of the e\u000bect. We suggest an\nexperimental setup where the KTO sample is exposed to4\n0 2 4 6 8 10−0.1000.0000.1000.200 K\nO\nTa\ntime (ps)Mz(µN)\nFIG. 3. Site resolved dynamically induced moments within\nthe unit cell. We used the same pulse as in Fig. 2, a driving\nfrequency of 3.17 THz and a damping of \u0011= 0:05 THz.\na circularly polarized laser \feld in the terahertz range\nto excite phonons resonantly. The dynamically induced\nmagnetization due to locally oscillating dipoles could be\nmeasured by the time-resolved Faraday e\u000bect using a\nfemtosecond laser pulse in the visible range. The esti-\nmated scale of the e\u000bect for an experimentally feasible\nsetup is in the order of 10\u00002\u0016Nper unit cell, with \u0016N\nbeing the nuclear magneton. In Eq. (7) we show that in\nan asymptotic limit, the induced moment scales quadrat-\nically with the electric \feld strength and to the the third\npower in the charge. It also scales inversely with the\nthird power in driving frequency and the mass squared.\nIn particular the latter feature could be of interest.\nHere we discussed the ionic movement as a driver for\nthe induced magnetism. We now point out an inter-\nesting possibility of induced electron motion that also\nwould produce the magnetic moment. We expect the an-\ngular momentum transfer from the moving ions to the\nelectronic charge cloud in the solid. While the exact\nmicroscopic details need to be worked out the qualita-\ntive argument goes as follows. To estimate the gyro-\nmagnetic ratio for the coupling we follow Refs. [13, 30]\nin a modi\fed form. The position of a charged ion is\ndenoted by u+, the average displacement of the elec-\ntron cloud is u\u0000. The respective masses are m+and\nm\u0000. We introduce average and relative coordinates\nU= (m+u\u0000+m\u0000u+)=(m++m\u0000) and u=u+\u0000u\u0000. We\nfocus on the relative coordinate, having the momentum\np=\u0016_uwith\u0016=m+m\u0000=(m++m\u0000). It follows for the\nangular momentum of the relative coordinate\nL=u\u0002p=m+m\u0000\nm++m\u0000u\u0002_u: (11)\nSettingm\u0000u+=m+u\u0000we obtain for the dynamically\ninduced moment according to Eq. (1)\nM=m++m\u0000=q\n2m+\u0000m\u0000\nm++m\u0000u\u0002_u: (12)\nTaking M=\rL, we obtain for the gyromagnetic ratio\n\r=q\n2\u00121\nm\u0000\u00001\nm+\u0013\n: (13)For nonequal charges, this equation generalizes to\n\r=m+\nm\u0000q+\nm++m\u0000\u0000m\u0000\nm+q\u0000\nm++m\u0000: (14)\nHence, from Eqs. (13) and (14) it becomes apparent that\nthe total gyromagnetic ratio of ion and electron is domi-\nnated by the electron mass ( mi=me\u0018103:::105). Here\nwe need to distinguish between a direct coupling of the\nelectron to the external \feld \u0018\u000f0(\u000f\u000b\f\u0000\u000e\u000b\f)E\fas well\nas an induced motion of the electrons due to the ionic\nmovement. While the former contribution to the total\nmagnetization should vanish with vanishing electric \feld,\nthe latter should be present as long as the ionic move-\nment persists. More precise analysis will be a topic of a\nseparate publication.\nWe propose KTO as a prominent candidate for the\nobservation of the dynamical multiferroicity. Our \fnd-\nings open up a route for the experimental detection of\nthe entangled dynamical orders. They should also mo-\ntivate further studies of the candidate materials for the\nrealization of the e\u000bect.\nAcknowledgment. We are grateful to G. Aeppli, U.\nAschauer, M. Basini, M. Pancaldi, O. Tjernberg, I.\nSochnikov, N. Spaldin and J. Weissenrieder for useful\ndiscussions. We acknowledge support from VILLUM\nFONDEN via the Centre of Excellence for Dirac Mate-\nrials (Grant No. 11744), the European Research Coun-\ncil under the European Union Seventh Framework ERS-\n2018-SYG 810451 HERO, the Knut and Alice Wallen-\nberg Foundation KAW 2018.0104. V.J. acknowledges\nthe support of the Swedish Research Council (VR 2019-\n04735) and J.-X.Z. was supported by the Los Alamos\nNational Laboratory LDRD Program. SB acknowledges\nsupport from the Swedish Research Council (VR 2018-\n04611). The computational resources were provided\nby the Swedish National Infrastructure for Computing\n(SNIC) via the High Performance Computing Centre\nNorth (HPC2N) and the Uppsala Multidisciplinary Cen-\ntre for Advanced Computational Science (UPPMAX).\n[1] D. M. Juraschek, M. Fechner, A. V. Balatsky, and N. A.\nSpaldin, Physical Review Materials 1, 014401 (2017).\n[2] H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev.\nLett. 95, 057205 (2005).\n[3] J. D. Jackson, Classical Electrodynamics (J. Wiley &\nsons, 1999).\n[4] D. E. Khmel'nitskii and V. L. Shneerson, Sov. Phys.\nJETP 37, 164 (1973).\n[5] S. E. Rowley, L. J. Spalek, R. P. Smith, M. P. M. Dean,\nM. Itoh, J. F. Scott, G. G. Lonzarich, and S. S. Saxena,\nNat. Phys. 10, 367 (2014).\n[6] P. Chandra, G. G. Lonzarich, S. E. Rowley, and J. F.\nScott, Rep. Prog. Phys. 80, 112502 (2017).\n[7] R. Roussev and A. J. Millis, Phys. Rev. B 67, 014105\n(2003).5\n[8] J. M. Edge, Y. Kedem, U. Aschauer, N. A. Spaldin, and\nA. V. Balatsky, Phys. Rev. Lett. 115, 247002 (2015).\n[9] C. W. Rischau, X. Lin, C. Grams, D. Finck, S. Harms,\nJ. Engelmayer, T. Lorenz, Y. Gallais, B. Fauque, J. Hem-\nberger, and B. Kamran, Nat. Phys. 13, 643 (2017).\n[10] A. Narayan, A. Cano, A. V. Balatsky, and N. A. Spaldin,\nNat. Mater. 18, 223 (2018).\n[11] J. Arce-Gamboa and G. Guzman-Verri, Phys. Rev.\nMater. 2, 104804 (2018).\n[12] K. Dunnett, J.-X. Zhu, N. A. Spaldin, V. Juri\u0014 ci\u0013 c, and\nA. V. Balatsky, Physical Review Letters 122, 057208\n(2019).\n[13] A. Khaetskii, V. Juri\u0014 ci\u0013 c, and A. V. Balatsky, Journal of\nPhysics: Condensed Matter 33, 04LT01 (2021).\n[14] D. M. Juraschek, M. Fechner, and N. A. Spaldin, Phys.\nRev. Lett. 118, 054101 (2017).\n[15] M. E. Lines and A. M. Glass,\nPrinciples and applications of ferroelectrics and related materials\n(Oxford university press, 2001).\n[16] I. S. Golovina, S. P. Kolesnik, V. P. Bryksa, V. V.\nStrelchuk, I. B. Yanchuk, I. N. Geifman, S. Khainakov,\nS. V. Svechnikov, and A. N. Morozovska, Physica B:\nCondensed Matter 407, 614 (2012).\n[17] R. L. Prater, L. L. Chase, and L. A. Boatner, Phys. Rev.\nB23, 221 (1981).\n[18] M. Tyunina, J. Narkilahti, M. Plekh, R. Oja, R. M.\nNieminen, A. Dejneka, and V. Trepakov, Phys. Rev.\nLett. 104, 227601 (2010).\n[19] E. A. Zhurova, Y. Ivanov, V. Zavodnik, and V. Tsirelson,Acta Crystallographica Section B: Structural Science 56,\n594 (2000).\n[20] A. Togo and I. Tanaka, Scripta Materialia 108, 1 (2015).\n[21] G. Kresse and J. Furthm uller, Physical Review B 54,\n11169 (1996).\n[22] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical\nreview letters 77, 3865 (1996).\n[23] R. M. Geilhufe and W. Hergert, Frontiers in Physics 6,\n86 (2018).\n[24] W. Hergert and R. M. Geilhufe,\nGroup Theory in Solid State Physics and Photonics: Problem Solving with Mathematica\n(Wiley-VCH, 2018) isbn: 978-3-527-41133-7.\n[25] E. Farhi, A. Tagantsev, R. Currat, B. Hehlen,\nE. Courtens, and L. Boatner, The European Physical\nJournal B-Condensed Matter and Complex Systems 15,\n615 (2000).\n[26] P. Ghosez, J.-P. Michenaud, and X. Gonze, Phys. Rev.\nB58, 6224 (1998).\n[27] K. Persson, \\Materials data on KTaO 3(SG:221) by Ma-\nterials Project,\" (2014), accessed, March 3rd 2021.\n[28] A. Cartella, T. F. Nova, M. Fechner, R. Merlin, and\nA. Cavalleri, Proceedings of the National Academy of\nSciences 115, 12148 (2018).\n[29] P. Sal\u0013 en, M. Basini, S. Bonetti, J. Hebling, M. Krasil-\nnikov, A. Y. Nikitin, G. Shamuilov, Z. Tibai, V. Zhauner-\nchyk, and V. Goryashko, Physics Reports 836-837 ,\n1 (2019), matter manipulation with extreme terahertz\nlight: Progress in the enabling THz technology.\n[30] Y. T. Rebane, Zh. Eksp. Teor. Fiz 84, 2323 (1983)."    },    {        "title": "2004.12243v1.Pulse_assisted_magnetization_switching_in_magnetic_nanowires_at_picosecond_and_nanosecond_timescales_with_low_energy.pdf",        "content": "Pulse -assisted magnetization switching in magnetic nanowires at picosecond and \nnanosecond timescales with low energy  \n \nFurkan Şahbaz, Mehmet C. Onbaşlı*  \nKoç University, Department of Electrical and Electronics Engineering, Sarıyer, 34450 Istanbul  \n*corresponding author: monbasli@ku.edu.tr   \n \nDetailed understanding of spin dynamics in magnetic nanomaterial s is necessary for develop ing \nultrafast , low -energy and high -density  spintronic logic and memory. Here, we develop  \nmicro magnetic models and analytical solutions to elucidate  the effect of increasing damping and \nuniaxial anisotropy on magnetic field pulse -assisted switching time, energy and field requirements \nof nanowires with perpendicular magnetic anisotropy and yttrium iron garnet -like spin transport \nproperties. A nanowire is initially magnetized using an  external magnetic field pulse (write) and \nself-relaxation. Next, magnetic moments exhibit deterministic switching upon receiving 2.5 ns -\nlong external magnet ic pulses in both vertical polarities . Favorable  damping (α~ 0.1-0.5) and  \nanisotropy energies  (104-105J·m-3) allow for as low as picosecond magnetization switching times. \nMagnetization reversal with fields below coercivity was observed using spin precession \ninstabilities. A competition or a nanomagnetic trilemma  arises among the switching rate, energy \ncost and external field required. Developing magnetic nanowires  with optimized damping and \neffective anisotropy could  reduce the switching energy barrier down to 3163 ×kBT at room \ntemperature . Thus, pulse -assisted picosecond and low energy switching in nanomagnets could \nenable ultrafast nanomagnetic logic and cellular automata . \n I. Introduction  \nAn in-depth understanding of spin relaxation in magnetic nanostructures is necessary to \ndevelop highly efficient and ultrafast switching methods . The interplay between external magnetic \nfield and  magnetic material properties remains to be understood at sub -100 nanosecond and \nnanometer length scales for fundamental studies of spin -spin, spin -electric field and spin -magnetic \nfield interactions for developing future spintronic devices . The effects of external field amplitude \n[1], frequency [ 2] and polarization [3] on the spin relaxation of nano magnetic media have been \ninvestigated. Previous studies indicate that switching field decreases when the polarization \ndirection and frequency of the circularly polarized microwave field matches that of the \nferromagnetic re sonance  of the nanomagnet [3 ]. Applying a microwave magnetic field with \noptimal frequency at or near ferromagnetic resonance  reduces the coercive field by helping \novercome the effective energy barrier of the domain nucl eation [1 ]. The coercivity reduction is \nlarger than the microwave field magnitude H rf at a cert ain frequency and input power [2 ].  \nWhile microwave, laser or heat -assisted switching effects that determine switching energy \nhave been investigated  [4,5], the intrinsic magnetic material property dependence of spin \nrelaxation has not been studied extensively. Previous spin relaxation studies include nanodot \nmodels [ 6], permalloy rectangle models [ 7], nan owire models [ 8], and ferromagnetic nano particles \n[9]. The key magnetic properties that affect spin relaxation include Gilbert damping constant, \nsaturation magnetic moment,  exchange stiffness, anisotropy, dimensions and aspect ratios . In this \nstudy, we use analytical and numerical micromagnetic models to qu antify the regimes  under which \nincreasing damping, uniaxial anisotropy and external pulse field can switch magnetism in  sub-100 \nnm nanowires . The results of these analyses prompted us to  propose a n external  magnetic field \npulse -assisted magnetization rever sal mechanism that could enable sub -coercivity and sub -nanosecond nonvolatile switching with low energy (a few thousand k BT per bit at room \ntemperature) . \nPrevious studies show  that damping plays a key role in magnetization dynamics of \nnanostructures [ 10,11]. Ref. [ 10] inspected the effect of damping on reversal time without \nanisotropy and showed that magnetization reversal time can increase (decrease) with increasing \n(decreasing) damping constant. Since realistic materials have nonzero intrinsic magnetic  \nanisotropy, magnetization reversal models must include anisotropy . A generalized analysis of \nmagnetization reversal  [12] highlights the significa nce of demagnetization factors and anisotropy \nparameters . In nanostructures, magnetoelastic [1 3], magnetocrystalline [ 14], off -stoichiometry \n[15], growth -induced anisotropy [ 16, 17] and surface -induced anisotropy (especially for large \nsurface area -to-volume ratio nanostructures ) can be modeled with an overall uniaxial anisotropy \nterm, which alters switching times significantly [18,19 ]. One could engineer these terms to achieve \nperpendicular  magnetic anisotropy (PMA) preferred in high-density memory  [20]. Large \nperpendicular anisotropy increases the effective field and causes precession -driven magnetization \ndynamics  with high precession frequencies [ 21].  \nWe present  the results of our analytical spin relaxation model  and numerical methods in \nSection II.  In section III , we present Gilbert damping constant and uniaxial anisotropy dependence \nof spin relaxation and magnetization reversal time. In section IV, magnetization switching time \nand energy are modeled function s of external magnetic field pulse intensi ty and width.  \nII. Numerical Modeling and Analytical Solutions of Spin Relaxation  \n1. Numerical model details  \nNumerical models were developed  to understand the magnetic relaxation and reversal in \nnanowires . We used Object -oriented Micromagnetic Framework (OOMMF) to obtain magnetic nanowire hysteresis loops and  spin relaxation dynamics  as function of damping  (α) and uniaxial \nanisotropy constant (K u). A rectangular 20×100×10 nm3 Y3Fe5O12 (YIG) nanowire (width, length, \nthickness)  was used for all models  in this study . The time evolution of m x, my and m z vectors were \ncalculated with minimum temporal step sizes of 2.14 fs. These nanowire dimensions were chosen \nto elucidate the effect of damping (α= 10-4–10) and uniaxial anisotropy (K u=103–106J·m-3) in the \nnear single dom ain regime . These dimensions are experimentally feasible with state -of-the-art \nfabrication techniques [22-26]. We chose YIG (exchange stiffness A ex=3.65± 0.38pJ·m-1, \nsaturation magnetization M s=140 kA·m-1) due to its very low and tunable damping  [27-29] and \ndue to its lower exchange stiffness compared with permalloy (13pJ·m-1) [7], cobalt -platinum \nmultilayers as well as Heusler alloys (15pJ·m-1) [30, 31]. We focus on m agnetic insulators (MI)  \nlike YIG  over metals  due to their reduced Joule dissipation , lower damping  and lower exchange \nstiffness. L ower exchange stiffness and exchange energy in MI  allow write energy per bit could \nbe lower for MI than for metals. The magnetic field pulses applied on nanowire  were chosen to be \n2 ns wide, as Si CMOS can operate at similar periods for read/write memory pulses.  \nThe switching models were prepared in three steps:   \n1) Self-relaxation  (0-15ns) . First, nanowires with different uniaxial anisotropy constants but \nidentical geometries were allowed to equilibrat e into minimum energy states  in absence of external \nmagnetic field or initial magnetization. The magnetization profiles after self-relaxation  for \nnanowires  with increasing uniaxial anisotropy constants were calculated and are shown in Fig. 1 . \nWhen uniaxial anisotropy constant is low (103 J·m-3), shape anisotropy renders the nanowire  an \nin-plane easy axis material. When uniaxial anisotropy is large enough to overcome shape \nanisotropy, the nanowire  becomes PMA. For lower field and lower energy switching, \nperpendicular ma gnetic anisotropy (PMA) in the nanowires is desired.  2) Initialization  (15-45ns) . In the second  step, we applied an external magnetic field pulse of 2.2 \nTesla for initialization of magnetic moment s along + z axis . \n3) Deterministic switching  (45-100 ns) . In this third and final step, 2 ns -wide and apart field pulses \nwere applied to investigate the effect of anisotropy and damping on switching time and energy  of \nthe nanowire .  \n \n2. Analytical model results  \nThe Landau -Lifshitz -Gilbert equation (Supplementary Materials Part 1) captures the time \nevolution in nanomagnets. Its analytical solutions yield three general cases based on the main  \nparameters, which determine switching likelihood and time constants: prec ession -driven, \ndamping -driven and effective field -driven regimes. In the precession -driven regime (α ≪ 1), \nmagnetization reversal cannot settle  since the nanomagnet  undergoes precession indefinitely:  \n𝐌(𝐫,t)=Ms𝑒−𝜅𝑡(−𝐲̂sin(γ̅Hefft)+𝐱̂cos(γ̅Hefft)) (10) \n \nIn the damping -driven regime ( α ≫ 1), the spins dissipate the injected pulse energy before \ntriggering any magnetization reversal :  \n∂𝐦\n∂t≈(−|γ̅|α𝐦×(𝐦×𝐇𝐞𝐟𝐟)) (14) \n𝐌(𝐫,t)=𝑒−|γ̅|αΔ∗𝑡(𝐱̂Mx0+𝐲̂My0)+𝐳̂Mz0 (15) \nThe effective field -driven case contains multiple in and out -of-plane anisotropy field terms that \nassist magnetization reversal. Here, the reversal time constant is determined by the external field, \ndemagnetizing field and damping constant . Overall, an o ptimal window of damping and uniaxial \nanisotropy constant s were found to enable deterministic magnetization reversal  in picoseconds . \n \n III. Uniaxial anisotropy and Gilbert Damping dependence of magnetization reversal  \nIn this section, we investigate the effect of Ku on the self-relaxation and pulse -assisted \nswitching. Fig. 1(a) -(d) show the time evolution for magnetic moments of the nanowires with K u \n= 103, 104, 105 and 106 J·m-3, respectively, during self -relaxation (no external field applied: 0 -15 \nns) and during applied external magnetic field pulse (15 -45 ns) and after the pulse is applied (45 -\n100 ns). The nanowires  were first set  to an infinitesimally small magnetic moment and  they were \nallowed to relax their magnetic moments in absence of external magnetic field until 15 ns. This \ninitialization numerical ly demonstrates  the easy axis  for each case before applying the magnetic \nfields. The magnetic moment of the nanowire  in Fig. 1(a) with K u = 103 J·m-3 self-relaxes towards \n+y direction, which indicates that its magnetic easy axis is along the long axis (y) of the structure \nand that K u < K shape. For Fig. 1(b), the structure relaxes to –z direction, indicating that uniaxial \nanisotr opy now overcomes shape and renders the nanowire  PMA. In Fig. 1(c), although Ku = 105 \nJ·m-3 > K shape, the nanowire cannot relax to a vertical direction since the spins form  a transient \ndomain wall (Spin profiles in Supplementary Figure S1). When an additional external pulse was \napplied, the multi -domain structure overcomes the domain wall energy barrier , aligns and \nstabilizes along +z direction. In Fig. 1(d), the structure is clearly PMA and it relaxes to –z direction \nin 40 ps. Increasing uniaxial anisotropy energy from 103 to 106 J·m-3 changes  self-relaxation times \nfrom 2 ns (in -plane) down to 4 ns (PMA,  single domain), 2 ns (PMA but  two transient domains) \nto 40 ps (PMA, single domain), respectively.  \nWhen an external magnetic field pulse along +z axis has been applied for initialization, in \neach case except Fig. 1(d), the magnetic moment aligns with the external field first. Since the \nstructure in Fig. 1(a) has in -plane easy axis, it cannot retain its moment along +z and it relaxes to \nsurface plane. Since the structure in Fig. 1(b) is PMA, it switches to +z and retains its remanent state (Hexternal  = 2.2 T > Hsat ~ 2K u/Ms = 0.143 T ). The applied field on the nanowire  in Fig. 1(c) \nhelps overcome the domain wall energy and helps align the domains along +z axis as the saturation \nfield for this structure (H sat ~ 2K u/Ms = 1.43 T) is less than the applied pu lse intensity. Since the \nstructure is intrinsically PMA, the structure retains its magnetic moment along +z. In Fig. 1(d), \nsince the calculated Hsat is about 14.3 T, the structure is not magnetically saturated  and does not \nswitch  although it is PMA. The nanowire  size determines the shape anisotropy and the minimum \nuniaxial anisotropy energy needed for PMA. When PMA is achieved with sufficiently large K u, \nincreasing Keff reduces the self -relaxation time down to sub -100 ps ranges although i ncreasing K u \nto as high as 106 J·m-3 increases the saturation field beyond feasible magnetic field intensities.  \n \nFIG. 1. Magnetic initialization steps of the nanowires with α = 0.1 for (a -d) K u = 103, 104, 105, \nand 106 J·m-3, respectively.   \nDeterministic switching has been shown by applying six consecutive positive and negative \nswitching pulses with 2200 mT intensity and 2 ns width each. Fig. 2 shows the corresponding \nswitching time as a function of K u and α. The colored regions indicate deterministic switching with \ntimes corresponding to their color codes. The gray regions indicate no deterministic switching. \nSwitching time is defined as the time it takes for transitioning from m z = -1 to +1 (or vice versa) \nupon receiving an external magnetic field pulse. For α > 0.1, a s uniaxial anisotropy increases, the \ntotal effective field H eff of nanowire increases and the switching time increases due to longer  \nprecession. As α increases, switching time decreases as the damping term starts balancing the \nprecession term in the LLG eq uation. In the ideal case of no damping, the spins would have \nprecessed indefinitely at ⍵ = γH eff without aligning with the applied external magnetic field. With \nfinite or increasing damping term, the precession energy is absorbed and the spins equilibrate  \nfaster. Deterministic switching was not observed for materials with low damping (α < 10-1) as \nprecession prevented switching.   \nFor α > 10-1, relaxation timescales are reduced to below mostly 400 ps. When uniaxia l \nanisotropy energy exceeds 5× 104 Jᐧm-3 until 1.5 ×105 Jᐧm-3, the nanowire  starts forming domain \nwalls, which prevents from or delays reaching steady state reversal. As the uniaxial anisotropy \nenergy increases towa rds 5 ×104 Jᐧm-3, the domain wall width δDW = 2√(A/K u), decreases to 17 nm \nwhich is  below the nanowire width (20 nm). With higher uniaxial anisotropy energies, domain \nwall width decreases and domains  form within the nanowire.   For Ku > 1.5× 105 Jᐧm-3, saturation \nfield exceeds the applied external field pulse (2200 mT) and the nanowires  are not fully saturated. \nTherefore, for multi -domain grains or nanostructures, anisotropy energy must be large enough to \nachieve PMA and K u should be sufficiently small such that realistic ex ternal field pulse intensities \ncould reverse magnetic  orientatio n.  \nFIG. 2. Gilbert damping and uniaxial anisotropy constant dependence of switching  time \n(Hext=2200 mT) . \n \nIV. Dependence of switching energy and rate on pulse width and intensity  \nFig. 3(a) shows the switching energy of the nanowire in units of k BT (T = 300 K) for pulse \nwidths between 50 and 3000 ps and external magnetic field intensities between 1000 and 10000 \nmT. The energy barriers were calculated in micromagnetic models based on the energy magnitudes  \nthe nanowire overcomes after applying the ex ternal magnetic field pulse. In these calculations, the \nenergy difference accounts for the Zeeman, demagnetizing, exchange and uniaxial anisotropy \nenergies. In the Hamiltonian ( Suppl. eqn. 3 -6), the time evolution of the energy is driven mainly \nby the chan ges in the Zeeman energy due to reorientation of the  nanowire  spins upon applying \nfield pulse and the demagnetizing field of the geometry. In this figure, the gray regions show no \nswitching (N/S) and the other regions have switching energies corresponding to their color codes. \nThe figure indicates that the switching energy of the nanowire could be lowered from over 12000 \nkBT to 3163  kBT by tuning the applied field pulse . This effect indicates low Zeeman energy (due \nto its reduced volume ), low uniaxial aniso tropy and low exchange stiffness (of YIG ) make \nmagnetization reversal  energetically favorable. Thus, d ecreasing the field intensity decreases the \nswitching energy.  \nThe hysteresis loop calculated for the mz component  indicate that nanowires have a \ncoercivity of 1950 mT with PMA  (Supplementary Figure S4) . The nanowire switches its magnetic \norientation even below this coercivity with the external field pulse . Decr easing the pulse width \nhelps reduce switching energy unti l applied field pulses of 4394 mT, since it reduces the average \nZeeman energy injected into the nanowire. Therefore, Fig. 3(a) indicates that pulse -assisted and \nsub-coercivity switching with lower energy costs can be achieved for magnetic nanowires.  Fig. \n3(b) shows  nanowire switching time for the same pulse widths and intensities used in Fig. 3(a). \nThe switching time was calculated based on the time the nanowire takes for a complete steady -\nstate reversal of its vertical magnetization. The fastest magnetizat ion reversal occurs at 0.150 ns \nfor 10000 mT and 368.4 ps pulse width. The large field intensity and short pulses enable fast \nmagnetization reversal and minimal time spent in transient precession motion. Decreasing the field \nintensity increases the switchi ng time. While the nanowire switches faster for fields above its \ncoercive field (1950 mT), one could achieve complete magnetization reversal with sub -coercivity \npulses with longer switching times. Switching with sub -coercivity pulses relies on the dynamic \ninsta bilities of the magnetic moment and  most sub -coercivity switching cases in Fig. 3(b) have \nextended switching times. Low field intensities cause precession for extended periods  (damping -\ndominated regime) , thus preventing or significantly delaying reversal . Decreasing the pulse width \nhelps reduce switching time as it reduces the interaction time between the magnetic moment and \nthe field. Pulse-assisted and sub -coercivity switching could be achieved for magnetic nanowires if longer transient reversal  times are allowed. Thus , a trade -off between optimal  switching energy/ \ntime and pulse width/ field intensity could be established.  \nFIG. 3. Pulse width and intensity dependence of (a) s witching energy (units o f kBT at T = \n300K ) and (b) switching time for the nanowire with Ku = 104 Jᐧm-3 and α = 0.01. \n \n \nFIG. 4. Deterministic switching for cases with short and long er switching times. External field \ndependence of relaxation rate for (a) and (b) with external field intensity: 2276 mT, pulse width: \n2564.3 ps (faster) ( 𝛂 = 0.01, Ku = 104 Jᐧm-3), and for (c and d) external field intensity: 1638 mT, \npulse width: 2564.3 ps (sub -coercivity, slower) ( 𝛂 = 0.01, Ku = 104 Jᐧm-3), \nBased on the deterministic switching results shown on Fig. 3, we investigate further two \ncases from Fig. 3(a) and (b): (pulse intensity, pulse width) = (2276 mT, 2564.3 ps) and (1638 mT, \n2564.3 ps). For this condition, the calculated hysteresis loops indicate that the saturation field is \n1950 mT ( Suppl.  Fig. S4). These two cases were chosen to investigate the deterministic switching \ndynamics for above and below -coercivity switching, respectively. The switching dynamics of the \nfirst and second cases are shown on Fig. 4(a,b) and Fig. 4(c,d), respectively. Fig. 4(a,b) \ndemon strate shorter switching times due to the higher external field intensity on  the nanowire. Fig. \n4(c,d) show  deterministic magnetization switching at sub -coercivity external field s (below 1950 \nmT). As shown on Fig. 3, sub -coercivity deterministic switching requires the pulse duration to be \ngreater than a minimum threshold (i.e. 2192 ps for 1931 mT). This threshold depends on both the \nextrinsic factors (external field intensity , nanowire dimensions ) and intrinsic factors (Ku, α, A ex \nand M s). Magnetic r eversal is delayed  due to precession -driven  switching dynamics . These results \nshow that deterministic sub -coercivity switching in magnetic nanowires is feasible and allows for \nreduced switching fields with lower energy barrier materials and geometries.  \n \nV. Conclusions  \nThe temporal and spatial evolution of magnetization switching in nanowires were \ninvestigated  as functions of pulse width, pulse intensity, uniaxial anisotropy constant and damping.  \nDamping, precession and effective field -driven regimes have been identified in the analytical \nmodels of magnetization reversal in nanowires. These simulations and models indicate that the \nmagnetization states of these magnetic nanowires could be reversed under external pulses with \nsufficient pulse intensity and width for optimal damping ( 𝛂 > 0.1) and uniaxial anisotropy (K u < \n105 Jᐧm-3). In high aspect ratio nanowires (in plane x:y = 100:20), sufficiently high uniaxial \nanisotropy constants K u (at least 104 Jᐧm-3) are needed to obtain perpendicular magnetic anisotropy \nby overcoming shape anisotropy. When Ku becomes too high (≥ 105 Jᐧm-3), the effective anisotropy \nof nanowire increases beyond feasible magnetic field pulse intensities (2.2 T or less). For optimal \ndamping, anisotropy and pulse properties ( 𝛂 ∈ [0.1, 0.5], Ku ∈ [104, 105 Jᐧm-3), 0.5  to 3ns-wide \npulses), the nanowires could switch with picosecond timescales and low energy consumption per \nbit as low as 3.163 ×103 kBT at T = 300 K. In all switching cases, PMA is necessary for \ndeterministic pulse -assisted magnetization reversal and dense memory bits. The effective field \nprovides the energy barrier needed for both stable memory and low -power logic functionalities.   Two key outcome s emerge from this study : First is the observation of a nanomagnetic \nswitching trilemma or the competition between nanowire (i) switching rate, (ii) energy cost of \nswitching per bit and (iii) external field required for switching. In this trilemma, high switching \nrate requires either high e xternal field or high switching energy. Lower switching energy requires \nan optimal external magnetic field intensity and pulse width per bit. Minimizing the external \nmagnetic field and reducing switching time requires optimal pulse width at the cost of inc reasing \nenergy per bit. This trilemma originates from a more general competition between the energy -\ndelay product between the external magnetic field and the damping -driven magnetization reversal \n(Suppl . Fig. S3). The second key outcome is s ub-coercivity s witching observed under appropriate \nuniaxial anisotropy (i.e. K u=104Jᐧm-3), damping ( 𝛂 = 0.1) and pulse properties (163 8<intensity \n<1931 mT, 2564< pulse width< 3000 ps). The results shown in Fig. 3(b) demonstrate that the \nnanowire geometry is appropriate for deterministic switching with external magnetic field pulses \nwith peak intensities below the coercive field. Engineering the nanowire geometry and \nperpendicular effective anisotropy (K eff) reduces the switching time to sub -150 ps ranges with sub -\ncoerciv ity switching.  Engineering the energy barrier through nanostructure geometry optimization  \nand operating at the optimal point of the nanomagnetic trilemma could pave the way for efficient \nmemory, logic and cellular automata at high bit rates at room tempera ture. \n \nAcknowledgements  \nF. Ş. acknowledges Koç University Scholarship. M.C.O. acknowledges BAGEP 2017 Award, \nTÜBA -GEBİP Award by Turkish Academy of Sciences and TUBITAK Grant No. 117F416.   \n \n \n References  \n1. P. M. Pimentel, B. Leven, B. Hillebrands, and H. Grimm, Kerr microscopy studies of \nmicrowave assisted switching, J. Appl. Phys.  102, 063913 (2007).   \n2. T. Moriyama, R. Cao, J. Q. Xiao, J. Lu, X. R. Wang, Q. Wen, and H. W. Zhang, Microwave -\nassisted magnetization switching of Ni80Fe20 in magnetic tunnel junctions, Appl. Phys. \nLett. 90, 152503 (2007).   \n3. H. Suto, T. Kanao, T. Nagasawa, K. Kudo, K. Mizushima, and R. Sato, Subnanosecond \nmicrowave -assisted magnetization switching in a circularly polarized microwave magnetic \nfield, Appl. Phys. Lett.  110, 262403 (2017).   \n4. K. Rivkin, M. Benakli, N. Tabat, and H. Yin, Physical principles of microwave assisted \nmagnetic recording, J. Appl. Phys.  115, 214312 (2014).  \n5. R. E. Rottmayer et al. , Heat -Assisted Magnetic Recording , IEEE T. Magn.  42, 2417 (2006).  \n6. S. Okamoto, M. Furuta, N. Kikuchi, O. Kitakami, and T. Shimatsu, Theory and Experiment of \nMicrowave -Assisted Magnetization Switching  in Perpendicular Magnetic Nanodots , IEEE T. \nMagn.  50, 83 (2014).   \n7. R. Hertel, Thickness dependence of magnetization structures in thin Permalloy rectangles, Z. \nMetallkd . 93, 957 (2002).   \n8. C. Tannous, A. Gha ddar, and J. Gieraltowski, Geometric signature of reversal modes in \nferromagnetic nanowires, EPL 91, 17001 (2010).   \n9. Y. Nozaki, M. Ohta, S. Taharazako, K. Tateishi, S. Yoshimura, and K. Matsuyama, Magnetic \nforce microscopy study of microwave -assisted magnetization reversal in submicron -scale \nferromagnetic particles,  Appl . Phys. Lett.  91, 082510 (2007).   \n10. R. Kikuchi, Magnetization Reversal by Rotation , J. Appl. Phys.  27, 1352 (1956).   \n11. J.-H. Moon, T. Y. Lee, and C. -Y. You, Relation between switching time distribution and \ndamping constant in magnetic nanostructure, Sci. Rep. 8, 13288 (2018).   \n12. P. R. Gillette and K. Oshima, Magnetization Reversal by Rotation , J. Appl. Phys.  29, 529 \n(1958).  \n13. M. Kubota, et al., Stress -induced perpendicular magnetization in epitaxial iron garnet thin \nfilms, Appl. Phys. Express  5, 103002 (2012),   \n14. D.B. Huber. Magnetoelastic Properties of the Iron Garnets, Springer, Berlin, Heidelberg \n(1970), pp. 346 -348 in Part A.   15. H. Guo, et al. Strain doping: Reversible single -axis control of a complex oxide lattice via \nhelium implant ation. Phys. Rev. Lett., 114 256801 (2015),    \n16. M. Kubota, et al. Systematic control of stress -induced anisotropy in pseudomorphic iron garnet \nthin films, J. Magn. Magn. Mater.  339, 63 (2013) .  \n17. T. Goto, M.C. Onbaşlı, C.A. Ross, Magneto -optical properties of cerium substituted yttrium \niron garnet films with reduced thermal budget for monolithic photonic integrated circuits, Opt. \nExpress, 20, 28507  (2012). \n18. S. Kanai, M. Yamanouchi, S. Ikeda, Y. Nakatani, F. Matsukura, and H. Ohno, Electric field -\ninduced magnetization reversal  in a perpendicular -anisotropy CoFeB -MgO magnetic tunnel \njunction. Appl. Phys. Lett. 101, 122403 (2012) . \n19. T. Devolder, P. Crozat, J. -V. Kim, C. Chappert, K. Ito, J. A. Katine, and M. J. Carey, \nMagnetization switching by spin torque using subnanosecond current pulses assisted by hard \naxis magnetic fields, Appl. Phys. Lett. 88, 152502 (2006).  \n20. S. Yakata, H. Kubot a, Y. Suzuki, K. Yakushiji, A. Fu kushima, S. Yuasa, and K. Ando, \nInfluence of perpendicular magnetic anisotropy on spin -transfer switching current in \nCoFeB∕MgO∕CoFeB magnetic tunnel junctions , J. Appl. Phys.  105, 07D131 (2009).  \n21. A. Barman, S. Wang, O. Hellwig, A. Berger, E. E. Fullerton, and H. Schmidt, Ultrafast \nmagnetization dynamics in high perpendicular anisotropy [Co∕Pt] n multilayers, J. Appl. Phys. \n101, 09D102 (2007).   \n22. G. M. Whitesides, Self-Assembly At All Scales, Science  295, 2418 (2002).  \n23. A. Imre, G. Csaba, L. Ji, A. Orlov, G. H. Bernstein, and W. Porod, Majority Logic Gate for \nMagnetic Quantum -Dot Cellular Automata , Science  311, 205 (2006).  \n24. S. Y. Chou, M. Wei, P. R. Krauss, and P. B. Fischer, Large area high density qua ntized \nmagnetic disks fabricated using nanoimprint lithography, Journal of Vacuum Science & \nTechnology B: Microelectronics and Nanometer Structures  12, 3695 (1994).  \n25. M. A. Imtaar, A. Yadav, A. Epping, M. Becherer, B. Fabel, J. Rezgani, G. Csaba, G. H. \nBern stein, G. Scarpa, W. Porod, P. Lugli, Nanomagnet Fabrication Using Nanoimprint \nLithography and Electrodeposition, IEEE Trans . Nanotechnol . 12, 547 (2013).  \n26. N. Singh, S. Goolaup, and A. O. Adeyeye, Fabrication of large area nanomagnets, \nNanotechnology  15, 1539 (2004).  27. S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. \nHillebrands and A. Conca, Measurements of the exchange stiffness of YIG films using \nbroadband ferromagnetic resonance techniques, J. Phys. D: Appl. Phys.  48, 015001 (2015).  \n28. A. Kehlberger, K. Richter, M. C. Onbasli, G. Jakob, D. H. Kim, T. Goto, C. A. Ross, G. Götz, \nG. Reiss, T. Kuschel, and M. Kläui, Enhanced Magneto -optic Kerr Effect and Magnetic \nProperties of CeY 2Fe5O12 Epitaxial Thin Films, Phys. Rev. Appl.  4, 014008 (2015).   \n29. M. C. Onbasli, A. Kehlberger, D. H. Kim, G. Jakob, M. Kläui, A. V. Chumak, B. Hillebrands, \nC. A. Ross, Pulsed laser deposition of epitaxial yttrium iron garnet films with low Gilbert \ndamping and bulk -like magnetization, APL Mater. 2, 106102 (2014).  \n30. H. T. F ook, W. L. Gan, and W. S. Lew, Gateable Skyrmion Transport via Field -induced \nPotential Barrier Modulation , Sci. Rep.  6, 21099 (2016).   \n31. S. Mizukami, D. Watanabe, M. Oogane, Y. Ando, Y. Miura, M. Shirai, and T. Miyazaki, Low \ndamping constant for Co 2FeAl Heusler alloy films and its correlation with density of states, J. \nAppl. Phys. 105, 07D306 (2009).  Supplementary Information for “Pulse -assisted magnetization switching in rectangular \nmagnetic nanowires at picosecond and nanosecond timescales with low energy”  \n \nFurkan Şahbaz, Mehmet C. Onbaşlı*  \nKoç University, Department of Electrical and Electronics Engineering, Sarıyer, 34450 Istanbul  \n*corresponding author: monbasli@ku.edu.tr   \n \nIn this supplementary information, we provide the additional numerical modellin g results \nand theoretical analyses in two sections: (1) Numerical Modelling and (2) Theoretical \nAnalysis. The appendix to these sections consists of our OOMMF source codes (mif files).  \n \n1. Numerical Modelling  \nTransient domain wall formation:  In Fig. 1(c) of t he main manuscript, we observed that the spins \ncannot relax to a vertical direction despite uniaxial energy exceeding shape anisotropy energy \n(Ku>K shape). We attribute the origin of this effect in the main manuscript to a transient domain wall \n(DW) formati on. In Suppl ementary  Figure S1, we present the calculated spin profile for K u = 105 \nJ·m-3. Fig. S1 shows a DW , which traverses the nanowire along its short axis.  \n \nSupplementary Figure S1. For K u = 105 J·m-3, a domain wall prevents relaxation to the \nvertical axis (z) (the units are in nm for all three axes).  \nNondeterministic reversal of nanomagnets over a wider window of uniaxial anisotropy and Gilbert \ndamping values:  Fig. 2 of the main manuscript includes results on Gilbert damping (α = 0.01 -0.5) \nand uniaxial anisotropy (K u = 104-105 J·m-3) dependence of relaxation rate. We calculated the \nrelaxation rates for the range of α = 10-4-10-1 and K u = 103-106 J·m-3. We observed that switching \nis not deterministic or cannot happen at all under these conditions. In Supplementary Fig ure 2, we \nindicate this result as a lack of deterministic switching case (N/S: no switching). For K u > 106 J·m-\n3, the applied external magnetic field is not sufficient to overcome the effective magnetic field. For \nKu < 106 J·m-3, damping constant is not sufficient to stabilize the spin precession during \nmagnetization reversal.  \n \nSupplementary Figure S2. The calculated relaxation rates for the range of α = 10-4-10-1 and \nKu = 103-106 J·m-3. For K u > 1.5 × 105 J·m-3, the saturation field exceeds the applied external \nfield pulse and the nanostructures are not saturated. For lower anisotropy values, although \nmagnetization reversal occurs, the damping is not sufficient to end the precession motion for \ndeterministic switching.  \n  \n  \n \nEnergy -delay product and the nanomagnetic trilemma:  In the main text, we mentioned the \ncompetition between (i) switching rate, (ii) energy cost of switching per bit and (iii) external field \nrequired for switching. We named this competition the nanomagnetic trilemma since this effect \noriginates from a more general competition between the energy -delay product between the external \nmagnetic field intensity and the internal precession/damping -driven reversal mechanisms of \nmagnetic nanostructures. In Supplementary Fig S3, we provide the calculated energy -delay \nproduct (units of fJ·ps) for different pulse widths and external field intensities.   \n \nSupplementary Figure S3. Energy -delay product (EDP, fJ·ps) for the external field -driven \nmagnetization rever sal and the internal precession/damping -driven reversal mechanisms of \nmagnetic nanostructures.  \n \n \n \n \n \n \n \nEnergy -delay product and the nanomagnetic trilemma:  In the main text, we mentioned that the \ncalculated hysteresis loops indicate that the saturation field  is 1950 mT. In Supplementary Fig. S4, \nwe present the calculated hysteresis loop for the normalized vertical magnetic moment component \nmz as a function of applied magnetic field H z. \n \nSupplementary Figure S4. Simulated magnetic hysteresis loop of the nanow ires when an \nexternal magnetic field is applied on the nanowire along vertical axis (z). The hysteresis loop \nshows the normalized vertical magnetization component (m z), a perpendicular magnetic easy \naxis with a vertical remanent  state and coercive and saturation field of H c = H sat = 1950 mT.  \n \n \n \n \n \n \n \n1. Theoretical Analysis  \nDynamic evolution of spin vector components in a magnetic material is described using \nthe Landau -Lifshitz -Gilbert ( LLG ) shown in equation s (1) and (2 ). Here, m is the normalized \nmagnetization vector  with 𝐦= 𝐌(x,y,z,t)\nMs, where  M(x,y,z ,t) is the magnetization profile throughout \nthe magnetic nanostructure and M s is the saturation magnetic moment of the material.   \n∂𝐦\n∂t=−γ𝐦×𝐇𝐞𝐟𝐟+𝛼𝐦×∂𝐦\n∂t (1) \nor  \n∂𝐌\n∂t=−|γ̅|𝐌×𝐇𝐞𝐟𝐟−|γ̅|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟) (2) \nwhere γ̅:Landau −Lifshitz  gyromagnetic  ratio ,α:damping  coefficient   \n \nIn this equation, time evolution of magnetic moment vectors sampled within a rectangular \ngrid of 5  nm size is calculated over the rectangular magnetic nanostructure presented above. Heff \nis the effective magnetic field vector. As the total energy of the nanostructure is minimized along \nperpendicular axis, Equation (3) describes the magnetic anisotropy and perpendicular easy axis of \nthe magnetic nanostructure:  \n𝐇𝐞𝐟𝐟=𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 +𝐇𝐝𝐞𝐦𝐚𝐠 +𝐇𝐞𝐱𝐜𝐡𝐚𝐧𝐠𝐞 +𝐇𝐮𝐧𝐢𝐚𝐱𝐢𝐚𝐥  (3) \n𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 =𝐳̂H0sin(ωt) (4) \n𝐇𝐝𝐞𝐦𝐚𝐠 =𝐲̂H1 (5) \n𝐇𝐞𝐱𝐜𝐡𝐚𝐧𝐠𝐞 =𝐳̂2A0\nμ0Ms∇2𝑚 (6) \n𝐇𝐮𝐧𝐢𝐚𝐱𝐢𝐚𝐥 =𝐳̂(K1sin2θ+K2sin2θsin2α)≈𝐳̂K1sin2θ (6) \n \nThe first term of th e right hand side of Equation (2 ) is known as the precession term that drives \noscillations within nanowires  under magnetic fields. The second term , also called the damping \nterm,  drives the alignment rate of the magnetic moment with the external magnetic field. This \ndissipative term is one of the energy loss channels in the relaxation process. The “Hamiltonian” \nfor the LLG equation or the effective field is defined in Equation 3. This equation includes the \ncontributions from external magnetic field, demagnetization field (shape anisotropy term), \nexchange field term and uniaxial anisotropy field. The external field, H external , is applied perpendicular to the nanostructure surface al ong z axis. The demagnetization term, H demag, is the \nfield, which originates due to the absence of magnetic monopoles (divergence -free magnetic flux \ndensity) and the resulting field distribution within the nanowire geometry  along its long in -plane \ny axis . Demagnetizing field is one of the terms , which capture s the effect of geometry on \nanisotropy  and magnetization dynamics . The exchange term, H exchange , is a field generated due to \nthe Heisenberg exchange interaction between adjacent spins. This field can be come particularly \nimportant when metallic magnetic nanowires are used (permalloy: A ex = 13 pJ·m-1) [1] and Cobalt -\nPlatinum multilayers (A ex = 15 pJ·m-1) [2]) and less significant for YIG nanostructures with A ex = \n3.65 ± 0.38 pJ·m-1 [3-5] although the presence of exchange  interaction is not essential for spin \nwave propagation or magnetization reversal [ 6]. The last term is the uniaxial anisotropy energy \nterm, which indicates the vertical directional preference of magnetic moment during relaxation \nand switching. This term could originate from a variety of sources includin g magnetocrystalline \nanisotropy , magnetoelastic anisotropy fo r thin epitaxial nanostructures , strain doping as well as \nother gro wth-induced uniaxial anisotropy . \nOne can a nalyze magnetic relaxation and switching in nanostructures in three regimes:  \n(1) precession -driven (when damping term is negligible ),  \n(2) damping -driven (α is large such that the damping term prevents magnetization reversal)  \n(3) effective field -driven  \nConsidering  the combined effects of material constants and the anisotropy terms, we derive and \ninvestigate these regime s in further detail  below .  \nI. Precession -driven magnetization dynamics   \nWhen Gilbert damping α is small such that damping term is much smaller than the precession \nterm, magnetic relaxation and r eversal is driven by precession:   \n|−|γ̅|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟)|≪|−|γ̅|𝐌×𝐇𝐞𝐟𝐟| (7) \n|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟)|≪|𝐌×𝐇𝐞𝐟𝐟|  \n|α𝐌\nMs||(𝐌×𝐇𝐞𝐟𝐟)||sin(90°)|≪|𝐌×𝐇𝐞𝐟𝐟|  \nα|m|≪1  \nα≪1 (8) Here, the angle between the magnetization vector M and the M×Heff vector  is 90° . Since the \nmagnitude of the normalized magnetization vector m is always one, precession term dominates \nwhen the damping coefficient is much smaller than one. When magnetization dynamics is driven  \nby precession, relaxation or reversal processes continue indefinitely or much longer than otherwise \nin absence of damping . As a result, we do not observe any magnetization reversal :  \n∂𝐌\n∂t≈−|γ̅|𝐌×𝐇𝐞𝐟𝐟 (9) \n𝐌(𝐫,t)=Ms𝑒−𝜅𝑡(−𝐲̂sin (γ̅Hefft)+𝐱̂cos (γ̅Hefft)) (10) \n \nFor κ = 0 (zero damping limit), the oscillation continues indefinitely. For nonzero and small κ, the \noscillations continue for extended p eriods with an evanescent decaying envelope. Gilbert damping \nparameter α is low for YIG  [4,5]  (3-7×10-4), magnetostrictive spinel ferrites  [7] (< 3×10-3), Heusler \n[8] (10-3) or other low -damping metallic alloys such as CoFe  [9] (10-4-10-3). A very small damping \ncoefficient, regardless of the H eff magnitude, triggers the precession -driven regime . The terms \npresent in the effective field determine the Larmor precession frequency (around few GHz for \nferromagnets and potentially towards THz for antiferromagnets).  \n \nII. Damping -driven magnetization dynamics  \nWhen Gilbert damping α is large such that the damping term is much larger than the precession \nterm, magnetic relaxation an d reversal is driven by damping:   \n \n|−|γ̅|𝐌×𝐇𝐞𝐟𝐟|≪|−|γ̅|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟)| (11) \n|𝐌×𝐇𝐞𝐟𝐟|≪|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟)|  \n|𝐌×𝐇𝐞𝐟𝐟|≪|α\nMs𝐌||𝐌×𝐇𝐞𝐟𝐟||sin(θ)|  \n1≪α|𝑚||sin(θ)|  \n1≪α (12) \n \nWhen the magnetization dynamics is driven by damping  due to large damping coefficient , \nrelaxation or reversal processes occur with very short  evanescent lifetimes : ∂𝐌\n∂t≈−|γ̅|α\nMs𝐌×(𝐌×𝐇𝐞𝐟𝐟) (13) \n∂𝐌\n∂t≈−|γ̅|α 𝐦×(𝐌×𝐇𝐞𝐟𝐟)  \n1\nMs∂𝐌\n∂t≈1\nMs(−|γ̅|α 𝐦×(𝐌×𝐇𝐞𝐟𝐟))  \n∂𝐦\n∂t≈(−|γ̅|α 𝐦×(𝐦×𝐇𝐞𝐟𝐟)) (14) \n  \nEquation 14 shows that the decay rate in the damping -driven regime is driven by the gyromagnetic \nratio, damping constant , effective field  and the orientation of the effective field with respect to the \ninitial magnetization orientation. For a magnetic moment initially oriented along +z, large damping \ndecays the in -plane excitations due to the pulse and the initial magnetization along +z is retained:  \n \n𝐌(𝐫,t)=𝑒−|γ̅|α Δ∗𝑡(𝐱̂Mx0+𝐲̂My0)+𝐳̂Mz0 (15) \n \nIn order to trigger damping -driven regime described by equations 11, 14  and 15, Gilbert damping \nconstant α should  be larger than 1. The effective field should also not be parallel to the initial \nmagnetization, since this configuration would not trigger any reversal.  \n \nWhen a material has ultralow damping and has high effective fields as in Yttrium iron garnet \nnanowires , precession regime prevails and relaxation timescales could be extended indefinitely as \nlong as  damping is negligible or compensated. For ultrafast magnetization reversal,  damping must \nnot be negligible  or effective field (anisotropy and external field) must not be too high.  \n \nIII. Effective field -driven magnetization dynamics  \nWhen neither damping nor precession term dominates time -dependent magnetic relaxation, \ndynamic control of individual terms in the effective field determines the time evolution of \nmagnetization reversal process. In th is case, the LLG equation follows the standard form in \nequation 2 . The vectors in the effective field term determine the switching time scales together \nwith damping. The external field or the uniaxial anisotropy determine  the timescales in the LLG \nequation when they are much large r with respect to the other terms in the  effective field . The other in-plane terms in the effective field, such as the demagnetizing fields, are necessary to trigger \nmagnetization reversal : \n \n∂𝐌\n∂t=−|γ̅|𝐌×(𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 +𝐇𝐝𝐞𝐦𝐚𝐠 +𝐇𝐞𝐱𝐜𝐡𝐚𝐧𝐠𝐞 +𝐇𝐮𝐧𝐢𝐚𝐱𝐢𝐚𝐥 )−|γ̅|α\nMs𝐌×(𝐌\n×(𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 +𝐇𝐝𝐞𝐦𝐚𝐠 +𝐇𝐞𝐱𝐜𝐡𝐚𝐧𝐠𝐞 +𝐇𝐮𝐧𝐢𝐚𝐱𝐢𝐚𝐥 )) (16) \n∂𝐌\n∂t≈−|γ̅|𝐌×(𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 +𝐇𝐢𝐧 𝐩𝐥𝐚𝐧𝐞 )−|γ̅|α\nMs𝐌×(𝐌×(𝐇𝐞𝐱𝐭𝐞𝐫𝐧𝐚𝐥 +𝐇𝐢𝐧 𝐩𝐥𝐚𝐧𝐞 ) (17) \n \nHere, achieving ultrashort reversal time constants depend on the damping co nstant and the external \nfield intensity along the direction of the new magnetic state. Large external field and some nonzero \nbuilt-in in -plane field drives a faster preces sion and magnetization reversal, while a sizable \ndamping is necessary to stabilize the moments along the final orientation.  \n \nAppendix  1. OOMMF Source Code  \n(rect_structure_field .mif and rect_structure_hysteresis .mif) \nAppendix  2. Calculated domain wall movie for Supplementary Fig. S1 & Figure 1(c)   \n(1E5 Sample .mov)  \n \nReferences  \n1. R. Hertel, Thickness dependence of magnetization structures in thin Permalloy rectangles, Z. \nMetallkd.  93, 957 (2002).  \n2. H. T. Fook, W. L. Gan, and W. S. Lew, Gateable Skyrmion Transport via Field -induced \nPotential Barrier Modulation , Sci. Rep.  6, 21099 (2016).  \n3. S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. \nHillebrands and A. Conca, Measurements of the exchange stiffness of YIG films using \nbroadband ferromagnetic resonance techniques, J. Phys. D: Appl. Phys. 48, 015001 (2015).  \n4. A. Kehlberger, K. Richter, M. C. Onbasli, G. Jakob, D. H. Kim, T. Goto, C. A. Ross, G. Götz, \nG. Reiss, T. Kuschel, and M. Kläui, Enhanced Magneto -optic Kerr Effect and Magnetic \nProperties of CeY 2Fe5O12 Epitaxial Thin Films, Phys. Rev. Appl.  4, 014008 (2015).  5. M. C. Onbasli, A. Kehlberger, D. H. Kim, G. Jakob, M. Kläui, A. V. Chumak, B. Hillebrands, \nC. A. Ross, Pulsed laser deposition of epitaxial yttrium iron garnet films with low Gilbert \ndamping and bulk -like magnetization, APL Mater. 2, 1061 02 (2014).  \n6. K. Oyanagi, S. Takahashi, L. J. Cornelissen, J. Shan, S. Daimon, T. Kikkawa, G. E. W. Bauer, \nB. J. van Wees & E. Saitoh, Spin transport in insulators without exchange stiffness, Nat. \nCommun. 10, 4740 (2019).  \n7. S. Emori, B. A. Gray, Jeon, H.‐M.,  J. Peoples, M. Schmitt, K. Mahalingam, M. Hill, M. E. \nMcConney, T. M. Gray, U. S. Alaan, A. C. Bornstein, P. Shafer, A. T. N'Diaye, E. Arenholz, \nG. Haugstad, K.‐Y. Meng, F.  Yang, F.,  D. Li, S. Mahat, D. G. Cahill, P. D hagat, A. Jander, \nN. X. Sun, Y. Suzuki, B. M. Howe, Coexistence of Low Damping and Strong Magnetoelastic \nCoupling in Epitaxial Spinel Ferrite Thin Films , Adv. Mater.  29, 1701130 (2017).  \n8. A. Conca, A. Niesen, G. Reiss, and B. Hillebrands, Low damping magnet ic properties and \nperpendicular magnetic anisotropy in the Heusler alloy Fe1.5CoGe, AIP Advances 9, 085205 \n(2019) . \n9. M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J. Silva, H. T. Nembach, O. Eriksson, O. \nKaris & J. M. Shaw, Ultra -low magnetic damping of a metallic ferromagnet, Nat. Phys. 12, \n839 (2016).  "    },    {        "title": "1806.01002v2.Monte_Carlo_simulation_of_equilibrium_and_dynamic_phase_transition_properties_of_an_Ising_bilayer.pdf",        "content": "arXiv:1806.01002v2  [cond-mat.stat-mech]  6 Jun 2018Monte Carlo simulation of equilibrium and dynamic\nphase transition properties of an Ising bilayer\nYusuf Y¨ uksel\nDepartment of Physics, Dokuz Eyl¨ ul University, Tr-35160 ˙Izmir, Turkey\nAbstract\nMagnetic properties of an Ising bilayer system defined on a honeyco mb lat-\ntice with non-magnetic interlayers which interact via an indirect exch ange\ncoupling have been investigated by Monte Carlo simulation technique. Equi-\nlibrium properties of the system exhibit ferrimagnetism with P-,N- and\nQ- type behaviors. Compensation phenomenon suddenly disappears with\ndecreasing strength of indirect ferrimagnetic interlayer exchang e coupling.\nQualitative properties are in a good agreement with those obtained b y effec-\ntive field theory. In order to investigate the stochastic dynamics o f kinetic\nIsing bilayer, we have introduced two different types of dynamic mag netic\nfields, namely a square wave, and a sinusoidally oscillating magnetic field\nform. For both field types, compensation point and critical temper ature de-\ncrease with increasing amplitude and field period. Dynamic ferromagn etic\nregion in the presence of square wave magnetic field is narrower tha n that\nobtained for sinusoidally oscillating magnetic field when the amplitude an d\nthe field period are the same for each type of dynamic magnetic fields .\nKeywords: Dynamic phase transitions, Ferrimagnetism, Magnetic bilayer,\nMonte Carlo\n1. Introduction\nNowadays, magnetic properties of low dimensional systems in forms of\ngraphene-like structures have attracted significant amount of in terest. The\nreason is due to the fact that two dimensional graphene [1, 2, 3, 4, 5], and\n∗Corresponding author. Tel.: +902323019544; fax: +9023245341 88.\nEmail address: yusuf.yuksel@deu.edu.tr (Yusuf Y¨ uksel)\nPreprint submitted to xxx June 7, 2018its variants [6] defined on a honeycomb lattice exhibit a variety of inte resting\nelectric and magnetic properties which are significantly affected by v ary-\ning system size. After experimental realization of exactly two dimen sional\nmonocrystalline graphitic films [7] which are only a few atoms thick but s ta-\nble under environmental conditions, theoretical and experimenta l research\ninterests have been directed to the studies of two-dimensional lay ered struc-\ntures. For instance, in order to reveal the finite-temperature p roperties of\nhoneycomb iridates with general formula A 2IrO3which exhibit strong spin-\norbit coupling (SOC), Price and Perkins [8, 9] have performed Monte Carlo\n(MC) simulations based on the classical Heisenberg-Kitaev (HK) mod el [10]\non a honeycomb lattice where the interactions between nearest ne ighbors are\nofXX,YYorZZtype. Very recently, it has been shown that transition\nmetal trihalides (MX 3) defined on a two dimensional honeycomb lattice may\nexhibit magnetic order below a finite critical temperature [11, 12].\nImportance of honeycomb lattice not only originates as a conseque nce\nof experimental research on graphene, but resides also on the th eoretical\ngrounds. Namely, it offers reduced mathematical complexity, and t here are\nalso some exact results regarding the magnetic properties for this structure\n[13, 14, 15, 16, 17, 18]. From the experimental point of view, single la yer,\ndouble layer and few (3 to 10) layer honeycomb structures are clas sified as\nthree different types of 2D crystals, and thin film limit is reached for t hicker\nsystems [7]. In this regard, investigation of magnetic properties of graphene-\nlike multilayers gained particular attention, and a wide variety of such sys-\ntems have been successfully modeled within the framework of Ising m odel\nand its variants [19, 20, 21, 22, 23]. For instance, using the effectiv e field\ntheory (EFT) formalism, Jiang and coworkers [21] investigated the magnetic\nproperties such as magnetization and the magnetic susceptibility of a nano-\ngraphene bilayer. For a trilayer Ising nanostructure, EFT calculat ions have\nbeen performed and from the thermal variations of the total mag netization,\nsix distinct compensation types have been reportedby Santos and S´ a Barreto\n[22]. In a recent paper, Kaneyoshi [23] investigated the magnetic behavior\nof an Ising bilayer with non-magnetic inter-layers. Based on EFT met hod,\nsome characteristic features of ferrimagnetism have also been re ported in this\nstudy. In that work, a realistic case has also been considered by as suming a\ndistance-dependent indirect exchange interaction between the t wo magnetic\nlayers.\nOn the other hand, after experimental realization of dynamic phas e tran-\nsitions [24, 25] in uniaxial cobalt films [26], stochastic dynamics of kinet ic\n2systems gained renewed interest [27, 28]. In such systems, a dyn amic phase\ntransition between dynamically ordered and disordered phases tak es place\nwhich is characterized by a dynamic symmetry breaking. Depending o n the\ntwo competing time scales, namely, the period of the externally applie d oscil-\nlating magnetic field and relaxation time of the system, kinetic Ising mo del\nmay exhibit dynamic ferromagnetic (FM) or dynamic paramagnetic (P M)\ncharacter. Winner of the competition of the above mentioned time s cales is\ndetermined by another complicated competition between the field am plitude,\nfield period, temperature, and exchange coupling.\nThe effective field theory [29] partially takes into account the spin flu c-\ntuations, and it is superior to conventional mean field theory [30] w here the\nspin-spin correlations are completely ruled out. Despite its mathema tical\nsimplicity, mean field predictions are only valid for the systems with dime n-\nsionality d≥4. In a recent work, we have shown that EFT and MC results\nqualitatively agree well with each other for a particular ternary spin sys-\ntem [31]. In this regard, EFT method promises reasonable results wit h less\ncomputational cost.\nThe aim of the present paper is two fold: First, a direct comparison o f\nMC results obtained within the present work with the available EFT res ults\nof Ref. [23] will be presented for the Ising bilayer system. As will be s hown\nin the following discussions, qualitatively plausible agreement exists be tween\nEFT and MC results. Second, we will present some results regarding the\nstochastic dynamics and compensation behavior of the kinetic Ising bilayer\ninthe presence oftwo different formsof the oscillating magnetic field , namely\na square wave form and a sinusoidal wave form. The rest of the pap er can\nbe outlined as follows: In Section 2, we will present the formulation an d\nsimulation details of our model. Section 3 contains numerical results a nd\nrelated discussions. Finally, Section 4 is devoted to our conclusions.\n2. Model and Formulation\nOur bilayer model consists of successive stacking of 2D honeycomb mono-\nlayers forming a 3D graphite structure (Fig 1a). The bottom layer, i.e. the\nsublattice Aconsists of Ising spins with σi=±1\n2whereas the topmost layer\n(sublattice B) consists of tightly packed magnetic atoms with a pseudo spin\nvariable Si=±1,0. The number of nonmagnetic layers between the sub-\nlatticesAandBis denoted by n. The intra-layer exchange couplings are\nrespectively denoted by JA(>0) andJB(>0) whereas the interlayer ex-\n3(a) (b)\nFigure 1: (a) Schematic representation of the simulated magnetic b ilayer. Sublattice A\n(B) is occupied by σ=±1/2 (S=±1,0) spins. (b) Equivalent of honeycomb lattice\nin the brick lattice representation. Each pseudo spin has three nea rest neighbors, and is\nlocated on the nodes of a L×Lsquare lattice.\nchange coupling is represented by JR(>0). This selection of interaction\nconstants allows us to study the ferrimagnetic behavior of the mod el. We\nconsider an indirect exchange coupling between the layers AandB. Hence,\nfollowing the same notation with Ref. [23], we assume\nJR=Jexp[−λ(n+1)]/(n+1)δ, (1)\nwhere the parameter λis related to the disorder and δis related to the\ndimensionality of the system, and nis the number of nonmagnetic layers\nbetween the sublattices AandB, (please see Ref. [23] for details). The\nHamiltonian of the model represented by Fig. 1 is given by\nH=−JA/summationdisplay\n<ij>σiσj−JB/summationdisplay\n<kl>SkSl+JR/summationdisplay\n<ik>σiSk−DB/summationdisplay\nk(Sk)2,(2)\nwhere the spin-spin coupling terms in the first three sums are taken over\nonly the nearest-neighbor spin pairs whereas the last summation is c arried\nout over all the lattice sites of sub-lattice BwithDBbeing the single ion\nanisotropy parameter of spin-1.\n4In order to implement the MC simulation procedure for the present s ys-\ntem, each pseudo spin variable σiandSkis assigned on the lattice sites of a\nbrick lattice [32, 33] with lateral dimension Lwhich is topologically equiva-\nlent of the honeycomb lattice (Fig. 1b). Periodic boundary condition s have\nbeen imposed in both lateral and vertical directions. During the simu lations,\nwe have monitored the quantities of interest over 250000 Monte Ca rlo steps\nper lattice site for equilibrium system, after discarding the first 500 00 steps.\nOn the other hand, for the calculation of kinetic properties, we hav e obtained\ntime series of magnetization over 2000 cycles of external magnetic field, and\nallowed the system to relax during the first 1000 periods.\nIn the equilibrium case, the thermal average of sub-lattice ( MAandMB)\nand total ( MT) magnetizations have been calculated according to\nMα=/angbracketleftBigg/summationdisplay\ntmα(t)/angbracketrightBigg\n, α=A,B,T (3)\nwheremα(t) is the time series of corresponding sub-lattice (or total) mag-\nnetization per spin. Then the definition of magnetic susceptibility and the\nalternative description of the total magnetization can also be given by\nχ=NT\nkBT\n/angbracketleftBigg/summationdisplay\nt(mT(t))2/angbracketrightBigg\n−/angbracketleftBigg/summationdisplay\ntmT(t)/angbracketrightBigg2\n, (4)\nMT= [MA+MB]/2.0, (5)\nwhereNTis the total number of lattice sites. Some of the simulation param-\neters have been fixed as JA= 1.0J,JB= 0.5J. For simplicity, we also set\nkB= 1.\n3. Results and Discussion\nIn section 3.1, we will present the magnetic properties of Ising bilaye r\nin the absence of magnetic field. However, section 3.2 is devoted for the\ndiscussions regarding the nonequilibrium stochastic behavior of the system\nin the presence of time dependent oscillating magnetic field.\n5/s45/s50/s46/s48 /s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s84\n/s67/s47/s74\n/s68/s47/s74/s32 /s61/s48/s46/s48\n/s32 /s61/s48/s46/s50/s53\n/s32 /s61/s48/s46/s53\n/s74\n/s65/s61/s49/s46/s48/s74\n/s74\n/s66/s61/s48/s46/s53/s74\n/s110/s61/s49/s46/s48\n/s61/s51/s46/s48\n(a)\n/s32/s76/s61/s50/s53/s54\n/s84/s47/s74\n(b)\nFigure 2: (a) Phasediagram of the Ising bilayerwith L= 128 plotted in a ( DB/JvsTc/J)\nplane for three different values of λ. (b) Magnetic properties such as the total magneti-\nzationMTand magnetic susceptibility χforDB=−1.5 withn= 1,λ= 0 and δ= 3.0.\nDifferent symbols correspond to different lattice size L.\n3.1. Equilibrium properties\nWe start our investigation by examining the phase diagram of the pre sent\nmodel in a ( DB/JvsTc/J) plane for three values of disorder parameter λ\nwhere the numerical value of the transition temperature has been estimated\nfrom the peak point of susceptibility curves. Here, we consider one mono-\nlayer of nonmagnetic sites. According to Eq. (1), antiferromagne tic interface\nexchange coupling JRexponentially approaches to zero with increasing λ.\nHence, for large values of this parameter, we have JR→0, and in this limit,\n6the two sublattices AandBbecome magnetically independent of each other.\nFor moderate values such as λ≤0.5, ferrimagnetic character is adopted in\nthe system, and both sublattices undergo a phase transition at th e same\ncritical temperature. For λ= 0.0,JRapproaches its maximum value, and\nfor positive DB/J, critical temperature becomes reduced with increasing λ.\nOn the other hand, for large negative values of DB/J, onlySi= 0.0 state\nis allowed in sublattice B. Therefore, if we define a threshold value D∗\nB/J\nfor single ion anisotropy parameter then the sublattice Bbecomes nonmag-\nnetic for DB/J < D∗\nB/J. In this case, the horizontal line in the phase\ndiagram is the sole contribution of sublattice Ato the transition tempera-\nture. For spin-1 Blume-Capel model, MC calculations predict a tricrit ical\npoint atDt/J=−1.446 for the same phase diagram [34] whereas EFT result\nisDt/J=−1.41 [29, 35]. We note that, the selection of exchange coupling\nparameters, namely, JA= 1.0JandJB= 0.5Jhelps us to omit the first or-\nder phase transitions in the present system. This can be seen from Fig. 2b,\nwhere we plot the magnetization and magnetic susceptibility as a func tion\nof temperature for several lattice sizes ranging from L= 64 toL= 256. As\nshowninthisfigure, themagnetizationexhibits acontinuous phaset ransition\nin the vicinity of critical temperature and magnetic susceptibility cur ves ex-\nhibit a size dependent positive divergence around Tc. All these observations\nclearly demonstrate that the phase transition is always of second o rder for\nthe whole range of DB/Jvalues. Besides, the ground state magnetization\nsaturates at MT= 0.25, since the magnetization of sublattice Bis zero for\nDB=−1.5J. As a final note regarding this figure, we should point out that\na qualitatively similar phase diagram has been obtained in Ref. [23] whe re\nthe author used EFT. This fact again shows that the models solved b y EFT\nmethod exhibit the same topology as those obtained from the Monte Carlo\n(MC) simulation.\nNext in Fig. 3, we present some ferrimagnetic properties of the sys tem\nwhere the total magnetization MThas been plotted as a function of tem-\nperature for some selected values of DB/J. The other system parameters\nhave been fixed as displayed in the figure. In a recent work [21], six dif -\nferent compensation types [36, 37] have been observed for an Is ing trilayer\nsystem. On the other hand, Ref. [23] reports that the total mag netization of\nIsing bilayer with indirect interlayer exchange exhibits P-,N- andQ- type\nbehaviors which have also been observed in our calculations. Moreov er, the\nunclassified curve corresponding to d=−0.85 of Ref. [23] is identical to the\ncurve corresponding to DB/J=−0.88 in Fig. 3 of the present study [38].\n7/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s74\n/s65/s61/s49/s46/s48/s74\n/s74\n/s66/s61/s48/s46/s53/s74\n/s110/s61/s49/s46/s48\n/s61/s51/s46/s48\n/s61/s48/s46/s48\n/s48/s46/s48/s77\n/s84\n/s84/s47/s74/s48/s46/s56/s45/s48/s46/s53/s45/s48/s46/s56/s45/s48/s46/s56/s53/s45/s48/s46/s56/s56/s45/s48/s46/s57/s48/s45/s49/s46/s48\nFigure 3: Total magnetization MTas a function of DB/Jfor fixed system parameters\nwhich are shown the figure. The system size has been fixed as L= 128.\nThis observation again supports the consistency of the results ob tained by\nEFT and MC methods.\nAs shown in Fig. 3, a compensation behavior may originate in the syste m\nfor a narrow range of DB/Jvalues. Compensation temperature is peculiar\nto the systems exhibiting ferrimagnetism at which the sublattice mag netiza-\ntions cancel each other below the transition temperature. The infl uence of\nvaryingλ,δandnon the magnetisation profile has been depicted in Fig. 4.\nAs shown in this figure, N- type magnetization curve evolves towards the Q-\ntype behavior with increasing λ,δ, andn. This is an expected result, since\nJRrapidly decays towards zero with increasing values of these parame ters.\nTherefore, ferrimagnetism is destructed, and we obtain two indep endent fer-\nromagnetic layers.\n3.2. Kinetic properties\nUp to now, we have considered the ferrimagnetic properties of Isin g bi-\nlayer in the absence of magnetic field. From now on, we will discuss the\nvariation of magnetic properties of the system in the presence of t ime de-\npendent oscillating magnetic field for the following set of system para meters:\nJA= 1.0J,JB= 0.5J,DB=−0.85J,n= 1.0,δ= 3.0, andλ= 0.0. This\nset of parameters not only allows us to avoid the first order phase t ransitions,\nbut also provides information about how the compensation behavior varies\nin the presence of oscillating magnetic field. For this aim we consider tw o\ndistinct types of magnetic field: (i) sinusoidal wave, (ii) square wave . In this\n8/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51 /s74\n/s65/s61/s49/s46/s48/s74\n/s74\n/s66/s61/s48/s46/s53/s74\n/s110/s61/s49/s46/s48\n/s61/s51/s46/s48\n/s68/s61/s45/s48/s46/s56/s53/s74\n/s84/s47/s74/s77\n/s84/s45/s48/s46/s50\n/s48/s46/s48/s32 /s61/s48/s46/s51\n/s32 /s61/s48/s46/s52\n/s32 /s61/s48/s46/s53\n/s48/s46/s49\n(a)/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s74\n/s65/s61/s49/s46/s48/s74\n/s74\n/s66/s61/s48/s46/s53/s74\n/s110/s61/s49/s46/s48\n/s61/s48/s46/s48\n/s68/s61/s45/s48/s46/s56/s53/s74/s52/s46/s48\n/s51/s46/s53\n/s51/s46/s48\n/s50/s46/s48\n/s49/s46/s53\n/s84/s47/s74/s77\n/s84\n/s50/s46/s53\n(b)\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s74\n/s65/s61/s49/s46/s48/s74\n/s74\n/s66/s61/s48/s46/s53/s74\n/s61/s51/s46/s48\n/s61/s48/s46/s48\n/s68/s61/s45/s48/s46/s56/s53/s74\n/s84/s47/s74/s77\n/s84/s32/s110/s61/s49\n/s32/s110/s61/s50\n/s32/s110/s61/s51\n/s32/s110/s61/s53\n(c)\nFigure 4: Influence of (a) λ, (b)δ, and (c) non the compensation behavior of the total\nmagnetization of the Ising bilayer with L= 128.\ncase, the Hamiltonian equation can be written as\nH=H0+h(t)(/summationdisplay\niσi+/summationdisplay\nkSk), (6)\nwhereH0is the Hamiltonian equation in the absence of dynamic magnetic\nfield, and the second and the third summations correspond to dyna mic Zee-\nman energy terms. As we have underlined in the preceding sections, the\nsystem can exhibit a field induced dynamic phase transition between o rdered\nand disordered phases. Such a situation is shown in Fig. 5 where we re -\nspectively select the field amplitude and the temperature as h0/J= 0.4, and\n9/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s45/s49/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s109\n/s65/s109\n/s65/s32/s104/s40/s116/s41/s109/s40/s116/s41/s44/s104/s40/s116/s41\n/s116/s105/s109/s101/s32/s40/s77/s67/s83/s83/s41/s80/s61/s50/s48/s109\n/s66\n(a)/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s80/s61/s50/s48/s48/s32/s104/s40/s116/s41\n/s109\n/s65/s109\n/s66/s109/s40/s116/s41/s44/s104/s40/s116/s41\n/s116/s105/s109/s101/s32/s40/s77/s67/s83/s83/s41\n(b)\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s45/s49/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s109\n/s65/s109\n/s65/s32/s104/s40/s116/s41/s109/s40/s116/s41/s44/s104/s40/s116/s41\n/s116/s105/s109/s101/s32/s40/s77/s67/s83/s83/s41/s80/s61/s50/s48/s109\n/s66\n(c)/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s80/s61/s50/s48/s48/s32/s104/s40/s116/s41\n/s109\n/s65/s109\n/s66/s109/s40/s116/s41/s44/s104/s40/s116/s41\n/s116/s105/s109/s101/s32/s40/s77/s67/s83/s83/s41\n(d)\nFigure 5: Time series of magnetizations mA,mBand magnetic field h(t) for the system\nsizeL= 128. The time evolution of magnetic field is either in sinusoidal form (( a),(b))\nor in square wave form ((c),(d)). The leftmost plots have been obt ained for P= 20\nwhereas the rightmost curves correspond to high period case P= 200. The magnetic field\namplitude has been fixed as h0= 0.4J.\nT= 0.8Tc. HereTcdenotes the critical temperature in the absence of any\nmagnetic field. Oscillation period of the magnetic field is denoted by P. In\nFig. 5, the top and bottom panels respectively correspond to sinus oidal and\nsquare wave forms of the oscillating magnetic field. In the high frequ ency\nregime (i.e. the left panels) the sublattice magnetizations mAandmBoscil-\nlate around a nonzero value. This corresponds to the dynamically or dered\nphase. On the other hand, in the low frequency regime, the sublatt ice mag-\nnetizations mAandmBcan follow the external perturbation with a small\n10phase lag, and the time average of the magnetization is very close to zero\nwhere the system is in the dynamically disordered phase. In this proc ess, it\nis possible to trigger a field induced dynamic phase transition by prope rly\nadjusting the field period P.\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s45/s48/s46/s51/s48/s45/s48/s46/s50/s53/s45/s48/s46/s50/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s104\n/s48/s61/s48/s46/s51/s74\n/s104\n/s48/s61/s48/s46/s49/s74/s81\n/s84\n/s84/s47/s74\n/s104\n/s48/s61/s48/s46/s49/s74\n/s74\n/s65/s61/s49/s46/s48/s74\n/s74\n/s66/s61/s48/s46/s53/s74/s81\n/s65/s44/s81\n/s66\n/s84/s47/s74/s110/s61/s49/s46/s48\n/s61/s51/s46/s48\n/s61/s48/s46/s48/s104\n/s48/s61/s48/s46/s51/s74\n/s80/s61/s53/s48\n(a)/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s104\n/s48/s61/s48/s46/s49/s74\n/s104\n/s48/s61/s48/s46/s51/s74/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s45/s48/s46/s51/s48/s45/s48/s46/s50/s53/s45/s48/s46/s50/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s81\n/s84\n/s84/s47/s74/s104\n/s48/s61/s48/s46/s49/s74/s104\n/s48/s61/s48/s46/s51/s74/s81\n/s65/s44/s81\n/s66\n/s84/s47/s74/s110/s61/s49/s46/s48\n/s61/s51/s46/s48\n/s61/s48/s46/s48/s74\n/s65/s61/s49/s46/s48/s74\n/s74\n/s66/s61/s48/s46/s53/s74\n/s80/s61/s53/s48\n(b)\nFigure 6: Variation of dynamic order parameters QAandQBas functions of temperature\nforL= 128. The magnetic field h(t) varies in (a) sinusoidal (b) square wave form with\ntime. System parameters accompany each figure. In the inset, th e total dynamic order\nparameter QThas been depicted.\nCompensation behavior in the presence of dynamic magnetic fields ca n\nbe examined by calculating the thermal average of dynamic order pa rame-\nters corresponding to sublattices, as well as the total magnetiza tion. These\nmagnetic properties are defined as the time averaged magnetizatio ns over the\nsuccessive cycles of the oscillating field [39],\nQα=1\nNP/contintegraldisplay\nmα(t)dt, α=A,BorT (7)\nwherePis the field period, and Ndenotes the number of magnetic field\ncycles. In Fig. 6, in order to compare the stochastic behavior of th e system\ninthepresenceofsinusoidal andsquarewavemagneticfield, wehav edepicted\nthe thermal variationof sublattice magnetizations QAandQBasfunctions of\nthe temperature. It can be seen from this figure that transition t emperature,\nas well as the compensation point Tcompreduces with increasing magnetic\nfield amplitude h0. Moreover, in the presence of square wave magnetic field,\nnumerical values of TcandTcompare clearly lower than those obtained for the\n11sinusoidally oscillating magneticfields. The insets inFig. 6show thether mal\nvariation of total magnetization when the field amplitude is varied. Fo r both\nforms of the magnetic field, QTexhibits N- type behavior. Therefore, we\ncan conclude that although the compensation temperature is redu ced with\nincreasing h0,QTmaintains its Ne´ el classification scheme.\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s45/s48/s46/s51/s48/s45/s48/s46/s50/s53/s45/s48/s46/s50/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s74\n/s65/s61/s49/s46/s48/s74\n/s74\n/s66/s61/s48/s46/s53/s74\n/s68\n/s66/s61/s45/s48/s46/s56/s53/s74\n/s110/s61/s49/s46/s48\n/s61/s51/s46/s48\n/s61/s48/s46/s48/s81\n/s84\n/s84/s47/s74/s32/s80/s61/s53/s48/s32/s40/s99/s111/s115/s105/s110/s101/s41\n/s32/s80/s61/s50/s48/s48/s32/s40/s99/s111/s115/s105/s110/s101/s41\n/s32/s80/s61/s53/s48/s32/s40/s115/s113/s117/s97/s114/s101/s41\n/s32/s80/s61/s50/s48/s48/s32/s40/s115/s113/s117/s97/s114/s101/s41\n/s104\n/s48/s61/s48/s46/s51/s74\n(a)\nFigure 7: Variation of dynamic order parameter QTas a function of temperature for\nL= 128. The magnetic field h(t) varies with time either in cosine or in square wave form\n. System parameters accompany each figure.\nFinally, let us conclude our investigation for the Ising bilayer system b y\nexamining the variation of compensation phenomenon as a function o f vary-\ning field period P. In figure 7, termal variation of QThas been depicted\nfor both sinusoidal and square wave forms of magnetic field. Here, the field\namplitude has been fixed as h0= 0.4J, and we consider two different values\nof field period P. Either for square and sinusoidal wave forms of magnetic\nfield, the order parameter QTmaintains its N- type profile for high and low\nfrequency perturbations. Our simulation results also show that inc reasing\nmagnetic field period causes a decline in critical and compensation tem pera-\nture values. However, in Ref. [40], it has been reported that the fi eld period\ndoes not alter the compensation behavior of a mixed ferrimagnetic b ulk sys-\ntem. In this regard, it can be concluded that the mechanism behind t he\nvariation of the compensation behavior with respect to the stocha stic dy-\nnamics in low dimensional systems such as magnetic bilayers may be rat her\ndifferent from those originated in bulk systems.\n124. Conclusion\nWehaveperformedMonteCarlosimulationsregardingthemagneticp rop-\nerties of an Ising bilayer system defined on a couple of stacked hone ycomb\nlatticeswherethesublattices AandBinteract via indirect exchange coupling\nJR. In the first part of our analysis, we have investigated the equilibriu m\nferrimagneticpropertiesofthesystem, andweobtained P-,N-,Q-typemag-\nnetization profiles which have been classified according to Ne´ el clas sification\nscheme. Compensation phenomenon suddenly disappears with decr easing\nstrength of indirect ferrimagnetic interlayer exchange coupling. W e have also\ncompared the obtained results with those reported in the literatur e, and\nfound that MC simulations qualitatively reproduce the magnetization curves\nobtained from EFT. In this regard, we have concluded that EFT met hod\nexhibits the same topology as those obtained from the MC simulation w ith\nless computational time. In the second part of our analysis, we hav e focused\non the evolution of compensation behavior observed in the system in the\npresence of a time depending magnetic field. Two different forms for the\ntime dependence of the dynamic magnetic field has been considered a s si-\nnusoidal oscillations, and square wave form. For both cases, comp ensation\npointTcompand transition temperature Tctend to decrease with increasing\nfield amplitude h0. The increasing field period Palso causes to the same\nconsequence. For the fixed values of h0andP, obtained TcompandTcvalues\nfor a square wave are clearly lower than those obtained for the sinu soidally\noscillating magnetic fields.\nInvestigation of dynamical critical properties of magnetic spin sys tems\nrevealed very rich physical phenomena, and these systems promis e even more\ninteresting andnovel features. Forinstance, whether thecritic al exponents of\nmagnetization and magnetic susceptibility exhibit any dimensional cro ssover\nas the geometry of the kinetic Ising bilayer system evolves from gra phene-\nlike structure to a graphite-like topology seems to be an interesting problem.\nHowever, this will be the subject of our near future work.\nAcknowledgements\nThe numerical calculations reported in this paper were performed a t\nTUBITAK ULAKBIM High Performance and Grid Computing Center (TR -\nGrid e-Infrastructure).\n13References\nReferences\n[1] K. S. Novoselov et al., “Two-dimensional atomic crystals“, Proc. Natl\nAcad. Sci. USA.\n[2] P. R. Wallace, “The band theory of graphite”, Phys. Rev. 71 (19 47) 622.\n[3] J. W. McClure, “Diamagnetism ofgraphite”, Phys. Rev. 104(195 6)666.\n[4] J. C. Slonczewski, P. R. Weiss, “Band structure of graphite”, P hys. Rev.\n109 (1958) 272.\n[5] A. K. Geim, K. S. Novoselov, “The rise of graphene”, Nat. Mater . 6\n(2007) 183.\n[6] S. Cahangirov, M. Topsakal, E. Akt¨ urk, H. S ¸ahin, S. Ciraci, “T wo- and\nOne-Dimensional Honeycomb Structures of Silicon and Germanium”,\nPhys. Rev. Lett. 102 (2009) 236804.\n[7] K. S. Novoselov et al., “Electric field effect in atomically thin carbon\nfilms”, Science 306 (2004) 666.\n[8] C. C. Price, N. B. Perkins, “Critical Properties of the Kitaev-He isenberg\nModel”, Phys. Rev. Lett. 109 (2012) 187201.\n[9] C. C. Price, N. B. Perkins, “Finite-temperature phase diagram o f the\nclassical Kitaev-Heisenberg model”, Phys. Rev. B 88 (2013) 02441 0.\n[10] A. Kitaev, “Anyons inanexactly solved model andbeyond”, Ann . Phys.\n(N.Y.) 321 (2006) 2.\n[11] S. Sarikurt, Y. Kadioglu, F. Ersan, E. Vatansever, O. ¨Uzengi Akt¨ urk,\nY. Y¨ uksel, ¨U. Akinci, E. Akt¨ urk, “Electronic and magnetic properties of\nmonolayer α−RuCl3: a first-principles and Monte Carlo study”, Phys-\nical Chemistry Chemical Physics 20 (2018) 997.\n[12] F. Ersan, E. Vatansever, S. Sarikurt, Y. Y¨ uksel, Y. Kadioglu , H. D.\nOzaydin, O. Uzengi Akt¨ urk, ¨U. Akinci, E. Akt¨ urk, “Exploring the Elec-\ntronicandMagneticProperties ofNewMetal HalidesfromBulktoTwo -\nDimensional Monolayer: RuX 3(X=Br, I)”, arXiv:1804.01560v1.\n14[13] T. Horiguchi, “A spin-one Ising model on a honeycomb lattice”, P hys.\nLett. 113 A (1986) 425.\n[14] V. Urumov, “The magnetisation of the spin-one Ising model on a hon-\neycomb lattice”, J. Phys. C: Solid State Phys. 20 (1987) L 875.\n[15] V. Urumov, “The dilute spin-one Ising model on a honeycomb latt ice”,\nJ. Phys.: Condensed Matter 1 (1989) 1159;\n[16] V. Urumov, “Errata: The dilute spin-one Ising model on a honey comb\nlattice”, J. Phys.: Condensed Matter 2 (1990) 2497.\n[17] V. Urumov, “Exact results for a spin-one Ising model with rand om crys-\ntal field”, J. Phys.: Condensed Matter 1 (1989) 7037.\n[18] Zhen-Li Wang, Zhen-Ya Li, “The randomly diluted site-bond, sp in-1\nIsing model on a honeycomb lattice”, J. Phys.: Condensed Matter 2\n(1990) 8615.\n[19] R. Masrour, A. Jabar, “Magnetic properties of bilayer graphe ne: a\nMonte Carlo study”, J. Comput. Electron 16 (2017) 12.\n[20] A. Mhirech, S. Aouini, A. Alaoui-Ismaili, L. Bahmad, “Study of RKK Y\nInteractions in a Bilayer Graphene Structure with Non-equivalent\nPlanes”, J. Supercond. Nov. Magn. 30 (2017) 3189.\n[21] W. Jiang, Y. Y. Yang, A. B. Guo, “Study on magnetic properties of a\nnano-graphene bilayer”, Carbon 95 (2015) 190.\n[22] J. P. Santos, F. C. S´ a Barreto, “An effective-field theory st udy of tri-\nlayer Ising nanostructure: Thermodynamic and magnetic propert ies”,\nJ. Magn. Magn. Mater. 439 (2017) 114.\n[23] T. Kaneyoshi, “Phase Transition in a Spin-1/2 and Spin-1 Ising Bila yer\nFilm with Non-magnetic Inter-layers”, J. Supercond. Nov. Magn (2 018),\nhttps://doi.org/10.1007/s10948-018-4606-y.\n[24] M. Acharyya, “Nonequilibrium phase transition in the kinetic Ising\nmodel: Existence of a tricritical point and stochastic resonance”, Phys.\nRev. E 59 (1999) 218.\n15[25] G. M. Buendia, P. A. Rikvold, “Dynamic phase transition in the two -\ndimensional kinetic Ising model in an oscillating field: Universality with\nrespect to the stochastic dynamics”, Phys. Rev. E 78 (2008) 051 108.\n[26] A. Berger, O. Idigoras, and P. Vavassori, “Transient Behavio r of the\nDynamically Ordered Phase in Uniaxial Cobalt Films”, Phys. Rev. Lett.\n111 (2013) 190602.\n[27] Y. Y¨ uksel, “Monte Carlo study of magnetization dynamics in unia xial\nferromagnetic nanowires in the presence of oscillating and biased ma g-\nnetic fields”, Phys. Rev. E 91 (2015) 032149.\n[28] E. Vatansever, N. G. Fytas, “Dynamic phase transition of the Blume-\nCapel model in an oscillating magnetic field”, Phys. Rev. E 97 (2018)\n012122.\n[29] T. Kaneyoshi, “Differential operator technique in the Ising spin sys-\ntems”, Acta Physica Polonica A 83 (1993) 703.\n[30] J.Strecka, M.Jascur, “AbriefaccountoftheIsingandIsing -likemodels:\nMean-field, effective-field and exact results”, Acta Physics Slovac a 65\n(2015) 235.\n[31] Y. Y¨ uksel, U. Akinci, “A comparative study of critical phenome na and\nmagnetocaloric properties of ferromagnetic ternary alloys”, J. P hys.\nChem. Solids 112 (2018) 143.\n[32] S. Y. Kim, “Honeycomb-lattice antiferromagnetic Ising model in a mag-\nnetic field”, Phys. Lett. A 358 (2006) 245.\n[33] S. Morita, S. Suzuki, “Phase Transition of Two-Dimensional Isin g Mod-\nels on the Honeycomb and Related Lattices with Striped Random Im-\npurities”, J. Stat. Phys. 162 (2016) 123.\n[34] R. J. C. Booth, L. Hua, J. W. Tucker, C.M. Care, I. Halliday, “Mo nte\nCarlo study oftheBEG model onahoneycomb lattice”, J. Magn. Mag n.\nMater. 128 (1993) 117.\n[35] J. W. Tucker, “Two-site cluster theory for the spin-one Ising model”, J.\nMagn. Magn. Mater. 87 (1990) 16.\n16[36] L. Ne´ el, “Magnetic properties of ferrites: ferrimagnetism an d antiferro-\nmagnetism”, Ann. Phys. 3 (1948) 137.\n[37] J. Streˇ cka, “Exact results of a mixed spin-1/2 and spin-S Isin g model\non a bathroom tile (48) lattice: Effect of uniaxial single-ion anisotrop y”\nPhysica A 360 (2006) 379 (and references therein).\n[38] For instance compare Fig. 5 of Ref. [23] with Fig. 3 of the presen t work.\n[39] T. Tom´ e, M. J. de Oliveira, “Dynamic phase transition in the kinet ic\nIsing model under a time-dependent oscillating field”, Phys. Rev. A 4 1\n(1990) 4251.\n[40] E. Vatansever, H. Polat, “Nonequilibrium dynamics of a mixed spin -1/2\nand spin-3/2 Ising ferrimagnetic system with a time dependent oscilla t-\ning magnetic field source”, J. Magn. Magn. Mater. 392 (2015) 42.\n17"    },    {        "title": "1903.09499v1.Learning_magnetization_dynamics.pdf",        "content": "Learning magnetization dynamics\nAlexander Kovacs1, Johann Fischbacher1, Harald Oezelt1, Markus\nGusenbauer1, Lukas Exl2, Florian Bruckner3, Dieter Suess3, and\nThomas Schre\r1\n1Department for Integrated Sensor Systems, Danube University\nKrems, Austria\n2WPI c/o Faculty of Mathematics, University of Vienna, A-1090\nVienna, Austria\n3Christian Doppler Laboratory for Advanced Magnetic Sensing\nand Materials, Faculty of Physics, University of Vienna, Austria\nMarch 25, 2019\nAbstract. Deep neural networks are used to model the magnetization dy-\nnamics in magnetic thin \flm elements. The magnetic states of a thin \flm\nelement can be represented in a low dimensional space. With convolutional au-\ntoencoders a compression ratio of 1024:1 was achieved. Time integration can\nbe performed in the latent space with a second network which was trained by\nsolutions of the Landau-Lifshitz-Gilbert equation. Thus the magnetic response\nto an external \feld can be computed quickly.\nKeywords: micromagnetics, magnetic sensors, machine learning, model or-\nder reduction\n1 Introduction\nMagnetic thin \flm elements are a key building block of magnetic sensors [1].\nIn order to compute the magnetic response of thin \flm elements, the Landau-\nLifshitz-Gilbert (LLG) equation is solved numerically. The \fnite di\u000berence [2]\nor \fnite element [3] computation of the demagnetizing \felds and the time inte-\ngration of the Landau-Lifshitz-Gilbert equation requires a considerable compu-\ntational e\u000bort. On the other hand electronic circuit design and real time process\ncontrol need models that provide the sensor response quickly. A possible route\nto build reduced order models that give the magnetic state as function of ap-\nplied \feld and time is the use of deep neural networks. Machine learning has\nbeen successfully used in \ruid dynamics in order to speed up simulations [4, 5].\nThese methods \frst learn a representation of the \ruid in reduced dimensions\n1arXiv:1903.09499v1  [cond-mat.mtrl-sci]  22 Mar 2019by convolutional neural networks. With the compressed \ruid states a second\nneural network is trained for the time integration in the latent space. Finally,\nthe velocity or pressure \felds along the trajectory are reconstructed.\nIn this letter we propose a convolutional neural network to reduce the di-\nmensionality of thin \flm magnetization and show how latent space dynamics\ncan be applied to predict the magnetic response of magnetic thin \flm elements.\nThe concept is demonstrated for the micromagnetic standard problem 4 [6].\nTable 1: Layout of the autoencoder. Here we use the names as used in Keras [7]\nto specify the type of the layer and the activation function. The \frst convolution\nlayer and the last convolution layer use a kernel width 4 \u00024. For all other\nconvolution layers the kernel width is 2 \u00022. For all convolution layers we use a\nstride 2\u00022. The drop out rate of the dropout layers is 0.1.\nLayer Activation Output shape\nInput - 64 \u0002256\u00023\nConv2D elu 32 \u0002128\u000216\nConv2D elu 16 \u000264\u000232\nConv2D elu 8 \u000232\u000264\nConv2D elu 4 \u000216\u0002128\nConv2D elu 2 \u00028\u0002256\nConv2D elu 1 \u00024\u0002512\nFlatten - 2048\nDropout - 2048\nDense elu 16\nDense elu 2048\nDropout - 2048\nReshape - 1 \u00024\u0002512\nConv2DTrans elu 2 \u00028\u0002256\nConv2DTrans elu 4 \u000216\u0002128\nConv2DTrans elu 8 \u000232\u000264\nConv2DTrans elu 16 \u000264\u000232\nConv2DTrans elu 32 \u0002128\u000216\nConv2DTrans tanh 64 \u0002256\u00023\n2 Methods\nIn order to build a neural network based reduced order model for e\u000bective time-\nintegration of the Landau-Lifshitz-Gilbert equation we require the following\nbuilding blocks:\n(i) Magnetic states obtained from Landau-Lifshitz-Gilbert numerical time-\nintegration used for training the neural networks,\n2Table 2: Layout of the predictor. Here we use the names as used in Keras [7]\nto specify the type of the layer and the activation function. The drop out rate\nof the dropout layer is 0.02.\nLayer Activation Output shape\nInput - 66\nDense elu 64\nDense elu 64\nDropout - 64\nDense elu 32\nDense elu 32\nDropout - 32\nDense elu 16\nDense elu 16\n(ii) Models to compute the latent space representation of a magnetic state and\nto reconstruct a full magnetization state from its compressed state, and\n(iii) A model to predict a future magnetic state in the latent space from pre-\nvious states in the latent space.\nLet us consider the discretized magnetization vector at the nTdiscrete time\npointsti; i= 1;:::;n Twithmi2(RN)3, whereNdenotes the number of\nspatial discretization points. Let Ebe the encoder model that compresses a\nmagnetic state mi, that is,E(mi) =ci2Rm, wherem\u001cNis the number\nof units in the output layer of the encoder model. Note that the compressed\nstates cihave much lower dimensionality than the states mi. The decoder\nmodelDbuilds a full magnetization emifrom the compressed state represen-\ntation, that is, D(ci) =emi. Our goal is to build a model Pthat predicts\nthe future time evolution in latent space from previous points in time, that is,\nP(h;ci\u0000n;:::;ci\u00001;ci) =eci+1, where the prediction eci+1should be as close as\npossible to ci+1andhdenotes the external \feld. In our simulations we set\nn= 3. Once we have trained a neural network that represents Pwe can loop in\ntime and follow the dynamics of the system in latent space. Finally, we decode\nthe compressed states along the trajectory to obtain an approximate solution\nof the Landau-Lifshitz-Gilbert equation.\nNeural networks require data for training. We use \fdimag [8] to generate\nmagnetic states via solving the Landau-Lifshitz-Gilbert equation. We apply an\nexternal \feld to the initial state de\fned in the speci\fcations of the micromag-\nnetic standard problem 4 and integrate the Landau-Lifshitz-Gilbert equation for\ndi\u000berent applied \felds. The grid resolution is 256 \u000264\u00021 which gives a mesh\nsize of 1:95 nm in-plane and 3 nm in the out-of-plane direction and N= 16 384.\nFor each \feld we integrate for one nanosecond and store the magnetic state\nevery 0:01 nanoseconds, so nT= 100. We compute trajectories for 200 di\u000berent\n\felds which gives a total of 200 \u0002nT= 20 000 magnetic states for training.\nWe apply \felds that trigger switching of the magnetic thin \flms. The \felds\n3oppose the initial magnetization and are applied in-plane. We randomly sample\nthe \felds from a segment with opening angle of 44 degrees. The \feld strengths\nvaries from 22 mT/ \u00160to 41 mT/\u00160.\nWe use unsupervised learning to \fnd compressed magnetic states. We train\na convolutional autoencoder [9]. An autoencoder learns to copy their inputs\nto their outputs. Thereby they learn representing the input state in lower di-\nmensionality. Autoencoders consist of several layers of neurons. The layers are\nsymmetric with respect to the central hidden layer. In our case the central hid-\nden layer has 16 units. Thus the dimension of a vector in latent space is m= 16.\nFrom the inputs to the hidden layer (encoder) the number of units decreases\nfrom layer to layer, from the hidden layer to the outputs (decoder), the number\nof units increases from layer to layer. The layout of the autoencoder is given in\nTable 1. The input to the neural network are the magnetization vectors at the\ncomputational grid points. Thus the shape of the input is 64 \u0002256\u00023. Convo-\nlution layers learn local patterns in a small two-dimensional window whose size\nis de\fned by the kernel width. The distance between two successive windows\nis called stride. With a 2 \u00022 stride each convolution layer reduces the number\nof features by a factor of 1 =2. The activation function determines the output\nof each unit of a layer. Clevert and co-workers [10] show that the exponential\nlinear unit (elu) speeds up learning of autoencoders. Dropout randomly sets to\nzero a number of output units of the layer during training. The dropout rate\nis the fraction of units beeing dropped. Dropout is an e\u000ecent means to avoid\nover\ftting in neural networks [11].\nTo train the autoencoder we minimize the following loss function (\fxed time\npointtiand omitting the index i):\nLED =L1+L2;where\nL1=X\nj(jmj;x\u0000emj;xj+jmj;y\u0000emj;yj+ 10jmj;z\u0000emj;zj) and\nL2=X\nj\u0012q\n(emj;x)2+ (emj;y)2+ (emj;z)2\u00001\u0013\n:\nPlease note that for training the autoencoder we do not include the external\n\feld as input. Here jrefers to the index of the computational cell; and x,y,z\nrefer to the Cartesian components of the unit vector of the magnetization. The\ninput and output of the autoencoder are the components mj;x; mj;ymj;zand\nemj;x;emj;y;emj;z, respectively. In soft magnetic thin \flms the magnetization is\npreferably in-plane. In order to train the network also for the small out-of-plane\ncomponent of the magnetization we weight the error in the z-component with\na factor of 10. The term L2is a penalty term that tries to keep the length of\nreconstructed magnetization vectors to 1. We split the 20 000 magnetic states\ninto 16 000 states used for training, 2 000 states used for validation, and 2 000\nused for testing the neural network. We tuned the hyper parameters per hand in\norder to minimize the loss function computed for the validation set. We obtain\ngood results by using the Nadam optimizer [12] for training with an initial\nlearning rate of 0 :0001. Nadam is a gradient descent optimization algorithm\n4which is supposed to converge quickly. The learning rate determines the step\nsize of the algorithm.\nFigure 1: Magnetization components in x,y, andzdirection integrated over\nthe sample as function of time for \feld 1 (left) and \feld 2 (right). Dashed lines:\nmicromagnetics, dots: reconstruction after encoding and decoding.\nThe second neural network is trained for predicting future magnetic states.\nHere we use a feed-forward neural network whose layout is given in Table 2.\nWe use 4 magnetic states from the past to predict the magnetic state at the\nnext time step. In addition to the magnetic states of the past the external \feld\nis an important input. Thus the input vector of the network has a length of\n2+4\u000216 = 66. In order to de\fne a loss function for the predictor we recursively\napply the predictor to 8 future magnetic states:\nLP=jci+1\u0000P(h;ci\u00003;ci\u00002;ci\u00001;ci)j2\n2+\njci+2\u0000P(h;ci\u00002;ci\u00001;ci;eci+1)j2\n2+\njci+3\u0000P(h;ci\u00001;ci;eci+1;eci+2)j2\n2+\njci+4\u0000P(h;ci;eci+1;eci+2;eci+3)j2\n2+\njci+5\u0000P(h;eci+1;eci+2;eci+3;eci+4)j2\n2+\njci+6\u0000P(h;eci+2;eci+3;eci+4;eci+5)j2\n2+\njci+7\u0000P(h;eci+3;eci+4;eci+5;eci+6)j2\n2+\njci+8\u0000P(h;eci+4;eci+5;eci+6;eci+7)j2\n2:\nTraining the neural network by looking ahead in time improves the predictive\ncapability. This way of training the predictor was originally applied by Kim and\nco-workers [4] for \ruid simulations. In order to generate the training data we\ncompress the magnetic states computed micromagnetically with the encoder.\nWe split the data into a training set, a validation set, and a test set. Again, we\nuse the Nadam optimizer [12] for training with a learning rate of 0.0001.\n5Figure 2: Magnetic states at di\u000berent times for an external \feld of \u00160Hx=\n\u000024:6 mT and\u00160Hy= 4:3 mT (\feld 1). Left: Micromagnetic result. Right:\nReconstructed magnetization after encoding and decoding.\n3 Results\nWe used the micromagnetic standard problem 4 to demonstrate the dimen-\nsionality reduction achieved by the autodecoder and the prediction of magneti-\nzation dynamics using latent space integration by a trained neural network.\nThe standard problem treats a 500 nm \u0002125 nm\u00023 nm permalloy ele-\nment. For computing the magnetization dynamics two di\u000berent external \felds\nshould be applied. Field 1 is \u00160Hext;1= (\u000024:6 mT;4:3 mT;0); and \feld 2 is\n\u00160Hext;2= (\u000035:5 mT;\u00006:3 mT;0). Please note that \feld 1 and \feld 2 were\nnot included in the training set and the validation set which were used to train\nthe neural networks. The dashed lines in Figure 1 show the magnetization com-\nponents as function of time. The dots give the magnetization obtained after\ncompression and reconstruction of the magnetic states with the autoencoder.\nThe results show that the autoencoder perfectly found a very low dimensional\nrepresentation of the magnetic states. Only for \feld 2 at around 0.5 ns there is\na small deviation of Mxfrom the reconstruction from Mxcomputed by micro-\nmagnetic simulations (see right hand side of Figure 1). To give a fair estimate\nof the compression rate we ignore the out-of-plane component of the magneti-\nzation. Then the magnetization vector at a grid point can be described by one\nmagnetization angle. For 64 \u0002256 computational grid points and 16 units in\nthe hidden layer of the autoencoder we achieve a compression ratio of 1024:1.\nFigure 2 compares magnetic states obtained from micromagnetic simulations\nand after reconstruction from the compressed states. Whereas no signi\fcant\ndi\u000berence is seen for the integrated quantities (see left hand side of Figure 1),\nslight di\u000berence in the local magnetization con\fguration can be observed at the\n6right side of the slab for 0 :6 ns.\nFigure 3: Examples for the representation of the magnetization in the hidden\nlayer of the autoencoder. The images give 4 examples reconstructed magneti-\nzation states when just one neuron of the hidden layer is activated.\nWe speculated whether the representation by the hidden layer of the autoen-\ncoder is related to the spectral modes of the sample [13]. Therefore we activated\njust one of the neurons of the hidden layer and decoded this state. Thus we\ncan see the magnetic state that corresponds to an active neuron of the hidden\nlayer. Figure 3 shows four out of the possible sixteen representations. In con-\ntrast to the spectral modes reported in [13] for the very same sample, some of\nthe 16 hidden states are clearly asymmetric. Thus we conclude that the sparse\nrepresentation achieved by the convolutional autoencoder is di\u000berent from those\nobtained by mode analysis [14] of magnetic samples. In fact, the notion of modal\nsubspace approximation is through linear combination of (di\u000berent) eigenmodes\nwhere in principle convergence is reached through expanding the basis subset\nsu\u000eciently. In the case of arti\fcial neural networks the approximation of con-\ntinuous functions is through a \fnite linear combination of a nonlinear activation\nfunction, like exponential linear unit (elu), which exhibits convergence due to\nthe universal approximation theorem [15][Th. 2].\nFor applying the neural network to time integration of the Landau-Lifhsitz-\nGilbert equation we computed the magnetic states at 0 :01 ns, 0:02 ns, 0:03 ns,\nand 0:04 ns using the micromagnetic solver. Using these four precomputed\nstates all states from 0 :05 ns to 1 ns were predicted using the neural network\npredictor. Figure 4 compares the micromagnetic results with the prediction from\nlatent space integration. For \u00160Hx=\u000024:6 mT and\u00160Hy= 4:3 mT (\feld 1)\nthe predictions are almost perfect. For \u00160Hx=\u000035:5 mT and\u00160Hy=\u00006:3 mT\n(\feld 2) some deviation occur between the ground truth and the prediction from\nthe neural network at around 0 :4 ns. This is not surprising when considering\nthat also di\u000berent conventional micromagnetic solvers diverge at approximately\nthe same time [16]. This divergence might be related to the annihilation of a\n360 degree domain wall and the resulting dynamics on a \fne length scale [6]. In\norder to learn this \fne-scale dynamics a larger training set might be required.\nComparing the ground truth and the predicted magnetic states for \feld 1\nat di\u000berent points in time time (see Figure 5), we see that some local details of\nthe magnetization distribution are lost at 0.6 ns and 0.8 ns.\n7Figure 4: Magnetization components in x,y, andzdirection integrated over the\nsample as function of time. The external \feld is \u00160Hx=\u000024:6 mT and\u00160Hy=\n4:3 mT (left) and \u00160Hx=\u000035:5 mT and\u00160Hy=\u00006:3 mT (right). Dashed\nlines: micromagnetics, dots: predicted magnetization from neural network based\nintegration in latent space.\n4 Discussion\nNeural network autodecoders may be an alternative to spectral modes for di-\nmensionality reduction of magnetic states in sensor elements. Although there\nis a computational e\u000bort associated with training the network for the speci\fc\ngeometry, the high compression rate may be bene\fcial for developing reduced\norder models of the magnetization dynamics. Solutions of the Landau-Lifshitz-\nGilbert equation for a range of di\u000berent external \felds were encoded and used\nto train a neural network for time integration in the latent space. The neural\nnetwork model is used to predict future magnetic states in compressed form.\nFinally, the time evolution of the magnetization is obtained by decoding the\npredictions. Though we did not do any measurement of CPU time, magneti-\nzation dynamics with pretrained neural networks is orders of magnitude faster\nthan the direct integration of the Landau-Lifshitz-Gilbert equation.\n5 Conclusions\nA machine learning approach for modeling magnetization dynamics in magnetic\nthin \flms elements was presented. Deep neural networks were applied for dimen-\nsionality reduction and for time integration in the latent space. The potential\nof this approach was demonstrated with the micromagnetic standard problem\n4. In summary, we show that neural network based reduced order models may\nhelp to simulate magnetization dynamics e\u000bectively for a prescribed range of\nparameters, like for the external \feld in our case. These models may be useful\nfor applications where computation time matters.\n8Figure 5: Magnetic states at di\u000berent times for an external \feld of \u00160Hx=\n\u000024:6 mT and\u00160Hy= 4:3 mT (\feld 1). Left: Micromagnetic result. Right:\nPrediction from latent space integration by a neural network model.\nAcknowlegement\nThe support from the Christian Doppler Laboratory Advanced Magnetic Sens-\ning and Materials (\fnanced by the Austrian Federal Ministry of Economy, Fam-\nily and Youth, the National Foundation for Research, Technology and Develop-\nment) is acknowledged. LE is supported by the Austrian Science Fund (FWF)\nvia the project \"ROAM\" under grant No. P31140-N32.\nReferences\n[1] D. Suess, A. Bachleitner-Hofmann, A. Satz, H. Weitensfelder, C. Vogler,\nF. Bruckner, C. Abert, K. Pr ugl, J. Zimmer, C. Huber, et al., Topologically\nprotected vortex structures for low-noise magnetic sensors with high linear\nrange, Nature Electronics 1 (6) (2018) 362 (2018).\n[2] J. E. Miltat, M. J. Donahue, Numerical Micromagnetics: Finite Di\u000berence\nMethods, American Cancer Society, 2007 (2007). arXiv:https://\nonlinelibrary.wiley.com/doi/pdf/10.1002/9780470022184.hmm202 ,\ndoi:10.1002/9780470022184.hmm202 .\nURL https://onlinelibrary.wiley.com/doi/abs/10.1002/\n9780470022184.hmm202\n[3] T. Schre\r, G. Hrkac, S. Bance, D. Suess, O. Ertl, J. Fidler, Numerical Meth-\nods in Micromagnetics (Finite Element Method), American Cancer Society,\n2007 (2007). arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.\n91002/9780470022184.hmm203 ,doi:10.1002/9780470022184.hmm203 .\nURL https://onlinelibrary.wiley.com/doi/abs/10.1002/\n9780470022184.hmm203\n[4] B. Kim, V. C. Azevedo, N. Thuerey, T. Kim, M. Gross, B. Solenthaler, Deep\n\ruids: A generative network for parameterized \ruid simulations, arXiv\npreprint arXiv:1806.02071 (2018).\n[5] S. Wiewel, M. Becher, N. Thuerey, Latent-space physics: Towards learn-\ning the temporal evolution of \ruid \row, arXiv preprint arXiv:1802.10123\n(2018).\n[6] R. D. McMichael, M. J. Donahue, D. G. Porter, J. Eicke, Switching dy-\nnamics and critical behavior of standard problem no. 4, Journal of Applied\nPhysics 89 (11) (2001) 7603{7605 (2001).\n[7] F. Chollet, et al., Keras, last accessed: Feb 27, 2019, https://keras.io\n(2015).\n[8] M.-A. Bisotti, D. Cort\u0013 es-Ortu~ no, R. Pepper, W. Wang, M. Beg, T. Kluyver,\nH. Fangohr, Fidimag{a \fnite di\u000berence atomistic and micromagnetic sim-\nulation package, Journal of Open Research Software 6 (1) (2018).\n[9] A. Gron, Hands-on machine learning with Scikit-Learn and TensorFlow:\nconcepts, tools, and techniques to build intelligent systems, OReilly Media,\nSebastopol, 2017 (2017).\n[10] D.-A. Clevert, T. Unterthiner, S. Hochreiter, Fast and accurate deep\nnetwork learning by exponential linear units (elus), arXiv preprint\narXiv:1511.07289 (2015).\n[11] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, R. Salakhutdinov,\nDropout: a simple way to prevent neural networks from over\ftting, The\nJournal of Machine Learning Research 15 (1) (2014) 1929{1958 (2014).\n[12] T. Dozat, Incorporating nesterov momentum into adam, ICLR Workshop\n(2016).\n[13] M. dAquino, C. Serpico, G. Bertotti, T. Schre\r, I. Mayergoyz, Spectral\nmicromagnetic analysis of switching processes, Journal of Applied Physics\n105 (7) (2009) 07D540 (2009).\n[14] F. Bruckner, M. dAquino, C. Serpico, C. Abert, C. Vogler, D. Suess, Large\nscale \fnite-element simulation of micromagnetic thermal noise, Journal of\nMagnetism and Magnetic Materials 475 (2019) 408{414 (2019).\n[15] G. Cybenko, Approximation by superpositions of a sigmoidal function,\nMathematics of Control, Signals and Systems 2 (4) (1989) 303{314 (Dec\n1989). doi:10.1007/BF02551274 .\nURL https://doi.org/10.1007/BF02551274\n10[16] R. McMichael, J. Eicke, \u0016MAG Standard Problem #4 results, last ac-\ncessed: Feb 27, 2019, https://www.ctcms.nist.gov/ ~rdm/mumag.org.\nhtml (2000).\n11"    },    {        "title": "0706.1746v2.Triplet_Josephson_effect_with_magnetic_feedback.pdf",        "content": "arXiv:0706.1746v2  [cond-mat.supr-con]  13 Jun 2007Triplet Josephson effect with magnetic feedback\nV. Braude and Ya. M. Blanter\nKavli Institute of Nanoscience, Delft University of Techno logy, 2628 CJ Delft, The Netherlands\n(Dated: October 31, 2018)\nWe study AC Josephson effect in a superconductor-ferromagne t heterostructure with a variable\nmagnetic configuration. The system supports triplet proxim ity correlations whose dynamics is\ncoupledto themagnetic dynamics. This feedback dramatical ly modifies the behavior ofthe junction.\nThe current-phaserelation becomes double-periodic at bot hverylow andhigh Josephson frequencies\nωJ. At intermediate frequencies, the periodicity in ωJtmay be lost.\nPACS numbers: 74.78.Fk, 74.50.+r, 72.25.Ba\nSpin-dependent transport through hybrid structures\ncombining ferromagnets (F) and normal metals has at-\ntracted a lot of interest in the recent years. This interest\nis motivated by the prospect of potential technological\napplications in the field of spintronics [1]. Particular at-\ntention is given to two related effects involving mutual\ninfluencebetweenthe electriccurrentthroughastructure\nand its magnetic configuration. The first is giant magne-\ntoresistance [2] in which the conductance is much larger\nwhen different magnetic regions have their magnetic mo-\nments aligned than when they are anti-aligned. The op-\nposite effect is the appearance of torques acting on mag-\nnetic moments when an electric current flows through\nthe system [3]. These non-equilibrium current-induced\ntorques appear due to non-conservation of spin currents\naccompanying a flow of charge through ferromagnetic re-\ngions. They allow manipulation of the magnetic config-\nuration, including switching between the opposite direc-\ntions or steady-state precession, without application of\nmagnetic fields [4]. The two effects combined promise\nimportant practical applications in nonvolatile memory,\nprogrammable logic, and microwave oscillators.\nWhen the multilayer is coupled to a superconductor\n(S), an additional constraint is added, viz. that the spin\ncurrent through the superconducting part vanishes [5].\nThis modifies the non-equilibrium torques, opening the\npossibility of perpendicular alignment of magnetic mo-\nments. A very different situation arises when a magnetic\nstructure is contacted by twosuperconductors. In this\ncase, the proximity effect may be present, leading to a\nfinite Josephson current through the structure at equilib-\nrium. The torques generated by this current correspond\nto anequilibrium effective exchange interaction between\nthe magnetic moments which can be controlled by the\nphase difference between the superconductors [6]. The\nsame mechanism enables the reciprocal effect in which\nthe supercurrent depends on the magnetic configuration.\nNaive considerations might suggest that the proximity\neffect should be suppressedat shortdistances in the pres-\nence of ferromagnets. However, recently it was shown\nthat a long-range effect can exist due to triplet super-\nconducting correlations [7]. This triplet proximity ef-\nfect (TPE), and in particular, the associated Josephsoncurrent, depend essentially on the magnetic configura-\ntion of the system [8]. Hence S/F multilayers exhibiting\nTPE are especially suitable for studying the Josephson-\ninduced magnetic exchange interaction.\nBy varying the relative magnetization directions of dif-\nferent magnetic regions, one can control the supercurrent\nflowingthroughthestructure. Then, ifthemagneticcon-\nfiguration is allowed to respond to the Josephson-current\ninduced torques, it creates feedback for the supercurrent\nand considerably modifies it. We show that its main\nsignature is frequency doubling in the current-phase re-\nlation.\nBelow, we consider this feedback for a Josephson junc-\ntion biased by a dc voltage. In the AC Josephson ef-\nfect, the time dependence is normally determined by the\ncurrent-phase relation. TPE in diffusive systems usu-\nally leads to the conventional J=Jcsinφrelation, ex-\ncept for a special magnetic configuration with mutually\nperpendicular directions where a transition between ”0”\nand ”π” states occurs [8]. Then the first harmonic van-\nishes and the current is given by the second harmonic\n∼sin2φ; however, its amplitude is relatively small. In\ngeneral, Josephson junctions exhibiting double-periodic\nbehavior, besides being interesting objects in proximity-\neffect studies, may be useful in flux-qubit design schemes\n[9]. Josephson frequency doubling was predicted in other\ntypesofjunctions involvingunconventionalsuperconduc-\ntors, such as s-p [10], s-d-s [9], p-p and d-d junctions with\nspecific misorientationanglesofthe orderparameter[11].\nIt was observed in experiments involving d-wave grain-\nboundary junctions [12]. It should be stressed, however,\nthat in all these cases the frequency doubling occurs at\nisolated points in the parameter space where the first\nharmonic vanishes. Moreover, the magnitude of the cur-\nrent is suppressed in comparison with the usual value\n∼∆/eRn, whereRnis the normal-state resistance.\nIn this work, we consider the magnetic exchange in-\nteraction induced by Josephson currents in a dirty S/F\nheterostructureexhibiting TPE. We show that this inter-\naction may prefer non-collinear magnetic configurations\nand the preferred direction depends continuously on the\nsuperconducting phase difference. Thus, the static mag-\nneticconfigurationcanbecontrolledbytheappliedphase2\ndifference. We then consider the influence of feedback\nfrom the magnetic moments on the AC Josephson effect.\nThe magnetic system exhibits a range of different be-\nhaviors, from simple harmonic oscillations to fractional-\nfrequency periodic behavior and chaotic motion. A fi-\nnite zero-frequency deviation from the equilibrium con-\nfiguration may appear, allowing control of the direction\nof the average magnetization also by an applied volt-\nage. The magnetic feedback complicates the behavior\nof the current in the time domain, making it generally\nimpossible to express it in terms of a current-phase re-\nlation. On the other hand, we find that both in the\nlow- and high frequency limit such a relation becomes\nmeaningful, with the current exhibiting a double-phase\ndependence, J∼sin2φ(t) orJ∼cos2φ(t). The criti-\ncal current in the low-frequency regime is of the order of\nthe value ETh/eRn, characteristic for diffusive systems.\nThe unusual cosine dependence of the Josephson current\nappears when Gilbert damping is important in the mag-\nnetic dynamics, breakingthe time-reversalsymmetry. At\nhighfrequencies,themagnetizationcannoteffectivelyfol-\nlow the phase variation, leading to a ∼1/ω2suppression\nof the effective Josephson coupling. At even higher fre-\nquencies, the damping is dominant, and the frequency\ndependence becomes ∼1/ω. The presence of damping\nis expressed in the appearance of a dc component of the\ncurrent leading to a finite resistance.\nThe system . We consider an S/F heterostructure\ndescribed in Fig. 1 which is a minimal discrete setup ex-\nhibiting the triplet proximity effect. Two magnetic re-\ngions 1 and 3 are adjacent to the superconducting reser-\nvoirs that induce proximity mini-gaps ∆ 1,3in them. Be-\ntween these regions there is an additional magnetic re-\ngion 2 whose length is much larger than ξhand where\ntriplet superconducting correlations are induced. This\nregion is assumed to be weakly polarized (metallic), so\nthat both spin directions are present at the Fermi sur-\nface. The magnetic regions are characterized by the ex-\nchange energies hi, while the magnetization directions\nniare specified by the angles θ1,θ3andχas shown in\nS\nF□1 F□2 F□3Sn1 n2n3z\nxyn1n3q1q3\ncn2\nFIG. 1: The experimental setupFig. 1. Assuming that the conductances of these regions\nare much higher than the conductances gn\n1,3of the con-\nnectors between them, our system can be described by a\ncircuit-theory model for the triplet proximity effect used\nin Ref. 8. Magnetization directions ofregions1 and 2 are\nassumed fixed, e.g.by pinning to an antiferromagnetic\nsubstrate, or by geometrical shaping, with the angle be-\ntween them being θ1. On the other hand, magnetization\nn3is free to rotate, with region 3 separated by a normal\nspacer from region 2 in order to avoid exchange coupling\nbetween them.\nIn accordance with the model assumptions, regions 1\nand 3 act as effective S-F reservoirs, hence their energies\nare independent of the magnetic configuration. On the\nother hand, triplet superconducting correlations extend-\ning through region 2, are very sensitive to the magnetiza-\ntion directions. Hence the configuration-dependent part\nof the energy can be found by integrating over the den-\nsity of states (DOS) in region 2. The DOS for each spin\ndirection is given by [8]\nν↑,↓(ε) =ν0\n2Re/parenleftbigg\n1−a2\n1+a2\n3+2a1a3cos(φ±χ)\n(b1+b3−iǫ/ETh)2/parenrightbigg−1\n2\n,\n(1)\nwhere ν0is the normal-state DOS, ak=\ngn\nk|∆k|sinθk/(gn\n1+gn\n3)/radicalbig\nh2\nk−|∆k|2,bk=gn\nkhk/(gn\n1+\ngn\n3)/radicalbig\nh2\nk−|∆k|2,φis the superconducting phase differ-\nence, and EThis the Thouless energy of the structure.\nUsing this expression, one can see that the energy is\ngiven by a logarithmic integral and the main contribu-\ntion comes from ǫ≫ETh. In the leading order one\nobtains\nE=ν0v2\n2log∆cut\nEThE2\nTh/parenleftbig\na2\n1+a2\n3+2a1a3cosφcosχ/parenrightbig\n,\n(2)\nwherev2is the volume of the magnetic region 2 and\n∆cut≃min(∆ i,hi−∆i) is a cutoff energy. This ex-\npression can be written in a form presenting explicitly\nthe dependence on the orientation angles θ3andχ,\nE=p2\n3sin2θ3+2p1p3sinθ3cosφcosχ,(3)\nwithp1,3being effective exchange couplings for the mag-\nnetic vector n3. The stable configuration is achieved\nwhen all magnetization directions are in the same plane,\ndenoted in the following as the x−zplane, and n3is\ntilted with respect to n2by a finite angle satisfying\nsinθ3=p1\np3|cosφ|. (4)\nThis angle depends continuously on the applied super-\nconducting phase difference φ, while the angle χassumes\nthe values 0 or πso that the product cos φcosχis nega-\ntive. In fact, there are two stable directions, given by the\nanglesθ3andπ−θ3. In what follows we will treat them3\n−1 −0.5 0 0.5 1−0.2−0.100.10.20.3\nnxnyωJ=0.1\n T= T0a)\n−1 −0.5 0 0.5 1−1−0.500.51\nnxωJ=0.3 b)\n−0.8 −0.6 −0.4 −0.2−1−0.500.51\nnxnyωJ=0.8\n T=2T0c)\n−1 −0.5 0 0.5 1−0.8−0.6−0.4−0.200.20.40.60.8\nnx ωJ=1.3; \n  T=3T0d)\nFIG. 2: Trajectories of the magnetization vector in the x−y\nplane for different Josephson frequencies (given in units of\nωm). Trajectory (b) is chaotic, while trajectory (c) has a\nfinite zero-frequency component for nx. HereTis the period\nof the trajectory, and T0= 2π/ωJ. For comparison, the low-\nfrequency trajectory lies entirely on the xaxis.\nas equivalent, since they correspond to the same current.\nThe energy of the stable configuration is given by\nEmin=−p2\n1cos2φ . (5)\nHence allowing the magnetization direction n3to orient\nitselfalongthe stable directionleadsto the current-phase\nrelationJ=Jcsin2φ.\nLow frequencies . When a small voltage Vis applied\nto the structure, such that the corresponding frequency\nωJ= 2eV/¯his much smaller than the characteristic fre-\nquencyofthemagneticsystem ωm(seebelow), thevector\nn3follows the stable direction given by Eq. (4), perform-\ning slow oscillations in the x−zplane. The alternating\nJosephson current oscillates with the double frequency\nJ=2e\n¯hp2\n1sin4eV\n¯ht , (6)\nwhile the critical current remains of the same order of\nmagnitude as in the case with a fixed magnetic configu-\nration.\nFor higher Josephson frequencies, the variation of n3\nis no more limited to the x−zplane. Instead, the mag-\nnetization performs a variety of non-harmonic motions\nwhose frequency may be a multiple or a fraction of the\ndriving frequency ωJ[Fig. 2 (a), (c), (d)]. For certain\ntrajectories the time average of θ3is finite [Fig. 2 (c)],\ncorresponding to a tilt of n3away from the equilibrium\ninresponsetoanappliedvoltage. Within somefrequency\nintervals, the motion is chaotic, asshown in Fig. 2 (b). In\nthese intermediate regimes, the Josephson current showsa complicated time dependence which is generally not\nperiodic in 2 π/ωJ. Hence this dependence cannot be pa-\nrameterizedin termsofthephase. Instead, onecanspeak\nof a Josephson current with a time-dependent coupling.\nHigh frequencies . When applied voltage is high, the\nJosephsonfrequencybecomesmuchhigherthanthe mag-\nnetic frequencies. In this case the magnetic vector n3\ncannot effectively follow the fast oscillations of the po-\ntential, and the time-averaged potential seen by n3has\na minimum for n3/bardblz. The motion of n3can be deter-\nmined by expanding n3=z+δnand using a linearized\nLandau-Lifshits-Gilbert (LLG) equation,\nδ˙n=z(−γ×Heff+αδ˙n), (7)\nwhereγis the gyromagnetic ratio, αis the effective\ndamping coefficient, Heff=−∂E/∂m3is the effective\nfield, and m3is the magnetization density of magnetic\nregion 3.\nWhen Gilbert damping is negligible, the trajectory of\nn3has a very low aspect ratio, so that the motion is\nalmost completely confined to the yaxis. It is given by\nδnx=γ2p1p3\n3¯h2\ne2V2m2\n3cos2eVt\n¯h;\nδny=γp1p3¯h\neVm3sin2eVt\n¯h. (8)\nThus at high frequencies, n3precesses in phase with the\nvoltagepumping. ThisleadstoanincreaseintheJoseph-\nson energy, and, correspondingly, a negative Josephson\ncurrent,\nJ=−2¯h\ne/parenleftbiggγp1p2\n3\nVm3/parenrightbigg2\nsin4eVt\n¯h. (9)\nHence in the high-frequencyregime the system showsnot\nonly frequency doubling, but also an effective π-junction\nbehavior. The magnitude of the current is suppressed as\n∼V−2as shown in Fig. 3.\nThe neglect of damping is justified as long as αωJ≪\nωm=γp2\n3/m3. When the voltage is high enough, this\ncondition is not satisfied anymore, and the dissipation\nstarts to be important. As the Josephson frequency be-\ncomes so large that the opposite inequality holds, the\nmotion of n3is determined by the driving against the\ndamping force,\nδn=γp1p3¯h\neVm3(−αˆx+ˆy)sin2eVt\n¯h.(10)\nThen the Josephson current is given by\nJ=2αγp2\n1p2\n3\nm3V/parenleftbigg\n1−cos4eVt\n¯h/parenrightbigg\n. (11)\nNote the unusual cosine dependence on the phase. It oc-\ncurs since the time-reversal symmetry is broken by the4\n10010110−410−310−210−1\nωJ/ωmamplitude∼ ωJ−1 ↑← ∼ ωJ−2 \nFIG. 3: The absolute value of the Josephson current harmon-\nics proportional to sin2 ωJt(asterisks) and to 1 −cos2ωJt\n(dots). Solid lines are fits ∼1/ωand 1/ω2. The data\npoints are obtained from numerical integration of the full\n(non-linear) LLG equation.\ndissipation in this regime. Due to the same reason, a\nzero-frequency component of the current appears, signi-\nfying the onset of a finite nonlinear resistance across the\nstructure. Since this regime is governed by the damp-\ning, the current amplitude is proportional to α, while\nthe suppression ∼1/Vis weaker in this regime (Fig. 3).\nTo estimate the magnetic dynamics frequency ωm, we\nuse typical values ETh∼1 meV, ν0∼1/(eV/atom),\nandm3∼1µB/atom, where µBis the Bohr magneton.\nThenωm∼v2/v3GHz, where v2,3are the volumes of\nthe corresponding magnetic regions. As this frequency\nis quite low, observation of the high-frequency regimes\nshould present no difficulty. On the other hand, the low-\nfrequency AC regime would require extremely low volt-\nages, below 1 µV. A reasonable alternative would be in-\ncorporating the structure in a superconducting loop and\nmeasuring the Josephson current as a function of the ap-\nplied flux.\nApplicability of our model requires that any mag-\nnetic anisotropy of part 3 should be smaller than the\nproximity-induced energy, Eq. (2). With the above val-\nues of the parameters it is of the order of 104×v2J/m3,\nso one should choose materials with low value of the\ncristalline anisotropy, such as permalloy. Finally, we em-\nphasize that the properties discussed above are specific\nfor metallic systems. In half-metals, the behavior will\nbe very different. Thus, in the low-frequency regime n3\nprecesses around n2at a constant angle θ3, while the\nJosephson current vanishes.\nConclusions . We have considered the AC Joseph-\nson effect in a S/F/S structure with magnetic dynam-\nics coupled to the dynamics of superconducting correla-\ntions. The magnetic configuration in the structure was\nassumed to be non-uniform so that the structure ex-hibits a triplet proximity effect. Variation of the mag-\nnetic configuration is shown to essentially modify the\ncurrent behavior that can be observed in the appearance\nof fractional Shapiro steps. Thus measurement of the\nJosephson current would provide information about the\ncoupling and self-consistent feedback dynamics between\nthe superconducting and magnetic degrees of freedom.\nThe coupling also allows to control the magnetization\ndirection by means of applied voltage or superconduct-\ning phase. In the low-frequency limit, the magnetization\nfollows the immediate potential minimum, leading to a\n∼sin2φcurrent-phase relation. The critical current has\nthe same order of magnitude ETh/eRnas that due to the\nusual singlet proximity effect in dirty structures. In the\nhigh-frequency regime, as long as the damping is not im-\nportant, the Josephson current is negative, correspond-\ning to aπ-junction behavior. It is suppressed by a factor\n∼(ωm/ωJ)2relative to the low-frequency regime. At\neven higher frequencies, Gilbert damping starts playing\nthe major role in the dynamics. Then the time-reversal\nsymmetry is broken and the current-phase relation takes\nan unusual cosine form. In addition, a DC component\nof the current appears, manifesting itself in a finite resis-\ntance. The current suppression becomes weaker in this\nregime.\nThe authors are grateful to G. E. W. Bauer and\nYu. V. Nazarov for useful discussions. This work was\nsupported by EC Grant No. NMP2-CT2003-505587\n(SFINX).\n[1] M. A. M. Gijs and G. E. W. Bauer, Adv. in Phys. 46,\n285 (1997); I. ˇZuti´ c, J. Fabian, and S. Das Sarma,\nRev. Mod. Phys. 76, 323 (2004).\n[2] F. J. Himpsel , J. E. Ortega, G. J. Mankey, and R. Willis,\nAdv. Phys. 47, 511 (1998).\n[3] J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989);\nJ. Magn. Magn. Matter, 159, L1 (1996); L. Berger, Phys.\nRev. B54, 9353 (1996).\n[4] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n[5] X. Waintal and P. W. Brouwer, Phys. Rev. B 63, 220407\n(2001).\n[6] X. Waintal and P. Brouwer, Phys. Rev. B 65, 054407\n(2002).\n[7] For a review, see F. S. Bergeret, A. F. Volkov, and\nK. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005).\n[8] V. Braude and Yu. V. Nazarov, Phys. Rev. Lett. 98,\n077003 (2007).\n[9] L. B. Ioffe, V. B. Geshkenbein, M. V. Feigel’man,\nA. L. Fauch` ere, and G. Blatter, Nature 398, 679 (1999).\n[10] Y. Asano, Y. Tanaka, M. Sigrist, and S. Kashivaya, Phys.\nRev. B67, 184505 (2003).\n[11] T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev.\nB75, 094514 (2007).\n[12] E. Il’ichev et al., Phys. Rev. B 60, 3096 (1999)."    },    {        "title": "1103.4492v1.Zero_temperature_spin_glass_freezing_in_self_organized_arrays_of_Co_nanoparticles.pdf",        "content": "arXiv:1103.4492v1  [cond-mat.mes-hall]  23 Mar 2011epl draft\nZero-temperature spin-glass freezing in self-organized a rrays of\nCo nanoparticles\nR. L´opez-Ruiz1, F. Luis∗1, J. Ses´e2, J. Bartolom ´e1, C. Deranlot3andF. Petroff3\n1Instituto de Ciencia de Materiales de Arag´ on - CSIC-Univer sidad de Zaragoza, 50009 Zaragoza, Spain, and\nDepartamento de F´ ısica de la Materia Condensada, Universi dad de Zaragoza, 50009 Zaragoza, Spain\n2Instituto de Nanociencia de Arag´ on, Universidad de Zarago za, and Departamento de F´ ısica de la Materia Conden-\nsada, Universidad de Zaragoza, 50009 Zaragoza, Spain\n3Unit´ e Mixte de Physique CNRS/Thales - Route D´ epartementa le 128, 91767 Palaiseau Cedex, France, and Universit´ e\nParis-Sud - 91405 Orsay Cedex, France\nPACS75.50.Tt – Fine-particle systems; nanocrystalline materials\nPACS75.40.Gb – Dynamic properties\nPACS75.50.Lk – Spin glasses and other random magnets\nAbstract. - We study, by means of magnetic susceptibility and magnetic aging experiments,\nthe nature of the glassy magnetic dynamics in arrays of Co nan oparticles, self-organized in N\nlayers from N= 1 (two-dimensional limit) up to N= 20 (three-dimensional limit). We find no\nqualitative differences between the magnetic responses mea sured in these two limits, in spite of\nthe fact that no spin-glass phase is expected above T= 0 in two dimensions. More specifically,\nall the phenomena (critical slowing down, flattening of the fi eld-cooled magnetization below the\nblocking temperature and the magnetic memory induced by agi ng) that are usually associated\nwith this phase look qualitatively the same for two-dimensi onal and three-dimensional arrays. The\nactivated scaling law that is typical of systems undergoing a phase transition at zero temperature\naccounts well for the critical slowing down of the dc and ac su sceptibilities of all samples. Our data\nshow also that dynamical magnetic correlations achieved by aging a nanoparticle array below its\nsuperparamagnetic blocking temperature extend mainly to n earest neighbors. Our experiments\nsuggest that the glassy magnetic dynamics of these nanopart icle arrays is associated with a zero-\ntemperature spin-glass transition.\nIntroduction. – Dense arrays of magnetic nanopar-\nticles contain the physical ingredients of spin-glasses [1].\nDisorder in the positions and orientations of the particles\nleads to disorder and frustration of the dipolar interac-\ntions, usuallydominant, betweentheirmagneticmoments.\nIn contrast with ”canonical” spin-glasses, the slow mag-\nnetic relaxation introduced by interactions coexists with\nthe slow magnetization reversal associated with the high\nanisotropy energy barriers. Many experiments performed\nondense nanoparticulatematerialsshowphenomena, such\nas magnetic aging [2] and the slowing down of the ac\nsusceptibility [3], which are typicalof spin-glasses [4–11].\nHowever,someofthesephenomenaarenot exclusive ofthe\nspin-glass phase [12,13]. The question is, then, whether\nreal nanoparticulate materials show a true (super)spin-\nglass phase.Experimental studies are often hindered by the lack of\ncontrol over the sample parameters that determine the\nnature and strength of dipolar interactions, such as inter-\nparticle distances, spatial organization, etc. This usually\nmakes it difficult to know a priori if a particular system\nis expected to show a spin-glass phase. Perhaps the most\nclear-cutsituation to discuss the existence ofa phase tran-\nsitionand its experimentalmanifestations is offeredby the\nstudy of a single layer of nanoparticles. In contrast with\nthree-dimensional systems [14], it is generally accepted\n[15,16] that the transition temperature Tgvanishesin two-\ndimensions. Results of tempered Monte Carlo simulations\nseem to confirm the same conclusion also for Ising spins\ncoupled by dipolar interactions [17].\nBased on these considerations, our work was aimed to\nelucidate the nature of the glassy magnetic dynamics, i.e.\np-1R. L. L´ opez et al.\nwhether it is associated with a superspin-glass phase at\nTg>0 or if, by contrast, Tg= 0, in self-organized\nnanoparticlearrays. Forthis, we compareresults obtained\non very well-characterized three- and two-dimensional ar-\nrays of Co nanospheres. Previous experiments reveal that\nthe superparamagnetic blocking temperature Tb, defined\nas the temperature of the in phase χ′ac susceptibility\ncusp, increases as additional layers are deposited on a\ntwo-dimensional sample [18, 19]. Since the number of\nlayers modifies the number of nearest neighbors in the\nnanoparticlearray,that result indicates that dipolarinter-\nactions slow down the magnetic relaxation processes. In\nthe present study, we haveinvestigated how the number of\nlayersmodifies the critical slowing down and the magnetic\naging, properties that are usually associated with the spin\nglass behavior [20]. Our results show that no qualitative\nchanges in these quantities occur as the two-dimensional\nlimit is approached. The control over the number of lay-\ners and their separation has also enabled us to directly\nprobe the magnetic correlation length and show that it is\nmainly restricted to a first shell of nearest neighbors and,\nin any case, shorter than what would be expected for a\nconventional spin glass.\nExperimental details. – Samples made of Nlay-\ners of Co nanoparticles with average diameter D≃2.6\nnm were prepared by the sequential sputtering of N=\n1,2,3,4,5,7,10,15,and 20 Co and Al 2O3layers on sili-\ncon substrates [18,19,21,22]. The particle’s shape and\naverage size (thus also the average magnetic moment µp\nper particle), as well as the width of the size distribu-\ntion (σD= 0.26D) are approximately independent of N\n[19]. Nanoparticles deposited on adjacent layers tend\nto self-organize in a structure that resembles a closed-\npacked hexagonal lattice of nanospheres [22]. The separa-\ntion between the Co layers is determined by the thickness\ntAl2O3= 3 nm of the alumina layer. Nearest neighbors\nseparations are dnn,/bardbl≃4.6 nm, within a given layer, and\ndnn,⊥≃4.2nm, betweenadjacentlayers. Theycorrespond\nto dipolar energies Edip=µ2\np/d3\nnn≈13 K and 17 K, re-\nspectively. Asdescribedin[23], theanisotropyenergybar-\nrierU0for the magnetization reversal was estimated from\nac susceptibility experiments performed under sufficiently\nstrong magnetic fields, which dominate over dipolar inter-\nactions. This method gives U0≃430 K. In the same way,\nwe estimate an attempt time τ0∼10−13s, of the same\norder of that found for samples of very small Co nanopar-\nticles (D∼1 nm), prepared by the same technique [18],\nfor which interactions are expected to become neglibible.\nA multilayer with N= 20 layers but a larger interlayer\nseparation tAl2O3= 10 nm, and thus also a much smaller\ninterlayer Edip≈1.6 K, was prepared under identical ex-\nperimental conditions.\nAc susceptibility and magnetizationmeasurementswere\nperformed with a commercial SQUID magnetometer.\nSamples were rectangular plates with approximate dimen-\nsions 9×3×0.5 mm3. Ac and dc magnetic fields wereparallel to the plane of the sample to minimize demagne-\ntizing effects. In our study of aging [6,7], we measured the\ntime-dependent relaxation of the zero-field cooled (ZFC)\nmagnetization on samples aged, at zero field, for a time tw\nat temperatures Tw< Tb. In addition, we employed a dif-\nferent method which consists on measuring magnetization\ncurves (zero-field and field cooled (FC), and remanence)\nusing the waiting time protocol described in [24,25].\nResults and discussion. – A typical method to\ncharacterize the spin-glass behavior is by measuring the\nfrequency-dependent ac magnetic susceptibility [3–5]. At\nany fixed frequency ωwe define a characteristic relaxation\ntimeτcsuch that ωτc(T) = 1 at T=Tb(ω). For spin\nglassesτcdiverges at Tgaccording to a power law, reflect-\ning the growth of magnetic correlations [4]\nτc=τ∗|1−T/Tg|−zν(1)\nIn Fig. 1, we plot τcversus the reduced temperature for\nNranging from 1 to 20 layers. The experimental data are\ncompatible with a critical slowing down of the magnetiza-\ntion dynamics at a finite Tg. In order to limit the number\nof fitting parameters, we took Tg, for each sample, as the\ntemperature of the ZFC susceptibility cusp (i.e. equal to\ntheTbcorresponding to a typical timescale of the order of\n170 s). The microscopic time scale τ∗and the dynamical\ncritical exponent zνare found to be nearly the same for\nallsamples. The criticalexponent is closeto typical values\nfound for spin glasses [5] as well as for some nanopartic-\nulate materials [26]. The present results are remarkable\nbecause it is generally believed that Tg= 0 in two dimen-\nsions. Notice however that, as often happens with plots of\nthis type obtained fornanoparticles[3,26,27], experiments\ndo not explore the close vicinity of the critical region. For\nthis reason, the data are relatively easy to fit. Fits of sim-\nilar quality can be obtained by scaling all Tg’s by a factor\nin between 1 and 0 .75. The characteristic τ∗increases\nthen from about 10−6s to 10−4s while, at the same time,\nthe exponent zνincreases from 7 .3 to 14. In fact, if one\nwishes to include also in the analysis the freezing tem-\nperature extracted from ZFC susceptibility data (getting\ncloser to Tg), the best fits with Eq. (1) are obtained then\nfor the largest zνandτ∗values (thus also for the lowest\nTg). Such large zνvalues are not uncommon in systems\nof magnetic nanoparticles [3,27] but they are significantly\nlarger than what it is expected for a canonical spin glass\nphase transition (of the order of zν= 7 [28]).\nAn alternative theoretical framework to describe the\nfrequency-dependent susceptibility, which seems very ap-\npropriatein the caseof a layeredmaterial with a markedly\ntwo-dimensionalcharacter,istheactivateddynamicschar-\nacteristic of glassy systems undergoing a phase transition\natTg= 0 [4,16]. In the latter situation, the critical slow-\ning down of τcobeys the following expression\nτc=τ0exp(Ea/kBT)σ(2)\np-2Zero-temperature spin-glass freezing in self-organized arrays o f Co nanoparticles\n/s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50\n/s32/s78 /s32/s61/s32/s49\n/s32/s78 /s32/s61/s32/s50\n/s32/s78 /s32/s61/s32/s51\n/s32/s78 /s32/s61/s32/s53\n/s32/s78 /s32/s61/s32/s49/s48\n/s32/s78 /s32/s61/s32/s49/s53\n/s32/s78 /s32/s61/s32/s50/s48\n/s32/s102/s105/s116\n/s32/s32/s99/s40/s115/s41\n/s32/s61/s32/s40 /s84 /s47/s84\n/s103/s45/s49/s41/s122 /s32/s61/s32/s55/s46/s51/s40/s51/s41\n/s108/s111/s103/s32\n/s42/s32/s61/s32/s45/s54/s46/s49/s40/s51/s41\nFig. 1: (Color online). Critical slowing down of the charac-\nteristic relaxation time extracted from ac susceptibility exper-\niments. Results are shown for samples with varying number of\nlayersN.\nwhereEais an effective activation energy and σis a crit-\nical exponent. As it is shown in Fig. 2, we find a good\nagreement with our data, including also the temperature\nof the ZFC magnetization cusp, for σ= 1.3 (to be com-\npared with σ= 3.2 found for 2 −Dspin-glasses [16]) and\nEagradually increasing with the number of layers from\n345 K up to 471 K. From these frequency-dependent sus-\nceptibility experiments, we conclude that the nature of\nthe slow magnetic dynamics of two-dimensional (i.e. with\nNequal to or close to unity) and three-dimensional (with\nlargeN) nanoparticle arrays is the same. The description\nbased on a zero-temperature phase transition is appeal-\ning, because it is consistent with the behavior expected\nfor a single layer. By themselves, however, these experi-\nments cannot discriminate between the two alternatives,\ni.e., whether the underlying physics corresponds to the ex-\nistence of a second-order phase transition at a finite Tgor\nif, by contrast, Tg= 0.\nAging experiments can shed some light and help decid-\ning between these two alternatives, since they probe how\ndynamical magnetic correlations grow with time [6,8,11].\nWehavecarriedouttwodifferentexperiments, whichmea-\nsure the magnetic memory effects associated with the\naging of the sample at a given temperature. In the\nfirst of these, the quantity of interest is the difference\n∆M=M−Mwbetween the magnetizations (ZFC, FC\nor remanent) measured after cooling the sample without\nor with a pause at an intermediate temperature Tw< Tb\n[24]. Results measured for tw= 104s are shown in Fig. 3.\n∆MZFCshows a peak centered near Tw. IfTwis varied,\nthe peak shifts accordingly. In addition, the relationship/s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s32/s78 /s32/s61/s32/s49/s44/s32 /s69\n/s97 /s47/s107\n/s66 /s32/s61/s32/s51/s52/s53/s32/s75\n/s32/s78 /s32/s61/s32/s50/s44/s32 /s69\n/s97 /s47/s107\n/s66 /s32/s61/s32/s51/s57/s51/s32/s75\n/s32/s78 /s32/s61/s32/s51/s44/s32 /s69\n/s97 /s47/s107\n/s66 /s32/s61/s32/s52/s49/s55/s32/s75\n/s32/s78 /s32/s61/s32/s53/s44/s32 /s69\n/s97 /s47/s107\n/s66 /s32/s61/s32/s52/s53/s55/s32/s75\n/s32/s78 /s32/s61/s32/s49/s48/s44/s32 /s69\n/s97 /s47/s107\n/s66 /s32/s61/s32/s52/s54/s48/s32/s75\n/s32/s78 /s32/s61/s32/s50/s48/s44/s32 /s69\n/s97 /s47/s107\n/s66 /s32/s61/s32/s52/s55/s49/s32/s75\n/s32/s32/s108/s110/s40\n/s99/s47\n/s48/s41\n/s84 /s40/s75/s41/s32/s61/s32/s49/s46/s51\nFig. 2: (Color online). Log-log plot showing the variation w ith\ntemperature of ln( τc/τ0), where τcis a characteristic relaxation\ntime extracted from ac susceptibility data and τ0= 10−13s.\nThe lines are fits of the law τc=τ0exp(Ea/kBT)σ, character-\nistic of a spin-glass transition at Tg= 0\n[24]∆MFC= ∆Mr+∆MZFCisfulfilled, showingthatthey\nare associated with the aging of the sample at Twand not\nwith experimental artifacts.\nFigure 3 compares results obtained on a single layer\nN= 1 with those measured on a multilayer made of\nN= 20 layers. The aging was performed at Tw= 0.7Tb\nfor the two samples. Besides the obvious difference in\nthesignal-to-noiseratios,theylookqualitativelythesame.\nThe maximum in ∆ MZFCvsTis just about 25% largerin\nthe case of the multilayer. A first conclusion is, therefore,\nthat the magnetic memory induced by aging a nanopar-\nticle array does not show any abrupt change as the two-\ndimensional (2 D) limit is approached. Notice also that, as\nwe have seen with the critical slowing down, the analogy\nis not restricted to aging. The FC curves measured on the\ntwo samples show also the same degree of “flattening” be-\nlowTb, a property that has been considered as a signature\nof the superspin glass phase [25].\nBy gradually changing the number of layers Nwe can\nstudyhowmagneticcorrelationsgrow. In Fig. 4, weshow,\nas a function of N, the relative amplitude of the mag-\nnetic memory effect ∆ MZFC/MZFCmeasured after aging\natTw= 15.8 K fortw= 104s. Within the droplet picture\nofthespin-glassphase[29], thisquantityisconnectedwith\nthe size that domains of correlated spins attain after time\ntw[11]. We seethat∆ MZFC/MZFCincreasesrapidlywhen\none or two layers are added to a two-dimensional sample,\nnearly saturating as Nincreases further. The right-hand\npanel of Fig. 4 shows that, within the relatively large ex-\nperimental uncertainties, ∆ MZFC/MZFCis approximately\nproportionaltotheincreaseintheaveragenumberofnear-\nestneighbors N⊥= 6(N−1)/Nthatis associatedwith the\naddition of extra layers. The same linear dependence was\nalso found for the blocking temperature [19]. This behav-\nior suggests that the enhancement in the amplitude of the\nmagnetic memory is provided mainly by correlations with\np-3R. L. L´ opez et al.\n/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48\n/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s49/s48/s49/s50/s51/s52\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s49/s48/s49/s50/s51/s52/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s45/s49/s48/s49/s50/s51/s32 /s32/s32/s77\n/s90/s70/s67/s32/s40/s101/s109/s117/s47/s99/s109/s51\n/s41\n/s32 /s32/s32/s77\n/s70/s67/s32/s40/s101/s109/s117/s47/s99/s109/s51\n/s41\n/s32\n/s84 /s40/s75/s41\n/s32/s32/s77\n/s114/s32/s40/s101/s109/s117/s47/s99/s109/s51\n/s41/s78 /s32/s61/s32/s49\n/s84\n/s98/s32/s61/s32/s50/s50/s46/s53/s32/s75/s32\n/s84\n/s119/s32/s61/s32/s49/s53/s46/s56/s32/s75\n/s116\n/s119/s32/s61/s32/s49/s48/s52\n/s32/s115\n/s32\n/s84 /s40/s75/s41\n/s32/s32/s77\n/s90/s70/s67/s32/s40/s101/s109/s117/s47/s99/s109/s51\n/s41\n/s32\n/s84 /s40/s75/s41\n/s32/s32/s77/s32\n/s70/s67/s32/s40/s101/s109/s117/s47/s99/s109/s51\n/s41\n/s32\n/s84 /s40/s75/s41\n/s32/s32/s77\n/s114/s32/s40/s101/s109/s117/s47/s99/s109/s51\n/s41\n/s32 /s32/s32\n/s32\n/s84 /s40/s75/s41\n/s32/s32/s77/s32\n/s90/s70/s67/s40/s101/s109/s117/s47/s99/s109/s51\n/s41/s32 /s32/s32\n/s78 /s32/s61/s32/s50/s48\n/s84\n/s98/s32/s61/s32/s51/s53/s46/s50/s32/s75\n/s84\n/s119/s32/s61/s32/s50/s53/s32/s75\n/s116\n/s119/s32/s61/s32/s49/s48/s52\n/s32/s115\n/s32\n/s84 /s40/s75/s41\n/s32/s32/s77\n/s70/s67/s32/s40/s101/s109/s117/s47/s99/s109/s51\n/s41/s32\n/s84 /s40/s75/s41\n/s32/s32\n/s32\n/s84 /s40/s75/s41\n/s32/s32/s77\n/s114/s32/s32 /s40/s101/s109/s117/s47/s99/s109/s51\n/s41\nFig. 3: (Color online). ZFC, FC and thermoremanence curves\nmeasured with (red crosses) and without (black solid line)\npause. The insets show absolute values of ∆ MZFC, ∆MFCand\n∆Mr. Left: sample with N= 1;Twwas 15.8 K. Right: sample\nwithN= 20;Twwas 25 K. The waiting time was 104s and\nthe applied magnetic field was 10 Oe.\nthe first one or two nearest layers. Another result suggest-\ning that magnetic correlations remain rather short-ranged\nis shown in Fig. 5. There, we compare the magnetic mem-\nory ∆MZFCobtained for a single layer of 2 .6 nm particles\nwith that obtained for a N= 20 multilayer in which the\ninterlayer separation is tAl2O3= 10 nm, i.e. more than\ntwicednn,/bardbl∼4.6 nm. Within their respective experimen-\ntal uncertainties, these two quantities are found to be the\nsame. Also Tband other quantities agree. It seems then\nthat no measurable magnetic correlations are established\nbetween layers of nanoparticles located 10 nm far form\neach other.\nWe also studied the effects of aging using a different\nexperimental method, which enables a more quantitative\ndetermination of magnetic correlation lengths. For this,\nwe measured the magnetic relaxation of the ZFC mag-\nnetization of a N= 15 multilayer at Tw= 15.8 K. The\nsample was first cooled from 100 K to Twin zero field.\nAfter aging the sample for tw= 104s, a magnetic field\nHwas applied and the ensuing magnetization measured\nas a function of time. This method has been applied to\nestimate the number of correlated spins in spin glasses\n[30,31], and recently applied also to investigate the slow\ndynamics of frozen ferrofluids [26]. Its basic idea is as\nfollows. During the waiting time tw, at zero field, mag-\nnetic correlations between nanoparticles grow [6]. Typi-\ncal free energy barriers ∆( tw) for the flip of Ns(tw) cor-/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s46/s48/s51/s48/s48/s46/s48/s51/s53/s48/s46/s48/s52/s48/s48/s46/s48/s52/s53/s48/s46/s48/s53/s48\n/s48 /s49 /s50 /s51 /s52 /s53 /s54\n/s32/s32/s77\n/s90/s70/s67/s77\n/s90/s70/s67/s32\n/s78\n/s32\n/s32/s32\n/s78\nFig. 4: (Color online). Left: Variation with the number N\nof layers of the memory ∆ MZFC/MZFCmeasured after aging\nthe sample for tw= 104s atTw= 15.8 K (•). Right: Same\ndata as a function of the average number of nearest neighbors\nN⊥= 6(N−1)/Nthat a nanoparticle has in adjacent layers\n[19]. The lines represent 0 .033+0.015N⊥.\nrelated spins increase also with the age twof the sys-\ntem. This growth of dynamical correlations reflects itself\nin the appearance of a maximum in the relaxation rate,\ndefined as ∂MZFC/∂log(t), when the experimental time\ntapproaches the age of the system teff\nw∼tw(see Fig.\n6). A magnetic field Hreduces the free energy barriers,\nfrom its zero-field value to ∆( tw)−EZ[H,Ns(tw)], where\nEZ[H,Ns(tw)] =µ[H,Ns(tw)]Handµis the magnetic\nmoment of a ”drop” of Nscorrelated spins. The energy\nshift induced by this Zeeman term effectively reduces the\n”age” of the system according to\nteff\nw(H) =teff\nw(H= 0)exp/braceleftbigg\n−EZ[H,Ns(tw)]\nkBT/bracerightbigg\n(3)\ntherefore shifting the relaxation rate maximum towards\nshorter times with increasing H, as it is indeed observed\nexperimentally (Fig. 6). From a series of experiments\nperformed at different fields, ranging from 5 Oe up to 100\nOe, we have extracted the Zeeman energy EZ[H,Ns(tw)],\nwhich we plot in the main panel of Fig. 6. The field\ndependence of this energy can be fitted using a quadratic\nfunction of H, compatible with the following expression\nEZ[H,Ns(tw)] =Ns(tw)χZFCH2(4)\nwhichwasfound to agreealsowith experiments performed\non frozen ferrofluids [26]. Inserting in Eq. (4) the mea-\nsured FC susceptibility per particle χFC, we estimate the\nnumberofcorrelatedspins Nsto be approximately17, i.e.,\nrather close to the average number of nearest neighbors\n(12) in a multilayer of nanoparticles [19].\nOur experimental findings suggest therefore that mag-\nnetic correlationsachieved after agingthe sample at low T\nextend mainly to nearest neighbors. The detailed charac-\nterizationofoursamplesenablesus tomakeaquantitative\np-4Zero-temperature spin-glass freezing in self-organized arrays o f Co nanoparticles\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s32/s83/s105/s110/s103/s108/s101/s32/s108/s97/s121/s101/s114\n/s32/s50/s48/s32/s108/s97/s121/s101/s114/s115/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s115/s101/s112/s97/s114/s97/s116/s101/s100/s32/s98/s121/s32/s49/s48/s32/s110/s109\n/s32/s32/s77\n/s90/s70/s67/s32\n/s32/s40/s101/s109/s117/s47/s99/s109/s51\n/s67/s111/s41\n/s84/s32 /s40/s75/s41/s116\n/s87 /s61/s49/s48/s52\n/s32/s115\n/s84\n/s87 /s61/s49/s53/s46/s56/s32/s75\nFig. 5: (Color online). Magnetic memory for a single layer of\n2.6 nm particles (black thin line) compared with data obtain ed\nfor aN= 20 multilayer in which the interlayer separation is\ntAl2O3= 10 nm (red thick line).\ncomparison of the present results with predictions for the\ngrowth of correlations in spin-glasses. Theoretical con-\nsiderations as well as experiments support the idea that\ncorrelations grow approximately as a power law of time\n[26,30,32]:\nξ(t∗,Tw)/dnn∼(t∗)α(Tw)(5)\nwherednn∼4.4 nm is the distance to nearest neighbors,\nt∗=tw/τ(Tw) is a dimensionless timescale, the exponent\nα= 0.17(Tw/Tg), andτis the relaxation time of individ-\nual spins at the given temperature. We have estimated\nτ=τ0exp(U/kBT) using parameters estimated, as de-\nscribed above, for the noninteracting case: τ0∼10−13s\nandU≃430 K [23]. For Tw= 15.8 K and tw= 104s,\nEq. 5 gives 4 .2dnn< ξ <7dnn, i.e., between 19 and 31\nnm for a single layer and 3 dnn< ξ <4.2dnn(13−19 nm)\nfor a multilayer. The upper and lower limits of ξcorre-\nspond to, respectively, the lower and upper limits of the\nfreezing temperatures Tgthat are compatible with the ac\nsusceptibility experiments described above. Our magnetic\nmemory experiments point to significantly shorter corre-\nlation lengths ξ∼4.4 nm.\nConclusions. – The central result of the present\nstudy is that we observe the same magnetic memory and\ncritical slowing down in two-dimensional nanoparticle ar-\nrays, as well as in multilayers, suggesting that the un-\nderlying physical behavior is also the same. The slowing\ndown of the ac (and dc) susceptibility curves measured on\nall these samples can, in fact, be accounted for using the\nactivated law [Eq.(2)] that is typical of two-dimensional\nspin-glasses. These results suggest, therefore, that the\nglassy magnetic dynamics observed in these materials is\nassociated with a phase transition occurring at Tg= 0,/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49/s48/s32/s45/s69\n/s90/s47/s107/s84\n/s72 /s40/s79/s101/s41/s84\n/s119/s32/s61/s32/s49/s53/s46/s56/s32/s75\n/s116\n/s119/s32/s61/s32/s49/s48/s52\n/s32/s115\n/s50 /s51 /s52/s48/s52/s56/s49/s50/s49/s54/s100 /s77 /s47/s100/s108/s111/s103/s40 /s116/s41\n/s108/s111/s103/s40 /s116/s47/s115/s41\nFig. 6: (Color online). Zeeman energies estimated from the\nrelaxation of ZFC magnetization curves measured after agin g\nthe sample at zero field and at Tw= 15.8 K for a waiting time\ntW= 104s, before the application of a magnetic field H. The\ninset shows the time dependence of the magnetic viscosity, d e-\nfinedas∂M/∂log(t)withMbeingthesample’s magnetization,\nmeasured for (from bottom to top curves) H= 5 15, 20, 30,\n50, 60, and 80 Oe. This quantity shows maxima, marked by\ngrey dots, at the effective age of the system that decreases wi th\nincreasing H. The dotted line is a guide to the eye.\nrather than with a conventional spin-glass transition with\na finiteTg. This conclusion is supported by the results of\nmagnetic memory experiments, which show that dynami-\ncal magnetic correlations are rather short ranged and, in\nany case, shorter than expected for canonical spin glasses.\nAs mentioned in the introductory section above, the\nobserved facts disagree with the prediction, derived from\nMonte Carlo simulations, that Ising-like spins interacting\nvia dipole-dipole interactions should undergo a spin-glass\ntransition below a finite temperature [17]. Establishing\nthe origin for this discrepancy is beyond the scope of the\npresent work. Here, we content ourselves with discussing\npossible deviations of real materials from the ideal con-\nditions set by such models. In our opinion, an important\naspect to be consideredis the, unavoidable, distribution in\nparticle sizes. In our multilayers, the distribution is prop-\nerly described by a Gaussian function of width σD≃0.7\nnm, which is equivalent to roughly ±1 atomic layer and\nprovides an indication of the good homogeneity of these\nsamples. This narrow size distribution leads, however, to\nan extremely large dispersion in the relaxation times τ\nassociated with the magnetic anisotropy of the nanoparti-\ncles. Using the parameters Uandτ0given above (see also\nthe second reference in [19] for further details), it follows\nthat intrinsic timescales separated by more than 13 orders\nof magnitude can coexist at temperatures near or below\nTb. A possible consequence of this enormous dispersion\np-5R. L. L´ opez et al.\nis the following. Smaller, and therefore faster, relaxing\nnanoparticles are able to immediately react to spin flips of\nlarger (slower) ones, minimizing their mutual interaction\nenergy. Wehavepreviouslyshownthatthiseffectaccounts\nfor the modification of the average relaxation times by in-\nteractions, at temperatures close to Tb[19]. We might\nspeculate with the possibility that the disorder in relax-\nation times also hinders the growth of magnetic correla-\ntions at lower temperatures. For instance, the formation\nof negatively polarized magnetic clouds surrounding the\nlargest nanoparticles can screen dipolar interactions be-\ntween them. Clearly, further theoretical studies that in-\nclude effects of disorder and nonequilibrium dynamics are\nrequired to clarify the nature of the collective magnetic\nresponse in nanoparticle arrays.\n∗∗∗\nThis work was partly funded under grants\nMAT08/1077, MAT2009-13977-C03 and ”Molecular\nNanoscience” (CSD2007-00010) from Spanish MICINN\nand PI091/08 ”NABISUP” from DGA.∗Corresponding\nauthor. Email: fluis@unizar.es\nREFERENCES\n[1] Mydosh J. A., Spin glasses : an experimental introduction\n(Taylor and Francis) 1993.\n[2] Jonsson T.,Mattsson J. ,Djurberg C., Khan F. A., Nord-\nblad P. and SvedlindhP., Phys.Rev. Lett. ,75(1995) 4138.\n[3] Djurberg C., Svedlindh P., Nordblad P., Hansen M. F.,\nBodkerF. andMørupS., Phys. Rev. Lett. ,79(1997) 5154.\n[4] Mulder C. A. M., van Duyneveldt A. J. and Mydosh J.\nA.,Phys. Rev. B ,23(1981) 1384.\n[5] Souletie J. and Tholence J. L., Phys. Rev. B ,32(1985)\n516.\n[6] Lundgren L., Svedlindh P., Nordblad P. and Beckman O.,\nPhys. Rev. Lett. ,51(1983) 911.\n[7] Chamberlin R. V., Phys. Rev. B ,30(1984) 5393.\n[8] Granberg P., Sandlund L., Nordblad P., Svedlindh P. and\nLundgren L., Phys. Rev. B ,38(1988) 7097.\n[9] Lefloch F., Hammann J., Ocio M. and Vincent E., Euro-\nphys. Lett. ,18(1992) 647.\n[10] Jonason K., Vincent E., Hammann J., Bouchaud J. P.\nand Nordblad P., Phys. Rev. Lett. ,81(1998) 3243.\n[11] J¨ onsson P. E., Yoshino H., Nordblad P., Aruga Katori H.\nand Ito A., Phys. Rev. Lett. ,88(2002) 257204.\n[12] Mattsson J., Djurberg C., Nordblad P., Hoines L., Stubi\nR. and Cowen J. A., Phys. Rev. B ,47(1993) 14626.\n[13] Schins A. G. et al.,Phys. Rev. B ,48(1993) 16524.\n[14] Palassini M. and Caracciolo S., Phys. Rev. Lett. ,82\n(1999) 5128.\n[15] Young A. P., Phys. Rev. Lett. ,50(1983) 917.\n[16] Dekker C., Arts A. F. M., de Wijn H. W., van Duyneveldt\nA. J. and Mydosh J. A., Phys. Rev. Lett. ,61(1988) 1780;\nSandlund L. et al.,Phys. Rev. B ,40(1989) 869.\n[17] Fern´ andez J. F., Phys. Rev. B ,78(2008) 064404.\n[18] Luis F., et al.,Phys. Rev. B ,65(2002) 094409.[19] Luis F., Petroff F., Torres J. M., Garc´ ıa L. M., Bartolom ´ e\nJ., Carrey J. and Vaur` es A., Phys. Rev. Lett. ,88(2002)\n217205;ibid Phys. Rev. Lett. ,90(2003) 059706.\n[20] Petracic O., Chen X., Bendata S., Kleeman W., Sahoo\nS., Cardoso S. and Freitas P. P., J. Magn. Magn. Mater. ,\n300(2006) 192.\n[21] Maurice J. L., Bri´ atico J., Carrey J., Petroff F., Schel p\nL. F. and Vaur` es A., Philos. Mag. A ,79(1999) 2921.\n[22] Babonneau D., Petroff F., Maurice J.L., FettarF., Vaur` es\nA. and Naudon A., Appl. Phys. Lett. ,76(2000) 2892.\n[23] Luis F., Bartolom´ e J., Petroff F., Garc´ ıa L. M., Vaur` e s A.\nand Carrey J., J. Appl. Phys. ,93(2003) 7032.\n[24] Mathieu R., J¨ onsson P., Nam D. N. H. and Nordblad P.,\nPhys. Rev. B ,63(2001) 092401.\n[25] Sasaki M., J¨ onsson P. E., Takayama H. and Mamiya H.,\nPhys. Rev. B ,71(2005) 104405.\n[26] Wandersman E., Dupuis V., Dubois E., Perzynski R.,\nNakamae S., Vincent E., Europhys. Lett. ,84(2008)\n37011.\n[27] Petracic O., Kleemann W., Binek Ch., Kakazei G. N.,\nPogorelov Yu. G., Sousa J. B., Cardoso S., and Freitas P.\nP.,Phase Transitions ,75(2002) 73.\n[28] Ogielsky A. T., Phys. Rev. B ,32(1985) 7384.\n[29] Fisher D. S. and Huse D. A., Phys. Rev. B ,38(1988),\n373.\n[30] Joh Y. G., Orbach R., Wood G. G., Hammann J. and\nVincent E., Phys. Rev. Lett. ,82(1999) 438.\n[31] Bert F., Dupuis V., Vincent E., Hamman J., and\nBouchaud J.-P., Phys. Rev. Lett. ,92(2004) 167203.\n[32] Komori T., Yoshino H. and Takayama H., J. Phys. Soc.\nJpn.,68(1999) 3387; 69(2000) 1192.\np-6"    },    {        "title": "1308.2283v1.Measurements_of_Coronal_and_Chromospheric_Magnetic_Fields_using_Polarization_Observations_by_the_Nobeyama_Radioheliograph.pdf",        "content": "arXiv:1308.2283v1  [astro-ph.SR]  10 Aug 2013Measurements of Coronal and Chromospheric Magnetic\nFields using Polarization Observations by the Nobeyama\nRadioheliograph\nKazumasa Iwaiand Kiyoto Shibasaki\nNobeyama Solar Radio Observatory, National Astronomical O bservatory of Japan, Nobeyama,\nMinamimaki, Minamisaku, Nagano 384-1305, Japan\nkazumasa.iwai@nao.ac.jp\n(Received ; accepted )\nAbstract\nCoronal and chromospheric magnetic fields are derived from polariz ation\nand spectral observations of the thermal free-free emission us ing the Nobeyama\nRadioheliograph (NoRH). In magnetized plasma, the ordinary and ex traordinary\nmodes of free-free emission have different optical depths. This cr eates a circularly\npolarized component in an atmosphere with a temperature gradient . We observed an\nactiveregiononApril13, 2012toderiveitscoronalandchromosph ericmagneticfields.\nThe observed degree of circular polarization was between 0.5 % and 1 .7 %. The radio\ncircular polarization images were compared with ultraviolet images obs erved by the\nAtmospheric Imaging Assembly and the photospheric magnetic field o bserved by the\nHelioseismic and Magnetic Imager, both on board the Solar Dynamic Ob servatory.\nAt the edge of the active region, the radio circular polarization was e mitted mainly\nfrom coronal loops, and the coronal magnetic field was derived to b e about 70 G. At\nthe center of the active region, the chromospheric and coronal c omponents cannot be\nseparated. The derived magnetic field is about 20 % to 50 % of the cor responding\nphotospheric magnetic field, which is an emission-measure-weighted average of the\ncoronal and chromospheric magnetic fields.\nKey words: Sun: chromosphere - Sun: magnetic fields - Sun: radio radiation\n-@methods: data analysis\n1. Introduction\nThe magnetic field of the solar atmosphere is an important clue to und erstanding many\nsolarphenomena suchasflaresandcoronalmassejections. Itha sbeenobservedintheopticalor\ninfrared ranges by using the Zeeman effect at the photosphere. O ptical or infrared observations\n1can also be used to measure the coronal and chromospheric magne tic fields using the Zeeman\nand/or Hanle effect (Trujillo Bueno et al 2005; Lin et al. 2004), altho ugh it is much more\ndifficult because of the weaker magnetic fields in hot, turbulent plasm a. The coronal magnetic\nfields are also derived by extrapolations of the photospheric magne tic fields using the potential\nfield (Sakurai 1982; Shiota et al 2008) and linear or nonlinear force -free field approximations\n(e.g., Inoue et al. 2012). However, the magnetic field in the photosp here is neither potential\nnor force-free due to the high gas pressure. Hence, these obse rvation and modeling methods\nare still being developed and should be verified by comparison with eac h other. In this study,\nwe derived the chromospheric and coronal magnetic fields by microw ave radio observations.\nSeveral methods are used to derive the magnetic field using radio ob servations (Gary\n& Keller 2004; Shibasaki et al 2011a for a review). The magnetic field above sunspots can\nbe derived from the gyro-resonance emission (Dulk 1985; Gary & Hu rford 1994). Polarization\nreversal by quasi-transverse propagation of radio emission is also used for estimating coronal\nmagnetic fields (Cohen 1960; Ryabov et al. 1999).\nThe longitudinal component of the magnetic field can be derived by th ermal\nbremsstrahlung or so-called thermal free-free emission. Bogod & Gelfreikh (1980) observed\nthe radio polarization using one-dimensional scanning spectral obs ervations of RATAN-600.\nCircularly polarized emission is observed in plage regions at wavelength s between 2.3 and 4.0\ncm. The observed circular polarization degrees were up to 1.5 % and w ere converted to a\ncoronal magnetic field of 20 to 60 G using the observed radio spectr al index.\nObservations in a higher frequency range enable us to derive the ma gnetic field at a\ndeeper layer. Grebinskij et al (2000) observed circular polarizatio n using single-frequency ob-\nservations with the Nobeyama Radioheliograph (NoRH) at 17 GHz. Th ey used a radiation\ntransfer model simulation to deduce the magnetic field from the obs erved circular polarization.\nThe ratio of the radio-deduced magnetic field and the correspondin g photospheric magnetic\nfield is 0.5 to 0.6. Their radiation transfer model simulation showed tha t both the coronal and\nchromospheric components are included in the circular polarization a t 17 GHz.\nMicrowave radio observation is advantageous because it can be use d to derive both the\nchromospheric andcoronal magnetic fields. Onthe other hand, se parationof the chromospheric\nand coronal components requires spectral observation or mode ling. In particular, spectral\nobservation is essential for examining the chromospheric compone nt because the spectral index\nof the chromosphere is unknown. Hence, the circular polarization o bservation and spectral\nobservation should be combined (see Equation 1 in Section 2.2).\nThe purpose of this study is to derive the chromospheric and coron al magnetic fields\nby combining two-dimensional radio polarization and radio spectral im aging observations using\nNoRH at 17 and 34 GHz. Then, the derived radio magnetic fields are ev aluated by comparing\nthem with the coronal loop structure and photospheric magnetic fi eld. The instrument and\ntheory to derive magnetic fields are described in Section 2. The data analysis results are\n2presented in Section 3 and discussed in Section 4.\n2. Observation\n2.1. Instrument\nNoRH is a radio interferometer dedicated to solar observation; it ha s 84 antennas, each\nwith a diameter of 80 cm (Nakajima et al. 1994). NoRH observes the f ull solar disk every 1\ns at 17 GHz (intensity and circular polarization) and 34 GHz (intensity ). Radio images are\nsynthesized every 1 s.\nAn active region observed on April 13, 2012 (NOAA 11455), is analyz ed in this study.\nFigure 1 shows the radio intensity image observed with NoRH at 17 GHz and the magnetic field\nobserved with the Helioseismic and Magnetic Imager (HMI: Scherrer et al. 2012) on board the\nSolar Dynamics Observatory (SDO) on April 13, 2012. There was a s mall active region near\nthe disk center (N06W19). This situation is suitable for observing th e longitudinal component\nof the magnetic field.\nThree main emission processes which produce microwave solar radio e missions are; free-\nfree emission, gyro-resonance emission, and gyro-synchrotron emission. Gyro-resonance emis-\nsion is emitted from sunspots at lower harmonics (2 or 3) of their loca l gyro-frequency. The\nthird harmonic of the gyro-frequency at 2000 G is about 17 GHz. In Figure 1, the largest\nmagnetic field in the active region observed with HMI is less than 2000 G . Hence, there is no\ngyro-resonance emission component from the observed active re gion. Gyro-synchrotron emis-\nsion is emitted from non-thermal electrons during flares. In this st udy, flare-quiet periods are\nselected to avoid contamination by gyro-synchrotron emission. Th erefore, the radio emission\nobserved from the active region in this study is purely thermal free -free emission.\n2.2. Derivation of the magnetic field\nIn magnetized plasma, the ordinary and extraordinary modes of fr ee-free emission have\ndifferent optical depths. That makes a circular polarized componen t which is inverted to obtain\nthe longitudinal component of the magnetic field Blas follows (Bogod & Gelfreikh 1980):\nBl[G]=10700\nnλ[cm]V\nI\nn≡d(logI)\nd(logλ)(1)\nwhereIis the brightness temperature, Vis the brightness temperature of the circularly po-\nlarized component, λis the wavelength in cm, and nis the power-law spectral index of the\nbrightness temperature. For an optically thin case, the spectral index is close to 2. This\nmethod is used to derive the magnetic fields of coronal loops in limb obs ervations (Shibasaki\net al 2011b).\nIn the microwave range, the opacity of the free-free emission bec omes thick around the\n3upper chromosphere. Equation 1 shows that an optically thick regio n of uniform temperature\n(n≈0) cannot produce the circularly polarized component regardless o f the existence of the\nmagnetic field. However, a difference in opacity between the ordinar y and extraordinary modes\nmeans that these two modes can penetrate into different layers. I f a temperature gradient exists\nbetween these two layers, the brightness temperatures of the o rdinary and extraordinary modes\ndiffer. Therefore, the magnetic field in the chromosphere in the pre sence of a temperature\ngradient produces a nonzero spectral index and hence also produ ces the circularly polarized\ncomponent.\n3. Data Analysis\n3.1. Data Accumulation Time and Noise Level\nBecause the circular polarization degree of free-free emission is ve ry small, hundreds to\nthousands of synthesized images should be averaged to reduce th e noise level of the images. In\nthis study, the averaging period is determined by the standard dev iation of the polarized signal\nin the quiet region. The Stokes parameters of IandVin this study are given by,\nV=(R−L)/2\nI=(R+L)/2 (2)\nwhere,RandLare the brightness temperatures of the right- and left-handed c ircular polarized\ncomponent, respectively. A degree of polarization ( P) is given by,\nP=V\nI=R−L\nR+L(3)\nThe standard deviation of P(=σP) is given by the law of propagation of errors as follows,\nσ2\nP=/parenleftBigg∂P\n∂R/parenrightBigg2\nσ2\nR+/parenleftBigg∂P\n∂L/parenrightBigg2\nσ2\nL (4)\nwhere,σRandσLare the standard deviations of R and L, respectively. Equation 4 is s olved as\nfollows,\nσ2\nP=/parenleftBigg2L0\n(R0+L0)2/parenrightBigg2\nσ2\nR+/parenleftBigg−2R0\n(R0+L0)2/parenrightBigg2\nσ2\nL (5)\nwhere,R0andL0are the averages of R and L, respectively. We assume R0=L0andσR=σL\nfor non-polarized emission. Then, σPis derived as follows,\nσP=σR√\n2R0(6)\nThe standard deviation of V(=σV) is also given by the law of propagation of errors as\nfollows,\nσ2\nV=/parenleftBigg∂V\n∂R/parenrightBigg2\nσ2\nR+/parenleftBigg∂V\n∂L/parenrightBigg2\nσ2\nL=σ2\nR\n2(7)\n4Hence,σPis derived by Equations 2, 6, 7 as follows,\nσP=σV\nI0(8)\nwhereI0is the average intensity, and I0=R0=L0for non-polarized emission.\nThe white rectangle in Figure 1 shows the radio-quiet region used in th is study, and\nTable 1 shows the standard deviation of the circularly polarized signa l (σV) in this region. The\nstandard deviations of the polarization component were 21 K after the images were averaged\nover 2 min and 11 K after averaging over 20 min. The average intensit y (I0) of the solar disk\nis about 10,000 K. We define five-sigma as the minimum detectable signa l level. Hence, the\nminimum detectable level of a degree of polarization corresponds to 1.0 % and 0.5 % for 2 min\nand 20 min averaging, respectively. Synthesized images are averag ed for 20 min in the following\ndata analysis to derive weaker magnetic fields. The solar rotation du ring 20 min is about 3\narcsec, which is smaller than the beam size of NoRH (10 arcsec at 17 G Hz). Hence, the solar\nrotation effect is neglected in the averaged data.\n3.2. Radio Polarization, Spectra, and Magnetic Field\nFigure 2a shows a radio circular polarization map superimposed on the optical magne-\ntogram observed with HMI on April 13, 2012. The red and blue conto urs show the positive and\nnegative components of the radio circular polarization, respective ly. The location and polarity\nof the radio circular polarization correspond to those of the optica l magnetogram. The degree\nof polarization is up to 1.7 % for negative polarity. Figure 2c shows an E UV image at 304 ˚A\nobserved by the Atmospheric Imaging Assembly (AIA: Lemen et al. 2 012) on board the SDO.\nThe white contours show the radio intensity at 17 GHz. The radio inte nsity corresponds to the\nbright region at 304 ˚A.\nThe calibration sequence of NoRH image synthesis uses the sky and t he quiet Sun levels.\nA histogram of the pixel counts included in a synthesized image is deriv ed for each image. The\nmost frequent count level is defined as the background sky level, w hich is assumed to be\nat 0 K because the cosmic microwave background radiation is negligible compared with the\nradiation from the Sun. The second most frequent count level is de fined as the quiet Sun level.\nThe brightness temperature of the quiet Sun at microwave to millimet er wavelengths has been\nobserved and modeled in several studies (Linsky 1973; Beckman et al. 1973; Kuseski & Swanson\n1976; Zirin et al 1991; Selhorst et al 2005). From these studies, t he brightness temperatures\nof the quiet Sun at 17 GHz and 34 GHz are 10,000 K and 9000 K, respec tively. The radio\nspectral index of the quiet region between 17 and 34 GHz is about 0.1 5 using this model. A\nlinear approximation between the background sky and the quiet Sun enables us to calibrate\nthe observed brightness temperatures. The spectral index of t he quiet Sun at microwave range\nis assumed to be constant (e.g Selhorst et al 2005). Hence, we defi ne that the spectral index\nbetween 17 and 34 GHz is same as the local spectral index at 17 GHz. The green contours in\nFigure 2b show the radio spectral index of the active region, which is between about 0.4 and\n50.6 around the active region.\nThe chromospheric temperature structure of active regions migh t be different from that\nof the quiet region. Thus, the magnetic field that is derived from the spectral index between\n17 and 34 GHz contains an error. Although it is difficult to estimate the extent of this error,\nit is unlikely that the spectral index of the active region at 17 GHz is fa r from that of between\n17 and 34 GHz.\n3.3. Magnetic Fields in the Photosphere and Chromosphere\nFigure 2b shows the magnetic field strength derived by substituting the radio circular\npolarization, intensity, andradiospectralindexinEquation1. Ther adiomagneticfieldstrength\nis derived only in regions that have circular polarization degrees of gr eater than 0.5 % and\nspectral index of greater than 0.15. Red and blue contours show t he positive and negative\ncomponents of the radio magnetic field, respectively. The location o f the radio magnetic field\ncorresponds to that of the photospheric magnetic field within the b eam size of NoRH ( ∼10\narcsec at 17 GHz).\nOptical magnetograms usually have higher spatial resolutions than the beam size of\nNoRH. Hence, the area-averaged magnetic fields within 20 arcsec s quare regions are compared.\nThe radio magnetic fields at the center of the positive and negative p olarity regions [footpoint\n(FP) regions in Figure 2a] are 116 G and -217 G, respectively. The co rresponding photospheric\nmagnetic fields are 568 G and -456 G in the positive and negative polarit y regions, respectively.\nThe ratio between the photospheric and radio magnetic fields is 0.20 in the positive polarity\nregion and 0.47 in the negative polarity region.\n4. Interpretation and Discussion\n4.1. Coronal and Chromospheric Components of the brightnes s temperature\nThe microwaves observed in this study (17 and 34 GHz) are emitted m ainly from the\nchromosphere. However, the coronal component exists especia lly in coronal loops around ac-\ntive regions. For simplicity, we adopt a two-component atmosphere model consisting of the\ncorona and chromosphere (Zirin et al 1991; Grebinskij et al 2000) . In this model, the observed\nbrightness temperature ( Tb) is given by\nTb=Te,chr(λ)exp(−τc(λ))+Te,cor(λ)(1−exp(−τc(λ))) (9)\nwhereτc(λ) is the optical depth of the corona at a given wavelength, and Te,corandTe,chrare\nthe electron temperatures of the corona and chromosphere, re spectively.\nThe corona and chromosphere are assumed to be optically thin and t hick, respectively.\nIn addition, the electron temperature of the corona ( ∼106) is about 100 times larger than that\nof the chromosphere ( ∼104). Hence, Equation 9 is simplified as follows (Grebinskij et al 2000):\nTb,obs(λ)=Tb,chr(λ)+Tb,cor(λ) (10)\n6whereTb,obs(λ) is the observed radio intensity, and Tb,chr(λ) andTb,cor(λ) are the chromospheric\nand coronal components, respectively. In this study, the bright ness temperature of the quiet\nSun is assumed to be 10,000 K at 17 GHz and 9,000 K at 34 GHz. These ca n be considered\nas the base brightness temperature of the chromosphere. Henc e, the increment relative to\nthis chromospheric component is estimated to be the coronal comp onent if we neglect the\ntemperature variation in the chromosphere.\nThe assumption in Equation 10 can be checked by plugging the observ ational results\ninto Equation 9. In the FP- region in Figure 2a, for example, the obse rved total radio in-\ntensity at 17 GHz Tb,obs(1.76cm) is about 14,100 K. The coronal and chromospheric electron\ntemperatures Tb,cor(1.76cm) andTb,chr(1.76cm) are assumed to be 106and 104K, respectively.\nThese observational results and assumptions yield exp( −τc(1.76cm))≈0.99 orτc(1.76cm)≈0.\nHence, Equation 10 is approximately accurate when τc(λ)≪1 andTb,chr(λ)≪Tb,cor(λ), which\nare usually true in the solar atmosphere.\n4.2. Coronal and Chromospheric Components of the polarized emission\nThe polarized component Vobs(λ) is also derived as Tb,obs(λ) is, by using a simple model\nwith a constant coronal magnetic field,\nVobs(λ)=Vchr(λ)+Vcor(λ) (11)\nHowever, these two components cannot be separated because t he coronal and chromospheric\nmagnetic fields are both unknown.\nFigure 3 shows the relationship between the radio circular polarizatio n degree at 17\nGHz and the photospheric magnetic field observed by HMI. The radio circular polarizations\nat the center of the active region (FP+ and FP-) are correlated wit h the magnetic field in\nthe photosphere. In the EA regions, however, the radio circular p olarization is observed even\nthough the corresponding photospheric magnetic fields are almost 0 G. Each area is averaged\nwithin 20 arcsec2, which is sufficiently larger than the beam size of NoRH at 17 GHz ( ∼10\narcsec). Hence, the circularly polarized component at the center of the active region cannot\naffect the high degree of circular polarization in the EA regions.\nFigure 2d shows an EUV image at 171 ˚A observed by AIA. The coronal loop structures\nexhibit broader structures than the photospheric magnetic field. The red and blue contours\nshow the radio circular polarization degree at 17 GHz. The circularly p olarized component is\nobserved between the foot points and loop tops of the coronal EU V loops. In particular, it is\nclear that the 0.5 % contour of the positive circular polarization degr ee lies along the envelope\nof the tops of the coronal loops.\nFigure 4 shows the radio intensity, circular polarization, and photos pheric magnetic field\nalong the white line in Figure 2d. The circularly polarized component of 1 7 GHz is observed\nto the north of 170 arcsec. On the other hand, photospheric mag netic field is observed only\nnorth of 195 arcsec. The separation of the two is larger than the b eam size of NoRH at 17\n7GHz (∼10 arcsec). The location of the chromospheric magnetic field is thou ght to be similar\nto that of the photosphere. Hence, there should be no chromosp heric component of circularly\npolarized radio emission around 170 arcsec. Therefore, the coron al component is the source of\nthe polarized signal, even though the coronal component of intens ity is only 1000 to 2000 K as\nshown in the top panel.\nNow we consider the circular polarization degree in the EA region, whic h does not\ncontain chromospheric component. In the EA+ region in Figure 2a, t he coronal emission is\n4040 K, and the circularly polarized component is 96 K. The optically th in coronal component\nis expected to have a spectral index of 2. These observational va lues and Equation 1 yield a\ncoronal magnetic field of 72 G in the EA+ region. This magnetic field is co nsistent with the\nresults of Bogod & Gelfreikh (1980) that also derived coronal magn etic field using the radio\nfree-free observation at the longer wavelength.\n4.3. Separation of the Chromospheric and the Coronal compon ents\nAt the foot points of the active region, the observed circularly pola rized signal includes\nboth the chromospheric and coronal components. These two com ponents are emission measure\nweighted. Because NoRH observes the circular polarization in only on e observation band, we\ncannot separate the circularly polarized component from these tw o layers.\nCircular polarization observations at multiple frequency bands are e ffective for distin-\nguishing the chromospheric and coronal circularly polarized compon ents. The use of multiple\ncircular polarization observation bands enables better inversion, e specially for coronal three-\ndimensional magnetic fields. Observations in higher frequency rang es are also effective for\nreducing contamination by the coronal component and deriving only the chromospheric mag-\nnetic field accurately.\nInaddition, itisalsopossible toimprove theseparationofthechromo spheric andcoronal\nmagnetic fields by combining radio observations with other observat ions or models. Optical\nor infrared observations can be used to measure the chromosphe ric magnetic fields using the\nZeeman or Hanle effect (Trujillo Bueno et al 2005; Hanoka 2005). Th e given chromospheric\nor coronal magnetic fields may enable the separation of the chromo spheric and coronal radio\ncircular polarization.\n5. Summary\nWe derived the coronal and chromospheric magnetic fields using circ ular polarization\nobservationsat17GHzandspectralobservationsat17and34GH zbyNoRH.Theobservational\nresults are summarized as follows:\n•Thesynthesized imageswere averagedtoreduce their noiselevel. T he minimum detectable\nlevel of the circular polarization degree is 1.0 % and 0.5 % for 2 min and 20 min accumu-\nlations, respectively. The observed circular polarization degree wa s between 0.5 % and 1.7\n8Table 1. Standard deviation of circularly polarized signal ( V) and average radio intensity ( I) at 17 GHz inside the white\nrectangle in Figure 1.\nAveraging time (min) 2 20\nStandard deviation of V(K) 21 11\nSignal threshold (5 σ) ofV(K) 105 55\nAverage of I(K) 10,448 10,452\n%.\n•Theradiointensitywascalibratedusingthebrightnesstemperatur eofthequietSun, which\nis defined as 10,000 K and 9000 K at 17 GHz and 34 GHz, respectively fr om Selhorst et al\n(2005). The spectral index of the brightness temperature in the quiet region is about 0.15\nusing this model. The observed spectral index is between 0.4 and 0.6 a round the active\nregion.\n•The magnetic fields are derived from the observed radio circular pola rization and the\nspectral index of the brightness temperature. The ratio of the o bserved radio magnetic\nfield and the corresponding photospheric magnetic field is about 0.2 t o 0.5 at the center\nof the active region.\nThe observed radio magnetic fields contain both the coronal and ch romospheric com-\nponents. We assume that the solar atmosphere observed in the mic rowave range has two\ncomponents: the optically thin corona and the optically thick chromo sphere. The radio circu-\nlar polarizationimages were compared withEUV images observed by AI Aandthe photospheric\nmagnetic field observed by HMI. The results are summarized as follow s:\n•At the edge of the active region, radio circular polarization is observ ed even though the\ncorresponding magnetic field at the photosphere is almost 0 G. At th e same time, the\nlocation of the observed radio circular polarization corresponds to that of the coronal loop\nstructures. Therefore, we can assume that there is almost no ch romospheric circularly\npolarized component there and derive the pure coronal magnetic fi eld strength.\n•At the center of the active region, the chromospheric and corona l components cannot\nbe separated, and the derived magnetic field is emission-measure-w eighted. It requires\nadditional information for its separation.\nFor the next step, it is necessary to develop an instrument for two -dimensional polariza-\ntion observations with high spatial and spectral resolutions in a high er frequency range. These\nrequirements will be achieved by recent and future radio interfero meters such as the SSRT\n(Lesovoi et al. 2012), CSRH (Yan et al. 2009), FASR (Bastian 2004 ), and ALMA. These radio\nobservations will be more useful for cooperation with ground- and space-based magnetic field\nobservations such as Solar-C.\n9Fig. 1.(Left) Radio intensity at 17 GHz observed with NoRH at 03:00:15 UT on April 13, 2012. Solid\nrectangle shows radio-quiet region described in Table 1. (Right) Pho tospheric magnetic field observed\nwith SDO/HMI at 03:00:35 UT. Dotted rectangles in right and left pane ls show active region described in\nFigure 2.\nSDO data are courtesy of NASA/SDO and the AIA and HMI science te rms. This work\nwas conducted by a joint research program of the Solar-Terrest rial Environment Laboratory,\nNagoya University.\n10HMI 13-Apr-2012 03:00:35 UT\n-50050100150200250\nX (arcsecs)50100150200250300350Y (arcsecs)(a)\nFP-\nEA-EA+LT\nFP+\nHMI 13-Apr-2012 03:00:35 UT\n-50050100150200250\nX (arcsecs)50100150200250300350Y (arcsecs)0.60.40.2 (b)\nAIA 304A 13-Apr-2012 03:00:08 UT\n-50050100150200250\nX (arcsecs)50100150200250300350Y (arcsecs)(c)\nAIA 171A 13-Apr-2012 03:00:00 UT\n-50050100150200250\nX (arcsecs)50100150200250300350Y (arcsecs)(d)\nFig. 2.(a) Magnetic fields observed by HMI at 03:00 UT on April 13, 2012. Ra dio circular polarization\nat 17 GHz is superimposed as contours: positive components in red, 0.5 %, 1.0 %; negative components\nin blue, 0.5 %, 1.0 %, 1.5 %. LT: loop top, FP: footpoint, EA, edge of act ive region. (b) Magnetic fields\nobserved by HMI at 03:00 UT. Radio magnetic fields at 17 GHz are supe rimposed as contours: positive\ncomponents in red, 50, 150 G; negative components in blue, 50, 150 , 250 G. Green contours: radio spectral\nindex of the brightness temperature spectrum between 17 and 34 GHz, 0.2, 0.4, 0.6. (c) EUV image at\n304˚A observed by AIA. White contours: radio intensity at 17 GHz. (d) E UV image at 171 ˚A observed\nby AIA. Red and blue contours indicate radio circular polarization deg ree at 17 GHz.\n11-600 -400 -200 0 200 400 600\nHMI magnetic field (G)-0.02-0.010.000.010.02Radio circular polarization degreeFP-EA-EA+\nLTFP+\nFig. 3.Relationship between the radio circular polarization degree at 17 GHz and photospheric magnetic\nfield observed by HMI.\nReferences\nBastian, T.S.2004, inSolarandSpaceWeatherRadiophysics (AstrophysicsandSpaceScienceLibrary,\nVol. 314; Berlin: Springer), 47\nBeckman, J. E., Clark, C. D., & Ross, J. 1973, Sol. Phys., 31, 3 19\nBogod, V. M., & Gelfreikh, G. B. 1980, Sol. Phys., 67, 29\nCohen, M.H., 1960, ApJ, 131, 664.\nDulk, G. A., 1985, ARA&A, 23, 169\nGary, D. E., & Hurford, G. J. 1994, ApJ, 420, 903\nGary, D. E., & Keller, C. U. (ed.) 2004, Solar and Space Weathe r Radiophysics: Current Status and\nFuture Developments (Astrophysics and Space Science Libra ry, Vol. 314; Dordrecht: Kluwer)\nGrebinskij, A., Bogod, V., Gelfreikh, G., Urpo, S., Pohjola inen, S., & Shibasaki, K. 2000, A&AS, 144,\n169\nHanoka, Y. 2005, PASJ, 57, 235\nInoue, S., Shiota, D., Yamamoto, T. T., Pandey, V. S., Magara , T., & Choe, G. S. 2012, ApJ, 760, 17\nKuseski, R. A., & Swanson, P. N. 1976, Sol. Phys., 48, 41\nLemen, J. R., et al. 2012, Sol. Phys., 275, 17\nLesovoi, S. V., Altyntsev, A. T., Ivanov, E. F., & Gubin, A. V. 2012, Sol. Phys., 280, 651\nLin, H., Kuhn, J. R., & Coulter, R. 2004, ApJ, 613, L177\nLinsky, J. L. 1973, Sol. Phys., 28, 409\nNakajima, H., et al. 1994, IEEEP, 82, 705\nRyabov, B. I., Pilyeva, N. A., Alissandrakis, C. E., Shibasa ki, K., Bogod, V. M., Garaimov, V. I., &\nGelfreikh, G. B. 1999, Sol. Phys., 185, 157\nSakurai, T. 1982, Sol. Phys., 76, 301\n1217 GHz Intensity (I)\n150 200 250 300\nY (arcsec)1.0·1041.2·1041.4·1041.6·1041.8·1042.0·104Brightness Temperature (K)\n17 GHz Circular Polarization (V)\n150 200 250 300\nY (arcsec)-50050100150200Brightness Temperature (K)\nHMI Magnetic Field\n150 200 250 300\nY (arcsec)-2000200400Magnetic Field (G)\nFig. 4.(Top) radio intensity at 17 GHz, (middle) circular polarizationat 17 GH z, and (bottom) magnetic\nfield observed by HMI, along the white line in Figure 2d.\nScherrer, P. H., et al. 2012, Sol. Phys., 275, 207\nSelhorst, C. L., Silva, A. V. R., & Costa, J. E. R. 2005, A&A, 43 3, 365\nShibasaki, K., Alissandrakis, C. E., & Pohjolainen, S. 2011 a, Sol. Phys., 273, 309\nShibasaki, K., Narukage, N., & Yoshimura, K. 2011b, in ASP Co nf. Ser. 437, Solar Polarization 6,\neds. J. Kuhn et al. (San Francisco, CA: ASP), 433\nShiota, D., Kusano, K., Miyoshi, T., Nishikawa, N., & Shibat a, K. 2008, J.Geophys. Res., 113, A03S05\nTrujillo Bueno, J., Merenda, L., Centeno, R., Collados, M., & Landi Degl’Innocenti, E. 2005, ApJ,\n619, L191\nYan, Y. H., et al. 2009, Earth Moon Planets, 104, 97\nZirin, H., Baumert, B. M., & Hurford, G. J. 1991, ApJ, 370, 779\n13"    },    {        "title": "1502.08011v2.Magnetic_catalysis_and_inverse_magnetic_catalysis_in_QCD.pdf",        "content": "Magnetic catalysis and inverse magnetic catalysis in QCD\nNiklas Mueller1and Jan M. Pawlowski1, 2\n1Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany\n2ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für\nSchwerionenforschung mbH, Planckstr. 1, D-64291 Darmstadt, Germany\nWe investigate the effects of strong magnetic fields on the QCD phase structure at vanishing\ndensity by solving the gluon and quark gap equations, and by studying the dynamics of the quark\nscattering with the four-fermi coupling. The chiral crossover temperature as well as the chiral\ncondensate are computed. For asymptotically large magnetic fields we find magnetic catalysis, while\nwe find inverse magnetic catalysis for intermediate magnetic fields. Moreover, for large magnetic\nfields the chiral phase transition for massless quarks turns into a crossover.\nThe underlying mechanisms are then investigated analytically within a few simplifications of the\nfull numerical analysis. We find that a combination of gluon screening effects and the weakening of\nthe strong coupling is responsible for the phenomenon of inverse catalysis. In turn, the magnetic\ncatalysis at large magnetic field is already indicated by simple arguments based on dimensionality.\nPACS numbers: 11.15.Tk, 11.30.Rd, 12.38.Aw, 12.38.Gc\nI. INTRODUCTION\nIn recent years there has been a growing interest in the\nQCD phase structure in the presence of strong magnetic\nfields, see e.g. [1–8]. Such fields may play an important\nroleforthephysicsoftheearlyuniverse,incompactstars,\nand in non-central heavy ion collisions [4, 7, 9, 10].\nDespite the rich phenomenology, theoretical predic-\ntionsarechallenging. StartingfromQED,e.g.[11–14]the\ninfluence of magnetic fields onto QCD was investigated\nin both model calculations, e.g. [15–29], such as quark-\nmeson, Nambu-Jona-Lasinio models and AdS/QCD, e.g.\n[30–36], with functional renormalisation group methods,\ne.g. [37–43], Dyson-Schwinger equations, e.g. [6, 44–46]\nand in lattice calculations, e.g. [47–53].\nThe importance of magnetic fields for chiral symmetry\nbreaking has been pointed out in [11]. It has been ar-\ngued that chiral symmetry breaking is enhanced due to\nan effective dimensional reduction, the magnetic cataly-\nsis. This effect has been linked to an increase of the chi-\nral condensate as well as that of the critical temperature\nTcin model studies. Recent lattice results, [47–49, 53],\nhave shown that while the chiral condensate indeed is\nincreased, the critical temperature is decreasing with an\nincreasing magnetic field, at least for small enough mag-\nnetic field strength. This effect has been called inverse\nmagnetic catalysis ormagnetic inhibition , [54].\nContinuum studies have mainly been performed in low\nenergy fermionic models, such as the (Polyakov loop en-\nhanced) quark-meson– or NJL–model. Hence the reason\nfor the discrepancy has to relate to the full dynamics of\nQCD, and in particular the back-reaction of the matter\nsector to the gluonic fluctuations. There have been a\nnumber of improvements to these model studies to in-\nclude QCD dynamics [24, 26–28, 55, 56]. Input parame-\nters of low energy effective models, such as the four-fermi\ncoupling, should be determined from the QCD dynam-\nics at larger scales. At these scales they are sensitive\nto sufficiently large external parameters such as temper-ature, density, or magnetic fields. This has been em-\nphasized and used in functional renormalisation group\n(FRG) studies, see [57–60]. The dependence of the four-\nfermi coupling on temperature and magnetic field effects\nincluding gluon screening has been investigated in the\nrecent FRG-work [43] of QCD in strong magnetic fields,\nwhere inverse magnetic catalysis at small magnetic fields\nanda delayed magnetic catalysis atlargefieldswasfound,\nsee also [36] for an AdS/QCD computation.\nIn the present work we investigate (inverse) magnetic\ncatalysis by solving the coupled quark and gluon gap\nequations within the Dyson-Schwinger (DSE) approach\nto QCD, and within a FRG study of the four-fermi cou-\npling based on QCD flows and low energy effective mod-\nels. We find magnetic catalysis at large magnetic fields,\nwhileinversemagneticcatalysistakesplaceatsmallmag-\nnetic fields.\nThe present work is organized as follows: The gap\nequations for quark and gluon propagators at finite tem-\nperature and magnetic field in two flavor QCD are dis-\ncussed in Section II. We discuss the dependence of the\nchiral transition temperature Tcon the magnetic field as\nwell as the magnetic field dependence of the chiral con-\ndensate. In Section III the mechanisms behind the phe-\nnomena of magnetic and inverse magnetic catalysis are\nevaluated within analytically accessible approximations\nto the gap equations as well as to the dynamics of the\nfour-fermi coupling. In this set-up we are also able to\nreproduce the lattice results at eB < 1GeV2. In sum-\nmary this provides a complete picture of chiral symmetry\nbreaking in the presence of magnetic fields in QCD.\nII. CHIRAL SYMMETRY BREAKING IN\nLARGE MAGNETIC FIELDS\nWe investigate chiral symmetry breaking in the pres-\nence of large magnetic fields within a functional contin-\nuum approach. To this end we calculate the chiral con-arXiv:1502.08011v2  [hep-ph]  13 May 20152\nFIG. 1: Quark Dyson-Schwinger equation. Lines with\nblobs stand for fully dressed propagators, vertices with\nlarge blobs stand for fully dressed vertices. Lines with-\nout blobs stand for classical propagators, vertices with\nsmall blobs stand for classical vertices.\ndensate for the two lightest quark flavors and obtain the\ncritical temperature Tcat finite magnetic field. This is\ndonebysolvingthegapequationsforthequarkandgluon\npropagator in the presence of a magnetic field using the\nRitus method [61–67]. The computations are performed\nin the Landau gauge.\nA. Quark and gluon gap equations\nThe gap equation for the quark propagator, see Fig. 1,\ndepends on the gluon propagator and the quark-gluon\nvertex. The former is expanded about the quenched\npropagator. This expansion has been successfully used\nat vanishing temperature, e.g. [68, 69], and at finite tem-\nperature in e.g. [70–73], the reliability of this expansion\nhas been discussed in [74]. The quark-gluon vertex is es-\ntimated with the help of Slavnov-Taylor identities (STIs)\nfrom the quark and gluon propagators. The systematic\nerror of the latter estimate gives rise to the dominating\nsystematic error, at vanishing temperature this has been\ninvestigated in [75], a related upgrade of the vertex will\nbe used in a subsequent work.\nThe inverse quark and gluon propagators, Gq(q)and\nGA(q)respectively, read in a tensor decomposition at fi-\nniteeBandT\nG\u00001\nq(q) =Zq(q) (i\r3q3+i\r0q0Z0+i\r?q?Z?+M);\nG\u00001\u0016\u0017\nA (q) =\u0010\nZkP\u0016\u0017\nk+Z?P\u0016\u0017\n?\u0011\nq2+1\n\u0018q\u0016q\u0017\nq2;(1)\nwithP\u0016\u0017\nk= (g\u0016\u0017\nk\u0000p\u0016\nkp\u0017\nk=p2\nk)andP?=P\u0000Pk, where\nP\u0016\u0017is the transverse projector. The projection operator\ng\u0016\u0017\nkhas the property g\u0016\u0017\nkp\u0016\nk=p\u0017\nk. The Ritus represen-\ntation Eq. (1) for the quark propagator is equivalent to\nthe Schwinger proper time method, see e.g. [76]. In the\nfollowing we will denote ZA\u0011Zkand concentrate on the\nLandau gauge, \u0018= 0. The STIs-induced parametrisation\nof the quark-gluon vertex is introduced as\n\u0000\u0016(q;p) =\r\u0016zDSE\n\u0016qAq(q;p); (2)\nwithzDSE\n\u0016qAq(q;p)discussed in Appendix A. The quark gap\nFIG. 2: Gluon Dyson-Schwinger equation. The gluon\nline with the yellow dot represents the pure glue loops.\nequation can be written in a compact notation as\nG\u00001\nq(p) =G\u00001\nq;0(p) +CfXZ\nq(g\r\u0016)Gq(q)\u0000\u0017(q;p)G\u0016\u0017\nA(q0);\n(3)\nwithq0=q\u0000pandGq;0as the bare propagator. The\nintegrationPR\nqstands for an integration over momenta,\nas well as sums over Matsubara frequencies and Landau\nlevels. The gluon propagator can be expanded about its\npure glue part,\nG\u00001\u0016\u0017\nA (p) =G\u00001\u0016\u0017\nglue (p) + \u0005\u0016\u0017\nf(p); (4)\nwhere we have written the fermionic part of the gluon self\nenergy explicitly, while the gluon and ghost loop contri-\nbutions are contained in Gglue. The corresponding DSE\nforthegluonpropagatorwithinthisexpansionisdepicted\nFig. 2. In the following we consider the back-reaction of\nthe vacuum polarisation on the pure glue part as small,\nand approximate\nG\u00001\u0016\u0017\nglue (p)\u0019G\u00001\u0016\u0017\nYM (p): (5)\nAt vanishing temperature this has been shown to hold\nquantitatively for momenta q&4GeV, while for smaller\nmomentathisapproximationstillholdsqualitativelywith\nan error of less than 20%, see Fig. 6 in [74]. Note\nthat for momenta q&4GeV the dominant effect of the\nunquenching is the modification of the scales ( \u0003YM!\n\u0003QCD) and the momentum dependence induced by the\ndifferent\f-functions. This is well-captured with the\nabove procedure. In turn, at lower momentum scale the\nnon-perturbative mass-gap related to confinement comes\ninto play. The magnetic field leads to a shift in the mo-\nmentum dependence such as that of the running cou-\npling, as well as (additional) mass-gaps in propagators.\nFor both asymptotic regimes ( eB!0andeB!1)\nthese effects are well-captured semi-perturbatively and\nwe expect that the approximation (5) holds well. For the\nintermediate regime we rely on the error estimate at zero\ntemperature of about 20% deduced from [74].\nThe fermionic vacuum polarisation part \u0005\u0016\u0017\nf(P)reads\n\u0005\u0016\u0017\nf(p) =1\n2trXZ\nq(g\r\u0016)Gq(q)\u0000\u0017(q;p)Gq(q0);(6)\nwhere the trace includes a sum over the quark flavors.\nDetails of this expansion can be found in [6]. Here we\nproceed in the lowest Landau level approximation, where3\nFIG. 3: Relation of the quark DSE interaction kernel\nto a 1PI skeleton expansion, which in effect induces an\neffective momentum dependent four-fermi vertex.\nwe write down the most general tensor decomposition for\ngluon and quark propagators. Projecting onto different\ntensor compositions, we obtain a coupled set of equa-\ntions for the dressing functions of the different tensor\ncomponents. In the next section we will comment on the\nrelation of the Dyson-Schwinger equations to other func-\ntional expansions and discuss the numerical solutions to\nthese equations.\nB. Skeleton expansion\nBeforeproceedingtothenumericalanalysis, wediscuss\nthe standard approximation schemes for the quark-gluon\nvertex used in the Dyson-Schwinger framework from a\nmore general point of view. This allows us to connect\nthe present ansätze to the approximations used in gap\nequations derived within other functional approaches,\nsuch as functional renormalisation group (FRG) or nPI-\napproaches.\nDSE studies have made extensively use of the specific\ninput for the quark-gluon vertex and the YM-gluon prop-\nagator in (A2) and (A4) and similar truncations with\ngreat success. Since the quark and gluon self energy dia-\ngrams, depictedinFig.1andFig.2, containonebarever-\ntex, the correct renormalisation group behavior and mo-\nmentum dependence of the equations must be discussed\ncarefully. The truncations to the gap equations (3) and\n(4) can actually be very well motivated from a skeleton\nexpansion of the 1PI effective action, which would yield\nsimilar diagrams as in Fig. 1 and Fig. 2, but with both\nvertices dressed. Fig. 3 serves to strengthen this motiva-\ntion as it becomes clear that all approximations should\nencode the correct behavior of the four-fermi interaction,\nwhich is at the heart of chiral symmetry breaking. This\nallows to consistently reshuffle functional dependencies\nin the interaction kernels of the above equations.\nIn turn, the FRG-approach (or nPI effective action)\ncan be used to systematically derive gap equations in\nterms of full propagators and vertices respectively, see\ne.g. [77]. Here, we simply note that the 1PI effective\naction can be written as\n\u0000[\u001e] =1\n2Tr ln \u0000[\u001e] +Z\nt@t\u0000k[\u001e]\u0000terms;(7)\nwhere\u001eencodes all species of fields, the trace in (7) sums\nover momenta, internal indices and all species of fields in-\ncluding relative minus signs for fermions (  and yarecounted separately), and a logarithmic RG-scale t= lnk.\nThe RG-scale in (7) is an infrared scale. Momenta\np2.k2are suppressed in \u0000k[\u001e], and \u0000[\u001e] = \u0000k=0[\u001e].\nThe second term on the right hand side of (7) is a RG-\nimprovement term which only contains diagrams with\ntwo loops and more in full propagators and vertices. To\nsee this we discuss the gap equation derived from (7). It\nfollows by taking the second derivative of (7) w.r.t. to\nthe fields. The first term of the right hand side gives\nthe diagrams as in Fig. 1 and Fig. 2 with only full ver-\ntices (and additional tadpole diagrams). These diagrams\ncan be iteratively re-inserted into the RG-improvement\nterm, systematically leading to higher loop diagrams in\nfull propagators and vertices. Due to its sole dependence\non dressed correlation functions such a diagrammatics\nnaturally encodes the momentum- as well as the RG-\nrunning on an equal footing. This also facilitates the con-\nsistent renormalisation. Note however that it comes at\nthe price of an infinite series of loops diagrams which can\nbe computed systematically. Here we take the simplest\nnon-trivial approximation which boils down to Fig. 1 and\nFig. 2 with only full vertices. In terms of the original gap\nequation this leads to the relation\nzDSE\n\u0016qAq\u0019\u0000\nz1PI\n\u0016qAq\u00012; (8)\nwherez1PI\n\u0016qAqis the dressing function of the 1PI-quark\ngluon vertex. This immediately leads to the standard\nDSE-dressing in (A2). Moreover, in our numerical study\nthe vertices are evaluated at their symmetric momentum\npoint.\nNote that, while the ansatz for zDSE\n\u0016qgqis indeed consis-\ntent when used in the quark and gluon gap equations, it\ncannot be used in functional equations for higher vertices\nsuch as the four-fermi vertex. It is already clear from the\ndiscussionabovethataconsistentevaluationofrenormal-\nisation group running and momentum dependence must\nbe considered separately for each vertex equation.\nC. Results\nWe numerically solve the coupled system of quark and\ngluon functional equations in the lowest Landau level ap-\nproximation at finite temperature. This approximation\nis valid in the presence of a clear scale hierarchy with\neB\u001d\u0003QCD. We use an ansatz for \u0000\u0016similar to that\nused in Dyson-Schwinger studies, e.g. [6, 78], discussed\nin appendix A, but adapted for temperature and mag-\nnetic field effects.\nWhileatlargemomentumtheinfluenceoftemperature\nand magnetic fields is very small, at large temperatures\nand magnetic fields the system is effectively dimension-\nally reduced and hence the momentum dependencies cor-\nresponding to the absent dimensions vanish. This can be\naccounted for if we replace Q2\n?by2jeBjonceQ2\n?<2jeBj\nandQ0by2\u0019TforQ0<2\u0019Tas the relevant scale in the\nquark gluon vertex, which is consistent with renormali-4\nææææææææææææææææ\nææò\nò\nò\nòòòò\nò\nòò\nò\nò\nò\nòòòòææææææ\næ\næææææææææææò\nò\nò\nò\nò\nòòòòòòòòòòòòæeB=12GeV2\nòeB=24GeV2\n1201301401501601701801900.00.51.01.52.02.5\nT@MeVD-<qq>@GeV3D\nFIG. 4: Comparison of the chiral condensate (scenario\n1) for up (continuous lines) and down quark (dashed\nlines) ateB= 12GeV2andeB= 24GeV2.\nsation group arguments. Within this parametrisation we\nare still left to decide what exact momentum scale to\nchoose, at which the influence of the external scales T\nandeBis small already. We investigate this question in\ndetail in section IIIA.\nThe gluon propagator deserves some additional atten-\ntion. It is decomposed in different polarisation compo-\nnents in the presence of an external magnetic field, see\ne.g. [6]. Apart from the splitting into longitudinal and\ntransverse components with respect to the heat bath,\nthere is an additional splitting transverse and longitu-\ndinal to the magnetic field. In the lowest Landau level\napproximation only the polarisation subspace projected\nonto byP\u0016\u0017\nk= (g\u0016\u0017\nk\u0000p\u0016\nkp\u0017\nk=p2\nk)receives contributions\nfrom the quark loop in the self energy, see [6]. Note that\nin analogy to temperature effects, also the other gluon\ncomponents must receive contributions from the interac-\ntionwiththemagneticfield, asgluonandghostloopsmix\ndifferent polarisation components. This is an important\ndifference between QCD and QED. From dimensionality\nthese contributions are linear in eBat least for asymp-\ntotically large magnetic fields, leaving aside implicit B-\ndependencies via the vertices. Their full computation is\nbeyond the scope of the present work. Here we investi-\ngate the following two limiting cases.\n1.Scenario 1 We simply neglect the screening effect\nof the magnetic field onto those polarisation com-\nponents that feel magnetic effects only through the\nYang-Mills sector in a QED-type approximation.\nThis leads to underestimating the effects leading\nto inverse magnetic catalysis and hence an upper\nlimit forTc.\n2.Scenario 2 For the large magnetic fields discussed\nhere, the gluon and ghost loops contributions to\nthe self energy must have a similar dependence on\neBas the fermionic part. Since this sector does\nnot directly contain charged particles, the effect of\nthemagneticfieldontotheYM-sectorissuppressed\nby powers of the involved couplings. Hence, most\næææææ\næ\næææææ\næ\næ\næææòòòòòòò\nòòò\nò\nò\nò\nòòòæææææ\næ\nææææææææææòòòòòò\nò\nò\nòòòòòòòòæeB=12GeV2\nòeB=24GeV2\n80901001101201301400.00.51.01.52.02.5\nT@MeVD-<qq>@GeV3DFIG. 5: Comparison of the chiral condensate for sce-\nnario 2 ateB= 12GeV2andeB= 24GeV2.\nlikely theB-dependence is much smaller than that\nfromthefermionicsector. Asalimitingcasewewill\nassumethesamemagnitudeoftheselfenergyforall\ngluon components, which is given by the fermionic\ncontributions. Withthatweoverestimatethegluon\nscreening effect and obtain a lower limit for Tc.\nBoth scenarios give consistent limiting cases for the trun-\ncation used here.\nAs an order parameter for chiral symmetry breaking\nwe calculate the chiral condensate as a function of tem-\nperature and magnetic field in two flavor QCD in the\nlimit of vanishing bare quark masses mu\u0019md\u00190. The\nRitus method is not reliable for rather small values of\nqfeB, withqf+ 2=3and\u00001=3for up and down quark\nrespectively. We expect the lowest Landau level approx-\nimation to be a good estimate once eB&4GeV2(see\n[6]) which is also the regime where the approximation\n(5) works well for vanishing temperature.\nThe numerical computation is very demanding in the\nvicinity of the phase transition due to the diverging cor-\nrelation length. This translates into a numerical error\nin the critical temperature indicated by the error bars\nin the plots. Fig. 4 and Fig. 5 show the up- and down-\nquark condensate for different values of eB. The inverse\nmagnetic catalysis effect described in [48, 49] is evident.\nWhile the chiral condensate still rises with the exter-\nnal field in the low temperature limit, the transition be-\ntween chiral broken and symmetric phase drops. This\nsignals inverse magnetic catalysis as observed on the lat-\ntice, [47, 48]. Furthermore the phase transition, which is\nsecond order at zero magnetic field turns into a crossover\nwith growing eB, even for vanishing bare quark masses.\nThis can be understood as magnetic screening: the mag-\nnetic field effectively serves as an infrared cutoff, which\ninhibits an infinite correlation length.\nInthepresentcomputationintwo-flavorQCD,aneven\nmore intricate effect is observed. Up and down quarks\ncome with different electric charges, therefore the pres-\nence of a strong electromagnetic field breaks isospin ex-\nplicitly. This results in a non-degenerate chiral phase5\næææææ\næ\næ\næ\næ\nææææ\nææ\næ\næ\næ\nææààà\nà\nààà\nà\nà\nà\nà\nàààà\nà\nà\nà\nààààààààààm=10MeV HuL\nm=10MeV HdLm=0HuL\nm=0HdL\n1601701801902002100.00.10.20.30.40.5\nT@MeVD-<qq>@GeV3D\nFIG. 6: Comparison of the chiral condensate at zero\nbare mass and at a finite bare quark mass of mu=\nmd= 10MeV ateB= 4GeV2in scenario 1.\ntransition for the two flavors. Because gluons travel\nthrough a medium filled with both virtual up and down\nquarks, isospin breaking effects the self interactions of\nthe quarks, which leads to interference between the chi-\nral transitions of the two flavors as seen in Fig. 4 and\nFig. 5.\nThis interference can be interpreted as follows. Virtual\nquarkfluctuationscontributingtothegluonscreeningare\nsuppressed in the chiral broken phase by the quark mass.\nSince the down quark undergoes the chiral phase transi-\ntion already at lower scales, its fluctuations are suddenly\nenhanced due to the vanishing mass in the symmetric\nphase. The up quark, while still in its chirally broken\nphase, is drastically effected by these enhanced fluctua-\ntions, which lead to reduction of the up quark condensate\neven below the real phase transition.\nIt can be seen from Fig. 4 and Fig. 5 that this effect\nis more prominent in scenario 2, which should come as\nno surprise, as the coupling of the magnetic field to the\ngauge sector is probably overestimated here. Neverthe-\nless the isospin induced chiral transition substructure is\nobservable in the limiting scenario 1 as well, which is a\nstrong indication of its validity. Therefore this important\nphysical effect might be observable in lattice calculations,\nas well. In [48, 49] the averaged chiral condensate was\ninvestigated at finite quark mass. However when we in-\nvestigate the chiral transition at a bare quark mass of\n10MeV we find that the interference effect is completely\nmaskedbythecrossoverbehaviorascanbeseeninFig.6.\nNotethatheretheunregularizedcondensateatfinitebare\nmass is plotted, hence the offset between the curves.\nIn analogy with lattice calculation we define Tcat the\ninflection points of the curves shown. In Fig. 7 and Fig. 8\nthe obtained values for Tcfor the limiting cases described\nby scenario 1 and 2 are shown. The two curves give lower\nand upper limits for Tc, as discussed before. The chiral\ntransition temperature is decreasing for a large range in\neBbefore it seems to saturate for intermediate values in\nboth scenarios. At very large fields it rises again.\næ\næ\nææææà\nà\nà\nà\nààò\nò\nòòòòæupquark\nàdown quark\nòflavor average\n01020304050100120140160180200\neB@GeV2DTc@MeVDFIG. 7: Critical temperature obtained from scenario 1\nfor up quark, down quark and from the flavor averaged\ncondensate.\nInaccordancewithourpreviousdiscussionsweseethat\nthe up and down quark chiral transitions do not coin-\ncide. The transition temperature from the flavor aver-\naged quark condensate is given in Fig. 7 and Fig. 8 as\nwell. As can be seen from Fig. 4 and Fig. 5 the tran-\nsition temperature of the flavor averaged condensate is\nessentially determined by the up quark.\nBoth scenarios give estimates for the chiral transi-\ntion temperature, which differ only quantitatively. Sce-\nnario 1, which underestimates the magnetic field effects\nin the gluon sector extrapolates to a critical tempera-\nture ateB= 0between 170\u0000210MeV with a turning\npoint between catalysis and inverse catalysis of about\neB\u001930GeV2. On the other hand scenario 2 gives Tc\nat zero magnetic field of about 140\u0000165MeV with a\nturning point slightly higher than in scenario 1. This is\nin accordance with the fact that scenario 2 overestimates\nthegluonicsector, whichisthesourceoftheinversecatal-\nysis effects. At B= 0the chiral phase transitions for up\nand down quark coincide. While the continuous lines in\nFig. 7 and Fig. 8 are obtained from a fit with a simple\nquadratic polynomial, reflecting the turnover behavior at\nlarge fields, these should not be mistaken as extrapola-\ntions towards zero. Furthermore the computations have\nbeenperformedinthelowestlandaulevelapproximation.\nThis leads to an uncertainty of about 10% for Bsmaller\nthan 10GeV2, while the qualitative behavior is not ef-\nfected, as discussed in [6, 8]. In the following section we\nwill see that the behavior of Tcat smallBis steeper than\njust quadratic.\nIt is well known that within approximation schemes\nsuch as the one discussed here, relative fluctuation scales\nare usually well accounted for, whereas absolute scales\nhave to be fixed. The position of TcateB= 0gives us\nthe possibility of identifying absolute scales and allows to\nadjust our truncation. We will not be concerned about\nmatching the exact scale of Tcat zero magnetic field with\nthe lattice, moreover we will investigate the mechanisms\nbehind the B\u0000Tphase structure in greater detail. We6\næ\næ\næ\næææà\nà\nà\nà\nà\nàò\nò\nò\nòòòæupquark\nàdown quark\nòflavor average\n010203040506080100120140160\neB@GeV2DTc@MeVD\nFIG. 8: Critical temperature obtained from scenario 2.\nwill discuss the issue of scales in the following sections.\nIII. ANALYTIC APPROACHES\nIn the present Section we are specifically interested in\nthe mechanisms at work in magnetic and inverse mag-\nnetic catalysis. To that end we discuss approximations\nto the quark gap equation in Section IIIA, as well as to\nthe dynamics of the four-fermi coupling or quark scat-\ntering kernel in Section IIIB, that allow for an analytic\napproach to chiral symmetry breaking. While the quark\ngap equation can be straightforwardly reduced to an an-\nalytic form from that used for the numerical study, the\nfour-fermi coupling is studied in a renormalisation group\napproach to QCD, that reduces to an NJL-type model\nfor low momentum scales.\nA. Quark gap equation\nThe mechanisms behind the phenomena observed in\nour numerical study can be analyzed within approxima-\ntions detailed below, that allow for an analytic access.\nThese approximations to the gap equation have been in-\ntroducedin[13]forQED,andcanbeextendedtoQCDat\nfinite temperature. The self-consistent Dyson-Schwinger\nequation for the mass functions reads in lowest Landau\nlevel approximation with zero bare mass\nM(pk) = 4\u0019CFZX\nqkM(qk)Tr(\u0001(sgn(eB))\r\u0016\nk\r\u0017\nk)\nM2(qk) +q2\nk\nZ\nk?\u000bsexp\u0012\n\u0000k2\n?\n2jeBj\u0013P\u0016\u0017(k)\nk2+ \u0005(k2):(9)\nHerePR\n=TP\nnR\ndqk=(2\u0019)3and\u0001(s) = (1 +s\u001b3)=2.\nThe quark gap equation (9) is obtained from a skeleton\nexpansion of the effective action, e.g. [79], and is nothingbut a manifestly renormalisation group invariant approx-\nimation of the above Dyson-Schwinger equations, see the\ndiscussion in Section IIB. It includes only dressed ver-\ntices. In appendix A we discuss how the interaction ker-\nnelscanberelatedinbothpictures. The1PIquark-gluon\nvertex is parametrized as\n\u0000\u0016\n\u0016qAq(q2) =Z1=2\nA(q2)p\n4\u0019\u000bs(q2)\r\u0016\nk;(10)\nThe gluon propagator is transversal due to the Lan-\ndau gauge, and we allow for a gluonic mass via thermal\nand magnetic effects. M(pk)is a function that is ap-\nproximately constant in the IR but falls of rapidly for\np2\nk\u00152jeBj. Hence, if we are interested in M(0) =MIR\nwe can write, dividing the equation by its trivial solution,\n1\u00004\u00192CFTZ2eBX\nqk1\nM2\nIR+q2\nk;f\n\u0002Z\ndx\u000bsexp (\u0000x=2jeBj)\nq2\nk;b+x+ \u0005(x;qk;b) \n2\u0000q2\nk;b\nq2\nk;b+x!\n= 0:\n(11)\nIn (11) we have introduced qk;b\u0011(q3;2n\u0019T)andqk;f\u0011\n(q3;2\u0019T(n+ 1=2)). Chiral symmetry breaking is realized\nonceasolution M2\nIR>0exists. Duetotheshapeof M(q)\nand the exponential factor in (11), the integrand only\nhas support for x.2jeBj. In the following we carefully\ninvestigate the ingredients to this self consistent equation\nand the physical mechanisms, which are responsible for\nthe intriguing behavior seen in the previous section.\nDue to the finite support of the integrand, the mo-\nmenta running through the vertices are comparable or\nsmaller than the relevant dimensionful quantities eBand\nT2. Note that in our numerical study we have used an\nansatz for the quark gluon vertex, that includes generic\neBandTdependencies. Here we utilize the fact that\nthe running of \u000bsis dominated by the temperature and\nmagnetic field scales. We resort to a simple ansatz for\n\u000bs(Q2=\u00032\nQCD)based on the analytic coupling \u000bs;HQsug-\ngested in [80, 81], see [82] for an investigation within the\npresent context. This coupling yields a linear potential\nsuch as seen in the heavy quark limit.\n\u000bs(z) =\u000bs;HQ(z)rIR(z); (12)\nwhere\n\u000bs;HQ(z) =1\n\f0z2\u00001\nz2log(z2); (13)\nwith\f0= (33\u00002Nf)=12\u0019and\nz2=\u0015B2eB+\u0015T(2\u0019T)2\n\u00032\nQCD; (14)\nwith coefficients \u0015T,\u0015B, which are of order one. These7\ncoefficients determine the point at which eBorTdomi-\nnate momentum scales. For the relevant magnetic fields\nand temperatures the running of the coupling with tem-\nperature is very small compared to the running with eB.\nWe use an ansatz for the infrared behavior of the vertex,\nwhich is parametrized in rIR. Here we use\nrIR(z2) =z4\n(z2+b2)2\u0012\n1 +c2\nz2+b2\u0013\n;(15)\nwhich scales with /z4forz!0, and approaches unity\nin the perturbative regime. Eq. (12) reproduces the cor-\nrect behavior of the full quark gluon vertex in (10). We\nleavebandcas parameters which allow us to model the\ninfrared behavior of the quark gluon vertex. Our ansatz\nfor (15) is motivated from the quantitative renormalisa-\ntion group study of quenched QCD in [75], which we use\nto determine bandc. We get\nb= 1:50; c = 7:68; (16)\nfrom the fit to Fig. 4 in [75].\nFurthermore we discuss the gluon self energy in the\npresence of magnetic fields at finite temperature in this\nsimplified setup. It is important to notice that we can\nfacilitateourcalculationsbythefollowingargument. The\nfunctionontherighthandsideof(11)isacontinuousreal\nfunction of MIRand approaches +1asMIR!1. Hence\nit is sufficient to check whether the expression is negative\nforMIR= 0, because then it had to pass through zero at\nsome point, which means that a solution exists.\nThe gluon self energy receives two important contri-\nbutions. The first is through the appearance of fermion\nloops, which are also present in an abelian calculation.\nThe fermionic self energy part in lowest Landau level ap-\nproximation with MIR= 0factorizes\n\u0005\u0016\u0017\nf(p) =\u000beB exp\u0000\n\u0000p2\n?=2eB\u0001\n\u0005\u0016\u0017(pk;T):(17)\nContracting with P\u0016\u0017in the Landau gauge, we can write\nthe second term as\n\u0005f(pk;T) =\u00008\u00192h\n3\u00002(1\u0000p2\nk=p2)i1\n\u001c2\n\u00021Z\n0dxZX\n~qkx(x\u00001)\n(~q2\n3+ (2\u0019)2(n+ 1=2)2+x(1\u0000x)=\u001c2)2;\n(18)\nwherewedefined \u001c2\u0011T2=p2\nk. Thefunctioncanbeevalu-\nated numerically and is very well described by the simple\nfunction\n\u0005f(pk;T) = (1=2\u0019)h\n3\u00002(1\u0000p2\nk=p2)i1\n1 + (4\u00192=3)\u001c2:\n(19)\nEq. (17) and Eq. (19) state that the relevant contribu-\nlB=3lB=1\nlB=2\n010203040100150200250\neB@GeV2DTc@MeVDFIG. 9: Analytic calculation of the critical temperature\nfor the chiral phase transition. The bands indicated\ncorrespond to \u0015T= 1and\u0015T= 0. Arrows indicate the\ndirection from \u0015T= 1to\u0015T= 0.\næ\næ\næ\næææ\n01020304050100120140160\neB@GeV2DTc@GeVD\nFIG. 10: Comparison of the critical temperature ob-\ntained with our full numerical procedure to the simple\nanalytic estimate for \u0015B= 1:1,\u0015T= 1and\u0014= 1:19.\ntionstotheselfenergystemfrom p2\n?\u00192eBandp2\nk\u0019T2.\nSimilar as before, the influence of the magnetic field onto\ntheYang-Millssectorisnoteasilyaccountedfor. Herewe\nfocus on the abelian-like part of the gluon self energy. As\nwe have investigated before numerically, this is qualita-\ntively correct and we will use Eq. (14) to account for the\ncorrect scales. It is well known from Dyson-Schwinger\nstudies [83], that approximations similar to this semi-\nbare vertex ansatz underestimate the strength of chiral\nsymmetry breaking, due to the negligience of important\ntensor structures in the vertex, especially those struc-\ntures that break chiral symmetry explicitly [75]. In order\nto compensate the overall weakness of the interaction, we\nallow for a phenomenological parameter \u0014in front of the\nintegral in Eq. (11).\nUsing our simple ansatz we can investigate chiral sym-\nmetry. In Fig. 9 a family of solutions to Eq. (11) is shown\nfor various values of \u0015Band\u0015T, using the ansatz de-\nscribed above with \u0014= 1:2for the two upper curves and\n\u0014= 1:4for the lower curves. The choice of \u0014is for better8\nFIG. 11: Diagrams contributing to the renormalisation\ngroup flow of the four-fermi coupling.\nvisualisation only, as the curves can be shifted up and\ndown using this parameter.\nThe observed behavior agrees with that in our numer-\nical study. It can be seen from Fig. 9, that for small eB\ninverse magnetic catalysis is present, while at large eB\nthethecriticaltemperaturerisesagainwiththemagnetic\nfield, with\nTc(B=\u00032\nQCD!1 )/p\neB; (20)\nas one would anticipate from dimensional considerations.\nThis behavior is universal for all \u0015Band\u0015T. We see that\nthe choice of \u0015Beffects the position of the turning point\nof the chiral phase boundary.\nWith the present analytical considerations the numer-\nical results in Fig. 7 and Fig. 8 are readily explained:\nthey roughly correspond to \u0015B\u00191, which explains the\nrelatively large value of eBat the turning point. We see\nthat already small changes in \u0015Bhave a huge effect on\nthis quantity, see Fig. 9.\nIn Fig. 10 we have plotted the analytic result with\n\u0015B= 1:1,\u0015T= 1and\u0014= 1:19, which agrees well\nwith the numerical results from scenario 2. Based on\nthe present work we estimate that \u0015B\u00192\u00003is a realis-\ntic choice for the B-dependence of the running coupling,\nas in our numerical study quark and gluon propagator\nturn into their corresponding B= 0-propagators at this\nmomentum scale.\nThe present analysis reveals the following mechanism:\nThe gauge sector acquires a B-dependence through the\nfeedback of the fermionic sector. This dependence is\nresponsible for the phenomena called inverse magnetic\ncatalysis, as has been also observed recently in a FRG-\nstudy within QCD, [43]. This also explains why it cannot\nbe seen in model calculation without explicit QCD input.\nFrom Eq. (11), Eq. (12) and Eq. (19) we see that the\ngluon screening and the running of the strong coupling\n(both by thermal and magnetic effects) are competing\nwith the generic fermionic enhancement of chiral sym-\nmetry breaking in a dimensionally reduced system. We\nsee from Fig. 9 that at small magnetic field screening ef-\nfects dominate the behavior of the fermionic self energy,\nwhile at asymptotically large fields, thermal fluctuations\nare negligible and hence eB, as the dominating scale,\ndrives the phase transition towards higher Tc(magnetic\ncatalysis).B. Four-fermi coupling\nFor a further analytical grip we also resort to a low\nenergy effective theory point of view: integrating-out the\ngappedgluonsleadstoaneffectivefour-fermitheory, that\nis initialized at about the decoupling scale of the glue\nsector of \u0003\u00191GeV. Previously there have been phe-\nnomenological approaches in low energy effective models\nto include QCD dynamics as the source of the inverse\nmagnetic catalysis effect [24, 25, 55]. From the point of\nview of the FRG for QCD this can be seen as follows\n[60, 70, 74, 75, 84–86]: At a large momentum scale k\nQCD is perturbative, and the 1PI effective action \u0000kin\n(7) is well-described perturbatively. A four-fermi cou-\npling is generated from the one-loop diagrams (in full\npropagators and vertices) encoded in (7), the related di-\nagrams are depicted in Fig. 11. In the present discussion\nwe have dropped diagrams that depend on the q\u0016q\u0000AA\nvertex,qq\u0016q\u0016q\u0000AA-vertex and the qqq\u0016q\u0016q\u0016q-vertex. Fur-\nthermore we assume a classical tensor structure for the\n\u0016qAq-vertex with a couplingp4\u0019\u000bs;k, and only consider\nthe scalar–pseudo-scalar four-fermi vertex\n\u0000four-fermi [q;\u0016q;B] =1\n2\u0016qa\u000b\niqb\u000b\nj\u0000abcd\nk;ijlm \u0016qc\f\nlqd\f\nm;(21)\nwith the scalar–pseudo-scalar tensor structure\n\u0000abcd\nk;ijlm =\u0015k\u0002\n\u000eij\u000elm\u000eab\u000ecd+ (i\r5)ij(i\r5)lm(\u001cn)ab(\u001cn)cd\u0003\n:\n(22)\nThe four-fermi term in (21) can be viewed as the inter-\naction term of a NJL-type model. Within the approxi-\nmation to QCD outlined above the flow of the four-fermi\ncoupling,@t\u0015k, has the form\n@t\u0015k=\u0000k2\u00152\nkF\u0015(Gq)\u0000\u0015k\u000bs;kF\u0015\u000bs(Gq;GA)\n\u0000\u000b2\ns;k\nk2F\u000b2s(Gq;GA); (23)\nwith positive coefficients F\u0015;F\u0015\u000bs;F\u000bs. The respective\ndiagrams are depicted in Fig. 11. The different classes\nof diagrams in Fig. 11 depend on combinations of gluon\nand quark propagators, GAandGqrespectively.\nThe four-fermi coupling \u0015kin two-flavor QCD at\nT= 0has been quantitatively computed (including\nits momentum-dependence) in quenched QCD with the\nFRG in [75], and in a more qualitative approximation\n(without its momentum-dependence) in fully dynami-\ncal QCD in [74]. The respective results are depicted in\nFig. 12. As expected, the couplings have a similar de-\npendence and maximal strength. However, the slope of\nthe coupling in the qualitative computation in the peak\nregime relevant for chiral symmetry breaking is bigger for\nthe qualitative computation. This can be traced back to\nthe missing momentum-dependencies, whose lack artifi-\ncially increases the locality in momentum space and in\nthe cutoff scale. Hence, guided by the experience gained9\nfullQCD\nquenched\n0.00.51.01.52.0050100150200\nk@GeVDl\nFIG. 12: Scalar–pseudo-scalar four-fermi coupling in\nthe vacuum, T= 0,B= 0, computed with quantita-\ntively reliable QCD-flows in quenched QCD, [75], and\nwith qualitative full QCD flows, [74].\nin the DSE-computations we expect the slope to play a\nlarge r ^ ole and we shall use the quantitative quenched re-\nsultsfor\u0015kand\u000bsinourpresentcomputations. Weshall\nfurther comment on the differences in the next Section.\nFor large cutoff scales kthe propagators approach the\nclassical propagators. The current quark mass at these\nscales is negligible and only the cutoff scale is present, if\ntemperature and magnetic field are considered small rel-\native to the cutoff scale. Then the dimensionless Fsare\nsimple combinatorial factors. For optimized regulators,\n[87], they are given as\nF\u0015= 4Nc; F\u0015\u000bs= 12N2\nc\u00001\n2Nc; F\u000b2s=3\n169N2\nc\u000024\nNc;\n(24)\nin the vacuum, see e.g. [74, 75, 86] for more details. For\nsmall enough cutoff scales kthe gluonic diagrams decou-\nple due to the QCD mass gap. In the Landau gauge\nthis can be directly seen with the gapping of the gluon\npropagator. For T= 0; B= 0this entails\np2GA(p2.\u00032)/p2=m2\ngap: (25)\nwith \u0003\u00191GeV. We emphasize that (25) only reflects\nthe mass gap present in the Landau gauge gluon prop-\nagator, the gluon propagator is not that of a massive\nparticle, see e.g. [88]. For momentum scales p2.\u00032this\napproximately leaves us with an NJL-type model with\nthe action\n\u0000NJL[q;\u0016q;B] =Z\nx\u0016qi/@q+ \u0000four-fermi [q;\u0016q;B];(26)\nwith the scalar–pseudo-scalar four-fermi interaction de-\nfined in (21). In the presence of a magnetic field this\nmodel including fermionic fluctuations has been inves-\ntigated in [38] within the FRG. Here we shall use the\nrespective results within the lowest Landau level approx-\nimation. Then Tcshows an exponential dependence onthe dimensionful parameter eB\nTc= 0:42\u0003 exp0\nB@\u00002\u00192\nNc\u0015\u0003P\nfjqfeBj1\nCA:(27)\nThe well-known exponential dependence of Tcon the\nfour-fermi coupling \u0015\u0003already explains the large sensi-\ntivity of the scales of magnetic calatysis and inverse mag-\nnetic catalysis to details of the computation. Eq. (27) is\nvalid for large magnetic field and for \u0003\u001cm2\ngap, that is\ndeep in the decoupling regime of the gluons. An estimate\nthat also interpolates to small magnetic fields is given by\nTc= 0:42\u0003 exp\u0012\n\u00001\nc\u0003\u0015\u0003\u0013\n; (28)\nwith\nck(B) =Nc\n2\u001920\n@X\nfjqfeBj+c1k21\nA;withc1= 3;\n(29)\nwherec1has been adjusted to reproduce Tc(B= 0)\u0019\n158MeV. While Eq. (29) resembles a lowest Landau level\napproximation, it is actually an expansion in B. Using\nthis ansatz we can describe the behavior of the phase\ntransition on scales below 1GeV2qualitatively, while the\nB= 0limit is fixed.\nIt is also well-known that for k\u001dmgapthe flow of\nthe four-fermi coupling is driven by the gluonic diagrams\nsummed-up in F\u000bs: for large scales we can set \u0015k\u001dmgap\u0019\n0. The gauge coupling is small, \u000bs;k\u001dmgap\u001c1and the\nflow gives\u0015k/\u000b2\ns. This entails that the diagrams with\nfour-fermi couplings are suppressed by additional powers\nof\u000bs, and the four-fermi coupling obeys\n@t\u0015glue;k=\u0000\u000b2\ns;k\nk2F\u000bs(Gq;GA); (30)\nwherethesubscript’glue’indicatesthattheflowisdriven\nby glue fluctuations. As discussed before, for k\u001dmgap\nwe have classical dispersions for quark and gluon, and\nthe diagrammatic factor F\u000bsis a constant, see (24). The\nstrong coupling \u000bs;khas the form (12) with z/k. Inte-\ngrating (30) with (12) gives\n\u0015glue;k/\u000b2\ns;k\n2k2F\u000bs(Gq;GA): (31)\nwhere an estimate for the B-dependence of the gluonic\ndiagram in F\u000bsis given in Appendix B.\nAt vanishing magnetic field \u0015glue;kagrees well with the\nfull result for the four-fermi coupling in [75] for k&2\nGeV, see Fig. 13. Below k\u00192GeV,\u0015glue;kis increas-\ninglysmallerthanthefullscalar–pseudo-scalarfour-fermi\ncoupling in quenched QCD. In this intermediate range,\nwherealldiagramscontribute, wewritetheresultingcou-10\nlHquenched QCD L\nlglue\n1.01.52.02.53.03.54.00.00.20.40.60.8\nk@GeVDl,lglue\nFIG. 13: Scalar–pseudo-scalar four-fermi coupling at\nT= 0; B= 0computed with quantitatively reliable\nQCD-flows in quenched QCD, [75], in comparison to\n\u0015gluecomputed from (31).\npling within a resummed form that captures already the\nfermionic diagram proportional to F\u0015,\n\u0015k=\u0016\u0015k\n1\u0000\u0016ck\u0016\u0015k;with \u0016ck=Z\u0003\nkdk0k0F\u0015(Gq):(32)\nThe resummed form in (32) already reflects the matter\npart of the flow in (23) which is the term proportional to\n@t\u0015k. The other terms add up to\n@t\u0016\u0015k=\u0000(1\u0000\u0016ck\u0016\u0015k)2 \n\u0015k\u000bs;kF\u0015\u000bs+\u000b2\ns;k\nk2!\n:(33)\nFor\u0016ck\u0016\u0015k\u001c1the flow of \u0016\u0015kboils down to (30). For\n\u0016ck\u0016\u0015!1the flow in (33) tends towards zero. In this\nregime the four-fermi coupling grows large and the mat-\nter flow dominates. Hence, for the present qualitative\nanalysis we simply identify \u0016\u0015with the glue \u0015glue, (31), up\nto a prefactor,\n\u0016\u0015k=Z\u0015\u0015glue;k: (34)\nThe prefactor Z\u0015accounts for the fact that we have used\nresults of quantitative QCD-flows [75] for the strong cou-\npling which also includes wave function renormalisations\nfor the quarks. In the current model considerations with-\nout wave function renormalisation and further simplifica-\ntions this has to be accounted for. For the same reason\nthe normalisation 0:42 \u0003related to a four-fermi flow with\nan optimised regulator has to be generalised. Moreover,\ntheprefactor \u0016c\u0015;kistheintegratedfour-fermiflowalready\npresent in (28) up to an overall normalisation accounting\nfor the model simplifications. We choose\n\u0016ck(B) =c3ck(B);and 0:42 \u0003!0:42 \u0003 exp (c2\u0000c3);\n(35)and arrive at\nTc= 0:42\u0003 exp\u0012\n\u00001\nc\u0003\u0016\u0015\u0003+c2\u0013\n; (36)\nwithc\u0003as given in (29) and \u0016\u0015in (34) and (31). Note\nthat the parameter c3has dropped out. Its value can be\nadjusted to achieve a quantitative agreement of (31)with\nthe QCD result in [75] with\nc3=1\n2Z\u0015; (37)\nwherethefactor 1=Z\u0015simplyremovesthemappingfactor\nadjusting for the missing wave function renormalisations\nin the model computation. This quantitative agreement\nstrongly supports the reliability of the approximate solu-\ntiontotheflowequationgivenby(32)intheintermediate\nmomentum regime that is of importance for the current\nconsiderations. The remaining parameters are fixed as\nfollows,\nZ\u0015= 2:2; c 1= 3; c 2= 1:4: (38)\nThe parameter c1has already been adjusted to meet\nTc(B= 0)\u0019158MeV, see (28) and (29). The parameter\nc2re-adjusts the overall scale 0:42 \u0003!0:42 \u0003 expc2=\n1:7\u0003. As already discussed above, it depends on the reg-\nulator and the approximation at hand. It reflects the\ndependence on the renormalisation group scheme. Simi-\nlarly toc1it is fixed with Tc(B= 0)\u0019158MeV, and is\na function of the overall normalisation of the four-fermi\ncouplingZ\u0015. The latter is the only free parameter left.\nIn (38) we use the value that reproduces the lattice re-\nsults, see Fig. 14. We emphasise that no other parameter\nis present that allows to shift the minimum in Tc, the lat-\nter being a prediction.\nObviously, the effect seen in our numerical and ana-\nlytic DSE-study, is also present in the analytic approach\nto the dynamics of the four-fermi coupling, including a\ndirect grip on the underlying mechanisms. We see that\nthe non-monotonous behavior, i.e. the delayed magnetic\ncatalysis, [43, 52], is already present at smaller scales\ncompared to Fig. 7 and Fig. 8, while the lattice results\nare reproduced.\nIn turn, for asymptotically large magnetic field, the\ncritical temperature runs logarithmically with B,\nTc(B=\u00032\nQCD!1 )/lnB=\u0003QCD;(39)\nrelated to a double-log–dependence on Bof the expo-\nnent. Due to the qualitative nature of the approximation\noftheB-dependenceofthegluonpropagatoritcannotbe\ntrustedforasymptoticallylarge B. Indeed, (39)hastobe\ncompared to (20) within the analytic DSE-approach pre-\ndicting a square root dependence. Note that in the latter\ncomputation the quark vaccum polarisation is included\nselfconsistently at large Beven though the backreaction\non the pure glue loops in Fig. 2 is neglected. Still this11\n012345120130140150160170\neB@GeV2DTc@MeVD\nFIG. 14: Comparison of the chiral transition temper-\nature obtained within the simple mean field NJL es-\ntimate Eq. (14) to the lattice results of [47] (see their\nFig. 10).\nindicates the validity of the square root dependence, even\nthough a definite answer to this question requires more\nwork.\nC. Discussion of scales & mechanisms\nWith the findings of the last two sections we have\nachieved an analytic understanding of the mechanisms\nat work. The decrease of Tcfor small magnetic fields,\nthe increase of Tcfor larger fields, as well as the related\nmagnetic field regimes can now be understood. In partic-\nularthisconcernsthemagneticfield Bmin, whereTc(Bmin)\nis at its minimum. This is the turning point between in-\ncreasing and decreasing Tc(B).\nMagneticcatalysisrelatestothedimensionalreduction\ndue to the magnetic field in diagrams with quark corre-\nlation functions leading to an increase of the condensate.\nAt finite temperature the catalysis due to the dimen-\nsional reduction is accompanied by a thermal gapping of\nthe quarks that counteracts against the magnetic cataly-\nsis effects. In total this leads to a rise of both, the chiral\ncondensate and the critical temperature, if the magnetic\nfield dependence of the involved couplings is sufficiently\nsmall. As the magnetic field also sets a momentum scale\nof the physics involved, this scenario holds true for suf-\nficiently large magnetic field strength eB=\u00032\nQCD\u001d1,\nwhere theB-dependence of the couplings can be com-\nputed (semi-)-perturbatively. This explains the regime\nof delayed magnetic catalysis.\nThe above discussion of the standard scenario already\nentails that rapidly changing couplings are required for\na decreasing Tc. The couplings involved are the scalar–\npseudo-scalar four-fermi coupling \u0015kand the strong cou-\npling\u000bs;k, whereksets the momentum scale. Both are\nrising rapidly towards the infrared for momentum scales\nk.4\u000010GeV, for\u0015ksee Fig. 13. In this regime chi-\nral symmetry breaking and confinement is triggered andtakes place in QCD at vanishing magnetic field. Switch-\ning on the magnetic field increases the relevant momen-\ntum scalek2/eBand hence decreases \u0015and\u000bs. The\ncondensate still grows with Bas theB-enhancement in\nthe broken phase is still present, only Tcdecreases.\nOur results from the analytic approach to the quark\ngap equation, presented in Fig. 9, support these findings.\nThe position of the turning point Bminin both the full\nnumericalaswellastheanalyticanalysisofthegapequa-\ntion depends crucially on the magnetic field and temper-\nature dependence of the quark gluon vertex, see Fig. 9.\nWhen contrasted with the quantitative FRG results of\n\u000bsin [75], the strong coupling in (A2) decays consider-\nably slower towards the UV. In turn, the couplings in the\nqualitative FRG study for full QCD, [74] have a steeper\ndecay, for the four-fermi coupling see Fig. 13. Seemingly,\nthis already explains the large value of Bminin the cur-\nrent DSE-study as well as the small value of Bminin [43],\nwhich uses approximations similar to [74]. Note however,\nthat we have used the quenched quantitative \u000bsin the\nanalytic DSE-study which agrees well with the numerical\nDSE result for \u0015B\u00191.\nIn summary we have identified the physics mechanisms\nbehind the T\u0000Bphase diagram from our full QCD cal-\nculations. Moreover, Fig. 14 suggests a turning point for\neBmin\u00191:5\u000010GeV2, the large regime for eBminbeing\nrelated to the exponential dependence on the couplings.\nEvidently, the effects observed depend on a sensitive bal-\nance of different scales and parameters. Hence, further\nstudiesarerequiredtofullyuncovertheintricateunderly-\ning dynamics. Very recent findings in AdS/QCD models,\n[36], indicate an inverse magnetic catalysis behavior up\ntoeB\u00194GeV2, which supports our findings.\nIV. CONCLUSIONS\nWe have investigated the chiral phase structure of\nQCD at finite temperature in the presence of an external\nmagnetic field. Our study resolves the discrepancy be-\ntween recent lattice and continuum calculations at mag-\nnetic fields below 1GeV2, see also [43]. We confirm the\ninverse magnetic catalysis effect seen in lattice studies at\nsmallB. At larger Bwe see that magnetic catalysis is\nrestored, with Tc/p\neB. Indications for the turnover\nbehavior have already been found in [43], and in [52]\nwithin two-color lattice-QCD. We hope that further lat-\ntice calculations in full QCD at the scales discussed here\nwill become feasible soon.\nThe reason for this non-monotonous behavior are\nscreening effects of the gauge sector, i.e. modifications\nof the gluon self energy, as well as the strong coupling\n\u000bsin the presence of magnetic fields. Moreover we have\ninvestigated the nature of the chiral transition at finite\nmagnetic field.\nApart from the B-dependence of the critical tempera-\nture, we observe that the phase transition in the chiral\nlimit turns smoothly into a crossover with rising B. No-12\ntably, we find a non-degeneracy in the phase transition\nwhich is due to the explicit isospin breaking caused by\nthe different electric charges of up and down quark. This\nnon-degeneracy might lead to phenomenological conse-\nquences in experimental studies of the QCD phase di-\nagram with non-central heavy-ion collisions, as there\nmight be a mixed phase between the up and down quark\ntransitions. Recent lattice calculations [89] support the\npossibility of a non-degenerate chiral phase transition.\nIn addition, our calculations show that, due to this\nisospin breaking, there is a step-like behavior in the up\nquark condensate triggered by the chiral transition of the\ndown quark. While this is an significant effect in the chi-\nral limit it smoothens out rapidly with increasing cur-\nrent quark mass. Physical current quark masses are in\nthe transition regime, and this effect might have phe-\nnomenological consequences. To our knowledge, this is a\nnovel effect in the QCD phase diagram and it certainly\ndeserves further investigation.\nWe have used analytic studies of the quark gap equa-\ntion and the dynamics of the four-fermi coupling for an\ninvestigation of the physics mechanisms behind (inverse)\nmagnetic catalysis. The results are discussed at length in\nthe previous Section IIIC, leading to a rough prediction\noftheturningpointat eBmin\u00191:5\u000010GeV.Ourinvesti-\ngations highlight the rich phenomenology of QCD matter\nin external magnetic fields, which motivates further stud-\nies, e.g. at finite chemical potential, towards more realis-\ntic descriptions of matter under extreme conditions. Re-\ncent studies [90] have suggested even richer QCD phase\nstructures in the presence of magnetic fields.\nACKNOWLEDGMENTS\nWe thank J. Braun, C.S. Fischer, K. Fukushima,\nW.A. Mian, M. Mitter, S. Rechenberger, F. Rennecke\nand N. Strodthoff for discussions and work on related\nsubjects. This work is supported by the Helmholtz Al-\nliance HA216/EMMI and the grant ERC-AdG-290623.\nNM acknowledges support by the Studienstiftung des\nDeutschen Volkes.\nAppendix A: Gluon Propagator and Quark Gluon\nvertex from Dyson Schwinger studies\nHere we discuss the truncation scheme for the quark\ngapequationandthegluonpropagator, basedon[71,78].\nThe quark gluon vertex is taken as \u0000\u0016=zqgq\r\u0016, with\nzqgq(Q2) =d1\nd2+Q2(A1)\n+Q2\n\u00032+Q2\u0012\f0\u000b(\u0016) logQ2=\u00032+ 1\n4\u0019\u00132\u000e\n;\n(A2)containing the parameters\nd1= 7:9GeV2d2= 0:5GeV2;\n\u000e=\u000018=88; \u0003 = 1:4GeV:(A3)\nHere the scales must be identified correctly in order to\ncapture the correct dependence with TandeB. We take\nQto be the symmetric momentum Q2= (q2+p2+\n(q\u0000p)2)=3at the vertex with Q2=Q2\n3+Q2\n0+Q2\n?,\nwhereQ2\n0= (2\u0019T)2ifQ2\n0<(2\u0019T)2andQ2\n?= 2jeBjif\nQ2\n?<2jeBj. Wenotethatthisroughlycorrespondstoan\nidentification of scales as in section IIIA with \u0015B\u00191, al-\nthough the present vertex is clearly more sophisticated as\nit includes momentum dependencies and thereby generic\neBeffects. For a current overview of the quark gluon\nvertex in Dyson-Schwinger truncations see [91, 92]. Fur-\nthermore in order to be able to solve the gluon Dyson-\nSchwinger equation we rely on lattice input for the Yang-\nMills part, which we then \"dress\" with magnetic field\neffects, as described above. The reliability of this trunca-\ntion was already discussed in detail at finite temperature\n[71] and utilized in the presence of magnetic fields before\n[6]. The lattice fit is given by\nZ\u00001\nYM(Q2) =Q2\u00032\n(Q2+ \u00032)2h\u0012c\nQ2+a\u00032\u0013b\n+Q2\n\u00032\u0012\f0\u000b(\u0016) logQ2=\u00032+ 1\n4\u0019\u0013\ri\n;(A4)\nwith\n\u0003 = 1:4GeV; c= 11:5GeV2;\n\f0= 11Nc=3; \r =\u000013=22; (A5)\nwhere\u000b(\u0016) = 0:3andaandbare temperature dependent\nparameters, which can be found in [78]. As discussed\nbefore the Dyson-Schwinger truncation scheme can be\nrelated to the skeleton expansion done in our analytic\nestimate, which was motivated by renormalisation group\ninvariance\n4\u0019\u000bs(Q2)rIR(Q2)P\u0016\u0017\nQ02+ \u0005\u0011P\u0016\u0017\nZYMQ02+ \u0005fzqgq;(A6)\nwhere the sum over different polarisation tensor compo-\nnents is implied. The right hand side actually serves as\nthe input to our numerical study, while the different com-\nponents of \u0005are determined dynamically from solving\nthe gluon Dyson-Schwinger equation.\nAppendix B: Magnetic field dependence of the\nfour-fermi coupling from QCD\nAs we have discussed in Section IIIB the value of the\nNJL coupling \u0015at the intrinsic cutoff scale of the model13\nis determined by QCD dynamics. At large scales the dy-\nnamics of\u0015is driven by the rightmost diagram shown\nin Fig. 11. Within simplifications we will motivate the\nfunctional dependence of this diagram on temperature\nand the magnetic field. In the lowest Landau level ap-proximation the quarks are constraint to the t-z plane\ndenoted by (k), whereas the gluons propagate in all four\ndimensions (k;?). We write the gluon box diagram in\nFig. 11 at zero external momentum as\nF\u000bs(eB\u00150:3 GeV)'4:5eBZ1\n0dqk;qk\nq2\nk+m2q+\u000bseBcqZ1\n0dq?;q?\n[q2\n?+q2\nk+m2\nA+eB\u000bscA]2;(B1)\nwhere\u000bsis given as Eq. (12).\nForeB < 0:3(B1) is smoothly (quadratic fit) extrap-\nolated toeB= 0with minimising the eB-dependence.\nThe flavor, color and Dirac tensor indices have been con-\ntracted, and the comparison with the results for \u0015in\nquenched QCD shown in Fig. 13 shows that the prefactor\nresulting from the tensor contract is approximately 4:5.\nWe have written the propagators in a semi-perturbative\nform with medium dependent mass terms. Further we\nhave taken mA\u00191GeV as the decoupling scale, mq\u0019\n300MeV in the chiral broken phase and cA=cq= 1.\nStrictly speaking both masses are larger than 1 GeV as\nwe have to add the cutoff masses /\u00032. We have chosen\nsmaller masses in order to also potentially have access tothe infrared domain k!0, where the constituent quark\nmassisoftheorder 0:3GeVandthegluonicmassgapisof\nthe order 1GeV. Furthermore we have approximated the\nMatsubara sum by an integration, due to the small level\nspacing compared to the magnetic field. This approxi-\nmation does not hold small eB, but (B1) is only used for\neB\u00150:3GeV. Eq. (B1) includes the correct dependence\non\u000bsas well and thus captures eBandTeffects qual-\nitatively. The model parameters in Section IIIB allow\nus to reproduce the quantitative behavior of the chiral\ntransition temperature and a more elaborate version of\nEq. (B1) does not give much greater insight. Apart from\nthe agreement with the Tcresults from lattice calcula-\ntion, Fig. 13 shows that quantitatively reliable results\nfrom QCD-flows in quenched QCD [75] are reproduced.\n[1] I. A. Shovkovy, Lect.Notes Phys. 871, 13 (2013),\narXiv:1207.5081 [hep-ph].\n[2] M. D’Elia, Lect.Notes Phys. 871, 181 (2013),\narXiv:1209.0374 [hep-lat].\n[3] K. Fukushima, Lect.Notes Phys. 871, 241 (2013),\narXiv:1209.5064 [hep-ph].\n[4] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa,\nNucl.Phys. A803, 227 (2008), arXiv:0711.0950 [hep-ph].\n[5] K. Fukushima, D. E. Kharzeev, and H. J. Warringa,\nPhys.Rev. D78, 074033 (2008), arXiv:0808.3382 [hep-\nph].\n[6] N. Mueller, J. A. Bonnet, and C. S. Fischer, Phys.Rev.\nD89, 094023 (2014), arXiv:1401.1647 [hep-ph].\n[7] J. O. Andersen, W. R. Naylor, and A. Tranberg, (2014),\narXiv:1411.7176 [hep-ph].\n[8] V. A. Miransky and I. A. Shovkovy, (2015),\narXiv:1503.00732 [hep-ph].\n[9] G. Basar, D. Kharzeev, and V. Skokov, Phys.Rev.Lett.\n109, 202303 (2012), arXiv:1206.1334 [hep-ph].\n[10] G. Basar, D. E. Kharzeev, and E. V. Shuryak, Phys.Rev.\nC90, 014905 (2014), arXiv:1402.2286 [hep-ph].\n[11] V. Gusynin, V. Miransky, and I. Shovkovy, Nucl.Phys.\nB462, 249 (1996), arXiv:hep-ph/9509320 [hep-ph].\n[12] V. Gusynin, V. Miransky, and I. Shovkovy,\nPhys.Rev.Lett. 83, 1291 (1999), arXiv:hep-th/9811079\n[hep-th].\n[13] V. Gusynin, V. Miransky, and I. Shovkovy, Nucl.Phys.\nB563, 361 (1999), arXiv:hep-ph/9908320 [hep-ph].[14] V. Gusynin, V. Miransky, and I. Shovkovy, Phys.Rev.\nD52, 4747 (1995), arXiv:hep-ph/9501304 [hep-ph].\n[15] T. Inagaki, D. Kimura, and T. Murata,\nProg.Theor.Phys. 111, 371 (2004), arXiv:hep-\nph/0312005 [hep-ph].\n[16] J. K. Boomsma and D. Boer, Phys.Rev. D81, 074005\n(2010), arXiv:0911.2164 [hep-ph].\n[17] M. D’Elia, S. Mukherjee, and F. Sanfilippo, Phys.Rev.\nD82, 051501 (2010), arXiv:1005.5365 [hep-lat].\n[18] A. J. Mizher, M. Chernodub, and E. S. Fraga, Phys.Rev.\nD82, 105016 (2010), arXiv:1004.2712 [hep-ph].\n[19] R. Gatto and M. Ruggieri, Phys.Rev. D82, 054027\n(2010), arXiv:1007.0790 [hep-ph].\n[20] R. Gatto and M. Ruggieri, Phys.Rev. D83, 034016\n(2011), arXiv:1012.1291 [hep-ph].\n[21] B. Chatterjee, H. Mishra, and A. Mishra, Phys.Rev.\nD84, 014016 (2011), arXiv:1101.0498 [hep-ph].\n[22] R. Gatto and M. Ruggieri, Lect.Notes Phys. 871, 87\n(2013), arXiv:1207.3190 [hep-ph].\n[23] M. Frasca and M. Ruggieri, Phys.Rev. D83, 094024\n(2011), arXiv:1103.1194 [hep-ph].\n[24] M. Ferreira, P. Costa, O. Lourenco, T. Frederico,\nand C. Providencia, Phys.Rev. D89, 116011 (2014),\narXiv:1404.5577 [hep-ph].\n[25] E. Ferrer, V. de la Incera, and X. Wen, (2014),\narXiv:1407.3503 [nucl-th].\n[26] R. Farias, K. Gomes, G. Krein, and M. Pinto, Phys.Rev.\nC90, 025203 (2014), arXiv:1404.3931 [hep-ph].14\n[27] A. Ayala, M. Loewe, A. J. Mizher, and R. Zamora,\nPhys.Rev. D90, 036001 (2014), arXiv:1406.3885 [hep-\nph].\n[28] A. Ayala, M. Loewe, and R. Zamora, Phys.Rev. D91,\n016002 (2015), arXiv:1406.7408 [hep-ph].\n[29] L. Yu, J. Van Doorsselaere, and M. Huang, (2014),\narXiv:1411.7552 [hep-ph].\n[30] O. Bergman, G. Lifschytz, and M. Lippert, JHEP 0805,\n007 (2008), arXiv:0802.3720 [hep-th].\n[31] C. V. Johnson and A. Kundu, JHEP 0812, 053 (2008),\narXiv:0803.0038 [hep-th].\n[32] V.G.Filev, C.V.Johnson, andJ.P.Shock,JHEP 0908,\n013 (2009), arXiv:0903.5345 [hep-th].\n[33] F. Preis, A. Rebhan, and A. Schmitt, JHEP 1103, 033\n(2011), arXiv:1012.4785 [hep-th].\n[34] F. Preis, A. Rebhan, and A. Schmitt, Lect.Notes Phys.\n871, 51 (2013), arXiv:1208.0536 [hep-ph].\n[35] A. Ballon-Bayona, JHEP 1311, 168 (2013),\narXiv:1307.6498 [hep-th].\n[36] K. A. Mamo, (2015), arXiv:1501.03262 [hep-th].\n[37] V. Skokov, Phys.Rev. D85, 034026 (2012),\narXiv:1112.5137 [hep-ph].\n[38] K. Fukushima and J. M. Pawlowski, Phys.Rev. D86,\n076013 (2012), arXiv:1203.4330 [hep-ph].\n[39] K. Kamikado and T. Kanazawa, JHEP 1403, 009 (2014),\narXiv:1312.3124 [hep-ph].\n[40] K. Kamikado and T. Kanazawa, (2014), arXiv:1410.6253\n[hep-ph].\n[41] J.O.AndersenandA.Tranberg,JHEP 1208,002(2012),\narXiv:1204.3360 [hep-ph].\n[42] J. O. Andersen, W. R. Naylor, and A. Tranberg, JHEP\n1404, 187 (2014), arXiv:1311.2093 [hep-ph].\n[43] J. Braun, W. A. Mian, and S. Rechenberger, (2014),\narXiv:1412.6025 [hep-ph].\n[44] T. Kojo and N. Su, Phys.Lett. B720, 192 (2013),\narXiv:1211.7318 [hep-ph].\n[45] T. Kojo and N. Su, Phys.Lett. B726, 839 (2013),\narXiv:1305.4510 [hep-ph].\n[46] P. Watson and H. Reinhardt, Phys.Rev. D89, 045008\n(2014), arXiv:1310.6050 [hep-ph].\n[47] G. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. Katz,\net al., JHEP1202, 044 (2012), arXiv:1111.4956 [hep-lat].\n[48] G. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. Katz,\net al., Phys.Rev. D86, 071502 (2012), arXiv:1206.4205\n[hep-lat].\n[49] Bali, G.S. and Bruckmann, F. and Endrodi, G. and\nGruber, F. and Schäfer, A., JHEP 1304, 130 (2013),\narXiv:1303.1328 [hep-lat].\n[50] F. Bruckmann, G. Endrodi, and T. G. Kovacs, JHEP\n1304, 112 (2013), arXiv:1303.3972 [hep-lat].\n[51] Bali, G.S. and Bruckmann, F. and Endrödi, G. and\nKatz, S.D. and Schäfer, A., JHEP 1408, 177 (2014),\narXiv:1406.0269 [hep-lat].\n[52] E. M. Ilgenfritz, M. Muller-Preussker, B. Petersson,\nand A. Schreiber, Phys.Rev. D89, 054512 (2014),\narXiv:1310.7876 [hep-lat].\n[53] V. Bornyakov, P. Buividovich, N. Cundy, O. Ko-\nchetkov, and A. Schaefer, Phys.Rev. D90, 034501\n(2014), arXiv:1312.5628 [hep-lat].\n[54] K. Fukushima and Y. Hidaka, Phys.Rev.Lett. 110,\n031601 (2013), arXiv:1209.1319 [hep-ph].\n[55] E. Fraga, B. Mintz, and J. Schaffner-Bielich, Phys.Lett.\nB731, 154 (2014), arXiv:1311.3964 [hep-ph].[56] J. O. Andersen, W. R. Naylor, and A. Tranberg, (2014),\narXiv:1410.5247 [hep-ph].\n[57] J. M. Pawlowski, AIP Conf.Proc. 1343, 75 (2011),\narXiv:1012.5075 [hep-ph].\n[58] L. M. Haas, R. Stiele, J. Braun, J. M. Pawlowski,\nand J. Schaffner-Bielich, Phys.Rev. D87, 076004 (2013),\narXiv:1302.1993 [hep-ph].\n[59] T. K. Herbst, M. Mitter, J. M. Pawlowski, B.-J.\nSchaefer, and R. Stiele, Phys.Lett. B731, 248 (2014),\narXiv:1308.3621 [hep-ph].\n[60] J. M. Pawlowski, Nucl.Phys. A931, 113 (2014).\n[61] D. Lee, C. N. Leung, and Y. Ng, Phys.Rev. D55, 6504\n(1997), arXiv:hep-th/9701172 [hep-th].\n[62] V. Miransky and I. Shovkovy, Phys.Rev. D66, 045006\n(2002), arXiv:hep-ph/0205348 [hep-ph].\n[63] C. N. Leung and S.-Y. Wang, Annals Phys. 322, 701\n(2007), arXiv:hep-ph/0503298 [hep-ph].\n[64] A. Ayala, A. Bashir, A. Raya, and E. Rojas, Phys.Rev.\nD73, 105009 (2006), arXiv:hep-ph/0602209 [hep-ph].\n[65] E. Rojas, A. Ayala, A. Bashir, and A. Raya, Phys.Rev.\nD77, 093004 (2008), arXiv:0803.4173 [hep-ph].\n[66] V. Ritus, Annals Phys. 69, 555 (1972).\n[67] V. Ritus, Sov.Phys.JETP 48, 788 (1978).\n[68] C. S. Fischer, P. Watson, and W. Cassing, Phys. Rev.\nD72, 094025 (2005), arXiv:hep-ph/0509213.\n[69] D.Nickel, R.Alkofer, andJ.Wambach,Phys.Rev. D74,\n114015 (2006), arXiv:hep-ph/0609198.\n[70] J. Braun, L. M. Haas, F. Marhauser, and\nJ. M. Pawlowski, Phys.Rev.Lett. 106, 022002 (2011),\narXiv:0908.0008 [hep-ph].\n[71] C. S. Fischer and J. Luecker, Phys.Lett. B718, 1036\n(2013), arXiv:1206.5191 [hep-ph].\n[72] C. S. Fischer, L. Fister, J. Luecker, and J.M. Pawlowski,\nPhys.Lett. B732, 273 (2014), arXiv:1306.6022 [hep-ph].\n[73] C. S. Fischer, J. Luecker, and C. A. Welzbacher,\nPhys.Rev. D90, 034022 (2014), arXiv:1405.4762 [hep-\nph].\n[74] J. Braun, L. Fister, J. M. Pawlowski, and F. Rennecke,\n(2014), arXiv:1412.1045 [hep-ph].\n[75] M. Mitter, J. M. Pawlowski, and N. Strodthoff, (2014),\narXiv:1411.7978 [hep-ph].\n[76] T.-K. Chyi, C.-W. Hwang, W. Kao, G.-L. Lin, K.-W.\nNg,et al., Phys.Rev. D62, 105014 (2000), arXiv:hep-\nth/9912134 [hep-th].\n[77] J. M. Pawlowski, Annals Phys. 322, 2831 (2007),\narXiv:hep-th/0512261 [hep-th].\n[78] C. S. Fischer, A. Maas, and J. A. Muller, Eur.Phys.J.\nC68, 165 (2010), arXiv:1003.1960 [hep-ph].\n[79] C. S. Fischer and J. M. Pawlowski, Phys. Rev. D80,\n025023 (2009), arXiv:0903.2193 [hep-th].\n[80] A. Nesterenko, Phys.Rev. D62, 094028 (2000),\narXiv:hep-ph/9912351 [hep-ph].\n[81] A. Nesterenko, Phys.Rev. D64, 116009 (2001),\narXiv:hep-ph/0102124 [hep-ph].\n[82] N. Christiansen, M. Haas, J. M. Pawlowski, and\nN. Strodthoff, (2014), arXiv:1411.7986 [hep-ph].\n[83] R. Alkofer, C. S. Fischer, F. J. Llanes-Estrada,\nand K. Schwenzer, Annals Phys. 324, 106 (2009),\narXiv:0804.3042 [hep-ph].\n[84] J. Braun and H. Gies, JHEP 0606, 024 (2006),\narXiv:hep-ph/0602226 [hep-ph].\n[85] J. Braun and H. Gies, Phys. Lett. B645, 53 (2007),\narXiv:hep-ph/0512085.15\n[86] J. Braun, J.Phys. G39, 033001 (2012), arXiv:1108.4449\n[hep-ph].\n[87] D. F. Litim, Phys.Lett. B486, 92 (2000), arXiv:hep-\nth/0005245 [hep-th].\n[88] C. S. Fischer, A. Maas, and J. M. Pawlowski, Annals\nPhys.324, 2408 (2009), arXiv:0810.1987 [hep-ph].[89] G. Endrodi, (2015), arXiv:1504.08280 [hep-lat].\n[90] K. Nishiyama, S. Karasawa, and T. Tatsumi, (2015),\narXiv:1505.01928 [nucl-th].\n[91] R. Alkofer, G. Eichmann, C. S. Fischer, M. Hopfer,\nM. Vujinovic, et al., PoSQCD-TNT-III , 003 (2013),\narXiv:1405.7310 [hep-ph].\n[92] R. Williams, (2014), arXiv:1404.2545 [hep-ph]."    },    {        "title": "1403.6885v2.Driving_ferromagnets_into_a_critical_region_of_a_magnetic_phase_diagram.pdf",        "content": "Driving ferromagnets into a critical region of a magnetic phase diagram\nB. Y. Mueller∗and B. Rethfeld\nDepartment of Physics and Research Center OPTIMAS, University of Kaiserslautern,\nErwin-Schr¨ odinger-Str. 46, 67653 Kaiserslautern, Germany\n(Dated: October 12, 2018)\nExciting a ferromagnetic sample with an ultrashort laser pulse leads to a quenching of the magne-\ntization on a subpicosecond timescale. On the basis of the equilibration of intensive thermodynamic\nvariables we establish a powerful model to describe the demagnetization dynamics. We demonstrate\nthat the magnetization dynamics is mainly driven by the equilibration of chemical potentials. The\nminimum of magnetization is revealed as a transient electronic equilibrium state. Our method iden-\ntifies the slowing down of ultrafast magnetization dynamics by a critical region within a magnetic\nphase diagram.\nPACS numbers: 75.78.-n, 75.78.Jp, 05.70.Ln\nThe strong increase of computational power within the\nlast thirty years has also boosted the need for large and\nfast data storage. However, the physical speed limits of\nconventional magnetic recording, which are on the order\nof nanoseconds, are nowadays reached [1]. A promising\nenhancement lies in a subpicosecond change of magneti-\nzation, as has been found in 1996 by exciting a ferromag-\nnetic material with an ultrashort laser pulse [2]. Though,\na detailed understanding of the underlying physical pro-\ncesses of this ultrafast demagnetization is still lacking and\nseveral models compete, hampering the further develop-\nment [3–6].\nThe most promising concepts are based on superdiffu-\nsive spin transport [7–10] or Elliott-Yafet (EY) spin-flip\nprocesses [5, 6, 11–21]. It has been shown experimentally,\nthat both processes contribute to the magnetization dy-\nnamics, depending on the sample geometry [3, 9]. On\nthe one hand, superdiffusive transport dominates on bulk\nand multilayer systems and has been successfully com-\npared to experiments [8, 9]. On the other hand, EY spin-\nflip scattering has been investigated with kinetic mod-\nels and reproduces the magnetization dynamics for thin\nfilms [14–16]. Due to the complexity of the methods,\ntemperature-based models have been proposed, like the\nmicroscopic three temperature model (M3TM) [12, 21].\nRecently, it has been shown, that this simplification is\njustified, despite of an ultrafast laser excitation [14].\nIn this Letter, we derive a µT model (µTM), which\ntraces the dynamics and the equilibration of temper-\natures and chemical potentials of the electron subsys-\ntems simultaneously. The essential concepts of the µTM\nare based on a kinetic approach [13–15], including EY-\ntype spin-flip scattering and a dynamic exchange split-\nting [11, 14]. The µTM reproduces the experimental\nmagnetization curves for different laser fluences. We find\nthat the equilibration of chemical potentials drives the\ndynamics of the magnetization and the magnetization\nminimum is revealed as a transient equilibrium state\nwithin a magnetic phase diagram. We identify a critical\nregion within this phase diagram: For certain fluences,the material is driven into this region, causing an ex-\ntreme deceleration of the magnetization dynamics. This\nfinding confirms the experimental observation of a crit-\nical slowing down [12, 21]. Unlike the M3TM, we trace\nthe dynamics of minority and majority electron densi-\nties explicitly, which opens the possibility to extend the\nmodel for superdiffusive transport effects.\nA general matrix formulation of a time- and space-\ndependent coupled transport equation is given by\nCd\ndt/vectorX=∇K∇/vectorX+G/vectorX+/vectorS , (1)\nwhere/vectorXis the vector of transient variables, C,Kand\nGare matrices of capacities, transport and coupling, re-\nspectively, and /vectorSis the source vector. A representative\nof such equation system is the well-known two tempera-\nture model (TTM) [22] of two coupled heat conduction\nequations. In that case the vector of interest /vectorXcon-\nsists of the respective electron and lattice temperature,\nTeandT/lscript, the source vector contributes to the equa-\ntion for the electron energy, and the capacity matrix as\nwell as the transport matrix are diagonal matrices. The\ntemperatures are coupled through an equilibration term,\n±g(Te−T/lscript), thus the coupling matrix Gcontains also\noff-diagonal elements. Here, gis the electron–lattice cou-\npling parameter.\nIn itinerant ferromagnets, the electrons of majority and\nminority spins can be treated separately. The temper-\natures of both electron types, denoted by T↑\neandT↓\ne,\nrespectively, may differ. Moreover, the respective parti-\ncle densities may change due to EY spin-flip processes\nand only their sum n=n↑+n↓is constant. Therefore,\nthe chemical potentials µ↑andµ↓have to be considered\nas further variables of /vectorXin Eq. (1). Further, in the\nframe of an effective Stoner model [14, 23], the densities\nof statesD↑(E) andD↓(E) of up- and down electrons,\nrespectively, are shifted by an exchange splitting ∆. This\nexchange splitting is not constant but is directly coupled\nwith the magnetization mthrough the effective Coulomb\ninteraction U[23]. In Ref. 14 it was shown that the in-arXiv:1403.6885v2  [cond-mat.mtrl-sci]  7 Apr 20142\nstantaneous feedback of the transient magnetization on\nthe exchange splitting,\n∆(t) =Um(t), (2)\nis essential for the quantitative description of demagne-\ntization dynamics. The normalized magnetization m(t)\nresults from the transient particle density of each electron\nreservoir as\nm(t) =/parenleftbig\nn↑(t)−n↓(t)/parenrightbig\n/n . (3)\nThe particle density nσ(Tσ\ne,µσ,m) and internal en-\nergy density uσ\ne(Tσ\ne,µσ,m) of the spin σ∈ {↑,↓}are\ncalculated by the zeroth and first moment of the current\nFermi distribution f(E,Tσ\ne,µσ). Thus, under the given\nconditions both, particle density and internal energy den-\nsity, depend on the two intrinsic variables Tσ\neandµσand\non the magnetization which determines the energy shift\nof the exchange splitting ∆, see Eq. (2). The tempo-\nral derivatives of the energy density uσ\neand the particle\ndensitynσinclude partial derivatives, e.g.\nduσ\ne\ndt=cσ\nT∂Tσ\ne\n∂t+cσ\nµ∂µσ\n∂t+cσ\nm∂m\n∂t, (4)\ndefining the capacity equivalents cσ\nx≡∂uσ\ne\n∂x. Analogously,\npartial derivatives of the particle density are defined as\npσ\nx≡∂nσ\n∂x. This allows us to mathematically separate the\nvariablesTσ\ne,µσandm.\nTo demonstrate the power of the µT model and to sep-\narate the time-dependent effects from transport effects,\nwe restrict ourselves here to the temporal dependence of\nthe decisive variables, which is capable to predict and\nexplain important characteristics of the magnetization\ndynamics of thin ferromagnetic films. The temporal evo-\nlution ofT↑\ne,T↓\ne,T/lscript,µ↑,µ↓andmis expressed with an\nequation of type (1):\n\nc↑\nT0 0c↑\nµ0c↑\nm\n0c↓\nT0 0c↓\nµc↓\nm\n0 0c/lscript0 0 0\np↑\nT0 0p↑\nµ0p↑\nm\n0p↓\nT0 0p↓\nµp↓\nm\n−p↑\nTp↓\nT0−p↑\nµp↓\nµn↑+n↓\nd\ndt\nT↑\ne\nT↓\ne\nT/lscript\nµ↑\nµ↓\nm\n=\n\n−γ−g↑γ g↑0 0 0\nγ−γ−g↓g↓0 0 0\ng↑g↓−g↑−g↓0 0 0\n0 0 0 −ν ν 0\n0 0 0 ν−ν0\n0 0 0 0 0 0\n\nT↑\ne\nT↓\ne\nT/lscript\nµ↑\nµ↓\nm\n+\nS↑(t)\nS↓(t)\n0\n0\n0\n0\n.(5)\nThe first three equations determine the energy of spin-up\nand spin-down electrons as well as of the lattice, respec-\ntively. Equations four and five trace the densities of both\nelectron systems. The last equation defines the transientmagnetization, Eq. (3). In the spirit of the TTM [22], we\nintroduce an respective equilibration term for the elec-\ntron temperatures, ±γ/parenleftbig\nT↑\ne−T↓\ne/parenrightbig\n, and chemical poten-\ntials,±ν(µ↑−µ↓). The laser excitation of each electron\nsystem is described by the source term Sσ(t). To con-\nserve the total energy with a dynamic exchange splitting,\nthe correlation energy [23] uCorr(t) =−Un↑(t)n↓(t)/nis\ntaken into account in c↑,↓\nT,c↑,↓\nµandc↑,↓\nm.\nWe solve the µTM for nickel, with the density of\nstates from Ref. 24. The effective Coulomb interaction\nU= 5.04 eV reproduces the experimental [25] equilib-\nrium magnetization curve well [14]. The lattice heat ca-\npacity is taken as c/lscript= 3.776×106J/Km3[26]. For sim-\nplicity, we introduce the same electron-lattice coupling\ngσ=g/2 = 1×1018W/Km3[27] for both electron sys-\ntems. The coupling parameters between chemical poten-\ntials,ν= 5.80×10601/Jsm3, and the inner-electronic\ntemperature coupling, γ= 163.8×gσ, are newly intro-\nduced in this work. They are determined through a fit of\nthe transient magnetization curve obtained by the µTM\nto experimental data of Ref. 21. With the same laser\nparameters as in Ref. 21, and a reflectivity of R= 0.44,\ntheµTM reproduces the magnetization curve for differ-\nent fluences. A comparison between the experiment and\ntheµT model is depicted in the upper panel of Fig. 1.\nFigure 1 shows from top to bottom the dynamics of\nthe magnetization, of the chemical potentials µ↑andµ↓\nand of the temperatures T↑\ne,T↓\neandT/lscript. Two different flu-\nences were applied for the calculations, F0= 2.5 mJ/cm2\n(blue curves) and 2 ×F0(red curves). The minima of\nthe magnetization curves are marked with vertical lines\nthrough all three panels of Fig. 1. The chemical poten-\ntials (central panel) of majority and minority electrons\ndiffer strongly during irradiation, equal each other for\nan instant cross-over and equilibrate on later timescales.\nThe electron temperatures (lower panel) both grow fast\nduring irradiation, however majority and minority tem-\nperatures differ due to the different heat capacities. After\nexcitation, both electron temperatures equilibrate with\neach other and later also with the lattice temperature.\nInverting the capacity matrix Cin Eq. (5) leads\nto a direct formulation for the temporal derivatives of\nT↑\ne,T↓\ne,T/lscript,µ↑,µ↓andm. In particular, the change of\nmagnetization is given by\ndm\ndt=−2ν\nn/parenleftbig\nµ↑−µ↓/parenrightbig\n, (6)\nwhere the time-dependence occurs only in the difference\nof the chemical potentials. Thus, the µTM directly iden-\ntifies the equilibration of chemical potentials of majority\nand minority electrons as the driving force of magneti-\nzation dynamics, as proposed in Ref. 13.\nFive characteristic points appear in the magnetization\ndynamics. They are indicated in the magnetization curve\nfor the lower excitation in Fig. 1. Their origins are ex-\nplained with the µT model in the following:3\n(I)(II)(III)(IV)(V)0102030405060708090100magnetization [%]m(F=1F0)\nm(F=2F0)\n8.658.6558.668.6658.67chemical potential [eV]µ↑(F=1F0)\nµ↓(F=1F0)\nµ↑(F=2F0)\nµ↓(F=2F0)\n40060080010001200140016001800temperature [K]\n0.5 1.0 1.5 2.0 2.5 3.0\ntime [ps]T↑\ne(F=1F0)\nT↓\ne(F=1F0)\nT/lscript(F=1F0)\nT↑(F=2F0)\nT↑(F=2F0)\nT/lscript(F=2F0)F=1×F0F=2×F0\nFIG. 1. Typical results of the µT model, transient magneti-\nzation (upper panel), chemical potentials (central panel) and\ntemperatures (lower panel). The blue curves correspond to a\nlow fluence F0= 2.5 mJ/cm2, whereas the red curves are cal-\nculated after excitation with twice of that fluence, 2 ×F0. In\nthe upper panel, experimental results [21] are shown for com-\nparison. The vertical lines indicate the respective time where\nthe magnetization dynamics suffer a minimum. Characteristic\npoints (I) to (V), as marked for the blue solid demagnetization\ncurve, are analyzed in the text.\n(I) We analyze the magnetization dynamics directly at\nthe time when the laser hits the sample. Recent ab initio\ncalculations did this as well [5, 6], concluding that the\ninitial change of magnetization, dm/dt|t=0, is too small\nto induce a reasonable demagnetization. This is in ac-\ncordance with the µTM, that predicts even a vanishing\nfirst derivative, dm/dt , for the initial time step, when\nthe chemical potentials are still in equilibrium µ↑=µ↓,\nsee Eq. (6). The feedback effect, induced by a dynamic\nexchange splitting only occurs at later times, when the\nchemical potentials are driven out of equilibrium. The\nµTM explicitly accounts for the feedback effect and its\ninfluence can be illustrated by calculating the secondderivative of Eq. (5) during a constant laser excitation\nd2m\ndt2=−/parenleftBigg\np↑\nT\nc↑\nµp↑\nT−c↑\nTp↑\nµ−p↓\nT\nc↓\nµp↓\nT−c↓\nTp↓\nµ/parenrightBigg\nνS\nn,\nassumingGandCas constant over the considered time\ninterval. Initially, d2m/dt2∝S(t= 0) holds for very\nshort times and the transient magnetization is deter-\nmined mainly by m(t)≈m0+1\n2d2m\ndt2t2. Thus, even for\nvanishingdm\ndt/vextendsingle/vextendsingle\nt=0the description of demagnetization is\npossible by including a feedback effect, and ab initio cal-\nculations as in Ref. [4–6, 11] do not contradict the EY\npicture.\n(II) After the excitation, the magnetization decreases\nrapidly, reaching the maximum change at the inflection\npoint of the magnetization curve. Eq. (6) proposes, that\nalso the nonequilibrium of chemical potentials is at its\nmaximum, which is supported by Fig. 1.\n(III) At the minimum of magnetization a transient equi-\nlibrium between the electron subsystems is observed.\nHere, the chemical potentials µ↑=µ↓(as expected from\nEq. (6)) and also the temperatures T↑\ne=T↓\neare equili-\nbrated, both confirmed by Fig. 1. However, the lattice\nis still not in equilibrium with the electron system. In\nthis transient equilibrium state, the µTM shows that the\nparabola approximation of the minimum,\nd2m\ndt2=2gν\nn/parenleftBigg\np↑\nT\nc↑\nµp↑\nT−c↑\nTp↑\nµ−p↓\nT\nc↓\nµp↓\nT−c↓\nTp↓\nµ/parenrightBigg\n(Te−T/lscript),\nis mainly determined by the temperature difference be-\ntween the electrons and the lattice.\n(IV) After the transient equilibrium state of the elec-\ntron subsystems, the chemical potentials are driven out\nof equilibrium again. This is due to the relaxation with\nthe lattice. At the maximum difference between both\nchemical potentials, the second inflection point in the\nmagnetization curve occurs.\n(V) For larger times, the chemical potentials and tem-\nperatures of the electrons and the lattice equilibrate, see\nFig. 1, and the magnetization reaches its equilibrium\nvaluem(Te).\nThe strength of the µTM is the possibility of analyti-\ncal predictions about many relevant physical processes in\nultrafast magnetization dynamics. In particular, we ob-\nserve in Fig. 1 a so-called critical slowing down of mag-\nnetization dynamics [28, 29] for the high laser fluence.\nThe reason is explained with Fig. 2, which depicts the\nphase diagram of mandTe. For long excitations, in the\norder of nanoseconds, we expect that the magnetization\nfollows the equilibrium magnetization m(Te) which is in-\ndicated as a black curve. However, the ultrashort laser\npulse drives the system out of equilibrium and the mag-\nnetization becomes a function of T↑\ne,T↓\ne,µ↑andµ↓. In\nparticular, in these nonequilibrium states, the chemical\npotentials differ strongly, which is reflected in the central4\n0102030405060708090100\n400 600 800 1000 1200 1400 1600 1800 2000 22000.0 ps 0.2 ps 0.4 ps 0.6 ps 0.8 ps 1.0ps\nmagnetization (%)\nelectron temperature (K)≥\nequilibrium\nincr. fluence\nFIG. 2. Phase diagram of ultrafast magnetization dynamics.\nThe black curve is the equilibrium magnetization cuve m(T).\nThe gray curves result from the µT model for the fluences\nF/F 0= 1.0,1.2,...,2.4,2.8,3.2,3.6 withF0= 2.5 mJ/cm2.\nThe dots mark several times at t= 0,0.3,0.5,1,2,5,10,25 ps.\nThe background color labels the relaxation time towards the\nequilibrium magnetization.\npanel of Fig. 1. The temperatures T↑\neandT↓\neare close\nto each other and are approximated by their mean value\nTefor the following discussion. In Fig. 2, the parametric\n(m,Te) curves of magnetization dynamics after different\nlaser fluences are indicated by gray lines. All curves start\nat room temperature, top left of the diagram. Further\ndots up to 25 ps show the dynamical behavior on the\nparametric curves. The laser drives the system to high\nelectronic temperatures, however, due to the nonequilib-\nrium situation, the magnetization is still finite even for\nT > TC. The first cross-over of the equilibrium mag-\nnetization curve, observed for fluences up to 2 .2×F0,\ncorresponds to the transient equilibrium state (III) and\ncoincides with the minimum of the respective magneti-\nzation curve.\nImportantly, the time to reach the final state on the\nequilibrium magnetization curve m(Te) differs for differ-\nent fluences. For each pair ( m,Te) both chemical poten-\ntialsµ↑,µ↓can be determined by simultaneously solv-\ning Eq. (3) and the equation of particle conservation,\nn= const. We can estimate the time τeqto reach the\nequilibrium magnetization (black curve) for each point\nofmandTein the phase diagram by a relaxation time\napproximation of Eq. (6),\nm/parenleftbig\nTe,µ↑,µ↓/parenrightbig\n−m(Te)\nτeq=−2ν\nn/parenleftbig\nµ↑−µ↓/parenrightbig\n.(7)\nThe relaxation time to equilibrium, τeq, is depicted in the\nbackground color code of Fig. 2. Under strong nonequi-librium conditions, especially at high temperatures, this\nrelaxation occurs very fast: The large difference in chemi-\ncal potentials rapidly drives the magnetization to its equi-\nlibrium value. However, around the Curie temperature\nat 631 K [26], the chemical potentials are nearly equal\nand the equilibration time according to Eq. (7) reaches\nrather high values up to nanoseconds, thus, the magne-\ntization dynamics is extremely decelerated. The fluences\nF/F 0≈2.0−2.4 drive the system into this critical re-\ngion, appearing red in Fig. 2. For low and very high laser\nfluences this region is circumvented. Thus, the µTM di-\nrectly illustrates the origin of a critical slowing down and\nexplains why experiments show a maximum in demag-\nnetization time [12], by utilizing basic thermodynamical\nconcepts.\nIn conclusion, we derived the µT model for itinerant\nferromagnets. The description traces the dynamics of\nspin-resolved electron temperatures andchemical poten-\ntials simultaneously and combined with the coupling to\nthe lattice. The demagnetization process can be de-\nscribed based on a few fundamental physical concepts,\nlike dynamic exchange splitting and the relaxation to-\nwards thermodynamic equilibrium. Our method iden-\ntifies the minimum of the magnetization as a transient\nequilibrium state of the electron systems. We explain\nthe experimentally observed slowing down of the mag-\nnetization dynamics by a critical region in the magnetic\nphase diagram, Fig. 2. For certain fluences, the system\nis driven into this region and the time to reach the equi-\nlibrium magnetization increases considerably.\nFinancial support of the Deutsche Forschungsgemein-\nschaft through the Heisenberg project RE 1141/15 “Ul-\ntrafast Dynamics of Laser-excited Solids” is gratefully\nacknowledged.\n∗bmueller@physik.uni-kl.de\n[1] T. W. McDaniel, Journal of Physics: Condensed Matter\n17, R315 (2005)\n[2] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot,\nPhysical Review Letters 76, 4250 (1996)\n[3] A. J. Schellekens, W. Verhoeven, T. N. Vader, and\nB. Koopmans, Applied Physics Letters 102, 252408\n(2013)\n[4] A. J. Schellekens and B. Koopmans, Physical Review Let-\nters110, 217204 (2013)\n[5] C. Illg, M. Haag, and M. F¨ ahnle, Physical Review B 88,\n214404 (2013)\n[6] K. Carva, M. Battiato, and P. M. Oppeneer, Physical\nReview Letters 107, 207201 (2011)\n[7] M. Battiato, K. Carva, and P. M. Oppeneer, Physical\nReview Letters 105, 027203 (2010)\n[8] A. Eschenlohr, M. Battiato, P. Maldonado, N. Pon-\ntius, T. Kachel, K. Holldack, R. Mitzner, A. F¨ ohlisch,\nP. M. Oppeneer, and C. Stamm, Nature materials 12,\n332 (2013)\n[9] E. Turgut, C. La-o vorakiat, J. M. Shaw, P. Grychtol,5\nH. T. Nembach, D. Rudolf, R. Adam, M. Aeschlimann,\nC. M. Schneider, T. J. Silva, M. M. Murnane, H. C.\nKapteyn, and S. Mathias, Physical Review Letters 110,\n197201 (2013)\n[10] S. Kaltenborn, Y.-H. Zhu, and H. C. Schneider, Physical\nReview B 85, 235101 (2012)\n[11] S. Essert and H. C. Schneider, Physical Review B 84,\n224405 (2011)\n[12] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. F¨ ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann,\nNature Materials 9, 259 (2010)\n[13] B. Y. Mueller, T. Roth, M. Cinchetti, M. Aeschlimann,\nand B. Rethfeld, New Journal of Physics 13, 123010\n(2011)\n[14] B. Y. Mueller, A. Baral, S. Vollmar, M. Cinchetti,\nM. Aeschlimann, H. C. Schneider, and B. Rethfeld, Phys-\nical Review Letters 111, 167204 (2013)\n[15] M. Krauss, T. Roth, S. Alebrand, D. Steil, M. Cinchetti,\nM. Aeschlimann, and H. C. Schneider, Physical Review\nB80, 180407 (2009)\n[16] D. Steil, S. Alebrand, T. Roth, M. Krauß, T. Kubota,\nM. Oogane, Y. Ando, H. C. Schneider, M. Aeschlimann,\nand M. Cinchetti, Physical Review Letters 105, 217202\n(2010)\n[17] B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and\nW. J. M. de Jonge, Physical Review Letters 95, 267207\n(2005)\n[18] M. F¨ ahnle, J. Seib, and C. Illg, Physical Review B 82,144405 (2010)\n[19] J. Walowski, G. M¨ uller, M. Djordjevic, M. M¨ unzenberg,\nM. Kl¨ aui, C. A. F. Vaz, and J. A. C. Bland, Physical\nReview Letters 101, 237401 (2008)\n[20] E. Carpene, E. Mancini, C. Dallera, M. Brenna, E. Pup-\npin, and S. De Silvestri, Physical Review B 78, 174422\n(2008)\n[21] T. Roth, A. J. Schellekens, S. Alebrand, O. Schmitt,\nD. Steil, B. Koopmans, M. Cinchetti, and M. Aeschli-\nmann, Physical Review X 2, 021006 (2012)\n[22] S. I. Anisimov, B. L. Kapeliovich, and T. L. Perel’man,\nSov. Phys. JETP 39, 375 (1974)\n[23] W. Nolting and A. Ramakanth, Quantum Theory of Mag-\nnetism (Springer, Berlin Heidelberg, 2009)\n[24] Z. Lin, L. V. Zhigilei, and V. Celli, Physical Review B\n77, 075133 (2008)\n[25] F. Tyler, Philosophical Magazine Series 7 11, 596 (1931)\n[26] D. R. Lide, G. Baysinger, L. I. Berger, R. N. Gold-\nberg, H. V. Kehiaian, K. Kuchitsu, G. Rosenblatt, D. L.\nRoth, and D. Zwillinger, CRC Handbook of Chemistry\nand Physics (CRC Press, 2005)\n[27] B. Y. Mueller and B. Rethfeld, Physical Review B 87,\n035139 (2013)\n[28] O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell, and\nD. Garanin, Physical Review B 74, 094436 (2006)\n[29] M. G. M¨ unzenberg, Nature Materials 9, 184 (2010)"    },    {        "title": "1103.4443v1.Nonlinear_Dynamics_of_Magnetic_Islands_Imbedded_in_Small_Scale_Turbulence.pdf",        "content": "arXiv:1103.4443v1  [physics.plasm-ph]  23 Mar 2011Nonlinear Dynamics of Magnetic Islands Imbedded in Small Sc ale Turbulence\nM. Muraglia1, O. Agullo1, S. Benkadda1, X. Garbet2, P. Beyer1, A. Sen3\n1France-Japan Magnetic Fusion Laboratory, LIA 336 CNRS, Mar seille, France\n2CEA, IRFM, 13108, St-Paul-Lez-Durance, France\n3Institute for Plasma Research, Bhat, Gandhinagar 382428, I ndia\nThe nonlinear dynamics of magnetic tearing islands imbedde d in a pressure gradient driven tur-\nbulence is investigated numerically in a reduced magnetohy drodynamic model. The study reveals\nregimes where the linear and nonlinear phases of the tearing instability are controlled by the proper-\nties of the pressure gradient. In these regimes, the interpl ay between the pressure and the magnetic\nflux determines the dynamics of the saturated state. A second ary instability can occur and strongly\nmodify the magnetic island dynamics by triggering a poloida l rotation. It is shown that the complex\nnonlinear interaction between the islands and turbulence i s nonlocal and involves small scales.\nMagnetic reconnection is a complex phenomenon in-\nvolving plasma flows and a rearrangement of the mag-\nnetic field lines inside a narrow region (the reconnection\nlayer) where topologically different magnetic flux tubes\ncan get interconnected and reconfigure themselves. It\nplays an important role in fusion experiments and in\nmany astrophysical events [1]. In a complex fusion de-\nvice, such as a tokamak, the plasma is susceptible to\nmany kinds of instabilities which can occur concurrently\nat various space and time scales. Such a coexistence of\nmicroturbulence and magnetohydrodynamic (MHD) ac-\ntivities has been observed in many experiments [2] with\nsome evidence of correlated effects arising from their si-\nmultaneous existence. An important question to address\nis therefore the nature and amount of mutual interaction\nbetween microturbulence and large-scale MHD instabil-\nities - an issue that is at the heart of multi-scale phe-\nnomena of complex systems in astrophysics, geophysics,\nnonlinear dynamics and fluid turbulence. Early ana-\nlytic attempts at investigation of this important ques-\ntion have relied on ad-hoc modeling of turbulence effects\nthrough anomalous transport coefficients [3]. More re-\ncently a minimal self-consistent model based on wave ki-\nnetics and adiabatic theory has been used in [4] to study\nthe interaction of a tearing mode with drift wave tur-\nbulence. Numerical simulation studies in [5] have di-\nrectly addressed the problem of multiscale interactions\nand have taken into account the nonlinear modifications\nof the equilibrium profiles due to turbulence. Such stud-\nies have been extended in [6] to investigate the interac-\ntion between double tearing modes and micro-turbulence\nthrough the excitation of zonal flows. Finally in [7] a\nnumerical investigation of the interaction of a 2D elec-\ntrostatic turbulence with an island whose dynamics is\nnot fully self-consistent but is governed by a generalized\nRutherford equation has been carried out. In this paper\nwe report on self-consistent simulations of the multiscale\ninteraction between microturbulence driven by pressure\ngradients and magnetic islands with a focus on regimes\nwhere the growth of the latter is essentially due to pres-\nsure effects and where small-scale dynamics appear to\nbe important. The background microturbulence is foundto induce a nonlinear rotation of the island as well as\nto significantly alter its final quasi-equilibrium state by\nthe excitation of a secondary instability. We discuss the\ncharacteristics of the various stages of the nonlinear evo-\nlution and also delineate the role of small scales in the\noverall dynamics of the system.\nWe consider a minimalist two-dimensional plasma\nmodel based on the two fluid Braginskii equations in the\ndrift approximation [8, 9] with cold ions and isothermal\nelectrons. The model includes magnetic curvature effects\nand electron diamagnetic effects but neglects electron in-\nertia and Hall effect contributions. The evolution equa-\ntions are\n∂\n∂t∇2\n⊥φ+/bracketleftbig\nφ,∇2\n⊥φ/bracketrightbig\n=/bracketleftbig\nψ,∇2\n⊥ψ/bracketrightbig\n−κ1∂p\n∂y+µ∇4\n⊥φ,(1)\n∂\n∂tp+[φ,p] =−v⋆/parenleftbigg\n(1−κ2)∂φ\n∂y+κ2∂p\n∂y/parenrightbigg\n+ ˆρ2/bracketleftbig\nψ,∇2\n⊥ψ/bracketrightbig\n+χ⊥∇2\n⊥p,(2)\n∂\n∂tψ= [ψ,φ−p]−v⋆∂ψ\n∂y+η∇2\n⊥ψ, (3)\nwhere the dynamical field quantities are the electrostatic\npotentialφ, the electron pressure pand the magnetic flux\nψ. The equilibrium quantities are a constant pressure\ngradient and a magnetic field corresponding to a Harris\ncurrent sheet model [1]. Further, κ1= 2ΩiτAL⊥\nR0and\nκ2=10\n3Lp\nR0are the curvature terms with R0representing\nthe major radius of a toroidal plasma configuration. Lp\nis the gradient scale length, τAis the Alfvén time based\non a reference perpendicular length scale L⊥andΩiis\nthe ion cyclotron frequency. Equations (1-3) are nor-\nmalized using the characteristic Alfvén speed vAand the\nlength scale L⊥.µis the viscosity, χ⊥the perpendicular\ndiffusivity, ηis the plasma resistivity, v⋆=βe/ΩiτAis\nthe normalized electron diamagnetic drift velocity with\nβebeing the ratio between the electronic kinetic pressure\nand the magnetic pressure. ˆρ=ρS\nL⊥is the normalized ion2\n−5 −4 −3 −2 −1−2−10\nlog(η)log(γ)\n  \n p = 0\nφ = 0\nTheoretical points\nNumerical points\nFigure 1: Numerical and theoretical results of the linear\ngrowth rate γversusηatν=χ⊥= 0,∆′= 6andˆρ= 10−1.\nsound Larmor radius. In the limit R0→ ∞, we recover\nthe drift tearing model [9], and when magnetic fluctua-\ntions are weak, ψ∼0 (κ1/negationslash= 0), the system describes\nthe electrostatic interchange instability. Conversely, t he\nlarge island limit, ˆρ=v⋆= 0withΩiτA∼1, gives the\nhighβmodel which was originally introduced by H. R.\nStrauss [10]. The minimalist model used here is, in fact,\na reduced version of the four fields model of [11] where\nwe have ignored the parallel ion dynamics and thereby\nneglected its effect on the transversal pressure balance\n[11].\nAs a preliminary to the numerical study of eqs.(1-3),\nwe first look at some linear results of a simplified set of\nequations with µ=χ⊥=κi= 0where it is possible to\nobtain analytic relations for the linear growth rate of the\ntearing mode under the constant ψapproximation [1, 12],\nnamely,\n∆′=γ2\nk2yα−3/integraltext+∞\n−∞χ′′(z)\nzdz,\nz=−z2χ(z)+(1+ ˆρ2α−2z2)χ′′(z)(4)\nwhereφ(x) =−α−1ψ(0)χ(z),z=αx,α=/parenleftbig\nηγ/k2\ny/parenrightbig1/4\nand∆′is the standard stability parameter. The per-\nturbed pressure is given by p(x) = ˆρ2φ′′(x)(withˆρ= 0\ncorresponding to the classical tearing situation). Fig. 1\nshows the dependence of the linear growth rate γ(η)of\nthe instability on the resistivity with ˆρ= 10−1. The\nnumerical results (circles) are seen to agree quite well\nwith the values (diamonds) of the solution to the an-\nalytic relation Eq.(4). We observe that γ(η)exhibits\na change of slope after a certain value of η. The two\nregimes correspond to the two limiting cases Te= 0or\np= 0(solid line) and φ= 0(dashdot line). The in-\ntersection of the two lines gives the critical value of the\nresistivityηc= 0.58 ∆′−1/2ˆρ5/2∼8×10−4for∆′= 6.\nWhenη>ηc, the linear growth rate given by the classical\ntearing case is higher than the other limiting case. The\nsystem chooses the more unstable case and the classical\ntearing mode is recovered with the scaling laws γ∼η3/5\nandδ∼η2/5. Whenη < ηc, the coupling between p0 1000 2000 3000 4000 5000 6000 7000−25−20−15−10−50\nt/τalog(Energy)\n  \nMagnetic Energy\nPressure Energy\nKinetic Energy\nFigure 2: Time evolution of the magnetic, pressure and kinet ic\nenergies.\nandψis strong and the island formation is driven by the\npressure perturbation. The resistive layer becomes thin-\nner or more singular than in the case Te= 0where only\na(φ,ψ)−coupling exists. Further the disagreement ob-\nserved between the numerical and the theoretical results\nforη >5×10−2is a consequence of the breaking down\nof the constant ψapproximation in this regime.\nWe now discuss the full nonlinear numerical simulation\nof Eqs.(1-3) that explores the mutual interaction between\nsmall scale interchange modes and a small magnetic is-\nland. A semi-spectral code with a 2/3-dealiasing rule in\nthe poloidal direction, a resolution of 128grid points in\nthe radial direction, 96poloidal modes and that main-\ntains conservation properties of the nonlinear terms to\na high degree, has been used. The computational box\nsize isLx=Ly= 2π. In order to isolate the nonlinear\nmechanisms responsible for the island rotation, the lin-\near diamagnetic effect has been turned off in Eq.(3). The\neffect of the latter on the evolution of the tearing mode is\nwell known, namely that it leads to a real frequency and\nconsequently a rotation of the island in the diamagnetic\ndrift direction. Note that we have checked à posteriori\nby turning on the linear diamagnetic term in eq.(3) that\nthe amount of induced nonlinear rotation (obtained by\nsubtracting the linear diamagnetic frequency from the\ntotal rotation) remains the same. In eq.(3) we also set\nκ2= 0, since we find from our simulations that the κ2\ncontribution is rather weak. ˆρandv⋆are taken to be\nequal to 1andβe= 10−2. The parameter related to the\ninterchange instability is κ1= 10−2. The shape of the\nequilibrium magnetic field is chosen to allow a tearing in-\nstability to develop with a poloidal mode number ky= 1\nwith∆′= 6[1]. Fig. 2 shows, for µ=χ⊥=η= 10−4, the\ntime evolution of the magnetic ( Eψ), pressure ( Ep) and\nkinetic (Eφ) energies of the fluctuations for η<ηc, corre-\nsponding to a regime where the magnetic island genera-\ntion is pressure driven. Four phases are observed. First, a\nexponential growth of the magnetic island ( t/lessorsimilar1300τA),\nfollowed by a quasi-plateau phase with however an in-\ncrease of the energies of the three fields ( t/lessorsimilar4500τA).3\nMagnetic Flux, t = 3000 τA\nxy\n  \n−3.14 0 3.1403.146.28\n−2.5−2−1.5−1−0.50Pressure, t = 3000 τA\nxy\n  \n−3.14 0 3.1403.146.28\n−6−4−20246x 10−4\nMagnetic Flux, t = 6000 τA\nxy\n  \n−3.14 0 3.1403.146.28\n−2.5−2−1.5−1−0.50Pressure, t = 6000 τA\nxy\n  \n−3.14 0 3.1403.146.28\n−0.8−0.6−0.4−0.200.2\nFigure 3: Snapshots of the magnetic flux and the pressure at\nt= 3000τA(Upper panel) and t= 6000τA(Lower panel).\nNext, a phase characterized by an abrupt growth of the\nkinetic and pressure energies in which the kinetic energy\nlevel equals the energy of pressure perturbations and fi-\nnally, the system reaches a new quasi-plateau phase for\n(t/greaterorsimilar5100τA). During the linear and first plateau phases,\nthe energy associated with the pressure perturbations is\nhigher than the kinetic energy, i.ethe dynamics is con-\ntrolled by an interplay between the magnetic flux and the\npressure. In the second phase, t/lessorsimilar4500τA, the magnetic\nisland is maintained by adjacent pressure cells similar to\nwhat is usually observed for flow cells in the nonlinear\nregime of a tearing island [1]. This is illustrated in Fig. 3\n(upper panel, t= 3000τA). During this phase, the kinetic\nenergy piles up in the flow cells which are located in the\nvicinity of the island. After t∼3600τA, the flow cells are\nno longer located in the vicinity of the magnetic island.\nAtt/greaterorsimilar4500τA, a sharp growth of the kinetic and pressure\nenergies occurs. Far from the island the current is not\nsignificant, and for t/τA∈[4500,5000], a dominant in-\nterchange mode outside the sheet (φ11,p11)is enhanced.\nThe associated kinetic and pressure energies of the lat-\nter are equal. Here, φ11meansφ(kx= 1,ky= 1). The\ncompetition between the interchange and tearing modes\nlead to the generation of small scale pressure structures\nin the vicinity of the island that suffer further desta-\nbilization leading to a drastic modification of the dy-\nnamics. Indeed, in less than 200Alfven times, around\nt∼5000τA, an abrupt growth of the energy contained\nin the pressure perturbation is observed and the system\ndynamics changes, i.e, a bifurcation occurs. At larger\ntimes,t>5100τA, the pressure dominates over the flow,\nEp≫Eφ, and the size of the magnetic island finally sat-\nurates. Fig. 4 shows the energy spectra of the fields just\nbefore and after the bifurcation . Before the bifurcation,\nthe interchange mode is observed at ky= 1, and as long\nasky≥2, the pressure energy is much higher than the\nkinetic energy. After this dynamical bifurcation, we ob-\nserve a persistence of small scales as well as an enhance-0 20 40 60−70−60−50−40−30−20−100\nkyln(Energy)(a)    t = 4800 τA\n  \nMagnetic Energy\nPressure Energy\nKinetic Energy\n0 20 40 60−40−35−30−25−20−15−10−5\nkyln(Energy)(b)    t = 5200 τA\n  \nMagnetic Energy\nPressure Energy\nKinetic Energy\nFigure 4: Spectral energy densities as functions of the polo idal\nmode number ky, just (a) before ( at t= 4800τA) and (b) after\n(att= 5200τA), the bifurcation.\n−4 −2 0 2 4−1.5−1−0.500.511.52x 10−3 (a)    t = 3000 τA∂ <p>y/∂ x\nx−4 −2 0 2 4−0.6−0.4−0.200.20.40.60.8 (b)    t = 6000 τA∂ <p>y/∂ x\nx\nFigure 5: Plots of the poloidal diamagnetic velocity vdiaat\nt= 3000τA(a) andt= 6000τA(b).\nment of the energies (Fig. 4b). We also find that a mean\npoloidal pressure and flow have been generated, and as\nwe will see below, it is linked to the rotation properties\nof the island. For 8≤ky≤50, there is a trend towards\nan equipartition of the magnetic and pressure spectra. It\nis worth noting that even though the magnetic island is\nstill, at this point, in a quasilinear stage (the magnetic\nenergy being concentrated on the mode ky= 1), the pres-\nsure perturbation has a fully nonlinear structure and is\nmade up mainly of the modes ky<7. An interesting\nfeature clearly observed in the snapshot shown in Fig. 3\natt= 6000τAis the generation of an island structure in\nthe pressure field containing almost 90%of the pressure\nenergyEp.\nIn the final stage where the energies reach a new quasi-\nplateau, as observed in Fig. 2, the change of dynamics\nis characterized by two important macroscopic features.\nFirst, there is a change of symmetry - the poloidal dia-\nmagnetic velocity vdia=∂\n∂x< p >yhaving even par-\nity fort/lessorsimilar5100τA(brackets mean an average over the\npoloidal direction), loses this property after the bifurca -\ntion and has in fact an odd parity in the vicinity of the\ncurrent sheet. This change of parity is clearly shown in\nFig. 5) where vdiais plotted, before and after the transi-\ntion, att= 3000τAandt= 6000τArespectively. The sec-\nond macroscopic change is the inversion of the poloidal\nrotation direction of the magnetic island together with4\nFigure 6: Time evolution of the poloidal position of the cent er\nof the magnetic island yisland(t)(solid blue line), and the\nmodelsydia(t)(dashed red line), yp,φ(t)(dashdot green line).\nan amplification of the velocity. The amplified velocity\narising from the nonlinear interactions is of the order of\nthe linear diamagnetic velocity as verified from á posteri-\noriruns made with the linear diamagnetic term retained\nin the equations. The change of direction in the island\nrotation can be observed in the zoomed frame of Fig. 6\nwhich shows the time evolution of the poloidal position of\nthe island. We find that the increase of vdiaat the transi-\ntion is linked to the coincident growth of the interchange\nmode (φ11,p11) which feeds the angular momentum. The\ndetailed mechanism of this nonlinear generation of an-\ngular momentum is however not known at this time and\nremains an open question.\nSome insights into the origin of the island poloidal ro-\ntation can be obtained from Eq.(3) where one notes that\nboth the self generated zonal and diamagnetic flow terms,\nvzon=∂\n∂x<φ>yandvdia, can produce a poloidal rota-\ntion of the island. To investigate the role of these flows,\nwe have plotted in Fig. 6 the poloidal position of the\ncenter of the island yisland, and the poloidal positions\nrelated to the contributions of the diamagnetic velocity\nvdiaand the zonal flow velocity vzon. More precisely, we\nhave plotted ydia(t) = (1/δ)/integraltextt\n0dt/integraltextδ/2\n−δ/2dxvdia(x,t)and\nyp,φ= (1/δ)/integraltextt\n0dt/integraltextδ/2\n−δ/2dx(vdia−vzon). We observe\nthat the model ydiareproduces well the time evolution of\nthe rotation of the island, before and after the bifurca-\ntion. At larger times, t/greaterorsimilar6000τA, we observe that the\ncontribution of the zonal flow cannot be neglected, even\nif the rotation of the island is mainly governed by the\nnonlinear generation of poloidal diamagnetic velocity. In\n[9], a similar approach was taken, without an averaging\nover the sheet, and a value of the velocity at the center\nof the sheath was used.\nTo summarize, we have shown that the dynamics of\nmagnetic islands can be strongly affected by the presence\nof a background of interchange modes. In the low resis-tivity and/or small ∆′limit, the coupling between the\nmagnetic flux and the pressure is dominant compared to\nthat between the magnetic flux and the plasma poten-\ntial. In the asymptotic nonlinear regime, a pressure is-\nland structure builds up and the pressure pattern is not a\nflux function except in the center of the magnetic island.\nIn fact this regime is a result of a novel nonlinear transi-\ntion that is observed for a wide range of parameters when\nthe condition η < ηcis satisfied. It is initiated by elec-\ntrostatic interchange modes which compress the magnetic\nstructure and generate small scales inside the island. An-\nother noteworthy finding is that the bifurcation leads to a\nchange of symmetry of the diamagnetic velocity that oc-\ncurs when the energy of the large scale interchange mode\nis of the same order of magnitude as the thermal energy\ncontained in the cell maintaining the magnetic structure.\nThe destabilization leads to a poloidal rotation of the is-\nland that is linked to the nonlinearly generated diamag-\nnetic velocity in the current sheet. The basic phenomena\nhighlighted by our results are reproducible over a large\nregion of parametric space and in that sense appear to\nbe generic albeit within the constraints of our minimal-\nist model. Effects ignored in our model including parallel\nheat conduction, parallel ion dynamics and contributions\nof the Hall and electron inertia terms may bring about\nsome modifications. Investigation of such effects in an en-\nlarged model are therefore necessary to provide a more\nglobal perspective of this complex phenomenon and for\nwhich our present studies provide a minimalist and basic\ndescription.\n[1] D. Biskamp, Magnetic Reconnection in Plasmas (Cam-\nbridge University Press, Cambridge, England, 2000).\n[2] K. Tanaka et al,Nucl. Fusion 46, 110 (2006); E. Joffrin\net al,Nucl. Fusion 43, 1167 (2003)\n[3] P.K. Kaw et al,Phys. Rev. Lett. 43, 1398 (1979); A. K.\nSundaram and A. Sen, Phys. Rev. Lett. 44, 322 (1980);\nA. Furuya et al,J. Phys. Soc. Japan 71, 1261 (2002)\n[4] C. J. McDevitt and P. H. Diamond, Phys. of Plasmas\n13, 032302 (2006)\n[5] A. Thyagaraja et al,Phys. of Plasmas 12, 090907 (2005)\n[6] A. Ishizawa and N. Nakajima, Nucl. Fusion 47, 1540\n(2007)\n[7] F. Militello et al,Phys. Plasmas 15, 050701-1 (2008)\n[8] B.D. Scott et al,Phys. Rev. Lett. 54, 1027 (1985).\n[9] M. Ottaviani et al,Phys. Rev. Lett. 93, 075001 (2004)\n[10] H. R. Strauss, Phys. Fluids 20,1354–1360 (1977)\n[11] R. D. Hazeltine et al,Phys. of Fluids 28, 2466 (1985);\nR. D. Hazeltine and H. R. Strauss, Phys. Rev. Lett. 37,\n102 (1976)\n[12] R.B. White, The Theory of Toroidally Confined Plasmas\n(Imperial College Press, London, England, 2001)."    },    {        "title": "0705.1803v1.Comment_on_six_papers_published_by_M_A__El_Hakiem_and_his_co_workers_in_International_Communications_in_Heat_and_Mass_Transfer__Journal_of_Magnetism_and_Magnetic_Materials_and_Heat_and_Mass_Transfer.pdf",        "content": "1\nComment on \nsix papers published by M.A. El-Hakiem and his co-workers  in\nInternational Communications in Heat and Mass Transfer, Journal of \nMagnetism and Magnetic Materials and Heat and Mass Transfer\nAsterios Pantokratoras\nAssociate Professor of Fluid Mechanics \nSchool of Engineering, Democritus University of Thrace,\n67100 Xanthi – Greece\ne-mail:apantokr@civil.duth.gr\n1. “Joule heating effects on magneto hydrodynamic free convection \nflow of a micro polar fluid”, by M.A. El-Hakiem, A.A. \nMohammadein and S.M.M. El-Kabeir [International \nCommunications in Heat  and Mass Transfer , 26, 1999, pp. 219-\n227]\n2. “Viscous dissipation  effects on MHD free convection flow over a \nnonisothermal surface in a micro polar fluid”, by M.A. El-Hakiem\n[International Communications in Heat  and Mass Transfer , 27, \n2000, pp. 581-590]\n3. “Heat and mass transfer in magneto hydrodynamic flow of a micro \npolar fluid on a circular cylinder with uniform heat and mass flux”, \nby M.A. Mansour, M.A. El-Hakiem and S.M. El-Kabeir [ Journal of \nMagnetism and Magnetic Materials , 220, 2000, pp. 259-270]\n4. “Thermal radiation effect on non-Darcy natural convection with \nlateral mass transfer”, by M.A. El-Hakiem and M.F. El-Amin [ Heat \nand Mass Transfer , 37, 2001, pp. 161-165]2\n5. “Mass transfer effects on the non-Newtonian fluids past a vertical \nplate embedded in a porous medium with non-uniform surface heat \nflux”, by M.A. El-Hakiem and M.F. El-Amin [ Heat and Mass \nTransfer , 37, 2001, pp. 293-297]\n6. “Combined convection in  non-Newtonian fluids along a \nnonisothermal vertical plate in a porous medium with   lateral mass \nflux”, by M.A. El-Hakiem  [ Heat and Mass Transfer , 37, 2001, pp. \n379-385]\nIn all the above papers there is a problem in some figures. It is \nknown in boundary layer theory that velocity and temperature \nprofiles approach the ambient fluid conditions asymptotically and do \nnot intersect the line which represents the boundary conditions. In \nthe following figure we show schematically one correct profile and \none wrong profile. 3\n4\nIn the above six papers there are many profiles which are similar to \nprofile 2 as follows:\nFirst paper: All profiles  included in figure 1 \nSecond paper: All profiles included in figures 1 and 4\nThird paper: All profiles included in figures 7 and 10. \nFourth paper: Some profiles included in figures 2 and 3.\nFifth paper: Some profiles included in figure 6\nSixth paper: Some profiles included in figures 3, 5, 6 and 7.  \nAll the above profiles  are probably truncated due to a small  \ncalculation domain used and are wrong.   "    },    {        "title": "1403.3188v2.Monte_Carlo_simulated_dynamical_magnetization_of_single_chain_magnets.pdf",        "content": "arXiv:1403.3188v2  [cond-mat.mes-hall]  22 Oct 2014Monte Carlo simulated dynamical magnetization of single-c hain magnets\nJun Li and Bang-Gui Liu∗\nBeijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n(Dated: February 13, 2020)\nHere, a dynamical Monte-Carlo (DMC) method is used to study t emperature-dependent dynam-\nical magnetization of famous Mn 2Ni system as typical example of single-chain magnets with st rong\nmagnetic anisotropy. Simulated magnetization curves are i n good agreement with experimental\nresults under typical temperatures and sweeping rates, and simulated coercive fields as functions of\ntemperature are also consistent with experimental curves. Further analysis indicates that the mag-\nnetization reversal is determined by both thermal-activat ed effects and quantum spin tunnelings.\nThese can help explore basic properties and applications of such important magnetic systems.\nPACS numbers: 75.75.-c, 05.10.-a, 75.78.-n, 75.10.-b, 75. 90.+w\nVarious nanoscale spin chains have been attracting\ngreat attention because of their important properties\nand potential applications in information science and\ntechnology[1–5]. The single-chain magnet (SCM) is a\nnew member of such nanoscale spin chains, and its basic\nspin unit come from some transition-metal or rare-earth\nions combined with appropriate organic molecules[2–\n14]. A famous SCM is the [Mn 2Ni] system[3–7], with\nC62H64N10O14Cl2Mn2Ni and C 60H66N12O14Cl2Mn2Ni\nas two typical formula units with spin S= 3. A well-\nknown Arrhenius law has been observed for their spin\nrelaxation at high enough temperature [15, 16]. On\nthe other hand, at low enough temperature, quantum\nLandau-Zener (LZ) spin tunneling should play impor-\ntant roles in their spin reversal[17, 18]. Such phenomena\nacn be investigated by using some methods for single-\nmolecule magnets[19–28]. As for SCM systems, inter-\nspin exchangeinteractionsplay important rolesand ther-\nmal effects can cause Glauber spin dynamics[5, 6], which\nwas originally proposed for one-dimensional Ising spin\nmodel[29–31]. Furthermore, a systematical experimental\nstudy showsthat quantum nucleation canbecome impor-\ntant to reverse single spins, create domains of reversed\nspins, and reverse the whole SCM[7]. Therefore, it is\nuseful to elucidate what roles these play in determining\ndynamical magnetization of SCM systems.\nHere, we use the hybrid DMC method and thereby in-\nvestigate the [Mn 2Ni] SCM system as a typical example\nof SCMs, taking both classical and quantum effects into\naccount. Our results for typical temperatures and sweep-\ning rates are consistent with corresponding experimental\ncurves. It is very interesting that we can satisfactorily\nfit the simulated and experimental Bc-Tcurves by one\nsimple function. These means that the DMC method\nand simulated results are both reasonable and reliable\nfor such SCM systems. Furthermore, we explain mag-\nnetization reversal modes for different temperatures on\nthe basis of our simulated results and analyses. More\ndetailed results will be presented in the following.\nThe single-chain magnet can be considered a one-dimensional composite spin lattice whose spins can be\nconstructedby repeating abasic unit of[Mn 2Ni]: Mn-Ni-\nMn (or Mn3+-Ni2+-Mn3+). The antiferromagentic Ni-\nMn interaction is much stronger than the ferromagnetic\nMn-Mn one so that the low-temperatures physics of this\nspin chain can be modelled by an effective ferromagentic\nchain of the units of [Mn 2Ni] (S= 3) with spin interac-\ntion only between the nearest units[3, 4, 6–12].\nThe ferromagnetic spin Hamiltonian can be expressed\nas[7, 9, 14]\nˆH=ˆH0−N−1/summationdisplay\ni=1Jˆ/vectorSi·ˆ/vectorSi+1−N/summationdisplay\ni=1gµBBzˆSz\ni(1)\nwheregistheLande gfactor(g= 2isused), µBtheBohr\nmagneton, J(>0) the ferromagnetic exchange constant.\nˆ/vectorSi={ˆSx\ni,ˆSy\ni,ˆSz\ni}is the spin vector operator for the i-th\nMn2Ni unit, and\nˆH0=N/summationdisplay\ni=1{−D(ˆSz\ni)2−E[(ˆSx\ni)2−(ˆSy\ni)2]}(2)\nis the Hamiltonian for the isolated ferromagnetic spin.\nDandEare the anisotropic parameters. As for the\nparametersofthe spin interaction and on-site anisotropy,\nwe use J/kB=1.56K and D/kB=2.5K from thermo-\ndynamical measurements[7, 9, 14]. The transverse\nanisotropic parameter Eis much smaller, but necessary\nto realize the Landau-Zener spin tunnelling. We take\nE/kB=0.1K by comparing our simulated results with ex-\nperimental ones.\nWe use a dynamical Monte Carlo method to simulate\nthe spin dynamics of the interacting spin system under\nsweeping magnetic field[28, 32, 33]. At the beginning, we\nset all of the spins at the state Sz=−3.\nWe divide the time tinto small time steps with a step\nlength ∆ tand describe the Monte Carlo time points with\nt(n), wherentakes0, 1, 2, 3,.... The magnetic field starts\nfrom−B0and increases by an increment of ∆ t·νuntil\nB0. The spin can be reversed within a Monte Carlo step\n(MCS) through the two reversal mechanisms.2\nΔ\u0001\n\u0000 \u0002 \u0003\u0004 \u0005\u0006\n\u0007\b \t\nΔ\n\u000b \f \r\u000e \u000f \u0010\n\u0011\n\u0012 \u0013 \u0014 \u0015\u0016\u0017\u0018\n\u0019 \u001a\n\u001b\u001c \u001d\u001e \u001f\n \n! \" # $% &'\n( ) * +, -. / 0\nFIG. 1: A schematic of the three spin reversal mechanisms:\nThermal-activated barrier hurdling (a), direct and therma l-\nassisted LZ tunnelings (b). The horizontal solid line with\narrow means that the two energy levels satisfy the resonance\nconditions. Theprobabilities, energylevels, barrier, an dother\nsymbols are defined in the text.\nFor the classical thermal activation, we can obtain the\nfollowing probability Pththat within the time decrement\n∆t[15, 16].\nPth= 1−exp(−Ri∆t) (3)\nWhereRi=R0exp(−∆Ei\nkBT) is the transition rate, kB\nis the Boltzmann constant, Ttemperature, and R0the\ncharacteristic frequency for the spin system (3 ×108s−1).\n∆Eiis the potential barrier of the i-th spin between\nSz\ni=−3 andSz\ni= 3, as shown in Fig. 1.\nThere is a necessary condition for a LZ tunnelling\nof a spin to occur: one of the spin energy levels on\nthe side must be equivalent to another, for example,\nEm(t) =Em′(t), as shown in Fig. 1. With the neighbor-\ning spins taken into account, such conditions are satisfied\nat the given magnetic fields[28]. The corresponding LZ\ntransition probability is given by\nPm,m′\nLZ= 1−exp/bracketleftBigg\n−π(∆m,m′)2\n2/planckover2pi1gµB|m−m′|ν/bracketrightBigg\n(4)\nwhere the tunnelling splitting ∆ mm′is the energy gap\nat the avoidedcrossingofstates mandm′, andνdenotes\nthe sweeping rate of the magnetic field.\nWhenmequalsto S=−3andm′isS′, weobtainadi-\nrect LZtunnelling with the probability Pd\nLZ=PS,S′\nLZ. For\nother possible LZ tunnelling to happen, the spin at first\nmust be excited from Sz=−3 to the mvalues through\nsome thermal activations, as shown in Fig. 1. Consider-\ning the thermal probability PS,m\nthwhich can be obtained\nbyusingtheexpressions(3), theprobabilityofspinrever-\nsal in this channel, Pm\nLZ, is given by Pm\nLZ=PS,m\nth·Pm,m′\nLZ.\nAll the three spin-reversal channels are combined to give\nthe total probability for a spin reversal[28]:\nPtot= 1−(1−Pth)·/parenleftbig\n1−Pd\nLZ/parenrightbig\n·/productdisplay\nm(1−Pm\nLZ) (5)\n1 2 3 4 5 6 7 8 9 :\n; < = >? @ A B\nC D EF G HI J K\nL M NO P Q RS T U VWX Y Z [ \\ ] ^ _\n`abc\nd e f g h i jk l m n o p qr s t u v w x yz { | } ~          \nFIG. 2: Hysteresis loops (normalized magnetization ( M/Ms)\ncurves against the sweeping field B) for three temperatures:\n0.5, 1.5, and 2.5 K. For every temperature, five magnetizatio n\ncurves are plotted with five field sweeping rates: 0.001, 0.00 4,\n0.017, 0.07, and 0.28 tesla/s (from the innermost loop to the\noutermost for each temperature).\nIn our simulations, we take ∆ t= 0.1ms and use 100\nunits of Mn 2Ni with free boundary condition. The mag-\nnetization is calculated by averaging Sz\nioverthe 100 spin\nsites. Each data point is calculated by averaging 10000\nindependent runs to reduce possible errors. The value of\nB0is made large enough to obtain complete hysteresis\nloops with the help of a symmetrization treatment.\nPresented in Fig. 2 are our typical simulated mag-\nnetization curves for five different field sweeping rates ν\n(0.001, 0.004, 0.017, 0.07, and 0.28 tesla/s) at three dif-\nferenttemperatures T: 2.5, 1.5,and0.5K.Thesimulated\nresults show that the hysteresis loops are strongly depen-\ndent on both temperature Tand field sweeping rate ν.\nOur simulation shows that there is no hysteresis loop for\nall the field sweeping rates when temperature reaches 3\nK, and at 2.5 K, the thermal effects are dominant and\nspins can be easily reversed, which results in very small\nhysteresis loops. Our data analysis indicates that when\nthe temperature further decreases, the thermal-activated\nspin reversal becomes less important and the thermal-\nassistedLZspintunnellingalreadytakesplacefrequently.\nAt 1.5 K, another typical temperature, these two chan-\nnels are available for the spin being reversed, but the\ntotal reversal probability is less than that of 2.5 K, and\nhence the coercive fields is substantially larger than that\nof 2.5 K. When the temperature becomes very low, for\nexample down to 0.5 K, our probability analysis reveals\nthat the thermal activation is almost frozen and the spin\nreversal can be realized only through the direct LZ spin\ntunnelling, and as a result, the coercive fields are large\nbecause the transverse parameter Eis very small. Even\nat this low temperature, there is no clear step structure\nin the magnetization curves, which should be attributed\nto the strong spin exchange interaction in the J-term.3                    \n  ¡¢ £ ¤¥ ¦ §¨ © ª« ¬ ® ¯ °± ² ³´ µ ¶\n· ¸ ¹ º » ¼ ½ ¾¿ À Á Â Ã Ä Å ÆÇ È É Ê Ë Ì ÍÎ Ï Ð Ñ\nÒÓÔÕÖ×ØÙÚFIG. 3: Temperature ( T) dependence of the coercive fields\n(Bc) for the three field sweeping rates: 0.004, 0.035, and 0.14\ntesla/s. The curves are well fitted with Eq. (6) using the set\nof parameters ( A,T1,T2) presented in Table I.\nThis is in contrast to those in the cases of Mn 12and\nFe8systems[19, 20, 28]. Our simulated magnetization\ncurves also show that the larger the field sweeping rate,\nthe larger the hysteresis loop. This trend can be ex-\nplained by considering that larger sweeping rate means\nshorter time for spins to try towards reversal, as shown\nin classical nanoscale spin systems[32, 33].\nFurthermore, we have done more simulations with\nmore field sweeping rates and more temperatures. In\nFig. 3 we present our systematical results on the co-\nercive fields Bcas functions of temperature Tfor three\nsweeping rates ν: 0.004, 0.035, and 0.14 tesla/s. For all\nthe three field sweeping rates, it is clear that the coercive\nfields decrease with temperature increasing. It is very\ninteresting that these Bc-Tcurves can be well fitted by\nthe following simple function.\nBc=A\n1+exp(T\nT1−T2\nT)(6)\nFor the three Bc-Tcurves in Fig. 3, the fitting pa-\nrameters ( A,T1,T2) are summarized in Table I. When\nthe temperature is high, the Bc-Tcurves are dominated\nby theT1term in the exponential, Bc∼Aexp(−T\nT1),\nwhich should be naturally attributed to thermal activa-\ntions. When the temperature is below 1 K, the coercive\nfields substantiallydeviate fromclassicalbehavior. Espe-\ncially when the temperature decreases below 0.5 K, the\ncoercive fields tend to saturate, Bc∼A. This means\nthat the low-temperature saturation behavior is consis-\ntent with quantum LZ effect, in contrast with Glauber\ndynamics[29–31].\nIt is very surprising that the simple function (6) can\nsatisfactorily describe the experimental Bc-Tcurves for\nsuch sweeping rates, too. Our fitted parameters for the\nexperimental curves are summarized in Table II.\nIn high-temperature region, the magnetization rever-TABLE I: Fitting parameters of the three theoretical Bc-T\ncurves in Fig. 3 in terms of the function defined in Eq. (6).\nν(tesla/s) A(tesla) T1(K) T2(K)\n0.004 3.22 0.62 2.54\n0.035 3.27 0.65 3.27\n0.14 3.31 0.67 3.63\nsal is characterized by easy classical end-site nucleation\nand fast classicalwall-movinggrowth ofthe reversedspin\ndomain. At intermediate temperature such as 1.5 K, the\nmagnetization reversal is realized by many-site quantum\nnucleations and classical wall-moving growth of the re-\nversed spin domains. In the low-temperature region, the\nmagnetizationreversalisduetofrequentmany-sitequan-\ntum nucleations of the reversed-spin domains and these\ndomains are effectively merged by subsequent spin tun-\nnelings. Importantly, it can leads to crossover between\nthese three modes to changetemperature. Therefore, the\nthree modes have already been unified into one mecha-\nnism in terms of our theory (our model treatment plus\nour simulation).\nTABLE II: Fitting parameters of the three experimental[7, 9 ,\n14]Bc-Tcurves in terms of the function defined in Eq. (6).\nν(tesla/s) A(tesla) T1(K) T2(K)\n0.004 2.63 0.63 2.85\n0.035 2.82 0.70 2.99\n0.14 2.97 0.75 3.08\nAs are clearly shown in Table II, our theoretical pa-\nrameters are in good agreement with those from experi-\nmental data[7, 9, 14]. In addition, we consider a three-\ndimensional spin system by introducing a very weak\ninter-chain spin exchange coupling. Our Monte Carlo\nsimulation indicates that its sublattice magnetization as\na function of temperature is consistent with experimen-\ntal results concerned[11]. Our simulated magnetization\ncurves shown Fig. 2 and Bc-Tcurves in Fig. 3 are both\nin good agreement with experimental curves[7]. These\nshow that our model treatment and simulation methods\nare reliable and our simulated results, with parameters\nfrom experiment, are reasonable.\nAs for spin dynamics in SCM systems, Glauber be-\nhavior, usually with some modifications due to finite size\neffects, is frequently observed, and on the other hand,\nthere are convincing evidences that quantum nucleation\nplays some important roles in the magnetization dynam-\nics. Our simulated results show that both the classical\nthermal activation and quantum spin tunneling play im-\nportant roles in determining the spin dynamics. For high\ntemperatures, the classical thermal activation is domi-\nnating, but at very low temperatures the classical effect4\nbecomes less important, even is frozen, so that the spin\ndynamicsis determined mainly by the quantum spin tun-\nneling effect.\nIn summary, we have made the hybrid DMC method\nsuitable to studying the spin dynamics of SCMs with\nstrong magnetic anisotropy, and used it to investigate\ntemperature-dependent dynamical magnetization behav-\niors of the famous [Mn 2Ni] SCM system. Our DMC\nsimulated magnetization curves are in good agreement\nwith experimental results under typical temperatures\nand sweeping rates. We have also calculated the coer-\ncive fields as functions of temperature and plotted Bc-T\ncurves for typical sweeping rates. It is interesting and\nsurprising that our simulated Bc-Tcurves are well con-\nsistentwith experimentalones, and bothofthe simulated\nand experimental curves can be satisfactorily fitted with\nthe simple function in Equ. (6). These means that our\ntheory and simulated results are reasonable and reliable\nto other SCM systems and those made from adatoms on\nsurfaces[1, 2, 34].\nThis work is supported by Nature Science Foun-\ndation of China (Grant No. 11174359), by Chinese\nDepartment of Science and Technology (Grant No.\n2012CB932302), and by the Strategic Priority Research\nProgram of the Chinese Academy of Sciences (Grant No.\nXDB07000000).\n∗Corresponding author: bgliu@iphy.ac.cn\n[1] S. Loth, S. Baumann, C. P. Lutz, D. M. Eigler, and A.\nJ. Heinrich, Science 335, 196 (2012).\n[2] E. Heintze, F. E. Hallak, C. Clauss, A. Rettori, M. G.\nPini, F. Totti, M. Dressel, and L. Bogani, Nat. Mater.\n12, 202 (2013).\n[3] R. Clerac, H. Miyasaka, M. Yamashita, and C. Coulon,\nJ. Am. Chem. Soc. 124, 12837 (2002)\n[4] H.Miyasaka, R.Cleac, K.Mizushima, K.Sugiura, M.Ya-\nmashita, W. Wernsdorfer, and C. Coulon, Inorg. Chem.\n42, 8203 (2003)\n[5] W.-X. Zhang, R. Ishikawa, B. Breedlovea and M. Ya-\nmashita, RSC Advances 3, 3772 (2013).\n[6] C. Coulon, R. Clerac, L. Lecren, W. Wernsdorfer, and H.\nMiyasaka, Phys. Rev. B 69, 132408 (2004).\n[7] W. Wernsdorfer, R. Clerac, C. Coulon, L. Lecren, and H.\nMiyasaka, Phys. Rev. Lett. 95, 237203 (2005)\n[8] J. Kishine, T. Watanabe, H. Deguchi, M. Mito, T. Sakai,\nT. Tajiri, M. Yamashita, and H. Miyasaka, Phys. Rev. B\n74, 224419 (2006)[9] L. Lecren, W. Wernsdorfer, Y. Li, A. Vindigni, H.\nMiyasaka, and R. Clerac, J. Am. Chem. Soc. 129, 5045\n(2007)\n[10] C. Coulon, R. Clerac, W. Wernsdorfer, T. Colin, A.\nSaitoh, N. Motokawa, and H. Miyasaka, Phys. Rev. B\n76, 214422 (2007)\n[11] C. Coulon, R. Clerac, W. Wernsdorfer, T. Colin, and H.\nMiyasaka, Phys. Rev. Lett. 102, 167204 (2009)\n[12] O. V. Billoni, V. Pianet, D. Pescia, and A. Vindigni,\nPhys. Rev. B 84, 064415 (2011).\n[13] M. G. Pini, A. Rettori, L. Bogani, A. Lascialfari, M.\nMariani, A. Caneschi, and R. Sessoli, Phys. Rev. B 84,\n094444 (2011).\n[14] K. Bernot, L. Bogani, A. Caneschi, D. Gatteschi, and R.\nSessoli, J. Am. Chem. Soc. 128, 7948 (2006)\n[15] S. Arrhenius, Z. Phys. Chem. 4, 226 (1889).\n[16] R. D. Kirby, J. X. Shen, R. J. Hardy, and D. J. Sellmyer,\nPhys. Rev. B 49, 10810 (1994).\n[17] L. Landau, Phys. Z. Sowjetunion 2, 46 (1932).\n[18] C. Zener, Proc. R. Soc. London, Ser. A 137, 696 (1932).\n[19] D. Gatteschi, R. Sessoli, and J. Villain, Molecular Nan o-\nmagnets, Oxford University Press, New York, 2006.\n[20] D. Gatteschi and A. Vindigni, Single-chain magnets, in :\nJ. Bartolome, F. Luis, and J. F. Fernandez, Ed., Molecu-\nlar Magnets - Physics and Applications, Springer-Verlag\nBerlin Heidelberg 2014.\n[21] H. De Raedt, S. Miyashita, K. Saito, D. Garcia-Pablos,\nand N. Garcia, Phys. Rev. B 56, 11761 (1997).\n[22] Q. Niu and M. G. Raizen, Phys. Rev. Lett. 80, 3491\n(1998).\n[23] W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi,\nand A. Cornia, Europhys. Lett. 50, 552 (2000).\n[24] V. L. Pokrovsky and N. A. Sinitsyn, Phys. Rev. B 65,\n153105 (2002).\n[25] M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J .\nH. Muller, E. Courtade, M. Anderlini, and E. Arimondo,\nPhys. Rev. Lett. 91, 230406 (2003).\n[26] E. Rastelli and A. Tassi, Phys. Rev. B 64, 064410 (2001).\n[27] P. F¨ oldi, M. G. Benedict, J. M. Pereira, Jr., and F. M.\nPeeters, Phys. Rev. B 75, 104430 (2007).\n[28] G.-B. Liu and B.-G. Liu, Appl. Phys. Lett. 95, 183110\n(2009); Phys. Rev. B 82, 134410 (2010).\n[29] R. J. Glauber, J. Math. Phys. (N.Y.) 4, 294 (1963).\n[30] G. Korniss, J. White, P. A. Rikvold, and M. A. Novotny,\nPhys. Rev. E 63, 016120 (2000).\n[31] K. Park, P. A. Rikvold, G. M. Buendia, and M. A.\nNovotny, Phys. Rev. Lett. 92, 015701 (2004).\n[32] Y. Li and B.-G. Liu, Phys. Rev. B 73, 174418 (2006);\nPhys. Rev. Lett. 96, 217201 (2006).\n[33] B.-G. Liu, K.-C. Zhang, and Y. Li, Front. Phys. China\n2, 424 (2007).\n[34] A. Spinelli, B. Bryant, F. Delgado, J. Fernandez-Rossi er,\nand A. F. Otte, Nat. Mater. 13, 782 (2014)."    },    {        "title": "1504.01162v1.Magnetic_states_in_multiply_connected_flat_nano_elements.pdf",        "content": "arXiv:1504.01162v1  [cond-mat.mes-hall]  5 Apr 2015Magnetic states in multiply-connected flat nano-elements.\nAndrei B. Bogatyr¨ ev\nInstitute for Numerical Mathematics, Russian Academy of Sc iences, 8 Gubkina str., Moscow GSP-1, Russia 119991\nKonstantin L. Metlov\nDonetsk Institute of Physics and Technology, 72 R. Luxembou rg str., Donetsk∗\n(Dated: July 9, 2021)\nFlat magnetic nano-elements are an essential component of c urrent and future spintronic devices.\nBy shaping an element it is possible to select and stabilize c hosen metastable magnetic states,\ncontrol its magnetization dynamics. Here, using a recent si gnificant development in mathematics of\nconformal mapping, complex variable based approach to the d escription of magnetic states in planar\nnano-elements is extended to the case when elements are mult iply-connected (that is, contain holes\nor magnetic anti-dots). We show that presence of holes impli es a certain restriction on the set of\nmagnetic states of nano-element.\nPACS numbers: 75.60.Ch, 75.70.Kw, 85.70.Kh\nWhile existence of topological solitons1as metastable\nstates in infinite 2-d ferromagnets was known theoreti-\ncally for quite a long time,2–4real systems have finite\nsize. Their boundary imposes restrictions on the set of\nlow-energytopologicalstates, makingit equivalent to the\nset of rational functions of complex variable with real\ncoefficients,5as opposed to complex coefficients in the\nlaterally unconstrained thin film case.2Also, restricted\ngeometry implies the possibility of formation of half-\nvortex states, pinned at the side of the planar magnet,5,6\nwhich are topologically similar to the boundary states\nin the fractional quantum Hall systems and topological\ninsulators. The shape of nano-element enters this com-\nplex description5via its conformal mapping to the unit\ndisk. It is also possible to express Landau-Lifshitz mag-\nnetization dynamics directly in terms of these complex\nfunctions using Lagrangian approach.7The complex de-\nscription of magnetization states5and their dynamics7,8\nconstitutes a complete set of analytical tools to study\nmagnetic textures in simply-connected nanomagnets far\nfrom magnetic saturation. Here this description is ex-\ntended forthe casewhen planarnano-magnetis multiply-\nconnected .\nLet us briefly introduce the description for simply-\nconnected case.5In small enough magnets the surface ef-\nfectsdominatethevolumeones,andalsotheexchangein-\nteraction is more important than the magnetostatic one.\nThe former is typical for any small systems, while the\nlatter is easy to understand based on the representation\nof the magnetostatic energy in terms of the interaction\nof fictitious magnetic charges of opposite signs. The to-\ntal magnetic charge is always zero, thus, whatever the\ndistribution of the magnetization is, as the size of the\nmagnet decreases, the positive and negative charges are\nbrought closer together, so that their positive self-energy\ngets more and more compensated by their negative mu-\ntual interaction energy. The scaling of the exchange en-\nergy of the set of such an arbitrary magnetization distri-\nbutions follows the volume of the magnet and does not\nhave such an additional reduction. Thus, in small mag-nets the exchange interaction becomes more important.\nTomakeuse ofthis energyhierarchy,webuild anapprox-\nimate expressions for magnetic states by minimizing the\nenergy terms sequentially (as opposed to minimization\nof their sum, which would result in the exact theory).\nThe process of sequential minimization is analogous to\nsieving.\nFirst, consider all possible magnetization distributions\nwith fixed length of magnetization vector |/vector m|= 1:\nmX+ımY=2w(z,z)\n1+w(z,z)w(z,zc)(1)\nmZ=1−w(z,z)w(z,zc)\n1−w(z,z)w(z,zc),\nwhere/vector m={mX,mY,mZ}=/vectorM/MSis the local magne-\ntization vector, expressed in units of saturation magne-\ntizationMS;z=X+ıY, withX,YandZbeing the\nCartesian coordinates (the element is assumed to be the\na flat cylinder with axis, parallel to Zand magnetization\ndistributions are assumed to be Z-independent), w(z,z)\nis a function (not necessarily meromorphic) of complex\nvariablez.\nSecond, out of all these functions select the ones,\nwhich, additionally to having the constant length of the\nmagnetization, minimize the exchange energy (these are\nthe famous Belavin and Polyakov solitons2). Then fur-\nther restrict the set of the remaining functions twice by\nselecting those, which give minimum for the energies of\nfacemagneticchargesandthose, whichminimize (in fact,\ntotallyavoid)thesidemagneticcharges. Thefinalresult5\nof such a selection will be the following representationfor\nw(z,z)\nw(z,z) =\n\nf(z)/e1 |f(z)| ≤e1\nf(z)//radicalBig\nf(z)f(z)e1<|f(z)| ≤e2\nf(z)/e2 |f(z)|> e2,(2)\nwheree1ande2are real positive constants. The complex\nfunction f(z) is a solution of Riemann-Hilbert boundary2\nc)a) b)\nd)\nY\nX\nFIG. 1. Equilibrium and transient magnetization states in\nferromagnetic nano-disk following from the Eq. 3: a) center ed\nmagneticvortex; b)“leaf” state; c)displacedmagneticvor tex;\nd) “C”-like state.\nvalue problem of finding the meromorphic function in\nthe domain D, which corresponds to the face of the pla-\nnar nanoelement, having no normal components to the\ndomain’s boundary. This problem usually has many so-\nlutions. For simply-connected case their set is equivalent\nto the set of rational functions with real coefficients (con-\nstants), whose zeros(poles) correspond to vortex(anti-\nvortex) centers.5\nFor example, in a unit disk the subset of states with\nno antivortices can be expressed6as\nfdisk(z) =ızc+A−Az2, (3)\nwherecandAare a real and a complex constants respec-\ntively. While the equilibrium values of these constants\ncan be found only by minimizing the total (including the\nenergy of the volume magnetic charges) energy of the\nnanomagnet,9the expression (3) can already be useful to\npinpoint thatthereareseveraltypesofmagneticstatesin\nthe disk, such as centered magnetic vortex(when A= 0),\nthe so-called“leaf” state (when c= 0) and C-likemagne-\ntization state (when 4 AA > c). These states are shown\nin Figure 1. The solution (3) can be extended when there\nare additional vortex-antivortex pairs present,5allowing\nto describe both transient dynamical states as well as\nfinite (simple and topologically charged domain walls).\nBefore proceeding to the consideration of multiply-\nconnected case, let us note a general property of the\nabove Riemann-Hilbert boundary value problem. If it\nhas a solution f(z) in the region z∈ D, the correspond-\ning solution u(t) in another region t∈ D′, connected to\nDvia the conformal mapping t=T(z), can be expressedC0\nC1C2\nC3\nFIG. 2. An example of a quadruply-connected circular do-\nmain with an outer circular boundary C0and inner bound-\nariesCj,j= 1,2,3.\nas\nu(t) =f(z)T′(z)|z=T(−1)(t), (4)\nwhereT(−1)(t) is the inverse of the conformal mapping\nt=T(z), which is always defined. This expression ap-\nplies both to simply- and multiply-connected cases and\nallows to express u(t) in an arbitrarily shaped cylinder,\nbasedonits expressions f(z)forsomecanonicaldomains.\nIn the simply-connected case these canonical domains\ncan be chosen, based on convenience only, to be the unit\ndisk,6the half-plane5or any other simply connected re-\ngion for which the solution can be written explicitly. Se-\nlectionofthecanonicaldomainistrickierin the multiply-\nconnected case, since not every pair of regions with the\nsame connectivity can be conformally mapped into each\nother.\nThanks to the Koebe’s theorems, it is well known that\nit is possible to define parametrized canonical families of\nmultiply-connected regions, which, after parameter ad-\njustment, can be mapped to an arbitrary region of the\nsame connectivity. Among them is the family10of circu-\nlar domains with cut out inner circles shown in Fig. 2.\nPositions and radii of the circular holes in these domains\nare not arbitrary and are dictated by the shape of the\ntarget multiply-connected region D′to which f(z) may\nbe mapped. Because such canonical regions by them-\nselves have physical significance (such as the simplest\ncase of concentric ring, which is doubly-connected), let\nus choose them as the basis and set the goal of express-\ning the solutions f(z) of the Riemann-Hilbert problem in\nthese regions.\nBut first we need to address the question of exis-\ntence of such solutions and their general properties. For-\nmally, given afinitely connected flat domain Dwith good\nenough boundary (say, piecewise analytic), the problem3\nis to find a meromorphic function f(z) in the domain,\nsatisfying the boundary condition\nRe(f(z)n(z)) = 0, z∈∂D, (5)\nwheren(z) is a normal to the boundary of the domain.\nThe solution f(z) may have zeros and polynomial singu-\nlarities at the boundary. This (Riemann-Hilbert) bound-\nary condition can be reformulated as follows:\nImdz\nf(z)= 0, z∈∂D, (6)\nwhich means that meromorphic differential (d z)/f(z) is\nreal(i.e. the restriction of this differential to the bound-\nary is a real differential). The latter condition is confor-\nmally invariant, which, in particular, implies (4).\nTo describe the set of all real differentials in D, the\nuseful notion is the doubleof the domain. The latter is a\ncompact Riemann surface X(D) of genus gequal to the\nnumber of the boundary components of Dminus one.\nIt is made of two copies of the domain glued along its\nboundaries, zbeingtheconformalcoordinateononecopy\nand ¯z– on the other. This surface admits natural an-\nticonformal involution ¯J(orreflection ), the interchange\nof copies, which is stationary exactly on the boundary\ncomponents of the domain D. The reflection acts on the\ndifferentials and the condition for the differential dηto\nbe real is exactly the following:\n¯Jdη=dη.\nCorollary: The following topological phenomenon\ncan be observed:\n♯{f(z)poles}−♯{f(z)zeroes}=♯{Dboundaries }−2\nwhere the poles and the zerosarecounted with their mul-\ntiplicities and for those at the boundary the multiplicity\nshould be divided by two.\nProof: The solution f(z) corresponds to the mero-\nmorphic differential d η:= (dz)/f(z) on the double of\nD. The degree of the divisor of a differential is 2 g−2.\nThis implies that we can’t control independently num-\nber of zeros and poles of f(z).\nThe natural recipe to cook a real differential is sym-\nmetrization: take any meromorphic differential d ηon\nthe surface, then its symmetrization d η+¯Jdηis mero-\nmorphic and real. This recipe is easy to use once we\nhave a representation of the double X(D) as an algebraic\ncurve. This representation, for the case of circular mul-\ntiply connected domains can be represented via a series\noverthe elements of the correspondingSchottky group.11\nTheevaluationofsuchseries, however,canbeveryincon-\nvenient if approached directly. That’s why in the follow-\ningwewill givetherepresentationofthe solution f(z)via\nthe Schottky-Klein prime function,12,13which not only\nadmits an efficient numerical evaluation,14but can also\nbe directly used in building the conformal map t=T(z)ofDtoD′if the latter is a multiply-connected polyg-\nonal domain.15It also has a number of applications in\nproblems of optimization and computation.16,17\nInformally (for the formal definition see e.g. Ref. 14),\nthe Schottky-Klein prime function w(z,ζ) for a specific\nmultiply-connected circular domain Dcan be thought as\na generalization of the difference\nz−ζ=w1(z,ζ), (7)\nwhich in multiply connected case becomes\nw=w(z,ζ). (8)\nIn a simply-connected case products of such differences\ncan be used to build the rational functions of complex\nvariable, such as those equivalent to the set of Belavin-\nPolyakovsolitons2ora similar set ofstates offinite nano-\nmagnet,5or those, entering the Schwarz-Christoffel for-\nmula for the conformal mapping of polygons. General-\nization of the latter to the multiply-connected case was\ndone by Crowdy,15here we shall outline a generalization\nof the former.\nIn the simplest doubly-connected case of a concentric\nringwith anouterradiusof1andaninnerradiusof q <1\nthe Schottky-Klein prime function can be expressed as a\nproduct\nwr(z,ζ) = (z−ζ)/producttext∞\nk=1(1−q2kz/ζ)(1−q2kζ/z)\n(/producttext∞\nk=1(1−q2k))2(9)\nor written via the q-Pochhammer symbols ( a;q)nas\nwr(z,ζ) =zζ\nz−ζ(ζ/z,q2)∞(z/ζ,q2)∞\n((q2,q2)∞)2.(10)\nThis representation will be used in some of the following\nexamples.\nTo build the solutions of the Riemann-Hilbert problem\nwith the specified positions of vortices (or anti-vortices)\nletusdefinetwoauxiliaryfunctions, followingfromthose,\nintroduced in the Section 5 of Ref. 15:\nF1(z,ζ1,ζ2) =w(z,ζ1)\nw(z,ζ2), (11)\nF2(z,ζ) =w(z,ζ)w(z,1/ζ)\nw(z,ζ)w(z,1/ζ), (12)\nwhich, provided ζ1andζ2belong to the same inner\nor outer circle Cj(withj= 0,1,2,...) of the multiply-\nconnected circular domain and ζis any point inside of\nit, have constant argument (complex phase) on all the\nboundaries Cjof the said domain. This means that any\nproduct of these functions will have the constant argu-\nment onthe boundariesofthe circulardomaintoo, which\nimplies that the logarithmic derivative of this product\ng(z) =∂\n∂zlog/parenleftBigg/productdisplay\nmF1(z,ζ1,m,ζ2,m)/productdisplay\nnF2(z,ζn))/parenrightBigg\n(13)4\nY\nX\nFIG. 3. Finite domain walls in a ring of q= 0.3,\ngiven by (1) and (2) with e1= 0,e2→ ∞,g(z) =\n(∂/∂z)logF1(z,e−8πı/9,eπı/9) andf(z) =g(1/z).\nY\nX\nFIG. 4. A domain structure with several finite domain\nwalls in a ring of q= 0.7, with g(z) containing now\nthe product of three F1functions: F1(z,e8πı/9,e−πı/9),\nF1(z,qe4πı/3,qe−2πı/3) andF1(z,qeπı/3,qe−πı/3). The rest of\nparameters are the same as in Figure 3.\nwill be an analytic function of zsatisfying the condi-\ntion (5) at the boundary of the multiply-connected cir-\ncular domain. That is, f(z) =g(z) will be a solution\nof the Riemann-Hilbert problem with the specified (by\nthe parameters ζ1,m,ζ2,m,ζn) positions of zeros and\npoles. It will have some additional zeros and poles too,\nsothatthe previouslymentionedtopologicalconstraintisY\nX\nFIG. 5. Vortex domain walls in a ring of q= 0.6,\nwithg(z) containing the product of F1(z,e4πı/3,eπı/3) and\nF1(z,qe11πı/9,qe2πı/9). The rest of parameters are the same\nas in Figure 3.\nY\nX\nFIG. 6. Antivortex domain walls in a ring. The parameters\nfor this plot are the same as in Figure 5, except that now\nf(z) =g(z).\nalwayssatisfied. Also, it is easytoshowthat the function\nf(z) =g(1/z) corresponds to a “mirror” solution of the\nproblem, where the vortices, specified by the parameters\nζ, correspond to the antivortices and vice versa. This\nis illustrated by several examples in Figures 3-7, some\nof which closely resemble well known magnetization tex-\ntures, observed in ferromagnetic nanorings.\nFrom these, the corresponding magnetization textures5\nY\nX\nFIG. 7. A vortex in a triply connected domain, with the cut\nout circles of the radii 0.2 and 0.3, located at x=0.5 and x=-\n0.5 respectively. The function g(z) = (∂/∂z)logF2(z,−0.52ı)\nandf(z) =g(z). The Schottky-Klein prime function was\nevaluated numerically, using the Matlab package developed\nin Ref. 14.\nin the nano-magnets of the same high connectivity but\nother shapes can be derived according to the Eq. 4 us-\ning the conformal mapping (e.g. the mapping to the\nmultiply-connected polygonal domains, which are also\nexpressed in terms of Schottky-Klein prime function15).\nThus, we have built an approximate analytical repre-\nsentation of the low-energy magnetization states in pla-\nnar multiply-connected nanomagnets. It is parametrized\nvia the positions of some of vortices or antivortices in the\nmagnet. Not all topological singularities can be freely\nplaced. There are additional restrictions on their posi-tions, whose number is equal to the number of boundary\ncomponents minus one. These restrictions will be dis-\ncussed in our forthcoming paper. Moreover, there is a\n“soft” constraint on the number of vortices and antivor-\ntices in relationto the connectivity ofthe system, present\nin or near the equilibrium. It is of the same “soft” nature\nas the topological solitons themselves in finite systems.\nReally, only in the infinite 2-d magnet the topological\ncharge is conserved and there is an infinitely high energy\nbarrier,2separating the states with different topological\ncharge. Infinite magnetsthisbarrierisfinite andvortices\nmayenter/exittheparticlethroughtheboundary, chang-\ning the topological charge of the magnetization texture\ninside. Yet, oncethis barrieris holding(andthe magneti-\nzation vector is pinned to the boundary of the nanomag-\nnet) the vortices inside the planar nanoelements behave\nlike true topological solitons.\nComplete solution of the micromagnetic problems, us-\ning these parametrized magnetization textures as trial\nfunctions, will require computation and minimization of\nthe total energy of the magnet (including the exchange,\nmagnetostatic and other energy terms), or, solution of\nthe equations of the magnetization dynamics using the\nparameters as collective variables.7In many cases this\ncan be a very complex task, which is probably still easier\nto approach numerically (as it was done in the frame-\nworkof the Magnetism@home distributed computational\nproject18). Yet, even without the total energy computa-\ntion, these analytical trial functions can still be useful to\nunderstand, classify and interpret the magnetization tex-\ntures, obtained in the experiment and simulations. Also,\nthey can lead to beautiful analytical results, with the\ngenerality well beyond what numerical micromagnetics\nmay offer. Like, for example, the formula for the mag-\nnetic vortex precession frequency,19based on the dis-\nplaced vortex model (3). The presented “complex vari-\nables” approach might generate more useful models, now\nin multiply-connected case as well.\n∗metlov@fti.dn.ua\n1T. H. R. Skyrme, Proc. Roy. Soc. A 247, 260 (1958)\n2A. A. Belavin and A. M. Polyakov, ZETP lett. 22, 245\n(1975), (in Russian)\n3G. Woo, Journal of Mathematical Physics 18, 1264 (1977)\n4D. J. Gross, Nuclear Physics B 132, 439 (1978)\n5K. L. Metlov, Phys. Rev. Lett. 105, 107201 (2010)\n6K. L. Metlov, “Two-dimensional topological soli-\ntons in soft ferromagnetic cylinders,” (2001),\narXiv:cond-mat/0102311\n7K. L. Metlov, Phys. Rev. B 88, 014427 (2013)\n8K. L. Metlov, J. Appl. Phys. 114, 223908 (2013)\n9K. L. Metlov and Y. P. Lee, Appl. Phys. Lett. 92, 112506\n(2008)\n10P. Koebe, Acta Mathematica 41, 305 (1914)11F. Schottky, J. Reine Angew. Math. 83, 300 (1877)\n12F. Schottky, J. Reine Angew. Math. 101, 227 (1887)\n13F. Klein, Math. Ann. 36, 1 (1890)\n14D. Crowdy and J. Marshall,\nComputational Methods and Function Theory 7, 293\n(2007)\n15D. Crowdy, Proc. R. Soc. A 461, 2653 (2005)\n16A. B. Bogatyrev, Computational Methods and Function\nTheory7, 309 (2007)\n17A. B. Bogatyrev, Computational Methods and Function\nTheory9, 47 (2009)\n18K. L. Metlov, J. Appl. Phys. 113, 223905 (2013)\n19K. Y. Guslienko, X. F. Han, D. J. Keavney, R. Divan, and\nS. D. Bader, Phys. Rev. Lett. 96, 067205 (2006)"    },    {        "title": "1407.2184v1.Competing_magnetic_phases_and_field_induced_dynamics_in_DyRuAsO.pdf",        "content": "arXiv:1407.2184v1  [cond-mat.str-el]  8 Jul 2014Competing magnetic phases and field-induced dynamics in DyR uAsO\nMichael A. McGuire, V. Ovidiu Garlea, Andrew F. May, and Brian C. Sale s\nOak Ridge National Laboratory, Oak Ridge, Tennessee 37831 U SA\n(Dated: August 12, 2021)\nAnalysis of neutron diffraction, dc magnetization, ac magne tic susceptibility, heat capacity, and\nelectrical resistivity for DyRuAsO in an applied magnetic fi eld are presented at temperatures near\nand below those at which the structural distortion ( TS= 25 K) and subsequent magnetic ordering\n(TN= 10.5 K) take place. Powder neutron diffraction is used to det ermine the antiferromagnetic\norder of Dy moments of magnitude 7.6(1) µBin the absence of a magnetic field, and demonstrate\nthe reorientation of the moments into a ferromagnetic config uration upon application of a magnetic\nfield. Dy magnetism is identified as the driving force for the s tructural distortion. The magnetic\nstructure of analogous TbRuAsO is also reported. Competiti on between the two magnetically\nordered states in DyRuAsO is found to produce unusual physic al properties in applied magnetic\nfields at low temperature. An additional phase transition ne arT∗= 3 K is observed in heat\ncapacity and other properties in fields /greaterorsimilar3 T. Magnetic fields of this magnitude also induce spin-\nglass-like behavior including thermal and magnetic hyster esis, divergence of zero-field-cooled and\nfield-cooled magnetization, frequency dependent anomalie s in ac magnetic susceptibility, and slow\nrelaxation of the magnetization. This is remarkable since D yRuAsO is a stoichiometric material\nwith no disorder detected by neutron diffraction, and sugges ts analogies with spin-ice compounds\nand related materials with strong geometric frustration.\nPACS numbers:\nI. INTRODUCTION\nThe common crystallographic feature of layered iron\nsuperconductorsistheFe Xlayer,composedofFebonded\ntoX, a pnictogen or chalcogen, in edge-sharing tetrahe-\ndral coordination. There are several related structural\nfamiliesofsuchcompounds, whicharedifferentiatedfrom\none another by what is found between the Fe Xlayers\n[1,2]. In manycases, isostructuralcompounds areknown\nwith other elements in place of Fe. Among 1111-type\nmaterials (ZrCuSiAs structure type) with compositions\nLnFeAsO (Ln= trivalent lanthanide) [3], Fe can be fully\nreplaced to form LnMAsO where M= Mn [4], Ru [3],\nCo [3], Rh [5], Ir [6], Ni [7], Zn [8], and Cd [9].\nThe physical properties of LnMAsO materials reflect\na wide range of behaviors and associated ground states.\nThese range from insulating to superconducting, and in-\nclude many types of magnetism. LnMnAsO compounds\nare semiconductors displaying giant magnetoresistance\nand antiferromagnetic (AFM) ordering of Mn moments\nat temperatures often exceeding room temperature, and\noften accompanied by spin reorientation transitions at\nlower temperatures [10–13]. The well known Fe com-\npoundsLnFeAsO exhibit spin-density-wave-like AFM\nwhich is at least partly itinerant, and is strongly coupled\nto the lattice distortion which occurs near the magnetic\nordering transition[2, 14–16]. Ferromagnetism (FM) as-\nsociated with itinerant magnetic moments on Co is ob-\nserved in LnCoAsO, and a transition to AFM order is\nobserved at lower temperatures due to effects of local-\nized 4fmoments on the magnetic lanthanides [17–20].\nStudies of the 4 dand 5dtransition metal analogues of\ntheCocompounds( LnRhAsOand LnIrAsO)reportonly\nrare-earth magnetic ordering and only in the case of Ln\n= Ce [5, 6]. Some LnNiAsO compounds are supercon-ducting at low temperatures ( Ln= La[7, 21], Pr [22]),\nwhile others are not. CeNiAsO shows two magnetic or-\ndering transitions associated with Ce moments, and is\ndescribed as a dense Kondo lattice metal [23]. With a\nfilled 3dshell,LnZnAsO compounds are semiconductors\nwith band gaps near 2 eV, and form as transparent crys-\ntals with colors varying from yellow-orange to red [24].\nClearly this structure type provides fertile grounds for\ninteresting physics accessible by simple chemical substi-\ntutions.\nAmong the many studied substitutions, Ru is unique\nin that it is isoelectronic with Fe. Partial replacement\nof Fe with Ru in the related 122 materials SrFe 2As2\nand BaFe 2As2produces superconductivity with transi-\ntion temperatures near 20 K [25, 26], while partial sub-\nstitution of Ru into 1111-type materials only suppresses\nthe magnetism without the appearance of superconduc-\ntivity [27, 28]. LnRuAsO compounds are metals, and\nshow magnetic ordering at low temperatures when mag-\nnetic lanthanides are included [29–31].\nOur previous study of Ru containing 1111 materials\nuncovered particularly interesting behaviors in DyRu-\nAsO [31]. This material undergoes a structural phase\ntransition from tetragonal to orthorhombic near TS= 25\nK, but adopts a different low temperature structure (Fig.\n1c)thanthatobservedin LnFeAsO.Thedistortionwhich\noccurs in DyRuAsO involves a stretching of the unit cell\nalong the a-axis, maintaining a single Ru −Ru distance\nwithin the Ru net, but Ru −Ru−Ru angles which devi-\nate from 90◦. This is unlike the distortion which occurs\nin the parent phases of the layered iron superconductors,\nwhich shears the unit cell and results in a rectangular\nnet of Fe atoms [14]. In addition, anomalies in heat ca-\npacityandmagnetizationofDyRuAsO indicatemagnetic\nordering below TN= 10.5 K.2\nThe temperature and magnetic field dependence of the\nphysical properties indicated complex physics related to\nmagnetism is at play in DyRuAsO, and indications of\nstrong magnetoelastic coupling were observed. A meta-\nmagnetictransition wasobservedbelow TN, and the heat\ncapacity anomaly at the structural transition responded\nstrongly to a magnetic field. The present work aims to\nimprove our understanding of the underlying physics re-\nlated to these phenomena by identifying the magnetic\nstructures, determining the possible role of Ru mag-\nnetism, and further examining the effects of the com-\npetition between ordered ground states on the thermal,\ntransport, and magnetic properties of this material.\nHere we report results of neutron powder diffraction\nexperiments which reveal the low temperature magnetic\norderings, and the nature of the field induced transition,\nwhich involves competing AFM and FM phases. Effects\nof the competition between these phases include thermal\nand magnetic hysteresis in the magnetization and electri-\ncal resistivity, divergence between zero field cooled and\nfield cooled magnetization data, frequency dependence\nof the dynamical (ac) susceptibility, and time dependent\nproperties over a range of magnetic fields and tempera-\ntures. These behaviors highlight the complexity of the\nmagnetic interaction in DyRuAsO, and are reminiscent\nof spin-glass physics under some conditions. Dy mag-\nnetism is identified as the driving force for the structural\nphase transition, with little or no influence from Ru. In\naddition, detailed heat capacity, electrical resistivity, and\nmagnetization measurements provide evidence of a third\nphase transition which appears to be related to the com-\npeting magnetic ground states. The transition occurs\nnearT∗= 3 K and is strongest at magnetic fields of 3 −4\nT, diminished at higher fields, and absent at fields of 0 −2\nT.\nII. EXPERIMENTAL DETAILS\nPolycrystalline samples were synthesized from Dy,\nDy2O3, and RuAs as described in Ref. 31. Rietveld\nrefinement of powder diffraction patterns (neutron and\nx-ray) of the samples used in this study show them to be\n/greaterorsimilar95% pure, with Dy 2O3as the main impurity. Neutron\ndiffraction experiments were conducted at the High Flux\nIsotope Reactor at Oak Ridge National Laboratoryusing\nthe Neutron Powder Diffractometer (beamline HB-2A).\nData were collected at multiple temperatures and mag-\nnetic fields and using two neutron wavelengths, 1.538 ˚A\nand 2.41 ˚A. A collection of rectangular bars (4 ×4×7\nmm3) were cut from a sintered polycrystalline pellet for\nthe neutron diffraction measurement to prevent rotation\nof the powder grains in the applied magnetic field. A\nvanadium can with inner diameter of 6 mm was used to\ncontain the sample, which was loaded into a 5 T vertical-\nfield cryomagnet. Thefield wasdirected perpendicularto\nthe scattering plane. Dy has a high thermal neutron ab-\nsorption cross section; however, this did not preclude thecollection of data of sufficient quality for Rietveld anal-\nysis, which was performed using FullProf [32]. Similar\nneutron powder diffraction measurements, with no ap-\nplied magnetic field, were also performed on TbRuAsO,\nalso prepared as described in Ref. 31.\nMeasurements of the temperature and magnetic field\ndependence of the electrical resistivity, ac and dc mag-\nnetization, and heat capacity were performed using a\nQuantum Design Physical Property Measurement Sys-\ntem. Electrical contacts were made using platinum wires\nand conducting silver paste.\nIII. RESULTS AND DISCUSSION\nA. Magnetic structure\nFigure1 showsneutron powderdiffraction (NPD) data\ncollected in the paramagnetic, tetragonal state at 40 K,\nin the paramagnetic, orthorhombic state at 15 K, and in\nthe magnetically ordered, orthorhombic state at 1.5 K.\nRietveld analysis(not shown)ofthe patterns at 40Kand\n15Kareconsistentwiththetetragonalandorthorhombic\nstructures [31], respectively, with no indication of mixed\nor partial occupancy of any of the atomic sites. No indi-\ncations of long range magnetic order accompanying the\nstructural transition is seen. Strong magnetic reflections\nare observed at 1.5 K. All of the magnetic scattering oc-\ncurs at the positions of nuclear Bragg reflections, indi-\ncating the magnetic and nuclear unit cells are identical\n[propagation vector k= (0 0 0)]. The temperature de-\npendence of the intensity of the 001 reflection is shown in\nthe inset of Figure 1. The onset of magnetic order occurs\nnear 10 K, consistent with magnetization measurements\nwhich identify TN= 10.5 K in Ref. 31 and below, with\nsaturation of the ordered moment occurring near 5 K.\nRepresentational analysis was used to determine the\nsymmetry-allowed magnetic structures that can result\nfrom a second-order magnetic phase transition, given the\ncrystal symmetry before the transition ( Pmmn) and the\npropagation vector of the magnetic ordering, k=( 0 0\n0) . These calculations were carried out using the pro-\ngram SARA h-Representational Analysis.[33] The decom-\nposition of the magnetic representation (i.e. irreducible\nrepresentations (IRs)) for the Dy site (0 .25,0.25, z) is\nΓMag= 1Γ2+1Γ3+1Γ4+1Γ5+1Γ6+1Γ7. The labeling\nof the propagation vector and the IRs follows the scheme\nused by Kovalev[34]. Each representation contains only\none basis vector meaning that the magnetic moments are\nconfined to one direction, while the two Dy atoms of the\nprimitive cell can carry parallel or antiparallel moments.\nStrong magnetic contributions to the 00 ℓreflections in-\ndicate moments with large components in the ab-plane.\nRietveld refinement results of the diffraction data col-\nlected below TNusing wavelength 1.538 ˚A are shown in\nFigure 1(b). The orthorhombic distortion ( a>b) allows\nthe distinction of the directions in the ab-plane, and the\nbest fits were obtained with the Dy moments along the3\n(c)10 20 30 40 50 60 70 80 90 0 10 20 30 0.6 0.8 1.0 1.2 1.4 \n1.5 K 15 K intensity (arbitrary units) \n2θ (deg) T = 40 K DyRuAsO \nλ = 1.538 Å\n001 \nI (10 3 cts.) \nT (K) 001 \n (a)\n20 40 60 80 100 -50 050 100 150 200 250 300 \n  DyRuAsO \nT = 1.5 K \nλ = 1.538 Å\n2θ (deg.) intensity (b)99.5 100.5 2θ (deg)  40K  15K 400-tet \nFIG. 1: (Color online) (a) Neutron powder diffraction pat-\nterns from DyRuAsO collected in the tetragonal state at 40\nK, the orthorhombic state at 15 K, and the magnetically or-\ndered state at 1.5 K. The insets in (a) show the splitting\nof the tetragonal 400 reflection resulting from the structur al\ntransition and the intensity of the 001 reflection (labeled i n\nthe main panel) as a function of temperature, illustrating t he\nonset of magnetic order below about 10 K. (b) Rietveld fits\nof the nuclear and magnetic structures of DyRuAsO at 1.5 K.\nThe lower ticks locate reflections from the Dy 2O3impurity.\nThe AFM arrangement of the Dy moments determined from\nthe diffraction analysis is shown in (c).b-axis, corresponding to the representation Γ 4(or Shub-\nnikov magnetic space group Pm′mn). In this model, the\nmoments on the two Dy atoms in the primitive cell are\naligned AFM. This produces FM layers of Dy in the ab-\nplane, with AFM alignment between neighboring Dy lay-\ners.The resulting magnetic structure is shown in Figure\n1(c). Because of the compression of the ODy 4tetrahe-\ndral units along the c-axis(Fig. 1c), the shortest Dy −Dy\ndistances (3.5 ˚A) are those between neighboring layers\nwithin the DyO slabs. The shortest distance between Dy\natoms within a single layer are considerably longer (4.0\n˚A).\nNeutron powder diffraction analysis of isostructural\nTbRuAsO was also performed, and the same magnetic\nstructure as DyRuAsO was determined for the Tb mo-\nments. In this case, however, no distinction between the\naandbdirections can be made, since TbRuAsO remains\ntetragonal within experimental resolution to at least 1.5\nK. Structural information and magnetic moments deter-\nmined from the refinements of both compounds at the\nlowest temperatures investigated are collected in Table\nI. No conclusive evidence for ordered magnetic moments\non Ru is seen in the data. Small, non-zero values ( /lessorsimilar0.5\nµB) are obtained when Ru moments are included in the\nrefinements at the lowest temperatures, but the quality\nof the fit is not improved by their addition. The refined\nvalues of the rare-earth moment at 1.5 K are 7.6(1) µB\nfor Dy in DyRuAsO and 5.76(8) µBfor Tb in TbRuAsO.\nThe refined ordered moments are reduced from their\nfree ion values of gJ, which are 9 µBfor Tb and 10 µB\nfor Dy, as commonly found in related Fe-based materi-\nals [35–38]. This is attributed to crystalline electric field\neffects, the details of which are not known at this time.\nIn these materials the rare earths are in somewhat irreg-\nular coordination environments. The nearest neighbors\nof the Dy and Tb sites form distorted square-antiprisms,\nwith the squares formed by As on one side and O on\nthe other (distorted squares in the case of orthorhombic\nDyRuAsO). At 1.5 K the site symmetry is mm2for Dy\n(Wyckoffposition2 a) and 4mmforTb(Wyckoffposition\n2c). It is likely that the temperature and magnetic field\ndependence of the relative positions of the crystal field\nlevels plays an important role in the unusual magnetic\nproperties described below.\nSince evidence for a metamagnetic transition and a\nstrongmagnetic field effect on the heat capacity has been\nobservedin DyRuAsO [31], NPD data werealsocollected\nin applied magnetic fields. Results of these experiments\nare shown in Figure 2. As seen in the difference curves in\nthe middle of Fig. 2a, application of a 2 T magnetic field\natT= 3 K has little effect on the diffracted intensities;\nhowever, significant changes are observed as the field is\nincreased to 5 T. It is important to note that many re-\nflections show little or no response to the magnetic field.\nThis shows that the texture of the pelletized sample is\nnot affected. The magnetic field dependence of the inten-\nsity of two diffraction peaks, measured upon decreasing\nthe magnetic field, is shown in Fig. 2b. These peaks4\nµ0H\nFIG. 2: (Color online) (a) Neutron powder diffraction pat-\nterns collected at 3 K with applied magnetic field of µ0H=\n0, 2, and 5 T. The difference between the data collected at\n2 T and zero field, and between the data collected at 5 T\nand 2 T are also shown. The lowest two patterns are simu-\nlations including only magnetic scattering for AFM and FM\nmoments on Dy. Patterns are offset vertically for clarity. (b )\nThe field dependence of the relative scattered intensity at t he\npeaks marked by the square and circle in (a), showing a di-\nvergence for fields above about 2 T. The data are labeled by\nMiller indices of the overlapping reflections which contrib ute\nmagnetic scattering intensity to the measured peaks. (c) Ri -\netveld refinement results for µ0H= 5 T and T = 3 K using\na predominately FM model (see text for details) with all Dy\nmoments along the b-axis.TABLE I: Results and agreement factors from Rietveld re-\nfinement of neutron ( λ= 1.538 ˚A) powder diffraction data\nfor DyRuAsO and TbRuAsO at 1.5 K with no applied mag-\nnetic field. Dy/Tb and As occupy sites at (1/4, 1/4, z), while\nRu and O occupy sites at (3/4, 1/4, z).\nDyRuAsO TbRuAsO\nspace group Pmmn P 4/nmm\na (˚A) 4.0222(1) 4.0215(1)\nb (˚A) 4.0070(1) = a\nc (˚A) 8.0092(3) 8.0558(3)\nzDy/Tb 0.1311(5) 0.1332(8)\nzRu 0.500(2) 1/2\nzAs 0.665(1) 0.664(1)\nzO 0.013(2) 0\nmDy/Tb(µB) 7.6(1) 5.76(8)\nχ21.03 3.56\nRmag 8.29 4.77\nare labeled by the Miller indices of the reflections which\ncontribute magneticscatteringintensity to them, and are\nidentified in Fig. 2a by the data markers used in Fig. 2b.\nThere is a clear change in the field dependence which\nonsets near 2 T, which is identified as a transition from\nAFM to FM order. At the bottom of Fig. 2a, simulated\ndiffraction patterns including only the magnetic contri-\nbution are shown for the AFM structure determined at\nzero applied field (Fig. 1c) and for the FM structure ob-\ntained by aligning all the Dy moments along the b-axis\n(corresponding to IR Γ 5and Shubnikov group Pm′mn′).\nComparing these simulations with the difference curve\nbetween the 5 T and 2 T data reveals that the peaks\nwhich are strongly suppressed at high fields are associ-\nated only with the AFM structure and those which are\nenhanced at high field are associated only with the FM\nstructure.\nResults from Rietveld refinement of data collected at\nµ0H= 5 T and T= 3 K are shown in Fig. 2c. The\nmajority of the magnetic scattering is accounted for us-\ning the FM model with Dy moments of 7.3(3) µBalong\nthe b-axis; however, the data indicates the presence of a\nsmall AFM component as well. The fit shown includes\nboth FM and AFM contributions, and the fraction of\nthe AFM phase is estimated to be about 13% at 5 T.\nThe refinement is relatively insensitive to the direction\nof the moment within ab-plane, and similar results are\nobtained when the FM moment is constrained along the\nb-axis,orallowedtohavecomponentsalongboth aandb.\nNo indication of a c-component is observed. The tran-\nsition between the AFM and FM structures involves a\nchange in the relative orientation of moments on nearest\nneighbor Dy atoms. In addition, the data is not consis-\ntent with a fully polarized powder, in which every grain,\nregardless of crystallographic orientation, would have a\nmoment directed perpendicular to the scattering plane.\nThese results arein agreementwith the magnetic proper-\nties discussed below, in which a preference for moments\nin theab-plane is inferred, and a field of 5 T is seen to be5\ninsufficient to fully polarize the polycrystalline material.\nSimilar fits (not shown) were performed for data col-\nlected at fields of 2 and 5 T and temperatures of 15 K\n(below the structural transition) and 40 K (above the\nstructural transition). Of these, only the pattern from 15\nK at 5 T indicated the presence of magnetic scattering,\nwhich was well modeled with FM ordering of 5.5(3) µB\nmoments on Dy aligned along the b-axis. This tempera-\nture is above TN, and no indication of an AFM compo-\nnent was observed at any field at this temperature. This\nshows that FM order emerges out of the orthorhombic,\nparamagnetic state ( TS> T > T N) when a large mag-\nnetic field is applied.\nIt is expected that the competing FM and AFM states\nmay strongly affect the physical properties of DyRuAsO,\nand consideration of this competition is required in un-\nderstanding the behavior of magnetic, transport, and\nthermal properties presented below.\nB. dc magnetization\nFigure 3a shows the results of dc magnetization mea-\nsurements as a function of applied field for a wide range\nof temperatures. Similar results restricted to lower fields\nand fewer temperatures were previously reported [31].\nAt temperatures below TN, a rapid increase in the mag-\nnetic moment ( m) is observed as the field is increased\nbeyond 2 −3 T. This is consistent with the analysis of\nthe neutron diffraction data presented above. At 2 K,\nthe magnetic moment approaches a saturation value of\n6.8µBper formula unit at 12 T similar to but less than\nthe ordered moment on Dy of 7.6(1) µBdetermined by\nneutron diffraction in zero applied field. An approach to\nsaturation near the same value can be inferred from the\ndata above TNin the paramagnetic state as well, as ex-\npected for large moments in high fields at relatively low\ntemperatures.\nIt is interesting to compare the measured magnetiza-\ntion with the results of the neutron diffraction measure-\nments. At 3 K, in a field of 5 T, the refined value of the\nFM moment on Dy is 7.3(3) µB, and the data suggest\nthe moments are constrained to lie in the ab-plane. The\nmeasuredmagneticmomentatthis temperatureandfield\nis 5.1µBper Dy. This is about 2/3 of the refined mo-\nment. Such a suppression of the measured moment rela-\ntive to the ordered moment is expected due to the mag-\nnetic anisotropy; some crystallites in the magnetization\nsample will have their c-axes along the field direction,\nand thus not contribute to the measured magnetization.\nResults of dc magnetization vs. temperature measure-\nments are summarized in Fig. 3b, which shows the tem-\nperature dependence of M/Hat low temperatures in ap-\nplied fields ranging from 1 to 6 T. Data were collected\nusing both zero field cooled (ZFC) and field cooled (FC)\nprocedures. At temperatures above about 50 K, similar\nbehavioris observedin all of the applied fields. The large\ndecreasein M/Hupon coolingthrough TNforµ0H <3T/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50/s48/s50/s52/s54/s56\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s68/s121 /s82/s117/s65/s115/s79\n/s32/s32/s84/s32 /s61/s32/s50/s32/s75\n/s32/s53/s32/s75/s109 /s32/s40\n/s66 /s32/s47/s32/s70/s46/s85/s46/s41\n/s48/s72 /s32/s40/s84/s41/s84 /s32/s61/s32/s49/s48/s44/s32/s49/s53/s44\n/s50/s48/s32/s46/s46/s46/s32/s53/s48/s32/s75\n/s32\n/s48/s72 /s32/s61/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84\n/s32/s54/s32/s84/s40/s98/s41/s40/s97/s41\n/s32/s77/s47/s72 /s32/s40/s99/s109/s51\n/s32/s47/s32/s109/s111/s108/s32/s70/s46/s85/s46/s41\n/s84 /s32/s40/s75/s41/s115/s111/s108/s105/s100/s32/s61/s32/s90/s70/s67\n/s111/s112/s101 /s110/s32/s61/s32/s70/s67\nFIG. 3: (Color online) (a) The field dependence of the mag-\nnetic moment per formula unit determined from dc magneti-\nzation measurements. (b) Low temperature behavior of M/H\nmeasured on warming after zero field cooling (ZFC) and field\ncooling (FC), showing a divergence which is strongest at in-\ntermediate fields.\nin Figure 3b is noteworthy. At the lowest applied fields,\nthe moment decreases by approximately two-thirds rel-\native to the value observed just above TN. In a typical\nAFM, the powder-averaged moment decreases by only\none third below TN[39]. This is indicative of anisotropic\nsusceptibility in the paramagnetic state, with a larger\nthan average value in the direction along which the mo-\nments order below TN, theb-axis in this case. Since\nthe orthorhombic distortion is small, it may be expected\nthatχa≈χb, which then would imply that χcis small\ncompared to the in-plane susceptibility. This suggests\nthat the Dy moments prefer to lie in the ab-plane above\nTN, as well as in the ordered state. Similar easy-plane\nanisotropy has been observed in the paramagnetic state\nof CeFeAsO [40], which has ordered Ce moments lying\nnearly in the ab-plane at low temperatures [41]. A di-\nvergence between the ZFC and FC data is observed in\nFigure 3b near 5 K for magnetic fields larger than 2 T,\nassociated with the emerging FM.\nAspreviouslynoted[31], ananomalyoccursin M/Hat6\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s48/s50/s48/s51/s48\n/s50 /s51 /s52 /s53/s48/s49/s50\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s50 /s50/s52 /s50/s54 /s50/s56/s49/s50/s49/s52/s49/s54\n/s40/s98/s41\n/s32/s32\n/s32\n/s48/s72 /s32/s61/s32/s48\n/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84\n/s32/s54/s32/s84/s99\n/s80 /s40/s74/s32/s47/s32/s75/s32/s109/s111/s108/s32/s70/s46/s85/s46/s41\n/s84 /s32/s40/s75/s41/s68/s121 /s82/s117/s65/s115/s79/s40/s97/s41\n/s32/s32/s99\n/s80/s40/s74/s32/s47/s32/s75/s32/s109/s111/s108/s32/s70/s46/s85/s46/s41\n/s84 /s32/s40/s75 /s41\n/s84/s32/s42/s84\n/s78/s32/s32\n/s48 /s72 /s32/s61/s32/s48\n/s32/s32/s50/s32/s84\n/s32/s32/s52/s32/s84\n/s32/s32/s54/s32/s84/s32\n/s32/s32/s83 /s32/s40/s82/s41\n/s84 /s32/s40/s75/s41/s84\n/s83/s32\n/s99\n/s80/s40/s74/s32/s47/s32/s75/s32/s109/s111/s108/s32/s70/s46/s85/s46/s41/s32\n/s84 /s32/s40/s75 /s41\nFIG. 4: (Color online) (a) Heat capacity of DyRuAsO at\nthe indicated magnetic fields. The insets shows the behav-\niors near TS= 25KandT∗= 3K. (b) The entropy change\ndetermined by integration of cP/T, after subtracting a esti-\nmated background contribution determined by scaling data\nfrom LaRuAsO to match the data measured at zero field and\n50 K.\nTS, most easily seen at low fields. No long ranged mag-\nnetic order is seen in the diffraction data, so this could\nsignal a change in the crystal field levels of the Dy ion ac-\ncompanying the structural distortion. This could explain\nwhy the effective moment determined by the Curie-Weiss\nmodel in Ref. 31 agree well with the expectations for the\nfree Dy ion, while the ordered moment at low tempera-\ntures does not. The evolution of the crystal field levels\nwith temperature and magnetic field below this struc-\ntural phase transition is expected to be complex, but\nmay prove important in understanding this material.\nC. Heat capacity\nApplication of a 6 T magnetic field has been shown\nto affect strongly the heat capacity anomalies at TSand\nTN[31]. Heat capacity data collected at µ0H= 0−6 T\nin 1 T increments are shown in Fig. 4a. As the mag-netic field is increased, the peak at TNis gradually sup-\npressed and broadened toward lower temperatures, with\nonly a small anomaly remaining at 6 T. The insets show\nthe behaviors near TS(upper inset) and at low tempera-\ntures (lower inset). For fields up to 2 T, the heat capac-\nity anomaly at TSis nearly unchanged. When the field\nis increased to 3 T and above, the peak is suppressed\nand skewed toward higher temperatures. The suppres-\nsion and skewing increase with field for µ0H≥3T. This\nfield effect on the anomaly at TSindicates a magnetic\ncomponent to the phase transition occurring at this tem-\nperature, or at minimum supports strong magnetoelastic\ncoupling. The field dependence is similar to that ex-\npected for FM ordering, although no long range ordering\naboveTNis discernable in the neutron diffraction data\ndiscussed above. To gain some insight into the origin of\nthestructuraldistortion, samplesofthe transition-metal-\nfree analog DyZnAsO were synthesized and preliminary\nheat capacity and diffraction measurements were per-\nformed (see Supplemental Material). The tetragonal to\northorhombic distortion indeed occurs in DyZnAsO at a\ntemperaturenear 30K.This eliminates Ru magnetismor\norbital ordering as a source for the structural distortion,\nand implicates Dy magnetism.\nAt the lowest temperatures (Fig. 4a, lower inset), the\neffect of increasing the field from 0 to 2 T is an over-\nall increase in magnitude, caused at least in part by the\nbroadening of the anomaly at TN. However, a qualita-\ntive change in the low temperature heat capacity occurs\nasµ0His increased to 3 T, as also noted near TS. At\nthis field an additional peak appears below T∗= 3 K, in-\ndicating an additional phase transition in this material.\nThe strong field response suggests that this transition is\nmagnetic in nature, and appears to be related to several\nunusual behaviors of the magnetic and transport proper-\nties which will be discussed below.\nThe entropy change (∆ S) up to 50 K, estimated by in-\ntegration of cP/Tafter subtraction of a background cP,\nare shown in Fig. 4b. The background data were esti-\nmated by scaling the heat capacity of LaRuAsO [30] to\nmatch the measured heat capacity of DyRuAsO at zero\nfield and 50 K. Isostructural LaRuAsO is not known to\nundergo any magnetic or crystallographic phase transi-\ntions in this temperature range. For purposes of integra-\ntion, the background-subtracted cP/Tdata were linearly\ninterpolated from the lowest measurement temperature\nto the origin. In all of the studied magnetic fields, the\ntotal entropy change up to 50 K is similar. Although in-\ncreasingthe fieldfromzeroto2Tsignificantlysuppresses\nthe sharpness of the peak at TN, the total entropy asso-\nciated with it is not changed and is about 80% of Rln(2).\nIncreasing the field beyond 2 T results in a suppression\nof the entropy obtained upon integration up to TN. The\nchange in ∆ Sbetween TNand 50 K is similar in all the\nfields studied (0.8 −0.9R). This suggests the total en-\ntropy associated with the structural transition at TSnot\nstronglydependentontheappliedfield, thoughtheshape\nof the anomaly is significantly changed.7\nThe phasetransitions observedin DyRuAsO appear to\nbesecondorderinnature. Thisisindicatedbytheshapes\nof the heat capacity anomalies in Fig. 4a, the lack of any\nanomalous behavior in the raw heat capacity data [42],\nthe temperature dependence of the magnetic order pa-\nrameter (Fig. 1a) and the absence of thermal hysteresis\nin the physical properties measured at zero field.\nD. Magnetoresistance\nMagnetic field effects on the temperature dependence\nof the electrical resistivity ( ρ) of DyRuAsO are depicted\nin Fig. 5. An abrupt decrease in ρis observed upon cool-\ning through TNforµ0H/lessorsimilar2T. This feature is diminished\nsignificantly at higher fields. The effect of the structural\ntransitionisnotdirectlyapparentfromtheobservedtem-\nperature dependence; however, a slope change in dρ/dT\nis seen at TS(Fig. 5b). Though it is subtle, this anomaly\npersists up to 10 T, suggesting the structural transition\noccurs at all fields investigated here.\nEffects of the field induced transition at T∗are also\nobserved in the resistivity data. This is manifested as\nan abrupt upturn in ρoccurring below 3 K for µ0H≥\n3 T (Fig. 5a). This is clearly observed in the derivative\n(Fig. 5b). The upturn in ρis strongest for µ0H= 3 and\n4 T, the fields at which the heat capacity anomaly at\nT∗is also strongest. In the inset of Fig. 5a, ρdata are\nshown for cooling in the applied field followed immedi-\nately by warming in the applied field. In addition to the\nupturn upon cooling already noted, an apparent thermal\nhysteresis is observed below T∗forµ0H≥3 T, and is\nobserved most clearly at 3 T. In the following discussion,\nthis will be shown to be related to slow relaxation of the\nmagnetic state ofthe material upon movingthrough that\nparticular region of the H−Tphase diagram.\nE. Time and frequency dependent phenomena\nCompetition between the AFM state stable at low\nfields and the FM state stable at high fields leads to\nseveral unusual properties occurring near the associated\ncritical temperatures and magnetic fields. The ZFC-FC\ndivergence in dc magnetization below about 5 K (Fig.\n3b) and the apparent thermal hysteresis in the electrical\nresistivity (Fig. 5a) were noted above when the applied\nfield was ≥3 T. The dynamics in DyRuAsO were fur-\nther examined by measurement of the time dependence\nof the magnetization, with results shown in Fig. 6. The\nsample was cooled to 2 K in zero field, and then the field\nwas increased in 1 T increments up to 6 T. At each field,\nthe magnetic moment ( M) was measured every minute\nfor 60 minutes. The time required to ramp and stabi-\nlize the field was approximately 3 minutes. The data\nare shown in Fig. 6a, plotted as a percentage difference\nrelative to the value measured just after the magnetic\nfield stabilized ( t= 0). When the field is increased from/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s54/s55/s56/s57/s49/s48\n/s48 /s53 /s49/s48/s53/s46/s56/s54/s46/s48/s54/s46/s50/s54/s46/s52\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s68/s121 /s82/s117/s65/s115/s79\n/s32/s32\n/s32\n/s48/s72 /s32/s61/s32/s48\n/s32/s49/s32/s84\n/s32/s50/s32/s84\n/s32/s51/s32/s84\n/s32/s52/s32/s84\n/s32/s53/s32/s84\n/s32/s54/s32/s84\n/s32/s49/s48/s32/s84/s32/s40/s109 /s32/s99/s109/s41\n/s84/s32 /s40/s75/s41\n/s40/s98/s41/s54\n/s53\n/s52\n/s51\n/s49/s50\n/s32/s32\n/s99/s111/s111/s108/s105/s110/s103\n/s119/s97/s114/s109/s105/s110/s103/s32/s40/s109 /s32/s99/s109/s41\n/s84 /s32/s40/s75 /s41/s48 /s72 /s32/s40/s84/s41\n/s48/s40/s97/s41\n/s84/s32/s42/s84\n/s78/s49/s48/s32/s84\n/s54/s32/s84\n/s53/s32/s84\n/s52/s32/s84\n/s51/s32/s84\n/s50/s32/s84\n/s32/s32/s100 /s100 /s84 /s32/s40/s109 /s32/s99/s109/s32/s75/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s48/s72 /s32/s61/s32/s48/s49/s32/s84/s84\n/s83\nFIG. 5: (Color online) (a)Resistivity ( ρ) below 50 K in the\nmainpanel, withthebehaviornearandbelow TNshowninthe\ninset. Data were collected on cooling in the indicated appli ed\nfield (open markers), followed immediately by warming in the\nsame field (solid markers). (b) Temperature derivative of ρ\nwith phase transition temperatures marked on the plot. The\ncurves in (b) are offset vertically for clarity.\n0→1 T and 1 →2 T, no relaxation of the magnetization is\nobserved; the measured moment is independent of time.\nHowever, when the field is increased further, a time de-\npendent moment is observed. The time dependence is\nstrongest after increasing the field from 2 →3 T and 3 →4\nT. A small change with time occurs for higher fields as\nwell.\nThe time evolution of the moment was also examined\nat higher temperatures. The results after increasing the\nfield from 2 →3T at 2, 4, and 6 K are shown in Fig. 6b.8\nThe relaxation seen at 2 K is strongly suppressed, but\nstill observable, at 4 K and absent at 6 K. Similar behav-\nior is seen in the magnetoresistance (not shown). This\nis likely related to the ZFC-FC divergence in the mag-\nnetization (Fig. 3b) and the divergence of the resistivity\nmeasuredupon coolingandthen warming(Fig. 5a)when\nthe field is 3 T or higher.\nThe dynamics resulting from the competition between\nthe AFM andFM states isalsodemonstratedin the mag-\nnetic moment and magnetoresistance ( MR) measured at\nfixed temperature upon increasing and then decreasing\nthe field. Figure 6c shows the field dependence of the\nmoment (field sweep rate of 15 Oe ·s−1), which displays\na divergence which is strongest between about 2 and 4\nT at 2 K, but no detectable divergence at 4 K. Similar\nresults are seen for the magnetoresistance (field sweep\nrate of 25 Oe ·s−1) in Fig. 6d. The local maximum in\nMRupon decreasing the field at 2 K indicates a sig-\nnificant enhancement in charge carrier scattering under\nthese conditions.\nTheobservationofslowdynamicsatintermediatemag-\nneticfieldsfortemperaturesbelow T∗butnotabove(Fig.\n6), suggests that the dissipation is related to the changes\nin other physical properties near in this temperature and\nfield range. For comparison, thermal, transport, and\nmagnetic properties measured at 3 T near T∗are re-\nplotted together in Fig. 7. These anomalies suggest a\nphase transition occurs near this temperature for mag-\nnetic fields exceeding about 2 T, the field above which\nthe FM phase fraction appears to increase most rapidly\nat low temperatures. The heat capacity (Figs. 7a and 4)\nshows a small but relatively sharp anomaly at T∗for\nµ0H= 3−5 T. It is significantly suppressed at higher\nfields. Theelectricalresistivity(Figs. 7b and 5) increases\nsharply upon cooling through this transitions in fields\ngreater than 3 T. This behavior is still clearly observed\nat 6 T, but is suppressed at 10 T. In the dc magnetiza-\ntion (Fig. 3), the strong signal from Dy moments over-\nwhelm subtle features at low temperature, but a marked\nFC-ZFC divergence onsets just above T∗in this same\nrange of magnetic fields. In addition, a subtle downturn\nis observed in FC data at T∗forµ0H= 3 T (Fig. 7c).\nAnomalies near T∗are observed in both components of\nthe ac susceptibility at µ0H= 3 T (Fig. 7d), which is\npresented in more detail below.\nThe shape and relative sharpness of the heat capac-\nity anomaly at T∗and the observation of anomalies at\nthis temperature in other physical properties are indica-\ntive of a thermodynamic phase transition, and not, for\nexample, a Schottky anomaly. Better understanding of\nthis transitionmay come fromadditional neutron diffrac-\ntion studies to investigate how the crystal and magnetic\nstructures evolve with temperature near T∗at different\nmagnetic fields. From the present data, it can be con-\ncluded that magnetism is involved in the transition di-\nrectly or indirectly (through for example magnetoelastic\ncoupling). The observation of an increase in resistivity\nupon cooling through the transition suggests that either/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s53/s49/s48/s49/s53\n/s48/s50/s52/s54\n/s48 /s50 /s52 /s54 /s56/s48/s50/s52/s54/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s53/s49/s48/s49/s53\n/s48/s53/s49/s48\n/s48 /s50 /s52 /s54 /s56/s48/s53/s49/s48/s48/s72 /s32/s61/s32/s48/s32 /s32/s49/s32/s84/s32\n/s49/s32 /s32/s50/s32/s84/s32\n/s50/s32 /s32/s51/s32/s84/s32\n/s51/s32 /s32/s52/s32/s84\n/s52/s32 /s32/s53/s32/s84\n/s53/s32 /s32/s54/s32/s84\n/s32/s32/s77 /s32/s40/s37/s41\n/s116/s105/s109/s101/s32/s40/s109/s105/s110/s41/s68/s121 /s82/s117/s65/s115/s79\n/s84 /s32/s61/s32/s50/s32/s75\n/s32/s32\n/s32/s32\n/s84 /s32/s61/s32/s52/s32/s75/s40/s99/s41\n/s32/s32/s109 /s32/s40\n/s66 /s32/s47/s32/s70/s85/s41\n/s48/s72 /s32/s40/s84/s41/s84 /s32/s61/s32/s50/s32/s75/s48/s72 /s32/s61/s32/s50/s32 /s32/s51/s32/s84/s40/s98/s41\n/s32/s32\n/s32/s84 /s32/s61/s32/s50/s32/s75\n/s32/s52/s32/s75\n/s32/s54/s32/s75/s77 /s32/s40/s37/s41\n/s116/s105/s109/s101/s32/s40/s109/s105/s110/s41/s40/s97/s41\n/s40/s100/s41/s32/s77/s82 /s32/s40/s37/s41/s84 /s32/s61/s32/s52/s32/s75\n/s84 /s32/s61/s32/s50/s32/s75/s32\n/s48/s72 /s32/s40/s84/s41\nFIG. 6: (Color online) (a) Time dependence of the magnetic\nmoment of DyRuAsO after increasing the field as indicated\nin the legend at a temperature of 2 K. (b) Time dependence\nof the magnetic moment after increasing the field from 2 to 3\nT at T = 2, 4 and 6 K. (c) Field dependence of the magnetic\nmoment measured upon increasing then decreasing the field\nat T = 2 and 4 K. (d) Field dependence of the magnetore-\nsistance relative to the zero field resistivity values measu red\nupon increasing then decreasing the field at T = 2 and 4 K.\nthe electronic structure is altered, or the scattering rate\nis increased. The former could be due to a subtle struc-\ntural distortion or orbital ordering involving Ru, and the\nlatter could be related to magnetic domain walls which\nform in the mixed AFM/FM state.\nThe dynamics of the low temperature magnetism in\nDyRuAsO were also investigated using ac magnetic sus-\nceptibility measurements. The frequency, temperature,\nandfield dependence ofthe real( m′) andimaginary( m′′)9\n/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s48/s46/s51/s48/s48/s46/s51/s53/s48/s46/s52/s48/s48/s46/s52/s53/s48/s46/s53/s48/s54/s46/s48/s53/s54/s46/s49/s48/s54/s46/s49/s53/s54/s46/s50/s48\n/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s48/s46/s53/s49/s46/s48\n/s102/s32/s61/s32/s49/s48/s48/s32/s72/s122/s32/s99\n/s80 /s32/s40/s74/s32/s47/s32/s75/s32/s47/s32/s109/s111/s108/s41/s68/s121 /s82/s117/s65/s115/s79\n/s48/s72 /s32/s61/s32/s51/s32/s84\n/s84/s42/s32/s61/s32/s51/s46/s48/s32/s75/s40/s97/s41\n/s100/s40 /s77 /s47/s72 /s41/s47/s100 /s84/s77 /s47/s72/s40/s99/s41/s32/s77 /s47/s72 /s32/s40/s99/s109/s51\n/s32/s47/s32/s109/s111/s108/s41/s40/s98/s41/s32/s40/s109 /s32/s99/s109/s41\n/s109 /s39/s39/s40/s100/s41/s32\n/s84/s32 /s40/s75/s41/s109 /s39/s44/s32/s109 /s39/s39 /s32/s40/s49/s48/s45/s52\n/s32/s101/s109/s117/s41/s109 /s39\nFIG. 7: (Color online) (a) Heat capacity, (b) electrical res is-\ntivity, (c) dc “susceptibility” ( M/H), and (d) ac magnetiza-\ntion of DyRuAsO near T∗in an applied magnetic field of 3\nT. In (c) the temperature derivative of M/His also shown, in\narbitrary units.\nparts of the ac magnetization were studied near the mag-\nnetic orderingtemperatures. The results aresummarized\nin Fig. 8.\nIn zero applied dc field (Fig. 8a,b), the ac magne-\ntization resembles closely the low field dc magnetization\ndata (Fig. 3b), with a sharp drop just below TN. A weak\nfrequency dependence is seen for 4.5 < T <8.0 K, well\nbelowTN. A corresponding anomaly is seen in m′′at\nµ0H= 0 (Fig. 8b) and increases in both magnitude and\ntemperature as frequency increases. The temperatures\nspanned by the m′′peak position is similar to the tem-\nperature range over which m′is found to be frequency\ndependent.\nWhen the applied dc field is increased to 3 T, near the\nmeta-magnetic transition, the features noted in the zero-\nfield data are enhanced. A strong frequency dependence\nis observed in m′(Fig. 8c), and m′′is about one order\nof magnitude larger at 3 T (Fig. 8d) than in zero field\n(Fig. 8b). The temperature of the cusp in m′(Fig. 8c)\nincreases with frequency from 6.5 K at 10 Hz to 8.5 K at\n1 kHz, while the peak magnitude of m′decreases. The(e) (f)\n(g) (h)µ0H = 0 µ0H = 0\nµ0H = 3 T µ0H = 3 Tµ0H\nµ0H = 3 T\nµ0H = 3 TT T\nT T\nf Tfm’ m’’ \nm’ ’ m’ ’ m’ ’ m’ m’\nFIG. 8: (Color online) (a-d) Temperature dependence of the\nreal (m′) and imaginary ( m′′) parts of the ac magnetization\nmeasured at µ0H= 0 and 3 T using frequencies indicated\nin panel (a). (e,f) Contour plots of the real part ( m′) and\nimaginary part ( m′′) of the ac magnetization measured at\n100 Hz as functions of temperature and applied dc magnetic\nfield. (g) Arrhenius fit using the temperatures ( TP) at which\nm′peaks at different frequencies for H= 3 T. (h) Frequency\ndependenceof m′′atH=3Tandtheindicatedtemperatures.\nAll measurements were conductedusing asample of mass 27.1\nmg and an ac excitation field of 10 Oe.\ntemperature at which m′′peaks clearly increases with\nfrequency, while its magnitude shows a more subtle in-\ncrease. Contour plots of the real and imaginary parts of\nthe the ac susceptibility measured at 100 Hz are shown\nin Fig. 8e and 8f. The m′′data indicate that the dissipa-\ntion is strongest below TNand for magnetic fields near\n3 T, the region where AFM-FM competition is expected\nto be strongest.\nThe behaviors shown in Fig. 8c,d are precisely those\nexpected for a spin-glass near its freezing temperature\n[43, 44]. In fact, it is interesting to note the similarities10\nof many of the behaviors of DyRuAsO at low temper-\natures and fields near 3 T to those of a spin glass, in-\ncluding frequency dependent ac susceptibility, FC-ZFC\ndivergence in dc magnetization, and slow relaxation of\nthe magnetic moment when the applied field is changed.\nHowever, there is no chemical disorder detected by neu-\ntron diffraction in this material.\nGlass-likebehaviorwithout chemicaldisordercan arise\nfrom strong geometrical frustration, as realized, for ex-\nample, inDy-pyrochlorespin-icesystems[45,46]. Similar\nbehavior has been reported in the related Ising antiferro-\nmagnet Dy 2Ge2O7, which does not adopt the pyrochlore\nstructure, and in which the glass-like behavior is spec-\nulated to arise from collective relaxation of short-range\nspin correlations [47]. The magnetic structure adopted\nby DyRuAsO (Fig. 1c) does not suggest strong frustra-\ntion in this compound, due to the FM coupling within\neach Dy net in the ab-plane. For purely AFM inter-\nactions, however, the structure does support geometrical\nfrustration. Comparingthe ac susceptibility and heat ca-\npacity behavior of DyRuAsO and Dy 2Ge2O7[47], strong\nsimilarities are observed. An important exception is the\nZFC-FC divergence of dc magnetization seen in DyRu-\nAsO (Fig. 3b). This is absent in Dy 2Ge2O7, and its\nabsence is used to distinguish this material from a spin-\nglass. In this respect, DyRuAsO appears to behave more\nlike a spin-glass than does the pyrogermanate. The prox-\nimity ofthe glass-likebehaviorin DyRuAsO to the meta-\nmagnetic transition suggests that domain walls separat-\ning FM and AFM domains may also play a role in the\nobserved dynamics. Similarly, it has been suggested that\nAFM domain walls may contribute to the frequency de-\npendent phenomena observed in Dy 2Ge2O7[47].\nFurther analysisof the ac magnetization data collected\nin a dc field of 3 T are shown in Fig. 8g,h. The relation-\nship between the temperature at which m′peaks (TP)\nand the measurement frequency is seen to follow an Ar-\nrhenius law f=f0e−EA/kBT(Fig. 8g), as typically seen\nin spin-glasses [43, 44], but not spin-ices [48]. The acti-\nvation energy determined from the fit is EA/kB= 110\nK, and the attempt frequency is f0= 360 MHz. A simi-\nlar activation energy of 162 K is reported for Dy 2Ge2O7,\nwhich wasfound to correspondto the separationbetween\nthe ground and first excited crystal field states [47]. The\nfrequency dependence of m′′at temperatures from 4.0 to\n7.5 K is shown in Fig. 8h. Plotted in this way, a peak\ncorresponds to characteristic spin relaxation frequencies\n[45, 49]. The uniform shift in peak position with temper-\nature is consistent with classical thermal relaxation.\nIV. SUMMARY AND CONCLUSIONS\nNeutron diffraction has been used to identify the mag-\nnetic ordering of Dy moments in DyRuAsO at low tem-\nperature, and how the moments respond to application\nof magnetic fields. The results provide a framework nec-\nessary for understanding the peculiar physical propertiesof this material, which is structurally and electronically\nrelated to the 1111 Fe superconductor systems. In the\nabsence of an applied magnetic field, AFM ordering oc-\ncurs at 10.5 K. The magnetic unit cell is the same as\nthe orthorhombic crystallographic unit cell. Magnetic\nmoments on Dy of magnitude 7.6(1) µBlie long the b-\naxis. The moments are arranged FM within sheets in\nthe ab-plane, with AFM stacking along the c-axis. This\nsame magnetic structure describes the diffraction data\ncollected in a magnetic field of 2 T at T= 3 K. A re-\nlated FM structure was determined when the field was\nincreased to 5 T, with moments of magnitude 7.3(3)\nµBaligned in the ab-plane. The neutron diffraction re-\nsults distinguish the meta-magnetic transition occurring\nin DyRuAsO from the more commonly observed spin-\nflop.\nAt intermediate fields, the competition between the\nAFM and FM states is evident in the physical proper-\nties of DyRuAsO, and results in several unusual behav-\niors. Many of the field induced phenomena appear to\nbe related to a thermal anomaly identified at T∗= 3 K,\nwhich is evident in the heat capacity for magnetic fields\nnear 3 T. The resistivity increases upon cooling through\nT∗. The dc magnetization shows a subtle inflection near\nT∗, and develops a FC-ZFC divergence slightly above\nT∗. In addition, the transport and magnetic properties\ndevelop a time dependence below T∗for fields near 3\nT, producing apparent thermal and magnetic hysteresis\nin magnetization and magnetoresistance measurements.\nThe observed slow relaxation of magnetization, as well\nas frequency dependent ac magnetic susceptibility val-\nues, are reminiscent of behaviors associated with spin-\nglasses [43] and Dy-based spin-ice and related materials\n[45–47,49]. Thisissomewhatsurprising; nochemicaldis-\norder is detected in this material, and the geometry does\nnot indicate strong magnetic frustration [31]. Movement\nof magnetic domain walls related to the competing AFM\nandFMphasesprovideonepossiblesourceofdissipation.\nSince the magnetism in DyRuAsO is dominated by\nthe Dy atoms, crystalline electric field effects likely play\nan important role in determining the magnetic proper-\nties. Indeed, it has been noted that the energy barrier to\nspin-relaxation in Dy 2Ge2O7corresponds to the energy\nsplitting of the lowest crystal field levels [47].The cur-\nrent experimental data show a saturation moment near\n7µB(similar to the refined ordered moment from the\ndiffractionresults), whichissignificantlysmallerthanthe\nfree ion value of 10 µB, and the magnetic entropy deter-\nmined from the heat capacity is relatively small. The\nCurie Weiss behavior of the magnetization for tempera-\ntures just above TSis consistent with the free ion value\nof the effective moment. Detailed calculations and in-\nelastic scattering experiments (complicated by relatively\nstrong neutron absorption by Dy) would be desirable in\ndeveloping an understanding of these effects. Since this\nmaterial undergoes a structural phase transition and dis-\nplays meta-magnetic behavior, the dependence of the Dy\ncrystalfieldlevelsontemperature,field,andcoordination11\ndetails will be required to develop a complete picture.\nIron magnetism is closely linked to the structural dis-\ntortion that occurs in the isoelectronic Fe compounds\n[2, 15, 16]. In the present data, no conclusive evidence\nfor ordered magnetic moments on Ru atoms is seen, and\nno magnetic order is observed between TNandTSin the\nabsence of an applied magnetic field. The driving force\nfor the structural transition at TS, which also occurs in\nDyZnAsO, is identified as Dy magnetism. The distortion\nmustaltertheDycrystalfieldlevels,whichprovidessome\ndegree of magnetoelastic coupling. In addition, TbRu-\nAsOisisostructuralwithDyRuAsOatroomtemperature\nand was found here to have a low temperature magnetic\nstructure similar to DyRuAsO, but no structural distor-\ntion occurs in TbRuAsO at temperatures as low as 1.5\nK.\nZrCuAsSi-type oxyarsenides incorporating heavy\ntransition metal atoms have been relatively little studied\ncompared to the 3 dmetal analogues or the relatedThCr2Si2-type arsenides. The structure type shows a\nlarge degree of chemical flexibility. Many interesting\nmaterials and behaviors have already been identified;\nhowever, many compounds and phenomena likely remain\nundiscovered or understudied. The surprisingly complex\nand glass-like magnetic behavior of DyRuAsO suggest\nfurther study of such compounds will uncover other new\nand interesting cooperative phenomena, and motivates\nfurther study of dynamic effects in rare-earth magnetic\nmaterials.\nResearch sponsored by the US Department of Energy,\nOffice of Basic Energy Sciences, Materials Sciences and\nEngineering Division. Neutron scattering at the High\nFlux Isotope Reactor was supported by the Scientific\nUser Facilities Division, Basic Energy Sciences, US De-\npartment of Energy. The authors thank J.-Q. Yan for\nhelpful discussions.\n[1] D. Johrendt, H. Hosono, R.-D. Hoffmann, and\nR. P¨ ottgen, Z. Kristallogr. 226, 435 (2011).\n[2] M. A. McGuire, in Handbook of Magnetic Materials ,\nedited by K. H. J. Buschow (North-Holland, Amsterdam,\n2014), vol. 22.\n[3] P. Quebe, L. J. Terb¨ ue, and W. Jeitschko, J. Alloys\nCompd. 302, 70 (2000).\n[4] A. T. Nientiedt, W. Jeitschko, P. G. Pollmeier, and\nM. Brylak, Z. Naturforsch. 52b, 560 (1997).\n[5] S. Muir, A. W. Sleight, and M. A. Subramanian, Mater.\nRes. Bull. 45, 392 (2010).\n[6] S. Muir, A. W. Sleight, and M. A. Subramanian, J. Solid\nState Chem. 184, 1972 (2011).\n[7] T. Watanabe, H. Yanagi, Y. Kamihara, T. Kamiya,\nM. Hirano, and H. Hosono, J. Solid State Chem. 181,\n2117 (2008).\n[8] A. T. Nientiedt and W. Jeitschko, Inorg. Chem. 37, 386\n(1998).\n[9] D. O. Charkin, P. S. Berdonosov, V. A. Dolgikh, and\nP. Lightfoot, J. Alloys Compd. 292, 118 (1999).\n[10] A. Marcinkova, T. C. Hansen, C. Curfs, S. Margadonna,\nand J.-W. G. Bos, Phys. Rev. B 82, 174438 (2010).\n[11] N. Emery, E. J. Wildman, J. M. S. Skakle, G. Giriat,\nR. I. Smith, and A. C. Mclaughlin, Chem. Commun. 46,\n6777 (2010).\n[12] Y. Tsukamoto, Y. Okamoto, K. Matsuhira, M.-H.\nWhangbo, and Z. Hiroi, J. Phys. Soc. Jpn. 80, 094708\n(2011).\n[13] N. Emery, E. J. Wildman, J. M. S. Skakle, A. C.\nMclaughlin, R. I. Smith, and A. N. Fitch, Chem. Com-\nmun.46, 6777 (2010).\n[14] C. de la Cruz, Q. Huang, J. W. Lynn, J. Li, W. Ratcliff\nII, J. L. Zarestky, H. A. Mook, G. F. Chen, J. L. Luo,\nN. L. Wang, et al., Nature 453, 899 (2008).\n[15] D. C. Johnston, Advances in Physics 59, 803 (2010).\n[16] M. D. Lumsden and A. D. Christianson, J. Phys.: Con-\ndens. Matter 22, 203203 (2010).\n[17] H. Yanagi, R. Kawamura, T. Kamiya, Y. Kamihara,\nM. Hirano, T. Nakamura, H. Osawa, and H. Hosono,Phys. Rev. B 77, 224431 (2008).\n[18] H. Ohta and K. Yoshimura, Phys. Rev. B 80, 184409\n(2009).\n[19] M. A. McGuire, D. J. Gout, V. O. Garlea, A. S. Sefat,\nB. C. Sales, and D. Mandrus, Phys. Rev. B 81, 104405\n(2010).\n[20] A. Marcinkova, D. A. M. Grist, I. Margiolaki, T. C.\nHansen, S. Margadonna, and J.-W. G. Bos, Phys. Rev.\nB81, 064511 (2010).\n[21] Z. Li, G. Chen, J. Dong, G. Li, W. Hu, D. Wu, S. Su,\nP. Zheng, T. Xiang, N. Wang, et al., Phys. Rev. B 78,\n060504(R) (2008).\n[22] S. Matsuishi, A. Nakamura, Y. Muraba, and H. Hosono,\nSupercond. Sci. Technol. 25, 084017 (2012).\n[23] Y.Luo, H. Han, H. Tan, X. Lin, Y. Li, S. Jiang, C. Feng,\nJ. Dai, G. Cao, Z. Xu, et al., J. Phys.: Condens. Matter\n23, 175701 (2011).\n[24] H. Lincke, R. Glaum, V. Dittrich, M. H. M¨ oller, and\nR. P¨ ottgen, Z. Anorg. Allg. Chem. 635, 936 (2009).\n[25] W. Schnelle, A. Leithe-Jasper, R. Gumeniuk,\nU. Burkhardt, D. Kasinathan, and H. Rosner, Phys.\nRev. B79, 214516 (2009).\n[26] S. Sharma, A. Bharathi, S. Chandra, V. R. Reddy,\nS. Paulraj, A. T. Satya, V. S. Sastry, A. Gupta, and\nC. S. Sundar, Phys. Rev. B 81, 174512 (2010).\n[27] M. A. McGuire, D. J. Singh, A. S. Sefat, B. C. Sales, and\nD. Mandrus, J. Solid State Chem. 182, 2326 (2009).\n[28] I. Pallecchi, F. Bernardini, M. Tropeano, A. Palenzona ,\nA. Martinelli, C. Ferdeghini, M. Vignolo, S. Massidda,\nand M. Putti, Phys. Rev. B 84, 134524 (2011).\n[29] G.-F. Chen, Z. Li, D. Wu, J. Dong, G. Li, W.-Z. Hu,\nP. Zheng, J.-L. Luo, and N.-L. Wang, Chin. Phys. Lett.\n25, 2235 (2008).\n[30] M. A. McGuire, A. F. May, and B. C. Sales, J. Solid\nState Chem. 191, 71 (2012).\n[31] M. A. McGuire, A. F. May, andB. C. Sales, Inorg. Chem.\n51, 8502 (2012).\n[32] J. Rodriguez-Carvajal, Physica B 192, 55 (1993), pro-\ngram available at www.ill.fr/dif/Soft/fp/.12\n[33] A. Wills, Physica B 276, 680 (2000), program available\nfrom www.ccp14.ac.uk.\n[34] O. V. Kovalev, Representations of the Crystallographic\nSpace Groups (Gordon and Breach, Switzerland, 1993),\n2nd ed.\n[35] J. Zhao, Q. Huang, C. de la Cruz, S. Li, J. W. Lynn,\nY. Chen, M. A. Green, G. F. Chen, G. Li, Z. Li, et al.,\nNature Materials 7, 953 (2008).\n[36] J. Zhao, Q. Huang, C. de la Cruz, J. W. Lynn, M. D.\nLumsden, Z.A. Ren, J. Yang, X. Shen, X.Dong, Z. Zhao,\net al., Phys. Rev. B 78, 132504 (2008).\n[37] Y. Qiu, W. Bao, Q. Huang, T. Yildirim, J. M. Simmons,\nM. A.Green, J. W.Lynn, Y.C. Gasparovic, J. Li, T. Wu,\net al., Phys. Rev. Lett. 101, 257002 (2008).\n[38] Y. Xiao, M. Zbiri, R. A. Downie, J.-W. G. Bos,\nT. Br¨ uckel, and T. Chatterji, Phys. Rev. B 88, 214419\n(2013).\n[39] J. S. Smart, Effective Field Theories of Magnetism (W.\nB. Saunders Company, Philadelphia and London, 1966),\n1st ed.\n[40] A. Jesche, C. Krellner, M. de Souza, M. Lang, and\nC. Geibel, New J. Phys. 11, 103050 (2009).\n[41] J. Zhao, Q. Huang, C. de la Cruz, S. Li, J. W. Lynn,Y. Chen, M. A. Green, G. F. Chen, G. Li, Z. Li, et al.,\nNature Materials 7, 953 (2008).\n[42] J. C. Lashley, M. F. Hundley, A. Migliori, J. L. Sarrao,\nP. G. Paliuoso, T. W. Darling, M. Jaime, J. C. Coo-\nley, W. L. Hults, L. Morales, et al., Cryogenics 43, 369\n(2003).\n[43] J. A. Mydosh, Spin Glasses: An Experimental Introduc-\ntion(Taylor and Francis, London, 1993).\n[44] K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801\n(1986).\n[45] K. Matsuhira, Y. Hinatsu, and T. Sakakibara, J. Phys.:\nCondens. Matter 13, L737 (2001).\n[46] K. Matsuhira, Y. Hinatsu, K. Tenya, H. Amitsuka, and\nT. Sakakibara, J. Phys. Soc. Jpn. 71, 1576 (2002).\n[47] X. Ke, M. L. Dahlberg, E. Morosan, J. A. Fleitman, R. J.\nCava, and P. Schiffer, Phys. Rev. B 78, 104411 (2008).\n[48] J. Snyder, B. G. Ueland, J. S. Slusky, H. Karunadasa,\nR. J. Cava, and P. Schiffer, Phys. Rev. B 69, 064414\n(2004).\n[49] J. Snyder,J.S.Slusky,R.J.Cava, andP.Schiffer, Natur e\n413, 48 (2001)."    },    {        "title": "0807.4546v1.Static_and_dynamic_properties_of_vortices_in_anisotropic_magnetic_disks.pdf",        "content": "arXiv:0807.4546v1  [cond-mat.mtrl-sci]  28 Jul 2008Static and dynamic properties of vortices in anisotropic ma gnetic disks\nTiago S. Machado1, Tatiana G. Rappoport2, and Luiz C. Sampaio1\n1Centro Brasileiro de Pesquisas F´ ısicas, Xavier Sigaud,\n150, Rio de Janeiro, RJ, 22.290-180, Brazil and\n2Instituto de F´ ısica, Universidade Federal do Rio de Janeir o, Rio de Janeiro, RJ, 68.528-970, Brazil\n(Dated: November 8, 2018)\nWe investigate the effect of the magnetic anisotropy ( Kz) on the static and dynamic properties\nof magnetic vortices in small disks. Our micromagnetic calc ulations reveal that for a range of Kz\nthere is an enlargement of the vortex core. We analyze the infl uence of Kzon the dynamics of\nthe vortex core magnetization reversal under the excitatio n of a pulsed field. The presence of Kz,\nwhich leads to better resolved vortex structures, allows us to discuss in more details the role played\nby the in-plane and perpendicular components of the gyrotro pic field during the vortex-antivortex\nnucleation and annihilation.\nPACS numbers:\nThe manipulation of magnetization in nanostructured\nmaterials by means of magnetic field and/or spin polar-\nized current has attracted substantial attention in the\nlast decade. More recently, a special focus on the study\nof magnetization reversal dynamics in magnetic disks [1]\nandlines[2]hasbeenmotivatedmainlybytheirpotential\nimportance in the implementation of memory and logical\noperations.\nMicro-sizedPermalloy(Py, FeNi)diskscanexhibit, de-\npending on their size and aspect ratio, a magnetic vortex\nwith a core ( ∼10−20 nm) magnetized perpendicular\nto the disk plane [3]. Due to the magnetic bi-stability\nof the vortex structure, they have been considered for\ntechnological applications. The vortex core magnetiza-\ntion reversal can be achieved by applying an in-plane\nmagnetic field or spin polarized current in the form of\nshort pulses [4] and/or alternating (AC) resonant excita-\ntion[1,5, 6]-bothhaveanequivalentroleondetermining\nthe vortex core dynamics.\nIt is usually considered that the size of the vortex core\ndepends on parameters such as exchange constant, thick-\nness and diameter of the magnetic disk. Most of the\nresearch on vortices in magnetic systems neglects the ef-\nfect of magnetic anisotropy. On the other hand, it has\nbeen demonstrated that a uniaxial magnetic anisotropy\nin Permalloy particles can be induced by the deposi-\ntion process [7]. Experiments and simulations have also\nshown that the presence of anisotropy in thick magnetic\ndisks gives rise to a diversity of domain patterns [7, 8].\nIn this letter, we study the role played by the mag-\nnetic anisotropy in the magnetic properties of disks. Us-\ning micromagnetic simulations, we consider a magnetic\nanisotropy ( Kz) perpendicular to the disk plane and an-\nalyze how it modifies the magnetization pattern and the\ndynamics of the vortex core under action of an in-plane\npulsed magnetic field. We also discuss in detail the core\nreversalprocessandthe influence of Kzon the gyrotropic\nfield and the magnetization reversal time.\nThe simulations were performed with a code we devel-\noped, which employs the Landau-Lifshitz-Gilbert (LLG)\nequation. We used the typical magnetic parameters ofPy: the saturation magnetization is given by Ms=\n8.6×105A/m, the exchange coupling is A= 1.3×10−11\nJ/m and the Gilbert damping constant is α= 0.2. The\nmagnetic anisotropy was included in the total effective\nmagnetic field heffand is given by (2 Kz/µ0M2\ns)mzˆ z [9].\nKzvaries from 0 to 106J/m3. We have simulated Py\ndisks with the diameter of 300 nm and thickness of 12\nnm. The disk is discretized in cells of 3 ×3×3 nm3.\nLet us first consider the vortex core structure in static\nequilibrium(intheabsenceofanexternalmagneticfield).\nThe magnetization pattern of the Py disk presents three\ncharacteristic regimes, as can be seen in Fig. 1 (b-f): for\nKzbelow 2.5×105J/m3the main consequence of in-\ncreasing the anisotropy is an increase of the vortex core\ndiameter. The size of the vortex core is measured at half\nof the maximum value of mzand its dependence with\nKzis shown in Fig. 1(a). For Kzbetween 2 .5×105and\n4.0×105J/m3,mzexhibits concentric regions with + Ms\nand−Ms, still preserving the core in the center of the\ndisk. For Kzbetween 4 ×105and 6×105J/m3the core\ndisappears and the number of concentric rings increases,\nwhich also happens for larger disks. Moreover, in these\ntwo ranges, the in-plane magnetization still preserves the\nvorticity (see Figs.1d-e). The similarity between these\npatterns and previous observed patterns in Co nanomag-\nnets [10] and thick NiFe nanodisks [7] is noteworthy. In\nboth cases, the concentric rings are clearly seen in the\nimages obtained by Magnetic Force Microscopy (MFM).\nForKzabove 6×105J/m3, the vortex structure is lost\nand we see what amount to a single domain in mz(Fig.\n1f). For larger and thinner disks, the magnetization pat-\ntern is composed by stripes.\nThe prospect of having large vortex cores has clear ad-\nvantages for magnetization detection. Still, in order to\nconsider the practical aspects it is necessary to study\nthe stability of these vortex cores and the possibility\nof switching their magnetization. For this purpose, we\nstudied the dynamics of the system under short in-plane\nmagnetic field pulses. We simulated the magnetic rever-\nsal process and constructed a switching diagram. Simi-\nlar diagrams for pulse parameters have been constructed2\nFIG.1: (a)Thediameterofthevortexcoreasafunctionof Kz\nfor a disk with diameter of 300 nm and thickness of 12 nm.\nPanels (b-f) show the magnetization pattern for increasing\nvalues of Kz, illustrating the different regimes of (a). The\ncolors represent the direction of thein-plane component of the\nmagnetization mxy. Avortexstructureis givenbyaclockwise\ncolor sequence of blue-green-red.\nrecently for magnetic disks [4]. They show that the op-\nerating field range is narrow: low fields do not produce\ncore switching. On the other hand, higher fields can give\nrise to multiple switchings.\nFIG. 2: (a) Switching vortex core magnetization diagram for\nmagnetic field pulse strength and Kz. The pulse duration\nis fixed (263 ps). Red, green, and blue colors represent no,\none, and multiple switches, respectively. In the grey area t he\nvortex core is expelled from the disk. (b) The switching time\nas a function of Kzfor a fixed field strength B0= 64 mT.\nWith the intention of discussing the influence of Kz\non this process, we built the diagram sketched in Fig.\n2a. We use a pulse of Gaussian form with a fixed pulse\nduration (263 ps) and a variable field strength B0. For\neach value of Kz, we count the number of core magne-\ntization inversions during a single pulse length. Fig. 2a\nshows three different dynamical regimes in response to\nthe exciting field. In the absence of magnetic anisotropy,\nthe field strength that is necessary to switch the core\nmagnetization is B0= 60 mT. However, for fields higher\nthan 95 mT undesirable multiple switches are produced.\nFor disks with Kz/negationslash= 0, the pulse necessary to induce\nthe switching process has a strength comparable to the\nKz= 0 case. As can be seen in Fig. 2a, an importantconsequence of increasing Kzis the decrease of the mini-\nmum field necessary to produce a single switch. Such de-\npendence on Kz, for the minimal field for the switching,\nopens the possibility of producing selective vortex inver-\nsions in a group of magnetic disks with different Kz’s.\nNote that under the action of a pulsed field, the core\nmoves towards the disk border in a curved (or spiraling)\ntrajectory. When the pulse is over, the core returns back\nto the disk center. Depending on the pulse strength and\nduration, the vortex core can be expelled from the disk\nduring this process. Such behavior is not shown in our\ndiagram and can be avoided by increasing the disk diam-\neter.\nAnother interesting aspect of the influence of Kzcan\nbe seen in the switching time τs.τsis the interval be-\ntween the pulse start and the complete reversal of the\nvortex core magnetization. We have calculated τsfor a\ngiven pulse strength ( B0= 64 mT) in the single switch-\ning regime (see the horizontal line in the Fig.2a). τsfor\nKz= 0 is 213 ps and it increases monotonically with\nKz, reaching values ≃1.4 times larger than τs(Kz= 0)\n(Fig.2b). For thinner disks (6nm) τscan reach up to\n2.5τs(Kz= 0).\nFIG. 3: (a) Time evolution of the gyrotropic field (perpen-\ndicular component). Snapshots of vortices and antivortex f or\n(b)Kz= 0 at the maxima of the gyrotropic field and (c)\nKz= 2.5×105J/m3andt= 263 ps. The color map repre-\nsents the perpendicular component of the magnetization ( mz)\nand the arrows show the in-plane component. The inset of\npanel (a) gives a transverse view of mz.\nA closer look at the core reversal dynamics for disks\nwithKz/negationslash= 0 shows that the intermediate processes lead-\ning to the switching are similar to the ones obtained\nin previous analysis for Kz= 0 [1, 4, 5, 11, 12]: the\ncore shape changes during the magnetization reversal,3\nthrough the formation of the adjacent V-AV (vortex-\nantivortex) pair and the subsequent V-AV annihilation\nand nucleation of the reversed vortex. However, in op-\nposition to what is normally observed in micromagnetic\ncalculations, here, due to the increase of the vortex core\ndiameter, the formation of the adjacent V-AV pair dur-\ning the reversal process can be well resolved (see inset of\nFig. 3 and Figs. 3(b) and (c)).\nFIG. 4: In-plane component of the gyrotropic field (arrows)\njust before (a) and after (b-d) the nucleation and separatio n\nof the vortex and antivortex with negative magnetizations.\nThe color map represents the out-of-plane magnetization.\nThe vortex core magnetization reversal, its switching\ntime, the core motion and its deformations can be under-\nstood in terms of the gyrotropic field [1, 11], which acts\non the vortex core only during its movement. The gy-\nrotropicfield is givenby hg=1\nγm×[v·∇m][1, 11], where\nγandvare the gyromagnetic factor and the core veloc-\nity, respectively. Indeed, using Thiele’s equation [13] in\nits original form, hgcan be simplified using ˙minstead of\nv, and it is reduced to hg=1\nγm×˙m. All calculations\nshown below were performed with this last expression of\nhg.\nPrevious discussions on the gyrotropic field have em-\nphasized the role of the perpendicular component of this\neffective field in the reversal process[11]. To further in-\nvestigate this, we have calculated the perpendicular hz\ng\nandthein-plane hxy\ngcomponentaswell. First, wediscuss\nthe time evolutionofthe perpendicularcomponent hz\ngfor\nKz= 0 and 2 .5×105J/m3, which is shown Fig. 3a. Fol-\nlowing the application of the field pulse, the core moves\nandhz\ngacts on the core at the side opposite to the move-\nment direction leading to the formation of a peak with\nnegative magnetization. At this stage, it is just a peak\nwithout a vortex structure. hz\ngincreases (in modulus) up\nto the point where the peak is so wide that it leads to\nthe nucleation of a V-AV pair, labelled as V−-AV−, re-\nspectively. In conjunction with this V−-AV−nucleation\nthere is a decrease (in modulus) of hz\ng. Subsequently, the\nAV−-V+annihilation takes place at the hz\ngdivergence.\nThis divergence is in agreement with Ref. 11. Such be-havior is independent of Kzbut it is worth noticing that\nthe separationof the V−-AV−is better resolved spatially\nand takes place after an interval that is longer than the\none forKz= 0. This can be seen in the snapshots in the\ninset of Fig. 3, and in the comparison of Figs. 3 (b) and\n(c). On the right one sees the V+with a close neighbor,\nat its left, which is the AV−. Further on the left, one\nsees the V−. As can be gathered from the figures, hz\ngde-\ncreases for increasing values of Kzand that is the reason\nfor the increase of τs.\nOne of the main advantages of considering Kz/negationslash= 0 in\nour calculations is the large separation between the Vs\nandAVs involvedintheswitchingprocess. Thisallowsus\nto analyze in more details the structure of the gyrotropic\nfieldinbetweenthesevortexstructures. Normally,dueto\nthe small distance between them, we cannot resolve the\ndifferences in the gyrotropic field produced by the move-\nment of vortices and antivortices and the nucleation of\nthe V-AV pair cannot be fully understood from this type\nof analysis. For that, we calculated the hxy\ngcomponent\nduring the V−-AV−nucleation and separation, which are\nfundamental steps in the process of core magnetization\nreversal. Fig. 4a shows hxy\ngfor an instant just before\nthe V−-AV−nucleation and separation. The color map\nand the arrows illustrate the zmagnetization component\nandhxy\ng, respectively. The red circle is the original V+\nand the blue one is the negative peak. We can see from\nFigs. 4a-b that hxy\ngis responsible for the transformation\nof this wide negative peak to a V−-AV−pair and acts\nas a driven force pushing V−. As can be seen Fig. 4b,\nhxy\ngpushes the AV−in the direction of V+, producing\nthe pair annihilation. In Fig. 4c it is also possible to\nobserve the spin waves generated by the AV−-V+anni-\nhilation. In the last snapshot, we can see the remaining\nV−(see Fig. 4d). We would like to stress that to our\nknowledge, this is the first analysis that shows explicitly\nthe dynamics responsible for both the AV−-AV−nucle-\nation and separation and V−-V+annihilation. For in-\nstance, considering only the perpendicular component of\nhg, together with the vorticity conservation, one is able\ntoexplain the AV−-V+annihilationbut not the V−-AV−\nnucleation and separation process.\nIn conclusion, we presented a detailed analysis of the\ninfluence of Kzon static and dynamic properties of mag-\nnetic vortices in disks. We showed that increasing values\nofKzproduce a growth of the vortex core. In addition,\nhigh values of Kzcause a change in the magnetization\npattern. We then showed, by means of dynamical cal-\nculations using in-plane magnetic field pulses, that both\nin-plane and perpendicular components of the gyrotropic\nfield,hz\ngandhxy\ng, contribute and are fundamental to the\nunderstanding of the vortex core magnetization reversal\nprocess.\nWe thank Fl´ avio Garcia for useful discussions and the\nBrazilian agencies CNPq and FAPERJ for financial sup-\nport.4\n[1] K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H.\nKohno, A. Thiaville, and T. Ono, Nature Materials, 6,\n269 (2007).\n[2] S.S.P. Parkin, M. Hayashi, and L. Thomas, Science, 320,\n190 (2008), and References therein.\n[3] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto and T.\nOno, Science 289, 930 (2000); A. Wachowiak J. Wiebe,\nM. Bode, O. Pietzsch, M. Morgenstern, and R. Wiesen-\ndanger, Science 298, 577 (2002).\n[4] R. Hertel, S. Gliga, M. F¨ ahnle, and C. M. Schneider,\nPhys. Rev. Lett. 98, 117201 (2007).\n[5] B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T.\nTyliszczak, R. Hertel, M. Fhnle, H. Brckl, K. Rott, and\nG. Reiss, Nature 444, 461 (2006).\n[6] K.-SLee, K.Y.Guslienko, J.-YLee, andS.-K.Kim, Phys.Rev. B76, 174410 (2007).\n[7] P. Eames and E. Dan Dahlberg, J. Appl. Phys. 91, 7986\n(2002).\n[8] C. Moutafis, S. Komineas, C. A. F. Vaz, J. A. C. Bland,\nand P. Eames, Phys. Rev. B 74, 214406 (2006).\n[9] J. Fidler, and T. Schrefl, J. Phys. D: Appl. Phys. 33, 135\n(2000).\n[10] M. Hehn et al., Science 272, 1782 (1996).\n[11] K.Y. Guslienko, Ki-Suk Lee, and Sang-Koog Kim, Phys.\nRev. Lett. 100, 027203 (2008).\n[12] Q.F. Xiao, J. Rudge, B. C. Choi, Y. K. Hong, and G.\nDonohoe, Appl. Phys. Lett. 89, 262507 (2006).\n[13] A.A. Thiele, Phys. Rev. Lett. 30, 230 (1973)."    },    {        "title": "1910.01413v1.Finite_size_effects_on_the_ultrafast_remagnetization_dynamics_of_FePt.pdf",        "content": "Finite size effects on the ultrafast remagnetization dynamics of FePt\nL. Willig,1, 2A. von Reppert,1M. Deb,1F. Ganss,3O. Hellwig,3, 4and M. Bargheer1, 2, \u0003\n1Institut für Physik & Astronomie, Universität Potsdam,\nKarl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany\n2Helmholtz Zentrum Berlin, Albert-Einstein-Str. 15, 12489 Berlin, Germany\n3Institut für Physik, Technische Universität Chemnitz,\nReichenhainer Str. 70, 09126 Chemnitz, Germany\n4Institut für Ionenstrahlphysik und Materialforschung,\nHelmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstrasse 400, 01328 Dresden, Germany\n(Dated: October 4, 2019)\nWe investigate the ultrafast magnetization dynamics of FePt in the L 10phase after an optical\nheating pulse, as used in heat assisted magnetic recording. We compare continuous and nano-\ngranular thin films and emphasize the impact of the finite size on the remagnetization dynamics.\nThe remagnetization speeds up significantly with increasing external magnetic field only for the\ncontinuous film, where domain wall motion governs the dynamics. The ultrafast remagnetization\ndynamics in the continuous film are only dominated by heat transport in the regime of high magnetic\nfields, whereas the timescale required for cooling is prevalent in the granular film for all magnetic\nfield strengths. These findings highlight the necessary conditions for studying the intrinsic heat\ntransport properties in magnetic materials.\nThefascinatingfieldofultrafastmagnetizationdynam-\nics has developed rapidly from the first demonstration\nof femtosecond demagnetization[1] towards all-optical\nswitching[2, 3], magnetization reversal by ultrashort\nelectron pulses[4] and heat-assisted magnetic recording\n(HAMR)[5–7]. Future light-based data-recording ap-\nplications motivate an understanding of the fundamen-\ntal light-matter-interactions in magnetic materials with\nnanoscale bit-dimensions, which are a key to satisfy the\nincreasing demand for high-density information storage\ntechnology.\nInformation storage devices using the HAMR scheme\nhave demonstrated that data densities beyond 1.4Tb/in2\nare feasible since long-term data stability of nanoscopic\nbits can be achieved by using materials with large per-\npendicular anisotropy[8, 9]. FePt in the highly or-\ndered L1 0phase is a promising material since it com-\nbines a large uniaxial magneto-crystalline anisotropy\nKu\u0018=7\u0001106J/m3, with a relatively low Curie tempera-\nture (TC\u0014750K) and the possibility to grow nanograins\nwith diameters down to 3nm by commercially viable\nsputtering techniques[9].\nBesides nanoscopic bit volumes for high information\ndensity, it is of utmost importance to write and read\ninformation at the fastest possible speed and with the\nhighest efficiency. This has triggered many research\nprojects studying the ultrafast magnetization response\nof thin films to rapid heating via pulses of optical pho-\ntons, photoinduced hot electrons or few picosecond long\nelectrical currents.[1, 10–12]. Much of the fundamen-\ntal research has focused on the timescales and mecha-\nnisms of the spin-angular momentum transfer during the\ndemagnetization process[13–15]. The subsequent field-\nassisted remagnetization has received less experimental\nattention despite its equal importance for future workingdevices[16].\nMagnetizationswitchingincontinuousthinfilmsunder\nthermal equilibrium requires the nucleation, propagation\nand finally coalescence of reversed magnetic domains.\nThe substantial laser heating used in the HAMR-scheme\naddstheneedforunderstandingnanoscalethermaltrans-\nport and potential changes of material properties such as\nthe magnetic anisotropy that occur upon heating close\nto and above TC, as in fascinating all-optical switching\nexperiments, which report toggle switching of the mag-\nnetization in the absence of an external field[10].\nCurrent modeling approaches for the remagnetization\ndynamics[11, 17–19] often employ a three temperature\nmodel for the equilibration of electron-, phonon- and\nspin-temperatures where the local spin-temperature de-\nfines a stochastic magnetic field, that enters in an atom-\nisticLandau-Lifshitz-Gilbert(LLG)equation. Extending\nthe atomistic LLG-model to micromagnetic simulations\nof the macrospin evolution using the Landau Lifshitz-\nBloch equation computes the magnetization dynam-\nics on the relevant picosecond to nanosecond timescale\neven for macroscopic specimens[11, 18]. The incorpora-\ntion of temperature dependent material parameters[19,\n20], quantized spin-states[18] and the proper quantum-\nthermodynamics noise-distribution function[21] further\nimprove the comparison to experiments. However, spa-\ntial confinement effects and interfaces that alter the do-\nmain propagation and the thermal transport are often\nnot implemented in the modeling.\nHere we present an experimental comparison of\nthe magnetization dynamics of equally thin continu-\nous and nano-granular FePt after medium to high-\nfluence femtosecond laser-excitation. The observed\nremagnetization-process speeds up by a factor of three\nwith increasing external field for the continuous film.arXiv:1910.01413v1  [cond-mat.mtrl-sci]  3 Oct 20192\nThis effect is significantly reduced for the nano-granular\nFePt and for medium fluences the timescale of remagne-\ntization is independent of the external field. We argue\nthat domain wall propagation is irrelevant in the grains\nwhich are magnetically decoupled by the surrounding\namorphous carbon-matrix. In the continuous film the\nremagnetization rate saturates for high external fields of\nabout 0.7T and it approaches the timescale observed in\nthegranularfilm, wheretheremagnetizationspeedislim-\nitedbytheheatdissipationrate. Wediscusstheinfluence\nof the carbon matrix, which rapidly absorbs about 30 %\nof the deposited energy and partially stores it for about\n1ns.\nThe investigated continuous and granular FePt films\nare grown in the ordered L1 0phase onto MgO (100)\noriented substrates, which aligns the easy magnetic axis\nout of plane. A more detailed description of the growth\nconditions and the structural properties of the samples\ncan be found in Ref[22]. In particular, we mention that\nthe granular film consists of segregated FePt-nanograins\nembedded in amorphous carbon with a size distribution\nof FePt particles centered at approximately 10nm (see\nFig. 1(d)). The continuous film of FePt is capped by a\n1nm Al layer, which is oxidized to Al 2O3(see Fig. 1(c)).\nThe static magnetic properties of the samples were char-\nacterized using a superconducting quantum interference\ndevice - vibrating sample magnetometer under an exter-\nnal magnetic field ( Bext) applied normal to the sample\nplane. The measured hysteresis loops of the continuous\nand granular films are shown in Fig. 1(a). Their square\nshape shows that the magnetic easy axis is oriented per-\npendicular to the plane of the films. The coercive field\nof the granular film is around 5T, which is very large\ncompared to the coercive field of the continuous film of\napproximately 0.4T. This is due to the different mag-\nnetization reversal mechanisms. Domain wall nucleation\nandpropagationgovernsthecontinuousfilm, whereasthe\nmagnetization dynamics in the nanoparticles are domi-\nnated by quasi-coherent magnetization reversal [23]. In\naddition, we observed a reduction of the saturation mag-\nnetization ( MS) by 30% in the granular film, which can\nbe related to the volume occupied by the nonmagnetic\ncarbon matrix in that film together with a small contri-\nbution related to the well-known effect of finite size[24]\nonMS. The temperature dependence of the saturation\nmagnetization was also measured in a wide range of tem-\nperatures between 300 and 850K (see Fig. 1(b)) for sim-\nilarly prepared samples. The Curie temperature of the\ngranular film is about \u0018=660K, approximately 30K lower\nthan the value obtained for the continuous film, which\nis also in agreement with previous investigations of the\nfinite size effect on TC[25]. The laser induced ultrafast\nmagnetization dynamics was investigated using the time-\nresolved magneto-optical Kerr effect (TR-MOKE) setup\nsketched in Fig. 1(e). The pump and probe pulses were\ngenerated from an amplified Ti:Sapphire laser system de-\na)\n-6 -3 0 3 6-1-0.500.51\nMagnetic Field (T)Magnetization (emu/cm3· 103)\n350 500 650 80000:20:40:60:81b)\nTemperature (K)\nMagnetization (M/M(300 K)\ngranular continuous\nc) d)\ne)\nBeam Splitter Mirror\nElectromagnetλ/2\nPolarizer\nBBO\nPolarizerλ/2λ/2 Wollaston\nPrism Balanced\nPhotodiodeBoxCarDAQSampleBextTi:Sa laser system\n1 kHz p-pol\nλ = 800nm \nEpulse=1.5mJ\nτFWHM = 130fs\nDelay Stage\nChopperFigure 1. Static magnetic properties of the FePt samples and\nsketch of the TR-MOKE setup. (a),(b) Magnetization hys-\nteresis loops (a) and temperature dependence of the magne-\ntization (b) of the granular (open symbols) and continuous\n(plain symbols) samples. The magnetic field was applied per-\npendicular to the plane of the samples.(c),(d) Sketch of the\ncontinuous (c) and the granular (d) FePt samples. (e) Sketch\nof the TR-MOKE setup.\nlivering 130fs pulses centered at 800nm at the repetition\nrate of 1kHz. The pump beam is kept at the funda-\nmental of the amplifier at 800nm and excites the sample\nunder a small angle from the surface normal to the sam-\nple (\u0018=2°), while the probe beam is frequency doubled\nto 400nm with a nonlinear Beta-Barium-Borate (BBO)\ncrystal and incident onto the sample at almost perpen-\ndicular incidence ( \u0018=1°). A chopper reduces the repeti-\ntion rate of the pump pulses to 500Hz enabling pulse-to-\npulse comparison of the pumped and unexcited states of\nthe sample at the full 1kHz repetition rate. Both light\npulses are linearly polarized and focused through one of\nthe pole-shoes of an electromagnet onto the sample on a\nspot of 1000 \u0016m for the pump and 300 \u0016m for the probe.\nThe reflected probe pulses allow measuring the differ-\nential change of the polar Kerr rotation ( \u0001\u0012K) using a\npolarization bridge consisting of a \u0015/2 waveplate, a Wol-\nlaston prism and a balanced photodiode. The detected\nsignal is analyzed via a Boxcar integrator and a data ac-\nquisition (DAQ) card. The Bextof the electromagnet is\napplied perpendicular to the surface of the sample.3\nResults and discussion\nFigure 2 shows the TR-MOKE measured for the\ncontinuous (Fig. 2(a)) and the granular (Fig. 2(b))\nsamples as a function of Bextat a pump fluence of\nFpump=5mJ/cm2. To compare the field effect on the\nultrafast magnetization dynamics, \u0001\u0012K(t) signals were\nscaled by the \u0001\u0012Kamplitude corresponding to the maxi-\nmum demagnetization. In both samples, the pump laser\npulses induce a subpicoseond demagnetization process,\nwhich is independent of Bext. We focus on the remagne-\ntization process following this ultrafast demagnetization.\nInterestingly, a clear difference in the effect of Bexton\nthis remagnetization is observed for the continuous and\ngranularfilms. Indeed, theremagnetizationofthecontin-\nuous film speeds up significantly when the external field\nincreases from 0.1 to 0.7T (see Fig 2(a)), while the re-\nmagnetization of the granular one is independent of Bext\n(see Fig. 2(b)). In order to study this phenomenon in\nmore detail, we measured the TR-MOKE at high pump\nfluence ofFpump=10mJ/cm2and over a wide range of\nBextup to 1.2T. The results of this study are summa-\nrized in Fig. 2(c). For both samples and at the two Fpump\nvalues of 5mJ/cm2and 10mJ/cm2we show the field de-\npendence of the time t1/2corresponding to the time in\nwhich half of the demagnetization amplitude has recov-\nered. Interestingly, the effect of Bexton the remagneti-\nzation becomes more pronounced in the continuous film\nat higher fluence and it reaches a saturation around 0.6\nand 0.7T, while the remagnetization of the granular film\nremains weakly sensitive to Bext, even though the range\nof applied fields is nearly twice as large. It is straightfor-\nward to assign the large field dependence of the contin-\nuous film to domain-wall propagation that is well-known\nto govern the magnetization dynamics in continuous thin\nfilms and very sensitive to the amplitude of Bext[26–28].\nOn the other hand, the domain wall propagation is irrel-\nevant in nanosized magnets where the remagnetization\nshould be governed by the cooling of the grain. Further-\nmore, we show that t1/2in the continuous film converges\nunder high Bextto a value of \u0018=100ps, similar to the one\ncharacterizing the granular film at high fluence. Such\nsaturation can be explained by the fact that the remag-\nnetization in the continuous film at high fields is essen-\ntially governed by the dissipation of heat which cools the\nFePt spin system. Indeed, when the external field is large\nenough, it can keep the entire film essentially in a mon-\nodomain state such that domain wall propagation does\nnot play a role. Fig. 2(a) clearly shows that only at high\nmagnetic fields the heat transport is the dominant pro-\ncess for the ultrafast remagnetization dynamics in the\ncontinuous magnetic film. Therefore, in order to study\nthe intrinsic heat transport properties in magnetic ma-\nterials, it is necessary to investigate the magnetization\ndynamics in nanosized structures or under a high exter-\na)\ncontinuous FePt\nBe\nxt(T)0 100 200 300 400 500 600\n1.0\n0.5\n0Time (ps)∆\u0012K(norm)\n0.1 0.2 0.3 0.4\n0.5 0.6 0.7\nb) granular FePt\nBe\nxt(T)\n0 100 200 300 400 500 6001.0\n0.5\n0\nTime (ps)∆\u0012K(norm)\n0.1 0.2 0.3\n0.4 0.5c)\nF (mJ/cm2)\n0 0:5 1150300\nBe\nxt(T)t1/2(ps)5,\ncont\n9,\ncont\n5,\ngran\n10,\ngranFigure 2. Magnetic field dependence of laser induced mag-\nnetization dynamics. (a,b) Normalized Kerr rotation \u0001\u0012K\nmeasured in the (a) continuous and (b) granular samples at\ndifferent applied magnetic fields for a fixed pump fluence of\n5mJ/cm2. The inset (c) shows the magnetic field dependence\nof the time t1/2at which half of the demagnetization ampli-\ntude has recovered for this pump fluence of 5mJ/cm2and an\napproximately doubled fluence of 9 and 10mJ/cm2for the\ncontinuous and granular sample, respectively. Solid lines are\na guide to the eye.\nnal magnetic field.\nIn order to quantify the vertical axis of the TR-MOKE\ntraces and to study the ultrafast magnetization dynam-\nics in more detail, we performed hysteresis measurements\non both samples using a maximum field of 0.75T at\ndifferent pump fluences and delays between the pump\nand the probe pulses (Figure. 3). The two samples\nexhibit very different fluence and time-dependent hys-\nteresis loops. Let us first focus on the continuous film\nfor which the amplitude of the hysteresis was normal-\nized to the one measured without excitation (Fig. 3(a)).\nThe excitation at a low fluence of Fpump =2.0mJ/cm2\nshows that the absorbed energy of the pump pulse in-\nduces only a very small decrease in the saturation mag-\nnetization at short timescales ( t= 1ps) without any\nchange of the coercive field. At a medium fluence of\nFpump = 5:0mJ/cm2, the saturation magnetization is re-\nduced significantly, and the coercive field is smaller com-\npared to the one measured without excitation. At high\nfluence ofFpump = 8:5mJ/cm2the hysteresis is fully\nclosed over a significant amount of time of at least 25ps.4\na) continuous FePt\n-1 ps 1 ps 25 ps 300 ps\n2.0mJ\ncm2\n5.0mJ\ncm2\n8.5mJ\ncm2Ms\n-Ms\nMs\n-Ms\nMs\n-Msno pump\nb) granular FePt\n-1 ps 1 ps 15 ps 150 ps 500 ps\n2.5mJ\ncm2-1 ps 1 ps 15 ps 150 ps 500 ps\n5.0mJ\ncm2-1 ps 1 ps 15 ps 150 ps 500 ps\n8.5mJ\ncm2Ms\n-Ms\nMs\n-Ms\nMs\n-Ms\nFigure3. Hysteresismeasuredinthecontinuous(a)andgran-\nular (b) samples at different pump fluence and delay between\nthe pump and the probe pulses. In all measurements the\nappliedBextranges between \u00060:75T. The vertical scaling is\nfixed for all hysteresis curves and the saturation magnetiza-\ntion is normalized as described in the text a) to the hysteresis\nloop without pump and b) by the signal observed at the high-\nest fluence.\nFor the granular sample, no hysteresis loop is observed\nat the low fluence of Fpump = 2:5mJ/cm2(Fig. 3(b)),\nsince the coercive field (cf. Fig. 1(a)) is larger than the\nmaximum applied field of 0.75T. At this low fluence, the\ntransient hysteresis at the time delay of 1ps shows the\nsame reduction of the magneto-optical signal for the full\nBextfield range. This vertical shift indicates a reduced\ntransient magnetization. No grains are switched in their\nmagnetization when Bextis reversed and we only access\nthe upper hysteresis branch of the granular film. A clear\nopen hysteresis is observed at negative time delays for\na medium fluence of Fpump = 5:0mJ/cm2and by fur-\nther increasing Fpumpits amplitude becomes more and\nmore pronounced and its coercivity is continuously re-\nduced. The increasing hysteresis amplitude with increas-\ningFpumpimplies a larger fraction of switched particles.\nThis phenomenon is related to the size distribution of the\nFePt grains, which leads to a large spread in the temper-\nature changes proportional to the inhomogeneous light\nabsorption that varies between 10 and 30 %according to\nfiniteelementsimulationsofthefieldenhancementeffects\nintheopticalabsorption[29]. Thisinhomogeneousdistri-bution of heat cannot be washed out by electronic heat\ntransport through the insulating carbon matrix, which\nrapidly cools down the FePt particles but does not trans-\nport the heat efficiently from grain to grain. Only the\ngrains which experience a temperature rise close to the\nCurie temperature TCparticipate in the switching. We\nestimatethatmorethan90 %oftheparticlesareswitched\nat 8.5 mJ/cm2, since for 11 mJ/cm2the demagnetization\nonly increases marginally. As a good approximation we\nthus calibrate the vertical axis in Fig. 3(b) by the am-\nplitude of the hysteresis loop measured at negative delay\nwithhighfluencesincethisvalueisthesaturationmagne-\ntization of the granular film. This calibration in Fig. 3(b)\nemphasizes that more and more particles switch with in-\ncreasingFpump. Wementionthatatthetimedelayof1ps\nand forFpump = 8:5mJ/cm2the hysteresis measured in\nboth – continuous and granular – films are flat and have\nzero amplitude, which indicates that for both films the\ntemperature exceeds TCand therefore they are in the\nparamagnetic phase. On the other hand, we observe that\nfor time delays larger than \u0018=15ps for the granular film\nand\u0018=25ps for the continuous one the hysteresis ampli-\ntudes gradually increase (Fig. 3), nicely visualizing the\nremagnetization dynamics of the films.\nTo better illustrate the remagnetization dynamics as a\nfunction of the pump fluence, TR-MOKE measurements\nat selected Fpumpare shown for the continuous and the\ngranular film in Fig. 4(a) and Fig. 4(b), respectively. The\ndata were recorded for an applied field of 0:75T. In both\ncases an increasing pump fluence causes the remagnetiza-\ntion to slow down. In addition, with the exception of the\nlowestfluencemeasurementofbothsamples, thegranular\nfilm recovers significantly faster than the continuous film\natequalincidentfluencevalues. Toillustratethisfeature,\nFig.4(c)showsthepumpfluencedependencesofthetime\nt1/2of the continuous and granular film. To directly vi-\nsualize the faster recovery of the granular film, we plot\nby a dash-dotted line in Fig. 4(b) the \u0001\u0012K(t) measured\nfor the continuous film at Fpump = 4:5mJ/cm2. The\nearly dynamics of the granular film are similar to this\ndash-dotted line when excited at almost twice the flu-\nence. For 8.5mJ/cm2excitation of the continuous film\nthe dynamics are much slower (dashed line), suggesting\na significantly reduced light absorption in the granular\nfilm. However, for both fluences the continuous film ap-\nproaches the saturation magnetization faster than the\ngranular one beyond 300ps time delay (see Fig. 4(b)).\nThe similarity between the initial remagnetization of the\ncontinuous film and the granular one when exposed to\ntwice the incident fluence suggests that a difference in\nthe absorption for two films plays a role on the observed\nbehavior. In order to examine the validity of this hy-\npothesis, we have employed a transfer matrix calcula-\ntion to estimate the optical absorption profile for the\ntwo samples[30]. We have used in our numerical cal-\nculation experimental values for the optical constants of5\na) continuous FePt\nF (mJ/cm2)0 100 200 300 400 500 600\n1.0\n0.5\n0Time (ps)∆θK/∆θsat\n2.0 3.5 4.0 4.5 5.5\n6.5 7.5 8.5 9.0\nb) granular FePt\nF (mJ/cm2)\n0 100 200 300 400 500 6001.0\n0.5\n0\nTime (ps)∆θK/∆θsat\n2.5 5.0\n6.0 8.0\n8.52 4 6 8050100\nF (mJ/cm2)t1/2(ps)c) B ext=0.75 T\ncont\ngran\nFigure 4. Pump fluence dependence of laser induced ultrafast\nmagnetization dynamics. (a,b) \u0001\u0012Kmeasured in the (a) con-\ntinuous and (b) granular samples as a function of the pump\nfluence at high external magnetic field of Bext= 0:75T. The\ndashed dotted and dashed lines in (b) are the \u0001\u0012Ksignal in-\nduced in continuous film by Fpumpof 4.5 and 8.5mJ/cm2, re-\nspectively. The inset (c) shows the pump fluence dependence\nof the time t1/2at which half of the demagnetization ampli-\ntude has recovered for 0.75T with the solid lines as guide to\nthe eye.\nthe continuous (n= 3.30 + 2.63i) and granular (n= 2.98\n+ 1.78i) FePt films[29, 31], which are measured in sam-\nples with comparable thickness, size of particles and car-\nbon matrix. We note that the smaller extinction coeffi-\ncient of the granular film (i.e. the imaginary part) not\nonly indicates that it absorbs less energy compared to\nthe continuous FePt layer. It also reveals that the ab-\nsorption in the FePt grains is considerably larger than in\nthe carbon, since otherwise the effective medium should\nhave an increased imaginary part. The continuous FePt\nlayer absorbs approximately Acont\u0018=33%of the incident\nfluenceFpump, while the granular one absorbs approxi-\nmatelyAgran\u0018=27%intheeffectivemediumthatconsists\nof FePt immersed in amorphous carbon. Thus, the ab-\nsorbed light energy heating the electron system of the\ngranular FePt is only about Agran=Acont= 81%of the\ncontinuous film.\nHeat conduction to the surrounding carbon matrix is\na second factor which reduces the energy density in the\nFePt grains. A four temperature model was recently\nproposed to capture this energy transfer for the nano-granular FePt-carbon composite[32]. Due the compara-\nble specific heat per volume of both materials, we esti-\nmate that the fraction of energy that will flow from FePt\ngrains (cV= 3:8\u0001106J/m3K) to the carbon matrix is\nroughly proportional to its volume fraction of 0.3. Since\nthe Debye temperature of amorphous carbon is consid-\nerably above room temperature[33], cVincreases from\ncV= 2to 3.5\u0001106J/m3K between room temperature and\n800K and thus becomes comparable to the FePt value\nfor our strong excitation regime. Hence, combining the\ntwo arguments, after heat flow to the carbon matrix the\naverage FePt unit cell of the granular film contains only\na fraction \u0018=70% of the absorbed energy, which is, more-\nover only \u0018=81% of the absorbed energy in the continuous\nfilm. This reconciles that about only half ( 57%) of the\nincident fluence is needed in the continuous film to trig-\nger the same magnetization dynamics as in the granular\none.\nFinally, we address the observation that for long\ntimescales beyond 300ps the continuous film always ap-\nproaches the saturation magnetization faster than the\ngranular film, although it has absorbed twice the energy.\nThe relatively weak van-der-Waals bonds of carbon was\nshown to exhibit a significantly reduced interface conduc-\ntance as compared to metal-oxide interfaces with strong\nbinding. [34] Therefore, the carbon matrix should store\nthe heat energy longer than the FePt and it can serve as\nan additional heat bath, which heats the FePt grains on\nlong timescales up to 1ns and beyond. The temperature\ngradient across the interface has then reversed compared\nto the initial situation, where carbon cools the FePt par-\nticles, after they have been optically heated to very high\ntemperatures.\nConclusion\nWehavecomparedthelaserinducedmagnetizationdy-\nnamicsofacontinuousandagranularFePtthinfilmwith\na similar thickness of about 10nm under various excita-\ntion fluences and external magnetic fields. Our experi-\nmental results show that the granular nature of the film\ninfluences the observed dynamics in several ways. First\nof all, the laser energy absorbed by the grains shows a\nbroad distribution, where the average FePt unit cell in\nthe continuous sample absorbs about twice the energy\ncompared to the granular sample. Moreover, the carbon\nmatrix changes the dynamics in three ways: It i) rapidly\ntakes up approximately 30 %of the absorbed energy and\nthusinitiallyspeedsuptheremagnetization,ii)givesheat\nback to FePt after cooling and therefore slows down the\nremagnetization at later times iii) the grain boundaries\nprevent domain wall motion and therefore strongly re-\nduce the impact of an external field on the remagnetiza-\ntion dynamics. We believe that this thorough compari-\nson of the two morphologies of the L 10phase of FePt is6\nuseful as a reference for the laser-induced magnetization\ndynamics, especially on the timescale of remagnetization\nand cooling.\nAcknowledgements\nWe acknowledge the BMBF for the financial support\nvia 05K16IPA and the DFG via BA 2281/11-1. MD\nthanks the Alexander von Humboldt foundation for fi-\nnancial support.\n\u0003bargheer@uni-potsdam.de;\nhttp://www.udkm.physik.uni-potsdam.de\n[1] Beaurepaire, Merle, Daunois, and Bigot, “Ultrafast spin\ndynamicsinferromagneticnickel,” Physicalreviewletters\n76, 4250–4253 (1996).\n[2] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA.Tsukamoto, A.Itoh, andThRasing,“All-opticalmag-\nnetic recording with circularly polarized light,” Physical\nreview letters 99, 047601 (2007).\n[3] C-H Lambert, S. Mangin, B. S. D. Ch S. Varaprasad,\nY. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski,\nK.Hono, Y.Fainman, M.Aeschlimann, andE.E.Fuller-\nton, “All-optical control of ferromagnetic thin films and\nnanostructures,” Science (New York, N.Y.) 345, 1337–\n1340 (2014).\n[4] Yang Yang, Richard B. Wilson, Jon Gorchon, Charles-\nHenri Lambert, Sayeef Salahuddin, and Jeffrey Bokor,\n“Ultrafastmagnetizationreversalbypicosecondelectrical\npulses,” Science advances 3, e1603117 (2017).\n[5] W. A. Challener, Chubing Peng, A. V. Itagi, D. Karns,\nWei Peng, Yingguo Peng, XiaoMin Yang, Xiaobin\nZhu, N. J. Gokemeijer, Y.-T. Hsia, G. Ju, Robert E.\nRottmayer, Michael A. Seigler, and E. C. Gage, “Heat-\nassisted magnetic recording by a near-field transducer\nwith efficient optical energy transfer,” Nature Photonics\n3, 303 EP – (2009).\n[6] Dieter Weller, Gregory Parker, Oleksandr Mosendz, An-\ndreas Lyberatos, Dmitriy Mitin, Nataliia Y. Safonova,\nand Manfred Albrecht, “Review Article: FePt heat as-\nsisted magnetic recording media,” Journal of Vacuum\nScience & Technology B, Nanotechnology and Microelec-\ntronics: Materials, Processing, Measurement, and Phe-\nnomena 34, 060801 (2016).\n[7] Francesca Casoli and Gaspare Varvaro, “Ultrahigh-\ndensity magnetic recording,” in Ultrahigh-density mag-\nnetic recording , Vol. Chapter 7, edited by Francesca Ca-\nsoli and Gaspare Varvaro (Pan Stanford Publishing, Sin-\ngapore and Roca Raton, FL, 2016).\n[8] K. Hono, Y. K. Takahashi, Ganping Ju, Jan-Ulrich\nThiele, Antony Ajan, XiaoMin Yang, Ricardo Ruiz, and\nLei Wan, “Heat-assisted magnetic recording media mate-\nrials,” MRS Bulletin 43, 93–99 (2018).\n[9] Dieter Weller, Oleksandr Mosendz, Gregory Parker, Si-\nmone Pisana, and Tiffany S. Santos, “L1 0 FePtX-Y me-\ndia for heat-assisted magnetic recording,” physica status\nsolidi (a) 210, 1245–1260 (2013).[10] Jon Gorchon, Charles-Henri Lambert, Yang Yang, Ak-\nshayPattabi, RichardB.Wilson, SayeefSalahuddin, and\nJeffrey Bokor, “Single shot ultrafast all optical magne-\ntization switching of ferromagnetic Co/Pt multilayers,”\nApplied Physics Letters 111, 042401 (2017).\n[11] J.Mendil, P.Nieves, O.Chubykalo-Fesenko, J.Walowski,\nT. Santos, S. Pisana, and M. Münzenberg, “Resolving\nthe role of femtosecond heated electrons in ultrafast spin\ndynamics,” Scientific reports 4, 3980 (2014).\n[12] Jiaqi Zhao, Boyin Cui, Zongzhi Zhang, B. Ma, and\nQ. Y. Jin, “Ultrafast heating effect on transient mag-\nnetic properties of L10-FePt thin films with perpendicu-\nlar anisotropy,” Thin Solid Films 518, 2830–2833 (2010).\n[13] B.Koopmans, G.Malinowski, F.DallaLonga, D.Steiauf,\nM. Fähnle, T. Roth, M. Cinchetti, and M. Aeschlimann,\n“Explaining the paradoxical diversity of ultrafast laser-\ninduced demagnetization,” Nature materials 9, 259 EP –\n(2009).\n[14] Jean-Yves Bigot, Mircea Vomir, and Eric Beaurepaire,\n“Coherent ultrafast magnetism induced by femtosecond\nlaser pulses,” Nature Physics 5, 515–520 (2009).\n[15] M. Battiato, K. Carva, and P. M. Oppeneer, “Superdif-\nfusive spin transport as a mechanism of ultrafast demag-\nnetization,” Physical review letters 105, 027203 (2010).\n[16] N. Kazantseva, U. Nowak, R. W. Chantrell, J. Hohlfeld,\nand A. Rebei, “Slow recovery of the magnetisation after\na sub-picosecond heat pulse,” EPL (Europhysics Letters)\n81, 27004 (2008).\n[17] Raghuveer Chimata, Anders Bergman, Lars Bergqvist,\nBiplab Sanyal, and Olle Eriksson, “Microscopic model\nfor ultrafast remagnetization dynamics,” Physical review\nletters 109, 157201 (2012).\n[18] P.NievesandO.Chubykalo-Fesenko,“ModelingofUltra-\nfastHeat-andField-AssistedMagnetizationDynamicsin\nFePt,” Physical Review Applied 5, 173 (2016).\n[19] Andreas Lyberatos and Gregory J. Parker, “Model of\nballistic-diffusive thermal transport in HAMR media,”\nJapanese Journal of Applied Physics 58, 045002 (2019).\n[20] Philippe Scheid, Gregory Malinowski, Stéphane Man-\ngin, and Sébastien Lebègue, “Ab initio study of elec-\ntronic temperature effects on magnetic materials prop-\nerties,” Physical Review B 99(2019), 10.1103/Phys-\nRevB.99.174415.\n[21] Joseph Barker and Gerrit E. W. Bauer, “Quantum\nthermodynamics of complex ferrimagnets,” arXiv ,\n1902.00449 (2019).\n[22] A. von Reppert, L. Willig, J.-E. Pudell, M. Rössle,\nW. Leitenberger, M. Herzog, F. Ganss, O. Hellwig, and\nM.Bargheer,“Ultrafastlasergeneratedstrainingranular\nand continuous FePt thin films,” Applied Physics Letters\n113, 123101 (2018).\n[23] Francesca Casoli and Gaspare Varvaro, “Ultrahigh-\ndensity magnetic recording,” in Ultrahigh-density mag-\nnetic recording , Vol. Chapter 5, edited by Francesca Ca-\nsoli and Gaspare Varvaro (Pan Stanford Publishing, Sin-\ngapore and Roca Raton, FL, 2016).\n[24] A Lyberatos, D Weller, and GJ Parker, “Finite size\neffects in l1 o-fept nanoparticles,” Journal of Applied\nPhysics 114, 233904 (2013).\n[25] O.Hovorka, S.Devos, Q.Coopman, W.J.Fan, C.J.Aas,\nR. F. L. Evans, Xi Chen, G. Ju, and R. W. Chantrell,\n“The Curie temperature distribution of FePt granular\nmagnetic recording media,” 052406, 1–5 (2014).7\n[26] R. P. Cowburn, J. Ferré, S. J. Gray, and J. A. C.\nBland, “Domain wall mobility in ultrathin epitaxial\nAg/Fe/Ag(001)films,” AppliedPhysicsLetters 74,1018–\n1020 (1999).\n[27] P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier,\nJ. Ferré, V. Baltz, B. Rodmacq, B. Dieny, and R. L.\nStamps, “Creep and flow regimes of magnetic domain-\nwall motion in ultrathin Pt/Co/Pt films with perpen-\ndicular anisotropy,” Physical review letters 99, 217208\n(2007).\n[28] Alex Hubert and Rudolf Schäfer, Magnetic domains:\nThe analysis of magnetic microstructures , corr. print.,\n[nachdr.] ed. (Springer, Berlin, ca. 2011).\n[29] PatrickW.Granitzka,EmmanuelleJal,LoïcLeGuyader,\nMatteo Savoini, Daniel J. Higley, Tianmin Liu, Zhao\nChen, Tyler Chase, Hendrik Ohldag, Georgi L. Dakovski,\nWilliam F. Schlotter, Sebastian Carron, Matthias C.\nHoffman, Alexander X. Gray, Padraic Shafer, Elke Aren-\nholz, Olav Hellwig, Virat Mehta, Yukiko K. Takahashi,\nJian Wang, Eric E. Fullerton, Joachim Stöhr, Alexan-\nderH.Reid, andHermannA.Dürr,“MagneticSwitching\nin Granular FePt Layers Promoted by Near-Field Laser\nEnhancement,” Nano letters 17, 2426–2432 (2017).[30] L. Le Guyader, A. Kleibert, F. Nolting, L. Joly, P. M.\nDerlet, R. V. Pisarev, A. Kirilyuk, Th Rasing, and A. V.\nKimel, “Dynamics of laser-induced spin reorientation in\nCo/SmFeO3 heterostructure,” Physical Review B 87, 1–\n11 (2013).\n[31] Z.H.Cen, B.X.Xu, J.F.Hu, J.M.Li, K.M.Cher, Y.T.\nToh, K. D. Ye, and J. Zhang, “Optical property study of\nFePt-C nanocomposite thin film for heat-assisted mag-\nnetic recording,” Opt. Express 21, 9906–9914 (2013).\n[32] A. H. Reid, X. Shen, P. Maldonado, T. Chase, E. Jal,\nP. W. Granitzka, K. Carva, R. K. Li, J. Li, L. Wu,\nT. Vecchione, T. Liu, Z. Chen, D. J. Higley, N. Hart-\nmann, R. Coffee, J. Wu, G. L. Dakovski, W. F. Schlot-\nter, H. Ohldag, Y. K. Takahashi, V. Mehta, O. Hellwig,\nA. Fry, Y. Zhu, J. Cao, E. E. Fullerton, J. Stöhr, P. M.\nOppeneer, X. J. Wang, and H. A. Dürr, “Beyond a phe-\nnomenological description of magnetostriction,” Nature\ncommunications 9, 388 (2018).\n[33] J. Krumhansl and H. Brooks, “The lattice vi-\nbration specific heat of graphite,” The Jour-\nnal of Chemical Physics 21, 1663–1669 (1953),\nhttps://doi.org/10.1063/1.1698641.\n[34] W.-P. Hsieh, A.S. Lyons, E. Pop, P. Keblinski, and D.G.\nCahill, “Pressure tuning of the thermal conductance of\nweak interfaces,” Physical Review B 84, 184107 (2011)."    },    {        "title": "1407.1092v1.Mixed_spin__1_2_1__hexagonal_Ising_nanowire__some_dynamic_behaviors.pdf",        "content": "Mixed spin (1/2-1) hexagonal Ising nanowire: some dynamic behaviors \n \nErsin Kantar1, * and Yusuf Kocakaplan2 \n1Department of Physics, Erciyes University, 38039 Kayseri, Turkey \n2Graduate School of Natural and Applied Sciences, Erciyes University, 38039 Kayseri, Turkey  \n \nAbstract \n \nThe dynamic behaviors of a mixed spin (1/2-1) hexagonal Ising nanowire (HIN) with core-shell structure in the presence of a time dependent magnetic field are investigated by using the \neffective-field theory with correlations based on the Glauber-type stochastic dynamics (DEFT).  \nAccording to the values of interaction parameters, temperature dependence of the dynamic \nmagnetizations, the hysteresis loop areas and the dynamic correlations are investigated to characterize the nature (first- or second-order) of the dynamic phase transitions (DPTs). Dynamic \nphase diagrams, including compensation points, are also obtained. Moreover, from the thermal \nvariations of the dynamic total magnetization, the five compensation types can be found under \ncertain conditions, namely the Q-, R-, S-, P-, and N-types.   \n \nKeywords:  Hexagonal Ising nanowire, Effective-field theory, Glauber-type stochastic dynamics, \nDynamic phase diagram, Compensation behavior.  \n \n \n1. Introduction \n \nRecently, magnetic nanomaterials (nanoparticles,  nanofilms, nanorods, nanobelts, nanowires and \nnanotubes etc.) have attracted a great interest both theoretically and experimentally. The reason is that these materials can be used technological area, such as biomedical applications [1, 2], \nsensors [3], nonlinear optics [4], permanent magn ets [5], environmental remediation [6], and \ninformation storage devices [7–9]. In particular, magnetic nanowire systems have attracted considerable attention not only because of their academic interest, but also the technological \napplications. In the experimental area, the magnetic nanowires have been synthesized and their \nmagnetic properties have been investigated, such as Fe–Co [10], Co–Pt [11], Ni [12], Ga1-xCuxN [13], Fe [14], Fe3O4 [15], Co [16], Fe–Pt [17], Ni-Fe [18], Co-Cu [19] etc. In theoretical \narea, the magnetic nanowires have been investigated within the various theoretical methods, such \nas effective-field theory (E FT) with correlations [20-24], Monte Carlo Simulations (MC\nS) [25]. \nOn the other hand, the mixed spin Ising systems have attracted a great deal of attention and \nintensively investigated within the concept of statistical physics. These systems have observed \nmany new phenomena that cannot be exhibit in single-spin Ising systems. The most extensively \nmixed system is mixed spin (1/2-1) Ising system. This model has been studied by the mean-field \napproximation (MFA) [26–29], MC S [30, 31] and EFT with correlation [32-35]. Moreover, some \n\n*Corresponding author. \nTel: + 90 352 2076666 # 33136 \nE-mail address: ersinkantar@erciyes.edu.tr  (Ersin Kantar) \nfurther works about magnetic nanomaterials of mixed spin (1/2-1) Ising system were given [36-\n40]. \nFinally, we should also mentioned that the dynamic phase transition (DPT) temperature has \nattracted much attention in recent years, both theoretically [41–49] and experimentally [50–54]. \nHowever, as far as we know, the DPT temperatures of the magnetic nanostructured materials have only been investigated a few works by usin EFT [55-58] and MCs [59-60]. Therefore, the aim of this paper is to investigate temperature dependence of the dynamic magnetizations and the \ndynamic phase diagrams, including compensation points, of the HIN in an oscillating magnetic \nfield within the DEFT.  \nThe paper is organized as follows. In Section II, we define the model and give briefly the \nformulation of a mixed spin (1/2-1) HIN by using the DEFT. In Section III, we present the numerical results and discussions. Finally, Section IV contains the summary and conclusions.  \n \n2. Model and Formulation \n \nThe Hamiltonian of the hexagonal Ising nanowire  (HIN) includes nearest neighbor interactions \nand the crystal field is given as follows:  \n \n2\nSi j C m n 1i m i i m\nij mn im i i mHൌ‐J SS ‐J σ σ ‐J Sσ ‐D S ‐h t S  σ\n       (1)  \n  \nwhere 1/2  and S1 , 0 . The J S, JC and J 1 are the exchange interaction parameters between \nthe two nearest-neighbor magnetic particles at the shell surface, core and between the shell \nsurface and core, respectively (see Fig. 1). D is a Hamiltonian parameter and stands for the \nsingle-ion anisotropy (i.e. crystal field). The surface exchange interaction SC SJJ 1  and \ninterfacial coupling 1C rJ / J are often defined to clarify the effects of the surface and interfacial \nexchange interactions on the physical pr operties in the nanosystem, respectively. hሺtሻ is a time-\ndependent external oscillating magnetic field and is given as   .0htൌ h s i nw t  In here, h0 and \nωൌ2πν  are the amplitude and the angular frequency of the oscillating field, respectively. The \nsystem is in contact with an isothermal heat bath at absolute temperature T A. \n \nFor the mixed spin (1/2-1) HIN sy stem, within the framework of the EFT with correlations, one \ncan easily find the magnetizations m S, m C and quadrupole moment q S as coupled equations as \nfollows: \n \n  \nmS\nSS S 1 C4x=0 2\nS\nx=1\nS q0m1 m sinh J cosh J -1 cosh(J / 2)+2m sinh(J / 2F\n,qm)x\nFx      \n  (2a) \n \n   2\nCCC C S62\nSmx=101 m cosh(J / 2)+2m sinh(J / 2) 1 m sinh J cosh x JG , -1 m                    (2b) \n \nwhere x  is the differential operator. The functions F(x) and G(x) are defined as \n m2sinh[β(x+h)]Fx =exp(-βD)+2cosh[ β(x+h)]                         (3a) \n \nq2cosh[β(x+h)]Fx =exp(-βD)+2cosh[ β(x+h)]           ( 3 b )  \n \n m11Gx  =  t a n h βx + h22 \n  .             ( 3 c )  \n \nwhere BA1/ k T , TA is the absolute temperature and k B is the Boltzman factor.  \n \nIn this point, we can obtain th e set of the dynamical effective-field equations by means of the \nGlauber-type stochastic dynamics. We employ the Glauber transition rates, which the system \nevolves according to the Glauber-type stochastic process at a rate of 1 transitions per unit time. \nHence, the frequency of spin flipping, f, is 1. After some manipulations the set of dynamic \nequations of motion for the magnetizations are obtained as: \n \n SS CSmfdm=dtm, m                            (4a) \n  \n CC SCmdm=dmtgm, ,                        (4b) \n \nIn here, f and g functions came from the expanding right-hand side of Eqs. 2(a) and 2(b), \nrespectively. These functions consist long coeffi cients that and can be easily calculated by \nemploying differential operator technique, namely  e x p a fx fxa . But, these \ncoefficients will not be expressed here because of complicate and long expressions. The dynamic \norder parameters or dynamic magnetizations as the time-averaged magnetization over a period of \nthe oscillating magnetic field are given as  \n \n    wM( ) ,2mt d t\u0000                                              (5) \n \nwhere α = S (Shell), C (Core), T (Total) which corre spond to the dynamic magnetizations for the \nshell, core, and the dynamic total magnetization, respectively. On the other hand, the hysteresis \nloop area is defined by Acharyya [43] as \n \n0 A (t) dh h w (t) cos(wt)dt, mm    \u0000 \u0000   (6) \n \nwhich corresponds to the energy loss due to the hysteresis. The dynamic correlations are \ncalculated as \n \n0wh wC (t)h(t)dt (t)sin(wt)dt.22mm \u0000\u0000 (7)  \nWe should also mention that in the numerical calculations, the hysteresis loop areas A α and the \ndynamic correlations C α are also measure in units J C. The dynamic total magnetization M T \nvanishes at the compensation temperature T comp. Then, the compensation point can be determined \nby looking for the crossing point between the absolute values of the surface and the core \nmagnetizations. Therefore, at the compensation point, we must have \n \n Sc o m p Cc o m pMT = MT ,       ( 8 )  \nand \n         Sc o m p Cc o m p sgn M T = -sgn M T .               (9) \n \nWe also require that T comp < T C, where T C is the critical point temperature. In the next section we \nwill give the numerical results of these equations. \n \n3.  Numerical Results and Discussions \n \nIn this section, we investigate behavior of time variations of average order parameters to find phases in this system. Then, we calculated phase diagrams in the different planes, namely the (T, h), (D, T), (r, T) and ( Δ\nS, T) planes. Finally, we obtain the dynamic compensation points and \ndetermine different dynamic compensation behaviors. We has fixed J C = 1.0 throughout of the \npaper.  \n3.1. The phases in the system: Time variations of average order parameters \n \nAt first, the time variations of the average shell and core magnetizations are investigated to obtain \nthe phases in the system. In order to determine the behaviors of time variations of the average \nmagnetizations, the stationary solutions of the dy namic effective-field coupled equations, namely \nEqs. (4a)-(4b), have been studied for various values of the system parameters.  The stationary \nsolutions of these equations will be a periodic function of \n with period 2 ; that is, \nSSmξ+2π= mξ and CCmξ+2π=mξ. Moreover, they can be one of the three types \naccording to whether they have or do not have the properties \n \n      SSmξ+π= -mξ,          (10a) \nand \n      CCmξ+π= - mξ.         ( 1 0 b )  \n \nwhere ξ = ωt. By utilizing the Adams-Moulton predictor-corrector method, we can solve Eqs. \n(10a) and (10b) for a given set of parameters and initial values. The first type solution of Eqs. \n(10a) and (10b) is a symmetric solution and it corresponds to a paramagnetic (p) phase. In the \nsymmetric solution, average shell and core magnetizations delayed with respect to the external magnetic field. The second type solution of Eqs. (10a) and (10b) is called a non-symmetric \nsolution that corresponds to a ferrimagnetic (i) solution. In this case, average shell and core \nmagnetizations do not follow the external magnetic field any more, but instead of oscillating around zero value. The results of these solutions ar e presented in Fig. 2. Fi g. 2(a)-(c) display p, i \nand nonmagnetic (nm) fundamental phases for different physical parameters and initial values, respectivelly. In Fig. 2(a), the initial values of average shell magnetization m S = 1.0 and -1.0,  and \naverage core magnetization m C = 0.5 and -0.5 and oscillate  around zero value, namely\nCSmξ= mξ =0. Hence, the system shows symetric solution, namely p phase. In Fig. 2(b), \naverage shell and core magnetizations have di fferent initial values. The shell magnetization \noscillates around 1.0 value, core magnetization oscillates around 0.5 value and system illustrates i phase. In Fig. 2(c), shell magnetization oscill ates around the zero value and is delayed with \nrespect to the external magnetic field and core magnetization does not follow the external \nmagnetic field anymore, but instead of oscillating around a zero value, it oscillates around 0.5 value and system illustrates nm phase. These soluti ons do not depend on the initial values, seen in \nFig. 2(a)-(c) explicitly. \n  \n3.2. Thermal behaviors of the dynamic magnetizations \n \nThe dynamic order parameters or the dynamic shell and core magnetizations as the time-averaged magnetization over a period of the oscillating magnetic field have given Eq. (5). With the combination of the Adams-Moulton predictor corrector and Romberg integration numerical \nmethods, we solve Eq. (5) and examine the thermal behavior of dynamic magnetizations M\nα (α = \nS (shell), C (core) and T (total)) for different va lues of system parameters. The thermal behaviors \nof dynamic magnetizations gives the dynamic phas e transition (DPT) point and the type of the \ndynamic phase transition. Figs. 3(a)-(d) are presen ted for obtained numerical results of Eqs. (5), \n(6) and (11). In Fig. 3, T C and T t display the critical or the second -order phase tran sitions and the \nfirst-order phase transition temperatures, respectively. The A α, is dynamic hysteresis loop area \nand Cα is dynamic correlations. Fig. 3(a) shows the thermal behavior of dynamic magnetizations, \ndynamic hysteresis loop area and dynamic correlation for r = 1.0, ∆S = 0.5, D = 0.0 and h 0 = 2.0 \nvalues. At zero temperature, M S = 1.0 and M C = 0.5 and with the increase of temperature they \ndecrease to zero continuously; thus the system undergoes a second order phase transition from \nthe ferrimagnetic (i) phase to the paramagnetic (p) phase at T C = 3.9. We have checked the \nstability of dynamical phase transition between the phases of the system by examine the dynamic \nhysteresis loop areas A α and the dynamic correlations C α. The dynamic hysteresis loop areas and \nthe dynamic correlations become a maximum a nd a minimum (negative) at the second-order \nphase transition temperature T C, respectively. For r = -0.1, ∆S = -0.5, D = 0.0 and h 0 = 1.0 values, \nthe dynamic behavior of M α, Aα and C α is obtained in Fig. 3(b). In this figure, M S and M C take \n1.0 and -0.5 values at zero temperature, and they exhibits a continuous move to zero from these values. Hence, the system undergoes a second-order phase transition from the i phase to the p \nphase at T\nC = 1.08. The A S and A T dynamic hysteresis loop areas and the C S and C T dynamic \ncorrelations become a maximum and a minimum (negative) at the second-order phase transition temperature T\nC, respectively. Moreover, the A C does not become a maximum and the C C, become \na maximum (positive) at T C. Fig. 3(c) is plotted for r = 1.0, ∆S = 0.0, D = 0.0 and h 0 = 3.7 values. \nIn this figure, at zero temperature M S = 1.0 and M C = 0.5 and they decrease zero discontinuously \nas the temperature increases; hence, the system undergoes a first-order phase transition from the i \nphase to the p phase at T t = 0.53. Therefore, T t is the first-order phase transition temperature \nwhere the discontinuity or jump occurs. We also checked this dynamic discontinuous transition to \ninvestigate the thermal behavior of the dynamic hysteresis loop areas A α and dynamic \ncorrelations C α, as seen in figure. As temperature increase from zero, the A α and C α increase from \nzero to a certain positive nonzero values, and A α and C α suddenly jump to the higher positive and \nlower negative values, respectively. Fig. 3(d) is obtained for r = 1.0, ∆S = 0.0, D = 0.0 and h 0 = \n3.4 and different initial values. We can see that the system undergoes two successive phase transitions; the first is a first-order phase tran sition from the p phase to the i phase at T t=0.78, the \nsecond is a second-order one from the i phase to p phase at T C=1.36. While the A α decrease from \nzero as the temperature increase, the C α increase from zero. They suddenly jump at T t=0.78 \nvalues. Then, with the temperature increase the A α become a maximum and C α become a \nminimum (negative) at the second-order phase transition temperature T C = 1.36.  \n \n3.3.  Dynamic phase diagrams \n \nNow, we can obtain the dynamic phase diagrams of the system. The dynamic phase diagrams are \nrepresented in the (h, T), (D, T), ( ΔS, T) and (r, T) planes for different values of the physical \nparameters of the system. In Fig. 4, the solid and dashed lines stand for the second- and first-\norder phase transition lines, respectively. The dashed-dotted line illustrates the behavior of \ncompensation temperatures. The dynamic tricritical point (TCP) is represented by a filled circle. \nThe phase diagram in the (h, T) plane are illustrated in Fig. 4(a) for r = -1.0, ΔS = -0.9 and D = -\n1.5 values. In Fig. 4(a), the system displays one dynamic TCP where signals the change from a \nfirst- to a second-order phase transition. Phase di agram contains i and p phases. Fig. 4(b) show \nthe phase diagram in the (D, T) plane for the h 0 = 0.1, r = -1.0 and ΔS = -0.9 values. As clearly \nseen from Fig. 4(b), the phase diagram include only second-order phase transition. The phase diagram contains i, p and nonmagnetic (nm) phas es as well as compensation temperatures. One \ncan clearly see that in low temperature and crystal field values, the system show nm phase. For h\n0 \n= 0.1, r = -1.0 and D = -1.5 values, the phase diagram is plotted in the ( ΔS, T) plane as seen in \nFig. 4(c). Similar to Fig. 4(b), Fig. 4(c) also contains only second-order phase transition and compensation temperature. Phase diagram displays i and p phases. With the increase of surface exchange interaction parameter ( Δ\nS), the phase transition temperature is increase. Finally, Fig. \n4(d) is obtained to show the phase diagram in (r, T) plane for h 0 = 0.1, ΔS = -0.9 and D = -1.5 \nvalues. The phase diagram contains i, p and nm phases, second-order phase transition lines as well as compensation temperatures. \n \n3.4. The total magnetization behavior of mixed spin (1/2-1) HIN system  \n  \nFig. 5(a) displays the effect of the core-shell  interfacial coupling on the total magnetization M\nT \nbehavior. Fig. 5(a) is obtained for h 0 = 0.5, D = 0.0, ΔS = -0.5 fixed values and r = -0.01, -0.5 and \n-1.0. In this figure, the P- and Q-type of compensation behaviors are obtained for r = -0.01, and -\n0.5 and -1.0 values, respectively. For the same va lues, the total dynamic hy steresis loop area A T \nand total dynamic correlations C T are obtained, as seen in Fig. 5(b) and 5(c), respectively.  Fig. \n6(a) is plotted for h 0 = 0.5, r = -1.0, D = 0.0 fixed values and for ΔS = -0.99, -0.5, and 0.0 values \nto investigate the effect of the surface shell coupling on the M T behavior. For ΔS = -0.99, and -0.5 \nand 0.0 values, the S- and Q-type of compensation behaviors are observed, respectively. In Fig. 6(b) and 6(c), the total dynamic hysteresis loop area A\nT and total dynamic correlations C T are \npresented, respectively. It can be easily seen from  Fig. 6(b) that phase tr ansition temperature is \ngrowing with the increase of the ΔS values. Fig. 7(a) illustrates th e influence of the crystal field \non the M T behavior. Fig. 7(a) is obtained for h 0 = 0.5, r = -1.0, ΔS = -0.5 fixed values and D = -\n1.0, -0.5 and 0.0. While the R-type is obta ined for D = -1.0, the Q-type of compensation \nbehaviors is observed for D = -0.5 and 0.0 values.  \n \n  3.5. The Compensation types of mixed spin (1/2-1) HIN system \n \nAs known, the existence of the compensation temperature in a magnetic nanoparticle has \nimportant applications in the field of thermo-magnetic recording. In this purpose, we also studied the temperature variation of the total magnetization for various values of physical parameters of \nthe system to obtain the compensation temperature and determine compensation types by using \nEqs. (8) and (9).  Fig. 8(a) shows the Q-type behaviors for the curve labeled h\n0 = 0.1, r = 0.75, ΔS \n= -0.5 and D = 0.25. The R-type behavior is obtained in Fig. 8(b) for h 0 = 0.1, r = 1.0, ΔS = -0.75 \nand D = -0.25. Fig. 8(c) indicates the P-type behaviors for h 0 = 0.1, r = 1.0, ΔS = -0.99 and D = \n0.0 values. For h 0 = 0.5, r = -0.01, ΔS = -0.5 and D = 0.0 values, the S-type behaviors is obtained \nas seen in Fig. 8(d). For h 0 = 0.1, r = -0.75, ΔS = -0.99 and D = -1.0 values, the N-type behaviors \nhave observed as seen in Fig. 8(e). The Q-, R-, P- and N- types of compensations behaviors \nclassified in the Néel theory [61] and S-type was obtained by Strecka [62]. It is also worth noting that recently the Q-, R-, S- and N-type [36] and the Q-, R-, N-, M-, P-, and S- type [35, 63] \nbehaviors have been obtained in the mixed Ising nanoparticles and mixed hexagonal Ising \nnanowire systems, respectively.  \n4. Summary and Conclusion \n \nWithin the DEFT with correlations the dynamic phase transition points (DPTs), dynamic phase \ndiagrams and dynamic compensation behaviors of the mixed spin (1/2-1) HIN system under a \ntime oscillating longitudinal magnetic field were  investigated. By utilizing the Glauber-type \nstochastic process, the EFT equations of motion for the average shell and core magnetizations are obtained for the system. We were presented the dynamic phase diagrams in the (h, T), (D, T), \n(Δ\nS, T)  and (r, T)  planes. Our results show that  the dynamic phase diagrams contain the i, p and \nnm fundamental phases as well as TCP point and compensation temperature. The Q-, R-, P-, S- and N-types of compensation behaviors [35, 61, 62] have obtained in the system. We found that \nthe dynamic behavior of the system strongly depends on the values of the interaction parameters. \nFinaly, although the equilibrium phase transitions  of the nanosystems have been conspicuously \nstudied, the dynamic or nonequilibrium properties of these systems have not been investigated \nconsiderably. Hence, we hope that our work c ontributes to close this shortcoming in the \nliterature.   \nReferences \n \n[1] C. Alexiou, A. Schmidt, R. Klein, P. Hu llin, C. Bergemann, W. Arnold, J. Magn. Magn. \nMater., 252 (2002) 363. \n[2] N. Sounderya, Y. Zhang, Recent Patents on Biomedical Engineering 1 (2008) 34. [3] 22. G.V. Kurlyandskaya, M.L. Sanchez, B. Hernando, V.M. Prida, P. Gorria, M. Tejedor, \nAppl. Phys. Lett., 82 (2003) 3053. \n[4] S. Nie, S.R. Emory, Science, 275 (1997) 1102. [5] H. Zeng, J. Li, J.P. Liu, Z.L. Wang, S. Sun, Nature, 420 (2002) 395. \n[6] D.W. Elliott, W.-X. Zhang, Environ. Sci. Tech., 35 (2001) 4922. \n[7] J.E. Wegrowe, D. Kelly, Y. Jaccard, Ph. Gu ittienne, J.Ph. Ansermet, EPL, 45 (1999) 626. \n[8] A. Fert, L. Piraux, J. Magn. Magn. Mater., 200 (1999) 338. \n[9] R.H. Kodama, J. Magn. Magn. Mater., 200 (1999) 359. \n[10] H.L. Su, G.B. Ji, S.L. Tang, Z. Li, B.X. Gu, Y.W. Du, Nanotechnology 16 (2005) 429;           X. Lin, G. Ji, T. Gao, X. Chang, Y. Liu, H. Zhang Y. Du, Solid State Commun. 151        (2011) 1708. \n[11] W. Chen, Z. Li, G.B. Ji, S.L. Tang, M. Lu, Y.W. Du, Solid State Commun. 133 (2005) \n      235;  \n      J.Y. Chen, H.R. Liu, N. Ahmad, Y.L. Li, Z.Y. Chen, W.P. Zhou, X.F. Han, J. Appl. Phys.          109 (7) (2011) 07E157. \n[12] H. Pan, B.H. Liu, J.B. Yi, C. Poh, S. Lim, J. Ding, Y.P. Feng, C. Huan, J.Y. Lin, J. Phys. \n      Chem. B 109 (2005) 3094. [13] K-H. Seong, J-Y. kim, J-J. Kim, S-C. Lee, S-R. Kim, U. Kim, T-E. Park, H-J. Choi, Nano   \n       Letters 7 (2007) 3366. \n[14] B. Hamrakulov, I.S. Kim, M.G. Lee, et  al., Trans. Nonferrous Metals Soc. China \n      19 (2009) 83. \n[15] L. Zhang, Y. Zhang, J. Magn. Magn. Mater. 321 (2009) L15. \n[16] N. Ahmad, J.Y. Chen, J. Iqbal, W.X. Wang, W.P. Zhou, X.F. Han, J. Appl. Phys. 109 \n      (2011) 07A331; \n       S. Pal, S. Saha, D. Polley, A. Barman, Solid State Commun., in press. \n[17] J.P. Xu, Z.Z. Zhang, B. Ma, Q.Y.  Jin, J. Appl. Phys. 109 (2011) 07B704. \n[18] C. Rousse, P. Fricoteaux, J. Mater. Sci. 46 (2011) 6046. \n[19] Z. H. Yang, Z. W. Li, L. Liu, L. B.  Kong, J. Magn. Magn. Mater. 323 (2011) 2674. \n[20] T. Kaneyoshi, J. Magn. Magn. Mater. 322 (2010) 3410;  322 (2010) 3014; 323 (2011)   \n3014; Physica A 390 (2011) 3697; Phys. Stat. Sol. B 248 (2011) 250. \n[21] M. Keskin, N. Şarlı, B. Deviren, Solid State Commun. 151 (2011) 2483. \n[22] M. Boughrara, M. Kerouad, A. Zaim, J. Magn. Magn. Mater. 360 (2014) 222; 368 (2014) \n169. \n[23] Y. Kocakaplan, E. Kantar, Chinese Phys. B 23 (2014) 046801. \n[24] E. Kantar, M. Keskin, J. Magn. Magn. Mater.\n349 (2014) 165. \n[25] A. Feraoun, A. Zaim, M. Kerouad, Physica B 445 (2014) 74. [26] T. Kaneyoshi, E.F. Sarm ento, I.F. Fittipaldi, Phys. Stat. Sol. B 150 (1988) 261. \n[27] T. Kaneyoshi, J.C. Chen, J. Magn. Magn. Mater. 98 (1991) 201. \n[28] J.A. Plascak, Physica A 198 (1993) 665. [29] M. Keskin, M. Erta ş, J. Stat. Phys. 139 (2010) 333. \n[30] G.M. Zhang, C.Z. Yang, Phys. Rev. B 48 (1993) 9452. \n[31] G.M. Buendia, M.A. Novotny, J. Zhang, in : D.P. Landau, K.K. Mon, H.B. Schuttler (Eds.), \nSpringer Proceeds in Physics 78, Computer Si mulations in Condensed Matter Physics, Vol. \nVII, Springer, Heidelberg, 1994, p. 223. \n[32] S.L. Yan, L. Liu, J. Magn. Magn. Mater. 312 (2007) 285. [33] Y.F. Zhang, S.L. Yan, Solid State Commun. 146 (2008) 478; Phys. Lett. A 372 (2008) 2696. \n[34] W. Jiang, G.Z. Wei, A. Du, J. Magn. Magn. Mater. 250 (2002) 49. \n[35] C. Ekiz, J. Strecka, M. Jascur, Cent. Eur. J. Phys. 7 (2009) 509. [36] E. Kantar, Y. Kocakaplan, Solid State Commun. 177 (2014) 1. \n[37] Y. Kocakaplan, E. Kantar, Eur. Phys. J. B 87 (2014) 135. \n[38] A. Zaim, M. Kerouad, Y. E. Amraoui, J. Magn. Magn. Mater. 321 (2009) 1077. [39] N. Şarlı, Physica B 411 (2013) 12.  \n[40] O. Canko, A. Erdinç, F. Ta şkın, A. F. Y ıldırım, J. Magn. Magn. Mater. 324 (2012) 508.  \n[41] T. Tomé, M.J. de Oliveira, Phys. Rev. A 41 (1990) 4251. [42] G.M. Buendía, E. Machado,\nPhys. Rev. E 58 (1998) 1260. \n[43] B.K. Chakrabarti, M. Acharyya, Rev. Mod. Phys. 71 (1999) 847. \n[44] S.W. Sides, P.A. Rikvold, M.A. Novotny, Phys. Rev. E 59 (1999) 2710. \n[45] M. Keskin, O. Canko, U. Temi zer, Phys. Rev. E 72 (2005) 036125. [46] M. Keskin, E. Kantar, O. Canko, Phys. Rev. E 77 (2008) 051130. \n[47] M. Keskin, E. Kantar, J. Magn. Magn. Mater.  322 (2010) 2789. \n[48] M. Erta ş, B. Deviren, M. Keskin, Phys. Rev. E 86 (2012) 051110. \n[49] M. Erta ş, Y. Kocakaplan, M. Keskin, J. Magn. Magn. Mater.  348 (2013) 113. \n[50] Q. Jiang, H.-N. Yang, G.C. Wang, Phys. Rev. B 52 (1995) 14911. \n[51] J.S. Suen, J.L. Erskine, Phys. Rev. Lett . 78 (1997) 3567-3570, May. 1997. \n[52] D.T. Robb, Y.H. Xu, O. Hellwig, J. McCord, A. Berger, M.A. Novotny, P.A. Rikvold, Phys. \nRev. B 78 (2008) 134422. \n[53] N. Gedik, D.-S. Yang, G. Logvenov, I. Boz ovic, and A.H. Zewail, Science 316 (2007) 425. \n[54] K. Kanuga, M. Cakmak, Polymer 48 (2007) 7176. [55] B. Deviren, M. Erta ş, M. Keskin, Phys. Scripta 85 (2012) 055001.  \n[56] B. Deviren, E. Kantar, M. Keskin, J. Magn. Magn. Mater. 324 (2012) 2163. \n[57] M Erta ş, Y Kocakaplan, Phys. Lett. A 378 (2014) 845. \n[58] E Kantar, M Erta ş, M Keskin, J. Magn. Magn. Mater.361 (2014) 61 \n[59] V.S. Leite, W. Figueirredo, Physica A 350 (2005) 379. \n[60] Y. Yüksel, E. Vatansever, H. Polat, J. Phys.: Condens. Matter 24 (2012) 436004. [61] L. Néel, Ann. Phys. 3 (1948) 137.   \n[62]\nJ. Strečka, Physica A, 360 (2006) 379. \n[63] Y. Kocakaplan, E. Kantar, and M. Keskin, Eur. Phys. J. B 86 (2013) 420.  \n \n \nList of Figure Captions \n \nFig. 1. (Color online) Schematic presentation of hexagonal Ising nanowire. The blue and red \nspheres indicate magnetic atoms at the surface shell and core, respectively. \nFig. 2.  (Color online) Time variations of the core and shell magnetizations (m C and m S):  \n(a) Paramagnetic phase (p), r = 1.0, ∆S = 0.5, D = 0.0 and h 0 = 2.0 and T = 4.5. \n(b) Ferrimagnetic phase (f), r = -0.1, ∆S = -0.5, D = 0.0 and h 0 = 1.0, and T = 0.5. \n(c) Nonmagnetic phase (nm), r = -0.25, ∆S = -0.9, D = -1.5 and h 0 = 0.1, and T = 0.1.  \nFig. 3. (Color online) Thermal behaviors of the dynamic core and shell magnetizations with the \nvarious values of r and ∆S. \n   (a) r = 1.0, ∆S = 0.5, D = 0.0 and h 0 = 2.0 \n   (b) r = -0.1, ∆S = -0.5, D = 0.0 and h 0 = 1.0 \n   (c) r = 1.0, ∆S = 0.0, D = 0.0 and h 0 = 3.7 \n            (d)  r = 1.0, ∆S = 0.0, D = 0.0 and h 0 = 3.4 \nFig. 4.The dynamic phase diagrams in the (h, T), (D, T), ( ΔS, T) and (r, T) planes of the \nhexagonal Ising nanowire. The solid and dash ed lines stand for the second- and first-\norder phase transition lines, respectively. The dashed-dotted line illustrates the behavior \nof compensation temperatures. The dynamic tricritical point (TCP) is represented by a \nfilled circle. \n   (a) r = -1.0, ΔS = -0.9 and D = -1.5      (b) r = -1.0, ΔS = -0.9 and h 0 = 0.1  \n   (c)  r = -1.0, h 0 = 0.1 and D = -1.5   \n   (d) ΔS = -0.9, h 0 = 0.1 and D = -1.5 \nFig. 5.  (Color online) For h 0 = 0.5, D = 0.0, ΔS = -0.5 fixed values and r = -0.01, -0.5 and -1.0 \nvalues\n   (a) Total dynamic magnetization \n    (b) Total dynamic correlations  \n   (c)  Total dynamic hysteresis loop area. \nFig. 6.  Same as with Fig. 5, but for h 0 = 0.5, r = -1.0, D = 0.0 fixed values and for ΔS = -0.99, -\n0.5, and 0.0 values \n   (a) Total dynamic magnetization \n    (b) Total dynamic correlations  \n   (c)  Total dynamic hysteresis loop area. \nFig. 7.  Same as with Fig. 5, but for h 0 = 0.5, r = -1.0, ΔS = -0.5 fixed values and D = -1.0, -0.5 \nand 0.0. values \n   (a) Total dynamic magnetization \n    (b) Total dynamic correlations  \n   (c)  Total dynamic hysteresis loop area. \nFig. 8.  The type of compensation behaviors for:  \n(a) h0 = 0.1, r = 0.75, ΔS = -0.5 and D = 0.25. \n(b) h0 = 0.1, r = 1.0, ΔS = -0.75 and D = -0.25. \n(c) h0 = 0.1, r = 1.0, ΔS = -0.99 and D = 0.0. \n(d) h0 = 0.5, r = -0.01, ΔS = -0.5 and D = 0.0. \n(e) h0 = 0.1, r = -0.75, ΔS = -0.99 and D = -1.0. \n\nJS\nJCJ1\nJSmS\nmSmC0 50 100 150 200 250mS(), mC()\n-1.0-0.50.00.51.0\n0 50 100 150 200 250mS(), mC()\n-1.0-0.50.00.51.0\nnm phasep phase\ni phase\n0 50 100 150 200 250mS(), mC()\n-1.0-0.50.00.51.0(a)\n(b)\n(c)\nFig. 2012345Mk\n0.00.20.40.60.81.01.2\nMC\nMS\nMT\nT012345Ck\n-0.00050.00000.00050.00100.00150.00200.0025 CC\nCS\nCT(a)\nFig. 3aAk\n0.000.050.100.150.200.250.300.35AC\nAS\nATT0.00 .51 .01 .5Ck\n-0.00050.00000.00050.00100.00150.00200.0025\nCC\nCS\nCT0.0 0.5 1.0 1.5Mk\n-0.6-0.30.00.30.60.91.2\nMC\nMS\nMT(b)\nFig. 3bAk\n0.00.10.20.3\nAC\nAS\nATT0.0 0.2 0.4 0.6 0.8 1.0Ck\n-0.05-0.04-0.03-0.02-0.010.000.01\nCC\nCS\nCT0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.01.2MC\nMS\nMTAk\n0.00.51.01.52.02.53.0\nAC\nAS\nATMk(C)\nFig. 3c0.0 0.5 1.0 1.5Mk\n-0.20.00.20.40.60.81.0MC\nMS\nMT\n0.0 0.5 1.0 1.5Ck\n-0.05-0.04-0.03-0.02-0.010.000.01\nCC\nCS\nCTAk\n0.00.51.01.52.02.5\nAC\nAS\nAT(d)\nFig. 3dh0.0 0.5 1.0 1.5 2.0 2.5T\n0.00.20.40.60.81.01.2\nD-4 -3 -2 -1 0 1 2T\n0.00.20.40.60.81.01.21.41.61.8\nS-1.0 -0.5 0.0 0.5 1.0T\n01234\nr-2.0 -1.5 -1.0 -0.5 0.0T\n0.00.51.01.52.02.5\nFig. 4ip\nip\nnm\np\nip\nnmi(a) (b)\n(c)(d)0.0 0.5 1.0 1.5 2.0 2.5MT\n0.00.20.40.60.81.0\nr=-0.01\nr=-0.5\nr=-1.0\n0.0 0.5 1.0 1.5 2.0 2.5CT\n0.00000.00010.00020.00030.00040.00050.0006\nT0.0 0.5 1.0 1.5 2.0 2.5AT\n0.000.020.040.06(a)\n(b)\n(c)01234MT\n0.00.20.40.60.8\nCol 1 vs Col 8 \nCol 1 vs Col 19 \nCol 1 vs Col 30 \n01234CT\n0.00000.00050.00100.00150.00200.0025\nCol 1 vs Col 9 \nCol 1 vs Col 20 \nCol 1 vs Col 31 \nT01234AT\n0.000.020.040.060.080.10S = -0.99\nS = -0.5\nS = 0.0(a)\n(b)\n(c)0.0 0.5 1.0 1.5 2.0 2.5CT\n0.00000.00010.00020.00030.00040.0 0.5 1.0 1.5 2.0 2.5MT\n0.00.20.40.60.81.0\nT0.0 0.5 1.0 1.5 2.0 2.5AT\n0.000.010.020.030.04(a)\n(b)\n(c)D=0.0\nD=-0.5\nD=-1.0N-type\nT0.0 0.5 1.0MT\n0.000.020.040.060.08P-type\n0 . 00 . 51 . 01 . 52 . 0MT\n0.00.20.40.60.81.00.0 0.5 1.0 1.5 2.0MT\n0.00.20.40.60.81.0\nQ-type\n0 . 00 . 51 . 01 . 52 . 00.00.20.40.60.81.0\nS-type0 . 00 . 51 . 01 . 52 . 00.00.20.40.60.81.0\nR-type(a) (b)\n(c) (d)\n(e)\n      Fig. 8"    },    {        "title": "2205.04795v2.Non_hermitian_off_diagonal_magnetic_response_of_Dirac_fermions.pdf",        "content": "arXiv:2205.04795v2  [cond-mat.mes-hall]  27 Sep 2022Non-hermitian off-diagonal magnetic response of Dirac ferm ions\nRoberta Zs´ ofia Kiss,1Doru Sticlet,2C˘ at˘ alin Pa¸ scu Moca,3,4and Bal´ azs D´ ora1,5,∗\n1Department of Theoretical Physics, Institute of Physics,\nBudapest University of Technology and Economics, M˝ uegyet em rkp. 3., H-1111 Budapest, Hungary\n2National Institute for R&D of Isotopic and Molecular Techno logies, 67-103 Donat, 400293 Cluj-Napoca, Romania\n3MTA-BME Quantum Dynamics and Correlations Research Group, Institute of Physics,\nBudapest University of Technology and Economics, M˝ uegyet em rkp. 3., H-1111 Budapest, Hungary\n4Department of Physics, University of Oradea, 410087, Orade a, Romania\n5MTA-BME Lend¨ ulet Topology and Correlation Research Group ,\nBudapest University of Technology and Economics, M˝ uegyet em rkp. 3., H-1111 Budapest, Hungary\n(Dated: September 28, 2022)\nWe perform a comparative study for the magnetization dynami cs within linear response theory\nof one and two dimensional massive Dirac electrons, after sw itching on either a real (hermitian)\nor an imaginary (non-hermitian) magnetic field. While hermi tian dc magnetic fields polarize the\nspins in the direction of the external magnetic field, non-he rmitian magnetic fields induce only off\ndiagonal response. An imaginary dc magnetic field perpendic ular to the mass term induces finite\nmagnetization inthethirddirection onlyaccording tother ighthandrule. Thiscan beunderstoodby\nanalyzing the non-hermitian equation of motion of the spin, which becomes analogous to a classical\nparticle in crossed electric and magnetic fields. Therein, t he spin expectation value, the mass term\nand imaginary magnetic field play the role of the classical mo mentum, magnetic and electric field,\nrespectively. The latter two create a drift velocity perpen dicular to them, which gives rise to the\noff-diagonal component of the dc spin susceptibility, simil arly to how the Hall effect develops in the\nclassical description.\nI. INTRODUCTION\nWith the advent of graphene and topological\ninsulators[1, 2], the Dirac equation has been essentially\nrediscovered in condensed matter physics, giving rise\nthe plethora of interesting effects in various dimensions\nand under several conditions. Due to the (pseudo)-spin\nstructure in the Dirac equation[3], a variety of pecu-\nliar phenomena such as the anomalous quantum Hall ef-\nfect, electron chirality and Klein paradox has been ob-\nserved and this degree of freedom has been suggested to\nbe useful for possible applications in spintronics[4] and\npseudospintronics[5].\nIn order for the (pseudo)-spin structure of the Dirac\nequation to be useful for applications, one needs to be\nable to control it with external field. While much is\nknown about this within hermitian quantum mechanics,\nthe effect of non-hermitian external perturbations has\nbeen largely unexplored. Non-hermitian systems have\nbeen extensively investigated[6–14] and present many\nunusual features, such as unidirectional invisibility[15],\nexceptional points[16], supersonic modes[17], the non-\nhermitian linear response theory[18–20] reveals unex-\npected features. These include measuring the anticom-\nmutators of observables instead of commutators, thus\nopening the door to access novel physical quantities ex-\nperimentally as well as containing additional terms due\nto non-unitary dynamics of non-hermitian systems.\nIn order to shed light and investigate non-hermitian\nspin dynamics, we focus on the gapped hermitian Dirac\n∗dora.balazs@ttk.bme.huequation in various dimensions and evaluate the real\npart of the magnetic susceptibility when the system is\nperturbed with a hermitian or a non-hermitian external\nmagnetic field. We find that in response to the hermi-\ntian magnetic fields, the spin susceptibility is a diagonal\ntensor in the zero frequency limit, indicating that the in-\nduced magnetization always develops in the direction of\nthe applied external field. On the other hand, for imagi-\nnary magnetic field, the response is anisotropic and only\nthexycomponent of the real part of the susceptibility\nis finite in the dc limit for a mass term in the zdirec-\ntion. This can be understood by mapping the dynamics\nofthe spins ontothe classicalNewton equationofa parti-\ncle moving in a Lorentz force from magnetic field and an\nelectric field. The former originates from the mass term\nwhile the latter stems from the non-hermitian magnetic\nfield. Within the context of the classical Newton equa-\ntion, these fields induce a drift velocity perpendicular to\nthem and give rise to a finite momentum for the classical\nmotion, which in turn is responsible for the development\nof the Hall effect. In our case, in complete analogy to\nthe classical scenario, a finite spin component is induced\nperpendicular to the mass term and imaginary magnetic\nfield, according to the right hand rule.\nII. ONE DIMENSIONAL DIRAC EQUATION\nWe start with the one dimensionalmassiveDirac equa-\ntion, whose Hamiltonian is\nH0=vpσx+∆σz, (1)2\nwhereσ’s are Pauli matrices, denoting the (pseudo-)spin\nof the particles, vis the Fermi velocity and ∆ is the mass\nterm, which couples to σz. This is readily diagonalized\nto yield the spectrum E±=±/radicalbig\n(pv)2+∆2.\nWe perform a comparative analysis and study the in-\nduced magnetization of the system in the long time limit.\nThe system is initially prepared in the ground state of\nH0at half filling with all E−energies occupied. At t= 0\nthe system is perturbed with a weak real (hermitian) or\nimaginary (non-hermitian) external magnetic field. To\nthis end, we evaluate the realpart of the frequency de-\npendent susceptibility using the Kubo formula from her-\nmitian and non-hermitian linear response theory, whose\ndc,ω→0 limit is responsible for the value of the mag-\nnetization in the long time limit.\nLet us also note that the particle current operator\nfor the one dimensional Dirac equation is vσx, therefore\nthrough measuring the magnetic response in the xdirec-\ntion, implicitly the current correlation function is probed\nthrough the magnetoelectric effect[2]. However, for the\notherdirection, σyorσzcannotbe identified asa particle\ncurrent operator. Nevertheless, all three Pauli matrices\ncan be coupled to by magnetic fields.\nA. Hermitian magnetic field\nThe magnetic response is evaluated using the Kubo\nformula for the spin susceptibility\nχij(t,t′) =−iΘ(τ)/an}bracketle{t[σi(τ),σj]/an}bracketri}ht0 (2)\nin response to an external perturbation of the form Bσj.\nHere, the expectation value is taken with respect to the\nground state wavefunction of H0, and [A,B] denotes\nthe commutator. Here we introduced τ=t−t′and\nσ(τ) =eiH0τσe−iH0τ. Then, the time dependence of the\nmagnetization follows from\n/an}bracketle{tσi(t)/an}bracketri}ht=/integraldisplayt\n0χij(t,t′)B(t′)dt′. (3)\nUsing the fact that the susceptibility is time transla-\ntional invariant, χij(t,t′) =χij(t−t′), and performing\nthe Fourier transformation, we get\n/an}bracketle{tσi(ω)/an}bracketri}ht=χij(ω)B(ω). (4)\nSince our focus is mostly on the possible finite mag-\nnetization in the long time limit, we need to evaluate\nReχ(ω→0), whose non-vanishing value would signal fi-\nnite magnetic response to a static magnetic field. Using\nAppendix B, the dc, ω→0 limit of the real part of the\nmagnetic susceptibilities are\nχ(0) =1\nπv\n|sgn(∆)|0 0\n0 ln2W\n|∆|0\n0 0 ln2W\n|∆|\n,(5)0 0.5 1 1.5 2 2.5 3-101234567\nFIG. 1. The real part of the hermitian susceptibility (blue\nsolid line), χxxand the non-hermitian one (red dashed line),\nχxyis visualized for the one dimensional Dirac equation.\nwhich is a diagonal matrix and Wrepresents the high\nenergycutoff. Thisimpliesthataconstantmagnetization\nin the long time limit develops only in the direction of\nthe applied external magnetic field.\nB. Non-hermitian magnetic field\nIn the case of imaginary magnetic field, the full prob-\nlem becomes non-hermitian and the weak perturbation\nisiBσjwhich is apparently non-hermitian due to the i\nprefactor. The magnetic field is applied along the j=x,\nyorz, directions and Bis assumed real and denotes the\nstrength of the non-hermitian magnetic field. Then, the\noperator to which the external perturbation couples to\nbecomes non-hermitian as iσj. The corresponding real\npart of the susceptibility can be obtained from the non-\nhermitian Kubo formula[18–20] and Eq. ( 4) still applies\nfor the non-hermitian setting. In particular, in Ref. 19,\nsome of the authors worked out the non-hermitian linear\nresponsetheory and applied it to a non-hermitian system\n(i.e. tachyons) in the presence of hermitian perturbation\n(vector potential). In contrast to that, we now focus\non hermitian systems, namely the Dirac equation, in the\npresence of a non-hermitian perturbation, an imaginary\nmagnetic field.\nAdapting this to the present case of hermitian initial\nsystem in non-hermitian, imaginaryperturbation, we ob-\ntain\nχij(t,t′) =−Θ(τ)/parenleftbig\n/an}bracketle{t{σi(τ),σj}/an}bracketri}ht0−2/an}bracketle{tσi/an}bracketri}ht0/an}bracketle{tσj/an}bracketri}ht0/parenrightbig\n,(6)\nwhere{A,B}denotes the anticommutator, which arises\ninstead of the commutator due to the non-hermitian op-\neratoriσj, which the external field couples to. After\nsome straightforward algebra (see Appendix A) and per-\nforming the Fourier transformation to frequency space,\nthe momentum integrals of the non-zero elements for the3\nreal parts are evaluated as\nχxx(ω) =−∆2\n2/integraldisplay\ndp1\nE2δ(2E−|ω|) =\n=−|∆|/radicalBig\nω2\n4∆2−1|ω|vΘ(ω2−4∆2), (7)\nχxy(ω) =2∆\nπ−/integraldisplay\ndp1\n4E2−ω2=\n=sgn(∆)\n2vΘ(4∆2−ω2)/radicalBig\n1−/parenleftbigω\n2∆/parenrightbig2, (8)\nχyy(ω) =−1\n2/integraldisplay\ndpδ(2E−|ω|) =\n=−1\n2/radicalBig\n1−4∆2\nω2vΘ(ω2−4∆2), (9)\nand\nχzz(ω) =−1\n2/integraldisplay∞\n−∞dp(vp)2\nE2δ(2E−|ω|) =\n=−1\n2v/radicalbigg\n1−4∆2\nω2Θ(ω2−4∆2). (10)\nThe only component which does not exhibit gapped be-\nhaviour is χxy, whose frequency dependence is plotted in\nFig.1. The dc limit of the real part of these susceptibil-\nities is evaluated as\nχ(0) =sgn(∆)\n2v\n0 1 0\n−1 0 0\n0 0 0\n, (11)\nwhich is an off-diagonal matrix, and the only finite el-\nements follows the ”right hand rule”, namely that the\ninduced magnetization in the long time limit is perpen-\ndicular to both the mass term ( zdirection in the present\ncase)andthedirectionoftheappliedimaginarymagnetic\nfield. This is explained in Sec. V.\nIII. TWO DIMENSIONAL DIRAC EQUATION\nThe two dimensional gapped Dirac equation is written\nas\nH0=vxpxσx+vypyσy+∆σz, (12)\nwhose spectrum is E±=±/radicalbig\n(vxpx)2+(vypy)2+∆2,\nand the system is initially prepared in its ground state at\nhalf filling, i.e. the E−energies are occupied. Similarly\nto the one dimensional case, the particle current opera-\ntors in the xandydirections are vxσxandvyσy, while\nσzcannot be identified as a current.A. Hermitian magnetic field\nWe use the conventional Kubo formula again from\nEq. (2) and the Appendices. Eventually, in the ω→0\nlimit, we get\nχ(0) =W\n4πvxvy\n1 0 0\n0 1 0\n0 0 2\n, (13)\nwhich is again a diagonal matrix, similarly to the one\ndimensional case.\nB. Non-hermitian magnetic field\nSimilarly to the one dimensional case, we use the non-\nhermitian Kubo formula in Eq. ( 6). The finite elements\nof the real part of the frequency dependent susceptibility\nfrom the Appendix are\nχxx(ω) =−1\n4π/integraldisplay\nd2p(vypy)2+∆2\nE2δ(2E−|ω|) =\n=−|ω|\n16vxvy/parenleftbigg\n1+4∆2\nω2/parenrightbigg\nΘ(ω2−4∆2),(14)\nχxy(ω) =∆\nπ2−/integraldisplay\nd2p1\n4E2−ω2=\n=∆\n4πvxvyRe/parenleftBigg\nln/parenleftBigg/parenleftbigW\n∆/parenrightbig2\n1−/parenleftbigω\n2∆/parenrightbig2+1/parenrightBigg/parenrightBigg\n,(15)\nχyy(ω) =χxx(ω) and\nχzz(ω) =−1\n4π/integraldisplay\nd2p(vxpx)2+(vypy)2\nE2δ(2E−|ω|) =\n=−|ω|\n8vxvy/parenleftbigg\n1−4∆2\nω2/parenrightbigg\nΘ(ω2−4∆2).(16)\nWhilethediagonalcomponentsaregapped,theonlynon-\nvanishing off-diagonal element, χxyis cutoff dependent,\nwhich is expected to dominate over the additional fre-\nquency dependence. This is shown in Fig. 2.\nAfter taking the ω→0 limit, we find that the real part\nof the magnetic susceptibility is off-diagonal, similarly to\nthe one dimensional case as\nχ(0) =∆\n2πvxvyln/parenleftbiggW\n|∆|/parenrightbigg\n0 1 0\n−1 0 0\n0 0 0\n.(17)\nThe ensuing structure of Eq. ( 17) is explained in Sec. V.\nIV. THREE DIMENSIONAL DIRAC-WEYL\nEQUATION\nThe three dimensional Dirac-Weyl equation is written\nas\nH0=vxpxσx+vypyσy+vzpzσz,(18)4\n0 0.5 1 1.5 2 2.5 30.70.80.911.11.21.31.4\nFIG. 2. The real part of the hermitian susceptibility (blue\nsolid line), χxxand the non-hermitian one (red dashed line),\nχxyis visualized for the two dimensional Dirac equation with\nW/∆ = 100.\nwhose spectrum is E± =\n±/radicalbig\n(vxpx)2+(vypy)2+(vzpz)2. This equation cannot\nbe gapped out, a term of the form ∆ σzdoes not open a\ngap in the spectrum.\nIn orderto calculatethe correspondingsusceptibilities,\nwe realize that the three dimensional Dirac-Weyl equa-\ntion in Eq. ( 18) can be obtained from the two dimen-\nsional gapped Dirac equation from Eq. ( 12) by replac-\ning ∆ with vzpz. Consequently, the frequency dependent\nsusceptibility of the former is obtained from that of the\nlatter in Eqs. ( A3) and (A4) after the same ∆ →vzpz\nreplacement and integrating over pz. When moving to\nthe dc limit, the corresponding dc susceptibility is ob-\ntained by performing the same replacement in Eqs. ( 13)\nand (17). For the hermitian case, the dc susceptibility\nis finite and independent from ∆ in Eq. ( 13) for the\ntwo dimensional case. By moving to the Dirac-Weyl case\nwith the ∆ →vzpzreplacement and integrating over pz,\nthe dc limit of the hermitian susceptibility is finite as\nχ(0)∼W2, as already identified for three dimensional\nDirac semimetals in Ref. 21.\nIn contrast, Eq. ( 17) is odd in ∆ for the non-hermitian\ncase in two dimensions. Due to this, by moving into the\nDirac-Weyl case with ∆ →vzpzchange and momentum\nintegration, it will vanish. Therefore, we find that the dc\nlimit of the non-hermitian susceptibilities for Dirac-Weyl\nsystems are zero as\nχ(0) = 0. (19)\nNofinitemagnetizationcanbeinducedbyastatic, imag-\ninary magnetic field at half filling (when the E−band is\nfilled.\nBy applying the same procedureto Eqs. ( 14), (15) and\n(16), we obtain the frequency dependent non-hermitian\nsusceptibilitiesaswell. Theoff-diagonalcomponentsvan-\nish (Eq. ( 15) is odd in ∆) and the diagonals are equals\nto each other as χ(ω) =−ω2/24πvxvyvz.V. EQUATION OF MOTION FOR THE SPIN\nThe off-diagonal nature of the real part of the non-\nhermitian dc magnetic susceptibility can be understood\nby inspecting the equation of motion for the spins[22].\nWe consider the full non-hermitian Hamiltonian with\nH= (Ap+iB)·σ, whereAp= (vp,0,∆) for the one\ndimensional case and Ap= (vxpx,vypy,∆) for the two\ndimensional case and iBdenotes the imaginary time in-\ndependent magnetic field, that is switched on at t= 0.\nThe expectation value of the spin for a given momentum\nis evaluated from[12, 23–25]\n/an}bracketle{tσp(t)/an}bracketri}ht=/an}bracketle{tΨp|eiH+tσe−iHt|Ψp/an}bracketri}ht\n/an}bracketle{tΨp|eiH+te−iHt|Ψp/an}bracketri}ht,(20)\nand the system starts from lowest energy eigenstate, Ψ p,\nofAp·σ. The equation of motion for the spin for a given\nmomentum preads as\n∂t/an}bracketle{tσp(t)/an}bracketri}ht= 2Ap×/an}bracketle{tσp(t)/an}bracketri}ht−2B+2/an}bracketle{tσp(t)/an}bracketri}ht[/an}bracketle{tσp(t)/an}bracketri}ht·B],\n(21)\nwhich resembles closely to the Newton’s equation of a\nclassical particle in a crossed electric and magnetic field.\nHere,/an}bracketle{tσp(t)/an}bracketri}htrepresents the classical momentum, Ap\nplays the role of the magnetic field and the first term\non the r.h.s. of Eq. ( 21) represents the Lorentz force,\nBrepresents an electric field and the last term is the\nrelativistic correction.\nIn a crossedelectric and magneticfield, the particle ex-\nperiencesa drift velocityforthe guiding center[26], which\nis perpendicular to both the electric and magnetic field.\nIn this case, this effective ”drift velocity” points towards\nB×Ap, and after averaging over momentum, in order\nto get the total spin as/summationtext\np/an}bracketle{tσp(t)/an}bracketri}ht, it becomes perpen-\ndicular to both the zdirection (the direction of the mass\nterm, which does not average out) and the direction of\nthe imaginary magnetic field. This results in an effective\nmagnetization in the perpendicular direction, in agree-\nment with Eqs. ( 11) and (17), similarly to how the Hall-\neffect develops in the classical case.\nThe equation of motion method also allows us to an-\nalyze the three dimensional Dirac-Weyl case with Ap=\n(vxpx,vypy,vzpz), which can be obtained from the two\ndimensional case after the ∆ →vzpzchange and inte-\ngration over pz, similarly to Sec. IV. For a given fix pz,\nthere will be a finite magnetization developing perpen-\ndicular to both the non-hermitian magnetic field and z\ndirection, analogouslyto the two dimensional case. How-\never, this will be compensated exactly by the contribu-\ntion of the −pzterm, which arises from the momentum\nintegration. This will cancel the magnetization from the\n+pzterm and yield Eq. ( 19).\nVI. EXPERIMENTAL POSSIBILITIES\nIn terms of experimental realization, various forms of\nthe Dirac equation in various spatial dimensions have al-5\nready been realized[1, 27–32] both in condensed matter\nand cold atomic systems as well as in photonic crys-\ntals. In an open quantum system, interacting with its\nenvironment, the non-hermitian term (i.e. the imaginary\nmagnetic field) arises from an effective Lindblad equa-\ntion without the recycling term[23, 24] through continu-\nous monitoring of the system and postselection[23, 24],\nusing jump operators for bonds[17, 33, 34]. This results\nin the appropriate imaginary magnetic fields for the one,\ntwo and three dimensional Dirac equations. By prepar-\ning the system in a given initial state with the E−band\nfilledandcouplingitweaklytoenvironment,the(pseudo-\n)magnetization in a given direction can be measured,\nwhich would directly yield the calculated susceptibilities.\nOne can also profit from the recent non-hermitian real-\nizationofspin-orbitcoupledfermions[35]. Byconsidering\nweak spin dependent173Yb atom losses, a non-hermitian\nmagnetic field can be engineered. By monitoring the en-\nsuingtime dependent spin profile, the non-hermitianspin\nsusceptibility is directly accessible after Fourier transfor-\nmation to frequency space.\nPhotonic waveguides[8, 31, 32, 36] are also used to em-\nulate the non-hermitian Dirac equation, with the com-\nplex refractive index due to losses representing the non-\nhermitian term. Weak imaginary magnetic fields are cre-\nated by weak losses, and by measuring light propagation\nacross the experimental setup yields the non-hermitian\npseudo-magneticsusceptibilitieswithinthevalidityrange\nof our linear response calculations.\nAdditionally, single photoninterferometryis alsoavail-\nable to realize Dirac equation in the presence of non-\nhermitian magnetic fields[37] directly in momentum\nspace. By preparing the system in the lower energy band\nandcontrollingthetimeevolutioninthepresenceofweak\nnon-hermitian terms by a variety of optical elements, the\ntime dependent magnetization can be obtained.VII. CONCLUSIONS\nWestudiedthemagnetizationdynamicsintermsofthe\nreal part of the frequency dependent spin susceptibility\nof one, two and three dimensional gapped Dirac elec-\ntrons, in response to hermitian or non-hermitian mag-\nnetic fields. By focusing on the long time limit of the\nmagnetization, we find that a hermitian magnetic field\ninduces diagonal response and the ensuing spin expec-\ntation value points in the direction of the external per-\nturbation. In sharp contrast, a non-hermitian magnetic\nfield triggers off-diagonal response according to the right\nhand rule: a constant magnetization develops in the di-\nrection perpendicular to both the direction of the mass\nterm and that of the non-hermitian magnetic field. This\nis understood by mapping the equation of motion of the\nspintoaNewtonequationofaclassicalparticleinelectric\nand magnetic fields, the latter giving rise to the Lorentz\nforce. In the classical case, a finite drift velocity develops\nperpendicular to both the electric and magnetic fields.\nAnalogously for the spin, a finite spin expectation value\nisonlyexpectedperpendiculartoboththemasstermand\nthe imaginarymagnetic field. Our results could be useful\nfor further manipulation of the spin of Dirac particles in\nopen quantum systems and in dissipative environment.\nACKNOWLEDGMENTS\nUseful discussions with Ferenc Simon, J´ anos Asb´ oth\nare gratefully acknowledged. This research is supported\nby the National Research, Development and Innova-\ntion Office - NKFIH within the Quantum Technology\nNational Excellence Program (Project No. 2017-1.2.1-\nNKP-2017-00001), K142179, K134437, by the BME-\nNanotechnology FIKP grant (BME FIKP-NAT), and by\na grant of the Ministry of Research, Innovation and\nDigitization, CNCS/CCCDI-UEFISCDI, under Projects\nNo. PN-III-P4-ID-PCE-2020-0277 and PN-III-P1-1.1-\nTE-2019-0423.\n[1]A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.\nNovoselov, and A. K. Geim, The electronic properties of\ngraphene , Rev. Mod. Phys. 81, 109 (2009).\n[2]M. Z. Hasan and C. L. Kane, Colloquium: Topological\ninsulators , Rev. Mod. Phys. 82, 3045 (2010).\n[3]D. Song, V. Paltoglou, S. Liu, Y. Zhu, D. Gallardo,\nL. Tang, J. Xu, M. Ablowitz, N. K. Efremidis, and\nZ. Chen, Unveiling pseudospin and angular momentum\nin photonic graphene , Nature Communications 6, 6272\n(2015).\n[4]M. He, H. Sun, and Q. L. He, Topological insulator:\nSpintronics and quantum computations , Front. Phys. 14,\n43401 (2019).\n[5]D. P. amd Allan H. MacDonald, Spintronics and pseu-\ndospintronics in graphene and topological insulators , Na-ture Materials 11, 409 (2012).\n[6]T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D.\nFraser, S. Brodbeck, M. Kamp, C. Schneider, S. H¨ ofling,\nY. Yamamoto, F. Nori, Y. S. Kivshar, et al.,Observa-\ntion of non-hermitian degeneracies in a chaotic exciton-\npolariton billiard , Nature 526, 554 (2015).\n[7]I. Rotter and J. P. Bird, A review of progress in the\nphysics of open quantum systems: theory and experiment ,\nRep. Prog. Phys. 78, 114001 (2015).\n[8]J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer,\nS. Nolte, M. S. Rudner, M. Segev, and A. Szameit, Ob-\nservation of a topological transition in the bulk of a non-\nhermitian system , Phys. Rev. Lett. 115, 040402 (2015).\n[9]L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang,\nSingle-mode laser by parity-time symmetry breaking , Sci-6\nence346(6212), 972 (2014).\n[10]H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia,\nR. El-Ganainy, D. N. Christodoulides, and M. Kha-\njavikhan, Enhanced sensitivity at higher-order excep-\ntional points , Nature 548, 187 (2017).\n[11]E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex-\nceptional topology of non-hermitian systems , Rev. Mod.\nPhys.93, 015005 (2021).\n[12]Y. Ashida, Z. Gong, and M. Ueda, Non-hermitian\nphysics, Advances in Physics 69, 3 (2020).\n[13]R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H.\nMusslimani, S. Rotter, and D. N. Christodoulides, Non-\nhermitian physics and pt symmetry , Nat. Phys. 14(1), 11\n(2018).\n[14]M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli,\nNon-reciprocal phase transitions , Nature 592, 363(2021).\n[15]Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao,\nand D. N. Christodoulides, Unidirectional invisibility in-\nduced by PT-symmetric periodic structures , Phys. Rev.\nLett.106, 213901 (2011).\n[16]W. D. Heiss, The physics of exceptional points , J. Phys.\nA: Math. Theor. 45(44), 444016 (2012).\n[17]Y. Ashida and M. Ueda, Full-counting many-particle dy-\nnamics: Nonlocal and chiral propagation of correlations ,\nPhys. Rev. Lett. 120, 185301 (2018).\n[18]L. Pan, X. Chen, Y. Chen, and H. Zhai, Non-hermitian\nlinear response theory , Nature Physics 16, 767 (2020).\n[19]D. Sticlet, B. D´ ora, and C. P. Moca, Kubo formula for\nnon-hermitian systems and tachyon optical conductivity ,\nPhys. Rev. Lett. 128, 016802 (2022).\n[20]K. T. Geier and P. Hauke, From non-hermitian lin-\near response to dynamical correlations and fluctuation-\ndissipation relations in quantum many-body systems ,\nPRX Quantum 3, 030308 (2022).\n[21]Y. Ominato and K. Nomura, Spin susceptibility of three-\ndimensional dirac-weyl semimetals , Phys. Rev. B 97,\n245207 (2018).\n[22]R. Botet and H. Kuratsuji, The duality between a non-\nhermitian two-state quantum system and a massless\ncharged particle , Journal of Physics A: Mathematical and\nTheoretical 52(3), 035303 (2018).\n[23]A. J. Daley, Quantum trajectories and open many-body\nquantum systems , Advances in Physics 63, 77 (2014).\n[24]H. Carmichael, An Open Systems Approach to Quantum\nOptics(Springer-Verlag, Berlin, 1993).\n[25]E. M. Graefe, H. J. Korsch, and A. E. Niederle, Mean-\nfield dynamics of a non-hermitian bose-hubbard dimer ,\nPhys. Rev. Lett. 101, 150408 (2008).\n[26]T. G. Northrop, The guiding center approximation to\ncharged particle motion , Annals of Physics 15(1), 79\n(1961).\n[27]R. Gerritsma, G. Kirchmair, F. Z¨ ahringer, E. Solano,\nR. Blatt, and C. F. Roos, Quantum simulation of thedirac equation , Nature 463(7277), 68 (2010).\n[28]L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and\nT. Esslinger, Creating, moving and merging Dirac points\nwith a Fermi gas in a tunable honeycomb lattice , Nature\n483, 302 (2012).\n[29]L. Lamata, J. Le´ on, T. Sch¨ atz, and E. Solano, Dirac\nequation and quantum relativistic effects in a single\ntrapped ion , Phys. Rev. Lett. 98, 253005 (2007).\n[30]T. E. Lee, U. Alvarez-Rodriguez, X.-H. Cheng,\nL. Lamata, and E. Solano, Tachyon physics with trapped\nions, Phys. Rev. A 92, 032129 (2015).\n[31]B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer,\nA. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljacic,\nSpawning rings of exceptional points out of dirac cones ,\nNature525, 354 (2015).\n[32]W. Song, S. Gao, H. Li, C. Chen, S. Wu, S. Zhu,\nand T. Li, Demonstration of imaginary-mass parti-\ncles by optical simulation in non-hermitian systems\nArXiv:2011.08496.\n[33]Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi-\ngashikawa, and M. Ueda, Topological phases of non-\nhermitian systems , Phys. Rev. X 8, 031079 (2018).\n[34]Y. Takasu, T. Yagami, Y. Ashida, R. Hamazaki,\nY. Kuno, and Y. Takahashi, PT-symmetric non-\nHermitian quantum many-body system using ultracold\natoms in an optical lattice with controlled dissipa-\ntion, Progress of Theoretical and Experimental Physics\n2020(12) (2020), 12A110.\n[35]Z. Ren, D. Liu, E. Zhao, C. He, K. K. Pak, J. Li, and G.-\nB. Jo,Chiral control of quantum states in non-hermitian\nspin–orbit-coupled fermions , Nat. Phys. 18, 385 (2022).\n[36]S. Longhi, Optical realization of relativistic non-\nhermitian quantum mechanics , Phys. Rev. Lett. 105,\n013903 (2010).\n[37]L. Xiao, D. Qu, K. Wang, H.-W. Li, J.-Y. Dai, B. D´ ora,\nM. Heyl, R.Moessner, W. Yi, andP. Xue, Non-hermitian\nkibble-zurek mechanism with tunable complexity in single-\nphoton interferometry , PRX Quantum 2, 020313 (2021).\n[38]T. Giamarchi, Quantum Physics in One Dimension (Ox-\nford University Press, Oxford, 2004).\n[39]L. Zhao, H. Deng, I. Korzhovska, Z. Chen, M. Kon-\nczykowski, A. Hruban, V. Oganesyan, and L. Krusin-\nElbaum, Singular robust room-temperature spin response\nfrom topological dirac fermions , Nature Materials 13, 580\n(2014).\n[40]A. Scholz and J. Schliemann, Dynamical current-current\nsusceptibility of gapped graphene , Phys. Rev. B 83,\n235409 (2011).\nAppendix A: Calculation of the susceptibility\nIn one dimension with hermitian magnetic field, we get\nχ(t,t′) =1\n2π/integraldisplay\ndp2\nE2\n∆2sin(2Eτ) ∆Ecos(2Eτ)−∆vpsin(2Eτ)\n−∆Ecos(2Eτ)E2sin(2Eτ)vpEcos(2Eτ)\n−∆vpsin(2Eτ)−vpEcos(2Eτ) (vp)2sin(2Eτ)\nΘ(τ). (A1)\nwhereE=/radicalbig\n(vp)2+∆2.7\nIn one dimension with imaginary magnetic field, we obtain\nχ(t,t′) =1\n2π/integraldisplay\ndp2\nE2\n−∆2cos(2Eτ) ∆Esin(2Eτ) ∆vpcos(2Eτ)\n−∆Esin(2Eτ)−E2cos(2Eτ)vpEsin(2Eτ)\n∆vpcos(2Eτ)−vpEsin(2Eτ)−(vp)2cos(2Eτ)\nΘ(τ). (A2)\nIn two dimensions with hermitian magnetic field, the susceptibility is\nχ(t,t′) =1\n4π2/integraldisplay\ndpxdpy2\nE2\n((vypy)2+∆2)−(vypy)(vxpx)−(vxpx)∆\n−(vypy)(vxpx) ((vxpx)2+∆2)−(vypy)∆\n−(vxpx)∆ −(vypy)∆ ((vxpx)2+(vypy)2)\nsin(2Eτ)Θ(τ)\n+1\n4π2/integraldisplay\ndpxdpy2\nE\n0 ∆ −(vypy)\n−∆ 0 ( vxpx)\n(vypy)−(vxpx) 0\ncos(2Eτ)Θ(τ), (A3)\nwhereE=±/radicalbig\n(vxpx)2+(vypy)2+∆2. The two dimensional case with imaginary magnetic field yields\nχ(t,t′) =1\n4π2/integraldisplay\ndpxdpy2\nE2\n−((vypy)2+∆2) (vypy)(vxpx) ( vxpx)∆\n(vypy)(vxpx)−((vxpx)2+∆2) ( vypy)∆\n(vxpx)∆ ( vypy)∆−((vxpx)2+(vypy)2)\ncos(2Eτ)Θ(τ)\n+1\n4π2/integraldisplay\ndpxdpy2\nE\n0 ∆ −(vypy)\n−∆ 0 ( vxpx)\n(vypy)−(vxpx) 0\nsin(2Eτ)Θ(τ). (A4)\nThe susceptibility of three dimensional Dirac-Weyl\nfermions follows from Eqs. ( A3) and (A4) after replacing\n∆ withvzpz.\nAppendix B: Frequency dependent susceptibilities\nfor hermitian magnetic field\n1. One dimension\nBased on Appendix A, the non-vanishing components\nof the real part of the susceptibility (with Wthe high\nenergy cutoff) are\nχxx(ω) =2∆2\nπ−/integraldisplay\ndp1\nE(4E2−ω2)=−1\nπv|ω\n2∆|×\n×Re\n1/radicalBig/parenleftbigω\n2∆/parenrightbig2−1atanh\n1/radicalBig\n1−/parenleftbig2∆\nω/parenrightbig2\n\n(B1)\nwithE=/radicalbig\n(vp)2+∆2and−/integraltext\ndenoting Cauchy’s princi-\npal value of an integral,\nχxy(ω) =∆\n2/integraldisplay\ndp1\nEδ(2E−|ω|) =\n=∆√\nω2−4∆2vΘ(ω2−4∆2), (B2)χyy(ω) =2\nπ−/integraldisplay\ndpE\n4E2−ω2=1\nπv/parenleftbigg\nln/parenleftbigg2W\n|∆|/parenrightbigg\n−\n−Re\n1/radicalBig\n1−/parenleftbig2∆\nω/parenrightbig2atanh\n1/radicalBig\n1−/parenleftbig2∆\nω/parenrightbig2\n\n\n,(B3)\nand\nχzz(ω) =2\nπ−/integraldisplay\ndp(vp)2\nE(4E2−ω2)=1\nπv/parenleftbigg\nln/parenleftbigg2W\n|∆|/parenrightbigg\n−\n−Re\n/radicalBigg\n1−/parenleftbigg2∆\nω/parenrightbigg2\natanh\n1/radicalBig\n1−/parenleftbig2∆\nω/parenrightbig2\n\n\n(B4)\nInthecontinuumlimit, theallowedmomentaregionisex-\ntended to infinity, but in order to make contact with the\noriginalmodel, acutoffneedsto be introducedforcertain\nnon-universal physical quantities. The cutoff presence is\nrather natural in effective low energy theories[38], and\naccounts for the finite bandwidth, which is present in the\noriginal tight binding Hamiltonian, and stems from the\nfinite Brillouin zone.\nAmong these components, χxyexhibits a gapped be-\nhaviour (i.e. |ω|>2∆ is required), while χyyandχzzare\ndominated by the cutoff dependent term, which domi-\nnates over the additional frequency dependences. The\nfrequency dependence of the cutoff independent χxxis\nshown in Fig. 1.8\n2. Two dimensions\nBased on Appendix A, we get\nχxx(ω) =1\nπ2−/integraldisplay\nd2p(vypy)2+∆2\nE(4E2−ω2)=(W−|∆|)\n4πvxvy+\n+|ω|\n8πvxvy/parenleftBigg\n1+/parenleftbigg2∆\nω/parenrightbigg2/parenrightBigg\nRe/parenleftbigg\natanh/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2∆\nω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg/parenrightbigg\n,(B5)\nwhereE=±/radicalbig\n(vxpx)2+(vypy)2+∆2,χyy(ω) =\nχxx(ω) and\nχxy(ω) =∆\n4π/integraldisplay\nd2p1\nEδ(2E−|ω|) =\n=∆\n4vxvyΘ(ω2−4∆2), (B6)\nand\nχzz(ω) =1\nπ2−/integraldisplay\nd2p(vxpx)2+(vypy)2\nE(4E2−ω2)=(W−|∆|)\n2πvxvy+\n+|ω|\n4πvxvy/parenleftBigg\n1−/parenleftbigg2∆\nω/parenrightbigg2/parenrightBigg\nRe/parenleftbigg\natanh/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2∆\nω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg/parenrightbigg\n.(B7)The diagonal components scale with the cutoff, which\nis expected to overwhelm the additional frequency de-\npendences, while the off-diagonal piece, χxyexhibits\ngapped behaviour and contributes only for |ω|>2∆.\nWe note that similar cutoff dependent spin susceptibil-\nity,χ(ω→0)∼W, was observed experimentally for two\ndimensional topological Dirac fermions[39]. Therein, by\nmeasuring the singular spin response of Dirac fermions,\nthe effective cutoff was extracted from the experimental\ndata in Fig. 3 of 39. In addition, χxx(ω) agrees with\nthe dynamical current-current susceptibility of gapped\ngraphene in Ref. 40.\n3. Three dimensions\nBy replacing ∆ with vzpzin Eq. ( A3) and perform-\ning the momentum integrals, we find that the resulting\nsusceptibility is diagonal and the diagonal elements are\nequal as\nχ(ω) =1\n3π2vxvyvz/parenleftbigg\nW2+ω2\n4ln/parenleftbigg4W2+ω2\nω2/parenrightbigg/parenrightbigg\n.(B8)\nIt is dominated by the first, frequency independent term\n∼W2."    },    {        "title": "2401.10594v1.Modelling_and_dynamics_of_pendulum_systems_subjected_to_a_nonstationary_magnetic_field.pdf",        "content": "Politechnika Łódzka\nWydział Mechaniczny\nKatedra Automatyki,\nBiomechaniki i Mechatroniki\nmgr inż. Krystian Polczyński\nRozprawa Doktorska\nw dyscyplinie Inżynieria Mechaniczna\nModelowanie i dynamika układów wahadeł\npoddanych działaniu niestacjonarnego pola\nmagnetycznego\nPromotor pracy:\nprof. dr hab. inż. Jan Awrejcewicz\nPromotor pomocniczy:\ndr inż. Adam Wijata\nŁódź 2023 r.Dziękuję mojemu promotorowi i opiekunowi naukowemu,\nprof. dr. hab. inż. Janowi Awrejcewiczowi za zaufanie, poświęcony czas oraz\npomoc merytoryczną.\nSkładam serdeczne podziękowania moim rodzicom, dziadkom, siostrze oraz\nbliższej i dalszej rodzinie za cierpliwość i wspieranie mnie w trudnych\nmomentach.\nPragnę osobno podziękować mojej ukochanej Annie, za nieocenione wsparcie\ni wyrozumiałość, jakimi obdarzyła mnie w czasie powstawania tej pracy.\nDziękuję serdecznie wszystkim obecnym i byłym pracownikom Katedry\nAutomatyki, Biomechaniki i Mechatroniki, w szczególności dr. hab. inż. Jerzemu\nMrozowskiemu za wiarę we mnie oraz dr. inż. Adamowi Wijacie za ważne\nwskazówki i możliwość konsultacji podczas prowadzonych badań.Niniejsza rozprawa doktorska zawiera badania zrealizowane w ra-\nmach grantów naukowych PRELUDIUM 20 No. 2021/41/N/ST8/01019,\npt. „Theoretical-numerical-experimental analysis of nonlinear dynamics of pen-\ndulums subjected to a non-stationary magnetic field ” przyznanego przez Na-\nrodowe Centrum Nauki i kierowanego przez autora oraz OPUS 14 No.\n2017/27/B/ST8/01330, pt. „ Modelling and nonlinear dynamics of magneto-\nelectro-mechanical systems ” również przyznanego przez Narodowe Cen-\ntrumNauki,któregokierownikiembyłprof.drhab.inż.JanAwrejcewicz,\na w którym autor był stypendystą.\nW trakcie trwania studiów trzeciego stopnia autor wspierany był rów-\nnież finansowo z projektu POWER \"Wdrożeniowa Szkoła Doktorancka\"\no numerze POWR.03.02.00-00-I042/16-00 przyznanego przez Narodowe\nCentrum Badań i Rozwoju.Spis treści\nWykaz symboli 3\nStreszczenie 10\nAbstract 12\n1 Wstęp 14\n1.1 Dotychczasowy stan wiedzy o układach wahadeł magne-\ntycznych . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n1.1.1 Układy o jednym stopniuswobody . . . . . . . . . . . 16\n1.1.2 Układy o wielu stopniach swobody . . . . . . . . . . . 23\n1.2 Geneza i uzasadnienie tematu pracy . . . . . . . . . . . . . . 27\n1.3 Cel naukowy, teza i zakres pracy . . . . . . . . . . . . . . . . 28\n1.4 Wkład wyników pracy w dyscyplinę naukową . . . . . . . . 29\n1.5 Struktura pracy . . . . . . . . . . . . . . . . . . . . . . . . . . 30\n2 Układ pojedynczego wahadła magnetycznego 32\n2.1 Stanowisko badawcze . . . . . . . . . . . . . . . . . . . . . . . 32\n2.2 Modelowanie matematyczne . . . . . . . . . . . . . . . . . . . 36\n2.2.1 Równanie ruchu . . . . . . . . . . . . . . . . . . . . . . 36\n2.2.2 Modele oporów ruchu . . . . . . . . . . . . . . . . . . 37\n2.2.3 Model gumowego elementu podatnego . . . . . . . . 38\n2.2.4 Empirycznemodelemomentusiłyoddziaływaniama-\ngnetycznego . . . . . . . . . . . . . . . . . . . . . . . . 39\n2.2.5 Modelprostokątnegopulsującegosygnałuprądowe-\ngo płynącego w cewce elektrycznej . . . . . . . . . . . 46\n2.2.6 Identyfikacja parametrów . . . . . . . . . . . . . . . . 47\n2.3 Dynamika nieliniowa wahadła magnetycznego . . . . . . . . 50\n2.3.1 Metodauśrednianiadlasłabonieliniowegorównania\nruchu i wykresy rezonansowe . . . . . . . . . . . . . . 50\n2.3.2 Badanie zjawiska drgań w jednym„dołku” potencjału 62\n3 Układ dwóch sprzężonych wahadeł 79\n3.1 Dynamika nieliniowa dwóch sprzężonych wahadeł . . . . . 79\n3.1.1 Stanowisko badawcze . . . . . . . . . . . . . . . . . . 79\n3.1.2 Modelowanie matematyczne . . . . . . . . . . . . . . 80\n1SPIS TREŚCI\n3.1.3 Analiza bifurkacyjna . . . . . . . . . . . . . . . . . . . 82\n3.2 Sterowanie przepływem energii między wahadłami . . . . . 85\n3.2.1 Stanowisko badawcze . . . . . . . . . . . . . . . . . . 85\n3.2.2 Model fizyczny i matematyczny układu . . . . . . . . 87\n3.2.3 Badania wstępne . . . . . . . . . . . . . . . . . . . . . 88\n3.2.4 Adaptacja modelu matematycznego . . . . . . . . . . 90\n3.2.5 Procedura uśredniania . . . . . . . . . . . . . . . . . . 93\n3.2.6 Numeryczna weryfikacja uśrednionego modelu . . . 94\n3.2.7 Analizajakościowatrajektoriifazowychnapłaszczyź-\nnie𝑃∆. . . . . . . . . . . . . . . . . . . . . . . . . . . 97\n3.2.8 Eksperymentalnawalidacjamodeluostrategiiotwar-\ntego sterowania . . . . . . . . . . . . . . . . . . . . . . 99\n3.2.9 Dalszaredukcjaukładuiuwagidotyczącezamkniętej\nstrategii sterowania . . . . . . . . . . . . . . . . . . . . 100\n3.2.10 Strategia zamkniętego sterowania . . . . . . . . . . . . 105\n4 Podsumowanie i wnioski 110\nZałączniki 114\nA Metoda wariacji stałych dowolnych . . . . . . . . . . . . . . . 114\nB Energia potencjalna i częstości drgań własnych zlinearyzo-\nwanegoukładuzachowawczegodwóchsprzężonychwahadeł115\nC Metoda Van der Pola dla dwóch wahadeł magnetycznych . . 116\nD Procedura linearyzacji dla przypadku drgań o warunkach\npoczątkowych bliskich antyfazie . . . . . . . . . . . . . . . . 119\nLiteratura 122\n2Wykaz symboli\n𝐴1,𝐴2amplituda drgań wahadła 1 i 2\n𝐶𝜇stała zależna od przenikalności magnetycznej ośrodka\n𝐸11całkowita energia wahadła 1 odniesiona do jego momentu bez-\nwładności\n𝐸12,\n𝐸21czynnikenergetycznyodniesionydomomentubezwładnościwa \nhadła i informujący o przesunięciu fazowym między nimi\n𝐸22— całkowita energia wahadła 2 odniesiona do jego momentu bez-\nwładności\n𝐸 całkowita energia wahadeł 1 i 2\n𝐸𝑖𝑗— elementsymetrycznej\"macierzyenergii\"wukładziesprzężonych\nwahadeł magnetycznych ( 𝑖,𝑗= 1,2)\n𝐸′\n𝑢pochodna po czasie całkowitej energii wahadeł 1 i 2\n𝐹𝑒𝑥𝑝siławcięgniepodczaseksperymentuwyznaczaniamomentuod \ndziaływania magnetycznego pomiędzy magnesem a cewką elek-\ntryczną\n𝐹𝑚𝑎𝑔siławzajemnegooddziaływaniapomiędzyodseparowanymibie-\ngunami dipola magnetycznego w modelu Gilberta\n𝐺 prawastronarównaniaruchuoryginalnegoukładupojedynczego\nwahadła\n𝐺𝑎prawastronarównaniaruchuprzybliżonegoukładupojedyncze-\ngo wahadła\n𝐼0 amplituda prostokątnego pulsującego sygnału prądowego cewki\n𝐼𝐵0zmodyfikowana funkcja Bessela pierwszego rodzaju rzędu 0\n𝐼𝐵1— zmodyfikowana funkcja Bessela pierwszego rodzaju rzędu 1\n𝐽;𝐽1,𝐽2masowy moment bezwładności wahadła względem osi obrotu;\nmasowy moment bezwładności wahadła 1 i 2 względem osi ob-\nrotu\n3SPIS TREŚCI\n𝑀𝐴𝑚𝑎𝑔 2;\n𝑀𝐴𝑚𝑎𝑔 4;\n𝑀𝐴𝑚𝑎𝑔 6modele wielomianowe momentów magnetycznych o następują \ncych stopniach wielomianu mianownika: 2, 4 i 6\n𝑀𝐶𝑀modelcoulombowskioporówruchuztłumieniemwiskotycznym\n𝑀𝐹;\n𝑀𝐹1,\n𝑀𝐹2— moment oporów ruchu; moment oporów ruchu wahadła 1 i 2\n𝑀𝐹𝑒𝑥𝑝moment oporów ruchu (eksperymentalny)\n𝑀𝐹𝑚𝑎𝑔— moment magnetyczny wg modelu Gilberta\n𝑀𝐾moment generowany przez gumowy element podatny\n𝑀𝑅𝑚𝑎𝑔model wielomianowego momentu magnetycznego\n𝑀𝑆𝐸— modelmomentuoporówruchuzefektemStribeckaitłumieniem\nwiskotycznym\n𝑀𝑆𝑒𝑥𝑝momentskręcającygumowyelementpodatny(eksperymentalny)\n𝑀𝑐;\n𝑀𝑐1,\n𝑀𝑐2— staławartośćmomentuoporówcoulombowskich(kinetycznych);\nstała wartość momentu oporów coulombowskich (kinetycznych)\ndla wahadła 1 i 2\n𝑀𝑒𝑥𝑝moment siły oddziaływania magnetycznego pomiędzy magne-\nsem a cewką elektryczną (eksperyment)\n𝑀𝑚𝑎𝑔model gaussowski momentu magnetycznego\n𝑀𝑠stała wartość statycznego momentu oporów ruchu\nb𝑀𝐴𝑚𝑎𝑔 2;\nb𝑀𝐴𝑚𝑎𝑔 4;\nb𝑀𝐴𝑚𝑎𝑔 6— modele wielomianowe momentów magnetycznych dla zmienne-\ngo prądu cewki o następujących stopniach wielomianu mianow-\nnika: 2, 4 i 6\nb𝑀𝑅𝑚𝑎𝑔model wielomianowego momentu magnetycznego dla zmienne-\ngo prądu cewki\nb𝑀𝑚𝑎𝑔modelgaussowskiegomomentmagnetycznydlazmiennegoprą \ndu cewki\n𝑃 współczynnik podziału energii pomiędzy wahadłami 1 i 2\n𝑃0— położenie stacjonarne współczynnika 𝑃podziału energii\n𝑃′\n𝑢pochodna po czasie współczynnika podziału energii pomiędzy\nwahadłami 1 i 2\n𝑄 — \"wskaźnikkoherencji\"zależnyodprzesunięciafazowegomiędzy\nwahadłami 1 i 2\n𝑄𝑚𝑎𝑔moment pochodzący od oddziaływania magnetycznego pomię-\ndzy cewką a magnesem\n4SPIS TREŚCI\n𝑅1;𝑅2prawastronarównanianapochodnąpoczasieamplitudy 𝑘;prawa\nstronarównanianapochodnąpoczasieprzesunięciafazowego 𝑢\n𝑅𝐷promień polimerowego dysku\n𝑇𝑝okres drgań swobodnych wahadła\n𝑇𝑡— okres wymuszenia\n𝑉 całkowita energia potencjalna układu\n𝑉𝑀𝐹𝑚𝑎𝑔potencjał momentu magnetycznego dla modelu Gilberta\n𝑉𝑀𝑚𝑎𝑔— potencjał momentu magnetycznego dla modelu gaussowskiego\n𝑉𝐽energia potencjalna zachowawczego zlinearyzowanego układu\nsprzężonych wahadeł odniesiona do masowego momentu bez \nwładności\n𝑉𝑔𝑟𝑎𝑤energia potencjalna pochodząca od grawitacji dla zlinearyzowa \nnego wahadła\n𝑉𝑝𝑀zlinearyzowana energia potencjalna pola magnetycznego\n𝑉𝑠suma energii potencjalnej pola grawitacyjnego oraz elementu\nsprężystego\n𝑉𝑠𝑝𝑟energia potencjalna zgromadzona w sprężynie sprzęgającej wa \nhadła\n𝑎 parametrmodelugaussowskiegomomentumagnetycznegoijego\npotencjału\n𝑎𝐼prądowy współczynnik kierunkowy\n𝑏 parametrmodelugaussowskiegomomentumagnetycznegoijego\npotencjału\n𝑐;𝑐𝑤1,𝑐𝑤2współczynniktłumieniawiskotycznego;współczynnikitłumienia\nwiskotycznego dla wahadła 1 i 2\n𝑐1parametr funkcji piłokształtnej 𝑔𝑢\n𝑐2parametr funkcji piłokształtnej 𝑔𝑘\n𝑐𝑒— współczynniktłumieniawiskotycznegogumowegoelementupo-\ndatnego\n𝑐𝑒𝑡współczynnik tłumienia wiskotycznego stalowej sprężyny\n𝑑 odległość między środkami odseparowanych biegunów dipola\nmagnetycznegomierzonawpołożeniuinnymniżdolnepołożenie\nrównowagi wahadła\n𝑑1parametr funkcji piłokształtnej przesunięcia fazowego 𝑢\n𝑑2— parametr funkcji piłokształtnej amplitudy 𝑘\n5SPIS TREŚCI\n𝑑𝑔odległość między środkami odseparowanych biegunów dipola\nmagnetycznego mierzona w dolnym położeniu równowagi wa-\nhadła\n𝑓 częstotliwość prostokątnego pulsującego sygnału prądowego\ncewki\n𝑓𝑗,𝑓1,\n𝑓2funkcjeopisującetłumienie,sprzężeniemechaniczneiwpływpo-\nla magnetycznego na wahadło 1 i 2 ( 𝑗= 1,2)\n𝑓𝑁 częstotliwość drgań swobodnych pojedynczego wahadła magne-\ntycznego\n𝑔 przyspieszenie grawitacyjne\n𝑔𝑘funkcja piłokształtna aproksymująca amplitudę 𝑘\n𝑔𝑢— funkcja piłokształtna aproksymująca przesunięcie fazowe 𝑢\n𝑖;𝑖1,𝑖2sygnał prądu płynącego w cewce elektrycznej; sygnał prądu pły-\nnącego w cewce elektrycznej wahadła 1 i 2\n𝑖0 — parametrmodeluprostokątnegopulsującegosygnałuprądowego\n𝑖𝐴 amplituda sygnału prądowego\n𝑖𝑝 sygnałprostokątnegopulsującegoprądupłynącegowcewceelek \ntrycznej\n𝑘 amplituda rozwiązania opisującego ruchu drgający wahadła\n𝑘𝑒współczynnik sztywności gumowego elementu podatnego\n𝑘𝑒𝑡współczynnik sztywności stalowej sprężyny\n𝑘𝑛liczba naturalna\n⟨𝑘⟩wartość średnia amplitudy 𝑘\n𝑚 masa wahadła\n𝑚𝑜𝑑𝑤masa odważnika\n𝑛 wartość okresowości drgań wahadła w jednym dołku potencjału\n𝑝 parametr modelu wielomianowego momentu magnetycznego\n𝑝𝐼prądowy współczynnik kierunkowy\n𝑞 parametr modelu wielomianowego momentu magnetycznego\n𝑞𝑚1;\n𝑞𝑚2wartości ładunków odseparowanych biegunów dipola magne-\ntycznego\n𝑞𝑡współczynnik skalujący czas\n𝑟 odległość pomiędzy biegunami dipola magnetycznego\n6SPIS TREŚCI\n𝑠;𝑠1,𝑠2odległośćpomiędzyśrodkiemciężkościwahadłaajegoosiąobro-\ntu;odległośćpomiędzyśrodkiemciężkościwahadła1i2aosiami\nobrotu\n𝑡 czas\n𝑡0 — parametrmodeluprostokątnegopulsującegosygnałuprądowego\n𝑡𝑚𝑎𝑥𝑂𝐹𝐹 czas przez jaki cewka powinna być niezasilana przy maksymal \nnym wypełnieniu sygnału prądowego, aby spełniony został ruch\nukładu według scenariusza I\n𝑡𝑚𝑎𝑥𝑂𝑁 — czas przez jaki cewka powinna być zasilana przy maksymalnym\nwypełnieniusygnałuprądowego,abyspełnionyzostałruchukła \ndu według scenariusza I\n𝑡𝑚𝑖𝑛𝑂𝐹𝐹 czasprzezjakicewkapowinnabyćniezasilanaprzyminimalnym\nwypełnieniusygnałuprądowego,abyspełnionyzostałruchukła-\ndu według scenariusza I\n𝑡𝑚𝑖𝑛𝑂𝑁 czasprzezjakicewkapowinnabyćzasilanaprzyminimalnymwy \npełnieniu sygnału prądowego, aby spełniony został ruch układu\nwedług scenariusza I\n𝑢 przesunięcie fazowe rozwiązania opisującego ruch drgający wa \nhadła\n⟨𝑢⟩— wartość średnia przesunięci fazowego 𝑢\n𝑣𝑗,𝑣1,\n𝑣2prędkość kątowa wahadła 1 i 2, ( 𝑗= 1,2)\n𝑤 — wypełnienieprostokątnegopulsującegosygnałuprądowegocew-\nki\n𝑤𝑚𝑎𝑥maksymalnawartośćwypełnieniasygnałuprądowegodającajed \nnookresowe oscylacje wahadła według scenariusza I\n𝑤𝑚𝑖𝑛— minimalna wartość wypełnienia sygnału prądowego dająca jed-\nnookresowe oscylacje wahadła według scenariusza I\n𝑤𝑛𝑇wartość wypełnienia sygnału prądowego dająca wielookresowe\noscylacje wahadła według scenariusza I\n∆ przesunięcie fazowe między wahadłem 1 i 2\n∆0położenie stacjonarne przesunięcia fazowego\n∆′\n𝑢pochodna po czasie przesunięcia fazowego ∆\nΩ częstość drgań własnych zlienaryzowanego wahadła\nΩ𝑛𝑇okres rozwiązania wielookresowego dla wahadła drgającego w\njednym dołku potencjału\nΩ𝑡częstość drgań rozwiązania opisującego ruch drgający wahadła\n7SPIS TREŚCI\n𝛼 znormalizowany parametr tłumienia wiskotycznego stalowej\nsprężyny\n𝛽 znormalizowany parametr sztywności stalowej sprężyny\n𝛿 \"szybka\" faza\n𝛿′\n𝑢— pochodna po czasie \"szybkiej\" fazy 𝛿\n𝜖 mały parametr zaburzenia\n𝜀𝐼parametr regularyzacyjny w modelu prostokątnego pulsującego\nsygnału prądowego\n𝜀𝑐parametr regularyzacyjny\n𝜁1,𝜁2znormalizowane parametry oporów ruchu wahadeł 1 i 2\n𝜂 — numeryczny parametr\n𝜃 faza rozwiązania opisującego ruch drgający wahadła\n𝜅 parametr modelu aproksymującego momentu oporów ruchu z\nefektem Stribecka\n𝜆1,𝜆2,\n𝜆3,𝜆4wykładniki Lapunowa\n𝜆𝑎1znormalizowana energia wahadła 1\n𝜆𝑎2znormalizowana energia wahadła 2\n𝜇 parametr modelu aproksymującego momentu oporów ruchu z\nefektem Stribecka\n𝜈𝑠— prędkość Stribecka\n𝜈𝑤parametr zależny od wypełnienia prostokątnego pulsującego sy-\ngnału prądowego\n𝜎 — parametr modelu aproksymującego momentu oporów ruchu z\nefektem Stribecka\n𝜏 okres prostokątnego pulsującego sygnału prądowego cewki\n𝜏𝑂𝐹𝐹czas przez jaki cewka powinna być niezasilana, aby otrzymać\nrozwiązanie wielookresowe według scenariusza I\n𝜏𝑂𝑁czas przez jaki cewka powinna być zasilana, aby otrzymać roz \nwiązanie wielookresowe według scenariusza I\n𝜏𝑡— bezwymiarowy czas\n𝜏𝑤czasstanuniskiegoprostokątnegopulsującegosygnałuprądowe-\ngo cewki\n𝜏𝑧— czasstanuwysokiegoprostokątnegopulsującegosygnałuprądo-\nwego cewki\n8SPIS TREŚCI\n𝜙 przesunięcie fazowe rozwiązania harmonicznego\n𝜑;𝜑𝑗,\n𝜑1,𝜑2położeniekątowewahadłamagnetycznego;położeniekątowewa \nhadła 1 i 2 ( 𝑗= 1,2)\n𝜑0amplitudawychyleniawahadłapodczasoscylacjiwjednymdołku\npotencjału (kąt początkowy)\n𝜑𝐴kątgranicznywystępowaniastrefyaktywnejpolamagnetycznego\n𝜑𝑆kąt położenia równowagi wahadła znajdującego się w stanie 2\n𝜑𝑘— kąt wychylenia wahadła, przy którym następuje włączenie zasi-\nlania cewki i zmiana stanu układu z 1 na 2\n𝜑𝑙𝑘kątwychyleniawahadła,przyktórymnastępujewyłączeniezasi \nlania cewki i zmiana stanu układu z 2 na 1\n𝜑𝑚𝑎𝑥— maksymalna wartość wychylenia kątowego wahadła\n¤𝜑;¤𝜑𝑗,\n¤𝜑1,¤𝜑2prędkość kątowa wahadła; prędkość kątowa wahadeł 1 i 2 ( 𝑗=\n1,2)\n𝜑′;𝜑”— prędkość kątowa; przyspieszenie kątowe wahadła przy bezwy-\nmiarowym czasie\n𝜒 parametr regularyzacyjny\n𝜔,𝜔1,\n𝜔2— częstości drgań własnych zachowawczego zlinearyzowanego\nukładu sprzężonych wahadeł\n𝜔0początkowa prędkość kątowa wahadła podczas oscylacji w jed \nnym dołku potencjału\n𝜔𝑖— częstość kołowa funkcji sinusoidalnej w modelu prostokątnego\npulsującego sygnału prądowego\n𝜔𝑘prędkość kątowa wahadła, przy której następuje włączenie zasi \nlania cewki i zmiana stanu układu z 1 na 2\n𝜔𝑙𝑘prędkośćkątowawahadła,przyktórymnastępujewyłączenieza \nsilania cewki i zmiana stanu układu z 2 na 1\n𝓁 długośćramieniasiłyoddziaływaniamagnetycznegomiędzybie-\ngunami dipola magnetycznego\n9Streszczenie\nPrzygotowanarozprawadoktorskapoświęconajestbadaniomdynamikiukładów\nzłożonychzwahadełmagnetycznychpoddanychdziałaniuniestacjonarnegopola\nmagnetycznego.Przezwahadłomagnetycznerozumiesięwahadłofizycznezma-\ngnesem zamocowanym na jego końcu, które umieszczone jest w zewnętrznym\npolu magnetycznym. Niestacjonarne zewnętrzne pole magnetyczne generowane\njest przez cewkę elektryczną umieszczoną pod wahadłem i zasilaną zmiennym\nwczasiesygnałemprądowym.Przeprowadzonebadaniadotycząprzedewszyst-\nkim drgań pojedynczego wahadła magnetycznego odbywających się w jednym\n„dołku”potencjałuorazsterowaniaprzepływemenergiimiędzydwomasprzężo-\nnymi skrętnie wahadłami magnetycznymi.\nWpierwszymrozdzialepracydokonanoprzegląduliteraturydotyczącejdrgań\nukładów mechanicznych wykorzystujących pole magnetyczne. Przegląd ten zo-\nstał podzielony na dwie części, z której pierwsza dotyczyła układów o jednym\nstopniuswobody,natomiastdrugaukładówowielustopniachswobody.Omawia-\nnepracestaranosięuporządkowaćzewzględunachronologięorazpodobieństwo\nanalizowanych problemów badawczych. Badania te najczęściej dotyczyły waha-\ndeł magnetycznych znajdujących się w stacjonarnym polu magnetycznym, a jeśli\nnie, to niestacjonarne pole magnetyczne generowane było przez sinusoidalny sy-\ngnałprądowypłynącywcewceelektrycznej.Przeglądliteraturywskazujenabrak\nbadań naukowych dotyczących sterowania przepływem energii między sprzężo-\nnymiwahadłamimagnetycznymipoddanymioddziaływaniupólmagnetycznych.\nCelem naukowym pracy jest opracowanie nowych modeli matematycznych\ndla pojedynczego wahadła magnetycznego oraz układu dwóch skrętnie sprzę-\nżonych wahadeł magnetycznych, a także zbadanie ich dynamiki nieliniowej pod\nwzględempraktycznegozastosowaniauzyskanychwyników.Wpracypostawio-\nno dwie tezy badawcze. Pierwsza odnosi się do przypadku drgań okresowych\npojedynczego wahadła magnetycznego odbywających się w jednym „dołku” po-\ntencjału,dla którychmożliwejest zmienianieokresowościtych drgańbezwyraź-\nnego naruszenia przebiegu ich trajektorii fazowej. Zmiana okresowości odbywa\nsię poprzez zmianę częstotliwości sygnału prądowego cewki elektrycznej. Okre-\nsowośćdrgańwahadłarozumianajestjakoliczbaokresówwymuszenia(sygnału\nprądowego)przypadającanajedenokresdrgańwahadła.Drugatezawskazujena\nmożliwość kontrolowania przepływu energii pomiędzy sprzężonymi wahadła-\nmimagnetycznymiprzyużyciupolamagnetycznegogenerowanegoprzezcewki\nelektryczne znajdujące się pod nimi.\nOpracowanomodeledynamiczneukładówojednymidwóchstopniachswo-\nbody wykorzystujące różne modele oporów ruchu oraz empiryczne modele od \n10SPIS TREŚCI\ndziaływaniamagnetycznego.Badaniadynamikiprzeprowadzonowykorzystując\nmetodynumeryczneorazanalityczno-numeryczneopartenametodzieuśrednia-\nnia. Analiza oparta na klasycznych metodach stosowanych w dynamice nielinio-\nwej pozwoliła na wykrycie bogatej dynamiki obu układów z uwzględnieniem\ndrgań chaotycznych i quasiokresowych, co zostało potwierdzone w sposób eks-\nperymentalny na specjalnie zbudowanych stanowiskach badawczych.\nPrzeprowadzona analiza numeryczna oraz teoretyczna udowodniła pierw-\nszą z postawionych tez badawczych. Badania wykazały, że w przypadku drgań\nokresowych pojedynczego wahadła magnetycznego odbywających się w jednym\n„dołku” potencjału, możliwe jest zmienianie okresowości tych drgań bez wyraź-\nnegonaruszeniaprzebieguichtrajektoriifazowej.Zachowanietojestmożliweze\nwzględunaszczególnycharakteroddziaływaniamagnetycznegoukładuiistnie-\nnie tzw. strefy aktywnej (strefy faktycznego oddziaływania magnetycznego pary\ncewka magnes).\nWprzypadkusprzężonychwahadełzaproponowanodwiemetodysterowania\nprzepływemenergiimiędzynimi:otwartąbezsprzężeniazwrotnegoizamkniętą\nze sprzężeniem zwrotnym. Podczas drgań w antyfazie energia przemieszcza się\nzwahadłapoddanegoodpychającemupolumagnetycznemudowahadłaznajdu-\njącego się w przyciągającym polu. Natomiast w przypadku drgań w fazie prze-\npływ energii jest odwrotny. Na podstawie obserwacji oraz analizy numerycznej\ndynamikiukładuwykazano,żepodczassterowaniabezsprężeniazwrotnegoko-\nniecznymjestznanieaprioritypudrgańwahadeł(tj.wfazielubwantyfazie)oraz\nczasu po jakim energia układu zostanie całkowicie rozproszona. W przypadku\nsterowania ze sprzężeniem zwrotnym koniecznym jest określenie przesunięcia\nfazowego między wahadłami na podstawie pomiaru ich położeń. Przedstawio-\nne badania numeryczne oraz eksperymentalne pokazały, że przy odpowiednim\nsterowaniu polami magnetycznymi cewek, a z mechanicznego punktu widze-\nnia nieliniową sztywnością poszczególnych wahadeł, możliwe jest zapewnienie\nkierunkowegotransferuenergiimiędzysprzężonymiwahadłami.Badaniateudo-\nwodniły drugą z postawionych tez badawczych.\nBadania przeprowadzone dla pojedynczego wahadła magnetycznego mogą\nstanowić podstawy do nowego sposobu modelowania silników krokowych. Do-\ndatkowo opracowane metody sposobu sterowania przepływem energii w ukła-\ndach połączonych wahadeł magnetycznych, mogą stanowić bazę do przyszłych\nbadańwzakresietłumieniadrgańbądźodzyskiwaniaenergiizdrgającychstruk-\ntur składających się z łańcuchów oscylatorów.\n11Abstract\nTheprepareddoctoraldissertationfocusesonstudyingdynamicsofsystemscom-\nposedofmagneticpendulumssubjectedtoanon-stationarymagneticfield.Ama-\ngneticpendulumisaphysicalpendulumwithamagnetattachedtoitsendandis\nplacedinanexternalmagneticfield.Thenon-stationaryexternalmagneticfieldis\ngenerated by anelectric coil placed under thependulum and powered by atime-\nvarying current signal. The presented research mainly concerns the oscillations\nin one potential well of a single magnetic pendulum, as well as the control of the\nenergy flow between two torsionally coupled magnetic pendulums.\nIn the theoretical introduction, a review of works on mechanical systems\nusing a magnetic field was carried out. This introduction was divided into two\nparts,thefirstonedescribedsystemswithonedegreeoffreedom,andthesecond\none concerned systems with multiple degrees of freedom. The discussed papers\nwere organized in terms of chronology and similarity of the analyzed research\nproblems.Theanalyzedstudiesaremostlyaimedatmagneticpendulumsplaced\nin a stationary magnetic field, and if not, a non stationary magnetic field was\nproduced by the sinusoidal current signal flowing in an electric coil. In addition,\nthere are no scientific studies focused on the control of the energy flow between\ncoupled magnetic pendulums as a result of generating magnetic fields in their\nvicinity.\nThe scientific goal of the work is to develop new mathematical models for\na single magnetic pendulum and a system of two torsionally coupled magnetic\npendulums, as well as to study their non-linear dynamics in terms of the ap-\nplicability of the obtained results. Two research theses were formulated in the\ndissertation.Thefirstoneclaimsthatinthecaseofperiodicoscillationsofasingle\nmagnetic pendulum taking place in one potential well, it is possible to change\ntheperiodicityoftheseoscillationswithoutdisturbingtheirphasetrajectory.The\nchangeinperiodicityiscausedbyachangeinthefrequencyofthecurrentsignal\nflowing in the electric coil. The periodicity of pendulum oscillations is understo-\nod in a classical way, i.e. as the number of excitation periods (current signal) per\none period of pendulum oscillation. The second thesis indicates that it is possi-\nble to control the energy flow between coupled magnetic pendulums using the\nmagnetic field generated by electric coils located underneath them.\nMathematicalmodelsofsingleandcoupledpendulumssystemsweredevelo-\nped.Variousmodelsofmotionresistanceaswellasempiricalmodelsofmagnetic\ninteraction were tested. Dynamics studies were carried out using numerical and\nsemi analytical methods based on the averaging method. Using classical me-\nthods appliedin non linear dynamics,a basic dynamicanalysis was obtainedfor\n12SPIS TREŚCI\nboth systems, showing their rich dynamics including chaotic or quasi periodic\nphenomena. These behaviors were confirmed experimentally on specially built\nexperimental rigs.\nNumerical and theoretical analysis explained why a significant change in the\ndutycycleandfrequencyof therectangularcurrentsignaldoesnothavetoaffect\nthe phase trajectory of the one well oscillations of a single magnetic pendulum.\nItwasconcludedthatduetothespecialnatureofthemagneticexcitationandthe\nexistence of the so-called active zone (the zone of the actual magnetic interaction\nof the coil magnet pair), it is possible to change the periodicity of pendulum\noscillation taking place in one potential well while phase trajectory remains the\nsame.\nIn the case of coupled pendulums, two methods of controlling the energy\nflow between them were proposed: open (without feedback) and closed (with\nfeedback). During an antiphase oscillation, energy moves from a pendulum in\na repulsive magnetic field towards a pendulum in an attractive field. In contrast,\nduringin-phaseoscillations,theflowofenergyisreversed.Basedontheobserva-\ntions and numerical analysis of the system dynamics, it was found that during\ncontrol without feedback, it is necessary to know a priori the type of pendulum\noscillation (i.e. in-phase or antiphase) and the time after which the energy of the\nsystem will be completely dissipated. However, in the case of control with fe-\nedback, it is necessary to determine the phase shift between the pendulums via\nmeasurements of their positions. Presented numerical and experimental studies\nshown that with appropriate control of the coil magnetic fields (from the me-\nchanical point of view, the non-linear stiffness of each pendulums is controlled),\nit is possible to provide directional energy transfer between coupled oscillators.\nThe studies carried out for a single magnetic pendulum can be the basis for\nanewmethodofmodellingsteppermotors.Additionally,thedevelopedmethods\nofcontrollingtheenergyflowinsystemsofconnectedmagneticpendulumscanbe\nthebasisforfutureresearchinthefieldofvibrationdampingorenergyharvesting\nof structures consisting of oscillator chains.\n13Rozdział 1\nWstęp\nW teorii mechaniki rozróżnia się dwa rodzaje wahadeł: wahadło matematyczne\n(proste) i wahadło fizyczne. Pierwsze z nich definiowane jest jako punkt mate-\nrialny zawieszony na idealnie wiotkiej, nierozciągliwej i nieważkiej nici (pręcie)\nmogący wykonywać ruchy w płaszczyźnie pionowej. Drugie rozumiane jest jako\nciało mogące wykonywać swobodny ruch obrotowy dookoła poziomej osi nie-\nprzechodzącej przez środek masy tego ciała [1, 2]. Wynika stąd, że każde ciało\nwrzeczywistościmożepełnićrolęwahadłafizycznego,natomiastwahadłomate-\nmatyczne może stanowić jego wyidealizowany model.\nWahadła towarzyszą ludziom od wieków, chociażby pod postacią huśtawki\ndo zabawy. Jednakże w przeszłości oprócz bycia zabawkami potrafiły one pełnić\nrównież bardziej zaawansowane i użytkowe role. W starożytnych Chinach (ok.\n132 r.) zaprojektowano mechanizm oparty na wahadle, którego zadaniem było\nostrzeganieludziozbliżającymsiętrzęsieniuziemi[3,4].Mechanizmpełniłwięc\nrolę urządzenia nazywanego dzisiaj sejsmografem i został uznany jako jedno\nz największych osiągnięcie technologicznych tamtej epoki.\nNajwiększą popularność wahadło zyskało jednak jako element wykorzysty-\nwany do pomiaru czasu. Jednym z pierwszych badaczy, który zainteresował się\nteorią ruchu wahadeł był Galileusz. Zaobserwował on i opisał przybliżony izo-\nchronizm 1wahań i na tej podstawie w 1637 r. wykorzystał go do odmierzania\nczasu. Próby budowy zegara przez Galileusza oraz jego syna odniosły fiasko ze\nwzględunatrudnościwykonaniamechanizmuwyzwalania[5].Dopierow1673r.\nholenderski uczony C. Huygens opracował pierwszy projekt mechanizmu przy-\npominającego ten, który znajduje się we współczesnych zegarach wahadłowych\n[5, 6].\nObserwacje ruchu wahadła przysłużyły się również do potwierdzenia teorii\nobrotu ziemi dookoła własnej osi. W 1851 r. francuski fizyk Foucault przymo-\ncował do kopuły Panteonu w Paryżu wahadło, którego ramię miało długość 67\nmetrów, a na jego końcu zamocowano mosiężną kulę o masie 28 kilogramów.\nTrajektorie wykreślone przez poruszającą się kulę dały do zrozumienia, że pio-\nnowapłaszczyznawahańzmieniaswojepołożeniewzględemZieminaskutekjej\nobrotu [7].\n1izochronizm [gr.], własność układu drgającego polegająca na niezależność okresu\ndrgań własnych od amplitudy tych drgań.\n141.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\nObecnie wahadła wykorzystywane są np. jako tłumiki drgań wysokich bu-\ndynków, które szczególnie podatne są na działanie obciążeń dynamicznych po-\nchodzących od wiatru czy trzęsień ziemi. Największe drapacze chmur takie jak\nTaipei 101 (509,2 m) czy Burj Al Arab (321 m) posiadają w swoich konstrukcjach\nogromne wahadła mające tłumić ich drgania wywołane wyżej wymienionymi\nczynnikami środowiskowymi [8, 9].\nWahadłostanowiprzykładnajprostszegooscylatoranieliniowego,azewzglę-\ndu na swoja prostą budowę jest ono wykorzystywane przez naukowców w ba-\ndaniach empirycznych jako analogia innych oscylatorów spotykanych w fizyce,\nnp. sprężyn z nieliniowościami [10], drgających atomów [11] czy zjawiska tune-\nlowania Josephsona występującego między nadprzewodnikami lub w układach\nelektrycznych [5, 12, 13].\nZewzględunaszybkirozwójtechnologiiichęćtworzeniapołączeńsynergicz-\nnych pomiędzy różnymi dziedzinami techniki, coraz więcej układów czysto me-\nchanicznychpoddawanychjestdziałaniupolaelektromagnetycznego.Najbardziej\npowszechnym przykładem takiego urządzenia mechatronicznego, które zrewo-\nlucjonizowało przemysł jest silnik elektryczny [14, 15]. W silniku elektrycznym\nruch wirnika spowodowany jest powstaniem siły oddziaływania magnetyczne-\ngopomiędzyuzwojeniemwzbudzeniaauzwojeniemtwornika.Układyprostych\nwahadełmechanicznychrównieżpoddanorozwojowitechnologicznemu.Odpo-\nczątku pierwszej połowy XX wieku zaczęły pojawiać się pierwsze teoretyczne\nprace naukowe, których przedmiotem badań były wahadła wykonane z ferro-\nmagnetyków bądź wyposażone w magnesy, a następnie umieszczane w polu\nmagnetycznymgenerowanymprzezcewkęelektrycznąlubinnymagnes.Układy\ntego rodzaju otrzymały nazwę wahadeł magnetycznych [16].\nPrzedmiotembadańniniejszejpracyjestukładpojedynczegowahadłamagnetycznego\noraz układ dwóch torsyjnie sprzężonych wahadeł magnetycznych. Przeprowadzone bada-\nniaobejmujązarównosymulacjenumerycznejakieksperymenty.Wwynikubadańopra-\ncowane zostały modele matematyczne wyżej wymienionych układów z uwzględnieniem\nużytkowychmodelioddziaływaniamagnetycznego,atakżewykazanoistnienieciekawych\nzjawisk dynamicznych oraz procesów, które mogą znaleźć zastosowanie w technice.\n1.1 Dotychczasowy stan wiedzy o układach\nwahadeł magnetycznych\nPrzedstawiona rozprawa poświęcona jest dynamice układu pojedynczego waha-\ndła magnetycznego (układ o jednym stopniu swobody) i układu dwóch sprzężo-\nnychwahadełmagnetycznych(układodwóchstopniachswobody).Jakjużzostało\nwspomniane wcześniej we Wstępie, wahadło magnetyczne to wahadło fizyczne\nz zamocowanym na końcu ramienia magnesem lub ferromagnetykiem i umiesz-\nczonewzewnętrznympolumagnetycznympochodzącymodinnegomagnesulub\ncewki elektrycznej. Ze względu na różne stopnie swobody układów analizowa-\nnychwpracy,przeglądliteraturyzawartywtympodrozdzialezostałpodzielony\nna układy o jednym i wielu stopniach swobody.\n151.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\n1.1.1 Układy o jednym stopniu swobody\nJedną z najstarszych prac jaką udało się znaleźć autorowi dysertacji na temat\npojedynczego wahadła magnetycznego jest praca francuskiego inżyniera J. Be-\nthenoda [17]. Przedmiotem jego badań teoretycznych było wahadło, na końcu\nktóregozamocowanoferromagnetycznąkulkę.Podwahadłemumieszczonocew-\nkę elektryczną, której oś rdzenia pokrywała się z osią wahadła w spoczynku.\nCewka zasilana była prądem o wysokiej częstotliwości. Badany układ został roz-\ndzielony na część elektryczną i mechaniczną. Układ elektryczny opisany został\npoprzezzwyczajnerównanieróżniczkowepierwszegorzędu,natomiastrównanie\nruchu wahadła zamodelowano poprzez zlinearyzowane równanie różniczkowe\ndrugiegorzędu.Czynnikiemsprzęgającymdwaukładybyłaindukcyjnośćcewki\nbędąca funkcją wychylenia wahadła. Jego badania wykazały, że istnieje pewien\nprógnapięciaorazczęstotliwośćprąducewki,dlaktórychwahadłobędziewyko-\nnywało drgania o zadanej amplitudzie. Doprowadziło to do wniosku, że układ\nten mógłby służyć jako „transformator” zamieniający drgania elektryczne o wy-\nsokiej częstotliwości na drgania mechaniczne o niskiej częstotliwości. Badania te\nbyły jednak przeprowadzone z dużą liczbą przybliżeń oraz rozpatrywały tylko\nkilkawybranychprzypadkówruchuwstosunkudoprzyjętychparametrówukła-\ndu. W kolejnych latach pojawiły się prace rozszerzające nieznacznie te badania\n[18, 19]. Autorzy tych prac przybliżyli indukcyjność cewki szeregami potęgowy-\nmi, tworząc proste nieparzyste funkcje zależne od położenia wahadła. Ponadto,\npodali dodatkowe warunki jakie muszą spełniać parametry układu (rezystancja,\npojemność, indukcyjność, tłumienie wiskotyczne, częstość prądu, częstość wła-\nsna wahadła), aby drgania utrzymywały zadaną amplitudę. Inny pogląd na ten\nproblem zaprezentowany został przez Minorsky’iego [20–22], który zbadał sy-\nmetrycznyrozkładindukcyjnościcewkiwzględempołożeniazerowegowahadła.\nIntuicyjnie sprowadził on problem do równania Mathieu zakładając, że parame-\ntrem zmieniającym się periodycznie w czasie jest długość wahadła. Pozwoliło\ntonawyznaczeniewystarczającegowarunkuwystępowaniastacjonarnychdrgań\nwahadła.\nPraca Minorsky’iego spotkała się jednak z krytyką ze strony Kesavamurthy\niin.[23].AutorzyzarzuciliMinorsky’iemu,żebłędniesformułowałrównanieprą-\ndu cewki poprzez pominięcie składowych o wysokich częstotliwościach oraz, że\nstosującrównanieMathieuniemógłpoprawniezbadaćzjawiskaindukowaniasię\ndodatniego tłumienia w przypadku dominującej rezystancji układu elektryczne-\ngo. Zaproponowali oni również paraboliczny rozkład indukcyjności cewki, a na-\nstępnie przeprowadzili analizę teoretyczną i eksperymentalną ruchu wahadła.\nW latach 70. i 80. XX wieku pojawiło się kilka prac, które bezpośrednio nie\ndotyczyływahadełmagnetycznych,alebyłyichanalogicznymiodpowiednikami.\nWpracy[24]skupionosięnaopracowaniuogólnegosposobuanalizydrgańpara-\nmetrycznych urządzeń elektromechanicznych, w których występują siły magne-\ntyczne.JakoprzykładytychurządzeńpodanorezonatoryVHFużywanedoprzy-\nspieszania cząsteczek oraz obrotowy parametryczny silnik elektryczny. W rezul-\ntacie wykazano, że układy te maja tendencję do podtrzymywania swoich drgań.\nDopodobnychwnioskówdoszedłautorpracy[25]badającteoretycznieiekspery-\nmentalnie silnik reluktancyjny opisany nieliniowym równaniem różniczkowym\n161.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\ndrugiego rzędu z okresowo zmiennymi parametrami. Russel i in. [26] zbadali\njakie warunki musi spełniać taki silnik, aby mógł poprawnie pracować oraz wy-\nznaczylijegooptymalneczęstotliwościpracy.Blakley[27]natomiastskupiłsięna\nzachowaniu energii w tego rodzaju układach. Zauważył on, że energia pomię-\ndzy układem elektrycznym a mechanicznym wymieniana jest tylko w wąskim\nzakresie ich wzajemnego położenia, dlatego transfer energii pomiędzy układami\naproksymował przy pomocy impulsu. Pozwoliło to na zapisanie układu nieli-\nniowychnieautonomicznychrównańruchuwpostaciliniowegoautonomicznego\nrównaniadrugiegorzęduiwykreślenieprzybliżonychtrajektoriinapłaszczyźnie\nfazowej.\nWystępowanie nieokresowego ruch w układzie wahadła magnetycznego za-\nprezentowane zostało przez Moona i in. [28, 29]. Badane przez nich wahadło\nmagnetyczne składało się z ferromagnetycznej belki wspornikowej umieszczo-\nnej pomiędzy dwoma magnesami, której ruch wymuszany był przez harmo-\nniczne przemieszczanie się całego układu w poziomie. Umieszczenie wahadła\npomiędzymagnesamigenerowałosymetrycznydwudołkowypotencjałpolama-\ngnetycznego. W wyniku przeskakiwania układu pomiędzy dwoma lub trzema\nstanami równowagi wykazywał on zachowania chaotyczne potwierdzone teore-\ntycznieieksperymentalnie,atakżeautorzywykrylidziwneatraktorychaotyczne\nwraz z ich fraktalną budową. Zmodyfikowany układ takiego wahadła analizo-\nwano w pracy [30]. Autorzy tej pracy opisali ruch układu równaniem Duffinga\ni wprowadzić zmienną magnetyzację belki modelując ją nieliniowością Preisa-\ncha. W wyniku badań numerycznych doszli do wniosku, że histereza w postaci\nnieliniowościPreisachawwiększościprzypadkówodgrywałarolędodatkowego\nczynnika tłumiącego. Radons i in. [31] również analizowali prototypowy układ\nwahadłamagnetycznegocharakteryzującysięzłożonąnaturąhisterezowąopartą\nnaoperatorzePreisacha.Układtenmiałsymulowaćzmianyzachodzącewmikro-\nstrukturze materiałów magnetycznych, stopów z pamięcią kształtu i materiałów\nporowatych. Wyniki badań numerycznych wykazały, że regularne i chaotyczne\nzachowaniaukładuwykazująfraktalnezależnościodparametrów.Strukturafrak-\ntalna występuje również w zależności od warunków początkowych, co według\nautorów pracy wydaje się być unikatową cechą dla tego rodzaju układów. Ku-\nmar i in. [32] uznali, że zwykłe równanie Duffinga nie uwzględnia wszystkich\nnieliniowych efektów oddziaływania pomiędzy magnesami a wahadłem magne-\ntycznymwanalizowanymukładzie.Dlategoopracowalianalitycznymodeltakiej\ninterakcji, zapewniając możliwość zbadania wpływu nieliniowych efektów szó-\nstego stopnia na dynamikę układu. Przeprowadzili badania numeryczne oraz\neksperymentalne w celu wyjaśnienia bifurkacji położeń równowagi. Następnie,\ninformacje pozyskane z analizy bifurkacyjnej zostały wykorzystane do śledze-\nnia zmian konfiguracji stabilności oscylatora z monostabilnego do tristabilnego,\nz tristabilnego do bistabilnego, itd.\nSiahmakoun [33] także zaobserwował dwudołkowy charakter potencjału ba-\ndanegoprzezsiebiewahadłamagnetycznego,którewymuszanebyłododatkowo\nzewnętrzną siłą harmoniczną. Eksperymentalne i numeryczne portrety fazowe\nwykazały istnienie atraktorów jednodołkowych (odpowiadającym prawemu i le-\nwemudołkowipotencjału),atakżeatraktorówmiędzydołkowych[34,35].Wwy-\nnikach pojawiły się również typowe dla układów nieliniowych skoki amplitudy\n171.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\nihisterezarezonansowa.Kwuimyiin.[36]rozważaliwpływasymetrycznegoroz \nkładupotencjałunadynamikęwahadłamagnetycznego.Asymetrycznypotencjał\nuzyskano dzięki umieszczeniu wahadła pomiędzy magnesami na równi pochy-\nłej, której powierzchnia poruszała się ruchem harmonicznym. Stosując metodę\nMielnikowa wyprowadzili analitycznie warunek na przejście dynamiki układu\nzokresowejnachaotyczną.Wynikitychanalitycznychprzewidywańzostałyprze-\ntestowane i zweryfikowane poprzez analizę fraktalnych i regularnych kształtów\nbasenów przyciągania. Baseny o regularnym kształcie wskazywały na okreso-\nwą dynamikę układu, podczas gdy baseny o nieregularnym (fraktalnym) kształ-\ncie związane były z dynamiką nieokresową. Kryterium energetyczne „ucieczki”\nwahadła magnetycznego z asymetrycznych studni potencjału podano w pracy\n[37]. Procedura zawierała określenie poziomu energii jaką musi osiągnąć układ\ndo przedostania się przez lokalną barierę potencjału. Ze względu na asymetrię\npotencjału, konieczne były obliczenia kryterium „ucieczki” dla każdej z dwóch\nróżnychstudnitegopotencjału.Analityczneieksperymentalnebadaniadynamiki\nnieliniowejwahadłamagnetycznegoumieszczonegomiędzydwomamagnesami\niwymuszanegosilnikiemDCprzedstawionowpracach[38,39].Autorzyanalizo-\nwaliokresoweichaotyczneruchywodniesieniudoróżnychwartościparametrów\nkontrolnych takich jak amplituda i częstotliwość zewnętrznego wymuszenia. Na\nwykresach bifurkacyjnych widoczne było, że chaos w układzie powstaje poprzez\npodwajanie się okresu. Wskazano, że układ ten mógłby znaleźć zastosowanie\nwmieszaniucząstekmaterii[40],gdziewystępowaniedynamikichaotycznejpo-\nzwoliłoby na równomierne rozproszenie cząstek i uniknięcie kumulowania się\nw jednym miejscu.\nTeoretyczne,numeryczneieksperymentalnebadanianadchaosemtłumione-\ngo wahadła magnetycznego przeprowadził Khomeriki [41]. Wykorzystując rów-\nnanieMathieudoopisudynamikiukładuwyznaczyłgraniceistnieniarezonansu\nparametrycznego,atakżewykazałwoparciuoobliczeniewartościwykładników\nLapunowa, że w jego układzie niestabilność parametryczna zawsze prowadzi do\nzachowań chaotycznych.\nElementysterowaniaruchemwahadłamagnetycznegozostałyprzedstawione\nwpracach[42,43].Sterowaniepolegałonaodpowiednimzasilaniudwóchcewek\numieszczonych w dolnym i górnym położeniu równowagi wahadła magnetycz-\nnego w zależności od jego położenia kątowego. Eksperymentalne i teoretyczne\nbadania dotyczyły drgań samowzbudnych, wymuszenia parametrycznego oraz\ndrgań w trzech studniach potencjału. Dzięki wprowadzeniu sterowania z dodat \nnim i ujemnym sprzężeniem zwrotnym zbadane zostały drgania samowzbudne,\ngdzie straty energii kompensowane były energią pochodzącą od pola magne-\ntycznego. Do analizy teoretycznej tego zjawiska użyto równania Van der Pola.\nZjawisko wymuszenia parametrycznego wytłumaczono opierając się na równa-\nniuMathieuzezmiennąsztywnościąukładu.Przyodpychającymoddziaływaniu\npomiędzy cewką a magnesem, dla dużych wartości prądu cewki obserwowano\ndrgania w dwóch studniach potencjału, natomiast dla małych wartości prądu\ndrgania odbywały się w trzech studniach potencjału.\nTran i in. [44] wykryli dwa atraktory chaotyczne w układzie wymuszanego\nwahadła magnetycznego oraz poprzez porównanie danych eksperymentalnych\nz numerycznymi zademonstrowali jakościowe i ilościowe zdolności przewidy-\n181.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\nwania struktury atraktora chaotycznego poza zakresami ruchu opracowanego\nmodelu układu. Wykazali też zależność wartości parametrów układu od często-\ntliwościjegowymuszeniaorazwykazali,żezjawiskotowystępujewprzypadku,\ngdy model układu zawiera tarcie suche.\nWahadło magnetyczne dzięki występowaniu zachowań chaotycznych speł-\nnia obecnie rolę taniego, popularnego i łatwego w budowie układu do badań\nnaukowych i edukacyjnych. W pracach [45–49] przedstawione zostały szczegóły\ndotyczące budowy takich stanowisk oraz programów do analizy symulacyjnej\ni doświadczalnej. Wahadła z oddziaływaniem magnetycznym znalazły również\nzastosowaniejakoaktuatory[50]mającenaceluwykonywanieruchuwahadłowe-\ngozróżnymiograniczeniamiiwymogamiokreślonymiprzezoperatora.Tsubono\ni in. [51] zaprojektowali potrójne wahadło magnetyczne jako system wibroizola-\ncyjny dla interferometru laserowego. W tym przypadku autorzy skupili się na\ntłumieniu drgań ramion wahadeł siłami magnetycznymi, a nie na ich wymusza-\nniu.OpierającsięnareguleLenzaizjawiskuindukcjiprądówwirowychwmetalu\nporuszającym się w sąsiedztwie magnesu opracowali formułę tłumienia magne-\ntycznego. Eksperymentalne i obliczeniowe badania udowodniły dobre własności\ntłumiące układu, zapobiegające niepożądanemu zjawisku sprzęgania się piono-\nwego i poziomego ruchu luster interferometru.\nStanowiskawahadełmagnetycznychużywanesąrównieżdoweryfikacjispeł-\nnieniaprawaindukcjielektromagnetycznej.Jangiin.[52]wyprowadzilianalitycz-\nnie dwie funkcje aproksymujące indukcję magnetyczną dla dowolnego punktu\nz otoczenia metalowej obręczy oraz solenoidu, w których płynie prąd. Następ-\nnie opierając się na tych funkcjach, wyznaczyli siły magnetyczne działające na\nmagnes przemieszczający się nad obręczą lub solenoidem. Obliczyli też moment\nparysiłgenerowanypomiędzymagnesemacewkąznajdującychsięwpołożeniu\nniewspółosiowym. Do sprawdzenie poprawności swoich obliczeń wykorzysta-\nli model wahadła magnetycznego, gdzie doświadczalnie [53] wykonali pomiary\nnapięciaindukowanegowcewceprzyróżnejintensywnościpolamagnetycznego,\npoczątkowej prędkości wahadła oraz liczby zwojów cewki.\nW pracy [54] dokonano analizy stabilności wahadeł znajdujących się w po-\nlu magnetycznym. Badane wahadło zamodelowano poprzez jedną lub dwie za-\nmkniętepętleelektryczneumieszczonewzmiennympolumagnetycznym.Teore-\ntycznebadaniaopartebyłynaasymptotycznymrozwiązaniurównańLagrange’a-\nMaxwella, opisujących dynamikę układów. W zależności od parametrów i wa-\nrunków początkowych układu, ruch wahadła z jedną pętlą elektryczną dążył do\njednegozpołożeńrównowagilubdocykligranicznychusytuowanychbliskotych\npołożeń. Badania układu z dwoma pętlami elektrycznymi pokazały, że istnieje\nmożliwośćobracaniasięwahadła.Jeślidolnepołożenierównowagijestniestabil-\nnetomożliwejestwystąpieniedwóchstabilnychruchówobrotowych(wróżnych\nkierunkach).Jeślizaśjestonostabilne,towahadłomożesięnieobracaćlubmoże\nwykazywać cztery przypadki ruchu obrotowego, gdzie dwa z nich są stabilne,\na pozostałe dwa niestabilne.\nStabilnościąwahadłamagnetycznegozajmowalisięrównieżnaukowcywpra-\ncy [55, 56]. Badany przez nich układ składał się z wahadła z zamocowanym na\nkońcu magnesem i harmonicznie przybliżającą się (oddalającą się) od dołu me-\ntalową płytą. Poruszająca się płyta jest w stanie przekazać energię do wahadła\n191.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\npoprzez pole magnetyczne prądów wirowych indukujących się w jej wnętrzu.\nProblemstabilnościukładurozważanybyłjakdlaklasycznegorównaniaMathieu\nz współczynnikiem tłumienia. Wykorzystując analityczną metodę bilansu har-\nmonicznychokreślonezostaływarunkiwystąpieniarezonansuparametrycznego\nw zależności od amplitudy drgań płyty oraz jej odległości od magnesu wahadła.\nBadania wykazały, że niestabilność układu jest możliwa tylko wtedy, gdy am-\nplitudadrgańiodległośćmiędzypłytąamagnesemspełniająpewnewymagania\nzwiązanezjednoczesnymwzbudzaniemitłumieniemwahadłaprzezporuszającą\nsię płytę.\nW pracy [57] analizowano dynamikę przejściową niewymuszanego i tłumio-\nnego przestrzennego wahadła magnetycznego. Na wahadło działały siły przy-\nciągania magnetycznego pochodzące od trzech magnesów położonych pod nim\ni rozmieszczonych w wierzchołkach trójkąta równobocznego. Wykazano, że ze\nwzględu na brak długotrwałego ruchu, zachowanie analizowanego układu jest\nzupełnie inne niż to, które obserwowano wcześniej dla układów z wymusze-\nniem.Pracępodsumowująnastępującewnioski:i)obliczonywymiargranicbase-\nnu przyciągania może być niecałkowity, a wykładniki Lapunowa w skończonym\nczasie mogą być dodatnie we wszystkich skalach; ii) granice basenu mają frak \ntalne kowymiary 1; iii) prawdopodobieństwo utrzymania się trajektorii z dala od\natraktorów spada superwykładniczo w czasie.\nAnaliza dynamiki globalnej przestrzennego wahadła magnetycznego została\nprzeprowadzona przez Qin i in. [58]. Dotyczyła ona przede wszystkim wrażli-\nwości układu na warunki początkowe, oszacowania liczby położeń równowagi\nw zależności od różnych odległości magnesów oraz analizy stabilności. Ewolu-\ncjafraktalnychbasenówprzyciąganiazostałaprzeprowadzonanumerycznieipo-\ntwierdzonaeksperymentalnie.Pokazano,żepołożeniemagnesuwahadławczasie\njego ruchu wprost wpływa na topologię basenu przyciągania.Wynika to z faktu,\nżezakresprzyciąganiaatraktora,któryjestnajbliżejzostajeznaczącozwiększony,\nnatomiast wpływ strefy innych atraktorów jest pomniejszany.\nPewną modyfikację wahadła magnetycznego typu Bethenoda zaproponował\nDuboshinski [21, 59]. Postanowił on zmienić orientację cewki znajdującej się pod\nwahadłem w taki sposób, że jej oś była prostopadła do osi wahadła w spoczyn-\nku. W układzie tym zaobserwowano zjawisko „kwantyzacji amplitudy” [60, 61].\nPolegało ono na tym, że dla danej częstotliwości prądu i położenia początkowe-\ngo,wahadłowykazywałosiękilkomastabilnymidrganiamioróżnejamplitudzie,\nprzy czym położenie początkowe nie mogło być mniejsze od tego, które odpo-\nwiadało drganiom o najmniejszej amplitudzie [62]. Za stabilizację tych drgań\nodpowiedzialne jest występowanie w układzie tłumienia, którego straty energii\nsąrównoważoneprzezenergiępolamagnetycznego.Energiategopolamusijed \nnak znajdować się w ściśle określonym przedziale. Zbyt mała wartość energii\npola magnetycznego dostarczona do układu nie będzie w stanie zrównoważyć\nstrat energii tłumienia i drgania zgasną. Natomiast zbyt duża wartość energii\nspowodujechaotyczne„przeskakiwanie”pomiędzyróżnymistabilnymidrgania-\nmi wahadła. Układ zazwyczaj wybiera najbliższą stabilną amplitudę drgań dla\ndanych warunków początkowych. Według Luo i in. [63] występowanie wielu\n„dyskretnych” („skwantowanych”) rozwiązań okresowych w tym układzie jest\nwynikiem powstawania rezonansu subharmonicznego. Częstotliwość rezonansu\n201.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\nsubharmonicznego związana jest z symetrią siły wymuszającą. Nieparzyste re-\nzonansesubharmonicznewystępują,gdysymetrycznafunkcjasiływymuszającej\njest parzysta i odwrotnie. W pracy [64, 65] wahadło to zakwalifikowano do kla-\nsy tzw. samoadaptujących układów wzbudzanych impulsowo (ang. kick-excited\nself adaptive dynamical systems ). Wynika to z faktu, że magnetyczna siła wymu-\nszająca układ działa tylko w pewnym, ograniczonym zakresie położeń wahadła\nwzględem cewki. Przeprowadzone badania analityczne i numeryczne dotyczyły\npunktówosobliwychorazbasenówprzyciągania.Analizowanebyłyrównieżwie-\nlokrotne bifurkacje zbiorów atraktorów, a także ich ewolucje i fraktalne postacie.\nDrgania, które charakteryzują się skwantowanymi amplitudami nazywane\nsą argumentacyjnymi (ang. argumental oscillations ). Badania nad nimi prowadził\nCintraiin.[66,67]wykorzystująckilkaróżnychwahadełDuboshinskiego.Nazwa\nwynika z faktu, że wymuszenie oscylatora jest zależne od „argumentu”, któ-\nrym jest jego położeniem w przestrzeni. Traktując układ jak oscylator Duffinga,\nopracowali analityczne wyrażenie na kryterium jego stabilności. Wykorzystując\nmetodę uśredniania, autorzy wyznaczyli rozwiązanie analityczne równania ru-\nchu dla układu bez tłumienia, a w przypadku układu z tłumieniem podali tylko\nrozwiązanie przybliżone tego równania. Eksperymentalne, numeryczne i anali-\ntycznewynikiswoichbadańprzedstawiliwpostacibiegunowychwykresówVan\nderPola,gdziewolnozmieniającasięamplitudadrgaństanowiłapromień,afaza\ndrgań była kątem obrotu.\nCiekawa analogia do wahadła magnetycznego przedstawiona została w pra-\ncach [68, 69]. Autorzy tych prac zaobserwowali eksperymentalnie drgania tzw.\nściany domenowej (ang. domain wall ) i porównali je do drgań wahadła podda-\nnego zmiennej grawitacji. Ściana domenowa powstaje w materiale, gdy w je-\ngo strukturze spotkają się dwa przeciwnie namagnetyzowane regiony. Badana\nściana miała szerokość 50nm i została wytworzona w zakrzywionym drucie\nniklowo-żelazowym, przez który przepuszczano bardzo mały prąd. Ruch ścia-\nny jest równoważny drganiom wahadła, gdzie masa ściany domenowej odpo-\nwiada masie wahadła. Natomiast oddziaływanie magnetostatyczne zachodzące\npomiędzy polem magnetycznym wytwarzanym przez płynący w drucie prąd\na polem magnetycznym ściany domenowej odpowiada grawitacji. Analizowane\nbyły drgania ściany domenowej podczas przepuszczania przez drut zmiennego\nprądu o różnych częstotliwościach. Autorzy wspomnianych prac na podstawie\nbadańdoświadczalnychwykazali,żedrganiarezonansoweścianydomenowejsą\ndostatecznie duże, a zarazem pobierają bardzo mało prądu, co daje możliwości\ndoopracowaniaistworzenienowychukładówelektronicznychoniskimpoborze\nmocy stanowiących konkurencję dla technologii CMOS.\nKolejną analogię do wahadła magnetycznego opisano w pracy [70]. Autorzy\nwykazali, że wahadło magnetyczne może stanowić klasyczny przykład zdegene-\nrowanej teorii zaburzeń w mechanice kwantowej. Degeneracją (zwyrodnieniem)\nnazywasięsytuację,gdziejednejwartościenergiiukładuodpowiadająróżnesta-\nny kwantowe. Zwykłe wahadło przestrzenne ma cylindryczną symetrię wokół\ncięgna zawieszenia oraz dwie postacie drgań normalnych, tzn. wahadło może\noscylować w dowolnym kierunku z tą samą częstotliwością. Naukowcy posta-\nnowili „złamać” tę symetrię poprzez dołączenie magnesu do ramienia wahadła\ni umieszczenie go w niejednorodnym polu magnetycznym innego magnesu. In-\n211.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\nterakcja magnetyczna przesunęła położenie równowagi układu w wyniku czego\nczęstotliwościdrgańwdwóchprostopadłychkierunkach,niebyłyjużtakiesame.\nEksperymentalne i analityczne badania pokazały, że drgania wahadła mierzone\nwzdłużdanegokierunkucharakteryzująsiędudnieniamiwynikającyztransferu\nenergii na drgania w kierunku prostopadłym. Wykazano również, że okres dud \nnień w stosunku do okresu drgań wahadła jest proporcjonalny do piątej potęgi\nodległości między środkami magnesów.\nWartowtymmiejscuwspomnieć,żewdotychczasowejliteraturzeistniejeteż\nszeregprac,którychobiektembadańniejestbezpośredniowahadłomagnetyczne,\naleukładliniowegooscylatorapoddanegodziałaniusiłmagnetycznych.Zewzglę-\ndu na charakter równania ruchu wspomnianego oscylatora uznać go można za\nukładopisującyruchuproszczonego(zlinearyzowanego)wahadłamagnetyczne-\ngo. Z tego powodu warto również prześledzić dotychczasowe prace z zakresu\ntego rodzaju układów dynamicznych.\nDarula i in. [71] dokonali szerokiej analizy zjawiska rezonansu w układzie\nelektromagnesuprzyciągającegoferromagnetycznyelement.Stosującanalityczną\nmetodę wielu skal określili stany ustalone oscylatora wymuszanego harmonicz-\nniezczęstościąbliskąrezonansowej.Nawykresachamplitudowo-częstościowych\nwidoczny był wpływ nieliniowego oddziaływania magnetycznego, który „prze-\nchylał” krzywą rezonansową (ang. softening effect ). Dodatkowo wykazano ana-\nlitycznie w jaki sposób parametry obwodu elektrycznego mogą zmieniać bądź\ntłumić drgania elementu ferromagnetycznego. Rozszerzone badania nad tłumie-\nniem elektromagnetycznym w układzie nieliniowego oscylatora ze sztywnością\ntypu Duffinga i wymuszanego zewnętrzną siłą zaprezentowano w pracy [72].\nW rozważanym układzie, do drgającej masy przymocowano na stałe cewkę elek-\ntryczną, która w czasie ruchu „przechodziła” przez nieruchomy magnes. Nume-\nrycznie przeanalizowano wpływ na dynamikę układu takich parametrów jak:\nrezystancja,indukcyjnośćipojemnośćcewki.Okazałosię,żetorezystancjacewki\nma największy wpływ na tłumienie elektromagnetyczne, a wartość pojemno-\nści znacząco wpływa na zachowanie dynamiczne układu, co zostało wykazane\nprzy pomocy wykresów bifurkacyjnych. Ruch heterokliniczny i kryteria wystą-\npieniachaosuMielnikowaprzedstawionowpracy[73].Ponadtozaobserwowano,\nżeukładwykazywałruchokresowyichaotycznyodpowiednioponiżejipowyżej\nprogu Mielnikowa.\nEksperymentalne i analityczne badania dynamiki oscylatora w postaci meta-\nlowejmasyzawieszonejnasprężynieiporuszającejsięwewnątrzcewkielektrycz \nnej zostały zaprezentowane przez Ho i in. [74]. Korzystając z metody wielu skal,\nautorzy otrzymali przybliżone rozwiązanie wskazujące na bliską relację między\nczęstotliwościądrgańmetalowejmasyaczęstotliwościąnapięciaprzyłożonegodo\ncewkiwprzypadkuinnymniżwzbudzenierezonansowe.Rozwiązanieanalitycz \nne ułatwiło identyfikację krytycznych wartości parametrów układu, które miały\nwpływnajegoodpowiedź.WpracyBednarkaiin.[75]przedstawionyzostałukład\ndo aktywnego tłumienia drgań oscylatora łożyskowanego aerostatycznie i pod \ndanego działaniu sprężyny elektromagnetycznej składającej się z pary cewka-\nmagnes. Na podstawie doświadczenia wyznaczono formuły matematyczne opi-\nsujące sztywność sprężyny elektromagnetycznej w funkcji prądu cewki. Dzięki\nspecjalnieopracowanemusterowaniuprądemcewkimożliwebyłonatychmiasto-\n221.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\nwe wytłumienie drgań oscylatora, co potwierdzono w badaniach symulacyjnych\ni eksperymentalnych.\nWitkowskiiin.[76]opracowalimodelematematyczneopisująceruchdrgający\nwózka z oddziaływaniami magnetycznymi pozwalające wykonywać względnie\nszybkie symulacje numeryczne ruchu układu. Wózek oprócz przymocowanych\ndo niego sprężyn mechanicznych wyposażono w magnes, który był odpychany\nprzezmagnesutwierdzonydopodstawystanowiska.Odpychającedziałaniema-\ngnesów wprowadzało do równania ruchu sztywność typu Duffinga, dla której\nautorzyopracowalipięćróżnychmodelimatematycznych.Jakościowaiilościowa\nanaliza bifurkacyjna dowiodła, że użycie modelu oddziaływania magnetyczne-\ngo w formie nieparzystej funkcji wymiernej prowadzi do najlepszej zgodności\nwyników pomiędzy symulacją a eksperymentem.\nUkładmagnetycznegooscylatorazuderzeniamizostałzaprezentowanywpra-\ncach[77 79].Oscylatorzbudowanybyłzmasyzamocowanejdodwóchbelekpeł-\nniącychrolęanalogicznądosprężyn.Zjednejstronymasyzamocowanomagnes,\nktóry podczas oddziaływania z nieruchomą cewką wymuszał ruch układu. Na-\ntomiast druga strona masy mogła uderzać w nieruchomą przeszkodę. W pracy\nzaproponowano analityczną metodę obliczania dużych ugięć belki, przy czym\nzagadnienie brzegowe pozwoliło na wyznaczenie wyrażenia na siłę sprężysto-\nści. Wykorzystując prawo Biota-Savarta i metodę analizy nieskończenie małych\nelementów, autorzy opracowali wzór na oddziaływanie magnetyczne pomiędzy\nmagnesemacewkąookreślonejliczbiezwojów.Pracęuzupełniająwykresyrezo-\nnansowe i bifurkacyjne obejmujące stabilne oraz niestabilne rozwiązania układu,\na także przeprowadzono analizy różnych typów bifurkacji.\n1.1.2 Układy o wielu stopniach swobody\nLiczba prac naukowych odnosząca się do badań nad dynamiką układów o wielu\nstopniachswobodyopartychnawahadłachmagnetycznychjestznaczniemniejsza\nniż miało to miejsce w przypadku układów o jednym stopniu swobody. W tym\nparagrafie omówione zostaną prace, których wyniki są najbardziej zbliżone do\ntematu niniejszej rozprawy doktorskiej.\nFradkov i in. [80] analizowali problemem wzbudzenia i synchronizacji drgań\nw układzie mechatronicznym złożonym z dwóch torsyjnie sprzężonych waha-\ndeł magnetycznych. Skomplikowany układ składał się z dwóch obracających się\nobręczy, które sprzężono skrętnie elastycznym elementem. Na dole obręczy za-\nmontowano magnesy oddziałujące z cewką elektryczną umieszczoną pod nimi.\nDodatkowo,wewnątrztychobręczyzamocowanoobracającesiędźwignie,nakoń-\ncach których przyczepiono magnesy stałe odpychające się z magnesami obręczy.\nDo zbudowanego stanowiska eksperymentalnego opracowane zostały algoryt-\nmy do estymacji wektora stanu układu, a także identyfikacji parametrów oraz\nokreślenia poziomu energii układu. Stosując metodę szybkiego spadku opraco-\nwano sterowanie pozwalające na osiągnięcie przez wahadła zadanego poziomu\nenergetycznego z wymogiem drgań w przeciwfazie. Dokonano również analizy\nukierunkowanejnawyznaczeniewartościwspółczynnikawzbudzenia,któryjest\nwersją asymptotycznego wzmocnienia mierzącą właściwości rezonansowe ukła-\ndu.\n231.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\nW pracy [81] autorzy skupili się na polepszeniu zdolności odzyskiwania\nenergii z ruchu dwóch skrętnie sprzężonych wahadeł poddanych zewnętrzne-\nmu wymuszeniu. Na końcu wałów wahadeł zamocowane były kołowe magnesy\nz dwoma przeciwstawnymi biegunami, które podczas obracania się generowa-\nły napięcie wewnątrz rozmieszczonych wokół nich cewek. Naukowcy wykazali,\nżeukładtenjestodpowiednidoodzyskiwaniaenergiizdrgańoniskichczęstotli-\nwościach ( <5Hz). Stosując analityczną metodę bilansu harmonicznych wykryto\nrozwiązaniaomałejamplitudziedrgańorazwspółistniejącerozwiązaniaoznacz-\nnie większej amplitudzie, które były niewykrywalne w obliczeniach numerycz \nnych.Zaprezentowanewynikisymulacyjnepokazały,żezwiększenieodzyskiwa-\nnia energii dla szerokiego spektrum częstotliwości drgań wahadeł może zostać\nzapewnione poprzez wybór odpowiednich warunków początkowych.\nSurganova i in. [82] badali wpływ parametrów takich jak masa, moment bez-\nwładnościczysztywnośćsprzężeniananieliniowepostaciedrgańukładudwóch\nsprzężonych wahadeł magnetycznych. Korzystając z metody małego parametru\noraz metod numerycznych przeprowadzili szereg badań w odniesieniu do róż-\nnych warunków początkowych układu. Wykazali, że drgania w fazie występują\ndla wszystkich wychyleń początkowych wahadeł zarówno przy silnym jak i sła-\nbym oddziaływaniu magnetycznym. Dodatkowo, przy wychyleniach początko-\nwych mniejszych niż 4 5◦wykryto drgania układu, których trajektorie fazowe\nodpowiadały rozwiązaniom quasiokresowym.\nRussell i in. [83] analizowali zachowanie się łańcucha osiemnastu wahadeł\nmagnetycznychrozmieszczonychwzdłużokręgu.Czynnikiemsprzęgającymwa-\nhadławukładziebyłysiłymagnetyczne.Przedstawionewynikinumeryczneieks-\nperymentalne dowiodły, że w układzie powstają poruszające się nieliniowe fale\n(znane w literaturze pod angielską nazwą moving breathers ), których energia kon-\ncentrujesięwsposóbzlokalizowanyiwykazujecharakteroscylacyjny.Faletemają\nstrukturęobwiednisolitonówiczasamisąnazywanesolitonamiobwiedniowymi.\nBadaniadotyczyłyrównieżstabilnościtychfalwodniesieniudomałychprzypad \nkowych zakłóceń układu. Według autorów pracy, otrzymane wyniki są ważne\nz punktu widzenia transferu energii zachodzącego podczas zderzeń pomiędzy\natomami (lub jonami) poruszającymi się ze stosunkowo dużymi prędkościami\na atomami ciała stałego.\nDynamikę chaotyczną i okresową podwójnego wahadła magnetycznego ana-\nlizował Wojna i in. [84]. Na końcu dolnego wahadła zamocowano silny magnes,\nktóryodpychałsięzdrugimmagnesemzamocowanymwpodstawiestanowiska.\nWymuszenie realizowane było przez specjalnie skonstruowany silnik elektrycz-\nny. Autorzy przedstawili równania ruchu układu uwzględniając złożony model\noporów ruchu generowany przez łożyska toczne, a także moment magnetyczny\nw funkcji różnych położeń kątowych wahadeł. Eksperymentalne i numeryczne\nwykresy bifurkacyjne, trajektorie fazowe i przekroje Poincarégo wykazały dużą\nzgodność opracowanego modelu z badaniami doświadczalnymi. Chaos w ukła-\ndzie powstawał przez podwajanie się okresu, co zaobserwowano na wykresach\nbifurkacyjnych. W pracy [85] skorzystano z występowania w tym układzie zja-\nwiskachaosuwceluefektywniejszegoodzyskiwaniaenergii.Magnesznajdujący\nsięwpodstawiestanowiskazastąpionoszeregiemcewekelektrycznychrozmiesz-\nczonychrównomierniewzdłużłukudolnejtrajektoriiwahadła.Dziękitemuauto-\n241.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\nrzyodzyskiwalienergiemechanicznądrgańpodłożawpostacienergiielektrycz \nnejgenerowanejwcewkach.Przeprowadzonoanalizęnumerycznądookreślenia\noptymalnych wartości parametrów układu (masy i długości wahadeł oraz masy\nmagnesu) względem odzyskiwanej ilości energii. Badania numeryczne i ekspe-\nrymentalnewykazały,żenagływzrostodzyskiwaniaenergiinastępujewówczas,\ngdy ruch dolnego wahadła staje się chaotyczny, a także gdy zwiększona zostaje\nliczba cewek elektrycznych.\nZhangiin.[86]zaproponowaliautoparametrycznypochłaniaczdrgańodwóch\nstopniach swobody oparty na wahadle magnetycznym. Badany model fizyczny\nskładał się z drgającej pionowo masy poddanej harmonicznemu wymuszeniu\ni wewnątrz której znajdowało się wahadło magnetyczne. Wahadło umieszczone\nbyłopomiędzydwomaprzyciągającymigomagnesami.Numerycznieorazanali-\ntycznie zbadano odpowiedzi układu na siłę wymuszającą. Korzystając z metody\nmałego parametru wyznaczony został optymalny zakres tłumienia drgań w od \nniesieniu do parametrów układu. Wyniki badań pokazały, że zaprezentowany\nukład efektywniej tłumi drgania niż układ wyposażony w zwykłe wahadło ze\nwzględu na szerszy zakres częstotliwości pracy.\nZaawansowany układ pochłaniacza drgań i jednocześnie urządzenia odzy-\nskującegoenergięprzedstawionowpracach[87–89].Układskładałsięzdrgającej\nna sprężynach masy, do której przymocowany był silnik elektryczny z waha-\nłem zawieszonym na jego wale. Wzdłuż ramienia wahadła zamocowano cewkę\nelektryczną, wewnątrz której poruszał się magnes. Układ w czasie ruchu tłumił\ndrgania masy odzyskując przy tym energię na dwa sposoby: (i) przy pomocy\nsamej cewki (układ o 4 stopniach swobody) lub (ii) przy pomocy silnika i cewki\n(układ o 5 stopniach swobody). Przeprowadzona analiza dynamiczna wykazała,\nże tzw. „języki” niestabilności parametrycznej występują dla szerokiego pasma\nczęstotliwości wymuszenia i są spowodowane przez bifurkację typu Neimarka-\nSackera. Obszary tych niestabilności mogą być kontrolowane przez odpowiednie\nzmienianie rezystancji w obwodzie silnika. Zaproponowano trzy wskaźniki opi-\nsujące skuteczność odzyskiwania energii oraz jeden wskaźnik tłumienia drgań.\nAnalizując wszystkie wskaźniki znaleziono obszar zgodności, w którym wystę-\npuje jednoczesne tłumienie drgań i odzyskiwanie energii. Badania numeryczne\nieksperymentalnewykazały,żesilnikelektrycznyefektywniejodzyskujeenergię\nniżzespółcewka-magneszamocowanynawahadle.Zdrugiejstronyjednakzna-\nczącoobniżaonzdolnośćukładudorozpraszaniaenergiidrgańmasy.Natomiast\nzespół cewka-magnes pomimo, że jest mniej skuteczny w odzyskiwaniu energii\npraktycznie nie wpływa na tłumienie drgań masy.\nAutorzypracy[90]zbadalidynamikęzabawkinazywanejLevitronem,wktó-\nrej znaleźli analogię do wahadła magnetycznego. Levitron jest to zabawka o sze-\nściu stopniach swobody, składająca się z wirującego „bąka lewitującego w polu\nmagnetycznym. Na podstawie analizy stabilności ruchu tej zabawki w kierunku\npionowymstwierdzono,żerównanieruchuodpowiadającetemukierunkowijest\nanalogią do wahadła magnetycznego i jego drgań wokół położenia równowagi.\nNa podstawie zlinearyzowanych równań dynamicznych wyznaczone zostały za-\nkresystabilnościwzależnościodprędkościobrotowej„bąka”ijegoodległościod\npodstawy zabawki.\nUkładyowielustopniachswobodyzłożonezliniowychoscylatorówipodda-\n251.1. DOTYCHCZASOWY STAN WIEDZY O UKŁADACH WAHADEŁ MAGNETYCZNYCH\nne działaniu sił magnetycznych, podobnie jak w przypadku układów o jednym\nstopniu swobody, można uznać za uproszczone analogie układów sprzężonych\nwahadeł magnetycznych poruszających się w zakresie małych kątów. Dlatego\npodobnie jak wcześniej, opisane zostaną wyniki uzyskane w pracach, których\nprzedmiotem badań są tego rodzaju obiekty.\nW pracy [91] przedstawiony i zamodelowany został układ dwóch szerego-\nwo połączonych oscylatorów. Oprócz liniowych sprężyn łączących oscylatory, na\njeden z nich działała silnie nieliniowa sprężyna o ujemnej sztywności zbudowa-\nna z odpychających się magnesów. Siła oddziaływania magnetycznego została\nopisanawpostaciwielomianutrzeciegostopnia.Badaniateoretycznedrgańswo-\nbodnych i wymuszonych przeprowadzone zostały w oparciu o analizę równania\nDuffinga.Badaniaeksperymentalnewykonanodlaróżnychnieliniowościisztyw-\nnościsprężynymagnetycznej,wynikającychzezmianyodległościmiędzymagne-\nsami. Obliczono, że istnieje pewna wartość sztywności sprężyny magnetycznej,\npowyżej której występuje niestabilność układu. Podczas badań z wymuszeniem\nukładuzaobserwowano,żedladrgańprzedrezonansowychsprężynamagnetycz-\nna wykazuje tylko liniowe zachowanie. Ponadto wartość częstotliwości rezonan-\nsowej układu z nieliniową sprężyną magnetyczną zmniejszyła się w odniesie-\nniu do przypadku, gdyby zastąpić ja klasyczną sprężyną liniową. Zastosowanie\nsprzężny magnetycznej spowodowało zmniejszenie się własności wibroizolacyj-\nnychukładu.Pewnąmodyfikacjetegoukładuzaproponowaliautorzypracy[92].\nModyfikacja polegała na dodaniu kolejnej sprężyny magnetycznej do oscylatora,\nktóry wcześniej był pod działaniem tylko sprężyny liniowej. W nowym układzie,\njedynie sprężyna sprzęgająca dwa oscylatory była liniowa. Badania numeryczne\ni eksperymentalne skoncentrowane były głównie na dynamice nieliniowej oscy-\nlatorów. Zademonstrowano bifurkację będące wynikiem podwajania się okresu\ndrgań, chaotyczne atraktory, zachowania histerezowe, skoki amplitudy i drga-\nnia quasiokresowe. Autorzy w procesie walidacji modelu, uzyskali jego wysoką\nzgodność z danymi eksperymentalnymi. Ponadto nieciągłe funkcje występujące\nw modelu matematycznym zostały przybliżone funkcjami wymiernymi, co po-\nzwoliło na opracowanie rozwiązań analitycznych badanego układu przy użyciu\nmetody bilansu harmonicznych. W efekcie, znaleziono niestabilne okresowe roz-\nwiązania dynamiczne pominięte w symulacjach numerycznych.\nZhou i in. [93] zaproponowali inną konfigurację montażu sprężyn na spe-\ncjalnie skonstruowanym stanowisku, niż ta zaprezentowana w pracach [91, 92].\nRozważonoprzypadek,gdyoscylatorybyłysprzęgniętenieliniowąsprężynąma-\ngnetyczną, a zwykłe sprężyny linowe pełniły role łączników z nieruchomą pod-\nstawą stanowiska. Przeanalizowali dynamikę układu ze szczególnym uwzględ \nnieniem transferu energii z oscylatora wymuszanego do oscylatora pasywnego,\nw przypadku rezonansu wewnętrznego o stosunku częstości 1:3. Eksperyment\ni obliczenia analityczne wykazały występowanie zjawiska nasycenia amplitudy\ndrgań oraz jej przeskoków pomiędzy dwoma stanami ustalonymi. Główną kon-\ncepcjąbadańbyłozastosowaniezjawiskanasycenia,występującegowrezonansie\nwewnętrznym, do tłumienia rezonansu głównego oscylatora wymuszanego siłą\nzewnętrzną.\nUkład dwudziestu ośmiu sprzężonych ze sobą wahadeł magnetycznych za-\nprezentowanowpracy[94].Autorzyprzedstawilipomysłnabudowęnowegoro-\n261.2. GENEZA I UZASADNIENIE TEMATU PRACY\ndzaju mechanicznej anteny, pracującej na ultra niskich częstotliwościach (od 300\nHz do 3 kHz). Antena ta składa się z dwudziestu ośmiu walcowych magnesów\nneodymowych, namagnesowanych w kierunku diametralnym (tzn. magnesowa-\nnie odbywa się wzdłuż średnicy) i mogących obracać się wzdłuż osi symetrii.\nMagnesyumieszczonesąbliskosiebiewewnątrzprostokątnejcewkielektrycznej,\ntakżeichwłasnepolamagnetyczneoddziałująnasiebietworzącsprzężenie.Przy\nzałożeniu, że magnesy zachowują się jak wahadła magnetyczne, autorzy anali-\nzowali ich drgania oraz wydajność przy generowaniu fal elektromagnetycznych\nzzakresuultraniskichczęstotliwości.Wrezultacieokazałosię,żewydajnośćtego\nrodzaju anteny jest o 7 dB większa niż zwykłej anteny elektrycznej.\n1.2 Geneza i uzasadnienie tematu pracy\nBadaniadynamikiukładówzłożonychzwahadełmagnetycznychprowadzonesą\nprzeznaukowcówodconajmniejpierwszejpołowyXXwieku,ażdodniadzisiej-\nszego. Biorąc pod uwagę interdyscyplinarny charakter podjętej tematyki badań\ni wywodzącej się z połączenia dziedzin takich jak mechanika i elektrodynamika,\nprzyciąga ona zainteresowanie badaczy z obu tych obszarów nauki. Wynika to\nz faktu poszukiwania synergizmu i płynących z niego korzyści dla rozwoju tech-\nnologiiorazpoprawypoziomużycia.Pomimotegozainteresowania,wdostępnej\nliteraturzeodnoszącejsiędodynamikiukładówzłożonychzwahadełmagnetycz-\nnychwciążistniejąpewnebraki,któreprzyczyniłysiędopowstaniatejrozprawy.\nWniniejszympodrozdzialezostanąprzedstawionekierunkimożliwegodalszego\nrozwoju badań nad tymi układami, pozwalające poszerzyć obecny stan wiedzy.\nBiorącpoduwagęfakt,żewprawdziekonstrukcyjnieukładywahadełmagne-\ntycznych nie są skomplikowane to jednak wykazują one silnie nieliniową dyna-\nmikę. Dzięki temu wykorzystuje się je do badań podstawowych nad typowymi\nzjawiskaminieliniowymitakimijakbifurkacje[38,44,95],skokiamplitud[33,39],\ncyklegraniczne[65],zachowaniachaotyczne[29,45,47]czyruchyquasiokresowe\n[96],alerównieżdobadaniazjawiskarezonansuparametrycznego[36,41,55,56],\nczybadańnadmultistabilnością[37,97].Wdotychczasowychbadaniach,abywy-\njaśnić źródła powstawania tych zjawisk, naukowcy często opierali się na znanych\nmodelach matematycznych takich jak równanie Duffinga [28, 30, 32], równanie\nMathieu [20, 21, 41] czy równanie Van der Pola [42, 43]. Jednakże, ze względu\nna złożoność charakteru zjawiska oddziaływania magnetycznego, modele te nie\nzawsze się sprawdzają, a ponadto nie uwzględniają innych zjawisk, jak np. tarcie\nsuchewystępującew rzeczywistychukładachmechanicznych.Wydajesięwięc,że\ndokonanieanalizydynamikiprzywykorzystaniuzłożonychmodelimatematycz-\nnych może doprowadzić do wykrycia wcześniej nieobserwowanych zachowań.\nDlategowniniejszejpracywynikibadańpodstawowychzostałyopracowaneprzy\nwykorzystaniu modeli matematycznych, w których starano się jak najdokładniej\nodwzorować naturę oddziaływania magnetycznego oraz oporów ruchu, wyko-\nrzystując przy tym badania eksperymentalne.\nDostępnewliteraturzebadanianajczęściejprowadzonebyłydlawahadełma-\ngnetycznych umieszczonych w stacjonarnym polu magnetycznym [38, 57, 58].\nPole to pochodziło np. od stałych magnesów lub cewek elektrycznych zasilanych\n271.3. CEL NAUKOWY, TEZA I ZAKRES PRACY\nprądemstałym.Natomiastwymuszenietychukładówrealizowanowsposóbme-\nchaniczny, np. kinematycznie [91, 93] (poprzez układ elementów sprężystych),\nparametrycznie (poprzez ruch punktu zawieszenia [55, 86, 98]) bądź poprzez\nzewnętrznymomentgenerowanyprzezsilnik[44,84].Resztadostępnychartyku-\nłów poświęcona jest dynamice układów wymuszanych niestacjonarnym polem\nmagnetycznym, pochodzącym od cewki zasilanej zmiennym w czasie prądem\nelektrycznym. W znacznej większości tych prac analizie poddano układy, w któ-\nrychcewkazasilanabyłasinusoidalnymsygnałemprądowym[54,62,63,66].Inne\nsygnałytakiejaknp.prostokątnyniebyłybranepoduwagę.Brakwynikówtakich\nbadań w dotychczasowej literaturze skłonił autora pracy do zainteresowania się\ntą problematyką.\nKolejnymsłabympunktemobecnegostanuwiedzyjestmałailośćmateriałów\nnaukowych dotyczących zagadnień dynamiki układów wahadeł magnetycznych\nowielustopniachswobody.Odnoszącsiędopraczawartychwprzeglądzielitera-\ntury można zauważyć znaczącą dysproporcję pomiędzy liczbą prac dotyczących\nukładów o jednym stopniu swobody a liczbą prac dotyczących układów o wielu\nstopniach swobody. Dodatkowo tylko w dwóch pracach [80, 81] obiektem badań\nbył układ złożony z wahadeł magnetycznych, których sprzężenie odbywało się\nprzy pomocy skrętnego elementu sprężystego łączącego ich wały. W zamierze-\nniu autora, wspomniane luki w tym obszarze tematycznym zostały wypełnione\nbadaniami przedstawionymi w tej rozprawie doktorskiej.\n1.3 Cel naukowy, teza i zakres pracy\nCelem naukowym pracy jest opracowanie nowych modeli matematycznych dla\npojedynczego wahadła magnetycznego oraz układu dwóch skrętnie sprzężo-\nnych wahadeł magnetycznych, a także zbadanie ich dynamiki nieliniowej pod\nwzględem praktycznego zastosowania uzyskanych wyników. Opracowane mo-\ndele wykorzystywać będą empiryczne funkcje opisujące oddziaływanie magne-\ntycznemiędzykomponentamiukładuorazuwzględniaćbędąoporyruchuwystę-\npującewstanowiskachdoświadczalnych.Dziękitemustanowićbędąalternatywę\ndla prostych modeli proponowanych dotychczas w literaturze.\nZe względu na fakt, że w prezentowanej rozprawie doktorskiej przedmiotem\nbadańsądwaukładyskładającesięzwahadełmagnetycznych,dlakażdegoznich\nzostanieprzyjętainnatezabadawcza.Pierwszyukładzłożonyjestzpojedynczego\nwahadła,nakońcuktóregozamocowanyjestmagnes.Podwahadłemumieszczona\njestcewkaelektrycznazasilanaprostokątnymsygnałemprądowymoregulowanej\nczęstotliwości i tzw. współczynniku wypełnienia. Oddziaływanie magnetyczne\npomiędzy magnesem a cewką powoduje ich wzajemne odpychanie się. Drugi\nukładskładasięzdwóchwahadeł,nakońcachktórychzamocowanesąmagnesy,\na ponadto wały tych wahadeł połączone są ze sobą elementem sprężystym. Pod\nkażdymzwahadełznajdujesięcewkaelektrycznazasilanasygnałemprądowym\nozadanymkształcie.Sygnałyprądowepłynącewtychcewkach,mogągenerować\npola magnetyczne przyciągające jak i odpychające magnesy wahadeł.\nNa podstawie wstępnych badaniach eksperymentalnych i symulacyjnych po-\nczynionych dla układu pojedynczego wahadła magnetycznego, można postawić\n281.4. WKŁAD WYNIKÓW PRACY W DYSCYPLINĘ NAUKOWĄ\nnastępująca tezę:\nW przypadku drgań okresowych pojedynczego wahadła magnetycznego od-\nbywających się w jednym dołku potencjału, możliwe jest zmienianie okreso-\nwościtychdrgańbezwyraźnegonaruszeniaprzebieguichtrajektoriifazowej\npoprzezzmianęczęstotliwościprostokątnegosygnałuprądowegocewkielek \ntrycznej.\nOkresowość drgań wahadła rozumiana jest jako liczba okresów wymuszenia\n(sygnału prądowego) przypadająca na jeden okres drgań wahadła.\nWprzypadkuukładuzłożonegozdwóchsłabosprzężonychwahadełpodda-\nnych działaniu pól magnetycznych, można postawić następującą tezę badawczą:\nMożliwym jest kontrolowanie przepływu energii pomiędzy sprzężonymi\nwahadłami magnetycznymi przy użyciu pól magnetycznych generowanych\nprzez cewki elektryczne znajdujące się pod nimi .\nZakrespracpotrzebnychdorealizacjicelubadawczegoiuzasadnieniaprzyję-\ntych tez jest następujący:\n(i) gruntowny przegląd literatury w zakresie prowadzonych badań;\n(ii) opracowaniemodelimatematycznychbadanychukładówiidentyfikacjaich\nparametrów;\n(iii) modyfikacja stanowiska badawczego (powstałego jeszcze w ramach pracy\ninżynierskiej autora [99]) w zakresie układu zasilania cewek elektrycznych;\n(iv) symulacyjne badania dynamiki nieliniowej pojedynczego wahadła magne-\ntycznegodążącedoudowodnieniatezyoniezmiennymprzebiegutrajektorii\nfazowej drgań pomimo znaczących zmian parametrów sygnału prądowego\ncewki;\n(v) symulacyjnebadaniadynamikinieliniowejdwóchsprzężonychwahadełma-\ngnetycznychdążącedoudowodnieniatezyosterowaniuprzepływemenergii\nmiedzy wahadłami poprzez odpowiednie generowanie pól magnetycznych\ncewek;\n(vi) walidacja opracowanych modeli matematycznych na bazie przeprowadzo-\nnych badań eksperymentalnych.\n1.4 Wkład wyników pracy w dyscyplinę\nnaukową\nWydawaćbysięmogło,żedynamikawahadeł,któretowarzysząnamodwieków\nzostała przez te wszystkie lata skrupulatnie zbadana przez naukowców i niczym\nnowymnasniezaskoczy.Jednak,wprzedzialelat2010 2021baza Scopusodnoto-\nwała, aż17 066 artykułów,dla których słowemkluczowym było „wahadło”(ang.\n291.5. STRUKTURA PRACY\npendulum ). Liczba ta pokazuje, że pomimo powszechności wahadeł wciąż są one\nobiektembadańpodstawowych.Badaniateskupiająsięgłównienanieliniowych\naspektach dynamiki układów o jednym i wielu stopniach swobody, złożonych\nze sprzężonych wahadeł i poddanych różnym rodzajom wymuszeń [100–103].\nMniejszą popularnością wśród naukowców cieszy się „wahadło magnetyczne”\n(ang.magnetic pendulum ), które w bazie Scopusma 51 prac, a aż 33 opublikowa-\nne zostały w latach 2010-2022. Trzeba przyznać, że w dzisiejszym świecie nauki\ni przy obecnych możliwościach prowadzenia badań naukowych, liczba 51 prac\nnie jest zadowalająca.\nOkazujesię,żejużsamobogactwodynamikinieliniowejprezentowanejprzez\nwahadła magnetyczne może stanowić powód do rozpoczęcia badań naukowych\n[55,58,98,104,105],atakżebyćźródłemdanychwejściowychużywanychpodczas\nnp. testowania złożonych metod topologicznej analizy danych [106]. W naukach\ninżynieryjno-technicznychwahadłamagnetycznedobrzeodnajdująsięwprężnie\nrozwijającej się gałęzi nauki, jaką jest odzyskiwanie energii. Energia zgromadzo-\nna w ruchu wahadła będąca wynikiem drgań podłoża, może być przekształcana\npoprzez indukcję elektromagnetyczną lub piezoelektryki na energię elektryczną\ngromadzonąwakumulatorachigotowądoponownegoużycia[107 111].Ponadto\nukłady wahadeł magnetycznych ze względu na możliwość łatwej zmiany para-\nmetrów oddziaływania magnetycznego wykorzystywane są jako tłumiki niepo-\nżądanychdrgańmechanicznychnp.wsystemachgromadzącychenergięopartych\nnakołachzamachowych[112].Badaniadynamikiwahadełmagnetycznychmogą\nrównież stanowić źródło inspiracji dla nowych modeli matematycznych i symu-\nlacyjnych sprzęgieł elektromagnetycznych [113, 114] czy silników elektrycznych\n[115,116].Pakietysprzężonychwahadełmagnetycznychwykorzystywanesątak-\nże w radiotechnice do budowy nowoczesnych anten pracujących na ultra niskich\nczęstotliwościach [117 119].\n1.5 Struktura pracy\nPrzedstawionapracaskładasięzczterechrozdziałówpodzielonychtematycznie.\nW rozdziale pierwszym zawarto informacje o dotychczasowym stanie wie-\ndzy na temat układów wahadeł magnetycznych charakteryzujących się jednym\noraz wieloma stopniami swobody. Podano genezę i uzasadnienie podjęcia tema-\ntu pracy. Określono cele naukowe, sformułowano tezy pracy, a także jej zakres.\nW części końcowej rozdziału przedstawiono jaki wkład mogą mieć wyniki pracy\nw dyscyplinę naukową inżynieria mechaniczna.\nW rozdziale drugim przedstawiono badania symulacyjne i eksperymentalne\ndlapojedynczegowahadłamagnetycznego.Omówionazostałakonstrukcjaidzia-\nłanie stanowiska badawczego. W następnej kolejności opracowano model mate-\nmatycznyukładu.Przedstawionostosowanewbadaniachmodeleoporówruchu,\nmodelprostokątnegosygnałuprądowegoorazempirycznemodelemomentusiły\noddziaływania magnetycznego. Opisano też sposoby identyfikacji parametrów\nukładu. Dokonano numerycznych i analityczno-numerycznych badań dynamiki\nnieliniowej, a następnie skupiono się na przypadku drgań wahadła w jednym\ndołku potencjału.\n301.5. STRUKTURA PRACY\nW rozdziale trzecim przedstawiono badania symulacyjne i eksperymental-\nne układu dwóch sprzężonych torsyjnie wahadeł magnetycznych. Opisano kon-\nstrukcjęstanowiskabadawczegoopartegonastanowiskupojedynczegowahadła.\nOpracowanomodelmatematycznyukładuwoparciuowcześniejszemodeleuży-\nwane w układzie o jednym stopniu swobody. Przeprowadzono również analizę\nbifurkacyjną dynamiki układu oraz sterowania przepływem energii między wa-\nhadłami przy użyciu pola magnetycznego.\nWrozdzialeczwartympodsumowanouzyskanewynikipracyorazsformuło-\nwano wnioski wynikające z przeprowadzonych badań teoretycznych, numerycz \nnych, analityczno-numerycznych i doświadczalnych. Wyróżniono innowacyjne\nelementy pracy oraz podano dalsze kierunki możliwych badań.\n31Rozdział 2\nUkład pojedynczego wahadła\nmagnetycznego\nRozdział poświęcony jest opisowi stanowiska badawczego pojedynczego waha-\ndła magnetycznego, jego modelowaniu matematycznemu oraz wynikom badań\nudowadniających pierwszą z postawionych tez badawczych.\n2.1 Stanowisko badawcze\nPodrozdział ten poświęcony jest opisowi budowy stanowiska badawczego poje-\ndynczegowahadłamagnetycznego,naktórymwykonywanebyłybadaniaekspe-\nrymentalne.\nZdjęcie całego stanowiska badawczego, na którym odbywały się ekspery-\nmenty zarówno dla pojedynczego wahadła magnetycznego jak i układu dwóch\nsprzężonych wahadeł magnetycznych zostało pokazane na rys. 2.1. Jego pod \nstawowymi elementami są: komputer z oprogramowanie LabVIEW, zasilacz la-\nboratoryjny, generator sygnału, karta pomiarowa i przede wszystkim rekonfigu-\nrowalny układ wahadeł magnetycznych. Układ ten w podstawowej wersji daje\nmożliwość badania ruchu pojedynczego wahadła magnetycznego, a po stosun-\nkowo prostej rekonfiguracji można na nim badać układ składający się z dwóch\nskrętnie sprzężonych wahadeł magnetycznych. Rekonfiguracja polega na sprzę-\ngnięciuwałówwahadełpoprzezumieszczenieelementupodatnegonp.sprężyny\nw specjalnych uchwytach znajdujących się na ich końcach. Ponieważ rozdział\npoświęcony jest układowi pojedynczego wahadła magnetycznego, podczas ba-\ndaństanowiskoskonfigurowanebyłowsposóbprzedstawionynarys.2.2.Waha-\ndło oznaczone zostało numerem (1). Ramię wahadła zbudowane jest z elementu\nwykonanego z materiału kompozytowego (tekstolit), zaś uchwyt którym przy-\nmocowane jest ono do mosiężnego wału (3) wykonany został z aluminium. Na\nkońcu ramienia wahadła zamocowany został magnes neodymowy (2) o średnicy\n22 mm i wysokości 10 mm (niewidoczny na zdjęciu). Wał wahadła podparty jest\nprzy pomocy dwóch standardowych łożysk kulkowych, zamkniętych w oprawie\nprzymocowanej do aluminiowej ramy stanowiska (11). W specjalnym uchwycie\npodwahadłemumieszczonocewkęelektryczną(4)onastępującychparametrach:\n322.1. STANOWISKO BADAWCZE\nRys. 2.1. Rekonfigurowalne stanowisko eksperymentalne pojedynczego wahadła\nmagnetycznego.\nindukcyjność 22 mH, rezystancja drutu 10.6 Ω, średnica drutu 0.5 mm,\nśrednica zewnętrzna cewki 40 mm, średnica otworu cewki 17 mm i wyso-\nkość cewki – 31 mm. Uchwyt ten przymocowany jest do liniowego prowadnika\n(5),przy pomocyktórego włatwy sposóbmożna regulowaćodległość pomiędzy\ncewką a magnesem w kierunku pionowym. Podczas prowadzonych badań, od \nległość pomiędzy czołem cewki a powierzchnią magnesu ustawiona była na 1.6\nRys. 2.2. Stanowisko eksperymentalne pojedynczego wahadła magnetycznego,\ngdzie: 1 wahadło, 2 magnes neodymowy (niewidoczny na zdjęciu),\n3 mosiężnywał,4 cewkaelektryczna,5 prowadnikliniowy,6 gumo-\nwy element podatny, 7 czujnik momentu skręcającego, 8 polimerowy\ndysk,9-inkrementalnyczujnikoptyczny,10-tarczakodowa,11-alumi-\nniowa rama.\n332.2. MODELOWANIE MATEMATYCZNE\n2.2 Modelowanie matematyczne\n2.2.1 Równanie ruchu\nW celu napisania dynamicznego równania ruchu badanego układu posłużono\nsię modelem fizycznym zaprezentowanym na rys. 2.5. Wahadło poddane jest si-\nle grawitacji 𝑚𝑔przyłożonej w środku ciężkości oddalonym o 𝑠od osi obrotu,\nmomentowioporówruchu 𝑀𝐹,momentowi 𝑀𝐾pochodzącemuodelementupo-\ndatnego oraz momentowi 𝑄𝑚𝑎𝑔wynikającemu z oddziaływania magnetycznego\npomiędzy magnesem a cewką elektryczną zasilaną prądem 𝑖(𝑡). Wykorzystując\nprawaNewtonaorazsiłyimomentyuwzględnionewmodelufizycznym,równa-\nnie dynamiczne układu ma postać\n𝐽¥𝜑+𝑚𝑔𝑠 sin𝜑+𝑀𝐹(¤𝜑) +𝑀𝐾(𝜑,¤𝜑) =𝑄𝑚𝑎𝑔(𝜑,𝑖(𝑡)), (2.1)\ngdzie𝐽jest masowym momentem bezwładności wahadła względem osi obrotu,\nnatomiast¥𝜑i¤𝜑to przyspieszenie i prędkość kątowa wahadła.\nBadania symulacyjne i eksperymentalne prowadzone na rzecz tej pracy roz-\nciągałysięnaprzestrzeniponadczterechlat.Takdługiczaspowodował,żestano-\nwisko doświadczalne, w pewnym stopniu, zmieniało się na skutek zużycia oraz\nprowadzonych modernizacji. Zmiany te miały wpływ na badania symulacyjne,\nponieważ za każdym razem należało dostosowywać równanie ruchu oraz para-\nmetry układu w odniesieniu do obecnego stanu stanowiska, tak aby jak najlepiej\nodwzorowywały jego zachowanie. Najczęściej zmiany te dotyczyły czynników\nwpływających na opory ruchu w łożyskach, a w mniejszym stopniu parametrów\nelementupodatnegoczyoddziaływaniamagnetycznego.Dlategoteżposzczegól-\nne wyniki badań zawarte w tej pracy, będą opatrzone konkretnym równaniem\nruchuwrazzparametrami,dlaktórychzostałyotrzymane.Pomimoróżnychmo-\ndelidynamicznychstosowanychpodczasbadań,wynikiotrzymanenaichpodsta-\nwie są jakościowo i ilościowo podobne, dzięki czemu można je ze sobą zestawiać\ni porównywać.\nRys. 2.5. Model fizyczny układu pojedynczego wahadła magnetycznego.\n362.3. DYNAMIKA NIELINIOWA WAHADŁA MAGNETYCZNEGO\nTabela 2.1. Parametry układu pojedynczego wahadła magnetycznego przyjęte\npodczas badań analityczno-numerycznych i numerycznych.\n𝐽 6.786·10−4kgm2𝑣𝑠0.835rad\ns\n𝑚𝑔𝑠 5.707·102Nm𝜒 5.759s\nrad\n𝑘𝑒 2.264·102Nm\nrad𝑐 6.735·10−5Nm s\nrad\n𝑐𝑒 1.282·10−4Nm s\nrad𝑎𝐼3.615·10−2Nm rad\nA\n𝑀𝑠 3.874·104Nm𝑏 1.818·10−2rad2\n𝑀𝑐 2.371·10−4Nm𝜇 6.886·10−4 Nm\n𝜎 9.367·10−1[]𝜅 3.125·10−1 rad\n𝜖𝑖 200 [−]\ngdzieza𝑀𝐾podstawionoformułę(2.6).Wprowadźmyteraznastępującepodsta-\nwienie zmiennej czasowej 𝑡=𝜏𝑡/𝑞𝑡, gdzie𝑞𝑡jest współczynnikiem skalującym\nrównym𝑞𝑡=p\n(𝑚𝑔𝑠 +𝑘𝑒)/𝐽,a𝜏𝑡bezwymiarowymczasem.Wrzeczywistości,pa-\nrametr𝑞𝑡odpowiadaczęstościdrgańswobodnychukładu,natomiastichczęstotli-\nwośćwyrazićmożnajako 𝑓𝑁=𝑞𝑡/2𝜋[Hz].Zatem,zmiennepołożenia,prędkości\ni przyspieszenia kątowego wahadła pozostają w następujących zależnościach:\n𝜑(𝑡) =𝜑(𝜏𝑡),\n¤𝜑(𝑡) =𝑑𝜑(𝑡)\n𝑑𝑡=𝑑𝜑(𝑡)\n𝑑𝜏𝑡/𝑞𝑡=𝑞𝑡𝑑𝜑(𝑡)\n𝑑𝜏𝑡=𝑞𝑡𝑑𝜑(𝜏𝑡)\n𝑑𝜏𝑡=𝑞𝑡𝜑′,\n¥𝜑(𝑡) =𝑑2𝜑(𝑡)\n𝑑𝑡2=𝑑2𝜑(𝑡)\n𝑑(𝜏𝑡/𝑞𝑡)2=𝑞2\n𝑡𝑑2𝜑(𝑡)\n𝑑𝜏2\n𝑡=𝑞2\n𝑡𝑑2𝜑(𝜏𝑡)\n𝑑𝜏2\n𝑡=𝑞2\n𝑡𝜑”.(2.28)\nPodstawiając zależności (2.28) do równania (2.27) otrzymamy oryginalny układ\n𝜑” +𝜑=𝑀𝑆𝐸(𝑞𝑡𝜑′)\n𝑚𝑔𝑠 +𝑘𝑒𝑐𝑒\n𝐽𝑞𝑡𝜑′+b𝑀𝑚𝑎𝑔(𝜑,𝑖𝑝(𝜏𝑡/𝑞𝑡))\n𝑚𝑔𝑠 +𝑘𝑒≡𝐺(𝜑,𝜑′,𝜏𝑡).(2.29)\nPonadto korzystającz aproksymacjifunkcji momentuoporów wyrażonej równa-\nniem (2.4) oraz aproksymacji momentu magnetycznego (2.23) otrzymamy układ\nprzybliżony\n𝜑” +𝜑=𝑀𝑆𝐸𝑎(𝑞𝑡𝜑′)\n𝑚𝑔𝑠 +𝑘𝑒𝑐𝑒\n𝐽𝑞𝑡𝜑′+b𝑀𝐴𝑚𝑎𝑔 6(𝜑,𝑖𝑝(𝜏𝑡/𝑞𝑡))\n𝑚𝑔𝑠 +𝑘𝑒≡𝐺𝑎(𝜑,𝜑′,𝜏𝑡).(2.30)\nWzależnościodparametrówukładu,przybliżonaanalizadynamicznamożebyć\nzastosowana kiedy prawa strona równań, tj. 𝐺(𝜑,𝜑′,𝜏𝑡)i𝐺𝑎(𝜑,𝜑′,𝜏𝑡), osiąga\nmałe wartości. Parametry układu jakie przyjęto podczas prezentowanych badań\nzawarte są w Tabeli 2.1. Biorąc pod uwagę wartości parametrów układu można\nobliczyć, że 𝑞𝑡= 10.8381/s, a częstotliwość drgań swobodnych 𝑓𝑁= 1.725Hz.\n512.3. DYNAMIKA NIELINIOWA WAHADŁA MAGNETYCZNEGO\nDoopracowaniametodyanalityczno-numerycznejopartejnametodzieuśred \nniania, zastosowana zostanie zamiana zmiennych prowadząca do układu ekwi-\nwalentnegozapisanegowpostacinormalnej.Zgodnieztradycyjnąmetodąuśred \nniania[138],rozwiązanieodpowiadającejednorodnemu(homogenicznemu)rów-\nnaniu (2.29) można zapisać w następującej postaci\n𝜑=𝑘sin(Ω𝑡𝜏𝑡+𝑢), (2.31)\ngdzie𝑘jestamplitudą, 𝑢przesunięciemfazowym,natomiastczęstość Ω𝑡=𝜋𝑓/𝑞𝑡\njestdwukrotniemniejszaniżczęstośćwymuszająca 2𝜋𝑓/𝑞𝑡.Zakładając,żewielko-\nści𝑘i𝑢są funkcjami czasu, równanie oryginalnego układu (2.29) można zapisać\nwpostacidwóchrównańróżniczkowychzwyczajnychpierwszegorzędu(zobacz\nZałącznik A)\n𝑘′=cos𝜃\nΩ𝑡\u0002\n𝑘sin𝜃(Ω2\n𝑡1) +𝐺(𝑘sin𝜃,𝑘Ω𝑡cos𝜃,𝜏𝑡\u0003\n≡𝑅1(𝑘,𝑢),\n𝑢′=sin𝜃\n𝑘Ω𝑡\u0002\n𝑘sin𝜃(Ω2\n𝑡1) +𝐺(𝑘sin𝜃,𝑘Ω𝑡cos𝜃,𝜏𝑡\u0003\n≡𝑅2(𝑘,𝑢),\n𝜃= Ω𝑡𝜏𝑡+𝑢,(2.32)\ngdzie𝑅1i𝑅2wyrażają prawe strony równań.\nBadaniarozpoczętoodprzypadkusłabegowymuszeniawahadłamagnetycz-\nnego, tzn. gdy składnik b𝑀𝑚𝑎𝑔(𝜑,𝑖𝑝(𝑡))zawarty w modelu początkowym (2.26)\nosiągał małe wartości. Aby tak się stało, amplituda sygnału prądowego została\nprzyjęta na poziomie 𝐼0= 0.04A. Ponadto ustalono, że jego częstotliwość bę-\ndzie równa 𝑓= 2.1Hz, a wypełnienie 𝑤= 27%. Dla tak przyjętych wartości\nparametrów sygnału prądowego, obliczono numerycznie przebiegi 𝑘i𝑢ekwi-\nwalentnego układu (2.32) i pokazano je na rys. 2.22, przy pomocy niebieskich\ni czerwonych linii ciągłych. Przebiegi charakteryzują się nieliniowymi profila-\nmi, w szczególności dotyczy to przebiegu przesunięcia fazowego 𝑢. Tradycyjna\nmetoda uśrednianiapolega naaproksymacjiprzebiegów 𝑘i𝑢poprzez stałewar-\ntości, które są równe średniej wartości danego przebiegu. W prezentowanych\nbadaniach oprócz tradycyjnego podejścia, wartości amplitudy i przesunięcia fa-\nzowego zostaną aproksymowane przy użyciu funkcji piłokształtnych [139, 140].\nWybór funkcji piłokształtnej jako funkcji aproksymującej, pozwala w ogólnym\nprzypadku na możliwość stosowania analitycznego całkowania prawych stron\n𝑅1,𝑅2równań (2.32) oraz operowanie niewielką liczbą nieznanych parametrów.\nWartopodkreślić,żedlabadanegoukładumamyczterywariantywyborufunkcji\naproksymującej przebiegi 𝑘i𝑢, a mianowicie:\n(i) amplituda 𝑘i faza𝑢będą aproksymowane przez ich stałe wartości średnie\n⟨𝑘⟩i⟨𝑢⟩;\n(ii) amplituda 𝑘będzie aproksymowana przez jej stałą średnią wartość ⟨𝑘⟩, na-\ntomiast faza 𝑢przez funkcję piłokształtną;\n(iii) amplituda 𝑘będzie aproksymowana przez funkcję piłokształtną, a faza\n𝑢przez jej stałą średnią wartość ⟨𝑢⟩;\n(iv) amplituda 𝑘i faza𝑢będą aproksymowane przez funkcje piłokształtne.\n522.3. DYNAMIKA NIELINIOWA WAHADŁA MAGNETYCZNEGO\nW celu rozpoczęcia procedury uśredniania, nieliniowe funkcje 𝑀𝑆𝐸oraz𝑀𝑚𝑎𝑔\nwchodzące w skład oryginalnego równania (2.29) zastąpiono przez ich aproksy-\nmacje𝑀𝑆𝐸𝑎ib𝑀𝐴𝑚𝑎𝑔 6, co jest jednoznaczne z wykorzystaniem równania układu\nprzybliżonego (2.30) do dalszych obliczeń.\nWariant (i) . Zarówno amplituda 𝑘jak i faza𝑢są aproksymowane przez jej\nśrednie wartości⟨𝑘⟩i⟨𝑢⟩. Wartości te można otrzymać z przyrównania wyni-\nku uśredniania prawych stron ekwiwalentnego układu (2.32) do zera, zgodnie\nz równaniami\nZ𝑇𝑡\n0𝑅1(⟨𝑘⟩,⟨𝑢⟩)𝑑𝜏𝑡= 0,Z𝑇𝑡\n0𝑅2(⟨𝑘⟩,⟨𝑢⟩)𝑑𝜏𝑡= 0, (2.33)\ngdzie𝑇𝑡=𝑞𝑡/𝑓jest okresem wymuszenia.\nZe względu na to, że w prowadzonej analizie składniki 𝑅1i𝑅2opisane są\nwoparciuofunkcję 𝐺𝑎równania(2.30),toichpostacieanalityczneniesąmożliwe\ndouzyskania.Dlategoteżchcącobliczyćwartości ⟨𝑘⟩i⟨𝑢⟩korzystajączzależności\n(2.33) w programie Mathematica , wykorzystano metody całkowania numeryczne-\ngoorazfunkcjęnumerycznegoznajdowaniapierwiastków FindRoot[...].Wartości\njakieotrzymanopoobliczeniach sąnastępujące: ⟨𝑘⟩= 0.09537radi⟨𝑢⟩= 7.31994\nrad.Przerywanelinienarysunkach2.22a,bprzedstawiająobliczonewartościśred \nnie przebiegów 𝑘i𝑢.\nWariant (ii) . Amplituda 𝑘jest aproksymowana przez jej średnią wartość\n⟨𝑘⟩= 0.09537rad, podczas gdy przesunięcie fazowe 𝑢jest opisane przez funk-\ncję piłokształtną, która jest funkcją o postaci odcinkowej (ang. piecewise function ).\nBiorąc pod uwagę ciągłość i okresowość przebiegu fazy 𝑢, odcinkowa funkcja\npiłokształtna wyrażona została jako\n𝑔𝑢(𝜏𝑡) =(\n𝑐1𝜏𝑡+𝑑1+𝑐1𝜈𝑤𝑇𝑡\n𝜈𝑤−1,dla0≤𝜏𝑡< 𝜈𝑤𝑇𝑡,\n𝑐1𝜈𝑤𝜏𝑡\n𝜈𝑤−1+𝑑1, dla𝜈𝑤𝑇𝑡≤𝜏𝑡≤𝑇𝑡,(2.34)\ngdzie𝜈𝑤=𝑤/100%, a𝑐1i𝑑1są stałymi parametrami. Łatwo zauważyć, że speł-\nnionajestzależność 𝑔𝑢(0) =𝑔𝑢(𝑇𝑡)nakrańcachdziedzinyorazzależnośćciągłości\nfunkcji w punkcie łączenia 𝜈𝑤𝑇𝑡, tj. wartość 𝑔𝑢obliczona „idąc” od lewej strony\ntegopunktujesttakasamajakwartość 𝑔𝑢obliczona„idąc”odprawejstronytego\npunktu,𝑔𝑢(𝜈𝑤𝑇𝑡) =𝑔𝑢(𝜈𝑤𝑇𝑡+).\nDo obliczenia wartości parametrów 𝑐1i𝑑1należy narzucić trzy warunki.\nPierwszywarunektozaprezentowanewcześniejrównania(2.33),zktórychwyni-\nkająwartościśrednie ⟨𝑘⟩i⟨𝑢⟩.Drugiwarunekpochodzizprzyrównaniawartości\nśredniej⟨𝑢⟩do wyrażenia matematycznego na wartość średnią funkcji 𝑔𝑢, co\nmożna zapisać jako\n⟨𝑢⟩=⟨𝑔𝑢⟩, (2.35)\nnatomiast samo wyrażenie na wartość średnią funkcji odcinkowej 𝑔𝑢ma postać\n⟨𝑔𝑢⟩=1\n𝑇𝑡Z𝑇𝑡\n0𝑔𝑢(𝜏𝑡)𝑑𝜏𝑡=𝑑1+𝑐1𝜈𝑤(1 +𝜈𝑤)\n2(𝜈𝑤1)𝑇𝑡. (2.36)\nZrównań(2.35)i(2.36)otrzymamyrównaniezdwomaniewiadomymi 𝑐1i𝑑1.Trze-\nci warunek wynika z podstawienia funkcji 𝑔𝑢, zdefiniowanej równaniem (2.34),\n532.3. DYNAMIKA NIELINIOWA WAHADŁA MAGNETYCZNEGO\ndo drugiego równania układu (2.32) z założeniem, że funkcja ta wyzeruje wynik\ndla wszystkich 𝜏𝑡. Prowadzi to do następującej zależności\nZ𝜈𝑤𝑇𝑡\n0{𝑐1𝑅2(⟨𝑘⟩,𝑔𝑢)}𝜏𝑡𝑑𝜏𝑡+Z𝑇𝑡\n𝜈𝑤𝑇𝑡\u001a𝜈𝑤𝑐1\n𝜈𝑤1𝑅2(⟨𝑘⟩,𝑔𝑢)\u001b𝜈𝑤𝜏𝑡\n𝜈𝑤1𝑑𝜏𝑡= 0.(2.37)\nObliczonenapodstawieukładurównań(2.36)i(2.37)wartościparametrów 𝑐1i𝑑1są\nnastępujące: 𝑐1=1.39569radi𝑑1= 5.62819rad.Odpowiadająceimprzybliżenia\nzostały pokazane na rys. 2.22c,d.\nWariant(iii) .Terazprzyjmijmy,żeprzesunięciefazowe 𝑢jestaproksymowane\nprzez średnią wartość ⟨𝑢⟩= 7.31994rad obliczoną na podstawie równań (2.33),\npodczas gdy amplituda 𝑘jest opisana przez funkcję piłokształtną\n𝑔𝑘(𝜏𝑡) =(\n𝑐2𝜏𝑡+𝑑2+𝑐2𝜈𝑤𝑇𝑡\n𝜈𝑤−1,dla0≤𝜏𝑡< 𝜈𝑤𝑇𝑡,\n𝑐2𝜈𝑤𝜏𝑡\n𝜈𝑤−1+𝑑2, dla𝜈𝑤𝑇𝑡≤𝜏𝑡≤𝑇𝑡.(2.38)\nDo obliczenia wartości parametrów 𝑐2i𝑑2posłużą nam równania podobne do\n(2.35) i (2.37), których wyrażenia są następujące\n⟨𝑘⟩= 0.09537rad=𝑑2+𝑐2𝜈𝑤(1 +𝜈𝑤)\n2(𝜈𝑤1)𝑇𝑡,\nZ𝜈𝑤𝑇𝑡\n0{𝑐2𝑅1(𝑔𝑘,⟨𝑢⟩)}𝜏𝑡𝑑𝜏𝑡+Z𝑇𝑡\n𝜈𝑤𝑇𝑡\u001a𝜈𝑤𝑐2\n𝜈𝑤1𝑅1(𝑔𝑘,⟨𝑢⟩)\u001b𝜈𝑤𝜏𝑡\n𝜈𝑤1𝑑𝜏𝑡= 0.\n(2.39)\nPo obliczeniach otrzymujemy, że 𝑐2= 0.02028rad i𝑑2= 0.11996rad, a odpowia-\ndające im przybliżenia zostały pokazane na rys. 2.22e,f.\nWariant (iv) . Ostatni wariant zakłada, że zarówno 𝑢i𝑘są aproksymowane\nprzezfunkcjepiłokształtnezdefiniowaneprzez(2.34)i(2.38).Wtedyniewiadome\n𝑐1,2i𝑑1,2muszą spełniać układ poniższych równań\nZ𝑇𝑡\n0𝑅1(⟨𝑔𝑘⟩,⟨𝑔𝑢⟩)𝑑𝜏𝑡= 0,Z𝑇𝑡\n0𝑅2(⟨𝑔𝑘⟩,⟨𝑔𝑢⟩)𝑑𝜏𝑡= 0,\nZ𝜈𝑤𝑇𝑡\n0{𝑐1𝑅2(𝑔𝑘,𝑔𝑢)}𝜏𝑡𝑑𝜏𝑡+Z𝑇𝑡\n𝜈𝑤𝑇𝑡\u001a𝜈𝑤𝑐1\n𝜈𝑤1𝑅2(𝑔𝑘,𝑔𝑢)\u001b𝜈𝑤𝜏𝑡\n𝜈𝑤1𝑑𝜏𝑡= 0,\nZ𝜈𝑤𝑇𝑡\n0{𝑐2𝑅1(𝑔𝑘,𝑔𝑢)}𝜏𝑡𝑑𝜏𝑡+Z𝑇𝑡\n𝜈𝑤𝑇𝑡\u001a𝜈𝑤𝑐2\n𝜈𝑤1𝑅1(𝑔𝑘,𝑔𝑢)\u001b𝜈𝑤𝜏𝑡\n𝜈𝑤1𝑑𝜏𝑡= 0,(2.40)\ngdzie wyrażenie⟨𝑔𝑢⟩=𝑑1+𝑐1𝜈𝑤(1+𝜈𝑤)\n2(𝜈𝑤1)𝑇𝑡, natomiast⟨𝑔𝑘⟩=𝑑2+𝑐2𝜈𝑤(1+𝜈𝑤)\n2(𝜈𝑤1)𝑇𝑡. W re-\nzultacie obliczone wartości są następujące: 𝑐1=1.37382rad,𝑑1= 5.65469rad,\n𝑐2= 0.02184rad,𝑑2= 0.12185rad, a przybliżone przebiegi amplitudy i fazy\nzostały pokazane na rys. 2.22g,h.\nRysunek 2.23 przedstawia porównanie przebiegów ruchu wahadła, otrzy-\nmanych na podstawie równania (2.31) z zaimplementowanymi aproksymacjami\namplitudy 𝑘i przesunięcia fazowego 𝑢(według czterech analizowanych powy-\nżejwariantów)orazrozwiązanianumerycznegootrzymanegodlaprzybliżonego\nukładu (2.30). Spośród czterech zaproponowanych wariantów aproksymacji dla\nprzebiegówamplitudyiprzesunięciafazowegoprzyjęto,żetowariant(iv)powi-\nnien w najlepszy sposób odwzorowywać wynik rozwiązania numerycznego ze\n552.3. DYNAMIKA NIELINIOWA WAHADŁA MAGNETYCZNEGO\nczęstotliwości 𝑓∈(3.2; 3.5)Hz,gdziepojawiająsiętrzywspółistniejącerozwiąza-\nnia.Ponadtoukładcharakteryzujesięnagłymiskokamiamplitud.Różnicepomię-\ndzy obliczonymi krzywymi a eksperymentem mogą mieć kilka przyczyn. Przy-\nbliżonyukładzewzględunaswojeuproszczenia,przedewszystkimlinearyzację\nfunkcji sinusoidalnej, znacznie bardziej odbiega od eksperymentu szczególnie\ndla dużych kątów wychylenia wahadła. Różnice mogą być również spowodowa-\nne zbyt krótkimi czasami przejściowymi przyjętymi podczas eksperymentu lub\nwynikiem niedokładności wykonania stanowiska, np. luzów w łożyskach czy\nlekko uginającej się podstawy cewki elektrycznej, co nie było brane pod uwagę\npodczas modelowania matematycznego.\nPodsumowując, warto zauważyć, że wyniki otrzymane przy pomocy opraco-\nwanej metody analityczno-numerycznej są jakościowo bardzo zbliżone do wyni-\nkówotrzymanychprzypomocymetodnumerycznychorazeksperymentu.Wpo-\nrównaniu z tradycyjną metodą uśredniania, która ogranicza się do obliczenia\nuśrednionej i stałej w czasie wartości przebiegu amplitudy i fazy rozwiązania\nokresowego, opracowana procedura pozwala na aproksymację tych przebiegów\nfunkcjąpiłokształtną.Wpływatonadokładniejszeopisanieprzebiegudynamicz-\nnego wspomnianej amplitudy i przesunięcia fazowego dla modeli z pulsującym\nwymuszeniem,kiedytowrozwiązaniuokresowympojawiająsię„podskoki”am-\nplitudy(rys.2.23).Przedstawionewtymparagrafiebadaniazostałyopublikowane\nwpracy[141].Dodatkowowartykuleprzeanalizowanoprzypadekzastosowania\nzaprezentowanej metody analityczno-numerycznej dla silnie nieliniowego ukła-\ndu, w którym nie przeprowadzono linearyzacji funkcji sinus.\nSzersza analiza odpowiedzi układu na zmiany parametru kontrolnego, za\nktóryprzyjętoczęstotliwość 𝑓sygnałuprądowego 𝑖𝑝(𝑡),zostałapokazananaeks-\nperymentalnychisymulacyjnychwykresachbifurkacyjnych(rys.2.25i2.26).Wy-\nkresy symulacyjne obliczone zostały dla układu startowego (2.26) i parametrów\npodanych w tabeli 2.2.\nBadania przeprowadzone były dla silnego wymuszenia, gdzie amplituda sy-\ngnału prądowego wynosiła 𝐼0= 1A, a wypełnienie 𝑤= 30%. Podczas ekspery-\nmentuczęstotliwośćsygnałuprądowegonarastała(lubmalała)wsposóbliniowy\nwczasie,przezcosygnałprądowywtymprzypadkumożnauznaćzatzw.sygnał\nświergotowy (ang. sweep chrip signal ). Wspomniane narastanie i zmniejszanie się\nczęstotliwości w czasie, odbywało się wolno ( 25·104Hz/s), tak aby zapewnić\nTabela 2.2. Parametry układu pojedynczego wahadła magnetycznego przyjęte\npodczas analizy bifurkacyjnej.\n𝐽 6.786·104kgm2𝑣𝑠0.733rad\ns\n𝑚𝑔𝑠 5.800·10−2Nm𝜒 5.759s\nrad\n𝑘𝑒 1.742·10−2Nm\nrad𝑐 7.369·10−5Nm s\nrad\n𝑐𝑒 1.282·104Nm s\nrad𝑎𝐼3.615·10−2Nm rad\nA\n𝑀𝑠 4.436·10−4Nm𝑏 1.818·10−2rad2\n𝑀𝑐 2.223·10−4Nm𝜖𝑖200 [−]\n582.3. DYNAMIKA NIELINIOWA WAHADŁA MAGNETYCZNEGO\nmodelumatematycznegoukładudladanejstałejwartościczęstotliwośćiwczasie\nodpowiadającym1200okresomsygnałuwymuszającego.Pierwsze1000okresów\nuznawanozaczasprzejściowyipomijano,apunktyzostatnich200okresówpre-\nzentowano na wykresie. Ponadto podczas symulacji prowadzonej dla ustalonej\nczęstotliwości jako warunki początkowe przyjmowano wartość położenia i pręd \nkości kątowej odpowiadające ostatniemu punktowi obliczonemu dla poprzedza-\njącejczęstotliwości.Biorącpoduwagęwynikianalizybifurkacyjnejzauważalnym\njest,żeruchukładumożebyćzarównochaotycznyjakiregularny.Ruchregularny\n(rozumiany jako okresowy) może przyjmować różne okresowości: 1-okresowość,\n2-okresowość, 4-okresowość, 6-okresowość, a także 10-okresowość. przypomnij-\nmy,żepojęcieokresowościdrgańrozumianejestjakoliczbaokresówwymuszenia\n(sygnałuprądowego)przypadającanajedenokresrozwiązania.Obliczonenume-\nryczniewykresybifurkacyjneodwzorowująprawiewszystkiezachowaniazareje-\nstrowane podczas eksperymentu. Zaobserwować można również występowanie\nwspółistniejących atraktorów np. dla częstotliwości 𝑓= 2.25Hz na obu wykre-\nsachzrys.2.25bi2.26bpojawiająsięrozwiązania2-okresowe,alesątodwaróżne\nrozwiązania.Rodzajrozwiązania(atraktora)zależyodwarunkówpoczątkowych.\nNa wykresach symulacyjnych mających odzwierciedlać eksperyment (rys. 2.25c\ni2.26c),istnienieniektórychokienprzypominającychrozwiązaniechaotycznemo-\nżebyćspowodowanezbytdługimczasemtrwaniaruchuprzejściowego,którynie\nzostał pominięty. Problem ten nie występuje na wykresach bifurkacyjnych wy-\nkonanych w sposób „klasyczny” (rys. 2.25d i 2.26d), dla przykładu 2-okresowe\nrozwiązanie widoczne dla 𝑓= 2.85Hz na rys. 2.26d zostało zweryfikowane eks-\nperymentalnie dla stałej częstotliwości i pokazane na portrecie fazowym z prze-\nkrojem Poincarégo na rys. 2.27f.\nPawiewszystkieodpowiedziukładuzarejestrowaneeksperymentalniezosta-\nły znalezione podczas symulacji numerycznych. Brakujące odpowiedzi zdołano\nujawnićdopieropozmianiewarunkówpoczątkowych;dotakichodpowiedzina-\nleżątepokazanenarys.2.25dwzakresieczęstotliwości 4÷4.5Hz(skrajnyprawy\nwykres) oraz 2.26d w zakresie częstotliwości 5÷5.5Hz (skrajny prawy wykres).\nPrzykładowe symulacyjne i eksperymentalne wykresy fazowe z przekrojami Po-\nincarégoprzedstawionezostałynarys.2.27.Naróżnicepomiędzysymulacyjnymi\na eksperymentalnymi trajektoriami fazowymi wpływa w pewnym stopniu spo-\nsóbprzetwarzaniadanycheksperymentalnych,ponieważwczasieeksperymentu\nrejestrowano tylko położenie kątowe, natomiast prędkość otrzymywana była po-\nprzez jego numeryczne różniczkowanie, co generowało pewne zakłócenia, które\nnastępnie należało odfiltrować. Wpływ tego błędu można zaobserwować np. na\nrys. 2.27b, gdzie punkt Poincarégo powinien znajdować się dokładnie w miejscu\nnagłego spadku prędkości (tak jak na symulacji) spowodowanego pojawieniem\nsię bariery w postaci pola magnetycznego. W przypadku rys.2.27j,n, na różnice\npomiędzy symulacją a eksperymentem największy wpływ miał fakt, że dla ba-\ndanych częstotliwości układ posiada różne współistniejące rozwiązania. Dlatego\nwprzypadkupojawieniasięnajmniejszychniedokładności(zaburzeń)wukładzie\nrzeczywistymwynikającychnp.zluzówwłożyskach,rozwiązaniatebyłybliskie\n„przeskakiwania” między sobą, co objawiało się brakiem idealnego pokrycia tra-\njektorii fazowej w kolejnych okresach drgań.\nZaprezentowana analiza bifurkacyjna potwierdziła znakomitą różnorodność\n612.3. DYNAMIKA NIELINIOWA WAHADŁA MAGNETYCZNEGO\nmuszą spełniać następującą zależność\n𝑉𝑠(𝜑𝐴)\n𝑉𝑠(𝜑𝐴) +𝑉𝑀𝑚𝑎𝑔(𝜑𝐴)= 0.999. (2.42)\nNa podstawie tej zależności określono wartość 𝜑𝐴=±0.348rad.\nPrzejście z układu o potencjale jednodołkowym do dwudołkowego (i na od-\nwrót) dyktowane jest przez sygnał prądowy, który w badanym przypadku jest\nprostokątny i pulsujący. Ze względu na skokowy charakter tego sygnału uznać\nmożna,żeprzejściepomiędzypotencjałamiukładunastępujeprawienatychmiast.\nFakt ten umożliwia wprowadzenie pewnego uproszczenia w opisie dynamiki\nukładu. Mianowicie można przyjąć, że układ posiada dwa stany:\n(1) kiedy cewka nie jest zasilana prądem, wtedy mamy do czynienia z układem\njednodołkowym;\n(2) kiedy cewka jest zasilana prądem o natężeniu 𝐼0, wtedy mamy do czynienia\nz układem dwudołkowym.\nModel matematyczny takiego układu można zapisać jako układ dwóch równań\nróżniczkowych,zktórychkażdeodpowiadainnemustanowi.Pierwszerównanie\nmodelujące stan 1 układu zapisać można jako\n𝐽¥𝜑+𝑚𝑔𝑠 sin𝜑+𝑀𝑆𝐸(¤𝜑) +𝑀𝐾(𝜑,¤𝜑) = 0, (2.43)\nnatomiast drugie równanie odpowiadające stanowi 2 wyraża się\n𝐽¥𝜑+𝑚𝑔𝑠 sin𝜑+𝑀𝑆𝐸(¤𝜑) +𝑀𝐾(𝜑,¤𝜑)b𝑀𝑚𝑎𝑔(𝜑,𝐼0) = 0.(2.44)\nOba stany w odniesieniu do ich energii potencjalnej zostały zobrazowane na rys.\n2.29a,b.Zdynamicznegopunktuwidzenia,dolnepołożenierównowagiwahadła\nbędącepunktemosobliwymdlakażdegozdwóchanalizowanychstanówmainny\ncharakter. Gdy przez cewkę elektryczną nie płynie prąd (stan 1), punkt ten jest\nogniskiemstabilnym,copokazanonarys.2.29c.Wsytuacji,gdycewkaelektryczna\njest zasilana i wahadło jest odpychane, punkt osobliwy staje się siodłem tak jak\nto przedstawiono na 2.29d. Przełączanie pomiędzy równaniami (2.43) i (2.44)\nnastępuje w odpowiednich chwilach czasowych zależnych od okresu 𝜏= 1/𝑓\ni wypełnienia 𝑤sygnału prądowego, co zobrazowano na rys. 2.30a i zapisać\nmożna w następujący sposób:\nStan 1→Stan 2 dla 𝑡=𝑘𝑛𝜏, 𝑘𝑛∈N;\nStan 2→Stan 1 dla 𝑡=𝜏𝑤\n100%+𝑘𝑛𝜏, 𝑘𝑛∈N. (2.45)\nJak już wspomniano, układ będąc w stanie 1 posiada jedno stabilne położenie\nrównowagi (𝜑,¤𝜑) = (0,0), podczas gdy przechodząc do stanu 2 położenie to za-\nmienia się w niestabilne siodło. W sąsiedztwie tego siodła powstają dwa stabilne\npołożenia równowagi symetryczne względem kąta 𝜑= 0rad. Z tego względu\ndynamikę układu można rozpatrywać w kategorii periodycznie zmieniającego\n642.3. DYNAMIKA NIELINIOWA WAHADŁA MAGNETYCZNEGO\nzostały sztucznie zmienione, w celu lepszej wizualizacji ich częstotliwości i wy-\npełnień. Czerwony przebieg odpowiada sygnałowi prądowemu z minimalnym\nwypełnieniem 𝑤𝑚𝑖𝑛(czerwona linia na rys. 2.35a). Natomiast zielony przebieg\nodwzorowuje sygnał prądowy o maksymalnym wypełnieniu 𝑤𝑚𝑎𝑥(zielona linia\nnarys.2.35a).Niebieskiprzebiegodzwierciedlasygnałprądowyzwypełnieniem\n𝑤𝑛𝑇,któryprowadzićmadowielookresowegorozwiązaniaotakiejsamejtrajekto-\nriifazowejjakdlaruchujednookresowego.Abytaksięstało,wartośćwypełnienia\n𝑤𝑛𝑇rozwiązania wielookresowego musi spełniać następujące warunki:\n•okres𝜏rozwiązaniajednookresowegojestwielokrotnościąokresu Ω𝑛𝑇roz-\nwiązania wielookresowego, dlatego\n𝜏=𝑛·Ω𝑛𝑇dla𝑛∈N+\\{1}. (2.47)\n•zależność pomiędzy czasami sygnałów o wypełnieniach 𝑤𝑚𝑖𝑛,𝑤𝑚𝑎𝑥i𝑤𝑛𝑇\nsą następujące:\n𝑡𝑚𝑖𝑛𝑂𝑁≤𝜏𝑂𝑁,\n𝑡𝑚𝑎𝑥𝑂𝐹𝐹≤𝜏𝑂𝐹𝐹.(2.48)\nTeraz można zapisać równania, które pozwolą na przejście z wartości czasów na\nwartości wypełnień:\n𝜏𝑂𝑁=𝑤𝑛𝑇·Ω𝑛𝑇,\n𝜏𝑂𝐹𝐹= Ω𝑛𝑇𝜏𝑂𝑁= Ω𝑛𝑇(1𝑤𝑛𝑇),\n𝑡𝑚𝑖𝑛𝑂𝑁 =𝑤𝑚𝑖𝑛·𝜏,\n𝑡𝑚𝑎𝑥𝑂𝑁 =𝑤𝑚𝑎𝑥·𝜏,\n𝑡𝑚𝑎𝑥𝑂𝐹𝐹 =𝜏𝑡𝑚𝑎𝑥𝑂𝐹𝐹 =𝜏(1𝑤𝑚𝑎𝑥).(2.49)\nKorzystając z równań (2.47) (2.49), otrzymujemy następujące zależności jakim\npodlega wypełnienie 𝑤𝑛𝑇rozwiązania wielookresowego w stosunku do wypeł-\nnień granicznych 𝑤𝑚𝑖𝑛i𝑤𝑚𝑎𝑥rozwiązania jednookresowego:\n𝑤𝑛𝑇≥𝑛·𝑤𝑚𝑖𝑛,\n𝑤𝑛𝑇≤1𝑛(1𝑤𝑚𝑎𝑥).(2.50)\nObliczenianumeryczneprzeprowadzonedlanierówności(2.50)wykazały,żedla\nbadanego układu są one spełnione tylko dla 𝑛= 2i𝑛= 3, co odpowiada ruchom\ndwu- i trój-okresowym w jednym dołku potencjału.\nZakresyparametrów 𝑓i𝑤dladwu itrój okresowychrozwiązańpokazanezo-\nstałynarys.2.39a.Wceluweryfikacjiobliczeńzobszarówdwu itrój-okresowości\nwybrano po dwa punkty odpowiadające różnym 𝑓i𝑤(kołowe znaczniki na rys.\n2.39a), a następnie przeprowadzono dla nich symulacje, których wyniki zesta-\nwiono z eksperymentem, co pokazano rys. 2.39b,c. Warto podkreślić, że w stre-\nfie aktywnej znajduje się tylko jeden punkt Poincarégo, natomiast kolejne punk-\nty zwiększające okresowość znajdują się poza nią. Wnioskować więc można, że\nw tego typu układzie ze zlokalizowanym wymuszeniem magnetycznym, można\nwsposób„sztuczny”zmieniaćokresowośćrozwiązaniabezwpływunaprzebieg\ntrajektorii fazowej tego rozwiązania.\n722.3. DYNAMIKA NIELINIOWA WAHADŁA MAGNETYCZNEGO\nzwizualizowananarys.2.42,przypomocykrzywychoznaczonychjakoKAiKA’.\nKrzywaKAjestzbioremwartościpunktów (𝜑𝑘,𝜔𝑘),podczasgdykrzywaKA’jest\nzbiorem wartości punktów (𝜑𝑙𝑘,𝜔𝑙𝑘). Ton koloru krzywych przyporządkowuje\nkonkretny punkt (𝜑𝑘,𝜔𝑘)do konkretnego punktu (𝜑𝑙𝑘,𝜔𝑙𝑘). Dodatkowo, punkt\n𝜑𝑆= 0.274rad znajdujący się na granicy obszarów i w miejscu, gdzie styka się\nona z linią¤𝜑= 0rad/s jest punktem równowagi dla wahadła znajdującego się\nw stanie 2.\nRysunek 2.43 przedstawia przykładowe przecięcia się trajektorii fazowych na\npłaszczyźnie fazowej obliczone dla scenariuszy II i III. Czerwone linie odpowia-\ndają układowi znajdującemu się w stanie 2 (równanie (2.44)), a niebieskie odpo-\nwiadają układowi znajdującemu się w stanie 1 (równanie (2.43)). Na wykresach\nzaznaczone zostały punkty (𝜑𝑘,𝜔𝑘)i(𝜑𝑙𝑘,𝜔𝑙𝑘). Trajektorie fazowe pokazane na\nrys. 2.43a,b odzwierciedlają drgania układu według scenariusza II, natomiast te\npokazane na rys. 2.43c,d według scenariusza III. Na wszystkich czterech wykre-\nsach fazowych, trajektorie wyznaczone dla układu w stanie 2 (czerwone linie)\nsą takie same, podczas gdy trajektorie wyznaczone dla układu w stanie 1 (nie-\nbieskie linie) są różne. Wykresy na rys. 2.43a,c pokazują dwa różne przypadki\nruchówokresowychotrzymanychdlatychsamychpunktów (𝜑𝑘,𝜔𝑘),alerożnych\npunktów (𝜑𝑙𝑘,𝜔𝑙𝑘), to samo dotyczy rys. 2.43b,d. Zauważyć można, że zgodnie\nz wcześniejszym założeniem, trajektorie obliczone dla układu w stanie 1 i 2 poza\nstrefa aktywną są niemal identyczne, a wewnątrz niej znacząco się od siebie róż-\nnią.Uwagęmogąprzyciągnąćdrganiapokazanenarys.2.43a,c,którychtrajektorie\nfazoweposiadają„ostre”przejściapomiędzyfragmentamiodpowiadającymiróż \nnym stanom układu. Z punktu widzenia analizy dynamicznej opierającej się na\nprzełączaniu pomiędzy równaniami (2.43)-(2.44); takie rozwiązania są możliwe,\naczkolwiekniezawszesąonemożliwedootrzymanianapodstawiepoczątkowe-\ngo „ciągłego” modelu matematycznego (2.26). Przyczyna ta tkwi prawdopodob-\nnie w pominięciu przez model „dyskretny” tj. opisany równaniami (2.43)-(2.44),\nciągłości zjawiska fizycznego jakim jest narastanie i opadanie sygnału prądowe-\ngo w obwodzie cewki elektrycznej. W przeciwieństwie do modelu dyskretnego,\nw którym występują tylko dwa stany zmieniające się w sposób nagły, w modelu\nciągłym ze względu na stopniową zmianę sygnału prądowego można wyróżnić\nstany pośrednie, co zostało zobrazowane na rys. 2.44a. Zauważyć można, że dla\nchwiliprzełączeniasięstanówz1na2wmodeludyskretnym(krzywa2),sygnał\nprądowydlamodeluciągłego(krzywa1)osiągadopieropołowęswojejamplitudy,\nco zostało zaznaczone czerwonym punktem na krzywej 1. Punkt ten traktowany\njest w modelu ciągłym jako punkt wymuszenia. Fakt ten powoduje, że podczas\nużywaniamodeludyskretnego,częśćinformacjiowpływienarastaniaiopadania\nsygnałuprądowegonadynamikęukładujestpomijana.Narys.2.44b,c,duwypu-\nklonowpływstopniowejzmianysygnałuprądowegonatrajektoriefazoweukładu\ndla analizowanych wcześniej scenariuszy ruchu. Można zauważyć, że trajektorie\nodpowiadającemodelowiciągłemu(krzywe1)są„gładsze”wsąsiedztwiepunk-\ntu wymuszenia niż w przypadku modelu dyskretnego (krzywe 2). Dlatego też,\nrozwiązanie obliczone z modelu dyskretnego i przedstawione na rys. 2.43c jest\nniemożliwe do otrzymania (w takiej formie) z modelu ciągłego układu. Ponad \nto na rys. 2.44c widać wyraźnie, że zmiana stanu układu z 1 na 2 dla modelu\ndyskretnego powoduje nagłą zmianę prędkości wahadła i następuje dokładnie\n76Rozdział 3\nUkład dwóch sprzężonych\nwahadeł\nRozdział poświęcony jest badaniom dwóch układów składających się z dwóch\nsprzężonych torsyjnie wahadeł. Pierwszy układ zbudowany jest z wahadła ma-\ngnetycznego połączonego z drugim wahadłem, które jest wahadłem fizycznym.\nDrugi układ składa się z dwóch słabo sprzężonych wahadeł magnetycznych.\nPrzedstawione zostaną badania numeryczne i eksperymentalne dynamiki tych\nukładów, a ponadto dla układu z dwoma wahadłami magnetycznymi opraco-\nwane zostanie sterowanie przepływem energii między nimi przy wykorzystaniu\npola magnetycznego. Wyniki zawarte w tym rozdziale udowadniają drugą z po-\nstawionych w pracy tez badawczych.\n3.1 Dynamika nieliniowa dwóch sprzężonych\nwahadeł\nPodrozdział zawiera opis stanowiska badawczego, modelowanie matematyczne\noraz badania dynamiki układu składającego się z dwóch sprzężonych torsyjnie\nwahadeł, z których tylko jedno jest wahadłem magnetycznym.\n3.1.1 Stanowisko badawcze\nZdjęciestanowiskabadawczegopokazanezostałonarys.3.1.Stanowiskotobazuje\nna stanowisku pojedynczego wahadła magnetycznego (rys. 2.1). Z tego względu\nopisbudowystanowiskazostanieograniczonytylkodonajważniejszychelemen-\ntów potrzebnych do zrozumienia jego działania. Wahadła oznaczone zostały nu-\nmerami(1 2).Wahadło(2)wyposażonezostałowmagnesneodymowy(3)ośred \nnicy 22 mm i wysokości 10 mm, podczas gdy wahadło (1) wyposażone zostało\nwelementmosiężny(4)otakichsamychwymiarachimasiejakmagneswahadła\n(2). Pod wahadłami umieszczone zostały dwie takie same cewki elektryczne (5).\nZe względu na fakt, że wahadło (1) nie posiada magnesu, tylko element niema-\ngnetyczny,cewkaznajdującasiępodnimniebędziezasilanasygnałemprądowym\n793.1. DYNAMIKA NIELINIOWA DWÓCH SPRZĘŻONYCH WAHADEŁ\nRys. 3.1. Stanowisko eksperymentalne dwóch sprzężonych wahadeł, z których\njedno jest magnetyczne; gdzie: 1,2 wahadła, 3 magnes neodymowy,\n4 mosiężny element niemagnetyczny, 5 cewka elektryczna, 6 wał\ni 7 - gumowy element podatny.\npodczas prowadzonych badań. Na końcach mosiężnych wałów (6) wahadeł za-\nmontowano specjalne uchwyty i połączono je ze sobą przy pomocy gumowego\nelementupodatnego(7).Jesttotensamelement,któryzostałużytywstanowisku\npojedynczego wahadła magnetycznego.\nPodczas badań, cewka wahadła 2 zasilana była takim samym prostokątnym\npulsującym sygnałem prądowym jak w przypadku pojedynczego wahadła ma-\ngnetycznego (rys. 2.3).\n3.1.2 Modelowanie matematyczne\nModel fizyczny analizowanego układu zaprezentowano na rys. 3.2. Przyjmując,\nżewahadłapodwzględemmasyigeometriisątakiesameuznano,żeichsiłycięż \nkości wynoszą 𝑚𝑔i leżą w odległościach 𝑠1,2=𝑠od osi obrotu. Wstępne badania\neksperymentalne pokazały, że pomimo działań mających na celu utrzymanie sy-\nmetrycznychwłasnościmechanicznychpomiędzywahadłami,momentyoporów\nruchu𝑀𝐹1,2nie były takie same pod względem wielkości tłumienia drgań. Gu-\nmowy element podatny sprzęga wały momentem oznaczonym jako 𝑀𝐾. Jak już\nwspomnianowopisiestanowiskaeksperymentalnego,tylkowahadło2poddane\njest działaniu momentu pola magnetycznego 𝑄𝑚𝑎𝑔, generowanego przez cewkę\nelektryczną zasilaną pulsującym prostokątnym sygnałem prądowym 𝑖𝑝.\nNapodstawieprzyjętegomodelufizycznegoorazwykorzystującprawaNew-\ntona zapisać można dynamiczne równania ruchu układu, w następującej postaci\n𝐽1¥𝜑1+𝑚𝑔𝑠 sin𝜑1+𝑀𝐹1(¤𝜑1)𝑀𝐾(𝜑1,𝜑2,¤𝜑1,¤𝜑2) = 0,\n𝐽2¥𝜑2+𝑚𝑔𝑠 sin𝜑2+𝑀𝐹2(¤𝜑2) +𝑀𝐾(𝜑1,𝜑2,¤𝜑1,¤𝜑2) =𝑄𝑚𝑎𝑔(𝜑2,𝑖𝑝(𝑡)),(3.1)\ngdzie𝐽1,2sąmasowymimomentamibezwładnościwahadełwzględemosiobrotu,\n803.1. DYNAMIKA NIELINIOWA DWÓCH SPRZĘŻONYCH WAHADEŁ\nRys. 3.2. Model fizyczny dwóch sprzężonych wahadeł, z których wahadło 1 jest\nwahadłem fizycznym, a wahadło 2 magnetycznym.\nnatomiast¥𝜑1,2i¤𝜑1,2toprzyspieszeniaiprędkościkątowewahadeł.Zewzględuna\nzwiększeniesięliczbystopniswobodyukładu,podczasbadańnumerycznychpo-\nstanowiono wykorzystać mniej skomplikowany model oporów ruchu 𝑀𝐹1,2niż\nten wykorzystywany w układzie pojedynczego wahadła magnetycznego. Mia-\nnowicie zaimplementowano prosty model Coulomba 𝑀𝐶𝑀wraz z tłumieniem\nwiskotycznym, który opisany jest równaniem (2.2). Ponadto zaimplementowa-\nno model (2.6) jako moment 𝑀𝐾elementu gumowego, gaussowski model (2.14)\nmomentu magnetycznego jako moment 𝑄𝑚𝑎𝑔i prostokątny model sygnału prą-\ndowego𝑖𝑝(𝑡)wyrażony równaniem (2.24). Po wstawieniu wymienionych modeli\nmatematycznychdorównańruchu(3.1),otrzymamynastępującymodelmatema-\ntyczny dwóch sprzężonych torsyjnie wahadeł\n𝐽1¥𝜑1+𝑚𝑔𝑠 sin𝜑1+𝑀𝑐1tgh(𝜀𝑐¤𝜑1)+𝑐𝑤1¤𝜑1+𝑐𝑒(¤𝜑1¤𝜑2) +𝑘𝑒(𝜑1𝜑2) = 0,\n𝐽2¥𝜑2+𝑚𝑔𝑠 sin𝜑2+𝑀𝑐2tgh(𝜀𝑐¤𝜑2)+𝑐𝑤2¤𝜑2+𝑐𝑒(¤𝜑2¤𝜑1) +𝑘𝑒(𝜑2𝜑1) =\n=b𝑀𝑚𝑎𝑔(𝜑2,𝑖𝑝(𝑡)).(3.2)\nWyrażenia 𝑐𝑤1,𝑐𝑤2pochodzą z modelu Coulomba i wyrażają tłumienia wisko-\ntyczne poszczególnych wahadeł.\nWartości parametrów użyte podczas obliczeń numerycznych zostały umiesz-\nczone w tabeli 3.1. Rys. 3.3 przedstawia porównanie wyników symulacji modelu\n(3.2) układu i eksperymentu dla przypadku ruchu swobodnego oraz wymuszo-\nnego, gdy parametry sygnału prądowego były następujące: 𝐼0= 1A,𝑓= 2Hz,\n𝑤= 25%. Zauważyć można, że pomimo zastosowania uproszczonego modelu\noporów ruchu w równaniach dynamicznych układu, zbieżność wyników symu-\nlacyjnych i eksperymentalnych jest wysoka.\n813.2. STEROWANIE PRZEPŁYWEM ENERGII MIĘDZY WAHADŁAMI\nnumerycznych, prace te zawierają spostrzeżenia dotyczące energii potencjalnej\nukładu, ale także uproszczone badania analityczne oparte na metodzie bilansu\nharmonicznych.\n3.2 Sterowanie przepływem energii między\nwahadłami\nWtympodrozdzialeprzedstawionyzostaniesposóbsterowaniawymianąenergii\npomiędzy dwoma słabo sprzężonymi wahadłami magnetycznymi spełniający-\nmi warunek rezonansu 1:1. Sterowanie to odbywać się będzie przy użyciu pól\nmagnetycznych generowanych przez cewki znajdujące się pod nimi. Wykazane\nzostanie,żeodpowiedniogenerowanepolamagnetycznecewekmogąskutecznie\nzmieniaćenergiepotencjalneposzczególnychwahadełwtakisposób,żekierunek\nprzepływuenergiipomiędzynimibędzieodbywałsięwsposóbzgóryzałożony.\n3.2.1 Stanowisko badawcze\nStanowiskobadawczedwóchsłabosprzężonychwahadełmagnetycznychpokaza-\nnezostałonarys.3.7.Stanowiskotojestprawietakiesamo,jaktowykorzystywane\ndobadańdynamicznychzrys.3.1,alezawierapewnemodyfikacje.Pierwszazmia-\nnadotyczyelementusprzęgającegowaływahadeł,bowomawianymstanowisku\nzaelementsprzęgającyposłużyłastalowasprężynatorsyjna(pozycja7narys.3.7).\nCharakteryzujesięonaznaczniemniejsząsztywnościąitłumieniem,niżwykorzy-\nstywanywcześniejgumowyelementpodatny.Sprężynazostaławykonanaręcznie\nz drutu o średnicy 0.6 mm i posiadała 4 zwoje o średnicy zewnętrznej 57 mm.\nDruga zmiana dotyczy wyposażenia wahadeł, ponieważ oba z nich wyposażone\nbyły w magnesy neodymowe. Dodatkowo podczas eksperymentów wykorzysty-\nwano dwie różne pod względem rozmiarów pary magnesów neodymowych \ndużych i małych. Duże magnesy cechowały się 22 mm średnicą, 10 mm wysoko-\nścią i masą wynoszącą 28.36 g, podczas gdy małe magnesy charakteryzowały się\n14 mm średnicą, 10 mm wysokością i 11.42 g masy.\nKolejnamodyfikacjadotyczysposobuzasilaniacewekelektrycznych(pozycje\n5i6)znajdującychsiępodwahadłamiigenerującychpolamagnetyczne.Wceluza-\npewnieniakontroliprzepływuenergiimiędzywahadłami,obiecewkielektryczne\nmusząmiećzdolnośćgenerowaniapólmagnetycznychzarównoprzyciągających,\njak i odpychających magnesy z określoną siłą. Z tego względu konieczna była\nzmiana sposobu zasilania cewek, tak aby sygnały prądowe mogły przyjmować\nróżne przebiegi o różnych znakach. Jako źródło zasilania cewek użyto sterowal-\nnegozasilaczalaboratoryjnegoRohde&Schwarz ®NGL202.Zasilacztenpozwala\nna wygenerowanie sygnału prądowego 𝑖(𝑡)o zadanym w programie LabView\nprzebiegu, jednakże sygnał ten może być tylko jednego znaku tzn. tylko dodatni\nlubtylkoujemny.Dlategozmianaznakusygnałuprądowegozrealizowanazostała\nprzy użyciu osobnego, specjalnie wykonanego układu elektronicznego opartego\nna przekaźnikach. Układ ten działa na zasadzie mostka H, którego zadaniem\njest zmienianie kierunku przepływu prądu w obwodzie cewki poprzez fizycz-\nną zmianę polaryzacji jej zasilania. Zmiana znaku sygnału prądowego odbywa\n853.2. STEROWANIE PRZEPŁYWEM ENERGII MIĘDZY WAHADŁAMI\nłającego na wahadła. Na podstawie przyjętych założeń, dynamiczne równania\nruchu badanego układu prezentują się w następującej postaci\n𝐽¥𝜑1=𝑀𝑐1sgn(¤𝜑1)𝑐𝑒𝑡(¤𝜑1¤𝜑2)𝑚𝑔𝑠\u0012\n𝜑11\n6𝜑3\n1\u0013\n𝑘𝑒𝑡(𝜑1𝜑2)+\nb𝑀𝑚𝑎𝑔(𝜑1,𝑖1(𝑡)),\n𝐽¥𝜑2=𝑀𝑐2sgn(¤𝜑2)𝑐𝑒𝑡(¤𝜑2¤𝜑1)𝑚𝑔𝑠\u0012\n𝜑21\n6𝜑3\n2\u0013\n𝑘𝑒𝑡(𝜑2𝜑1)+\n𝑀𝑚𝑎𝑔(𝜑2,𝑖2(𝑡)),(3.3)\ngdzie momenty magnetyczne 𝑀𝑚𝑎𝑔opisane są gaussowskim modelem (2.14).\nWartości parametrów występujących w równaniach (3.3) zawarto w tabeli 3.2,\nuwzględniając rozmiary zastosowanych magnesów.\nTabela 3.2. Parametry układu dwóch skrętnie sprzężonych wahadeł z podziałem\nna rozmiary magnesów.\nDuże magnesy Małe magnesy\n𝑎𝐼 8.036·10−3Nm·rad·A−1𝑎𝐼 3.635·10−3Nm·rad·A−1\n𝑏 30.810·10−3rad2𝑏 43.366·10−3rad2\n𝑀𝑐12.500·104Nm 𝑀𝑐13.114·104Nm\n𝑀𝑐21.600·10−4Nm 𝑀𝑐22.705·10−4Nm\n𝑐𝑒𝑡 9.615·10−6Nms·rad−1𝑐𝑒𝑡 9.615·10−6Nms·rad−1\n𝑘𝑒𝑡 3.999·103Nm·rad1𝑘𝑒𝑡 3.999·103Nm·rad1\n𝐽 6.787·10−4kgm2𝐽 5.675·10−4kgm2\n𝑚𝑔𝑠 5.840·102Nm 𝑚𝑔𝑠 5.018·102Nm\n3.2.3 Badania wstępne\nW tym paragrafie analizowane będą reguły jakimi powinny rządzić się sygnały\nprądowe𝑖1i𝑖2zasilającecewkiisterująceprzepływemenergiimiędzywahadłami.\nBadania wstępne rozpoczęto od wyznaczenia wyrażenia na energię potencjalną\nzlinearyzowanego wokół punktu równowagi układu zachowawczego wahadeł.\nW tym celu przyjęto, że w układzie nie występuje tłumienie oraz że wzięte pod\nuwagęzostanątylkoteczłonyenergiipotencjalnychpolagrawitacyjnegoimagne-\ntycznego, których stopień wielomianu jest nie większy niż dwa. Wyprowadzenie\nwzorów na poszczególne człony energii potencjalnej zawarto w Załączniku B.\nZatem, wyrażenie na energię potencjalną zlinearyzowanego układu zachowaw-\nczego odniesione do masowego momentu bezwładności wahadeł, przedstawia\n883.2. STEROWANIE PRZEPŁYWEM ENERGII MIĘDZY WAHADŁAMI\nsię następująco\n𝑉𝐽(𝜑1,𝜑2) =1\n2Ω2\u0002\n𝜑2\n1+𝛽(𝜑2𝜑2)+𝜑2\n2\u0003\n𝑎𝐼\n𝑏𝐽\u0010\n𝑖1𝜑2\n1+𝑖2𝜑2\n2\u0011\n,(3.4)\ngdzie Ω =p\n𝑚𝑔𝑠/𝐽jest częstością drgań własnych zlinearyzowanych wahadeł,\na parametr 𝛽=𝑘𝑒𝑡/𝑚𝑔𝑠odpowiada za względną „siłę” sprzężenia. Wartości\ntych parametrów jaki i innych pozostałych wykorzystywanych podczas badań\nsymulacyjnychzawartowtabeli3.3zpodziałemnarozmiarymagnesówwahadeł.\nTabela 3.3. Zredukowane parametry układu dwóch skrętnie sprzężonych waha-\ndeł z podziałem na rozmiary magnesów.\nDuże magnesy Małe magnesy\nΩ 9.276 s1Ω 9.403 s1\n𝛽 6.849·102rad1𝛽 7.969·102rad1\n𝜁11.985·10−2s−1𝜁12.918·10−2s−1\n𝜁21.270·102s1𝜁22.534·102s1\n𝛼 7.636·10−4rad−1𝛼 9.009·10−4rad−1\nFunkcja opisująca energię potencjalną 𝑉𝐽jest zależna od czasu przez wystę-\npowanie w niej prądów 𝑖1(𝑡)i𝑖2(𝑡). Wartości sygnałów prądowych wpływają\nna kształt powierzchni potencjału, a tym samym na lokalne zachowania dyna-\nmiczne układu wahadeł, tak jak pokazano to na wykresach z rys. 3.10. O ile\nobecność pola magnetycznego w układzie nie ma wpływu na położenie punktu\nstacjonarnego (𝜑1,𝜑2) = (0,0), o tyle wartości sygnałów prądowych mogą wpły-\nwaćnajegorodzaj.Diagramzrys.3.10pokazuje,jakwartościenergiipotencjalnej\n𝑉𝐽(𝜑1,𝜑2) =𝑐𝑜𝑛𝑠𝑡.układuewoluująwsąsiedztwiepunktustacjonarnego (0,0)na\nskutek zmian wartości sygnałów prądowych. Eliptyczne kształty energii poten-\ncjalnej w lewej dolnej części diagramu potwierdzają, że ujemne znaki sygnałów\nprądowych 𝑖1i𝑖2prowadządominimumenergiipotencjalnejukładuizapewnia-\nją stabilność położenia równowagi. Natomiast zwiększanie wartości jednego lub\ndwóchsygnałówprądowychmożeprowadzićdoprzekształceniasiępowierzchni\nenergiipotencjalnejukładuzwklęsłejnawypukłą,atakżesiodło.Oznaczatowte-\ndy,żeodpychającycharakteroddziaływaniamagnetycznegodominujewukładzie\nnad momentami przywracającymi układ do położenia równowagi, do których\nnależą momenty sił grawitacji oraz moment sprzęgający wahadła. Granice dla\ntrzech różnych rodzajów punktu stacjonarnego (0,0) z rys. 3.10 wyznaczono na\npodstawieczęstościdrgańwłasnychzlinearyzowanegoukładuprzyzałożeniu,że\nsygnały prądowe są stałe (obliczenia znajdują się w Załączniku B):\n𝜔2\n1,2= (1 +𝛽)Ω2𝑎𝐼\n𝑏𝐽(𝑖2\n1+𝑖2\n2)∓s\n𝛽2Ω4+𝑎2\n𝐼\n𝑏2𝐽2(𝑖2\n1𝑖2\n2). (3.5)\n893.2. STEROWANIE PRZEPŁYWEM ENERGII MIĘDZY WAHADŁAMI\nRys. 3.10. Ewolucje energii potencjalnej (3.4) układu w sąsiedztwie punktu rów-\nnowagi (0,0)obliczone dla różnych stałych wartości prądów 𝑖1i𝑖2ce-\nwek.Ciemnoniebieskiregionzeliptycznymikształtamiodpowiadami-\nnimum energii potencjalnej, podczas gdy jasnoniebieski region odpo-\nwiada siodłom. Jasny region znajdujący się w prawym górnym rogu\nodpowiada maksimum energii potencjalnej.\nObie z obliczonych częstości dla ciemnoniebieskiego regionu znajdującego się\nw lewym dolnym rogu rys. 3.10 muszą być rzeczywiste, wtedy funkcja (3.4) po-\nsiada minimum.\nPodsumowując ten paragraf, równania (3.3), (3.4) i (3.5) pokazują, że oba\nprądy płynące w cewkach mają parametryczny wpływ na zachowanie zlineary-\nzowanego układu poprzez skuteczną zmianę jego sztywność. Dlatego można\nprzyjąć, że dynamika analizowanego układu kontrolowana jest poprzez zmianę\njego parametrów, a nie poprzez dokładanie do niego zewnętrznych sił sterują-\ncych. Dodatkowo, sygnałami prądowymi można przesuwać wartości częstości\ndrgań własnych wahadeł w kierunku do lub od warunku rezonansu 1:1 w celu\nosiągnięcia pożądanego procesu wymiany energii między nimi.\n3.2.4 Adaptacja modelu matematycznego\nW tym paragrafie przedstawione zostanie podejście analityczne dla strategii ste-\nrowania przepływem energii między wahadłami. Podejście to opierać się będzie\nnaideiuśrednianiaukładuwodniesieniudojegotzw.„szybkiej”zmiennej,przy\nzałożeniu,żeobawahadłabędązlinearyzowaneicharakteryzowaćsiębędątąsa-\nmączęstośćdrgańwłasnych.Założenietojestkonieczne,abyrozważaćwymianę\nenergii podczas rezonansu 1:1.\nOpracowanie strategii sterowania przepływem energii wymaga odpowied-\nniego przejścia z układu opisanego podstawowymi zmiennymi stanu na układ\n903.2. STEROWANIE PRZEPŁYWEM ENERGII MIĘDZY WAHADŁAMI\ni𝜑2=𝐴2cos(Ω𝑡+ ∆), gdzie ∆jest przesunięciem fazowym. Podstawiając te rów-\nnania do (3.9) otrzymamy\n𝐸12=1\n2(¤𝜑1¤𝜑2+ Ω𝜑1𝜑2)=1\n2Ω2𝐴1𝐴2[sin(Ω𝑡)sin(Ω𝑡+ ∆)+\n+ cos(Ω𝑡)cos(Ω𝑡+ ∆)]=1\n2Ω2𝐴1𝐴2cos ∆,(3.10)\noraz\n𝐸11𝐸22=1\n4Ω4𝐴2\n1𝐴2\n2. (3.11)\nNa podstawie równań (3.10) i (3.11) można zauważyć, że przesunięcie fazowe\n∆jest zdefiniowane przez stosunek\n𝐸12√\n𝐸11𝐸22= cos ∆. (3.12)\nPowodem używania wielkości 𝐸𝑖𝑗jako zmiennych opisowych układu jest to,\nże takie zmienne opisują go w jednostkach energii, eliminując mniej informa-\ncyjną szybko zmieniającą się skalę czasową Ω𝑡. Z równań (3.13) poniżej widać,\nże algebraiczne kombinacje różnych elementów 𝐸𝑖𝑗dostarczają wystarczających\ninformacji o stanach dynamicznych dwóch identycznych oscylatorów. Niestety\nwielkości𝐸11i𝐸22tracą swoje znaczenie fizyczne z powodu rozpraszania energii\ni innych czynników. Niemniej jednak nadal mogą służyć do charakteryzowania\npoziomów wzbudzenia poszczególnych wahadeł, dlatego termin „energia” po-\nzostanie w użyciu oraz wprowadzone zostaną następujące kombinacje 𝐸𝑖𝑗:\n𝐸=𝐸11+𝐸22,\n𝑃=𝐸11𝐸22\n𝐸,1≤𝑃≤1,\n𝑄=𝐸12√\n𝐸11𝐸22= cos ∆,1≤𝑄≤1,(3.13)\ngdzie𝐸jest całkowitą energią wahadeł, 𝑃informuje o podziale energii między\nwahadłami, a 𝑄jest „wskaźnikiem koherencji” charakteryzującym przesunięcie\nfazowe ∆między wahadłami. Wartość 𝑃= 0oznacza, że energia jest równomier-\nnie rozłożona między wahadłami ( 𝐸11=𝐸22= 1/2𝐸). Wartość𝑃= 1wskazuje,\nże drga tylko pierwsze wahadło, podczas gdy drugie pozostaje w spoczynku\n(𝐸11=𝐸,𝐸22= 0). Sytuacja odwrotna ma miejsce, gdy 𝑃=1, wtedy drga tylko\nwahadło drugie, a pierwsze pozostaje w spoczynku ( 𝐸22=𝐸,𝐸11= 0). Wskaź-\nnik koherencji 𝑄określa rodzaj postaci synchronizacji drgań wahadeł, i został\nzobrazowany na rys. 3.11. Wartość 𝑄=1(∆ =𝜋) odpowiada postaci antyfazy,\nwartość𝑄= 0(∆ =𝜋/2)odpowiadapostaci„eliptycznej”,awartość 𝑄= 1(∆ = 0)\nodpowiada postaci w fazie. Następnie wykorzystując jawny parametr ∆przesu-\nnięcia fazowego, możemy dokonać transformacji z (3.13) do układu o oryginal-\nnych zmiennych stanu, tj. transformacji {𝐸,𝑃, ∆,𝛿}→{𝜑1,𝑣1,𝜑2,𝑣2}. W tym\ncelu, aby otrzymać bezpośrednie wyrażenia dla takiej transformacji podstawmy\n923.2. STEROWANIE PRZEPŁYWEM ENERGII MIĘDZY WAHADŁAMI\n(3.6)-(3.7)zwynikamiotrzymanymidlaukładuuśrednionego(3.17).Zewzględu\nna to, że wielkościami wyjściowymi układu oryginalnego są kąty i prędkości ką-\ntowe wahadeł, a modelu uśrednionego wielkości 𝑃i𝑄= cos ∆, kąty i prędkości\nzostały przeliczone na 𝑃i𝑄przy użyciu równań (3.9) i (3.13). Symulacje zostały\nprzeprowadzone dla następujących sygnałów prądowych\n𝑖1(𝑡) =𝐴+𝐵sin2\u0012\n𝜋𝑡\n𝑡𝑘\u0013\n, 𝑖 2(𝑡) =𝑖1(𝑡), (3.19)\ngdzie𝐴= 0.001A,𝐵= 0.055A i𝑡𝑘= 40s. Parametry sygnałów prądowych\n(3.19)zostałydobraneempirycznienapodstawiewstępnychbadańwtakisposób,\naby pożądany efekt wymiany energii między wahadłami stał się widoczny. Z te-\ngowzględuprzyjęto,żewartościsygnałówprądowychzaczynająsięodpewnego\nbardzoniskiegopoziomu 𝐴(tzw.offsetu),któryzawszemożebyćobecnypodczas\nbadań,anastępnieosiągająswojemaksimum 𝐴+𝐵wchwili𝑡=𝑡𝑘/2iostatecznie\nspadają do poziomu 𝐴w chwili𝑡=𝑡𝑘. Wartość𝑡𝑘jest czasem trwania proce-\nsu wymiany energii, zależnym od intensywności jej dyssypacji w układzie oraz\npoczątkowegopoziomuenergiicałkowitejukładu.Sygnałyprądoweopisanerów-\nnaniami (3.19) z zaimplementowanymi różnymi wartościami parametrów, będą\nwykorzystywane podczas symulacji numerycznych w dalszych częściach badań\nnadwymianąenergii.Wartoprzypomnieć,żeprądydodatniegenerująodpycha-\njące momenty magnetyczne, podczas gdy prądy ujemne wytwarzają momenty\nprzyciągające wahadła.\nPorównaniewynikówsymulacjiotrzymanychdlaukładuoryginalnegoiuśred-\nnionego z zaimplementowanymi parametrami odpowiadającymi dużym magne-\nsom pokazano na rys. 3.12. Lewa kolumna rysunku przedstawia wielkości uzy-\nskane na podstawie równań (3.6)-(3.7), natomiast prawa kolumna daje wgląd\nw dynamikę układu na podstawie wielkości współczynnika rozkładu energii 𝑃\noraz wskaźnika koherencji 𝑄= cos ∆. Jednocześnie, prawa kolumna potwierdza\ndobrązgodnośćpomiędzywielkościamiotrzymanyminapodstawieukładuory-\nginalnego i uśrednionego. Jak wynika z wykresów, obecność pól magnetycznych\nzasadniczo wpływa na dynamikę drgań układu w porównaniu z początkowy-\nmi swobodnymi drganiami w antyfazie, gdzie energia początkowa jest prawie\nrówno rozłożona pomiędzy wahadła (𝑃≈0). W przypadku drgań swobodnych,\ncałkowita energia stopniowo się rozprasza, podczas gdy jej bardzo mała część\nprzemieszcza się z jednego wahadła do drugiego w sposób „rytmiczny”. Nato-\nmiast stopniowo narastające sygnały prądowe łamią symetrię rozkładu energii\nw taki sposób, że energia jest prawie całkowicie przenoszona z wahadła będą-\ncego pod wpływem odpychającego momentu magnetycznego do wahadła, które\ndrgawprzyciągającympolumagnetycznym.Zjawiskotojestnajbardziejwidocz \nnenarys.3.12b,którypokazuje,żepoczątkowerównerozłożenieenergiimiędzy\nwahadłami (𝑃≈0)nie jest utrzymywane wraz ze wzrostem wartości sygnałów\nprądowych. Gdy prądy w cewkach osiągną określony poziom, energia zaczyna\n„płynąć” do wahadła znajdującego się w przyciągającym polu magnetycznym\nwytwarzanymprzezujemnyprąd.Dziejesiętakdlaok.12sekundy,kiedytoroz \nkład energii przyjmuje dolną wartość graniczną 𝑃= (𝐸11𝐸22)/(𝐸11+𝐸22) =1,\na wahadło oznaczone jako drugie absorbuje całą energię układu (rys. 3.12b,d).\nNależy zauważyć, że podczas całego procesu wymiany energii wskaźnik kohe-\nrencji (rys. 3.12f) pozostawał ujemny, 𝑄= cos ∆<0. Świadczy to o tym, że taki\n953.2. STEROWANIE PRZEPŁYWEM ENERGII MIĘDZY WAHADŁAMI\njednokierunkowy przepływ energii wiąże się z przewagą drgań w przeciwfazie.\nRys. 3.12g przedstawia zmiany postaci drgań, które występowały podczas oma-\nwianegoprocesuwymianyenergiimiędzywahadłami.Zmianyteprzedstawione\nsą na płaszczyźnie 𝜑1𝜑2i można je rozumieć jako przejście z drgań w antyfazie\ndodrgańozlokalizowanejnieliniowejpostaci,gdziedrganiaopostacieliptycznej\n𝑄≈0(∆≈𝜋/2) są drganiami przejściowymi. Uwagi te posłużą dalej jako pod-\nstawa do opracowania strategii sterowania przepływem energii. Z praktycznego\npunktuwidzeniaważnejest,abykontrolowaćprocesdomomentu,wktórymener-\ngia układu po raz pierwszy całkowicie przemieści się do docelowego oscylatora\nzakładając,żezostanieprzezniegowchłoniętalubpoprostuszybkorozproszona.\nTakiepodejściepozwalanadalszeuproszczeniaukładu(3.17)poprzezlinearyza-\ncjęwodniesieniudozmiennejrozkładuenergii 𝑃,takjaktoopisanowparagrafie\n3.2.9 i potwierdzono na rys. 3.12d.\n3.2.7 Analiza jakościowa trajektorii fazowych na\npłaszczyźnie 𝑃∆\nWyniki zawarte w paragrafie 3.2.6 ujawniły związek między współczynnikiem\nrozkładu energii 𝑃, a postacią drgań związaną z przesunięciem fazowym ∆. Dla-\ntego przeprowadzenie analizy dynamiki układu (3.17) poddanego zmieniającym\nsię polom magnetycznym przy wykorzystaniu płaszczyzny 𝑃∆, może być po-\nmocnedoopracowaniaadekwatnejstrategiisterowaniaprzepływemenergii.Rys.\n3.13 przedstawia trajektorie układu na płaszczyźnie 𝑃∆oraz punkty stacjonarne\nwyznaczone dla różnych kombinacji natężeń prądów cewek i przy założeniu, że\ncałkowita energia wahadeł 𝐸jest stała. Górny rząd (a, b, c) na rys. 3.13 pokazuje\newolucjętrajektoriifazowychwwynikuwzrostuodpychającegopolamagnetycz \nnego pod jednym z wahadeł, przy jednoczesnym braku działania pola magne-\ntycznego na drugie wahadło. Punkty stacjonarne ( ∆,𝑃)=(0,0) i (∆,𝑃)=(±𝜋,0) są\nzwiązane z drganiami układu w fazie i antyfazie (patrz równania (3.15)). Inte-\nresującym jest fakt, że czynniki rozpraszające energię mają efekt destabilizujący\npunkty stacjonarne; dla antyfazy są one zawsze niestabilnym ogniskiem, pod \nczasgdydladrgańwfaziezmieniająsięonezestabilnegoogniskananiestabilne\nwrazzespadkiemcałkowitejenergiiukładu(dyskusjaotymznajdujesięrównież\nw paragrafie 3.2.9). Środkowy rząd (d, e, f) daje wgląd w szczegóły zjawiska ani-\nhilacji antyfazy i postaci lokalnych pokazanych na rys. 3.13c. W wyniku zaniku\nwspomnianychpostacipowstajeefekttzw.„biegnącejfazy” ∆.Dolnyrząd(g,h,i)\npokazujesytuację,kiedyobawahadłapoddanesąpolommagnetycznymgenero-\nwanym przez prądy o różnych kombinacjach znaków. Z analizy tych wykresów\nmożna wnioskować, że najefektywniejsze przejście układu do drgań o postaci\nlokalnej, tj. gdy 𝑃≈±1, możliwe jest wtedy, kiedy prądy cewek mają przeciw-\nne znaki. Oczywistym jest też, że sygnały prądowe cewek w przypadku strategii\nzamkniętego sterowania opartej na pętli sprzężenia zwrotnego powinny zależeć\nod przesunięcia fazowego ∆(paragraf 3.2.10). Stopniowe rozpraszanie całkowitej\nenergiiukładuniewpływajakościowonatrajektoriezaprezentowanenarys.3.13\npodwarunkiem,żewielkościsygnałówprądowychzmniejszająsięwodpowied-\nniej proporcji. Należy jednak pamiętać, że wyniki z rys. 3.13 nie odzwierciedlają\ndokładniecałejzłożonościukładu(3.17),któryjestjeszczezależnyodzmieniającej\n973.2. STEROWANIE PRZEPŁYWEM ENERGII MIĘDZY WAHADŁAMI\nsię energii całkowitej 𝐸oraz zmiennych w czasie prądów cewek, a przestrzeń fa-\nzowa układu (3.17) jest czterowymiarowa: {𝐸,𝑃, ∆,𝑡}. Niemniej jednak, wykresy\nzrys.3.13wyjaśniajązwiązekmiędzysygnałamiprądowymicewekaefektemulo-\nkowywania energii układu na jednym z wahadeł. Związek pomiędzy kształtem\ntrajektorii fazowych wykreślonych na płaszczyźnie 𝑃∆a przebiegami wartości 𝑃\ni∆w czasie wyjaśniono na rys. 3.18.\n3.2.8 Eksperymentalna walidacja modelu o strategii\notwartego sterowania\nZ rezultatów zawartych w paragrafach 3.2.4 3.2.7 wynika, że informacja o prze-\nsunięciu fazowym ∆uzyskana w czasie rzeczywistym jest niezbędna do opra-\ncowaniastrategiizamkniętegosterowaniarozkłademenergii 𝑃.Niemniejjednak,\nw paragrafie 3.2.6 przeprowadzono numeryczną weryfikację modelu, poddane-\ngo strategii otwartego sterowania przepływem energii z wahadła 1 do wahadła\n2 (rys. 3.12b), gdzie sygnały prądowe miały z góry założone przebiegi opisane\nrównaniami (3.19). Stało się tak, ponieważ zależność (3.19) zawierała informację\no początkowej (antyfazowej) postaci drgań (rys. 3.12g). W przypadku, gdyby po-\nczątkową postacią drgań były drgania w fazie, energia płynęłaby z wahadła 2 do\nwahadła1.Sterowanieotwartemadużysenspraktycznyzewzględunaprostszą\nimplementacjęwporównaniuzesterowaniemzamkniętym,aletylkowtedy,gdy\nwarunki początkowe układu, a konkretnie postaci drgań są z góry znane.\nStrategia otwartego sterowania jednego wahadła\nRysunki3.14i3.15przedstawiająsymulacyjneieksperymentalnewykresyczaso-\nwedrgańirozkładówenergii,wykonanedlaukładuwykorzystującegosterowanie\notwarte.Wahadławyposażonesąwdużemagnesy,aletylkowahadło1poddane\njest działaniu odpychającego pola magnetycznego (na skutek przepływu prądu\n𝑖1),podczasgdycewkawahadła2jestniezasilana.Obawahadławprawianebyły\nw ruch z różnych początkowych położeń kątowych i zerowych prędkości, pro-\nwadzących układ swobodny do drgań w antyfazie lub bliskich postaci antyfazy.\nRys.3.16przedstawiawykresyczasoweprzesunięciafazowego ∆iwspółczynnika\nrozkładu energii 𝑃, odpowiadające przypadkowi drgań z rys. 3.15.\nPodsumowując,przedstawioneprzebiegiczasowewykazująwystarczającąna\npotrzebyprowadzonychbadańzgodnośćjakościowąiilościowąpomiędzyekspe-\nrymentami a wynikami symulacji.\nStrategia otwartego sterowania dwóch wahadeł\nWceluprzeprowadzeniabadańnadsterowaniemotwartymdladwóchwahadeł\nmagnetycznych, oba z nich wyposażono w małe magnesy. Następnie do cewek\nznajdujących się pod wahadłami, doprowadzono sygnały prądowe powodujące\nodpychanie wahadła 1 i przyciąganie wahadła 2. Podobnie jak wcześniej, oba\nwahadła wprawiane były w ruch z różnych początkowych położeń kątowych\ni zerowych prędkości, prowadzących układ swobodny do drgań w antyfazie lub\nbliskichantyfazie.Rys.3.17przedstawiasymulacyjneieksperymentalneprzebie-\ngi czasowe drgań i rozkładów energii w układzie, w przypadku zastosowania\nsterowania otwartego dla obu wahadeł magnetycznych. Rys. 3.18 przedstawia\n993.2. STEROWANIE PRZEPŁYWEM ENERGII MIĘDZY WAHADŁAMI\nLinearyzacja dla przypadku warunków początkowych bliskich postaci drgań\nw antyfazie\nZakładając,żezmniejszaniesięcałkowitejenergii 𝐸(𝑡)wczasieprocesuwymiany\nenergii między wahadłami jest powolne, dokonamy linearyzacji drugiego i trze-\nciego równania układu (3.17) w odniesieniu do przesunięcia fazowego bliskiego\n∆ =𝜋i gdy𝑃= 0. Zatem zakładając, że 𝐸jest stałe i eliminując przesunięcie\nfazowe, otrzymamy następujące równanie\n(3.20)𝑑2𝑃\n𝑑𝑡22Ω\u0012\n2𝛼+𝜁1+𝜁2√\n𝐸𝜋\u0013𝑑𝑃\n𝑑𝑡+\u001a1\n8𝐸𝛽+\u0014\n𝛽2+ 4𝛼\u0012\n𝛼+𝜁1+𝜁2√\n𝐸𝜋\u0013\u0015\nΩ2\n+𝑎𝛽\n𝐽𝑏2Ω2𝑒−𝐸\n2𝑏Ω2h\u0010\n𝑏Ω2+𝐸\u0011\n𝐼𝐵1𝐸𝐼𝐵0i\n(𝑖1+𝑖2)\u001b\n𝑃= 8𝛼Ω2𝜁1𝜁2√\n𝐸𝜋\n𝑎𝛽\n𝑏𝐽𝑒−𝐸\n2𝑏Ω2(𝐼𝐵0𝐼𝐵1)(𝑖1𝑖2),\n𝐼𝐵𝑛=𝐼𝐵𝑛\u0012𝐸\n2𝑏Ω2\u0013\n, 𝑛 = 1,2.\nPrzekształceniamatematyczneprowadzącedootrzymaniarównania3.20zawarto\nw Załączniku D.\nUjemny współczynnik tłumienia w równaniu (3.20) potwierdza uwagę z pa-\nragrafu3.2.7,dotyczącąlokalnejniestabilnościniektórychpunktówstacjonarnych\nwidocznych na płaszczyznach 𝑃∆z rys. 3.13. Fakt występowania różnicy sygna-\nłów prądowych po prawej stronie równania (3.20) pokazuje, że prądy o przeciw-\nnych znakach mają silniejszy wpływ na proces wymiany energii między waha-\ndłami.\nLinearyzacja dla przypadku warunków początkowych bliskich postaci drgań\nw fazie\nPostępując podobnie jak w przypadku postaci antyfazy, dokonamy teraz lineary-\nzacjiwzględemprzesunięciafazowegobliskiego ∆ = 0,wwynikuczegootrzyma-\nmy następujące równanie\n(3.21)𝑑2𝑃\n𝑑𝑡2+ 2Ω\u0012\n2𝛼+𝜁1+𝜁2√\n𝐸𝜋\u0013𝑑𝑃\n𝑑𝑡+\u001a\n1\n8𝐸𝛽+\u0014\n𝛽2+ 4𝛼\u0012\n𝛼𝜁1+𝜁2√\n𝐸𝜋\u0013\u0015\nΩ2\n𝑎𝛽\n𝐽𝑏2Ω2𝑒−𝐸\n2𝑏Ω2h\u0010\n𝑏Ω2+𝐸\u0011\n𝐼𝐵1𝐸𝐼𝐵0i\n(𝑖1+𝑖2)\u001b\n𝑃=\n8𝛼Ω2𝜁1𝜁2√\n𝐸𝜋+𝑎𝛽\n𝑏𝐽𝑒𝐸\n2𝑏Ω2(𝐼𝐵0𝐼𝐵1)(𝑖1𝑖2),\n𝐼𝐵𝑛=𝐼𝐵𝑛\u0012𝐸\n2𝑏Ω2\u0013\n, 𝑛 = 1,2.\nMożna zauważyć, że w tym przypadku współczynnik tłumienia może być chwi-\nlowo dodatni, dopóki całkowita energia 𝐸będzie wystarczająco duża, tj. będzie\nspełniała warunek√\n𝐸>(𝜁1+𝜁2)/(2𝛼𝜋).\nRys.3.19przedstawiaporównaniewynikówsymulacjizlinearyzowanychrów-\nnań (3.20)-(3.21), otrzymanych przy założeniu |𝑃|≪1, z numerycznymi rozwią-\nzaniamiukładuoryginalnego(3.6) (3.7).Naichpodstawiemożnawnioskować,że\n1043.2. STEROWANIE PRZEPŁYWEM ENERGII MIĘDZY WAHADŁAMI\nPrzypomnijmy,żewspółczynnikkoherencji(3.23)charakteryzujeprzesunięcie\nfazowe ∆między dwoma wahadłami jako 𝑄= cos ∆, a zatem jest niezależny od\ncałkowitejenergiiukładu 𝐸.Wrezultacie„siła”sprzężeniazwrotnegozależnaod\nprądów (3.22) utrzymywana jest pomimo stopniowego rozpraszania się energii\nukładu. Jak wynika z symulacji numerycznych, fakt ten nie ma niepożądanych\nskutków dla końcowej fazy dynamiki układu, co potwierdzają wyniki przedsta-\nwione na rys. 3.20. W rzeczywistości jednak, różne niedoskonałości strukturalne\ni nieuwzględnione czynniki fizyczne sprawiają, że układ wahadeł jest coraz bar-\ndziejwrażliwynazewnętrznezakłócenia,gdycałkowitaenergiaspadadopozio-\nmuodpowiadającegorównowadzewahadeł.Dlategonapotrzebyeksperymentu,\nw celu uniknięcia związanego z tym faktem zjawiska wzbudzania się wahadeł,\ndo równań (3.22) zostanie wprowadzony energetyczny składnik tłumiący\n𝑖1(𝑡) =𝑖𝐴\b\n1exp[𝜂𝐸(𝑡)]\t\ncos ∆(𝑡),\n𝑖2(𝑡) =𝑖𝐴\b\n1exp[𝜂𝐸(𝑡)]\t\ncos ∆(𝑡),(3.24)\ngdzie𝐸(𝑡)jest całkowitą energią wahadeł zdefiniowaną przez pierwsze równa-\nnie (3.13), a 𝜂jest numerycznym parametrem określanym w sposób empiryczny.\nWyniki eksperymentów przeprowadzonych dla układu z zaimplementowanymi\nsygnałamiprądowymi(3.24)otrzymanodlaparametrówzbliżonychdotychprzy-\njętychpodczassymulacjinumerycznych(rys.3.20)iprzedstawionojenarys.3.21.\nPotwierdzają one efekty przewidziane przez symulacje numeryczne dla trzech\nróżnych przypadków warunków początkowych. W szczególności, wyniki z rys.\n3.21a,c,e przedstawiają wyraźną i nieodwracalną tendencję dążenia współczyn-\nnika rozkładu energii do wartości 𝑃≈ 1, co świadczy o jednokierunkowym\nprzepływieenergiizpierwszegowahadładodrugiego.Należywziąćpoduwagę,\nże pojawiający się w końcowych fazach drgań układu wzrost wartości 𝑃, spowo-\ndowanyjestzanikaniemprądówsterujących.Trajektoriefazoweprzedstawionena\nrys. 3.21b,d,f potwierdzają dążenie układu do osiągnięcia lokalnej postaci drgań\n𝜑1≈0rad.\nPodsumowując,zaprezentowanewtymparagrafiestrategiesterowaniaoparte\nsą na następujących obserwacjach:\n•W przypadku drgań wahadeł w antyfazie, energia przemieszcza się z wa-\nhadła znajdującego się w odpychającym polu magnetycznym do wahadła\npoddanego działaniu przyciągającego pola magnetycznego.\n•W przypadku drgań wahadeł w fazie, energia przemieszcza się z waha-\ndła znajdującego się w przyciągającym polu magnetycznym do wahadła\npoddanego działaniu odpychającego pola magnetycznego.\nStrategia otwartego sterowania wymaga znajomości warunków początkowych\n(postacipoczątkowejdrgań)orazwartośćczasuruchuwahadełjakupłynie,ażdo\nchwilijegoustanianaskutekdyssypacjienergii.Strategiasterowaniazamkniętego\nnatomiastwymagatylkoinformacjioprzesunięciufazowymmiędzywahadłami,\nktóratojestpozyskiwanazpomiarówpołożeńkątowychwahadeł.Ztegowzględu\nmetodatadziękipętlisprzężeniazwrotnego,możebyćstosowanawprzypadkach,\ngdy ruch układu wynika nie tylko z niezerowych warunków początkowych, ale\nrównież z zakłóceń zewnętrznych takich jak np. siły czy momenty sił.\n1073.2. STEROWANIE PRZEPŁYWEM ENERGII MIĘDZY WAHADŁAMI\ntowane w podrozdziale wyniki badań zostały opublikowane w pracy [157].\n109Rozdział 4\nPodsumowanie i wnioski\nW pracy przedstawiono modelowanie matematyczne oraz badania symulacyjne\nieksperymentalnedynamikidwóchukładówmechatronicznych,którychelemen-\ntembazowymbyłowahadłomagnetyczne,tj.wahadłozzamocowanymnakońcu\nmagnesem. Pierwszy układ składał się z pojedynczego wahadła magnetyczne-\ngo umieszczonego w niestacjonarnym polu magnetycznym, generowanym przez\ncewkę elektryczną znajdują się pod nim. Drugi układ był rozszerzeniem pierw-\nszegoiskładałsięzdwóchtakichsamychwahadełmagnetycznych,którychwały\nsprzężone były ze sobą elementem podatnym. Ze względu na dwa układy o roż-\nnych stopniach swobody, pracę podzielono na dwie części i dla każdej z nich\npostawiono inną tezę badawczą.\nProces modelowaniamatematycznego dotyczył głównieopracowania modeli\noporówruchuioddziaływaniamagnetycznegopomiędzymagnesemacewką.Za-\nrównowukładzieojednymjakiodwóchstopniachswobodystosowanotesame\nmodele oporów ruchu oraz oddziaływania magnetycznego. W oparciu o ekspe-\nrymentwykazano,żeuzasadnionymbędzieuwzględnieniewrównaniachruchu\nwystępowaniaoporówstatycznychi/lubcoulombowskich.Modeloddziaływania\nmagnetycznegopomiędzycewkąamagnesem,sprowadzonyzostałdomomentu\nsił magnetycznych działającego na wahadło. Ponieważ literaturowe modele te-\ngo oddziaływania dawały wyniki dalekie od przeprowadzonego eksperymentu,\nopracowano dwa rodzaje nowych empirycznych modeli momentu magnetyczne-\ngo. Pierwszy z nich bazował na przekształconej funkcji Gaussa, natomiast drugi\noparty były na funkcjach wielomianowych. Ze względu na małe różnice mię-\ndzy przebiegami tych modeli, oba rodzaje wykorzystywane były podczas badań\nsymulacyjnych.\nZa ważne cechy charakterystyczne momentu magnetycznego, które miały\nwpływ na dynamikę badanych układów, uznano jego silną i nieliniową zależ-\nnośćodpołożeniamagnesuwahadławzględemcewkiorazzależnośćodsygnału\nprądowego. Im dalej magnes znajdował się od cewki, tym wpływ jej pola ma-\ngnetycznegonawahadłobyłmniejszy.Dziękitejceszeprzyjęto,żeistniejepewna\nstrefawychyleńkątowych,wewnątrzktórejwahadłoulegadziałaniupolamagne-\ntycznego cewki, a po opuszczeniu której zmiany tego pola będą miały znikomy\nwpływnajegodynamikę.Strefętęnazwanostrefąaktywną,ajejgranicezależały\nodwartościsygnałuprądowegopłynącegowcewce.Kieruneksygnałuprądowe-\n110godecydowałzaśoprzyciąganiubądźodpychaniumagnesuprzezcewkę.Dzięki\ntymdwómcechom,momentmagnetycznywrównaniachruchuwchodziłwskład\nczynnikaodpowiadającegozasztywnośćukładu,któraokazałasiębyćnieliniową,\nazewzględunaprzebiegsygnałuprądowegomogłasięzmieniaćwczasie.Wnio-\nskować więc można, że układy wahadeł magnetycznych należą do specyficznej\nrodziny oscylatorów parametrycznych.\nBadania dynamiki pojedynczego wahadła magnetycznego prowadzono dla\nprzypadku, gdy cewka zasilana była prostokątnym pulsującym sygnałem prą-\ndowym o stałej amplitudzie i regulowanej częstotliwości oraz wypełnieniu. Po-\nlaryzacja prądu cewki powodowała odpychanie od niej magnesu, przez co po-\ntencjałukładumiałcharakterdwudołkowy.Analizędynamicznąprzeprowadzo-\nno przy zastosowaniu metod numerycznych i opracowanej metody analityczno-\nnumerycznej opartej na metodzie uśredniania. Metoda analityczno-numeryczna\nwykorzystywałafunkcjepiłokształtnądoaproksymacjinieliniowychprzebiegów\nzmian amplitudy i przesunięcia fazowego w czasie dla przypadku drgań okre-\nsowych wahadła. W odniesieniu do tradycyjnej metody uśrednienia, wprowa-\ndzenie funkcji piłokształnych pozwoliło na jeszcze dokładniejsze opisanie dyna-\nmiki wspomnianej amplitudy i przesunięcia fazowego dla przypadku układów\nz pulsującym wymuszeniem, kiedy to w rozwiązaniu okresowym pojawiają się\n„podskoki” amplitudy (rys. 2.23). W ramach analizy dynamiki układu wykona-\nno szeregi wykresów czasowych, amplitudowo-czestotliwościowych, bifurkacyj-\nnych, fazowych oraz przekrojów Poincarégo. Pojedyncze wahadło magnetyczne\nwykazywało liczne drgań okresowe o różnych okresowościach, ale także typowe\ndladynamikinieliniowejzjawiskatakiejakmultistabilność,zachowaniachaotycz-\nne czy gwałtowne skoki amplitud.\nDokładnym badaniom poddany został specyficzny rodzaj jednookresowych\ndrgań wahadła magnetycznego odbywających się w jednym dołku potencjału.\nOkresowośćdrgańwahadłarozumianajestjakoliczbaokresówwymuszenia(sy-\ngnału prądowego) przypadająca na jeden okres drgań wahadła. Założenie ist-\nnienia wspomnianej strefy aktywnej, pozwoliło zastąpić ciągły nieautonomiczny\nmodelmatematycznyukładupoprzezmodeldyskretnywpostaciukładudwóch\nprzełączającychsięmiędzysobąautonomicznychrównańróżniczkowych.Przed-\nstawiono trzy różne scenariusze analizowanego ruchu jednookresowego oraz\nokreślono warunki ich istnienia w postaci warunków początkowych (położenie\niprędkośćkątowawahadła)dlarównańróżniczkowych.Następnieprzypomocy\ntych warunków, dla pierwszego ze scenariuszy wyznaczono parametry sygnału\nprądowegotakiejakwypełnienieiczęstotliwość,dlaktórychruchtenwystępuje,\natakżezbadanojegodwu itrój okresowemodyfikacje.Analizanumerycznaoraz\nteoretycznawyjaśniła,dlaczegoznacznazmianaparametruwypełnieniaiczęsto-\ntliwości sygnału prądowego nie musi wpływać na zmianę przebiegu trajektorii\nfazowej badanego ruchu. Na jej podstawie wywnioskowano, że ze względu na\nszczególnycharakteroddziaływaniamagnetycznegoukładuiistnienietzw.strefy\naktywnej, w przypadku drgań okresowych wahadła magnetycznego odbywają-\ncych się w jednym dołku potencjału, możliwe jest „sztuczne” zmienianie okre-\nsowości tych drgań bez wyraźnego naruszenia przebiegu ich trajektorii fazowej.\nWniosek ten udowadnia pierwszą z postawionych tez badawczych.\nBadaniadynamikiukładuodwóchstopniachswobodyprowadzonedlaprzy-\n111padku, gdy tylko jedno z wahadeł poddane było niestacjonarnemu polu magne-\ntycznemucewki.Układtencharakteryzowałsiębogatsządynamikąnieliniowąniż\nukładojednymstopniuswobody.Przedewszystkimopróczzachowańchaotycz-\nnychi okresowychpojawiły sięwnim nowezjawiska, takiejak quasiokresowość,\nktóra została potwierdzona zarówno na wykresach fazowych jak i przy pomocy\nwykładników Lapunowa.\nDalsze badania nad układem dwóch sprzężonych wahadeł magnetycznych\nskupiły się na możliwości sterowania przepływem energii między nimi, na sku-\ntekodpowiedniegogenerowaniapólmagnetycznychichcewek.Oprócztypowego\nmodelumatematycznegoukładuzawierającegoklasycznezmiennestanu,opraco-\nwanomodelzawierającyzmienneodnoszącesiędoenergiiicharakteryzującepro-\nces jej transferu. Zaproponowano dwie metody sterowania przepływem energii:\notwartą bez sprzężenia zwrotnego i zamkniętą ze sprzężeniem zwrotnym. Obie\nz nich oparto na obserwacjach, że w przypadku drgań w antyfazie energia prze-\npływa z wahadła poddanego odpychającemu polu magnetycznemu do wahadła\nznajdującego się w przyciągającym polu. Natomiast podczas drgań w fazie prze-\npływ energii jest odwrotny. Na podstawie obserwacji oraz analizy numerycznej\ndynamikiukładuwykazano,żepodczassterowaniabezsprężeniazwrotnegoko-\nniecznejestznanieaprioritypudrgańukładu(wfazie,wantyfazie)orazczasupo\njakim energia układu zostanie całkowicie rozproszona. Natomiast w przypadku\nsterowaniazesprzężeniemzwrotnymkoniecznejesttylkookreślenieprzesunięcia\nfazowego między wahadłami na podstawie pomiaru ich położeń. Opierając się\nnaprzeprowadzonychbadaniachnumerycznychorazeksperymentalnychmożna\nwnioskować,żeprzyodpowiednimsterowaniupólmagnetycznychcewek,azme-\nchanicznegopunktuwiedzenianieliniowąsztywnościąposzczególnychwahadeł,\nmożliwe jest zapewnienie kierunkowego transferu energii między sprzężonymi\noscylatorami.Wniosektenudowadniadrugązpostawionychwpracytezbadaw-\nczych.\nWedług autora, innowacyjnymi elementami pracy, dzięki którym jej cel na-\nukowy został osiągnięty są:\n1. Opracowanie modeli dynamicznych układów o jednym i dwóch stopniach\nswobody składających się z wahadeł magnetycznych oraz empirycznych\nmodelimatematycznychoddziaływaniamagnetycznegoparycewka-magnes.\n2. Opracowanieanalityczno-numerycznejmetodyopartejnametodzieuśred \nniania,wykorzystującejfunkcjepiłokształtnedoaproksymacjinieliniowych\nprzebiegów amplitudy i przesunięcia fazowego rozwiązania okresowego\ndla układu wahadła magnetycznego.\n3. Wykazanie, że podczas drgań wahadła magnetycznego w jednym dołku\npotencjału,któregocewkazasilanajestprostokątnymsygnałemprądowym,\nmożliwe jest zmienianie okresowości drgań bez zmieniania przebiegu ich\ntrajektorii fazowej.\n4. Opracowanie modelu matematycznego dla układu dwóch słabo sprzężo-\nnychwahadełmagnetycznych,wykorzystującegozmiennezwiązanezener-\ngią wahadeł oraz jej podziale między nimi.\n1125. Opracowanie dwóch metod sterowania przepływem energii między dwo-\nma słabo sprzężonymi wahadłami magnetycznymi, wykorzystujących od-\npowiednią zmianę pól magnetycznych ich cewek. Pierwsza metoda oparta\njestnaotwartejpętlisterowaniabezsprzężeniazwrotnego,natomiastdruga\nna zamkniętej pętli ze sprzężeniem zwrotnym.\nW perspektywie dalszych badań planowane jest rozszerzenie analizy dyna-\nmiki pojedynczego wahadła magnetycznego drgającego w jednym dołku poten-\ncjałuoprzypadekdrgańpomiędzydwomadołkamipotencjału.Zaprezentowane\nw rozprawie badania dotyczące pojedynczego wahadła magnetycznego, mogą\nstanowić podstawy do nowego sposobu modelowania silników krokowych.\nMechanizmy propagacji, pułapkowania (ang. trapping) i rozpraszania energii\nstanowią jedne z podstawowych problemów fizycznych zarówno na poziomie\nmikro jak i makro świata. Dlatego opracowane w rozprawie metody sposobu\nsterowania przepływem energii w układach połączonych wahadeł magnetycz \nnych, mogą stanowić bazę do przyszłych badań, w zakresie problemów wystę-\npujących podczas projektowania struktur molekularnych o pożądanych właści-\nwościach ukierunkowanego transferu energii (ang. targeted energy transfer TET )\n[149, 150], urządzeń odzyskujących energię z drgań mechanicznych [151, 152]\nlub mechanicznych pochłaniaczy energii wpływających na dynamikę konstrukcji\n[158].\n113Załączniki\nA Metoda wariacji stałych dowolnych\nAnalizując drgania układu opisane równaniem różniczkowym rzędu drugiego o\npostaci\n𝜑” +𝜑=𝐺(𝜑,𝜑′),(∗)′=𝑑(∗)\n𝑑𝜏𝑡(A.1)\nwiadomym jest, że przy dostatecznie małej wartości nieliniowej funkcji 𝐺(𝜑,𝜑′)\nopisujeonodrganiaquasi-liniowe.Przy 𝐺(𝜑,𝜑′) = 0rozwiązanietakiegorówna-\nnia jest następujące\n𝜑=𝑘sin(𝜃),𝜃= Ω𝑡𝜏𝑡+𝑢, (A.2)\na jego pierwsza pochodna pochodna wynosi\n𝜑′=𝑘Ω𝑡cos(𝜃). (A.3)\nPrzyjmując,żeprzymałym 𝐺(𝜑,𝜑′)drganiaukładuopisanesąrównaniem(A.2),\nwktórymmożnasięspodziewaćpowolnychzmianamplitudy 𝑘orazprzesunięcia\nfazowego𝑢wczasie.Biorąctopoduwagę,drganiatakiegoukładuopisaćmożna\npoprzez chwilowe wartości amplitudy 𝑘(𝜏𝑡)i fazy𝑢(𝜏𝑡).\nPrzyjmujączmienne 𝑘(𝜏𝑡)i𝑢(𝜏𝑡)orazbiorącpoduwagęzasadyróżniczkowa-\nnia funkcji złożonej, pochodna (A.2) wyrażona jest wzorem\n𝜑′=𝑘′sin(𝑢+𝜏𝑡Ω𝑡)+𝑘(𝑢′+ Ω𝑡)cos(𝑢+𝜏𝑡Ω𝑡) (A.4)\nPrzyrównując do siebie równania (A.3) i (A.4) otrzymamy zależność\n𝑘′sin(𝑢+𝑡Ω𝑡)+𝑘(𝑢′+ Ω𝑡)cos(𝑢+𝑡Ω𝑡)𝑘Ω𝑡cos(𝑢+𝑡Ω𝑡)= 0.(A.5)\nPonownieróżniczkującrównanie(A.3)przytakichsamychzałożeniach,tzn. 𝑘i𝑢\nsą funkcjami czasu otrzymamy\n𝜑′′= Ω𝑡𝑘′cos(𝑢+𝑡Ω𝑡)Ω𝑡𝑘(𝑢′+ Ω𝑡)sin(𝑢+𝑡Ω𝑡). (A.6)\nWstawiając równanie (A.6) i (A.2) do równania (A.1) otrzymamy\nΩ𝑡𝑘′cos(𝑢+𝑡Ω𝑡)Ω𝑡𝑘(𝑢′+ Ω𝑡)sin(𝑢+𝑡Ω𝑡)+𝑘sin(𝑢+𝑡Ω𝑡)=𝐺(𝜑,𝜑′)(A.7)\nRozwiązując układ równań liniowych (A.5) i (A.7) dla niewiadomych 𝑘′i𝑢′,\notrzymamy\n𝑘′=cos(𝑢+𝑡Ω𝑡)\u0000𝐺+\u0000Ω2\n𝑡1\u0001𝑘sin(𝑢+𝑡Ω𝑡)\u0001\nΩ𝑡,\n𝑢′=sin(𝑢+𝑡Ω𝑡)\u0000𝐺+\u0000Ω2\n𝑡1\u0001𝑘sin(𝑢+𝑡Ω𝑡)\u0001\nΩ𝑡𝑘,(A.8)\nktóry po przekształceniach da nam układ równań (2.32).\n114B. ENERGIA POTENCJALNA I CZĘSTOŚCI DRGAŃ WŁASNYCH ZLINEARYZOWANEGO\nUKŁADU ZACHOWAWCZEGO DWÓCH SPRZĘŻONYCH WAHADEŁ\nB Energia potencjalna i częstości drgań\nwłasnych zlinearyzowanego układu\nzachowawczego dwóch sprzężonych wahadeł\nRównanie(3.4)opisujeenergiępotencjalnąodniesionądomomentubezwładności\nwahadeł i obliczoną dla zlinearyzowanego, i zachowawczego układu. Całkowita\nenergia potencjalna układu dwóch słabo sprzężonych wahadeł jest sumą ener-\ngii potencjalnych pochodzących od pola grawitacyjnego, pola magnetycznego\ni sprężyny. Wzięte pod uwagę zostaną tylko te człony energii potencjalnych pola\ngrawitacyjnego i magnetycznego, których stopień wielomianu jest nie większy\nniż dwa. Poszczególne energie potencjalne można więc wyrazić następującymi\nformułami:\n•Energia potencjalna pochodząca od pola grawitacyjnego działającego na\nwahadło wynosi\n𝑉𝑔𝑟𝑎𝑤 =𝑚𝑔𝑠Z\u0012\n𝜑𝜑3\n6\u0013\n𝑑𝜑=𝑚𝑔𝑠\u0012𝜑2\n2𝜑4\n24\u0013\n≈1\n2𝑚𝑔𝑠𝜑2(B.1)\n•Energia potencjalna gromadzona w sprężynie zależy od względnego kąta\nskręcenia, który jest różnicą kątów wychyleń wahadeł. Wyraża się ją wzo-\nrem\n𝑉𝑠𝑝𝑟=1\n2𝑘𝑒𝑡(𝜑1𝜑2)2. (B.2)\n•Energia potencjalna pola magnetycznego opisana jest wzorem\n𝑉𝑝𝑀=𝑎𝐼\u0014\n1𝑒𝜑2\n𝑏\u0015\n𝑖, (B.3)\ngdzie𝑖toprądcewki.RozwijającwszeregTaylorawyrażenieeksponencjal-\nne zawarte we wzorze (B.3) otrzymamy\n𝑒𝜑2\n𝑏≈1𝜑2\n𝑏+𝜑4\n2𝑏2, (B.4)\na biorąc pod uwagę tylko składniki o stopniu nie większym niż dwa, przy-\nbliżoną wartość energii potencjalnej pola magnetycznego możemy zapisać\njako\n𝑉𝑝𝑀≈𝑎𝐼\u0014\n1\u0012\n1𝜑2\n𝑏\u0013\u0015\n𝑖=𝑎𝐼\n𝑏𝑖𝜑2. (B.5)\nOdnosząc przedstawione powyżej energie potencjalne do wartości masowego\nmomentu bezwładności wahadeł, otrzymamy wyrażenie na całkowitą energię\npotencjalną zlinearyzowanego układu zachowawczego wahadeł (3.4).\n115C. METODA V AN DER POLA DLA DWÓCH WAHADEŁ MAGNETYCZNYCH\nObliczeniakwadratówczęstości 𝜔1,2drgańwłasnychukładu,opartonanastę-\npującym układzie równań\n¥𝜑1=Ω2𝜑1Ω2𝛽(𝜑1𝜑2)+2𝑎𝐼\n𝑏𝐽𝑖1𝜑1,\n¥𝜑2=Ω2𝜑2Ω2𝛽(𝜑2𝜑1)+2𝑎𝐼\n𝑏𝐽𝑖2𝜑2.(B.6)\nRozwiązań tego układu równań poszukujemy w postaci [153]\n𝜑1=𝐴1cos\u0000𝜔𝑡𝜙\u0001,\n𝜑2=𝐴2cos\u0000𝜔𝑡𝜙\u0001,(B.7)\na po podstawieniu ich do równania (B.6) otrzymujemy\n𝐴1\u00002𝑎𝐼𝑖1+𝑏𝐽\u0000𝜔2(𝛽+ 1)Ω2\u0001\u0001\n𝑏𝐽+𝐴2𝛽Ω2= 0,\n𝐴1𝛽Ω2+𝐴2\u00002𝑎𝐼𝑖2+𝑏𝐽\u0000𝜔2(𝛽+ 1)Ω2\u0001\u0001\n𝑏𝐽= 0.(B.8)\nUkład ten zawsze ma rozwiązania zerowe, gdy 𝐴1= 0i𝐴2= 0, które odpowia-\ndają położeniu równowagi. Natomiast rozwiązanie niezerowe jest możliwe, gdy\nwyznacznik charakterystyczny układu (B.8) jest zerem\n\f\f\f\f\f\f(2𝑎𝐼𝑖1+𝑏𝐽(𝜔2−(𝛽+1)Ω2))\n𝑏𝐽𝛽Ω2\n𝛽Ω2(2𝑎𝐼𝑖2+𝑏𝐽(𝜔2−(𝛽+1)Ω2))\n𝑏𝐽\f\f\f\f\f\f= 0. (B.9)\nW efekcie otrzymujemy następujące równanie częstości\n4𝑎2\n𝐼𝑖1𝑖2\n𝑏2𝐽2+2𝑎𝐼(𝑖1+𝑖2)\u0000𝜔2(𝛽+ 1)Ω2\u0001\n𝑏𝐽+\u0010\n𝜔2Ω2\u0011 \u0010\n𝜔2(2𝛽+ 1)Ω2\u0011\n= 0,(B.10)\nktóreporozwiązaniuiprzekształceniachmatematycznychdadwaróżnekwadra-\nty𝜔2\n1,2częstości drgań swobodnych układu, odpowiadające równaniu (3.5).\nC Metoda Van der Pola dla dwóch wahadeł\nmagnetycznych\nW celu otrzymania równań (3.17 3.18) zastosowano metodę analogiczną do me-\ntody Van der Pola. Stosując metodę wariacji stałych dowolnych zażądajmy, aby\nrównania prędkości 𝑣1i𝑣2z (3.15) były równe prędkościom obliczonym z róż \nniczkowaniarównańna 𝜑1i𝜑2,gdy𝐸,𝑃,∆i𝛿sązależneodczasu.Wtensposób\notrzymamy następujące dwa równania\np\nE(𝑃+ 1) sin(𝛿)\"\n(E’𝑃+E’+P’E) cos(𝛿)2𝛿’E(𝑃+ 1) sin(𝛿)\n2Ωp\nE(𝑃+ 1)#\n= 0,\np\nE(1𝑃) sin(𝛿+ ∆)+\n\u00142E(𝑃1)(𝛿’+ ∆’) sin(𝛿+ ∆)+ (𝑃E’+E’P’E) cos(𝛿+ ∆)\n2Ω√\nEE𝑃\u0015\n= 0,(C.1)\n116C. METODA V AN DER POLA DLA DWÓCH WAHADEŁ MAGNETYCZNYCH\ngdzie (∗′) =𝑑\n𝑑𝑡(∗). Przy tym samym założeniu, że 𝐸,𝑃,∆i𝛿zależą od czasu,\nobliczamy przyspieszenia różniczkując równania na prędkości 𝑣1i𝑣2z (3.15),\na obliczone formuły wstawiamy do równań (3.6-3.7) otrzymując\n\"\n2𝛿’E(𝑃+ 1) cos(𝛿)(E’𝑃+E’+P’E) sin(𝛿)\n2p\nE(𝑃+ 1)#\n+ Ω2\u0010p\nE(𝑃+ 1) cos(𝛿)\u0011\nΩ+\n+𝑓1 p\nE(𝑃+ 1) cos(𝛿)\nΩ,p\nE(𝑃+ 1) sin(𝛿)!\n= 0,\n\u00142E(𝑃1)(𝛿’+ ∆’) cos(𝛿+ ∆)+ (E’(𝑃1) +P’E) sin(𝛿+ ∆)\n2√\nEE𝑃\u0015\n+\n+ Ω2p\nE(1𝑃) cos(𝛿+ ∆)\nΩ+𝑓2 p\nE(1𝑃) cos(𝛿+ ∆)\nΩ,p\nE(1𝑃) sin(𝛿+ ∆)!\n= 0,\n(C.2)\ngdzie𝑓1i𝑓2są funkcjami wychyleń i prędkości kątowych z (3.7).\nRozwiązującukładrównań(C.1 C.2)pozmiennych 𝑑{𝐸,𝑃,𝛿,∆}/𝑑𝑡otrzymu-\njemy następujący układ\n𝐸′\n𝑢=𝐸(𝑃+ 1)Ω sin (𝛿)cos(𝛿)©\n«𝛽2𝑎𝐼𝑖1𝑒𝐸(𝑃+1) cos2(𝛿)\n𝑏Ω2\n𝑏𝐽Ω2ª®\n¬𝐸(𝑃+ 1) cos3(𝛿)\n6Ω2\n2𝜁1p\nsgn(𝑃+ 1)sgn(sin(𝛿))\np\n𝐸(𝑃+ 1)\n+2𝛼√\n1𝑃2sin(𝛿+ ∆)+𝛽\u0010\n√\n1𝑃2\u0011\ncos(𝛿+ ∆)2𝛼(𝑃+ 1) sin (𝛿)\n𝑃+ 1\n𝐸(𝑃1) sin (𝛿+ ∆)\n6𝑏𝐽Ω√\n𝐸𝐸𝑃\u0014\n12𝑎𝐼𝑖2√\n𝐸𝐸𝑃cos(𝛿+ ∆)𝑒𝐸(𝑃1) cos2(𝛿+∆)\n𝑏Ω2\n12𝛼𝑏𝐽Ω2√\n𝐸𝐸𝑃sin(𝛿+ ∆)+ 12𝛼𝑏𝐽Ω2sin(𝛿)p\n𝐸(𝑃+ 1)\n+ 6𝑏𝛽𝐽Ω2√\n𝐸𝐸𝑃cos(𝛿+ ∆)6𝑏𝛽𝐽Ω2cos(𝛿)p\n𝐸(𝑃+ 1)\n𝑏𝐸𝐽√\n𝐸𝐸𝑃cos3(𝛿+ ∆) +𝑏𝐸𝐽𝑃√\n𝐸𝐸𝑃cos3(𝛿+ ∆)\n12𝑏𝜁2𝐽Ω2p\nsgn(1𝑃)sgn(sin(𝛿+ ∆))\u0015\n,\n117C. METODA V AN DER POLA DLA DWÓCH WAHADEŁ MAGNETYCZNYCH\n𝑃′\n𝑢=1\n48Ω\"\n1\n𝑏𝐽\u0012\n48𝑎𝐼𝑖1𝑃2sin(2𝛿)𝑒−𝐸(𝑃+1) cos2(𝛿)\n𝑏Ω248𝑎𝐼𝑖1sin(2𝛿)𝑒−𝐸(𝑃+1) cos2(𝛿)\n𝑏Ω2\n48𝑎𝐼𝑖2𝑃2sin(2(𝛿+ ∆))𝑒𝐸(𝑃−1) cos2(𝛿+∆)\n𝑏Ω2 + 48𝑎𝐼𝑖2sin(2(𝛿+ ∆))𝑒𝐸(𝑃1) cos2(𝛿+∆)\n𝑏Ω2\u0013\n48𝛼Ω2cos(2(𝛿+ ∆))+ 48𝛼Ω2cos(2𝛿)24𝛽Ω2sin(2(𝛿+ ∆))+ 24𝛽Ω2sin(2𝛿)\n+ 2𝐸sin(2(𝛿+ ∆))+𝐸sin(4(𝛿+ ∆))2𝐸sin(2𝛿)𝐸sin(4𝛿)+ 2𝐸𝑃3sin(2(𝛿+ ∆))\n+𝐸𝑃3sin(4(𝛿+ ∆))+ 2𝐸𝑃3sin(2𝛿)+𝐸𝑃3sin(4𝛿)2𝐸𝑃2sin(2(𝛿+ ∆))\n𝐸𝑃2sin(4(𝛿+ ∆))+ 2𝐸𝑃2sin(2𝛿)+𝐸𝑃2sin(4𝛿)\n+96𝜁2√\n1𝑃2Ω2p\n𝐸(𝑃+ 1)p\nsgn(1𝑃) sin(𝛿+ ∆)sgn(sin(𝛿+ ∆))\n𝐸\n2𝐸𝑃sin(2(𝛿+ ∆))𝐸𝑃sin(4(𝛿+ ∆))2𝐸𝑃sin(2𝛿)𝐸𝑃sin(4𝛿)\n+96𝜁1(𝑃1)Ω2sin(𝛿)p\n𝐸(𝑃+ 1)p\nsgn(𝑃+ 1)sgn(sin(𝛿))\n𝐸+ 48𝛼𝑃2Ω2cos(2(𝛿+ ∆))\n+96𝛼√\n1𝑃2𝑃2Ω2cos(2𝛿+ ∆)\n𝑃+ 1+96𝛼√\n1𝑃2𝑃Ω2cos(2𝛿+ ∆)\n𝑃+ 148𝛼𝑃2Ω2cos(2𝛿)\n96𝛼√\n1𝑃2𝑃2Ω2cos(∆)\n𝑃+ 196𝛼√\n1𝑃2𝑃Ω2cos(∆)\n𝑃+ 1+ 24𝛽𝑃2Ω2sin(2(𝛿+ ∆))\n+48𝛽√\n1𝑃2𝑃2Ω2sin(2𝛿+ ∆)\n𝑃+ 1+48𝛽√\n1𝑃2𝑃Ω2sin(2𝛿+ ∆)\n𝑃+ 124𝛽𝑃2Ω2sin(2𝛿)\n+48𝛽√\n1𝑃2𝑃Ω2sin(∆)\n𝑃+ 1+48𝛽√\n1𝑃2Ω2sin(∆)\n𝑃+ 1#\n,\n∆′\n𝑢=1\n48(𝑃+ 1)Ω(\n1\n𝑏𝐽(𝑃1)\u0014\n4\u0010\n𝑃21\u0011\ncos(2𝛿)𝑒−𝐸(𝑃+1) cos2(𝛿)\n𝑏Ω2\u0012\n12𝑎𝐼𝑖1\n+𝑏𝐽\u0010\n6𝛽Ω2+𝐸𝑃+𝐸\u0011\n𝑒𝐸(𝑃+1) cos2(𝛿)\n𝑏Ω2\u0013\n48𝑎𝐼𝑖1𝑃2𝑒𝐸(𝑃+1) cos2(𝛿)\n𝑏Ω2 + 48𝑎𝐼𝑖1𝑒𝐸(𝑃+1) cos2(𝛿)\n𝑏Ω2\n+ 48𝑎𝐼𝑖2𝑃2𝑒𝐸(𝑃1) cos2(𝛿+∆)\n𝑏Ω2 + 48𝑎𝐼𝑖2𝑃2cos(2(𝛿+ ∆))𝑒𝐸(𝑃−1) cos2(𝛿+∆)\n𝑏Ω2\n48𝑎𝐼𝑖2𝑒𝐸(𝑃1) cos2(𝛿+∆)\n𝑏Ω248𝑎𝐼𝑖2cos(2(𝛿+ ∆))𝑒𝐸(𝑃1) cos2(𝛿+∆)\n𝑏Ω24𝑏𝐸𝐽cos(2(𝛿+ ∆))\n𝑏𝐸𝐽cos(4(𝛿+ ∆))4𝑏𝐸𝐽𝑃3cos(2(𝛿+ ∆))𝑏𝐸𝐽𝑃3cos(4(𝛿+ ∆))6𝑏𝐸𝐽𝑃3\n+ 4𝑏𝐸𝐽𝑃2cos(2(𝛿+ ∆))+𝑏𝐸𝐽𝑃2cos(4(𝛿+ ∆))+ 4𝑏𝐸𝐽𝑃 cos(2(𝛿+ ∆))\n+𝑏𝐸𝐽𝑃 cos(4(𝛿+ ∆))𝑏𝐸𝐽(𝑃1)(𝑃+ 1)2cos(4𝛿)+ 6𝑏𝐸𝐽𝑃\n48𝛼𝑏𝐽Ω2sin(2(𝛿+ ∆))+ 48𝛼𝑏𝐽Ω2sin(2𝛿)+ 24𝑏𝛽𝐽Ω2cos(2(𝛿+ ∆))\n+ 48𝛼𝑏𝐽𝑃2Ω2sin(2(𝛿+ ∆))+ 96𝛼𝑏𝐽√\n1𝑃2𝑃Ω2sin(2𝛿+ ∆)\n48𝛼𝑏𝐽𝑃2Ω2sin(2𝛿)96𝛼𝑏𝐽√\n1𝑃2Ω2sin(∆)24𝑏𝛽𝐽𝑃2Ω2cos(2(𝛿+ ∆))\n48𝑏𝛽𝐽√\n1𝑃2𝑃Ω2cos(2𝛿+ ∆)48𝑏𝛽𝐽√\n1𝑃2𝑃Ω2cos(∆)\u0015\n+96𝜁2√\n1𝑃2Ω2p\n𝐸(𝑃+ 1)p\nsgn(1𝑃) cos (𝛿+ ∆)sgn(sin(𝛿+ ∆))\n𝐸(𝑃1)\n+ 96𝜁1Ω2cos(𝛿)r\n𝑃+ 1\n𝐸p\nsgn(𝑃+ 1)sgn(sin(𝛿)))\n,\n118D. PROCEDURA LINEARYZACJI DLA PRZYPADKU DRGAŃ O WARUNKACH\nPOCZĄTKOWYCH BLISKICH ANTYFAZIE\n𝛿′\n𝑢= cos2(𝛿)©\n«𝛽Ω2𝑎𝐼𝑖1𝑒𝐸(𝑃+1) cos2(𝛿)\n𝑏Ω2\n𝑏𝐽Ωª®\n¬𝐸(𝑃+ 1) cos4(𝛿)\n6Ω\n+ Ω2𝜁1Ω cos (𝛿)p\nsgn(𝑃+ 1)sgn(sin(𝛿))\np\n𝐸(𝑃+ 1)\nΩ cos (𝛿)\u0010\n2𝛼√\n1𝑃2sin(𝛿+ ∆)+𝛽√\n1𝑃2cos(𝛿+ ∆)+ 2𝛼(𝑃+ 1) sin (𝛿)\u0011\n𝑃+ 1.\n(C.3)\nTeraz należy uśrednić prawe strony równań (C.3) stosując operator (3.16),\nw efekcie otrzymamy\n𝑑𝐸\n𝑑𝑡=1\n2𝜋Z2𝜋\n0𝐸′\n𝑢𝑑𝛿=2𝛼𝐸√\n1𝑃2Ω cos (∆)\n2Ω\n𝜋\u0010\n𝜋𝛼𝐸+ 2𝜁1p\n𝐸(𝑃+ 1) + 2𝜁2√\n𝐸𝐸𝑃\u0011\n,\n𝑑𝑃\n𝑑𝑡=1\n2𝜋Z2𝜋\n0𝑃′\n𝑢𝑑𝛿=Ω\n𝜋𝐸\u0010\n𝜋𝐸√\n1𝑃2(𝛽sin(∆)2𝛼𝑃cos(∆))\n+ 4𝜁1(𝑃1)p\n𝐸(𝑃+ 1) + 4𝜁2(𝑃+ 1)√\n𝐸𝐸𝑃\u0011\n,\n𝑑∆\n𝑑𝑡=1\n2𝜋Z2𝜋\n0∆′\n𝑢𝑑𝛿=1\n8Ω𝑏𝐽\u001a\n𝑒−𝐸(𝑃+1)\n2𝑏Ω2\u0014\n8𝑎𝐼𝑖1\u0012\n𝐼𝐵0\u0012𝐸(𝑃+ 1)\n2𝑏Ω2\u0013\n𝐼𝐵1\u0012𝐸(𝑃+ 1)\n2𝑏Ω2\u0013\u0013\n8𝑎𝐼𝑖2𝑒𝐸𝑃\n𝑏Ω2\u0012\n𝐼𝐵0\u0012𝐸(𝑃1)\n2𝑏Ω2\u0013\n𝐼𝐵1\u0012𝐸𝐸𝑃\n2𝑏Ω2\u0013\u0013\u0015\u001b\n+𝐸𝑃\n8Ω2𝛼Ω sin (∆)√\n1𝑃2𝛽𝑃Ω cos (∆)\n√\n1𝑃2,\n𝑑𝛿\n𝑑𝑡=1\n2𝜋Z2𝜋\n0𝛿′\n𝑢𝑑𝛿=𝑎𝐼𝑖1𝑒𝐸(𝑃+1)\n2𝑏Ω2\u0010\n𝐼𝐵1\u0010\n𝐸(𝑃+1)\n2𝑏Ω2\u0011\n𝐼𝐵0\u0010\n𝐸(𝑃+1)\n2𝑏Ω2\u0011\u0011\n𝑏𝐽Ω\n+1\n2(𝛽+ 2)Ω𝐸(𝑃+ 1)\n16Ω1\n2r\n2\n𝑃+ 11Ω(𝛽cos(∆)2𝛼sin(∆)),\n(C.4)\nPodstawiającwartości 𝜆𝑎1=𝐸(1 +𝑃)/Ω2= 2𝐸11/Ω2i𝜆𝑎2=𝐸(1𝑃)/Ω2= 2𝐸22/Ω2\notrzymamy układ równań (3.17).\nD Proceduralinearyzacjidlaprzypadkudrgańo\nwarunkach początkowych bliskich antyfazie\nPoniżejprzedstawionyzostanietokpostępowaniaprowadzącydootrzymaniazli-\nnearyzowanego wzoru na 𝑃dla przypadku drgań o warunków początkowych\nbliskichantyfazie(równanie(3.20)).Przyjęto,żezmianaenergiicałkowitejwcza-\nsie𝑑𝐸/𝑑𝑡jestmałazczegowynika,że 𝐸jeststałe.Linearyzacjazostaławykonana\n119D. PROCEDURA LINEARYZACJI DLA PRZYPADKU DRGAŃ O WARUNKACH\nPOCZĄTKOWYCH BLISKICH ANTYFAZIE\nprzy założeniu, że przesunięcie fazowe ∆jest zaburzone wokół swojego położe-\nnia stacjonarnego ∆0, a współczynnik rozkładu energii 𝑃jest zaburzony wokół\nprzyjętego położenia stacjonarnego 𝑃0, co można zapisać jako\n∆ = ∆ 0+𝜖∆,\n𝑃=𝑃0+𝜖𝑃,(D.1)\ngdzie𝜖jest małym parametrem zaburzenia. Następnie zależności (D.1) zostały\nwstawione do drugiego i trzeciego równania uśrednionego układu (3.17) w wy-\nniku czego przyjęły następującą postać\n𝑃′=Ω©\n«\u0000𝛽sin(∆𝜖+ ∆ 0)2𝛼cos(∆𝜖+ ∆ 0) (𝑃𝜖+𝑃0)\u0001p\n1(𝑃𝜖+𝑃0)2\n+4Ω\u0012\n𝐸𝜁2q\n𝐸(−1+𝑃𝜖+𝑃0)\nΩ2 (1 +𝑃𝜖+𝑃0)+𝐸𝜁1(1 +𝑃𝜖+𝑃0)q\n𝐸(1+𝑃𝜖+𝑃0)\nΩ2\u0013\n𝐸2𝜋ª®®®®\n¬,\n∆′=1\n𝑏𝐽Ω𝑒−𝐸(1+𝑃𝜖+𝑃0)\n2𝑏Ω2\u0012\n𝑎𝑒𝐸(𝑃𝜖+𝑃0)\n𝑏Ω2𝑖2\u0012\n𝐼𝐵0\u0012𝐸(1 +𝑃𝜖+𝑃0)\n2𝑏Ω2\u0013\n+𝐼𝐵1\u0012𝐸(1 +𝑃𝜖+𝑃0)\n2𝑏Ω2\u0013\u0013\n+𝑎𝑖1\u0012\n𝐼𝐵0\u0012𝐸(1 +𝑃𝜖+𝑃0)\n2𝑏Ω2\u0013\n𝐼𝐵1\u0012𝐸(1 +𝑃𝜖+𝑃0)\n2𝑏Ω2\u0013\u0013\u0013\n+𝐸(𝑃𝜖+𝑃0)\n8ΩΩ\u00002𝛼sin(∆𝜖+ ∆ 0)+𝛽cos(∆𝜖+ ∆ 0) (𝑃𝜖+𝑃0)\u0001\np\n1(𝑃𝜖+𝑃0)2.\n(D.2)\nW kolejnym kroku, równania (D.2) zostały rozwinięte w szeregi Taylora wokół\n𝜖= 0. Podstawiając wartości ∆0=𝜋(warunek drgań w antyfazie) i 𝑃0= 0oraz\nbiorąc pod uwagę tylko dwa pierwsze wyrazy szeregu (odpowiadające 𝜖0i𝜖1)\notrzymamy\n(D.3)𝑃′=4(𝜁1+𝜁2)\n𝜋q\n𝐸\nΩ2+𝜖\u0012\n2𝑃𝜁1+ 2𝑃𝜁2+𝜋(2𝑃𝛼𝛽∆)q\n𝐸\nΩ2Ω\u0013\n𝜋q\n𝐸\nΩ2,\n∆′=𝑎𝑒−𝐸\n2𝑏Ω2(𝑖1𝑖2)\u0010\n𝐼𝐵0\u0010\n𝐸\n2𝑏Ω2\u0011\n𝐼𝐵1\u0010\n𝐸\n2𝑏Ω2\u0011\u0011\n𝑏𝐽Ω+𝜖©\n«𝐸𝑃\n8Ω+𝑃𝛽Ω + 2𝛼∆Ω\n+𝑎𝑒−𝐸\n2𝑏Ω2(𝑖1+𝑖2)𝑃\u0010\n𝐸𝐼𝐵0\u0010\n𝐸\n2𝑏Ω2\u0011\n+\u0000𝐸+𝑏Ω2\u0001𝐼𝐵1\u0010\n𝐸\n2𝑏Ω2\u0011\u0011\n𝑏2𝐽Ω3ª®®\n¬.\nOczywiście dla przypadku drgań w fazie, tok postępowania jest analogiczny,\nnależy tylko przyjąć ∆0= 0. Następnie przyjmując 𝜖= 1i upraszczając równania\n120D. PROCEDURA LINEARYZACJI DLA PRZYPADKU DRGAŃ O WARUNKACH\nPOCZĄTKOWYCH BLISKICH ANTYFAZIE\n(D.3) otrzymamy następujące wyrażenia\n(D.4)𝑃′=𝛽∆Ω4(𝜁1𝜁2)Ω√\n𝐸𝜋+𝑃\u0012\n2𝛼Ω +2(𝜁1+𝜁2)Ω√\n𝐸𝜋\u0013\n,\n∆′=2𝛼∆Ω +𝑎𝑒𝐸\n2𝑏Ω2(𝑖1𝑖2)\u0010\n𝐼𝐵0\u0010\n𝐸\n2𝑏Ω2\u0011\n𝐼𝐵1\u0010\n𝐸\n2𝑏Ω2\u0011\u0011\n𝑏𝐽Ω\n+𝑃©\n«𝐸\n8Ω+𝛽Ω +𝑎𝑒𝐸\n2𝑏Ω2(𝑖1+𝑖2)\u0010\n𝐸𝐼𝐵0\u0010\n𝐸\n2𝑏Ω2\u0011\n+\u0000𝐸+𝑏Ω2\u0001𝐼𝐵1\u0010\n𝐸\n2𝑏Ω2\u0011\u0011\n𝑏2𝐽Ω3ª®®\n¬.\nDootrzymaniawyrażeniana 𝑃zwyrugowanymprzesunięciemfazowym ∆należy\nobliczyć druga pochodną 𝑃korzystając z pierwszego równania (D.4)\n(D.5) 𝑃′′=\u0012\n2𝛼Ω +2(𝜁1+𝜁2)Ω√\n𝐸𝜋\u0013\n𝑃′𝛽Ω∆′\ni z tego samego równania wyznaczyć wzór na ∆\n(D.6) ∆ =4𝜁1Ω + 4𝜁2Ω + 2√\n𝐸𝜋𝛼Ω𝑃+ 2𝜁1Ω𝑃+ 2𝜁2Ω𝑃√\n𝐸𝜋𝑃′\n√\n𝐸𝜋𝛽Ω.\nPodstawiając do równania (D.5) wyrażenie na ∆’ z (D.4) otrzymamy\n𝑃′′=𝛽Ω𝑎𝑒−𝐸\n2𝑏Ω2(𝑖1𝑖2)\u0010\n𝐼𝐵0\u0010\n𝐸\n2𝑏Ω2\u0011\n𝐼𝐵1\u0010\n𝐸\n2𝑏Ω2\u0011\u0011\n𝑏𝐽Ω\n+©\n«𝐸\n8Ω+𝛽Ω +𝑎𝑒𝐸\n2𝑏Ω2(𝑖1+𝑖2)\u0010\n𝐸𝐼𝐵0\u0010\n𝐸\n2𝑏Ω2\u0011\n+\u0000𝐸+𝑏Ω2\u0001𝐼𝐵1\u0010\n𝐸\n2𝑏Ω2\u0011\u0011\n𝑏2𝐽Ω3ª®®\n¬𝑃\n+ 2𝛼Ω∆+ 2\u0012\n𝛼+𝜁1+𝜁2√\n𝐸𝜋\u0013\nΩ𝑃′,\n(D.7)\ngdzieostateczniemożemywyeliminować ∆wstawiającrównanie(D.6),otrzymu-\njąc\n(D.8)𝑃′′=8𝛼(𝜁1𝜁2)Ω2\n√\n𝐸𝜋+𝑎𝑒𝐸\n2𝑏Ω2(𝑖1𝑖2)𝛽\u0010\n𝐼𝐵0\u0010\n𝐸\n2𝑏Ω2\u0011\n+𝐼𝐵1\u0010\n𝐸\n2𝑏Ω2\u0011\u0011\n𝑏𝐽\n+©\n«1\n8(𝐸𝛽)\u0010\n4𝛼2+𝛽2\u0011\nΩ24𝛼(𝜁1+𝜁2)Ω2\n√\n𝐸𝜋\n+𝑎𝑒−𝐸\n2𝑏Ω2(𝑖1+𝑖2)𝛽\u0010\n𝐸𝐼𝐵0\u0010\n𝐸\n2𝑏Ω2\u0011\n\u0000𝐸+𝑏Ω2\u0001𝐼𝐵1\u0010\n𝐸\n2𝑏Ω2\u0011\u0011\n𝑏2𝐽Ω2ª®®\n¬𝑃\n+ 2\u0012\n2𝛼+𝜁1+𝜁2√\n𝐸𝜋\u0013\nΩ𝑃′.\nPo przekształceniach matematycznych (D.8) otrzymamy równanie (3.20).\n121Literatura\n[1] J. Awrejcewicz. Classical Mechanics. Dynamics. Springer ( 2012).\n[2] J. Leyko. Mechanika Ogólna Dynamika . Wydawnictwo Naukowe PWN,\nWarszawa, 9 wyd. ( 2008).\n[3] C. Science, A. W. Sleeswyk, N. Sivin. Dragons and Toads. The Chinese\nSeismoscope of A.D. 132. East Asian Science, Technology, and Medicine , 1 19\n(1983).\n[4] H.-S.S.Yan,K. H.H.Hsiao. Reconstructiondesignofthelostseismoscope\nof ancient China. Mechanism and Machine Theory , 42(12), 1601 1617 ( 2007).\n[5] G. L. Baker, J. A. Blackburn. The Pendulum. A Case Study in Physics . Oxford\nUniversity Press, New York ( 2005).\n[6] C.Huygens. HorologiumOscillatorium:Sive,deMotuPendulorumadHorologia\nAptato Demostrationes Geometricae . F. Muguet, Paris ( 1673).\n[7] J.Sommeria.FoucaultandtherotationoftheEarth. ComptesRendusPhysique ,\n18(9-10), 520–525 ( 2017).\n[8] V.J.García,E.P.Duque,J.A.Inaudi,C.O.Márquez,J.D.Mera,A.C.Rios.\nPendulum tuned mass damper: optimization and performance assessment\nin structures with elastoplastic behavior. Heliyon, 7(6) (2021).\n[9] M. Yucel, G. Bekdaş, S. M. Nigdeli, S. Sevgen. Estimation of optimum\ntuned mass damper parameters via machine learning. Journal of Building\nEngineering , 26 (2019).\n[10] J.Awrejcewicz. OrdinaryDifferentialEquationsandMechanicalSystems .Sprin-\nger International Publishing, Cham ( 2014).\n[11] A.Ugulava,Z.Toklikishvili,S.Chkhaidze. Onthedynamicsoftheangular\nmomentum of a quantum pendulum. Chaos, 30(6) (2020).\n[12] C.Beck. TestingaxionphysicsinaJosephsonjunctionenvironment. Modern\nPhysics Letters A , 26(38), 2841 2852 ( 2011).\n[13] C. Guarcello. Lévy noise effects on Josephson junctions. Chaos, Solitons and\nFractals, 153, 111531 ( 2021).\n[14] R. H. Bishop. The Mechatronics Handbook . CRC Press, Boca Raton ( 2002).\n122LITERATURA\n[15] J. Awrejcewicz, D. Lewandowski, P. Olejnik. Dynamics of Mechatronics Sys \ntems. World Scientific Publishing Co. Pte. Ltd., Singapore ( 2016).\n[16] V.H.Schmidt,B.R.Childers. Magneticpendulumapparatusforanalogde-\nmonstrationoffirst-orderandsecond-orderphasetransitionsandtricritical\npoints.American Journal of Physics , 52(1), 39–43 ( 1984).\n[17] J. Bethenod. Sur l’entretien du mouvement d’un pendule au moyen d’un\ncourant alternatif de fréquence élevée par rapport à sa fréquence propre.\nComptes rendus hebdomadaires des séances de l’Académie des sciences , 207(2),\n847–849 ( 1938).\n[18] Y. Rocard. General Dynamics of Vibrations . London, Crosby Lockwood and\nSon, 12 wyd. ( 1960).\n[19] H. P. Knauss, P. R. Zilsel. Magnetically Maintained Pendulum. American\nJournal of Physics , 19(5), 318–320 ( 1951).\n[20] N.Minorsky.Stationarysolutionsofcertainnonlineardifferentialequations.\nJournal of the Franklin Institute , 254(1), 21–42 ( 1952).\n[21] P.S.Landa. NonlinearOscillationsandWavesinDynamicalSystems . Springer,\nDordrecht ( 1996).\n[22] P. Mitkowski. Analysis of Bethenod’s phenomenon dynamics. Archives of\nElectrical Engineering , 57(224), 107 116 ( 2008).\n[23] N. Kesavamurthy, G. P. Rao. A Study of Bethenod’s Phenomenon. IEEE\nTransactions on Circuit Theory , 19(2), 215–218 ( 1972).\n[24] B. Z. Kaplan. Topological Considerations of Parametric Electromechanical\nDevices and Their Parametric Analysis. IEEE Transactions on Magnetics ,\n12(4), 373–380 ( 1976).\n[25] B. H. Smith. Some characteristics of a class of parametric reluctance ma-\nchines.Proceedings of the Institution of Electrical Engineers , 126(2), 162 166\n(1979).\n[26] A. P. Russell, I. E. Pickup. Principles of operation of the parametric reluc-\ntance motor. Electric Machines and Power Systems , 5(6), 485–496 ( 1980).\n[27] J. J. Blakley. On an Analysis of a Novel Tuned Circuit Electromechanical\nOscillator with a Long Traverse of Motion. Proceedings of the IEEE , 70(3),\n310–311 ( 1982).\n[28] F.C.Moon,P.J.Holmes. Amagnetoelasticstrangeattractor. JournalofSound\nand Vibration , 65(2), 275–296 ( 1979).\n[29] F. C. Moon, J. Cusumano, P. J. Holmes. Evidence for homoclinic orbits as a\nprecursortochaosinamagneticpendulum. PhysicaD:NonlinearPhenomena ,\n24(1 3), 383–390 ( 1987).\n123LITERATURA\n[30] M. Ó. M. Donnagáin, O. Rasskazov. Numerical modelling of an iron pen-\ndulum in a magnetic field. Physica B: Condensed Matter , 372(1-2), 37–39\n(2006).\n[31] G. Radons, A. Zienert. Nonlinear dynamics of complex hysteretic systems:\nOscillatorinamagneticfield. EuropeanPhysicalJournal:SpecialTopics ,222(7),\n1675–1684 ( 2013).\n[32] K.A.Kumar,S.F.Ali,A.Arockiarajan. Magneto-elasticoscillator:Modeling\nand analysis with nonlinear magnetic interaction. Journal of Sound and\nVibration , 393, 265–284 ( 2017).\n[33] A. Siahmakoun, V. A. French, J. Patterson. Nonlinear dynamics of a sinu-\nsoidallydrivenpenduluminarepulsivemagneticfield. AmericanJournalof\nPhysics, 65(5), 393–400 ( 1997).\n[34] W. Szemplińska-Stupnicka, E. Tyrkiel. Bifurkacje, Chaos i Fraktale w Dynami-\nce Wahadła . Instytut Podstawowych Problemów Techniki PAN, Warszawa\n(2001).\n[35] W.Szemplińska-Stupnicka,E.Tyrkiel. Bifurkacje,DrganiaChaotyczneiFrakta-\nlewOscylatorze\"Dwudołkowym\" . InstytutPodstawowychProblemówTech-\nniki PAN, Warszawa ( 2001).\n[36] C. A. Kitio Kwuimy, C. Nataraj, M. Belhaq, C. Kwuimy, C. Nataraj, C. A.\nKitio Kwuimy, C. Nataraj, M. Belhaq. Chaos in a magnetic pendulum\nsubjectedtotiltedexcitationandparametricdamping. MathematicalProblems\nin Engineering , 2012, 1 18 ( 2012).\n[37] B.P.Mann. Energycriterionforpotentialwellescapesinabistablemagnetic\npendulum. Journal of Sound and Vibration , 323(3 5), 864–876 ( 2009).\n[38] B. Nana, S. B. Yamgoué, R. Tchitnga, P. Woafo. Dynamics of a pendulum\ndriven by a DC motor and magnetically controlled. Chaos, Solitons and\nFractals, 104, 18 27 ( 2017).\n[39] B. Nana, S. B. Yamgoué, R. Tchitnga, P. Woafo. Nonlinear dynamics of a\nsinusoidally driven lever in repulsive magnetic fields. Nonlinear Dynamics ,\n91(1), 55–66 ( 2018).\n[40] B. Nana, S. B. Yamgoué, P. Woafo. Dynamics of an autonomous electrome-\nchanical pendulum like system with experimentation. Chaos, Solitons and\nFractals, 152, 111475 ( 2021).\n[41] G. Khomeriki. Parametric resonance induced chaos in magnetic damped\ndriven pendulum. Physics Letters, Section A: General, Atomic and Solid State\nPhysics, 380(31 32), 2382 2385 ( 2016).\n[42] Y. Kraftmakher. Experiments with a magnetically controlled pendulum.\nEuropean Journal of Physics , 28, 1007 1020 ( 2007).\n124LITERATURA\n[43] Y.Kraftmakher. Demonstrationswithamagneticallycontrolledpendulum.\nAmerican Journal of Physics , 78(5), 532–535 ( 2010).\n[44] V. Tran, E. Brost, M. Johnston, J.Jalkio. Predicting the behavior of achaotic\npendulumwithavariableinteractionpotential. Chaos,23(3),033103( 2013).\n[45] J.A.Blackburn,S.Vik,B.Wu,H.J.T.Smith. Drivenpendulumforstudying\nchaos.Review of Scientific Instruments , 60(3), 422–426 ( 1989).\n[46] J. A. Blackburn, G. L. Baker. A comparison of commercial chaotic pendu-\nlums.American Journal of Physics , 66(9), 821 830 ( 1998).\n[47] J.P.Berdahl,K.VanderLugt. Magneticallydrivenchaoticpendulum. Ame \nrican Journal of Physics , 69(9), 1016–1019 ( 2001).\n[48] M.García-Sanz. Designofanexperimentalsystemforstudyingthecontrol\nof multivariable, non linear and chaotic processes. International Journal of\nElectrical Engineering Education , 38(1), 26–44 ( 2001).\n[49] H. B. Pedersen, J. E. V. Andersen, T. G. Nielsen, J. J. Iversen, F. Lyckegaard,\nF.K.Mikkelsen. Anexperimentalsystemforstudyingtheplanependulum\nin physics laboratory teaching. European Journal of Physics , 41(1), 015701\n(2020).\n[50] N.Ida. Designandcontrolofamagneticpendulumactuator. Proceedingsof\nthe International Conference on Optimisation of Electrical and Electronic Equip \nment, OPTIM , 439–443 ( 2012).\n[51] K. Tsubono, A. Araya, K. Kawabe, S. Moriwaki, N. Mio. Triple-pendulum\nvibration isolation system for a laser interferometer. Review of Scientific\nInstruments , 64(8), 2237 2240 ( 1993).\n[52] T. Jang, H. J. Ha, Y. K. Seo, S. H. Sohn. Off axis magnetic fields of a cir \ncular loop and a solenoid for the electromagnetic induction of a magnetic\npendulum. Journal of Physics Communications , 5(6) (2021).\n[53] Y.K.Seo,S.H.Sohn,T.Jang. Astudyontheelectromagneticinductionlaw\nusing a simple pendulum. New Physics: Sae Mulli , 70(3), 254 261 ( 2020).\n[54] D. Skubov, D. Vavilov. Dynamics of the conductivity bodies of pendulum\ntypes in alternating magnetic field. ZAMM - Journal of Applied Mathematics\nand Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik , 94(11),\n951–956 ( 2014).\n[55] T. Boeck, S. L. Sanjari, T. Becker. Parametric instability of a magnetic pen-\nduluminthepresenceofavibratingconductingplate. NonlinearDynamics ,\n102(4), 2039–2056 ( 2020).\n[56] T. Boeck, S. L. Sanjari, T. Becker. Dynamics of a magnetic pendulum in the\npresenceofanoscillatingconductingplate. ProceedingsinAppliedMathema \ntics and Mechanics , 20(1), 2020–2021 ( 2021).\n125LITERATURA\n[57] A. E. Motter, M. Gruiz, G. Károlyi, T. Tél. Doubly transient chaos: Generic\nform of chaos in autonomous dissipative systems. Physical Review Letters ,\n111(19), 194101 ( 2013).\n[58] B.Qin,H.L.Shang,H.M.Jiang.Globaldynamicbehavioranalysisoftypical\nmagnetic pendulum. Acta Physica Sinica , 70(18) (2021).\n[59] D.I.Penner,D.Duboshinskii,M.I.Kozakov,A.S.Vermel’,Y.V.Galkin,D.B.\nDoubochinski, M. I. Kozakov, A. S. Vermel’, Y. V. Galkin. Asynchronous\nexcitation of undamped oscillations. Uspekhi Fizicheskih Nauk , 109(2), 402\n(1973).\n[60] J. Tennenbaum. Amplitude Quantization As an Elementary Property of\nMacroscopic Vibrating Systems. 21st Century Science and Technology , 50–63\n(2005).\n[61] D.B.Doubochinski,J.Tennenbaum,E. m.Conference,D.B.Doubochinski,\nJ. Tennenbaum, R. D. Wattignies. Theory and Applications of the Macro-\nscopic Quantization Effect in Nonlinearly-Coupled Vibrating Systems. 1st\nEuro-Mediterranean Conference on Structural Dynamics and Vibroacoustics , 23–\n26. Marrakech, Marocco ( 2013).\n[62] A.I.Shumaev,Z.A.Maizelis. Distributionfunctionsofargumentaloscilla-\ntions of the Duboshinskiy pendulum. J. Appl. Phys , 121(15), 154902 ( 2017).\n[63] Y.Luo,W.Fan,C.Feng,S.Wang,Y.Wang.Subharmonicfrequencyresponse\nin a magnetic pendulum. American Journal of Physics , 88(2), 115 ( 2020).\n[64] V. Damgov, I. Popov. “Discrete” Oscillations and multiple attractors in\nkick-excited systems. Discrete Dynamics in Nature and Society , 4(2), 99–124\n(2000).\n[65] V. Damgov. \"Quantized\" oscillations and irregular behavior of a class of\nkick-excited self-adaptive dynamical systems. Progress of Theoretical Physics\nSupplement , 139, 344 352 ( 2000).\n[66] D. Cintra, P. Argoul. Nonlinear argumental oscillators: A few examples\nof modulation via spatial position. Journal of Vibration and Control , 23(18),\n2888–2911 ( 2017).\n[67] D. Cintra, P. Argoul. Non-linear argumental oscillators: Stability criterion\nand approximate implicit analytic solution. International Journal of Non \nLinear Mechanics , 94, 109–124 ( 2017).\n[68] C. Chappert, T. Devolder. A magnetic pendulum. Nature, 432(7014), 162\n(2004).\n[69] E. Saitoh, H. Miyajima, T. Yamaoka, G. Tatara. Current-induced resonance\nandmassdeterminationofasinglemagneticdomainwall. Nature,432(7014),\n203–206 ( 2004).\n126LITERATURA\n[70] G.Dolfo,J.Vigué.Apendulumwithabrokencylindricalsymmetry:Normal\nmodes and beats. European Journal of Physics , 43(1) (2022).\n[71] R. Darula, S. Sorokin. On non linear dynamics of a coupled electro-\nmechanical system. Nonlinear Dynamics , 70(2), 979–998 ( 2012).\n[72] B.R.Pontes,M.Silveira,A.C.Mazotti,P.J.Gonçalves,J.M.Balthazar. Con-\ntributionofelectricalparametersonthedynamicalbehaviourofanonlinear\nelectromagnetic damper. Nonlinear Dynamics , 79(3), 1957–1969 ( 2014).\n[73] M. S. Siewe, C. N. D. Buckjohn. Heteroclinic motion and energy transfer in\ncoupled oscillator with nonlinear magnetic coupling. Nonlinear Dynamics ,\n77(1-2), 297–309 ( 2014).\n[74] J. H. Ho, K. C. Woo. Approximate analytical solution to oscillations of a\nconductor in a magnetic field. Nonlinear Dynamics , 64(4), 315–330 ( 2011).\n[75] M.Bednarek,D.Lewandowski,K.Polczyński,J.Awrejcewicz. Ontheactive\ndampingofvibrationsusingelectromagneticspring. MechanicsBasedDesign\nof Structures and Machines , 49(8), 1131–1144 ( 2021).\n[76] K. Witkowski, G. Kudra, G. Wasilewski, J. Awrejcewicz. Mathematical\nmodelling, numerical and experimental analysis of one-degree-of-freedom\noscillatorwithDuffing-typestiffness. InternationalJournalofNon-LinearMe \nchanics, 138, 103859 ( 2022).\n[77] M. Wiercigroch, S. Kovacs, S. Zhong, D. Costa, V. Vaziri, M. Kapitaniak,\nE.Pavlovskaia. Versatilemassexcitedimpactoscillator. NonlinearDynamics ,\n99(1), 323–339 ( 2020).\n[78] D. Costa, V. Vaziri, M. Kapitaniak, S. Kovacs, E. Pavlovskaia, M. A. Savi,\nM. Wiercigroch. Chaos in impact oscillators not in vain: Dynamics of new\nmass excited oscillator. Nonlinear Dynamics , 102(2), 835–861 ( 2020).\n[79] Z. Hao, D. Wang, M. Wiercigroch. Nonlinear dynamics of new magneto-\nmechanical oscillator. Communications in Nonlinear Science and Numerical\nSimulation , 105, 106092 ( 2022).\n[80] A. Fradkov, B. Andrievsky, K. Boykov. Control of the coupled double pen-\ndulums system. Mechatronics , 15(10), 1289–1303 ( 2005).\n[81] P.V.Malaji,M.I.Friswell,S.Adhikari,G.Litak. Enhancementofharvesting\ncapability of coupled nonlinear energy harvesters through high energy or-\nbits.AIP Advances , 10(8), 085315 ( 2020).\n[82] Y. E. Surganova, Y. V. Mikhlin. Localized and non localized nonlinear nor-\nmal modes in a system of two coupled pendulums under a magnetic field.\nInternational Journal of Non Linear Mechanics , 147(July), 104182 ( 2022).\n[83] F.Russell,Y.Zolotaryuk,J.Eilbeck,T.Dauxois. Movingbreathersinachain\nof magnetic pendulums. Physical Review B Condensed Matter and Materials\nPhysics, 55(10), 6304–6308 ( 1997).\n127LITERATURA\n[84] M. Wojna, A. Wijata, G. Wasilewski, J. Awrejcewicz. Numerical and expe-\nrimental study of a double physical pendulum with magnetic interaction.\nJournal of Sound and Vibration , 430, 214 230 ( 2018).\n[85] R. Kumar, S. Gupta, S. F. Ali. Energy harvesting from chaos in base exci-\nted double pendulum. Mechanical Systems and Signal Processing , 124, 49–64\n(2019).\n[86] A. Zhang, V. Sorokin, H. Li. Dynamic analysis of a new autoparametric\npendulum absorber under the effects of magnetic forces. Journal of Sound\nand Vibration , 485 (2020).\n[87] K.Kecik,A.Mitura. Energyrecoveryfromapendulumtunedmassdamper\nwithtwoindependentharvestingsources. InternationalJournalofMechanical\nSciences, 174, 105568 ( 2020).\n[88] K.Kecik. Numericalstudyofapendulumabsorber/harvestersystemwith\nasemi activesuspension. ZAMMZeitschriftfurAngewandteMathematikund\nMechanik , 101(1), 1 12 ( 2021).\n[89] K. Kecik. Simultaneous vibration mitigation and energy harvesting from a\npendulum-typeabsorber. CommunicationsinNonlinearScienceandNumerical\nSimulation , 92, 105479 ( 2021).\n[90] E.Bonisoli,C.Delprete.NonlinearandlinearisedbehaviouroftheLevitron.\nMeccanica , 51(4), 763–784 ( 2016).\n[91] A. O. Oyelade. Experiment study on nonlinear oscillator containing ma-\ngnetic spring with negative stiffness. International Journal of Non Linear Me-\nchanics, 120, 103396 ( 2020).\n[92] K. Witkowski, G. Kudra, S. Skurativskyi, G. Wasilewski, J. Awrejcewicz.\nModeling and dynamics analysis of a forced two-degree-of-freedom me-\nchanical oscillator with magnetic springs. Mechanical Systems and Signal\nProcessing , 148, 107138 ( 2021).\n[93] J. W. Zhou, W. Zhang, X. S. Wang, B. P. Mann, E. H. Dowell. Primary reso-\nnance suppression of a base excited oscillator using a spatially constrained\nsystem: Theory and experiment. Journal of Sound and Vibration , 496, 115928\n(2021).\n[94] M.N.SrinivasPrasad,R.U.Tok,F.Fereidoony,Y.E.Wang,R.Zhu,A.Propst,\nS.Bland. MagneticPendulumArraysforEfficientULFTransmission. Scien \ntific Reports , 9(1), 1–13 ( 2019).\n[95] S.Y.Kim,S.H.Shin,J.Yi,C.W.Jang. Bifurcationsinaparametricallyforced\nmagnetic pendulum. Physical Review E Statistical Physics, Plasmas, Fluids,\nand Related Interdisciplinary Topics , 56(6), 6613–6619 ( 1997).\n[96] P. V. Malaji, S. F. Ali. Analysis and experiment of magneto-mechanically\ncoupled harvesters. Mechanical Systems and Signal Processing , 108 (2018).\n128LITERATURA\n[97] S. Fang, S. Zhou, D. Yurchenko, T. Yang, W. H. Liao. Multistability phe-\nnomenon in signal processing, energy harvesting, composite structures,\nand metamaterials: A review. Mechanical Systems and Signal Processing , 166,\n108419 (2022).\n[98] T.Becker,S.L.Sanjari. Parametricinstabilityofaverticallydrivenmagnetic\npendulum with eddy-current braking by a flat plate. Nonlinear Dynamics ,\n109, 509–529 ( 2022).\n[99] K.Polczyński. Konstrukcjairejestracjaruchuwahadełfizycznychpoła ¸czonychpo \ndatnymelementemiwymuszanychzmiennympolemmagnetycznym .Inżynierska\npraca dyplomowa, Politechnika Łódzka ( 2017).\n[100] S.A. Abdelhfeez, T. S. Amer, R. F. Elbaz, M. A. Bek. Studying the influence\nof external torques on the dynamical motion and the stability of a 3DOF\ndynamic system. Alexandria Engineering Journal , 61(9), 6695–6724 ( 2022).\n[101] J. H. He, T. S. Amer, A. F. Abolila, A. A. Galal. Stability of three degrees-\nof-freedom auto-parametric system. Alexandria Engineering Journal , 61(11),\n8393–8415 ( 2022).\n[102] I. Jawarneh, Z. Altawallbeh. A Heteroclinic Bifurcation in a Motion of\nPendulum:Numerical TopologicalApproach. InternationalJournalofApplied\nand Computational Mathematics , 8(3), 1 14 ( 2022).\n[103] V. Puzyrov, J. Awrejcewicz, N. Losyeva, N. Savchenko. On the stability of\nthe equilibrium of the double pendulum with follower force: Some new\nresults.Journal of Sound and Vibration , 523, 116699 ( 2022).\n[104] L.M.Ramos,C.R.Reis,L.B.Calheiro,A.M.Goncalves. UseofArduinoto\nobserve the chaotic movement of a magnetic pendulum. Physics Education ,\n56(1) (2021).\n[105] U. B. Pili. Modeling damped oscillations of a simple pendulum due to\nmagnetic braking. Physics Education , 55(3), 6–11 ( 2020).\n[106] A. Myers, F. A. Khasawneh. Dynamic State Analysis of a Driven Magnetic\nPendulum Using Ordinal Partition Networks and Topological Data Analy-\nsis.Proceedings of the ASME Design Engineering Technical Conference , tom 7.\nAmerican Society of Mechanical Engineers Digital Collection ( 2020).\n[107] B. Ambrozkiewicz, G. Litak, P. Wolszczak. Modelling of electromagnetic\nenergy harvester with rotational pendulum using mechanical vibrations to\nscavenge electrical energy. Applied Sciences , 10(2), 1–14 ( 2020).\n[108] B. Bao, S. Zhou, Q. Wang. Interplay between internal resonance and non-\nlinear magnetic interaction for multi directional energy harvesting. Energy\nConversion and Management , 244, 114465 ( 2021).\n129LITERATURA\n[109] J.Y.Cho,S.Jeong,H.Jabbar,Y.Song,J.H.Ahn,J.H.Kim,H.J.Jung,H.H.\nYoo, T. H. Sung. Piezoelectric energy harvesting system with magnetic\npendulum movement for self-powered safety sensor of trains. Sensors and\nActuators, A: Physical , 250, 210–218 ( 2016).\n[110] J. H. Li, M. J. Cai, L. H. Xie. Develop a magnetic pendulum to scavenge\nhuman kinetic energy from arm motion. Applied Mechanics and Materials ,\n590, 48 52 ( 2014).\n[111] M. A. Karami, D. J. Inman. Hybrid rotary-translational energy harvester\nformulti axisambientvibrations. ASME2012ConferenceonSmartMaterials,\nAdaptive Structures and Intelligent Systems, SMASIS 2012 , 2, 1–8 (2016).\n[112] Y. Qiu, S. Jiang. Suppression of low-frequency vibration for rotor-bearing\nsystem of flywheel energy storage system. Mechanical Systems and Signal\nProcessing , 121, 496–508 ( 2019).\n[113] B. Dolisy, T. Lubin, S. Mezani, J. Leveque. Three-dimensional analytical\nmodel for an axial-field magnetic coupling. Progress In Electromagnetics\nResearch M , 35, 173 182 ( 2014).\n[114] N. C. Tsai, C. T. Yeh, H. L. Chiu. Linear magnetic clutch to automatically\ncontrol torque output. ISA Transactions , 76, 224 234 ( 2018).\n[115] M.Wciślik,K.Suchenia.Holonomicityanalysisofelectromechanicalsystem\nontheexampleofsimpleswitchedreluctancemotor. ComputerApplications\nin Electrical Engineering , 13, 330–339 ( 2015).\n[116] X. Wang, R. Palka, M. Wardach. Nonlinear digital simulation models of\nswitched reluctance motor drive. Energies, 13(24), 1 17 ( 2020).\n[117] F. Fereidoony, S. P. M. Nagaraja, J. P. D. Santos, Y. E. Wang. Efficient ULF\nTransmission Utilizing Stacked Magnetic Pendulum Array. IEEE Transac \ntions on Antennas and Propagation , 70(1), 585–597 ( 2022).\n[118] M. N. Srinivas Prasad, F. Fereidoony, Y. E. Wang. 2D Stacked Magnetic\nPendulum Arrays for Efficient ULF Transmission. 2020 IEEE International\nSymposium on Antennas and Propagation and North American Radio Science\nMeeting,IEEECONF2020 Proceedings ,1305–1306.InstituteofElectricaland\nElectronics Engineers Inc., Toronto ( 2020).\n[119] S.P.MysoreNagaraja,R.U.Tok,R.Zhu,S.Bland,A.Propst,Y.E.Wang. Ma-\ngnetic Pendulum Arrays for Efficient Wireless Power Transmission. Journal\nof Physics: Conference Series , 1407(1) ( 2019).\n[120] S. Babic, C. Akyel. Magnetic Force Calculation Between Thin Coaxial Cir-\ncular Coils in Air. IEEE Transactions on Magnetics , 44(4), 445–452 ( 2008).\n[121] G.Lemarquand,V.Lemarquand,S.Babic,C.Akyel,G.Lemarquand,V.Le-\nmarquand, S. Babic, C. Akyel. Magnetic Field Created by Thin Wall So-\nlenoids and Axially Magnetized Cylindrical Permanent Magnets. PIERS\nProceedings , 614–618 ( 2009).\n130LITERATURA\n[122] A.Shiri,A.Shoulaie.Anewmethodologyformagneticforcecalculationsbe-\ntweenplanarspiralcoils. ProgressInElectromagneticsResearch,PIER ,tom95,\n39–57 (2009).\n[123] R.Ravaud,G.Lemarquand,S.Babic,V.Lemarquand,C.Akyel. Cylindrical\nMagnets and Coils: Fields, Forces, and Inductances. IEEE Transactions on\nMagnetics , 46(9), 3585–3590 ( 2010).\n[124] R. Ravaud, G. Lemarquand, V. Lemarquand, S. Babic, C. Akyel. Mutual\ninductanceandforceexertedbetweenthickcoils. ProgressInElectromagnetics\nResearch, PIER , 102, 367 380 ( 2010).\n[125] S.I.Babic,C.Akyel. Torquecalculationbetweencircularcoilswithinclined\naxes in air. International Journal of Numerical Modelling , 24, 230–243 ( 2011).\n[126] W. Robertson, B. Cazzolato, A. Zander. Axial Force Between a Thick Co-\nil and a Cylindrical Permanent Magnet: Optimizing the Geometry of an\nElectromagnetic Actuator. IEEE Transactions on Magnetics , 48(9), 2479 2487\n(2012).\n[127] J. M. Camacho, V. Sosa. Alternative method to calculate the magnetic field\nofpermanentmagnetswithazimuthalsymmetry. RevistaMexicanadeFisica\nE, 59(1), 8 17 ( 2013).\n[128] ZhongJianWang,YongRen,Z.J.Wang,Y.Ren,ZhongJianWang,YongRen,\nZ.J.Wang,Y.Ren. MagneticForceandTorqueCalculationbetweenCircular\nCoilswithNonparallelAxes. IEEETransactionsonAppliedSuperconductivity ,\n24(4), 1–5 ( 2014).\n[129] D.J.Griffiths,V.Hnizdo. Mansuripur’sparadox. AmericanJournalofPhysics ,\n81(8), 570–574 ( 2013).\n[130] S.Babic,C.Akyel. Magneticforcebetweeninclinedcircularloops(Lorentz\napproach). Progress In Electromagnetics Research B , 38, 333 349 ( 2012).\n[131] S.Babic,C.Akyel. NewFormulasforCalculatingTorquebetweenFilamen-\ntary Circular Coil and Thin Wall Solenoid with Inclined Axis Whose Axes\nare at the Same Plane. Progress In Electromagnetics Research M , 73, 141 151\n(2018).\n[132] S. P. Hendrix, S. G. Prilliman. Measuring the Force between Magnets as an\nAnalogy for Coulomb’s Law. Journal of Chemical Education , 95(5), 833–836\n(2018).\n[133] M. Kwon, J. Jung, T. Jang, S. Sohn. Magnetic Forces Between a Magnet and\na Solenoid. The Physics Teacher , 58(5), 330–334 ( 2020).\n[134] D. J. Griffiths. Introduction to Electrodynamics . Pearson Education, 4 wyd.\n(1981).\n131LITERATURA\n[135] F. Marques, P. Flores, J. C. Claro, H. M. Lankarani, J. Pimenta Claro, H. M.\nLankarani,J.C.Claro,H.M.Lankarani,J.PimentaClaro,H.M.Lankarani.\nModeling and analysis of friction including rolling effects in multibody\ndynamics: a review. Multibody System Dynamics , 45(2), 223–244 ( 2019).\n[136] J.Harris,A.Stevenson.OntheRoleofNonlinearityintheDynamicBehavior\nof Rubber Components. Rubber Chemistry and Technology , 59(5), 740–764\n(1986).\n[137] G. Terenzi. Dynamics of SDOF system with nonlinear viscous damping.\nJournal of Engineering Mechanics , 125, 972–978 ( 1999).\n[138] J. Awrejcewicz, V. A. Krysko. Wprowadzenie do Współczesnych Metod Asymp-\ntotycznych . Wydawnictwo Naukowo Techniczne, Warszawa ( 2004).\n[139] V. N. Pilipchuk. Application of special nonsmooth temporal transforma-\ntions to linear and nonlinear systems under discontinuous and impulsive\nexcitation. Nonlinear Dynamics , 18(3), 203–234 ( 1999).\n[140] R. A. Ibrahim. Vibro-impact Dynamics: Modeling, Mapping and Applications ,\ntom 43. Springer, Berlin ( 2009).\n[141] S.Skurativskyi,K.Polczyński,M.Wojna,J.Awrejcewicz.Quantifyingperio-\ndic, multi periodic, hidden and unstable regimes of a magnetic pendulum\nvia semi-analytical, numerical and experimental methods. Journal of Sound\nand Vibration , 524, 116710 ( 2022).\n[142] A.Wijata,K.Polczyński,J.Awrejcewicz. Theoreticalandnumericalanalysis\nof regular one-side oscillations in a single pendulum system driven by a\nmagneticfield. MechanicalSystemsandSignalProcessing ,150,107229( 2021).\n[143] G.Benettin,L.Galgani,A.Giorgilli,J.-M.M.Strelcyn.LyapunovCharacteri-\nsticExponentsforsmoothdynamicalsystemsandforhamiltoniansystems;\na method for computing all of them. Part 1: Theory. Meccanica , 15(1), 9 20\n(1980).\n[144] M. Sandri. Numerical calculation of Lyapunov exponents. Mathematica\nJournal, 6(3), 78–84 ( 1996).\n[145] J. Awrejcewicz. Drgania Deterministyczne Układów Dyskretnych . Wydawnic-\ntwo Naukowo Techniczne, Warszawa ( 1996).\n[146] K. Polczyński, A. Wijata, J. Awrejcewicz, G. Wasilewski. Numerical and\nexperimentalstudyofdynamicsoftwopendulumsunderamagneticfield.\nProceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems\nand Control Engineering , 233(4), 441–453 ( 2019).\n[147] K. Polczyński, A. Wijata, G. Wasilewski, G. Kudra, J. Awrejcewicz. Model-\nlingandAnalysisofBifurcationDynamicsofTwoCoupledPendulumswith\naMagneticForcing. I.Kovacic,S.Lenci(red.), IUTAMSymposiumonExplo \niting Nonlinear Dynamics for Engineering Systems , tom 37, 213–223. Springer\nInternational Publishing, Cham ( 2020).\n132LITERATURA\n[148] K.Polczyński,S.Skurativskyi,M.Bednarek,J.Awrejcewicz.Nonlinearoscil-\nlations of coupled pendulums subjected to an external magnetic stimulus.\nMechanical Systems and Signal Processing , 154, 107560 ( 2021).\n[149] S. Aubry, G. Kopidakis, A. M. Morgante, G. P. Tsironis. Analytic condi-\ntions for targeted energy transfer between nonlinear oscillators or discrete\nbreathers. Physica B: Condensed Matter , 296(1-3), 222–236 ( 2001).\n[150] P.Maniadis,S.Aubry. TargetedenergytransferbyFermiresonance. Physica\nD: Nonlinear Phenomena , 202(3-4), 200–217 ( 2005).\n[151] F.Cottone,H.Vocca,L.Gammaitoni.NonlinearEnergyHarvesting. Physical\nReview Letters , 080601(102), 1–4 ( 2009).\n[152] L.Gammaitoni,I.Neri,H.Vocca. Nonlinearoscillatorsforvibrationenergy\nharvesting. Applied Physics Letters , 94(16), 1–4 ( 2009).\n[153] Z. Parszewski. Drgania i Dynamika Maszyn . WNT, Warszawa ( 1982).\n[154] V.N.Pilipchuk.Nonlinearinteractionsandenergyexchangebetweenliquid\nsloshing modes. Physica D: Nonlinear Phenomena , 263, 21–40 ( 2013).\n[155] V.N.Pilipchuk. Frictioninducedpatternformationsandmodaltransitions\nin a mass-spring chain model of sliding interface. Mechanical Systems and\nSignal Processing , 147, 107119 ( 2021).\n[156] V.N.Pilipchuk.EffectiveHamiltoniansforResonanceInteractionDynamics\nand Interdisciplinary Analogies. Procedia IUTAM , 19, 27 34 ( 2016).\n[157] V.N.Pilipchuk,K.Polczyński,M.Bednarek,J.Awrejcewicz.Guidanceofthe\nresonance energy flow in the mechanism of coupled magnetic pendulums.\nMechanism and Machine Theory , 176, 105019 ( 2022).\n[158] T.Li,S.Seguy,A.Berlioz. Onthedynamicsaroundtargetedenergytransfer\nfor vibro-impact nonlinear energy sink. Nonlinear Dynamics , 87(3), 1453–\n1466 (2017).\n133"    },    {        "title": "1108.0378v1.A_non_isothermal_phase_field_model_for_the_ferromagnetic_transition.pdf",        "content": "A NON ISOTHERMAL PHASE-FIELD MODEL FOR THE\nFERROMAGNETIC TRANSITION\nV. BERTI AND D. GRANDI\nAbstract. We propose a model for non isothermal ferromagnetic phase transition based\non a phase \feld approach, in which the phase parameter is related but not identi\fed with\nthe magnetization. The magnetization is split in a paramagnetic and in a ferromagnetic\ncontribution, dependent on a scalar phase parameter and identically null above the Curie\ntemperature. The dynamics of the magnetization below the Curie temperature is gov-\nerned by the order parameter evolution equation and by a Landau-Lifshitz type equation\nfor the magnetization vector. In the simple situation of a uniaxial magnet it is shown\nhow the order parameter dynamics reproduces the hysteresis e\u000bect of the magnetization.\n1.Introduction\nThe peculiar feature of ferromagnetic materials is the behaviour of the magnetiza-\ntion vector below a characteristic value \u0012cof temperature, named the Curie temperature .\nFirstly, at temperatures \u0012<\u0012 c, a non zero value M0(\u0012) of the magnetization is stable even\nat zero external magnetic \felds; this magnetization is said the spontaneous magnetization .\nOn the contrary, in the paramagnetic regime, that is for \u0012>\u0012 c, the magnetization vanishes\nat zero external \feld. Moreover, the way in which external \felds in\ruence the magnetiza-\ntion vector di\u000bers in the two cases. While in the paramagnetic state the magnetization at\na point is a function of the magnetic \felds at that point (with a proportionality relation\nat su\u000eciently low \felds), in the ferromagnetic state the magnetization shows well known\nhysteresis phenomena [1]. In the ferromagnetic materials the external \felds in\ruence\nprimarily the direc tion of the magnetization vector, and a model for the magnetization\nevolution in saturation conditions has been proposed long ago by Landau and Lifshitz\n[2, 3].\nThese qualitative di\u000berences in the behaviour of the magnetization on the two sides of\nthe Curie temperature can be understood in the framework of the phase transitions. The\nclari\fcation of this issue is a fundamental contribution of the Landau theory of phase\ntransitions [4, 5]. The approach of Landau is based on the concept of an order parameter ,\nwhich is a physical (macroscopical) observable quantity whose behaviour is able to account\n1arXiv:1108.0378v1  [cond-mat.other]  1 Aug 20112 V. BERTI AND D. GRANDI\nfor the microscopical change of structure which generally characterizes phase transitions.\nStrictly speaking, the Landau theory accounts for second order (or continuous) phase\ntransitions, which, according to Landau, are properly understood in terms of symmetry\nbreaking . So, in the case of ferromagnetic transition, for example, the order parameter of\nthe Landau theory is the magnetization vector: the transition manifests itself as rotational\nsymmetry breaking due to the set up of a non-zero magnetization\nM0under otherwise isotropic conditions, that is in zero external \feld. Perhaps, it is not\nsuper\ruous to point out that the value of the order parameter is not in itself an indicator\nof the phase of the material, except in the particular case of null external \feld.\nAlong these lines, in [6] a three dimensional evolutive model is proposed, using the whole\nmagnetization vector with the order parameter, ruled by a vectorial Ginzburg-Landau\ntime-dependent equation.\nIn this paper we propose a model di\u000berent from the original Landau setting and closer to a\ngeneral phase \feld approach. That is, we introduce a scalar phase \feld, which (unlike the\nmagnetization vector) vanishes above the Curie temperature, even in presence of external\n\felds. The order parameter vanishes in a continuous way when the Curie temperature\nis approached, according to the second order character of the transition. The magneti-\nzation will be decomposed in two contributions, one which is of paramagnetic character,\nnamely it is a direct function of the external \feld, and an other one, depending also on\nthe order parameter, which is considered as an independent \feld with its own evolution\nequation (the time-dependent Ginzburg-Landau equation). The sense of this separation\nis not, of course, that of a physical distinction between two di\u000berent sources of magneti-\nzation, as well as, for example, the two \ruid theory of super\ruidity [7] is not the theory\nof a mixture of \ruids. The two magnetization contributions re\rect the di\u000berent way in\nwhich the magnetization evolves under the external \feld in the ferromagnetic and in the\nparamagnetic regime. In particular, the hysteresis phenomena manifesti ng themselves\nin the ferromagnetic phase are determined by the coupling with the phase \feld evolution\nequation. Under this respect, the model draws on the internal variable models, in which\na history-dependent constitutive equation for a physical quantity (in our case the mag-\nnetization) is obtained through the coupling with an internal variable obeying its own\ndi\u000berential evolution equation ([8]). Nevertheless, in this model, the phase \feld is not a\nmere internal variable, nor its evolution equation is a constitutive one. We assume thatA NON ISOTHERMAL PHASE-FIELD MODEL FOR THE FERROMAGNETIC TRANSITION 3\nthe phase \feld equation is a true balance equation associated with its own power balance\n[9].\nThe phase \feld of this model is related to the modulus of the spontaneous magnetization\n(nevertheless it is in\ruenced by the external \feld); loosely speaking, we can say that its\nphysical meaning is related to the microscopic order set up by the microscopic exchange\ninteractions, responsible of spontaneous magnetization.\nThe model we are proposing couples the Maxwell equations for the electromagnetic \feld\nwith a scalar time-dependent Ginzburg-Landau equation for the order parameter and a\nheat balance equation for the temperature. Two constitutive equations de\fne the rela-\ntion between the magnetization and the order parameter: the magnetization is split in\na paramagnetic and a ferromagnetic contribution and the direction of the last one, in\nthe tridimensional case, is ruled by a Landau-Lifshitz-Gilbert equation [2, 10], suitably\nmodi\fed in order to describe the non saturated regime. The constitutive choices of the\nmodel are veri\fed to be consistent with the second law of thermodynamics in the form of\nthe Clausius-Duhem inequality.\nFinally, we present a particular case of the model when the direction of the magnetization\nis \fxed, as in the case of uniaxial ferromagnets (one-dimensional model). We show in this\nsituation how the dynamics of the order parameter in the ferromagnetic phase gives the\nusual shape of the hysteresis cycle of the magnetization.\n2.Three dimensional model\nLet us consider a ferromagnetic material occupying a bounded domain \n \u001aR3. Denot-\ning by E;H;D;Bthe electric \feld, the magnetic \feld, the electric displacement and the\nmagnetic induction, the behavior of the material is ruled by Maxwell's equations\nr\u0002E=\u0000_B;r\u0002H=_D+J; (1)\nr\u0001B= 0;r\u0001D=\u001a; (2)\nwhere Jis the current density and \u001ais the charge density. We assume the constitutive\nequations\n(3) D=\"E; B=\u00160H+M; J=\u001bE;\nwhere\";\u0016 0;\u001bare respectively the dielectric constant, the magnetic permeability and the\nconductivity, while Mdenotes the magnetization. As known, in paramagnetic materials,4 V. BERTI AND D. GRANDI\nthe magnetization is a function of the magnetic \feld H. On the contrary, ferromagnetic\nsystems are characterized by a time-non local relation between magnetization and the\nmagnetic \feld. Therefore, in order to describe paramagnetic-ferromagnetic transitions,\nwe write\n(4) B=\u0016H+^M; \u0016 =\u0016(H;\u0012):\nThis amounts to split the magnetization as M= (\u0016\u0000\u00160)H+^M, where ^Mis the part of\nthe magnetization whose value at a given time cannot be expressed as function of the \feld\nHat the same time. This contribution exists only in the ferromagnetic phase and it is\nhistory-dependent, so ^M= 0 in the paramagnetic state and ^M6= 0 in the ferromagnetic\nregime. The model we propose is set in the general context of the Ginzburg-Landau\ntheory by de\fning, as order parameter, a scalar phase variable 'such that'= 0 in\nthe paramagnetic phase and ' > 0 in the ferromagnetic state. As a consequence the\nmagnetization is related to '. More precisely, we assume that\n(5) ^M=M(';\u0012)m; M (';\u0012)\u00150\nwhere mis a unit versor and the modulus M(';\u0012) depends also on the temperature.\nThe evolution of the phase 'is given by the Ginzburg-Landau equation typical of\nphase transition models. We introduce the classical potentials describing second-order\nphase transitions ([9])\nF(') =1\n4'4\u00001\n2'2; G (') =1\n2'2\nand assume the following equation\n(6) \u001c_'=1\n\u00142\u0001'\u0000\u0012cF0(')\u0000\u0012G0(')\u0000A(';\u0012)I(\u0012c\u0000\u0012)H\u0001m\nwhere\u001c,\u00142are positive constants, Ais a generic function whose de\fnition will be speci\fed\nlater, andIis the unit step function, i.e.\nI(x) =(\n0 ifx<0\n1 ifx\u00150\nNotice that for large values of the temperature, \u0012>\u0012 c, last term of (6) vanishes and the\nfunction\nW(') =\u0012cF(') +\u0012G(')\nadmits the minimum value '= 0 which characterizes the paramagnetic phase.A NON ISOTHERMAL PHASE-FIELD MODEL FOR THE FERROMAGNETIC TRANSITION 5\nThe evolution equation for 'should preserve the de\fning condition '\u00150 , which is not\nautomatic in eq. (6). So this is a further constraint which has to be enforced, for example,\nin any numerical solution of the equation, and amounts to add a singular contribution in\nthe potential function F(') such that F(') = +1for'<0.\nConcerning the evolution of the direction of the magnetization, we assume that the\nunit versor msatis\fes the Landau-Lifshitz equation ([2])\n(7) '2_m=\u0000\r'2m\u0002H\u0000\u0015m\u0002(m\u0002H); \r;\u0015> 0:\nNotice that\n'2j_mj= (\r2'4+\u00152)1\n2jm\u0002Hj:\nAs a consequence, when 'approaches zero, the direction of mmoves toward the direction\nofH. Moreover, by multiplying (7) by mwe obtain\n'2_m\u0001m= 0;\nwhich is consistent with the condition jm(x;t)j= 1 for any t>0, provided thatjm(x;0)j=\n1.\nLike other models of phase transitions (see [9]), equation (6) can be interpreted as a\nbalance law of the order structure. Indeed it can be written in the form\nk=r\u0001p;\nwhere\nk=\u001c_'+\u0012cF0(') +\u0012G0(') +A(';\u0012)I(\u0012c\u0000\u0012)H\u0001m;\np=1\n\u00142r':\nThis formulation allows us to de\fne the internal power related to the phase variable as\nP'=k_'+p\u0001r_'\n=\u001c_'2+\u0012c_F(') +\u0012_G(') +A(';\u0012)I(\u0012c\u0000\u0012) _'m\u0001H+1\n\u00142r'\u0001r_':\nFrom (4) and (5) we deduce the relation\n_B=\u0016_H+\u0012@\u0016\n@\u0012_\u0012+@\u0016\n@H\u0001_H\u0013\nH+\u0012@M\n@'_'+@M\n@\u0012_\u0012\u0013\nm+M(';\u0012)_m:\nHence the electromagnetic power\nPel=_B\u0001H+_D\u0001E+\u001bE26 V. BERTI AND D. GRANDI\ncan be written as\nPel=\u0016_H\u0001H+\u0012@\u0016\n@\u0012_\u0012+@\u0016\n@H\u0001_H\u0013\nH2+\u0012@M\n@'_'+@M\n@\u0012_\u0012\u0013\nm\u0001H+M(';\u0012)_m\u0001H\n+\"_E\u0001E+\u001bE2:\nWe denote by ethe internal energy and hthe thermal power. The \frst law of thermody-\nnamics reads\n(8) _ e=Pel+P'+h:\nwherehsatis\fes the thermal balance law\n(9) h=\u0000r\u0001q+r;\nandq,rare respectively the heat \rux and the heat source.\nIn order to prove the consistence of the model with the second law of thermodynamics\nwe look for the constitutive relations for the entropy function \u0011and the heat \rux qthat\nensure the ful\fllment of Clausius-Duhem inequality\n_\u0011\u0015\u0000r\u0001\u0010q\n\u0012\u0011\n+r\n\u0012:\nThermal balance law (9) yields\n\u0012_\u0011\u0015q\n\u0012\u0001r\u0012+h:\nHence, by introducing the free energy  =e\u0000\u0012\u0011, the previous inequality leads to\n_ +\u0011_\u0012\u0014P el+P'\u0000q\n\u0012\u0001r\u0012:\nBy substituting the expressions of the powers, we obtain\n_ +\u0011_\u0012\u0014\u0016_H\u0001H+\u0012@\u0016\n@\u0012_\u0012+@\u0016\n@H\u0001_H\u0013\nH2+\u0012@M\n@'_'+@M\n@\u0012_\u0012\u0013\nm\u0001H+M(';\u0012)_m\u0001H\n+\"_E\u0001E+\u001bE2+\u001c_'2+\u0012c_F(') +\u0012_G(') +A(';\u0012)I(\u0012c\u0000\u0012) _'m\u0001H\n+1\n\u00142r'\u0001r_'\u0000q\n\u0012\u0001r\u0012:\nBy means of (7), we deduce\n_ +\u0011_\u0012\u0014\u0016_H\u0001H+\u0012@\u0016\n@\u0012_\u0012+@\u0016\n@H\u0001_H\u0013\nH2+\u0012@M\n@'_'+@M\n@\u0012_\u0012\u0013\nm\u0001H\n+\u0015M(';\u0012)'\u00002jm\u0002Hj2+\"_E\u0001E+\u001bE2+\u001c_'2+\u0012c_F(') +\u0012_G(')\n+A(';\u0012)I(\u0012c\u0000\u0012) _'m\u0001H+1\n\u00142r'\u0001r_'\u0000q\n\u0012\u0001r\u0012:A NON ISOTHERMAL PHASE-FIELD MODEL FOR THE FERROMAGNETIC TRANSITION 7\nWe assume that the free energy  depends on the variables ( ';r';\u0012;E;H), so that\nthe previous inequality yields\n\u0014@ \n@'\u0000\u0012\nA(';\u0012)I(\u0012c\u0000\u0012) +@M\n@'\u0013\nm\u0001H\u0000\u0012cF0(')\u0000\u0012G0(')\u0015\n_'\n+\u0014@ \n@r'\u00001\n\u00142r'\u0015\n\u0001r_'+\u0014@ \n@\u0012+\u0011\u0000@\u0016\n@\u0012H2\u0000@M\n@\u0012m\u0001H\u0015\n_\u0012+\u0014@ \n@E\u0000\"E\u0015\n\u0001_E\n+\u0014@ \n@H\u0000\u0016H\u0000@\u0016\n@HH2\u0015\n\u0001_H\n\u0014\u0015M(';\u0012)'\u00002jm\u0002Hj2+\u001bE2+\u001c_'2\u0000q\n\u0012\u0001r\u0012: (10)\nThe previous inequality is ful\flled if we choose the constitutive relations\nA(';\u0012)I(\u0012c\u0000\u0012) +@M\n@'= 0\nq=\u0000k0(\u0012)r\u0012 k 0(\u0012)>0:\nUsual arguments of thermodynamics based on the arbitrariness of ( _ ';r_';_\u0012;_E;_H) lead\nto the following expressions of the free energy and entropy\n = 0(\u0012) +\"\n2E2+\u0016H2+1\n2\u00142jr'j2\u0000Z\n\u0016(H;\u0012)H\u0001dH+\u0012cF(') +\u0012G(')\n\u0011=\u0000@ \n@\u0012+@\u0016\n@\u0012H2+@M\n@\u0012m\u0001H\n=\u0000 0\n0(\u0012) +Z@\u0016\n@\u0012H\u0001dH\u0000G(') +@M\n@\u0012m\u0001H:\nSubstitution into (10) yields\n\u0015M(';\u0012)'\u00002jm\u0002Hj2+\u001bE2+\u001c_'2+k0(\u0012)\n\u0012jr\u0012j2\u00150;\nwhich guarantees that Clausius-Duhem inequality is satis\fed.\nThe evolution equation for the temperature follows from the thermal balance law (9),\nby substituting the expression of hdeduced from the \frst law (8). Since the internal\nenergy is written as\ne= +\u0012\u0011=e0(\u0012) +\"\n2E2+\u0016H2+1\n2\u00142jr'j2\n\u0000Z\u0012\n\u0016\u0000\u0012@\u0016\n@\u0012\u0013\nH\u0001dH+\u0012cF(') +\u0012@M\n@\u0012m\u0001H;\nwhere\ne0(\u0012) = 0(\u0012)\u0000\u0012 0\n0(\u0012);8 V. BERTI AND D. GRANDI\nsubstitution into (8) yields\nh=e0\n0(\u0012)_\u0012\u0000\u0012_G(') +\u0012@\u0016\n@\u0012H\u0001_H+\u0012d\ndt\u0012@M\n@\u0012m\u0001H\u0013\n+\u0012_\u0012Z@2\u0016\n@\u00122H\u0001dH\n\u0000\u001c_'2\u0000\u001bE2\u0000\u0015M(';\u0012)'\u00002jm\u0002Hj2: (11)\nHence the temperature satis\fes the equation\n(12) h=\u0000r\u0001 [k0(\u0012)r\u0012] +r:\nIn this model we assume the following constitutive equation\nA(';\u0012) =\u0012\u0000\u0012c:\nThis choice leads to a continuous temperature dependence for the modulus of ^M, namely\nM='(\u0012\u0000\u0012c)\u0000;\nwhere the subscript \u0000denotes the negative part of a function, i.e. f\u0000= maxf\u0000f;0g.\nTherefore the Ginzburg-Landau equation for the phase \feld reads\n\u001c_'=1\n\u00142\u0001'\u0000\u0012cF0(')\u0000\u0012G0(') + (\u0012\u0000\u0012c)\u0000H\u0001m:\n3.One dimensional model\nIn this section we will consider a one-dimensional model, obtained by assuming that\nthe magnetic and electric \felds have constant and orthogonal directions, say y,z, and\nthat the components of the unknown \felds depend only by the variable x, namely\nE=E(x)k; H=H(x)j:\nIn the description of uniaxial ferromagnets, we modify the de\fnition of the order param-\neter, by requiring that '6= 0 in the ferromagnetic phase and '= 0 in the paramagnetic\nstate. Therefore 'is allowed to take negative values and the vector mis de\fned as\nm=sign(')j:\nWe assume the constitutive equations\n(13) M(';\u0012) =j'j(\u0012\u0000\u0012c)\u0000:\nTherefore\n^M='(\u0012\u0000\u0012c)\u0000jA NON ISOTHERMAL PHASE-FIELD MODEL FOR THE FERROMAGNETIC TRANSITION 9\nand equations (1)-(4) imply\n\"_E=@xH\u0000\u001bE\n\u0016_H+@\u0016\n@\u0012H_\u0012+@\u0016\n@HH_H=@xE\u0000_'(\u0012\u0000\u0012c)\u0000+'I(\u0012c\u0000\u0012)_\u0012:\nThe evolution of the magnetization 'is governed by the Ginzburg-Landau equation\n(14) \u001c_'=1\nk2@xx'\u0000\u0012cF0(')\u0000\u0012G0(') + (\u0012\u0000\u0012c)\u0000H:\nFinally, the evolution equation for the temperature is deduced by (11) and (12)\n\u0014\ne0\n0(\u0012) +\u0012Z@2\u0016\n@\u00122H\u0001dH\u0015\n_\u0012\u0000\u0012_G(') +\u0012@\u0016\n@\u0012H_H\u0000\u0012d\ndt[I(\u0012c\u0000\u0012)'H]\n=\u001c_'2+\u001bE2\u0000@xq+r: (15)\nA constitutive equation of \u0016(\u0012;H) has to be given. For example, in the classical Landau\nmodel of ferromagnetism, the (total) magnetization Mas a function of the magnetic \feld\nand the temperature is given by\n(16) b0M3+a0(\u0012\u0000\u0012c)M\u0000H= 0;\nfrom which it is obtained the permeability at \u0012>\u0012 c\n(17)\u0016\n\u00160\u00001 =1\nb0M(H;\u0012)2+a0(\u0012\u0000\u0012c):\nForH= 0 this equation provides the well known Curie-Weiss law for the susceptibility,\n\u001f0/1\n\u0012\u0000\u0012c\u0012>\u0012 c:\nForH6= 0 the resultant permeability is a regular function of the temperature. In our\nmodel, it is required that, whatever the constitutive relation for \u0016is taken, the integral\nJ(\u0012;H) =ZH\n0@2\u0016\n@\u00122H0\u0001dH0\nexists \fnite. We observe that for H!1 ,@2\u0016=@\u00122is expected to tend at zero for satu-\nration reasons, while, in this respect, the permeability resulting from (17) is reasonable\nonly at small \felds. Moreover, the function e0\n0(\u0012) has to satisfy e0\n0(\u0012) +\u0012J(\u0012;H)>0 for\nevery\u0012andHto have a standard parabolic heat equation.\nWe see that equation (14) is able to recover the hysteresis diagram typical of the phe-\nnomenon of ferromagnetism. To this purpose, we will consider a spatially homogeneous\nmaterial in isothermal conditions, with \u0012<\u0012 c. Then (14) reduces to\n(18) \u001c_'=\u0000\u0012cF0(')\u0000\u0012G0(')\u0000(\u0012\u0000\u0012c)H:10 V. BERTI AND D. GRANDI\nMoreover, from (4) and (5) we obtain\n(19) B=\u0016H\u0000(\u0012\u0000\u0012c)':\nHere we assume \u0016as approximately H-independent in the considered range of the mag-\nnetic \feld. If His a known function of time, equations (18)-(19) allow us to obtain\ntheB\u0000Hdiagram. In particular if H=H0sin(!t),t2[0;T] and the initial condi-\ntion is'(0) = 0, we deduce the following hysteresis diagrams for di\u000berent values of the\ntemperature.\n-4 -2 2 4\n-0.1-0.050.050.1\nFigure 1. B\u0000Hdiagram with the numerical constants \u0012= 0:9; \u0012 c=\n1; ! =\u0019; \u001c = 0:01; H 0= 4; \u0016 = 0:01; T = 2:5.\n-4 -2 2 4\n-0.6-0.4-0.20.20.40.6\nFigure 2. B\u0000Hdiagram with the numerical constants \u0012= 0:5; \u0012 c=\n1; ! =\u0019; \u001c = 0:01; H 0= 4; \u0016 = 0:01; T = 2:5.A NON ISOTHERMAL PHASE-FIELD MODEL FOR THE FERROMAGNETIC TRANSITION 11\n-4 -2 2 4\n-1.5-1-0.50.511.5\nFigure 3. B\u0000Hdiagram with the numerical constants \u0012= 0:1; \u0012 c=\n1; ! =\u0019; \u001c = 0:01; H 0= 4; \u0016 = 0:01; T = 2:5.\nReferences\n[1] M. Brokate, J. Sprekels. Hysteresis and phase transitions . Springer: New York; 1996.\n[2] L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii. On the theory of the dispersion of magnetic permeability\nin ferromagnetic bodies. Physikalische Zeischrift der Sowjetunion 1935; 8: 153-169.\n[3] L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii. Electrodynamics of continuous media . Pergamon Press:\nOxford; 1984.\n[4] L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii. Statistical Physics, Part 1 , Pergamon Press: Oxford;\n1984.\n[5] N. Goldenfeld. Lectures on phase transitions and the normalization group . Addison-Wesley: Reading,\nMass.; 1992.\n[6] V. Berti, M. Fabrizio, C. Giorgi. A three dimensional phase transition model in ferromagnetism:\nexistence and uniqueness. Journal of Mathematical Analysis and Applications . 2009; 335(2): 661{\n674. DOI:10.1016/j.jmaa.2009.01.065.\n[7] D. R. Tilley, J. Tilley. Super\ruidity and superconductivity . Institute of physics publishing: Bristol,\nPhiladelphia; 1990.\n[8] A. Visintin. Di\u000berential Models of Hysteresis . Applied Mathematical Sciences, vol. 111. Springer:\nBerlin; 1994.\n[9] M. Fabrizio. Ginzburg-Landau equations and \frst and second order phase transitions. International\nJournal of Engineering Science . 2006; 44(8-9): 529-539. DOI:10.1016/j.ijengsci.2006.02.006\n[10] T.L. Gilbert. A phenomenological theory of damping in ferromagnetic materials. IEEE Transactions\non Magnetics 2004; 40(6): 3443-3449. DOI:10.1109/TMAG.2004.836740.\n[11] C. Kittel. Introduction to solid state physics . John Wiley & Sons: New York, 1961.\n(V. Berti) University of Bologna, Department of Mathematics, Piazza di Porta S. Do-\nnato 5, 40126 Bologna, Italy.\nE-mail address :berti@dm.unibo.it\n(D. Grandi) University of Bologna, Department of Mathematics, Piazza di Porta S.\nDonato 5, 40126 Bologna, Italy.\nE-mail address :grandi@dm.unibo.it"    },    {        "title": "1509.02836v1.Inertial_terms_to_magnetization_dynamics_in_ferromagnetic_thin_films.pdf",        "content": "arXiv:1509.02836v1  [cond-mat.mtrl-sci]  9 Sep 2015Inertial terms to magnetization dynamics in ferromagnetic thin\nfilms\nY. Li1,4, A.-L. Barra2, S. Auffret3,4,5, U. Ebels3,4,5, and W. E. Bailey1∗\n1Materials Science & Engineering, Dept. of Applied Physics &\nApplied Mathematics, Columbia University, New York NY 10027, USA\n2Laboratoire National des Champs Magn´ etiques Intenses,\nCNRS/UJF/UPS/INSA, 38042 Grenoble Cedex, France\n3Univ. Grenoble Alpes, INAC-SPINTEC, F-38000 Grenoble, France\n4CNRS, SPINTEC, F-38000 Grenoble, France\n5CEA, INAC-SPINTEC, F-38000 Grenoble, France\n∗Correspondence to: web54@columbia.edu\n(Dated: July 9, 2021)\nAbstract\nInertial magnetization dynamics have been predicted at ult rahigh speeds, or frequencies ap-\nproachingtheenergy relaxation scaleof electrons, inferr omagnetic metals. Hereweidentifyinertial\nterms to magnetization dynamics in thin Ni 79Fe21and Co films near room temperature. Effective\nmagnetic fields measured in high-frequency ferromagnetic r esonance (115-345 GHz) show an addi-\ntional stiffening term which is quadratic in frequency and ∼80 mT at the high frequency limit of\nour experiment. Our results extend understanding of magnet ization dynamics at sub-picosecond\ntime scales.\n1The magnetization M(t) in ferromagnetic materials is generally understood to evolve\nwithout memory of its prior motion. As described by the Landau-Lifs hitz (LL) equation[1],\nmagnetization dynamics dM(t)/dtcan be written in terms of the magnetization M(t) alone,\nexcluding any temporal derivatives dnM(t)/dtn. Magnetization is then ’inertialess’: it re-\nsponds to a step magnetic field H(t) with an instantaneous change in speed and with infinite\nacceleration.\nAs pointed out by Ciornai et al.and F¨ ahnle et al.[2–4], followed by other theoreticians[5,\n6], the absence of inertia is questionable for magnetization dynamics at very high frequen-\ncies. The high frequency behavior, >100 GHz, becomes relevant in ultrafast ( <ps-scale)\nmagnetization switching schemes envisaged for magnetic informatio n storage technology[7–\n11]. In metallic ferromagnets, the electrons which comprise the mag netization themselves\npossess inertia, expressed through a finite (Bloch-state) lifetime τand cannot change their\nmomenta infinitely quickly. These authors[2–4] have proposed that the LL equation should\nbe amended to include an inertial term:\ndm\ndt=−µ0γ[m×Heff+αm×(m×Heff)]±αm×τd2m\ndt2(1)\nHeremis the unit magnetization vector, γis the gyromagnetic ratio, Heffis the ef-\nfective field, and αis the Gilbert damping coefficient. The additional nonlinear term\n±αm×τd2m/d2tis second-order in time. It is straightforward to show that this iner tial\nterm leads to an effective field H′\neff=∓αω2τ/γ, quadratic in frequency, which acts on\nmagnetization dynamics. Increasing the frequency limit for ferrom agnetic resonance (FMR)\nexperiments by a factor of 10 will therefore increase the magnitud e of the inertial effective\nfield by a factor of 100.\nThe sign of the inertial term in Eq. (1) refers to two different origins for the proposed\ninertial dynamics. Ciornai et al.[2, 12] argue that additional energy terms for the precession,\ntypically neglected, come about from transverse magnetization mo tion. Finite rotational\nmoments of inertia about the two transverse axes were introduce d semiclassically to account\nfor this energy in Ref. [2]. On the other hand, F¨ ahnle et al.[3, 4] have pointed out that\nsimilar dynamics arise from a more complete consideration of damping. Here the inertia\ncan be associated with the linear momentum of electrons: electrons cannot change their\nmomenta kmore rapidly during precession than their Bloch state relaxation time τB, and\ntend to remain at higher energy states. In this sense, inertia is pro portional to τB. The\n2rotational moment of inertia (”+” in Eq. 1) softens the resonance frequency while the linear\ninertia due to relaxation (”-” in Eq. 1) stiffens the resonance frequ ency.\nIn this work, we test for the existence of inertial effective fields wh ich act on the mag-\nnetization of the device ferromagnets permalloy (Ni 79Fe21, or Py) and Co. To do so, we\nmeasure FMR up to 345 GHz, roughly an order of magnitude higher th an is accessible\nusing conventional sources. Through measurements of the freq uency-dependent resonance\nfieldµ0Hres(ω), we find in all samples a negative nonlinear shift quadratic in frequen cy\nwhich reflects a positive effective field added by the inertial term. Th e sign of the shift\nshows the dominance of inertia due to the finite Bloch state relaxatio n time[3, 4] over the\ninertia due to finite rotational moment of inertia of electrons[2]. In a ll samples the extracted\nvalues of τare comparable with low-fluence limits of remagnetization times τEmeasured in\nultrafast demagnetization experiments. Our results show that hig h-frequency FMR (HF-\nFMR) provides a near-ground-state ( ∼1 meV) view of the processes involved in ultrafast\nmagnetization dynamics, heretofore accessible through far-fro m-equilibrium techniques.\nPy and Co thin films were deposited on Si/SiO 2substrates using ultra-high vacuum mag-\nnetron sputtering. The capping layers are Cu(5 nm)/SiO 2(5 nm) for Py and Cu(5 nm)/Ta(3\nnm) for Co; the seed layers are Ta(5 nm)/Cu(5 nm) in both series. T he thicknesses tFM=\n6, 10 and 30 nm for Py and 6, 10, 15, 30 nm for Co, respectively. Two series of field-swept,\nvariable-frequency FMR experiments were carried out, both with a pplied field and static\nmagnetization perpendicular to film plane. At low fields and frequencie s (3-26 GHz, 1-2.5\nT, 298 K), a coplanar waveguide was used to measure the power tra nsmission through the\nthin films. These measurements were carried out at Columbia Univers ity using an Fe-core\nelectromagnet[13]. At high fields and frequencies (115, 230 and 345 GHz, 4-13 T, 273-280\nK) continuous-wave high-field electron paramagnetic resonance w as used to measure the\npower reflection from the samples. These measurements were car ried out at the LNCMI-G\nusing a superconducting magnet[14]. Six field sweeps were taken for HF-FMR to reduce\nmeasurement and hysteresis errors.\nSeveral steps have been taken to ensure the consistency of low- and high-frequency mea-\nsurements. These are essential to detect the nonlinear term in ma gnetization dynamics.\n3(a)Fi/g427ng residual \nlow-f\nHF-FMRextrapola/g415onsignal x3\n230 GHz\n115 GHz345 GHz\nCo 15 nm (b)\nFIG. 1. (a) Low-frequency resonance fields for Py and Co <26 GHz. Lines and dashed lines are\nfits toω/γ=µ0(Hres−Meff). The upper panel shows the fitting residuals of the lower pan el.\n(b) High-frequency FMR absorption spectra for Co 15 nm. The ( horizontal) field axis shows the\ndifference of measured high-frequency resonance fields with e xpected values based on extrapolation\nof the low-frequency resonance fields without inertial effect s. The solid data points indicate the\nresonance field µ0Hres, with fits shown in the dashed curve.\nFirst, the magnetic fields of both low- and high-frequency FMR were calibrated in-situ\nusing paramagnetic markers[15]. An accuracy of 0.01% is obtained, c orresponding to ∼1\nmT at 345 GHz ( ∼13 T). Second, for low-frequency FMR, each sample normal has be en\nindependently aligned along the applied field by rotating the waveguide with respect to two\naxes (with <0.2◦precision) to maximize the value of µ0Hresat 3 GHz. The samples were\naligned optically, to lower precision, in HF-FMR. Third, we have correc ted for the small ( ∼\n20K)temperaturedifferences between thetwo apparati, throug hvariable-temperatureFMR\nmeasurements at Columbia University, which allow us to correct for t emperature-dependent\nmagnetization µ0Meff(T). See the Supplemental Information for details.\nFig. 1(a) shows the low-frequency µ0Hresas a function of ω/2πfor Py and Co. Contin-\nuous and dashed lines are fits to the linear Kittel equation ω/γ=µ0(Hres−Meff). The\ntop panel shows the fitting residuals, which are randomly distribute d within ±1 mT. In\norder to examine whether resonance field offset exists at high freq uencies, we plot in Fig.\n1(b) the HF-FMR lineshapes (average of six measurements) for Co 15 nm, as a function of\nµ0(H−Hextrap\nres), withµ0Hextrap\nresthe resonance field extrapolated from low-frequency FMR.\n4The solid circles indicate the resonance field µ0Hresextracted from the FMR lineshapes for\nthe three frequencies. The dashed lines are a fit of the resonance positions to a quadratic\nnonlinear term to frequency. A negative curvature is found in the d ashed lines. This implies\na negative sign in Eq. (1) with positive effective inertial field. Similar effe cts are found in\nother Py and Co films.\nFi/g427ng residual \nFi/g427ng residual \n(a) (b)\nFIG. 2. Deviation of µ0Hresfrom the linear Kittel equation for (a) Py and (b) Co. Main pan els:\nµ0Hres|exp−(µ0Meff+ω/γ)|fitin solid data points; α0ω2τ/γ|fitin curved lines. Meff,γandτ\nare the fit parameters from Eq. (2). Upper panels: fitting resi duals of main panels as a function\nof frequency. Insets: full plots of µ0Hresand fits to Eq. (2).\nTo quantify the nonlinear term from Eq. (1), we fit the resonance fi elds of both low- and\nhigh-frequency FMRmeasurements together into the linear Kittel equation, plus a quadratic\nterm:\nµ0Hres=µ0Meff+ω/γ−α0ω2τ/γ (2)\ntakingMeff,γandτas fit parameters. The parameter τgoverns the magnitude of the\ninertial term. The Gilbert damping coefficients α0are extracted by fitting the low-frequency\nFMR linewidths into µ0∆H1/2=µ0∆H0+ 2α0ω/γ, whereµ0∆H0is the inhomogeneous\nlinewidth. To estimate errorbars for HF-FMR, we take the scatter of all six measurements\n5into the data sequence.\nFigure 2 contains the central result of this manuscript. In the main panels we show\nexperimental resonance fields µ0Hresand fits to Eq. (2) as a function of frequency ω/2π\nfor (a) Py and (b) Co. The conventional effective field terms µ0Meff+ω/γhave been\nsubtracted away; plotting the data as µ0Hres|exp−(µ0Meff+ω/γ)|fit(solid data points)\nandα0ω2τ/γ|fit(curves) isolates the inertial contribution. The fit values are tabu lated in\nTable 1; error bars denote 3 σof uncertainty. We find a reasonable fit to the data through\nthe introduction of the quadratic term. For both Py and Co, the va lues ofτstay roughly\nat the same level between 10-30 nm: ∼0.1 ps for Py and ∼0.3 ps for Co. For Py 06 nm\nnegligible shifts in µ0Hresis found, as will be discussed.\nSample µ0Meff geff α0 τ(µ0Hres)τ(µ0∆H1/2)\nPy 06 nm 0.915 T 2.106 0.0075∗0.28±0.06 ps 0.09 ±0.09 ps\nPy 10 nm 0.935 T 2.100 0.0072 0.13 ±0.03 ps 0.11 ±0.04 ps\nPy 30 nm 0.993 T 2.101 0.0069 0.12 ±0.05 ps 0.06 ±0.08 ps\nCo 06 nm 1.409 T 2.147 0.0067 0.47 ±0.08 ps 0.19 ±0.07 ps\nCo 10 nm 1.547 T 2.153 0.0053 0.26 ±0.09 ps 0.14 ±0.07 ps\nCo 15 nm 1.596 T 2.154 0.0051 0.32 ±0.03 ps 0.15 ±0.05 ps\nCo 30 nm 1.672 T 2.155 0.0056 0.26 ±0.05 ps 0.21 ±0.09 ps\nTABLE I. Fit parameters of experimental resonance fields µ0Hres(ω) to the inertial term Eq. (2)\nand linewidths µ0∆H1/2to Eq. (3) for Py and Co. Values of low-frequency Gilbert damp ing\nα0are found from linear fits of low-frequency linewidths. Erro rbars of τrepresent 3 σdeviations.\n∗Misaligment analysis has been applied; see text for details .\nA primary source of uncertainty may come from sample misalignment in HF-FMR measure-\nments. Given the experimental geometry, the samples cannot be a ligned independently of\nmounting. Canting the sample off normal alignment in HF-FMR will also lea d to reduced\nresonance fields as in Fig. 1(b). Misalignment will be manifested as a lineardeviation of\nµ0Hres|exp−(µ0Meff+ω/γ)|fitfrom being zero at low frequencies ( ω/2π <30 GHz) in\n6Fig. 2. For Co samples, the deviations at low frequencies are negligible , indicating that\nCo samples are well aligned perpendicular to biasing field. For Py the sa me is true, with\nthe exception of the 6 nm sample. This sample shows a significant linear deviation at low\nfrequencies (green circles), indicating that the sample plane may no t be perfectly perpendic-\nular to the biasing field. According to simulations, shown in the Supplem ental Information,\nsuch a deviation would be produced by a misalignment of ∼6◦for Py 06 nm, which will\nadd an additional negative µ0Hresshift of∼20 mT. With this correction, the estimated τ\nis increased from 0.01 ps to 0.28 ps in Py 06 nm, consistent with the larg er value for Co 6\nnm. The corrections are much smaller for other samples and have be en neglected.\nThe influence of the inertial dynamics which we observe is a stiffening, rather than\na softening, of the resonance. We conclude therefore that the in ertial term from linear\nmomentum[3, 4, 6] dominates over the inertial term from angularmomentum proposed in\nRef. [2]. The former term is a direct product of the breathing Fermi surface model causing\nconductivity-like Gilbert damping[16, 17]. In the conductivity-like mod el the value of ob-\nservedτ, 0.1-0.5 ps, represent the average of Bloch state lifetime τBnear the Fermi surface\nfor Py and Co.\nLow-frequency \nextrapola/g415on Data \n& fitsLow-frequency \nextrapola/g415on Data \n& fits(a) (b)\nFIG. 3. Ferromagnetic resonance half-power linewidths µ0∆H1/2as a function of ω/2πfor (a) Py\nand (b) Co, between 4 to 345 GHz. Dashed lines are linear extra polations of the low-frequency\nFMR linewidths; solid curves are fits to Eq. (3). Each point wi th error bar represents a summary\nof six measurements.\n7Note that Gilbert damping coefficient α, calculated by the breathing Fermi surface model,\nshould also be frequency-dependent. A relation of α=α0/(1+ω2τ2\nB) can be obtained[16–\n18] with α0the zero-frequency damping. Fig. 3 shows the linewidths of both low -frequency\nand high-frequency ranges for Py and Co. Here points with errorb ars (one σ) are used for\nfrequencies >100 GHz to reflect the scattering of six repeats. The dashed lines a re low-\nfrequency linear extrapolations from α0. Compared with extrapolated linewidths from the\nlow-frequency ∆ H1/2(ω), we observe reduced linewidths at high frequencies (115-345 GHz ).\nNo explicit prediction has been made for the effect of rotational iner tia on the linewidth in\nRef. [2, 12], but the observed behavior matches well with the predic tion of linear inertial\nterms in Ref. [16–18]. The solid curves are fits to the form:\nµ0∆H1/2=µ0∆H0+2α0ω\nγ1\n1+ω2τ2(3)\ntaking the α0from the low-frequency linewidths. The fitted τare listed in Table 1. The\nvalues of τextracted from µ0∆H1/2are close but slightly smaller than from µ0Hres. We\nnote here that the linewidth measurements are less precise than µ0Hresand more sensitive\nthan resonance fields to various sources of inhomogeneities in the s amples.\nAt room temperature, the relaxation rate of Bloch states (1 /τB) is determined by the\nelectronscatteringwithphononsandimpurities. Inthissense, τBissimilar totheremagneti-\nzation time τEin the ultrafast demagnetization experiments[19–21], where optica lly excited\nelectrons also relax through electron-phonon and electron-impur ity interactions. In the limit\nof zero laser fluence, nonlinear effects due to high occupation numb er of excited states are\nreduced[21, 22]. The zero-fluence τEhas been reported to be 0.2-0.25 ps for Py 10-30 nm[22]\nand∼0.4 ps for Co 15 nm[21], close to the value of τin this work.\nAt high frequencies the (nonmagnetic) skin depths δsin the ferromagnetic films become\nsmaller and the enhanced eddy current effect may influence the res onance field[23]. The res-\nonance field will be enhanced by ∼µ0Ms/δ4\nsk4\n0, wherek0is the lowest-energy wavenumber\ndetermined by the surface anisotropy. Because δ2\ns∝1/ω, the resonance field enhancement\nis proportional to ω2and may influence the quadratic term in Eq. (2). Our calculations\nshow that this term is negligible, about 0.4 mT for Py 30 nm and 0.09 mT fo r Co 30 nm,\ncompared with the observed effects up to 80 mT (See the Supplemen tal Information for\ndetails).\nWe do not believe that interfacial effects, including spin pumping, play an important role\n8in the observed high-frequency behavior. Both Gilbert damping α0and the two inertial\ndynamics lifetimes τ(µ0Hres) andτ(µ0∆H1/2) in Table 1 show little thickness dependence\nfor either Py or Co, indicating that bulk relaxation is dominant. The we ak thickness de-\npendence of α0is consistent with the very low spin pumping effect of Py/Ta identified in\nRef. [24], in any case negligible for 30 nm films and without quadratic fre quency depen-\ndence. Theoretical predictions of resonance shifts from imaginar y spin mixing conductance\nare three orders of magnitude lower than observed here[25]. Only τ(µ0Hres) measured from\nresonance shifts for 6 nm films, not matched in τ(µ0∆H1/2) measured from linewidths, differ\nsignificantly. This enhancement may be structural in origin.\nA technological implication of our results is that effective field require ments for preces-\nsional switching will be reducedas switching times in magnetic storage decrease into the\nfew-picosecond range. In this sense, the inertial dynamics ease u ltrafast switching, if the\nbehaviors of Py and Co are representative of other metallic ferrom agnets. The effective field\nreduction, here up to 80 mT in Co, is not small in an absolute sense, an d might according\nto Eq. (2) be enhanced significantly in ferromagnets with higher Gilbe rt damping. On the\nother hand, prior switching experiments with high-field relativistic ele ctron bunches seem\nto indicate that nonlinear damping increases effective field requireme nts by a rather larger\namount for large-angle dynamics in CoCrPt[8], underscoring the utilit y of HF-FMR to iden-\ntify the inertial effect.\nIn summary, we identify a novel term to magnetization dynamics in th e ferromagnetic\nmetal films Py and Co which is quadratic in frequency and becomes sign ificant above 100\nGHz. Thetermstiffensthefrequency, aidingappliedfieldsindriving ult rafastmagnetization\nmotion. The behavior is best explained by dynamics retarded throug h a finite Bloch-state\nrelaxation time τBas proposed in Refs. [3, 4, 6]. Extracted relaxation times are 0.1-0.2 ps\nfor Py and 0.2-0.4 ps for Co, close to the remagnetization times meas ured in optical pump-\nprobe demagnetization experiments. Our findings extend underst anding of magnetization\ndynamics at picosecond time scales and may open up new possibilities fo r high-speed inertial\nswitching in ferromagnetic materials, previously demonstrated only in antiferromagnets[10].\nWe thank M. F¨ ahnle, J. E. Wegrowe and J. Fransson for discussion s and X. Yang for\nsuggestions on statistical analysis. We acknowledge fundings by th e EC through CRONOS\n(N◦280829), NSF-DMR-1411160, and the Chair of Excellence Program of the Nanosciences\nFoundation in Grenoble France for support.\n9[1] Eq. (1) is equivalent to the Landau-Lifshitz-Gilbert (L LG) equation with renormalized gyro-\nmagnetic ratio γ. See: T. L. Gilbert, IEEE Trans. Magn. 6, 3443 (2004).\n[2] M.-C. Ciornei, J. M. Rubi and J.-E. Wegrowe, Phys. Rev. B 83, 020410(R) (2011).\n[3] M. F¨ ahnle, D. Steiauf and C. Illg, Phys. Rev. B 84, 172403 (2011).\n[4] M. F¨ ahnle, D. Steiauf and C. Illg, Phys. Rev. B 88, 219905(E) (2013).\n[5] D. B¨ ottcher and J. Henk, Phys. Rev. B 86, 020404(R) (2012).\n[6] S. Bhattacharjee, L. Nordstr¨ om and J. Fransson, Phys. Rev. Lett. 108, 057204 (2012).\n[7] C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Wel ler, E. L. Garwin and H. C.\nSiegmann, Science285, 864 (1999).\n[8] I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann , J. St¨ ohr, G. Ju, B. Lu and\nD. Weller, Nature428, 831 (2004).\n[9] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Ts ukamoto, A. Itoh and Th. Rasing,\nPhys. Rev. Lett. 99, 047601 (2007).\n[10] A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev, A. Kirilyuk and Th. Rasing, Nature\nPhys.5, 727 (2009).\n[11] S. Mangin, M. Gottwald, C.-H. Lambert, D. Steil, V. Uhl´ ıˇ r, L. Pang, M. Hehn, S. Alebrand,\nM. Cinchetti, G. Malinowski, Y. Fainman, M. Aeschlimann and E. E. Fullerton, Nature Mater.\n13, 286 (2014).\n[12] J.-E Wegrowe and M.-C. Ciornei, Am. J. Phys. 80, 607 (2012).\n[13] Y. Li and W. E. Bailey, arXiv1401.6467\n[14] A.-L. Barra, A. K. Hassan, A. Janoschka, C. L. Schmidt an d V. Sch¨ unemann, Appl. Magn.\nReson.30, 385 (2006).\n[15] Paramagnetic markers used: BDPA complex with benzene( 1:1) free radical for low-frequency\nFMR,geff=2.0025; MnO diluted in the diamagnetic host MgO for high-fr equency FMR,\ngeff=2.00101\n[16] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).\n[17] V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007).\n[18] A. M. Clogston, Bell Syst. Tech. J. 34, 739 (1955).\n[19] E. Beaurepaire, J.-C. Merle, A. Daunois and J.-Y. Bigot ,Phys. Rev. Lett. 76, 4250 (1996).\n10[20] B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa and W. J. M. d e Jonge, Phys. Rev. Lett. 95,\n267207 (2005).\n[21] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. F¨ ahnle, T. Roth, M. Cinchetti\nand M. Aeschlimann, Nature Mater. 9, 259 (2009).\n[22] C. A. C Bosco, A. Azevedo and L. H. Acioli, Appl. Phys. Lett. 83, 1767 (2003).\n[23] P. Pincus, Phys. Rev. 118, 658 (1960).\n[24] S. Mizukami, Y. Ando and T. Miyazaki, J. Magn. Magn. Mater. 226, 1640 (2001).\n[25] M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas a nd G. E. W. Bauer, Phys. Rev. B\n71, 064420 (2005).\n11"    },    {        "title": "2002.06982v1.Periodic_Magnetic_Geodesics_on_Heisenberg_Manifolds.pdf",        "content": "arXiv:2002.06982v1  [math.DG]  17 Feb 2020PERIODIC MAGNETIC GEODESICS ON HEISENBERG\nMANIFOLDS\nJONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nAbstract. We study the dynamics of magnetic flows on Heisenberg groups.\nLetHdenote the three-dimensional simply connected Heisenberg Lie group\nendowed with a left-invariant Riemannian metric and an exac t, left-invariant\nmagnetic field. Let Γ be a lattice subgroup of H,so that Γ \\His a closed\nnilmanifold. We first find an explicit description of magneti c geodesics on\nH, then determine all closed magnetic geodesics and their len gths for Γ \\H.\nWe then consider two applications of these results: the dens ity of periodic\nmagnetic geodesics and marked magnetic length spectrum rig idity. We show\nthat tangent vectors to periodic magnetic geodesics are den se for sufficiently\nlarge energy levels. We also show that if Γ 1,Γ2< Hare two lattices such that\nΓ1\\Hand Γ2\\Hhave the same marked magnetic length spectrum, then they\nare isometric as Riemannian manifolds. Both results show th at this class of\nmagnetic flows carries significant information about the und erlying geometry.\nFinally, weprovide anexampleto showthatextending thisan alysisofmagnetic\nflows to the Heisenberg type setting is considerably more diffi cult.\n1.Introduction\nFrom the perspective of classical mechanics, the geodesics of a Rie mannian man-\nifold (M,g) are the possible trajectories of a point mass moving in the absence of\nany forces and in zero potential. A magnetic field can be introduced b y choos-\ning a closed 2-form Ω on M. A charged particle moving on Mnow experiences a\nLorentz force, and its trajectory is called a magnetic geodesic. As with Riemannian\ngeodesics, they can be handled collectively as a single object called th e magnetic ge-\nodesic flow on TMorT∗M(see Section 2.1 for precise definitions). Many classical\nquestions concerning geodesic flows have corresponding analogs f or magnetic flows.\nIndeed, magnetic flows display a number of remarkable properties. See [Gro99],\n[Pat06], [BM06], [BP08], and [AMMP17] for a sampling of results.\nOne can interpret magnetic flows as a particular type of perturbat ion of the\nunderlying geodesic flow. Much is known about the the underlying geo desic flow of\nnilmanifolds, and we are interested in what properties persist or fail to persist for\nmagnetic flows. This perspective is adopted for the property of to pological entropy\nin[Eps17]in the settingoftwo-stepnilmanifoldsandin [BP08] inthe set tingofSOL\nmanifolds; and for topological entropy and the Anosov property in [PP96], [PP97]\nand [BP02]. In [PS03] the authors show that at high enough energy le vels the\nmagnetic geodesics are quasi-geodesics with respect to the under lying Riemannian\nstructure. An important classical question of geodesic flows conc erns the existence\nof closed geodesics and related properties such as their lengths an d their density.\nThispaperfocuses onthesepropertiesin the contextofmagnetic flowsgeneratedby\nDate: February 18, 2020.\n12 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nleft-invariant magnetic fields on Riemannian two-step nilmanifolds. Alt hough this\nsetting is more complicated than the Euclidean setting (i.e. 1-step nilm anifolds),\nmany explicit computations are still tractable, and it has been a rich s ource of\nconjectures and counter-examples.\nLetHdenote a simply connected (2 n+ 1)-dimensional Heisenberg group en-\ndowed with a left-invariant Riemannian metric. The Lie group Hadmits cocom-\npact discrete subgroups (i.e. lattices) Γ and, because the Riemann ian metric is\nleft-invariant, the quotient inherits a metric such that Γ \\His a compact Riemann-\nian manifold and H→Γ\\His a Riemannian covering. A geodesic σ(t) inHis\nsaid to be translated by an element γ∈Hifγσ(t) =σ(t+ω) for alltand for\nsomeω >0. A geodesic that is translated by γis said to be γ-periodic. When\nγ∈Γ, each geodesics translated by γwill project to a smoothly closed geodesic in\nΓ\\H. Geodesic behavior in Γ \\Hand, more generally, in Γ \\N, whereNdenotes\na simply connected two-step nilpotent Lie group with a left-invariant metric, is\nfairly well understood. In the general Riemannian two-step case, it is possible to\ndescribe precisely the set of smoothly closed geodesics in Γ \\N, along with their\nlengths. See Eberlein [Ebe94] for the Heisenberg case and Gornet- Mast [GM00] for\nthe more general setting. Our main result is a complete analysis of lef t-invariant,\nexact magnetic flows on three-dimensional Heisenberg groups.\nTheorem (SeeSection3,Lemma4.5andTheorem4.9) .LetHbe a three-dimensional\nsimply connected Heisenberg group, ga left-invariant metric on H, andΩa left-\ninvariant, exact magnetic field on H. For anyγ∈H, there is an explicit description\nof all theγ-periodic magnetic geodesics of the magnetic flow generated by(H,g,Ω)\nsatisfyingσ(0) =e, the identity element. The lengths of closed magnetic geode sics\nmay be explicitly computed in terms of metric Lie algebra inf ormation.\nThis theorem allows for the explicit computation of all closed magnetic geodesics\nin the free homotopy class determined by each γ∈Γ. Unlike the Riemannian\ncase, closed magnetic geodesics exist in all nontrivial homotopy clas ses only for\nsufficiently large energy. In addition, there exist closed and contra ctible magnetic\ngeodesics on sufficiently small energy levels.\nWe give two applications of our main result. The first concerns the de nsity of\ntangent vectors to closed magnetic geodesics. Eberlein analyzes t his property for\nRiemannian geodesic flows on two-step nilmanifolds with a left-invarian t metric,\nshowing that for certain types of two-step nilpotent Lie groups (in cluding Heisen-\nberggroups), thevectorstangenttosmoothlyclosedunit speed geodesicsinthe cor-\nresponding nilmanifold are dense in the unit tangent bundle [Ebe94]; Ma st [Mas94]\nand Lee-Park [LP96] broadened this result. In Theorem 4.17, we sh ow that the\ndensity property continues to hold for magnetic flows on sufficiently high energy\nlevels on the Heisenberg group. The second is a marked length spect rum rigidity\nresult (see Section 4.4 for the definition). It known that within cert ain classes of\nRiemannian manifolds, if two have the same marked length spectrum t hen they\nare isometric. This is true in the class of negatively curved surfaces (see [Cro90]\nand [Ota90b, Ota90a]) and compact flat manifolds (see [BGM71], [B ´86], [MR03]).\nIn [Gro05], S. Grognet studies marked length spectrum rigidity of ma gnetic flows\non surfaces with pinched negative curvature. In Theorem 4.19, we show that the\nmarked magnetic magnetic length spectrum of left-invariant magne tic systems onPERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 3\ncompact quotients of the Heisenberg group determine the Riemann ian metric. Al-\nthough it’s a perturbation of geodesic flow, the magnetic flow still ca rries informa-\ntion about the underlying Riemannian manifold.\nThis paper is organized as follows. In Section 2, we present the nece ssary pre-\nliminaries in order to state and prove the main theorems. The definitio n and basic\nproperties of magnetic flows are given in Section 2.1 and the necessa ry background\non nilmanifolds is given in Section 2.4. Next, we show how a left-invariant Hamil-\ntonian system on the cotangent bundle of a Lie group reduces to a s o-called Euler\nflow on the dual to the Lie algebra. Such Hamiltonians are known as co llective\nHamiltonians, and this process is outlined in Section 2.3. Section 2.5 spe cializes\nthe preceding to the case of exact, left-invariant magnetic flows o n two-step nilpo-\ntent Lie groups. In Section 3, the magnetic geodesic equations on t he (2n+ 1)-\ndimensional Heisenberg group are solved. In Section 4, we apply the se formulas to\nobtain our main theorem and the applications described above. Many geometric\nresults for the Heisenberg group have been shown to hold for the la rger class of\nHeisenberg type manifolds. In Section 5, we use a specific example to show why\nour analysis of magnetic flows on Heisenberg type manifolds is conside rably more\ndifficult. Lastly, the so-called j-maps are a central part of the theory of two-step\nRiemannian nilmanifolds. In the appendix, we provide an alternative ap proach to\nstudying the magnetic geodesics using j-maps instead of collective Hamiltonians.\n2.Preliminaries\n2.1.Magnetic flows. Amagnetic structure on a Riemannian manifold ( M,g) is\na choice of closed 2-form Ω on M, called the magnetic 2-form. Themagnetic flow\nof (M,g,Ω) is the Hamiltonian flow Φ tonTMdetermined by the symplectic form\n̟mag= ¯̟+π∗Ω (1)\nand the kinetic energy Hamiltonian H0:TM→R,given by\nH0(v) =1\n2g(v,v) =1\n2|v|2. (2)\nHereπ:TM→Mdenotes the canonical projection and ¯ ̟denotes the pullback\nvia the Riemmanian metric of the canonical symplectic form on T∗M.\nThe magnetic flow models the motion of a charged particle under the e ffect of a\nmagnetic field whose Lorentz force F:TM→TMis the bundle map defined via\nΩx(u,v) =gx(Fxu,v)\nfor allx∈Mand allu,v∈TxM. The orbits of the magnetic flow have the form\nt/ma√sto→˙σ(t), whereσis a curve in Msuch that\n(3) ∇˙σ˙σ=F˙σ.\nIn the case that Ω = 0, the magnetic flow reduces to Riemannian geod esic flow. A\ncurveσthat satisfies (3) is called a magnetic geodesic . The physical interpretation\nof a magnetic geodesic is that it is the path followed by a particle with un it mass\nand charge under the influence of the magnetic field. Because Fis skew-symmetric,\nthe acceleration of the magnetic geodesic is perpendicular to its velo city.\nRemark 2.1.It is straightforward to show that magnetic geodesics have const ant\nspeed. In contrast to the Riemannian setting, a unit speed repara metrization of a\nsolution to (3) may no longer be a solution. To see this, let σ(s) be a solution that4 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nis not unit speed and denote energy E=|˙σ|>0.Defineτ(s) =σ(s/E), which is\nunit speed. Then\n∇˙τ˙τ=1\nE2∇˙σ˙σ=1\nE2F˙σ=1\nEF˙τ/\\e}atio\\slash=F˙τ,\nin general. Therefore, one views a magnetic geodesic as the path, n ot the param-\neterized curve. (Observe that τis a solution to the magnetic flow determined by\nthe magnetic form1√\nEΩ.)\nRecall that the tangent and cotangent bundles of a Riemannian man ifold are\ncanonically identified, and the Riemannian metric on TM→Minduces a non-\ndegenerate, symmetric 2-tensor on T∗M→M. We will present most of the theory\ninthesettingofthecotangentbundle, whileoccasionallyindicatingho wtotranslate\nto the tangent bundle. Note that many authors use the tangent b undle approach.\nSee for example [BM06].\nSlightly abusing notation, we now let πdenote the basepoint map of the cotan-\ngent bundle, let gdenote the metric on the cotangent bundle, and define H0:\nT∗M→RasH0(p) =1\n2g(p,p) =1\n2|p|2. Accordingly, the magnetic flow of ( M,g,Ω)\nis the Hamiltonian flow Φ ton the symplectic manifold ( T∗M,̟+π∗Ω) determined\nby the Hamiltonian H0. Regardless of approach, the projections of the orbits to\nthe base manifold will be the same magnetic geodesics determined by ( 3).\nOn the cotangent bundle\n̟mag=̟+π∗Ω (4)\ndefines a symplectic form as long as Ω is closed; Ω may be non-exact or exact.\nIn the former case, Ω is referred to as a monopole . In the latter case, when Ω is\nexact, the magnetic flow can be realized either as the Euler-Lagran ge flow of an\nappropriate Lagrangian, or (via the Legendre transform) as a Ha miltonian flow on\nT∗Mendowed with its canonical symplectic structure. Note that even if two mag-\nnetic fields represent the same cohomology class, they generally de termine distinct\nmagnetic flows.\nSuppose that Ω = dθfor some 1-form θ. A computation in local coordinates\nshows that the diffeomorphism f:T∗M→T∗Mdefined by f(x,p) = (x,p−θx)\nconjugates the Hamiltonian flow of ( T∗M,̟+π∗Ω,H0) with the Hamiltonian flow\nof (T∗M,̟,H 1) where\nH1(x,p) =1\n2|p+θx|2. (5)\n2.2.Example: Magnetic Geodesics in the Euclidean Plane. Before intro-\nducingtwo-stepnilmanifoldsinthefollowingsubsection,wefirstprov ideanexample\nof a left-invariant magnetic system in a simpler context.\nLetM=R2endowed with the standard Euclidean metric g. Let Ω =B dx∧dy\ndenote a magnetic 2-form, where ( x,y) denote global coordinates and Bis a real\nparameter that can be interpreted as modulating the strength of the magnetic field.\nLetσv(t) = (x(t),y(t)) denote the magnetic geodesic through the identity e=\n(0,0)withinitialvelocity v= (x0,y0) =x0∂\n∂x+y0∂\n∂y/\\e}atio\\slash= 0andenergy E=/radicalbig\nx2\n0+y2\n0.\nThe Lorentz force FsatisfiesF(1,0) =B(0,1) andF(0,1) =−B(1,0). By (3)\nσv(t) satisfies\n(¨x,¨y) =F(˙x,˙y) =B(−˙y,˙x).PERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 5\nThe unique solution satisfying σv(0) =eand ˙σv(0) =vis\nx(t) =−y0\nB(1−cos(tB))+x0\nBsin(tB)\ny(t) =x0\nB(1−cos(tB))+y0\nBsin(tB).\nThenσv(t) is acircleof radiusE\n|B|and center/parenleftbig\n−y0\nB,x0\nB/parenrightbig\n. It is immediate that\nmagnetic geodesics cannot be reparameterized. For if σv′(t) is another magnetic\ngeodesicthroughtheidentitywith v′paralleltovbutwith |v| /\\e}atio\\slash=|v′|, thenσv′(t)will\ndescribe a circle of different radius. Furthermore magnetic geodes ics are not even\ntime-reversible. The magnetic geodesic σ−v(t) is a circle of radiusE\n|B|and center/parenleftbigy0\nB,−x0\nB/parenrightbig\n; in particular, σ−v(t) andσv(t) are both circles of the same radius but\ntrace different paths. Note that every magnetic geodesic in this se tting is periodic.\nThis will not be the case for two-step nilmanifolds.\n2.3.Left-invariant Hamiltonians on Lie groups. LetGbe a Lie group with\nLie algebra g. On the one hand, T∗G( =G×g∗) is a symplectic manifold and each\nfunctionH:T∗G→Rgenerates a Hamiltonian flow with infinitesimal generator\nXH. On the other hand, g∗is a Poisson manifold and each function f:g∗→R\ndetermines a derivation of C∞(g∗) and hence a vector field Ef, called the Euler\nvectorfieldassociatedto f. Whenthefunction Hisleft-invariant,i.e. H((Lx)∗α) =\nH(α) for allx∈Gand allα∈T∗G, it induces a function h:g∗→Rand the flow\nofXHfactors onto the flow of Eh. Moreover, the flow of XHcan be reconstructed\nfromEhand knowledge of the group structure of G. Note that this is a special\ncase of a more general class of Hamiltonians, called collective Hamilton ians. More\ndetails and physical motivation can be found in Sections 28 and 29 of [G S90]. We\noutline below how we will use this approach to study magnetic flows.\nA Poisson manifold is a smooth manifold Mtogether with a Lie bracket {·,·}\non the algebra C∞(M) that also satisfies the property\n{f,gh}={f,g}h+g{f,h} (6)\nfor allf,g,h∈C∞(M). Hence, for a fixed function h∈C∞(M), the map\nC∞(M)→C∞(M) defined by f/ma√sto→ {f,h}is a derivation of C∞(M). Therefore,\nthere is an Euler vector field EhonMsuch thatEh(·) ={·,h}.\nAn important source of Poisson manifolds is the vector space dual t o a Lie\nalgebra. We will make use of the standard identifications Tpg∗≃g∗andT∗\npg∗≃\n(g∗)∗≃g, and/a\\}b∇acketle{t ·,· /a\\}b∇acket∇i}htwill denote the natural pairing between gandg∗. For a\nfunctionf∈C∞(g∗), its differential dfpatp∈g∗is identified with an element of\nthe Lie algebra g. The Lie bracket structure on ginduces the Poisson structure on\ng∗by\n{f,g}(p) =−/a\\}b∇acketle{tp,[dfp,dgp]/a\\}b∇acket∇i}ht=−p([dfp,dgp]). (7)\nAntisymmetry and the Jacobi Identity follow from the properties o f the Lie bracket\n[·,·], while the derivation property (6) follows from the Leibniz rule for th e\nexterior derivative.\nIt is useful to express the Euler vector field Ehin terms of hand the represen-\ntation ad∗:g→gl(g∗) dual to the adjoint representation, defined as\n/a\\}b∇acketle{tad∗\nXp,Y/a\\}b∇acket∇i}ht=−/a\\}b∇acketle{tp,adXY/a\\}b∇acket∇i}ht. (8)6 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nFrom the definition of the differential of a function,\n/a\\}b∇acketle{tEh(p),dfp/a\\}b∇acket∇i}ht=Eh(f)(p) ={f,h}(p) =−/a\\}b∇acketle{tp,[dfp,dhp]/a\\}b∇acket∇i}ht=−/a\\}b∇acketle{tad∗\ndhpp,dfp/a\\}b∇acket∇i}ht.\nFrom this we conclude that\nEh(p) =−ad∗\ndhpp. (9)\nNowconsider T∗G≃G×g∗trivializedvialeft-multiplication. Let r:G×g∗→g∗\nbe projection onto the second factor. If h:g∗→Ris any smooth function, then\nH=h◦ris a left-invariant Hamiltonian on T∗G. Conversely, any left-invariant\nHamiltonian Hfactors asH=h◦r. Recall that the canonical symplectic structure\n̟onT∗G≃G×g∗is\n̟(x,p)((U1,α1),(U2,α2)) =α2(U1)−α1(U2)+p([U1,U2]) (10)\nwhere we identify T(g,p)T∗G≃g×g∗(see section 4.3 of [Ebe04] for more details).\nTo find an expression for the Hamiltonian vector field XH(x,p) = (X,λ) of a\nleft-invariant Hamiltonian, first consider vectors of the form (0 ,α) in the equation\n̟(XH,·) =dH(·). We have\n̟(x,p)((X,λ),(0,α)) =dH(x,p)(0,α) =d(h◦r)(x,p)(0,α),\nα(X)−λ(0)+p([X,0]) =dhp(α),\nα(X) =α(dhp).\nSince this is true for all choices of α, we getX=dhp. Next consider vectors of the\nform (U,0). SinceHis left-invariant,\n̟(x,p)((dhp,λ),(U,0)) =dH(x,p)(U,0),\n−/a\\}b∇acketle{tλ,U/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tp,[dhp,U]/a\\}b∇acket∇i}ht= 0,\n/a\\}b∇acketle{tλ,U/a\\}b∇acket∇i}ht=−/a\\}b∇acketle{tad∗\ndhpp,U/a\\}b∇acket∇i}ht.\nSince this must be true for every U, we have that λ=−ad∗\ndhpp=Eh(p). For a\nleft-invariant Hamiltonian, the equations of motions for its associat ed Hamiltonian\nflow are\nXH(x,p) =/braceleftBigg\n˙x= (Lx)∗(dhp)\n˙p=Eh(p) =−ad∗\ndhpp. (11)\n2.4.The Geometry of Two-Step Nilpotent Metric Lie Groups. Ourobjects\nof study in this paper are simply connected two-step nilpotent Lie gr oups endowed\nwith a left-invariant metric. For an excellent reference regarding t he geometry of\nthese manifolds, see [Ebe94].\nLetgdenote a two-step nilpotent Lie algebra with Lie bracket [ ,] and non-\ntrivial center z. That is, gis nonabelian and [ X,Y]∈zfor allX,Y∈g. LetG\ndenote the unique, simply connected Lie group with Lie algebra g; thenGis a\ntwo-step nilpotent Lie group. The Lie group exponential map exp : g→Gis a\ndiffeomorphism, with inverse map denoted by log : G→g. Using the Campbell-\nBaker-Hausdorff formula, the multiplication law can be expressed as\nexp(X)exp(Y) = exp/parenleftbigg\nX+Y+1\n2[X,Y]/parenrightbigg\n. (12)PERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 7\nFor anyA∈gand anyX∈TAg≃g, the push-forwardof the Lie group exponential\natAis\n(exp∗)A(X) = (Lexp(A))∗/parenleftbigg\nX+1\n2[X,A]/parenrightbigg\n.\nUsing this, the tangent vector to any smooth path σ(t) = exp(U(t)) inGis given\nby\nσ′(t) = (Lσ(t))∗/parenleftbigg\nU′(t)+1\n2[U′(t),U(t)]/parenrightbigg\n. (13)\nWhen a two-step nilpotent Lie algebra gis endowed with an inner product g, then\nthere is a natural decomposition g=v⊕z, wherezis the center of gandvis the\northogonal complement to zing. Every central vector Z∈zdetermines a skew-\nsymmetric linear transformationof v(relative to the restriction of g), denotedj(Z),\nas follows:\ng(j(Z)V1,V2) =g([V1,V2],Z) (14)\nfor any vectors V1,V2∈v. In fact, this correspondenceis a linear map j:z→so(v).\nThese maps, first introduced by Kaplan [Kap81], capture all of the g eometry of a\ntwo-step nilpotent metric Lie group. For example, the j-maps provide a very useful\ndescription of the Levi-Civita connection. For V1,V2∈vandZ1,Z2∈z,\n∇X1X2=1\n2[X1,X2],\n∇X1Z1=∇Z1X1=−1\n2j(Z)X, (15)\n∇Z1Z2= 0.\n2.5.Exact, Left-Invariant Magnetic Forms on Simply Connected T wo-\nStep Nilpotent Lie Groups. We use the formalism of Subsection 2.3 to express\nthe equations of motion for the magnetic flow of an exact, left-inva riant magnetic\nform on a simply connected two-step nilpotent Lie group. Througho ut this section,\ngdenotes a two-step nilpotent Lie algebra with an inner product and Gdenotes\nthe simply connected Lie group with Lie algebra gendowed with the left-invariant\nRiemannian metric determined by the inner product on g.\nAs a reminder, angled brackets denote the natural pairing of a vec tor space and\nits dual. Recall that any (finite dimensional) vector space Vis naturally identified\nwithV∗∗by sending any vector v∈Vto the linear functional V∗/ma√sto→Rdefined\nby evaluation on v. Using this identification, we can and do view elements of\nVsimultaneously as elements of V∗∗. The inner product on g∗is specified by a\nchoice of linear map ♯:g∗→gsuch that (a) /a\\}b∇acketle{tp,♯(p)/a\\}b∇acket∇i}ht>0 for allp/\\e}atio\\slash= 0 and (b)\n/a\\}b∇acketle{tp,♯(q)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t♯(p),q/a\\}b∇acket∇i}htfor allp,q∈g∗. The inner product of p,q∈g∗is then given by\n/a\\}b∇acketle{tp,♯(q)/a\\}b∇acket∇i}htand the induced norm is |p|=/radicalbig\n/a\\}b∇acketle{tp,♯(p)/a\\}b∇acket∇i}ht. Conversely any inner product\nong∗induces a map ♯:g∗→g∗∗≃gwith the properties (a) and (b). Of course,\n♯−1=♭is then the flat map andthe innerproduct of XandYingcan be computed\nas/a\\}b∇acketle{tX,♭(Y)/a\\}b∇acket∇i}ht.\nLetg=v⊕zbe the decomposition of ginto the center and its orthogonal\ncomplement. Let g∗=v∗⊕z∗be the corresponding decomposition where v∗is the\nset of functionals that vanish on zand vice versa.8 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nLemma 2.2. IfΩis an exact, left-invariant 2-form on G, then there exists B∈R\nandζm∈z∗such that |ζm|= 1andΩ =d(Bζm).\nProof.By hypothesis, Ω = dθfor some left-invariant 1-form θ. By left-invariance,\nθcan be expressed as θ=θv+θz, whereθv∈v∗andθz∈z∗, anddθv(X,Y) =\n−θv([X,Y]) for anyX,Y∈g. Because [X,Y]∈z,dθv= 0. Hence\nΩ =dθ=d(θv+θz) =dθz.\nLastly, setζm=θz/|θz|andB=|θz|. /square\nGivenB∈Randζm∈g∗, we define the function H:T∗G→Rby\nH(x,p) =1\n2|p+Bζm|2. (16)\nBy the previous lemma, we may assume ζmis a unit element in g∗that vanishes on\nv. Becauseζmis left-invariant, His left-invariant and factors as H=h◦r, where\nh:g∗→Ris the function\nh(p) =1\n2|p+Bζm|2. (17)\nNote that when B= 0, the Hamiltonian flow of His the geodesic flow of the chosen\nRiemannian metric.\nLemma 2.3. The differential of hisdhp=♯(p+Bζm).\nProof.For anyp∈gand anyq∈Tpg∗≃g∗, we compute\n/a\\}b∇acketle{tq,dhp/a\\}b∇acket∇i}ht=d\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0h(p+tq)\n=1\n2d\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0|p+tq+Bζm|2\n=1\n2d\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0/a\\}b∇acketle{tp+Bζm+tq,♯(p+Bζm+tq)/a\\}b∇acket∇i}ht\n=1\n2d\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0/parenleftbig\n|p+Bζm|2+2t/a\\}b∇acketle{tp+Bζm,♯(q)/a\\}b∇acket∇i}ht+t2|q|2/parenrightbig\n=/a\\}b∇acketle{tp+Bζm,♯(q)/a\\}b∇acket∇i}ht.\nThe Lemma now follows from the properties of ♯. /square\nWe now prove that the Euler vector field on g∗is independent of the choice of\nexact magnetic field, including the choice Ω = 0.\nLemma 2.4. Leth∈C∞(g∗)be any function of the form (17)and define the\nfunctionh0∈C∞(M)byh0(p) =1\n2|p|2. ThenEh0=Eh.\nProof.For anyζ∈z∗and anyV∈v,/a\\}b∇acketle{tV,♭(♯(ζ))/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tV,ζ/a\\}b∇acket∇i}ht= 0 shows that ♯(z∗) =z.\nFor anyX∈g, by the previous lemma,\n/a\\}b∇acketle{tad∗\ndhpp,X/a\\}b∇acket∇i}ht=−/a\\}b∇acketle{tp,[♯(p+Bζm),X]/a\\}b∇acket∇i}ht=−/a\\}b∇acketle{tp,[♯p,X]/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tad∗\n(dh0)pp,X/a\\}b∇acket∇i}ht.\nHence ad∗\ndhp= ad∗\n(dh0)pand the proof follows from the expression (9) for the Euler\nvector field. /squarePERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 9\nWe now describe the structure of the Euler vector field. Much of th is can be\ngleaned from the results of [Ebe94]. However, we include it here for t he sake of self-\ncontainment. For any X∈gandp∈g∗, we writeX=Xv+Xzandp=pv+pzfor\nthe respective orthogonal decomposition according to g=v⊕zandg∗=v∗⊕z∗.\nLemma 2.5. The integral curves of the Euler vector field Ehare of the form\np(t) =pv(t)+ζ0whereζ0∈z∗andpv(t)∈v∗is a path that satisfies p′\nv(t) =A(pv(t))\nfor some skew-symmetric transformation of v∗.\nProof.From (8), the dual adjoint representation clearly has the following proper-\nties: ad∗\nZ= 0 for every Z∈z, ad∗\nX(g∗)⊂v∗for allX∈g, and ad∗\nX(v∗) ={0}\nfor everyX∈g. From this, if p(t) =pv(t)+pz(t) is an integral curve of Eh, then\npz(t) =pz(0) =ζ0is constant, and, using Lemmas 2.3 and 2.4, pv(t) must satisfy\nthe system\np′\nv(t) =Eh(p(t)) =−ad∗\ndhp(t)p(t) =−ad∗\n♯(pv(t))pz(t) =−ad∗\n♯(pv(t))ζ0.\nSinceA:v∗→v∗is skew-symmetric with respect to the inner product restricted\ntov∗, this completes the Lemma. /square\nLet (G,g,Ω) be a magnetic system, where Gis a simply connected two-step\nnilpotent Lie group, gis a left-invariant metric, and Ω an exact, left-invariant\nmagnetic form. Let ♭:g→g∗and♯=♭−1be the associated flat and sharp maps,\nand letζmbe as in Lemma 2.2. The magnetic flow can be found as follows. First,\ncompute the coadjoint representation of ad∗:g→gl(g∗) and integrate the vector\nfieldE(p) =−ad∗\ndhpp. It follows that the curves σ(t) satisfying σ′(t) =dhp(t),\nwherep(t) is an integral curve of E, will be magnetic geodesics. To make this\nstep more explicit, let g=v⊕zbe the decomposition of gwherezis the center\nandvis its orthogonal complement. Suppose that p(t) =p1(t)+ζ0is an integral\ncurve ofE, wherep1(t)∈v∗andζ0∈z∗, andσ(t) = exp( X(t) +Z(t)) is a path\ninG, whereX(t)∈vandZ(t)∈z. Using (13), we can decompose the equation\nσ′(t) =dhp(t)=♯(p(t)+Bζm) as\nX′(t) =♯(p1(t)), (18)\nZ′(t)+1\n2[X′(t),X(t)] =♯(ζ0+Bζm). (19)\nAssuming that the path satisfies σ(0) =e, the first equation can be integrated to\nfindX(t), which then allows the second equation to be integrated to find Z(t).\nRemark2.6.The presence of the magnetic field can be thought of as a perturba tion\nof the geodesic flow of ( G,g), modulated by the parameter B. In the procedure\noutlined here for two-step nilpotent Lie groups, the magnetic field o nly appears in\nthe final step. The Euler vector field, and hence its integral curve , is unchanged\nby the magnetic field. In addition, the non-central component of t he magnetic\ngeodesics is the same as that of the Riemannian geodesics. The pres ence of a left-\ninvariant exact magnetic field only perturbs the geodesic flow in cent ral component\nof the Riemannian geodesics.\nRemark 2.7.For a magnetic geodesic σ(t), we will call |σ′(t)|itsenergy. Note that\nthis is a conserved quantity for magnetic flows. Since we are not con sidering a\npotential, the total energy of a charged particle in a magnetic syst em is its kinetic\nenergy|σ′(t)|2/2. Although this would be commonly referred to as the energy in10 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nthe physics and dynamics literature, we find our convention to be mo re convenient\nfrom our geometric viewpoint.\nRemark 2.8.Althought/ma√sto→(σ(t),p(t)) is an integral curve of the Hamiltonian\nvector field, the Hamiltonian his not the kinetic energy, and hence the energy of\nthe magnetic geodesic is not equal to |p(0)|. Instead, by (18) and (19), the energy\nsquared is\n|σ′(t)|2=|♯(p(0))+B♯(ζm)|2=|♯(p1(0))|2+|♯(ζ0+Bζm)|2. (20)\n3.Simply Connected (2n+1)-Dimensional Heisenberg Groups\nLethn= span{X1,...,X n,Y1,...,Y n,Z}and define a bracket structure on hn\nby declaring the only nonzero brackets among the basis vectors to be [Xi,Yi] =Z\nand extending [ ·,·] to all of hn×hnby bilinearity and skew-symmetry. Then hn\nis a two-step nilpotent Lie algebra called the Heisenberg Lie algebra of dimension\n2n+ 1, and the simply connected Lie group Hnwith Lie algebra hnis called the\nHeisenberg group of dimension 2 n+1. Let {α1,β1,...,α n,βn,ζ}be the dual basis\nofh∗\nn. The following Lemma, proven in Lemma 3.5 of [GW+86], shows that to\nconsider every inner product on hn, we need only consider inner products on hn\nthat have a simple relationship to the bracket structure.\nLemma 3.1. Letgbe any inner product on hn. There exists ϕ∈Aut(hn)such\nthat/braceleftbiggX1√A1,...,Xn√An,Y1√A1,...,Yn√An,Z/bracerightbigg\n(21)\nis an orthonormal basis relative to ϕ∗g, whereAi>0, i= 1...n,are positive real\nnumbers.\nProof.Consider the linear map defined by\nXi/ma√sto→Xi\n|Z|Yi/ma√sto→YiZ/ma√sto→Z\n|Z|.\nThis is an automorphism of hnandZis a unit vector relative to the pullback of\nthe metric. Hence we can and will assume that |Z|= 1.\nLetψ1be the linear map defined by ψ1(Xi) =Xi−g(Xi,Z)Z,ψ1(Yi) =\nYi−g(Yi,Z)Z, andψ1(Z) =Z. Nowψ1∈Aut(hn) andv= span{X1,...,X n,\nY1,...,Y n}is orthogonal to z= span{Z}relative toψ∗\n1g.\nNext consider the map j(Z)∈so(v,ψ∗\n1g). Because it is skew-symmetric, there\nexists aψ∗\n1g-orthonormal basis {/tildewideX1,...,/tildewideXn,/tildewideY1,...,/tildewideYn}ofvsuch thatj(Z)/tildewideXi=\ndi/tildewideYiandj(Z)/tildewideYi=−di/tildewideXifor some real numbers di>0. Because\n(ψ∗\n1g)(Z,[/tildewideXi,/tildewideYi]) = (ψ∗\n1g)(j(Z)/tildewideXi,/tildewideYi) = (ψ∗\n1g)(di/tildewideYi,/tildewideYi) =di\nwe see that [/tildewideXi,/tildewideYi] =diZ. Define the linear map ψ2by\nψ2(Xi) =1√di/tildewideXiψ2(Yi) =1√di/tildewideYiψ2(Z) =Z.\nThenψ2∈Aut(hn) because\n[ψ2(Xi),ψ2(Yi)] =Z=ψ2(Z) =ψ2([Xi,Yi])\nand, setting Ai=di, it is clear that the basis (21) is orthonormal relative to\nψ∗\n2(ψ∗\n1g). Henceϕ=ψ1◦ψ2is the desired automorphism of hn. /squarePERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 11\nWhen (21) is an orthonormal basis of hn, the sharp and flat maps are given by\n♭(Xi/√Ai) =√Aiαi, ♯(√Aiαi) =Xi/√Ai,\n♭(Yi/√Ai) =√Aiβi, ♯(√Aiβi) =Yi/√Ai,\n♭(Z) =ζ, ♯ (ζ) =Z.\nRelative to the basis {X1,...,X n,Y1,...,Y n,Z}, the adjoint representation is\nadU=\n0··· 0 0\n.........\n0··· 0 0\n−y1··· −ynx1···xn0\n\nwhereU=/summationtextxiXi+/summationtextbiyi+zZ. Relative to the dual basis, the coadjoint repre-\nsentation is the negative transpose\nad∗\nU=−(adU)T=\n0··· 0y1\n.........\n0··· 0yn\n0··· 0−x1\n.........\n0··· 0−xn\n0··· 0 0\n.\nBecause the center of hnis one-dimensional, ζm=ζ, whereζmis as specified in\nLemma 2.2. Letting p=/summationtext\niaiαi+/summationtext\nibiβi+cζbe a point in h∗\nn, the differential of\nthe Hamiltonian is\ndhp=♯(p+Bζ) =/summationdisplay\niai\nAiXi+/summationdisplay\nibi\nAiYi+(c+B)Z\nand the Euler vector field is\nEh(p) =−ad∗\ndhpp=/summationdisplay\ni−cbi\nAiαi+/summationdisplay\nicai\nAiβi.\nTo integrate the system p′=Eh(p), note that the central component of the Euler\nvector field is constant by Lemma 2.5. Suppose that p(t) =/summationtextai(t)αi+/summationtextbi(t)βi+\nc(t)ζis a solution that satisfies the initial condition p(0) =/summationtextuiαi+/summationtextviβi+z0ζ.\nThenc(t) =z0and the remaining components form a linear system,\na′\ni(t) =−z0\nAibi(t)b′\ni(t) =z0\nAiai(t)\nthat is directly integrated to find\nai(t) =uicos/parenleftbiggz0t\nAi/parenrightbigg\n−visin/parenleftbiggz0t\nAi/parenrightbigg\n,\nbi(t) =uisin/parenleftbiggz0t\nAi/parenrightbigg\n+vicos/parenleftbiggz0t\nAi/parenrightbigg\n.\nWith an expression for the integral curves of the Euler vector field now estab-\nlished, we use equations (18) and (19) to obtain a coordinate expre ssion for the12 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nmagnetic geodesics through the identity. Let X(t) =/summationtextxi(t)Xi+/summationtextyi(t)Yi. If\nz0/\\e}atio\\slash= 0, a direct integration of (18) together with X(0) = 0 yields\nxi(t) =ui\nz0sin/parenleftbiggz0t\nAi/parenrightbigg\n+vi\nz0cos/parenleftbiggz0t\nAi/parenrightbigg\n−vi\nz0,\nyi(t) =−ui\nz0cos/parenleftbiggz0t\nAi/parenrightbigg\n+vi\nz0sin/parenleftbiggz0t\nAi/parenrightbigg\n+ui\nz0.\nIfz0= 0,we obtain\nxi(t) =ui\nAit,\nyi(t) =vi\nAit.\nBecause the center is one-dimensional, the central component Z(t) in (19) can be\nexpressed as Z(t) =z(t)Z. To integrate (19) in the case that z0/\\e}atio\\slash= 0, first compute\n[X′(t),X(t)] =/summationdisplay\n(x′\niyi−xiy′\ni)Z=/summationdisplayu2\ni+v2\ni\nAiz0/parenleftbigg\ncos/parenleftbiggz0t\nAi/parenrightbigg\n−1/parenrightbigg\nZ\nso that\nZ′(t) =z′(t)Z=♯(z0ζ+Bζ)−1\n2[X′(t),X(t)]\n= (z0+B)Z−/summationdisplayu2\ni+v2\ni\n2Aiz0/parenleftbigg\ncos/parenleftbiggz0t\nAi/parenrightbigg\n−1/parenrightbigg\nZ\n=/parenleftbigg\nz0+B+/summationdisplayu2\ni+v2\ni\n2Aiz0/parenrightbigg\nZ−/summationdisplayu2\ni+v2\ni\n2Aiz0cos/parenleftbiggz0t\nAi/parenrightbigg\nZ\nand hence\nz(t) =/parenleftbigg\nz0+B+/summationdisplayu2\ni+v2\ni\n2Aiz0/parenrightbigg\nt−/summationdisplayu2\ni+v2\ni\n2z2\n0sin/parenleftbiggz0t\nAi/parenrightbigg\n.\nIn summary, when z0/\\e}atio\\slash= 0, every magnetic geodesic σ(t) = exp(/summationtextxi(t)Xi+/summationtextyi(t)Yi+z(t)Z) satisfying σ(0) =ehas the form\nxi(t) =ui\nz0sin/parenleftbiggz0t\nAi/parenrightbigg\n−vi\nz0/parenleftbigg\n1−cos/parenleftbiggz0t\nAi/parenrightbigg/parenrightbigg\n, (22)\nyi(t) =ui\nz0/parenleftbigg\n1−cos/parenleftbiggz0t\nAi/parenrightbigg/parenrightbigg\n+vi\nz0sin/parenleftbiggz0t\nAi/parenrightbigg\n, (23)\nz(t) =/parenleftbigg\nz0+B+/summationdisplayu2\ni+v2\ni\n2Aiz0/parenrightbigg\nt−/summationdisplayu2\ni+v2\ni\n2z2\n0sin/parenleftbiggz0t\nAi/parenrightbigg\n. (24)\nWhenz0= 0,we obtain\nxi(t) =ui\nAit, (25)\nyi(t) =vi\nAit, (26)\nzi(t) =Bt. (27)\nRemark 3.2.A magnetic geodesic σ(t) will be a one-parameter subgroup if and\nonly ifz0= 0 orz0/\\e}atio\\slash= 0 andui=vi= 0 for alli. We will sometimes call a magnetic\ngeodesic spiraling if it is not a one-parameter subgroup, and non-spiraling if it is.PERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 13\nWe will also call a magnetic geodesic centralif it is of the form σ(t)∈Z(Hn) for\nallt.\nThe initial velocity of the magnetic geodesic σ(t) is\nσ′(0) =/summationdisplay/parenleftbiggui\nAiXi+vi\nAiYi/parenrightbigg\n+(z0+B)Z.\nBecause |Xi|2=|Yi|2=Ai, we can compute the square of the energy E=|σ′(t)|=\n|σ′(0)|as (see Remark 2.7)\nE2=|σ′(0)|2=/summationdisplay\niu2\ni+v2\ni\nAi+(z0+B)2(28)\nNote that this expression is valid for all values of z0.\nTheorem 3.3. There exist periodic magnetic geodesics with energy Eif and only if\n0<E <|B|. For any 0<E <|B|, letz0=−sgn(B)√\nB2−E2and letuiandvi\nbe any numbers satisfying (28). Then the spiraling magnetic geodesics determined\nbyu1,v1,...,u n,vn,z0will be periodic of energy E. Moreover, the period of such a\ngeodesic is ω= 2πA/z0.\nProof.Recall that non-spiraling magnetic geodesics cannot be periodic. In spection\nof the coordinate functions (22)-(24) of a spiraling magnetic geod esic shows they\nwill yield a periodic magnetic geodesic if and only if the coefficient of tin (24) is\nzero. This condition is\n0 =z0+B+/summationdisplayu2\ni+v2\ni\n2Aiz0=z0+B+1\n2z0/parenleftbig\nE2−(z0+B)2/parenrightbig\nor\nz2\n0=B2−E2.\nIt can only be satisfied when E <|B|. To obtain a spiraling magnetic geodesic we\nneed to require that ( z0+B)2<E2or, equivalently, z0∈(−B−E,−B+E). Since\nthis interval contains only negative or positive numbers, depending on the sign\nofB, we must choose z0=−sgn(B)√\nB2−E2. Finally, to see that z0is indeed\ncontained in this interval, note that/radicalbig\n(B−E)(B+E) =/radicalbig\n(−B+E)(−B−E) is\nthe geometric mean of the endpoints of interval. /square\nExample 3.4.For convenience we state the component functions of a magnetic\ngeodesicσ(t) = exp(x(t)X+y(t)Y+z(t)Z) in the 3-dimensional Heisenberg group\n(i.e.n= 1) with σ(0) =e. To ease notation, we use the dual bases {α,β,ζ}\nand{X,Y,Z}forh∗\n1andh1, respectively, and we let A=A1. Given a point\np(0) =u0α+v0β+z0ζ,z0/\\e}atio\\slash= 0, the correspondingmagnetic geodesic has component\nfunctions\nx(t) =u0\nz0sin/parenleftbiggz0t\nA/parenrightbigg\n−v0\nz0/parenleftbigg\n1−cos/parenleftbiggz0t\nA/parenrightbigg/parenrightbigg\n,\ny(t) =u0\nz0/parenleftbigg\n1−cos/parenleftbiggz0t\nA/parenrightbigg/parenrightbigg\n+v0\nz0sin/parenleftbiggz0t\nA/parenrightbigg\n,\nz(t) =/parenleftbigg\nz0+B+u2\n0+v2\n0\n2Az0/parenrightbigg\nt−u2\n0+v2\n0\n2z2\n0sin/parenleftbiggz0t\nA/parenrightbigg\n.14 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nWhenz0= 0,we obtain\nx(t) =u0\nAt y(t) =v0\nAt z(t) =Bt.\nRemark 3.5.It is instructive to compare the magnetic geodesics on R2given in\nSection 2.2 and the magnetic geodesics on H1given in Example 3.4. In the former,\nall magnetic geodesics are closed circles with radii that depend on th e energy. In\nthe latter, the paths x(t)X+y(t)Ythrough the complement to the center are also\ncircles whose radii depend on both the energy and z0. It is also worth noting\nsome qualitative differences between Riemannian geodesics and magn etic geodesics\non Heisenberg groups. In Riemannian case, one-parameter subgr oups of the form\nexp(t(x0X+y0Y)) are always geodesics. In contrast, the central component z(t)\nof a magnetic geodesic can never be zero. Finally, note that in the Rie mannian\nsetting there are never closed geodesics in Hn(compare with Theorem 3.3).\n4.Compact Quotients of Heisenberg Groups\nA geodesic σ:R→Min a Riemannian manifold Mis called periodic or\n(smoothly) closed if σ(t+ω) =σ(t) for allt∈R. A periodic or closed magnetic\ngeodesic is defined similarly, and we now investigate the closed magnet ic geodesics\non manifolds of the form Γ \\Hn, where Γ is a cocompact (i.e., Γ \\Hncompact),\ndiscrete subgroup of the (2 n+1)-dimensional simply connected Heisenberg group\nHn. As is common, we proceed by considering γ-periodic magnetic geodesicson the\nuniversal cover Hn. An important distinction between the magnetic and Riemann-\nian settings is that in the latter one needs to address each energy le vel separately\nbecause magnetic geodesics cannot be reparameterized.\n4.1.γ-Periodic Magnetic Geodesics.\nDefinition4.1. LetNbeasimplyconnectednilpotentLiegroupwithleftinvariant\nmetric and magnetic form. For any γ∈Nnot equal to the identity, a magnetic\ngeodesicσ(t) is calledγ-periodic with period ωifω/\\e}atio\\slash= 0 and for all t∈R\nγσ(t) =σ(t+ω). (29)\nWe also say that γtranslates the magnetic geodesic σ(t)by amountω. The number\nωis called a period ofγ.\nWhen Γ<Nis a cocompact discrete subgroup and γ∈Γ, aγ-periodic magnetic\ngeodesic will project to a smoothly closed magnetic geodesic under t he mapping\nN→Γ\\Nand will be containedin the free homotopyclassrepresented by γ. Every\nperiodic magnetic geodesic on Γ \\Narises as the image of a γ-periodic magnetic\ngeodesic on N.\nLemma 4.2. Letγ= exp(Vγ+Zγ)∈Hn, whereZγ∈Z(hn)andVγis orthogonal\ntoZ(hn), and letσ(t) = exp( X(t) +Z(t))be aγ-periodic magnetic geodesic. If\nVγ/\\e}atio\\slash= 0, thenσis a noncentral 1-parameter subgroup (see Remark 3.2).\nProof.Repeateduseof (29)showsthat γkσ(t) =σ(t+kω). Usingthemultiplication\nformula (12) on each side of the equation, the non-central compo nents must satisfy\nkVγ+X(t) =X(t+ω). IfVγ/\\e}atio\\slash= 0, then the vector-valued function X(t+ω)−X(t)\nmust be unbounded. Inspection of the magnetic geodesic equation (22)-(27) shows\nthat this can only happen if z0= 0, i.e.σis a 1-parameter subgroup. Moreover σPERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 15\ncannot be a central 1-parameter subgroup because then the lef t-hand side of (29)\nwould be noncentral and right-hand side would be central, a contra diction.\n/square\nTheorem 4.3. Letγ= exp(Vγ+zγZ)∈Hn, withVγ/\\e}atio\\slash= 0. For each E >|B|,\nthere exist two γ-periodic magnetic geodesic σ(t)with energy Eand periods ω=\n±|Vγ|/√\nE2−B2. There do not exist any γ-periodic magnetic geodesics with energy\nE≤ |B|.\nProof.ByLemma4.2, weneedonly considernon-spiralingmagneticgeodesics . The\nenergy of any such magnetic geodesic satisfies\nE2=/summationdisplayu2\ni+v2\ni\nAi+B2≥B2\nIf equality holds, then σis a central 1-parameter subgroup, which is excluded by\nLemma 4.2. Hence E >|B|.\nFixV0∈vsuch that its magnitude satisfies |V0|2+B2=E2and its direction\nis parallel to Vγ,V0= (B/k)Vγfor somek∈R/negationslash=0. Defineγ∗= exp(Vγ+kZ) and\nσ∗(t) = exp(t(V0+BZ)). Then\nγ∗σ∗(t) = exp/parenleftbiggk\nB/parenleftbiggB\nkVγ+BZ/parenrightbigg/parenrightbigg\nexp(t(V0+BZ))\n= exp/parenleftbiggk\nB(V0+BZ)/parenrightbigg\nexp(t(V0+BZ))\n= exp/parenleftbigg/parenleftbigg\nt+k\nB/parenrightbigg\n(V0+BZ)/parenrightbigg\n=σ∗/parenleftbigg\nt+k\nB/parenrightbigg\nshows that σ∗is aγ∗-periodic magnetic geodesic of energy Ewith period ω=k/B.\nUsing the multiplication formula (12) and the fact that Z(hn) is one-dimensional,\nit is straightforward to see that γandγ∗are conjugate in Hn. Thus, there exists\na∈Hnsuch thataγ∗a−1=γ. Nowσ=a·σ∗is a magnetic geodesic of energy E\nand\nγ·σ(t) =aγ∗a−1σ(t) =aγ∗σ∗(t) =aσ∗(t+ω) =σ(t+ω)\nshows that it is γ-periodic of period ω. The expression for ωfollows from ±k/B=\n|Vγ|/|V0|,and|V0|=√\nE2−B2. /square\nHaving dealt with the periods of a non-central element of Hn, we now consider\nthe case when γ= exp(zγZ) is central. In this case, there exist γ-periodic magnetic\ngeodesics starting at the identity of energy both greater than an d less than |B|. For\na fixed energy E >|B|, there will be finitely many distinct periods associated with\nγ-periodic magnetic geodesics, while there will be infinitely many distinct periods\nwhenE <|B|.\nLemma 4.4. Letγ= exp(zγZ)for somezγ∈R∗and suppose that σ(t)is aγ-\nperiodic magnetic geodesic and a 1-parameter subgroup. The nσ(t) = exp(tz0Z)\nfor somez0∈R∗. Moreover, for every E >0, there exist two γ-periodic magnetic\ngeodesics of energy E,σ(t) = exp(t(±E)Z), with period ω=zγ/(±E).16 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nProof.Sinceσis a 1-parameter subgroup by hypothesis, σ(t) = exp(tV0+BtZ).\nOn the one hand γσ(t) = exp(tV0+ (Bt+zγ)Z) and on the other σ(t+ω) =\nexp((t+ω)V0+B(t+ω)Z). HenceωV0= 0 and since ω/\\e}atio\\slash= 0, we conclude that\nV0= 0, showing the first claim.\nFor each energy E >0, letz0=−B±Eand letσ(t) be the magnetic geodesic\nσ(t) = exp(t(±E)Z). Thenσis a magnetic geodesic of energy Eand\nγσ(t) =σ((zγ±Et)Z) = exp/parenleftbigg\n±E/parenleftbiggzγ\n±E+t/parenrightbigg\nZ/parenrightbigg\n=σ(t+ω)\nshows that it is γ-periodic of period ω. /square\nNext suppose that σ(t) is a spiraling magnetic geodesic, so that the compo-\nnent functions of σ(t) have the form (22)-(24). Comparing the coefficients of\nX1,...,X n,Y1,...,Y ninγσ(t) andσ(t+ω) give conditions\nsin/parenleftbiggz0\nAi(t+ω)/parenrightbigg\n= sin/parenleftbiggz0\nAit/parenrightbigg\ncos/parenleftbiggz0\nAi(t+ω)/parenrightbigg\n= cos/parenleftbiggz0\nAit/parenrightbigg\n(30)\nfor eachi= 1,...,nsuch thatu2\ni+v2\ni/\\e}atio\\slash= 0.\nWe now specialize to case of the three-dimensional Heisenberg grou p and obtain\na complete description of the spiraling γ-periodic magnetic geodesics through the\nidentity. Since the left-invariant metric is determined by one parame ter, and a\nmagnetic geodesic through the identity is determined by z0and only one pair of\nui,vi, we write A=A1,u0=u1andv0=v1to ease notation. In general, the\nanalysis will depend on the relative size of EandB, and hence breaks up naturally\ninto the three cases E >|B|,E <|B|andE=|B|. In each case, we first establish\nthe range of permissible integers ℓ. Next, for each permissible ℓ, we describe the\nmagnetic geodesics through the identity translated by γalong with their respective\nperiods.\nIn this case, the period ωand the coordinate z0must be related by ωz0= 2πAℓ,\nwhereℓ∈Z. Comparing the central components in γσ(t) andσ(t+ω) gives the\nconditionz(t)+zγ=z(t+ω). That is,\n/parenleftbigg\nz0+B+u2\n0+v2\n0\n2Az0/parenrightbigg\nt−u2\n0+v2\n0\n2z2\n0sin/parenleftbiggz0t\nA/parenrightbigg\n+zγ\n=/parenleftbigg\nz0+B+u2\n0+v2\n0\n2Az0/parenrightbigg\n(t+ω)−u2\n0+v2\n0\n2z2\n0sin/parenleftBigz0\nA(t+ω)/parenrightBig\n.\nThis simplifies to\nzγ=/parenleftbigg\nz0+B+u2\n0+v2\n0\n2Az0/parenrightbigg\nω, (31)\nand using (28) to eliminate the fraction and ωz0= 2πAℓto eliminate ωthis can be\nwritten as\nzγ=/parenleftbigg\nz0+B+1\n2z0(E2−(z0+B)2)/parenrightbigg2πAℓ\nz0. (32)\nIfE=|B|, then the above simplifies to zγ=πAℓ. IfE/\\e}atio\\slash=|B|, then after clearing\ndenominators and solving for z0, we obtain the expression\nz2\n0=E2−B2\nzγ\nπAℓ−1. (33)PERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 17\nLemma 4.5. Letγ= exp(zγZ)be a central element of the Heisenberg group. For\neach nonzero energy level, the range of admissible integers ℓand the correspond-\ning choices of z0for which there exists a γ-periodic magnetic geodesic through the\nidentity are given by the following table.\nℓ z0\n(1a)E >|B|1<2E\nE+B<zγ\nπAℓ−/radicalBig\nE2−B2\nzγ\nπAℓ−1\n(1b)E >|B|1<2E\nE−B<zγ\nπAℓ+/radicalBig\nE2−B2\nzγ\nπAℓ−1\n(2a)0<E <B2E\nE−|B|<zγ\nπAℓ<2E\nE+|B|<1−/radicalBig\nE2−B2\nzγ\nπAℓ−1\n(2b)B <E < 02E\nE−|B|<zγ\nπAℓ<2E\nE+|B|<1+/radicalBig\nE2−B2\nzγ\nπAℓ−1\n(3a)E=Bℓ=zγ\nπA−2B<z0<0\n(3b)E=−Bℓ=zγ\nπA0<z0<−2B\nIn all cases, the associated period is ω= 2πAℓ/z 0and one can choose any u0and\nv0such thatu2\n0+v2\n0=A(E2−(z0+B)2).\nProof.The condition ( z0+B)2<E2is equivalent to\n−E−B <±/radicalBigg\nE2−B2\nzγ\nπAℓ−1<E−B. (34)\nIn case (1), −E−B <0 andE−B >0, so this leads to the two inequalities\n−E−B <−/radicalBigg\nE2−B2\nzγ\nπAℓ−1<0, 0</radicalBigg\nE2−B2\nzγ\nπAℓ−1<E−B.\nAfter squaring both inequalities and isolating zγ/(πAℓ), these become\n1<2E\nE+B<zγ\nπAℓ1<2E\nE−B<zγ\nπAℓ\nyielding cases (1a) and (1b), respectively. Notice that one of thes e ranges for ℓis a\nsubset of the other. We keep them separate as they affect the ch oice of sign for z0.\nIn case (2), either −B−E<−B+E <0 ifB >0, or 0<−B−E <−B+E<0\nifB <0. A similar computation as above leads to the inequalities\n2E\nE−B<zγ\nπAℓ<2E\nE+B<1,2E\nE+B<zγ\nπAℓ<2E\nE−B<1.\nBoth of these ranges can be expressed simultaneously in terms of |B|as in the\nLemma statement. However, in case (2a), when B >0,z0is chosen according to\nthe negative branch, and vice versa in case (2b).\nFor case (3), it was noted above (33) that if E=|B|, thenzγ=πAℓ. Choosez0\nso that (z0+B)2<E2. WhenB >0, this inequality is the same as −2B <z0<0.\nSettingω= 2πAℓ/z 0= 2zγ/z0, it is straightforward to check thatz0\nA(t+ω) =z0\nAt\nand that (31) holds. The case when B <0 is handled similarly. /square18 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nRemark 4.6.In case (2), the condition that E <|B|ensures that E2−B2<0,\nwhile the conditions on ℓensure that zγ/(πAℓ)−1<0. Hence the expression under\nthe radical in z0will be positive.\nRemark 4.7.In every case, for each admissible z0there is a 1-parameter family of\nγ-periodic magnetic geodesics.\nRemark 4.8.The cases where E=|B|are to be interpreted as follows. When zγ\nandAare such that zγ/πA∈Z, then there exist γ-periodic magnetic geodesics\nwith energy Eandz0as described in the table. Otherwise, the collection of such\nmagnetic geodesics is empty.\n4.2.Lengths of Closed Magnetic Geodesics. We are now in a position to\ncompute the lengths of closed magnetic geodesics on Γ \\Hin the free homotopy\nclass ofγ∈Γ. If Γ< His a cocompact discrete subgroup, and γ∈Γ, then the\nlength of the corresponding closed magnetic geodesic on the compa ct quotient Γ \\H\nwill be\n/integraldisplay|ω|\n0|σ′(t)|dt=E|ω|. (35)\nPreviousresultsresultsconcerningthelengthsofclosedgeodesic sintheRiemannian\ncase include [GW+86], [Ebe94], [GM00], [GM03]. Unlike the Riemannian case,\nmagnetic geodesics cannot be reparamterized to have a different e nergy. So it is\nmore natural to consider the collection of lengths of closed geodes ics of a fixed\nenergy. Let L(γ;E) denote the set of distinct lengths of closed magnetic geodesics\nin the free homotopy class of γ.\nTheorem 4.9. LetΓ<Hbe a cocompact discrete subgroup of the Heisenberg group\nHand letγ= exp(Vγ+zγZ)∈Γ.\n•Ifγ=eis the identity ( Vγ= 0andzγ= 0), then\nL(e;E) =\n\n∅ ifE≥ |B|/braceleftBigg\n2πA/radicalBig\nB2\nE2−1/bracerightBigg\nif0<E <|B|(36)\n•Ifγis not central ( Vγ/\\e}atio\\slash= 0) then\nL(γ;E) =\n\n∅ if0<E≤ |B|/braceleftBigg\n|Vγ|/radicalBig\n1−B2\nE2/bracerightBigg\nifE >|B|(37)\n•Ifγis central (Vγ= 0andzγ/\\e}atio\\slash= 0) then\nL(γ;E) = (38)\n\n\n/braceleftBigg√\n4πAℓ(zγ−πAℓ)/radicalBig\n1−B2\nE2:ℓ∈Z,2E\nE+|B|<zγ\nπAℓ/bracerightBigg\n∪{|zγ|} E >|B|\n/braceleftBigg√\n4πAℓ(πAℓ−zγ)/radicalBig\nB2\nE2−1:ℓ∈Z,2E\nE−|B|<zγ\nπAℓ<2E\nE+|B|/bracerightBigg\n∪{|zγ|}0<E <|B|\n/braceleftBig\n2E|zγ|\n|z0|:z0∈R,(z0+B)2<E2/bracerightBig\n∪{|zγ|} E=|B|PERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 19\nProof.The case when γ=efollows from Theorem 3.3. The lengths of closed\nmagnetic geodesics obtained in that theorem is\nE|ω|=E/vextendsingle/vextendsingle/vextendsingle/vextendsingle2πA\n−sgn(B)√\nB2−E2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=2πAE√\nB2−E2\nThe case when γ= exp(Vγ+zγZ) is not central follows from Theorem 4.3. The\nlength of closed magnetic geodesics obtain in that theorem is\nE|ω|=E/vextendsingle/vextendsingle/vextendsingle/vextendsingle|Vγ|√\nE2−B2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=E|Vγ|√\nE2−B2.\nThe case when γis central follows from Lemma 4.4 and Lemma 4.5. In the former\ncase, which applies to every energy, the length of the closed magne tic geodesic is\nE|ω|=E/vextendsingle/vextendsingle/vextendsingle/vextendsinglezγ\n±E/vextendsingle/vextendsingle/vextendsingle/vextendsingle=|zγ|.\nIn the latter case, when E >|B|the lengths are\nE|ω|=E/vextendsingle/vextendsingle/vextendsingle/vextendsingle2πAℓ\nz0/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 2πAEℓ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle±/radicalBigg\nzγ\nπAℓ−1\nE2−B2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=2E/radicalbig\nπAℓ(zγ−πAℓ)√\nE2−B2\nand whenE <|B|the lengths are\nE|ω|=E/vextendsingle/vextendsingle/vextendsingle/vextendsingle2πAℓ\nz0/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 2πAEℓ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle±/radicalBigg\n1−zγ\nπAℓ\nB2−E2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=2E/radicalbig\nπAℓ(πAℓ−zγ)√\nB2−E2.\nThe lengths when E=|B|depend not on ℓ(which must be ℓ=zγ/(πA)) but\ninstead onz0and are given by\nE|ω|=E/vextendsingle/vextendsingle/vextendsingle/vextendsingle2πAℓ\nz0/vextendsingle/vextendsingle/vextendsingle/vextendsingle=E/vextendsingle/vextendsingle/vextendsingle/vextendsingle2zγ\nz0/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\n/square\nRemark 4.10.AsE→ ∞orB→0, the denominator/radicalbig\n1−B2/E2→1. Roughly\nspeaking, the cases E≤ |B|will be eliminated, and the collection of lengths in the\ncaseE >|B|will approach the length spectrum in the Riemannian case, which\nwas computed in [GM00]. This reflects the following physical intuition: w hen the\nmagnetic field is very weak charged particles will behave more like they would in\nthe absence of any forces, and when a particle is very energetic th e magnetic field\nwill have less of an effect on its trajectory.\nRemark 4.11.The dynamics of the magnetic flow on the various energy levels splits\nroughly into three regimes:\n•For fixed energy levels E >|B|, there exist closed magnetic geodesics in\nevery free homotopy class and the set of their lengths is finite.\n•For fixed energy levels E <|B|, there exist free homotopy classes without\nany closed magnetic geodesics, and in the case that there are close d mag-\nnetic geodesics, the set of their lengths is countably infinite. This re flects\nthe paradigm that the dynamics on high energy levels will resemble tha t of\nthe underlying geodesic flow.\n•Finally, when E=|B|,γis central, and zγ∈πAZ(i.e. the set of lengths\nis nonempty), then the infinite set of lengths is not discrete.20 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nThe following three lemmas address bounds on the collection of length s of closed\nmagnetic geodesics in a given central free homotopy class.\nLemma 4.12. Consider the case |B|<Ein(38). The set\n\n\n/radicalbig\n4πAℓ(zγ−πAℓ)/radicalBig\n1−B2\nE2:ℓ∈Z,2E\nE+|B|<zγ\nπAℓ\n\n\nis bounded above by |zγ|//radicalbig\n1−B2/E2,which is larger than |zγ|.The example below\nshows that this upper bound is the best possible.\nProof.Without loss of generality, we assume zγ>0.The condition on ℓimplies\n0<ℓ<zγ\n2πA/parenleftBig\n1+|B|\nE/parenrightBig\n.We define\nλ(ℓ) =4πAℓ(zγ−πAℓ)/parenleftbig\n1−B2\nE2/parenrightbig.\nThe parabola λ(ℓ) opens downward and has zeroes at ℓ= 0 andℓ=zγ/πA,hence\nachieves a maximum of z2\nγ//parenleftbig\n1−B2/E2/parenrightbig\natℓ=zγ/2πA.See the example below for\nvalues ofA,B,z γsuch that this maximum is achieved. The result follows. /square\nExample 4.13.Consider the particular example where A= 1,B= 1 andE= 2.\nChoose the central element γ= exp(20πZ) so thatzγ= 20π. In this case, for each\nℓsuch that 0 < ℓ <15, there is a closed magnetic geodesic with length given by\n(38). In particular, when ℓ= 10 the corresponding length is (2 /√\n3)20π>zγ.\nRemark 4.14.In the setting of Riemannian two-step nilmanifolds, the maximal\nlength of a closed magnetic geodesic in a central free homotopy clas s is the length\nof the central geodesic. In fact, the maximal length spectrum de termines the length\nspectrum for central free homotopy classes (see Proposition 5.1 5 of [Ebe94]). Ex-\nample 4.13 shows that this is no longer true in the magnetic setting.\nLemma 4.15. Consider the case |B|>Ein(38). The set\n\n\n/radicalbig\n4πAℓ(πAℓ−zγ)/radicalBig\nB2\nE2−1:ℓ∈Z,2E\nE−|B|<zγ\nπAℓ<2E\nE+|B|\n\n(39)\nis bounded below by |zγ|.\nProof.Without loss of generality, we assume zγ>0.We define\nλ(ℓ) =4πAℓ(πAℓ−zγ)/parenleftbigB2\nE2−1/parenrightbig.\nThe parabola λ(ℓ) opens upward and has zeroes at ℓ= 0 andℓ=zγ/πA.The\ncondition on ℓimpliesℓ >zγ\n2πA/parenleftBig\n1+|B|\nE/parenrightBig\n>zγ\nπAorℓ <zγ\n2πA/parenleftBig\n1−|B|\nE/parenrightBig\n<0.A\nlower bound of the set is thus provided by the minimum of/radicalbigg\nλ/parenleftBig\nzγ\n2πA/parenleftBig\n1+|B|\nE/parenrightBig/parenrightBig\nand/radicalbigg\nλ/parenleftBig\nzγ\n2πA/parenleftBig\n1−|B|\nE/parenrightBig/parenrightBig\n.However, both of these evaluate to zγ,and the result\nfollows. /squarePERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 21\nLemma 4.16. Consider the case |B|=Ein(38). If the set\n/braceleftbigg2E|zγ|\n|z0|:z0∈R,(z0+B)2<E2/bracerightbigg\nis nonempty (see Remark 4.8), then it is unbounded above and h as an infimum of\n|zγ|.\nProof.IfB >0, thenz0can be chosen in the interval −2B<z0<0. Asz0→0−,\nthe length diverges to infinity, and as z0→(−2B)+the lengths converge to |zγ|.\nThe case when B <0 is analogous. /square\n4.3.Density of Closed Magnetic Geodesics. Given a Riemannian manifold\nM, defineSEM={V∈TM:|V|=E}and letSE\nγMdenote the tangent sphere\nof radiusEat the point γ. Given a vector V∈TM, letσVdenote the magnetic\ngeodesicsuchthat σ′\nV(0) =V. Weareinterestedinthe sizeofthe setofvectorsthat\ndetermine periodic magnetic geodesics. In the Riemannian case, this set is scale\ninvariant. That is, if Vdetermines a periodic geodesic, then so does cVfor any\nc/\\e}atio\\slash= 0. So it is natural in this case to restrict attention to unit vectors . However,\nthis property does not hold for magnetic geodesics. Therefore, in the following\ndefinition we include a dependence on the energy of the vectors.\nPerE(M) :={V∈SEM:σVis periodic } ⊂SEM. (40)\nIn the context of Riemannian two-step nilmanifolds, the density of t his set was first\ninvestigated in [Ebe94], and subsequently in [Mas94], [Mas97], [LP96], [De M01],\n[DeC08]. The following result shows that for magnetic flows on the Heis enberg\ngroup, density persists for sufficiently high energy.\nTheorem 4.17. For eachE >|B|,PerE(Γ\\H)is dense in SE(Γ\\H).\nProof.We begin with a series of reductions. First, it suffices to show that th e set\nofV∈SE(H) such that σVisγ-periodic for some γ∈Γ is dense in SE(H). For\nanyV∈SE(Γ\\H), letW∈π−1(V) and let {Wi} ⊂SE(H) be such that σWiis\nγi-periodic for some γi∈Γ andWi→W. Then{Vi=π(Wi)} ⊂SE(Γ\\H) is a\nsequence of tangent vectors such that σViis periodic and Vi→V.\nNext, we claim that it suffices to show that the set of W∈SE\neHsuch thatσWis\nperiodic for some γ∈Z(Γ) is dense in SE\neH. For ifσWis such a magnetic geodesic\nandφ∈His any element, then φ·σW(t) is a (φγφ−1)-periodic magnetic geodesic\nsatisfyingφ·σV(0) =φand (φ·σV)′(0) =Lφ∗(V). Becauseγis central,φγφ−1=γ\nandφ·σWis aγ-periodic magnetic geodesic. Because Lφ∗SE\ne(H)→SE\nφ(H) is a\ndiffeomorphism, this proves the claim.\nLastly, we claim that it suffices to show that set z0∈[−B−E,−B+E] chosen\naccording to cases (1a) and (1b) in Lemma 4.5 (for some choice of γ∈Z(Γ)) is\ndense in [ −B−E,−B+E]. As noted in Lemma 4.5, for any such z0there is a\none parameter family of γ-periodic magnetic geodesics given by any choice of u0,v0\nsuch thatu2\n0+v2\n0=A(E2−(z0+B)2). Hence if the resulting z0are dense in\n[−B−E,−B+E], then there is a dense set of latitudes in the ellipsoid E2=\n((u2\n0+v2\n0)/A)+(z0+B)2⊂R3such that those vectors yield γ-periodic magnetic\ngeodesics for some γ∈Γ. The initial conditions ( u0,v0,z0)∈R3determine the\nmagnetic geodeisc σVwhereV= (u0/A)X+(v0/A)Y+(z0+B)Z, showing that\nthe set ofV∈SE\neHtangent to γ-periodic magnetic geodesics ( γ∈Γ) is dense in\nSE\neH.22 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nBy Proposition 5.4 of [Ebe94], Γ ∩Z(H) =Z(Γ) is a lattice in Z(H). Hence\nthere exists ¯ z∈R∗such that Γ ∩Z(H) ={exp(h¯zZ) :h∈Z}. By replacing ¯ z\nwith−¯z, if necessary, we can assume that ¯ z>0. Consider the set of numbers\n/braceleftbiggh\nℓ:h,ℓ∈Z+and/parenleftbigg2πAE\n¯z(E+B)/parenrightbigg\nℓ<h/bracerightbigg\n.\nThis set is dense in the interval (2 πAE/(¯z(E+B)),∞). Via a sequence of contin-\nuous mappings of R, each of which preserves density,\n/braceleftBigg\n−/radicalBigg\nE2−B2\nh¯z\nπAℓ−1:h,ℓ∈Z+and/parenleftbigg2E\nE+B/parenrightbigg\nℓ<h/bracerightBigg\nis dense in the interval ( −E−B,0). These are preciesly the values for z0appearing\nin case (1a) of Lemma 4.5. Starting instead with the set\n/braceleftbiggh\nℓ:h,ℓ∈Z+and/parenleftbigg2πAE\n¯z(E−B)/parenrightbigg\nℓ<h/bracerightbigg\nand using a parallel sequence of transformations shows that\n/braceleftBigg/radicalBigg\nE2−B2\nh¯z\nπAℓ−1:h,ℓ∈Z+and/parenleftbigg2E\nE−B/parenrightbigg\nℓ<h/bracerightBigg\nis dense in (0 ,E−B). These numbers are the z0appearing in case (1b) of Lemma\n4.5. This shows the density of permissible z0in the interval [ −E−B,E−B] and\nhence the theorem.\n/square\n4.4.Rigidity and the Marked Magnetic Length Spectrum. We begin by\nrecalling the notion of marked length spectrum for a compact Rieman nian manifold\nM. For each nontrivial free homotopy class C, there exists at least one smoothly\nclosed Riemannian geodesic. Let L(C) denote the collection of all lengths of smooth\nclosedgeodesicsthat belong to C. Recall that free homotopy classesofclosed curves\nonMare in bijection with conjugacy classes of π1(M). If¯Mis another compact\nRiemannian manifold and φ:π1(M)→π1(¯M) is an isomorphism, then φmaps\nconjugacy classes of π1(M) bijectively onto conjugacy classes of π1(¯M). Henceφ\ninduces a bijection φ∗of the set of free homotopy classes of closed curves on Monto\ntheset offreehomotopyofclassesofclosedcurveson ¯M. TwocompactRiemannian\nmanifoldsMand¯Maresaidtohavethe same marked length spectrum ifthereexists\nan isomorphism φ:π1(M)→π1(¯M) such that L(φ∗C) =L(C) for all nontrivial\nfree homotopy classes of closed curves on M. Specializing to the case at hand,\nletG1andG2be two simply connected 2-step nilpotent Lie groups and Γ 1< G1\nand Γ 2< G2cocompact discrete subgroups. Then π1(Γi\\Gi)≃Γi. With these\nidentifications, we say the nilmanifolds Γ 1\\G1and Γ 2\\G2have the same marked\nlength spectrum if there is an isomorphism φ: Γ1→Γ2such thatL(φ∗C) =L(C)\nfor all nontrivial free homotopy classes of closed curves on Γ 1\\G2. See [Ebe94]\nand [GM04] for previous results on marked length spectrum rigidity o f Riemannian\ntwo-step nilmanifolds.\nWhile the above definition could be used in the context of magnetic flow s on\nnilmanifolds, it seems more natural to modify it in light of the dependen ce of the\ndynamics on the relative magnitudes of Eand|B|. For a fixed homotopy class\nC, the collection lengths of closed magnetic geodesics of any energy c ould be anPERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 23\ninfinite open interval. Therefore, let L(C;E) denote the collection of all lengths\nof smoothly closed magnetic geodesics that belong to Cand have energy E. By\nTheorem 4.9, Lis not well-defined for E≤ |B|. In order to avoid this, we only\ndefineLforE >|B|.\nDefinition 4.18. LetHbe the simply connected three-dimensional Heisenberg\ngroup. Let g1andg2be two left-invariant Riemannian metrics on Hwith param-\netersA1andA2. Let Ω 1and Ω 2be two left-invariant magnetic forms on Hwith\nparameters B1andB2. Let Γ 1,Γ2<Hbe two cocompact discrete subgroups, and\nφ: Γ1→Γ2anisomorphism. ThenilmanifoldsΓ 1\\HandΓ2\\Hwithcorresponding\nmagnetic structures are said to have the same marked magnetic length spectrum if\nL(φ∗C;E2) =L(C;E1) for someE1>|B1|and someE2>|B2|for each nontrivial\nfree homotopy class C.\nEven though the magnetic flow is a perturbation away from the unde rlying ge-\nodesic flow, it reflects enough of the underlying Riemannian geometr y to exhibit a\ndegree of geometric rigidity.\nTheorem 4.19. LetHbe the simply connected, three-dimensional Heisenberg gro up\nendowed with left-invariant Riemannian metric gand left-invariant magnetic form\nΩ, with corresponding parameters AandBrespectively. Let Γ1,Γ2< Hbe two\ncocompact lattices. Suppose that for some E >|B|, the two manifolds Γ1\\Hand\nΓ2\\Hhave the same marked magnetic length spectrum at energy E. ThenΓ1\\H\nandΓ2\\Hare isometric.\nThe proof of Theorem 4.19 is similar to the proof of Theorem 5.20 in [Ebe 94],\nwith one notable exception. The latter uses the maximal marked leng th spec-\ntrum, i.e. only the length longest closed geodesic in each free homoto py class. For\nRiemannian geodesics in central free homotopy classes (on two-st ep nilpotent Lie\ngroups), this is always length of the one-paramter subgroup. Exa mple 4.13 and\nRemark 4.14 show that the maximal magnetic marked length spectru m is not so\nwell behaved. To circumvent this, we consider all the lengths of clos ed magnetic\ngeodesics in central free homotopy classes. This argument is given in the following\nLemma.\nLemma 4.20. Under the same hypotheses as Theorem 4.19, let exp(¯z1Z)and\nexp(¯z2Z)be generators for the central lattices Γ1∩HandΓ2∩H, respectively.\nThen|¯z1|=|¯z2|.\nProof.First, we claim that\nsup\nh∈Z/braceleftbiggmax(L([exp(h¯z1Z)]1;E))\n|h|/bracerightbigg\n= sup\nh∈Z/braceleftbiggmax(L([exp(h¯z2Z)]2;E))\n|h|/bracerightbigg\n(41)\nwhere [γ]idenotes the free homotopy class of closed curves on Γ i\\Hdetermined by\nγ∈Γi. Letφ: Γ1→Γ2be an isomorphism. Since φis an isomorphism of Z(Γ1)\nontoZ(Γ2),φ(exp(h¯z1Z)) = exp( ±h¯z2Z), andsoφ∗[exp(h¯z1Z)]1= [exp(±h¯z2Z)]2.\nBy hypothesis, the sets of lengths of closed magnetic geodesics in t hese two classes\nare equal. Moreover, the positive integer |h|is the same for both free homotopy\nclasses. Hence the sets over which the supremums are taken are e qual.\nNext we evaluate the supremums in (41). By Lemma 4.12, the set of le ngths\nof smoothly closed magnetic geodesics in the free homotopy class de termined by\nan element of the form exp( h¯ziZ) is bounded above by |h¯zi|//radicalbig\n1−B2/E2. After24 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\ndividing all the lengths in each set by |h|, respectively, we obtain a uniform upper\nbound,\nsup\nh∈Z\n\n/radicalbig\n4πAℓ(h¯zi−πAℓ)\n|h|/radicalBig\n1−B2\nE2:ℓ∈Z,2E\nE+|B|<h¯zi\nπAℓ\n\n≤|¯zi|/radicalBig\n1−B2\nE2. (42)\nWe now claim that the inequality in (42) is actually an equality. If the qua ntity\n¯zi/(2πA)∈Q, then forhlarge enough ℓ= (h¯zi)/(2πA) will be an allowable integer\nvalue forℓ, and max ℓ(/radicalbig\n4πAℓ(h¯zi−πAℓ)) =|h¯zi|. If the quantity ¯ zi/(2πA)/∈Q,\nthen the numbers ( h¯zi)/(2πA) will will come arbitrarily close to an integer. In\neither case, the supremum is |¯zi|//radicalbig\n1−B2/E2. The lemma now follows. /square\nWe now proceed with the proof of Theorem 4.19.\nProof.First, extend the marking φ: Γ1→Γ2to an automorphism φ:H→H.\nBecauseφ∗is a Lie algebra automorphism of h=v⊕z, we can decompose it as\nφ∗=R1+R2+S, whereR1:v→v,R2:v→z, andS:z→zare linear maps.\nUsing Lemma 4.20,\n|S(¯z1Z)|=|±¯z2Z|=|¯z2|=|¯z1|=|¯z1Z|\nshows that Sis an isometry of z. Letπv:h→vdenote the projection. For any\nV∈πvlogΓ1, there is some ξ∈Γ1such thatξ=V+ZandZ∈z. By hypothesis,\nexp(ξ) and\nφ(exp(V+Z)) =φ(exp(ξ)) = exp(φ∗ξ) = exp(R1(V)+R2(V)+S(Z))\nhave the same lengths of closed magnetic geodesics. By (37), we ha ve\n|V|/radicalBig\n1−B2\nE2=|R1(V)|/radicalBig\n1−B2\nE2\nand we conclude that R1is an isometry of v. It is straightforward to check that\nR1+S:h→his an isometric Lie algebra isomorphism. Let φ1:H→Hbe\nthe Lie group isomorphism such that ( φ1)∗=R1+S. DefineT:h→hby\nT(V+Z) =V+Z+(S−1◦R2)(V). Once can verify directly that Tis an inner\nautomorphism of hand (φ1)∗◦T=φ∗. Letφ2be the inner automorphism of H\nsuch that (φ2)∗=T.\nNow we have that φ=φ1◦φ2, whereφ1is an isometric automorphism of H\nandφ2is an in inner automorphism of H. Becauseφ2is an inner automorphism,\nΓ1\\His isometric to φ2(Γ1)\\H(via a left-translation), and φ2(Γ1)\\His isometric\ntoφ1(φ2(Γ1))\\H=φ(Γ1)\\H= Γ2\\H. /square\nRemark 4.21.ForE <|B|, the set of lengths in any noncentral free homotopy class\nis empty. Hence magnetic length spectrum does not determine\nRemark4.22.The isometryΓ 1\\H→Γ2\\Hpreservesthe magneticformΩ = d(Bζ)\nup to sign. Since the isometry is realized by φ1◦Lxfor somex∈H,\n(φ1◦Lx)∗ζ=L∗\nx(φ∗\n1ζ) =L∗\nx(±ζ) =±ζ.PERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 25\n5.Heisenberg Type Manifolds\nHeisenberg type manifolds are Riemannian manifolds that generalize t he Heisen-\nberg group endowed with a left-invariant metric. A metric two-step nilpotent Lie\nalgebrahis ofHeisenberg type if\nj(Z)2=−|Z|2Iv\nfor everyZ∈z(see (14)). A simply connected two-step nilpotent Lie group with\nleft-invariant metric is of Heisenberg type if its metric Lie algebra is of Heisen-\nberg type. It is often the case that theorems concerning the Heis enberg group\nendowed with a left-invariant metric, or their analogous formulation s, are also true\nfor Heisenberg type manifolds. For an example, see the results of [G M00]. This\nparadigm does not appear to extend to the setting of Heisenberg t ype manifolds\nendowed with a left-invariant magnetic field. In this section, we show by way of a\nsimple example that the computation of the lengths of closed magnet ic geodesics\nbecomes significantly more complex for Heisenberg type manifolds.\nLeth= span{X1,...,X 4,Z1,Z2}and define a bracket structure by\n[X1,X2] =Z1[X1,X3] =Z2[X2,X4] =−Z2[X3,X4] =Z1\nand extending by bilinearity and skew-symmetry to all of h. Define the met-\nric onhby declaring {X1,X2,X3,X4,Z1,Z2}to be an orthonormal basis. It\nis straightforward to check that his of Heisenberg type with two dimensional\ncenterz= span{Z1,Z2}. Let{α1,α2,α3,α4,ζ1,ζ2}be the basis of h∗dual to\n{X1,X2,X3,X4,Z1,Z2}. LetHbe the simply connected Lie group with left-\ninvariant metric with metric Lie algebra h. As in Lemma 2.2, any exact, left-\ninvariant 2-form is of the form Ω = d(ζm) for some ζm∈z∗. For simplicity we\ntake as magnetic field Ω = d(Bζ1). For each γ∈H, we wish to understand the\nγ-periodic geodesics and the associated periods.\nProceedingasin theHeisenbergcasein Section 4, let p(t) =/summationtextai(t)αi+/summationtextci(t)ζi\nbe the integral curve of the Euler vector field on h∗with initial condition p(0) =/summationtextuiαi+/summationtextziζi. Then the component functions are\na1(t) =u1cos(ˆzt)+/parenleftbigg−z1u2−z2u3\nˆz/parenrightbigg\nsin(ˆzt)\na2(t) =u2cos(ˆzt)+/parenleftbiggz1u1+z2u4\nˆz/parenrightbigg\nsin(ˆzt)\na3(t) =u3cos(ˆzt)+/parenleftbiggz2u1−z1u4\nˆz/parenrightbigg\nsin(ˆzt)\na4(t) =u4cos(ˆzt)+/parenleftbigg−z2u2+z1u3\nˆz/parenrightbigg\nsin(ˆzt)\nc1(t) =z1\nc1(t) =z2\nwhere ˆz=/radicalbig\nz2\n1+z2\n2. Next, let σ(t) be the magnetic geodesic through the identity\ndetermined by the integral curve p(t). Henceσ(t) solvesσ′(t) =dhp(t), where\nh:h∗→Ris the Hamiltonian (see (17)). Writing σ(t) = exp( X(t) +Z(t)),\nwhereX(t) =/summationtextxi(t)XiandZ(t) =/summationtextzi(t)Zi, we have on the one hand under26 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\ntrivialization by left-multiplication,\nσ′(t) =X′(t)+Z′(t)+1\n2[X′(t),X(t)] =/summationdisplay\nx′\ni(t)Xi+Z′(t)+1\n2[X′(t),X(t)],\nand on the other hand\ndhp(t)=♯(p(t)+Bζ1) =/summationdisplay\nai(t)Xi+(z1+B)Z1+z2Z2.\nMatching up the non-central components shows that\nx1(t) =u1\nˆzsin(ˆzt)+−z1u2−z2u3\nˆz2(1−cos(ˆzt))\nx2(t) =u2\nˆzsin(ˆzt)+z1u1+z2u4\nˆz2(1−cos(ˆzt))\nx3(t) =u3\nˆzsin(ˆzt)+z2u1−z1u4\nˆz2(1−cos(ˆzt))\nx4(t) =u4\nˆzsin(ˆzt)+−z2u2+z1u3\nˆz2(1−cos(ˆzt))\nwhile the central components satisfies\nZ′(t) = (z1+B)Z1+z2Z2−1\n2[X′(t),X(t)].\nA tedious computation shows\n[X′(t),X(t)] =/parenleftbigg\n−z1ˆu2\nˆz2+z1ˆu2\nˆz2cos(ˆzt)/parenrightbigg\nZ1+/parenleftbigg\n−z2ˆu2\nˆz2+z2ˆu2\nˆz2cos(ˆzt)/parenrightbigg\nZ2\n=−z1ˆu2\nˆz2(1−cos(ˆzt))Z1−z2ˆu2\nˆz2(1−cos(ˆzt))Z2\n= (1−cos(ˆzt))/parenleftbigg\n−z1ˆu2\nˆz2Z1−z2ˆu2\nˆz2Z2/parenrightbigg\nwhere ˆu=/radicalbig\nu2\n1+···+u2\n4. A final integration now provides the central compo-\nnents:\nz1(t) =/parenleftbigg\nz1+B+z1ˆu2\n2ˆz2/parenrightbigg\nt−z1ˆu2\n2ˆz3sin(ˆzt)\nz2(t) =/parenleftbigg\nz2+z2ˆu2\n2ˆz2/parenrightbigg\nt−z2ˆu2\n2ˆz3sin(ˆzt).\nLetγ= exp(ξ1Z1+ξ2Z2) be a central element of H. Comparing the components\nofγσ(t) =σ(t+ω) shows that ω= 2πk/ˆz,k∈Z. With this choice of ω, the non-\ncentral components are equal, while the central component yield t he system\n/parenleftbigg\nz1+B+z1ˆu2\n2ˆz2/parenrightbigg2πk\nˆz=ξ1 (43)\n/parenleftbigg\nz2+z2ˆu2\n2ˆz2/parenrightbigg2πk\nˆz=ξ2. (44)\nEach choice of u1,...,u 4,z1,z2satisfying this system and the energy constraint\n1 = ˆu2+(z1+B)2+z2\n2= ˆu2+ ˆz2+2z1B+B2PERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 27\nwill yield a unit speed magnetic geodesic translated by γ. In the case that the\nmagnetic field and γare “parallel”, i.e. ξ2= 0, then (44) becomes\nz2/parenleftbigg\n1+ˆu2\n2ˆz2/parenrightbigg2πk\nˆz= 0.\nThe second and third factor are necessarily nonzero, so z2= 0. This reduces (43) to\nan equation that can be solved in the same way as the Heisenberg cas e, according\nto the strength of the magnetic field relative to the energy. When γis an arbitrary\nelement of the center, it it is much more difficult to completely solve (43 ) and (44),\nand hence obtain an explicit description of all the γ-periodic geodesics.\n6.Appendix: Tangent Bundle Viewpoint: Periodic Magnetic\nGeodesics in Heisenberg manifolds\nIn[Kap81], A.Kaplanintroducedso-called j-mapstostudyCliffordmodules(see\nSection2.4forthe definition). Ametric two-stepnilpotent Liealgebr ais completely\ncharacterized by its associated j-maps. Since being introduced, they have proven\nvery useful in the study of two-step nilpotent geometry. In this a ppendix we show\nhow the magnetic geodesic equations can be characterized in terms of thej-maps.\nLetGbe a two-step nilpotent Lie group endowed with a left-invariant metr icg\nand an exact, left-invariant magnetic form Ω. Let g=v⊕zbe the decomposition\nof the Lie algebra into the center and its orthogonal complement. B y Lemma 2.2,\nthere isζm∈z∗such that Ω = d(Bζm).\nLemma 6.1. The Lorentz force associated to the magnetic field ΩsatisfiesFv=\nj(−BZm)andFz= 0, whereZm=♯(ζm).\nProof.LetX∈g,V∈vandZ∈z. Then\ng(F(Z),X) = Ω(Z,X) =d(Bζm)(Z,X) =−Bζm([Z,X]) = 0,\nand\ng(F(V),X) = Ω(V,X) =d(Bζm)(V,X) =−Bζm([V,X])\n=−Bg(♯(ζm),[V,X]) =−Bg(j(Zm)V,X)\n=g(j(−BZm)V,X).\n/square\nBecause of Lemma 6.1, we will write F=j(−BZm) with the understanding that\nFvanishes on central vectors and agrees with j(−BZm) on vectors in v. Letγ(t) =\nexp(X(t)+Z(t))beamagneticgeodesicon GwhereX(t)∈vandZ(t)∈z. By(13),\nwe can express the velocity vector of γasγ′(t) =X′(t) +1\n2[X′(t),X(t)] +Z′(t).\nThe condition for γto be a magnetic geodesic is ∇γ′(t)γ′(t) =F(γ′(t)). Using\n(15) to expand this condition and imposing the initial conditions γ(0) =eand\nγ′(0) =X0+Z0, the geodesic equations on vandzseparately are\nX′′(t) =j(Z0−BZm)X′(t) (45)\nZ′(t)+1\n2[X′(t),X(t)] =Z0. (46)\nWe restrict to the three-dimensional Heisenberg case and conside r the magnetic\ngeodesics in this context. Following the approach as illustrated in Pro p. 3.5 on\npages 625–628 of [Ebe94], and reducing to the three-dimensional H eisenberg case,\na straightforward calculation gives the following result.28 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\nCorollary 6.2. Ifz0−B= 0, (or ifz0−B/\\e}atio\\slash= 0andx0=y0= 0) thenσ(t)is the\none parameter subgroup\n(47)σ(t) = exp(x(t)X+y(t)Y+z(t)Z) = exp(t(x0X+y0Y+z0Z)).\nIfz0−B/\\e}atio\\slash= 0, the solution is\n(48)/parenleftbigg\nx(t)\ny(t)/parenrightbigg\n=A\nz0−B/parenleftbigg\nsin/parenleftbig\nt/parenleftbigz0−B\nA/parenrightbig/parenrightbig\n−/parenleftbig\n1−cos/parenleftbig\nt/parenleftbigz0−B\nA/parenrightbig/parenrightbig/parenrightbig\n1−cos/parenleftbig\nt/parenleftbigz0−B\nA/parenrightbig/parenrightbig\nsin/parenleftbig\nt/parenleftbigz0−B\nA/parenrightbig/parenrightbig/parenrightbigg/parenleftbigg\nx0\ny0/parenrightbigg\n,\nand\n(49) z(t) =/parenleftbigg\nz0+A(x2\n0+y2\n0)\n2(z0−B)/parenrightbigg\nt−A2(x2\n0+y2\n0)\n2(z0−B)2sin/parenleftbig\nt/parenleftbigz0−B\nA/parenrightbig/parenrightbig\n.\nRemark6.3.Thecoordinatefunctions(48)and(49) areequivalenttheoneobt ained\nin (22)-(24) in the followingsense. In orderto obtainthe magneticg eodesicthrough\nthe origin determined by ( u0,v0,z0) as in section 3 take as initial tangent vector in\n(48) and (49) to be ( x0,y0,z0) = (u0/A,v0/A,z0+B).\nWe nowpresentsomeofthe main resultsaboutthe three-dimension alHeisenberg\nmanifold proved in the body of the paper, but expressed using the t angent bundle,\nrather than the cotangent bundle.\nContinuing the notation from the previous sections, we fix energy E,magnetic\nstrengthB,and metric parameter A.Let (H,gA,Ω) denote a simply connected\nHeisenberg manifold. The theorems in this section state precisely th e set of periods\nωsuch that there exists an intial velocity vp∈THsuch thatσvp(t) is periodic\nwith period ω. We also precisely state the set of initial velocities vp, hence the set\nof geodesics, that produce each period ω.\nLet Γ denote a cocompact discrete subgroup of Hand, as above, denote the\nresulting compact Heisenberg manifold by (Γ \\H,gA,Ω).For allγ∈Γ,we state\nbelow preciselythe set ofperiods ωsuch that there exists an intial velocity vp∈TH\nsuchthatσvp(t)isγ-periodicwith period ω. We alsopreciselystatethe setofinitial\nvelocitiesvp, hence the set of geodesics, that produce each period ω.\n6.1.Periodic Magnetic Geodesics on the Simply Connected Heisen berg\nGroup. We now consider the existence of periodic geodesics in ( H,gA,d(Bζ)), the\nthree-dimensional Heisenberg Lie group Hwith left-invariant metric determined by\nthe orthonormal basis/braceleftBig\n1√\nAX,1√\nAY,Z/bracerightBig\nand magnetic form Ω = −Bα∧β. Recall\nthat for a vector v∈h,σv(t) denotes the magnetic geodesic through the identity\nwith initial velocity v. Note that if vp∈TpH, thenσvp(t) denotes the magnetic\ngeodesic through p=σvp(0) with initial velocity vp. Also note that because gA\nand Ω are left-invariant, that σvp(t) =Lpσv(t), wherevp=Lp∗(v); i.e., magnetic\ngeodesics through p∈Hare just left translations of magnetic geodesics through\nthe identity. Clearly, a magnetic geodesic through p∈His periodic with period ω\nif and only if its left translation by p−1is a magnetic geodesic through the identity\nwith period ω.\nTheorem 6.4. With notation as above, fix energy E,magnetic strength B,and\nmetric paramter A.\n(1)IfB2> E2then there exists a one-parameter family of vectors v∈h,\n|v|=E,such thatσv(t)is periodic. In particular, σv(t)is periodic if andPERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 29\nonly ifz0=B−sgn(B)√\nB2−E2and\nx2\n0+y2\n0=−2z0(z0−B)/A.\nThe set of periods of σv(t)is2π√\nB2−E2Z/negationslash=0and the smallest positive period\nisω=/vextendsingle/vextendsingle/vextendsingle2πA\nz0−B/vextendsingle/vextendsingle/vextendsingle=2πA√\nB2−E2.\n(2)IfB2≤E2then there does not exist a vector vwith|v|=Esuch thatσv(t)\nis periodic.\n6.2.Periodic Geodesics on Compact Quotients of the Heisenberg g roup.\nWe ultimately wish to consider closed magnetic geodesics on Heisenber g manifolds\nof the form Γ \\H, where Γ is a cocompact discrete subgroup of H.As above, we\nproceed by considering γ-periodic magnetic geodesics on the cover H.\nThe purpose of this section is stating more precisely, and proving, t he following,\nwhich is divided into several cases. See Theorem 6.6, Theorem 6.7, an d Theorem\n6.8 below.\nTheorem 6.5. Consider the three-dimensional Heisenberg Lie group Hwith left\ninvariant metric gAdetermined by the orthonormal basis/braceleftBig\n1√\nAX,1√\nAY,Z/bracerightBig\nand\nmagnetic form Ω =d(Bζ) =−Bα∧β. Fixγ∈Hand fix energy E,magnetic\nstrengthBand metric parameter A.Then we can state precisely the set of periods\nωsuch that there exists an initial velocity vp∈THwith|vp|=Esuch thatσvp(t)\nisγ-periodic with period ω. We can also precisely state the set of initial velocities\nvp, hence the set of geodesics, that produce each period ω.\n6.2.1.Noncentral Case. Letγ= exp(xγX+yγY+zγZ)∈Hwithx2\nγ+y2\nγ/\\e}atio\\slash= 0.\nLeta= exp(axX+ayY+azZ)∈H. From (12), the conjugacy class of γinHis\nexp(ˆxX+ ˆyY+RZ).\nTheorem 6.6. Fix energy E,magnetic strength Band metric parameter A.Let\nγ= exp(xγX+yγY+zγZ)∈Hwithx2\nγ+y2\nγ/\\e}atio\\slash= 0.\n(1)IfE2> B2(ie, ifµ >1) then there exists a two-parameter family of\nelementsa∈Hsuch thataγa−1= exp/parenleftbig\nxγX+yγY+z′\nγZ/parenrightbig\nwherez′\nγ=\n±B/radicalBig\nA/parenleftbig\nx2γ+y2γ/parenrightbig\n/(E2−B2). Lettingv=B\nz′\nγ/parenleftbig\nxγX+yγY+z′\nγZ/parenrightbig\n, which\nsatisfies |v|=E, thenγtranslates the (non-spiraling) magnetic geodesic\na−1exp(tv)with period ω=±/radicalBig\nA/parenleftbig\nx2γ+y2γ/parenrightbig\n/(E2−B2). These are the\nonly magnetic geodesics with energy Etranslated by γ.\n(2)IfE2≤B2(ie, ifµ≤1) then neither γnor any of its conjugates in H\ntranslate a magnetic geodesic with energy E.\n6.2.2.Central Case. Throughout this subsection, we assume that xγ=yγ= 0; i.e.,\nthatγlies inZ(H),the center of three-dimensional Heisenberg group H.Recall\nthat sinceγis central,γtranslates a magnetic geodesic σ(t) through the identity\ne∈Hwith period ωif and only if for all a∈H,γtranslates a magnetic geodesic\nthroughawith period ω. That is, without loss of generality, if γlies in the center,\nwe may assume that σ(0) =e.\nRecall that magnetic geodesics in Hare either spiraling or one parameter sub-\ngroups. We first consider the case of one-parameter subgroups .\nTheorem 6.7. Fix energy E,magnetic strength Band metric parameter A.Let\nγ= exp(zγZ)∈Z(H), withzγ/\\e}atio\\slash= 0. The element γtranslates the magnetic30 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\ngeodesicsσ(t) = exp( ±tEZ)with initial velocities v=±EZand periods ω=\n±zγ/E. This pair of one-parameter subgroups and their left transl ates are the only\nstraight magnetic geodesics translated by γ.\nTheorem 6.8. Fix energy E,magnetic strength B,and metric parameter A.De-\nnoteµ=E\n|B|. Letγ= exp(zγZ)∈Z(H),zγ/\\e}atio\\slash= 0. If there exists a vector\nv=x0X+y0Y+z0Zand a period ω/\\e}atio\\slash= 0such that the spiraling geodesic σv(t)\nisγ-periodic with period ω, then there exists ℓ∈Z/negationslash=0such thatζℓ=zγ\nπℓsatisfies\nthe conditions relative to µspecified in the following six cases and A/parenleftbig\nx2\n0+y2\n0/parenrightbig\n,z0,\nandωare as expressed below. Conversely, for every choice of ℓ∈Z/negationslash=0such that\nζℓ=zγ\nπℓsatisfies the conditions in one of the cases below, there exis ts at least one\nvectorvas given below such that σv(t)isγ-periodic (spiraling) geodesic with period\nωas given below. Note that Case 1 requires E2<B2. Cases 2 through 5 require\nE2>B2,and Case 6 requires E2=B2.Note that in all cases, ζℓ/\\e}atio\\slash= 0.\n(1)−2µ\n1−µ<ζℓ\nA<2µ\n1+µ<1,\n(2) 1<2µ\n1+µ<ζℓ\nA<2,\n(3) 2<ζℓ\nA≤2µ\nµ−1,\n(4) 2<2µ\nµ−1<ζℓ\nA,\n(5)ζℓ\nA= 2andµ>1,\n(6)ζℓ\nA= 1andµ= 1.\nIn Cases 1 through 4, we choose any x0,y0∈Rso that\n(50) A/parenleftbig\nx2\n0+y2\n0/parenrightbig\n=B2/parenleftBigg\nµ2−1\nζℓ\nA−1/parenleftbiggζℓ\nA−2/parenrightbigg\n+2/radicalBigg\nµ2−1\nζℓ\nA−1/parenrightBigg\nand let\n(51) z0=−B/parenleftBigg\n−1+/radicalBigg\nµ2−1\nζℓ\nA−1/parenrightBigg\nand\nω=2zγA\nζℓ(z0−B)=2πℓ/radicalbigg/vextendsingle/vextendsingle/vextendsingleζℓ\nA−1/vextendsingle/vextendsingle/vextendsingle\n√\nE2−B2.\nIn Case 4, we may also choose any x0,y0∈Rso that\n(52) A/parenleftbig\nx2\n0+y2\n0/parenrightbig\n=B2/parenleftBigg\nµ2−1\nζℓ\nA−1/parenleftbiggζℓ\nA−2/parenrightbigg\n−2/radicalBigg\nµ2−1\nζℓ\nA−1/parenrightBigg\nand let\n(53) z0=−B/parenleftBigg\n−1−/radicalBigg\nµ2−1\nζℓ\nA−1/parenrightBigg\nand\nω=2zγA\nζℓ(z0−B)=−2πℓ/radicalbigg/vextendsingle/vextendsingle/vextendsingleζℓ\nA−1/vextendsingle/vextendsingle/vextendsingle\n√\nE2−B2.\nThe conditions on µ,ζℓ,x0,y0andz0implyµ2−1\nζℓ\nA−1>0,x2\n0+y2\n0>0,E2=A/parenleftbig\nx2\n0+y2\n0/parenrightbig\n+z2\n0, and the (spiraling) magnetic geodesic through the identit yσv(t)with initialPERIODIC MAGNETIC GEODESICS ON HEISENBERG MANIFOLDS 31\nvelocityv=x0X+y0Y+z0Zisγ= exp(zγZ)-periodic with energy Eand period\nωas given.\nIn Case 5, which only occurs ifzγ\nA∈2πZ/negationslash=0, we choose any x0,y0∈Rso that\nA/parenleftbig\nx2\n0+y2\n0/parenrightbig\n= 2|B|/radicalbig\nE2−B2\nand\nz0=B−A/parenleftbig\nx2\n0+y2\n0/parenrightbig\n2B.\nThen the conditions on µ,ζℓ,x0,y0andz0imply that E2=A/parenleftbig\nx2\n0+y2\n0/parenrightbig\n+z2\n0and\nthe (spiraling) magnetic geodesic σv(t)starting at the identity with initial velocity\nv=x0X+y0Y+z0Zisγ-periodic with energy Eand period\nω=−sgn(B)zγ√\nE2−B2.\nIn Case 6, which only occurs ifzγ\nA∈πZ/negationslash=0, we choose any x0,y0,z0∈Rso that\nE2=B2=A/parenleftbig\nx2\n0+y2\n0/parenrightbig\n+z2\n0andz0/\\e}atio\\slash=±B. The conditions on µandζℓimply that\nthe (spiraling) magnetic geodesic σv(t)with intial velocity v=x0X+y0Y+z0Z\nwill yield a γ-periodic magnetic geodesic with energy Eand period\nω=2zγ\nz0−B.\nRemark 6.9.In Case 1, there are infinitely many values of ℓthat satisfy the con-\nditions, hence infinitely many distinct periods ω.In particular, if µ<1 and there\nexistsℓ0∈Z>0such thatζℓ0∈/parenleftBig\n−2µ\n1−µ,2µ\n1+µ/parenrightBig\n, then for all ℓ>ℓ0,ζℓ∈/parenleftBig\n−2µ\n1−µ,2µ\n1+µ/parenrightBig\n.\nLikewise if there exists ℓ0∈Z<0such thatζℓ0∈/parenleftBig\n−2µ\n1−µ,2µ\n1+µ/parenrightBig\n, then for all ℓ < ℓ0,\nζℓ∈/parenleftBig\n−2µ\n1−µ,2µ\n1+µ/parenrightBig\n.\nRemark 6.10.In Case 6, the magnitude of the periods take all values in the interva l\n(|zγ|/E,∞). The period ω=|zγ|/Eis achieved when v=−BZ, which implies\nσvis a one-parameter subgroup; i.e., non-spiraling. The magnitude of t he period\napproaches ∞asv→BZ. This behavioris in contrastto the Riemannian case; i.e.,\nthe caseB= 0. In the Riemannian case, there are finitely many periods associat ed\nto each element γ.However, if there exists γ∈Γ such that log γ∈2πZ,then\nΓ\\Hdoes not satisfy the Clean Intersection Hypothesis, so the fact t hat unusual\nmagnetic geodesic behavior occurs in this case is not unprecedente d (see [Gor05]).\nReferences\n[AMMP17] Alberto Abbondandolo, Leonardo Macarini, Marco M azzucchelli, and Gabriel P. Pa-\nternain. Infinitely many periodic orbits of exact magnetic fl ows on surfaces for almost\nevery subcritical energy level. J. Eur. Math. Soc. (JEMS) , 19(2):551–579, 2017.\n[B´86] Pierre H. B´ erard. Spectral geometry: direct and inverse problems , volume 1207 of\nLecture Notes in Mathematics . Springer-Verlag, Berlin, 1986. With appendixes by\nG´ erard Besson, and by B´ erard and Marcel Berger.\n[BGM71] Marcel Berger, Paul Gauduchon, and Edmond Mazet. Le spectre d’une vari´ et´ e rie-\nmannienne . Lecture Notes in Mathematics, Vol. 194. Springer-Verlag, Berlin-New\nYork, 1971.\n[BM06] Keith Burns and Vladimir S. Matveev. On the rigidity o f magnetic systems with the\nsame magnetic geodesics. Proc. Amer. Math. Soc. , 134(2):427–434, 2006.\n[BP02] Keith Burns and Gabriel P. Paternain. Anosov magneti c flows, critical values and\ntopological entropy. Nonlinearity , 15(2):281–314, 2002.32 JONATHAN EPSTEIN, RUTH GORNET, MAURA B. MAST\n[BP08] Leo T Butler and Gabriel P Paternain. Magnetic flows on sol-manifolds: dynamical\nand symplectic aspects. Communications in mathematical physics , 284(1):187–202,\n2008.\n[Cro90] Christopher B. Croke. Rigidity for surfaces of nonp ositive curvature. Comment. Math.\nHelv., 65(1):150–169, 1990.\n[DeC08] Rachelle C. DeCoste. Closed geodesics on compact ni lmanifolds with Chevalley ratio-\nnal structure. Manuscripta Math. , 127(3):309–343, 2008.\n[DeM01] Lisa DeMeyer. Closed geodesics in compact nilmanif olds.Manuscripta Math. ,\n105(3):283–310, 2001.\n[Ebe94] Patrick Eberlein. Geometry of 2-step nilpotent gro ups with a left invariant metric.\nAnnales scientifiques de l’Ecole normale sup´ erieure , 27(5):611–660, 1994.\n[Ebe04] Patrick Eberlein. Left invariant geometry of Lie gr oups.Cubo, 6(1):427–510, 2004.\n[Eps17] Jonathan Epstein. Topological entropy of left-inv ariant magnetic flows on 2-step nil-\nmanifolds. Nonlinearity , 30(1):1–12, 2017.\n[GM00] Ruth Gornet and Maura B. Mast. The length spectrum of R iemannian two-step nil-\nmanifolds. Ann. Sci. ´Ecole Norm. Sup. (4) , 33(2):181–209, 2000.\n[GM03] Ruth Gornet and Maura B. Mast. Length minimizing geod esics and the length spec-\ntrum of Riemannian two-step nilmanifolds. J. Geom. Anal. , 13(1):107–143, 2003.\n[GM04] Ruth Gornet and Maura B. Mast. The minimal marked leng th spectrum of Riemann-\nian two-step nilmanifolds. Michigan Math. J. , 52(3):683–716, 2004.\n[Gor05] Ruth Gornet. Riemannian nilmanifolds and the trace formula. Trans. Amer. Math.\nSoc., 357(11):4445–4479, 2005.\n[Gro99] St´ ephane Grognet. Flots magn´ etiques en courbure n´ egative. Ergodic Theory Dynam.\nSystems, 19(2):413–436, 1999.\n[Gro05] Stephane Grognet. Marked length spectrum of magnet ized surfaces. arXiv preprint\nmath/0502424 , 2005.\n[GS90] Victor Guillemin and Shlomo Sternberg. Symplectic techniques in physics . Cambridge\nuniversity press, 1990.\n[GW+86] Carolyn S Gordon, Edward N Wilson, et al. The spectrum of t he laplacian on rie-\nmannian heisenberg manifolds. The Michigan mathematical journal , 33(2):253–271,\n1986.\n[Kap81] Aroldo Kaplan. Riemannian nilmanifolds attached t o Clifford modules. Geom. Dedi-\ncata, 11(2):127–136, 1981.\n[LP96] Kyung Bai Lee and Keun Park. Smoothly closed geodesic s in 2-step nilmanifolds.\nIndiana Univ. Math. J. , 45(1):1–14, 1996.\n[Mas94] Maura B. Mast. Closed geodesics in 2-step nilmanifo lds.Indiana Univ. Math. J. ,\n43(3):885–911, 1994.\n[Mas97] Maura B. Mast. Low-dimensional 2-step nilpotent Li e groups in resonance. Algebras\nGroups Geom. , 14(3):321–337, 1997.\n[MR03] R. J. Miatello and J. P. Rossetti. Length spectra and p-spectra of compact flat man-\nifolds.J. Geom. Anal. , 13(4):631–657, 2003.\n[Ota90a] Jean-Pierre Otal. Le spectre marqu´ e des longueur s des surfaces ` a courbure n´ egative.\nAnn. of Math. (2) , 131(1):151–162, 1990.\n[Ota90b] Jean-Pierre Otal. Sur les longueurs des g´ eod´ esi ques d’une m´ etrique ` a courbure\nn´ egative dans le disque. Comment. Math. Helv. , 65(2):334–347, 1990.\n[Pat06] Gabriel P. Paternain. Magnetic rigidity of horocyc le flows. Pacific J. Math. ,\n225(2):301–323, 2006.\n[PP96] Gabriel P. Paternain and Miguel Paternain. Anosov ge odesic flows and twisted sym-\nplectic structures. In International Conference on Dynamical Systems (Montevide o,\n1995), volume 362 of Pitman Res. Notes Math. Ser. , pages 132–145. Longman, Har-\nlow, 1996.\n[PP97] Gabriel P. Paternain and Miguel Paternain. First der ivative of topological entropy for\nAnosov geodesic flows in the presence of magnetic fields. Nonlinearity , 10(1):121–131,\n1997.\n[PS03] N. Peyerimhoff and K. F. Siburg. The dynamics of magnet ic flows for energies above\nMa˜ n´ e’s critical value. Israel J. Math. , 135:269–298, 2003."    },    {        "title": "2106.13102v2.Modeling_Magnetic_Particle_Imaging_for_Dynamic_Tracer_Distributions.pdf",        "content": "arXiv:2106.13102v2  [eess.IV]  25 Aug 2021Sensing and Imaging manuscript No.\n(will be inserted by the editor)\nModeling Magnetic Particle Imaging for Dynamic Tracer\nDistributions\nChristina Brandt ·Christiane Schmidt\nReceived: date / Accepted: date\nAbstract Magnetic Particle Imaging (MPI) is a promising tracer-base d, functional medi-\ncal imaging technique which measures the non-linear magnet ization response of magnetic\nnanoparticles to a dynamic magnetic field. For image reconst ruction, system matrices from\ntime-consuming calibration scans are used predominantly. Finding modeled forward oper-\nators for magnetic particle imaging, which are able to compe te with measured matrices in\npractice, is an ongoing topic of research. The existing mode ls for magnetic particle imaging\nare by design not suitable for arbitrary dynamic tracer conc entrations. Neither modeled nor\nmeasured system matrices account for changes in the concent ration during a single scanning\ncycle.\nIn this paper we present a new MPI forward model for dynamic co ncentrations. A static\nmodel will be introduced briefly, followed by the changes due to the dynamic behavior of\nthe tracer concentration. Furthermore, the relevance of th is new extended model is exam-\nined by investigating the influence of the extension and exam ple reconstructions with the\nnew and the standard model.\nKeywords magnetic particle imaging ·model-based reconstruction ·dynamic inverse\nproblems ·motion artifacts ·motion compensation\n1 Introduction\nMagnetic Particle Imaging (MPI) is a relatively new medical imaging modality invented by\nWeizenecker and Gleich in 2005 [ 9]. In this tomographic imaging technique, the non-linear\nmagnetization response of the superparamagnetic tracer ma terial to an external magnetic\nfield induces a potential in the receive coils of the scanner. The spatial distribution of the\nmagnetic particles is reconstructed from these measuremen ts. MPI allows for a rapid data\nacquisition with high temporal resolution which makes it a p romising imaging device for\ndifferent imaging applications, see [ 21] for an overview. In many of these applications, vi-\nsualization of tracer dynamics is highly relevant, such as p hysiological diagnosis like stroke\nChristina Brandt ·Christiane Schmidt\nUniversit¨ at Hamburg, Department of Mathematics, Bundess traße 55, 20146 Hamburg, Germany\nE-mail: christiane.schmidt@uni-hamburg.de2 Christina Brandt, Christiane Schmidt\ndetection [ 24], visualization of blood flow [ 31] or localization of medical instruments in\nvascular interventions [ 14].\nThe MPI forward operator can be described by model- or measur ement-based approaches\n[13]. In a measurement-based approach the forward operator is r epresented by a calibration\nscan [ 27,30]. Therefore, the signal generated by a delta sample of trace r material is measured\nfor a finite number of spatial positions. The modeling approa ch describes the measurement\nprocess with physical laws [ 22]. Unfortunately, models usually idealize the physical set ting\nto limit the complexity of the model. These simplifications c an lead to large modeling er-\nrors and give reasons for the time consuming measurement app roach being still dominant in\npractice.\nThe state of the art model underlies the assumption of a (near ly) static concentration dur-\ning the signal acquisition. This assumption is not always fu lfilled. MPI is able to visualize\nthe distribution of a liquid tracer. It can accumulate, diss ipate or move e.g. with the blood\nflow. The behavior of the particle concentration is not stati c in these cases. Also time-series\nmeasurements imply a dynamic tracer distribution such that the static model is only true\nfor piecewise constant concentrations. The same problem is valid for the measurement ap-\nproach. Since the delta sample is static during each cycle of the calibration scan, the mea-\nsured system matrix does not cover dynamic behavior. Curren tly, the only way to reconstruct\nnon-periodic dynamic concentrations is to reconstruct a ti me-series of images under the as-\nsumption of static behavior during the scan [ 28,14]. Reconstruction of periodic dynamics\nin magnetic particle imaging is investigated in [ 8] in order to reduce of artifacts induced by\ncardiac- or respiratory motion in multi-patch MPI. The auth ors use the measurement-based\napproach and assume limits on the velocity and periodicity o f the motion. By rearranging\nmeasurements from the same motion phase into virtual frames , dynamic tracer distributions\ncan be reconstructed by static reconstructions from the vir tual frames.\nThe model-based approach gives rise to various directions o f research covering all com-\nponents of the signal generation chain and analyses of the mo dels. One of these direc-\ntions is modeling of the magnetic behavior of magnetic nanop articles which was studied\nby Kluth [ 17,19] and Weizenecker [ 29]. The most frequently used magnetization model is\nthe Langevin- or equilibrium model, which is also the basis f or the derivations in the fol-\nlowing articles. The equilibrium model does not respect mag netic relaxation effects. In [ 17],\nthe model is extended for different kinds of relaxation. The author presents forward models\nincorporating either Brownian rotation or N´ eel relaxatio n in the cases of mono- and poly-\ndisperse tracers under the assumption of single domain part icles with uniaxial anisotropy.\nBased on the equilibrium model the authors of [ 26] derive analytical reconstruction formu-\nlae as well as numerical reconstruction schemes for two- and three-dimensional MPI. They\nexamine and compare the ill-posedness of the reconstructio n problem for different dimen-\nsions. A mathematical analysis of the 1D model is provided by Erb, Weinmann et al. [ 7].\nThey investigate properties like the ill-posedness and dis cover an exponential singular value\ndecay of the reconstruction problem. Goodwill and Conolly [ 10] follow the X-space ap-\nproach. They consider the dependence of the spatial positio n of the field free point (FFP),\nwhich is the time-dependent volume of vanishing magnetic fie ld strength, and the drive field\nof the scanner. As a result the forward operator of 1D MPI can b e identified as a convolu-\ntion operator. The authors extend their approach to multipl e dimensions in [ 11]. The more\nanalytically focused article by Maass and Mertins deals wit h closed-form expressions for\nthe Fourier transform of the system function for multiple di mensions [ 25]. The system func-\ntion is related to tensor products of Chebychev polynomials of the second kind and tensor\nproducts of Bessel functions. This result might allow for an alytical insights into the system\nfunction and more efficient reconstruction techniques in th e future.Modeling Magnetic Particle Imaging for Dynamic Tracer Dist ributions 3\nAnother common model simplification is the assumption of ide al magnetic fields. In prac-\ntice magnetic fields can be distorted which is influencing spa tial signal encoding. In [ 5],\nthe authors use spherical harmonics to achieve more realist ic representations of magnetic\nfields for FFP and field free line (FFL) scanners. A 3D forward m odel for non-ideal mag-\nnetic fields which can be reconstructed with the algebraic re construction technique (ART)\nis presented. Artifacts caused by distorted magnetic fields have also been investigated in the\ncontext of the X-space approach [ 32]. These distortions especially affect multi-patch MPI,\nsince distortions increase with the distance from the cente r. In the case of measurement-\nbased reconstruction, compensation methods for displacem ent artifacts in multi-patch scans\nare studied in [ 4,3].\nIn this paper we present an extended MPI forward model for dyn amic tracer distributions,\nin the discrete and continuous case, both in time- and Fourie r domain. While the initial the-\noretical setup presented in Sec. 3is identical to the one in [ 8], our model is not limited to\nperiodic motion and covers dynamic tracer distributions wi th high velocities. Furthermore,\nwe provide simulation experiments to show the importance of the extension relative to the\ntracer dynamics and the impact on reconstruction quality co mpared to the static model. The\npresented approach is of special interest for blood flow meas urements [ 16] because the speed\nof the motion is part of the model and can be reconstructed sim ultaneously.\nThe remainder of this paper starts with a brief introduction to the principles of an ideal\nMPI system in Sec. 2and is followed by presenting the standard modeling approac h in\nSec.2.1which we will extend to arbitrary dynamic tracer distributi ons in Sec. 3. Based on the\nLangevin model, FFP scanners and Lissajous trajectories we investigate the influence of the\nextension to the signal for different kinds of dynamics in Se c.4, while we compare recon-\nstructions of simulated dynamic measurements with the stan dard and the extended model in\nSec.5. We close with a discussion of the results in Sec. 6.\n2 Basic principles of magnetic particle imaging\nThe aim of magnetic particle imaging is the reconstruction o f the multi-dimensional spatial\nconcentration of the particles. Spatial encoding of the sig nal is realized by applying a spa-\ntially and temporally varying magnetic field H∈L2(Ω×R+,R3)where Ω⊂R3denotes\nthe field of view (FOV). The magnetic field\nH(r,t)=HS(r)+HD(t)\nconsists of a spatially inhomogeneous selection field HSand a temporally varying drive\nfield HD. The selection field is a gradient field that has a point of zero field strength in the\ncenter, the so-called field free point (FFP), and a linearly i ncreasing field strength to the pe-\nriphery (see Fig. 1a). The drive field HD(t)=[ alsin(2πflt+ϕl)]l=1,...,3is spatially constant\nbut changes its magnetization over time according to a sine f unction in each dimension. It\nhas three parameters per dimension, the amplitude al∈Rdetermining the size of the field of\nview, the frequency fl∈Rdefining the density of the scan trajectory and a phase shift ϕl∈R\nsetting the starting point of the scan trajectory. Choosing the parameters appropriately, the\noverlay of HSandHDforms the field Hwhich has a FFP moving through the volume of\ninterest along a so-called Lissajous curve (see Fig. 1b).\nSuperparamagnetic means that the magnetic nanoparticles b ehave like tiny magnets, while\nan external magnetic field is applied. They have their own mag netic moments which are\nlarger than their atomic moments. There is no remanent magne tization after the applied\nmagnetic field is removed [ 2].4 Christina Brandt, Christiane Schmidt\n(a) The selection field Hsis a static gradient field\nwith vanishing field strength in the center and lin-\nearly increasing field strength to the border of the\nfield of view.(b) The trajectory of the field free\npoint forms a Lissajous curve dur-\ning a single scan cycle.\nFig. 1: Setup of the magnetic field\nThe magnetic moment of the particles responds to temporal ch anges of magnetic fields.\nThere are different models describing the magnetic behavio r of the particles which where\nstudied in [ 17,19,29]. In Sec. 4and5the Langevin or equilibrium model is used but could\nbe replaced by more complex models. When the field free point m oves over a position r,\nit causes a change in the mean magnetic moment at this locatio n. The magnitude of the\nmagnetization\nM(r,t)=c(r)¯m(r,t), M∈H1(Ω×R+),c∈L2(Ω), (1)\nis proportional to the tracer concentration c:Ω/mapsto→R+and the mean magnetic moment\n¯m:Ω×R+/mapsto→R3. The Sobolev space\nH1(D)={f∈L2(D):Dγf∈L2(D)for 0≤|γ|≤1,γ∈Nd,D⊂Rd}\ndenotes the space of L2-functions whose first weak derivatives are also functions i nL2on the\ndomain D⊂Rdwith γ∈Ndbeing a multi-index with |γ|=∑d\ni=1γi, see [ 1]. The change in\nmagnetization induces a current in the receive coils of the s canner. Due to the construction of\nthe magnetic field, a measured voltage at time point tcan be connected to a magnetization\nchange and thus to a certain concentration at a position r. The measurement process is\ndescribed by a forward model in the following section.\n2.1 MPI forward model\nThe static forward model\nu(t)=−µ0d\ndt/integraldisplay\nΩp(r)/parenleftbig\nc(r)¯m(r,t)+H(r,t)/parenrightbig\ndr (2)\ndescribes the magetic particle imaging process in time doma in [20]. The sensitivity of the\nreceive coils pis multiplied with the permeability constant µ0and the change in magneti-\nzation which is caused by the superparamagnetic particles a nd the applied magnetic field HModeling Magnetic Particle Imaging for Dynamic Tracer Dist ributions 5\nfrom the send-coils.\nAssuming that the signal generated by the excitation field is removed by a filter yields\nu(t)=/integraldisplay\nΩS(r,t)c(r)dr, with S(r,t)=−µ0p(r)∂\n∂t¯m(r,t). (3)\nAs defined in the previous section, the excitation field is mul ti-dimensional which means\nthat the system function S(r,t)maps toR3, thus u(t)∈R3is a voltage vector, where each\nvalue is measured by the respective receive coil. In the disc rete models in the remainder of\nthis paper we will refer to a single component of u(t)since the computations are analogous\nfor all channels.\nDiscretization The formulation of discrete forward models is motivated by m easured sys-\ntem matrices and the use of numerical reconstruction method s. Therefore, we use a basis\n{φi}i=1,...,R⊂L2of a finite-dimensional subspace XR⊂L2. An intuitive choice are piece-\nwise constant basis functions on equisized, pairwise disjo int quadratic or cubic domains as\nthey are a reasonable representation of both the pixels or vo xels in an image and the delta\nprobe used for the calibration scans.\nUsing the basis functions, we obtain piecewise constant app roximations of the concentration\nand system function\n˜c(r)=R\n∑\ni=1ciφi(r)and ˜S(r,t)=R\n∑\ni=1Si(t)φi(r)∈XR,\nevaluated at equidistant time sampling points {tj}j=1,...,nTwith tj=(j−1)Tc/(nT−1)and\nTcbeing the repetition time for one Lissajous cycle. Insterti ng ˜cin (3) yields the following\ndiscrete forward problem\nu(tj)=R\n∑\ni=1Si(tj)ci,j=1,..., nT. (4)\nIt can also be written as a matrix vector multiplication of a c oncentration vector and the\nsystem matrix S\nu=Sc with u∈CnT,S∈CnT×R,c∈RR. (5)\nReconstructing the concentration vector cfrom a given measurement vector uis a classic\ninverse problem. In [ 18,26], it was shown that the multidimensional MPI reconstructio n\nproblem is severely ill-posed. Thus, computing a stable and unique solution requires regu-\nlarization. Two common regularization methods in MPI are Ti khonov- and iterative regular-\nization. The former defines a Tikhonov functional by adding a penalty term with a regular-\nization parameter. The resulting minimization problem can then be solved by the Kaczmarz\nalgorithm or other iterative schemes adapted to the applied regularization term. The latter\noption regularizes the iterative method directly by choosi ng a maximum number of itera-\ntions. In both cases, the standard iterative method used in M PI is the Kaczmarz algorithm\n[15]. The information from reconstructions of several channel s can be combined to improve\nimage quality.6 Christina Brandt, Christiane Schmidt\n3 Dynamic forward model\nThe forward model presented in the preceding section underl ies the assumption of a (nearly)\nstatic concentration during the signal acquisition which m ight be violated in case of dynam-\nically changing tracer distributions. MPI is able to visual ize the distribution of a liquid tracer\nwhich can accumulate, dissipate or move e.g. with the blood fl ow. In these situations, the\nbehavior of the particle concentration is clearly not stati c.\nIn practice oftentimes measured system matrices are used fo r MPI reconstruction. These\nmatrices are the results of calibration scans which measure the induced voltage of a delta\nsample during a scanning cycle for each spatial position. Th is approach yields good results\nfor static concentrations because the matrices also incorp orate the transfer function of the\nsystem. Since the delta sample is static during the complete cycle the measured system ma-\ntrix does not cover dynamic behavior.\nIn order to adapt the model to dynamic tracer concentrations , the magnetization function ( 1)\nis modified such that it contains a time-dependent concentra tion\nM(r,t)=c(r,t)¯m(r,t), c∈H1(Ω×R+),M,¯m∈H1(Ω×R+). (6)\nThus, the forward model ( 2) changes to\nu(t)=−µ0d\ndt/integraldisplay\nΩp(r)/parenleftbig\nc(r,t)¯m(r,t)+H(r,t)/parenrightbig\ndr.\nAssuming a constant coil sensitivity pand that the signal generated by the excitation field is\nremoved by a filter results in the dynamic forward model\nu(t)=ηd\ndt/integraldisplay\nΩc(r,t)¯m(r,t)dr, with η:=−µ0p∈R (7)\n=η/integraldisplay\nΩ∂¯m\n∂t(r,t)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=S1c(r,t)+¯m(r,t)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=S2∂c\n∂t(r,t)dr (8)\n=η/integraldisplay\nΩS1(r,t)c(r,t)+S2(r,t)∂c\n∂t(r,t)dr, c,S2∈H1(Ω×R+). (9)\nIt describes a measurement u:R+/mapsto→R3in time domain and contains a sum of two system\nfunctions S1,S2:Ω×R+/mapsto→R3multiplied with the tracer concentration c:Ω×R+/mapsto→R+\nand its time derivative. The derivatives∂c\n∂t,S1=∂¯m\n∂tand the measurement uareL2-functions\nbecause the concentration candS2=¯mare in the Sobolev space H1(Ω×R+).\nDynamic forward model in frequency domain MPI measurements are usually given in fre-\nquency domain. Due to the time dependence of the concentrati on the static model\nˆu(k)=η/integraldisplay\nΩc(r)F/braceleftBig∂¯m\n∂t/bracerightBig\n(r,k)dr\nin frequency domain changes to\nˆu(k)=η/integraldisplay\nΩF/braceleftbig\nc/bracerightbig\n(r,k)∗F/braceleftBig∂¯m\n∂t/bracerightBig\n(r,k)+F/braceleftBig∂c\n∂t/bracerightBig\n(r,k)∗F/braceleftbig\n¯m/bracerightbig\n(r,k)dr, (10)\n=η/integraldisplay\nΩˆc(r,k)∗ˆS1(r,k)+/hatwider∂c\n∂t(r,k)∗ˆS2(r,k)dr, ˆc,ˆS2∈H1(Ω×R+). (11)Modeling Magnetic Particle Imaging for Dynamic Tracer Dist ributions 7\nThe measurement in frequency space ˆ u:R+/mapsto→C3and the derivatives/hatwider∂c\n∂t:Ω×R+/mapsto→C,\nˆS1:Ω×R+/mapsto→C3areL2-functions because ˆ c,ˆS2∈H1(Ω×R+). The convolution is only\napplied to the frequency components.\nDiscretization Using the same pixel-basis {φi}i=1,...,R⊂L2for discretization as in Sec. 2.1\nyields the following representation of a piecewise constan t dynamic concentration\n˜c(r,t)=R\n∑\ni=1ci(t)φi(r)∈XR\nand analogously for the derivative∂c\n∂tand system functions S1andS2. Together with the time\nsampling points {tj}j=1,...,nTfrom Sec. 2.1, we get the discretized dynamic forward problem\nu(tj)=ηR\n∑\ni=1S1,i(tj)ci(tj)+S2,i(tj)/parenleftBig∂c\n∂t/parenrightBig\ni(tj),j=1,..., nT. (12)\nFor measurements with F≥1 cycles the time sampling for the measurement and concentra -\ntion changes to {τj}j=1,...,FnTwith τj=(j−1)FTc/(FnT−1)while the system functions\nare evaluated at tjmod nT. Eq.( 12) is no longer a matrix-vector multiplication as in ( 5) but a\nsum of element-wise matrix multiplications\nu=η/bracketleftbig\nS1⊙c+S2⊙Dc/bracketrightbig\n·1R\nwith u∈RnT,c,Dc∈RnT×R,S1,S2∈RnT×R,1R=[1,..., 1]⊤∈NRand⊙being an element-\nwise matrix multiplication, e.g. A⊙B=[ai jbi j]j=1,...,m\ni=1,...,n, with matrices A,B∈Rn×m.\nIn frequency space the same approach yields the following di screte forward problem\nˆu=η/bracketleftbigˆS1∗ˆc+ˆS2∗ˆDc/bracketrightbig\n·1R\nwith ˆu∈CnK,ˆc,ˆDc∈CnK×R,ˆS1,ˆS2∈CnK×R,1R=[1,..., 1]⊤∈NR. Again the convolution\nis only applied to the frequency components, i.e. the respec tive matrix columns. Note that\na frequency domain reconstruction computes ˆ c. To see the behavior of the concentration in\ntime, the inverse Fourier transform needs to be applied. Rec onstruction becomes a decon-\nvolution problem in frequency space. A typical solution app roach for this ill-posed inverse\nproblem is to make use of the convolution theorem of the Fouri er transform which in this\ncase results in time domain reconstruction.\nThe dynamic model ( 8) is also mentioned in [ 8] but followed by strong restrictions of the\ndynamics such that there are no further consequences in the r econstruction process. In con-\ntrast, the models proposed in this section are valid for a bro ad range of dynamics, e.g. rapid\nchanges or non-periodic behavior. The tracer distribution is required to be differentiable in\ntime and integrable in space.\n4 Relevance of the dynamic model\nAs mentioned in Sec. 2.1, the concentration is usually assumed to be constant. The ti me\nderivative of the concentration would therefore be nearly z ero and the second summand of\nthe extended model ( 9) would thus be small such that it can be neglected.\nWe investigate the structure of S2in comparison to S1and consider a set of simulated dy-\nnamic concentrations to survey whether neglecting the seco nd term in the new model may8 Christina Brandt, Christiane Schmidt\nTable 1: Physical parameters used for the simulations\nParameter Value cf.\nConstants\nPermeability constant µ0 4π·10−7N/A2\nBoltzmann constant kB 1.38064852 ·10−23J/K\nParticles\nTemperature T 310 K [ 22]\nSaturation magnetization MC0.6\nµ0T [ 22]\nParticle core diameter D 20·10−9m [ 6]\nParticle core volume VC1\n6πD3m3[22]\nParticle magnetic moment α MCVCAm2[22]\nParameter of Langevin function β (kBT)−1N−1m−1[22]\nScanner [23]\nExcitation frequencies [fx,fy,fz] [ 2.5/102,2.5/96,2.5/99]MHz\nExcitation amplitudes [ax,ay,az] [ 0.012,0.012,0.0]T\nExcitation phase shifts [ϕx,ϕy,ϕz] [π\n2,π\n2,π\n2]\nGradient strengths [gx,gy,gz] [−1,−1,2]T/m\nExcitation repetition time Tc 652.8·10−6s\nbe justified. Before looking at example reconstructions to c ompare reconstructions with the\nold and extended model in Sec. 5, we are looking at the influence of the new summand in the\ndynamic model.\nFor the simulation, the system functions with 19 ×19×1 voxels and 1632 sampling points\nin time are modeled according to the Langevin model using the parameters listed in Tab. 1.\nLangevin model In the Langevin model the particles are assumed to be in therm al equi-\nlibrium and the applied magnetic fields to be static. The mean magnetic moment at spatial\nposition rand time point tis given by\n¯m(r,t)=Lα,β(/bardblH(r,t)/bardbl2)H(r,t)\n/bardblH(r,t)/bardbl2, ¯m∈H1(Ω×R+),Lα,β∈H1(R),\nwithLα,β:R/mapsto→Rbeing the Langevin function\nLα,β(z)=/braceleftBigg\nαcoth(αβz)−1\nβz,ifz/ne}ationslash=0\n0, ifz=0\nwith α,β∈Rbeing particle dependent parameters.\nEq.(11) shows a sum of two convolutions. In a first step, we are intere sted in the shape of\nthe convolution kernels. Therefore, we compute\nmax\nr∈Ω{|ˆSl(r,k)|}, l∈{1,2}\nwhich are shown in Fig. 2together with an approximation of their convex hulls. The ap -\nproximation of the convex hull was calculated by determinin g and connecting the maximum\nvalues within the next 15 frequency steps to include all peak s of the function. Both matrices\nexhibit a similar structure. The peaks have the same distanc es (≈15 frequency steps) andModeling Magnetic Particle Imaging for Dynamic Tracer Dist ributions 9\nthe convex hull (orange line) has a similar shape. The full-w idth-at-half-maximum (FWHM)\nof the convex hull is the same ( ≈33 frequency steps) while the maximum of the second sys-\ntem matrix ˆS2is 104smaller than the maximum of the first system matrix ˆS1. Thus, on a first\nview, the assumption that the second term S2in the dynamic model is negligible might be\nreasonable.\n(a) Shape of system matrix 1 in Fourier\nspace\n(b) Shape of system matrix 2 in Fourier\nspace\nFig. 2: Comparing the influence of the two system matrices by a nalyzing the maximum\nover all voxels of the absolute values of the matrices in Four ier space. The orange line is an\napproximation of the convex hull.\nIn a second step, we looked at four types of dynamic concentra tions during one cycle for a\nsingle voxel. The plots in Fig. 3show the concentration over the scan time, its time derivati ve\nand the respective Fourier transforms for each example conc entration.\nExample concentration 1 is depicted in Fig. 3awhich shows one peak at the beginning of\nthe scan. The tracer is flowing through the voxel for a short pe riod of time. This could be\na small tracer bolus moving fast through the volume of the vox el. In the second example,\nshown in Fig. 3b, the concentration increases strongly in the beginning, re mains constant for\na short period of time and decreases again. The tracer flows th rough the voxel for a longer\nperiod of time. Example 2 represents a larger bolus moving fa st through the voxel. Example\n3, shown in Fig. 3c, shows a slow increase and decrease of the concentration. Th is represents\na slowly moving small bolus. Example 4 is a periodic version o f the first example. Fig. 3d\nshows two peaks within the scan time. The tracer flows two time s through the voxel with a\nhigh velocity. This represents a small bolus with fast perio dic motion.\nLooking at the Fourier transformations shows that the maxim al absolute values of the Fourier\ntransformed concentrations ˆ care about 104smaller than the maximal absolute values of the\nFourier transformed time derivatives/hatwidedc\ndtfor all 4 examples. This demonstrates that for these\ndynamic concentrations the magnitude of the two summands of the new dynamic model is\nthe same.\nThe imaging process in Fourier space is a convolution of the F ourier transformed system\nmatrices with the Fourier transformed concentration and it s time derivative. Thus, the con-\ncentration is smoothed by the system matrix. The kernels S1andS2have the same width\nmeaning that the concentration and its derivative are smoot hed equally.10 Christina Brandt, Christiane Schmidt\n(a) Example 1: The tracer flows through the voxel for a short pe riod of time\n(b) Example 2: The tracer flows through the voxel for a longer p eriod of time\nFig. 3: Example concentrations and their time derivatives i n time and frequency domain.\nThe dynamics appear within one scanning cycle.Modeling Magnetic Particle Imaging for Dynamic Tracer Dist ributions 11\n(c) Example 3: The tracer accumulates and dissipates slowly in the voxel\n(d) Example 4: The tracer flows through the voxel for a short pe riod of time for two times\nFig. 3: Example concentrations and their time derivatives i n time and frequency domain.\nThe dynamics appear within one scanning cycle.12 Christina Brandt, Christiane Schmidt\nTo further examine these effects, we split up the discrete fo rward model such that the signal\nˆu(kj)=ηR\n∑\ni=1ˆS1(ri,kj)∗ˆc(ri,kj)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=a(i,j)+ηR\n∑\ni=1ˆS2(ri,kj)∗/hatwider∂c\n∂t(ri,kj)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\n=b(i,j)\n=ηR\n∑\ni=1a(i,j)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nA+ηR\n∑\ni=1b(i,j)\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nB, j=1,..., nK,\nis now the sum of AandB, where Adenotes the signal component generated by the first\nsystem matrix S1andBthe signal component generated by the second system matrix S2.\nThe convolution of the frequency components of system matri x 1 and the tracer distribution\nis named aand the convolution of system matrix 2 with the derivative of the concentration\nis named b.\n(a)a=ˆS1(ri,kj)∗ˆc(ri,kj)\n(b)b=ˆS2(ri,kj)∗/hatwider∂c\n∂t(ri,kj)\nFig. 4: Absolute values of the convolution of the system matr ices with the concentration and\nits derivative. Each curve shows the frequency amplitudes f or one of the 192voxels.\nUsing the dynamic example concentration 3 shown in Fig. 3c, Fig. 4shows aandbfor each\nvoxel. As expected, one can see that the shape and the maximum values of both terms are\nsimilar. Both plots show maximum values of about 1 .2·10−4. Fig. 5shows the plots of A\nandBfor the frequencies k∈[0.08,1.25]MHz. Again, both terms have the same order of\nmagnitude. For Athe frequencies with high amplitudes have a small variance w hile for B\nhigh amplitudes can be observed in the whole frequency range . This shows that even for this\nexample with slower dynamics the second component of the for ward model has a significant\nimpact on the signal. Thus, the second summand in Eq.( 9) should not be generally neglected\nfor dynamic concentrations.\n5 Comparing reconstructions with the dynamic and static mod el\nThe challenge in solving the dynamic inverse problem ( 12) is the high number of degrees\nof freedom. We therefore use a minimalist setup with a grid of 3×3×1 voxels. We use\ntwo computational phantoms to simulate an MPI measurement w ith the dynamic model and\nreconstruct it with the dynamic and the static model. They ar e named one-peak phantom andModeling Magnetic Particle Imaging for Dynamic Tracer Dist ributions 13\n(a) Signal part A=∑R\ni=1ˆS1(ri,kj)∗ˆc(ri,kj)\ngenerated by the first system matrix(b) Signal part B=∑R\ni=1ˆS2(ri,kj)∗\n/hatwider∂c\n∂t(ri,kj)generated by the second system\nmatrix\nFig. 5: Comparison of the two summands of the dynamic forward model ˆ u(kj)=η(A+B).\nA frequency selection with k∈[0.08,1.25]MHz is shown.\nthree-peak phantom and their spatial setup can be seen in Fig. 6.\nr1\nr2r7\nr8\nr9 r3r4\nr5\nr6\nFig. 6: Spatial setup of the One- and three-peak phantom. The y consist of 3 ×3×1 voxels\nindexed from 1 to 9.\n5.1 Parameterized concentration curve\nTo reduce the degrees of freedom and computational cost, the tracer concentration of the\none-peak phantom is described by parametric curves\nc(ri,t)=∑\nmbm,riψm(t)=c(λB(ri),t)∈L2(R3)×C2(R)\nfori=1,..., R, with parameter set λB=[bm,ri]m=1,...,M\ni=1,...,R,M∈Nand basis functions ψmwhich\nare cubic B-splines. This means that for each voxel rithere is a set of coefficients bmwhich\ntogether with the spline basis form a continuous concentrat ion curve in time. Consequently,\nwe assume that the concentration is twice differentiable wi th respect to twhich is a stronger\ncondition than previously assumed in the dynamic model.\nCubic B-spline curves are well suited to model the dynamics o f the magnetic tracer. In [ 12],\nthe authors deal with the reconstruction of spatiotemporal tracer distributions in Single Pho-\nton Emission Computed Tomography (SPECT). Cubic B-spline c urves are used to describe14 Christina Brandt, Christiane Schmidt\nand reconstruct the dynamic distribution of the radioactiv e tracer from gated cardiac SPECT\nsequences. We generate three variants of the one-peak phant om 1F, 2F and 4F. For one-peak\nTable 2: Reconstruction parameters\nParameter Value\nV oxel size (phantoms and system matrices) 0 .0107×0.0107×0.0107 m3\nField of view [0.0320,0.0320,0.0107]m\nTime sampling per cycle nT 408\nTransition time between frames ∆f 0 s\nNumber of frames Sec. 5.1 F 4\nNumber of frames Sec. 5.2 F 10\nphantom 1F the coefficients for all R=9 voxels except r5are zero. The concentration is\nnon-zero within the scan time of one frame. Fig. 7ashows the development of the tracer\ndistribution for the total scan time where each curve descri bes the concentration within one\nvoxel. The plot shows a peak at t=0.4128ms with a concentration of 2 .67 for voxel r5.\nVersions 2F and 4F differ only in the width of the concentrati on peak. The concentration\npeak for r5lasts for the scan time of 2 frames in version 2F and 4 frames in version 4F (see\nFig.7b). The three variants can be related to boluses with differen t velocities. The bolus in\nversion 1F is twice as fast as in 2F and four times faster than i n version 4F. Measurements\n(a) The tracer distribution of the central\nvoxel r5of one-peak phantom 1F changes\nduring the scan of the first frame. The remain-\ning voxels have a constant tracer concentra-\ntion of zero.\n(b) The tracer distribution of the central\nvoxel r5of the three versions of the one-peak\nphantom.\nFig. 7: The three versions of the one-peak phantom differ onl y in the width of the concen-\ntration peak of voxel r5, while the remaining voxels have a constant tracer concentr ation of\nzero.\nwith 4 frames which are each sampled at 408 time points and the dynamic forward model\nwith S1=∂¯m\n∂tandS2=¯mare simulated according to ( 12). The parameters used for the sim-\nulations are listed in Tab. 1and Tab. 2. There is no transition time in between the frames.\nWe reconstruct the concentrations by minimization with res pect to the parameter set λB\nso that we get continuous concentration curves. The solutio n set is restricted to parametricModeling Magnetic Particle Imaging for Dynamic Tracer Dist ributions 15\n(a) Reconstruction of 1F with the dynamic\nmodel using S1andS2.\n(b) Reconstruction of 1F using only S1.\n(c) Reconstruction of 2F with the dynamic\nmodel using S1andS2.\n(d) Reconstruction of 2F using only S1.\n(e) Reconstruction of 4F with the dynamic\nmodel using S1andS2.\n(f) Reconstruction of 4F using only S1.\nFig. 8: Measurements of the dynamic one-peak phantoms 1F, 2F and 4F are simulated with\nthe dynamic forward model ( 12). They are reconstructed with either both S1andS2(left)\nor only S1(right). All plots show averages of x- and y-channel reconst ructions. The dashed\nlines outline the true concentration in voxel r5and the vertical grid lines mark the start and\nend of frames.16 Christina Brandt, Christiane Schmidt\nspline curves in L2(R3)×C2(R)which is an implicit regularization. The curves are re-\nconstructed with two different settings. In the first experi ment both matrices are used for\nreconstruction which corresponds to minimizing\nmin\nλB/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftBigR\n∑\ni=1S1(ri,tj)c(λB(ri),tj)+S2(ri,tj)∂c\n∂t(λB(ri),tj)/bracketrightBig\nj=1,...,nT−u/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\nwith u∈RnT. The problem is minimized with 200 iterations of a conjugate gradient algo-\nrithm and no further regularization. Fig. 8ashows the average of reconstructions of the x- and\ny-channel of one-peak phantom 1F. The peak for voxel r5is located at t=0.4448ms with\na concentration of 2 .96 which is very close to ground truth. In the same period of ti me also\nthe concentration of the remaining voxels is non-zero. The p eaks of the voxels with even\nindices have concentration values of about 0 .9 and the peaks of the voxels with odd indices\nhave even smaller values of about 0 .4. Even if these voxels have a non-zero concentration,\nit is significantly lower than the value of r5, so that we can expect sufficient contrast in\nthe reconstructed images. The values for the off-diagonal v oxels (voxels with even indices,\ncf. Fig. 6) show higher concentration values than the ones on the diago nal. The x-channel\nreconstruction locates the concentration correctly in x an d the y-channel reconstruction lo-\ncates the concentration correctly in y. Thus, the off-diago nal voxels are masked by the high\nconcentration in the central voxel.\nIn the next experiment the same measurement is reconstructe d only with S1which corre-\nsponds to minimizing\nmin\nλB/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftBigR\n∑\ni=1S1(ri,tj)c(λB(ri),tj)/bracketrightBig\nj=1,...,nT−u/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\nwith u∈RnT. The problem is again minimized with 200 iterations of a conj ugate gradient\nalgorithm and no further regularization. The result for the average of x- and y-channel recon-\nstructions is shown in Fig. 8b. The reconstruction shows a peak for voxel r5att=0.3808ms\nwhich is close to the ground truth but with a significantly sma ller concentration of 0 .85.\nAgain there are concentration peaks for all remaining voxel s with values of about 0 .4. This\nmeans that the reconstructed images will show reduced contr ast. And the true concentration\nis underestimated.\n(a) Phantom\n (b) Reconstruction\nusing S1andS2\n(c) Reconstruction using\nonly S1\nFig. 9: Reconstructions of the one-peak phantom 1F at the tim e point of the concentration\npeak t=0.4128ms\nTo get a more intuitive impression of the impact of the discus sed curves on the reconstruc-\ntion quality, Fig. 9shows a frame of the phantom and the two reconstructed time-s eries atModeling Magnetic Particle Imaging for Dynamic Tracer Dist ributions 17\nthe time point of the maximum concentration ( t=0.4128ms). Looking at the image of the\nfirst experiment in Fig. 9bone can observe a good contrast and slightly higher concentr ation\nvalues for the off-diagonal voxels. As discussed above, the image from the second exper-\niment shown in Fig. 9cexhibits poor contrast and significantly lower concentrati on values\ncompared to the phantom.\n(a) Phantom\n (b) Reconstruction\nusing S1andS2\n(c) Reconstruction using\nonly S1\nFig. 10: Reconstructions of one-peak phantom 4F at the time p oint of the concentration peak\nt=1.304ms\nVersion 2F and 4F of the one-peak phantom are reconstructed a nalogously to version 1F.\nWhile also for version 2F reconstructed with both matrices t he concentration peaks show\nthe correct location and 83% of the true amplitude (see Fig. 8c), the peak in the reconstruc-\ntion using only S1is less than 50% of ground truth (see Fig. 8d). The reconstructions for\nversion 4F with and without S2are shown in Fig. 8eand8f. The location is correct for both\nreconstruction methods and the amplitude of the peak for r5reaches 87% of the ground truth\nfor reconstruction with both matrices and 73% for using only S1. Also the concentrations\nfor the remaining voxels are sufficiently low in both cases. L ooking at a frame of one-peak\nphantom 4F and the two reconstructed time-series at the time point of the maximum concen-\ntration ( t=1.304ms) in Fig. 10, one can see that the contrast in the reconstruction without\nS2(see. Fig. 10c) is improved compared to version 1F and almost comparable to the recon-\nstruction using S1andS2(see. Fig. 10b).\nFig. 11: Simplified bolus moving through a voxel with constan t velocity v.\nIn order to get an impression on the strength of the dynamics i n one-peak phantom 4F, we\nrelate these values to a 2 ×2×2 mm3bolus with constant concentration cmaxmoved through\na 2×2×2 mm3voxel with a constant velocity v(see Fig. 11). The one-peak phantom has\na maximum concentration of cmax=2.667 and a maximum time derivative of ˙ cmax=3065.\nThe average change rate yields a bolus velocity of vav=2·10−3/(2·Tc)=1.53 m/s and the\nmaximum change rate results in a velocity of vmax=2·10−3/(cmax/˙cmax)=2.3 m/s. Thus,\nwe state vdyn=1.53 m/s as a preliminary threshold velocity. For reconstruct ions with an18 Christina Brandt, Christiane Schmidt\naverage flow v>vdynthe dynamic model will improve the reconstruction quality i n com-\nparison to the static model.\n5.2 Frame-by-frame reconstruction\nAnother way to investigate the impact of the new model is to re construct a dynamic mea-\nsurement frame-by-frame with the assumption of a static con centration within each frame.\nTherefore we use the three-peak phantom.\nAs the one-peak phantom, the tracer concentration of the thr ee-peak phantom is described\nby parametric curves c(ri,t)=∑mbm,riψm(t)=c(λB(ri),t)with parameter set λBand basis\nfunctions ψmwhich are cubic B-splines meaning that for each voxel rithere is a set of co-\nefficients bmwhich together with the spline basis form a continuous conce ntration curve in\ntime.\nFor the three-peak phantom only the coefficients for the voxe lsr4,r5andr6are non-zero.\nThe tracer distribution during the total scan time is shown i n Fig. 12awhere each curve de-\nscribes the concentration within one voxel. There is a conce ntration peak of 6 .67 for voxel\nr4,r5andr6. The peaks are shifted in time, such that this dynamic can be s een as an object\nor tracer bolus that moves from voxel r4to voxel r6considering the location of the voxels in\nFig.6. The peaks are located in the scan time of frame 3, 4 and 5 and ha ve a temporal width\nof about 4 frames. The concentration of the remaining voxels is zero.\nA measurement with F=10 frames which are each sampled at 408 time points and the\ndynamic matrix model ( 12) with S1andS2is simulated. The parameters used for the simu-\nlation can be found in Tab. 1and Tab. 2.\nThe dynamic tracer distribution is reconstructed with two d ifferent settings. The first one\nuses information about the tracer dynamics from the reconst ructions of previous frames and\nthe second one reconstructs each frame independently. In fa ct the reconstructions are piece-\nwise constant functions over time. For better comparison th e results depicted in Fig. 12show\nlinear interpolations of the static reconstructions of 10 f rames.\nIn the first experiment both matrices are used for reconstruc tion of each frame while the time\nderivative∂c\n∂t=cf−cf−1\n∆tis the divided difference of the concentration vector of the current\nand the preceding frame. This corresponds to minimizing\nmin\ncf/vextenddouble/vextenddouble/vextenddouble/parenleftBig\nS1cf+S2cf−cf−1\n∆t/parenrightBig\n−uf/vextenddouble/vextenddouble/vextenddouble2\n2, f=1,..., F\nwith uf∈RnT,cf∈RR,S1,S2∈RnT×R. Fig. 12bshows the average of x- and y-channel\nreconstructions which were reconstructed in time domain wi th 100 iterations of a gradient\ndescent algorithm and no further regularization. It can be s een that the peaks are correctly\nlocated in frame 3, 4 and 5. The amplitude of the peaks is sligh tly lower than the ground truth\nand decreasing, 5 .41 for r4, 4.96 for r5and 4.82 for r6. There is a non-zero concentration\nfor the remaining voxels in the first 5 frames of less than 0 .5. So the reconstructed images\nwill exhibit sufficient contrast.\nIn the next experiment the same measurement is reconstructe d using only S1, i.e. minimizing\nmin\ncf/vextenddouble/vextenddoubleS1cf−uf/vextenddouble/vextenddouble2\n2, f=1,..., F\nwith uf∈RnT,cf∈RR,S1∈RnT×Rin time domain with 100 iterations of a gradient descent\nalgorithm and no further regularization. The result is show n in Fig. 12c. Again the peaks areModeling Magnetic Particle Imaging for Dynamic Tracer Dist ributions 19\nlocated correctly in frame 3, 4 and 5. The amplitudes 5 .47 for r4, 5.07 for r5and 5.49 for r6\nare also slightly lower than in the phantom and differ less th an in the first experiment. The\nremaining voxels show non-zero concentrations up to 1 .2 being more than twice as high as\nfor the first experiment.\n(a) The concentration of the three-peak phan-\ntom changes in time only in voxel r4,r5and\nr6. In the remaining voxels the concentration\nis zero. The time-shifted concentration peaks\nform a motion from r4tor6.\n(b) Average of x- and y-channel frame-\nby-frame reconstructions with the dynamic\nmodel using S1,S2\n(c) Average of x- and y-channel frame-by-\nframe reconstructions with the static model\nusing only S1\nFig. 12: A measurement of the dynamic three-peak phantom is s imulated with the dynamic\nforward model ( 12). Each frame is reconstructed separately assuming a static tracer dis-\ntribution within each frame. The frames are reconstructed w ith the dynamic and the static\nmodel.\n6 Discussion and conclusion\nWe introduced a new extended forward model for dynamic magne tic particle imaging. It\nwas shown that the standard forward model does not account fo r dynamic tracer distribu-\ntions which is corrected by the extended model presented in t his paper.20 Christina Brandt, Christiane Schmidt\nOne of the main differences is that the dynamic model contain s a second summand with\na second system matrix. For different kinds of dynamic conce ntrations the two summands\nhave been examined. The order of magnitude of the summands is the same for the chosen\ndynamic examples. This emphasizes the importance of the new model for dynamic tracer\ndistributions.\nFurthermore, we simulated measurements from dynamic conce ntrations with the extended\nmodel and reconstructed them with both the dynamic and the st atic model. In the experi-\nments in Sec. 5.1three simple phantoms with different change rates are exami ned. For one-\npeak 4F, the phantom with the lowest change rates, the static approach using only one system\nmatrix provided an acceptable reconstruction quality. For the phantoms with higher change\nrates, one-peak phantom 2F and 1F, the static approach resul ted in reconstructions with low\ncontrast and significantly lower amplitudes than ground tru th while the dynamic approach\nperformed well on all three phantoms. While a quantitative s tudy of this is beyond the scope\nof this article, we can state that for higher change rates tha n in one-peak phantom 4F the\ndynamic model should be considered for reconstruction.\nThe presented dynamic model is more general than existing ap proaches for dynamic con-\ncentration reconstruction as it is not limited to periodic m otion and can be applied to motions\nwith high velocities. While in this paper the equilibrium mo del is used, the dynamic model\nallows to incorporate more advanced magnetization models w hich could improve the recon-\nstruction quality in the future. The reconstruction approa ch with parametric concentration\ncurves features an implicit dynamic regularization. Addit ional spatial or temporal regular-\nization, like sparsity in time and space, can be included eas ily. Moreover, the model allows\nfor joint reconstructions of the particle concentration an d its time-derivative which might be\nof special interest for blood-flow diagnostics.\nIt remains future research to develop new reconstruction te chniques for dynamic tracer dis-\ntributions based on this model and extend it to multi-patch i maging sequences for larger\nvolumes. Furthermore, the methods need to be evaluated for s imulations with phantoms\nof realistic size and physical phantom measurements. Pract ice-oriented scenarios might re-\nquire improved minimization schemes for reconstruction. T he dynamic model might also\nbe combined with measurement-based reconstruction. While using a calibration scan for\nS1, the second system matrix can be modeled and corrected with t he transfer function. Al-\nternatively, S2might be learned from its measured time-derivative S1. In addition to our\npreliminary statement, a quantitative study of phantoms wi th different velocities is required\nfor a more precise proposition about when the dynamic model i s necessary based on the level\nof dynamics and the desired reconstruction quality. A furth er theoretical research direction\nis the analysis of features like the ill-posedness of the dyn amic reconstruction problem.\nDeclarations\nFunding No funds, grants, or other support was received.\nConflict of interest The authors have no conflicts of interest to declare that are r elevant to\nthe content of this article.\nAvailability of Data The data sets generated during the experiments are availabl e from the\ncorresponding author on reasonable request.Modeling Magnetic Particle Imaging for Dynamic Tracer Dist ributions 21\nCode availability The code used for the simulations is available from the corre sponding\nauthor on reasonable request.\nReferences\n1. Adams, R.A., Fournier, J.J.: Sobolev spaces. Elsevier (2 003)\n2. Bean, C., Livingston, u.D.: Superparamagnetism. Journa l of Applied Physics 30(4), S120–S129 (1959)\n3. Boberg, M., Knopp, T., M¨ oddel, M.: Reducing displacemen t artifacts by warping system ma-\ntrices in efficient joint multi-patch magnetic particle ima ging. International Journal on\nMagnetic Particle Imaging 6(2), 1–3 (2020). DOI 10.18416/IJMPI.2020.2009030. URL\nhttps://journal.iwmpi.org/index.php/iwmpi/article/v iew/292 . Inproceedings, multi-\npatch, artifact\n4. Boberg, M., Knopp, T., Szwargulski, P., M¨ oddel, M.: Gene ralized mpi multi-patch reconstruction using\nclusters of similar system matrices. IEEE Transactions on M edical Imaging 39(5), 1347–1358 (2020).\nDOI 10.1109/TMI.2019.2949171. Article, multi-patch, art ifact\n5. Bringout, G., Erb, W., Frikel, J.: A new 3D model for magnet ic particle imaging using realistic magnetic\nfield topologies for algebraic reconstruction. arXiv prepr int arXiv:2004.13357 (2020)\n6. Eberbeck, D., Dennis, C.L., Huls, N.F., Krycka, K.L., Gru ttner, C., Westphal, F.: Multicore magnetic\nnanoparticles for magnetic particle imaging. IEEE Transac tions on Magnetics 49(1), 269–274 (2012)\n7. Erb, W., Weinmann, A., Ahlborg, M., Brandt, C., Bringout, G., Buzug, T.M., Frikel, J., Kaethner, C.,\nKnopp, T., M¨ arz, T., et al.: Mathematical analysis of the 1D model and reconstruction schemes for\nmagnetic particle imaging. Inverse Problems 34(5), 055012 (2018)\n8. Gdaniec, N., Boberg, M., M¨ oddel, M., Szwargulski, P., Kn opp, T.: Suppression of motion artifacts\ncaused by temporally recurring tracer distributions in mul ti-patch magnetic particle imaging. IEEE\nTransactions on Medical Imaging 39(11), 3548–3558 (2020). DOI 10.1109/TMI.2020.2998910. UR L\nhttps://ieeexplore.ieee.org/document/9104762 . Article, multi-patch, artifact, opendata\n9. Gleich, B., Weizenecker, J.: Tomographic imaging using t he nonlinear response of magnetic particles.\nNature 435(7046), 1214–1217 (2005)\n10. Goodwill, P.W., Conolly, S.M.: The X-space formulation of the magnetic particle imaging process: 1-\nD signal, resolution, bandwidth, SNR, SAR, and magnetostim ulation. IEEE Transactions on Medical\nImaging 29(11), 1851–1859 (2010)\n11. Goodwill, P.W., Conolly, S.M.: Multidimensional x-spa ce magnetic particle imaging. IEEE Transactions\non Medical Imaging 30(9), 1581–1590 (2011)\n12. Gravier, E., Yang, Y ., Jin, M.: Tomographic reconstruct ion of dynamic cardiac image sequences. IEEE\nTransactions on Image Processing 16(4), 932–942 (2007)\n13. Gr¨ uttner, M., Knopp, T., Franke, J., Heidenreich, M., R ahmer, J., Halkola, A., Kaethner, C., Borgert,\nJ., Buzug, T.M.: On the formulation of the image reconstruct ion problem in magnetic particle imaging.\nBiomedical Engineering/Biomedizinische Technik 58(6), 583–591 (2013)\n14. Haegele, J., Rahmer, J., Gleich, B., Borgert, J., Wojtcz yk, H., Panagiotopoulos, N., Buzug, T.M.,\nBarkhausen, J., V ogt, F.M.: Magnetic particle imaging: vis ualization of instruments for cardiovascular\nintervention. Radiology 265(3), 933–938 (2012)\n15. Kaczmarz, S.: Angen¨ aherte Aufl¨ osung von Systemen line arer Gleichungen. Bull. Int. Acad. Pol. Sci.\nLett. Class. Sci. Math. Nat pp. 355–7 (1937)\n16. Kaul, M.G., Salamon, J., Knopp, T., Ittrich, H., Adam, G. , Weller, H., Jung, C.: Magnetic particle imag-\ning for in vivo blood flow velocity measurements in mice. Phys ics in Medicine & Biology 63(6), 064001\n(2018)\n17. Kluth, T.: Mathematical models for magnetic particle im aging. Inverse Problems 34(8), 083001 (2018)\n18. Kluth, T., Jin, B., Li, G.: On the degree of ill-posedness of multi-dimensional magnetic parti-\ncle imaging. Inverse Problems 34(9), 095006 (2018). DOI 10.1088/1361-6420/aad015. URL\nhttps://doi.org/10.1088/1361-6420/aad015\n19. Kluth, T., Szwargulski, P., Knopp, T.: Towards accurate modeling of the multidimensional MPI physics.\nInternational Journal on Magnetic Particle Imaging 6(2), 1–3 (2020). DOI 10.18416/IJMPI.2020.\n2009004. URL https://journal.iwmpi.org/index.php/iwmpi/article/v iew/318 . Inpro-\nceedings\n20. Knopp, T., Buzug, T.: Magnetic Particle Imaging: An Intr oduction to Imaging Principles and Scanner\nInstrumentation. Springer Berlin Heidelberg (2012)\n21. Knopp, T., Gdaniec, N., M¨ oddel, M.: Magnetic particle i maging: from proof of principle to preclinical\napplications. Physics in Medicine & Biology 62(14), R124–R178 (2017). DOI 10.1088/1361-6560/\naa6c9922 Christina Brandt, Christiane Schmidt\n22. Knopp, T., Sattel, T.F., Biederer, S., Rahmer, J., Weize necker, J., Gleich, B., Borgert, J., Buzug, T.M.:\nModel-based reconstruction for magnetic particle imaging . IEEE Transactions on Medical Imaging\n29(1), 12–18 (2009)\n23. Knopp, T., Szwargulski, P., Griese, F., Gr¨ aser, M.: Ope nMPIData: An initiative for freely accessible\nmagnetic particle imaging data. Data in Brief 28, 104971 (2020). DOI https://doi.org/10.1016/j.dib.2019 .\n104971. URL http://www.sciencedirect.com/science/article/pii/S2 352340919313265\n24. Ludewig, P., Gdaniec, N., Sedlacik, J., Forkert, N.D., S zwargulski, P., Graeser, M., Adam, G., Kaul,\nM.G., Krishnan, K.M., Ferguson, R.M., et al.: Magnetic part icle imaging for real-time perfusion imaging\nin acute stroke. ACS nano 11(10), 10480–10488 (2017)\n25. Maass, M., Mertins, A.: On the representation of magneti c particle imaging in fourier space. International\nJournal on Magnetic Particle Imaging 6(1) (2020)\n26. M¨ arz, T., Weinmann, A.: Model-based reconstruction fo r magnetic particle imaging in 2D and 3D. arXiv\npreprint arXiv:1605.08095 (2016)\n27. Rahmer, J., Weizenecker, J., Gleich, B., Borgert, J.: An alysis of a 3-d system function measured for\nmagnetic particle imaging. IEEE Transactions on Medical Im aging 31(6), 1289–1299 (2012)\n28. Storath, M., Brandt, C., Hofmann, M., Knopp, T., Salamon , J., Weber, A., Weinmann, A.: Edge pre-\nserving and noise reducing reconstruction for magnetic par ticle imaging. IEEE Transactions on Medical\nImaging 36(1), 74–85 (2017). DOI 10.1109/TMI.2016.2593954\n29. Weizenecker, J.: The Fokker–Planck equation for couple d Brown–N´ eel-rotation. Physics in Medicine &\nBiology 63(3), 035004 (2018)\n30. Weizenecker, J., Borgert, J., Gleich, B.: A simulation s tudy on the resolution and sensitivity of magnetic\nparticle imaging. Physics in Medicine & Biology 52(21), 6363 (2007)\n31. Weizenecker, J., Gleich, B., Rahmer, J., Dahnke, H., Bor gert, J.: Three-dimensional real-time in vivo\nmagnetic particle imaging. Physics in Medicine & Biology 54(5), L1 (2009)\n32. Yagiz, E., Cagil, A.R., Saritas, E.U.: Non-ideal select ion field induced artifacts in x-space mpi. Interna-\ntional Journal on Magnetic Particle Imaging 6(2) (2020)"    },    {        "title": "1907.12306v1.Magnetization_Dynamics_in_Holographic_Ferromagnets__Landau_Lifshitz_Equation_from_Yang_Mills_Fields.pdf",        "content": "Magnetization Dynamics in Holographic\nFerromagnets:\nLandau-Lifshitz Equation from Yang-Mills Fields\nNaoto Yokoi1;2\u0003, Koji Sato2y, and Eiji Saitoh1;2;3;4;5z\n1Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan\n2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n3Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n4Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\nAbstract\nWe introduce a new approach to understand magnetization dynamics in ferro-\nmagnets based on the holographic realization of ferromagnets. A Landau-Lifshitz\nequation describing the magnetization dynamics is derived from a Yang-Mills\nequation in the dual gravitational theory, and temperature dependences of the\nspin-wave sti\u000bness and spin transfer torque appearing in the holographic Landau-\nLifshitz equation are investigated by the holographic approach. The results are\nconsistent with the known properties of magnetization dynamics in ferromagnets\nwith conduction electrons.\n\u0003yokoi@spin.t.u-tokyo.ac.jp\nykoji.sato.c5@tohoku.ac.jp\nzeizi@ap.t.u-tokyo.ac.jparXiv:1907.12306v1  [hep-th]  29 Jul 20191 Introduction\nThe Landau-Lifshitz equation [1] is the fundamental equation for describing the dynamics\nof magnetization (density of magnetic moments) in various magnetic materials. It has been\nalso playing a fundamental role in the development of modern spintronics [2]: For instance,\nits extension to the coupled systems of localized magnetic moments and conduction electrons\nhas led to the concepts of spin transfer torque [3, 4] and spin pumping [5]. So far, the\nsymmetries and reciprocity in electronic systems have been the guiding principles to develop\nsuch extensions. In this article, we introduce another guiding principle to explore the new\nextensions and magnetization dynamics on the basis of the holographic duality.\nThe holographic duality is the duality between the quantum many body system de\fned\nind-dimensional space-time and the gravitational theory (with some matter \felds) which\nlives in (d+1)-dimensional space-time [6, 7, 8].1We constructed a holographic dual model of\nthree-dimensional ferromagnetic systems, which exhibits the ferromagnetic phase transition\nwith spontaneous magnetization and the consistent magnetic properties at low temperatures\n[10].2In the holographic duality, \fnite temperature e\u000bect in ferromagnetic systems can\nbe incorporated as the geometrical e\u000bect of black holes in higher dimensional bulk gravity,\nand the Wick rotation at \fnite temperatures is not required for the analysis in the dual\ngravitational theory. Thus, the novel analysis for real-time dynamics of quantum many\nbody systems in nonequilibrium situations can be performed using the holographic approach\n(for a review, see [14]). In addition, the holographic duality is known to be a strong-weak\nduality, which relates strongly correlated quantum systems to classical gravitational theories.\nFrom these viewpoints, the holographic approach can provide new useful tools to analyze\nnonequilibrium and nonlinear dynamics of magnetization in ferromagnets.\nIn ferromagnets, spin currents are generated by magnetization dynamics. From the holo-\ngraphic dictionary between the quantities of ferromagnets and gravitational theory [10], the\nspin currents in ferromagnets correspond to the SU(2) gauge \felds in the dual gravitational\ntheory. This correspondence indicates that the dynamics of spin currents, consequently the\ndynamics of magnetization, can be described by the Yang-Mills equation for SU(2) gauge\n\felds [15] in the holographic dual theory. In the following, we derive a Landau-Lifshitz\nequation for magnetization dynamics from the Yang-Mills equation within the holographic\nrealization of ferromagnets. This derivation can provide novel perspectives for magnetization\ndynamics from the non-abelian gauge theory.\n1See [9] for a recent review on the applications of the holographic duality to condensed matter physics.\n2Other holographic approaches to ferromagnetic systems have been also discussed in [11, 12, 13].\n2This article is organized as follows. In Section 2, we summarize the results of the mag-\nnetic properties obtained from the holographic realization of ferromagnets in thermodynamic\nequilibrium. An extension to nonequilibrium situation including the \ructuations of magne-\ntization and spin currents is discussed in the dual gravitational theory, and the holographic\nequation of magnetization dynamics is derived in Section 3. In Section 4, temperature depen-\ndences of the parameters in the resulting holographic equation are investigated by numerical\ncalculations. Finally, we summarize the results in Section 5.\n2 Holographic Dual Model of Ferromagnets\nWe begin with a brief summary on the holographic dual model of ferromagnets [10]. The\ndual model is the \fve-dimensional gravitational theory with an SU(2) gauge \feld Aa\nMand a\nU(1) gauge \feld BM, whose action is given by\nS=Zp\u0000gd5x\u00141\n2\u00142(R\u00002\u0003)\u00001\n4e2GMNGMN\u00001\n4g2Fa\nMNFaMN\n\u00001\n2(DM\u001ea)2\u0000V(j\u001ej)\u0015\n: (1)\nHere,Ris the scalar curvature of space-time, and the \feld strength is de\fned by Fa\nMN=\n@MAa\nN\u0000@NAa\nM+\u000fabcAb\nMAc\nNandGMN=@MBN\u0000@NBM, respectively. The index alabels\nspin directions in the SU(2) space (a= 1\u00183), the index Mlabels space-time directions in\n\fve dimensions ( M;N = 0\u00184), and\u000fabcis a totally anti-symmetric tensor with \u000f123= 1.\nThe model also includes a triplet scalar \feld \u001eawith the covariant derivative DM\u001ea=\n@M\u001ea+\u000fabcAb\nM\u001ec, and theSU(2)-invariant scalar potential V(j\u001ej) with the norm j\u001ej2=\nP3\na=1(\u001ea)2. Note that the scalar \feld is neutral under the U(1) gauge transformation. In\norder to guarantee asymptotic Anti-de Sitter (AdS) backgrounds, the negative cosmological\nconstant \u0003 =\u00006=`2is introduced. The \feld-operator correspondence in the holographic\nduality [7, 8] leads to the following holographic dictionary between the \felds of the dual\ngravitational theory and the physical quantities of ferromagnets:\nDual gravity Ferromagnet\nScalar \feld \u001ea() Magnetization Ma\nSU(2) gauge \feld Aa\nM() Spin current Ja\ns\u0016\nU(1) gauge \feld BM() Charge current J\u0016\nMetricgMN() Stress tensor T\u0016\u0017\nTable 1: Holographic dictionary between the dual gravitational theory and ferromagnets.\n32.1 Black Hole as Heat Bath\nIn order to establish the holographic dictionary, thermodynamical properties of the physical\nquantities of ferromagnets should be calculated in the dual gravitational theory. In Ref. [10],\nthe temperature dependences of magnetic quantities and the behavior of ferromagnetic phase\ntransition are thoroughly discussed. In the context of the holographic duality, \fnite tem-\nperature e\u000bects in the ferromagnets can be incorporated by introducing the black holes into\nthe dual gravitational theory as the background space-time. Indeed, the dual gravitational\ntheory has the charged black hole solution which is a solution to the Einstein, Yang-Mills,\nand Maxwell equations derived from the action (1):\nRMN+\u0012\n\u0003\u00001\n2R\u0013\ngMN=\u00142\n2e2\u0012\n2GKMGKN\u00001\n2GKLGKLgMN\u0013\n+\u00142\n2g2\u0012\n2Fa\nKMFaKN\u00001\n2FaKLFaKLgMN\u0013\n; (2)\nrMFaMN+\u000fabcAb\nMFcMN= 0;rMGMN= 0; (3)\nwhererMis the covariant derivative for the a\u000ene connection, and the space-time indices\nM;N are raised or lowered by the bulk metric gMN. Here, we neglect the contribution from\nthe scalar \feld and set \u001ea= 0 for the background. The metric of the black hole3is given by\nds2=gMNdxMdxN=r2\n`2\u0000\n\u0000f(r)dt2+dx2+dy2+dz2\u0001\n+`2\nf(r)dr2\nr2; (4)\nwith the radial function,\nf(r) = 1\u0000(1 +Q2)\u0010rH\nr\u00114\n+Q2\u0010rH\nr\u00116\n: (5)\nHere, we de\fne the parameter Q:\nQ2=2\u00142\n3\u0012\u00162\ne\ne2+\u00162\ns\ng2\u0013\n: (6)\nTheU(1) charge\u0016eandSU(2) charge\u0016sof the black hole are supported by the time com-\nponents of the gauge \felds,\nB0=\u0016e\u0010rH\n`\u0011\u0012\n1\u0000r2\nH\nr2\u0013\nandA3\n0=\u0016s\u0010rH\n`\u0011\u0012\n1\u0000r2\nH\nr2\u0013\n: (7)\nNote that the black hole solution (4) is asymptotically AdS at r!1 , and has the (outer)\nhorizonr=rH.\n3This type of non-abelian black holes has been discussed in the context of the holographic duality, in the\nliterature such as [11, 16].\n4For the following discussion, we make a coordinate change of the radial coordinate rinto\nubyu= 1=r, and the black hole metric becomes\nds2=1\nu2\u0012\n\u0000f(u)dt2+dx2+dy2+dz2+du2\nf(u)\u0013\n; (8)\nand the transformed function f(u) is given by\nf(u) = 1\u0000\u0000\n1 +Q2\u0001\nu4+Q2u6; (9)\nwhere we have set the coupling parameters e=g= 1 and the black hole parameters rH=\n`= 1, for simplicity.\nIn the holographic dual model, the black hole (8) plays the role of the heat bath; due to\nthe Hawking radiation, the black hole temperature is given by\nT=2\u0000Q2\n2\u0019; (10)\nand the calculations on the black hole background lead to the thermodynamical properties\nof the corresponding ferromagnet. Since we focus only on the dynamics of magnetization\nand spin current, the background space-time is \fxed to be the black hole metric (8) in the\nfollowing.\n2.2 Thermodynamics of Ferromagnets from Scalar Dynamics on Charged\nBlack Hole\nIn order to investigate the thermodynamics of magnetization, we examine the equation of\nmotion for the scalar \feld \u001ea, which is also derived from the action (1):\n1p\u0000g@M\u0000p\u0000gDM\u001ea\u0001\n+\"abcAb\nMDM\u001ec=@V\n@\u001ea: (11)\nHere, we consider a static and homogeneous solution in the boundary coordinates, x\u0016=\n(t;x1;x2;x3), which corresponds to the homogeneous magnetization in ferromagnets. With-\nout loss of generality, the ansatz for such a scalar \feld, which is invariant under the transla-\ntions on the boundary, is given by\n\u001e1=\u001e2= 0; \u001e3= \b(u)6= 0: (12)\nInserting this ansatz, the metric (8), and the gauge \felds (7) into the equation (11), we obtain\nthe following equation for \b( u):\nu2f(u)d2\b\ndu2+\u0012\nu2df(u)\ndu\u00003uf(u)\u0013d\b\ndu=@V\n@\b: (13)\n5This equation governs the thermodynamics of magnetization in the dual gravitational theory.\nWe can analyze the solution to this equation numerically with a simple quartic potential\nV(j\u001ej) =\u0015\u0000\nj\u001ej2\u0000m2=\u0015\u00012=4, and the asymptotic behavior of the numerical solution near the\nboundaryu\u00180 (orr\u00181) is obtained:\n\b(u)'H0u\u0001\u0000+M(T)u\u0001+\u0010\n\u0001\u0006= 2\u0006p\n4\u0000m2\u0011\n: (14)\nAccording to the standard recipe in the holographic duality [17, 18], the coe\u000ecients H0and\nM(T) in the asymptotic expansion correspond to an external magnetic \feld and a magne-\ntization at temperature T(underH0), respectively. In Ref. [10], the resulting temperature\ndependences of magnetization, magnetic susceptibility, and speci\fc heat have been shown\nto reproduce the ferromagnetic phase transition in the mean \feld theory. Furthermore, the\ntemperature dependences at low temperatures are also consistent with the existence of the\nspin wave excitations (magnons) and conduction electrons in low-temperature ferromagnets.\nFor later convenience, we also comment on the solutions for the gauge \felds. Assuming\nthe translational and rotational invariance on the boundary, equilibrium solutions for the\ngauge \felds are given by the following form:\nB0=b(u) andA3\n0=a3(u); (15)\nwhere all the other components vanish. Inserting this ansatz and (12), the Maxwell and\nYang-Mills equations on the black hole are reduced to the following simple forms:\nd\ndu\u00121\nudb\ndu\u0013\n= 0 andd\ndu\u00121\nuda3\ndu\u0013\n= 0: (16)\nThe general solutions are given by the forms (7) in terms of u,\nb(u) =\u0016e\u0000\n1\u0000u2\u0001\nanda3(u) =\u0016s\u0000\n1\u0000u2\u0001\n: (17)\nHere, we impose the boundary conditions B0= 0 andA3\n0= 0 at the horizon ( u= 1), which\nguarantee the regularity of the gauge \felds on the horizon. The remaining integral constants,\n\u0016eand\u0016s, correspond respectively to the electrochemical potential of underlying electrons\nand the spin chemical potential (or spin voltage), through the holographic dictionary.\nTo summarize, the solutions (14) and (17) on the charged black hole describe the ther-\nmodynamical property of the holographic dual ferromagnets in the equilibrium.\n3 Magnetization Dynamics in Holographic Ferromagnets\nIn this section, we extend the holographic analysis in the equilibrium, summarized in the\nprevious section, to more general situations including the dynamics of magnetization and\n6spin currents. In order to discuss the dynamics of magnetization and spin currents, the\nstatic and homogeneous ansatze for the scalar \feld (12) and the gauge \felds (15) need to be\ngeneralized. Here, we focus on the dynamics with the long wave length in the ordered phase\n(symmetry broken phase) below the Curie temperature, where various phenomena in modern\nspintronics are intensively studied.\n3.1 Generalized Ansatz and E\u000bective Equations of Motion\nFor the scalar \feld, following the standard derivation of the equation for magnetization\ndynamics, we consider the generalized ansatz for the scalar \feld as a factorized form:\n\u001ea(u;t;x ) = \b(u)na(t;x) with3X\na=1nana= 1; (18)\nwhere \b(u) is a solution of the equation (13) with the asymtotic behavior (14). Note that,\nsince we focus only on the dynamics of spontaneous magnetization, we \fx H0= 0 throughout\nthis article. In this ansatz, na(t;x) corresponds to the (local) direction of magnetization in\nferromagnets.\nIn ferromagnetic systems, the magnetization dynamics generates various dynamics of spin\ncurrents [2]. In the holographic dual theory, the scalar dynamics is also expected to induce\nthe dynamics of the corresponding SU(2) gauge \feld, and thus we generalize the static and\nhomogeneous ansatz for the SU(2) gauge \felds to the following factorized forms:\nAk\n0(u;t;x ) = (1\u0000u2)ak\n0(t;x);\nA?\n0(u;t;x ) = (1\u0000u2)a?\n0(t;x);\nAk\ni(u;t;x ) =Gk(u)ak\ni(t;x);\nA?\ni(u;t;x ) =G?(u)a?\ni(t;x) (i= 1\u00183); (19)\nwhere we set the radial component Aa\nu\u00110 by using the gauge degrees of freedom. Due to the\nnontrivial scalar solution \b( u), corresponding to the spontaneous magnetization, the SU(2)\ngauge symmetry is broken to U(1). The gauge \felds can be correspondingly decomposed into\nan unbroken component Ak\n\u0016and two broken components A?\n\u0016, which are de\fned by Ak\n\u0016/na\nandn\u0001A?\n\u0016= 0, respectively. As in the case of the static solutions, the time components\nof gauge \felds should satisfy the horizon boundary condition, Aa\n0= 0 atu= 1, for the\nregularity. Although the spatial components Aa\niare not required to vanish on the horizon,\nthe regularity (or \fniteness) at the horizon is required. The asymptotic solutions to the\nlinearized Yang-Mills equation near the boundary ( u\u00180) give the asymptotic expansions for\n7the radial functions Gk(u), andG?(u),\nGk(u) = 1\u0000\u001bk\nsu2+O(u4);\nG?(u) = 1 +\u001b?\nsu2+O(u4): (20)\nWe discuss the concrete numerical solutions of Ga(u) and their physical implications in the\nnext section.\nSince the scalar \feld \u001eadoes not have the U(1) charge, the \ructuation (or dynamics) of\n\u001eadoes not induce further dynamics for the U(1) gauge \feld, which implies the solution for\nB\u0016in (7) is unchanged, and we can neglect the dynamics of B\u0016.\nAt \frst, we consider the equation of motion for the scalar \feld \u001ea. Inserting the generalized\nansatz (18) into the equation (11), we obtain the following equation for na:\n\u0014\nu5@u\u0000\nu\u00003f(u)@u\b\u0001\n\u0000@V\n@\b\u0015\nna=\u0014u2\nf(u)DtDtna\u0000u2DiDina\u0015\n\b: (21)\nHere, we have used the gauge condition Aa\nu= 0, and the gauge covariant derivative is de\fned\nasD\u0016na=@\u0016na+\"abcAb\n\u0016nc. The left-hand side of the equation (21) is proportional to the\nequation (13), and thus vanishes for the solution \b( u). Since \b( u) is a non-trivial solution,\nwhich is not identically zero, we have the e\u000bective equation of motion for na:\nf\u00001DtDtna\u0000DiDina= 0: (22)\nNext, the equation of motion for the gauge \felds is considered. The Yang-Mills equation\nfor theSU(2) gauge \feld Aa\nMis derived by the variation of the holographic action (1) and\ngiven by\n1p\u0000g@N\u0000p\u0000gFNMa\u0001\n+\u000fabcAb\nNFNMc=JMa; (23)\nwhere theSU(2) current is de\fned as\nJa\nM=\"abc\u001ebDM\u001ec=\"abc\u001eb\u0010\n@M\u001ec+\"cdeAd\nM\u001ee\u0011\n: (24)\nUnlike the static case, the generalized ansatz (18) and (19) give the non-vanishing currents:\nJa\n\u0016= \b2\u0010\n\"abcnb@\u0016nc+\"abc\"cdenbad\n\u0016ne+O\u0000\nu2\u0001\u0011\n: (25)\nNote that the radial component of the currents still vanishes, Ja\nu\u00110, due to the gauge \fxing\nconditionAa\nu\u00110. With this current, we can explicitly write down the Yang-Mills equations\non the charged black hole (8), in the boundary direction :\nJa\n0=u3f@u\u0000\nu\u00001Fa\nu0\u0001\n+u2(DiFa\ni0); (26)\nJa\ni=u3@u\u0000\nu\u00001fFa\nui\u0001\n\u0000u2f\u00001(D0Fa\n0i) +u2\u0000\nDjFa\nji\u0001\n; (27)\n8where the gauge covariant derivative for the \feld strength is de\fned as D\u0016Fa\n\u0017\u001a=@\u0016Fa\n\u0017\u001a+\n\"abcAb\n\u0016Fc\n\u0017\u001a. Inserting the ansatz (19), the Yang-Mills equations give the equations for naand\naa\n\u0016. In summary, using the generalized ansatze, we have obtained the coupled equations of\nmotion for naandaa\n\u0016, (22), (26), and (27).\n3.2 Landau-Lifshitz Equation from Yang-Mills Equation\nSince it is di\u000ecult to \fnd the general solutions for the coupled non-linear partial di\u000berential\nequations, we seek simple trial solutions for naandaa\n\u0016to obtain the e\u000bective equations of\nmotion. At \frst, instead of looking for general solutions to the equation (22), we consider\nthe solutions to the simpler equations:\nDtna= 0 and Dina= 0; (28)\nwhich are explicitly given by\n@\u0016na+\u000fabcab\n\u0016nc= 0 +O(u2): (29)\nThese equations lead to the ground state solutions for the e\u000bective Hamiltonian for na:\nHe\u000b=f\n2(\u0019a)2+1\n2(Dina)2; (30)\nwhere the conjugate momentum is de\fned by \u0019a=f\u00001Dtna. In this article, we wish to\ndiscuss the dynamics of magnetization and spin currents in the boundary ferromagnetic\nsystem, which is given by the leading terms in the asymptotic expansions at u\u00180. Hence,\nthe higher order terms in the expansion with respect to uare irrelevant, and we neglect them\nin the following. Dropping the O(u2) term, we can easily obtain the solution to (29) for aa\n\u0016\nin terms of na,\naa\n\u0016=C\u0016na\u0000\"abcnb@\u0016nc; (31)\nwhere we have introduced a vector \feld C\u0016which is arbitrary at this stage. This solution\ndemonstrates the clear separation of the gauge \felds:\nak\n\u0016=C\u0016naanda?\n\u0016=\u0000\"abcnb@\u0016nc: (32)\nThe relation for the broken components, a?\n\u0016, is nothing but a non-abelian analogue of the\nrelation between the gauge \feld and the quantum phase of Cooper pair, A\u0016=@\u0016\u0012, in super-\nconductivity, and also corresponds to the Maurer-Cartan one-form of G=H\u0018SU(2)=U(1)\nin terms of the Nambu-Goldstone modes na[19, 20]. Requiring the matching condition to\n9the static solution (17), aa\n0=\u0016s\u000ea3andaa\ni= 0 forna= (0;0;1), the vector \feld C\u0016should\nsatisfy the condition:\nC0=\u0016sandCi= 0; (33)\nin the static and homogeneous limit. Note that the relation (31) and the ansatz (18) do not\ninduce new contributions of the scalar \felds to the energy-momentum tensor TMNin the\nEinstein equations, and consequently the analysis in the probe approximation remain intact.\nNext, we consider the e\u000bective Yang-Mills equations, (26) and (27). It is not di\u000ecult to\nshow that the relation (31) leads to vanishing currents Ja\n\u0016up toO(u2), using the explicit\nform (25). Furthermore, the ansatz for gauge \felds (19) with Aa\nu= 0 implies\n@u\u0000\nu\u00001Fa\nu0\u0001\n= 0;and@u\u0000\nu\u00001fFa\nui\u0001\n= 0 +O(u4): (34)\nDropping the higher order terms such as O(u4), the remaining Yang-Mills equations reduce\nto\nDiFa\ni0= 0;andD0Fa\n0i+DjFa\nji= 0: (35)\nFrom the viewpoint of the boundary theory (on the ferromagnet side), the \frst equation\ncorresponds to a non-abelian version of Gauss's law, and the second corresponds to a non-\nabelian version of Ampere's law without source and currents, for the spin gauge \felds [21].\nUsing the relation (31), we obtain the SU(2) \feld strength,4\nFa\n\u0016\u0017=nah\n(@\u0016C\u0017\u0000@\u0017C\u0016)\u0000\"bcdnb@\u0016nc@\u0017ndi\n\u0011naf\u0016\u0017: (36)\nNote that a component of the \feld strength, f\u0016\u0017, parallel to the magnetization naonly\nremains. With the \feld strength (36), the e\u000bective Yang-Mills equations (35) and the Bianchi\nidentity for the SU(2) gauge \feld are reduced to the following equations:\n@\u0016f\u0016\u0017= 0 and \u000f\u0016\u0017\u001a\u001b@\u0017f\u001a\u001b= 0: (37)\nThe above equations are the same form as the Maxwell equations, and the terms depending\nonnain the gauge \feld f\u0016\u0017actually corresponds to the so-called spin electromagnetic \feld\ndiscussed in the study on ferromagnetic metals [22, 21]. The gauge \feld (36) also corre-\nsponds to the unbroken U(1) gauge \feld upon the symmetry breaking from SU(2) toU(1),\nwith a space-dependent order parameter, which is frequently discussed in the context of\nsolitonic monopoles in non-abelian gauge theories [23]. Since the unbroken gauge \felds in\n4We used the relation \"abc@\u0016nb@\u0017nc=na\"bcdnb@\u0016nc@\u0017nddue toP\nanana= 1.\n10the holographic dual theory are identi\fed as the (exactly) conserved currents in the bound-\nary quantum system, the gauge \feld C\u0016is naturally identi\fed as the spin current with the\npolarization parallel to the magnetization na, which originates from conduction electrons.\nSo far, we have obtained the relation between the gauge \feld aa\n\u0016and the (normalized)\nmagnetization na, which implies that the gauge \feld dynamics can be solely reduced to the\ndynamics of the magnetization and the spin electromagnetic \feld C\u0016. Finally, we consider\nthe remaining Yang-Mills equation in the radial u-direction, in the holographic dual theory :\n1p\u0000g@\u0016\u0000p\u0000gF\u0016ua\u0001\n+\u000fabcAb\n\u0016F\u0016uc=Jua: (38)\nThis equation is derived by the variation of the radial u-component of the SU(2) gauge \felds\nand speci\fes the dynamics of the gauge \felds in the \fve-dimensional bulk; this equation\ncannot be seen in the ferromagnetic system on the boundary. With the ansatz (18), the\nradial component of the current also vanishes ( Ja\nu\u00110), and the gauge \fxing condition\nAa\nu\u00110 leads to the simple SU(2) \feld strength Fa\n\u0016u=\u0000@uAa\n\u0016such as\nFk\n0u= 2uak\n0(t;x); F?\n0u= 2ua?\n0(t;x);\nFk\niu= 2u\u001bk\nsak\ni(t;x); F?\niu=\u00002u\u001b?\nsa?\ni(t;x); (39)\nwhere we used the ansatz (19) and discarded the irrelevant O(u3) terms. From these forms,\nthe second term in the left-hand side of (38) automatically vanishes due to \"abcab\n\u0016ac\n\u0016= 0.\nInserting the forms of \feld strength (39) and the relation (31), the equation can be recast as\nthe following form:\n@0\u0010\nC0na\u0000\"abcnb@0nc\u0011\n\u0000@i\u0010\n\u001bk\nsCina+\u001b?\ns\"abcnb@inc\u0011\n= 0; (40)\nwhere the subleading terms are neglected. Here, we can write down the e\u000bective equation of\nmotion for the magnetization nain our holographic dual model:\nC0_na\u0000\"abcnbnc\u0000\u001b?\ns\"abcnbr2nc\u0000\u001bk\nsCi@ina= 0; (41)\nwhere the dot denotes the time-derivative and r2=@i@i.5Here, we consider the condition,\n@0C0\u0000\u001bk\ns@iCi= 0, on the unbroken gauge \feld due to the constraintP\nanana= 1. This\ncondition implies the conservation of the spin current of conduction electrons, which corre-\nsponds to the unbroken gauge \feld C\u0016, as seen below. Note that, since the Maxwell equations\n(37) forC\u0016is gauge invariant, this condition can be consistently imposed as a gauge \fxing\ncondition.\n5A simlar analysis on e\u000bective equations at the linearized level in two-dimensional magnetic systems has\nbeen also discussed in [11].\n11Considering the matching condition (33), we decompose C0intoC0=\u0016s+~C0. Finally,\nwe obtain the holographic equation for magnetization dynamics:\n\u0016s_na\u0000\"abcnbnc\u0000\u001b?\ns\"abcnbr2nc+~C0_na\u0000\u001bk\nsCi@ina= 0: (42)\nHere, we take the spin chemical potential to be \u0016s=\u0000Ms=\rwith the magnitude of spon-\ntaneous magnetization Msand the gyromagnetic ratio \r(>0),6and also identify the spin\ncurrent and spin accumulation due to conduction electrons as Jk\nsi=\u0000\u001bk\nsCiand \u0001\u0016s=~C0,\nusing the holographic dictionary. Then, the holographic equation becomes the same form as\nthe Landau-Lifshitz equation (without damping terms),\nMs\n\r_na+\"abcnbnc=\u0000\u001b?\ns\"abcnbr2nc+ \u0001\u0016s_na+Jk\nsi@ina: (43)\nThe last two terms in the right-hand side can be interpreted as the well-known terms from spin\ntransfer torque, which describes the transfer of spin angular momentum between localized\nmagnetic moments and conduction electrons [21]. Furthermore, the holographic Landau-\nLifshitz equation (42) also naturally incorporate the spin inertia term proportional to the\nsecond time-derivative of the magnetization, which is discussed in metallic ferromagnets [24].\nIt should be noted that the holographic Landau-Lifshitz equation automatically incor-\nporates the spin transfer torque due to conduction electrons without introducing the cor-\nresponding \felds to electrons in the dual gravitational theory. This is consistent with the\nthermodynamical results at low temperatures, which was obtained in the previous paper [10].\n4 Phenomenology of Holographic Magnetization Dynamics\nIn the isotropic ferromagnets su\u000eciently below the Curie temperature ( T\u001cTc), the dynamics\nof magnetization vector (or density of magnetic moments), Ma, is described by the Landau-\nLifshitz equation [1, 26]:\n@Ma\n@t=\u0000\u000b\u000fabcMbr2Mcwith3X\na=1MaMa=M(T)2= const. (44)\nIn the following discussion, the external magnetic \feld and the damping term (or relaxation\nterm) are ignored for simplicity. From the quadratic constraint, the magnetization vector\ncan be represented as Ma(x;t) =M(T)na(x;t) with the unit vector na(x;t). In terms of\nna(x;t), the Landau-Lifshitz equation becomes\nM(T)@na\n@t=\u0000\u000bM(T)2\u000fabcnbr2nc: (45)\n6The negative sign is introduced due to the negative value of the gyromagnetic ratio for electrons.\n12Note that the equation has two parameters, the magnitude of spontaneous magnetization,\nM(T), at the temperature T, and the spin sti\u000bness constant, \u000b.\nComparing the holographic equation (42) with the Landau-Lifshitz equation (45), we \fnd\nthat the spin chemical potential, \u0016s, in the gauge \feld solution (17) should be proportional\nto the magnitude of magnetization, and the spin sti\u000bness constant is given by the coe\u000bcients\n\u001b?\nsin the gauge \feld solution (20) in the following way :\n\u0016s/\u0000M(T) and \u001b?\ns/\u000bM(T)2: (46)\nIn our holographic dual model, the magnitude of magnetization, M(T), at the temperature\nTis given by the static solution of the scalar \feld \b( u) through the formula (14). The\n\frst relation between the magnitude of magnetization and the spin chemical potential in\nferromagnets is well-known, and frequently used as the starting point to analyze the various\nspintronic phenomena [2].\nAlthough the spin chemical potential in the equilibrium, \u0016s, is an integration constant,\nthe coe\u000ecient, \u001b?\ns, is the derived quantity from the gauge \feld equation, and thus the second\nrelation in (46) on the spin sti\u000bness constant is a nontrivial consequence in the holographic\ndual model. In order to obtain the coe\u000ecient, \u001b?\ns, we consider the linearized equation of\nmotion for gauge \felds on the background solution, with the static and homogeneous ansatz,\nA?\ni=kG?(u), wherek= const.7Inserting this ansatz into the Yang-Mills equation (27),\nwe have the following linearized equation for G?(u):\nu3d\ndu\u0012f(u)\nu\u0012dG?\ndu\u0013\u0013\n+\u0000\nua3(u)\u00012\nf(u)G?= 0; (47)\nwhere the metric (8) and the SU(2) gauge \feld (17) are assumed to be the background.\nNote that this is a linear equation for G?, and the constant kis irrelevant. Here, we\nimpose the \frst relation in (46), \u0016s=\u0000M(T)=M(0), which is the magnetization normal-\nized by the saturated magnetization, M(T= 0).8Using the numerical results of the holo-\ngraphic spontaneous magnetization, M(T) in [10], which is obtained using the scalar poten-\ntialV(j\u001ej) =\u0015\u0000\nj\u001ej2\u0000m2=\u0015\u00012=4 with\u0015= 1 andm2= 35=9, we can numerically solve the\nequation (47) and obtain the asymptotic expansion (20) near the boundary ( u\u00180). The\nnumerical results of temperature dependences of the spin-wave sti\u000bness, D(T)'\u001b?\ns=M(T),\nwhich appears in the dispersion relation of spin-waves, !=D(T)k2, and the spin sti\u000bness\nconstant,\u000b(T)'\u001b?\ns=M(T)2, are shown in Figure 1.\n7The nontrivial pro\fle A?\nx(u) on the background does not contribute to the energy-momentum tensor in\nthe Einstein equation at the linearized level.\n8The proportionality constant is chosen for convenience in numerical calculations.\n13Figure 1: Temperature dependence of the spin-wave sti\u000bness is shown in Figure ( a): The\ndots are numerical results for D(T)=D(0), and the bold line is the magnetization curve,\nM(T)=M(0). Temperature dependence of the spin sti\u000bness constant, \u000b(T)=\u000b(0), is shown in\nFigure (b). All the results are calculated with the parameters, \u0015= 1 andm2= 35=9.\nThe results on the spin-wave sti\u000bness in Figure 1( a) clearly show that D(T)/M(T),\nwhich is consistent with the relation (46) based on the Landau-Lifshitz equation (44). Fur-\nthermore, the results in Figure 1( b) imply the slight temperature dependence of the spin\nsti\u000bness constant, \u000b=\u000b(T), which can be attributed to the nonlinear spin-wave e\u000bects [25].\nA similar argument also holds for the unbroken (or parallel) component of the gauge\n\felds,Ak\ni, and we can obtain the coe\u000ecient \u001bk\ns, which leads to the spin torque term in the\nholographic Landau-Lifshitz equation (42). The nontrivial pro\fle of gauge \feld, Ak\nx(u), which\nis the parallel component to the spin chemical potential, Ak\n0, leads to the non-vanishing o\u000b-\ndiagonal contribution in the right-hand side of the Einstein equation (2), and thus induces the\n\ructuation of the metric gtx(u) =htx(u)=u2, wherehtx(u) parameterizes the \ructuation \fnite\non the boundary. At the linearized level, two \ructuations, Ak\nx(u) andhtx(u), form the closed\nequations, which come from the Yang-Mills equation and Einstein equation, respectively\n[27, 28]:\nud\ndu \nf(u)\nu \ndAk\nx\ndu!!\n+\u0012da3(u)\ndu\u0013d\ndu\u0000\nu2htx\u0001\n= 0; (48)\nu\u00002d\ndu\u0000\nu2htx\u0001\n+ 2\u0012da3(u)\ndu\u0013\nAk\nx= 0: (49)\nDeleting the metric \ructuation, we can obtain the equation for Gk(u):\nud\ndu \nf(u)\nu \ndGk\ndu!!\n\u00002u2\u0012da3(u)\ndu\u00132\nGk= 0: (50)\nWe can numerically solve the equation, and obtain the coe\u000ecient \u001bk\nsfrom the asymptotic\nexpansion of the solution in (20). The resulting temperature dependence of the spin torque\ncoe\u000ecient, \u001cs(T) =\u001bk\ns=M(T), is shown in Figure 2.\n14Figure 2: Temperature dependence of the spin torque coe\u000ecient: The dots are numerical re-\nsults for\u001cs(T)=\u001cs(0), and the bold line is the magnetization curve M(T)=M(0). The dashed\nline is the \ftting curve, \u001cs(T)=\u001cs(0) =c(1\u0000T=Tc)2=5withc'1:41, near the Curie temper-\nature.\nThe results on the magnitude of the spin transfer torque, \u001cs(T), show that the spin torque\ne\u000bect is approximately constant at low temperatures (in comparison with magnetization\ncurve), and is vanishing towards the Curie temperature as \u001cs(T)/(1\u0000T=Tc)2=5. This\nproperty at low temperatures is consistent with the phenomenological form of the spin transfer\ntorque, ( Jk\ns\u0001rna)=M(T), whose magnitude is independent of the norm of magnetization due\ntojJk\nsj/M(T) at the leading order [21]. In addition, the \fnite spin torque coe\u000ecient is\na consequence of the both \ructuations of the gauge \feld and metric. In accordance with\nthe holographic dictionary [27], the metric \ructuation htxcorresponds to the temperature\ngradient,rxT=T, in the ferromagnetic system. This calculation implies that the e\u000bect of\nspin transfer torque appears only in the nonequilibrium situations, where spin transfer is\naccompanied by heat (or entropy) transfer.\n5 Summary and Discussion\nWe have discussed a novel approach to understand magnetization dynamics in ferromagnets\nusing the holographic realization of ferromagnetic systems. The Landau-Lifshitz equation de-\nscribing magnetization dynamics was derived from the Yang-Mills-Higgs equations in the dual\ngravitational theory. This holographic Landau-Lifshitz equation automatically incorporates\nnot only the exchange interaction but also the spin transfer torque e\u000bect due to conduction\nelectrons. Furthermore, we numerically investigated the temperature dependences of the\nspin-wave sti\u000bness and the magnitude of spin transfer torque in the holographic dual theory,\nand the results obtained so far are consistent with the known properties of magnetization\ndynamics in ferromagnets with conduction electrons.\n15This holographic approach to magnetization dynamics can be applied to more generic sit-\nuations. For instance, the holographic Landau-Lifshitz equation can incorporate the damping\nterm by considering more generic metric \ructuations, which correspond to phonon dynamics\nin the boundary ferromagnets. Moreover, the holographic dual theory may provide geomet-\nric approaches to spin caloritronics [29], where magnetization dynamics is considered under\ntemperature gradients, from higher dimensional perspectives. We thus believe that the holo-\ngraphic approach provides useful tools to analyze nonequilibrium and nonlinear dynamics\nof magnetization in ferromagnets, and also leads to new perspectives in spintronics from\ngravitational physics.\nAcknowledgement\nThe authors thank M. Ishihara for collaboration at the early stage of this work, and also K.\nHarii and Y. Oikawa for useful discussions. The works of N. Y. and E. S. were supported\nin part by Grant-in Aid for Scienti\fc Research on Innovative Areas \"Nano Spin Conversion\nScience\" (26103005), and the work of K. S. was supported in part by JSPS KAKENHI Grant\nNo. JP17H06460. The works of N. Y. and E. S. were supported in part by ERATO, JST.\nReferences\n[1] L. D. Landau and E. M. Lifshitz, \\On the theory of the dispersion of magnetic perme-\nability in ferromagnetic bodies,\" Phys. Z. Sowjet. 8, 153 (1935).\n[2] S. Maekawa, S. O. Valenzuela, E. Saitoh and T. Kimura (Eds.), \\Spin Current (Second\nEd.),\" Oxford University Press (2017).\n[3] L. Berger, \\Emission of spin waves by a magnetic multilayer traversed by a current,\"\nPhys. Rev. B 54, 9353 (1996).\n[4] J. C. Slonczewski, \\Current-driven excitation of magnetic multilayers,\" Journal of Magn.\nMagn. Mater. 159, L1 (1996).\n[5] Y. Tserkovnyak, A. Brataas, and G. E. Bauer, \\Enhanced Gilbert damping in thin\nferromagnetic \flms,\" Phys. Rev. Lett. 88, 117601 (2002).\n[6] J. M. Maldacena, \\The Large N limit of superconformal \feld theories and supergravity,\"\nInt. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231 (1998)] [hep-\nth/9711200].\n16[7] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, \\Gauge theory correlators from non-\ncritical string theory,\" Phys. Lett. B 428, 105 (1998) [hep-th/9802109].\n[8] E. Witten, \\Anti-de Sitter space and holography,\" Adv. Theor. Math. Phys. 2, 253\n(1998) [hep-th/9802150].\n[9] J. Zaanen, Y. W. Sun, Y. Liu and K. Schalm, \\Holographic duality in condensed matter\nphysics,\" Cambridge University Press (2015).\n[10] N. Yokoi, M. Ishihara, K. Sato and E. Saitoh, \\Holographic realization of ferromagnets,\"\nPhys. Rev. D 93, 026002 (2016) [arXiv:1508.01626 [hep-th]].\n[11] N. Iqbal, H. Liu, M. Mezei and Q. Si, \\Quantum phase transitions in holographic models\nof magnetism and superconductors,\" Phys. Rev. D 82, 045002 (2010) [arXiv:1003.0010\n[hep-th]].\n[12] R. G. Cai and R. Q. Yang, \\Paramagnetism-Ferromagnetism Phase Transition in a\nDyonic Black Hole,\" Phys. Rev. D 90, no. 8, 081901 (2014) [arXiv:1404.2856 [hep-th]].\n[13] R. G. Cai, R. Q. Yang, Y. B. Wu and C. Y. Zhang, \\Massive 2-form \feld and holographic\nferromagnetic phase transition,\" JHEP 1511 , 021 (2015) [arXiv:1507.00546 [hep-th]].\n[14] V. E. Hubeny and M. Rangamani, \\A Holographic view on physics out of equilibrium,\"\nAdv. High Energy Phys. 2010 , 297916 (2010) [arXiv:1006.3675 [hep-th]].\n[15] C. N. Yang and R. L. Mills, \\Conservation of Isotopic Spin and Isotopic Gauge Invari-\nance,\" Phys. Rev. 96, 191 (1954).\n[16] C. P. Herzog and S. S. Pufu, \\The Second Sound of SU(2),\" JHEP 0904 , 126 (2009)\n[arXiv:0902.0409 [hep-th]].\n[17] V. Balasubramanian, P. Kraus, A. E. Lawrence and S. P. Trivedi, \\Holographic probes\nof anti-de Sitter space-times,\" Phys. Rev. D 59, 104021 (1999) [hep-th/9808017].\n[18] I. R. Klebanov and E. Witten, \\AdS / CFT correspondence and symmetry breaking,\"\nNucl. Phys. B 556, 89 (1999) [hep-th/9905104].\n[19] H. Leutwyler, \\Nonrelativistic e\u000bective Lagrangians,\" Phys. Rev. D 49, 3033 (1994)\n[hep-ph/9311264].\n[20] J. M. Rom\u0013 an and J. Soto, \\E\u000bective \feld theory approach to ferromagnets and antiferro-\nmagnets in crystalline solids,\" Int. J. Mod. Phys. B 13, 755 (1999) [cond-mat/9709298].\n17[21] G. Tatara, \\E\u000bective gauge \feld theory of spintronics,\" Physica E: Low-dimensional\nSystems and Nanostructures 106, 208 (2019) [arXiv:1712.03489 [cond-mat.mes-hall]].\n[22] G. E. Volovik, \\Linear momentum in ferromagnets,\" J. Phys. C 20, L83 (1987).\n[23] For a review, J. A. Harvey, \\Magnetic monopoles, duality and supersymmetry,\" In\n*Trieste 1995, High energy physics and cosmology* 66-125 [hep-th/9603086].\n[24] T. Kikuchi and G. Tatara, \\Spin Dynamics with Inertia in Metallic Ferromagnets,\" Phys.\nRev. B 92, 184410 (2015) [arXiv:1502.04107 [cond-mat.mes-hall]].\n[25] U. Atxitia, D. Hinzke, O. Chubykalo-Fesenko, U. Nowak, H. Kachkachi, O. N. Mryasov,\nR. F. Evans and R. W. Chantrell, \"Multiscale modeling of magnetic materials: Temper-\nature dependence of the exchange sti\u000bness,\" Phys. Rev. B 82, 134430 (2010).\n[26] E. M. Lifshitz and L. P. Pitaevskii, \\Statistical physics: theory of the condensed state\n(Vol. 9)\", Elsevier (2013).\n[27] S. A. Hartnoll, \\Lectures on holographic methods for condensed matter physics,\" Class.\nQuant. Grav. 26, 224002 (2009) [arXiv:0903.3246 [hep-th]].\n[28] C. P. Herzog, K. W. Huang and R. Vaz, \\Linear Resistivity from Non-Abelian Black\nHoles,\" JHEP 1411 , 066 (2014) [arXiv:1405.3714 [hep-th]].\n[29] G. E. Bauer, E. Saitoh and B. J. Van Wees, \\Spin caloritronics,\" Nature materials, 11\n(5), 391 (2012).\n18"    },    {        "title": "2011.00326v1.Low_temperature_asymptotics_for_the_transverse_dynamical_structure_factor_for_a_magnetically_polarized__XX__chain_at_small_and_negative_frequencies.pdf",        "content": "arXiv:2011.00326v1  [cond-mat.str-el]  31 Oct 2020Low-temperature asymptotics for the transverse\ndynamical structure factor for a magnetically\npolarized XXchain at small and negative frequencies\nP.N. Bibikov\nRussian State Hydrometeorological University, Saint-Pet ersburg, Russia\nAbstract\nUsing the truncated form factor expansion the low-temperat ure asymptotics for\nthe transverse dynamical structure factor of the magnetica lly polarized XXchain\nis studied. Unlike the previous paper we do not use the repres entation of structure\nfactor in terms of the corresponding magnetic susceptibili ty. This enables to obtain\ncorrect results at small and negative frequencies.\n1 Introduction\nThe present paper is a supplement of the previous one [1] where the low-temperature\nasymptotics of the transverse dynamical structure factor (TD SF) was studied for the\nmagnetically polarized XXchain [2] related to the Hamiltonian\nˆH=−1\n2N/summationdisplay\nn=1/bracketleftBig\nJ/parenleftBig\nS+\nnS−\nn+1+S−\nnS+\nn+1/parenrightBig\n+h/parenleftBig\nSz\nn+Sz\nn+1−I/parenrightBig/bracketrightBig\n, (1)\nwhereS±\nnandSz\nnis the standard triple of spin-1\n2operators associated with n-th site\n[S+\nm,S−\nn] = 2δm,nSz\nn,[Sz\nm,S±\nn] =±δm,nS±\nn, (2)\n1Iis the identity 2N×2Nmatrix and his a magnetic field. As in [1] we postulate here the\nperiodic boundary conditions\nSN+1≡S1. (3)\nHamiltonian (1) acts in the tensor product Hilbert space\nH=N/productdisplay\nn=1⊗Vn, (4)\nwhere each Vnis the copy of C2generated by up | ↑/angbracketrightand down | ↓/angbracketrightpolarized states and\nattached to the n-th site.\nAll the calculations in [1] were based on the well known formula [3, 4]\nS(ω,q,T) =−1\nπ(1−e−βω)Imχ(ω,q,T), ω/negationslash= 0, (5)\nsupplementedwiththeDysonequationforthetransversemagnet icsusceptibility[3,5,6,7]\nχ(ω,q,T) =1\nχ−1\n0(ω,q)−Σ(ω,q,T). (6)\nHereχ0(ω,q) is the zero temperature susceptibility and Σ( ω,q,T) is the (thermally acti-\nvated) magnon self-energy\nχ0(ω,q) =χ(ω,q,0)⇐⇒Σ(ω,q,0) = 0. (7)\nThe guiding idea of the calculations in [1] was first suggested in [4]. It a sserts that\nin the gapped (massive) regime the low-temperature asymptotic of S(ω,q,T) completely\ndepends only onthe one- andtwo-magnon spectrums andhence ma y be obtained with the\nuse ofthe so called truncated formfactor expansion. It is well kno wn however [1, 4, 5] that\na direct derivation of this expansion for S(ω,q,T) (orχ(ω,q,T)) often yields a singular\nresult for the line shape of the resonance contour even at T >0. In order to avoid this\npathology it was suggested in [5] to search for the low temperature asymptotics not for\nχ(ω,q,T) but for Σ( ω,q,T). The latter task however is not simple if we work in the\nMatsubara temperature formalism where the Dyson equation (6) is usually proved by\nan analysis of the perturbation series. Namely in [5] (6) was only post ulated and the\nexpansion for Σ( ω,q,T) was obtained from the corresponding result for χ(ω,q,T) with\nthe use of the resummation procedure. Contrary in [1] the Dyson e quation for the related\ntoχ(ω,q,T) real two time retarded Green function and the spectral repres entation for the\ncorresponding Σ( ω,q,T) were rigorously proved within the approach previously suggested\nby N. M. Plakida and Yu. A. Tserkovnikov [6, 7, 8].\n2Being rather successful near resonance the approach [1, 4, 5] h owever fails at the\nvicinityω= 0 and at ω <0. Really according to (5) and the condition\nS(ω,q,T)≥0, (8)\n(whichdirectlyfollowsfromthewellknownexpansion(15)[3])ImΣ( ω,q,T)shouldchange\nits sign when ωpasses throw 0. This requirement however badly agrees with the fo rm\nfactor expansion for χ(ω,q,T) usually based on the standard spectral decomposition [1,\n3, 4, 5]\nχ(ω,q,T) = lim\nN→∞1\nZ(T,N)/summationdisplay\nµ,νe−βEν−e−βEµ\nω+Eν−Eµ+iǫ|/angbracketleftν|S+(q)|µ/angbracketright|2. (9)\nHereZ(T,N) is the partition function and\nS(q)≡1√\nNN/summationdisplay\nn=1e−iqnSn, (10)\nwhere according to the periodicity condition (3) it is implied\neiqN= 1. (11)\nReally the states related to the indices µandνin the sum (9) belong to different magnon\nnumber sectors (for example if νcorresponds to the ground state then µparameterizes\nthe one-magnon states). But the M-th order form factor expansion implies the cutoff of\ncontributions fromall m-magnon sectors with m > M. Hence if the index νin (9) belongs\nto theM-magnon sector we should reduce the corresponding term in (9) as follows\ne−βEν−e−βEµ\nω+Eν−Eµ+iǫ−→e−βEν\nω+Eν−Eµ+iǫ. (12)\nBut according to the well known formula\nIm1\nx+iǫ=−πδ(x), (13)\n(12) yields the reduction\ne−βEν/parenleftBig\n1−e−βω/parenrightBig\nδ(ω+Eν−Eµ)−→e−βEνδ(ω+Eν−Eµ), (14)\nin Imχ(ω,q,T) under which the factor 1 −e−βωin the denominator of (5) is not to be\ncanceled! Being negligible near the resonance peak this error becom es critical at ω→0\nand atω <0 results in the wrong answer S(ω,q,T)<0.\n3In order to obtain correct results for TDSF at the vicinity ω= 0 and at ω <0 we\nsuggest here the truncated form factor expansion directly for S(ω,q,T) basing on the\nspectral representation [3]\nS(ω,q,T) = lim\nN→∞1\nZ(T,N)/summationdisplay\nµ,νe−βEν|/angbracketleftν|S+(q)|µ/angbracketright|2δ(ω+Eν−Eµ).(15)\nSince this approach results in the resonance singularity at ω=Emagn(q) (hereEmagn(q)\nis the magnon energy, see (26)) [1] we shall decompose the TDSF on regular and singular\ncomponents\nS(ω,q,T) =S(reg)(ω,q,T)+S(sing)(ω,q,T), (16)\nimplying\nS(reg)(ω,q,T)/negationslash=∞, ω/negationslash=Emagn(q),\nS(sing)(ω,q,T) = 0, ω/negationslash=Emagn(q). (17)\nJustS(reg)(ω,q,T) will be asserted as a reliable approximation for TDSF at small and\nnegative ω.\nIn order to avoid manipulations with delta-functions we additionally su ggest an alter-\nnative method for evaluation of TDSF based on the formula\nS(ω,q,T) =−1\nπImξ(ω,q,T), (18)\nwhere the auxiliary quantity ξ(ω,q,T) is defined by the spectral decomposition\nξ(ω,q,T) =/summationdisplay\nµ,νe−βEν|/angbracketleftν|S+(q)|µ/angbracketright|2\nω+Eν−Eµ+iǫ, (19)\n(really (18) follows from (19), (15) and (13)).\nAs it will be shown in the paper the both approaches yield the same res ult for the\nlow-temperature asymptotics of S(reg)(ω,q,T).\n2 One- and two-magnon spectrums\nIn the present paper we study only the case\nh >|J|, (20)\n4under which all the terms in the sum (1) are non-negative operator s and the system has\nthe single zero-energy polarized ground state\n|∅/angbracketright=| ↑/angbracketright⊗...⊗| ↑/angbracketright, (21)\nwhich is the tensor product of Nvectors| ↑/angbracketright.\nThe one-magnon sector is spanned on the states\n|k/angbracketright=1√\nNN/summationdisplay\nn=1eikn| ↓n/angbracketright,| ↓n/angbracketright ≡S−\nn|∅/angbracketright, (22)\nwhere according to the periodicity condition (3)\neikN= 1. (23)\nIt may be readily proved that the system (22) is orthogonal and co mplete. Namely\nN−1/summationdisplay\nj=0|kj/angbracketright/angbracketleftkj|=1\nNN−1/summationdisplay\nj=0N/summationdisplay\nm,n=1eikj(m−n)S−\nm|∅/angbracketright/angbracketleft∅|S+\nn=N/summationdisplay\nn=1S−\nn|∅/angbracketright/angbracketleft∅|S+\nn,(24)\n/angbracketleftkj|kl/angbracketright=1\nNN/summationdisplay\nn=1ei(kl−kj)n=δjl. (25)\nThe corresponding to |k/angbracketrightenergy is\nEmagn(k) =h−Jcosk. (26)\nAccording to (26) and (20) the one-magnon sector really is gapped and\nEgap=Emagn(kgap) =h−|J|, (27)\nwhere\nkgap= 0, J > 0,\nkgap=π, J < 0. (28)\nA two-magnon state describes a pair of scattering magnons (ther e are not two-magnon\nbound states in the XXchain [1, 2]) and has the form (see (35) in [1])\n|k,κ/angbracketright=2\nN/summationdisplay\nn1<n2eik(n1+n2)/2sinκ(n2−n1)| ↓n1↓n2/angbracketright,| ↓n1↓n2/angbracketright ≡S−\nn1S−\nn2|∅/angbracketright,(29)\n5whereksatisfies (23) and without lost of generality one may put\n0< κ < π. (30)\nThe corresponding energy is\nEscatt(k,κ) = 2h−2Jcosk\n2cosκ. (31)\nAccording to the periodicity (3)\neik(n+N+1)/2sinκ(N+1−n) = eik(1+n)/2sinκ(n−1), n= 2,...,N,\neik(0+n)/2sinκ(n−0) = eik(n+N)/2sinκ(N−n), n= 1,...,N−1.(32)\nRepresenting (32) in an equivalent form\n/parenleftBig\nei(k/2+κ)N+1/parenrightBig\neiκ(1−n)+/parenleftBig\nei(k/2−κ)N+1/parenrightBig\neiκ(n−1), n= 2,...,N,\n/parenleftBig\nei(k/2+κ)N+1/parenrightBig\ne−iκn+/parenleftBig\nei(k/2−κ)N+1/parenrightBig\neiκn, n= 1,...,N−1,(33)\none readily reduce it to\nei(k/2±κ)N=−1. (34)\nThe solutions of (34) are slightly different for even and odd N. ForN= 2Mthe total\nset of wave numbers is\nkl=4πl\nN, l= 0,1,...M−1, κ λ=(2λ−1)π\nN, λ= 1,2,...M, (35)\nkm=2(2m−1)π\nN, m= 1,2,...M, κ µ=2πµ\nN, µ= 1,2,...M−1.(36)\nSince\neiklN/2= 1,eiκλN=−1,eikmN/2=−1,eiκµN= 1, (37)\n(34) is really satisfied. Moreover as it is shown in the Appendix A the sy stem of states\n(29), (35), (36) form a complete orthogonal basis of the two-ma gnon sector. Namely\n/summationdisplay\nl,λ|kl,κλ/angbracketright/angbracketleftkl,κλ|+/summationdisplay\nm,µ|km,κµ/angbracketright/angbracketleftkm,κµ|=/summationdisplay\n1≤n1<n2≤NS−\nn1S−\nn2|∅/angbracketright/angbracketleft∅|S+\nn1S+\nn2,(38)\n/angbracketleftk,κ|˜k,˜κ/angbracketright=δk˜kδκ˜κ. (39)\n63 Direct evaluation of S(reg)\n1(ω,q,T)\nThe truncated form factor expansion is based on the decompositio n of the total Hilbert\nspace (4) into the direct sum of the m-magnon sectors\nH=⊕N\nm=0HmˆQ/vextendsingle/vextendsingle/vextendsingle\nHm=m (40)\nThis follows from the commutativity of the Hamiltonian (1) with the mag non number\noperator\nˆQ=/summationdisplay\nn/parenleftBig1\n2−Sz\nn/parenrightBig\n. (41)\nThe one-dimensional subspace H0is spanned on |∅/angbracketright. The one-and two-magnon sectors\nare spanned on (22) and (29). Decomposition (40) results in the fo llowing expansions\nZ(T,N) = 1+N/summationdisplay\nm=1Zm(T,N),J(ω,q,T,N ) =N−1/summationdisplay\nm=0Jm(ω,q,T,N ),(42)\nfor the partition sum Zm(T,N) and an auxiliary one\nJ(ω,q,T,N )≡/summationdisplay\nµ,νe−βEν|/angbracketleftν|S+(q)|µ/angbracketright|2δ(ω+Eν−Eµ). (43)\nHere in (42)\nJm(ω,q,T,N )≡/summationdisplay\nµ,νe−βEν|/angbracketleftν|S+(q)|µ/angbracketright|2δ(ω+Eν−Eµ),|ν/angbracketright ∈ Hm,|µ/angbracketright ∈ Hm+1,\nZm(T,N)≡/summationdisplay\nνe−βEν,|ν/angbracketright ∈ Hm. (44)\nFrom (42) follows the form factor expansion for the TDSF (15)\nS(ω,q,T) =∞/summationdisplay\nm=0Sm(ω,q,T), S m(ω,q,T) =O/parenleftBig\ne−mβEgap/parenrightBig\n, (45)\nwhere\nS0(ω,q,T) = lim\nN→∞J0(ω,q,T,N ),\nS1(ω,q,T) = lim\nN→∞(J1(ω,q,T,N )−Z1(T,N)J0(ω,q,T,N )). (46)\nAccording to (44)\nZ1(T,N) =/summationdisplay\nke−βEmagn(k),\n7J0(ω,q,T,N ) =/summationdisplay\nk|/angbracketleft∅|S+(q)|k/angbracketright|2δ(ω−Emagn(k)) =δ(ω−Emagn(q)),\nJ0(ω,q,T,N ) =/summationdisplay\nk,κe−βEmagn(k−q)|/angbracketleftk−q|S+(q)|k,κ/angbracketright|2\n·δ(ω+Emagn(k−q)−Escatt(k,κ)), (47)\nand a substitution of (47) into (46) yields\nS0(ω,q,T) =δ(ω−Emagn(q)),\nS1(ω,q,T) = lim\nN→∞/summationdisplay\nke−βEmagn(k−q)/parenleftBig/summationdisplay\nκ|/angbracketleftk−q|S+(q)|k,κ/angbracketright|2\n·δ(ω+Emagn(k−q)−Escatt(k,κ))−δ(ω−Emagn(q))/parenrightBig\n. (48)\nAs it is shown in the appendix B\n/angbracketleftk−q|S+(q)|k,κ/angbracketright=2isinκ\nN(cosα−cosκ), (49)\nwhere\nα=k\n2−q. (50)\nAt the same time it may be readily checked according to (26), (31) an d (50) that\nEmagn(k−q)−Escatt(k,κ) =−Emagn(q)+2Jcosk\n2(cosκ−cosα).(51)\nSo the condition\nω+Emagn(k−q)−Escatt(k,κ) = 0, (52)\nyields\ncosα−cosκ=ω−Emagn(q)\n2Jcosk/2. (53)\nCorrespondingly\nsinκ=/radicalbig\n−D(k,ω,q)\n2|J|cosk/2, (54)\nwhere\nD(k,ω,q) = (ω−Emagn(q)−2Jcosk\n2cosα)2−4J2cos2k\n2. (55)\nIt may be readily proved from (51) and (54) that\n0<sinκ <1,−1<cosκ <1⇐⇒D(k,ω,q)<0. (56)\nIn order to study this conditions it is convenient to rewrite (55) in th e form\nD(k,ω,q) = (ω−Φdown(q,k))(ω−Φup(q,k)), (57)\n8where\nΦdown(q,k) =h+Jcos(k−q)−2|J|cosk\n2,\nΦup(q,k) =h+Jcos(k−q)+2|J|cosk\n2. (58)\nAccording to (57) the condition (56) reduces to\nD(k,ω,q)<0⇐⇒Φdown(q,k)< ω <Φup(q,k). (59)\nA substitution of (53) and (54) into (49) gives with the use of (51)\n|/angbracketleftk−q|S+(q)|k,κ/angbracketright|2δ(ω+Emagn(k−q)−Escatt(k,κ))\n=8|J|/radicalbig\n−D(k,ω,q)sinκ\nN2(ω−Emagn(q))2cosk\n2δ(ω−Emagn(q)+2Jcosk\n2(cosκ−cosα)).(60)\nExcluding nowthe singular term(proportional to δ(ω−Emagn(q)))andusing thestandard\nsubstitutions/summationdisplay\nk−→N\n2π/integraldisplay\ndk,/summationdisplay\nκ−→N\n2π/integraldisplay\ndκ, (61)\none readily gets from (48) and (60)\nS(reg)\n1(ω,q,T) =−ImΣ1(ω,q,T)\nπ(ω−Emagn(q))2, ω/negationslash=Emagn(q). (62)\nHere\nΣ1(ω,q,T) =−i\nπ/integraldisplayπ\n−πdke−βEmagn(k−q)/radicalbig\n|D(k,ω,q)|Θ(Φup(q,k)−ω)Θ(ω−Φdown(q,k)),\n(63)\nis the first term of the cluster expansion for the magnon self-ener gy obtained in [1].\n4 Alternative evaluation of S1(ω,q,T)\nIn the same manner as in (48) we may readily get the cluster expansio n forξ(ω,q,T)\nξ(ω,q,T) =ξ0(ω,q)+∞/summationdisplay\nm=1ξm(ω,q,T), ξ m(ω,q,T) =O/parenleftBig\ne−mβEgap/parenrightBig\n,(64)\n9where\nξ0(ω,q) = lim\nN→∞/summationdisplay\nk|/angbracketleft∅|S+(q)|k/angbracketright|2\nω−Emagn(k)+iǫ=1\nω−Emagn(q)+iǫ,\nξ1(ω,q,T) = lim\nN→∞/summationdisplay\nke−βEmagn(k−q)/parenleftBig/summationdisplay\nκ|/angbracketleftk−q|S+(q)|k,κ/angbracketright|2\nω+Emagn(k−q)−Escatt(k,κ)+iǫ\n−1\nω−Emagn(q)+iǫ/parenrightBig\n. (65)\nSince there are two types of two-magnon states (35) and (36) we have the decompo-\nsition\nξ1(ω,q,T) =ξ(1)\n1(ω,q,T)+ξ(2)\n1(ω,q,T), (66)\nwhere\nξ(1)\n1(ω,q,T) = lim\nN→∞M−1/summationdisplay\nl=0e−βEmagn(kl−q)/parenleftBigM/summationdisplay\nλ=1|/angbracketleftkl−q|S+(q)|kl,κλ/angbracketright|2\nω+Emagn(kl−q)−Escatt(kl,κλ)+iǫ\n−1\nω−Emagn(q)+iǫ/parenrightBig\n,\nξ(2)\n1(ω,q,T) = lim\nN→∞M/summationdisplay\nm=0e−βEmagn(km−q)/parenleftBigM−1/summationdisplay\nµ=1|/angbracketleftkm−q|S+(q)|km,κµ/angbracketright|2\nω+Emagn(km−q)−Escatt(km,κµ)+iǫ\n−1\nω−Emagn(q)+iǫ/parenrightBig\n(67)\nIntroducing the new variables\nx= cosκ, a =Jcosk\n2, b=ω−2h+Emagn(k−q,h), c= cosα,(68)\nwe readily obtain from (31) and (49) the following compact represen tation\n|/angbracketleftk−q|S+(q)|k,κ/angbracketright|2\nω+Emagn(k−q)−Escatt(k,κ)+iǫ=4(1−x2)\nN2(x−c)2(2ax+b+iǫ).(69)\nUsing the identity\n1−x2\n(x−c)2(2ax+b+iǫ)=1\n(2ac+b+iǫ)/parenleftBig1−c2\n(x−c)2−2c\nx−c/parenrightBig\n−1\n(2ac+b+iǫ)2/parenleftBig2a(1−c2)\nx−c+(b+iǫ)2−4a2\n2ax+b+iǫ/parenrightBig\n, (70)\nand taking into account that\n2ac+b+iǫ=ω−Emagn(q)+iǫ,(b+iǫ)2−4a2=D(k,ω+iǫ,q),(71)\n10we readily get the following expansion for (69)\n|/angbracketleftk−q|S+(q)|k,κ/angbracketright|2\nω+Emagn(k−q)−Escatt(k,κ)+iǫ=4\nN2/bracketleftBig/parenleftBigsin2α\n(cosκ−cosα)2−2cosα\ncosκ−cosα/parenrightBig\n·1\nω−Emagn(q)+iǫ−/parenleftBig2asin2α\ncosκ−cosα+D(k,ω+iǫ,q)\n2ax+b+iǫ/parenrightBig1\n(ω−Emagn(q)+iǫ)2/bracketrightBig\n.(72)\nAs it is shown in Appendix B\n4\nN2M/summationdisplay\nλ=1/parenleftBigsin2αl\n(cosκλ−cosαl)2−2cosαl\ncosκλ−cosαl/parenrightBig\n= 1+δαl,0+δ|αl|,π,\n4\nN2M−1/summationdisplay\nµ=1/parenleftBigsin2αm\n(cosκµ−cosαm)2−2cosαm\ncosκµ−cosαm/parenrightBig\n= 1−4\nN2,\n4\nN2M/summationdisplay\nλ=1sin2αl\ncosκλ−cosαl= 0,\n4\nN2M−1/summationdisplay\nµ=1sin2αm\ncosκµ−cosαm=−4cosαm\nN2, (73)\nwhere following (50) we introduced notations\nαl≡kl\n2−q, α m≡km\n2−q. (74)\nA substitution of (72) into (67) gives with the use of (73)\nξ(1)\n1(ω,q,T) =−lim\nN→∞4\n(ω−Emagn(q)+iǫ)2N2/summationdisplay\nkl,κλe−βEmagn(kl−q)D(kl,ω+iǫ,q)\n2acosκλ+b+iǫ\n+e−βEmagn(q)\nω−Emagn(q)+iǫ\nξ(2)\n1(ω,q,T) =−lim\nN→∞4\n(ω−Emagn(q)+iǫ)2N2/summationdisplay\nkm,κµe−βEmagn(km−q)D(km,ω+iǫ,q)\n2acosκµ+b+iǫ.(75)\nUsing now the N→ ∞substitutions\n1\nN/summationdisplay\nkl,1\nN/summationdisplay\nkm−→1\n4π/integraldisplay2π\n0dk,1\nN/summationdisplay\nκλ,1\nN/summationdisplay\nκµ−→1\n2π/integraldisplayπ\n0dκ, (76)\nand formula (66) one readily gets from (75)\nξ1(ω,q,T) =1\nπ(ω−Emagn(q)+iǫ)2/integraldisplay2π\n0dke−βEmagn(k−q)˜Γ(k,ω,q)\n+e−βEmagn(q)\nω−Emagn(q)+iǫ, (77)\n11where\n˜Γ(k,ω,q) =1\nπ/integraldisplayπ\n0D(k,ω+iǫ,q)dκ\n2acosκ+b+iǫ=1\n2πi/contintegraldisplay\n|z|=1dz4a2−(b+iǫ)2\na(z2+1)+(b+iǫ)z,(78)\nis the same as in equation (94) of [1]. So (77) and equation (93) in [1] yield\nξ1(ω,q,T) =Σ1(ω,q,T)\n(ω−Emagn(q)+iǫ)2+e−βEmagn(q)\nω−Emagn(q)+iǫ. (79)\nNow a substitution of (79) into (18) gives exactly the result (62) fo rS(reg)\n1(ω,q,T).\n5 ComparisonwiththeTDSFobtainedon theground\nof the Dyson equation\nIt is instructive to compare (62) with the result obtained in [1] on the ground of the Dyson\nequation\nS(DY)\n1(ω,q,T) =−1\nπ(1−e−βω)·ImΣ1(ω,q,T)\n(ω−Emagn(q)−ReΣ1(ω,q,T))2+(ImΣ 1(ω,q,T))2.\n(80)\nThe two expressions (62) and (80) has the two main differences. Fir st of all (80)\ncontains the denominator 1 −e−βωwhile (62) does not. This subject was already dis-\ncussed in the Introduction. Second (80) includes ReΣ 1(ω,q,T) while (62) does not. This\ndisagreement becomes clear if we notice that (80) follows from (5), (6) and the relation\nχ−1\n0(ω,q) =ω−Emagn(q)+iǫ. (81)\nHence Im χ1(ω,q,T) still contains ReΣ 1(ω,q,T) in the denominator. At the same time\nwe can not transfer Σ 1(ω,q,T) into the denominator of ξ0(ω,q)+ξ1(ω,q,T) postulating\nfor example the ”resummation procedure”\n1\nω−Emagn(q)+iǫ+Σ1(ω,q,T)\n(ω−Emagn(q))2−→1\nω−Emagn(q)−Σ1(ω,q,T),(82)\nbecauseξ(ω,q,T) a priori does not satisfy something like the Dyson equation. That is\nwhy ReΣ 1(ω,q,T) cancels when we evaluate Im ξ1(ω,q,T).\nWhat formula is more correct (62) or (80)? Obviously S(reg)\n1(ω,q,T) turns to infinity\natω→Emagn(q) whileS(DY)\n1(ω,q,T) may be singular at ω= 0 if\nωmin(q) =h−3|J|cos|q|+kgap−π\n3<0, (83)\n12(the latter condition guarantees that S(0,q,T)/negationslash= 0 [1]). Hence it seems natural to intro-\nduce the intermediate frequency 0 < ωs(q,T)< Emagn(q) defined by the condition\nS(reg)\n1(ωs(q,T),q,T) =S(DY)\n1(ωs(q,T),q,T). (84)\nAtω < ω s(q,T) one should use the formula (62) while at ω > ω s(q,T) the formula (80).\n6 Summary and discussion\nIn the present paper we evaluated the low-temperature asympto tic for TDSF of the mag-\nnetically polarized XXchain directly from the definition (15). We also confirmed the re-\nsult by alternative calculations according to the formula (18) with th e use of an auxiliary\nquantity ξ(ω,q,T) (19). We assert that the obtained formula (62) adequately desc ribes\nTDSF at very small and negative frequencies but becomes complete ly incorrect near the\nresonance. According to this result supplemented by the result of the paper [1] we have\nintroduced the frequency ωs(q,T) which separates between the small and resonance fre-\nquency regions. In the former one the TDSF is described by the for mula (62) suggested\nin the present paper while in the latter by the formula (80) obtained p reviously [1].\nAlso from (62) follows that at the O(e−βEgap) level the magnetically polarized XX\nchain does not possess a zero-frequency singularity in the TDSF [9]. So its isothermal\nand isolated transverse susceptibilities (on this level) coincide.\nA Orthogonalityand completenessofthetwo-magnon\nbasis\nLet us represent a two-magnon state in the form\n|k,κ/angbracketright=2√\nN/summationdisplay\nn1<n2eik(n1+n2)/2ϕn2−n1(κ)| ↓n1↓n2/angbracketright, (A.1)\nwhere\nϕn(κ) = ¯ϕn(κ) =2sinκn√\nN. (A.2)\nPeriodicity condition (34) may be represented now in an alternative f orm\nϕN−n(κ) = e±ikN/2ϕn(κ). (A.3)\n13According to (A.1) and (A.3) one has\n/angbracketleftk,κ|˜k,˜κ/angbracketright=1\nN/summationdisplay\n1≤n1<n2≤Nei(˜k−k)(n1+n2)/2¯ϕn2−n1(κ)ϕn2−n1(˜κ)\n=1\nNN−1/summationdisplay\nn=1¯ϕn(κ)ϕn(˜κ)N−n/2/summationdisplay\nm=n/2+1ei(˜k−k)m=1\nNN−1/summationdisplay\nn=1¯ϕN−n(κ)ϕN−n(˜κ)\n·N−n/2/summationdisplay\nm=n/2+1ei(˜k−k)(m+N/2)=1\nNN−1/summationdisplay\nn=1¯ϕn(κ)ϕn(˜κ)N−(N−n)/2/summationdisplay\nm=(N−n)/2+1ei(˜k−k)(m+N/2)\n=1\nNN−1/summationdisplay\nn=1¯ϕn(κ)ϕn(˜κ)N+n/2/summationdisplay\nm=N−n/2+1ei(˜k−k)m. (A.4)\nNow from (A.4) readily follows\n/angbracketleftk,κ|˜k,˜κ/angbracketright=1\n2NN−1/summationdisplay\nn=1¯ϕn(κ)ϕn(˜κ)N+n/2/summationdisplay\nm=1+n/2ei(˜k−k)m=δk,˜k\n2N−1/summationdisplay\nn=1¯ϕn(κ)ϕn(˜κ).(A.5)\nBut (A.2) yields\n1\n2N−1/summationdisplay\nn=1¯ϕn(κ)ϕn(˜κ) =δκ,˜κ−δκ,−˜κ=δκ,˜κ. (A.6)\nA combination of (A.5) and (A.6) gives (39).\nThe completeness condition\n/summationdisplay\nk,κ|k,κ/angbracketright/angbracketleftk,κ|=/summationdisplay\nn1<n2S−\nn1S−\nn2|∅/angbracketright/angbracketleft∅|S+\nn1S+\nn2, (A.7)\nis equivalent to the formula\nW≡4\nN2/summationdisplay\nk,κeik(n1+n2−˜n1−˜n2)/2sinκ(n2−n1)sinκ(˜n2−˜n1) =δn1˜n1δn2˜n2,(A.8)\nwhere it is implied that\nn2> n1,˜n2>˜n1. (A.9)\nIt is convenient to pass in (A.8) from kandκto the individual magnon wave numbers\nk1=k\n2−κ, k 2=k\n2+κ. (A.10)\nAccording to (30) and (34) k2> k1and\neik1N= eik2N=−1. (A.11)\n14Hence\nW=4\nN2/summationdisplay\nk1<k2...=1\n2N2/summationdisplay\nk1,k2/parenleftBig\nei[k1(n1−˜n1)+k2(n2−˜n2)]+ei[k2(n1−˜n1)+k1(n2−˜n2)]\n−ei[k1(n1−˜n2)+k2(n2−˜n1)]−ei[k2(n1−˜n2)+k1(n2−˜n1)]/parenrightBig\n=δn1˜n1δn2˜n2−δn1˜n2δn2˜n1.(A.12)\nNow (38) follows from (A.12), (A.9) and (A.8).\nB Evaluation of the form factor\nSince\n[S+(q),S−\nn] = 2e−iqn\n√\nNSz\nn,2Sz\nn|∅/angbracketright=|∅/angbracketright,/angbracketleftk−q|=1√\nN/summationdisplay\nn/angbracketleft∅|S+\nnei(q−k)n,(B.1)\none readily has\n/angbracketleftk−q|S+(q)|k,κ/angbracketright=2\nN√\nN/summationdisplay\n1≤n1<n2≤Nϕn2−n1(κ)cosα(n2−n1)\n=2\nN√\nNN−1/summationdisplay\nn=1(N−n)ϕn(κ)cosαn, (B.2)\nwhereϕn(κ) andαare given by (A.2) and (50). According to (11) and (50)\neiαN= e±ikN/2, (B.3)\nand hence (see (A.3))\nϕN−n(κ)cosα(N−n) =ϕn(κ)cosαn. (B.4)\nSo\nN−1/summationdisplay\nn=1(N−n)ϕn(κ)cosαn=N−1/summationdisplay\nn=1(N−n)ϕN−n(κ)cosα(N−n) =N−1/summationdisplay\nn=1nϕn(κ)cosαn,(B.5)\nand from (B.2) and (B.5) follows that\n/angbracketleftk−q|S+(q)|k,κ/angbracketright=1√\nNN−1/summationdisplay\nn=1ϕn(κ)cosαn=1\n2NN−1/summationdisplay\nn=1/parenleftBig\nei(κ+α)n+ei(κ−α)n\n−ei(α−κ)n−e−i(α+κ)n/parenrightBig\n=1\n2N/parenleftBig1+ei(κ+α)\n1−ei(κ+α)+1+ei(κ−α)\n1−ei(κ−α)−1+ei(α−κ)\n1−ei(α−κ)\n−1+e−i(κ+α)\n1−e−i(κ+α)/parenrightBig\n=1\nN/parenleftBig1+ei(κ+α)\n1−ei(κ+α)+1+ei(κ−α)\n1−ei(κ−α)/parenrightBig\n=2isinκ\nN(cosα−cosκ),(B.6)\nwhere we have put into account that according to (34) and (B.3)\nei(κ±α)N=−1. (B.7)\n15C Evaluation of sums\nFirst of all let us notice that according to (11) (37) and (50) one ha s\neiαlN= 1,eiαmN=−1,eiκλN=−1,eiκµN= 1, N= 2M.(C.1)\nHence\nsinαm/negationslash= 0, m= 1,...,M. (C.2)\nFirst of all we have to express the sums/summationtextM\nλ=1and/summationtextM−1\nµ=1in (73) from the sums/summationtextN\nλ=1\nand/summationtextN\nµ=1. According to (35) and (36) for an arbitrary function f(x) one has\nM/summationdisplay\nλ=1f(cosκλ) =1\n2N/summationdisplay\nλ=1f(cosκλ),\nM−1/summationdisplay\nµ=1f(cosκµ) =1\n2/parenleftBigN/summationdisplay\nµ=1f(cosκµ)−f(1)−f(−1)/parenrightBig\n. (C.3)\nAlso we shall need the identities\n1\ncosκ−cosα=i\nsinα/parenleftBig1\n1−ei(κ−α)−1\n1−ei(κ+α)/parenrightBig\n,cosα/negationslash=±1,\n1\ncosκ−cosα=2eiκ\n(cosα−eiκ)2= 2/parenleftBigcosα\n(cosα−eiκ)2−1\ncosα−eiκ/parenrightBig\n= 2cosα/parenleftBig1\n(1−cosαeiκ)2−1\n1−cosαeiκ/parenrightBig\n,cosα=±1. (C.4)\nLet\nS(z)≡N/summationdisplay\nj=11\nz−zj, zN\nj=−1. (C.5)\nSinceN= 2Mone has in (C.5) ( −zj)N=zN\nj=−1. Hence additionally to (C.5)\nN/summationdisplay\nj=11\nz+zj=S(z), zN\nj=−1. (C.6)\nObviously\nS(z) =d\ndzlogN/productdisplay\nj=1(z−zj) =d\ndzlog(zN+1) =NzN−1\nzN+1. (C.7)\nSince according to (C.1)\nei(κλ±αl)N= ei(κµ±αm)N=−1, (C.8)\n16one readily has from (C.4)\nN/summationdisplay\nλ=11\ncosκλ−cosαl=i(S(1)−S(1))\nsinαl(1−δαl,0−δ|αl|,π)\n−2cosαl(S′(1)+S(1))(δαl,0+δ|αl|,π) =−N2\n2cosαl(δαl,0+δ|αl|,π),\nN/summationdisplay\nµ=11\ncosκµ−cosαm=i(S(1)−S(1))\nsinαm= 0. (C.9)\nAt the same time according to (C.4) one has\nsin2α\n(cosκ−cosα)2=−/parenleftBig1\n1−ei(κ−α)−1\n1−ei(κ+α)/parenrightBig2\n. (C.10)\nAt sinα/negationslash= 0 this formula may be expanded to the form\nsin2α\n(cosκ−cosα)2=i\nsinα/parenleftBige−iα\n1−ei(κ−α)−eiα\n1−ei(κ+α)/parenrightBig\n−1\n(1−ei(κ−α))2−1\n(1−ei(κ+α))2.\n(C.11)\nHence\nN/summationdisplay\nλ=1sin2αl\n(cosκλ−cosαl)2= 2(S(1)+S′(1))(1−δαl,0−δ|αl|,π),=N2\n2(1−δαl,0−δ|αl|,π)\nN/summationdisplay\nµ=1sin2αm\n(cosκµ−cosαm)2= 2(S(1)+S′(1)) =N2\n2. (C.12)\nUsing now (C.9) into (C.12) one readily gets\nM/summationdisplay\nλ=1sin2αl\ncosκλ−cosαl=−N2\n4sin2αlcosαl(δαl,0+δ|αl|,π) = 0,\nM−1/summationdisplay\nµ=1sin2αm\ncosκµ−cosαm=sin2αm\n2/parenleftBig1\n1+cosαm−1\n1−cosαm/parenrightBig\n=−cosαm,\nM/summationdisplay\nλ=1/parenleftBigsin2αl\n(cosκλ−cosαl)2−2cosαl\ncosκλ−cosαl/parenrightBig\n=N2\n4/parenleftBig\n1−δαl,0−δ|αl|,π/parenrightBig\n+N2\n2/parenleftBig\nδαl,0+δ|αl|,π/parenrightBig\n=N2\n4/parenleftBig\n1+δαl,0+δ|αl|,π/parenrightBig\n,\nM−1/summationdisplay\nµ=1/parenleftBigsin2αm\n(cosκµ−cosαm)2−2cosαm\ncosκµ−cosαm/parenrightBig\n=N2\n4−1\n2/parenleftBigsin2αm\n(1−cosαm)2\n+sin2αm\n(1+cosαm)2/parenrightBig\n+cosαm\n1−cosαm−cosαm\n1+cosαm=N2\n4−1. (C.13)\nThe obtained system is equivalent to (73).\n17References\n[1] Bibikov P N 2020 Low-temperature asymptotic of the transvers e dynamical structure\nfactor for a magnetically polarized XXchainJ. Stat. Mech. 073106\n[2] Colomo F, Izergin A G, Korepin V E, Tognetti V 1993 Temperature correlation func-\ntions in the XX0 Heisenberg chain Theoret. Math. Phys. 9411-38\n[3] Mahan G D 2000 Many-Particle Physics 3rd ed. (Kluwer Academic/Plenum Publish-\ners)\n[4] Konik R M 2003 Haldane gapped spin chains: exact low temperature expansions of\ncorrelation functions, Phys. Rev. B68 104435\n[5] Essler F H L, Konik R M 2009 Finite-temperature dynamical correla tions in massive\nintegrable quantum field theories, J. Stat. Mech. P09018\n[6] Plakida N M 1973 Dyson equation for Heisenberg ferromagnet, Ph ys. Lett. A 43\n481-482\n[7] Plakida NM2011The two-timeGreen’s functionandthediagramtec hnique, Theoret.\nand Math. Phys. 1681303-1317\n[8] Tserkovnikov Yu A 1981 Method of solving infinite systems of equa tions for two-time\ntemperature Green functions, Theoret. and Math. Phys. 49993-1002\n[9] Kwok P C, Schultz T D 1969 Correlation functions and Green funct ions: zero-\nfrequency anomalies, J. Phys. C 21196-1205\n18"    },    {        "title": "1704.03749v1.Ultra_fast_magnetization_manipulation_using_single_femtosecond_light_and_hot_electrons_pulse.pdf",        "content": "1\t\tUltra-fast magnetization manipulation using single femtosecond light and hot-electrons pulse  Y. Xu1,2, M. Deb*1, G. Malinowski1, M. Hehn,1 W. Zhao2, S. Mangin1  1) Institut Jean Lamour, CNRS UMR 7198, Université de Lorraine, 54506 Vandœuvre-lès-Nancy, France 2) Fert Beijing Research Institute, BDBC Beihang University, Beijing 100191, China  Abstract:  Current induced magnetization manipulation is a key issue for spintronic application. Therefore, deterministic switching of the magnetization at the picoseconds timescale with a single electronic pulse represents a major step towards the future developments of ultrafast spintronic. Here, we have studied the ultrafast magnetization dynamics in engineered Gdx[FeCo]1-x based structure to compare the effect of femtosecond laser and hot-electrons pulses. We demonstrate that a single femtosecond hot-electrons pulse allows a deterministic magnetization reversal in either Gd-rich and FeCo-rich alloys similarly to a femtosecond laser pulse. In addition, we show that the limiting factor of such manipulation for perpendicular magnetized films arises from the multi-domain formation due to dipolar interaction. By performing time resolved measurements under various field, we demonstrate that the same magnetization dynamics is observed for both light and hot-electrons excitation and that the full magnetization reversal take place within 5 ps. The energy efficiency of the ultra-fast current induced magnetization manipulation is optimized thanks to the ballistic transport of hot-electrons before reaching the GdFeCo magnetic layer.     *Correspondence to:  M. Deb (Marwan.deb@univ-lorraine.fr)  .    2\t\tEnergy efficient and ultrafast magnetization manipulation without any applied magnetic field is of utmost importance for both future spintronic application and fundamental understanding of ultrafast magnetization dynamics. Indeed the dynamic response of magnetic order to ultrafast external excitation is a fascinating issue of modern magnetism1. Moreover an efficient mean of magnetization manipulation is of prime importance for information and memory storage devices. During the last ten years the effect of spin-transfer torque (SST)2–5, spin-orbit torque6,7, electric field8–10 and femto- or picosecond lasers pulses11–13 have motivated numerous theoretical and experimental investigations. The recent discoveries in the field of all optical switching have attracted a lot of attention especially the All Optical Helicity Independent Switching (AO-HIS) of GdFeCo using a single femtosecond laser pulse. The exact microscopic mechanisms still need to be understood, but both experimental and theoretical results indicate the pure thermal origin of AO-HIS in GdFeCo11,14,15. Concerning applications, the magnetization switches within 30 ps which allows pushing the speed of the recording technology to high frequencies above 10 GHz16. However in the context of spintronic devices, the magnetization switching induced by electronic currents and electric field are particularly interesting as they can be implemented in nanoscale devices17,18.  The technological importance of the ultrafast AO-HIS in GdFeCo immediately raised the question of the possibility to obtain the same reversal speed with a single electron pulse, which can be adapted for nanoscale technologies and open the field of ultrafast-spintronic. Hot-electrons transport has been recently presented as an alternative way to manipulate spins and has been demonstrated to offer an extensive control of the ultrafast demagnetization in metallic multilayers19, as initially demonstrated in Ni film by femtosecond laser pulses20. Furthermore, spin polarized hot-electrons allows controlling the speed of demagnetization21 and even initiate coherent magnetization precession by STT22. More recently, Williams et al. have shown the possibility to reverse the magnetization by hot-electrons in GdFeCo23. However, the timescale of such a magnetization reversal remains unknown. In addition, this magnetization reversal is only reported for one composition of GdFeCo. These results triggered different fundamental questions: What exactly is the timescale of this switching process? Is it similar to the one reported for the light induced AOS?  Since the demagnetization timescale of rare-earth and transition metal magnetic atoms is different, can hot-electrons reverse the magnetization in both Gd-rich and FeCo-rich GdFeCo alloys? If yes, does the two reversal timescales are the same? What is the hot-electrons transport mechanism?  3\t\tIn order to answer these questions, we have carefully studied the magnetization dynamics induced in Gdx[FeCo]1-x by hot-electrons and laser pulses as a function of the film composition, laser fluence, and external magnetic field (see suplementary material). In addition to switching induced by direct excitation of the laser pulse, we evidence that a single hot-electrons pulse reverses the magnetization in both Gd-rich and FeCo-rich samples without requiring any additional magnetic field. Furthermore, we demonstrate that the magnetization reversal process takes in a few picoseconds and that thanks to a ballistic hot-electrons transport ultra-short electrons pulse allows energy efficient magnetization manipulation.  I. EXPERIMENTAL METHOD AND STATIC SAMPLE PROPERTIES: Our study is based on a series of 5 nm thick Gdx[FeCo]1-x layers made of an amorphous alloy with different concentrations of Gd ranging from 20 to 30 %. The alloys are ferrimagnetic metallic materials, having two antiferromagnetically exchange coupled sublattices, formed by the magnetic moments of the transition metals Fe86Co14 (FeCo in the following) and the rare earth Gd. Therefore, by tuning the composition of the Gdx[FeCo]1-x it is possible to tune the magnetic properties, which paves the way for a general understanding of the physical phenomena. The investigated samples were deposited by dc magnetron sputtering onto a glass substrate according to the following artificial multilayer structure: Glass/Ta (3 nm)/Pt (5 nm)/ Cu (80 nm)/Gdx[FeCo]1-x (5 nm)/Ta (5 nm). The thin Ta capping layer prevents the oxidation of the magnetic film, while allowing probing the magnetic properties via the magneto-optical Kerr effect (MOKE).   Figure 1(a) shows the composition dependence of the coercive field (Hc) and the saturation magnetization (Ms) in the 5 nm Gdx[FeCo]1-x films, which are respectively measured at room temperature (RT) via MOKE in polar configuration and superconducting quantum interference device-vibrating sample magnetometer (SQUID-VSM). For all compositions and with the field applied perpendicular to film plane, the films have a square hysteresis loop revealing a perpendicular magnetic anisotropy and so a magnetization oriented perpendicular to the film plane. At room temperature, the net magnetization (Ms) reaches zero at x=26%, which corresponds to the compensation concentration (xcomp) of the system, where the magnetization of both Gd (MGd) and transition metal sublattices (MFeCo) are equal. Therefore, we show as expected at xcomp a clear divergence of HC. Ms is thus aligned in the direction of the CoFe moments below xcomp while it changes its sign and becomes aligned with the Gd 4\t\tmoments above xcomp. This is in agreement with the sign inversion observed between the hysteresis loops for Gd-rich and CoFe-rich [inset of Fig.1(a)] measured via the Kerr rotation (ΘK), since ΘK is mainly sensitive to the FeCo moments in the visible light (λ= 628 nm)24,25. Let us mention that with increasing the temperature of Gd-rich sample, Ms vanishes at the so-called compensation temperature (TM), while in CoFe-rich sample TM is below the RT 24. Using our engineered multilayer structures combined with femtosecond laser pulses, we can investigate the ultrafast magnetization dynamics induced by two types of ultrashort excitations: (i) An optical excitation as the laser pulse interacts directly with the magnetic film when the sample is irradiated from the front side [Fig. 1(b)] (ii) An indirect excitation induced only by hot-electrons, which can be performed by an optical pumping of the sample from the back side [Fig. 1(c)]. In this configuration, the magnetic film is buried under\tan 80 nm thick Cu layer, leading to a very low transmission at 800 nm (less than 0.1%)19. The ultrafast magnetization dynamics induced separately by femtosecond laser and hot-electrons pulses were investigated by the pump-probe configurations sketched in the Fig. 1(b) and Fig. 1(c), respectively. The pump and probe beams are generated from a 35 fs laser pulses delivered by a Ti:sapphire regenerative amplifier operating at 800 nm with a repletion rate of 5 KHz. The pump beam (800 nm) excites the sample at normal incidence, while the probe beam (400 nm) has a small angle of incidence of about 5°. They are both linearly polarized and focused onto the sample. The spot diameters of the pump are ~260 µm and ~330 µm for the back and the front sides of the sample, respectively, while the spot diameter of the probe is 60 µm. During the pump probe measurements, a static external magnetic field Hext of 1T is applied perpendicularly to the film plane, which allows saturating the samples and therefore reset the magnetic state before every laser pulse. The differential Kerr rotation ΘK signals (ΔΘK) are detected with a polarization bridge and synchronous detection scheme as a function of the delay between the pump and the probe. A static Kerr microscope is also used to observe the magnetic domains induced after the excitation of the sample by 800 nm femtosecond laser pulses with a spot size of ~50 µm.      \tII. RESULTS AND DISCUSSION: II.1 ULTRA-FAST MAGNETIZATION DYNAMICS FOR A BROAD RANGE OF CONCENTRATIONS:  Figure 2(a) displays the MOKE images obtained after one, then two, three and four direct light excitations of Gdx[FeCo]1-x films by femtosecond laser pulses. A single shot helicity independent all optical switching (AO-HIS) is observed in samples with an amount of Gd 5\t\t(23<x<28) corresponding to concentration above and below the compensation concentration (xcomp = 26). However, for the concentration far from xcomp, a multi-domain state is obtained (x = 20.5). This result tend to show that the single pulse (AO-HIS) can be observed only if the magnetic domain size is larger than the light spot, as it has been shown for multiple pulse All Optical helicity dependent switching (AO-HDS)26. This criteria is fulfilled for very thin films and/or low saturation magnetization (Ms) which can be obtained near xcomp26,27. This observation suggests that AO-HIS could be observed for a larger range of composition in the case of a smaller laser spot or a thinner film. A more detailed analysis is proposed in the supplementary material ( see Figure S1). The images obtained for various laser fluences (fig. 2a) prove that samples having a composition closer to compensation require less power to switch. Moreover, the influence of the laser fluence on the switching confirms the existence of a power threshold which depends on the concentration.  Figure 2b displays the MOKE images obtained after one, then two, three and four electron pulses excitation generated by a laser pulse of difference fluences. Again Both Gd-rich samples and FeCo-rich samples show clear single pulse switching. From the comparison between Fig. 2a and 2b we can conclude that a similar behavior is observed when the sample is excited with a hot-electrons pulse or with a direct laser pulse. Even the evolution as a function of the laser fluence is qualitatively similar for both excitations. Compared to the direct excitation, the laser fluence needed to create the hot electron pulse has to be 4 times higher than for the direct light pulse to observe similar switching. In the supplementary material Figure S1 present the comparison of the MOKE images obtained for light and electron pulses for six different concentrations. For a better understanding of the magnetization reversal mechanism, time-resolved pump-probe technique was employed to monitor the magnetization dynamics with a temporal resolution lower than 50 fs. The time-resolved MOKE (TR-MOKE) measurements of the magnetization dynamics induced by laser and hot-electrons pulses in GdxFeCo1-x films are displayed in Fig. 3(a) and Fig. 3(b), respectively. The amplitude of Hext is adjusted to be slightly larger than HC for each sample in order to ensure the same initial magnetic state. Let us first focus on the magnetization dynamics induced by the laser pulses. An ultrafast demagnetization occurring within 1 ps is observed for all samples. After this ultrafast demagnetization, we distinguish three different behaviors depending on the composition of Gdx[FeCo]1-x. The first is characterized by a rapid relaxation of the magnetization back to the initial state as can been seen for the sample with xGd=20.5%, which has a high Ms and didn’t 6\t\tshow AO-HIS using MOKE (Fig. 2a). The second occurs for samples with moderate Ms (xGd=22.5% and 28.6%) which have shown multi-domain state (Fig. 2a). It is characterized by a long lived fully demagnetized state. The third behavior is the switching toward negative values which is observed for the compositions near xcomp, where the samples are characterized by low Ms and consequently large magnetic domains. As it will be demonstrated the first behavior is observed when the Zeeman energy is large and that consequently magnetization tend to follow the applied field and relax back to its initial state. The second is observed when the magnetic layer tends to brake into small domains due to the dipolar field leading to a zero net magnetization. The third behavior is a clear demonstration of single pulse AO-HIS which is obtained for both CoFe-rich and Gd-rich samples and confirm the previous MOKE images. For both type of concentrations it occurs qualitatively on the same time scale. Indeed, after a full demagnetization within ~1 ps, the magnetic state shows a slow variation until ~3.2 ps, then a high speed reversal process is initiated until reaching almost 100% reversal within ~40 ps. All the previous steps are confirmed and presented in the supplementary material figure S4 by measuring the hysteresis loops at different delay times . A clear sign inversion of ΘK is observed after a time delay of 5 ps. The TR-MOKE induced by hot-electrons is displayed in Fig. 3b. Like for the optical excitation, a sub 5ps magnetization reversal is induced by hot-electrons for samples with a composition near xcomp. In addition, for the concentration far from xcomp, a long lived full demagnetization state or a rapid relaxation toward the initial state are observed depending on the sample. Let us mention that for the sample with xGd of 22.2%, showing single pulse AO-HIS in the static Kerr microscopy images, only a long lived full demagnetized state is obtained in TR-MOKE measurements. The main difference between the two measurements is the pump spot size. The large spot size in the TR-MOKE setup makes more likely the formation of magnetic domains and could explain the origin of the difference26. Figure 3c and 3d show the power dependence of TR-MOKE signal measured 100 ps after excitation of the sample by laser and hot-electrons pulse. For both excitations, three different behaviors can clearly be distinguished as a function of the laser fluence. First, a recovery of the magnetization toward its initial state is shown at low fluence for all samples (ΔΘK/ΘK at 100 ps ≈1). Second, a long lived partial demagnetization state is also observed for all samples (1>ΔΘK/ΘK at 100 ps >0). Finally, an AO-HIS is obtained above a threshold fluence within a certain composition range around xcomp (22<x<28). At a very high fluence, the signal measured for samples showing AO-HIS decreases (Fig. 3d, xGd= 25.6%). This is due to the creation of a multidomain state, which is located in the center of the pump beam as directly observed by Kerr microscopy. Those conclusions are in accord with the 7\t\tmagnetization dynamic as a function of the magnetic field amplitude presented in the supplementary materials. Finally, note that the value of laser fluence cannot be directly related to the energy absorbed by the GdFeCo films. Indeed, for example in the hot-electrons configuration, the Cu (80nm) layer reflects more than 85% of the incident light.  To further evaluate the importance of the magnetization switching by hot-electrons for nanoscale spintronic purposes, it is interesting to compare their characteristics with those obtained in the STT-MRAM technology. To determine the intrinsic hot-electrons switching energy, ES, we have used the transfer matrix calculation to estimate the absorption profile of the sample1. We find that only ~12% of the light is absorbed by the Pt/Cu layer. To determine the switching energy in different structure size, one can use the following formula which is adapted from ref28: 𝐸!=𝐹𝐴𝛼 Where F is the switching threshold fluence, A is the switched area, and α is the absorption coefficient at the pump wavelength. For a switching threshold fluence of ~10 mJ.cm-2, we estimate a hot-electrons switching energy ES  of ~4 fJ  for an area of 20 × 20 nm2. This energy value is one order of magnitude lower than the one required in STT-MRAM memories29,30. In addition, the switching speed obtained by hot-electrons is in the picosecond timescale, which is about thousand times faster than in the case of STT-MRAM29,30.              II.2 Cu THIKNESS DEPENDENCE OF THE ULTRAFST MAGNETIZATION DYNAMICS:  To reveal the mechanisms behind magnetization switching using hot-electrons, the ultrafast magnetization dynamics induced by hot electrons in Glass/Ta (3 nm)/Pt (5 nm)/ Cu (tCu)/Gd24.5[FeCo]75.5 (5 nm)/Ta (5 nm) films are studied as a function of the Cu thickness (tCu) which is varied from 5 to 200 nm. In order to determine the zero time delay with high accuracy, each sample is patterned in alternating areas with and without Cu, which gives the time for which direct excitation occurs. As tCu increases, the laser fluence needs to be increased to generate hot electrons and reverse the magnetization due to an increase in the sample reflection. To compare the magnetization dynamics for all the samples, we adjusted the laser fluence to obtain the same demagnetization amplitude as in the case of direct laser excitation for a fluence of 3.2mJ.cm-2(Fig. 3a). TR-MOKE measurements as a function of tCu are displayed in Fig. 4a. Two important features can be identified. First, a full demagnetization is reached within 2 ps for tCu =200 nm, which is totally opaque to the light. 8\t\tThis clearly establishes that hot-electrons transport is at the origin of the observed magnetization dynamics. We also emphasize that, for all Cu thickness, hot-electrons can reverse the magnetization of GdFeCo films. This result clearly confirms that the hot-electrons only are reversing the magnetization. Second, the onset time of the magnetization dynamics continuously increases as tCu increases. To determine the characteristic properties of the demagnetization process as a function of Cu thickness, we adjusted the TR-MOKE signals with the following empirical model:   𝛥𝛩! (𝑡)𝛩!=𝐴!−𝐴!/1+𝑒!!!!!!.!!!+𝐴! Where A1 is the amplitude of the signal at negative time delay, A2 is the value reached at maximum demagnetization, t0 is the time corresponding to 10 % of the maximum demagnetization, and τ is the demagnetization characteristic time. The results of this fits allow obtaining the relative time delay Δt0=[t0(tCu)-t0(d=5nm)] and the modification of the characteristic demagnetization time Δ τ =[τ(d)-τ(d=5nm)], which are displayed respectively in Fig. 4.b and Fig. 4.c. To compare with literature, we plot also in these figures the Cu thickness dependence of the induced Δt0 and Δτ in the ultrafast demagnetization of Cu(tCu)/[Co/Pt](4nm) multilayers19. One observes that Δt0 increases linearly with the Cu thickness. The slope of the linear fit yield a velocity v= 0.53 106  m/s, which is of the same order of magnitude as found in Cu (tCu)/[Co/Pt] (~0.68 106 m/s). The high velocity value is a direct imprint of hot-electrons ballistic transport. Furthermore, the ultrafast demagnetization in Cu(tCu)/[ Co/Pt] can be reproduce only by theoretical simulation based on hot-electrons ballistic transport19. These two properties rule out the contribution of pure thermal transport in the ultrafast process induced by hot-electrons. Furthermore, we show that Δτ increases linearly with Cu thickness. This can principally be related to the broadening of the hot-electrons pulse. On the other hand, the corresponding increases of Δτ is found more pronounced in GdFeCo than Co/Pt multilayers. One possible reason for this difference is that the demagnetization process induced by hot-electrons in the localized 4f moment in Gd is slower than in 3d magnetic moment in Co. Another reason could be that the hot-electrons attenuation within the ferromagnetic layer, which can be slightly higher in GdFeCo with thickness of 5nm than in Co/Pt multilayers with a total thickness of 4 nm.   III. CONCLUSION: In conclusion, our results clearly demonstrate that deterministic magnetization manipulation in thick GdFeCo films can be observed using either a single femto-second laser pulse or a 9\t\tsingle femto-second hot-electrons pulse without requiring any magnetic field. In both cases the effect is shown for a broad range of GdFeCo compositions above and below the compensation. One of the limiting factors for the observation of the magnetization switching is the presence of the stray field that generates multi-domain state in the case of large magnetization. Importantly we prove by studying the time-resolved magnetization dynamics that the reversal process induced by hot-electrons and light are similar and take place within 5 ps for both Gd-rich and FeCo-rich films. We suggest that the similar magnetization dynamics obtained by both type of excitations is related to the ballistic nature of hot-electrons transport which we clearly demonstrated in our especially engineer structure. Finally, we are convinced that such a fast magnetization switching using electron pulse opens the door to ultra-fast spintronic.  METHODS The AO-HDS measurements were performed using optical pulses, having a central wavelength of 800 nm (1.55 eV), a pulse duration of about 35 fs at the sample position and a 5 kHz repetition rate. The response of the magnetic film was studied using a static Kerr microscope to image the magnetic domains after the laser excitation. The present measurements are performed at room temperature (RT).  ACKNOWLEDGMENTS: The authors thanks C. Chang, M.S. El Hadri, T. Fache, E.E. Fullerton, T. Hauet, G. Kichin, G. Lengaigne, F. Montaigne, Y. Quessab, P. Scheid, P. Vallobra, and B. Zhang for fruit-full discussion and technical help.  The authors gratefully acknowledge the financial support of the Agence National de la recherche in France via the projects COMAG: # ANR-13-IS04-0008-01 and UMANI: # ANR-15-CE24-0009 and the International Collaboration Project B16001 and National Natural Science Foundation of China (Grant No. 61627813). Experiments were carried out on IJL Project TUBE-Davms equipments funded by FEDER (EU), PIA (Programme Investissemnet d’Avenir), Region Grand Est, Metropole Grand Nancy and ICEEL.    AUTHOR CONTRIBUTIONS: M. D, M.H, G.M and S.M., designed and coordinated the project; Y.X grew, characterized and optimized the samples. Y.X. M. D and G.M designed 10\t\tand operated the Faraday microscope and the pump laser set-up. M.D and S.M coordinated work on the paper with contributions from Y.X, M.H, G.M and W.Z and regular discussions with all authors      11\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tFigure 1: Quasi-static room-temperature magnetic properties of Glass/Ta (3 nm)/Pt (5 nm)/ Cu (80 nm)/Gdx[FeCo]1-x (5 nm)/Ta (5 nm) films for (20 %< xGd <30%) and the pump-probe experimental configurations. (a) Variation of the coercive field HC and the saturation magnetization Ms as a function of the Gd content (xGd). The inset shows a normalized polar hysteresis loops for the films with xGd of 23.9% and 27%. (b,c) Sketch of the time resolved experimental configurations allowing to separately study the ultrafast magnetization dynamics induced by single light pulse and hot-electron pulse.   12\t\t  \n Figure 2: Magneto-optical Kerr images obtained after the excitation of Gdx(FeCo)1-x films by (a) one then two, three and four direct light pulses for five different values of laser fluences ranging from 0.75   to 3.5 mJ.cm-2 and (b) one then two, three and four hot-electron pulses for five different values of laser fluences ranging from 4 to 12.25 mJ.cm-2.     \t\t\t\n13\t\t050100150200250-1.0-0.50.00.51.0050100150200250-1.0-0.50.00.51.0\n0510152025-1.0-0.50.00.51.0\n01234-1.0-0.50.00.51.0Hot-electron (a)xGd =  20.5 %  27.1 %  22.2 %  28.6 %    23.9 %                     25.6 %ΔΘK(t)/ΘKsatTime (ps)3.2 mJ/cm2  Laser \nΔΘK(t)/ΘKsat(b)xGd =  15 mJ/cm2   20.5 %  27.1 % 22.2 %   28.6 %   23.9 %                     25.6 %Time (ps)(d)ΔΘK(t=100 ps)/ΘKsat\nLaser fluence (mJ.cm-2)  20.5 %  22.2 %  23.9 %  25.6 %  27.1 %  28.6 %xGd = xGd = ΔΘK(t=100 ps)/ΘKsat\nLaser fluence (mJ.cm-2)  20.5 %  22.2 %  23.9 %  25.6 %  27.1 %  28.6 %(c)\t\t\t\t\t\t\t\t\t\t\t\tFigure 3: Comparison between the magnetization dynamics induced in 5 nm thick Gdx(FeCo)1-x films by femtosecond laser and hot-electron pulses for six different concentration above and below the compensation concentration. (a,b) ΔΘK (t)/ΘKsat induced by laser (a) and hot-electrons (b) pulses after excitation of the samples with fixed laser fluencies of 3.2 mJ/cm2 and 15 mJ/cm2, respectively. (c, d) Laser fluence dependence of the ΔΘK /ΘKsat signal measured at 100 ps after excitation of samples by laser (c) and hot-electron (d) pulses.            14\t\t     15\t\t-0.50.00.51.01.52.00.000.250.500.751.001.25\n05010015020025030001002000200400(a) 200 nm 160 nm 140 nm 80 nm 40 nm RefΔΘK/ΘKsatTime (ps)Cu thickness(c)\nΔτ ( fs )Cu thickness (nm)(b)\nΔt0 ( fs )                Figure 4: Magnetization dynamics in Glass/Ta (3 nm)/Pt (5 nm)/ Cu (tCu)/Gd24.5[FeCo]75.5 (5 nm)/Ta (5 nm) films measured as a function of Cu thickness (Glass/Ta (3 nm)/Pt (5 nm)/ Cu (tCu)/Gd24.5[FeCo]75.5 (5 nm)/Ta (5 nm)). (a) ΔΘK (t)/ΘKsat measured as a function of Cu thickness. The solid line is a fit using Eq. (1). (b,c) Comparison of the Cu thickness dependence of Δt0 (b) and Δτ (c) obtained in this work (open circles) and those reported for the ultrafast demagnetization in Cu(tCu)/[ Co/Pt] multilayers (adapted from ref. 19). The line is a linear fit to the data.          16\t\t1. Bigot, J. Y., Hübner, W., Rasing, T. & Chantrell, R. Ultrafast Magnetism I. Springer Proc. Phys. 159, (2015). 2. Katine, J. A. Current-Driven Magnetization Reversal and Spin-Wave Excitations in Co <span class. Phys. Rev. Lett. 84, 3149–3152 (2000). 3. Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353–9358 (1996). 4. Mangin, S. et al. Current-induced magnetization reversal in nanopillars with perpendicular anisotropy. Nat. Mater. 5, 210–215 (2006). 5. Mangin, S., Henry, Y., Ravelosona, D., Katine, J. A. & Fullerton, E. E. Reducing the critical current for spin-transfer switching of perpendicularly magnetized nanomagnets. Appl. Phys. Lett. 94, 012502 (2009). 6. Liu, L. et al. Spin-Torque Switching with the Giant Spin Hall Effect of Tantalum. Science 336, 555–558 (2012). 7. Liu, L., Lee, O. J., Gudmundsen, T. J., Ralph, D. C. & Buhrman, R. A. Current-induced switching of perpendicularly magnetized magnetic layers using spin torque from the spin Hall effect. Phys. Rev. Lett. 109, 096602 (2012). 8. Ohno, H. et al. Electric-field control of ferromagnetism. Nature 408, 944–946 (2000). 9. Chiba, D., Yamanouchi, M., Matsukura, F. & Ohno, H. Electrical manipulation of magnetization reversal in a ferromagnetic semiconductor. Science 301, 943–945 (2003). 10. Chiba, D. et al. Magnetization vector manipulation by electric fields. Nature 455, 515–518 (2008). 11. Kirilyuk, A., Kimel, A. V. & Rasing, T. Laser-induced magnetization dynamics and reversal in ferrimagnetic alloys. Rep. Prog. Phys. 76, 026501 (2013). 12. Mangin, S. et al. Engineered materials for all-optical helicity-dependent magnetic switching. Nat. Mater. 13, 286–292 (2014). 13. Lambert, C.-H. et al. All-optical control of ferromagnetic thin films and nanostructures. Science 345, 1337–1340 (2014). 14. Ostler, T. A. et al. Ultrafast heating as a sufficient stimulus for magnetization reversal in a ferrimagnet. Nat. Commun. 3, 666 (2012). 15. Radu, I. et al. Transient ferromagnetic-like state mediating ultrafast reversal of antiferromagnetically coupled spins. Nature 472, 205–208 (2011). 16. Vahaplar, K. et al. Ultrafast Path for Optical Magnetization Reversal via a Strongly Nonequilibrium State. Phys. Rev. Lett. 103, 117201 (2009). 17. Chen, X. et al. Nanoscale magnetization reversal caused by electric field-induced ion migration and redistribution in cobalt ferrite thin films. ACS Nano 9, 4210–8 (2015). 18. Zhang, C., Fukami, S., Sato, H., Matsukura, F. & Ohno, H. Spin-orbit torque induced magnetization switching in nano-scale Ta/CoFeB/MgO. Appl. Phys. Lett. 107, 012401 (2015). 19. Bergeard, N. et al. Hot-Electron-Induced Ultrafast Demagnetization in Co/Pt Multilayers. Phys. Rev. Lett. 117, 147203 (2016). 20. Beaurepaire, E., Merle, J.-C., Daunois, A. & Bigot, J.-Y. Ultrafast Spin Dynamics in Ferromagnetic Nickel. Phys. Rev. Lett. 76, 4250–4253 (1996). 21. Malinowski, G. et al. Control of speed and efficiency of ultrafast demagnetization by direct transfer of spin angular momentum. Nat. Phys. 4, 855–858 (2008). 22. Choi, G.-M., Min, B.-C., Lee, K.-J. & Cahill, D. G. Spin current generated by thermally driven ultrafast demagnetization. Nat. Commun. 5, 4334 (2014). 23. Wilson, R. B. et al. Ultrafast Magnetic Switching of GdFeCo with Electronic Heat Currents. arXiv: 1609, 05155 (2016). 17\t\t24. Hansen, P., Clausen, C., Much, G., Rosenkranz, M. & Witter, K. Magnetic and magneto\u0000optical properties of rare\u0000earth transition\u0000metal alloys containing Gd, Tb, Fe, Co. J. Appl. Phys. 66, 756–767 (1989). 25. Weller, D., Reim, W. & Schrijner, P. Electronic and magneto-optical properties of ion beam sputtered rare earth-FeCo thin films. IEEE Trans. Magn. 24, 2554–2556 (1988). 26. El Hadri, M. S. et al. Domain size criterion for the observation of all-optical helicity-dependent switching in magnetic thin films. Phys. Rev. B 94, 064419 (2016). 27. Hassdenteufel, A. et al. Low-remanence criterion for helicity-dependent all-optical magnetic switching in ferrimagnets. Phys. Rev. B 91, 104431 (2015). 28. Savoini, M. et al. Highly efficient all-optical switching of magnetization in GdFeCo microstructures by interference-enhanced absorption of light. Phys. Rev. B 86, 140404 (2012). 29. Apalkov, D. et al. Spin-transfer torque magnetic random access memory (STT-MRAM). ACM J. Emerg. Technol. Comput. Syst. JETC 9, 13 (2013). 30. Liu, J. P., Fullerton, E., Gutfleisch, O. & Sellmyer, D. J. Nanoscale magnetic materials and applications. (Springer, 2009).    "    },    {        "title": "2203.02145v2.Second_order_magnetic_responses_in_quantum_magnets__Magnetization_under_ac_magnetic_fields.pdf",        "content": "Second-order magnetic responses in quantum magnets:\nMagnetization under ac magnetic \felds\nTatsuya Kaneko1, Yuta Murakami2, Shintaro Takayoshi3, and Andrew J. Millis4;5\n1Computational Quantum Matter Research Team,\nRIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n2Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan\n3Department of Physics, Konan University, Kobe, Hyogo 658-8501, Japan\n4Department of Physics, Columbia University, New York, New York 10027, USA\n5Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA\n(Dated: May 23, 2022)\nWe investigate second-order magnetic responses of quantum magnets against ac magnetic \felds.\nWe focus on the case where the zcomponent of the spin is conserved in the unperturbed Hamiltonian\nand the driving \feld is applied in the xyplane. We \fnd that linearly polarized driving \felds induce a\nsecond-harmonic response, while circularly polarized \felds generate only a zero-frequency response,\nleading to a magnetization with a direction determined by the helicity. Employing an unbiased\nnumerical method, we demonstrate the nonlinear magnetic e\u000bect driven by the circularly polarized\n\feld in the XXZ model and show that the magnitude of the magnetization can be predicted by the\ndynamical spin structure factor in the linear response regime.\nI. INTRODUCTION\nNonlinear responses are important phenomena for\nprobing and controlling quantum states of matter under\nstrong electromagnetic \felds [1]. Nonlinear responses of\ncharge degrees of freedom to electric \felds have been ex-\ntensively investigated [2, 3]. Recent work along these\nlines includes high-harmonic generation (HHG) realized\nin solids [4{10] and bulk photovoltaic e\u000bect in noncen-\ntrosymmetric materials [11{17]. These nonlinear phe-\nnomena are closely related to electronic band structures\nand collective excitations [18{25].\nIn analogy to electronic charge responses, nonlinear\ne\u000bects driven by magnetic \felds in quantum magnets\nare directly related to structures of spin excitations. Al-\nthough the energy scale of spin excitations ( \u0018Jexchange\ncoupling) is much lower than a charge gap in a typi-\ncal magnetic insulator, the development of the terahertz\n(THz) laser technique opens a pathway to address non-\nlinear magnetic e\u000bects associated with low-energy mag-\nnetic excitations [26, 27]. In this context, several nonlin-\near magnetic phenomena in the THz regime have been\nproposed theoretically. For example, magnetic HHG un-\nder linearly polarized \felds [28, 29] and magnetization\ninduced by circularly polarized \felds [30, 31] have been\ndemonstrated numerically in driven quantum spin sys-\ntems. While these numerical studies address the higher-\norder e\u000bects nonperturbatively, it is important to formu-\nlate the nonlinear phenomena in the perturbative regime,\nwhere the connection to equilibrium formulas allows for\nphysical insight, which is most relevant to experiments.\nIn this paper, we investigate second-order magnetic re-\nsponses in quantum magnets, where the general structure\nof the theory can be elucidated. In particular, assuming\nthat thezcomponent of spin is conserved in the unper-\nturbed system, we derive the magnetic responses to ac\nmagnetic \felds applied perpendicular to the zaxis. We\n\fnd that while the linearly polarized \feld (with the drivefrequency \n) can produce a 2\n component of the mag-\nnetization, this 2\n oscillation is absent when circularly\npolarized \felds are applied. However, applied circularly\npolarized \felds produce a zero-frequency component of\nthe magnetization, whose direction depends on the helic-\nity of the applied \feld. Our results are consistent with\nthe numerical studies of Refs. [28] and [30]. Speci\fcally,\nwhen the total magnetization vanishes in equilibrium and\nthe excited states by the spin raising and lowering are\nsymmetric in the spectrum, the spin magnetization m(2)\nz\nat the second order is expressed as\ndm(2)\nz\ndt=i\r2\u001f+\u0000\ns(q= 0;\n) [B(\n)\u0002B(\u0000\n)]z;\nwhere\ris the gyromagnetic ratio, \u001f+\u0000\ns(q;\n) is the dy-\nnamical (transverse) spin structure factor at the momen-\ntumq, andB(\n) is the in-plane magnetic \feld at \n. We\ncan see that the magnetization direction is controlled by\nthe helicity of the applied magnetic \feld, and the mag-\nnitude is predicted by the dynamical spin structure fac-\ntor in the linear response regime. This nonlinear mag-\nnetic e\u000bect owing to low-energy spin excitations should\nbe contrasted with the inverse Faraday e\u000bect in metals,\nwhich is described by the cross product of the electric\n\feldE(\n)\u0002E(\u0000\n) [32], because the inverse Faraday\ne\u000bect in metals is essentially caused by electronic orbital\ndegrees of freedom [33, 34]. We also demonstrate the\nnonlinear magnetic e\u000bect in a driven XXZ model employ-\ning the in\fnite time-evolving block decimation (iTEBD)\nmethod [35] and con\frm the above relation numerically.\nThe rest of this paper is organized as follows. In Sec. II,\nwe introduce the spin model that we address. In Sec. III,\nwe derive the magnetization in the second order and dis-\ncuss the polarization dependence. Then, we focus on the\nmagnetization induced by the circularly polarized \feld.\nIn Sec. IV, we provide the results of the numerical demon-\nstration. Discussions and summary are given in Sec. V.arXiv:2203.02145v2  [cond-mat.str-el]  20 May 20222\nII. MODEL\nWe consider a system under a magnetic \feld;\n^H(t) =^H0\u0000~\rB(t)\u0001^S; (1)\nwhere ^H0is the Hamiltonian of the spin system and the\nsecond term is the Zeeman coupling between the total\nspin ^S(=P\nj^Sj) and the external magnetic \feld B(t).\n^S\u0017\nj(\u0017=x;y;z ) is the spin operator at site jand\r=\n\u0000g\u0016B=~is the gyromagnetic ratio, where gis thegfactor,\n\u0016Bis the Bohr magneton, and ~is the Planck constant.\nIn this paper, we assume [ ^H0;^Sz] = 0, i.e., the eigenstate\nj miof^H0satis\fes ^H0j mi=~!mj miand ^Szj mi=\nSz\nmj mi, where ~!mandSz\nmare the eigenenergy and\nquantum number of ^Sz, respectively. In this paper, we\nfocus on the magnetization Mz(t) =h^Sz(t)iunder the\nmagnetic \feld B(t) = (Bx(t);By(t);0). The magnetiza-tion in the ground state is M(0)\nz=h 0j^Szj 0i=Sz\n0.\nIII. SECOND-ORDER MAGNETIC EFFECTS\nA. Magnetization\nWe derive the \feld-induced magnetization using the\nperturbation theory at zero temperature (see details in\nAppendix A). Note that we do not assume M(0)\nz= 0\nat this stage. The magnetization at the \frst order in\nB(t) vanishes [i.e., M(1)\nz(t) = 0] because the perturba-\ntion ^V(t) =\u0000~\r[Bx(t)^Sx+By(t)^Sy] induces the spin\n\rip ( ^S\u0006) andh 0j^Sz^S\u0006j 0i= 0. Then, the lowest order\nof the \feld-induced magnetization is of the second or-\nder. Using the perturbative expansion (see Appendix A),\nthe time-dependent magnetization at the second order\nM(2)\nz(t) is given by\nM(2)\nz(t) =\r2\n4Zt\n\u00001dt1Zt\n\u00001dt2X\n\u0010=\u0006X\nm\u0010jh \u0010\nmj^S\u0010j 0ij2e\u0000i(!\u0010\nm\u0000!0)(t1\u0000t2)B(t1)\u0001B(t2)\n+\r2\n4iZt\n\u00001dt1Zt\n\u00001dt2X\n\u0010=\u0006X\nmjh \u0010\nmj^S\u0010j 0ij2e\u0000i(!\u0010\nm\u0000!0)(t1\u0000t2)[B(t1)\u0002B(t2)]z; (2)\nwhere ~!0is the ground-state energy of j 0iand ~!\u0010\nm(\u0010=\u0006) is the eigenenergy of j \u0006\nmi, in which ^Szj \u0006\nmi=\n(M(0)\nz\u00061)j \u0006\nmi. When the magnetic \feld B(t1) =P\n\n1B(\n1)e\u0000i\n1t1is applied adiabatically from t1=\u00001, the\nmagnetization M(2)\nz(t) is given by\nM(2)\nz(t) =\u0000\r2\n4X\n\n1;\n2X\n\u0010=\u0006X\nme\u0000i(\n1+\n2)t\n\n1+ \n 2+ 2i0+\"\n\u0010jh \u0010\nmj^S\u0010j 0ij2\n\n1+!\u0010\nm\u0000!0+i0++\u0010jh \u0010\nmj^S\u0010j 0ij2\n\n2\u0000!\u0010\nm+!0+i0+#\nB(\n1)\u0001B(\n2)\n\u0000\r2\n4iX\n\n1;\n2X\n\u0010=\u0006X\nme\u0000i(\n1+\n2)t\n\n1+ \n 2+ 2i0+\"\njh \u0010\nmj^S\u0010j 0ij2\n\n1+!\u0010\nm\u0000!0+i0++jh \u0010\nmj^S\u0010j 0ij2\n\n2\u0000!\u0010\nm+!0+i0+#\n[B(\n1)\u0002B(\n2)]z:(3)\nB. Second-harmonic generation\nThe oscillation of the magnetization with !=n\n (\n:\ndriving frequency) corresponds to high-harmonic gener-\nation from magnetic dipoles [28]. When a monochro-\nmatic \feld B(t) =B(\n)e\u0000i\nt+B(\u0000\n)ei\ntis applied,the magnetization at the second order M(2)\nz(t) in Eq. (3)\nhas the component of second-harmonic generation (SHG)\nat \n 1= \n 2= \n. Since B(\n)\u0002B(\n) = 0, the Fourier\ncomponent M(2)\nz(!= \n 1+ \n 2= 2\n) is given by\nM(2)\nz(2\n) =\u0000\r2\n4X\n\u0010=\u0006X\nm\u0010\n2(\n +i0+)\"\njh \u0010\nmj^S\u0010j 0ij2\n\n\u0000!\u0010\nm+!0+i0++jh \u0010\nmj^S\u0010j 0ij2\n\n +!\u0010\nm\u0000!0+i0+#\nB(\n)\u0001B(\n): (4)\nThe above Eq. (4) is written as\nM(2)\nz(2\n) =\u000bz(2\n; \n;\n)B(\n)\u0001B(\n): (5)When a linearly polarized magnetic \feld B(t) =3\nB B\n2ΩB\nMz= 0(0)Mz> 0(0)\nFigure 1. Schematic pictures of the second-order magnetic\ne\u000bects. Top panel: Magnetization under the linearly polar-\nized magnetic \feld when M(0)\nz>0 in the ground state, where\nM(2)\nz(2\n) is activated. Bottom panel: Helicity-dependent\nmagnetization under the circularly polarized magnetic \felds\nwhenM(0)\nz= 0.\nBL(cos \nt;cos \nt) [i.e., B(\n) =BL=2(1;1)] is applied,\nwe \fnd\nM(2)\nz(2\n) =1\n2\u000bz(2\n; \n;\n)B2\nL; (6)\nand thus magnetic SHG is allowed. Note that\n\u000bz(2\n; \n;\n) = 0 whenjh +\nmj^S+j 0ij2=jh \u0000\nmj^S\u0000j 0ij2and!+\nm=!\u0000\nm, implying that M(2)\nz(2\n) is nonzero only\nwhen the spin-\ripped states by ^S+and^S\u0000are asymmet-\nric. For example, M(2)\nz(2\n)6= 0 when the ground state\nj 0ihas net magnetization jM(0)\nzj>0 [28]. A schematic\npicture of this e\u000bect is shown in the top panel of Fig. 1.\nOn the other hand, when a circularly polarized \feld\nB(t) =BC(cos \nt;\u0006sin \nt) [i.e., B(\n) =BC=2(1;\u0006i)]|\nwhere\u0006indicates the right- and left-handed circularly\npolarization|is applied, we \fnd\nM(2)\nz(2\n) = 0; (7)\nbecause B(\n)\u0001B(\n) = 0. Hence, in contrast to the\nresponse under the linearly polarized \feld, M(2)\nz(2\n) is\nabsent under the circularly polarized \feld.\nC. Zero-frequency component\nThe second-order magnetization M(2)\nz(t) in Eq. (3) also\nhas a zero-frequency component at \n 1+\n2= 0. Because\n[\n1+\n2+2i0+]\u00001diverges at \n 1+\n2= 0, Eq. (3) is not\na well-de\fned formula for describing the zero-frequency\ncomponent. In order to get rid of the divergence arising\nfrom \n 1+ \n 2= 0, we consider the time derivative of\nM(2)\nz(t). When B(t) =B(\n)e\u0000i\nt+B(\u0000\n)ei\nt, the\ntime derivative of M(2)\nz(t) is given by\ndM(2)\nz(t)\ndt=T(2)\nz(0)\u0000iX\nn=\u000612n\nM(2)\nz(2n\n)e\u00002in\nt;\n(8)\nwith the zero-frequency ( != \n 1+ \n 2= 0) component\nT(2)\nz(0) =\u0019\r2\n2X\n\u0010=\u0006X\nm\u0010jh \u0010\nmj^S\u0010j 0ij2\u0002\n\u000e\u0000\n\n\u0000!\u0010\nm+!0\u0001\n+\u000e\u0000\n\n +!\u0010\nm\u0000!0\u0001\u0003\nB(\n)\u0001B(\u0000\n)\n+i\u0019\r2\n2X\n\u0010=\u0006X\nmjh \u0010\nmj^S\u0010j 0ij2\u0002\n\u000e\u0000\n\n\u0000!\u0010\nm+!0\u0001\n\u0000\u000e\u0000\n\n +!\u0010\nm\u0000!0\u0001\u0003\n[B(\n)\u0002B(\u0000\n)]z: (9)\nEquation (8) implies that the magnetization grows lin-\nearly withtwhenT(2)\nz(0)6= 0. Because matrix elements\nof the total spin raising ( \u0010= +) and lowering ( \u0010=\u0000)\noperators are involved, some properties of the \frst term\nin Eq. (9) are linked to the equilibrium magnetization\nM(0)\nz. For example, when the spin is fully polarized with\nM(0)\nz>0, the \frst term gives the negative contribution\nbecausejh +\nmj^S+j 0ij2= 0. On the other hand, this\nB(\n)\u0001B(\u0000\n) term vanishes if the \u0010= + (raising) and\n\u0000(lowering) contributions are equivalent. This condi-\ntion would require M(0)\nz= 0. Note that e\u000bects of re-laxation are not taken into account in the above formula.\nThe e\u000bects may be incorporated phenomenologically into\nEq. (3) by replacing 0+with a relaxation factor \u0000. In this\ncase, the magnetization M(2)\nzconverges to a \fnite value\nof the order of T(2)\nz(0)=\u0000.\nEquation (9) may be written as\nT(2)\nz(0) =\u000b0\nz(0; \n;\u0000\n)B(\n)\u0001B(\u0000\n)\n+i\f0\nz(0; \n;\u0000\n) [B(\n)\u0002B(\u0000\n)]z: (10)\nWhen a linearly polarized magnetic \feld B(t) =4\nBL(cos \nt;cos \nt) is applied, we \fnd\nT(2)\nz(0) =1\n2\u000b0\nz(0; \n;\u0000\n)B2\nL: (11)\nOn the other hand, when a circularly polarized \feld\nB(t) =BC(cos \nt;\u0006sin \nt) is applied,\nT(2)\nz(0) =1\n2[\u000b0\nz(0; \n;\u0000\n)\u0006\f0\nz(0; \n;\u0000\n)]B2\nC:(12)\nIn both cases, the B(\n)\u0001B(\u0000\n) term can be nonzero.\nIn contrast, the B(\n)\u0002B(\u0000\n) term can be nonzero\nonly for a circularly polarized \feld. In this case,\nthe magnetization exhibits helicity dependence. Hence,\nthe nonlinear magnetic responses under the right- and\nleft-handed circularly-polarized \felds are asymmetric if\n\u000b0\nz(0; \n;\u0000\n)6= 0. The bottom panel of Fig. 1 is a\nschematic picture of the e\u000bect when M(0)\nz= 0 and\n\u000b0\nz(0; \n;\u0000\n) = 0. As shown in Fig. 1, we can manip-\nulate the magnetization direction by the helicity of the\nmagnetic \feld.D. Magnetization by circularly polarized \felds\nIn the previous sections, we only assume the conserva-\ntion ofSzin the unbiased Hamiltonian ^H0, and thus the\nexpressions are general. Here, to see the nonlinear re-\nsponse can be connected to the dynamical spin structure\nfactor, we focus on speci\fc cases, where jh +\nmj^S+j 0ij2=\njh \u0000\nmj^S\u0000j 0ij2and!+\nm=!\u0000\nmare satis\fed in Eqs. (4)\nand (9). These conditions may be realized, e.g., when\nM(0)\nz= 0 and the Hamiltonian ^H0is invariant under\nthe time-reversal operation (or \u0019rotation around the y\naxis) ^S\u0006\nj!\u0000 ^S\u0007\njand ^Sz\nj!\u0000 ^Sz\nj. When the above con-\nditions are satis\fed, \u000bz(2\n; \n;\n) =\u000b0\nz(0; \n;\u0000\n) = 0\nand the response to a linearly polarized \feld vanishes\n[see Eqs. (6) and (11)]. However, even in this condition,\ntheB(\n)\u0002B(\u0000\n) term in Eq. (9) can be nonvanishing,\nimplying that a magnetization M(2)\nz(t) can be generated\nfromM(0)\nz= 0 by applying a circularly polarized \feld.\nSinceM(2)\nz(2\n) = 0 in Eq. (8), the second-order mag-\nnetic response is described by\ndM(2)\nz\ndt=i\u0019\r2X\nmjh mj^S\u0000j 0ij2[\u000e(\n\u0000!m+!0)\u0000\u000e(\n +!m\u0000!0)] [B(\n)\u0002B(\u0000\n)]z; (13)\nwhere we denote j \u0006\nmiand!\u0006\nmbyj miand!m.\nEquation (13) is related to the commonly-used dynam-\nical (transverse) spin structure factor\n\u001f+\u0000\ns(q;\n) =\u0019X\nmjh mj^S\u0000\nqj 0ij2\u000e(\n\u0000!m+!0);\n(14)\nwhere ^S\u0000\nq=1p\nNP\nj^S\u0000\nje\u0000iq\u0001Rj[N: number of lattice\nsites] is the spin-\rip operator in the momentum ( q)\nspace. Using \u001f+\u0000\ns(q;\n), the magnetization per unit\nm(2)\nz(=M(2)\nz=N) at \n>0 is given by\ndm(2)\nz\ndt=i\r2\u001f+\u0000\ns(q= 0;\n) [B(\n)\u0002B(\u0000\n)]z:(15)\nHence, by introducing the structure factor \u001f+\u0000\ns(q;\n),\nwe can describe the magnetization at the second order\nin the simple formula. Under a circularly polarized \feld\nB(\n) =BC=2(1;\u0006i), this magnetization exhibits the he-\nlicity (\u0006) dependence\ndm(2)\nz\ndt=\u00061\n2\r2\u001f+\u0000\ns(q= 0;\n)B2\nC: (16)\nFor the magnetic e\u000bect described by Eq. (15), the dy-\nnamical spin structure factor must be \u001f+\u0000\ns(q= 0;\n)6= 0.\nIn other words, once we know the dynamical spin struc-\nture factor \u001f+\u0000\ns(q;\n) in the linear response regime, wecan predict the main features of the magnetization at the\nsecond order. In the isotopic Heisenberg model [or spin-\nSU(2)-symmetric Hubbard model], \u001f+\u0000\ns(q= 0;\n) = 0\nat \n>0 and no magnetization m(2)\nzis induced. This\nimplies that \u001f+\u0000\ns(q= 0;\n)6= 0 may arise from magnetic\nanisotropies, e.g., Ising anisotropy in the XXZ model (see\nSec. IV) and the Dzyaloshinskii-Moriya interaction. As\ndiscussed in Appendix B, we can interpret this nonlinear\nmagnetic e\u000bect in the rotating frame, where the system\ncan be described by a static Hamiltonian [30, 31].\nEquation (15) is very similar to the formula of the cir-\ncular photogalvanic e\u000bect (CPGE) in which a generated\nsecond-order photocurrent J(2)\n\u0016under an electric \feld E\nis described byd\ndtJ(2)\n\u0016=i\u0011\u0016(\n)[E(\n)\u0002E(\u0000\n)]\u0016[18, 36].\nWhile the magnetization and electric current are di\u000ber-\nent, we may \fnd similar time-dependent properties to\nthe CPGE.\nIV. NUMERICAL DEMONSTRATION\nFinally, we numerically demonstrate the nonlinear\nmagnetic e\u000bect described by Eq. (15) using the spin-1/2\nXXZ model. As ^H0in Eq. (1), the Hamiltonian of the5\n-1 0 1\n 0  10  20  30  40(a) ×  10-  4\nRCP\nLCPmz(t)\ntΩ = 1\nΩ = 3\nΩ = 5\nΩ = 7\n0.00.10.20.30.4\n 1  2  3  4  5  6  7(b)\nχ +  −s   (q = 0, Ω)\nΩχ +  −s   (q = 0, Ω)\nmz(t)/(tγ 2B2\nC / 2)| t = 40\nFigure 2. (a) Time-dependent magnetization mz(t) under\nthe magnetic \feld B(t) =BC(cos \nt;\u0006sin \nt), whereJ= 1,\n\u0001 = 4, and ~\rBC= 0:005 ( ~=Jis a unit of time). The\nsolid and dotted lines indicate the magnetization under right-\nhanded circularly polarized (RCP) and left-handed circularly\npolarized (LCP) \felds, respectively. The data at \n = 1 and\n7 are overlapped around mz(t)\u00180. (b) Comparison between\nthe dynamical spin structure factor \u001f+\u0000\ns(q= 0;\n) (solid line)\nand the \n dependence of the magnetization normalized as\nmz(t)=(t\r2B2\nC=2)\f\f\nt=40(circles).\none-dimensional XXZ model is\n^HXXZ\n0=JX\njh\n^Sx\nj^Sx\nj+1+^Sy\nj^Sy\nj+1+ \u0001^Sz\nj^Sz\nj+1i\n;(17)\nwhereJ >0 is the antiferromagnetic exchange coupling\nand \u0001 is the magnetic anisotropy along the zdirection.\nHere, we set J(~=J) as a unit of energy (time). When\n\u0001>1, the magnetic excitation in the XXZ chain is\ngapped and \u001f+\u0000\ns(q= 0;\n) obtains the spectral weights\nabove the gap. Thus, the second-order magnetic ef-\nfect described by Eq. (15) is anticipated. To demon-\nstrate this e\u000bect, we employ the in\fnite time evolving\nblock decimation (iTEBD) [35] and calculate the time\ndependence of mz(t) under the circularly polarized \feld\nB(t) =BC(cos \nt;\u0006sin \nt).\nFigure 2(a) shows the magnetization mz(t) at \u0001 = 4\nin the XXZ model. Corresponding to \u001f+\u0000\ns(q= 0;\n)\n[see Fig. 2(b)], the magnetization mz(t) is generated\nat 2.\n.6. The sign of the magnetization is in-\nverted by switching the helicity ( \u0006) of the magnetic\n\feldB(t). While the linear growth of the magnetiza-tion is expected at t\u001d1 (see Appendix C), mz(t)\nalready grows up linearly with time up to t= 40.\nSincei\r2[B(\n)\u0002B(\u0000\n)]z=\r2B2\nC=2 in Eq. (15), we\nplot the \n dependence of the normalized magnetiza-\ntionmz(t)=(t\r2B2\nC=2)\f\f\nt=40in Fig. 2(b). As plotted in\nFig. 2(b), the magnetization shows good agreement with\n\u001f+\u0000\ns(q= 0;\n). Therefore, the second-order magnetic ef-\nfect in the gapped phase of the XXZ model is actually\ndescribed by Eq. (15). While a similar numerical simu-\nlation has been performed in Ref. [30], in our study, we\nformulate the nonlinear magnetic e\u000bect in a simple equa-\ntion (15) and identify the relation with the low-energy\nmagnetic excitation described by \u001f+\u0000\ns(q;\n).\nV. SUMMARY AND DISCUSSION\nIn this paper, we have investigated the second-order\nmagnetization perpendicular to the driving magnetic\n\felds. We have derived that while Mz(!= 2\n) can be\ninduced under the linearly polarized \feld, it is absent\nunder the circularly polarized \feld. Mz(!= 0) can be\ninduced by circularly polarized \felds and exhibits helic-\nity dependence. We have also discussed the speci\fc case\nwhen the ground state has no net magnetization, where\nwe have demonstrated the e\u000bect numerically in the driven\nXXZ model and have shown that the main features of\nthe magnetization are determined by the dynamical spin\nstructure factor \u001f+\u0000\ns(q= 0;\n).\nThis second-order magnetic e\u000bect emerges in a quan-\ntum magnet with magnetic anisotropy. For example,\nBaCo 2V2O8is described as an antiferromagnetic XXZ\nchain with \u0001 >1 [37{39], where we may \fnd a similar\nmagnetic e\u000bect demonstrated in Fig. 2. For J\u00183 meV\nclose to the value reported in BaCo 2V2O8[39], \n = 2\nand ~\rBC= 0:005 in Fig. 2 correspond to 1 :45 THz\nand 0:13 T [40], respectively, which may be accessi-\nble in experiments. In the recently realized twisted\nWSe 2that can be represented as a triangular lattice\nHubbard model [41{43], the displacement \feld leads to\na Dzyaloshinskii-Moriya-type anisotropic interaction in\nthe e\u000bective Heisenberg model in the strong-coupling\nlimit [42, 43]. Because of the gapped magnetic excitation\ndue to the anisotropic interaction, this moir\u0013 e Hubbard\nsystem may also be a candidate for the host of the second-\norder magnetization. While in Fig. 2 we used a model\nthat only has an excitation continuum, the relations we\nderived are exact regardless of the type of magnetic exci-\ntations. A magnetic collective mode, which gives a large\nresponse at a resonant excitation frequency in a dynam-\nical spin correlation function, can be a good source for\ne\u000ecient nonlinear magnetic e\u000bects.\nIn our study, e\u000bects of relaxation, which are present\nin any realistic systems (e.g., by spin-lattice relaxation),\nare not taken into account. When e\u000bects of relaxation\nare incorporated, the linear growth of the magnetization\n[e.g., in Fig. 2(a)] is observed until the relaxation time\n\u001c. The magnetization converges to a \fnite value in a6\nsteady state at t\u001d\u001c, where the magnitude of m(2)\nzmay\nbe proportional to \u001c.\nWhile we focus on the responses to the magnetic \feld\ncomponent of a THz \feld, the electric \feld component\nis usually larger than the magnetic \feld component [29].\nHence, if a spin-electric \feld coupling is crucial in a mag-\nnetic insulator, we might \fnd a larger nonlinear mag-\nnetic response, which is useful for electromagnetic \feld\nmanipulation of quantum materials. In order to address\nthis issue, one needs to consider a coupling term be-\ntween an electric \feld and a spin system via, e.g., spin-\nphonon or spin-orbit coupling. On the other hand, a\nrecent technique using a split-ring resonator enables us\nto selectively enhance the strength of the THz magnetic\n\feld [26, 44, 45], which may also open a pathway to re-\nalize a large nonlinear magnetic e\u000bect.\nACKNOWLEDGMENTS\nThis work was supported by Grants-in-Aid\nfor Scienti\fc Research from JSPS, KAKENHI\nGrants No. JP18K13509 (T.K.), No. JP20K14412,\nNo. JP20H05265, No. JP21H05017 (Y.M.), and\nNo. JP21K03412 (S.T.) and JST CREST Grant No. JP-\nMJCR1901 (Y.M.) and No. JPMJCR19T3 (S.T.). T.K.\nwas supported by the JSPS Overseas Research Fellow-\nship. A.J.M. was supported in part by Programmable\nQuantum Materials, an Energy Frontier Research Center\nfunded by the U.S. Department of Energy (DOE), O\u000ece\nof Science, Basic Energy Sciences (BES), under Award\nNo. DE-SC0019443. The Flatiron Institute is a division\nof the Simons Foundation.\nAppendix A: Perturbation theory\nWe employ the perturbation theory to derive a for-\nmula for the magnetization Mz(t). With respect to\nthe perturbation ^V(t) = ^H(t)\u0000^H0, the wave function\nj\t(t)i=e\u0000i^H0\n~tj\tI(t)ievolved from the ground state\nj 0iis obtained via\nj\tI(t)i=j 0i+1\ni~Zt\n\u00001dt1^VI(t1)j 0i (A1)\n+\u00121\ni~\u00132Zt\n\u00001dt1Zt1\n\u00001dt2^VI(t1)^VI(t2)j 0i+\u0001\u0001\u0001;\nwhere the subscript I indicates the interaction picture\nand ^OI(t) =ei^H0\n~t^O(t)e\u0000i^H0\n~t. Assuming a transverse\nmagnetic \feld, i.e., Bz(t) = 0, in the Hamiltonian (1),\nthe perturbation term is given by\n^VI(t) =\u0000~\rh\nBx(t)^Sx\nI(t) +By(t)^Sy\nI(t)i\n: (A2)\nUsing the interaction picture, the magnetization is\nMz(t) =h\tI(t)j^Sz\nI(t)j\tI(t)i: (A3)The magnetization in the ground state j 0iisM(0)\nz=\nh 0j^Szj 0i=Sz\n0. Although the magnetic \feld in\nEq. (A2) is applied, the magnetization at the \frst order\ninB(t) vanishes, i.e., M(1)\nz(t) = 0, because ^VI(t) induces\nthe spin \rip ( ^S\u0006) andh 0j^Sz\nI(t)^VI(t0)j 0i= 0.\nUsing Eq. (A1), the magnetization at the second order\nM(2)\nz(t) is given by\nM(2)\nz(t) =1\n~2Zt\n\u00001dt1Zt\n\u00001dt2h 0j^VI(t1)^Sz\nI(t)^VI(t2)j 0i\n\u00001\n~2Zt\n\u00001dt1Zt1\n\u00001dt2h 0j^Sz\nI(t)^VI(t1)^VI(t2)j 0i\n\u00001\n~2Zt\n\u00001dt1Zt1\n\u00001dt2h 0j^VI(t2)^VI(t1)^Sz\nI(t)j 0i:\n(A4)\nBecause ^VI(t) is comprised of the spin-\rip operators ^S\u0006,\nwe introduce the intermediate eigenstate j \u0006\nmiin which\nSz\nm=Sz\n0\u00061. Using ^Szj \u0006\nmi= (M(0)\nz\u00061)j \u0006\nmi, the\nintegrand of the \frst term in Eq. (A4) is given by\nh 0j^VI(t1)^Sz\nI(t)^VI(t2)j 0i\n=X\n\u0010=\u0006X\nm\u0010\nM(0)\nz+\u0010\u0011\nh 0j^VI(t1)j \u0010\nmih \u0010\nmj^VI(t2)j 0i:\n(A5)\nThe term involving M(0)\nzin Eq. (A5) cancels out the\nsecond and third terms in Eq. (A4). Hence, we obtain\nM(2)\nz(t) =1\n~2Zt\n\u00001dt1Zt\n\u00001dt2X\n\u0010=\u0006X\nm\u0010h 0j^VI(t1)j \u0010\nmi\n\u0002h \u0010\nmj^VI(t2)j 0i:\n(A6)\nCombining the relations\nh 0j^S\u0017j \u0006\nmih \u0006\nmj^S\u0017j 0i=1\n4jh \u0006\nmj^S\u0006j 0ij2; (A7)\nh 0j^Sxj \u0006\nmih \u0006\nmj^Syj 0i=\u00061\n4ijh \u0006\nmj^S\u0006j 0ij2;(A8)\nwhere\u0017=x;y, we \fnd\nh 0j^VI(t1)j \u0010\nmih \u0010\nmj^VI(t2)j 0i\n=~2\r2\n4jh \u0010\nmj^S\u0010j 0ij2e\u0000i(!\u0010\nm\u0000!0)(t1\u0000t2)B(t1)\u0001B(t2)\n+~2\r2\n4i\u0010jh \u0010\nmj^S\u0010j 0ij2e\u0000i(!\u0010\nm\u0000!0)(t1\u0000t2)[B(t1)\u0002B(t2)]z:\n(A9)\nHere, ~!0is the ground-state energy of j 0iand ~!\u0010\nm\n(\u0010=\u0006) is the eigenenergy of j \u0006\nmi. Then, applying\nEq. (A9) to Eq. (A6), we obtain Eq. (2).\nWhile the above formulas are the results at zero tem-\nperature, we may obtain the corresponding formulas\nat nonzero temperature by replacing h 0j\u0001\u0001\u0001j 0iwith\n1=ZP\nne\u0000\f~!nh nj\u0001\u0001\u0001j ni, where\fandZare the in-\nverse temperature and the partition function, respec-\ntively.7\nAppendix B: Magnetization in a rotating frame\nIn this appendix, we consider magnetization in a ro-\ntating frame. In this frame, we can discuss the magneti-\nzation under the circularly polarized \feld as the dynam-\nics described by the static Hamiltonian with an e\u000bective\nmagnetic \feld [30, 31].\nWith respect to the original Schr odinger equation\n[i~d\ndt\u0000^H(t)]j\t(t)i= 0, a state given by a unitary trans-\nformationj\t0(t)i=^U(t)j\t(t)isatis\fes [30]\n^U(t)\u0014\ni~d\ndt\u0000^H(t)\u0015\nU(t)yj\t0(t)i= 0: (B1)\nHere, assuming the U(1) spin rotational symmetry\naround the zaxis in the Hamiltonian ^H0, we apply\n^U(t) =ei\u0018\n^Szt; (B2)\nwhere\u0018=\u00061 denotes clock/anticlockwise rotation.\nThen,\n^U(t)i~d\ndt^U(t)y=i~d\ndt+\u0018~\n^Sz(B3)\nand\n^U(t)^H(t)^U(t)y=^H0\u0000~\rBx(t)h\n^Sxcos \nt\u0000\u0018^Sysin \nti\n\u0000~\rBy(t)h\n^Sycos \nt+\u0018^Sxsin \nti\n:\n(B4)\nHence, for the time-dependent equation\n\u0014\ni~d\ndt\u0000^H0(t)\u0015\nj\t0(t)i= 0; (B5)\nwe \fnd\n^H0(t) =^H0\u0000~\r[Bx(t) cos \nt+\u0018By(t) sin \nt]^Sx\n\u0000~\r[By(t) cos \nt\u0000\u0018Bx(t) sin \nt]^Sy\u0000\u0018~\n^Sz:\n(B6)\nHere, we consider the case under the circularly polar-\nized \feld. When frame rotation corresponds to the helic-\nity of the magnetic \feld as B(t) =BC(cos \nt;\u0018sin \nt),\nwe obtain the static Hamiltonian\n^H0\nC=^H0\u0000~\rBC^Sx\u0000\u0018~\n^Sz(B7)\nfor [i~d\ndt\u0000^H0\nC]j\t0(t)i= 0 [30, 31]. This static Hamilto-\nnian in the rotating frame indicates that the frequency \ngives the e\u000bective Zeeman term \u0000\u0018~\n^Szand the magne-\ntization direction depends on the helicity ( \u0018) of the cir-\ncularly polarized \feld. Since [ ^H0\u0000\u0018~\n^Sz;^Sz] = 0, the\ne\u000bective Zeeman term \u0000\u0018~\n^Szitself cannot change themagnetization Mz=h^Szifrom the ground state j 0iof\n^H0. However, the perturbation due to \u0000~\rBC^Sxbreaks\nthe conservation of Szand can modify the magnetization\nin anisotropic magnets [30, 31]. In this picture, when\nM(0)\nz= 0 att=\u00001 inj 0i, the magnetization at the\nsecond order is given by\nM(2)\nz(t)=\r2B2\nCZt\n\u00001dt1Zt\n\u00001dt2h 0j^Sx0\nI(t1)^Sz0\nI(t)^Sx0\nI(t2)j 0i;\n(B8)\nwhere ^O0\nI(t) =ei\n~(^H0\u0000\u0018~\n^Sz)t^Oe\u0000i\n~(^H0\u0000\u0018~\n^Sz)twith re-\nspect toj\t0(t)i=e\u0000i\n~(^H0\u0000\u0018~\n^Sz)tj\t0\nI(t)i. Assuming\njh +\nmj^S+j 0ij2=jh \u0000\nmj^S\u0000j 0ij2(!+\nm=!\u0000\nm), we \fnally\nobtain\ndM(2)\nz\ndt=\u0018\u0019\r2B2\nC\n2X\nmjh mj^S\u0000j 0ij2\u0002\n\u000e(\n\u0000!m+!0)\n\u0000\u000e(\n +!m\u0000!0)\u0003\n:\n(B9)\nThis is consistent with Eq. (13) since i[B(\n)\u0002\nB(\u0000\n)]z=\u0018B2\nC=2.\nAppendix C: Time evolution from t= 0\nIn the above derivations, we assumed adiabatic switch-\ning fromt=\u00001. Here, for real-time numerical\nsimulations, we derive the formula when the magnetic\n\feld is switched at t= 0. Whenjh +\nmj^S+j 0ij2=\njh \u0000\nmj^S\u0000j 0ij2and!+\nm=!\u0000\nmare satis\fed, the time-\ndependent magnetization in Eq. (2) under a monochro-\nmatic \feld B(t) =\u0012(t)B(\n)e\u0000i\nt+ c:c:is given by\nM(2)\nz(t) = 2i\r2X\nmjh mj^S\u0000j 0ij2\"\nsin2[(\n\u0000!m+!0)t=2]\n(\n\u0000!m+!0)2\n\u0000sin2[(\n+!m\u0000!0)t=2]\n(\n +!m\u0000!0)2#\n[B(\n)\u0002B(\u0000\n)]z;\n(C1)\nwherej \u0006\nmiand!\u0006\nmare denoted by j miand!m.\nM(2)\nz(t)/t4att\u00180. On the other hand, in the limit\nt!1 , we \fnd\nM(2)\nz(t) =it\u0019\r2X\nmjh mj^S\u0000j 0ij2h\n\u000e(\n\u0000!m+!0)\n\u0000\u000e(\n +!m\u0000!0)i\n[B(\n)\u0002B(\u0000\n)]z:\n(C2)\nHence,M(2)\nz(t!1 )/tand its time derivative is con-\nsistent with Eq. (13).8\n[1] Q. Ma, A. G. Grushin, and K. S. Burch, Nat. Mater. 20,\n1601 (2021).\n[2] J. Orenstein, J. Moore, T. Morimoto, D. Torchinsky,\nJ. Harter, and D. Hsieh, Annu. Rev. Condens. Matter\nPhys. 12, 247 (2021).\n[3] S. Ghimire and D. A. Reis, Nat. Phys. 15, 10 (2019).\n[4] S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini,\nL. F. DiMauro, and D. A. Reis, Nat. Phys. 7, 138 (2011).\n[5] O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek,\nC. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W.\nKoch, and R. Huber, Nat. Photonics 8, 119 (2014).\n[6] M. Hohenleutner, F. Langer, O. Schubert, M. Knorr,\nU. Huttner, S. W. Koch, M. Kira, and R. Huber, Nature\n523, 572 (2015).\n[7] G. Vampa, T. J. Hammond, N. Thir\u0013 e, B. E. Schmidt,\nF. L\u0013 egar\u0013 e, C. R. McDonald, T. Brabec, and P. B.\nCorkum, Nature 522, 462 (2015).\n[8] H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and\nD. A. Reis, Nat. Phys. 13, 262 (2017).\n[9] K. Kaneshima, Y. Shinohara, K. Takeuchi, N. Ishii,\nK. Imasaka, T. Kaji, S. Ashihara, K. L. Ishikawa, and\nJ. Itatani, Phys. Rev. Lett. 120, 243903 (2018).\n[10] N. Yoshikawa, K. Nagai, K. Uchida, Y. Takaguchi,\nS. Sasaki, Y. Miyata, and K. Tanaka, Nat. Commun.\n10, 3709 (2019).\n[11] S. M. Young and A. M. Rappe, Phys. Rev. Lett. 109,\n116601 (2012).\n[12] L. Z. Tan, F. Zheng, S. M. Young, F. Wang, S. Liu, and\nA. M. Rappe, npj Comput. Mater. 2, 16026 (2016).\n[13] A. M. Cook, B. M. Fregoso, F. de Juan, S. Coh, and\nJ. E. Moore, Nat. Comm. 8, 14176 (2017).\n[14] M. Nakamura, S. Horiuchi, F. Kagawa, N. Ogawa, T. Ku-\nrumaji, Y. Tokura, and M. Kawasaki, Nat. Comm. 8, 281\n(2017).\n[15] G. B. Osterhoudt, L. K. Diebel, M. J. Gray, X. Yang,\nJ. Stanco, X. Huang, B. Shen, N. Ni, P. J. W. Moll,\nY. Ran, and K. S. Burch, Nat. Mater. 18, 471 (2019).\n[16] M. Sotome, M. Nakamura, J. Fujioka, M. Ogino,\nY. Kaneko, T. Morimoto, Y. Zhang, M. Kawasaki, N. Na-\ngaosa, Y. Tokura, and N. Ogawa, Proc. Natl. Acad. Sci.\nUSA116, 1929 (2019).\n[17] T. Akamatsu, T. Ideue, L. Zhou, Y. Dong, S. Kitamura,\nM. Yoshii, D. Yang, M. Onga, Y. Nakagawa, K. Watan-\nabe, T. Taniguchi, J. Laurienzo, J. Huang, Z. Ye, T. Mo-\nrimoto, H. Yuan, and Y. Iwasa, Science 372, 68 (2021).\n[18] J. E. Sipe and A. I. Shkrebtii, Phys. Rev. B 61, 5337\n(2000).\n[19] G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug,\nP. B. Corkum, and T. Brabec, Phys. Rev. Lett. 113,\n073901 (2014).\n[20] N. Tsuji and H. Aoki, Phys. Rev. B 92, 064508 (2015).\n[21] T. Morimoto and N. Nagaosa, Sci. Adv. 2, e1501524\n(2016).\n[22] J. Ahn, G.-Y. Guo, and N. Nagaosa, Phys. Rev. X 10,\n041041 (2020).[23] Y. Murakami, S. Takayoshi, A. Koga, and P. Werner,\nPhys. Rev. B 103, 035110 (2021).\n[24] T. Kaneko, Z. Sun, Y. Murakami, D. Gole\u0014 z, and A. J.\nMillis, Phys. Rev. Lett. 127, 127402 (2021).\n[25] T. Tanabe, T. Kaneko, and Y. Ohta, Phys. Rev. B 104,\n245103 (2021).\n[26] Y. Mukai, H. Hirori, T. Yamamoto, H. Kageyama, and\nK. Tanaka, New J. Phys. 18, 013045 (2016).\n[27] J. Lu, X. Li, H. Y. Hwang, B. K. Ofori-Okai, T. Kurihara,\nT. Suemoto, and K. A. Nelson, Phys. Rev. Lett. 118,\n207204 (2017).\n[28] S. Takayoshi, Y. Murakami, and P. Werner, Phys. Rev.\nB99, 184303 (2019).\n[29] T. N. Ikeda and M. Sato, Phys. Rev. B 100, 214424\n(2019).\n[30] S. Takayoshi, H. Aoki, and T. Oka, Phys. Rev. B 90,\n085150 (2014).\n[31] S. Takayoshi, M. Sato, and T. Oka, Phys. Rev. B 90,\n214413 (2014).\n[32] R. Hertel, J. Magn. Magn. Mater. 303, L1 (2006).\n[33] M. Battiato, G. Barbalinardo, and P. M. Oppeneer,\nPhys. Rev. B 89, 014413 (2014).\n[34] M. Berritta, R. Mondal, K. Carva, and P. M. Oppeneer,\nPhys. Rev. Lett. 117, 137203 (2016).\n[35] G. Vidal, Phys. Rev. Lett. 98, 070201 (2007).\n[36] F. de Juan, A. G. Grushin, T. Morimoto, and J. E.\nMoore, Nat. Comm. 8, 15995 (2017).\n[37] S. Kimura, H. Yashiro, K. Okunishi, M. Hagiwara, Z. He,\nK. Kindo, T. Taniyama, and M. Itoh, Phys. Rev. Lett.\n99, 087602 (2007).\n[38] B. Grenier, S. Petit, V. Simonet, E. Can\u0013 evet, L.-P. Reg-\nnault, S. Raymond, B. Canals, C. Berthier, and P. Lejay,\nPhys. Rev. Lett. 114, 017201 (2015).\n[39] Q. Faure, S. Takayoshi, S. Petit, V. Simonet, S. Ray-\nmond, L.-P. Regnault, M. Boehm, J. S. White,\nM. M\u0017 ansson, C. R uegg, P. Lejay, B. Canals, T. Lorenz,\nS. C. Furuya, T. Giamarchi, and B. Grenier, Nat. Phys.\n14, 716 (2018).\n[40]g'2 and\u0016B= 0:0579 meV/T are used.\n[41] L. Wang, E.-M. Shih, A. Ghiotto, L. Xian, D. A. Rhodes,\nC. Tan, M. Claassen, D. M. Kennes, Y. Bai, B. Kim,\nK. Watanabe, T. Taniguchi, X. Zhu, J. Hone, A. Rubio,\nA. N. Pasupathy, and C. R. Dean, Nat. Mater. 19, 861\n(2020).\n[42] H. Pan, F. Wu, and S. Das Sarma, Phys. Rev. Research\n2, 033087 (2020).\n[43] J. Zang, J. Wang, J. Cano, and A. J. Millis, Phys. Rev.\nB104, 075150 (2021).\n[44] Y. Mukai, H. Hirori, T. Yamamoto, H. Kageyama, and\nK. Tanaka, Appl. Phys. Lett. 105, 022410 (2014).\n[45] T. Kurihara, K. Nakamura, K. Yamaguchi, Y. Sekine,\nY. Saito, M. Nakajima, K. Oto, H. Watanabe, and\nT. Suemoto, Phys. Rev. B 90, 144408 (2014)."    },    {        "title": "1905.03991v1.Lagrangian_formulation_for_emergent_elastic_waves_in_magnetic_emergent_crystals.pdf",        "content": "arXiv:1905.03991v1  [cond-mat.mes-hall]  10 May 2019Lagrangian formulation for emergent elastic waves in magne tic emergent crystals\nYangfan Hu∗\nSino-French Institute of Nuclear Engineering and Technolo gy, Sun Yat-sen University, 519082, Zhuhai, China\nMagnetic emergent crystals are periodic alignment of “part icle-like” spin textures that emerge\nin magnets. Instead of focusing on an individual spin or a mac roscopic magnetization field, we\nanalyze the dynamical behaviors of these novel states by tak ing a solid-state point of view. Based\non variational principles, the basic equations for lattice dynamics of any emergent crystalline states\nappearing in magnetic materials is established. For small a mplitude emergent elastic waves prop-\nagating in emergent crystals, the basic equations reduce to an eigenvalue problem, from which the\ndispersion relation and vibrational patterns for all emerg ent phonons are determined at the long\nwavelength limit.\nIntroduction\nSince the first successful experimental observa-\ntion of Bloch-type 2D magnetic skyrmion crystals\n(SkX) in bulk MnSi[1], SkX with different chirality[2],\ncommensurability[3, 4] and dimensionality[5], or even\ncrystalline states of other spin textures such as anti-\nskyrmion crystal [6] and bi-skyrmion crystal[7, 8] are\ndiscovered in various kinds of magnetic materials. As\nemergent crystals (ECs), the collective spin excitations\nof these macroscopic states and related properties have\nbeen extensively studied[9–21] due to their scientific\ninterest and application potential in magnonics. In\nterms of theoretical modeling, most of the studies start\nby numerically solving the well-known Landau-Lifshitz-\nGilbert[22, 23] (LLG) equation of the magnetization\nfield, where an explicit understanding of the dynam-\nical properties is difficult to be obtained. If we are\nonly interested in deriving the collective excitations of\nthese ECs around some equilibrium state, it is more\nconvenient to take into account the periodicity of these\nECs, and expand the magnetization Mas a Fourier\nseries[24, 25], where the dynamical behaviors around\nthe emergent crystalline state considered is described by\na vibrating emergent displacement field and vibrating\nFourier magnitudes. In previous studies, dynamics of\nthe emergent displacement field[17, 26] and the Fourier\nmagnitudes[10] have been studied individually, while\ntheir coupling has not been considered. Based on\nvariational principles, we establish the basic equations\ndescribing the coupling small amplitude vibration of the\nemergent displacement field and the Fourier magnitudes\nat long wavelength limit for any magnetic Ecs. Consid-\nering ECs as periodic alignment of emergent particles,\ni.e., localized field patterns instead of point masses, the\nequations obtained describe the lattice dynamics of these\nECs, where vibration of the emergent displacement field\ndescribes lattice vibration and vibration of the Fourier\nmagnitudes describes in-lattice vibration (i.e., vibration\nof the field pattern inside the lattice while keeping the\nlattice unchanged). Generally, the derived equations\nare hard to solve. Yet, for emergent elastic waves (long\n∗Corresponding author.huyf3@mail.sysu.edu.cnwavelength emergent phonons) propagation, we derive\nfrom the basic equations an eigenvalue problem which\ndetermines the dispersion relation of emergent phonons\nnear long wavelength limit.\nVariational principles for dynamics of the magneti-\nzation field\nThe Euler-Lagrange equation for dynamics of magne-\ntization can be derived from the principle of least action\nδS= 0, or equivalently\nδ/integraldisplay\nLdt= 0, (1)\nwhereSis the action of the system, and L=EK−Φ is\nthe Lagrangian of the system, with EKand Φ denoting\nrespectively the total kinetic energy induced by dynam-\nical behavior of the magnetization and the total free en-\nergy. It has been mentioned long ago that the form of\nEKis not unique. The most commonly used form de-\nrives from the Berry Phase action of a spin, which gives\nEK=M\nγ/integraltext\n˙ϕcosθdV[27, 28], where Mdenotes the aver-\nage modulus of magnetization, γdenotes the gyromag-\nnetic ratio, and ϕandθare Euler angles of the mag-\nnetization field. Alternatively, we can use the following\nvariational form of EK:\nδEK=M\nγ/integraldisplay\n(n×˙ n)δndV, (2)\nwherendenotes the unit magnetization vector. Sub-\nstitution of eq. (2) into eq. (1) yields the well known\nequation\n˙ n=γHeff×n, (3)\nwhereHeff=−1\nMδΦ\nδndenotes the effective magnetic\nfield. If dissipation is considered[23], eq. (1) becomes\nδ/integraldisplay\nLdt−/integraldisplay∂W\n∂˙ nδndt= 0, (4)\nwhereW=αM\n2γ/integraltext\n˙ n2dV. Substitution of eq, (2)\ninto eq. (4) yields the Landau-Lifshitz-Gilbert (LLG)\nequation[22, 23]\n˙ n=γHeff×n−α˙ n×n. (5)2\nWhen the free energy Φ can be expressed in terms of\nthe unit magnetization vector n, the upper equations are\ndirectly applicable. On the other hand, if we considerthe\nmodulation of modulus of magnetization due to a change\nofthetemperatureorduetothepresenceofECs,wehave\nΦ = Φ(M), where Mdenotes the magnetization vector.\nIn this case, eq. (2) should be replaced by\nδEK=1\nγM2/integraldisplay/parenleftBig\nM×˙M/parenrightBig\nδMdV, (6)\nwhereM=1\nV/integraltext|M|dV.\nFourier representation of deformable emergent crystals\nin magnetic materials\nDeformable ECs in helimagnets permit the following\nFourier expansion of the magnetization within the Eule-\nrian coordinates[25]\nM=/summationdisplay\nlMqleiql[I−Fe(r)]·r,(7)\nwhereqldenotes the reciprocal lattice vectors of the\nEC. For a d-dimensional EC ( d= 1,2,3),ql=\nl1q1+l2q2+···+ldqd,wherel= [l1, l2,···, ld]Tis\na vector of integers, and q1,q2, ,qdare the basic re-\nciprocal vectors. In eq. (7), Fe\nij(r) =εe\nij+ωe\nij, where\nεe\nij=1\n2/parenleftbig\nue\ni,j+ue\nj,i/parenrightbig\nare components of the emergent\nstrain tensor, ωe\nij=1\n2/parenleftbig\nue\ni,j−ue\nj,i/parenrightbig\nare components of the\nemergent rotation tensor, and ue\niare components of the\nemergent displacement vector. Disregarding rigid trans-\nlation of the EC considered, two types of deformation\nmay occur[25]: the lattice deformation described by εe\nij\nandωe\nijwhich transform the undeformed wave vectors\nqltoqe\nl/parenleftbig\nεe\nij, ωe\nij/parenrightbig\n= [I−Fe(r)]Tql, and the in-lattice\ndeformation described by variation of the Fourier mag-\nnitudesMql. Consider the dynamics of magnetization\nwhen the equilibrium state of the system is stabilized in\nan emergent crystalline state, eq. (7) becomes\nM=/summationdisplay\nl/bracketleftbig\n(Mql)st+(Mql(r,t))v/bracketrightbig\n×eiqe\nl((εe\nij)st,(ωe\nij)st)·[r−(ue(r,t))v],(8)\nwhere (P)stdenotes the static value of the quantity\nP, and (P)vdenotes time-dependent departure from\n(P)stdue to dynamical behavior of P. As intro-\nduced before[10, 25], one can take all components of\n(Mql(r,t))vfor all possible choices of ql, and construct\na large vector describing the vibration of all Fourier\nmagnitudes ( Mq(r,t))v. Eq. (8) shows that the dynamic\nbehavior of any magnetic EC around an equilibriumstate is determined by the vibrating emergent dis-\nplacement field ( ue(r,t))vand the vibrating Fourier\nmagnitudes ( Mq(r,t))v.\nBasic equations of emergent elastic waves in mag-\nnetic emergent crystals\nThe basic equations of emergent elastic wave propaga-\ntion in magnetic ECs can be derived by substituting eq.\n(8) into eq. (1), which gives\n−d\ndt/bracketleftbigg∂L\n∂(˙ ue)v/bracketrightbigg\n+∂L\n∂(ue)v−/summationdisplay\nid\ndri/bracketleftBigg\n∂L\n∂/parenleftbig\nue\n,i/parenrightbig\nv/bracketrightBigg\n= 0,(9)\n−d\ndt\n∂L\n∂/parenleftBig\n˙Mq/parenrightBig\nv\n+∂L\n∂(Mq)v−/summationdisplay\nid\ndri/bracketleftBigg\n∂L\n∂/parenleftbig\nMq\n,i/parenrightbig\nv/bracketrightBigg\n= 0.\n(10)\nMore conveniently, we have from eq. (2)\nδEK\nδ(ue)v−∂Φ\n∂(ue)v+/summationdisplay\nid\ndri/bracketleftBigg\n∂Φ\n∂/parenleftbig\nue\n,i/parenrightbig\nv/bracketrightBigg\n= 0,(11)\nδEK\nδ(Mq)v−∂Φ\n∂(Mq)v+/summationdisplay\nid\ndri/bracketleftBigg\n∂Φ\n∂/parenleftbig\nMq\n,i/parenrightbig\nv/bracketrightBigg\n= 0.(12)\nWhen damping is considered, eqs. (11-12) transform to\nδEK\nδ(ue)v−∂Φ\n∂(ue)v+/summationdisplay\nid\ndri/bracketleftBigg\n∂Φ\n∂/parenleftbig\nue\n,i/parenrightbig\nv/bracketrightBigg\n−∂W\n∂(˙ ue)v= 0,\n(13)\nδEK\nδ(Mq)v−∂Φ\n∂(Mq)v+/summationdisplay\nid\ndri/bracketleftBigg\n∂Φ\n∂/parenleftbig\nMq\n,i/parenrightbig\nv/bracketrightBigg\n−∂W\n∂/parenleftBig\n˙Mq/parenrightBig\nv= 0.\n(14)\nGenerally speaking, eqs. (11-14) are difficult to solve,\nsince the presence of ( ue(r,t))vand (Mq(r,t))vin eq.\n(8) breaks the orthogonality of the Fourier series expres-\nsion ofMintroduced in eq. (7). Hereafter we focus\non the solution of eqs. (11-12) for small amplitude vi-\nbration of ( ue(r,t))vand (Mq(r,t))vat long wavelength\nlimit, while eqs. (13-14) can be treated in a similar way.\nConsider the emergent elastic wave propagation in the\nEC with small amplitude, i.e., ( ue(r,t))vand (Mq(r,t))v\nhave small magnitudes and they both change smoothly\nin space. In this case, the orthogonality of the Fourier\nseries expression of Mis approximately maintained, and\nwe can expand Φ as3\nΦ = (Φ)st+1\n2/summationdisplay\ni,j/parenleftBigg\n∂2Φ\n∂(ue\ni)v∂/parenleftbig\nue\nj/parenrightbig\nv/parenrightBigg\nst(ue\ni)v/parenleftbig\nue\nj/parenrightbig\nv+1\n2/summationdisplay\ni,j,k,l\n∂2Φ\n∂/parenleftBig\nue\ni,k/parenrightBig\nv∂/parenleftBig\nue\nj,l/parenrightBig\nv\n\nst/parenleftbig\nue\ni,k/parenrightbig\nv/parenleftbig\nue\nj,l/parenrightbig\nv\n+/summationdisplay\ni,j,k\n∂2Φ\n∂/parenleftBig\nue\ni,k/parenrightBig\nv∂/parenleftbig\nue\nj/parenrightbig\nv\n\nst/parenleftbig\nue\ni,k/parenrightbig\nv/parenleftbig\nue\nj/parenrightbig\nv+1\n2/summationdisplay\ni,j/parenleftBigg\n∂2Φ\n∂(Mq\ni)v∂/parenleftbig\nMq\nj/parenrightbig\nv/parenrightBigg\nst(Mq\ni)v/parenleftbig\nMq\nj/parenrightbig\nv\n+1\n2/summationdisplay\ni,j,k,l\n∂2Φ\n∂/parenleftBig\nMq\ni,k/parenrightBig\nv∂/parenleftBig\nMq\nj,l/parenrightBig\nv\n\nst/parenleftBig\nMq\ni,k/parenrightBig\nv/parenleftBig\nMq\nj,l/parenrightBig\nv+/summationdisplay\ni,j,k\n∂2Φ\n∂/parenleftBig\nMq\ni,k/parenrightBig\nv∂/parenleftbig\nMq\nj/parenrightbig\nv\n\nst/parenleftBig\nMq\ni,k/parenrightBig\nv/parenleftbig\nMq\nj/parenrightbig\nv\n+/summationdisplay\ni,j,k\n∂2Φ\n∂/parenleftBig\nue\ni,k/parenrightBig\nv∂/parenleftbig\nMq\nj/parenrightbig\nv\n\nst/parenleftbig\nue\ni,k/parenrightbig\nv/parenleftbig\nMq\nj/parenrightbig\nv+/summationdisplay\ni,j,k\n∂2Φ\n∂/parenleftBig\nMq\ni,k/parenrightBig\nv∂/parenleftbig\nue\nj/parenrightbig\nv\n\nst/parenleftBig\nMq\ni,k/parenrightBig\nv/parenleftbig\nue\nj/parenrightbig\nv\n+/summationdisplay\ni,j/parenleftBigg\n∂2Φ\n∂(ue\ni)v∂/parenleftbig\nMq\nj/parenrightbig\nv/parenrightBigg\nst(ue\ni)v/parenleftbig\nMq\nj/parenrightbig\nv+/summationdisplay\ni,j,k,l\n∂2Φ\n∂/parenleftBig\nue\ni,k/parenrightBig\nv∂/parenleftBig\nMq\nj,l/parenrightBig\nv\n\nst/parenleftbig\nue\ni,k/parenrightbig\nv/parenleftBig\nMq\nj,l/parenrightBig\nv.(15)\nThe last term on the r.h.s. of eq. (15) is included when an AC magnetic fi eld is applied to the material. For the\nkinetic energy of the system, we have\nδEK\nδ(ue\ni)v=/summationdisplay\nj/parenleftBigg\n∂\n∂/parenleftbig\n˙ue\nj/parenrightbig\nvδEK\nδ(ue\ni)v/parenrightBigg\nst/parenleftbig\n˙ue\nj/parenrightbig\nv+/summationdisplay\nj\n∂\n∂/parenleftBig\n˙Mq\nj/parenrightBig\nvδEK\nδ(ue\ni)v\n\nst/parenleftBig\n˙Mq\nj/parenrightBig\nv, (16)\nδEK\nδ(Mq\ni)v=/summationdisplay\nj/parenleftBigg\n∂\n∂/parenleftbig\n˙ue\nj/parenrightbig\nvδEK\nδ(Mq\ni)v/parenrightBigg\nst/parenleftbig\n˙ue\nj/parenrightbig\nv+/summationdisplay\nj\n∂\n∂/parenleftBig\n˙Mq\nj/parenrightBig\nvδEK\nδ(Mq\ni)v\n\nst/parenleftBig\n˙Mq\nj/parenrightBig\nv, (17)\nSubstitution of eqs. (15-17) into eqs. (11, 12) yields\na set of linearized partial differential equations for\n(ue(r,t))vand (Mq(r,t))v. Consider the plane-wave\nform ofsolution ( ue(r,t))v=ue0ei(˜k·r−ωt), (Mq(r,t))v=\nMq0ei(˜k·r−ωt), a generalized eigenvalue problem of the\nfrequency ωcan be obtained as\n(Rω−K)/bracketleftbigg\nue0\nMq0/bracketrightbigg\n=0, (18)\nwhere\nR=/bracketleftbiggReReq\n(Req∗)TRq/bracketrightbigg\n, (19)\nK=/bracketleftbiggKeKeq\n(Keq∗)TKq/bracketrightbigg\n, (20)\nwhereReq∗andKeq∗denote complex conju-\ngate of ReqandKeq.Re\nij=−i/bracketleftBig\n∂\n∂˙ue\nj/parenleftBig\nδEBP\nδue\ni/parenrightBig/bracketrightBig\nst,Rq\nij=−i/bracketleftbigg\n∂\n∂˙Mq\nj/parenleftBig\nδEBP\nδMq\ni/parenrightBig/bracketrightbigg\nst, Req\nij=−i/bracketleftbigg\n∂\n∂˙Mq\nj/parenleftBig\nδEBP\nδue\ni/parenrightBig/bracketrightbigg\nst,\nKe\nij=/summationtext\np,s˜kp˜ks/bracketleftBig\n∂\n∂ue\nj,ps/parenleftBig\nd\ndrp/parenleftBig\n∂¯Φ\n∂ue\ni,p/parenrightBig/parenrightBig/bracketrightBig\nst, Keq\nij=\n/bracketleftBigg\n/summationtext\np,s˜kp˜ks∂\n∂Mq\nj,ps/parenleftBig\nd\ndrp∂Φ\n∂ue\ni,p/parenrightBig\n−/summationtext\npi˜kp∂\n∂Mq\nj,p/parenleftBig\nd\ndrp∂Φ\n∂ue\ni,p/parenrightBig/bracketrightBigg\nst,\nKq\nij=/bracketleftBigg\n∂\n∂Mq\nj/parenleftBig\n∂¯Φ\n∂Mq\ni/parenrightBig\n+/summationtext\npi˜kp∂\n∂Mq\nj,p/parenleftBig\n∂Φ\n∂Mq\ni/parenrightBig\n−/summationtext\npi˜kp∂\n∂Mq\nj,p\n/parenleftBig\nd\ndrp/parenleftBig\n∂Φ\n∂Mq\ni,p/parenrightBig/parenrightBig\n+/summationtext\np,s˜kp˜ks∂\n∂Mq\nj,ps/parenleftBig\nd\ndrp/parenleftBig\n∂Φ\n∂Mq\ni,p/parenrightBig/parenrightBig/bracketrightBigg\nst.Here\na subscript ” st” means that the term is calculated at\nthe equilibrium state ue= (ue)standMq= (Mq)st.\nWhen˜k→0, the stiffness matrix Kis completely de-\ntermined by the emergent elastic properties of the EC\nconsidered[11]. To be more specific Kq=µq,Keis de-\ntermined by Cethorough4\nKe\n11=−Ce\n11˜k2\n1−1\n4(Ce\n33+Ce\n44+2Ce\n34)˜k2\n2−(Ce\n13+Ce\n14)˜k1˜k2,\nKe\n22=−Ce\n22˜k2\n2−1\n4(Ce\n33+Ce\n44−2Ce\n34)˜k2\n1−(Ce\n23−Ce\n24)˜k1˜k2,\nKe\n12=Ke\n21=−1\n2(Ce\n13−Ce\n14)˜k2\n1−1\n2(Ce\n23+Ce\n24)˜k2\n2−1\n2(2Ce\n12+Ce\n33−Ce\n44)˜k1˜k2,(21)\nandKeqis determined by geqthrough\nKeq\n1i= i/parenleftbigg\ngeq\n1i˜k1+1\n2geq\n3i˜k2+1\n2geq\n4i˜k2/parenrightbigg\n,\nKeq\n2i= i/parenleftbigg\ngeq\n2i˜k1+1\n2geq\n3i˜k2−1\n2geq\n4i˜k2/parenrightbigg\n.(22)\nBased on the formulation established here, we systemati-\ncally study the long wavelength emergent phonon excita-tions for isotropic hexagonal SkX[29] and distorted SkX\ndue to presence of anisotropic effects[30].\nACKNOWLEDGMENTS\nThe author gratefully acknowledges J. D. Zang for\nhelpful discussion. Theworkwassupportedbythe NSFC\n(National Natural Science Foundation of China) through\nthe funds 11772360 and Pearl River Nova Program of\nGuangzhou (Grant No. 201806010134).\n[1] Mhlbauer, S. et al. Skyrmion Lattice in a Chiral Magnet.\nScience323915919 (2009).\n[2] Kezsmarki, I. et al. Neel-type skyrmion lattice with con -\nfined orientation in the polar magnetic semiconductor\nGaV4S8. Nature Materials 141116-+ (2015).\n[3] von Bergmann, K., Menzel, M., Kubetzka, A. & Wiesen-\ndanger, R. Influence of the Local Atom Configuration\non a Hexagonal Skyrmion Lattice. Nano Letters 15\n32803285 (2015).\n[4] Langner, M. C. et al. Coupled Skyrmion Sublattices in\nCu2OSeO3. Physical Review Letters 112167202 (2014).\n[5] Tanigaki, T. et al. Real-Space Observation of Short-\nPeriod Cubic Lattice of Skyrmions in MnGe. Nano Let-\nters1554385442 (2015).\n[6] Nayak, A. K. et al. Magnetic antiskyrmions above room\ntemperature in tetragonal Heusler materials. Nature 548\n561566 (2017).\n[7] Yu, X. Z. et al. Biskyrmion states and their current-\ndriven motion in a layered manganite. Nature Commu-\nnications 53198 (2014).\n[8] Wang, W. et al. A Centrosymmetric Hexagonal Magnet\nwith Superstable Biskyrmion Magnetic Nanodomains in\na Wide Temperature Range of 100340 K. Advanced Ma-\nterials2868876893 (2016).\n[9] Onose, Y., Okamura, Y., Seki, S., Ishiwata, S. & Tokura,\nY. Observation of Magnetic Excitations of Skyrmion\nCrystal in a Helimagnetic Insulator Cu2OSeO3. Physi-\ncal Review Letters 109037603 (2012).\n[10] Schwarze, T. et al. Universal helimagnon and skyrmion\nexcitations in metallic, semiconducting and insulating\nchiral magnets. Nature Materials 14478483 (2015).\n[11] Buettner, F. et al. Dynamics and inertia of skyrmionic\nspin structures. Nature Physics 11225228 (2015).\n[12] Cote, R., Luo, W., Petrov, B., Barlas, Y. & MacDonald,\nA. H. Orbital and interlayer skyrmion crystals in bilayer\ngraphene. Physical Review B 82245307 (2010).[13] Garst, M., Waizner, J. & Grundler, D. Collective spin ex -\ncitations of helices and magnetic skyrmions: review and\nperspectives of magnonics in non-centrosymmetric mag-\nnets. Journal of Physics D-Applied Physics 50293002\n(2017).\n[14] Roldn-Molina, A., Nunez, A. S. & Fernndez-Rossier,\nJ. Topological spin waves in the atomic-scale magnetic\nskyrmion crystal. New J. Phys. 18045015 (2016).\n[15] Takagi, R. et al. Spin-wave spectroscopy of the\nDzyaloshinskii-Moriya interaction in room-temperature\nchiral magnets hosting skyrmions. Physical Review B 95\n220406 (2017).\n[16] Mochizuki, M. Spin-Wave Modes and Their Intense Ex-\ncitation Effects in Skyrmion Crystals. Physical Review\nLetters108017601 (2012).\n[17] Petrova, O. & Tchernyshyov, O. Spin waves in a\nskyrmion crystal. Phys. Rev. B 84214433 (2011).\n[18] Tucker, G. S. et al. Spin excitations in the skyrmion hos t\nCu2OSeO3. Physical Review B 93054401 (2016).\n[19] Langner, M. C. et al. Nonlinear Ultrafast Spin Scatteri ng\nin the Skyrmion Phase of Cu2OSeO3. Physical Review\nLetters119107204 (2017).\n[20] Mochizuki, M. & Seki, S. Magnetoelectric resonances an d\npredicted microwave diode effect of the skyrmion crystal\nin a multiferroic chiral-lattice magnet. Physical Review\nB87(2013).\n[21] Mruczkiewicz, M., Gruszecki, P., Zelent, M. &Krawczyk ,\nM. Collective dynamical skyrmion excitations in a\nmagnonic crystal. Physical Review B 93174429 (2016).\n[22] Landau, L. D. & Lifshitz, E. M. Theory of the dispersion\nof magnetic permeability in ferromagnetic bodies. Phys.\nZ. Sowietunion 8153169 (1935).\n[23] Gilbert, T. L. Lagrangian formulation of the gyromag-\nnetic equation of the magnetization field. Phys. Rev. 100\n12431243 (1955).\n[24] Hu, Y. Wave nature and metastability of emergent crys-5\ntals in chiral magnets. Communications Physics 182\n(2018).\n[25] Hu, Y. & Wan, X. Thermodynamics and elasticity of\nemergent crystals. arXiv: 1905.02165 (2019).\n[26] Zang, J., Mostovoy, M., Han, J. H. & Nagaosa, N. Dy-\nnamics of Skyrmion Crystals in Metallic Thin Films.\nPhys. Rev. Lett. 107136804 (2011).\n[27] Tatara, G., Kohno, H. & Shibata, J. Microscopic ap-proach to current-driven domain wall dynamics. Physics\nReports 468213301 (2008).\n[28] Brown, W. F. Jr. Micromagnetics. (Interscience Tracts\non Physics and Astronomy Publishers, 1963).\n[29] Hu, Y. Emergent elastic waves in skyrmion crystals with\nfinite frequencies at long wavelength limit. ArXiv (2019).\n[30] Hu, Y. Long-wavelength emergent phonons in dis-\ntorted skyrmion crystals induced by intrinsic exchange\nanisotropy and a tilted magnetic field. ArXiv (2019)."    },    {        "title": "1302.0105v1.Entanglement_dynamics_in_finite_qudit_chain_in_consistent_magnetic_field.pdf",        "content": "arXiv:1302.0105v1  [quant-ph]  1 Feb 2013April 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\nInternational Journal of Quantum Information\nc/circlecopyrtWorld Scientific Publishing Company\nENTANGLEMENT DYNAMICS IN FINITE QUDIT CHAIN\nIN CONSISTENT MAGNETIC FIELD\nE. A. IVANCHENKO\nNational Science Center “Institute of Physics and Technolo gy”, Institute for Theoretical\nPhysics, Akademicheskaya str. 1, 61108 Kharkov, Ukraine\nyevgeny@kipt.kharkov.ua\nBased on the Liouville-von Neumann equation, we obtain a clo sed system of equations\nforthe description of a qutrit or coupled qutrits in an arbit rary,time-dependent, external\nmagnetic field. The dependence of the dynamics on the initial states and the magnetic\nfield modulation is studied analytically and numerically. W e compare the relative en-\ntanglement measure’s dynamics in bi-qudits with permutati on particle symmetry. We\nfind the magnetic field modulation which retains the entangle ment in the system of two\ncoupled qutrits. Analytical formulae for the entanglement measures in finite chains from\n2 to 6 qutrits or 3 quartits are presented.\nKeywords : Entanglement; qudit; multiqudit chain.\nPACS: 03.67.Bg, 03.67.Mg\n1. Introduction\nMulti-level quantum systems are studied extensively, since they ha ve wide appli-\ncations. Some of the existing analytical results1for spin 1 are derived in terms of\na coherent vector2. The class of exact solutions for a three-level system is given\nin Ref. 3. The application of coupled multi-level systems in quantum de vices is\nactively studied4. The study of these systems is topical in view of possible appli-\ncations for useful work in microscopic systems5. Exact solutions for two uncoupled\nqutrits interacting with the vacuum are obtained in Ref. 6. For the c ase of qutrits\ninteracting with a stochastic magnetic field, the exact solutions are obtained in\nRef. 7. The exact solutions for coupled qudits in an alternating magn etic field, to\nour knowledge, have not yet been found.\nThe entanglement in multi-particle coupled systems is an important re source for\nmany problems in the quantum information science, but its quantitat ive value com-\nputing is difficult because of different types of entanglement. Multi-d imensional\nentangled states are interesting both for the study of the found ations of quantum\nmechanics and for the topicality of developing new protocols for qua ntum commu-\nnication. For example, it was shown that for maximally entangled stat es of two\nquantum systems, the qudits break the local realism stronger tha n the qubits8,\nand the entangled qudits are less influenced by noise than the entan gled qubits.\n1April 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\n2E. A. Ivanchenko\nUsing entangled qutrits or qudits instead of qubits is more protectiv e from inter-\nception. From a practical point of view, it is clear that generating an d saving the\nentanglement in a controlled manner is the primary problem for the re alization of\nquantum computers. Maximally entangled states are best suited fo r the protocols\nof quantum teleportation and quantum cryptography.\nThe entanglement and the symmetry are two basic notions of quant um mechanics.\nWe study the dynamics of multipartite systems, which are invariant a t any subsys-\ntem permutation. The aim of this work is finding exact solutions for th e dynamics\nof coupled qudits interacting with an alternating magnetic field as well as the com-\nparative analysis of the entanglement measures in a finite chain of co upled qudits.\nThe paper is organized as following. The Hamiltonian of the anisotropic qutrit in\nan arbitrary alternating magnetic field is described in Sec. II. Then t he system of\nequations for the description of the qutrit dynamics is derived in the Bloch vector\nrepresentation. We introduce a consistent magnetic field, which de scribes an entire\nclass of field forms. In section III we find an analytical solution for t he density\nmatrix in the case of isotropic interaction. Analytical formulae, whic h describe the\nentanglement in finite spin chains of qutrits or quartits, are presen ted in Sec. IV.\nThe results are demonstrated graphically in Sec. V at specific param eters. The\nconclusions are given in Sec. VI.\n2. Qutrit\n2.1.Qutrit Hamiltonian and Liouville-von Neumann equation\nWe take the qutrit Hamiltonian (for the spin particle with s=1) in the sp ace of\none qutrit C3in the basis |1>= (1,0,0),|0>= (0,1,0),| −1>= (0,0,1), in an\nexternal magnetic field− →h= (h1,h2,h3) with anisotropy, in the form\nˆH(− →h) =h1S1+h2S2+h3S3+Q(S2\n3−s(s+1)\n3E2s+1×2s+1)+d(S2\n1−S2\n2),(1)\nwhereh1, h2, h3are the Cartesian components of the external magnetic field in\nfrequency units (we assume /planckover2pi1= 1, Bohr magneton µB= 1);S1, S2, S3are the\nspin-1 matrices9;E2s+1×2s+1is the unity matrix; Q, dare the anisotropy con-\nstants. When the constants Q, dare zeros, then the two Hamiltonian eigenvalues\nare symmetrically placed with respect to the zero level.\nThere exist many useful bases10. Allard and Hard11(AH) formed the Hermitian\nbasisCαfrom the linear combinations of the irreducible tensor operators. T his\nbasis is normalized so that S1=C1,x,S2=C1,y,S3=C1,z,irrespective of the\nspin quantum number s. It is convenient to construct the spin Hamiltonian for any\nspin. Hereinafter we use the Hermitian basis11. From the physical point of view, for\nimportant physical applications the basis11,12is preferred. It is not necessary for\nthe basis to be Hermitian since the results of the calculations are inde pendent of\nthe choice of base, but there is a significant advantage of the Herm itian basis. It is\nuseful that the Liouville-von Neuman equation does not involve comp lex numbersApril 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\nEntanglement dynamics in finite qudit chain 3\nand can be solved using real algebra. It makes numerical calculation s faster and\nsimplifies the interpretation of the equation system. The transition matrix deter-\nmines the coupling between the generalized Gell-Mann and (AH) Hermit ian matrix\nbases. This coupling for qutrit is presented in Ref. 9.\nThe qutrit dynamics in a magnetic field is described in the density matrix for-\nmalism using the Liouville-von Neumann equation\ni∂tρ= [ˆH, ρ], ρ(t= 0) =ρ0. (2)\nIt is convenient to rewrite Eq. (2) presenting the density matrix ρin the decomposi-\ntion with a full set11of orthogonal Hermitian matrices Cα(further the summation\nover the Greek indices will be from 0 to 8 and over the Latin ones from 1 to 8)\nρ=1√\n6CαRα. (3)\nSince TrCi= 0 for 1 ≤i≤8, then from the condition Tr ρ=R0it follows\nthatR0= 1. And although the results are independent of the basis choice, in\nthis basis the functions Ri= TrρCihave the concrete physical meaning11. The\nvaluesR1,R2,R3are the polarization vector Cartesian components; R4is the two-\nquantum coherencecontribution in R2;R5isthe one-quantumanti-phasecoherence\ncontribution in R2;R6is the contribution of the rotation between the phase and\nanti-phase one-quantum coherence; R7is the one-quantum anti-phase coherence\ncontribution in R1;R8is the two-quantum coherence contribution in R1.\nThe Liouville-von Neumann equation in terms of the functions Ritakes the form of\na closed system of 8 real differential first-order equations. This s ystem of equations\ncan be written in a compact form as following12,13:\n∂tRl=eijlhiRj, (4)\nwhereeijlarethe structureconstants, hi= 2(h1,h2,h3,0,0,Q√\n3,0,d)arethe Hamil-\ntonian components Eq. (1) in the basis Cα.\n2.2.The consistent field\nLet us consider the qutrit dynamics in an alternating field of the form\n/vectorh(t) = (ω1cn(ωt|k), ω1sn(ωt|k), ω0dn(ωt|k)), (5)\nwhere cn,sn,dn are the Jacobi elliptic functions14. Such field modulation under the\nchangingofthe elliptic modulus kfrom0to1describesthewholeclassoffield forms\nfrom trigonometric15(cn(ωt|0) = cosωt,sn(ωt|0) = sinωt,dn(ωt|0) = 1 ) to the\nexponentially impulse ones (cn( ωt|1) =1\nchωt,sn(ωt|1) = thωt,dn(ωt|1) =1\nchωt)16.\nThe elliptic functions cn( ωt|k) and sn(ωt|k) have the real period4K\nω, while the\nfunction dn( ωt|k) has a period of half the duration. Here Kis the full elliptic\nintegral of the first kind14. In other words, even though the field is periodic with a\ncommon real period4K\nω, but as we can see, the frequency of the longitudinal field\namplitude modulation is twice as high as that of the transverse field. W e call suchApril 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\n4E. A. Ivanchenko\nfield consistent.\nLet us make use of the substitution ρ=α−1\n1rα1with the diagonal matrix\nα1= diag(f,1,f−1), wheref(ωt|k) = cn(ωt|k) +isn(ωt|k).Then we obtain the\nequation for the matrix rin the form\ni∂tr= [α1ˆHα−1\n1−iα1∂t(α−1\n1),r]. (6)\nThe equation for the matrix rwithout taking into account the anisotropy can be\nwritten as following\ni∂tr= [ω1S1+δdn(ωt|k)S3,r], r(t= 0) =ρ0, δ=ω0−ω. (7)\nAtk= 0 equation (7) describes the dynamics of the qutrit in a circularly po larized\nfield15,17,18. The exact solutions of this equation are known, and under certain\ninitial conditions the explicit formulae are given in Ref. 19. At the exac t resonance,\nω=ω0it is straightforward to present (2) in the deformed field ( k/\\egatio\\slash= 0) (5) for the\ngiven initial condition ρ=ρ0:\nρ(t) =α−1\n1e−iω1tS1ρ0eiω1tS1α1. (8)\nExplicit solutions for specific initial conditions are given in Ref. 9.\n3. Bi-qutrit\nIn the space C3⊗C3the bi-qutrit density matrix can be written in the Bloch\nrepresentation\n̺=1\n6RαβCα⊗Cβ, R00= 1, ̺(t= 0) =̺0, (9)\nwhere⊗denotes the direct product. The functions Rm0,R0mcharacterize the in-\ndividual qutrits and functions Rmncharacterize their correlations.\nLet us consider the Hamiltonian of the system of two qutrits with anis otropic and\nexchange interaction in a magnetic field in the following form\nH2=ˆH(− →h)⊗E2s+1×2s+1+E2s+1×2s+1⊗ˆH(− →¯h)+JSi⊗Si,(10)\nwhere− →hand− →¯hare the magnetic field vectors in frequency units, which op-\nerate on the first and the second qutrits respectively, and Jis the constant\nof isotropic exchange interaction. We study the dynamics of two qu trits in the\nconsistent 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.\nLet us transform the matrix density ̺=α−1\n2r2α2with the matrix α2=α1⊗α1.\nThe equation for the matrix r2takes the form i∂tr2= [/tildewideH(dn(ωt|k)),r2] with the\ntransformed Hamiltonian /tildewideH(dn(ωt|k))9.\nSince dn(ωt|k)|k=0= 1, then the transformed Hamiltonian /tildewideHdoes not depend on\ntime, and the solution for the density matrix in the circularly polarized field has\nthe form\n̺(t) =α−1\n2e−i/tildewideHt̺0ei/tildewideHtα2|k=0. (11)April 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\nEntanglement dynamics in finite qudit chain 5\nIn the consistent field at resonance ω=̟0=ω0=hat equal̟1=ω1the\nHamiltonian eigenvalues equal to −2J,−J,J,J−2ω1,−J−ω1,J−ω1,−J+ω1,J+\nω1,J+2ω1. This allows to find the exact solution in the closed form for any initial\ncondition since the matrix exponent ei/tildewideHtin this case can be calculated analytically.\nFor a larger number of the qudits with a pairwise isotropic interaction , the general-\nizationis evident. Inthe caseofinteractionofqudits with a different dimensionality,\nthe reduction of the original system to the system with constant c oefficients can be\ndone by choosing, for example, the transformation matrix for spin -3/2 and spin-2\nin the form\ndiag(f3/2, f1/2, f−1/2, f−3/2)⊗diag(f2, f,1, f−1, f−2). (12)\nHowever, the Hamiltonian eigenvalues cannot be found in a simple analy tical form\nbecause of the lowering of the system’s symmetry.\n4. Analytical formulae for entanglement measures\n4.1.Entanglement in the bi-qutrit\nFor the initial maximally entangled state which is symmetrical at the pa rticle per-\nmutation\n|ψ>=1√\n31/summationdisplay\ni=−1|i>⊗|i>, (13)\nin the consistent field at the resonance ω=̟0=ω0=hat equal̟1=ω1,\nthe exact solution for the correlation functions is given in Ref. 9. Th e correlation\nfunctions have the property Rαβ=Rβα, i.e. the symmetry is conserved during the\nevolution, since the initial state and the Hamiltonian are symmetric wit h respect\nto the particle permutation.\nGiven the exact solution, one can find the negative eigenvalues of th e partly trans-\nposed matrix ̺pt= (T⊗E)̺(hereTdenotes the transposition): ǫ1=ǫ2=\n−1\n27√\n69+28cos3 Jt−16cos6Jt, ǫ3=−1\n27(5+4cos3Jt). The absolute value of\nthe sum of these eigenvalues\nmVW=3/summationdisplay\ni=1|ǫi| (14)\ndefines the entanglement measure (negativity) between the qutr its20.\nThe entanglement between the qudits can be described quantitativ ely with the\nmeasure21\nmSM=/radicalbigg\n1\nD−1(Rij−Ri0R0j)2, (15)\nwhere D is the basis dimension (for qutrit D= 9). This measure equals to 0 for the\nseparable state and to 1 for the maximally entangled state, and it is a pplicable for\nboth pure and mixed states.April 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\n6E. A. Ivanchenko\nThat is why for the maximally entangled initial state of two qutrits, th e entangle-\nment in the consistent field is defined by the formulae with the found s olution for\nthe density matrix\nmSM=1\n81√\n4457+2776cos3 Jt−632cos6Jt−56cos9Jt+16cos12Jt.(16)\nThis measure is numerically equivalent to the measure mVW20,22which is defined\nby the absolute value of the sum of the negative eigenvalues.\nAccording to the definition for N-qudit pure state23, the entanglement measure\nequals to\nηN=1\nNN/summationdisplay\ni=1Si, (17)\nwhereSi=−Trρilogbρiis the reduced von Neumann entropy, the index inumer-\nates the particles, i.e. the other particles are traced out. We use t he logarithm to\nthe basebto ensure that the maximal measure is normalizedto 1. The base bequals\nto 3 in the qutrit case.\nSince the qutrit reduced matrix eigenvalues equal to λ1=λ2=1\n27(5 +\n4cos3Jt), λ3=1\n27(17−8cos3Jt),then the entanglement measure in the bi-qutrit\ntakes the form\nη2=−3/summationdisplay\ni=1λilog3λi. (18)\nNormalized to unity the measure I-concurrence which is easy to calc ulate is defined\nby the formulae24\nmI=/radicalbigg\nd\nd−1/radicalBig\n(1−Trρ2\n1) =1\n9√\n57+32cos3 Jt−8cos6Jt, (19)\nwhered= 3 for a qutrit, ρ1=1√\n6CαRα0is the reduced qutrit matrix.\nThe measures mVW, mSM, η2, mIdo not depend on the parameters of the con-\nsistent field, the sign of the exchange constant at zero anisotrop y parameters. It\nshould be noted that the Wootters entanglement measure (the co ncurrence) in the\nsystem of two qubits with an isotropic interaction in a circularly polariz ed field at\nresonance is also independent of the alternating field amplitude25, but depends on\nthe exchange constant Jand the initial conditions only.\nThe numerical solution of the Liouville-von Neumann equation shows t hat if an\nidentical external field operates on every qudit, the free Hamilton ian and the inter-\naction Hamiltonian are commutative operators, the measures cons idered are deter-\nmined only by the symmetric two-body interaction with the interactio n constant\nofJ. If a different field operates on every qudit, there arises a broken permutation\nsymmetry of the total Hamiltonian, which changes the entanglemen t dynamics.\nThus it is possible to control entanglement by changing the paramet ers of an ex-\nternal field.April 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\nEntanglement dynamics in finite qudit chain 7\nIt is possible to show that the distance measure26,27/radicalbig\nTr(ρ(t)−ρ0)2depends\non the parameters of the consistent field.\nAt a zero external field the entanglement measure (15) takes the analytical form\nat equal non-zero anisotropy parameters Q=d=d=Q\nmSM(Q) =1\n(9J2+8QJ+16Q2)2/radicaltp/radicalvertex/radicalvertex/radicalbt4/summationdisplay\nk=0qkcos/parenleftBig\nk/radicalbig\n9J2+8QJ+16Q2t/parenrightBig\n,(20)\nwhereq0= 4457J8+ 11616QJ7+ 47392Q2J6+ 85888Q3J5+ 163072Q4J4+\n194560Q5J3+221184Q6J2+131072Q7J+65536Q8;\nq1= 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.\n4.2.Entanglement in the chain of qutrits\nWe consider the Hamiltonian of the chain of Nqutrits with the pairwise isotropic\ninteraction in the consistent field /vectorh(t) (5) at resonance in the following form\nHN=/summationdisplay\n(/vectorh(t)− →S⊗N−1/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nE⊗···⊗E+J− →S⊗− →S⊗N−2/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nE⊗···⊗E),(21)\nwherethe summationisoverdifferentpossiblepositionsof− →Sinthe directproducts.\nBecause the maximally entangled state of Nqutrits\n|φ>N=1√\n31/summationdisplay\ni=−1|i>⊗N(22)\nand the Hamiltonian (21) have a permutation symmetry, it follows tha t the density\nmatrix ofNqutrits has symmetric correlation functions.\nTheentanglementmeasuresformany-particlemulti-levelquantum systemshave\nnot been studied enough and are difficult to calculate in the analytical form, that\nis why we will present analytical formulae only for the entropy measu reηN, which\nis defined by the eigenvalues of the reduced one-particle matrices f or each qutrit.\nAs the result of the mentioned symmetry the reduced matrices are equal to each\nother. Therefore the entanglement measure for Nqutrits reads\nηN=−3/summationdisplay\ni=1rilog3ri. (23)\nThe eigenvalues of the reduced matrices for 3, 4, 5, and 6 qutrits a re presented inApril 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\n8E. A. Ivanchenko\nthe 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.\n(24)\nThe measures η3, η4, η5, η6do not depend on the sign of the exchange constant\nlike the measure η2.\n4.3.Entanglement in the bi-quartit\nThe applied approach for qutrits is translated to qudits. For a spin- 3/2 particle\nor a four-level system, also denoted as a quartit, we take the Ham iltonian in\nthe space C4in the basis |3/2>= (1,0,0,0),|1/2>= (0,1,0,0),| −1/2>=\n(0,0,1,0),| −3/2>= (0,0,0,1) and use the matrix representation of a complete\nset of the Hermitian orthogonal operators.\nWe will find the analytical formulae in the bi-quartit and in the 3 quartit s (in\nbi-pentit, see below) with a pairwise isotropic interaction of the initial maximally\nentangled state in a consistent magnetic field at resonance without taking into\naccount the anisotropy.\nThe negative eigenvalues of the partly transposed matrix ̺ptare\nequal to λ1=1\n100(−13−12cos5Jt),λ2=λ3=λ4=λ5=\n−1\n100√\n409+288cos5 Jt−72cos10Jt, λ6=1\n100(−37 + 12cos5 Jt).The entangle-\nment measure in the bi-quartit equals\nmbi−qrt\nVW=6/summationdisplay\ni=1|λi|. (25)\nThe entanglement between the quartits is described quantitatively with the\nmeasure21\nmbi−qrt\nSM=√\n1803365+191616cos5 Jt−35808cos10 Jt−6912cos15Jt+864cos20 Jt\n625√\n5.\n(26)\nSince the quartit reduced matrix eigenvalues equal to λ1=λ2=1\n100(13 +\n12cos5Jt), λ3=λ4=1\n100(37−12cos5Jt), hence the measure η2reads\nηbi−qrt\n2=−4/summationdisplay\ni=1λilog4λi. (27)\nThe I-concurrence is equal to\nmbi−qrt\nI=1\n25√\n553+96cos5 Jt−24cos10Jt. (28)April 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\nEntanglement dynamics in finite qudit chain 9\n4.4.Entanglement in the quartit chain\nThe eigenvalues of the reduced matrices for 3 quartits are equal t or1=r2=\n0.141+0.068cos5Jt\n2+0.04cos8Jt,r3=r4= 0.359−0.068cos5Jt\n2−0.04cos8Jt.\nTherefore the entanglement measure for 3 quartits is following\nηqrt\n3=−4/summationdisplay\ni=1rilog4ri. (29)\n4.5.Entanglement in the bi-pentit\nFor the spin-2 particle or a 5-level system, also denoted as a pentit , we take\nthe Hamiltonian in the space C5in the basis |2>= (1,0,0,0,0),|1>=\n(0,1,0,0,0),|0>= (0,0,1,0,0),|−1>= (0,0,0,1,0),|−2>= (0,0,0,0,1).\nThe entanglement between the pentits is described using the measu re21\nmbi−pnt\nSM= (0.802+0.106cos3Jt−0.019cos4Jt+0.242cos7Jt−0.098cos10Jt−\n0.088cos14Jt+0.067cos17Jt−0.014cos20Jt)1/2. The I-concurrenceis determined\nby the formulae mbi−pnt\nI= (0.791+0.114cos3Jt−0.018cos4Jt−0.005cos6Jt+\n0.230cos7Jt−0.079cos10Jt−0.079cos14Jt+0.060cos17Jt−0.015cos20Jt)1/2.\nWe have replaced the exact bulky rational coefficients by its decimal approxima-\ntions and the terms less than 0.001 have removed for inconvenience reduction.\nThe pentit reduced matrix eigenvalues are equal to p1=p2=1\n6125(1173−\n140cos3Jt+ 640cos7Jt−448cos10Jt),p3=p4=1\n6125(513 + 280cos3 Jt+\n320cos7Jt+ 112cos10 Jt),p5=1\n6125(2753−280cos3Jt−1920cos7Jt+\n672cos10Jt), hence the measure η2reads\nηbi−pnt\n2=−5/summationdisplay\ni=1pilog5pi. (30)\nAll the measures do not depend on the sign of the exchange consta nt and the\nparameters of the consistent field at zero anisotropy parameter s.\n5. Numerical results\nAlthough the analytical expressions for the measures in a bi-qutrit mVW, mSMare\ndifferent, but the numerical values are practically identical. The max imal deviation\nin the rectangle (1 ≥J≥0.01)×(100≥t≥0) equals 0.014.\nMeasuresη2andmIqualitatively coincide with the measures mVW, mSM.\nWe have found that the anisotropy of the qutrits disentangles the m, namely the\nentanglement is decreased down to 0.001 (see graphs 1 and 2 in Fig.1) .\nIn the constant longitudinal field− →h=−− →\nh= (0,0, ω0) (the bi-qutrit Hamiltonian\neigenvalues are equal to J,J,x1,x2,x3,−p,−p,p,p, wherex1,x2,x3are the roots of\nthe equation x3+2x2J−p2x−2J3= 0,p=/radicalbig\nJ2+ω2\n0) the Hamiltonian contains\ntheasymmetricpart,thusitfollowsthatthedensitymatrixforthe initialsymmetric\nstate will not be symmetric because of the breaking of the symmetr y of the particleApril 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\n10E. A. Ivanchenko\n/s48 /s49/s53 /s51/s48 /s52/s53/s48/s46/s53/s49/s46/s48\n/s113/s117/s116/s114/s105/s116/s115\n/s116/s49\n/s50/s51\n/s52/s53\n/s48 /s50/s48 /s52/s48 /s54/s48/s48/s46/s53/s49/s46/s48\n/s116\nFig. 1. Dynamics in the bi-qutrit: in the\nzero external field at equal anisotropy con-\nstantsQ= 0.0250, J=−0.1 (curve 1) and\nJ= 0.1 (curve 2); in the consistent field at\nJ= 0.1,Q= 0 the curve 3 shows complete\ncoincidence of mV WandmSM; the curve 4\nis the measure η2;mIis the curve 5.Fig. 2. Disentanglement of the maximally\nentangled state in the chain of 2,3,4,5,6\nqutrits with J= 0.1.\npermutations. The analytical solution is cumbersome. In the const ant longitudinal\nimpulse field− →h=−− →\nh= (0,0,2(θ((t−17)(t−60))+θ((40−t)(57−t)(t−60))))\nthe entanglement dynamics is blocked9atω0≫J. This points to the possibility\nto control the entanglement.\nIn Fig.2 we present the comparative dynamics of the entropy measu re in the finite\nqutrit chain. The disentanglement dynamics of the measures η3,η4,η5,η6is similar\nto the one in the case of two qutrits, but with smaller oscillation amplitu de, i.e.\nlarger number of the qutrits disentangles less than two qutrits (0 .889≤η3≤1).\nThe measures in bi-quartit, as shown in Fig.3, qualitative coincide, alm ost com-\npletelyηbi−qrt\n2andmbi−qrt\nVW. The disentanglement 3 quartits is insignificant less than\nin bi-quartit.\nThe disentanglement measures in bi-pentit, as shown in Fig.4, qualitat ive coin-\ncide, almost completely mbi−pnt\nIandmbi−pnt\nSM.\n6. Conclusion\nThe comparative analysis of the bi-qutrit entanglement measures o n the base of the\nanalyticalsolution for the density matrix demonstratesthat, in sp ite ofthe different\napproaches to the derivation of the formulae for the entanglemen t, all the formulae\nyield quite close results (Fig. 1), and the measures mV WandmSMare practically\nequal. This is in accordance with the general results for the entang lement in the\nsystems with a permutational symmetry22.\nThe analytical formulae for the measures η3,η4,η5,η6are similar to the measure for\ntwo qutrits η2, but with a numerically smaller oscillation amplitude, i.e. the larger\nnumber of the qutrits disentangles fewer than two qutrits.\nNevertheless, the comparison of measures in two coupled qutrits, quartits, andApril 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\nEntanglement dynamics in finite qudit chain 11\n/s48 /s49/s53 /s51/s48/s48/s46/s53/s49/s46/s48 /s113/s117/s97/s114/s116/s105/s116/s115/s32\n/s116/s49/s50\n/s51\n/s52/s53\n/s48 /s49/s53 /s51/s48/s48/s46/s53/s49/s46/s48\n/s112/s101/s110/s116/s105/s116/s115\n/s116/s51/s50/s49\nFig. 3. Disentanglement of the maximally\nentangled state in the chain of 2,3 quar-\ntits with J= 0.1. The measures mbi−qrt\nI,\nmbi−qrt\nSM,ηbi−qrt\n2,mbi−qrt\nV W, are presented by\nthe curves 1, 2, 3, 4 respectively; ηqrt\n3is the\ncurve 5.Fig. 4. Disentanglement of the maximally\nentangled state in the chain of 2 pentits\nwithJ= 0.1. The measures mbi−pnt\nI,\nmbi−pnt\nSM,ηbi−pnt\n2, are presented by the\ncurves 1, 2, 3 respectively.\npentits\nbi−qutrit m VW∼=mSM\nbi−quartit ηbi−qrt\n2∼=mbi−qrt\nVW\nbi−pentit mbi−pnt\nSM∼=mbi−pnt\nI(31)\non the base of analytical solutions shows the absence of a full coinc idence of the\nmeasures even in a particular case of disentangling a maximally entang led state.\nIn other words, it is impossible to prefer any measure, there remain s therefore the\nquestion concerning the quantitative determination of entangleme nt even in case of\ntwo multi-level particles.\nThe author is grateful to A. A. Zippa constant invaluable support.\nReferences\n1. F. T. Hioe, Phys. Rev. A 28(1983) 879.\n2. F. T. Hioe and J. H. Eberly, Phys. Rev. Lett. 47(1981) 838.\n3. A. M. Ishkhanyan, J. Phys. A 33(2000) 5041.\n4. V. E. Zobov et al., JETP Lett. 87(2008) 334.\n5. M. O. Scully et al., Science 299(2003) 862.\n6. L. Derkacz and L. Jakobczyk Phys. Rev. A 74(2006) 032313.\n7. A. Mazhar, Distillability sudden death in qutrit-qutrit systems under global decoher-\nence, quant-ph/0911.0767v1.\n8. D. Kaszlikowski, et al., Phys. Rev. Lett. 85(2000) 4418.\n9. E. A. Ivanchenko, Qutrit: entanglement dynamics in the fin ite qutrit chain in the\nconsistent magnetic field, quant-ph/1106.2297v1.\n10. R. A. Bertmann and P. Krammer, Bloch vectors for qudits an d geometry of entan-\nglement, quant-ph/0706.1743v1.\n11. P. Allard and T. Hard, Journal of Mag. Resonance 153(2001) 15.\n12. E. A. Ivanchenko J. Math. Phys. 50(2009) 042704.\n13. J. N. Elgin Phys. Lett. A 80(1980) 140.April 7, 2019 9:19 WSPC/INSTRUCTION FILE QUDITs-IJQI\n12E. A. Ivanchenko\n14. M. Abramovitz and I. A. Stegun (ed.), Handbook of Mathematical Functions (Dover,\nNew York, 1964).\n15. I. I. Rabi, Phys. Rev. 51(1937) 652.\n16. A. Bambini and P. R. Berman, Phys. Rev. A 23(1981) 2496.\n17. J. B. Miller, et al., Journal of Mag. resonance 151(2001) 228.\n18. M. Grifoni and P. Hanggi, Driven quantum tunneling, Physics Reports 304(1998)\n229.\n19. M. R. Nath, et al., Pramana-Journal of Physics 61(2003) 1089.\n20. G. Vidal and R. F. Werner, Phys. Rev. A 65(2002) 32314.\n21. J. Schlienz and G. Mahler, Phys. Rev. A 52(1995) 4396.\n22. G. Toth and O. G¨ uhne, Phys. Rev. Lett. 102(2009) 170503.\n23. F. Pan, et al., Int. J. Theor. Phys. 43(2004) 1241.\n24. F. Mintert, A. R. R. Carvalho, M. Kus and A. Buchleitner, M easures and dynamics\nof entangled states. Physics Reports 415(2005) 207.\n25. S. X. Zhang, et al., Commun. Theor. Phys. (Beijing, China) 50(2008) 883.\n26. V. Vedral and B. Plenio, Phys. Rev. A 57(1998) 1619.\n27. R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81(2009)\n865; Quantum entanglement, quant-ph/0702225v2."    },    {        "title": "2001.04044v1.Gate_tunable_spin_waves_in_antiferromagnetic_atomic_bilayers.pdf",        "content": " 1 Gate -tunable spin wave s in antiferromagnet ic atomic bilayers  \n \nXiao -Xiao Zhang1,2, Lizhong Li3, Daniel W eber4, Joshua Goldberger4, Kin Fai Mak1,3,5*, \nJie Shan1,3,5* \n \n1Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY 14853, USA  \n2Department of Physics, University of Florida  \n3School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA  \n4Department of Chemistry and Biochemistry , Ohio State University  \n5Laboratory of Atomic and Solid State Physics, Cornell Unive rsity, Ithaca, NY 14853, USA.  \n \n \nThe emergence of two -dimensional (2D) layered magnetic materials has open ed an exciting \nplayground for both fundamental studies of magnetism in 2D and exploration s of spin -\nbased applications  1-4. Remarkable properties , including spin filtering  in magnetic tunnel \njunctions  and gate control  of magnetic states , have recently been demonstrated  in 2D \nmagnetic materials  5-12. While these studies focu s on the static properties , dynamic \nmagnetic properties  such as excitation  and control of spin waves  have remain ed elusive. \nHere we excite spin waves and probe their dynamics in antiferromagnetic CrI 3 bilayers by \nemploying an ultrafast optical pump/ magneto -optical Kerr probe technique . We identify \nsub-terahert z magnetic resonances under an in -plane magnetic field , from which we \ndetermine the anisotropy and interlayer  exchange field s and the spin damping rates . We \nfurther show tuning of antiferromagnetic resonances by tens of gigahertz  through  \nelectrostatic gating . Our results shed light on magnetic excitations and spin dy namics in 2D \nmagnetic materials, and dem onstrate their unique potential for applications in ultrafast \ndata storage and  processing.  \n \nSpin waves, first predicted by F. Bloch in 1929, are propagating disturbance s in magnetic \nordering  in a magnetic material  13. The quanta of spin waves  are called magnons . The rich s pin-\nwave phenomena in magneti c materials have attracted fundamental  interest  and impacted on \ntechnology of telecommunication  systems , radars, and potentially  also low-power information \ntransmission and processing due to their decoupling from charge current  14,15. The main magnetic \nmaterial s of interest  have so far  been f erromagnets (FM) . The operation speed of FM-based \ndevice s is typically in  the GHz range , which is limited by  the ferromagnetic resonance (zero-\nmomentum resonance ) frequency . One of the major attractions of antiferromagnets (AFM ), a \nclass of much more common magnetic materials,  is the prospect of high -speed operation. The \nantiferromagnetic resonance s are in the frequency range of as high as THz due to the spin-\nsublattice exchange  16. The AFMs , however,  are difficult to access due to the absence of \nmacroscopic  magnetization.  \n \nThe recent discovery of t wo-dimensional (2D) layered magnetic materials  17-19, particularly A-\ntype AFMs  such as bilayer CrI 3 that are made of two antiferromagnetically coupled \nferromagnetic monolayers  17, present s new opportun ities to unlock the properties of AFMs. With \nfully uncompensated ferromagnetic surfaces , the magnetic state  can be easily accessed  and \ncontrolled  20. The van de r Waals nature allows their convenient integration into heterostructure s \nwith high -quality interface s 21. And t he atomic thickness  allows  the application of strong electric  2 field and large electrostatic doping to control the properties of 2D magnet ic material s. Although \nrapid progress has been made  in both fundamental understanding and potential applications  1-12,22, \nthe s pin dynamics , including basic properties such as magnetic resonances and damping , have \nremained unexplored  in these materials . A major technical challenge arises from the smal l \namount of spins present in atomically thin  samples of typical lateral dimensions of a few microns. \nThis makes studies with conventional probes,  such as  neutron scattering and microwave \nabsorption  23,24, extremely d ifficult or impossible. Microwave absorption  measurements are  \nfurther hindered by the high antiferromagnetic resonance frequencies.  \n \nHere we investigate spin-wave excitations in bilayer CrI 3 using  the time-resolved magneto -\noptical Kerr effect (MOKE). The sample consists of a heterostructure of bilayer CrI 3 and \nmonolayer WSe 2, which  is encapsulated in two hexagonal boron nitride (hBN) thin layers for \nprotect ion of  air-sensitive CrI 3 (Fig. 1a). While monolayer CrI 3 is a ferromagnetic semiconductor \nwith out -of-plane anisotropy  below the Curie temperature  of about 40 K, bilayer CrI 3 is an AFM \nwith spins in the two ferromagnetic monolayer s anti-aligned  below the N éel temperature of \nabout 45 K 17. Monolayer WSe 2 is a direct gap non-magnetic semiconductor  with strong spin-\norbit interaction  25. It is believed to have a type -II band alignment with CrI 3 26 (Fig. 1 b). The \nintroduction of WSe 2 significantly enhances  optical absorption of the pump and hot carrier  \ninjection into CrI 3 for magnetic excit ations . As will be discussed below , WSe 2 also breaks the \nlayer symmetry in bilayer CrI 3 to enable the detection of different oscillation  mode s of spin \nwaves in the polar MOKE  geometry . Figure 1 c is the magnetization of bilayer CrI 3 as a function \nof out -of-plane magnetic field at 4 K probed by  magnetic circular dichroism (MCD)  at 1.8 eV. \nThe antiferromagnetic behavior  is fully consistent with the reported results  17. The small nonzero \nmagnetization near zero field is a manifestation of the broken layer symmetry. The sharp turn -on \nof the magnetization around 0.75 T corresponds to a spin -flip transition, which provides a \nmeasure of the interlayer exchange field HE.  \n \nA pulsed laser (200 -fs pulse duration) was employed for the time -resolved  measurements . The \nheterostructure was excited by a  light pulse centered  near the WSe 2 fundamental exciton \nresonance  energy  (1.73 eV), and the change in CrI 3 magnetization is probed  by a time -\nsynchronized pulse  at a lower energy ( 1.54 eV ). Both t he pump and probe were  linearly \npolarized and at normal incidence. The polarization  rotation of the reflected probe beam  locked \nto the modulation frequency of the pump  was detected. In this configuration  the MOKE signal is \nsensitive only to the out -of-plane m agnetization.  An in -plane magnetic field H|| was applied, \nwhich causes the magnetization  of both the top and bottom layer s to cant (Fig. 1d). The field \nrequired to rotate the ordered moments into the  in-plane direct ion, which is referred to as the \nsaturation field HS, has been reported to be near 3.8 T for bilayer CrI 3 at 2 K8. Unless otherwise \nspecified, all measurements were  performed at 1.7 K.  (See Methods for details on the sample \nfabrication and the time -resolved MOKE set up.) \n \nFigure  2a displays the time evolution of the pump -induced change in the MOKE signal of bilayer \nCrI 3 under  H|| ranging from 0 – 6 T. For all fields , the MOKE signal shows a sudden change at  \ntime zero, followed by a decay  on the scale of 10’s – 100’s ps . This  reflects the incoherent  \ndemagnetization  process , in which the magnetic order is disturbed instantaneously by the pump \npulse and is slowly reestablishe d. Oscillations in the MOKE sig nal that are also instantaneous  3 with the optical excitation become clearly observable with increasing magnetic field . The \namplitude, frequency and damping of these oscillations evolve systematically with H||.  \n \nFigure 2b is the fast Fourier transform (FFT)  of the oscillat ory part  of the time traces in Fig. 2a.  \nTwo examples are shown in Fig. 3a and 3b for H|| at 1. 5 and 3.75 T, respectively. The \nexponential decay  of the incoherent demagnetization dynamics has been  subtracted from the \nMOKE signal before performing  FFT. At low magnetic field , a resonance around 70 GHz is \nobserved. As H|| increases, it splits into two  resonances  with one that redshifts significantly  and \nthe other  that exhibits minimal shifts in frequency until 3.3 T. Above  this field , both resonance s \nblueshift with  increasing  H||. While t he low -energy mode quickly becomes too weak to be \nobserved , the amplitude of the high -energy mode  does not depend strongly on  field.  \n \nWe performed a careful analysis of the MOKE dynamics directly in the time domain , fittting the \noscillations  with two damped harmonic waves  (red lines , Fig. 3a, b ). The extracted resonance \nfrequenc ies, damping rate s and amplitudes as a function of H|| are summarized in Fig. 3 d, 3e and \nSupplementary Fig. S 7, respectively. We first focus on the resonance frequencies. The field \ndependence  of the resonance frequencies  shows two di stinct regimes . Below about  3.3 T, the two \nnearly degenerate modes (at small fields) both soften with increasing field , one slightly and the \nother nearly to zero frequency . Above 3.3 T , both modes show a linear increase  in frequency \nwith a slope equal to the electron gyromagnetic ratio 𝛾/2𝜋 ≈ 28 GHz/T . The latter is \ncharacteristic of a ferromagnetic resonance  under high fields .  \n \nThe observed magnetic -field dispersion of the resonances is indicative of  their magnon origin \nwith 3.3 T correspond ing to the saturation field HS in bilayer CrI 3. The two modes are the spin \nprecession eigenmodes  of the coupled top and bottom layer magnetizations under an in -plane \nfield (Fig. 3c) . Above the saturation field , the spins are aligned  along the applied field and the \nspin waves become ferromagnetic -like. This interpretation is further supported by the  \ntemperature dependence  of the resonances (Supplementary Fig. S3-5). Clear mode softening is \nobserved with increasing temperature and the resonance feature disappears near the Néel \ntemperature  of bilayer CrI 3. The microscopic mechanism for the observed ultrafast excitation of \nspin waves in bilayer CrI 3 is not fully understood. A plausible  process is the exciton generation \nin WSe 2 by the optical pump , followed by ultrafast exciton dissociation and charge transfer at the \nCrI 3-WSe 2 interface  26, and an impulsive perturbation to  the magneti c interactions  27,28 in CrI 3 by \nthe hot carriers . Detail s on the supporting experiments of this mechanism are provided in \nMethods.  \n \nWe model the field dependent spin dynamics using  the coupled Landau -Lifshitz -Gilbert (LLG) \nequations, which d escribe  precession  of antiferromagnetically coupled top and bottom layer \nmagnetization s under an in -plane field 𝐻∥ 29 (Details are provided in Methods ). The effective \nmagnetic field responsible for spin precession  in each layer includes contributions from t he \napplied field H||, intralayer anisotropy  field HA, and the interlayer exchange field HE. In the \nsimple case of negligible damping and symmetr ic top and bottom layers , the frequency of the \nprecession eigenmodes are found as 𝜔𝑇= 𝛾[ 𝐻𝐴( 2𝐻𝐸+𝐻𝐴) + 2𝐻𝐸−𝐻𝐴\n 2𝐻𝐸+𝐻𝐴𝐻||2]1\n2, 𝜔𝐿=\n 𝛾[ 𝐻𝐴( 2𝐻𝐸+𝐻𝐴)− 𝐻𝐴\n 2𝐻𝐸+𝐻𝐴𝐻||2]1\n2  (before saturation ); and 𝜔𝑇= 𝛾√𝐻||(𝐻||−𝐻𝐴), 𝜔𝐿= 4  𝛾√(𝐻||−2𝐻𝐸)(𝐻||−2𝐻𝐸−𝐻𝐴)  (after saturation).  As shown schematically in Fig. 3c, the low-\nenergy mode  corresponds to the longitudinal (with respect to 𝐻∥) mode  𝜔𝐿, which  has net \nmoment oscillations only along the applied field direction  (the y-axis). The high-energy mode \ncorresponds to the transverse (with respect to 𝐻∥) mode 𝜔𝑇, which has net moment oscillations \nin the x-z plane. The longitudinal mode 𝜔𝐿 drops to zero at the saturation field 𝐻𝑆 = 2𝐻𝐸+𝐻𝐴.  \n \nThe simple solution fits the experimental data well for the entire magnetic field range (dashed \nlines, Fig. 3 d) with HA ≈ 1.77 T and HE ≈ 0.76 T . The interlayer  exchange  HE is in good \nagreement with th e value from  the spin-flip transition  measurement  under an out -of-plane field  \n(Fig. 1 c). The intralayer anisotropy HA or the saturation field ( HS  ≈ 3.3 T) is slightly smaller than \nthe reported value  8, likely due to the different doping  levels  present in different samples (see \ngate dependence studies  below) . In contrast to the simple model, t he measured 𝜔𝐿 is always \nfinite likely due to the layer asymmetry in  bilayer CrI 3 (caused by coupling to monolayer WSe 2), \nas well as inhomogeneous broadening (see below). The layer asymmetry also allows the \nobservation of the low-frequency mode  in the polar MOKE geometry, which would otherwise \nhave zero out-of-plane magnetization . \n \nNext  we discuss the damping of the spin waves  in 2D CrI 3. Figure 3 e is the magnetic -field \ndependence of the normalized damping rate 2𝜋\n𝜔𝜏 for both the transverse and longitudinal modes. \nOverall, damping is substantially higher below and near the saturation field for both modes . In \naddition, da mping of the longitudinal mode is generally higher than the transverse mode.  The \nhigh damping  observed below and near HS is likely originat ed from inhomogeneous broadening \nof the magnetic resonances  and spin wave dephasing . In this regime, the resonance frequencies \nare strongly dependent on internal magnetic interactions , which are sensitive to local doping and \nstrain  within the 2D layers . For instance, a ±10 % variation in the  interlayer exchange field alone  \n(which is comparable to the typical inhomogeneity reported in bilayer CrI 3 3) can account for the \nobserved damping  of the transverse mode  at HS. Inhomogeneous broadening also explains the \nseemingly larger damp ing for the longitudinal mode  near HS, where ωL has a steep dependence \non HE and HA. Above HS, the resonance  frequencies are basically determined by the  applied field \nand inhomogeneous broadening becomes insignificant , especially in the high-field limit ( e.g. at 6 \nT). Other damping mechanisms such as interfacial damping and  spin-orbit coupling of the i odine \natom  could be come  relevant  here. However,  our experiment on few -layer CrI 3 in the high -field \nlimit  show s weak dependence of  (𝜏𝑇)−1 on layer number  (Supplementary Fig. S 6), suggesting \nthat interfacial damping is not  important . Future systematic studies are warranted to fully \nunderstand the microscopic damping mechanisms.  \n \nFinally we demonstrate control of the spin waves by electrostat ic gating  using a dual -gate device \n(Methods) . Figure 4 a shows the FFT amplitude spectra of coherent spin oscillations under a \nfixed magnetic field  of 2 T at different gate voltages . The resonance shifts continuously  from ~ \n80 GHz to ~ 55 GHz when the gate voltage is var ied from -13 V to + 13 V  (corresponding to \nfrom ‘hole doping ’ to ‘electron doping ’). Figure 4b shows t he entire magnetic -field dispersion  of \nthe transverse mode at  varying gate voltages  (the longitudinal mode  is not studied because of its \nsmall amplitude) . As in the  zero gating  case, the initial redshift of the mode  is followed by a \nblueshift with increasing  magnetic  field at all gate voltages . The turning point, which is \ndetermine d by the sat uration field HS, is tuned by about  1 T by gate voltage . Furthermore, while \nthe dispersion  of 𝜔𝑇 is nearly unchanged by gating  above HS, it is strongly modified below HS.  5 In this regime the resonance frequency decreases by as much as 40 % when the gate voltage is \nvaried from -13 V to +13 V .   \n \nThe observed magnetic -field dispersion  of 𝜔𝑇 at all gate voltages  can be described by the simple \nsolution of the LLG equations  discussed above  (inset of Fig. 4 b) with doping dependent \ninterlayer exchange HE and intralayer anisotropy  HA (Fig. 4 c). Both fields decrease linearly with \nincreasing gate voltage , with HA at a faster rate than HE. A similar doping dependence  for HE has \nbeen reported previously from the spin-flip transition  measurement  under an out -of-plane field  6. \nSuch doping dependences of the magnetic interactions  can be understood as a consequence of \ndoping dependent electron occupancy of the magnetic Cr3+ ions and their wavefunction overlap . \nBased on this picture, increasing electron density weakens the magnetic interactions, and in turn \nthe effective magnetic fields responsible for spin precession below HS. Above HS, the \nmagnetization is fully saturated in the in -plane direction and  the spin resonance frequency is \nalmost solely determined by the applied field H|| and is therefore doping i ndependent.  A \nquantitative description  of the experimental result , however, would require ab initio  calculations \nand is beyond the scope of th e current  study.  \n \nIn conclusion, we have demonstrated the generation and detection of spin waves in a prototype \n2D magnetic material  of bilayer CrI 3 with a time-resolved optical pump -probe method . The \nresults allow the characterization  of important parameters such as the internal magnetic \ninteraction s and damping. We have also demonstrated widely gate tuna ble magnetic resonances \nin this 2D magnetic system ，revealing  the potential  of using 2D AFMs  to achieve  local gate \ncontrol of spin dynamics  for reconfigurable ultrafast spin-based devices  30,31.  \n \n \nMethods   \nSample and device fabrication  \nThe measured sample is a stack of 2D materials composed of (from top to bottom) few -layer \ngraphite, hBN , monolayer WSe 2, bilayer CrI 3, hBN, and few -layer graphite . The top and bottom \ngraphite/hBN pairs serve as gates.  An additional stripe of graphite is attached to the WSe 2 flake  \nfor grounding  and charge injection . The thickness of hBN layers is ~ 30 nm, and the graphite \nlayers , about  2-6 nm. Bulk crystals of hBN were purchased  from HQ graphene. Bulk CrI 3 \ncrystals were syn thesized by chemical vapor transport following methods described in previous \nreports32,33. These crystals crystallized into the C2/m space group with typical lattice constants \nof a=6.904Å,  b=11.899Å,  c=7.008Å and  β=108.74°, and Curie temperatures of 61 K. All \nlayer  material s were first exfoliated from their bulk crystals onto SiO 2/Si substrates and \nidentified by the ir color contrast under an optical microscope . The heterostructure was built by \nthe layer -by-layer d ry transfer technique  34. It was then released on to a substrate with pre -\npatterned gold electrodes, which contact the bottom gate, top gate, and grounding graphite flake.  \nThe steps involving CrI 3 before its full encapsulation in hBN layers were  performed inside a \nnitrogen -filled glovebox  because CrI 3 is air sensitive. In the gating experiment, equal top and \nbottom gate voltages were applied to the heterostructure and the gate voltage  shown in Fig. 4  \nwas the v oltage on each gate.  \n \nTime -resolved magneto -optical Kerr effect (MOKE)  and magnetic circular dichroism \n(MCD)   6 In the time-resolved MOKE setup, the probe beam  is the output of a Ti:Sapphire  oscillator  \n(Coherent Chameleon  with a repetition rate of 78 MHz  and pulse duration of 200 fs) centered at \n1.54 eV , and the p ump beam is the second harmonic of an optical parametric oscillator ( OPO ) \n(Coherent Chameleon compact OPO) output centered at 1.73 eV . The time delay between the \npump and probe pulses was controlled by a motorized linear delay stage. Both the pump and \nprobe beam  were  linearly polarized. The pump intensity was modulated at 100 kHz by a \ncombination of a half-wave photoelastic modulator (PEM ) and a linear polarizer whose \ntransmission axis is perpendicular to the original pump polarization. The pump and probe beam \nimpinge d on the sample at normal i ncidence. The reflected light was  first filtered to remove the \npump , passed through a half -wave Fr esnel rhomb  and a Wollaston prism , and detected by a pair \nof balanced photodiodes . The pump -induced change in Kerr rotation was determined  as the ratio \nof the intensity imbalance of the photodiodes obtained from a lock -in amplifier locked at the \npump modulation frequency and the intensity of each photodiode.   \n \nFor the MCD measurements, a single beam centered at  1.8 eV was used. The light beam  was \nmodulated at 50 kHz between the left and right circular  polarization  using a PEM . The reflected \nlight was focused onto a  photodiode . The MCD was determined as the ratio of the ac component  \nof the photodiode signal  measured by a lock -in amplifier at the polarization modulation \nfrequency  and the dc component of the photodiode signal measured by a voltmeter.  \n  \nFor a ll measurements samples were mounted in an optical cryostat (attoDry2100) with a base \ntemperature of 1.7 K and a  superconducti ng solenoid magnet  up to 9 Tesla. For measurements \nunder an out -of-plane field, the sample was mounted horizontally  and light was focused onto the \nsample at normal incidence  by a microscope objective . For measurements under an  in-plane field, \nthe sample was mounted vertically and the light beam was  guided by a  mirror at  45° and focused \nonto the sample at normal incident with a lens.    \n \nLandau -Lifshitz -Gilbert (LLG) equations  \nWe model the field  dependent spin dynamics in antiferromagnetic bilayer CrI 3 using coupled \nLandau -Lifshitz -Gilbert (LLG) equations  29,  \n \n𝜕𝑴𝑖\n𝜕𝑡=−𝛾𝑴𝑖×𝑯𝑖𝑒𝑓𝑓+𝛼\n𝑀𝑆𝑴𝑖×𝜕𝑴𝑖\n𝜕𝑡.  (1) \n \nwhere i = 1, 2. In Eqn. 1  𝑴𝑖 is the magnetization  of the top or bottom layer  (which are assumed \nto have  an equal magnitude 𝑀𝑆), 𝛾/2𝜋 ≈ 28 GHz/T is the electron gyromagnetic ratio , 𝛼 is the \ndimensionless damping factor , and 𝑯𝑖𝑒𝑓𝑓 is the effective magnetic field in each layer that is \nresponsible for spin precession . In the absence of applied magnetic field, 𝑴1 and 𝑴2 are anti-\naligned along the easy axis  (z-axis) . When  an in -plane field  𝑯∥ (along the y -axis) is applied, 𝑴1 \nand 𝑴2 are tilted symmetrically towards the y -axis, before fully turned into the applied field \ndirection at  the saturation field  𝐻𝑆 = 2𝐻𝐸+𝐻𝐴. Here 𝐻𝐸 and 𝐻𝐴 are the interlayer exchange  and \nintralayer anisotropy field s, respectively . A schematic is shown in Fig . 3c. The effective field  \n𝑯1,2𝑒𝑓𝑓=𝑯∥−𝐻𝐸\n𝑀𝑆𝑴2,1+𝐻𝐴\n𝑀𝑆(𝑴1,2)𝑧𝒛̂ has contributions from the applied field, the interlayer \nexchange  field,  and the intralayer anisotropy field . We search for solution in the form of a \nharmonic wave  𝑒𝑖𝜔𝑡 with angular frequency  ω. For the simpl e case of zero damping (𝛼 = 0), two \neigen mode frequencies 𝜔𝑇 and 𝜔𝐿 are given in the main text.   7  \nIn case of finite  but weak  damping, we find the following transverse and longitudinal modes  \nafter simplifying the LLG equations : \n \nBefore saturation (𝐻||<𝐻S), \n \n𝜔𝑇2(1+𝛼2)−𝑖𝛼𝜔𝑇𝛾(𝜔𝑇02𝛾2⁄\n2𝐻𝐸+𝐻𝐴+2𝐻𝐸+𝐻𝐴)−𝜔𝑇02=0; \n𝜔𝐿2(1+𝛼2)−𝑖𝛼𝜔𝐿𝛾(𝜔𝐿02𝛾2⁄\n𝐻𝐴+𝐻𝐴)−𝜔𝐿02=0; \nAfter saturation (𝐻||>𝐻𝑆), \n \n𝜔𝑇2(1+𝛼2)−𝑖𝛼𝜔𝑇𝛾(2𝐻||−𝐻𝐴)−𝜔𝑇02=0; \n \n𝜔𝐿2(1+𝛼2)−𝑖𝛼𝜔𝐿𝛾(2𝐻||−4𝐻𝐸−𝐻𝐴)−𝜔𝐿02=0. \n \nHere 𝝎𝑻𝟎 and 𝝎𝑳𝟎 correspond to  the solution  at zero damping  (𝜶 = 0). In particular, when 𝜶 << \n1, the oscillation frequency ( the real part of the solution for  𝝎𝑻 and 𝝎𝑳) becomes  𝝎𝟎\n√𝟏+𝜶𝟐, where \n𝝎𝟎 is the undamped solution  for the two modes . Overall,  the eigenmode frequencies are reduce d \ndue to damping , and t he two mode s will no longer be degenerate  at 𝑯||=𝟎 taking into  account  \nof higher order corrections of  𝜶.  At low temperature, we found this correction insignificant for \nthe high-frequency branch , which has a larger oscillation  amplitude and was measured with a \nhigher precision . Fitting the experimental data with the damped LLG solution yield ed similar \nvalues for 𝑯𝑬 and 𝑯𝑨.  \n \nMechanism for ultrafast excitation of coherent magnons   \nWe have investigated t he mechanism for the o bserved ultrafast excitation of magnons in bilayer \nCrI 3. A plausible picture involves exciton generation  in WSe 2 by the optical pump , ultrafast \nexciton dissociation and charge transfer at the CrI 3-WSe 2 interface, and  an impulsive \nperturbation to the magnetic anisotropy and exchange  fields  in CrI 3 by the injected hot carriers . \nSeveral  control experiments  were performed to test this picture . Pump-probe measurement s were \nperformed on both monolayer WSe 2 and bilayer CrI 3 areas alone  (non-overlapped  regions in  the \nheterostructure) under the same experimental conditions . Negligible pump -induced MOKE \nsignal  was observed . In addition , measurement was done on the heterostructure at different pump \nenergies . The magnetic resonance frequencie s were found unchanged, but the amplitudes follow \nthe absorption spectrum of WSe 2 (Supplementary Fig. S 1). These two experiments show that \nmagnons are generated through  optical excitation of  excitons in WSe 2. It has been reported \nearlier that CrI 3-WSe 2 heterostructure s have a type -II band alignment , which can facilitate \nultrafast exciton dissociation and charge transfer  26. Next t he onset of coherent oscillations  is \ninstantaneous with optical  excitation  in our experiment . This exclud es lattice heating in CrI 3 as a \ndominant mechanism  for the generation of magnons, which typically takes a longer time  to build \nup. Moreover , the resonance amplitude is independent of the pump laser polarization \n(Supplementary Fig. S 2), indicating that hot carriers, rather than the angular momentum  of the \ncarriers , are responsib le for the excitation of magnons. Finally, as we show in the main text , the  8 magnetic anisotropy and exchange can be effectively altered by carrier doping in CrI 3. These \nexperiments are all consistent with the proposed mechanism of ultrafast excitations  of magnons \nin CrI 3-WSe 2 heterostructures.  \n \nTemperature dependence of magnon modes  \nWe have performed the optical pump/MOKE probe experiment in CrI 3-WSe 2 heterostructures at \ntemperature ranging from 1.7 K to 50 K. No obvious oscillations can be measured above 50  K \nwhen bilayer CrI 3 is close to its N éel temperature.  The results at 1.7 K are presented in the main \ntext. Supplementary Fig. S3 and S 4 show the c orresponding measurements and analysis for 25  K \nand 45  K, respectively. With increasing temperature , the magnon frequency decreases and the \nsaturation field (estimated from the minimum of the frequency dispersion) also decreases . A \nsystematic temperature dependence is shown in Supplementary Fig. S 5 for the high -frequency \nmode 𝜔𝑇 at a fixed in -plane field of  2 T. The frequency has a negligible temperature dependence \nwell below the Néel temperature ( < 20 K), and decreases rapidly w hen the temperature \napproaches the N éel temperature . \n \nAdditiona l measurements on few -layer CrI 3   \nWe have measured the magnetic response from a few-layer CrI 3 (6-8 layer ) sample . Because of \nthe larger MOKE signal and higher optical absorption in thicker samples , magnetic oscillations \ncan be measured without the enhancement from monolayer WSe 2. The results are shown in \nSupplementary Fig. S 6. The comparison of results from samples of different thickness es \nprovides insight into the origin of magnetic damping. For instance, in the high -field limit  (6 T) , \nfew-layer and bilayer CrI 3 show a similar level of damping. This indicates that interfacial \ndamping is not the dominant contributor  to damping.   \n \n \nReferences  \n \n1 Gibertini, M., Koperski, M., Morpurgo, A. F. & Novoselov, K. S. Magnetic 2D materials \nand heterostructures. Nature Nanotechnology  14, 408 -419, (2019).  \n2 Gong, C. & Zhang, X. Two -dimensional magnetic crystals and emergent \nheterostructure devices. Science  363 , eaav4450, (2019).  \n3 Mak, K. F., Shan, J. & Ralph, D. C. Probing and controlling magnetic states in 2D \nlayered magnetic materials. Nature Reviews Physi cs 1, 646 -661, (2019).  \n4 Burch, K. S., Mandrus, D. & Park, J. -G. Magnetism in two -dimensional van der Waals \nmaterials. Nature  563 , 47-52, (2018).  \n5 Jiang, S., Shan, J. & Mak, K. F. Electric -field switching of two -dimensional van der \nWaals magnets. Nature M aterials  17, 406 -410, (2018).  \n6 Jiang, S., Li, L., Wang, Z., Mak, K. F. & Shan, J. Controlling magnetism in 2D CrI3 by \nelectrostatic doping. Nature Nanotechnology  13, 549 -553, (2018).  \n7 Huang, B.  et al.  Electrical control of 2D magnetism in bilayer CrI3. Nature \nNanotechnology  13, 544 -548, (2018).  \n8 Song, T.  et al.  Giant tunneling magnetoresistance in spin -filter van der Waals \nheterostructures. Science  360 , 1214, (2018).  \n9 Klein, D. R.  et al.  Probing magnetism in 2D van der Waals crystalline insulators via \nelectron tunneling. Science  360 , 1218, (2018).   9 10 Kim, H. H.  et al.  One Million Percent Tunnel Magnetoresistance in a Magnetic van der \nWaals Heterostructure. Nano Letters  18, 4885 -4890, (201 8). \n11 Song, T.  et al.  Voltage Control of a van der Waals Spin -Filter Magnetic Tunnel \nJunction. Nano Letters  19, 915 -920, (2019).  \n12 Wang, Z.  et al.  Very large tunneling magnetoresistance in layered magnetic \nsemiconductor CrI3. Nature Communications  9, 251 6, (2018).  \n13 Bloch, F. Zur Theorie des Ferromagnetismus. Zeitschrift für Physik  61, 206 -219, \n(1930).  \n14 Chumak, A. V., Vasyuchka, V. I., Serga, A. A. & Hillebrands, B. Magnon spintronics. \nNature Physics  11, 453 -461, (2015).  \n15 Kruglyak, V. V., Demokritov,  S. O. & Grundler, D. Magnonics. Journal of Physics D: \nApplied Physics  43, 264001, (2010).  \n16 Baltz, V.  et al.  Antiferromagnetic spintronics. Reviews of Modern Physics  90, 015005, \n(2018).  \n17 Huang, B.  et al.  Layer -dependent ferromagnetism in a van der Waal s crystal down to \nthe monolayer limit. Nature  546 , 270, (2017).  \n18 Gong, C.  et al.  Discovery of intrinsic ferromagnetism in two -dimensional van der \nWaals crystals. Nature  546 , 265, (2017).  \n19 Deng, Y.  et al.  Gate -tunable room -temperature ferromagnetism in two-dimensional \nFe3GeTe2. Nature  563 , 94-99, (2018).  \n20 Duine, R. A., Lee, K. -J., Parkin, S. S. P. & Stiles, M. D. Synthetic antiferromagnetic \nspintronics. Nature Physics  14, 217 -219, (2018).  \n21 Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto,  A. H. 2D materials and \nvan der Waals heterostructures. Science  353 , aac9439, (2016).  \n22 McCreary, A.  et al.  Quasi -Two -Dimensional Magnon Identification in \nAntiferromagnetic FePS3 via Magneto -Raman Spectroscopy. arXiv preprint \narXiv:1908.00608 , (2019).  \n23 MacNeill, D.  et al.  Gigahertz Frequency Antiferromagnetic Resonance and Strong \nMagnon -Magnon Coupling in the Layered Crystal ${ \\mathrm{CrCl}}_{3}$. Physical \nReview Letters  123 , 047204, (2019).  \n24 Lee, I.  et al.  Fundamental Spin Interactions Underlying the Magnetic Anisotropy in \nthe Kitaev Ferromagnet CrI $ _3$. arXiv preprint arXiv:1902.00077 , (2019).  \n25 Wang, G.  et al.  Colloquium: Excitons in atomically thin transition metal \ndichalcogenides. Reviews of Modern Physics  90, 021001, (2018).  \n26 Zhong, D.  et al.  Van der Waals engineering of ferromagnetic semiconductor \nheterostructures for spin and valleytronics. Science Advances  3, e1603113, (2017).  \n27 Kirilyuk, A., Kimel, A. V. & Rasing, T. Ultrafast optical manipulation of magnetic \norder. Reviews of Modern Phys ics 82, 2731 -2784, (2010).  \n28 van Kampen, M.  et al.  All-Optical Probe of Coherent Spin Waves. Physical Review \nLetters  88, 227201, (2002).  \n29 Gurevich, A. G. & Melkov, G. A. Magnetization oscillations and waves .  (CRC press, \n1996).  \n30 Vogel, M.  et al.  Optically reconfigurable magnetic materials. Nature Physics  11, 487, \n(2015).   10 31 Sadovnikov, A. V.  et al.  Magnon Straintronics: Reconfigurable Spin -Wave Routing in \nStrain -Controlled Bilateral Magnetic Stripes. Physical Review Letters  120 , 257203, \n(2018).  \n32 Shcherbakov, D.  et al.  Raman Spectroscopy, Photocatalytic Degradation, and \nStabilization of Atomically Thin Chromium Tri -iodide. Nano Letters  18, 4214 -4219, \n(2018).  \n33 Li, T.  et al.  Pressure -controlled interlayer magnetism in atomically thin CrI3.  Nature \nMaterials  18, 1303 -1308, (2019).  \n34 Wang, L.  et al.  One -Dimensional Electrical Contact to a Two -Dimensional Material. \nScience  342 , 614, (2013).  \n \n \nCompeting interests  \nThe authors declare no competing interests.  \n \n \nData availability  \nThe data that support the findings of this study are available within the paper and its \nSupplementary Information. Additional data are available from the corresponding authors upon \nrequest.  \n \n \n   11 Figures and figure captions  \n \n \nFigure 1 | Bilayer  CrI 3/monolayer WSe 2 heterostructure. a, Optical microscope image of the \nheterostruture . Bilayer CrI 3 is outlined with a purple line , and monolayer WSe 2, a black line. \nScale bar is 5 𝜇m. b, Schematic of a type-II band alignment between monolayer WSe 2 and CrI 3. \nOptically excited exciton in WSe 2 is dissocated at the interface and electron is transferred to CrI 3 \n26. c, MCD of the heterostrucutre as a function of out -of-plane magnetic field  at 4 K . Hysteresis \nis observed for field swe eping along two opposing  directions. Insets are schematics of the \ncorresponding  magnetizations in the top and bottom layers of  blayer CrI 3. The dashed line s \nindicate the spin -flip transition around  0.75 T. d, Schematic of bilayer CrI 3 under an in -plane \nmagnetic field H||. Below the saturation field, the magnetizations of the top and bottom layer are \nsymmetrically canted towards the applied field direction.  \n \n \n 12  \nFigure  2 | Time -resolved magn on oscillations. a, Pump-induced Kerr rotation as a function of \npump -probe delay time in bilayer CrI 3 under different in -plane magnetic field s. The curves are \ndisplaced vertically for clarity.  b, FFT amplitude spectra of the time dependences shown in a \nafter the demagnetization dynamics  (exponential decay ) were remov ed. The spectra are vertically \ndisplaced for clarity.  \n \n \n \n \n \n \n \n \n \n \n \n 13  \nFigure 3 |  Magnon dispersion and damping. a, b, Pump -induced MOKE dynamics in bilayer \nCrI 3 under two representative in -plane fields of 1.5  T (a) and 3.75 T  (b). Grey lines  are \nexperiment after subtracting the demagnetization dynamics , and red lines , fits to two damped  \nharmonic oscillations .  c, Illustration of two spin wave eigen modes in an  AFM : the transverse \nmode ( left) and the longitudinal mode  (right) . The dashed line s indicate the equilibrium top and \nbottom layer magnetization M 1 and M 2, which are  titled symmetrically from the z-direction \ntowards  the applied field direction ( y-axis) . The ma gnetization s precess follow ing the green and \nblue arrows  in the order  1 through 4.  d, e, Oscillation frequencies  (d) and da mping rates  (e) of \nthe transverse and longitudinal modes extracted from the two harmonic oscillation fit as a \nfunction of in-plane magnetic field. The error bars  are the fit uncertainties . The vertical d otted \nlines indicate the in -plane saturation magnetic field. Dashed lines in d are fits to the LLG \nequations as described in the text.  \n \n 14  \nFigure 4 | Gate  tunab le magnon  frequency . a, FFT amplitude spectra of the magnon s as a \nfunction of gate voltage  under a fixed in -plane field of  2 T. The dashed line is a guide to the eye \nof the evolution of the resonance frequency with gate voltage  and triangle s indicat e the peak of \nthe resonance . b, Magnetic -field dispersion of the transverse mode  at different gate voltage s. The \ninset shows the fits of the experimental data to the LLG equations. The same colored line s (LLG \nequation) and symbols (experiment) deno te the same gate voltage. c, Anisotropy  field HA and \nexchange field  HE extracted from the fits in b at different gate voltages . Error bars are the \nstandard deviation from the fitting. Dashed lines are linear fits.  \n \n \n \n \n \n \n \n  \n 15 Supplementary figures  \nFigure S1 | Amplitude  of the s pin wave s under a  fixed in-plane magnetic field of 2 T as a \nfunction of pump wavelength. The dependence  resembles that of the excitonic resonance in \nmonolayer WSe 2. The spectral broadening arises  from the additional WSe 2 trion absorption and \nthe linewidth of the light pulses  (~ 5 nm in full width at half maximum  (FWHM )) employed in \nthe pump -probe measurement . \n \n \nFigure S2 | Spin wave dynamics under H|| = 2 T excited by optical pump  of different  \npolarization s. The red, orange and blue lines correspond to left circularly polarized, linear \npolarized , and right circularly polarized pump , respectively . The curves were  vertically shifted \nfor easy comparison. The oscillation  amplitude does not depend on the pump polarization (i.e. \nphoton angular momentum ).  \n \n 16  \n \nFigure S3 | Magnon oscillations at 25  K. a, Spin dynamics in bilayer CrI 3 under  different \nmagnetic field s. The curves were  vertically displaced for clarity . b, FFT amplitude spectr a of a. \nc, In-plane field dispersion of the two magnon modes  extracted from fitting the time -resolved \nMOKE signal with two harmonic oscillations . \n \n \n \n \n \n \n \n \n \n  \n 17  \nFigure S4 | Magnon oscillations at 45 K. Same as in Supplementary Fig. S3. Due to the weak \nsignal, we can only identify the transverse mode 𝜔𝑇.  \n  \n 18  \nFigure S5 | Temperature dependence of the transverse magnon mode frequency under  a fixed in -\nplane magnetic field of 2  T. All other experimental conditions are the same  as in Fig. S3 and S4 .  \n \n \nFigure  S6 | Pump -probe m easurements on few -layer CrI 3 at 1.7  K. a, Spin wave dynamics \nunder  different in -plane magnetic field s. b, The corresponding FFT amplitude spectrum of a. The \ndamping at 6T is estimated to be ~0.04, which is similar to bilayer CrI 3.  \n \n 19  \nFigure S 7 | Amplitude of the longitudinal and transverse magnon modes extracted from  the \ntime-resolved MOKE measurement ( Fig. 2 of the main text ).  \n \n  \n"    },    {        "title": "1712.07280v1.Optical_Manipulation_of_Magnetic_Vortex_Visualized_in_situ_by_4D_Electron_Microscopy.pdf",        "content": "1 \n Optical Manipulation of Magnetic Vortex Visualized in situ by 4D Electron Microscopy  \n \nXuewen Fu1,*, Shawn D. Pollard2, Bin Chen3, Byung -Kuk Yoo4, Hyunsoo Yang2, Yimei Zhu1,* \n \nAffiliations:  \n1Condensed Matter Physics and Material Science Department, Brookhaven National Laboratory, Upton, \nNew York 11973, USA  \n2Department of Electrical and Computer Engineering, National University of Singapore, Singapore \n117576, Singapore  \n3Center for Ultrafast Science and Technology, School of Chemistry and Chemical Engi neering, \nCollaborative Innovation Center of IFSA, Shanghai Jiao Tong University, Shanghai 200240, China \n4Physical Biology Center for Ultrafast Science and Technology, Arthur Amos Noyes Laboratory of \nChemical Physics, California Institute of Technology, Pas adena, CA 91125, USA  \n*Corresponding author . E-mail:  zhu@bnl.gov ; xfu@bnl.gov  \n \nKeywords:  Magnetic vortex, Ferromagnetic  disk, Optical quenching , Lorentz  phase  imaging , \nMagnetization dynamics , 4D electron microscopy  \n  2 \n Abstract:  \nUnderstanding the fundamental dynamics  of topological  vortex and antivortex naturally \nformed in micro/nanoscale ferromagnetic building blocks under ex ternal perturbations is crucial  \nto magnetic vortex based information processing and spintronic devices. All previous studies have \nfocused on magnetic vortex -core switching  via external magnetic field s, spin -polarized current s, or \nspin wave s, which have largely prohibit ed the investigation o f novel spin configurations that could \nemerge from the ground states in ferromagnetic disks and their underlying  dynamics. Here, we \nreport in situ  visualization of femtosecond laser quenching induced magnetic vortex change in \nvarious symmetric ferromagneti c Permalloy  disks by Lorentz phase imaging using 4D electron \nmicroscop y. Besides the switching of magnetic vortex chirality and polarity,  we observed with  \ndistinct occurrence frequencies  a plenitude of complex magnetic  structure s that have  never been \nobserved by magnetic field or current assisted switching . These  complex  magnetic structure s \nconsist of a number of newly created topological  magnetic defects  (vortex and antivortex)  strictly \nconserv ing the topological winding number, demonstrat ing the direct impact of topological \ninvariant on the magnetization dynamics  in ferromagnetic disks . Their spin configurations show \nmirror or rotation symmetry due to the geometrical confinement of the disks. Combined \nmicromagnetic simulations with the experimental observations reveal the underlying \nmagnetization dynamics and formation mechanism of  the optical quenching induced complex \nmagnetic structures. The ir distinct  occurrence rates are pertinent to their formation -growth \nenerg etics  and pinning effects at the disk edge . Based on these findings , we propos e a paradigm of \noptical -quenching -assisted fast switching of vortex core s for the control of magnetic vortex based \ninformation recording  and spintronic devices.  3 \n Introduction  \nMagnetic vortex1,2 is one of the fundamental spin configurations occur ring in thin micro/nanometer -\nsized ferromagnetic disk  element s due to the confinement of spins imposed by geometric al \nrestrictions2,3. It is one kind of topological  magnetic defects characterized  by two degrees of freedom4: \n(1) “chirality” ( c = ±1), the in -plane curling magnetization that can be clockwise or anticlockwise along \nthe disk circumference; and (2) “polarity” ( p = ±1), the out -of-plane nanometer -sized core magnetization \nwhose direction is  either up or  down. This topologically protected magnetic vortex cannot be \ncontinuously transferred into a defect -free state , and is therefore regarded as very robust quasiparticles \nagainst thermal fluctuation5. Such unique properties make magnetic vortex a promising candidate for \nhigh density , non-volatile magnetic memories and spintronic devices6, because each  magnetic vortex can \nstore two bits of information by its chirality and polarity7,8.  \nFundamental u nderstanding of the magnetization dynamics associated with the precise manipulation \nof the chirality and polarity of the magnetic vortex in ferromagnetic building blocks is important for its \napplication  in magnetic data -storage7. It is well know n that the topological magnetic vortex in a \nferroma gnetic disk can be manipulated by external perturbations, such as pulsed magnetic fields7,9,10, \nalternating magnetic fields11,12, spin -polarized currents7,13-15, and fiel d-driven spin waves16,17. These \nexternal stimuli could drive the gyrotropic motion of the vortex core so that its polarization would be \nswitched through the creation and subsequent annihilation of a magnetic vortex -antivortex pair11,18 -20. \nNevertheless , due to  the gyrotropic motion of the vortex core prior to its switching, it is difficult to \nprecisely determine when the core switching occurs, and thus limit ing the ultimate  investigation of the \ncore switching dynamics that is important for designing vortex based data-storage devices. This \ngyrotropic motion also int rinsically restricts the speed of the magnetic vortex core  switching.  \nRecently, it has been demonstrated the possibility of ultrafast magnetic switching in ferromagnetic \nthin films via photothermal -assisted femtosecond (fs) laser pulse excitation21-25, where the  single  pulse 4 \n rapidly heats up the ferromagnetic films close to their Curie temperatures  and reduces the external \nmagnetic field required for the magnetic reversal . Especially by using  the inverse Faraday effect  of \ncircularly polarized fs -laser26,27, it is even possible to realize all -optical magnetization switching in \nferromagnetic films28-32. It has also been theoretically predicted the remarkable red uction of required \nmagnetic switching field for a topological magnetic vortex core at a temperature closely below the Curie \npoint33 and t he possibility of all -optical switching of magnetic vortex core34 in ferromagnetic disks . This \noptical ly associated  switching of magnetic vortex core does not involve the gyrotropic motion and thus \npossesses unique advantages in ultrafast magnetic recording. However, direct observation of magnetic \nvortex switching  or change  in geometrically con fined ferromagnetic disk s upon ultrafast laser quenching \nis rather challenging . Once achieved, it will provide a better fundamental understanding of the effects of  \ntopological  feature , magnetization relaxation  dynamics , and geometric al confinement on the magnetic \nvortex switching and its underlying  mechanism s.  \nIn this work, we report the in situ  visualization  of fs -laser quenching induced magnetic vortex \nconfigurations in symmetric ferromagnetic Permalloy  (Py) disks  by 4D electron  micro scopy (4D EM) \noperated in the Lorentz  phase imaging mode . Besides  the chirality and polarity switching of the \nmagnetic vortex, a  plenitude of complex metastable magnetic structures with distinct occurrence \nfrequencies were observed in the Py disks with a fs-laser pulse excitation above a fluence of 10 mJ/cm2. \nDifferent symmetric elements including circular, square and regularly triangular disks  were designed for \ninvestigating the geometrical confinement effect. The observed magnetic structures consist of a number \nof newly created topological defects (vortex and antivortex)  strictly restricted by  the topological winding \nnumber , and t heir spin configurations show mirror or rotation symmetry due to the geometrical \nconfinement  of the disks . Micromagnetic simulations reproduce all the observed magnetic structures, \nrevealing  the underlying magnetization dynamics and the formation mechanism s. Based on the results , 5 \n we propose a new paradigm of optical -quenching -assisted fast vortex core switching for the control of \nmagnetic vortex based information process ing and spintronic devices.  \nMetastable magnetic structures induced by fs -laser pulse  \nThe fs -laser pulse induced magnetic switching in three kinds of symmetric  Py (Ni81Fe19) disks, including \ncircle (diameter of 3 µm and 1.7 µm), square (edge length of 3 µm) and regular triangle (edge length of \n1.7 µm), were studied (see Methods) . To clearly resolve the magnetic vortex structures, we record the \nFresnel images by using continuous electron beam of 4D EM  in the Lorentz  phase imaging mode (see \nFig. 1a and Methods) . Under out -of-focus condition  of the Lorentz mode , the clockwise and counter -\nclockwise in -plane circling magnetizations of the magnetic vortices e xert opposite Lorentz force on the \nimaging electrons , result ing in black and white contrasts of the  vortex core35-38, as schematically shown \nin Fig. 1b . Upon a fs -laser pulse excitation , the Py disk is rapidly heated up and subsequently followed \nby a fast cooling at a quenching rate of ca.1012 K/s39 passing through  the thick  substrate below (see the \ninset of Fig. 1b ). Figure 1c present the typical Fresnel  images of the magnetic structures in the circular, \nsquare an d triangular Py disks before and after  fs-laser  pulse excitation with a fluence of 12 mJ/cm2 (see \nalso Movie S1 to S3). Based on the Lorentz contrast the corresponding spin configurations  are \nschematically depicted in the right panel of each Fresnel  image. After each fs -laser pulse excitation on \nthe initial single magnetic vort ex in all the three geometric al Py disks, the  Lorentz contrast in their \nFresnel images exhibit a high probability to reversal  (see the first three columns in Fig. 1c), implying  the \nrepeatable switch  of the chirality of the magnetic vort ex by the rapid optic al quenching. Below a \nthreshold fluence of about 10 mJ/cm2 a single fs -laser pulse excitation is insufficient to induce \nobservable change of the initial magnetic vortex in the  Py disks, since the pulse  induced transient \ntemperature of the Py disk is substantially lower than  its Curie point  (~ 850 K) . Note that , the observed \nreversal of the magnetic vortex chirality behaves randomly in the experiment . Interestingly, besides the \nrandom reversal of the magnetic vortex  chirality , we also  observe d, although less frequently,  some 6 \n complex magnetic structures consist ing of several newly generated vortices, antivortices and domains  in \nall the three geometries  (see typical ones in the las t column of Fig. 1c) , which were never been observed \nby magnetic field or spin current assisted magnetic vortex switching . The magnetic antivortex is the \ntopological counterpart of a magnetic vortex, which also contains a tiny core magnetized \nperpendicular ly to the plane in the center and enclosed by two adjacent vortex structures40,41. It is \ndiscernible as a saddle point  in the Fresnel  image , namely, the cross of two Néel walls showing  opposite \nLorentz contrast (white and black)39. As in our work we did not intend to determine the polarity of the \nvortex cores , it is possible that the vortex polarity switching may also occur during the ultrafast optical \nquenching25,34. \nTo understand the above interesting phenomenon , we repeated the same experiment on the three \ngeometrical Py disks more than hundreds of times, in order to get statistical ly meaningful measurements  \nof the fs -laser pulse induced metastable magnetic structures so that the underlying mechanism could be \nretrieved.  The typic al Fresnel  images  of the observed  metastable magnetic structures are displayed in the \nmiddle panel of each sub panel  in Fig. 2, while  their corresponding spin configurations and occurrence \nfrequency distributions are shown in the panels  below and above , respectively. Each Fresnel  image was \nacquired promptly  after each fs -laser pulse excitation (fluence of 12 mJ/cm2). In each statistic histogram, \nthe most frequent ly observed  single clockwise and counter -clockwise vortex structures  with opposite \nLorentz c ontrast were counted separately , while the other complex magnetic structures with opposite \nLorentz contrast but having the same spin configuration were added together. In all the three disks the \ncounter -clockwise and clockwise single vortex states occur randomly with the similar occurrence \nfrequency, which is overwhelmingly higher than that of other complex magnetic structures. This nearly \n90% occurrence frequency of the single vortex state ( including both clockwise and counter -clockwise  \nones) also verifi es the lowest energy and highest stability of it in the symmetric Py disks. All the other \ncomplex magnetic structures consist of a number of newly generated vortices, antivortices , domains and 7 \n pairs of Néel walls, where the antivortex is located between tw o vortices with the same chirality. \nIntriguingly, the spatial distribution of their spin configurations exhibit s striking mirror or rotation \nsymmetry  (see Fig. 2) , which is probably due to the confinement of the  geometrical  symmetr y of the Py \ndisks. For a magnetic structure  containing  a number of topological defects in a symmetric  ferromagnetic  \ndisk, the symmetric spin configuration will aid to reduce the total energy  of the system . Note that, these \ncomplex metastable magnetic structures ar e much more difficult to form in smaller circular Py disks (1.7 \nµm diameter) under the same quenching condition (see Movie S4), where  only single magnetic vortex \nstructure forms , implying the dimension ality affects the magnetization dynamics  and final magnetic \nstates . \nTo unravel the origin of the different occurrence s of the complex metastable magnetic structures, \nwe further carried out the fs -laser quenching experiments with a higher laser fluence of 16 mJ/cm2. At \nthis fluence, other more complex , symmetric metastable  magnetic structures consist ing of a larger \nnumber of topological defects were observed in all the three geometri es (see Fig. S1 to S3). For both the \ncircular and square disks, the magnetic structures comprise various vortices up to 6 were observed, \nwhile for the triangular one the most complex magnetic structure only contains 4 vortices, which is \nprobably due to its lower geometrical symmetry. Note that, most of these complex magnetic \nconfigurations have never been observed  in ferromagnetic disks with othe r external stimuli, such as \nannealing, magnetic field , spin-polarized current and spin wave. Basically, the more complex magneti c \nconfiguration s (with more topological defects) show lower occurrence frequency in the fs -laser \nquenching experiment (Fig. S1 t o S3). The occurrence of additional  more complex , symmetric magnetic \nstructures at this high fluence is mainly due to the strong laser heating induced crystallite change  in the \nPy disk, which will be discussed later.  \nThe magnetic vortex and antivortex are both topological defects with the local magnetization \nrotating by 360° on a closed loop around the tiny core, which can be characterized by a topological 8 \n winding number 𝑤=1/2𝜋∮𝛁𝛼∙𝑑𝑆 (w = ±1 for vortex and antivor tex, respectively), where 𝛼 is the \nlocal orientation of the magnetization vector, S is an arbitrary integral loop containing the  tiny core18. \nThe topological winding number has been theoretically predicted to have a direct impact  on the \nmagnetization dynamics40,42. Because of the spatial symmetry breaking at the edge  of the Py disk s, each \ncross of a pair of Néel walls with opposite Lorentz contrast at the edge can be considered as a half-\nantivortex and its topological winding number w turns out to be -1/2. One would find that the sum of the \ntotal topological winding numbers for each observed metastable magnetic structure in all the three \ngeometric al disks is equal to 1, which is the same as that of their initial single magnetic vortex state. \nNamely, the generation of new vortices and antivortices during the fs -laser quenching conserve s the \ntopological winding number of the Py disks, i.e., is strictly restricted by the topological invariant . This \ntopological feature is similar to that of the light induced magnetic network in homogeneous \nferromagnetic iron thin films, where the vortex -antivortex generates in pair s and follows the universal \nbehavior within the framework of Kibble -Zurek mechanism39.  \nMagnetization dynamics and formation mechanism  \nTo understand the formation mechanism and topological feature of the symmetric  magnetic structures \ninduced by fs-laser  pulse quenching, we performed finite -element micromagnetic simulations on these \nthree geometric al Py disks based on the Landau -Lifshitz -Gilbert equation with Langevin dynamics29 to \nreveal the underlying magnetization dynamics  (see Methods) . For the micromagnetic  simulation s we \nconsider follow ing scenarios : (1) the fs -laser pulse only interacts with the magnetization via the \nphotothermal effect; (2) the fs -pulse heats the Py disk above the Curie  point  and randomizes the local \nmagnetization, namely, melts the electronic spin  structures , but with out altering the integrity of the \nlattice ; and (3) each fs -laser pulse excitation may result in different random magnetization seeds in the \nmelted spin system. Under these conditions the optical  quenching induced magnetization dynamics in \nthe Py disks can be understood as follows. Upon a fs -laser pulse excitation  above the threshold fluence 9 \n for the Curie temperature , the thermal energy of the electronic system in the Py disk is rapidly increased, \ncreating a thermal bath for the spin system . This  sharp incre ase in thermal energy of the spin system \nleads to a rapid  and full demagnetization, namely,  spin melt of the initial magnetization in the Py disk \nwithin several picoseconds38. The subsequent energy transfer from the spin system to the lattice via \nelectron -phonon coupling29 leads to  a rapid decrease of the temperature with a cooling rate up to 1012 \nK/s39 to below Curi e point initiat ing the remagnetization process  of the spin system .  \nWe used t he exact dimensions of the samples in the micromagnetic simulations , and more than 25 \nruns of the numerical  simulation were performed on each Py disk. The bottom panel of each subfigure in \nFig. 3 presents the typical spin configurations  in each Py disk generated by the micromagnetic \nsimulation, and their corresponding  occurrence frequencies  (plotted in pink bars)  and energies are \nplotted together in the top panel. For comparison, t he corresponding occurrence frequency distributions \nof the magnetic structures measured by the experiments are also plotted in blue bars. Here the single \nmagnetic vortex states with opposite chirality were counted together. Clearly, all the magnetic struct ures \nobserved in the experiment  (laser fluence of 12 mJ/cm2) are well reproduced by the mi cromagnetic \nsimulations for all the three geometric al disks, and their occurrence frequencies agree  as well (Fig. 3) .  \nIn all three geometric al disks, the single magn etic vortex state always exhibits the highest \noccurrence due to its lowest energy (~ 6.98×10-17 J for circle, ~ 7.32×10-17 J for square, and ~ 9.52×10-17 \nJ for triangle). While the occurrence of other complex  magnetic structure s generally decrease s with their \nenerg y (see Fig. 3). Specifically, for the magnetic structures in circular disk, their energ y nearly \nincrease s linearly with increasing the number of the contained topological magnetic defects (vort ex) and \ntheir occurrence frequency decrease s monoton ously (Fig. 3a). While for both the square and triangular \ndisks, the energy increase levels off for the magnetic structures with more than three topological \nmagnetic defects (vortex); and counterintuitively, the magnetic structure containing two vortices w ith \nlower energy even exhibits smaller occurrence frequenc ies than that of the magnetic structure containing 10 \n three vortices with higher energy (Fig. 3b and 3c). This abnormal behavior is probably due to the much \nsmall er energy barrier  for the magnetic stru cture containing two vortices to overcome during the \nmagnetization relaxation  after the optical quenching . Based on the micromagnetic simulation, t he \nrelative  energy barriers for different magnetic structures during the magnetization relaxation in a square  \ndisk are schematically depicted in Fig. 4 . Because of th e much small er energy barrier , the metastable \nmagnetic structure containing two vortices strongly prefers  to relax to the single magnetic vortex s tate, \nresulting in the lower occurrence than the magnetic structure with three vortices in both square and \ntriangular disks (Fig. 3b and 3c ).  \nTo unravel the factor s that determine the final magnetic structures, the initial remagnetization \nprocess of the random magnetization  seeds, especially the rol e of the spin pinning at the edge defects of \nthe Py disk s, was considered  in our micromagnetic simulation. In a perfectly circular disk with smooth \nedge, the single magnetic vortex state is strongly preferred after  a fs-laser pulse excitation . Due to the \nconfinement of the geometric al symmetry  of the disk and the spin pinning at the edge defects (or edge \nroughness) , the finally magnetic structures prefer to form symmetric configuration to reduce the system \ntotal energy . For further  discussion, three  exempla ry time-dependent magnetization evolutions for the \nformation of the single magnetic vortex state , the metastable magnetic structure s with two  and three  \nvortices in a triangular  disk are respectively presented in Fig. 5 (see also Movie S5 to S7). After a fs-\nlaser pulse  induced randomization of the initial magnetization  in the disk , the melted spin system starts  \nto remagnetize when the temperat ure cools down to below the Curie point in several picoseconds and a \nnumber of vortices, antivortices, domains and  half antivortices at the disk edge are formed. Note that, \nthese initially formed topological magnetic defects converse the topological winding number (w = 1) of \nthe initial single magnetic vortex sate  in the Py disk . With time elapse, the adjacent vort ices and \nantivortices  inside the disk move spirally and approach each other, and then annihilate in pairs; while for \nthe half antivortices at the disk edge, a nearby inside vortex moves towards the center of two adjacent 11 \n half antivortice s and they annihilate once meet  together, as indicated by the colored circles in Fig. 5. \nDepending on the relative core polarization of the adjacent vortex -antivortex pair, their annihilation \nprocess is either a continuous transformation of the magnetization (parallel), or invo lving nucleation and \npropagation of a Bloch point causing a burstlike emission of spin waves (antiparallel)18. During this  \nincessant vortex -antivortex pair annihilation process , the energy of the  spin system continuously \ndecreases , until the system relaxes to the single magnetic vortex state with lowest energy  (see Fig. 5a).  \nBecause of the spin pinning at the disk edge (indicated by the blue and pink arrows  in Fig. 5b and \n5c; see also Fig. S4) the half-antivortices at these pinning sites cannot move freely, thus the two adjacent \nhalf antivortices at these pinning sites cannot annihilate with a nearby vortex. In such case, the spin \nsystem relaxes to a symmetric multivortex s tate due to the confinement of the symmetric geometry of \nthe disk (Fig. 5b and 5c ). The magnetization dynamics and the formation mechanisms of other \nmetastable complex magnetic structures , including those in the circular and square Py disks , have the \nsimilar features (see Movie S 8 to S1 3). The strong spin pinning effect at the disk edge could also \ninterpret the observation of additional more  complex , symmetric magnetic structures in our experiment \nat a higher laser fluence of 16 mJ/cm2 (see Fi g. S1  to S3). At this high fluence, the strong heating  effect \nwould cause lattice grain growth within  the Py disk (see Fig. S5), which would induce larger grain \nboundaries , i.e., roughness at the disk edge,  and thus more strong spin pinning sites, resultin g in more \ncomplex magnetic structures (Fig. S1  to S3).   \nOptical -quenching -assisted magnetic vortex switching for information recording  \nThermally assisted magnetization reversal has been proposed as one of the most promising way to \nenable high density magnetic  recording24,43, where the large  anisotropy values required for the stability \nof the recording media  films  are transiently reduced by laser heating and the require d magnetic \nswitching field is markedly declined. Recently, Lebecki and Nowak33 have theoretically  studied  the 12 \n temperature impact on the magnetic switching field of a magnetic vortex core in a circular ferromagnetic  \ndisk based on the  Landau -Lifshitz -Bloch equation44 incorporating with the thermal effect. As sh own in \ntheir prediction , the orthogonal magnetic field required for vortex core switching dramatically decreases \nwith increasing temperature due to the lower energy barrier at higher temperature , which may even \nvanish at a temperature slightly below the Cu rie point  (see Fig. 3 and 5 in Ref. 33). In contrast , our \nexperiment al results clearly demonstrate that both the magnetic vortex switching  and the final magnetic \nstate are random after a fs-laser pulse excitation  (see Fig. 2), implying  the purely photothermal \nswitching of a magnetic vortex is possible but uncontrollable , which  will seriously hinder the practical \napplication. Nevertheless , a fs -laser pulse with a proper fluence could  initiate a sharp increase of \ntemperature  in the Py disk above its Curie point  to induce a transient non -equilibrium paramagnetic s tate \n(spin melted state) . During this transient paramagnetic period , if a small external orthogonal magnetic \nfield is applied one could easily control the spin direction of the system  and initiate the polarization of \nthe magnetic vortex that formed in the subsequent magnetization relaxation  process  due to the  ultrafast \nquenching .  \nBased on our experimental observations together with the simulation results , we propose a new \nparadigm of optical -quenching -assisted fast switching of the magnetic vortex polarity for the control of \nmagnetic vortex based information recording and spintronic devices, as schematically shown in Fig. 6. \nA linear polarized fs -laser laser pulse to transiently demagn etize the initial magnetic vortex  in the Py \ndisk, and a small orthogonal magnetic field pulse  (with duration above tens of picoseconds)  to set the \npolarization of the newly formed  magnetic vortex  are simultaneously applied . In principle, the strength \nof th is external magnetic field pulse should be much smaller than that (500 mT) of the orthogonal \nmagnetic field required for the conventional quasistatic switching  of a magnetic vortex core45,46, which \ncould be handily determin ed by magnetic force microscope  measurements. It should be mentio ned that , \nfor this proposed optical -quenching -assisted magnetic vortex based information recording paradigm, the 13 \n following factors need to be considered carefully: (1)  the fluence of the fs -laser pulse is crucial, which \nshould be  able to instantaneously drive the system above the spin transition temperature , but below the \ntemperature that would cause the crystallite  damage  in the Py disk. Too strong fs -laser pulse excitation \nwould induce lattice change  and cause the growth of crys tallites in the Py disk  (see Fig. S5) , resulting in \nadditional spin pinning sites that would frustrate the magnetization relaxation. (2) The dimension and \nthe edge smoothness of the Py disk are also very important. As shown in our results, the small  circul ar \nPy disk s (1.7 µm diameter) strongly prefer to single magnetic vortex state after each fs-laser pulse \nquenching (see Movie  S4). The smooth disk edge would reduce the spin pinning effects during the \nmagnetization relaxation. Therefore, proper design of na noscale Py disk with smooth edge and uniform \nsmall crystallites would be greatly helpful to improve the stability and reliability  of the optical -\nquenching -assisted magnetic vortex switching , which  would offer the new possibility for magnetic \nvortex based high density information recording with fast writing rates, but consuming much less power.  \nIn conclusion, by using the unique ultrafast quenching  rate of up to  1012 K/s of a fs-laser pulse \nexcitation, we obser ved a plenitude of new metastable magnetic structures in three types of symmetric , \nmicrometer -sized Py disks  by 4D EM operated in the Lorentz phase  imaging mode . These  metastable \nmagnetic structures consist of a number of newly created topological magnetic  defects  strictly restricted \nby the topological invariant s, which were not observed previously in magnetic field or spin-current \nassisted magnetic vortex switching . Due to the confinement of the disk geometric al symmetry, their spin \nconfigurations show apparent  mirror or rotation symmetry. Micromagnetic simulations revealed the \nunderlying magnetization dynamics of all the observed magnetic structures, and the dependence of their \noccurrence frequencies on their energ etics and pinning effects at the disk e dge. Our result s provide  new \ninsights into the fundamental  spin switching dynamics in symmetric  Py disks under  fs-laser pulse \nquenching , which offers a guidance for  the design of optical -quenching -assisted fast switching of \ntopological vortices  for vortex based information recording  and spintronic devices.  14 \n  \nMethods  \nPreparation of Py disks  \nThe samples  studied  in our experiments were prepared by electron beam evaporation of a layer of Py \n(Ni81Fe19) film (30 nm thickness, ca. 10 nm grain size) onto silicon nitride membrane (100 nm thickness, \n300 µm × 300 µm window area) on silicon frame with prior prepared symmetric  patterns using \nphotolithography and liftoff process. Three kinds of symmetric disks were prepared : circle (diameter of \n3 µm and 1.7 µm), square (edge length of 3 µm) and regular triangle (edge length of 1.7 µm). The \ncorners of the square and regular triangle disks are  intentionally made of  an arc shape  to avoid artificial \nsingularity in s pin switching  (see Fig. 1c).  \nLorentz phase imaging of fs-laser  pulse  induced magnetic structures  \nTo image the fs-laser  quenching induced changes of magnetic structure in the Py disks, we performed \nout-of-focus Fresnel phase imaging in a 4D EM47-50 operated at Lorentz -mode  condition51,52. To obtain \nhigh Lorentz  contrast, the images were collected by using the continuous electron beam of the 4D EM \nrather than  pulsed electron s. Linearly polarized green fs -laser pulses (520 nm, 40 µm focal spot size, 350 \nfs pulse duration) were used for excitation, which were generated from infrared fs -laser pulses (1040 \nnm, 350 fs pulse duration) by a second harmonic generation. The in -plane circular magnetizations \n(clockwise or counter -clockwise) of the magnetic vortices exert opposite Lorentz force on the imaging \nelectrons , result ing in contrasts, or phase shift of the electron beam, related to the vortex core36-38, \nrespectively. The high throughput Fresnel  phase imaging allows to investigate the fs -laser quenching \ninduced magnetization changes at nanometer scale and the statistical properties of the resulting magnetic \nstructures.  \nMicromagnetic s imulation  15 \n Micromagnetic simulations were performed using micromagnetic software  (OOMMF ).53 A saturation \nmagnetization, Ms, of 800 kA/m, exchange stiffness, A, of 13 pJ/m, and a Gilbert damping, α, of 0.01, \nconsistent with ty pical values of Py, were used. A stopping condition of dM/dt = 0.1 was used to ensure \nconvergence. The cell size was set to 5 nm × 5 nm × 30 nm. The sample structure for each geometry was \ndetermined by creating a binary mask from TEM images for each. To determine possible metastable \ndomain configurations, 25 runs for each geometry were performed using different initial random \nmagnetic configurati ons, and then  allowed to relax.  Following relaxation, the energies and static domain \nconfigurations were recorded.  \n \nReferences  \n1 Shinjo, T., Okuno, T., Hassdorf, R., Shigeto, K. & Ono, T. Magnetic vortex core observation in \ncircular dots of permalloy . Science  289, 930 -932 (2000).  \n2 Wachowiak, A.  et al.  Direct observation of internal spin structure of magnetic vortex cores. Science  \n298, 577 -580 (2002).  \n3 Chou, K.  et al.  Direct observation of the vortex core magnetization and its dynamics. Appl. Phys. \nLett. 90, 202505 (2007).  \n4 Sugimoto, S.  et al.  Dynamics of coupled vortices in a pair of ferromagnetic disks. Phys. Rev. Lett.  \n106, 197203 (2011).  \n5 Braun, H. -B. Topological effects in nanomagnetism: from superparamagnetism to chiral quantum \nsolitons. Adv. Phys. 61, 1-116 (2012).  \n6 Bussmann, K., Prinz, G., Cheng, S. -F. & Wang, D. Switching of vertical giant magnetoresistance \ndevices by current through the device. Appl. Phys. Lett. 75, 2476 -2478 (1999).  16 \n 7 Yamada, K.  et al.  Electrical switching of the vor tex core in a magnetic disk. Nat. Mater.  6, 270 -273 \n(2007).  \n8 Höllinger, R., Killinger, A. & Krey, U. Statics and fast dynamics of nanomagnets with vortex \nstructure. J. Magn. Magn. Mater.  261, 178 -189 (2003).  \n9 Xiao, Q., Rudge, J., Choi, B., Hong, Y. & Don ohoe, G. Dynamics of vortex core switching in \nferromagnetic nanodisks. Appl. Phys. Lett.  89, 262507 (2006).  \n10 Lau, J. W., Beleggia, M. & Zhu, Y. Common reversal mechanisms and correlation between transient \ndomain states and field sweep rate in patterned P ermalloy structures. J. Appl . Phys . 102, 043906 \n(2007).  \n11 Van Waeyenberge, B.  et al.  Magnetic vortex core reversal by excitation with short bursts of an \nalternating field. Nature  444, 461 -464 (2006).  \n12 Curcic, M.  et al.  Polarization selective magnetic vo rtex dynamics and core reversal in rotating \nmagnetic fields. Phys. Rev. Lett. 101, 197204 (2008).  \n13 Liu, Y., Gliga, S., Hertel, R. & Schneider, C. M. Current -induced magnetic vortex core switching in \na Permalloy nanodisk. Appl. Phys. Lett.  91, 112501 (200 7). \n14 Yamada, K., Kasai, S., Nakatani, Y., Kobayashi, K. & Ono, T. Switching magnetic vortex core by a \nsingle nanosecond current pulse. Appl. Phys. Lett.  93, 152502 (2008).  \n15 Yamada, K., Kasai, S., Nakatani, Y., Kobayashi, K. & Ono, T. Current -induced sw itching of \nmagnetic vortex core in ferromagnetic elliptical disks. Appl. Phys. Lett.  96, 192508 (2010).  \n16 Kravchuk, V. P., Gaididei, Y. & Sheka, D. D. Nucleation of a vortex -antivortex pair in the presence \nof an immobile magnetic vortex. Phys . Rev. B 80, 100405 (2009).  \n17 Kammerer, M.  et al.  Magnetic vortex core reversal by excitation of spin waves. Nat. Commun.  2, \n279 (2011).  17 \n 18 Hertel, R. & Schneider, C. M. Exchange explosions: Magnetization dynamics during vortex -\nantivortex annihilation. Phys. Rev. Lett. 97, 177202 (2006).  \n19 Hertel, R., Gliga, S., Fähnle, M. & Schneider, C. Ultrafast nanomagnetic toggle switching of vortex \ncores. Phys. Rev. Lett.  98, 117201 (2007).  \n20 Guslienko, K. Y., Lee, K. -S. & Kim, S. -K. Dynamic origin of vortex core switching in soft magnetic \nnanodots. Phys. Rev. Lett.  100, 027203 (2008).  \n21 Beaurepaire, E., Merle, J. -C., Daunois, A. & Bigot, J. -Y. Ultrafast spin dynamics in ferromagnetic \nnickel. Phys. Rev. Lett.  76, 4250 (1996).  \n22 Ogasawara, T., Iwata, N., Murakami, Y., Okamo to, H. & Tokura, Y. Submicron -scale spatial feature \nof ultrafast photoinduced magnetization reversal in TbFeCo thin film. Appl. Phys. Lett.  94, 162507 \n(2009).  \n23 Radu, I.  et al.  Transient ferromagnetic -like state mediating ultrafast reversal of antiferroma gnetically \ncoupled spins. Nature  472, 205 -208 (2011).  \n24 Stipe, B. C.  et al.  Magnetic recording at 1.5 Pb m− 2 using an integrated plasmonic antenna. Nat. \nPhotonics  4, 484 -488 (2010).  \n25 Le Guyader, L.  et al.  Demonstration of laser induced magnetization re versal in GdFeCo \nnanostructures. Appl. Phys. Lett.  101, 022410 (2012).  \n26 Pitaevskii, L. Electric forces in a transparent dispersive medium. Sov. Phys. JETP  12, 1008 -1013 \n(1961).  \n27 Pershan, P., Van der Ziel, J. & Malmstrom, L. Theoretical discussion of th e inverse Faraday effect, \nRaman scattering, and related phenomena. Phy. Rev. 143, 574 (1966).  \n28 Stanciu, C.  et al.  All-optical magnetic recording with circularly polarized light. Phys. Rev. Lett.  99, \n047601 (2007).  18 \n 29 Ostler, T.  et al.  Ultrafast heating a s a sufficient stimulus for magnetization reversal in a ferrimagnet. \nNat. Commun.  3, 666 (2012).  \n30 Khorsand, A.  et al.  Role of magnetic circular dichroism in all -optical magnetic recording. Phys. Rev. \nLett. 108, 127205 (2012).  \n31 Mangin, S.  et al.  Enginee red materials for all -optical helicity -dependent magnetic switching. Nat. \nMater.  13, 286 -292 (2014).  \n32 Lambert, C. -H. et al.  All-optical control of ferromagnetic thin films and nanostructures. Science  345, \n1337 -1340 (2014).  \n33 Lebecki, K. M. & Nowak, U. Ferromagnetic vortex core switching at elevated temperatures. Phys. \nRev. B  89, 014421 (2014).  \n34 Taguchi, K., Ohe, J. -i. & Tatara, G. Ultrafast magnetic vortex core switching driven by the \ntopological inverse Faraday effect. Phys . Rev. Lett.  109, 127204 (2012).  \n35 Gomez, R., Luu, T., Pak, A., Kirk, K. & Chapman, J. Domain configurations of nanostructured \nPermalloy elements. J. Appl. Phys.  85, 6163 -6165 (1999).  \n36 Schneider, M., Hoffmann, H. & Zweck, J. Lorentz microscopy of circul ar ferromagnetic permalloy \nnanodisks. Appl. Phys. Lett.  77, 2909 -2911 (2000).  \n37 Raabe, J.  et al.  Magnetization pattern of ferromagnetic nanodisks. J. Appl. Phys. 88, 4437 -4439 \n(2000).  \n38 da Silva, N. R.  et al.  Nanoscale mapping of ultrafast magnetization dynamics with femtosecond \nLorentz microscopy. arXiv preprint arXiv:1710.03307  (2017).  \n39 Eggebrecht, T.  et al.  Light -induced metastable magnetic texture uncovered by in situ Lorentz \nmicroscopy. Phys. Rev. Lett.  118, 097203 (2017).  \n40 Gliga, S., Yan, M., He rtel, R. & Schneider, C. M. Ultrafast dynamics of a magnetic antivortex: \nMicromagnetic simulations. Phys. Rev. B  77, 060404 (2008).  19 \n 41 Huber Jr, E., Smith, D. & Goodenough, J. Domain‐Wall Structure in Permalloy Films. J. Appl. Phys.  \n29, 294 -295 (1958).  \n42 Thiele, A. Steady -state motion of magnetic domains. Phys. Rev. Lett. 30, 230 (1973).  \n43 Kryder, M. H.  et al.  Heat assisted magnetic recording. Proc. IEEE  96, 1810 -1835 (2008).  \n44 Garanin, D. A. Fokker -Planck and Landau -Lifshitz -Bloch equations for classica l ferromagnets. Phys. \nRev. B  55, 3050 (1997).  \n45 Okuno, T., Shigeto, K., Ono, T., Mibu, K. & Shinjo, T. MFM study of magnetic vortex cores in \ncircular permalloy dots: behavior in external field. J. Magn. Magn. Mater.  240, 1-6 (2002).  \n46 Thiaville, A., García, J. M., Miltat, J. & Schrefl, T. Micromagnetic study of Bloch -point -mediated \nvortex core reversal. Phys. Rev. B  67, 094410 (2003).  \n47 Zewail, A. H. Four -dimensional electron microscopy. Science  328, 187 -193 (2010).  \n48 Fu, X., Chen, B., Tang, J., Hassan, M. T. & Zewail, A. H. Imaging rotational dynamics of \nnanoparticles in liquid by 4D electron microscopy. Science  355, 494 -498 (2017).  \n49 Fu, X., Chen, B., Tang, J. & Zewail, A. H. Photoinduced nanobubble -driven superfast diffusion of \nnanoparticles imaged by 4D electron microscopy. Sci. Adv.  3, e1701160 (2017).  \n50 Chen, B.  et al.  Dynamics and control of gold -encapped gallium arsenide nanowires imaged by 4D \nelectron microscopy. Proc. Natl. Acad. Sci. U.S.A. , 201708761 (2017) . \n51 Park, H. S., Baskin, J. S. & Zewail, A. H. 4D Lorentz electron microscopy imaging: magnetic \ndomain wall nucleation, reversal, and wave velocity. Nano Lett.  10, 3796 -3803 (2010).  \n52 Rajeswari, J.  et al.  Filming the formation and fluctuation of skyrmion  domains by cryo -Lorentz \ntransmission electron microscopy. Proc. Natl. Acad. Sci. U.S.A.  112, 14212 -14217 (2015).  \n53 http://math.nist.gov/oommf/oommf_cites.html . 20 \n Acknowledgements  \nWe acknowledge Caltech for providing the access to 4D electron microscopy facility for this study. This \nwork is supported by the Materials Science and Engineering Divisions, Office of Basic Energy Sciences \nof the U.S. Department of Energy under Contract No . DESC0012704 . We wish to thank Dr. J. S. Baskin \nfor very helpful discussion and help on the Lorentz phase electron microscopy measurement with in situ  \nfs-laser excitation. We also wish to thank  J. A. Garlow for fruitful discussion on the Lorentz phase \nimaging measurement .  \nAuthor contributions:  \nY. Z. and X. F. conceived the research project. X. F., B. C. and B. K. Y. did the experimental \nmeasurements. S. D. P. prepared the samples. X. F. and S. D. P. did the data analysis with input from Y.  \nZ.  S. D. P. d eveloped the model and performed the numerical simulations. All the authors contributed to \nthe discussion and the writing of the manuscript . \nAdditional information  \nSupplementary information is available in the online version of the paper , including Fig. S1 to S5 and  \nMovie  S1 to S1 3.  \nCompeting financial interests  \nThe authors declare no competing financial interests.   21 \n Figure l egends  \nFig. 1. fs-laser pulse  quench ing of magnetic vortex in P y disks.  a, Sketch of imaging the fs-laser \npulse induced change  of spin configuration  in a ferromagnetic Py disk by 4D EM operated in Lorentz \nphase mode with continues electron beam. The green fs-laser pulse (520 nm, 350 fs pulse duration) is \nfocused to  40 µm on the sample. b, Schematic Lorentz contrast reverse mechanism of a magnetic vortex \nin a circular Py disk before and after a fs -laser pulse excitation  due to the change of spin chirality . \nBecause of  the opposite Lorentz force of the imaging electrons  impinging on the sample , the Lorentz \ncontrast of a vortex core can be either black or white . The inset depicts the typical transient temperature \nevolution s after a fs-laser excitation  (calculated by a two-temperature model39) in both the Py disk and \nthe silicon nitride substrate ( TC is the Curie point  of Py disk, TR is the room temperature,  laser  fluence  of \n12 mJ/cm2). c, fs-laser pulse induced variation  of magnetic vortex  in circular, square and regularly \ntriangular Py disks. The right panel of each Fresnel  image schematically depicts the correspond ing spin \nconfiguration . The blue and red dashed lines correspond to the white and black Lorentz contrasts, \nrespectively. While t he blue and red dots to the counter -clockwise and clockwise vortices, individually. \nThe gr een dot s mark the magnetic anti vortex. The same notes  are used in all the following figures .  \nFig. 2. Occurrence f requency distribution  of the fs-laser pulse  induced magnetic structures in \nthree geometr ical Py disks . a-c, Frequency distribution of the fs -laser pulse  (fluence of 12 mJ/cm2) \ninduced spin configurations  in circular, square and triangular Py disks, respectively. The bottom panel in \neach subfigure shows the typical Fresnel  images of the experiments , and the middle panel schematically \nshows the ir corresponding  spin configuration s. The most frequent single clockwise and counter -\nclockwise vortex structures with opposite Lorentz imaging contrast were counted separately, while the \nother magnetic structures with opposite Lorentz  imaging  contrast but having the same spin configuration \nwere added together.  The inset in each subfigure denotes the fs -laser pulse quench ing process  in the Py \ndisks .  22 \n Fig. 3. Comparison of micromagnetic simulations with experimental observations . a-c, Simulation \nresults of the occurrence frequency  distribution and energies of the fs -laser pulse  (fluence of 12 mJ/cm2) \ninduced magnetic structures in circular, square and triangular Py disks, respectively. The bottom panel \nin each subfigure shows the typical results of possible magnetic structures obta ined by  the \nmicromagnetic simulations  (pink bars) . The corresponding experiment  determined occurrence frequency  \ndistribution of the fs -laser pulse induced magnetic structures are also plotted in blue bars  for comparison . \nThe simulation results reproduce well the experimental results, except one magnetic structure in the \ntriangle disk ( indicated by the dashed red circle in Fig. 3 c). \nFig. 4. Schematic of r elative energy barriers for different metastable magnetic structures to \novercome during the magnetization  relaxation in a square Py disk. The magnetic structure \ncontaining two vortices has a smaller barrier to overcome comparing with other metastable \nmagnetic states.  \nFig. 5 . Typical magnetization dynamics in Py disk  after a fs -laser pulse quenching . Snapshots  of the  \nmagnetization dynamics during the formation of different magnetic structures in a triangular Py disk at \ndifferent times after a fs-laser pulse  excitation : a, formation a single magnetic vortex state;  b, formation \nof a magnetic structure with two vortices ; c, formation of a magnetic structure with three vortices. The \nlaser fluence is 12 mJ/cm2. The vortices  and antivortices (including the half -antivortices  indicated by the \ngreen half dots  at the disk edge)  in the different colored circles indicate the magnetic vortex -antivortex \npairs that annihilate during the magnetization relaxation process. The blue and pink arrows indicate the \nspin pining sites at the disk edge.  \nFig. 6 . A paradigm of optical -quenching -assisted magnetic vortex based information recording \nprocess. Left: Schematic of  the optical -quenching -assisted magnetic vortex based information recording \nsystem, where  a linear polarized fs -laser laser pulse is used to transiently demagnetize the initial 23 \n magnetic vortex and another synchronized orthogonal small magnetic field pulse is used to set the \npolarization of the newly formed  magnetic vortex. Right: sketch for th e working mechanism of the \noptical -assisted magnetic  vortex based information  recording process.  The data information “1” and “0” \nare recorded by the polarity (up and down) of the magnetic vortex.  The fluence of the fs -laser pulse \nshould be controlled above the threshold for spin melting , but below that for causing change of \ncrystallites in the ferromagnetic disk.   \n  24 \n Figures  \n \n \n \n \n \nFigure 1   \n25 \n  \n \nFigure 2  \n26 \n  \n \nFigure 3  \n27 \n  \n \n \n \nFigure  4 \n \n  \n28 \n  \n \n \nFigure  5  \n  \n29 \n  \n \n \nFigure  6 \n  \n30 \n Supplementary Information for  \nOptical Manipulation of Magnetic Vortex Visualized in situ by 4D Electron \nMicroscopy  \nXuewen Fu*, Shawn D. Pollard , Bin Chen , Byung -Kuk Yoo , Hyunsoo Yang , Yimei Zhu* \n*Corresponding author . E-mail:  zhu@bnl.gov ; xfu@bnl.gov  \n \n \nThis PDF file includes:  \nFigs. S1 to S 5 \nCaptions for Movies S1 to S 13 \n \nOther Supplementary Information  for this manuscript includes the following:  \nMovies S1 to S 13 \n \n \n  31 \n Figs. S1 to S5  \n \n \nFig. S1 . Occurrence frequency distribution  of fs -laser pulse  induced magnetic structures in a \ncircular  Py disk  at a fluence of 16 mJ/cm2. (a) Bottom panel: Fresnel images of the observed magnetic \nstructures in the circular Py disk (diameter of 3 µm); Top panel: corresponding schematic magnetization \nconfigurations. ( b) Occurrence  frequency distribution of the fs -laser pulse  induced different magnetic \nstructures .  \n32 \n  \nFig. S2.  Occurrence frequency distribution  of fs -laser pulse  induced magnetic structures in a \nsquare  Py disk  at a fluence of 16 mJ/cm2. (a) Bottom panel: Fresnel images of the observed magnetic \nstructures in the square Py disk (edge length of 3 µm); Top panel: corresponding schematic \nmagnetization configurations. ( b) Occurrence  frequency distribution of the fs -laser pulse induced \ndifferent magnetic structures . \n33 \n  \nFig. S3. Occurrence frequency distribution  of fs -laser pulse  induced magnetic structures in a \ntriangular  Py disk  at a fluence of 16 mJ/cm2. (a) Bottom panel: Fresnel images of the observed \nmagnetic structures in the triangular Py disk (edge length of 1.7 µm); Top panel: corresponding \nschematic magnetization configurations. ( b) Occurrence  frequency distribution of the fs -laser pulse  \ninduced different magnetic structures . \n34 \n  \nFig. S4. Typical magnetic structures in triangular Py disk s (edge length of 1.7 µm) determined by \nmicromagnetic simulation  to show the pinning sites at the disk edge . The colored arrows at the \ndisk edge show the pinning sites. The insets show their corresponding Fresnel images.  \n  \n35 \n  \nFig. S5. ABF images of a circular Py disk after a fs -laser pulse quenching with different fluences  to \nshow the change of the inside crystallites. The crystallites in the Py disk show n o obvious change \nunder the excitation fluence of 12 mJ/cm2 (a), while exhibit apparent growth after the fs -laser pulse \nexcitation with the fluence of 16 mJ/cm2 (c). The fs -laser pulse induced growth of crystallites  induces \nlarge grain boundaries in the Py disk, which  may cause  more pinning sites and result in more complex \nmagnetic structures.  \n  \n36 \n Captions for Movies S1 to S13  \nMovie S1.  Fresnel imaging of fs -laser pulse quenching induced magnetic structure change in a circular \nPy disk (diameter of 3 µm) at a fluence of 12 mJ/cm2. \nMovie S2. Fresnel  imaging of fs -laser pulse quenching induce d magnetic structure change in a  square \nPy disk (edge length of 3 µm) at a fluence of 12 mJ/cm2. \nMovie S3. Fresnel  imaging of fs -laser pulse quenching induced magnetic structure change in a \ntriangular Py disk (edge length of 1.7 µm) at a fluence of 12 mJ/cm2. \nMovie S4. Fresnel  imaging of fs -laser pulse quenching induced magnetic structure change in a circular \nPy disk ( diameter of 1.7 µm) at a fluence of 12 mJ/cm2. \nMovie S 5. Micromagnetic simulation on the m agnetization relaxation dynamics of the formation of a \nsingle magnetic vortex structure in the triangular Py disk (edge length of 1.7 µm) after a fs-pulse \nquenching ( fluence of 12 mJ/cm2). \nMovie S 6. Micromagnetic simulation on the m agnetization relaxation dynamics of the formation of a \nmagnetic structure with two magnetic vortices in the triangular Py disk (edge length of 1.7 µm) after a \nfs-pulse qu enching ( fluence of 12 mJ/cm2). \nMovie S 7. Micromagnetic simulation on the m agnetization relaxation dynamics of the formation of a \nmagnetic structure with three magnetic vortices in the triangular Py disk (diameter of 3.0 µm) after a fs -\npulse quenching (fluence of 12 mJ/cm2). \nMovie S 8. Micromagnetic simulation on the m agnetization relaxation dynamics of the formation of a \nsingle magnetic vortex structure in the circular Py disk (diameter  of 3.0 µm) after a fs -pulse quenching \n(fluence of 12 mJ/cm2). \nMovie  S9. Micromagnetic simulation on the m agnetization relaxation dynamics of the formation of a \nmagnetic structure with four magnetic vortices in the circular Py disk (diameter  of 3.0 µm) after a fs -\npulse quenching ( fluence of 12 mJ/cm2). \nMovie S 10. Micromagnetic simulation on the m agnetization relaxation dynamics of the formation of a \nmagnetic structure with four magnetic vortices in the circular Py disk (diameter  of 3.0 µm) after a fs -\npulse quenching ( fluence of 12 mJ/cm2). 37 \n Movie S 11. Micromagnetic simulation on the m agnetization relaxation dynamics of the formation of a \nsingle magnetic vortex structure in the square Py disk (edge length of 3 µm) after a fs -pulse quenching \n(fluence of 12 mJ/cm2). \nMovie S 12. Micromagnetic simulation on the m agnetization relaxation dynamics of the formation of a \nmagnetic structure with three magnetic vortices in the square Py disk (edge length of 3 µm) after a fs -\npulse quenching ( fluence of 12 mJ/cm2). \nMovie S 13. Micromagnetic simulation on the m agnetization r elaxation dynamics of the formation of a \nmagnetic structure with four magnetic vortices in the square Py disk (edge length of 3 µm) after a fs -\npulse quenching ( fluence of 12 mJ/cm2). \n "    },    {        "title": "2001.00417v1.Three_dimensional_dynamics_of_magnetic_hopfion_driven_by_spin_transfer_torque.pdf",        "content": "Three-dimensional dynamics of magnetic hop\fon driven by spin transfer torque\nYizhou Liu,1Wentao Hou,2Xiufeng Han,1, 3, 4,\u0003and Jiadong Zang2, 5,y\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n2Department of Physics and Astronomy, University of New Hampshire, Durham, New Hampshire 03824, USA\n3Center of Materials Science and Optoelectronics Engineering,\nUniversity of Chinese Academy of Sciences, Beijing 100049, China\n4Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China\n5Materials Science Program, University of New Hampshire, Durham, New Hampshire 03824, USA\nMagnetic hop\fon is three-dimensional (3D) topological soliton with novel spin structure that\nwould enable exotic dynamics. Here we study the current driven 3D dynamics of a magnetic hop\fon\nwith unit Hopf index in a frustrated magnet. Attributed to spin Berry phase and symmetry of the\nhop\fon, the phase space entangles multiple collective coordinates, thus the hop\fon exhibits rich\ndynamics including longitudinal motion along the current direction, transverse motion perpendicular\nto the current direction, rotational motion and dilation. Furthermore, the characteristics of hop\fon\ndynamics is determined by the ratio between the non-adiabatic spin transfer torque parameter\nand the damping parameter. Such peculiar 3D dynamics of magnetic hop\fon could shed light\non understanding the universal physics of hop\fons in di\u000berent systems and boost the prosperous\ndevelopment of 3D spintronics.\nIntroduction |Hop\fons are three-dimensional (3D)\ntopological solitons initially proposed in the Skyrme-\nFaddeev model [1{3]. The three spatial dimensions en-\ndow hop\fons with diverse con\fgurations such as rings,\nlinks, and knots that can be classi\fed by the Hopf in-\ndexQH, a topological index that characterizes the homo-\ntopy group \u0005 3(S2) classifying maps from S3toS2[4{7].\nAlthough hop\fons were \frst studied in the contents of\n\feld theories, they turn out to emerge in various physi-\ncal systems, such as optics, liquid crystals, Bose-Einstein\ncondensates, superconductors, etc [8{14]. Very recently,\ntheir magnetic counterparts have been theoretically pro-\nposed in frustrated magnets [15, 16] and con\fned chi-\nral magnetic heterostructures [17{19], further stimulat-\ning the study of hop\fon from a new respect.\nWhile the sophisticated con\fgurations of hop\fon could\ngive rise to fascinating physical phenomena [5, 20], many\nof their physical properties, especially their dynamics,\nare still largely unexplored. Low-dimensional magnetic\ntopological solitons like skyrmions and vortices have been\nextensively studied over the past few decades with long\nlasting interests in both their fundamental physical prop-\nerties and potential applications [21{23]. Therefore, it is\nalso important to unravel the dynamics of the magnetic\nhop\fon, especially its most essential dynamics driven by\nthe spin transfer torque (STT) under electric current.\nHop\fon dynamics have been recently studied in con\fned\nchiral magnetic heterostructures [24]. But in this case,\nhop\fons are only allowed to move in two spatial dimen-\nsions and the unique physics associated with the third\nspatial dimension is completely suppressed by the strong\nboundary condition.\n\u0003xfhan@iphy.ac.cn\nyJiadong.Zang@unh.eduIn this Letter, we investigate the current-induced dy-\nnamics of a magnetic hop\fon in frustrated magnet, where\nhop\fons are free to move in all directions and their full 3D\ndynamics can be explored. The hop\fon studied here has\nQH= 1 and its motion is driven by both the adiabatic\nand non-adiabatic STT e\u000bect [25{27]. Based on the sym-\nmetry of hop\fon's spin con\fguration (Fig. 1), two typical\ncases are studied, i.e., current in the torus midplane and\ncurrent perpendicular to the torus midplane. As mani-\nfested by its 3D con\fguration, hop\fon possesses various\ntypes of dynamics including translational motion, rota-\ntion, and dilation. The spin Berry phase of hop\fon hosts\nan entangled phase space, which further conjugates these\ndynamics and gives rise to more exotic dynamical prop-\nerties. All these dynamical behavior can be captured by\nan analytical model derived in terms of multi-dimensional\ncollective coordinates and generalized Thiele's approach.\nA phenomenological analysis is also employed to bridge\nthe dynamics of hop\fon and skyrmion string.\nSpin Berry phase and entangled phase space | We\nconsider here a hop\fon with QH= 1. A typ-\nical hop\fon con\fguration can be achieved by a\nstereographic projection from R3toS3:\u001f=\n((x=r) sinf;(y=r) sinf;(z=r) sinf;cosf), followed by the\nHopf map S0=hzj\u001bjzi, where the spinor jzi= (\u001f4+\ni\u001f3;\u001f1+i\u001f2)T,r2=x2+y2+z2, andfis a function\nofrsatisfying the boundary conditions f(0) =\u0019and\nf(1) = 0 [7]. Explicitly the con\fguration is given by:\nSx\n0=x\nrsin2f+yz\nr2sin2f;\nSy\n0=y\nrsin2f\u0000xz\nr2sin2f;\nSz\n0= cos2f+2z2\nr2sin2f:(1)\nThis is the simplest ansatz of a hop\fon with axial sym-\nmetry about z-axis. In this con\fguration, as shown inarXiv:2001.00417v1  [cond-mat.mes-hall]  2 Jan 20202\nFig. 1a, all the iso-spin contours with Sz= 0 form a\ntorus surface. Since QHis geometrically interpreted as\nthe linking number [28], we show in the inset of Fig. 1a\n(upper right corner) iso-spin contours of S=\u0006^x, which\nare indeed linked. This con\frms the nontrivial topology\nof the spin texture under investigation. Fig. 1b and c\nshow the cross-sectional view of the spin textures at xy\nandyzplane, respectively. A QH= 1 hop\fon can also be\nunderstood as a 2 \u0019twisted skyrmion string with its two\nends glued together. Therefore, a pair of skyrmion and\nantiskyrmion shows up at the yzcross-section as shown\nin Fig. 1c.\nAs a topological soliton, hop\fon has particle-like trans-\nlational dynamics. 3D anisotropic nature of the con-\n\fguration also allows the rotation of hop\fon. We can\ncapture the essential dynamics of hop\fon by analyzing\nthe collective coordinates of both translational and ro-\ntational motion. The spin con\fguration of hop\fon at\nposition r= (x;y;z ) and time tcan be expressed as\nS(r;t) =S0(^O(r\u0000R)), where R= (X, Y, Z ) charac-\nterizes the displacement, and ^Ois the rotation opera-\ntor. At in\fnitesimal rotation, ^O\u00191\u0000\u0002\u0001^L, where\n^Li=\"ijkrj@kis the angular momentum operator and\n\u0002= (\u0002x;\u0002y;\u0002z) is the rotation angle of hop\fon around\ndi\u000berent axes.\nThe dynamics of localized spins is in general de-\ntermined by the spin Berry phase term of the La-\ngrangian [29{31]\nLBP=Z\n(1\u0000cos\u0012)_\u001edV; (2)\nwhere\u0012and\u001eare the polar and azimuthal angle of the\nlocalized spin Swith unit length. By integrating out the\nspin con\fguration, the variation of the spin Berry phase\nterm\u000eLBP=R\nS\u0001\u000eS\u0002_SdVcan be written in terms of\nthe slow-varying collective coordinate as\n\u000eLBP=D(\u0002x_Y\u0000\u0002y_X) +I\u0002y_\u0002x; (3)\nwhereD=\u0000R\nS0\u0001(z@xS0\u0000x@zS0)\u0002@xS0dVand\nI=R\nS0\u0001(z@x\u0000x@z)S0\u0002(y@z\u0000z@y)S0dV. In Eq. 3,\nall other terms drop out due to parity of the spin con-\n\fguration. It clearly shows the rotations about x\u0000and\ny\u0000axes are canonical conjugate to each other. Through\nthe entanglement between the displacement and rotation,\ntranslations along x\u0000andy\u0000directions are intertwined\nas well. The longitudinal motion of hop\fon is thus ac-\ncompanied by transverse displacement and complex ro-\ntations.\nIt is noted that the z\u0000axis related displacement ( Z)\nand rotation ( \u0002z) are missing in Eq. 3 due to the sym-\nmetry of hop\fon con\fguration. To capture these dynam-\nics, it is necessary to include the auxiliary dilation of the\nhop\fon con\fguration S(r;t) =S0(\u0015r), where\u0015is a time-\ndependent dilation factor and at equilibrium \u0015= 1. The\nvariation with respect to \u0015then contributes an additional\nterm to the spin Berry phase\n\u000eLz\nBP= (\n_Z+\u0000_\u0002z)\u000e\u0015; (4)\nyxxyz(a)\nsxsy\nyz(b)torus midplane (c)\nFIG. 1. (a) Iso-spin contours with Sz= 0 for a magnetic\nhop\fon with QH= 1. Inset is the iso-spin contours of S= +^x\n(red) and S=\u0000^x(cyan) that demonstrate the unity linking\nnumber of the hop\fon. (b) and (c) are the cross-sections of\nhop\fon onto xy(b) andyz(c) planes, as depicted by the grey\narrows. At the initial state, the torus midplane lies in the xy\nplane. In the color scheme, black indicates Sz=\u00001 and\nwhite indicates Sz= 1. The color wheel stands for in-plane\nspin directions.\nwhere \n=R\nS0\u0001(r\u0001@rS0\u0002@zS0)dVand \u0000=R\nS0\u0001(r\u0001\n@rS0\u0002(x@y\u0000y@x)S0)dV. This additional term shows\nthe dilation is conjugated to both the displacement and\nrotation about z\u0000axis. Equation of motion taken from\nvariation of \u0015leads to the simultaneous translation and\nrotation. It should be noticed that the dilation is not\na collective coordinate since an energy change is associ-\nated with a dilation of the con\fguration. Nevertheless,\nit plays an important role in correctly determining the\ncorresponding hop\fon dynamics. Eq. 3 and Eq. 4 il-\nlustrate that the hop\fon moves in a phase space where\ndisplacement, rotation and dilation are all entangled to\neach other.\nCurrent-driven hop\fon dynamics | To validate our\nanalysis, numerical simulations were performed in order\nto precisely capture the hop\fon dynamics. We employ a\nfrustrated Heisenberg Hamiltonian H=\u0000P\n<i;j>JijSi\u0001Sj,\nin which the summation of the exchange interaction is\nextended up to fourth nearest neighbor. Here we choose\nthe following parameters J1= 1,J2=\u00000:164,J3= 0,\nJ4=\u00000:082, where the sub-indices represent the or-\nders of nearest neighbors and all the energy terms are\nnormalized to the value of J1, the nearest neighbor ex-\nchange [32]. We choose sin f= 2r\u0015=(r2+\u00152) as the\ninitial state where \u0015= 1. A stable hop\fon con\fgura-\ntion (Fig. 1) is obtained by a direct energy minimization.\nSymmetry of this stable con\fguration is the same as the\nprototype showing by Eq. 1.\nThe magnetization dynamics were calculated by solv-3\n0102030–8–6–4–200102030–3–2–1012\nxyzxzβ < α α = 0.1 β = 0.0 j // x ΔY (nm)\nΔZ (nm)ΔX (nm)ΔX (nm)(a)(b)(c)(d)\nβ = 0.1jx (1010 A/m2)vx (m/s)α = 0.1α = 0.1\nβ > α α = 0.1 β = 0.2 β = 0.0β = 0.20.51.01.52.02.51.03.05.07.09.0\nβ = 0.0β = 0.1β = 0.2\ny\nFIG. 2. Hop\fon dynamics in the presence of an in-plane cur-\nrent applied in x\u0000direction (jx= 0:5\u00021010Am\u00002). (a) and\n(b) show the displacements of hop\fon center in yandzdi-\nrection (\u0001Y and \u0001Z) versus the displacement in xdirection\n(\u0001X) for di\u000berent values of \f. Inset: Current density depen-\ndence of the longitudinal velocity ( vx) for di\u000berent values of\n\f. (c), (d) Rotational motion of hop\fon with \f <\u000b (c) and\n\f > \u000b (d). The red arrows represent the normal vector of\nthe hop\fon's midplane. The red dots are the corresponding\nangles of the torus midplane projected onto a unit sphere at\ndi\u000berent simulation time. The blue arrows indicate the direc-\ntion of rotation.\ning the Landau-Lifshitz-Gilbert (LLG) equation with the\nSTT terms:\ndS\ndt=\u0000\rS\u0002Be\u000b+\u000b\nSS\u0002dS\ndt\n+Pa3\n2eS(j\u0001r)S\u0000Pa3\f\n2eS2S\u0002(j\u0001r)S:(5)\nHere\ris the gyromagnetic ratio, \u000bis the damping con-\nstant,Pis the spin polarization, ais the lattice constant\nandjis the current density. Be\u000b=\u00001\n\u0016BS@H\n@Sis the e\u000bec-\ntive magnetic \feld and Sis the spin length, which is \fxed\nto be 1 here for simplicity. The last two terms in Eq. 5\ndescribe the STT e\u000bect induced by an applied current j\nand\fquanti\fes the non-adiabatic STT e\u000bect.\nWe begin with the current applied in the xyplane.\nThe simulation results for a current applied along the\nx\u0000axis are summarized in Fig.2. The hop\fon dynamics\ncan be better illustrated by using its center position and\nnormal vector of the torus midplane, as shown in Fig. 1a.\nAt the initial state, the center position is located at the\norigin, the midplane lies in the xyplane and its normal\nvector is aligned with z\u0000axis. In the case with \f= 0 and\n\u000b= 0:1 (\f <\u000b ), two transverse motions (\u0001 Yand \u0001Z)\nare associated with a longitudinal motion (\u0001 X) along\nthe current direction (Fig. 2a and b). Meanwhile, Fig. 2c\nshows the evolution (red dots) of the directional vector\nnormal to the midplane (red arrow), which describes therotation of the hop\fon.\nMore interestingly, the non-adiabatic \f-term signi\f-\ncantly a\u000bects the hop\fon dynamics. In the case with\n\f= 0:2 and\u000b= 0:1 (\f > \u000b ), sign of \u0001 Yis reversed\nwhile that of \u0001 Zis unchanged compared to \f <\u000b case\n(Fig. 2a and b). In contrast, for the rotational motion,\nthe sign of both \u0002xand \u0002yare reversed as shown in\nFig. 2d. However, once \f=\u000b, all transverse motions and\nrotations are suppressed, and the hop\fon moves straight\nahead along the current direction. For more comprehen-\nsive details of the hop\fon dynamics, see movies in the\nSupplemental Material.\nTo further understand the dynamics, we derive the\nequations of motion for hop\fon in the presence of STT\ne\u000bect. A conventional approach proposed by Thiele is to\n\frst apply the operator @S0=@ri\u0001(S0\u0002) on both sides of\nthe LLG equation, so that the velocity on the left hand\nside equals to the force density on the right [33, 34]. How-\never, such approach describes the translational motion\nonly. Notice that the term @S0=@rican be understood as\nthe momentum operator acting on the spins. Therefore,\nwe can generalize the Thiele's approach by applying the\noperator ^LS0\u0001(S0\u0002) on both sides of the LLG equation\nwhere ^Lis the angular momentum operator introduced\nin the spin Berry phase part. In this way, we can get ad-\nditional terms relating the angular velocity to the torque\ndensity. The full set of equations of motion are summa-\nrized as (for details of the derivations, see Supplementary\nMaterials [32]):\nD_\u0002x+\u000bKRR_Y+\u000bKR\u0002_\u0002y=\fKRR\u0018jy;\n\u0000D_\u0002y+\u000bKRR_X+\u000bKR\u0002_\u0002x=\fKRR\u0018jx;\nD_X\u0000I_\u0002x+\u000bKR\u0002_Y+\u000bK\u0002\u0002_\u0002y=\fKR\u0002\u0018jy\n+D\u0018jx;\n\u0000D_Y+I_\u0002y+\u000bKR\u0002_X+\u000bK\u0002\u0002_\u0002x=\fKR\u0002\u0018jx\n\u0000D\u0018jy;(6)\nwith\u0018=Pa3\n2e.KRR=R\n@xS0\u0001@xS0dV,KR\u0002=R\n@xS0\u0001\n(y@z\u0000z@y)S0dV, andK\u0002\u0002=R\n[(y@z\u0000z@y)S0]2dV\nare components of the dissipative tensor. The non-\ndissipative terms, namely the terms without \u000bon the\nleft hand side of each equation, are consistent with the\nBerry phase analysis, indicating such general approach\nis a proper method for handling hop\fon dynamics. By\nsolving Eq. 6 for a current applied along x\u0000axis (jx),\nwe have _X\u0018jx,_Y\u0018(\u000b\u0000\f)jx,_\u0002x\u0018(\u000b\u0000\f)jx, and\n_\u0002y\u0018(\u000b\u0000\f)jx._Y,\u0002x, and \u0002yall depend on ( \u000b\u0000\f)\nso that their signs depend on the ratio between \fand\n\u000b. Once\u000b=\f, only _Xhas non-zero value and only\na translational motion along the current direction is al-\nlowed. Finally, the longitudinal velocity vx=_Xis lin-\nearly proportional to the current density jx. All these\nresults are consistent with the hop\fon dynamics shown\nin Fig. 2.\nWhile Eqs. 6 can capture the hop\fon dynamics with\ncurrent in the midplane, the dynamics associated to the4\ncurrent component perpendicular to the midplane is com-\npletely missing. To imitate the discussion of spin Berry\nphase (Eq. 4), an auxiliary dilation term is included in or-\nder to fully understand the hop\fon dynamics. Under the\nsmall dilation approximation ( \u0015\u00181), the processional\nterm related to Be\u000bcan be still neglected [32]. In ad-\ndition to the linear momentum and angular momentum\napproaches applied before, we can apply ( r\u0001@r)S0\u0001(S0\u0002)\non both sides of the LLG equation and then get the equa-\ntions of motion along normal direction to the torus mid-\nplane,\n_Z=K1\n\u0000K2\u0000(\f=\u000b)\nK1\n\u0000K2\u0000\u0018jz;\n_\u0002z=\u0000K2\nK1\n\u0000K2\u0000(1\u0000\f=\u000b)\u0018jz;\n_\u0015z=\u0000\u000b\n\u0003\nK1\n\u0000K2\u0000(1\u0000\f=\u000b)\u0018jz;(7)\nwith \u0003 =Kz\nRRKz\n\u0002\u0002\u0000(Kz\nR\u0002)2,K1=\nKz\n\u0002\u0002\u0000\u0003Kz\nR\u0002,\nandK2=\nKz\nR\u0002\u0000\u0003Kz\nRR. The parameters Kz\nRR,\nKz\nR\u0002, andKz\n\u0002\u0002are de\fned as: Kz\nRR=R\n(@zS0)2dV,\nKz\nR\u0002=R\n@zS0\u0001(x@y\u0000y@x)S0dV, andKz\n\u0002\u0002=R\n[(x@y\u0000\ny@x)S0]2dV. It needs to be emphasized that in Eq. 6\nand Eq. 7, the current direction is relative to the mid-\nplane of the hop\fon's torus con\fguration. During the\nhop\fon dynamics, the coordinate must be co-rotating as\nwell.\nCombining these equations of motion, the hop\fon dy-\nnamics shown in Fig. 2 can be readily understood in the\nfollowing way. The current jx\frst induces an entan-\ngled dynamics including the longitudinal motion (\u0001 X),\ntransverse motion (\u0001 Y) and rotations ( \u0002xand \u0002y). As\nthe midplane of hop\fon starts to deviate from the xy\nplane, the current can be decomposed into two compo-\nnents, one in the midplane ( jk) and one normal to the\nmidplane (jz). While the former component is still re-\nsponsible for the entangled dynamics mentioned above,\nthe hop\fon motion \u0001 Zalong normal direction starts to\ndevelop according to Eq. 7.\nTo examine the dynamics in the normal direction, we\nstudy the hop\fon dynamics under jz. The correspond-\ning simulation results are summarized in Fig. 3a-c. The\ncurrent induces a translational motion of hop\fon along\nits direction in combination with a dilation and rotation\naboutz\u0000axis. The dilation type is determined by the\nratio between \fand\u000b(Fig. 3c). When \f <\u000b , the hop-\n\fon is compressed (expanded) by a negative (positive)\ncurrent, and the case is reversed for \f > \u000b . While for\n\f=\u000b, both dilation and rotation are absent. It is worth\nmentioning that the expansion and compression of hop-\n\fon are not quite symmetric since there is an energy bar-\nrier to prevent further compression of hop\fon in order to\nmaintain its topology. Similarly to the current in plane\ncase, the velocity of hop\fon vz=_Zhere is also linearly\nproportional to the current density (Fig. 3b). All these\ndynamics are well described by Eqs. 7.\nThe interesting dilation of hop\fon can be also un-\nderstood phenomenologically in terms of the skyrmion\n0.51.01.52.02.513579\n0.00.51.01.52.02.5–1.0–0.50.00.51.01.5(a)\ntime (ns)Δd (nm)(c)\njz > 0jz < 0β < α\nβ > α(d)Bx (a.u.)yz\nx(b)jz (1010 A/m2)vz (m/s)α = 0.1α = 0.1\nv (β < α)\nv (β > α)\nβ = 0.0β = 0.1β = 0.2\nβ = 0.0β = 0.1β = 0.2FIG. 3. (a) Hop\fon dynamics under out-of-plane applied cur-\nrent (jz). A translational motion along the current direction\nis associated with a dilation depending on the ratio \f=\u000b. (b)\nCurrent density dependence of the hop\fon velocity ( vz) for\ndi\u000berent values of \f. (c) Diameter change of hop\fon during\nits translational motion ( jz= 0:5\u00021010Am\u00002). (d) Calcu-\nlatedBxbased on the spin texture shown in Fig. 1c. The\narrows represent the velocities of skyrmion and antiskyrmion\nunder an applied current along positive zdirection for \f <\u000b\n(solid) and \f >\u000b (dashed).\nstring. As mentioned earlier, a hop\fon can be recog-\nnized as a 2 \u0019twisted skyrmion string with its two ends\nglued together and thus a skyrmion-antiskyrmion pair is\nformed in any cross-section plane including the z\u0000axis\n(e.g.,xzoryzplane), similar to that shown in Fig. 1c.\nTo further illustrate the hop\fon dynamics, the emergent\nmagnetic \feld Bi=1\n2\"ijkS\u0001(@jS\u0002@kS) is calculated\nbased on the hop\fon con\fguration in Fig. 1c and the\nBxis shown in color in Fig. 3d. It is known that the\ncurrent-driven motion of skyrmion has a transverse com-\nponent, i.e., the skyrmion-Hall e\u000bect [21, 31, 35, 36].\nThe corresponding skyrmion-Hall angle is determined by\nthe topological charge, or in identical terms, the emer-\ngent magnetic \feld of the skyrmion. More importantly,\nthe sign of the skyrmion-Hall angle depends the ratio\nbetween\fand\u000b[21, 37] as shown by the arrows in\nFig. 3d. As a result, the skyrmion-antiskyrmion pair\nshown in Fig. 3d respond to a current in z\u0000direction\nby moving towards or away from each other during their\nmotion along z. Same is true for any cross section slic-\ning the hop\fon. When the skyrmion and antiskyrmion\nmove towards (away from) each other, the hop\fon is com-\npressed (expanded). The hop\fon dynamics can thus also\nbe phenomenologically understood as a collective motion\nof skyrmion-antiskyrmion pairs\nConclusion | Current-driven 3D dynamics of magnetic\nhop\fon have been studied both analytically and numer-5\nically. The hop\fon exhibits rich dynamics of entangled\ntranslation, rotation and dilation. The theory built upon\nspin Berry phase and generalized Thiele's approach gives\nout simple equations of motion reproducing numerical\nresults. Our phenomenological analysis also reveals the\nvital role of skyrmion-antiskyrmion pair in hop\fon dy-\nnamics, and makes connection of soliton dynamics across\ndimensionality. Since our theory is built on the collec-\ntive coordinates that is independent of details of spin\ninteractions, it suggests the universality of the reported\ndynamics in all existing and forthcoming hop\fon mod-\nels, not only in magnetism, but also in other physicalsystems [5, 20, 38, 39]. The rich dynamics hosted by a\nQH= 1 hop\fon further foreshadows more exotic dynam-\nics for hop\fons with higher QHand their potentials in\nspintronic applications [40].\nAcknowledgement | J.Z. acknowldges the \fnical sup-\nport by the U.S. Department of Energy (DOE), O\u000ece\nof Science, Basic Energy Sciences (BES) under Award\nNo. de-sc0020221. Y.L and X.H. thank the \fnical\nsupports from the National Key Research and Develop-\nment Program of China (Grant No. 2017YFA0206200,\n2016YFA0300802), the National Natural Science Foun-\ndation of China (NSFC, Grants No.51831012, 11804380).\n[1] T. H. R. Skyrme, Proc. R. Soc. London Sect. A 260, 127\n(1961).\n[2] T. H. R. Skyrme, Nucl. Phys. 31, 556 (1962).\n[3] L. D. Faddeev, Lett. Math. Phys. 1, 289 (1976).\n[4] H. Hopf, Math. Ann. 104, 637 (1931).\n[5] L. Faddeev and A. J. Niemi, Nature 387, 58 (1997).\n[6] R. A. Battye and P. M. Sutcli\u000be, Phys. Rev. Lett. 81,\n4798 (1998).\n[7] J. Hietarinta and P. Salo, Phys. Lett. B 451, 60 (1999).\n[8] M. R. Dennis, R. P. King, B. Jack, K. O'Holleran, and\nM. J. Padgett, Nat. Phys. 6, 118 (2010).\n[9] P. J. Ackerman and I. I. Smalyukh, Nat. Mater. 16, 426\n(2017).\n[10] G. Volovik and V. Mineev, Sov. Phys. JETP 46, 401\n(1977).\n[11] E. Babaev, L. D. Faddeev, and A. J. Niemi, Phys. Rev.\nB65, 100512 (2002).\n[12] D. S. Hall, M. W. Ray, K. Tiurev, E. Ruokokoski, A. H.\nGheorghe, and M. M ott onen, Nat. Phys. 12, 478 (2016).\n[13] Y. Kawaguchi, M. Nitta, and M. Ueda, Phys. Rev. Lett.\n100, 180403 (2008).\n[14] E. Babaev, Phys. Rev. Lett. 88, 177002 (2002).\n[15] P. Sutcli\u000be, Phys. Rev. Lett. 118, 247203 (2017).\n[16] F. N. Rybakov, N. S. Kiselev, A. B. Borisov, L. D oring,\nC. Melcher, and S. Bl ugel, arXiv:1904.00250 (2019).\n[17] Y. Liu, R. K. Lake, and J. Zang, Phys. Rev. B 98, 174437\n(2018).\n[18] P. Sutcli\u000be, J. Phys. A: Math. Theor. 51, 375401 (2018).\n[19] J.-S. B. Tai and I. I. Smalyukh, Phys. Rev. Lett. 121,\n187201 (2018).\n[20] E. Radu and M. S. Volkov, Phys. Rep. 468, 101 (2008).\n[21] N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899\n(2013).\n[22] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8,\n152 (2013).\n[23] K. Y. Guslienko, J. Nanosci. Nanotechnol. 8, 2745 (2008).[24] X. Wang, A. Qaiumzadeh, and A. Brataas, Phys. Rev.\nLett.123, 147203 (2019).\n[25] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[26] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[27] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[28] P. Hilton, An Introduction to Homotopy Theory (Cam-\nbridge University Press, Cambridge, England, 1953).\n[29] A. Auerbach, Interacting Electrons and Quantum Mag-\nnetism (Springer-Verlag, New York, 1994).\n[30] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468,\n213 (2008).\n[31] J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys.\nRev. Lett. 107, 136804 (2011).\n[32] See Supplemental Material, for details of the derivations,\nnumerical simulations, and movies of hop\fon dynamics.\n[33] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n[34] O. A. Tretiakov, D. Clarke, G.-W. Chern, Y. B. Baza-\nliy, and O. Tchernyshyov, Phys. Rev. Lett. 100, 127204\n(2008).\n[35] W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. Ben-\njamin Jung\reisch, J. E. Pearson, X. Cheng, O. Heinonen,\nK. L. Wang, Y. Zhou, A. Ho\u000bmann, and S. G. E.\nte Velthuis, Nat. Phys. 13, 162 (2017).\n[36] K. Litzius, I. Lemesh, B. Kr uger, P. Bassirian, L. Caretta,\nK. Richter, F. B uttner, K. Sato, O. A. Tretiakov,\nJ. F orster, R. M. Reeve, M. Weigand, I. Bykova, H. Stoll,\nG. Sch utz, G. S. D. Beach, and M. Kl aui, Nat. Phys. 13,\n170 (2017).\n[37] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Com-\nmun.4, 1 (2013).\n[38] H. K. Mo\u000batt, Nature 347, 367 (1990).\n[39] N. Manton and P. Sutcli\u000be, Topological Solitons (Cam-\nbridge University Press, Cambridge, England, 2004).\n[40] A. Fern\u0013 andez-Pacheco, R. Streubel, O. Fruchart, R. Her-\ntel, P. Fischer, and R. P. Cowburn, Nat. Commun. 8,\nncomms15756 (2017)."    },    {        "title": "2204.04195v1.Particle_dynamics_in_non_rotating_Konoplya_and_Zhidenko_black_hole_immersed_in_an_external_uniform_magnetic_field.pdf",        "content": "arXiv:2204.04195v1  [gr-qc]  8 Apr 2022Prepared for submission to JHEP\nParticle dynamics in non-rotating Konoplya and\nZhidenko black hole immersed in an external uniform\nmagnetic field\nAqeela Razzaq and Rehana Rahim\nDepartment of Mathematics and Statistics, Riphah Internat ional University, Islamabad, Pakistan\nE-mail:aqeelarazzaq45@gmail.com ,rehana.rahim@riphah.edu.pk\nAbstract: In this paper, we investigate the dynamics of particles in th e background of\nnon-rotating Konoplya and Zhidenko black hole that is immer sed in an external uniform\nmagnetic field. The work involves circular motion of electri c and magnetic particles and\nparticle acceleration. First the motion of electric charge d particles is considered. The\neffective potential, energy and angular momentum expression s are obtained along with\ntheir graphs. The analysis of ISCO shows that radii of ISCO de crease with magnetic\ninteraction parameter. Motion of magnetically charged par ticles has also been studied.1 Introduction\nBlack holes are an interesting and important predictions of Einstein’s theory of general\nrelativity (GR). These are the objects having such an immens e gravitational force that\neven light cannot escape from them. They also serve as an exce llent laboratory for testing\nGR in the strong gravitational field regime. Event horizon ar ound the black holes acts as\na one way membrane from which things do not come out if they ent er the event horizon.\nMotion of photons and the matter in the close vicinity of a bla ck hole can help in direct\nand indirect observation of the event horizon.\nThe geometric structure of a spacetime can be studied throug h the analysis of particle\ndynamics around a black hole. The motion of charged particle s is affected by the presence\nof a test uniform magnetic field in the near vicinity of a black hole. As a black hole hole\ndoes not have a magnetic field, an external magnetic field can b e taken into account. Wald\ngave solution of the electromagnetic field equations for the Kerr black hole surrounded by\nan asymptotically uniform magnetic filed [ 1]. Afterward, many studies have been devoted\nfortheinvestigation ofelectromagnetic fieldsaroundblac k holessurroundedbytheexternal\nuniform and dipolar magnetic fields [ 2–5]. The strength of the magnetic field is assumed\nto be weak and particles are taken to be of mass which is neglig ible as compared to the\nblack hole’s mass.\nCurrently, GR is the best theory which describes gravity, ha ving passed the testing\nwith flying colors in the weak gravitational field regime. In G R, Kerr black hole describes\nthe metric around an astrophysical black hole. Kerr metric c ontains two parameters, which\nare mass and spin (and charge in case of Kerr-Newman black hol e). In alternate theories of\ngravity, numerous metrics have been developed which contai n deviations from Kerr [ 6–11].\nThe Kerr metric is obtained when the deviations vanish. In th is paper, we consider the\nmodified Kerr metric developed in Ref. [ 11] by Konoplya and Zhidenko (referred in this\nwork as KZ black hole). The main aim behind this metric was to s ee if the detection of\ngravitational waves lead to the possibility of modified theo ries of gravity [ 12]. Some studies\nalso suggest that a KZ spacetime might describe a real astrop hysical black hole [ 13].\nThe paper is arranged as: Section 2describes the Konoplya and Zhidenko black hole.\nIn Section 3, the magnetic field components are determined. In Sections 4and5, motion\nof electric and magnetic charged particles is discussed, re spectively. Center of mass energy\nfor the collision of two particles is studied in Section 6. The work has been concluded in\nthe last section.\n2 The Konoplya and Zhidenko black hole\nThe rotating Konoplya and Zhidenko black hole metric is give n as [11]\nds2=−/parenleftbigg\n1−2Mr2+η\nrΣ/parenrightbigg\ndt2+Σ\n∆dr2+Σdθ2+sin2θ/parenleftbigg\nr2+a2+(2Mr2+η)a2sin2θ\nrΣ/parenrightbigg\ndφ2\n−2(2Mr2+η)asin2θ\nrΣdtdφ, (2.1)\n– 2 –with\nΣ =r2+a2cos2θ,∆ =a2+r2−2Mr−η\nr, (2.2)\nwhereMis the mass and ais spin parameter of black hole. Deviations from Kerr metric\nare measured by parameter η. Equation ( 2.1) becomes Kerr metric when ηis set to zero.\nTo obtain the non-rotating form, the case of a= 0 is considered. This gives\nds2=−f(r)dt2+1\nf(r)dr2+r2dθ2+r2sin2θdφ2, (2.3)\nwith\nf(r) =r3−2Mr2−η\nr3. (2.4)\nThis article deals with the non-rotating form of metric ( 2.1) shown in Eq. ( 2.3).\n3 Magnetized Konoplya and Zhidenko black hole\nInthissection, weconsidermetric( 2.3)surroundedbyanexternal uniformmagneticfieldof\nstrength B. The magnetic field is taken to be static, axially symmetric a nd homogeneous\nat spatial infinity. It is also taken to be weak so that it does n ot effect the spacetime\ngeometry outside the black hole. Electromagnetic 4-potent ial determined through Wald\nmethod is [ 1]\nAµ=/parenleftbigg\n0,0,0,1\n2Br2sin2θ/parenrightbigg\n. (3.1)\nThe Maxwell tensor in terms of Aµis\nFαβ=Aβ,α−Aα,β, (3.2)\nwith the components\nFrφ=Brsin2θ, (3.3)\nFθφ=Br2sinθcosθ. (3.4)\nThe orthonormal components of magnetic field with respect to chosen frame are\nBˆr=Bcosθ, (3.5)\nBˆθ=/radicalbigg\nr3−2Mr2−η\nr3Bsinθ. (3.6)\nThe plot of Bˆθagainst various values of ηandθhas been shown in FIG. ( 1). From this\nfigure, it is observed that Bˆθincrease with decreasing value of η.\n4 The motion of the electric charged particles\nThis section deals with the circular motion of particles of m assmwith charge earound the\nKZ metric, surrounded by an external uniform magnetic field. Hamilton-Jacobi equation\nis employed for this purpose and it is given as\ngµν/parenleftbigg∂S\n∂xµ−eAµ/parenrightbigg/parenleftbigg∂S\n∂xν−eAν/parenrightbigg\n=−m2, (4.1)\n– 3 –Η/EΘual0\nΗ/EΘual0.5\nΗ/EΘual/Minus0.5\n2.0 2.5 3.0 3.5 4.0 4.5 5.00.00.10.20.30.4\nrBΘ/Hat\n/Slash1BM/EΘual1,Θ/EΘualΠ/Slash16\nΗ/EΘual0\nΗ/EΘual0.5\nΗ/EΘual/Minus0.5\n2.0 2.5 3.0 3.5 4.0 4.5 5.00.00.10.20.30.40.50.6\nrBΘ/Hat\n/Slash1BM/EΘual1,Θ/EΘualΠ/Slash13\nΗ/EΘual0\nΗ/EΘual0.5\nΗ/EΘual/Minus0.5\n2.0 2.5 3.0 3.5 4.0 4.5 5.00.00.20.40.60.8\nrBΘ/Hat\n/Slash1BM/EΘual1,Θ/EΘualΠ/Slash12\nFigure 1 : Radial profile of Bˆθfor some values of ηandθ.\nwhereSis the Hamilton-Jacobi action having the following equatio n\nS=−Et+Lφ+Sr(r)+Sθ(θ), (4.2)\nwithEbeing energy and Lbeing angular momentum of the particle, receptively. The\nmotion takes place on the equatorial plane ( θ=π/2). Equation ( 4.1) after putting the\nvalues, takes the form\n/parenleftbigg−r3\nr3−2Mr2−η/parenrightbigg\nE2+/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg\ng2\nrr˙r2+1\nr2(L−eAφ)2=−m2.(4.3)\nFurther simplification, leads to\n˙r2=ε2−/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/parenleftigg\n1+/parenleftbiggL\nr−ωBr/parenrightbigg2/parenrightigg\n, (4.4)\nwhereε=E/m,L=L/mbe the energy per unit mass, angular momentum per unit mass,\nrespectively, and\nωB=eB\n2m. (4.5)\nTheωBis the cyclotron frequency. It accounts for the magnetic int eraction between an\nelectric charge and an external magnetic field. Equation ( 4.4) can also be written as\n˙r2=ε2−Veff, (4.6)\n– 4 –where\nVeff=/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/parenleftigg\n1+/parenleftbiggL\nr−ωBr/parenrightbigg2/parenrightigg\n. (4.7)\nThe radial plot of Veffhas been shown in FIG. ( 2). The plots show that if we increase\nvalues of ωBandη, effective potential decreases.\nΩB/EΘual0\nΩB/EΘual0.2\nΩB/EΘual/Minus0.2\n2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.00.51.01.52.02.53.03.54.0\nrVeffL/EΘual6,M/EΘual1,Η/EΘual0.2\nη=0\nη\u00000.1\nη\u00010.2\nη\u0002 \u00030.1\nη\u0004 \u00050.2\n2 3 4 5 6-1.0-0.50.00.51.01.5\nrVeffL\u00066,ωB\n\u00070.01,M\b1\nFigure 2 : Radial plot of Veff. Graph on the left is shown for some values of the cyclotron\nfrequency with other parameters being fixed. The plot in the r ight panel shown Vefffor\nsome values of η.\nFor circular motion of the particles, one needs the conditio ns\n˙r= 0, (4.8)\ndVeff/dr= 0. (4.9)\nEquation ( 4.8) leads to\nVeff=ε2, (4.10)\nwhile Eq. ( 4.9) gives\nV′\neff=1\nr6/parenleftbigg\n−2Lr2ωB/parenleftbig\n3η+2Mr2/parenrightbig\n+ω2\nB/parenleftbig\n−2Mr6+2r7+ηr4/parenrightbig\n+L2/parenleftbig\n5η+6Mr2−2r3/parenrightbig\n+2Mr4+3ηr2/parenrightbigg\n= 0. (4.11)\nThis gives angular momentum Las\nL=1\n2(6Mr2−2r3+5η)/bracketleftbigg\n4Mr4ωB+6r2ωBη±2r/bracketleftbig\n3η/parenleftbig\n2r3−5η/parenrightbig\n−12M2r4\n+4M(r5−7ηr2)+4r2ω2\nB/parenleftbig\nη+2Mr2−r3/parenrightbig2/bracketrightbig1/2/bracketrightbigg\n. (4.12)\n– 5 –The energy per unit mass εis obtained as\nε2=/parenleftbigg\n1−2Mr2+η\nr3/parenrightbigg\n(4.13)\n\n1+\nrωB−4Mr4ωB+6r2ηωB−/parenleftigg/parenleftbig\n4Mr4ωB+6r2ηωB/parenrightbig2−4/parenleftbig\n−6Mr2+2r3−5η/parenrightbig\n/parenleftbig\n−2Mr4−3r2η+2Mr6ω2\nB−2r7ω2\nB−r4ηω2\nB/parenrightbig/parenrightigg1/2\n2r(6Mr2−2r3+5η)\n2\n.\nThe graphical behavior of angular momentum is shown in FIG. 3 which shows large values\nωB=0\nωB=0.1\nωB=0.2\n5 10 15 2051015202530\nrℒη=0.2,M=1\nFigure 3 : The graph of Lfor some values of ωB\nη=0\nη \t \n0.2\nη\u000b0.2\n3.03.54.04.55.05.56.00.70.80.91.0\nrϵM1\nωB=0.1\nωB=0.3\nωB=0.5\n3.03.54.04.55.05.56.01.01.52.0\nrϵ=0.2,M=1\nFigure 4 : Graph of energy of a charged particle. In the left panel ωB= 0.3 with varying\nvalues of η. In the right panel, graph is shown for some values of ωBwithηbeing held\nfixed.\nof angular momentum due to ωB. Energy is observed to decrease with ωBon the right\npanel of FIG. 4, while, in its left panel, energy is less than t he case of Schwarzschild metric.\n– 6 –4.1 The inner most stable circular orbits (ISCO)\nTo find inner most stable circular orbits or the ISCO, we have d2Veff/dr2= 0.This gives\n2\nr5(6Mr2−2r3+5η)2\n/parenleftigg\n/parenleftbig\n6Mr2−2r3+5η/parenrightbig/parenleftigg\n2M(6M−r)r4+\nr2(20M+3r)η+15η2/parenrightigg/parenrightigg\n+4r2/parenleftbig\n−2Mr2+r3−η/parenrightbig/parenleftigg\n24M2r4−22Mr5+4r6+\n40Mr2η−23r3η+10η2/parenrightigg\nω2\nB\n−2/parenleftbig\n2M(6M−r)r4+r2(20M+3r)η+15η2/parenrightbig\nωB\n×/radicaligg\nr2/parenleftbig\n2r2(−3M+r)−5η/parenrightbig/parenleftbig\n2Mr2+3η/parenrightbig\n+4r4/parenleftbig\n2Mr2−r3+η/parenrightbig2ω2\nB,\n/greaterorequalslant0.\nIt is impossible to have exact solution for risco, therefore, it is obtained numerically.\nThe numerical solution is shown for various values of ωBin Table 1. The table shows\ndecreasing riscowith increasing magnetic interaction parameter.\nωBrisco\n0.44.367406\n0.454.361155\n0.54.356620\n0.554.353231\n0.64.350632\n0.654.348598\n0.74.346976(4.14)\n5 Magnetized Particle Motion\nThis section deals with the dynamics of magnetized particle s around KZ black hole that\nis immersed in an external asymptotically uniform magnetic field. Modified form of Eq.\n(4.1) for motion of magnetized particles is\ngαβ∂S\n∂xα∂S\n∂xβ=−/parenleftbigg\nm−1\n2DαβFαβ/parenrightbigg2\n, (5.1)\nwithmbeing particle’s mass, Sdenotes action for magnetized particle in the curved space-\ntime background. Dµνrepresents the polarization tensor with the form\nDαβ=ηαβµνuµµν, (5.2)\nand has the following constraint\nDαβuβ= 0. (5.3)\nHereµνdenotes the 4-velocity of magnetic dipole moment and uνis the 4-velocity of the\nparticles inan arbitraryobserver’s restframeofreferenc e. Theproductof DµνFµνaccounts\nfor relationship between the external magnetic field and mag netized particles. In this work,\n– 7 –we assume that such an interaction is weak, thus one can negle ct (DµνFµν)2.The Maxwell\ntensor can be written as\nFαβ= 2u[µEν]−ηαβµνBµuν, (5.4)\nwhereEνandBνare the electric and magnetic field, respectively, and ηαβµνis obtained\nfrom the Levi-civita symbol ǫαβµνas\nηαβµν=ǫαβµν√−g, (5.5)\nwithgbeing the determinant of the metric. Taking into account Eqs . (5.2)-(5.5) leads to\nDαβFαβ= 2µαBα= 2µˆαBˆα= 2µB/radicalbig\nf, (5.6)\nwheref(r) is given in Eq. ( 2.4). The radial equation of motion is obtained from Eq. ( 5.1)\nand is given as\n/parenleftbigg−r3\nr3−2Mr2−η/parenrightbigg\nε2+/parenleftbiggr3\nr3−2Mr2−η/parenrightbigg\n˙r2+1\nr2L2=−/parenleftbigg\n1−µB√f\nm/parenrightbigg2\n,(5.7)\n˙r2=ε2−/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/bracketleftbigg\n(1−β/radicalbig\nf)2+L2\nr2/bracketrightbigg\n, (5.8)\nwhereβ=µB\nmis called magnetic coupling parameter that defines electrom agnetic interac-\ntion between magnetic dipole and external magnetic field. Eq uation (5.8) can be written\nas\n˙r2=ε2−Veff, (5.9)\nwithVeff\nVeff=/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/bracketleftbigg\n(1−β/radicalbig\nf)2+L2\nr2/bracketrightbigg\n. (5.10)\nFIG.5shows graph of Veffof Eq. (5.10). Both the panels show decreasing behavior with\nincreasing β(left panel) and η(right panel).\nTo determine the energy and angular momentum, Eqs. ( 4.8-4.10) are again employed. The\nderivative of Veffis\nV′\neff=1\nr7/bracketleftig\nL2(6Mr3−2r4+5rη)−(2Mr2+3η)/parenleftbig\n4Mr2β2+2β2η\n+r3/parenleftbig\n−1−2β2+3β/radicalbig\nf/parenrightbig/parenrightbig/bracketrightig\n(5.11)\nThis equation leads to angular momentum L2as\nL2=1\n6Mr3−2r4+5rη/bracketleftig\n−2Mr5+8M2r4β2−4Mr5β2−3r3η+16Mr2β2η−6r3β2η\n+6β2η2+6Mr2β/radicalbig\nf+9r3βη/radicalbig\nf/bracketrightig\n. (5.12)\nThe expression for energy is\nε2=(2Mr2−r3+η)/bracketleftigg\n4M2r4β2−β2η2+2r6/parenleftbig\n1+β−2β√f/parenrightbig\n+r3η/parenleftbig\n−2+β/parenleftbig\n−β+√f/parenrightbig/parenrightbig\n+2Mr5/parenleftbig\n−2+3β/parenleftbig\n−β+√f/parenrightbig/parenrightbig/bracketrightigg\n6Mr8−2r9+5r6η.\n(5.13)\n– 8 –β=0\nβ=0.2\nβ=0.4\n2 3 4 5 6 70.00.51.01.5\n2 \f \r\nrV\ne\u000e\u000f\nL=6 \u0010η=0 \u0011 \u0012\nη=0\nη=0.2\nη=-0.2\n2 3 4 5 6 70.00.51.01.5\n\u0013 \u0014 \u0015\nrV\n\u0016\u0017\u0018\n\u0019=\u001a \u001bβ=\u001c \u001d \u001e \u001f  =1\nFigure 5 : Graphical representation of Veffof radial motion of magnetized particles. On\nthe left, graph is drawn for varying β. The graph in the right panel has been plotted with\nvaryingη.\nThe graphs of angular momentum and energy are shown in FIGs. ( 6) and (7), respectively.\nThe behavior of angular momentum is increasing with increas ing values of ηand if we\nincrease values of β, angular momentum and energy decrease.\nΒ/EΘual0\nΒ/EΘual0.2\nΒ/EΘual0.4\n2 4 6 8 10246810\nr/ScriptCapLM/EΘual1,Η/EΘual0.1\nη=0\nη=0.15\nη! \"0.15\n2.0 2.5 3.0 3.5 4.0 4.5 5.00246810\nrℒβ# $ % & '1\nFigure 6 : Graph of Lwith varying β(left panel) and η(right panel).\nThe ISCO is given by the equation\n1\nr7\n−/parenleftbig\n2Mr2+3η/parenrightbig/parenleftig\n8Mrβ2+6Mr2β+9βη\n2r√f+3r2/parenleftbig\n−1−2β2+3β√f/parenrightbig/parenrightig\n−4Mr/parenleftbig\n4Mr2β2+2β2η+r3/parenleftbig\n−1−2β2+3β√f/parenrightbig/parenrightbig\n+(2Mr2+3η)(18Mr2−8r3+5η)(4Mr2β2+2β2η+r3(−1−2β2+3β√f))\n6Mr3−2r4+5rη\n≥0.\nAfter simplification, one gets\n1\nr7\n−/parenleftbig\n2Mr2+3η/parenrightbig/parenleftig\n8Mrβ2+6Mr2β+9βη\n2r√f+3r2ζ/parenrightig\n−4Mr/parenleftbig\n4Mr2β2+2β2η+r3ζ/parenrightbig\n+(2Mr2+3η)(18Mr2−8r3+5η)(4Mr2β2+2β2η+r3ζ)\n6Mr3−2r4+5rη\n≥0.\n– 9 –β=0\nβ=0.4\nβ=0.73 ( ) * + , 4.0 4.5 5.0 5.5 6.001\n-\n.456\nrϵM= 1 /=0.1\nFigure 7 : Graphical representation of energy for various values of β.\nwith\nζ=/parenleftig\n−1−2β2+3β/radicalbig\nf/parenrightig\n. (5.14)\n6 Center of mass energy in the equatorial plane\nThis section deals with center of mass energy for the collisi on of particles. The particles\nare assumed to be having equal masses and are assumed to be com ing from infinity with\nthe same initial energy E1/m1=E2/m2= 1 but with different angular momenta. The\ncenter of mass energy for the collision of two particles give n by Ba˜ nados, Silk and West\n(BSW) [14] as\nεcm=E2\ncm\n2m0= 1−gµνvµ\n1vν\n2, (6.1)\nwherevµ\ni= (˙ti,˙ri,˙θi,˙φi) fori= 1,2 represent the velocity of the particles.\n6.1 The center of mass energy for the collision of two neutral particles\nIn this subsection, collision of two neutral particles havi ng same rest mass energies will be\nconsidered at the equitorial plane. The velocity component s in this case are\n˙t=r3\nr3−2Mr2−η, (6.2)\n˙φ=l\nr2, (6.3)\n˙r2=ε2−/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/parenleftbigg\n1+l2\nr2/parenrightbigg\n. (6.4)\nThe center of mass energy is\nεcm= 1+r3\nr3−2Mr2−η−l1l2\nr2\n−r3\nr3−2Mr2−η/radicaligg\n1−/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/parenleftbigg\n1+l2\n1\nr2/parenrightbigg/radicaligg\n1−/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/parenleftbigg\n1+l2\n2\nr2/parenrightbigg\n,\n(6.5)\n– 10 –wherel1andl2, respectively, represent angular momentum of firstand seco nd particle. The\nradial plot of εcmis shown in FIG. 8. In FIG. 8the center of mass energy is decreasing with\nincreasing values of η(left) and increasing with increasing values of angular mom entum.\nη=0\nη=0.4\nη=0.99\n0.5 1.0 1.5 2.0 2.5 3.046810\n4 51416\nr\nc7M=8 9l1=: ; < > ?l@\nA B0.01\nl1/EΘual0.76,l2/EΘual/Minus0.76\nl1/EΘual3,l2/EΘual/Minus3\nl1/EΘual5;l2/EΘual/Minus5\n0.0 0.5 1.0 1.5 2.0 2.5 3.0020406080100\nrΕcmM/EΘual1,Η/EΘual0.99\nFigure 8 : Graph of εcmfor varying η(left panel) and angular momentum (right panel).\n6.2 The center of mass energy for the collision of two magnetized particles\nHere, the collision of two magnetized particles will becons idered. Theequations for motion\nin this case are\n˙t=r3\nr3−2Mr2−η, (6.6)\n˙φ=l\nr2, (6.7)\n˙r2=ε2−/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/parenleftbigg\n(1−β/radicalbig\nf)2+l2\nr2/parenrightbigg\n. (6.8)\nThe center of mass energy in this case is\nε2\ncm= 1+r3\nr3−2Mr2−η−l1l2\nr2−r3\nr3−2Mr2−η/radicalbigg\n1−/parenleftig\nr3−2Mr2−η\nr3/parenrightig/parenleftig\n(1−β1√f)2+l2\n1\nr2/parenrightig\n×\n/radicalbigg\n1−/parenleftig\nr3−2Mr2−η\nr3/parenrightig/parenleftig\n(1−β2√f)2+l2\n2\nr2/parenrightig\n.\nThe graphical behavior is shown in FIG. 9. In FIG. 9if we decrease values of deformation\nparameter and angular momentum then the center of mass energ y also decreases.\n6.3 The center of mass energy for the collision of a neutral and a magnetized\nparticle\nIn this subsection, collision of a magnetized and neutral pa rticle has been considered.\nThe equations of motion are given in sections 6.1and6.2. The particle 1 is taken to be\nmagnetized and particle 2 is assumed to be neutral. Using the se in center of mass energy\n– 11 –η=0.99\nη= C D E F\nη=-0.99\n2 3 4 5 623456\nr\nGHM=1lI=3,l2=-3\nl1=2,l2=-2\nl1=0.9,l2=-0.9\n2 4 6 8 10\nJ\nK456\nr\nMNM= O P=0.5\nFigure 9 : Center of mass energy for the two magnetized particles coll ision. On left,\nl1= 2.5,l2=−2.5,β1= 0.3 =β2.On the right, we have taken β1= 0.1 =β2.\nexpression\nε2\ncm=1+r3\nr3−2Mr2−η−l1l2\nr2−r3\nr3−2Mr2−η\n×/radicaligg\n1−/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/parenleftbigg\n(1−β1/radicalbig\nf)2+l2\n1\nr2/parenrightbigg/radicaligg\n1−/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/parenleftbigg\n1+l2\n2\nr2/parenrightbigg\n.\n(6.9)\nThe radial profile of εis shown in FIG. 10. In FIG. 10if we increase values of ηcenter of\nmass energy increases.\nη=0,l1=2,l2\nQ R2\nηS0.2,l1\nT2.5,l2\nU V2.5\nηW X0.2,l1\nY1.5,l2\nZ [1.5\n1 2 3 4 5 6 71\n\\\n]456\nrϵ\n^_\nFigure 10 : Radial profile of εwith varying ηandl1andl2. Here we have taken M=\n1,β1= 0.2.\n6.4 The center of mass energy for the collision of a charged and a magnetized\nparticle\nHere, collision of a magnetized and a charged particle has be en considered. The particle 1\nis taken to be charged and particle 2 is assumed to be magnetiz ed. Using these in center\nof mass energy expression, we obtain\n– 12 –ε2\ncm= 1+r3\nr3−2Mr2−η−/parenleftbiggl1\nr2−ωB/parenrightbigg\nl2−r3\nr3−2Mr2−η\n×/radicaltp/radicalvertex/radicalvertex/radicalbt1−/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/bracketleftigg\n1+/parenleftbiggl1\nr−ωBr/parenrightbigg2/bracketrightigg/radicaligg\n1−/parenleftbiggr3−2Mr2−η\nr3/parenrightbigg/parenleftbigg\n(1−β/radicalbig\nf)2+l2\n2\nr2/parenrightbigg\n.\n(6.10)\nThe center of mass energy is shown in FIG. 11. In FIG. 11if we increase values of ηcenter\nof mass energy increases.\nη=0,l1=2,l2\n` a2\nη b0.2,l1\nd2.5,l2\nf g2.5\nη h i0.2,l1\nj1.5,l2\nk m1.5\n2 3 4 5 6 7\nn\no456\nrϵ\npq\nFigure 11 : Radial profile of εwith varying ηandl1andl2. Here we have taken M=\n1,ωB= 0.1,β= 0.1.\n7 Summary and conclusion\nThe Kerr black hole solution is an axisymmetric, stationary and vacuum solution of the\nEinstein theory of general relativity. All the astrophysic al black holes are expected to be\ndescribed by the Kerr metric. There has been a lot of interest in modified Kerr black hole\nsolutions. Such solutions contain parameters which accoun t for possible deviations from\nKerr, which is obtained when deviations are set to zero. One s uch rotating metric has been\ndeveloped by Konoplya and Zhidenko [ 11], whose non-rotating case has been discussed\nin the present article. In this work, dynamics of charged and magnetized particles (in\nthe equatorial plane) have been discussed in the background of non-rotating Konoplya-\nZhidenko metric immersed in an external magnetic field.\nFirst, the effective potential, angular momentum and energy f or the circular motion of\ncharged test particles have been studied for the dependence on deformation parameter η\nand cyclotron frequency ωB. In FIG. 2, effective potential has been plotted for varying\nωB(left panel) and η(right panel). Both panels show deceasing trend with increa sing the\nrespective parameter. This can be explained as the decrease in the least distance between\nthe charged particles and the black hole with increase of ωBandη. In FIGs. 3and4,\nradial plots of angular momentum and energy are shown. If we t ake double derivative\nof effective potential it give us value of ISCO and it is not poss ible to find its analytical\n– 13 –solution so we just solve it numerically to check its behavio r. The table 1. shows the ISCO\nvalues at different values of ωBwithηfixed. A decreasing trend is observed for this table.\nDecreasing the ISCO radius is very important because the gra vitational potential near the\ncentral object can accelerate particles to high energies.\nSection5contains dynamics of magnetized particle in the background of non-rotating\nKZ black hole. First the effective has been shown in FIG. 5. The figure shows the same\ndecreasing trend for βandηas in the charged particles case. Next, angular momentum,\nenergy and ISCO have been studied. In the last section, cente r of mass energy was studied.\nThe exact expression were shown along with the graphical beh avior in each case.\nReferences\n[1] R. M. Wald, Phys. Rev. D 101680 (1974).\n[2] A. A. Abdujabbarov, B. J. Ahmedov and V. G. Kagramanova, Ge n. Relativ. Gravit. 402515\n(2008).\n[3] A. N. Aliev and D. V. Gal’tsov, Soviet Phys. Uspekhi 3275 (1989).\n[4] A. N. Aliev, D. V. Galtsov and V. I. Petukhov, Astrophys. Sp. Sc i.124137 (1986).\n[5] T. Oteev, A. Abdujabbarov, Z. Stuchlik and B. Ahmedov, Astro phys. Sp. Sci. 361269\n(2016).\n[6] T. Johannsen and D. Psaltis, Phys. Rev. D 83124015 (2011).\n[7] R. Rahim and K. Saifullah, Ann. Phys. 405220 (2019).\n[8] U. A. Gillani, R. Rahim and K. Saifullah, Astropart. Phys. 138102684 (2022).\n[9] R. Rahim and K. Saifullah, IJMPD Doi: 10.1142/S021827182150123 6 (2021).\n[10] K. Glampedakis and S. Babak, Class. Quantum Grav. 234167 (2006).\n[11] R. Konoplya and A. Zhidenko, Phys. Lett. B 756350 (2016).\n[12] F. Long. S. Chen, S. Wang and J. Jing, A. Zhidenko, Nucl. Phys. B.92683 (2018).\n[13] C. Bambi, S. Nampalliwar, Europhys. Lett. 11630006 (2016); Y. Ni, J. Jiang, C. Bambi, J.\nCosmol. Astropart. Phys. 914 (2016).\n[14] M. Ba˜ nados, J. Silk and S. M. West, Phys. Rev. Lett. 103(2009) 111102.\n– 14 –"    },    {        "title": "1305.5087v1.Self_consistent_calculation_of_spin_transport_and_magnetization_dynamics.pdf",        "content": "1 \n Self-consistent calculation of spin transport and magnetization dynamics \n \nKyung-Jin Leea,b,c,d*, M. D. Stilesc, Hyun-Woo Leee, Jung-Hwan Moona, Kyoung-Whan Kime, \nSeo-Won Leea \n \naDepartment of Materials Science and Engineeri ng, Korea University, Seoul 136-713, Korea \nbKU-KIST School of Converging Science and Techno logy, Korea University, Seoul 136-713, Korea \ncCenter for Nanoscale Science and Technology, National Institute of Standards and Technology, \nGaitherburg, Maryland 20899, USA \ndMaryland Nanocenter, University of Maryland, College Park, Maryland 20742, USA \neDepartment of Physics, Pohang University of Sc ience and Technology, Pohang, Kyungbuk 790-784, \nKorea \n * Corresponding author \n  E-mail address: kj_lee@korea.ac.kr (K. –J. Lee) \n 2 \n Abstract \nA spin-polarized current transfers its spin-angular momentum to a local magnetization, \nexciting various types of current-induced magnetiza tion dynamics. So far, most studies in this \nfield have focused on the direct effect of  spin transport on magnetization dynamics, but \nignored the feedback from the ma gnetization dynamics to the spin transport and back to the \nmagnetization dynamics. Although the feedback is usually weak, there are situations when it \ncan play an important role in the dynamics. In  such situations, simultaneous, self-consistent \ncalculations of the magnetization dynamics and the spin transpor t can accurately describe the \nfeedback. This review describes in detail the feedback mechanisms, and presents recent \nprogress in self-consistent calcu lations of the coupled dynamics. We pay special attention to \nthree representative examples, where the feedback generates non- local effective interactions \nfor the magnetization after the sp in accumulation has been integrated out. Possibly the most \ndramatic feedback example is the dynamic in stability in magnetic nanopillars with a single \nmagnetic layer. This instability does not occur without non-local feedback.  We demonstrate \nthat full self-consistent calculations generate simulation results in much better agreement with experiments than previous calculations th at addressed the feedback effect approximately. \nThe next example is for more typical spin valve nanopillars. Although the effect of feedback \nis less dramatic because even without feedba ck the current can make stationary states \nunstable and induce magnetization oscillation, the feedback can still have important \nconsequences. For instance, we show that the feedback can reduce the linewidth of \noscillations, in agreement with experimental observations. A key aspect of this reduction is \nthe suppression of the excitation of short wave  length spin waves by the non-local feedback. \nFinally, we consider nonadiabatic electron tran sport in narrow domain  walls. The non-local \nfeedback in these systems leads to a significan t renormalization of the effective nonadiabatic 3 \n spin transfer torque. These examples show that the self-consistent treatment of spin transport \nand magnetization dynamics is important fo r understanding the physics of the coupled \ndynamics and for providing a bridge between th e ongoing research fields of current-induced \nmagnetization dynamics and the newly emergi ng fields of magnetization-dynamics-induced \ngeneration of charge and spin currents.  \n 4 \n Contents  \n \n1. Introduction \n \n2. Non-local spin transfer torque in layered structures \n2.1.  Basic concept of non-local spin transfer  torque due to lateral spin diffusion \n2.2.  Previous studies on non-local spin transf er torque due to lateral spin diffusion \n2.3.  Self-consistent calculation in layered structures \n2.3.1.   Modeling scheme \n2.3.2.   Single ferromagnet \n2.3.3.   Spin valves \n2.4.  Summary \n \n3. Non-local spin transfer torque for a magnetic domain wall \n3.1.  Current-induced motion of a domain wall  \n3.2.  Non-local spin transfer to rque for a narrow domain wall \n3.3.  Previous studies on self-consistent cal culation for current-i nduced domain wall \nmotion \n3.4.  Semiclassical approach \n3.5.  Current-induced domain wall motion by non-local spin tr ansfer torque \n3.6.  Summary \n \n4. Conclusion and outlook 5 \n 1. Introduction  \n \nWhen electrons flow through systems that include a ferromagnetic region, the flowing \nelectrons become partially spin polarized due to the exchange interaction between conduction \nelectron spins and local magnetizat ions. Spin transfer torques [1 -4] then occur when the spin \npolarized current passes through another region with a magnetizati on non-collinear to that in \nthe first region. The spin-polarized current ex erts a torque on the non-collinear magnetization \nby transferring its transverse spin-angular mome ntum. Spin transfer to rques generate a wide \nvariety of magnetization dynamics such as full reversal of magnetization [5,6], steady-state \nprecession [7-10], domain wall motion [11,12], and modification of spin waves [13,14]. \nThese types of current-induced magnetization dynamics could potentially find use in novel \nfunctional spintronic devices [15] such as magnetic random access memories (MRAMs) [16], \nmicrowave oscillators [17,18], domain wall storag e devices [19], and spin wave logic devices \n[20].  \nIn order to investigate current -induced magnetic excitation, it is  essential to formulate the \nequation of motion of the magnetization affected by spin transport. To do so, spin transfer \ntorques are added to the Landau- Lifshitz-Gilbert (LLG) equation \n,ST eff 0ii\ni i ii\nt tNmm H mm                                     ( 1 )  \nwhere mi is the unit vector of the ith local magnetization, 0Hi\neff is the effective magnetic field \nacting on mi (it includes exchange, magnetostatic inte ractions, anisotropy, thermal fluctuation, \nand external fields),  is the gyromagnetic ratio of the ferromagnet,  ( / )Bg, g is the \nLandé g-factor,  is the Gilbert damping constant, and i\nSTN describes the spin transfer \ntorque acting on mi. For multilayers such as metallic spin  valves or magnetic tunnel junctions 6 \n where the current flows perp endicular to the interfaces, Ni\nST is taken as [21,22] \n)], ( ) ( [J J ST p m p m m N i i i ib a                                        ( 2 )  \nwhere aJ and bJ are the coefficients of the in-plane and out-of-plane spin transfer torques, \nrespectively, where the plane is defined to contain two vectors, m and p, and p is the \ndirection vector of the pinned-la yer magnetization, which is usua lly assumed to be fixed. On \nthe other hand, when the current flows with in a magnetic layer (or nanowire) with a \ncontinuously varying magnetization, e. g. domain walls and spin waves, Ni\nST for one-\ndimensional system is taken as [23-25] \n,0 0 ST \n\n\n\n\n\n\n\n\n\n\n\nii\nii\nxuxummmN                                         ( 3 )  \nwhere u0 (=S eB 2/eMPjg ) is the spin current velocity  corresponding to adiabatic spin \ntransfer torque, P is the spin polarization, je is the charge current density, MS is the saturation \nmagnetization, and  is the ratio of the nonadi abatic spin transfer tor que to the adiabatic one \n[24,25]. \nEquations (2)-(3) for the spin transfer torque  are based on the assumptions that the spin \ntransfer torque depends on the magnetization only instantaneously and locally. Using the instantaneity assumption, N\ni\nST is derived by solving the spin transport equation for relevant \nsystems with fixed (= time-independent) magne tization profiles and then applied to the \nmagnetization dynamics. This instantaneity assu mption depends on the ability to decouple the \nspin transport dynamics from the magnetizati on dynamics. The decoupling is justified based \non the difference in time scales [21,26]. Two char acteristic time scales for the spin transport \nare the spin-flip relaxation time sf and the spin precession time h/J where h is the Planck \nconstant and J is the interaction energy between conduction electron spins and local \nmagnetizations. Both time scales are of the orde r of picoseconds or less. On the other hand, 7 \n the characteristic time scale of the magnetization dynamics is  the inverse of the Larmor \nfrequency, which is typically of the order of nanoseconds. Because of this difference in time \nscales, the two dynamic equations do decouple and the instantaneity assumption is well \njustified. One can assume that the magnetizat ions do not vary with time, solve the spin \ntransport equation in the long-time limit to get Ni\nST, and feed the result into the equation of \nmotion for the magnetization.  \nThe local approximation is that in Eqs. (2) and (3), Ni\nST is determined by the local values \nof magnetization (= mi) and/or local spatial derivative of the magnetization (= ∂m/∂x|i). \nHowever, the local approximation is not alwa ys valid. For example, consider a system \nconsisting of a single ferromagnet (FM) layer sandwiched by two normal metal (NM) layers, \nwhere the charge current flows perpendicular to the FM|NM interfac es. The current through \nthe layers generates a spin accumulation, which in turn can generate a spin transfer torque \nwhenever it is not collinear w ith the magnetization at an inte rface. Although the direction and \nmagnitude of the spin transfer torque at a point on an interface depends locally on the spin \naccumulation at the same point, the spin accumulation has an inherently non-local \ndependence on the magnetization due to spin diff usion. Strictly speaking the spin transfer \ntorque remains local even in this case, but a local interaction betw een the spin accumulation \nand the magnetization leads to non-local effective interactions for the magnetization after spin accumulation has been integrated out. In this  paper, we call this feedback non-local spin \ntransfer torque because there is a non-local effective  relation between the spin torque and the \nmagnetization profile. For th e torque acting on the single FM layer, lateral spin diffusion in \nthe two neighboring NM layers [ 27,28] is an important source for non-locality of the torque. \nEven when net charge flow is perpendicular to the layers, spin diffusion occurs not only \nalong the perpendicular direction but also along the lateral dire ction (or in-plane direction). 8 \n Due to this lateral spin diffusion, spin accumulation at a point in the FM|NM interface \ndepends on the magnetization at other points on the interface within the reach of the spin \ndiffusion. Whenever the magnetization is inho mogeneous in the film plane, the non-local \ntorques will be non-zero. Even if the magnetizati on is initially in a single domain state, the \nconventional local spin transf er torques or thermal fluctu ations make the magnetization \ninhomogeneous [29-34] and the non-local torque s then becomes non-zero. This non-local \nspin transfer torque acts as a source of feedback from the magne tization to the spin transport, \nwhich, in turn, further affects the magnetization dynamics. \nA complete understanding of current-induced  magnetic excitations requires a careful \ntreatment of this non-local feedback. In this review, we do so by self-consistently  solving the \ntwo dynamic equations simultaneously, one fo r magnetization and the other for spin \naccumulation. In Secs. 2 and 3, we present exam ples where the self-consistent calculation is \nessential to capture properties of  the coupled dynamics. Section 2 presents the effect of lateral \nspin diffusion on the magnetization dynamics in la yered structures. We first analyze in detail \ncurrent-induced excitation of a single FM and then the current -driven magnetization \noscillation in spin valves th at contain two FM layers. Sec tion 3 presents current-induced \nmotion of a narrow domain wall. Here, we use a semiclassical approach to calculate spin transfer torques in the ballistic limit. We end the paper by remark ing on the prospects for \nfuture work on self-consistent calculation of  spin transport and magnetization dynamics.  \n 9 \n 2. Non-local spin transfer torque in layered structures \n \n   We consider two types of non-local spin tr ansfer torques in layered structures. One is \ncaused by lateral spin diffusion along the interf ace of FM|NM. The other is related to the \ncoupling of local spin accumulation along the vert ical (thickness) dir ection of the layers, \nwhich is effective when there are more than three ferromagnetic layers. In this section, we focus on the former and briefly disc uss the latter in Section 2.4. \n \n2.1. Basic concept of non-local spin transfer torque due to lateral spin diffusion \n   Spin transfer torques caused by lateral spin diffusion, which we will refer to as “lateral \nspin transfer torque”, were proposed by Polia nski and Brouwer [27]. The geometry of the \nsystem under consideration is shown in Fi g. 1: a ferromagnet (FM) of thickness t\nF is \nsandwiched by diffusive non-magnetic layers, NM 1 and NM 2, of thicknesses L 1 and L 2, \nrespectively. NM 1 and NM 2 are connected to reservoirs  and the magnetization in the \nferromagnet is inhomogeneous. When electrons flow from right to left (charge current j \nflowing from left to the right) the spin filter effect causes minority electrons to accumulate to the right of the ferromagnetic layer and majority  electrons to accumulate on the left. Majority \nelectrons have moments parallel to the magneti zation but spins that are antiparallel. This \ndifference requires some care in determining the sign of the spin transfer torques. \nThe two bottom panels in Fig. 1 describe the processes of spin transfer via lateral spin \ndiffusion in detail. On the side of the interface NM\n1|FM (bottom left panel), after passing \nthrough a local magnetization m1, a conduction electron spin s1 has its moment parallel to m1. \nThis s1 laterally diffuses along the interface, h its the interface at another point with \nmagnetization m3, and then scatters from the interf ace, transmitting with some probability 10 \n and reflecting with some probabili ty. The moment of the reflected s1 is anti-parallel to m3 and \nthat of the transmitted electron is parallel to m3. Since the spin angular momentum of s1 \nchanges due to this scattering process, the am ount of the change should be transferred to m3 \nto satisfy the conservation of the sp in angular momentum. As a result, m3 experiences spin \ntransfer torque 1 that pushes m3 to align with m1; i.e., spin transfer effect on the side of the \ninterface NM 1|FM where the majority spins accumulat e tends to suppress any inhomogeneity \nin the ferromagnetic magnetization. On the ot her hand, on the side of the interface FM|NM 2 \n(bottom right panel) where minority spins te nd to accumulate, the conduction electron spin s2 \nscattered by a local magnetization m1 initially has its moment anti-parallel to m1. Through \nthe lateral diffusion and th e backscattering process by m3, the moment of s2 becomes anti-\nparallel to m3. This backscattering process ge nerates spin transfer torque 2 whose direction is \nopposite to 1; i.e., spin transfer effect on the side of the interface FM|NM 2 where the spin \naccumulation is negative tends to enhance inhom ogeneity in the magnetization. Note that the \nlateral spin transfer torque is inherently  non-local because the ma gnetization everywhere \ncouples together through lateral spin diffusion.  \nIn symmetric systems, 1 and2 cancel each other and the late ral spin transfer torque has \nno net effect. Here we assume that the FM layer is sufficiently thin that the magnetization is \nuniform along the thickness direct ion. Making the thickness of NM 1 and NM 2 different; i.e., \nL1 ≠ L2, breaks the symmetry and removes this cancellation. The spin accumulation at the \ninterfaces NM 1|FM and FM|NM 2 can be found by solving the two second-order differential \nequations proposed by Valet and Fert [35]. It is st raightforward to use Valet-Fert theory in one \ndimension to show that asymmetric devi ces give asymmetric spin accumulation. \n2\nsfS\n2S2\nl z                                                           ( 4 )  11 \n 02e2\n\nz                                                            ( 5 )  \nwhere lsf is the spin diffusion length, e is the electrochemical potential for the electron \ndensity, and S is the spin chemical potential (that is  proportional to the spin accumulation \nnS through the Einstein relation    /2nDe , where  is the electri cal conductivity, D is \nthe diffusion constant, and n is the number density corres ponding to the spin accumulation). \nFigure 2 shows the profiles of S along the z-axis for symmetric (L 1 = L 2, Fig. 2(a)) and \nasymmetric (L 1 < L 2, Fig. 2(b)) structures. We use the boundary condition S = 0 at both \ninterfaces between the non-magnetic  layers and the reservoirs. This choice is motivated by \nthe idea that the reservoirs have an infinite density of states. That drives the spin accumulation to zero. Alternativel y, placing NM layers at the interfaces with a large spin-\norbit coupling, such as Pt or Pt -alloy, induces rapid spin-flip s cattering, which also drives the \nspin accumulation to zero. For a symmetric struct ure (Fig. 2(a)), the spin  accumulations at the \nleft and right interfaces of th e FM are of the same magnitude but with the opposite sign, \nwhereas for an asymmetric structure (Fig. 2(b) ), they are of different sign and magnitude. \nNote that Fig. 2(b) describes the ca se of charge current flowing from NM\n1 to NM 2, where the \nsum of the spin accumulations at the interf aces of FM|NM is negative. In this case, 2 \ndominates over 1; i.e., lateral spin transf er torque tends to increase any inhomogeneities in \nthe magnetization. Reversing the current polarit y reverses the spin accumulation so that 1 \ndominates over 2; i.e., lateral spin transfer torques suppress inhomogeneities. \n \n2.2. Previous studies on non-loca l spin transfer torque due to lateral spin diffusion \nBesides Ref. [27], several experimental [36- 38] and theoretical [28,39-41] studies have \nbeen performed to understand the lateral spin diffusion ef fect. Özyilmaz et al. [36] 12 \n experimentally observed current-induced excita tion of a single ferromagnetic layer. For an \nasymmetric Cu/Co/Cu nanopillar  structure, current-induced excitations were observed for \nonly one polarity of the current, where, according to the prediction [27], the lateral spin transfer torque should increase the magnetizati on inhomogeneity. In addition, they did not \nobserve such excitations in a symmetric struct ure, as expected from the discussion above. \nÖzyilmaz et al. [37] also repor ted experimental results indica ting that strong asymmetries in \nthe spin accumulation cause spin wave instabili ties in spin valve structures at high current \ndensities, similar to those observed for single magnetic layer junctions.  \nOne of us [28] theoretically extended the init ial calculation [27] of lateral spin transfer \ntorque to general situations to allow for varia tion of the magnetization in  the direction of the \ncurrent-flow. Such variation can give instabilit ies at a single interface, a possible explanation \nfor spin transfer effects seen in point contact experiments [38] . Brataas et al. [39] reported a \ntheoretical study on the mode dependence of cu rrent-induced magnetic excitations in spin \nvalves, and found agreement with the experimental  results of Ref. [37] . These calculations \n[27,28,39] are limited to the linear  regime. Even though they identif y the onset of instabilities, \nthey do not address the behavior of instabilities after the initial  nucleation. Adam et al. [40] \nperformed finite-amplitude self-c onsistent calculations of spin transport and magnetization \ndynamics for current-induced magnetic excitati ons of a thin ferromagnetic layer with \nasymmetric non-magnetic layers. Their work pr ovided an important proof-of-principle for \nlateral spin transfer torque, but  lacked the spatial resolution and sophist ication of full-scale \nmicromagnetic simulations. Hoefer et al. [41] performed a numerical study based on \nsemiclassical spin diffusion theory for a si ngle-layer nanocontact using a convolution \napproach to calculate the steady-state spin accumulation.  They found that directionally \ncontrollable collimated spin wave beams can be  excited by the interplay of the Oersted field 13 \n and the orientation of an applied field. Thes e self-consistent calcula tions [40,41] computed \nthe spin accumulation with either one-dimensi onal or two-dimensional steady-state solutions \nof the spin accumulation.  \nIn this section, we show numerical resu lts based on the three-dimensional dynamic \nsolutions of the spin accumula tion self-consistently coupled with the magnetization dynamics. \nSuch self-consistent treatments are essential to correctly describe the finite amplitude evolution of the spin wave modes excited by la teral spin transfer torque. They explain two \nimportant experimental results: spin wave instabi lities in a single FM [36] (Section 2.3.2) and \nlinewidths of precessional oscillations in spin -valves that are narrower than expected from \nlocal calculations of the magnetization dynamics [42]  (Section 2.3.3). \n \n2.3. Self-consistent calcula tion in layered structures \n 2.3.1. Modeling scheme \nWe self-consistently solve the equations of  motion of local magnetization (Eq. (6)) and \nspin accumulation n\nS (Eq. (7)) [27,28,39] \n \n2/zS,2/zS,\nFSFM\neff 0 FM\nF F tz tz tMt t  J Jmm H mm                   ( 6 )  \nsfS\next 0 S NM\n,,S,S\n\nnH n Jn\n zyx t                                  ( 7 )  \nwhere m is the unit vector of local magnetization,  is the gyromagnetic ratio, 0Heff is the \neffective field (including magnetostatic fiel ds, crystalline anisotropy, exchange, current-\ninduced Oersted fields, thermal fluc tuations, and external fields ( 0Hext)),  is the intrinsic \ndamping constant, MS is the saturation magnetization, tF is the thickness of ferromagnetic 14 \n layer, S S, n J D  is the spin (number) cu rrent density flowing in v direction \n( zyx,, ), D is the diffusion coefficient, sf = lsf2/D is the spin-flip scattering time, and lsf is \nthe spin diffusion length. The change  of charge and spin currents ( Je and JS) at the interface \nof FM|NM are related to the potentia l drop over the interfaces as [43,44] \n)/ () (/ ) ()2/(S e F e e G Ge G G tJ Δμm                            ( 8 )  \n\nm m mΔμ                       m Δμm J\n  \n \n)/ 2)(2/)( Re() ( ) ()2/( )2/(\nS2e S2\nF S\nt e GG G G Ge t\n  \n                      ( 9 )  \nwhere e is the electric potential, )02/( )02/(F F  t t    is the potential drop \nover the interface, Gs (s =  or ) is the spin-dependent interface conductance, and the last \nterm proportional to t/m  of Eq. (9) gives the spin- pumping contribution [44], which \ncouples the magnetization dynamics and the spin current. It is charact erized by the mixing \nconductance G. Generally, the mixing conductance has a real and an imaginary part, which \ncouple to the in-plane and out-of-plane terms in the dynamics respectiv ely. Although the out-\nof-plane spin transfer torque is important in magnetic tunnel j unctions [45-53], it is negligible \nin fully metallic multilayers [54,55]. Thus we neglect Im( G) and the associated out-of-plane \nspin transfer torque. At the interface FM|NM, Je and JSm are continuous under the condition \nof Sm =0 in the ferromagnet. S and m are related through Eqs. (7)-(9), and the spin-\nversion of Ohm ’s law with boundary conditions of e = -eV (0) and S = 0 (0) at the far-right \n(-left) end of the non-magnetic electrodes. We not e that the Eq. (9) is valid for a ferromagnet \nthinner than the exchange length but thicke r than the transverse penetration length. \nSince the spin accumulation in NM should be ta ken into account, the patterned part of Cu \nleads or spacer is also incl uded in the simulation. Thus, an additional boundary condition for \nthe spin accumulation is required at the side wall  of the nano-pillar. We assume that there is 15 \n no spin-current flow out of the system, i.e., 0 /n S r n , where rn is the surface normal \nvector at the side wall. All simulations repeat two alternating steps: (i) solve Eq. (6) with all \nboundary conditions to obtain a converged magnetiz ation configuration, and then (ii) solve \nEq. (7) to obtain the equilibrium spin accumulation configuration. These steps are repeated. The choice of boundary conditions at the side wa ll of the nanopillar gives different results \nthan the convolution method used in Ref. [41]. We s how that this difference is not important \nand discuss other differences between the two approaches in Appendix A. \n \n2.3.2. Single ferromagnet  \nIn this section, we show the main features  of current-induced single ferromagnetic layer \nexcitations, obtained from self-consistent  calculations. The la yer structure is Cu\n1 (10 nm) | Co \n(tCo nm) | Cu 2 (52 nm - tCo) where tCo varies from 2 nm to 8 nm. As explained above, \nasymmetric Cu leads provide asymmetric spin chemical potential S at each side of Co layer. \nThe average spin chemical potential μ at interfaces (= SCu1|Co + SCo|Cu2) is negative when \nthe electron flows from the thick to thin Cu layers, corresponding to a negative current. This \nnegative μ provides negative lateral spin transfer torques.  \nWe use the following geometric and magnetic parameters for the single ferromagnetic \nlayer of Co. We consider a nanopillar with  an elliptical shape of 60 nm × 30 nm, MS is 1420 \nkA/m, the exchange stiffness constant is 2×10-11 J/m, the gyromagne tic ratio of the \nferromagnet and non-magnet are 1.9×1011 T-1s-1 and 1.76×1011 T-1s-1 respectively, we assume \nthere is no anisotropy field,  is 0.01, the unit cell size is 3 nm, and the discretization \nthickness of the Cu layers varies depending on th e total Cu thickness and is not larger than 5 \nnm. Our results with these cell sizes are co nverged based on test calculations for a few \nconfigurations using smaller cells. The tran sport parameters for Cu, Co, Pt, and their 16 \n interfaces are summarized in Table 1. The non-local self-consistent calculation of the \ndynamics takes approximately 300 times  longer than a lo cal calculation. \nWe calculate magnetic excitations as a function of the out-of-plane field ( 0H = 0 T to 4.6 \nT) and current ( I = –15 mA to +15 mA) at 0 K. Initia l magnetic configurations are obtained \nwith applying the out-of-plane field for each case at zero current and zero temperature, and \nthen a current is applied. Figure 3(a) shows the time-averaged out-of-plane component of the \nmagnetization (= < Mz>/MS) as a function of the out-of-pl ane field and the current for tCo = 2 \nnm. For positive currents, the magnetization saturates along the out-of-plane direction when \nthe external field 0H exceeds the out-of-plane demagnetization field 0Hd (≈1.6 T). \nHowever, the magnetization does not saturate  at large negative cu rrents even though H is \nlarger than Hd, consistent with the data in Ref. [ 36]. The normalized modulus of the magnetic \nmoment (= | M|/MS) is smaller than 1 for those bias condi tions (Fig. 3(b)), indicating that the \nmagnetic state deviates considerably  from the single domain state.  \nFigure 4 shows snap shots of the magnetizat ion (Fig. 4(a)) and the spin accumulation \nprofiles (Fig. 4(b)-(d)) at 0H = 2 T and I = –5 mA ( tCo = 2 nm). The spin accumulation at the \nFM|NM interface approximately follows – M (Fig. 4(b)), whereas the spin accumulation \ninside of the Cu layer deviates significantly from the local magnetization pattern (Fig. 4(c) \nand 4(d)) because of spin diffusion. The effect  of spin diffusion on the spin accumulation is \nalso seen in Fig. 4(e). The out-of-plane compone nt of averaged spin chemical potential (= z) \nfollows Aexp(+ z/lsfCu)+Bexp(– z/lsfCu) where A and B are constant s, whereas the in-plane \ncomponent of averaged sp in chemical potential (= xy) rapidly decays with increasing the \ndistance from the interface because the spins are mixed during the diffusion process. The \ndecay constant in this case is determined by th e characteristic wave vector of the spatial 17 \n variation, i.e. ) 1/(22 2 2\nsf sf lk ll , where k is the wave vector characterizing the spatial \nvariation [28].  \nFigure 5(a) shows color plots of the microwav e power for various thicknesses of the Co \nlayer. The microwave power is obtained fr om the Fourier transformation of the time \nevolution of < Mz>/MS where < …> means spatial average. Th e microwave power is non-zero \nfor the bias conditions where | M|/MS is smaller than 1, indicati ng that the magnetizations are \nnot in stationary states at those bias conditions. The critical current IC for magnetic \nexcitations depends linearly on H (Fig. 5(a)). It is worthwhile comparing the IC values \nobtained from self-consistent calculation with those derived theoretically in the linear limit, which is given by [27,28,39] \n.2~2\nSd\n02\n1Co2\nS C \n\n\n  MJHHqSJtSMeI\nexex \n                                    ( 1 0 )  \nHere Jex is the spin stiffness, S is the area of free layer, 0Hd is the out-of-plane \ndemagnetization field, and ~ is the renormalized Gilbert damping constant,  \n\n   \n)( ) Re()(\n2) Re(~\n2\nCoS2\nCo\nqG GqG\netMG                                  ( 1 1 )  \nwhere q is the wave number of spin wave, and )(qG is given by \n. coth2)(2 2\nsf,Cu2 2\nsf,Cu q lL q l qGCu  \n\n                                ( 1 2 )  \nwhere ± reads the left and right (or top and bottom) Cu layer. \nS1 is the magnitude of the lateral spin transfer torque in dimensionless units, \n\n,)( ) Re()0()( )0( ) Re(\nsf,Cum1 \n     \nqG G GqG G\nlgGSCu                               ( 1 3 )  \nwhere gm is given by 18 \n .)/ tanh( 2 / ) (sf sf,Cu\nm\n  \n \n\nG GlL GG l G G\ngCu\n                            ( 1 4 )  \nIn Fig. 5(b), we compare the calculation results of the slope (= dIC/d0H) with those obtained \nfrom Eq. (10) for various q values. Here, we use the same spin transport parameters as those \nused in the self-consistent calculations to get the theoretical slopes. The simulation results are \nin reasonable agreement with analytic ones for q = /(60 nm). This good agreement is \nobtained only when the spin pumping term in Eq . (9) is included. Note that 60 nm is the \nlength of the device along the in-plane easy axis. It suggests that the wavelength of the lowest \nenergy spin wave mode is twice of the device  length, due to the ge ometry and the Oersted \nfield. However, the slopes from the simulations  and the analytic result s do not agree well with \nthose observed in the experiment (black solid sy mbols in Fig. 5(b)). This discrepancy may be \ndue to differences between the spin transport pa rameters used here and the true experimental \nvalues. \nOne aspect of the comparison between theory  and experiment that improves going from \nthe analytic model to the fu ll solution is the in tercept of the extr apolated boundary at I = 0. \nFrom Eq. (10), the theoretical intercept at I = 0 is the out-of-plane demagnetization field 0Hd. \nThe value of 0Hd slightly decreases from 1.6 T to 1.4 T as tCo increases from 2 nm to 8 nm, \ncaused by the change in the demagnetizing factor s depending on the geometry of FM junction. \nIn the experiment of Ref. [36], however, the intercept is found to be  much smaller than 0Hd. \nThe simulated results of the intercept are also considerably smaller than 0Hd, and the \nintercept decreases with increasing tCo, as shown in the inset of Fig.  5(b). Thus, the intercepts \nobtained from the self-consistent calculation ar e in better agreement with the experimental 19 \n observations than the theoretical ones. We attribute this better agreement to the fact that the \nself-consistent model more rea listically takes into account th e influence of the shape and \nfinite size of the nano-pillar on the spin  wave mode as we discuss below.  \nFigure 6(a) shows the time evolution of < Mz>/MS for various negative currents for 0H = \n2.5 T. The magnetization initially saturates al ong the out-of-plane direction because of the \nlarge out-of-plane field. When the current is turned on, a very small in-plane component of \nthe magnetization develops especially at the long  edges where the Oersted field is the largest. \nThe interplay between this la terally inhomogeneous magnetizati on and negative lateral spin \ntransfer torque excites spin waves,  resulting in a rapid decrease in < Mz> within a few \nnanoseconds.  \nTo understand spin wave mode excitation by late ral spin transfer torques, we perform an \neigenmode analysis for the magnetization dynamics  (Fig. 6(b)-(d)). To calculate eigenmodes, \nwe choose the bias condition of I = –11 mA and 0H = 2.5 T, which shows a periodic \noscillation of < Mz>. We note that such periodic oscillat ions are observed onl y for some bias \nconditions and the magnetic excita tion is highly nonlinear in genera l. The spectral density of \n<Mz> shows two peaks at two frequencies, fL (≈ 75 GHz) and 2 fL (≈ 150 GHz) (Fig. 6(b)), \nwhere fL satisfies fL = Co 0 H/2. On the other hand, for a singl e domain state, we expect the \nprecession frequency to be Co 0 (H–Hd<Mz>/MS)/2 because the effective magnetic field \nexperienced by the magnetizations is the summa tion of the external field and the internal \ndemagnetization field. At I = –11 mA, < Mz>/MS is about 0.6 (Fig. 6(a)); in this approximation, \nthe precession frequency would be 46 GHz, which is much smaller than the obtained precession frequency f\nL. This disagreement indicates that the precession frequency is mostly 20 \n determined by the external out-of-plane field 0H, and that contributions from 0Hd are \nnegligible. An eigenmode analysis of the spatial patterns (Fig. 6(c) and 6(d)) for the two peak \nfrequencies gives some insight in to this peculiar field dependence. The eigenmode images are \nobtained from local power spectrum Sz(r, f) [57] \n.) 2 exp(),( ),(2\n\njj j z z fti t M f S  r r                                       ( 1 5 )  \nThe precession region with a high er power is locali zed at the edges. These eigenmodes are \nunique features originating from lateral spin transfer torque and not expected for the field-driven excitation [57]. Figures 6(e)-6(h) show  the time evolution of the magnetic domain \npatterns at the same bias condition. The magnetiza tion near the edges is mostly in the plane, \nbut near the center of the cell, it is in vorte x-like states. The peculi ar frequency dependence \non the field could be explained by the form ation of vortex-like states where the \ndemagnetization field along the thickness direction is significantly reduced.   \n2.3.3. Spin valves \nIn this section, we apply the self-consistent non-local model to a spin-valve structure with \ntwo ferromagnetic layers experi mentally studied by Sankey et al. [42], Cu(80) | Py(20) | \nCu(6) | Py(2) | Cu(2) | Pt (Py = Permalloy), w ith all thicknesses in nm. They found that the \nresonances excited by current have narrower lin ewidths at low temperatures than expected \nfrom a finite temperature macrospin calculation.  This reduced linewidth indicates that some \nadditional effect can improve the cohere nce time of precession in nanomagnets.  \nWe use the same parameters for Cu as used  in the previous section and replace the \nparameters for Co by parameters for Py provid ed by the Cornell group. The pillar has an \nelliptical shape w ith 120 nm × 60 nm, M\nS is 645 kA/m [42], the excha nge stiffness constant is 21 \n 1.3×10-11 J/m,  is 0.025 [10], the unit cell size is 5 nm . The transport parameters for Py and \nPy|Cu are summarized in Table 1. We assume  the magnetization of the pinned layer (Py 20 \nnm) is fixed along the in-plane easy axis and that it gives no stray field. While the pinned \nlayer is likely not to be fixed in  reality, we keep it fixed to focus on the effect of lateral \ndiffusion. For finite temperature simulati ons, we add the Gaussian-distributed random \nfluctuation field [58] (mean = 0, standard deviation = 2 kBT/(MSVt), where t is the \nintegration time step, V is the volume of unit cell) to the effective field for magnetization. We \ntest convergence of the stochastic calculations  and find that the results are converged for t \nbelow 50 fs based on the average magnetization al ong the easy axis. For st ochastic simulation, \none may require temperature- and cell-size-depe ndent renormalization of  parameters in order \nto take into account effect of magnons having  a shorter wavelength than the unit cell size \nemployed in simulations. Several ways to renormalize the exchange constant and the \nsaturation magnetization have been proposed [59, 60]. However, we are not aware of any way \nto renormalize the damping constant and the spin  transfer torques. These parameters are of \ncritical importance for the calcu lation of current-induced magneti c excitations. In this work, \nwe do not consider temperature- and cell-size- dependent renormalization of parameters. We \nalso neglect any temperature dependence of the transport parameters.  \nTo investigate whether or not the reduced line width originates from lateral spin transfer \ntorques, we perform numerical simulations based on three different approaches: i) a \nmacrospin model (MACRO), ii) a conventiona l micromagnetic model without considering \nlateral spin transfer torque (CONV), and iii) a non-local, self-c onsistent model (SELF). Fig. 7 \nshows contours of sp ectral density of < MX> as a function of the current at the temperature T= \n4 K when a field of 50 mT is app lied along the in-plane easy axis (// x). Positive current \ncorresponds to the electron-flow from Cu(2) to Cu(6), and thus positive lateral spin transfer 22 \n torque. The macrospin simulations show the familiar red- and blue-shift depending on the \nbias current I (Fig. 7(a)). The conventional simulations  show only a red-shift up to a critical \ncurrent ( ICCONV ≈ 2 mA, Fig. 7(b)). Here, the magne tization dynamics becomes complicated \ndue to excitation of incoherent spin-waves when I > ICCONV. As indicated by an arrow, \nsecondary peaks are observed at about half of the frequency of main peaks, corresponding to \nthe precession of end domains [31]. In the non- local, self-consistent simulations, similar \nsecondary peaks are observed, indicating devia tions from a single domain state, but peak \nstructures are much clearer than  they are in the c onventional simulations up to about 2.4 mA, \nwhich is larger than ICCONV (Fig. 7(c)). The blue-shift followed by a transition region is also \nobserved. It indicates that positive lateral spin  transfer torques improve the coherence of the \nmagnetization dynamics.  \nFigure 8(a) shows the power spectra computed in the three models ( I = 1.4 mA and T = 10 \nK). It is evident that at low temperature, th e non-local, self-consistent simulations give the \nnarrowest linewidth. We calculate the temperature ( T) dependence of linewidth from \nLorentzian fits (Fig. 8(b )). At low temperatures ( T < 50 K), the non-local, self-consistent \nsimulations provide narrower linewidths than  other two approaches, consistent with \nexperimental observations [42]. However, we ob serve that the linewidths computed from the \nmacrospin simulation are wider than those co mputed from the conventional micromagnetic \nsimulation. This counterintuitive result may be due to the fact that the li newidth is affected by \nthe precession angle [8]. By estimating the pr ecession angle of micromagnetic results from \nthe spatial average of magnetization com ponent, we find that the macrospin and \nmicromagnetic models give different precessi on angles whereas two micromagnetic models \ngive similar angles at the bias current. Because  of these limitations, direct comparisons of the \nlinewidths between the macrospin and micromagne tic simulations may be limited. Below, we 23 \n discuss effect of the self-consistent feedback on the linewidth by comparing the two \nmicromagnetic modeling approaches; this comp arison would be relatively free from the \nabove-mentioned limitations and shows that the feedback reduces the linewidth.  \nFrom Fig. 8(b), we find that the non-local,  self-consistent mode l gives a narrower \nlinewidth than the conventional micromagnetic model for T < 50 K. It suggests that the \ncoupling among local magnetizations induced by pos itive lateral sp in transfer torque indeed \nresults in a substantial improvement of the cohe rence time of precession at a low temperature. \nFor T > 50 K, however, the linewidth in the non- local self-consistent simulation increases \nmore rapidly than in the conve ntional micromagnetic simulation.  We note that this does not \nmean that the positive lateral spin torque make s the linewidth very broad at high temperatures. \nAs shown in Fig. 8(c), the more rapid incr ease in the linewidth in the non-local self-\nconsistent simulations originates  from a mode splitting. We find that power spectra calculated \nfrom the non-local self-consistent simulations c onsist of two peaks; a narrow main peak at a \nhigher frequency indicated by up-arrows, and a s econdary broad peak at a lower frequency. \nThe frequency of the secondary peak does not ch ange much with temperature, whereas that \nof the main peak increases sli ghtly with temperature. This ki nd of mode splitting has been \nobserved in experiments [61] and numerical studies based on  a conventional micromagnetic \nmodel with no lateral spin tor que [62,63]. Because of this mode splitting, the linewidth \nobtained from the fit using a single Lorentzian function increases rapidly with temperature.  \nIn the low-temperature limit, two nonlinear effects of the positive lateral spin transfer \ntorques may cause the narrower linewidths in the non-local, self-consis tent simulations: an \nincrease of the effective exchange stiffness at  short range and an increase of the damping of \nincoherent spin-waves at long range. As a result , positive lateral spin tr ansfer torques provide \nan additional nonlinearity to the spin-wave da mping. For spin-torque nano-oscillators, the 24 \n linewidth  in the low-temperature limit (i.e. T < 10 K in our case) is given by [64,65] \n   \n\n\n\n\n\n\n\n\n\n\n\n2\neff 0B\n0 1 )(N\nETkP                                           ( 1 6 )  \nwhere ) 1( )( QP P  is the positive damping of the oscillator,  is the equilibrium \nlinewidth in the passive region, Q is a phenomenological coefficient characterizing the \nnonlinearity of the positive damping, P is the normalized power, kBT is the thermal \nenergy,  ) ( /) (C 0 C S 00 0 QII IIMV E       is the average energy of the stable auto-\noscillation, 0 is the ferromagnetic resonance frequency, IC is the critical current for the \nmagnetic excitation, dPPdN /)(  is the nonlinear frequency shift coefficient obtained \nfrom NP P0 )( , ) (C eff QIII  is the effective nonlinear damping, I is the bias \ncurrent, 0 B2/eMV gI ,  is the spin-polarization efficiency, V0 is the volume, \n) /()1(0 Q P   is the equilibrium oscillation power, and C/II  is the \nsupercriticality. \nEquation (16) predicts two important consequences of the nonlinearity. First, the \nlinewidth of an auto-oscillator w ith a nonlinear frequency shift (i.e. 0N ) increases by the \nfactor (1+( N/eff)2) from that of a linear oscillator (i.e. N = 0). Second, the linewidth of a \nnonlinear oscillator decreases w ith increasing n onlinear damping Q when N is large. The \nlinewidth is determined by nonlinear properties of the system where the normal linear damping is compensated by local spin transfer torques. In this case, an increase of the nonlinear damping can lead to a decrease of the linewidth, known as noise suppression due to \nnonlinear feedback [66,67] which has been widely  observed in various fields such as optics \n[68], mechanics [69], and biology [70]. While this non linear feedback typi cally requires an \nexternal feedback element, in spin-valves it is inherent. 25 \n Figure 8(d) shows that N is evidently nonzero and almo st identical in both the \nconventional micromagnetic simulations and the non-local, self-consistent simulations. Thus, \nin both approaches, the linewidth increases from that expected for a linear oscillator. Using \nequation (16), we fitted the values of Q from the calculated linewidths at T = 10 K and \nobtained Q = 0.12 in the conventional micromagnetic simulations and Q = 1.96 in the non-\nlocal, self-consistent simulations. The fitted value Q in the latter is consistent with the \nassumed  values ( Q = 1 to 3) in the Ref. [65,71] to explain experimental observations. It \nshould be noted that in Ref. [65,71], the large Q is purely phenomenological. Our non-local \nself-consistent treatment suggests that the large Q may be caused by the lateral spin diffusion. \nThus, the nonlinear spin-wave damping due to lateral spin transfer torque is probably \nresponsible for narrower linewidt hs in the non-local, self-c onsistent simulations at low \ntemperatures. For the opposite pola rity of the current (i.e. ne gative lateral spin transfer \ntorque), we observe an increase of the linewidth (not shown) as would be expected for the \ncase when lateral spin diffusion enhances inhomogeneity.  \n2.4. Summary \nTo summarize this section, we report non-local, se lf-consistent calculations for current-\ninduced excitation of a single ferromagnetic la yer and spin valves. The former are in good \nagreement with previous theoretical [27,39] a nd experimental studies [36]. They provide an \nimproved understanding of the coupled dynamics between magnetizations and spin transport, \nand the excitation of spin wave modes for negative  lateral spin transfer torques. In case of a \nsingle ferromagnetic layer, only a negative net lateral spin transfer torques lead to spin wave instabilities, while positive net lateral spin tran sfer torques do not. In spin valve structures, \nself-consistent calculations are crucial for corr ect evaluation of the os cillation linewidth. 26 \n Whereas the conventional spin transfer torque a nd its interplay with th e Oersted field tend to \ncause a large amplitude incoherent spin wave excitation [29-32], the positive lateral spin \ntransfer torque effect captured by the self -consistent calculation tends to reduce spatial \ninhomogeneities (suppress spin waves) and lead s to more coherent magnetization dynamics \nat low temperatures. This effect would be bene ficial for microwave oscillators utilizing spin \ntransfer torque, where a narrow linewidth is a key requirement.  \nLateral spin transfer torques are non-zero when the following three conditions are \nsatisfied: (i) the spin accumulation at the tw o interfaces of a ferromagnetic layer are \nasymmetric, (ii) at least one of  neighboring layers is diffusive,  and (iii) the magnetization is \ninhomogeneous. Condition (i) is generally satisf ied in multilayer structures (= NM | FM \n(pinned) | NM | FM (f ree) | NM) since there is a pinned ferromagnet on one side of the free \nferromagnet whereas there is no fe rromagnet on the other side. Condi tion (ii) is also generally \nsatisfied for fully metallic multilayers and ev en for magnetic tunnel ju nctions. In a typical \nmagnetic tunnel junction, the free ferromagnet is  sandwiched by an insulator and a diffusive \nnon-magnet (capping layer). The lateral spin diff usion is allowed only in  the capping layer, \nwhich maximizes the net lateral spin  transfer torque because the la teral spin transfer torque at \nthe other interface is essentiall y zero. Finally, condition (iii) is almost always satisfied \nbecause the current-induced Oersted field is  inhomogeneous and leads to inhomogeneous \nmagnetizations in all but the strongest saturating fields [31]. Furthermore, thermal \nfluctuations of the magnetization are spatially inhomogeneous. Therefore, lateral spin transfer \ntorques are usually non-zero.  \nFinally, we briefly comment on another type of non-local spin transfer torque in \nmultilayers. Let us consider a spin valve containing three FMs; i.e., FM 1 | NM 1 | FM 2 | NM 2 | \nFM 3, where FM 1 is pinned (= pinned layer) and other tw o FMs serve as a synthetic free layer. 27 \n Such multilayer structures with a synthetic free layer are of considerable interest for MRAM \napplications [72-77] and spin tran sfer torque-oscillators  [78,79]. In this structure, not only are \nthere spin transfer torques at the NM 1|FM 2 interface, but al so at the FM 2|NM 2 and NM 2|FM 3 \ninterfaces (Fig. 9). Furthermore, the spin tran sfer torques at each interface depend on the \norientation of both of the other magnetizations (Fig. 9(c)), because lo cal spin accumulations \nat each interface are vertically coupled through the whole layer structure. In this kind of \nstructure, the spin transfer to rque is non-local even  without the lateral spin diffusion, and \nrequires a self-consistent calculation to investigate current-induced magnetic excitation [80-82]. 28 \n 3. Non-local spin transfer tor que for a magnetic domain wall \n \n3.1. Current-induced motion of a domain wall \n   A magnetic domain wall is the transition region between two magnetic domains in which \nthe magnetization continuously varies. The inte rplay between the magnetic exchange on one \nhand and the crystalline anisotropy and the magn etostatic interactions on the other hand gives \na finite width to the wall. An electrical current passin g through a domain wall in a \nferromagnetic nanowire can move the wall, because the current creates a spin transfer torque. \nCurrent-induced domain wall motion has been in tensively studied both theoretically and \nexperimentally. Understanding this motion require s understanding the coupling between \nconduction electron spins and the continuously varying magnetizati on. It may also find use in \nstorage and logic devices in which the domain wall is used as the information unit (for comprehensive reviews about current-induced  domain wall motion based on local spin \ntransfer torque, please see Refs. [83-87]). \n \n3.2. Non-local spin transfer to rque for a narrow domain wall \nOne of the central issues for current-induced domain wall motion is how to reduce the \nthreshold current density to move the domain wa ll. The reason is twofold. A typical threshold \ncurrent density for a metal lic ferromagnet is about 10\n12 A/m2 [11,88]. Such high current \ndensities cause significant Joule heating, making it difficult to distinguish spin transfer effects \nfrom heating effects [89-93]. From an applicatio n point of view, devices need to operate with \ncurrent densities lower than this threshold cu rrent density to minimize electromigration. For \nthis reason, there has been substantial research  directed toward reducing the threshold current \ndensity. Several solutions have been proposed. On e approach is to use resonant dynamics of 29 \n domain wall motion by controlling current pulse widths [94] or injecting consecutive current \npulses [95]. Another approach is  to reduce the hard-axis anisot ropy that the spin transfer \ntorque must overcome to move a domain wall. Such reductions can be achieved by shaping \nnanowire geometries properly since the hard -axis anisotropy is caused by geometry-\ndependent demagnetizing effects, as predicted th eoretically [96] and ve rified experimentally \n[97].  \nYet another approach is to increase the nonadi abatic spin transfer torque, which controls \nthe wall motion for small currents in idea l nanowires. When electrons flow through a \nspatially slowly varying magnetizat ion configuration, their moment s tend to stay aligned with \nthe magnetization. Since this requires the moments to rotate, there must be a reaction torque on whatever is causing them to rotate, i.e. the magnetization. The reactio n torque has the form \nof the first term in Eq. (3) [1,2] and is referred to as the adiabatic spin transfer torque because \nit comes from the spins “adiabatically” following th e magnetization. The other term in Eq. (3), \nis perpendicular to the adiabatic  spin transfer torque and is referred to as the nonadiabatic \nspin transfer torque even though some contributions to it occur in  the adiabatic limit. Without \nthe nonadiabatic torque, the adiabatic torque in  combination with the other terms in the LLG \nequation leads to intrinsic pinning for curre nts below a threshold [23]. Intrinsic pinning \nhappens because the wall distorts as it moves and the distortion leads to torques that oppose \nthe motion. The nonadiabatic spin transfer torq ue acts like a magnetic field for domain walls \nand thus makes the threshold current density vanish for an ideal nanowire. The larger the nonadiabatic torque, the faster the do main wall motion for small currents. \nThe importance of the nonadiabaticity for cu rrent-induced domain wall motion, has led to \na number of theoretical [23-25,98-106] and expe rimental studies [94,107-112] to determine \nthe nonadiabatic spin torque parameter\n. Several mechanisms for the nonadiabatic spin 30 \n transfer torque have been proposed. One cla ss of mechanisms is based on the changes in \nprocesses that contribute to magnetic damping change in the presence of current. These \nchanges typically have the form of the nonadi abatic torque. For example, a phenomenological \ntreatment of the scattering of itinerant electrons  by spin-dependent impurities generates both \ndamping and a nonadiabatic spin transfer torque  in the presence of cu rrent [24]. Similarly, \nband structures with spin-orbit coupling and electron scattering give both damping [113] and \nnonadiabatic torques [103], both of which can be  calculated from firs t principles [104,105]. \nThe nonadiabatic torque due to these mechanis ms does not depend on the domain wall width. \nFor domain walls much wider than the characteri stic length scales of spin transport, these \nmechanisms are the only ones that make the spin current deviate from the magnetization \ndirection and give a non-adiabat ic spin transfer torque.  \nAdditional mechanisms become more signif icant as domain walls get narrower. For \nmoderately narrow domain walls (width ≈ 5 nm to 10 nm), spin diffusion can increase the \neffective nonadibaticity [114,115]. For narrower domain walls (width < 5 nm), the conduction \nelectron spins traversing the domain wall canno t follow a sharp change in the magnetization \nand thus contribute to the nona diabaticity [100,102]; i.e., ballis tic spin-mistracking. When the \ndomain wall is extremely narrow (i.e., one or two atomic layers), momentum transfer can \noccur due to the reflection of electron spin s from the domain wall [23]. This class of \nmechanisms (spin diffusion, spin mistracki ng, and momentum transfer) generally gives non-\nlocal spin transfer torques and their c ontributions depend on the domain wall width. \nInitial experiments for current-induced domain wall motion in metallic systems have used \nNi80Fe20 (Permalloy) for which domain wall widths are large ( ≈ 100 nm). The theoretical \npredictions for the enhanced nonadiabatic ity by reducing the domain wall width have \nencouraged experimentalists to  study systems with smaller do main wall widths by utilizing 31 \n materials with strong perpendicu lar anisotropy [110,112]. For na rrow domain walls, the role \nof non-local spin transfer  torque on the domain wall motion may be important.  \n \n3.3. Previous studies on self-c onsistent calculation for current-induced domain wall motion \nManchon et al. [114] theoretically predicted that  the spin diffusion generates an additional \nspin transfer torque that effectively enhances  the nonadiabatic torque. This new torque is \ninversely proportional to the square of the domain wall width and strongly depends on the \ndomain wall structure. For instance, it can increase the transverse velocity of vortex cores in vortex domain walls, whereas its influence rema ins negligible for transverse domain walls. \nThis dependence on the domain wall structure aris es from the fact that the spin diffusion \ncurrent transverse to the electron-flow direction is significant for a vortex wall but negligible for a transverse wall. Recently, Claudio-Gonzalez  et al. reported numerical results based on a \nself-consistent calculation of the drift-diff usion model and the LLG  equation [115]. They \nfound that an increase in the effective nonadi abaticity for a vortex wall but only minimal \nchanges for a transverse wall, consistent with  the theoretical prediction of Ref. [114]. \nFor Bloch or Nèel walls formed in perpe ndicularly magnetized nanowires, this spin \ndiffusion torque does not enhan ce the effective nonadiabaticity because the wall is a simple \none-dimensional domain wall in contrast to vo rtex walls. Then, unless the domain wall is \nextremely narrow, ballistic spin-mistracking wi ll be the important mechanism for changing \nthe nonadiabatic torque. Ohe et al. [116] perfor med self-consistent calculations to investigate \nthis effect based on a lattice model [117] where the conduc tion electrons are treated quantum \nmechanically and thus spin mixing in the states  of the conduction electrons is fully taken into \naccount. They found that when the Fermi energy of the electrons is larger than the exchange \nenergy (i.e., a typical situation for transiti on metals), spin precession induces spin-wave 32 \n excitations in the local magnetization. This sp in-wave excitation contributes to the domain \nwall displacement at low current densities but reduces the domain wall velocity for large \ncurrent densities as compared  to the adiabatic limit.  \nHere, we present self-consistent calculations of the non-local spin transfer torque based \non a semiclassical, free electron approach. Ou r approach differs from the previous self-\nconsistent calculation [116] in two aspects. One difference is the determination of which \nelectron states are occupied. In a Landauer pictur e, the Fermi levels of the leads are fixed and \ndifferent. The Fermi level of the material between the leads adjusts in response to the applied voltage to create local charge neutrality. This ad justment leads to current flow that is half \nexcess electrons moving forward a nd half a deficit of electrons moving backwards. Ref. [116] \nintroduced extra right-propagating electrons in the energy range E\nF < E < EF + eV where V is \nthe voltage drop across the nanowire (Fig. 10(a)). Since electron s were added to the \nequilibrium Fermi sea, charge neutrality was vi olated in their calculation. In contrast, we \ninduce extra right-propagating electrons in the energy range EF < E < EF + eV/2 and remove \nleft-propagating electrons in the energy range EF – eV/2 < E < EF (Fig. 10(b)), so that charge \nneutrality is preserved. The difference in occu pancy results in an important difference in the \nspatial distribution of non-local spin transfer  torque between Ref. [116] and our work. \nFigures 10(c) and (d) show the spatial distribution of spin transfer torque STN obtained from \nthe two approaches. Here, the spin transfer torque STN is separated into two vector \ncomponents, Nonadia\nSTAdia\nST ST N N N  , where Adia\nSTN  and Nonadia\nSTN  are aligned along xm/  \nand xm/ m , respectively. In Ref. [116] the osci llatory non-local spin  transfer torque \nappears at only one side of the domain wall, whereas in our work it ap pears at both sides of \nthe domain wall (see Fig. 10(c) and (d)). An additional difference between the calculations is that Ref. [116] assumes one-dimensional meso scopic transport by cons idering only a single-33 \n electron channel ( k-normal, k // x), whereas we treat the non-equilibrium spins over the full \nthree-dimensional Fermi surface. Treating the full Fermi surface generates spin dephasing \nbecause of the variation of the precession length over the Fe rmi surface. Figure 10(c) shows \nthat for a spin transfer torque calculation w ith a single-electron channel of Ref. [116], the \nnon-local oscillation of spin tran sfer torque is very significa nt and does not decay even far \nfrom the domain wall. In contrast, the oscillatio n is suppressed at large distances from the \nwall in our approach due to the st rong spin dephasing (Fig. 10(d)).  \n \n3.4. Semiclassical approach  \nHere we use a semiclassical approach proposed  by one of us [100], which is based on two \nmain approximations; i.e., ballistic transport and a parabolic band structure. With these approximations, we show that mistracking to rques can make important contributions to \ndomain wall dynamics. For all but extreme cases , these contributions can be captured through \neffective values of local parameters. This si mple model maximizes the importance of the non-\nlocal effects, but since the effects can be larg ely be accounted for by a local approximation, \nour use of the “best case” is appropriate. We expect a local parameterization to be even more appropriate when scattering and realistic band structures are taken into account. \nBefore explaining the model details, let us disc uss the relevance of this simple model. We \nexpect the ballistic limit to be appropriate for materials with very short domain wall widths \n(about 1 nm), which are shorter than the mean free path. A bal listic transport picture becomes \nless appropriate when domain wall widths are greater than mean free paths and precession \nlengths. In that case, we expect  that scattering will  reduce the non- local effects obtained from \na ballistic transport model. We also expect that non-local effect s will be weaker for realistic \nband structures than for parabolic band structur es because dephasing is stronger for realistic 34 \n band structures. Thus, we expect  that the results for a parabo lic band structure set an upper \nlimit for the importance of non-local effects. We show below that in most cases we consider, \nthe non-local effects can be accounted for by suitably renormalized local parameters. We \nexpect that conclusion to be even stronger for more realistic band structures in domain walls \nin which scattering is important. The Hamiltonian is  \n),(2ex22\nxmH Bσ                                               ( 1 7 )  \nwhere ),,(z y xσ  is a vector composed of the three Pauli matrices and Bg . \n)(exxB  is aligned along the local magnetization ev erywhere and describes the magnetic field \nexperienced by a conduction electr on spin through the s-d exchange coupling. Its magnitude \nis defined as \n./ 22\nB2\nex ex mk E  B                                                ( 1 8 )  \nThe Fermi wave vectors for up and down spins, \nFk and \nFk are given by \n,2\nB2\nF F k k k                                                       ( 1 9 )  \nwhere the Fermi energy is m k E 2/2\nF2\nF . The spatial evolution of the single-particle spin \ndensity ) ,(xkxs  for a given energy E is obtained from \n),(ˆ),(),(\nex2\nBx kxkk\ndxkxd\nxxB ss                                     ( 2 0 )  \nwhere 2/) ( kk k , and k and k for ) ,(xkxs  are defined by \n2\nB2 2/ 2 k mE k                                                         ( 2 1 )  \nwith m kk Ex 2/) (2\nB2 2 , and k and k for ) ,(xkxs  are defined by Eq. (21) with \nm k k Ex 2/) (2\nB2 2 . The semiclassical single-electron sp in-current density is then obtained \nfrom 35 \n . ),( ),(mkkx kxx\nx x\n s J                                                 ( 2 2 )  \nOne finds the spin current density )(xJ  by integrating ) ,(xkxJ  over the Fermi surface \nin the presence of an electric field xˆE, \n ,)2(),( ˆ )(33\n kJk x k Jdkx feEf xx    \n\n\n                            ( 2 3 )  \nwhere the spin-dependent Fermi-Dirac distribution function k k \n Fk f  implies that \nthe distribution of electrons outside the region of inhomogene ous magnetization are \ncharacteristic of the zero-temperature bulk [100]. Then, )( )( )( x x x  J J J  is the total \nspin current density, and the spin  transfer torque is given by \n.)()(STxxxJN                                                      ( 2 4 )  \nWe plug this semiclassical calculation of spin transfer torque in to the LLG equation, Eq. \n(1). At every integration time step, we compute the semiclassical calcul ation of the non-local \nspin transfer torque for a given magnetizat ion profile, and then update the magnetization \ndynamics using the spin tr ansfer torque for the ne xt time step. This pro cedure is repeated and \nas a result, the effect of spin transfer to rques on the magnetization dynamics and subsequent \nfeedback are taken into account self-consistently.  \nSeveral remarks on the computation are in orde r. First, the length of the nanowire treated \nin the calculation should be mu ch longer than the domain wa ll width. If not, unphysical \nnonequilibrium spin density can arise from disc ontinuities at the edges. Second, multiscale \nmodeling is important to reduce the computati on time. In this work, the unit cell size for \ncalculating the LLG equation is more than 10 times larger than for calculating the \nsemiclassical spin transport equation. The smaller cell size for the spin transport calculation is essential to ensure a convergence when solving Eq. (20).  36 \n  \n3.5. Current-induced domain wall moti on by non-local spin transfer torque \nA qualitative explanation for the origin of the non- local spin transfer torque is as follows. \nWhen the domain wall is sufficiently wide compared to the precession period of the spin density determined by k\nF and kB, the precession amplitude of the spin density is small and \naveraged out when integrated over the Fermi su rface. As a result, the lo cal spin direction of \nspin current is almost perfectly aligned along th e local magnetization direction, so that spin \ntransfer torque can be locally defined by the gradient of the lo cal magnetization. In contrast, \nwhen the domain wall is narrow and its widt h is comparable to the precession period, the \nprecession amplitude is considerable even at points far from the domain wall and the spin \ntransfer torque becomes non-local.  \nIn this work, we carry out micromagnetic  simulations for a semi-one dimensional \nnanowire (i.e., the nanowire is discretized along the length dire ction, but not along the width \nor the thickness direction), se lf-consistently coupled  with a semiclassical spin transport \ncalculation. We assume a perpendicularl y magnetized nanowire with the following \nparameters: the Fermi energy EF = 10 eV , the exchange splitting EB = 1 eV , the exchange \nconstant Aex = 110-11 J/m, the saturation magnetization MS = 1300 kA/m, the Gilbert \ndamping constant  = 0.03, the nanowire width = 200 nm , and the nanowire thickness = 4 nm. \nThe perpendicular crystalline anisotropy constant Ku is varied from 2 106 J/m3 to 1107 J/m3 \nin order to vary the domain wall width DW. The local nonadiabaticity ( ) caused by the spin \nrelaxation is assumed to be zero in order to fo cus on the non-local spin transfer torque caused \nby the ballistic spin-mistracking.  \nFigures 11(a) and (b) show three vector components of spin transfer torque for Ku = 2106 \nJ/m3 (DW ≈ 2.71 nm) and Ku = 107 J/m3 (DW ≈ 0.98 nm), respectively. The non-locality of 37 \n the spin transfer torque beco mes more pronounced for a smaller DW; i.e., the amplitude of \noscillatory spin transfer torque is larger, a nd the non-zero spin transfer torque is observed \nfurther away from the domain wall. Figures 11(c) and (d) show Adia\nSTN  and Nonadia\nSTN  for \nvarious DW values, respectively. Two observations are worth noting. Fi rst, the non-local \nnonadiabatic contribution of spin transfer torque (Fig. 11 (d)) becomes more  significant as  \nDW gets smaller. Second, both Adia\nSTN  and Nonadia\nSTN  are non-local (Fig. 11(c)). \nFigure 12 shows the do main wall velocity vDW as a function of the spin current velocity u0, \nobtained from the self-consistent  calculation. We did not obse rve any significant spin wave \nexcitations, in contrast to Ref. [116]. We attribut e this difference to the fact that the non-local \nspin transfer torque is not as significant as in Ref. [116] due to the strong spin dephasing (see \nFig. 10). Overall trends of vDW are similar to those expected from the local approximation \nwith nonzero local nonadiabaticity  [24,25]. When the spin current velocity u0 (proportional \nto the current density) is small, vDW is linearly proportional to u0, and the slope vDW/u0 in the \nlinear range increases with decreasing DW. When u0 exceeds a certain threshold ( uWB, \nindicated by down arro ws in Fig. 12), vDW deviates from the linear dependence. The threshold \nuWB corresponds to the Walker breakdown [83,118], above whic h the domain wall undergoes \na precessional motion. These overall trends of vDW indicate that the non-local spin transfer \ntorque indeed acts like an additional local  nonadiabatic spin transfer torque.  \nWe can understand the similarity of the doma in wall motion from a collective coordinates \napproach to analyze the calculation results obtained from the self-consistent model. \nFollowing Thiele’s work [119], we assume the domain wall structure is \n) cos, sin cos, sin (sin m  where sin  = sech[( x-X(t))/DW], cos = tanh[( x-\nX(t))/DW], and  = (t). Here, X is the domain wall position,  is the domain wall tilt angle, 38 \n DW is the domain wall width, and t is time. After some algebra,  one obtains the equations of \nmotion of the collect ive coordinates ( X, ) in the rigid domain wall limit (i.e., ∂DW/∂t = 0), \nDWJ\nDW~\n  c\ntX\nt                                                  ( 2 5 )  \n 2sin~1\nSd\nDWJ\nDW MK b\nt tX                                    ( 2 6 )  \n\n \n\n x\nx Xdx cmNmNonadia\nSTDW\nJ2~                                     ( 2 7 )  \n\n \n\n x\nxdx bmNmAdia\nST J21~                                           ( 2 8 )  \nwhere Kd is the hard-axis anisotropy of domain wall . In the local approximation, one can \nrecover 0 J~ub and 0 J~u c  using x u /0Adia\nST m N  and ) / (0Nonadia\nST x u  m m N  . \nIn our case, however, J~b and J~c can be obtained by integra ting Eqs. (27) and (28) \nnumerically, because of th e non-local nature of both Adia\nSTN  and Nonadia\nSTN . We define eff \n( )/~\n0 Jub  and eff (0 J/~uc ) that effectively describe the average adiabaticity ( ≈ effective \nspin polarization) and nonadiaba ticity of non-local spin transf er torque, respectively. The \ndependence of eff and eff on DW are summarized in Fig. 13. eff is close to 1 for a \nlarge DW and decreases with decreasing DW. In contrast, eff is close to 0 for a large DW \nand increases with decreasing DW. The changes in eff are much more significant than \nthose in eff. Given the uncertainty in the proper parameters to describe these systems, it is \nlikely that change in eff will be much more difficult to observe than those in eff. \nBased on Eqs. (25) to (28), one can define  several important phys ical quantities of \ndomain wall dynamics (see Appendix B for deta ils). The threshold spin current velocity uWB 39 \n for the Walker breakdown, the domain wall velocity vsteady for u0 < uWB, and the average \ndomain wall velocity v for u0 >> uWB are given by  \n,\neff eff SDWd\nWB\nMKu                                              ( 2 9 )  \n,0eff\nsteady u v                                                         ( 3 0 )  \n.10 2eff effu v\n                                                      ( 3 1 )  \nIn Fig. 14, we show how well this local approximation for eff and eff shown in Fig. \n13 can describe the self-consiste nt calculation results shown in Fig. 12. When they agree, \nthere is no need for the full self-consiste nt solution. Instead, one can calculate eff and eff \nbased on the semiclassical calculation in Eqs.  (27) and (28), and use them in the LLG \nequation with the local approximation for spin transfer torque. When it is valid, this procedure significantly reduces the computation time. The plots of \nvDW versus u0 are mostly \nsimilar in the two approaches (Fig. 14(a)-(e)), but there are some discre pancies. An important \ndiscrepancy is the Walker breakdown threshold, uWB. For instance, when Ku is 3106 J/m3 (the \nequilibrium DW ≈ 2.03 nm) (Fig. 14(b)), uWB for the self-consistent calculation is about 310 \nm/s whereas uWB for the local approximation is a bout 220 m/s. This difference in uWB is \ncaused by the fact that DW changes in the simulation but is treated as a constant in deriving \nthe local approximation. As the current increases, the domain wall tilt angle  also increases. \nThis change in  causes a change in Kd and in turn, a change in DW. Figure 14(f) shows DW \nin the steady state (   tatDWSteady\nDW ) versus u0 for Ku = 3106 J/m3. Steady\nDW  for a small \nu0 is close to its equilibrium value (= 2.03 nm), but decreases with increasing u0. As shown in \nFig. 13, the reduced domain wall width results in an increased eff; in this case, eff ≈ 40 \n 0.019 for u0 = 5 m/s whereas eff ≈ 0.022 for u0 = 300 m/s. Using these values of eff to \nEq. (29) with eff ≈ 1 and  = 0.03, one finds that uWB indeed changes substantially due to \nthis nonlinear effect as shown Fig. 14(b). We  conclude that the local approximation with eff \nand eff calculated from spin transport equations would capture the core effect of the non-\nlocal spin transfer torques quali tatively, but it cannot reproduce the results obta ined from the \nself-consistent calculation qua ntitatively unless they are artificially adjusted. \n \n3.6. Summary \n To summarize this section, we show self-consistent calculations for current-induced \ndynamics of narrow domain walls. We find that fo r narrow domain walls, the self-consistent \ncalculations predict the sp in transfer torque to be non-local and spatially oscillatory due to the \nballistic spin-mistracking mechanism. The non-lo cal spin transfer to rque generates domain \nwall motion and thus its effect is similar to the local nonadiabatic spin transfer torque. However some of its effect such as the Wa lker breakdown threshold value cannot be fully \ncaptured by the local nonadiabatic spin transf er torque approximation. Therefore when \nDW is \nclose to 1 nm, it is necessary to adopt the self-consistent calcula tions for quantitative \ndescription of current-driven domain wall motion. It is wo rth comparing our result to \navailable experimental ones. Thom as et al. [94], Heyne  et al. [109], and E ltschka et al. [111] \nhave found that vortex cores exhibit a much larger nonadiabaticity ( 8  to 10 ) \ncompared to transverse domain walls ( ). According to our result, this large \nnonadiabaticity of vortex cores is unlikely to be  caused by the ballistic spin-mistracking since \na typical width of a vortex core is a bout 10 nm. The large reported values of  in these \nsystems are more likely to be related to spin di ffusion effect [114,115] and/or anomalous Hall 41 \n effect [120]. On the other hand, Burrowes et al. [112] have tested very narrow Bloch-type \ndomain walls of about 1 nm us ing FePt nanowires and found th at such a narrow domain wall \ndoes not cause a significant increa se in the nonadiabaticity. This experimental result is \ninconsistent with our self-consistent calculation. Assuming that DW in the experiment is \nindeed around 1 nm, there are a few possible reasons for this discrepancy. Our model \nassumes a spherical Fermi surface with the free -electron approximation. However, the shape \nof a realistic Fermi surface usua lly deviates substantially from a sphere. If a realistic Fermi \nsurface was considered, the contribution from non-lo cal spin transfer torque is likely to be \nreduced because of additional spin dephasing du e to the complicated Fermi surfaces as we \nmention earlier in Sec. 3.4. Another possible reason for the inconsistency is that the experiment of Ref. [112] used a thermally activ ated depinning from a point defect to estimate \nthe nonadiabaticity. Since the width of FePt nanow ires in the experiment  is about 200 nm, it \nis reasonable to assume that a domain wall coul d bend when escaping from a point defect. If \nthis is the case, our one-dimensional model calculation should not be compared to this \nexperiment since a two-dimensional domain wa ll structure may cause an additional spin \ndephasing. Therefore, we believe that bett er defined measurements should be done to \nexperimentally test the role of the non-local spin transfer torque due to ballistic spin-\nmistracking for narrow domain walls. \nAlthough there are some ambiguities in dir ectly comparing our model calculation to \nexperiments, our result indica tes that it may be important to perform self-consistent \ncalculations to understand current-induced dynami cs of narrow domain walls in detail. Since \nmany recent experiments have utilized materi als systems with high perpendicular magnetic \nanisotropy, combining experimental measuremen ts and self-consistent calculations would be \nessential to understand the underlying physics and to design efficien t domain wall devices. 42 \n 4. Conclusion and outlook \n \n   In this review, we present self-consistent calculations of transport and magnetization \ndynamics for several representative examples. The self-consistent treatment allows us to \ncapture the core effect of the feedback from the magnetization dynamics to the spin transport \nand back to the magnetization dynamics th rough non-local spin transfer torques. The \nfeedback results in current- induced excitation of a single ferromagnetic layer, a narrower \nlinewidth for magnetization oscillation in spin valves, and an additional effective \nnonadiabatic spin tran sfer torque for domain wall dynami cs. These examples show the \nimportance of self-consistent treatments of spin transport and magnetization dynamics for \nunderstanding the physics of the coupled dynamics.    Before ending this review, we remark that the examples discussed so far are not the only cases for which a self-consistent treatment is required. In the following, we will briefly comment on other examples where the feedback mechanism is non-trivial. \n   Giant magnetoresistance is often considered as  an inverse effect of spin transfer torque. \nHowever, the generation of spin currents by magnetization dynamics would more aptly be \nconsidered the inverse process since the spin transfer torque is the excitation of magnetic \ndynamics by spin currents. These processes,  which generate spin currents by magnetic \ndynamics, are spin pumping [44,121,122] and the spin motive force [123-125]. Spin currents \ncannot be directly measured, bu t they couple to othe r processes that can. In ferromagnets, \nspin currents generate charge currents, which in turn generate elect ric voltages [126-135], and \nthe generation of spin currents e nhances magnetic damping [136-142].  \n    Just as spin transfer torques in multilayers and nanowires are similar processes in different geometries, so are spin pumping and th e spin motive force. Spin pumping occurs in 43 \n bilayer structures where a ferromagnetic layer is att ached to a non-magnetic layer \n[44,121,122]. A precessing magnetization in the ferromagnet pumps a spin current into the \nnon-magnet transferring energy and angular momentum from the ferromagnet to the \nconduction electrons of the non-magnet. This transf er increases the magnetic damping rate in \nthe ferromagnet. However, the pumped spin cu rrent generates a spin accumulation in the non-\nmagnet. This spin accumulation in turn generates a back-flow current back into the ferromagnet through diffusion processes. The quantitative enhancem ent of the Gilbert \ndamping [44] and the voltage drop across the in terface [126] requires pr oper treatment of the \nbalance between the pumping and back-flow curre nts. One approach for such calculations is \nthe magnetoelectronic circuit th eory used in Section 2.  \n   The spin motive force, on the other hand,  is found in systems with a single ferromagnet \n[123-125] like a magnetic nanowire. When the magnetization varies in both space and time, \nconduction electrons experience a sp in-dependent electric field that generates spin and charge \ncurrents. Early calculations of the spin mo tive force [123-125,128-130]  and the consequent \nenhancement of Gilbert damping [136-141] did not  consider other processes that might be \nimportant: spin accumulation, spin diffusion, and spin-flip scattering. However, just as it is \nnecessary to properly consider the backflow curr ent for a description of spin pumping, so is it \nnecessary to consider these processes for a calcu lation of the spin motive force. Several of us \nhave investigated these effects theoretically, and found that spin rela xation processes [142] \nsignificantly modify the spin motive force. Fo r example, charge currents are perfectly \ncanceled by diffusion currents in one-dimensi onal systems. Spin currents become non-local \nand become smaller depending on the characteri stic length of spatial variation of the \nmagnetization and the spin diffusion length. Fo r such one-dimensional systems, we provided \nan analytical expression of spin motive fo rce including spin relaxa tion processes [142]. For 44 \n two- or three-dimensional systems, however, su ch analytical solutions are not available so \nthat self-consistent calculations would be necessary to describe the coupled dynamics.  \n   Self-consistent calculations would also be ve ry important for descript ions of spin transfer \ntorques and spin motive forces in ferromagnetic systems with strong spin-orbit coupling, for \nexample, ferromagnets with Rashba interac tions. Obata and Tatara  [143], and Manchon and \nZhang [144] independently predicte d the existence of field-like spin transfer torque induced \nby in-plane current in  Rashba ferromagnets. A number of  experimental [145-150] and \ntheoretical [151-159] studies have followed this wo rk. Miron et al. report ed that an in-plane \ncurrent-induced field-like spin to rque is present for Pt|Co|AlO x structures where the inversion \nsymmetry is broken [145]. Miron et  al. also reported that a doma in wall in such structures \nmoves against the electron-flow  direction with high speed [146]. This reversed domain wall \nmotion with high speed cannot be  explained by conventional adia batic and nonadiabatic spin \ntransfer torques, but may be explained by a damp ing-like spin transfer torque in addition to \nall other spin transfer torques (i.e., adiabatic,  nonadiabatic, and the fi eld-like torques) [156] \nand the Dzyaloshinskii-Moriya interaction [159] . The damping-like spin transfer torque may \noriginate from a spin Hall effect in a hea vy metal layer like Pt [159-165] and/or a \nnonadiabatic correction to the field-like torque [155-158]. This  damping-like torque also \nallows switching the magnetization by in-plane currents [149,164,166].  \n   At present, the appropriate descri ption of this unconventional current-induced \nmagnetization dynamics is still cont roversial. It is not clear wh ether an explanation based on \nthe spin Hall effect, Rashba spin-orbit coupling, both, or something else, is appropriate for all \nexperiments or individual  experiments. To resolve this co ntroversy, it may be important to \ndevelop a model that takes into  account both types of spin-orb it effects and computes the \nproperties of spin transfer torques accurately . For instance, we have developed a Boltzmann 45 \n transport model considering the tw o sources of spin transfer tor ques (i.e., the spin Hall effect \nand Rashba spin-orbit coupling)  and found that both sources can  generate not only field-like \ntorques but also damping-like to rques for thin ferromagnets [165]. In a different approach, we \nhave found [167] that for two-dimensional el ectron gases and under th e assumption that the \nspin-orbit potential is comparable to the exchange  interaction, the field-like spin torque has a \ncomplicated dependence on the angle between th e current direction and the magnetization \ndirection. In this case, self-consistent calcula tions are needed to properly take into account \nthe effect of complicated angle-dependent  spin transfer tor que on current-induced \nmagnetization dynamics. Furthermore,  since spin transfer torque s and spin motive forces are \nclosely related, a sizable spin tr ansfer torque due to Rashba spin-orbit coupling suggests that \nthe magnetization dynamics in Rashba ferromagnets can generate a large spin motive force [168-170]. In this case, the spin motive force may require self-consistent calculations to \naccurately account for the spin relaxation pr ocess since the Rashba spin-orbit coupling \ncorrelates the spin directions with the wave vectors.    Up until this point, we have discusse d the coupled dynamics of charge, spin, and \nmagnetization. Another important degree of fr eedom is heat. Temperature gradients across \nstructures may also generate spin transfer torques just as voltage gradients do [171-178]. \nRecently, the existence of therma l spin transfer torques was e xperimentally demonstrated in \nmetallic spin valves [175] and theoretically pred icted in magnetic tunnel junctions [178]. This \ntype of torque mediated by magnon- and/or sp in-wave-spin current may find use in moving \ndomain walls [179-185]. It is closely related to sp in-dependent thermoelectric effects, such as \nspin-dependent Seebeck, Peltier, and Nern st effects [186-189]. These heat- and spin-\ndependent phenomena are unexplored largely at the moment and thus would require various \nself-consistent calculatio ns that couple heat, spin, and ma gnetization dynamics all together.  46 \n Appendix A. Comparison of the convolution method to a full so lution of the spin \naccumulation profiles in the lateral spin diffusion problem \n \nRef. [41] introduced a convolution method that leads to the speed up in  the calculation of \nthe lateral diffusion. Since the speed gain is subs tantial, it is important to test the validity of \nthe underlying approximations. Here we do so by examining our full solutions of the drift-diffusion equation.  \nIn the convolution method, the spin chemical potential \nS at a point r is given by \n)() (~)(S rmrrK rμ  dv , where the kernel ) (~rrK  is a 3 by 3 tensor that relates S at r \nto the magnetization  m at a different point r’. Its explicit form is given in Ref. [41]. In the \nconvolution method, the kernel K~ is assumed to depend on ( r–r’) but not explicitly on r \nitself. This assumption leads to substantial sp eed up in the computing time because the kernel \ncan be precomputed and the convolution can be done with fast numerical techniques.  \nSeveral approximations underlie this approach. It assumes that the kernel does not change \nnear boundaries in the structure and assumes that the magnetizati on only has small deviations \nfrom the average magnetization. \nHere we test the errors that are introdu ced by the convolution method in nanopillars in \nwhich all of the layers have been patterned . Figure A1(a) shows a schematic of a system \nconsisting of NM (10 nm) | FM (8 nm) | NM (32 nm). The layers have been patterned into a \nnanopillar 41 nm wide, and the spin diffusion lengths are chos en to be 200 nm in the non-\nmagnet and 10 nm in the ferromagnet. Other parame ters are similar to those of Py/Cu in the \nmain text. Arrows in the ferromagnet show lo cal magnetizations. The magnetization points in \nthe x direction except for the cell located at x = x 0 where it is in the z direction. Figure A1(b) 47 \n shows the z-component of spin chemical potential z, calculated by our a pproach, in the NM \nregion at the bottom interf ace of FM|NM for two cases; x0 = 0 (center of nanopillar) and x0 = \n18 nm (close to an edge). In case of x0 = 0, the spin chemical potential profile is symmetric \nalong the lateral direction (i.e., x-axis) whereas it is slightly asymmetric due to the boundary \neffect in case of x0 = 18 nm as indicated by arrows in Fig. A1(b). However, the two agree \nsurprisingly well. In part, this arises because the spin accumulation is much more local than \nwould be expected from  the long spin diffusion length. Th e spin accumulation is more local \nbecause the interface with the fe rromagnet and the interface with the reservoir acts as effect \nspin flip scattering sites. Unless the lead is  very thick, the spin diffusion length becomes \nlargely irrelevant compared to the sp in flip scattering at the interfaces. \nThe convolution approach will break down wh en the magnetization varies significantly \ncompared to its average value. We illustrate th is point in a spin-valve  structure with domain \nwalls in both layers. The problem  with the convolution method used in Ref. [41] for this \nsituation is that the kernel is for the tran sverse magnetization based on a solution for the \nlongitudinal transport that is uniform across the device. This assumption is clearly violated in \nthe structure considered here with domain walls (see Fig. A1(c)).  \nOverall, the convolution met hod is a convenient approximation to calculate the spin \naccumulation profiles in some cases because it uses significantly less computation time compared to full calculations. However, this method is not reliable in all situations. For \ninstance, it breaks down for calculations of ma gnetization reversal, pa rticularly when the \nreversal mode is non-uniform. In contrast, full ca lculations can be applied to all cases at the \ncost of time-consuming calculations. 48 \n Appendix B. Collective coordinates approach for non-local spin transfer torque in a \nnarrow domain wall \n \n   With eff and eff, Eqs. (25) and (26) can be rewritten as, \n,\nDW0 eff\nDW \n u\ntX\nt                                                ( B . 1 )  \n.2sin1\nSd\nDW0 eff\nDW  MK u\nt tX                                  ( B . 2 )  \nWhen u0 is smaller than uWB,  increases in the initial time stage and then becomes saturated \nto a certain value over time. In this limit (i.e., 0 / t  as t ), we find   \n . 2sin0 eff eff\nDWdSuKM                                           ( B . 3 )  \nuWB is determined from the maximum of R.H.S.  of Eq. (B.3), since the absolute value  of is \nmaximized at u0 = uWB. Thus, uWB is given as \n.\neff eff SDWd\nWB\nMKu                                             ( B . 4 )  \nWhen eff = 0 and eff = 1, Eq. (B.4) recovers the spin  current velocity for the Walker \nbreakdown (S DWd WB /M K u  ) driven by the local adiabatic spin transfer torque [23,190]. \nWhen u0 < uWB, the domain wall moves steadily. In this case, domain wall velocity vsteady \nis obtained from Eq. (B.1) with 0 / t  as \n.0eff\nsteady u v                                                        ( B . 5 )  \nWhen  u0 >> uWB, t/  is always nonzero and domain wall undergoes a continuous \nprecession motion. In the limit of very large current, one obtains the average velocity v by \naveraging Eqs. (B.1) and (B.2) over a period of the precession of  and using 0 2sin  49 \n where ... is the time-average over a period; \n.10 2eff effu v\n                                                      ( B . 6 )  50 \n Acknowledgements \n \nWe thank B. Dieny, A. Vedyayev, A., G.E.W. Baue r, A. Kent, J.Z. Sun, I.N. Krivorotov, A.N. \nSlavin, J.-V. Kim, S. Zhang, A. Manchon, R. D. McMichael, and T.J. Silva for fruitful \ndiscussions. This work was support ed by the NRF (2010-0023798, 2011-028163), by the \nMEST Pioneer Research Center Progr am (2011-0027905), and KU-KIST School Joint \nResearch Program. K.J.L. acknowledges suppor t under the Cooperative Research Agreement \nbetween the University of Maryland and the National Institute of Standards and Technology \nCenter for Nanoscale Science and Tec hnology, Award 70NANB10H193, through the \nUniversity of Maryland. 51 \n References \n[1] L. Berger, J. Appl. Phys. 3 (1978) 2156. \n[2] L. Berger, J. Appl. Phys. 3 (1979) 2137. \n[3] J. C. Slonczewski, J. Magn. Magn. Mater. 159 (1996) L1. \n[4] L. Berger, Phys. Rev. B 54 (1996) 9353. [5] E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, R.A. Buhrman, Science 285 (1999) 867. \n[6] J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, D.C. Ralph, Phys. Rev. Lett. 84 (2000) 3149. \n[7] M. Tsoi, A.G.M. Jansen, J. Bass, W.-C. Ch iang, V. Tsoi, P. Wyder, Nature 406 (2000) 46. \n[8] S. Kiselev, J. Sankey, I. Krivorotov, N. Emley,  R. Schoelkopf, R.A. Buhrman, D.C. Ralph, Nature \n425 (2003) 380. \n[9] W.H. Rippard, M.R. Pufall, S. Kaka, S.E. Russek , T.J. Silva, Phys. Rev.  Lett. 92 (2004) 027201. \n[10] I.N. Krivorotov, N.C. Emley, J.C. Sankey, S.I. Kiselev, D.C. Ralpha, R.A. Buhrman, Science \n307 (2005) 228. \n[11] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, T. Shinjo, Phys. Rev. Lett. 92 (2004) \n077205. \n[12] M. Yamanouchi, D. Chiba, F. Matsukura, H. Ohno, Nature 428 (2004) 539. [13] V. Vlaminck, M. Bailleul, Science 322 (2008) 410. \n[14] K. Sekiguchi, K. Yamada, S.-M. Seo, K.-J. Lee, D. Chiba, K. Kobayashi, T. Ono, Phys. Rev. Lett. \n108 (2012) 017203.  \n[15] A. Brataas, A. D. Kent, H. Ohno, Nature Mater. 11 (2012) 372. \n[16] J.A. Katine, E. E. Fullerton, J. Magn. Magn. Mater. 320 (2008) 1217. [17] F.B. Mancoff, N.D. Rizzo, B.N. Engel, S. Tehrani, Nature 437 (2005) 393. \n[18] S. Kaka, M.R. Pufall, W.H. Rippard, T.J. Silv a, S.E. Russek, J.A. Katine, Nature 437 (2005) 389. \n[19] S. S. P. Parkin, M. Hayashi, L. Thomas, Science 320 (2008) 190. [20] A. Khitun, D.E. Nikonov, M. Bao, K. Ga latsis, K.L. Wang, Nanotechnology 18 (2007) 465202. \n[21] S. Zhang, P.M. Levy, A. Fert, Phys. Rev. Lett. 88 (2002) 236601. \n[22] D.C. Ralph, M. D. Stiles, J. Magn. Magn. Mater. 320 (2008) 1190. 52 \n [23] G. Tatara, H. Kohno, Phys. Rev. Lett. 92 (2004) 086601. \n[24] S. Zhang, Z. Li, Phys. Rev. Lett. 93 (2004) 127204. \n[25] A. Thiaville, Y. Nakatani, J. Miltat,  Y. Suzuki, Europhys. Lett. 69 (2005) 990. \n[26] M.D. Stiles, A. Zangwill, Phys. Rev. B 66 (2002) 014407. \n[27] M.L. Polianski, P.W. Brouwer, Phys. Rev. Lett. 92 (2004) 026602. [28] M.D. Stiles, J. Xiao, A. Za ngwill, Phys. Rev. B 69 (2004) 054408. \n[29] J. Miltat, G. Albuquerque, A. Thiaville , C. Vouille, J. Appl. Phys. 89 (2001) 6982. \n[30] J. G. Zhu, X.C. Zhu, IEEE Trans. Magn. 40 (2004) 182. [31] K.-J. Lee, A. Deac, O. Redon, J.-P. Noziéres, B. Dieny, Nature Mater. 3 (2004) 877. \n[32] K.-J. Lee, B. Dieny, Appl. Phys. Lett. 88 (2006) 132506. \n[33] Y. Acremann, J.P. Strachan, V. Chembrolu,  S.D. Andrews, T. Tyliszczak, J.A. Katine, M.J. \nCarey, B.M. Clemens, H.C. Siegmann, J. Stöhr, Phys. Rev. Lett. 96 (2006) 217202. \n[34] J.P. Strachan, V. Chembrolu, Y. Acremann,  X.W. Yu, A.A. Tulapurkar, T. Tyliszczak, J.A. \nKatine, M.J. Carey, M.R. Scheinfein, H.C. Si egmann, J. Stöhr, Phys. Rev. Lett. 100 (2008) \n247201. \n[35] T. Valet, A. Fert, Phys. Rev. B 48 (1993) 7099. [36] B. Özyilmaz, A.D. Kent, J.Z. Sun, M.J. Rooks, R.H. Koch, Phys. Rev. Lett. 93 (2004) 176604. \n[37] B. Özyilmaz, A.D. Kent, M.J. Rooks, J.Z. Sun, Phys. Rev. B 71 (2005) 140403(R). \n[38] Y. Ji, C.L. Chien, M.D.Stiles, Phys. Rev. Lett. 90 (2003) 106601.  \n[39] A. Brataas, Y. Tserkovnyak, G.E.W. Bauer, Phys. Rev. B. 73 (2006) 014408. \n[40] S. Adam, M.L. Polianski, P.W. Brouwer, Phys. Rev. B 73 (2006) 024425. [41] M.A. Hoefer, T.J. Silva, M.D. Stiles, Phys. Rev. B 77 (2008) 144401. \n[42] J.C. Sankey, I.N. Krivorotov, S.I. Kiselev, P.M. Braganca, N.C. Emley, R.A. Buhrman, D.C. \nRalph. Phys. Rev. B 72 (2005) 224427. \n[43] A. Brataas, Y.V. Nazarov, G.E.W. Bauer, Eur. Phys. J. B 22 (2001) 99. \n[44] Y. Tserkovnyak, A. Brataas, G.E.W.  Bauer, Phys. Rev. Lett. 88 (2002) 117601. \n[45] I. Theodonis, N. Kioussis, A. Kalitsov, M. Chshiev, W.H. Butler, Phys. Rev. Lett. 97 (2006) 53 \n 237205. \n[46] C. Heiliger, M.D. Stiles, Ph ys. Rev. Lett. 100 (2008) 186805. \n[47] J.C. Sankey, Y.-T. Cui, J.Z. Sun, J.C. Slon czewski, R.A. Buhrman, D.C. Ralph, Nature Phys. 4 \n(2008) 67. \n[48] H. Kubota, A. Fukushima, K. Yakushiji, T. Nagahama, S. Yuasa, K. Ando, H. Maehara, Y. \nNagamine, K. Tsunekawa, D.D. Djayaprawira, N. Watanabe, Y. Suzuki, Nature Phys. 4 (2008) \n37. \n[49] Z. Li, S. Zhang, Z. Diao, Y. Ding, X. Tang, D.M. Apalkov, Z. Yang, K. Kawabata, Y. Huai, Phys. \nRev. Lett. 100 (2008) 246602. \n[50] S.-C. Oh, S.-Y. Park, A. Manchon, M. Chshiev, J.-H. Han, H.-W. Lee, J.-E. Lee, K.-T. Nam, Y. \nJo, Y.-C. Kong, B. Dieny, K.-J. Lee, Nature Phys. 5 (2009) 898. \n[51] O.G. Heinonen, S.W. Stokes, J.Y. Yi, Phys. Rev. Lett. 105 (2010) 066602. \n[52] C. Wang, Y.-T. Cui, J. A. Katine, R. A. Buhrman, D. C. Ralph, Nature Phys. 7 (2011) 496. [53] S.-Y. Park, Y. Jo, K.-J. Lee, Phys. Rev. B 84 (2011) 214417. \n[54] K. Xia, P. J. Kelly, G. E. W. Bauer, A. Brataas, I. Turek, Phys. Rev. B 65 (2002) 220401(R). \n[55] M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, G. E. W. BauerPhys. Rev. B 71 (2005) \n064420. \n[56] J. Bass, W. P. Pratt Jr., J. Magn. Magn. Mater. 200 (1999) 274. \n[57] R.D. McMichael, M.D. Stiles, J. Appl. Phys. 97 (2005) 10J901. \n[58] W. F. Brown, IEEE Trans. Magn. 15 (1979) 1196. \n[59] G. Grinstein, R. H. Koch, Phys. Rev. Lett. 90 (2003) 207201. [60] M. Kirschner, T. Schrefl, F. Dorfbauer, G. Hr kac, D. Suess, J. Fidler, J. Appl. Phys. 97 (2005) \n10E301. \n[ 6 1 ]  I .  N .  K r i v o r o t o v ,  D .  V .  B e r k o v ,  N .  L .  G o r n ,  N .  C .  E m l e y ,  J .  C .  S a n k e y ,  D .  C .  R a l p h ,  R .  A .  \nBuhrman; Phys. Rev. B 76 (2007) 024418. \n[ 6 2 ]  I .  N .  K r i v o r o t o v ,  D .  V .  B e r k o v ,  N .  L .  G o r n ,  N .  C .  E m l e y ,  J .  C .  S a n k e y ,  D .  C .  R a l p h ,  R .  A .  \nBuhrman; Phys. Rev. B 76 (2007) 024418. 54 \n [63] G. Siracusano, G. Finocchio, I. N. Krivorotov, L. Torres, G. Consolo, B. Azzerboni; J. Appl. Phys. \n105 (2009) 07D107. \n[64] A.N. Slavin, K. Pavel, IEEE Trans. Magn. 41 (2005) 1264. \n[65] J.-V . Kim, V . Tiberkevich, A. Slavin, Phys. Rev. Lett. 100 (2008) 017207. \n[66] J. Bechhoefer, J. Rev. Mod. Phys. 77 (2005) 783. [67] A. Grebennikov, RF and microwave: transistor oscillator design, Wiley, UK, 2007. \n[68] D. Zhao, L. Yicheng, X. Shu, L. Zhang, I. Bennion, Appl. Phys. Lett. 81 (2002) 4520. \n[69] J. Tamayo, A.D.L. Humphris, M.J. Miles, Appl. Phys. Lett. 77 (2000) 582. [70] A. Becksei, L. Serrano, Nature 405 (2000) 590. \n[71] J.-V . Kim, Q. Mistral, C. Chappert, V .S. Tiberkevich, A.N. Slavin, Phys. Rev. Lett. 100 (2008) \n0167201. \n[72] J. Hayakawa, S. Ikeda, Y .M. Lee, R. Sasaki,  T. Meguro, F. Matsukura, H. Takahashi, H. Ohno, \nJap. J. Appl. Phys. 45 (2006) L1057. \n[73] J. Hayakawa, S. Ikeda, K. Mirura, M. Yamanouchi, Y .M. Lee, R. Sasaki, M. Ichimura, K. Ito, T. \nKawahara, R. Takemura, T. Meguro, F. Matsukura,  H. Takahashi, H. Matsuoka, H. Ohno, IEEE \nTrans. Magn. 44 (2008) 1962. \n[74] C.-T. Yen, W.-C. Chen, D.-Y . Wang, Y .-J. Lee, C.-T. Shen, S.-Y . Yang, C.-H. Tsai, C.-C. Hung, \nK.-H. Shen, M.-J. Tsai, M.-J. Kao, Appl. Phys. Lett. 93 (2008) 092504. \n[75] S. Yakata, H. Kubota,T. Sugano, T. Seki, K. Yakushiji, A. Fukushima, S. Yuasa, K. Ando, Appl. \nPhys. Lett. 95 (2009) 242504. \n[76] S. Yakata, H. Kubota, T. Seki, K. Yakushiji, A. Fukushima, S. Yuasa, K. Ando, IEEE Trans. Magn. \n46 (2010) 2232. \n[77] T. Devolder, K. Ito, J. Appl. Phys. 111 (2012) 123914. \n[78] D. Gusakova, D. Houssameddine, U. Ebels, B. Dieny, L. Buda-Prejbeanu, M.C. Cyrille, B. \nDelaët, Phys. Rev. B 78 (2009) 104406. \n[79] T. Seki, H. Tomita, M. Shiraishi, T. Shin jo, Y . Suzuki, Appl. Phys . Express 3 (2010) 033001. \n[80] S.-W. Lee, K.-J. Lee, J. Magn. 15 (2010) 149. 55 \n [81] S.-W. Lee, K.-J. Lee, J. Appl. Phys. 109 (2011) 07C904. \n[82] P. Baláž and J. Barnas, Phys. Rev. B 83 (2011) 104422. \n[83] G. Tatara, H. Kohno, J. Shibata, Phys. Rep. 468 (2008) 213. \n[84] G.S.D. Beach, M. Tsoi, J.L. Erskine, J. Magn. Magn. Mater. 320 (2008) 1272. \n[85] Y . Tserkovnyak, A. Brataas, G.E.W. Baue r, J. Magn. Magn. Ma ter. 320 (2008) 1282. \n[86] H. Ohno, T. Dietl, J. Magn. Magn. Mater. 320 (2008) 1293. \n[87] M. Kläui, J. Phys.: Condense. Matter 20 (2008) 313001. \n[88] M. Kläui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G . Faini, E. Cambril, L. J. Heyderman, F. \nNolting, and U. Rüdiger, Phys. Rev. Lett. 94 (2005) 106601. \n[89] M. Laufenberg, W. Bührer, D. Bedau, P.-E. Melchy, M. Kläui, L. Vila, G. Faini, C.A.F. Vaz, J.A.C. \nBland, U. Rüdiger, Phys. Rev. Lett. 97 (2006) 046602. \n[90] C.-Y . You, S.-S. Ha, Appl. Phys. Lett. 91 (2007) 022507. \n[ 9 1 ]  G .  M e i e r ,  M .  B o l t e ,  R .  E i s e l t ,  B .  K r ü g e r ,  D . - H .  K i m ,  P .  F i s c h e r ,  ,  P h y s .  R e v .  L e t t .  9 8  ( 2 0 0 5 )  \n187202. \n[92] K.-J. Kim, J.-C. Lee, S.-B. Choe, K.-H. Shin, Appl. Phys. Lett. 92 (2008) 192509. \n[93] S.-M. Seo, K.-J. Lee, W. Kim, T.-D. Lee, Appl. Phys. Lett. 90 (2007) 252508. [94] L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, S.S.P. Parkin, Nature 443 (2006) 197. \n[95] L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, S.S.P. Parkin, Science 315 (2007) 1553. \n[96] S.-W. Jung, W. Kim, T.-D. Lee, K.-J. Lee, H.-W. Lee, Appl. Phys. Lett. 92 (2008) 202508. \n[97] T. Koyama, D. Chiba, K. Ueda, K. Kondou, H. Tanigawa, S. Fukami, T. Suzuki, N. Ohshima, N. \nIshiwata, Y . Nakatani, K. Kobayashi, T.  Ono, Nature Mater. 10 (2011) 194. \n[98] S.E. Barnes, S. Maekawa, Phys. Rev. Lett. 95 (2005) 107204. \n[99] Y . Tserkovnyak, H.J. Skadsem, A. Brataas, and G.E.W. Bauer, Phys. Rev. B 74 (2006) 144405. \n[100] J. Xiao, A. Zangwill, M.D. S tiles, Phys. Rev. B 73 (2006) 054428. \n[101] M.D. Stiles, W.M. Saslow, M.J. Donahue , A. Zangwill, Phys. Rev. B 75 (2007) 214423. \n[102] G. Tatara, H. Kohno, J. Shibata, Y . Lemaho, K.-J. Lee, J. Phys. Soc. Jpn. 76 (2007) 054707. \n[103] I. Garate, K. Gilmore, M.D. Stiles, A.H. MacDonald, Phys. Rev. B 79 (2009) 104416. 56 \n [104] K. Gilmore, Y .U. Idzerda, M.D. Stiles, Phys. Rev. Lett. 99 (2007) 027204. \n[105] K. Gilmore, I. Garate, A.H. MacDonald, M.D. Stiles, Phys. Rev. B 84 (2011) 224412. \n[106] S.-M. Seo, K.-J. Lee, H. Yang, T. Ono, Phys. Rev. Lett. 102 (2009) 147202. \n[107] M. Hayashi, L. Thomas, Ya.B. Bazaliy, C. Rettner , R. Moriya, X. Jiang, S.S.P. Parkin, Phys. Rev. \nLett. 96 (2006) 197207. \n[ 1 0 8]  R .  M o r i ya ,  L .  T h o m a s ,  M .  H a ya s h i ,  Y .  B .  Bazaliy, C. Rettner, S.S.P. Parkin, Nature Phys. 4 \n(2008) 368. \n[109] L. Heyne, M. Kläui, D. Backes, T.A. Moore, S. Krzyk, U. Rüdiger, L.J. Heyderman, A. F. \nRodríguez, F. Nolting, T.O. Mentes, M.Á. Niño, A. Locatelli, Phys. Rev. Lett. 100 (2008) 066603. \n[110] O. Boulle, J. Kimling, P. Warnicke, M. Kläui,  U. Rüdiger, G. Malinowski, H.J.M. Swagten, B. \nKoopmans, C. Ulysse, G. Faini, Ph ys. Rev. Lett. 101 (2008) 216601. \n[111] M. Eltschka, M. Wötzel, J. Rhensius, S. Krzy k, U. Nowak, M. Kläui, T. Kasama, R. E. Dunin-\nBorkowski, H. J. van Driel, R. A. Duine, Phys. Rev. Lett. 105 (2010) 056601. \n[112] C. Burrowes, A.P. Mihai, D. Ravelosona, J.-V . Kim, C. Chappert, L. Vila, A. Marty, Y . Samson, \nF. Garcia-Sanchez, L.D. Buda-Prejbeanu, I. Tudo sa, E E. Fullerton, J.-P. Attané, Nature Phys. 6 \n(2010) 17. \n[113] V . Kambersky, Czechoslovak Journal of Physics, Section B 26 (1976) 1366. \n[114] A. Manchon, W.-S. Kim, K.-J. Lee, arxiv:1110.3487. \n[115] D. Claudio-Gonzalez, A. Thiaville, J.  Miltat, Phys. Rev. Lett. 108 (2012) 227208. \n[116] J.-i. Ohe, B. Kramer, Phys. Rev. Lett. 96 (2006) 027204. \n[117] J.I. Ohe, M. Yamamoto, T. Ohtsuki, Phys. Rev. B 68 (2003) 165344. [118] N.L. Schryer, L.R. Walker, J. Appl. Phys. 45 (1974) 5406. \n[119] A.A. Thiele, Phys. Rev. Lett. 30 (1973) 230. \n[120] A. Manchon, K.-J. Lee, Appl. Phys. Lett. 99 (2011) 022504; ibid 99 (2011) 229905. [121] T. Tserkovnyak, A. Brataas, G.E.W. Bauer, Phys. Rev. B 67 (2003) 140404(R). \n[122] A. Brataas, G.E.W. Bauer, P. J. Kelly, Phys. Rep. 427 (2006) 157. \n[123] L. Berger, Phys. Rev. B 33 (1986) 1572. 57 \n [124] G.E. V olovik, J. Phys. C 20 (1987) L83. \n[125] S.E. Barnes, S. Maekawa, Phys. Rev. Lett. 98 (2007) 246601. \n[126] X. Wang, G.E.W. Bauer, B.J. van Wees, A. Brataas, Y . Tserkovnyak, Phys. Rev. Lett. 97 (2006) \n216602. \n[127] J.-i. Ohe, S.E. Barnes, H.-W. Lee, S. Maekawa, Appl. Phys. Lett. 95 (2009) 123110. [128] Y . Tserkovnyak, M. Mecklenburg, Phys. Rev. B 77 (2008) 134407. \n[129] W.M. Saslow, Phys. Rev. B 76 (2007) 184434. \n[130] R.A. Duine, Phys. Rev. B 79 (2009) 014407. [131] M.V . Costache, M. Sladkov, C.H. van der Wal, B.J. van Wees, Phys. Rev. Lett. 97 (2006) \n216603. \n[132] S.A. Yang, G.S.D. Beach, C. Knutson, D. Xiao, Q. Niu, M. Tsoi, J.L. Erskine, Phys. Rev. Lett. \n102 (2009) 067201. \n[133] Y . Yamane, K. Sasage, T. An, K. Harii, J. Oh e, J. Ieda, S.E. Barnes, E. Saitoh, S. Maekawa, \nPhys. Rev. Lett. 107 (2011) 236602. \n[134] M. Hayashi, J. Ieda, Y . Yamane, J. Ohe, Y .K. Takahashi, S. Mitani, S. Maekawa, Phys. Rev. Lett. \n108 (2012) 147202. \n[135] K. Tanabe, D. Chiba, J. Ohe, S. Kasai, H. Kohno, S.E. Barnes, S. Maekawa, K. Kobayashi, T. \nOno, Nature Comm. 3 (2012) 845. \n[136] S. Zhang, S.-L. Zhang, Phys. Rev. Lett. 102 (2009) 086601. \n[137] C.H. Wong, Y . Tserkovnyak, Phys. Rev. B 80 (2009) 184411. \n[138] C.H. Wong, Y . Tserkovnyak, Phys. Rev. B 81 (2009) 060404(R). [139] J.-H. Moon, K.-J. Lee, J. Magn. 16 (2011) 6. \n[140] S.-I. Kim, J.-H. Moon, W. Kim, K.-J. Lee, Curr. Appl. Phys. 11 (2011) 61. \n[141] J.-H. Moon, K.-J. Lee, J. Appl. Phys. 111 (2012) 07D120. [142] K.-W. Kim, J.-H. Moon, K.-J. Lee, H.-W. Lee, Phys. Rev. B 84 (2011) 054462. \n[143] K. Obata, G. Tatara, Phys. Rev. B 77 (2008) 214429. \n[144] A. Manchon, S. Zhang, Phys. Rev. B 78 (2008) 212405. 58 \n [145] I.M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. V ogel, P. Gamvardella, \nNature Mater. 9 (2010) 230. \n[146] I.M. Miron, T. Moore, H. Szambolics, L.D. Buda-Prejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, \nJ. V ogel, M. Bonfim, A. Schuhl, G. Ga udin, Nature Mater. 10 (2011) 419. \n[147] U.H. Pi, K.W. Kim, J.Y . Bae, S.C. Lee, Y .J. Cho, K.S. Kim, S. Seo, Appl. Phys. Lett. 97 (2010) \n162507. \n[148] T. Suzuki, S. Fukami, N. Ishiwata, M. Yamanouchi, S. Ikeda, N. Kasai, H. Ohno, Appl. Phys. \nLett. 98 (2011) 142505. \n[149] I.M. Miron, K. Garello, G. Gaudin, P. Zermatten, M.V . Costache, S. Auffret, S. Bandiera, B. \nRodmacq, A. Schuhl, P. Gambarde lla, Nature 276 (2011) 189. \n[150] J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, H. Ohno, \nNature Mater. 12 (2013) 240. \n[151] A. Matos-Abiague, R.L. Rodríquez-Suárez, Phys. Rev. B 80 (2009) 094424. [152] A. Manchon, S. Zhang, Phys. Rev. B 79 (2009) 094422. \n[153] K.M.D. Hals, A. Brataas, Y . Tserkovnyak, Europhys. Lett. 90 (2010) 47002. \n[154] J. Ryu, S.-M. Seo, K.-J. Lee, H.-W. Lee, J. Magn. Magn. Mater. 324 (2012) 1449. [155] X. Wang, A. Manchon, Phys. Rev. Lett. 108 (2012) 117201. \n[156] K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, H.-W. Lee, Phys. Rev. B 85 (2012) 180404(R). \n[157] D.A. Pesin, A.H. MacDonald, Phys. Rev. B 86 (2012) 014416. \n[158] E. van der Bijl, R.A. Duin e, Phys. Rev. B 86 (2012) 094406. \n[159] A. Thiaville, S. Rohart, E. Jue, V . Cr os, A. Fert, Europhys. Lett. 100 (2012) 57002. \n[160] J.E. Hirsh, Phys. Rev. Lett. 83 (1999) 1834. \n[161] S. Zhang, Phys. Rev. Lett. 85 (2000) 393. \n[162] L. Liu, T. Moriyama, D.C. Ralph, R.A. Buhrman, Phys. Rev. Lett. 106 (2011) 036601. [163] V .E. Demidov, S. Urazhdin, E.R.J. Edwards, M.D. Stiles, R.D. McMichael, S.O. Demokritov, \nPhys. Rev. Lett. 107 (2011) 107204. \n[164] L. Liu, C.-F. Pai, Y . Li, H.W. Tseng, D.C. Ralph, R.A. Buhrman, Science 336 (2012) 555. 59 \n [165] P.M. Hany , H.-W. Lee, K.-J. Lee, A. Manc hon, M.D. Stiles, Phys. Rev. B 87 (2013) 174411. \n[166] K.-S. Lee, S.-W. Lee, B.-C. Min, K.-J. Lee, Appl. Phys. Lett. 102 (2013) 112410. \n[167] K.-S. Lee et al., unpublished. \n[168] K.-W. Kim, J.-H. Moon, K.-J. Lee, H.-W. Lee, Phys. Rev. Lett. 108 (2012) 217202. \n[169] J.-H. Moon, K.-W. Kim, H.-W. Lee, K.-J. Lee, Appl. Phys. Express 5 (2012) 123002. [170] G. Tatara, N. Nakabayashi, K.-J. Lee, Phys. Rev. B 87 (2013) 054403. \n[171] M. Hatami, G.E.W. Bauer, Q. Zhang, P.J. Kelly, Phys. Rev. Lett. 99 (2007) 066603. \n[172] M. Hatami, G.E.W. Bauer, Q. Zhang, P.J. Kelly, Phys. Rev. B 79 (2009) 174426. [173] A.A. Kovalev, Y . Tserkovnyak, Phys. Rev. B 80 (2009) 100408(R). \n[174] Z. Yuan, S. Wang, K. Xia, Solid State Commun. 150 (2010) 548. \n[175] H. Yu, S. Granville, D.P. Yu, J. Ph. Ansermet, Phys. Rev. Le tt. 104 (2010) 146601. \n[176] G.E.W. Bauer, A.H. MacDonald, S. Maekawa, Solid State Commun. 150 (2010) 459. \n[177] J.C. Slonczewski, Phys. Rev. B 82 (2010) 054403. [178] X. Jia, K. Xia, G.E.W. Bauer, Phys. Rev. Lett. 107 (2011) 176603. \n[179] D. Hinzke, U. Nowak, Phys. Rev. Lett. 107 (2011) 027205. \n[180] P. Yan, X.S. Wang, X.R. Wang, Phys. Rev. Lett. 107 (2011) 177207. [181] X.S. Wang, P. Yan, Y .H. Shen, G.E.W. Bauer, X.R. Wang, Phys. Rev. Lett. 108 (2012) 167209. \n[182] D.S. Han, S.K. Kim, J.Y . Lee, S.J. Herms doerfer, H. Schultheiss, B. Leven, B. Hillebrands, \nAppl. Phys. Lett. 94 (2009) 112502. \n[183] M. Jamali, H. Yang, K.-J. Lee, Appl. Phys. Lett. 96 (2010) 242501. \n[184] S.-M. Seo, H.-W. Lee, H. Kohno, K.-J. Lee, Appl. Phys. Lett. 98 (2011) 012514. [185] W. Jiang, P. Upadhyaya, Y . Fan, J. Zhao, M. Wang, L.-T. Chang, M. Lang, K.L. Wong, M. \nLewis, Y .-T. Lin, J. Tang, S. Cherepov, X. Zhou, Y . Tserkovnyak, R.N. Schwartz, K.L. Wang, \nPhys. Rev. Lett. 110 (2013) 177202. \n[186] A. Slachter, F.L. Bakker, J.P. Adam, B.J. van Wees, Nature Phys. 6 (2010) 879. \n[187] N. Liebing, S. Serrano-Guisan, K. Rott, G. Re iss, J. Langer, B. Ocker, H.W. Schumacher, Phys. \nRev. Lett. 107 (2011) 177201. 60 \n [188] J. Flipse, F.L. Bakker, A. Slachter, F.K. Dejene, B.J. van Wees, Nature Nanotechnol. 7 (2012) \n166. \n[189] G.E.W. Bauer, E. Saitoh, B.J. va n Wees, Nature Mater. 11 (2012) 391. \n[190] A. Mougin, M. Cormier, J.P. Adam, P.J. Metaxas, J. Ferré, Europhys. Lett. 78 (2007) 57007. \n \n \n \n 61 \n Table 1 \nTransport parameters for numerical simula tions shown in section 2.3.2 and 2.3.3. \n Cua Coa Permalloyb \n(Py) Co|CuaCu|Pta Py|Cub \nBulk resistivity \n(nm) 6.0 75 255    \nBulk spin asymmetry, s 0 0.46 0.7    \nSpin diffusion length \nlsf (nm) 450 59 5.5    \nDiffusion coefficient \nD (×1015 nm2s-1) 41 1.7 1.7    \nInterfacial resistancec \nAR* (fm2)    0.51 0.12 0.97 \nInterfacial spin \nasymmetryc, s    0.77 0 0.77 \nSpin mixing conductance \nRe[G] (×1014 -1m-2)    5.5 - 6.0 \na Transport parameters for Cu, Co, Pt, and their interfaces are obtained from the literature [54, 56]. \nb Transport parameters for Py are provided by Cornell group. \nc Spin-dependent conductance Gs (s =  or ) in Eq. (9) is related with AR* and s through \n) 1( /1) (2 *\ns AR G G   and ) 1( / ) (2 *\ns sAR G G   .  \n 62 \n \n \n \nFig. 1. Top: schematic picture of a nanopillar  structure consisting of  a normal metal (NM 1) | a \nferromagnetic layer (FM) | a normal metal (NM 2). Bottom: cartoons of lateral spin diffusion \nat left and right interfaces of FM. 63 \n \n \nFig. 2. Spin chemical potential patterns in (a) a symmetric structure and (b) an asymmetric \nstructure, calculated from one-dim ensional Valet-Fert theory [35]. 64 \n \n \n \nFig. 3. Current-induced excitation of single ferromagnetic layer sandwiched by asymmetric \nCu layers: (a) Out-of-plane magnetization < Mz> as a function of the out-of-plane field and \ncurrent. (b) Normalized modulus of the magnetic moment as a f unction of the out-of-plane \nfield and current. 1 mA corresponds to about 7.07×1011 A/m2. 65 \n \n \n \nFig. 4. Vector plots of the magnetization pattern ( M) and the spin accumulation ( nS) patterns \nin a thick Cu layer at 0H = 2 T and I = -5 mA ( tCo = 2 nm). (a) M in Co layer. (b) nS (×1) at \ninterface ( z = 0 nm). (c) nS (×5) at z = 6 nm. (b) nS (×15) at z = 18 nm. (e) Normalized \naverage in-plane and out-of-plan e components of the spin accu mulation as a function of the \ndistance from the interface of Co|Cu. In (e), the lines are guides to the eye. 66 \n \n \n \nFig. 5. Dependence of the threshold current fo r current-induced excitati on on the thickness of \nsingle ferromagnetic layer: (a) Microwave power at various Co layer thicknesses. White lines \ncorrespond to phase boundaries . (b) Slope of the phase b oundary as a function of the \nthickness of Co layer. Inset of (b) shows th e intercept of the extr apolated boundary as a \nfunction of the thickness of Co layer. 67 \n \n  \n \nFig. 6. Frequency and eigenmode analysis  of current-induced excitation of single \nferromagnetic layer: (a) Time evolution of the out-of-plane component of magnetization \n<Mz> at 0H = 2.5 T and various negative cu rrents. (b) Power spectrum at 0H = 2.5 T and I \n= –11 mA. (c) and (d) Eigenmode images for the two peak frequencies i ndicated in (b). (e)-\n(h) Magnetic domain patterns at various times after the onset of curre nt: (e) 9.988 ns, (f) \n9.992 ns, (g) 9.996 ns, and (h) 10.000 ns.68 \n \n \n \nFig. 7. Comparisons of spectral densities of a spin valve, obtained from three different \nmodels: (a) Macrospin model (MACRO), (b) Conventional micromagnetic model without considering non-local sp in transfer torque (CONV), and (c ) Self-consistent model (SELF). \n(d) Main peaks of the microwave oscill ation obtained from the three models. 69 \n \n \n \nFig. 8. Effects of non-local spin transfer to rque on the linewidth. (a) Comparison of power \nspectra obtained in the three models at T = 10 K. (b) Linewidth as a function of the \ntemperature. (c) Power spectra obtained from no n-local, self-consistent model as a function \nof the temperature. The spectra are vertically  offset for clarity. Down-arrows indicate narrow \nsecondary peaks wherea s up-arrows indicate broad main peaks. Gray lines correspond to \nLorentzian fits. (d) The frequency versus the power normalized by | M|.  70 \n \n \n \nFig. 9. Angular dependence of spin torque in a multilayer of NM(10) | FM 1(5) | NM(5) | \nFM 2(3) | NM (1) | FM 3(3) (all in nanometers). We assume the following spin transport \nparameters: For FM and NM, respectively, the parameters are bulk resistivity   =  5 1  nm \nand 5 nm, bulk spin asymmetry  s = 0.51 and 0, spin diffusion length lsf = 60 nm and 1000 \nnm. For the interface parameters, FM | NM (or NM | FM), the parameters are interfacial \nresistance AR* = 0.52 f m2, interfacial spin asymmetry s = 0, and spin mixing conductivity \nRe(G ) = 5.42×1014 -1m-2. 1 and 2 represent the magnetization angles of FM 2 and FM 3 \nwith respect to the magnetization angle of FM 1, respectively. 71 \n \n \n \nFig. 10. Electron occu pation probabilities fR and fL for the right- and left-propagating \nelectrons as a function of the energy in (a) charge-neutrality-broken calculation and (b) \ncharge-neutrality-preserved calculation.  Spatial distribution of adiabatic (Adia\nSTN ) and \nnonadiabatic (Nonadia\nSTN ) spin transfer torques for a narrow domain wall centered at x = 0. (c) \nCharge-neutrality-broken calculation [116], and (d) charge-neutrality-preserved calculation \n(our work). Only the k-normal channel is considered ( )0,0 (F,kk ) in (c), whereas the \nintegration over the Fermi surface is performed in (d). Here, Ku is assumed to be 4.5×106 J/m3 \nand the upper panels of (c) and (d) show the domain wall profile. \n 72 \n \n \n \nFig. 11. Non-locality of spin transfer torque for narrow domain walls. Adiabatic (Adia\nSTN ) and \nnonadiabatic (Nonadia\nSTN ) vector components of spin  transfer torque for (a) Ku = 2106 J/m3 \n(DW ≈ 2.71 nm) and (b) Ku = 107 J/m3 (DW ≈ 0.98 nm). (c) Adia\nSTN  for various DW values. \n(d) Nonadia\nSTN  for various DW values. Here, Je is 1012 A/m2. 73 \n \n \n \nFig. 12. Domain wall velocity ( vDW) versus spin current velocity ( u0) for various domain wall \nwidth (DW). 74 \n \n \n \nFig. 13. Effective spin polarization (eff) and effective nonadiabaticity (eff) as a function of \nthe domain wall width ( DW). \n 75 \n \n \n \nFig. 14. Comparisons of domain wall ve locities between calculations with local \napproximation (red solid lines) and self-c onsistent calculations (symbols). (a) Ku = 2×106 \nJ/m3 (DW = 2.71 nm). (b) Ku = 3×106 J/m3 (DW = 2.03 nm). (c) Ku = 4.5×106 J/m3 (DW = \n1.57 nm). (d) Ku = 6.75×106 J/m3 (DW = 1.24 nm). (e) Ku = 1×107 J/m3 (DW = 0.98 nm). (f) \n DW in the steady state (   tatDWSteady\nDW ) versus u0 for Ku = 3106 J/m3. Vertical dotted \nlines correspond to the Walker breakdown thresholds. 76 \n \n \n \nFig. A1. Two tests of the diffus ion kernel. (a) Schematic of a model system consisting of NM \n(10 nm) | FM (8 nm) | NM (32 nm). We assume that the magnetization is in the x-direction \nexcept for the cell located at x = x0 is in z direction. (b) The z-component of spin chemical \npotential z, calculated by our approach, in the NM  region at the botto m interface of FM|NM \nfor two cases; x0 = 0 (center of nanopillar) and x0 = 18 nm (close to an edge). (c) Schematic \nof a model system consisting of NM (16 nm) | FM1 (6 nm) | NM (6 nm) | FM2 (6 nm) | NM (16 nm). In (c), arrows in the NM layers ( hollow head) show the spin accumulation vectors \nand the arrows in the FM layers (filled head) show local magnetization vectors.  "    }]