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Landau-Lifshitz Magnetodynamics As a Hamilton Model: Magnons in an Instanton Background

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Landau-Lifshitz Magnetodynamics As a Hamilton Model: Magnons in an Instanton Background PHYSICAL REVIEW B 82, 024410 ͑2010͒ Landau-Lifshitz magnetodynamics as a Hamilton model: Magnons in an instanton background Igor V. Ovchinnikov* and Kang L. Wang† Department of Electrical Engineering, University of California at Los Angeles, Los Angeles, California 90095-1594, USA ͑Received 22 March 2010; revised manuscript received 12 May 2010; published 13 July 2010͒ To take full advantage of the well-developed field-theoretic methods, Magnonics needs a yet-existing La- grangian formulation. Here, we show that Landau-Lifshitz magnetodynamics is a member of the covariant- Schrödinger-equation family of Hamilton models and apply the covariant background method arriving at the Ginzburg-Landau Lagrangian formalism for magnons in an instanton background. Magnons appear to be nonrelativistic spinless bosons, which feel instantons as a gauge field and as a Bose condensate. Among the examples of the usefulness of the proposition is the recognition of the instanton-induced phase shifts in magnons as the Berry phase and the interpretation of the spin-transfer-torque generation as a ferromagnetic counterpart of the Josephson supercurrent. DOI: 10.1103/PhysRevB.82.024410 PACS number͑s͒: 75.10.Ϫb, 75.30.Ds, 75.60.Ch I. INTRODUCTION LF deals with the problem of magnon spectrum in a non- trivial instanton background ͑Sec. IV C͒; discuss the enabled Nanometer-scale magnetization is believed to be one of dynamical diagrammatic technique ͑Sec. IV D͒, which can the most promising fabrics for the computational platforms take a detailed account of the geometrical nonlinearities, of the future. As long as the essence of computations is in the which, in turn, are dominant for magnons as we show by manipulation of the ͑nonlinear͒ correlations, it is the stable considering the three-magnon scattering10 in Sec. IV E; draw nonlinear magnetic entities—domain walls ͑DWs͒͑Ref. 1͒ a parallel between the spin-transfer-torque ͑STT͒ generation or more exotic instantons2,3—which could serve as state vari- and the Josephson phenomenon ͑Sec. IV F͒; and incorporate ables. Magnons, in turn, are natural tools to transfer informa- the soft-mode fluctuations into the proposed GL LF ͑Sec. tion within the system, to mediate the nonlocal exchange IV G͒. Section V concludes our work. coupling, or to probe the magnetic configuration. From this perspective, the studies of the instanton-magnon interaction, or more generally of Magnonics,4 are not only interesting II. COVARIANT FORM OF MAGNETODYNAMICS fundamentally but also have the potential for the technologi- A. Magnetodynamical Lagrangian formalism cal impact. ͑ ͒ Even though the Landau-Lifshitz ͑LL͒ equation5–10 is al- According to the LL equation LLE , the time evolution ជ ͑ ͒ most as old as the Schrödinger equation ͑SE͒, Magnonics is of the magnetization field, m r,t , where t is time and r ␮ ␮ still relatively immature in comparison with the other field =r , =1,...,D is a real-space point, is theories such as the nonrelativistic quantum mechanics itself mជ = Hជ ϫ mជ . ͑1͒ ץ−1␥ ͑i.e., the SE͒. The reason is the absence of the action mini- t eff ϫ ␥ ץ/ץ ץ ͒ ͑ mization principle Lagrangian formulation, LF underlying Here t = t, is the gyromagnetic ratio, denotes vector the dynamics of magnons. Only within the LF, the full po- ជ ͑ ͒ ␦H/␦ ͑ ͒ product, and Heff r,t = mជ r,t is the effective magnetic tential of the well-developed field-theoretic methods could field, given as a functional derivative of the GL energy func- be applied to Magnonics. In particular, the dynamical tional Ginzburg-Landau ͑GL͒ diagrammatic technique for magnons will become available. ͒ ͑ ជ ͒2 ץH ͵ E E ␬͑ In this paper, we construct this necessary but missing tool = , = ␮m + U, 2 r for Magnonics. In Sec. II A, we derive the covariant GL LF ␮ r , ␬ is the exchange constant, and U is theץ/ץ= ␮ץ for the LL magnetodynamics. In Secs. II B and II C,we where show, using the theory of Kähler manifolds,11 that the LL potential, which in its “minimal” form is the sum of the magnetodynamics belongs the family of Hamilton models, anisotropy energy and the interaction with the external and equations of motion of which are the covariant SEs ͑CSEs͒. the demagnetization magnetic fields. In Sec. III, we use the covariant background field ͑CBF͒ We do not consider the energy dissipation issues, which decomposition method12 to derive the GL action for mag- makes Eq. ͑1͒ a conservative time evolution preserving the nons in a background of instantons. Section IV is devoted to GL energy, H. When needed, Gilbert damping,13 or any the exemplification of the usefulness of our approach: we other form of energy dissipation due to the coupling to an interpret the instanton-induced magnon phase shift6 as the energy reservoir, can be incorporated into the LF either phe- Berry phase ͑Sec. IV A͒; show that in addition to the Berry nomenologically or systematically, e.g., within the Keldysh phase, instantons are also being felt by magnons as a Bose double-contour picture.14 condensate ͑Sec. IV B͒; consider the case of a single DW In its conventional form ͑1͒, the magnetodynamics is in a ͑Ref. 9͒ to demonstrate the ease with which the proposed GL sense overdetermined. It deals with mជ as though it has three 1098-0121/2010/82͑2͒/024410͑9͒ 024410-1 ©2010 The American Physical Society IGOR V. OVCHINNIKOV AND KANG L. WANG PHYSICAL REVIEW B 82, 024410 ͑2010͒ ͒ ¯⌿ ץ ⌿ ⌿2D ץ ⌿¯ ͒ ␮2 −2K͑D ץ⌿ ⌿2 ץ␮ −2K͑ i te t − t =− r e ␮ ␮ + ␮ ␮ + U˜ , ͑4a͒ ץ͑ ¯⌿ ץ ⌿ D ץ⌿͒ ץ 1⌫ ץ͑ ⌿ ץ D where ␮ ␮ = ␮ + 11 ␮ ␮ and ␮ ␮ = ␮ ¯ ¯ 1 1 −K 1 −K ␮⌿¯ , with ⌫ =−2e ⌿¯ and ⌫ =−2e ⌿¯ beingץ͒ ¯␮⌿ץ ⌫+ ¯1¯1 11 ¯1¯1 ͑ ͒ ␮ ␥−1 ␮2 2␬ the Christoffel symbols see below , t =2m0 , r =4m0 , and the potential term is ⌿aץ ⌿aץ ⌿aץ ˜ ͫ ͩ ͪ + ͬ␦ U = m + i − m ⌿aU, ץ ץ ץ z mx my mz a where ␦⌿a ϵ␦/␦⌿ ͑r,t͒. Further, we recall that vector mជ is ͉ ͉ FIG. 1. ͑Color online͒͑a͒ The ⌿ coordinates on S2 are obtained subject to the constraint mជ =0 and there are only two inde- 2 ⌿ m m 16 by the stereographic projection of S onto the complex plane. 0 is pendent components, say x and y. Therefore, one can ⌿a͑ ͒→⌿a͑ ͒ the static magnetization representing instantons and ⌿ is the dy- assume that mជ mx ,my so that the last term in the namical magnetization in the presence of magnonic excitations. ⌿គ definition of U˜ vanishes. The remaining part can be elabo- ͑straight arrow͒ is the “geodesicЉ coordinate, which is the vector rated by the following Jacobian of the coordinate transforma- ⌿ ⌿ touching the geodesic connecting 0 and and which belongs to tion: the linear tangent space, T⌿ . ␺ is the magnonic field, which is 0 2 ⌿a 1 1+⌿ 1+⌿¯ 2ץ essentially ⌿គ in the basis of vielbeins. ͑b͒ As a magnon ͑circled ͒ ͑ ͫ ͬ j x y arrows represent the magnonic gyration propagates in space path = , = , , m 2m ͑ ⌿2͒ ͑ ⌿¯ 2͒ץ l͒, it acquires the Berry phase, equal to the ͑oriented͒ area ͑light j z i 1− − i 1− ja gray area͒ formed by ⌿ ͑l͒෈S2 and the geodesic connecting its 0 leading to initial and final positions ͑dashed curve͒. ͑c͒ In the presence of a DW, besides the bulk continuum ͑a set of thinner curves͒, the mag- 2 U˜ = ␦⌿¯ U + ⌿ ␦⌿U. ͑4b͒ non spectrum has the GM. Also indicated the two possible DW- assisted three-magnon processes: the inelastic negative refraction of Now, we take the equation, which is a complex conjugate of an incident bulk magnon into another bulk magnon with the accom- Eq. ͑4͒, multiply it by ⌿2, subtract it from Eq. ͑4͒ itself, panying creation of a DW-GM boson ͑1͒→͑2͒+͑g͒; and the ab- →␮ →␮ rescale the coordinates as t tt, r rr, and drop out the sorption of an incident bulk magnon by the DW with the creation of common factor ͑1−͉⌿͉4͒e−2K. The result is the following ͑ Ј͒→͑ ͒ ͑ Ј͒ ͑ ͒ two DW-GM bosons: 1 g + g . d Graphical representation ͑gacg =␦a͒: of the two DW-assisted three-magnon processes. ␾ is the azimuthal cb b ͒ ͑ ␦⌿ 1a ץ ⌿ D ץ angle of the magnetization on the DW, fluctuation in which is the i t =− ␮ ␮ + g ⌿aU. 5a essence of the DW-GM. Combining Eq. ͑5a͒ with the similar equation for the antiho- ¯ independent components, whereas Eq. ͑1͒͑even in the pres- lomorphic field, ⌿, we arrive at ͒ ͉ ͉ ence of Gilbert damping preserves mជ =m0 so that mជ lives a b a ab ␮⌿ + g ␦⌿bU, ͑5b͒ץ⌿ =−D␮ ץ J on a two-dimensional ͑2D͒ sphere, S2, which is also known b t as the projective complex line CP1 ӍS2. Therefore, one can Jb ͑ ͒ ͑ where a =diag i,−i is the so-called complex structure see introduce the 2D coordinates on S2 by the stereographic pro- ͒ ͑ ͒ below . After lowering indices in Eq. 5b by gab we get jection onto a complex plane, C1 ͓see Fig. 1͑a͔͒ ͒ ͑ ␦ ⌿b ץ ⌿b D ץ ⍀ ab t =−gab ␮ ␮ + ⌿aU, 5c + −K⌿ −K͑ ⌿⌿¯ ͒ ⍀ Jc ͑ ͒ m = mx + imy =2m0e , mz = m0e 1− , where ab=gac b is the so-called Kähler form see below . As can be straightforwardly verified, Eq. ͑5͒ is the equa- tion of motion ͑EoM͒ of the action, S=͐ L, where the La- K ͑ ⌿⌿¯ ͒ 2 t,r where =ln 1+ is known as the Kähler potential on S grangian density is ͑see below͒.
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