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 nontrivial 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

͒ ¯⌿ 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 equation 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. The induced ͑Fubini-Study͒ metric on S2 is ͒͑͒ /⌿b ץ⌿a ץ ͑ ⌿a ץ L ␪ = a t − gab ␮ ␮ 2+U 6a ¯K ͑ / ͒ −K⌿ ץ͒ / ͑ g11¯ ␪ ␪ 0 ͑ ជ ͒2 2 ⌿a ⌿b ͩ ͪ ͑ ͒ with 1 = ¯1 = i 2 1 = i 2 e or dm =2m0gabd d , gab = , 3 g¯ 0 ͒ ͑ /K ץ␪ Jb 11 a = a b 2, 6b ¯ 2 where a,b=͑1,1¯͒, ⌿1 ϵ⌿, ⌿1 ϵ⌿¯ , and ⌿a ϵ͑⌿,⌿¯ ͒, the which is known as the Poincáre 1-form–a vector field on S ͒ ͑ ␪ ⍀ ץ ␪ ץ Einstein’s summation over the repeated indices is assumed obeying a b − b a = ab see below . The GL part of Eq. ͑6a͒͑the one in the parenthesis͒ is ץ K −2K ץ ץ throughout the paper, and g11¯ =g¯11 = 1 ¯1 =e with a ⌿a.15 merely Eq. ͑2͒ in terms of ⌿’s, whereas it can be shown thatץ/ץϵ 5 ץ ជ ͒ ͑ To obtain the LLE in terms of the new fields, we first the dynamical part the first term is equivalent to A· tmជ , write down the equation for m+, where Aជ is the vector potential of a Dirac string.17 Thus, Eq.

024410-2 LANDAU-LIFSHITZ MAGNETODYNAMICS AS A… PHYSICAL REVIEW B 82, 024410 ͑2010͒

͑6͒ is equivalent to the standard LF for magnetodynamics that if a form is closed, i.e., d⍀X =0, it is also ͑locally͒ exact, used, e.g., in Ref. 5. The advantage of our formulation is i.e., ⍀X =d␪X. solely in the covariance, which cuts deeper than it may seem Fortunately, all M’s are symplectic with ⍀ being the so- ⍀M MJc from the first sight. In particular, it is the covariance of Eq. called Kähler form, ab =gac b. As can be straightforwardly ͑6͒ which will allow to derive the magnonic action to all the verified, ⍀M =d␪M with orders of the magnon-magnon and instanton-magnon inter- ͒ ͑ /KM ץ␪M Jb actions in Sec. III below. a = a b 2. 10a Thus, the Hamilton-dynamics-defining part of the Lagrang- ⌿a ץTM ␪M B. Covariant-Schrödinger-equation family of Hamilton models ian density is = a t , which is holomorphically covariant, i.e., it preserves its form under the coordinate transfor- The above Lagrangian formalism for the LL magnetody- ⌿ ⌿¯ namics actually belongs to the entire family of Hamilton mations, which do not mix ’s and ’s and thus leave the models for fields, which may live on any Kähler manifold, complex structure unchanged. SM ͐ LM LM M, not only S2. The construction of this family of models Now we can construct the action = r,t , TM EM will allow firstly to clarify the geometrical meaning of the = − , where the last term has the standard nonlinear / ⌿b ץ⌿a ץEM M ͒ ͑ objects, introduced in the previous section, and secondary to sigma model NLSM form, =gab ␮ ␮ 2+U,so establish the covariance of Eq. ͑6͒, which is needed to justify that the application of the CBF method for the derivation of the M M a M a b ␮⌿ /2+U͒. ͑10b͒ץ ␮⌿ץ ⌿ − ͑g ץ L = ␪ magnonic action in Sec. III below. For this reason, let us a t ab digress here on this generalization of the LL magnetodynam- Note that it is the simple relation between gM and ⍀M, ics and on the brief review of the theory of M’s.11 which is unique for Ms and which assures the ͑holomor- M M ͒ TM EM Any has even number of real dimensions, dimR phic covariance of both and . M ͓ ͑ ͔͒ =2d, where d=dimC is its complex dimensionality. The The EoMs Eq. 9 now read M coordinates on can be split into the holomorphic and M b M b ␮⌿ + ␦⌿aU, ͑11a͒ץ⌿ =−g D␮ ץ ⍀ antiholomorphic sets: ⌿␩ and ⌿¯ ¯␩, ␩, ¯␩=1,...,d. The sets ab t ab ⌿b ץ⌿c͒ ץ a⌫ ץ⌿a ͑␦a ץ ␩ ¯␩ D are actually complex conjugates of each other ͑⌿ =⌿¯ ͒. where ␮ ␮ = b ␮ + bc ␮ ␮ is the Laplacian with ␩ However, it is convenient to treat them as independent and ⌫␩ ⌫¯ the only nonzero Christoffel symbols being ␯␰= ¯␯¯␰ M ͓͑ M͒ac M ␦a͔ 19 ץ¯⌿a ϵ͑⌿␩ ⌿¯ ¯␩͒ ␩ ͑ M͒␩␩ introduce the total Latin indices , , i.e., a= = g ␯g¯␩␰ g gcb = b . After rising indices in Eq. ¯␩, as opposed to the Greek indices, which run over only ͑11a͒ by ͑gM͒ab,weget either holomorphic or antiholomorphic components. The two ͒ ͑ ␦⌿a ͑ M͒ab ץ ⌿b D ץJa sets are geometrically separated by the so-called complex b t =− ␮ ␮ + g ⌿bU, 11b structure ͑a=␩ ¯␩, b=␯ ¯␯͒, which for only holomorphic fields reduces to ␩ ␩ 0␯ ␩ ␩ M ␩¯␯ ͒ ͑ ⌿ ͑ ͒ ␦ ␯ ץ ⌿ D ץ ¯ 1␯ Ja ͩ ͪ ͑ ͒ i t =− ␮ ␮ + g ⌿¯ ¯U. 11c b = i ¯␩ ¯␩ , 7 0 −1␯ ␯ ¯ Equations ͑10͒ and ͑11͒ are arranged in the exactly opposite existence of which makes M a complex manifold.18 order than that of Eqs. ͑5͒ and ͑6͒. This is meant to empha- Any M is Riemannian, i.e., it is provided with the con- size that unlike in the previous section, where the LF for the M 2 ⌿a ⌿b magnetodynamics was derived starting from the phenomeno- cept of the infinitesimal length, dl =gab d d . The metric tensor is ͑a=␩ ¯␩, b=␯ ¯␯͒, logical LLE ͓Eq. ͑1͔͒, the above construction of the M M general- LF is based on the covariance arguments and the 0␩␯ g␩¯␯ theory of M’s. M ͩ ͪ ͑ ͒ gab = M , 8 g␩␯ 0¯␩¯␯ ¯ C. Landau-Lifshitz equation as the covariant-Schrödinger where the nonzero components can be given via the Kähler equation ⌿¯ ¯␩ץ/ץ ץ ⌿␩ץ/ץ ץ KM ץ ץ M M potential: g␩¯␯ =g¯␩␯ = ␩ ¯␯ , with ␩ = , ¯␩ = . Equations ͑10͒ and ͑11͒ generalize the magnetodynamics Let us recall now that the Hamilton dynamics can be de- defined by Eqs. ͑5͒ and ͑6͒ and represent the entire family of fined only on a symplectic manifold, X, i.e., the one which Hamilton models. Each member of this family is uniquely ͑ ͒ ⍀X M admits a closed nondegenerate symplectic 2-form, ab= defined by the Kähler target space, , and the dimension- ͒ ͑ ⍀X ץ⍀X ⍀X ϵ − and d ͓a ͔ =0, where square brackets denote ality of the base real space. The “seed” representative is the ba bc 1 antisymmetrization and d is known as the exterior derivative. nonrelativistic quantum mechanics with ⌿෈C1 and KC a 1 The Hamilton EoMs for the fields, X ͑r,t͒෈X,is ¯ C C1 ¯ ,␮⌿+U is the Hamiltonianץ␮⌿ץ= ⌿⌿ so that g ¯ =1, E= M 11 b ␦HX/␦ a ͑ ͒ 1 ץ ⍀ ͒ ⌿ ץ ¯tX = X , 9 TC ͑ / ͒͑⌿ ab = i 2 t −c.c. , and the EoM is the conventional SE, X X which corresponds to the following action S =͐ ͓͑͐ T ͒ 2 .␮⌿ + ␦⌿¯ Uץ−= ⌿ ץt r i −HX͔, where HX is some energy functional of X’s and TX t ␪X ͑ ͒ M ץa ␪X ⍀X ␪X ϵ ץ␪X = a tX with a such that ab=d ͓a b͔. The local ex- From this perspective, the generic- EoMs, together with istence of ␪X is assured by the Poincaré lemma, which states the LLE, can be called the CSEs. It is worth mentioning also

024410-3 IGOR V. OVCHINNIKOV AND KANG L. WANG PHYSICAL REVIEW B 82, 024410 ͑2010͒

͒ ͑ ͒ ץ ⌼b ץ ⌼c D ץthat unlike for the SE, the nonlinearities enter into the CSEs L͑1͒ ⌼គ a͑ Jb = gab c t + gab ␮ ␮ − aU . 14a not only through the potential, U, but also through the non- vanishing ⌫’s. To simplify matters, we consider here only the case of the The mere identification of the LLE as the CSE is interest- local potential, though the generalization to the nonlocal U is ing aesthetically but does not explicitly lead to new physical possible. ͑n͒ results. However, it can be used for mapping some yet-to-be- The next task is to find L to a desired order using the S͑n−1͒ ץ S͑n͒ solved problems of magnetodynamics onto problems from recursion: = s . It is easier, however, to introduce 12 other areas of physics, which have longer history and are the covariant differentiation operator, Ds, which has the nS͑1͒ nS͑1͒ץ better understood. For example, the problem of the phase same effect on the action: s =Ds . The introduction of ͑ ͒ locking between STT generators,20 formulated in terms of Ds is a convenient way to use Eq. 12 and express all the n⌼ Ͼ ⌼ץ the “complex order parameter,” ⌿, seems similar in spirit to higher order s derivatives, s , n 1, in terms of only 21 ⌼គ ⌼ and . Ds acts on tensors, which are functions of only, as the Josephson phenomenon. In Sec. IV F, we will apply ␣ ␣ ␣ ␣ 1¯ n 1¯ n ⌼គ ␥ ͉ DsA␤ ␤ =A␤ ␤ ͉␥ , where denotes the covariant de- this picture to the problem of a single STT generator. 1¯ m 1¯ m 2 ⌼គ c rivative on S . In particular, Dsgab=gab͉c =0. Furthermore, III. LAGRANGIAN FORMALISM FOR MAGNONS IN AN ⌼គ a Ds acts on terms which involve as INSTANTON BACKGROUND D ⌼គ a =0, A. Covariant background field method s ⌼b͒⌼គ c ץ͑ ⌼គ a ⌫a ץ ⌼a ץ ⌼គ a ϵ Turning now to the construction of the Lagrangian for- Dx Ds x = x + bc x , malism for magnons, we note that magnons are the dynami- , ⌼d͒ =−Ja⌼គ bJ ץcal fluctuations of the magnetization around its stationary D D ⌼គ a = Ra ⌼គ b⌼គ c͑ ⌿ ͑ ͒ H s x bcd x b x configurations, 0 r , which is a “local” minimum of and which represents one of the infinite number of possible in- D2D ⌼គ a = Ra ⌼គ b⌼គ c͑D ⌼គ d͒ =−Ja⌼គ bj , stanton backgrounds, i.e., “magnon vacua.” The naive differ- s x bcd x b x ␦⌿͑ ͒ ⌿͑ ͒ ⌿ ͑ ͒ ␮ R ence, r,t = r,t − 0 r , however, is a bad choice for where the subscript x can be either or t, abcd=gadgbc 2 ␦⌿ 2 the magnon field. Indeed, S is not a linear space and is −gacgbd is the Riemann curvature tensor on S , and we intro- ill defined from the geometrical point of view. From the duced ⌼c ⌼គ b Jb ⌼គ c ץphysical point of view, magnons must exhibit such effects as ⌼គ a Jb interference, i.e., the magnon fields should observe the prin- Jx = gab c x and jx = gab cDx . ciple of superposition and thus must belong to a linear space, R Note that Ds abcd=0 as it should for the constant curvature while ␦⌿’s do not. The linear space under consideration is S2. Other useful relations needed in the following are D J 1 2 s x the tangent space, T⌿ ͑ ͒ ӍC —a complex plane touching S ␻ 0 r = jx, Dsjx =Jx , and consequently at point ⌿ ͑r͒෈S2 ͓see Fig. 1͑a͔͒. 0 2k+1 ⌼គ a Ja⌼គ b␻k The appropriate way to introduce the magnon field is Ds Dx =− b Jx, known as the CBF method, which is a basic ingredient of all 12 2k+2 ⌼គ a Ja⌼គ b␻k NLSMs. What is new in our case, compared with the con- Ds Dx =− b jx, ventional NLSMs, is the dynamical term in the Lagrangian, where we introduced ␻=⌼គ ag ⌼គ b ͑note that D ␻=0͒. T, which is only first order in the time derivative of the ab s In the new notations, Eq. ͑14a͒ can be rewritten as fields. However, by the virtue of the covariance of the mag- ͒ ͑ ⌼ ⌼គ a͒ ץL͑1͒ ͑ ⌼គ a ͔͒ ͑ ͓ netodynamical action Eq. 6 , this does not introduce any = Jt − D␮ ␮ a + U͉a . 14b additional complications and the CBF method is directly ap- ͑ ͒ plicable in our case. Acting once by Ds on Eq. 14b we arrive at the second ͑ ͒ One starts with defining the one-parameter geodesic flow, Gaussian term, ⌼͑ ͒ ෈͓ ͔ ͑2͒ a 2 a b r,t,s , s 0,1 , which connects the background con- L = j − ͑D␮⌼គ D␮⌼គ + J + U͉ ⌼គ ⌼គ ͒. ͑15͒ ⌼͑ ͒ ⌿ ͑ ͒ t a ␮ ab figuration, r,t,0 = 0 r , and the magnonically excited S͑2͒ configuration, ⌼͑r,t,1͒=⌿͑r,t͒, and satisfies Further acting recursively by Ds on , higher order terms in Eq. ͑13͒ turn out to be ⌼គ + ⌫1 ͑⌼͒⌼គ ⌼គ =0, ͑12͒ ץ s 11 ␻k−1͑␻ k ͒ ͑n͒ Jt −4J␮j␮ − U , ⌼͑r t s͒ L͑n͒ = ͭ ͮ ͑16͒ ץ⌼គ ͑r t s͒ϵ where , , s , , . Now, according to the physical ␻k−1͓␻ k͑ 2 ␻ 2 ͔͒ ͑n͒ essence of magnons as of fluctuations, the magnonic action jt −4 j␮ + J␮ − U , ⌬SϵS͑⌿͒ S͑⌿ ͒ S͑⌼͉͒s=1 S ͑ ͒ is − 0 = s=0, with from Eq. 6 .Itcan where the first line is for odd n=2k+1, the second line is for ͑n͒ a a be given as a Taylor series, even n=2k+2, and U =U͉ ⌼គ 1 ⌼គ n. To get the mag- a1¯an ¯ ϱ nonic action out of Eqs. ͑13a͒, ͑13b͒, ͑14a͒, ͑14b͒, ͑15͒, and ⌬S = ͚ ͑1/n!͒S͑n͒͑⌼͉͒ , ͑13a͒ n=1 s=0 ͑16͒, one allows s→0 by the following substitutions: ⌼͑ ͒ → ⌼͑ ͒ϵ⌿ ͑ ͒ r,t,s r,t,0 0 r,t , ͒ ͑ nS͑⌼͒ ͵ L͑n͒͑⌼͒͒ ץ͑ S͑n͒ = s = . 13b r,t ⌼គ ͑r,t,s͒ → ⌼គ ͑r,t,0͒ϵ⌿គ ͑r,t͒, The first s derivative of the action results,22 and

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͑ ͒ ͑ ͒ ͒ /L n ͓⌼͑r,t,s͔͒ →Ln ͓⌿ ͑r,t͒,⌿គ ͑r,t͔͒. ⌬L ¯␺ ٌ ␺ ٌ͉͑ ␺͉2 2 / ͑2͒ 0 = i t − ␮ + J␮ 2+U 2 ͑ ͒ ϱ Note also that the term in the parentheses in Eq. 14a is zero ͑ ͒ ͑ ͒ ͑1͒ ͑ ͉␺͉2 3 ͒/ L n / ͑ ͒ on s→0 due to the EoM ͓Eq. ͑5c͔͒. Therefore, L does not + 2 Jt −4J␮j␮ − U 6+͚ n!, 19 contribute to the magnonic action ͓Eq. ͑13͔͒, which thus n=4 starts with the second ͑Gaussian͒ term, as it should for a ͑ ͒ stable instanton background/magnon vacuum. where L n ’s ͑nϾ3͒ are given by Eq. ͑16͒ with the appropriate substitutions of Eq. ͑17͒. B. Magnon ﬁeld: Vielbeins

The predecessor of the magnon field is ⌿គ ͑r,t͒෈T⌿ ͑ ͒.It IV. DISCUSSION 0 r is the vector field touching S2 along the geodesic, ⌼͑r,t,s͒, connecting the background magnetization configuration, Previous sections contain the main results of this work, ⌿ ͑ ͒ ⌿͑ ͒ while here we would like to exemplify the advantages of our 0 r , and the magnonically excited configuration, r,t . The magnitude of ⌿គ ͑r,t͒ related to the “length” of this geo- approach, to discuss some new insights enabled by it, and to ⌿គ ͑ ͒ ͓⌿ ͑ ͔͒ outline possible future developments. Before we proceed desic. The metric for r,t , which for now is g11¯ 0 r ,is very inconveniently dependent on the position on S2, several comments are in order. Throughout the paper, we ͑ ͒ ͑ ͒ whereas S2 is a homogeneous space with all the points being kept referring to the two LFs given by Eq. 6 and Eq. 19 as equivalent to each other. the LL magnetodynamics and Magnonics, respectively. It is understood, however, that the two LFs are in fact equivalent By the virtue of that T⌿ ͑ ͒ is a linear space, one has the 0 r freedom to chose on it any linear basis. The basis has to be to each other. The question of which of the two to use is a matter of convenience. The use of Magnonic LF ͑19͒ is pref- chosen from the requirement that the metric on T⌿ ͑ ͒ is in- 0 r erable for those problems, which deal with relatively small dependent on ⌿ ͑r͒. Such basis is known as vielbeins in 0 fluctuations in the magnetization ͑Secs. IV A–IV E͒, while which for the “full-range-magnetization” problems the magnetody- a a −K a namical LF ͑19͒ is convenient ͑Secs. IV F and IV G͒. ⌿គ ͑r,t͒ → ␺ ͑r,t͒ = e 0⌿គ ͑r,t͒, ͑17a͒ Equation ͑19͒ is the Lagrangian density of the nonrelativistic spinless bosons, whereas magnons, being the excitation 01 → ͩ ͪ ͑ ͒ of the 3D-vector-Magnetization, are typically referred to as a gab , 17b vector ͑spin-1͒ excitation. The point here is that in the fixed- 10ab magnitude approsimation, which we have used so far ͑see, however, Sec. IV G for the generalization͒, the target space ͒ ͑ Ϯ ͒␺a ץ͑ ⌿គ a → ٌ ␺a Dx x = x iAx , 17c is 2D and the magnon field has only two real components, which being combined into a complex “boson wave func- K tion,” ␺, leave no other indices for the field, i.e., the zero- ⌿ + c.c., ͑17d͒ ץJ → ¯␺e− 0i x x 0 spin complex field. j → ¯␺iٌ ␺ + c.c., ͑17e͒ x x A. Berry phase

␻ → 2͉␺͉2, ͑17f͒ The above analogy between quantum particles and magnons allows to draw parallels between quantum mechanics and Magnonics. Indeed, it is a straightforward observation ͑n͒ nK a a U → e 0U͉ ␺ 1 ␺ n, ͑17g͒ ͑ a1¯an ¯ that the coupling to the gauge field A␮ via the “long” derivatives ٌ␮͒ results in the instanton-induced phase shift in ␮ K ϵK͓⌿ ͑ ͔͒ Ϯ where x again denotes either or t, 0 0 r , is for magnons. The shift is defined as the integral along the mag- a=1,¯ 1, respectively, and the gauge field is non propagation trajectory, l,

K ⌿a, ͑18͒ ץ ⌿ ͒ +c.c.=2␪ ץA = e− 0⌿¯ ͑i x 0 x 0 a x 0 ⌬͑ ␺͒ ͵ ͵ ␪ ͑⌿ ͒ ⌿a ͑ ͒ Im ln = i A␮dl␮ =2i a 0 d 0, 20 ␪ ͑ ͒ l CЈ with a being the Poincáre 1-form from Eq. 6b . Note also ⌿ ץ that At , Jt =0 as long as t 0 =0. ⌿͑ ͒ ⌿ ͑ ͒ ͑ ͒ Ј r,t and 0 r uniquely define the configuration of the where the second equality is due to Eq. 18 and C is the 2 ⌿ ͑ ͒ magnon field, ␺͑r,t͒෈T⌿ ͑ ͒, which thus can be used instead curve on S formed by 0 l as the magnon travels along l in 0 r of ␦⌿͑r,t͒ as discussed in the beginning of Sec. II A. the real space. It can be shown that the integral ͓Eq. ͑20͔͒ is zero along a geodesic. Therefore, CЈ can be closed into a loop C by a geodesic connecting the initial and final position C. Ginzburg-Landau magnonic action ⌿ ͓ ͑ ͔͒ of 0 see Fig. 1 b . Now, C=dV is a boundary of the ␺ ⌬S ͐ ⌬L 2 ͓ ͑ ͔͒ In terms of ’s, the magnonic action is = r,t , with corresponding 2D area, V,onS . The integral Eq. 20 can the Lagrangian density ͑up to the third order explicitly͒ being be further evaluated by the Stocks theorem

024410-5 IGOR V. OVCHINNIKOV AND KANG L. WANG PHYSICAL REVIEW B 82, 024410 ͑2010͒

a a b ␻ ␺ ͑z͒ = Hˆ ␺ ͑z͒ + Hˆ ¯␺ ͑z͒, ͵ ␪ ͑⌿ ͒ ⌿ ͵ ⍀ ⌿ ∧ ⌿ ͑ ͒ n,kʈ n,kʈ 1 n,kʈ 2 n,−kʈ 2i a 0 d 0 =2i abd d 21 dV V ⍀ ∧ with ab being the symplectic form from Sec. II B and − ␻ ¯␺ ͑z͒ = Hˆ †␺ ͑z͒ + Hˆ ¯␺ ͑z͒, being the so-called wedge product. We recall now that the n,kʈ n,−kʈ 2 n,kʈ 1 n,−kʈ invariant volume on S2 is ϳ⍀ d⌿a ∧d⌿b, ͑Ref. 11͒ so that i␾ 2 ˆ 2 2 2ץ ˆ ab the integral ͑21͒ is the solid angle corresponding to V. This is where H1 =− z +kʈ +1−sech z, H2 =e sech z, kʈ is the in- essentially the Berry phase,23 experienced by a magnon trav- plane momentum, and n is the quantum number of the spatial 2 ⌿ ͑ ͒ quantization in the z direction. The results of the numerical eling on S and following 0 l , which in our case plays the role of the adiabatic parameter. diagonalization, given in Fig. 1͑c͒, reveal the existence of the The instanton-induced phase shift in magnons was dem- Goldstone mode ͑GM͒.9 This is expectable, since away from ⌿ ϱ ͑ ͒ ͑ ͒ onstrated numerically in Ref. 6 and it is Eq. ͑19͒ which un- the DW, where 0 =0, , Eq. 19 is U 1 invariant, which covers its purely geometric origin. Note that the geometric is merely the rotational symmetry around the easy axis, phase is already known in magnetics, though in a different while the DW breaks this invariance leading to the appear- context of the phase acquired by electrons traveling through ance of the GM localized on the DW. The DW GM can be magnetic instantons.8 viewed as the Bogoliubov sound due to the presence of the magnon condensate, i.e., of the DW. The physical essence of the DW GM is the fluctuations in ␾ ͓see Fig. 1͑d͔͒ and the B. Instanton background as a Bose condensate fact that only one component of mជ is involved ͑unlike for ជ Some of the terms in Eq. ͑19͒ have unbalanced number of magnons in which the two components of m constitute the fluctuating complex field ␺͒ can be looked upon as the ͑␺͒ ͑¯␺͒ “annihilation” and “creation” fields, e.g., the term squeezing of the magnon quantum between the states with ⌿¯ ͒2 2 ץϳ͑␺ ␮ 0 contained in J␮. More terms of this kind may the opposite ͑in-plane͒ momenta—a feature pertinent to any also come from the potential, U,asinEq.͑22͒ below. Such homogeneous ͑in the in-plane sense in our case͒ Bose- terms explicitly break the gauge invariance—the symmetry condensed system. closely related to the particle number conservation. The physical picture is analogous to that of a Bose-condensed ⌿ D. Ginzburg-Landau perturbation series expansion system. Magnons can get in and out of the condensate, 0, while the total number of particles is conserved. Such pro- Nonlinearities experienced by magnons can be due to the cesses must lead to the quantum interference phenomena. interaction with other parts of the system, e.g., with phonons, For example, on passing through instantons a magnon should or “geometrical,” i.e., instanton assisted or due solely to the also experience what is known in quantum optics as nonlinearity of the target space, S2. At this, the geometrical 24 squeezing —a phenomenon useful for the phase-noise re- nonlinearities, such as the DW-assisted three-magnon scat- duction. In fact, the squeezing is not the only application of tering given in Figs. 1͑c͒ and 1͑d͒, are dominant as we dis- the quantum interference in photonics. Another example is cuss in the next section. Equation ͑19͒ is the first systematic 25 the so-called entanglement and the associated teleportation account of the geometric nonlinearities to all the orders and problem. Similar techniques could in principle be developed it serves as the starting point for the development of the for magnonics as well. Yet another point is that these cou- dynamical perturbation series expansion. In particular, Eq. plings may also contribute to the phase shift in magnons, in ͑19͒ can be used to renormalize the magnetodynamics given addition to the Berry phase. by Eq. ͑6͒. At this, ␺, must be viewed as the ͑dynamically or thermally͒ fluctuating field and the fluctuational corrections will modify the original action Eq. ͑6͒ for the instanton back- C. Magnon spectrum in an instanton background ⌿ ground, 0. In this line of thinking, it may be possible to get Equation ͑19͒ is a convenient tool for the systematic stud- some additional insights on the Bose condensation of ies of the magnon spectrum in various nontrivial instanon magnons,26 which thus must be viewed as the ͑thermal or ⌿ ͑2͒ ͒ backgrounds: for a given 0, one finds A␮, U , and diago- quantum -fluctuation-induced reconfiguration of the instan- ⌿ nalizes the Gaussian part of ͑i.e., the first line of͒ Eq. ͑19͒. ton background, 0. As an example, let us consider a DW in the local uniaxial anisotropy potential: Uϳm2 +m2 ϳe−K −e−2K ͑the anisotropy x y E. Three-magnon scattering constant can be set unity by the appropriate choice of units ͒ ⌿ ͑ ͒ i␾ z ␾ for energy . In this case, 0 z =e e , where is the azi- As we just mentioned in the previous section, the geo- muthal angle of mជ on the DW and z is the out-of-plane co- metrical nonlinearites are dominant in the magnon system. ⌿ ordinate, A␮ =0 because 0 follows a geodesic, and the To see this, consider the lowest-order nonlinearity—the Gaussian part of the potential term in Eq. ͑19͒ is three-magnon scattering, which is known to be the most pro- 10 K K K nounced nonlinear process in ferromagnets, and which as ͑ / ͒ 2 0 ␺a␺b ͑ ͉⌿ ͉2 −2 0͉͒␺͉2 −2 0͓͑⌿¯ ␺͒2 1 2 e U͉ab = 1−6 0 e − e 0 well can be phonon ͑or other “external” excitation͒ assisted. ͔ ͑ ͒ The geometrical three-magnon vertex in the first two terms ⌿ ץc.c. . 22 ͑ ͒ ϳ + of the second line of Eq. 19 is ␮ 0 while the vertex The linearized CSE to be diagonalized is coming from the potential is

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1 3K a b c −2K 2 2 e 0U͉ ␺ ␺ ␺ =2e 0͉͑⌿ ͉ −1͉͒␺͉ ͑⌿ ¯␺ + c.c.͒, 6 abc 0 0 ͑⌿ → ϱ͒ which vanishes when mជ is along the easy axis 0 0, . In other words, the geometrical three-magnon scattering is instanton assisted only. This fact will not be voided by the inclusion of the effects of the external and demagnetization magnetic fields if the both fields are parallel to the easy axis. In case of strong external fields, the ferromagnet is poled, there are no instantons, and the geometrical three-magnon scattering disappears. It is also understood that the phonon- assisted three-magnon scattering is not supposed to crucially depend on the external field. It is experimentally observed27 that in strong external fields the total three-magnon scattering weakens dramatically. Summing the above leads to the conclusion that the geometrical three-magnon scattering is indeed the strongest. This conclusion must hold as well for higher order nonlinearities, the full account of which is now enabled by Eq. ͑19͒.

F. STT generation as a Josephson supercurrent As we already mentioned in Sec. II C, the proposed La- grangian formulation for magnetodynamics in terms of the complex order parameter, ⌿, may provide a possibly useful alternative description of STT generators.28 Let us address here the problem of a single STT generator. The essence of the device operation is presented in Fig. ͑ ͒ ͑ ͒ FIG. 2. ͑Color online͒͑Top͒ An STT stack device and its 2 a . A dc bias, Vg, pushes electrons from the fixed layer FL ͑ ͒ into the free layer ͑fL͒. The magnetizations of the fL and FL energy-band profile. The FL, separated from the fL by a contact C , has a perpendicular magnetization ͑horizontal arrow͒ and higher are perpendicular to each other. Upon the thermalization to electrochemical potential raised by bias voltage, V . Being pushed the new equilibrium conditions, the arriving electrons give g from the FL to fL, an electron, e, brings in the out-of-equilibrium away their out-of-equilibrium energy and spin momentum to ͑ ͒ 29 energy and spin momentum short arrow , both being transferred to the magnetization of the fL. This is the “swaser” picture of the fL magnetization order parameter upon the thermalization the STT generation. ͑dashed arrows͒. If the energy drop, ␻, in the thermalization process The effective coupling between the magnetization of the of an electron is greater than the magnon gap, a magnon, ¯␺,is fL and the nonequilibrium electrons, which arrive from the created within its density of states ␯͑␻͒ and under certain condi- FL, is given to the lowest order in the Lagrangian density as tions the generation may occur. This picture of the STT generation L ϳ ␽ជ ⌿-el mជ , where mជ is the magnetization of the fL and corresponds to ␻→1 case below. ͑Bottom͒ The magnetization, ⌰, ␽ជ ͑r,t͒ is the appropriate nonequilibrium electron-density vs the effective pumping, ␽, as provided by Eq. ͑23͒ for ␣=0.1 and matrix. In the ⌿ coordinates for the fL magnetization, the for 0.1Ͻ␻Ͻ0.7. The upper branches of the curves correspond to effective coupling resembles that in the Josephson problem, the generation solutions. ͑Inset͒ The generation solutions exist if ͉␽͑␻͉͒2 ͑thick curve͒ is greater than f͑␻,␣͒͑a set of thin curves L −K͓␽͑ ͒⌿¯ ͑ ͒ ͔ ␣ ͒ ␻ ⌿-el = e r,t r,t + c.c. given for several ’s for some range of , as indicated by the shaded area for the case ␣=0.25. ␽ϳ␽ ␽ with x +i y being the appropriately redefined source term. In principle, ␽ can be obtained using the Keldysh non- of ͉⌿͉Ӷ1, this complex order parameter picture reduces to 14 equilibrium diagrammatic technique, and its exact form de- the coherent states picture of Ref. 30. pends on the physical picture adopted for the electron ͑spin͒ In case of the single-domain STT stack geometry, where injection. In particular, to make the full analogy with the the spatial dependence can be neglected, Eq. ͑23͒ provides a iV t Josephson phenomenon, one should chose ␪ϳe g .Wedo simple solution for the problem of the steady-state genera- not need, however, the exact form of ␽ for our qualitative tion, ⌿͑t͒→⌰e−i␻t, ͉␽͑␻͉͒2 =͉⌰͉2͕͓e−K͑1−͉⌰͉2͒−␻͔2 +␣2͖, consideration. ␽͑␻͒ ͐ ␽͑ ͒ i␻t where = t t e . To answer the question of whether Now, the magnetodynamics is governed by the nonhomo- the generation occurs, one has to find out if for a given ␽ geneous CSE ͓Eq. ͑5͔͒, there exists a range of ␻, corresponding to the generation −K 2 solutions ͓see Fig. 2͑b͔͒. This criterion can be written as ␮ + e ͑1−͉⌿͉ ͔͒⌿, ͑23͒ץi␣͒⌿ = ␽ + ͓− D␮ + ץi͑ t ͉␽͑␻͉͒2 Ͼ f͑␻,␣͒, where f is a complicated analytical func- ␣ ͑␻ ␣͉͒ ͑ / ͒͑ ␻͒/͑ ␻͒ where we phenomenologically included the damping, , and tion such that f , ␣→0 = 1 4 1− 1+ . Within this used again the uniaxial potential approximation. In the limit range of ␻, the sought ␻/⌰ pair is the one which extremizes

024410-7 IGOR V. OVCHINNIKOV AND KANG L. WANG PHYSICAL REVIEW B 82, 024410 ͑2010͒ the GL action corresponding to Eq. ͑23͒. Note also that the in choosing v⌽, which is thus case dependent. The same rotation in the “southern” hemisphere, ͉⌰͉Ͼ1, is in the op- could be said about function f͑⌽͒, which cannot be unam- posite temporal direction, ␻Ͻ0. biguously fixed by the covariance and which must be deter- In general, Eq. ͑23͒, accompanied by its parental action, is mined from some additional physical arguments. more trackable than the ordinary LLE and most importantly We believe that those must be the scaling arguments. For it provides the STT generation problem with the action mini- example, the requirement that it is only the potential U⌽, mization principle. It is also interesting to note that accord- which can break the scaling symmetry, leads to the conclu- ing to the above analogy with the Josephson phenomenon, sion that f =e−⌽. With this choice of f, the “potential-free the STT generation is the ferromagnetic counterpart of the action” ͑without U⌽͒ is invariant under the scaling transfor- ϳ ͑ ͒ → ␮ → ␮ ⌽→⌽ ␮ oscillating tunneling supercurrent, sin Vgt , between two mation: t t , r r , +log . Another possibility, superconducting order parameters externally biased by a which can be appropriate in case a ferromagnet is close to its 31 voltage, Vg. Hertz-Millis ferromagnetic quantum transition, is to use a separate scaling law for time as compared to space, with the G. Soft-mode fluctuations as a dilation field relative scaling exponent typically denoted as zϷ3. At this, however, in order for the potential-free action to be invariant In fact, magnons, which are the fluctuations in the orien- under: t→t␮z , r→r␮, ⌽→⌽+log ␮, the initially con- tation of the magnetization, are not the only excitations of stant v⌽ must now be a function of ⌽. the magnetization background. Where are also ͑gapped͒ fluctuations in the length of the magnetization known as the soft mode. These fluctuations can be incorporated into the LF by V. CONCLUSION allowing m0 to vary. This can be done, in turn, by the intro- In conclusion, we showed that the Landau-Lifshitz mag- ⌽ → ⌽ duction of the so-called dilation field, :m0 m0e . In other netodynamics is a member of the covariant-Schrödinger- words, this means we recall that the magnetization vector is equation family of Hamilton models. Then, using the cova- 3 3D, mជ ෈R and make the coordinate transformation mជ riant background field method we derived the Ginzburg- a 3 2 →͑⌿ ,⌽͒, R →S R. The invariant length of the magne- Landau Lagrangian formalism for magnons in an instanton tization vector takes the following form: background. The magnonic action turned out to be that of ͑ ͒ dmជ 2 ϳ e2⌽͑d⌽2 + g d⌿ad⌿b͒. ͑24͒ nonrelativistic spinless bosons coupled to the U 1 gauge ab field and to the Bose condensate, both representing instan- In terms of the dilation field, ⌽͑r,t͒, the soft-mode fluctua- tons. This allowed us to unveil the geometric origin of the tions are accounted for by turning to the following Lagrang- instanton-induced phase shifts in magnons; to develop a con- ian density, venient method of studying magnon spectrum in nontrivial instanton backgrounds and to demonstrate its usefulness by ͒ /⌿b ץ⌿a ץ ͑ ⌿a ץ L͑⌽ ⌿␣͒ 2⌽͕ ͑⌽͒␪ , = e f a t − gab ␮ ␮ 2+U⌿ considering the magnonic Goldstone mode on a domain -⌽͒2 ͑⌽͔͖͒ ͑ ͒ wall; to enable the Ginzburg-Landau dynamical diagram ץ͓͑ ⌽͒2 ץ−2͑ + v⌽ t − ␮ + U⌽ . 25 matic technique for magnons; to reconsider the problem of ␪ ͑ ͒ ͑ ͒ Here a and gab are, respectively, from Eqs. 3 and 6b , the spin-transfer-torque generation as the Josephson phenom- ␣ −2⌽ ␣ U⌿͑⌿ ͒=e U͑⌿ ͒ is the “anisotropy” potential, which re- enon; and to incorporate the soft-mode fluctuations into the ␣ −2⌽ ␣ lates to the previously used one as U⌿͑⌿ ͒=e U͑⌿ ͒, proposed approach. U⌽͑⌽͒ is the Landau part, which forces ⌽=0 be a ground- state value, v⌽ is a constant, which could be roughly called ACKNOWLEDGMENTS the ‘‘velocity’’ of the soft mode, and f͑⌽͒ is some function 2 ␮⌽͒ comes The work was supported in part by Western Institute ofץ͑ of ⌽. Note that according to Eq. ͑24͒, the term from the GL functional on the same footing with the term Nanoelectronics ͑WIN͒ funded by the NRI, Center on Func- ͒ ͑ ⌿b ץ⌿a ץ gab ␮ ␮ . Therefore, the constant in front of it is fixed tional Engineered NanoArchitectonics FENA funded by by the covariance arguments. In contrary, there is a freedom FCRP, and ARO MURI.

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