Quantum Brownian Motion of a Magnetic Skyrmion
QQuantum Brownian Motion of a Magnetic Skyrmion
Christina Psaroudaki, Pavel Aseev, and Daniel Loss Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland (Dated: April 22, 2019)Within a microscopic theory, we study the quantum Brownian motion of a skyrmion in a mag-netic insulator coupled to a bath of magnon-like quantum excitations. The intrinsic skyrmion-bathcoupling gives rise to damping terms for the skyrmion center-of-mass, which remain finite downto zero temperature due to the quantum nature of the magnon bath. We show that the quantumversion of the fluctuation-dissipation theorem acquires a non-trivial temperature dependence. As aconsequence, the skyrmion mean square displacement is finite at zero temperature and has a fastthermal activation that scales quadratically with temperature, contrary to the linear increase pre-dicted by the classical phenomenological theory. The effects of an external oscillating drive whichcouples directly on the magnon bath are investigated. We generalize the standard quantum theoryof dissipation and we show explicitly that additional time-dependent dissipation terms are generatedby the external drive. From these we emphasize a friction and a topological charge renormalizationterm, which are absent in the static limit. The skyrmion response function inherits the time pe-riodicity of the driving field and it is thus enhanced and lowered over a driving cycle. Finally, weprovide a generalized version of the nonequilibrium fluctuation-dissipation theorem valid for weaklydriven baths.
I. INTRODUCTION
The impact of the bath fluctuations on the dynam-ics of open nonequilibrium systems is commonly treatedby nonlinear stochastic differential equations for themacrovariables, known as generalized Langevin equations[1]. Within this description, the thermal bath exertsrandom fluctuating forces on the central system whicheventually undergoes a Brownian propagation [2–4]. Thesystem-bath coupling gives rise to non-Markovian mem-ory damping terms and random forces with a colored cor-relation [5]. In principle, both the noise and the damp-ing terms are determined by the system-bath interac-tion, a relation which is manifested in the well knownfluctuation-dissipation theorem [6].Quantum stochastic dynamics are present in a vari-ety of physical systems, ranging from quantum optics[7], transport processes in Josephson junctions [8], co-herence effects and macroscopic quantum tunnelling incondensed matter physics [9] and many more, whichform a large body of current active research. Here wefocus on the stochastic dynamics of particle-like mag-netic skyrmions, which similar to particle-like solitonictextures in quantum superfluids [10, 11], experience dis-sipative and stochastic forces from their environmentalsurroundings.Skyrmions are spatially localized two dimensional (2D)magnetic textures characterized by a topologically non-trivial charge Q [14, 15] given by Q = 14 π (cid:90) d r m · ( ∂ x m × ∂ y m ) , (1)where m is the normalized magnetization vector field and x and y are the spatial coordinates of the 2D magneticlayer. Besides their early theoretical prediction [12, 13],magnetic skyrmions have been observed in bulk metal-lic magnets [16–18], multiferroic insulators [19, 20] as well as ultrathin metal films on heavy-element substrates[21, 22]. Because of their protected topology, nanoscalesize, high mobility [23–27] and controllable creation [22],they are in the focus of current research as attractivecandidates for future spintronic devices [28].Classically, the dynamics of a magnetic skyrmion isgoverned by the Landau-Lifshitz- Gilbert (LLG) equa-tion [29, 30], which incorporates dissipation mechanismsby a phenomenological local in time Ohmic friction term,known as Gilbert damping. At finite temperatures, theskyrmion is subjected to thermal fluctuations that willrender its propagation stochastic, similarly to the Brow-nian motion of a particle. The conventional assumptionfor the fluctuating field acting on magnetic particles [31]as well as skyrmions [32–38], is that it is a Gaussianstochastic process with a white noise correlation functionproportional to the phenomenological Gilbert damping.In a magnetic insulator and at low enough tempera-ture, the skyrmion dynamics is dominated by the un-avoidable coupling of its center-of-mass with the mag-netic excitations generated by the skyrmion motion itself.Magnetic excitations are defined as fluctuations aroundthe classical skyrmion solution through a consistent sep-aration between collective (center-of-mass) and intrinsic(magnetic excitations) degrees of freedom. A descrip-tion of the dynamics of one-dimensional (1D) domainwalls [39] and 2D magnetic skyrmions [40] in a mag-netic insulator beyond the classical framework, demon-strated that the dissipation arising from the magneticexcitations is generally non-Markovian with a dampingkernel that is nonlocal in time. The quantum natureof the magnetic bath, naturally incorporated within thisapproach, becomes evident in the nontrivial temperature T dependence of the damping kernel which remains fi-nite even for vanishingly small T . A theory of dissipa-tion which ignores quantum effects based on the classicalphenomenological LLG equation is expected to be inade- a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r quate for atomic-size skyrmions observed in state-of-the-art experiments carried out at low temperatures of a fewK[21, 41, 42].In this paper we develop a microscopic descriptionof the skyrmion stochastic dynamics at finite tempera-ture using the functional Keldysh formalism for dissipa-tive quantum systems [43–45], as well as the Faddeev-Popov collective coordinate approach[39, 46] to promotethe skyrmion center-of-mass to a dynamic quantity. Wethen arrive at a Langevin equation of motion, which in-cludes a non-Markovian damping kernel and a stochasticfield with a colored autocorrelation function, as a resultof the skyrmion-magnon bath coupling. We demonstratethat the quantum version of the fluctuation-dissipationtheorem acquires a non-trivial temperature dependence.As an important consequence, the skyrmion mean squaredisplacement is finite at T = 0, and has a fast thermal ac-tivation being proportional to T for finite temperatures,in contrast to the linear T -increase obtained within theusual phenomenological theory [37].We also investigate the effects of an external oscillatingdrive which unavoidably couples with the magnon bath inan analogous fashion to many physical situations wherethe driving of the bath results in important contribu-tions to the dynamical response of the entire nanoscalesystem [47–50]. We demonstrate explicitly that addi-tional time-periodic dissipative terms are generated bythe driving field, in particular a friction and a topologi-cal charge renormalization term, which are both absentin the static limit. As a consequence, the skyrmion re-sponse function inherits the time periodicity of the drive,and it is thus enhanced and lowered over a driving cycle.Since the magnetic excitations are driven out of equilib-rium, a generalization of the fluctuation-dissipation theo-rem should not be expected in general. Quite remarkably,however, in the weak driving regime, we find a nonequi-librium fluctuation-dissipation relation, which reduces tothe equilibrium one in the static limit.For the efficient manipulation of skyrmions at thenanoscale it is important to understand how randomprocesses contribute to the skyrmion propagation, espe-cially in the presence of time-periodic microwave fieldswhich appear to be among the most efficient ways toinduce translational motion of skyrmions in magnetic in-sulators [51–53]. The microscopic understanding of thestochastic skyrmion motion becomes also important inview of proposed devices for stochastic computing basedon skyrmions [54, 55].The structure of the paper is as follows. In Sec. IIwe present a detailed derivation of the Langevin equa-tion for the skyrmion collective coordinate using thefunctional Keldysh formalism in the presence of a time-dependent magnetic field. In Sec. III we evaluate anddiscuss the damping kernel, while in Sec. IV we in-vestigate the skyrmion response function. The quan-tum fluctuation-dissipation theorem and its generalizednonequilibrium version in the presence of the oscillatingfield are presented in Sec. V, together with a discussion on the skyrmion mean square displacement. Our mainconclusions are summarized in Sec. VI, while some tech-nical details are deferred to four Appendices. II. LANGEVIN EQUATION
The purpose of this section is to present a derivation ofthe quantum Langevin equation for the skyrmion center-of-mass coordinate, by making use of a functional in-tegral approach for the magnetic degrees of freedom atfinite but low temperatures, combined with the Keldyshtechnique to include the effects of a time-dependent os-cillating magnetic field. To begin with, we note thatthe essential features of the dynamics of a normalizedmagnetization field in spherical parametrization m =[sin Θ cos Φ , sin Θ sin Φ , cos Θ] defined in the 2D space,are described by a partition function of the form Z = (cid:82) D Φ D Π e i S . Here, the functional integration is over allconfigurations and the field Π = cos Θ is canonically con-jugate to Φ. The Euclidean action S for a thin magneticinsulator in physical units of space ˜ r and time ˜ t is givenby S = (cid:90) d ˜ td ˜ r [ SN A α ˙Φ(Π − − N A W (Φ , Π)] , (2)where ˙Φ = ∂ ˜ t Φ denotes the real-time derivative of fieldΦ. The first term in Eq. (2) describes the dynamics andis known as the Wess-Zumino or Berry phase term [39],while the translationally symmetric energy term, W ( m ) = J ( ∇ ˜ r m ) + Dα m · ∇ ˜ r × m − Kα m z − gµ B Hα m z , (3)supports skyrmion configurations with nontrivial topo-logical number Q as metastable solutions dueto the presence of the Dzyaloshinskii-Moriya (DM)interaction[56, 57] of strength D . Here, ˜ r = (˜ x, ˜ y ), S isthe magnitude of the spin, N A is the number of magneticlayers along the perpendicular ˜ z axis and α is the latticespacing. The strength of the exchange interaction J , theeasy axis anisotropy K , and finally D are measured inunits of energy while the strength of the magnetic field H is given in units of Tesla (T).It is convenient to introduce dimensionless variables as r = ( D/Jα )˜ r , t = D ˜ t/J , and T = k B ˜ T J/D , where ˜ T is the temperature measured in Kelvin (K). Also, k B isthe Boltzmann constant and throughout this work we use (cid:126) = 1. The energy functional in reduced units is givenby F ( m ) = ( ∇ r m ) + m · ∇ r × m − κm z − hm z , (4)where κ = JK/D , h = Jgµ B H/D , and F ( m ) = J ( α ) /D W ( m ). The classical skyrmion field, denotedas Φ ( r ) and Π ( r ), is found by minimizing the energyfunctional F ( m ) [58, 59]. We then arrive at the fol-lowing rotationally symmetric solution in polar coordi-nates r = ( ρ cos φ, ρ sin φ ) given by Φ ( r ) = φ + π/ ⇢ ⇥ ( ⇢ ) h = 0 . h = 0 . h = 0 . x y ⇢ FIG. 1. Magnetization profiles Θ ( ρ ) of a skyrmion as func-tion of radial distance ρ for κ = 0 . of a skyrmion with Q = − xy plane for κ = 0 . h = 0 . while the skyrmion profile depends only on the radialcoordinate Θ ( r ) = Θ( ρ ). In Fig. 1 we depict themagnetization profile of the skyrmion Θ ( ρ ) for vari-ous values of the magnetic field h , using the trial func-tion Θ ( ρ ) = A cos − (tanh[( ρ − λ ) / ∆ ]), where A = π/ cos − (tanh[ − λ/ ∆ ]). The parameter λ , which de-notes the skyrmion size, and ∆ are calculated by fittingthe approximate function to the one obtained numeri-cally. This profile has a topological number Q = − and Θ incontact with the bath of magnetic excitations at finitetemperature driven by an external magnetic field thatoscillates in time. This is achieved by first promoting theskyrmion center-of-mass to a dynamical variable R ( t ),then treating the magnetic excitations as quantum fluc-tuations around the classical field, and finally obtainingan effective functional [1, 39, 40, 60] by integrating outthe magnon degrees of freedom. At the same time, thereal-time dynamics of the external field as well as thestochastic effects of the magnon bath at finite T are cap-tured by replacing the time integration by an integrationover the Keldysh contour which consists of two branches.The upper branch extends from t = −∞ to t = + ∞ ,while the lower branch extends backwards from t = ∞ to t = −∞ [45]. It is worth mentioning that the for-malism derived below is applicable to any general energyfunctional F as long as it satisfies the specified require-ments.We define two components of the fields as Φ + ≡ Φ( t + i
0) and Φ − ≡ Φ( t − i ± = Π( t ± i ± ( r , t ) = Φ ± ( r − R ± ( t )) + ϕ ± ( r − R ± ( t ) , t )Π ± ( r , t ) = Π ± ( r − R ± ( t )) + η ± ( r − R ± ( t ) , t ) , (5)where η and ϕ are the quantum fluctuations and thecoordinate R ( t ) is energy independent owing to the as-sumed translational invariance of the system. We there-fore expect the existence of a pair of zero modes Y i , with i = x, y , which need to be excluded from the functionalintegral to avoid overcounting degrees of freedom by im-posing proper gauge fixing conditions. We use the fol-lowing convenient spinor notation, χ ± = 12 (cid:18) ϕ ± sin Θ + iη ± / sin Θ ϕ ± sin Θ − iη ± / sin Θ (cid:19) , (6)and we also define linear transformations of the fieldsby performing a Keldysh rotation of the form χ c,q =( χ + ± χ − ) / √ R c,q = ( R + ± R − ) / √
2. Here, χ c ( R c ) and χ q ( R q ) denote the classical and quantumfluctuations (coordinate), respectively. Moreover, we in-troduce the field ζ = (cid:0) χ c χ q (cid:1) in order to obtain the action ina more compact form. Implementing all the above trans-formations in the action of Eq. (2), taking into accountthat time integration is now performed over the upperand lower time branches denoted by the symbol s = ± Z = (cid:82) D R c D R q e i S cl ˜ Z ,where˜ Z = (cid:90) D ζ † D ζ (cid:89) s = ± δ ( F sx ) δ ( F sy ) det( J F s ) e i S Q . (7)Here, F si = (cid:82) d r χ † s σ z Y i is the gauge condition and J s F ( t, t (cid:48) ) = d F s ( t ) /d R ( t (cid:48) ) is the Jacobian matrix of thecoordinate transformation and is treated as additionalperturbation to the N A -term in the action. The classicalpart of the effective action reads S cl = N A d (cid:90) t, r (cid:88) s = ± s [ − S ˙ R s Π s ∇ Φ s − b · m (Φ s , Π s )] , (8)where b ( t ) denotes a time-dependent external field, d =( J/D ) and we have also neglected an overall constantfrom the configuration energy of the classical skyrmion S = d (cid:82) r ,t F (Φ , Π ). The fluctuation-dependent part ofthe Keldysh action takes the form S Q = N A d ζ † ◦ ˆ G − ζ , (9)where ˆ G − = ( G − − V + √ K c ) σ x + √ K q . The magnonGreen function is G − = iSσ z ∂ t − H and the Hamilto-nian is defined as H = δ χ † δ χ F| χ = χ † =0 . The potential V ( r , t ) = b ( t ) · D with D = δ χ † δ χ m | χ = χ † =0 describes thecoupling of the external field with the magnons and it istreated as a time-dependent perturbation to the magnonHamiltonian. The magnetic fluctuations appear as solu-tions of the eigenvalue problem (EVP) H Ψ n = ε n σ z Ψ n ,solved in detail in Appendix D. Moreover, we define K s = − iSσ z ˙ R is Γ i , assuming that repeated indices, i, j = x, y ,are summed over and we also introduce the abbreviationΓ i = ∂ i − σ x cot Θ ∂ i Θ . The circular multiplicationsign in Eq. (9) implies convolution of the form ζ † ◦ G − ζ ≡ (cid:90) t, r (cid:90) t (cid:48) , r (cid:48) ζ † ( r , t ) G − ( r , r (cid:48) , t, t (cid:48) ) ζ ( r (cid:48) , t (cid:48) ) . (10)Note that Eq. 9 assumes the absence of potentials thatbreak translational symmetry which will generate ad-ditional classical dissipation terms [40] with interestingconsequences on the skyrmion dynamics in confined ge-ometries [53]. A considerable simplification is also pro-vided in the limit where the skyrmion configuration en-ergy S is much larger than the energy S B = d (cid:82) r ,t b ( t ) · m ( r , t ) added by the external applied field, S (cid:29) S B .In this case, m (Φ , Π ) is a good approximation for theskyrmion configuration, while terms linear in the fluctu-ations are negligibly small and do not appear in Eq. (9).To proceed we note that the functional ˜ Z is an inte-gral with a Gaussian form if we neglect terms O [1] in N A originating from the Jacobian determinant det( J F ).Thus, after integration, ˜ Z reduces to˜ Z = 1det (cid:48) ( − iN A d ˆ G − ) = e − Tr (cid:48) log[1+ G ( ˜ K− ˜ V )] det (cid:48) ( − iN A dG − ) , (11)with ˜ K = √ K c σ x + √ K q , G − = G − σ x , ˜ V = V σ x ,and the prime notation on the determinant and the traceexcludes the zero modes. By performing an expansionretaining terms up to the second order in ˙ R and first onein V , the effective action for the classical and quantumcoordinate is S eff = S cl − i (cid:48) [ G ˜ K G ˜ K − ∆ G ˜ K G ˜ K − G ˜ K ∆ G ˜ K ] , (12)where ∆ G = G ˜ V G . The advantage of the Keldyshrotation is that the operator G is identified with theGreen function of the fluctuations G = (cid:18) G K G R G A (cid:19) , (13)where G R,A = ( iSσ z ∂ t ± i − H ) − are the retarded andadvanced Green functions given in real time as G R,A ( t, t (cid:48) ) = ∓ iS σ z Θ( ± ( t − t (cid:48) )) T ± e − iσ z H ( t − t (cid:48) ) /S , (14)provided that T ± time orders in chronologi-cal/antichronological order. We parametrize theKeldysh Green function as G K = G R ◦ F − F ◦ G A ,where F = F ( t − t (cid:48) ) and in thermal equilibrium is givenby F ( ω ) = coth( βω/ β = 1 /T . The represe-nation in frequency space ω is obtained by the usualFourier transformation g ( t ) = (1 / π ) (cid:82) ∞−∞ dωg ( ω ) e − iωt . The standard way to calculate the quasiclassical equa-tion of motion for the skyrmion coordinate R c is to cal-culate the saddle point of the action (12) by extremizingwith respect to the quantum coordinate R q [63]. We notethat terms proportional to K q K c describe temperature-dependent dissipation due to magnon modes, while weshow explicitly that terms proportional to K q K q give riseto random forces. To distinguish between the contribu-tions from these terms we rewrite the effective action ofEq. (12) as S eff = S cl + S dis + S st , where the dissipativepart reads S dis = − i (cid:48) [ G K K c G A K q + G R K c G K K q + G K K q G R K c + G A K q G K K c ] , (15)where G i = G i − ∆ G i with i = R, A, K , ∆ G R,A = G R,A V G
R,A and ∆ G K = G R V G K + G K V G A . Similarly,the stochastic part is given by S st = − i (cid:48) [ G K K q G K K q + G R K q G A K q + G A K q G R K q ] ≡ − R iq ◦ C ij R jq . (16)The function C ij ( t, t (cid:48) ) is found by evaluating the traceappearing in Eq. (16) with the eigenstates Ψ ν ( r , t ) ofthe operator G and is given explicitly in Appendix A.To demonstrate that S st indeed gives rise to randomfluctuating forces, we introduce auxiliary fields ξ i via aHubbard-Stratonovich transformation, e i S st = det[(2 iC ) − ] (cid:90) D ξ D ξ † e i ( ξ † · (2 C ) − ξ + ξ † · ¯ R q + ¯ R † q · ξ ) , (17)where ¯ R q = (cid:0) R xq R yq (cid:1) / √ ξ = (cid:0) ξ x ξ y (cid:1) / √
2. Minimizing ther.h.s. of Eq.(17) with respect to R jq results in a randomforce term ξ j in the equation of motion characterized byan ensemble average of the form (cid:104) ξ ( t ) (cid:105) = 0 , (cid:104) ξ i ( t ) ξ j ( t (cid:48) ) (cid:105) = − iC ij ( t, t (cid:48) ) , (18)where (cid:104) . . . (cid:105) = det[(2 iC ) − ] (cid:82) D ξ D ξ † . . . e iξ † · (2 C ) − ξ . Byminimizing the effective action S eff , we obtain the dy-namical Langevin equation for the classical coordinate R c , ˜ Q (cid:15) ij ˙ R jc ( t ) + (cid:90) t −∞ dt (cid:48) ˙ R jc ( t (cid:48) ) γ ji ( t, t (cid:48) ) = ξ i ( t ) , (19)with ˜ Q = − πN A QSd , (cid:15) ij is the Levi-Civita tensor andthe time of preparation of the initial state is at t → −∞ .The first term in Eq. (19) is a Magnus force acting on theskyrmion and being proportional to the winding number[61, 62], while the nonlocal (in time) damping kernel isgiven by γ ji ( t, t (cid:48) ) = ∂ t [ γ RKji ( t, t (cid:48) )+ γ KAji ( t, t (cid:48) )+ γ KRij ( t (cid:48) , t )+ γ AKij ( t (cid:48) , t )] , (20)where γ abji ( t, t (cid:48) ) = − iS (cid:88) ν (cid:48) (cid:90) ¯ r , r , r (cid:48) Ψ ν (¯ r ) G a (¯ r , r , t, t (cid:48) ) σ z Γ j ( r ) × G b ( r , r (cid:48) , t (cid:48) , t ) σ z Γ i ( r (cid:48) ) σ z Ψ ν ( r (cid:48) ) , (21)with a, b = R, A, K .The damping kernel of Eq. (20) describes the dissipa-tion which originates from the coupling of the skyrmionto the quantum bath of magnetic excitations and hasan explicit temperature dependence through the KeldyshGreen function G K . Note that an external force actingon the skyrmion is absent, as a direct consequence ofthe spatial uniformity assumed for the external magneticfield. The translational motion of the skyrmion would beinduced by a spatially dependent magnetic field, for ex-ample a magnetic field gradient [64, 65], and its effect hasbeen studied in Ref. 53. Here, the external time-periodicfield acts on the quantum bath of magnons and is natu-rally incorporated in the stochastic Langevin equation ofEq. (19). This allow us to generalize the quantum theoryof dissipation to account for the effects of the driven bathin several observables related to the skyrmion dynamics. III. DAMPING KERNEL
Our next task is to analyze the damping kernel ofEq. (20) in the case of a driven bath. In Appendix Bwe obtain the real-time damping kernel γ ij ( t − t (cid:48) ) inthe absence of a drive, and thus establish agreementwith earlier results derived in Matsubara space using theimaginary-time functional integral approach [40]. Notethat, although the Laplace transform γ ij ( z ) is frequencydependent, we are usually interested in the long-timeasymptotic behavior of the skyrmion dynamics which isin turn determined by the low frequency part of the ker-nel. This low frequency regime is specified by the con-dition | ω | (cid:28) ε gap , with ω = (cid:60) ( iz ), ε gap = 2 κ + h , beingthe lowest magnon gap, while at the same time the tem-perature is limited to the quantum regime T (cid:28) ε gap .Thus, under the assumptions specified above the di-agonal damping kernel acquires the super-Ohmic powerlaw behavior γ ii ( z ) = z M ( T ) + O [( z/ε gap ) ]. Followingthe usual terminology [1], Ohmic friction is described bya damping term of the form zγ ( z ) ∝ z s with s = 1, whilefor s > T -dependent massis given by, M ( T ) = (cid:88) ν,ν (cid:48) (cid:48) (cid:60) ( B νν (cid:48) ii ) ¯ F νν (cid:48) ε ν (cid:48) − ε ν , (22)with ¯ F νν (cid:48) = F ( ε ν ) − F ( ε ν (cid:48) ) and F ( ε ν ) = coth( βε ν / ν = { q = ± , n } , where the index q distinguishes between parti-cle states ( q = 1), solutions of the eigenvalue problem H Ψ n = ε qn σ z Ψ n with positive eigenfrequency ε n = + ε n , . . . . ˜ T (K) . . . M ( ˜ T )( − k g ) . . . . ˜ T (K) ¯ W ii ( − N s / m ) FIG. 2. Temperature dependence of the quantum mass M ( ˜ T )of the skyrmion given in Eq. (22) for a static magnetic fieldof amplitude H = 216 mT, radius λ = 5 .
32 nm, and a choiceof J = 1 meV, S = 1 and J/D = 4, and α = 5 ˚A. Thedashed vertical line indicates the value of the magnon gap inunits of temperature ε gap = 0 .
435 K, up to which our resultis valid. The inset depicts the constant ¯ W ii of Eq. (30) foran oscillating magnetic field of amplitude b = 0 .
05 (27 mT), ω ext = 0 .
32 (4.8 GHz), and ϕ ext = π/ and antiparticle states ( q = −
1) with negative eigenfre-quency ε − n = − ε n [40]. The matrix elements are given by B νν (cid:48) ij = B n,q ; n (cid:48) ,q (cid:48) ij = ( qq (cid:48) / (cid:82) r Ψ † ν Γ i σ z Ψ ν (cid:48) (cid:82) r (cid:48) Ψ † ν (cid:48) Γ j σ z Ψ ν .Note that the expression of Eq. (22) is symmetric underthe exchange of indices ν and ν (cid:48) , and that there is nosingularity for ε ν = ε ν (cid:48) since lim ε ν → ε ν (cid:48) ¯ F ν (cid:48) ν / ( ε ν − ε ν (cid:48) ) = β/ ( βε ν / M ( T ) in the T → M (0) is finite andthat it is independent of the effective spin N A S , contraryto the magnus force proportional to ˜ Q = − πQN A Sd ,we refer to the mass of Eq. (22) as quantum mass . Thisterminology allows us to distinguish M ( T ) from the semi-classical mass already calculated in Ref. 40 in the pres-ence of spatial confinement, which scales linearly with N A S .The off-diagonal damping kernel has a super-Ohmiclow-frequency power law γ xy ( z ) ∝ z , irrelevant for theskyrmion dynamics at times t (cid:29) ε − . The T -dependenceof the quantum mass M ( T ) is depicted in Fig. 2 in physi-cal units for J = 1 meV, α = 5 ˚A, J/D = 4, h = 0 . λ = 2 .
67 (5.34 nm), κ = 0 .
1, and S = 1. Detailson the calculation are given in Appendix D.With this preparation, we are now in position to gen-eralize the damping kernel in the presence of the ex-ternal driving field turned on at time t = t , b ( t ) = b Θ( t − t ) cos( ω ext t )(sin ϕ ext , , cos ϕ ext ), tilted in the xz -plane with the angle ϕ ext away from the z -axis. In thepresence of b ( t ), the magnons are subjected to the po-tential V ( r , t ) = b Θ( t − t ) cos( ω ext t ) V ( r ), where V ( r )is given in Eq. (D6). The damping kernel of Eq. (20) ac-quires an additional correction due to the time-dependentfield, γ ji ( t, t (cid:48) ) = γ ji ( t − t (cid:48) ) + ∆ γ ji ( t, t (cid:48) ), where∆ γ ji ( t, t (cid:48) ) = ∂ t W ji ( t − t (cid:48) )[ g ji ext ( t ) + g ji ext ( t (cid:48) )] . (23)The function g ji ext ( t ) carries information on the externaldrive, g ji ext ( t ) = Θ( t − t ) b cos( ω ext t − | (cid:15) ji | π/ W ji ( t ) carries information about the magnon modes, W ji ( t ) = (cid:88) ν ,ν ,ν (cid:48) C ν ν ν ji [ w ν ν ( t ) − w ν ν ( t )]( ε ν − ε ν ) − ω ext , (24)where w ν ν ( t ) = Θ( t ) ¯ F ν ν sin[( ε ν − ε ν ) t ]. We also in-troduced the matrix elements C ν ν ν ii = q ν q ν q ν ( ε ν − ε ν ) (cid:60) ( b ν ν i V ν ν b ν ν i ) / S , C ν ν ν yx = q ν q ν q ν ω ext (cid:61) ( b ν ν y V ν ν b ν ν x ) / S , (25)where b ν ν i = (cid:82) r Ψ † ν Γ i σ z Ψ ν and V ν ν = (cid:82) r Ψ † ν V Ψ ν .We note that the triple summation over the magnonquantum numbers originates from the fact that the ex-ternal field induces a finite overlap, V ν ν (cid:54) = 0 for ν (cid:54) = ν .Note that Eq. (24) is valid only away from the resonancecondition ω ext = ε ν − ε ν , under the assumption thatthe external potential V induces only a small overlap0 < | V ν ν | (cid:28) ≤ | ε ν − ε ν | ≤ ε d and it also holds that ε d (cid:28) ω ext .In Fourier space with frequency ω , the equation of mo-tion given in Eq. (19) takes the form F ( t, ω ) = − iω [ ˜ Q (cid:15) ij + γ ji ( t, ω )] R jc ( ω ) − ξ i ( ω ) , (26)with F ( t, ω ) satisfying (cid:82) ∞−∞ dωe − iωt F ( t, ω ) = 0, andwhere γ ji ( t, ω ) = γ ji ( ω ) + ∆ γ ji ( t, ω ). It appears con-venient to calculate ∆ γ ji ( t, ω ) in Laplace space z with ω = (cid:60) ( iz ),∆ γ ji ( t, z ) = W ji ( z )[ zg ji ext ( t ) + ∂ t g ji ext ( t )] (27)+ b (cid:88) m = ± e − im ( ω ext t + π | (cid:15)ji | ) ( z + imω ext ) W ji ( z + imω ext ) . The correction to the damping kernel, ∆ γ ji ( t, z ), de-scribes the effects of the driven magnon bath on theskyrmion and is treated as a perturbation to γ ji ( z ).Here, W ji ( z ) is the Laplace transform of W ji ( t ) givenin Eq. (24). In Eq. (26) we assume that the time t co-incides with the preparation time of the initial state, i.e. t → −∞ , and we therefore neglect boundary terms thatdepend on t . A Taylor expansion around the origin, γ ji ( t, z ) (cid:39) γ ji ( t,
0) + z∂ z γ ji ( t, z ) | z =0 + O ( z ), valid forfrequencies ω (cid:28) ε gap , provides the low frequency power-law behavior of the damping kernel. For the diagonalpart we find∆ γ xx ( t, z ) (cid:39) D ( T ) sin( ω ext t ) + z δM ( T ) cos( ω ext t ) , (28) and similarly the off-diagonal corrections are∆ γ yx ( t, z ) (cid:39) δQ ( T ) cos( ω ext t ) + z G ( T ) sin( ω ext t ) . (29)Explicit expressions of the T -dependent coefficients ap-pearing in Eqs. (28) and (29) are given in Appendix C.As expected, in the static limit ω ext →
0, all the termsin Eqs. (28) and (29), except the mass renormalization,vanish. In the special case of ε d (cid:28) ω ext (cid:28) ε gap , where0 ≤ | ε ν − ε ν | ≤ ε d is the energy difference induced by theexternal potential V , we find the simplified expressions D ( T ) = − ω ext ¯ W ii , δM ( T ) = ¯ W ii , δQ ( T ) = ω ext ¯ W yx , and G ( T ) = ¯ W yx . The coefficient ¯ W ji is given by¯ W ji = (cid:88) ν ,ν ,ν (cid:48) C ν ν ν ji ω ext (cid:18) ¯ F ν ν ε ν − ε ν − ¯ F ν ν ε ν − ε ν (cid:19) , (30)where ¯ F νν (cid:48) is given after Eq. (22). From Eq. (30) and thestructure of the matrix elements of Eq. (25) it becomesapparent that ¯ W ji is symmetric under the exchange ofthe indices ν , ν , and ν . The temperature dependenceof the coefficient ¯ W ii is depicted in the inset of Fig. 2,for the choice ϕ ext = π/ b = 0 .
05 (27 mT), ω ext = 0 . h = 0 . γ xx ( t, z ) = ∆ γ yy ( t, z ) and ∆ γ xy ( t, z ) = − ∆ γ yx ( t, z ), thus the term δQ ( T ) cos( ω ext t ) can be con-sidered as a temperature- and time-dependent correctionto the topological charge ˜ Q , induced by the externaldrive. Similarly, the quantum mass acquires the correc-tion δM ( T ) cos( ω ext t ). The low-frequency linear depen-dence of the quantity zγ ji ( t, z ) signals a super-Ohmic toOhmic crossover behavior, with measurable consequenceson the skyrmion trajectory [53]. More specifically, the acdriving of the magnon bath at resonance displaces theskyrmion from its equilibrium position and results in aunidirectional helical propagation. IV. RESPONSE FUNCTION
In this section, we calculate the equilibrium skyrmionresponse function, which is then generalized to thenonequilibrium case of a driven bath of magnons. Thelinear response of the skyrmion to the fluctuating force ξ i ( t ) is encoded in the equilibrium response function χ ij ( t − t (cid:48) ) via the relation R ic ( t ) = (cid:90) t −∞ dt (cid:48) χ ij ( t − t (cid:48) ) ξ j ( t (cid:48) ) , (31)where the elements in Laplace space are χ ii ( z ) = γ ii ( z ) zπ ( z ) , χ yx ( z ) = ˜ Q + γ yx ( z ) zπ ( z ) , (32)with π ( z ) = [ ˜ Q + γ yx ( z )] + [ γ xx ( z )] and χ xy ( z ) = − χ yx ( z ). The response functions at finite frequency a ) z ˆ xx ( t, z ) ˜ Q ⇥ .
01 0 .
02 0 .
03 0 . z T ext T ext T ext T ext t .
005 0 . z .
005 0 . z z ˆ xx ( t , z ) ˜ Q ⇥ l l t = T ext / t = 3 T ext / l l / D ( T ) t = 2 T ext t = 5 T ext / h z = 0 h z = 0 b ) c ) / z ˜ Q a ) z ˆ xx ( t, z ) ˜ Q ⇥ .
01 0 .
02 0 .
03 0 . z T ext T ext T ext T ext t .
005 0 . z .
005 0 . z z ˆ xx ( t , z ) ˜ Q ⇥ l l t = T ext / t = 3 T ext / l l / D ( T ) t = 2 T ext t = 5 T ext / h z = 0 h z = 0 b ) c ) / z ˜ Q a ) z ˆ xx ( t , z ) ˜ Q ⇥ . . . . z T e x t T e x t T e x t T e x t t . . z . . z z ˆ xx ( t, z ) ˜ Q ⇥ ll t = T e x t / t = T e x t / l l / D ( T ) t = T e x t t = T e x t / h z = h z = b ) c ) / z ˜ Q FIG. 3. a ) The colored surface represents the time andLaplace frequency dependence of the diagonal response func-tion χ xx ( t, z ), given in Eq. (34), for T = 0 . ω ext =0 .
32 (4.8 GHz), ϕ ext = π/
4, and b = 0 .
05 (27 mT) the ampli-tude of the external field. The skyrmion is stabilized from auniform out-of-plane magnetic field of strength h = 0 . λ = 2 .
67 (5.32 nm), and we choose J = 1 meV, α = 0 . J/D = 4. Insets b ) and c ) depictthe frequency dependence of χ xx ( t, z ) at given times t , where T ext = 2 π/ω ext denotes the period of the external drive. χ ij ( z ) are dynamical observables carrying physical infor-mation on the skyrmion dynamics. By employing thelow-frequency power law behavior of γ ij ( z ), one findsthat χ ii ( z ) = M ( T ) / [ ˜ Q + M ( T ) z ] and χ yx ( z ) =˜ Q / [ z ( ˜ Q + M ( T ) z )]. The expansion at z = 0 yields χ ii ( z ) (cid:39) M ( T ) / ˜ Q + O ( z ). A finite static suscepti-bility χ = M ( T ) / ˜ Q implies that a free topologicalparticle with ˜ Q (cid:54) = 0 exhibits a different dynamical be-havior than the one with ˜ Q = 0. In particular, wenote that the static susceptibility χ is infinite for afreely moving and finite for a confined Brownian par-ticle [1]. For example, χ = 1 /ω for a damped har-monic oscillator of frequency ω [1]. Therefore, we seethat χ is finite due to the non-trivial ˜ Q , and as ex-pected, χ diverges for ˜ Q = 0. Moreover, the low fre-quency expansion for the off-diagonal response functionis χ yx ( z ) (cid:39) (1 / ˜ Q z ) + O ( z ), and in this case ˜ Q playsthe role of a velocity-dependent friction.The response of the skyrmion position R ( t ) when theexternal drive b ( t ) is turned on is encoded in the responsefunction χ ij ( t, t (cid:48) ) defined through the relation R ic ( t ) = (cid:90) t −∞ dt (cid:48) χ ij ( t, t (cid:48) ) ξ j ( t (cid:48) ) . (33)In an analogous fashion to the decomposition of thedamping kernel given in Eq. (23), we generalize the re- . . . . z T e x t T e x t T e x t T e x t t . . . . z . . . . z . . z [ ˆ yx ( t, z ) ˆ yx ( z )] ˜ Q ⇥ l l t = T e x t / t = T e x t / l l t = T e x t t = T e x t / z [ ˆ y x ( t , z ) ˆ y x ( z ) ] ˜ Q ⇥ / z ˜ Q ¯ / Q ( T ) b ) c ) a ) .
01 0 .
02 0 .
03 0 . z T ext T ext T ext T ext t .
01 0 .
02 0 .
03 0 . z .
01 0 .
02 0 .
03 0 . z . . z [ ˆ y x ( t , z ) ˆ y x ( z ) ] ˜ Q ⇥ l l t = T ext / t = 3 T ext / l l t = 2 T ext t = 5 T ext / z [ˆ yx ( t, z ) ˆ yx ( z )] ˜ Q ⇥ / z ˜ Q ¯ / Q ( T ) b ) c ) a ) .
01 0 .
02 0 .
03 0 . z T ext T ext T ext T ext t .
01 0 .
02 0 .
03 0 . z .
01 0 .
02 0 .
03 0 . z . . z [ ˆ y x ( t , z ) ˆ y x ( z ) ] ˜ Q ⇥ l l t = T ext / t = 3 T ext / l l t = 2 T ext t = 5 T ext / z [ˆ yx ( t, z ) ˆ yx ( z )] ˜ Q ⇥ / z ˜ Q ¯ / Q ( T ) b ) c ) a ) FIG. 4. a ) The colored surface represents the time andLaplace frequency dependence of the off-diagonal responsefunction χ yx ( t, z ), given in Eq. (35) for T = 0 . ω ext = 0 .
32 (4.8 GHz), ϕ ext = π/ b = 0 .
05 (27 mT)the amplitude of the external field. The skyrmion is stabi-lized from a uniform out-of-plane magnetic field of strength h = 0 . λ = 2 .
67 (5.32 nm),and we choose J = 1 meV, α = 0 . J/D = 4. In-sets b ) and c ) depict the frequency dependence of χ xx ( t, z ) atgiven times t , where T ext = 2 π/ω ext denotes the period of theexternal drive. sponse function as χ ij ( t, t (cid:48) ) = χ ij ( t − t (cid:48) ) + δχ ij ( t, t (cid:48) ).Starting from the equation of motion given in Eq. (19)and using Eq. (33), we solve for the function δχ ij ( t, ω ),defined as δχ ij ( t, t (cid:48) ) = (1 / π ) (cid:82) dωe − iω ( t − t (cid:48) ) δχ ij ( t, ω ),retaining first order terms in b . In Laplace space, byperforming an expansion of the full response function χ ij ( t, z ) = χ ij ( z ) + δχ ij ( t, z ) around z = 0 and keep-ing leading order terms in z , we find χ ii ( t, z ) (cid:39) D ( T )˜ Q z sin( ω ext t ) + χ + δχ cos( ω ext t ) , (34)with δχ = [ − M ( T ) δQ ( T ) + ˜ Q δM ( T )] / ˜ Q , and simi-larly χ yx ( t, z ) (cid:39) Q z − δQ ( T )˜ Q z cos( ω ext t )+ δ ¯ χ sin( ω ext t ) , (35)where δ ¯ χ = [ − M ( T ) D ( T ) − ˜ Q G ( T )] / ˜ Q . We observethat a new friction term emerges for the diagonal re-sponse function and a new static susceptibility term forthe off-diagonal one. The characteristic behavior of theresponse functions χ ji ( t, z ) is illustrated in Figs. 3–4. Tobegin with, an anticipated result is depicted in the col-ored surfaces plotted in Figs. 3(a) and 4(a), namely that χ ji ( t, z ) are periodic functions of time t , with a period T ext = 2 π/ω ext = 19 .
63 (1.3 ns). The z -dependence of χ ji ( t, z ) carries information on the memory effects thatoriginate from the skyrmion-magnon bath coupling, in-cluding the additional dissipative terms generated by theoscillating driving field. Thus we notice that the diag-onal χ ii ( t, z ) depends on the friction coefficient D ( T ),while the off-diagonal χ yx ( t, z ) has a dependence on thetopological charge renormalization δQ ( T ). V. FLUCTUATION-DISSIPATION THEOREM
In this section we turn our attention to the deriva-tion of the fluctuation-dissipation (FD) theorem, for askyrmion in contact to a bath of magnons at equilib-rium. An extension of the FD relation is also derived fora nonequilibrium bath of magnons which is weakly drivenby an oscillating magnetic field, a relation which reducesto the FD theorem in the static limit. We also calculatethe time and temperature dependence of the skyrmionmean square displacement (MSD).The FD theorem relates equilibrium thermal fluctu-ations and dissipative transport coefficients [1, 4]. Inthe absence of an external drive, the Fourier transform C ij ( ω ) of the quantum stochastic force correlation func-tion defined through Eq. (16) is related to the dampingkernel γ ij ( ω ) by the relation, C ij ( ω ) + C ji ( − ω ) = iω coth( βω γ ij ( ω ) + γ ji ( − ω )] . (36)Eq. (36) is the quantum mechanical version of the FDtheorem with the observation that quantum effects en-ter not only through the usual ω coth( βω/
2) term, butadditionally through the non-trivial ∝ coth( βε ν /
2) de-pendence of the damping kernel γ ij ( ω ).We now turn to the extension of the FD relation ofEq. (36) in the presence of an external field b ( t ). In gen-eral, the stochastic fluctuations of reservoirs driven outof equilibrium do not necessarily relate to their dissipa-tive properties, and a generalization of the FD theoremshould not be expected, except for some special cases[66, 67]. Following the same methodology as in Sec. III,we decompose the random force autocorrelation functionas follows, (cid:104) ξ j ( t ) ξ i ( t (cid:48) ) (cid:105) = − i [ C ji ( t − t (cid:48) ) + ∆ C ji ( t, t (cid:48) )] , (37)where the stochastic function ∆ C ji ( t, t (cid:48) ) satisfies∆ C ji ( t, t (cid:48) ) = ∂ t ∂ t (cid:48) U ji ( t − t (cid:48) )[ g ji ext ( t ) + g ji ext ( t (cid:48) )] . (38)Here, U ji ( t − t (cid:48) ) carries information about the magnonbath and is given by U ji ( t ) = (cid:88) ν ,ν ,ν (cid:48) i C ν ν ν ji [ u ν ν ( t ) − u ν ν ( t )]2[( ε ν − ε ν ) − ω ext ] , (39)where u ν ν ( t ) = [1 − F ( ε ν ) F ( ε ν )] cos[( ε ν − ε ν ) t ]. Weremind the reader that the damping kernel ∆ γ ji ( t, t (cid:48) ) ˜ T (K) . . . √ S ii [ α ] λ [ α ] . . S Q [ α ] FIG. 5. Root mean square displacement (RMSD) √ S ii givenin Eq. (42) as a function of temperature ˜ T at time t = 6 . λ = 2 . α , and Q = −
1. The RMSDis plotted for the choice J = 1 meV, N A S = 1, and d = 1, andgiven in units of the lattice constant α . Due to the quantummagnetic excitations, the RMSD at zero temperature, S Q ≡√ S ii ( ˜ T = 0), remains finite, while it scales linearly with ˜ T atfinite temperatures. The inset depicts the dependence of S Q on the skyrmion size λ . equals ∆ γ ji ( t, t (cid:48) ) = ∂ t W ji ( t − t (cid:48) )[ g ji ext ( t ) + g ji ext ( t (cid:48) )], with W ji ( t ) given in Eq. (24). The generalization of the FDtheorem is found to be independent of the form of theexternal drive and is expressed as a relation between thefunctions W ji ( t ) and U ji ( t ) in Fourier space, U ji ( ω )+ U ij ( − ω ) = coth( βω W ji ( ω ) − W ij ( − ω )] . (40)The non-equilibrium FD relation Eq. (40) is valid withinfirst order perturbation theory with respect to the am-plitude of the driving field, however we expect it willserve as a basis for future investigations of the effects oftime-dependent driving fields beyond first-order pertur-bation theory. In the special case of a static externalfield ω ext →
0, the FD theorem in equilibrium, Eq. (36),is recovered trivially, C ij ( ω ) + C ji ( − ω ) = iω coth( βω γ ij ( ω ) + γ ji ( − ω )] , (41)where C ij ( ω ) = C ij ( ω ) + 2 ω U ij ( ω ) g ji ext (0) and γ ij ( ω ) = γ ij ( ω ) + 2( − iω ) W ij ( ω ) g ji ext (0).We now focus on the temperature dependence of ther.h.s. of Eq. (36), which we expect to give rise to a finitezero-temperature mean squared displacement (MSD) ofthe skyrmion position. This motivates us to considerthe correlation function S ij ( t, t (cid:48) ) = (cid:104) [ R i ( t ) − R j ( t (cid:48) )] (cid:105) ,where (cid:104) . . . (cid:105) denotes ensemble average, and where (cid:104) R (cid:105) =0. From Eqs. (32) and (18) it follows that in the specialcase of b ( t ) = 0, the diagonal MSD S ii (¯ t ) = S ii ( t − t (cid:48) ) [ ↵ ] p S ii [ ↵ ] [ ↵ ] p S ii [ ↵ ] ˜ T = . K ˜ T = K ˜ T = K ˜ T = K ˜ T = K ˜ T = K FIG. 6. The RMSD √ S ii given in Eq. (43) as a function ofthe skyrmion size λ at three different temperatures, ˜ T = 5 , , and 15 K, for J = 1 meV, t = 50 ps, and N A S = 1. Thedashed vertical line indicates the value λ = α . The insetdepicts the λ -dependence of the RMSD at low temperaturesbelow 2 K. For a given temperature ˜ T , the RMSD has a localminimum at a critical radius λ cr ( ˜ T ) which signals a crossoverfrom short-time dynamical effects to long-time renormaliza-tion: For λ < λ cr , the RMSD decreases as 1 /λ , while for λ > λ cr it scales linearly with λ . reduces to S ii (¯ t ) = (cid:90) dω π ( e − iω ¯ t − χ il ( ω ) X lk ( ω ) χ ik ( − ω ) , (42)where X ij ( ω ) = − i [ C ij ( ω ) + C ji ( − ω )] is the symmetrizedautocorrelation function. Eq. (42) contains several con-tributions, of which we retain only the leading terms in˜ Q , under the assumption ˜ Q (cid:29)
1, to further simplifythe MSD to S ii (¯ t ) = 2 π ˜ Q (cid:88) ν,ν (cid:48) (cid:48) B ν ; ν (cid:48) ii [ F ( ε ν ) F ( ε ν (cid:48) ) −
1] sin [( ε ν (cid:48) − ε ν )¯ t/ . (43)First we focus on the temperature dependence of theroot mean square displacement (RMSD) (cid:112) S ii (¯ t ), whichis summarized in Fig. 5. As a result of the quantummagnetic excitations, the RMSD at ˜ T = 0, defined as S Q = √ S ii ( ˜ T = 0), remains finite. The dependence of S Q on the skyrmion size λ , illustrated in the inset ofFig. 5, implies that quantum fluctuations become impor-tant for very small skyrmions of a few lattice sites, whiletheir effect on the RMSD becomes negligible for largerskyrmions. We should emphasize that in this work weconsider a classical skyrmion coupled to a bath of quan-tum magnetic excitations, and disregard quantum effectsof the center-of-mass, which could increase the value of S Q further and make it experimentally more accessible.Such quantum effects are beyond the scope of this paper,and we leave it as a motivation for further studies. ˜ t (ps) S ii ( ˜ t ) [ α ] .
001 0 .
002 0 .
003 0 . ˜ t (ps) S ii ( ˜ t ) [ α ] FIG. 7. Mean squared displacement (MSD) S ii (˜ t ) given inEq. (43) as a function of time at temperature ˜ T = 14 . λ = 3 . α . The MSD is plotted forthe choice of J = 1 meV, N A S = 1, and d = 4. We observea ballistic regime at a very small time-scale, while for largertimes the MSD saturates quickly at the value obtained whenthe memory effects become negligible. Another important feature of Fig. 5 is the fast lin-ear thermal activation for temperatures ˜
T > (cid:112) S ii (¯ t ) (cid:39) .
14 ˜
T α /K. Such a behavior results from thenontrivial temperature dependence of the fluctuation-dissipation theorem Eq. (36) and stands in contrast tothe √ T dependence obtained in a classical description[37]. For a skyrmion with a radius 10 α , the RMSDis (3 . /N A S ) percentage of its radius at ˜ T = 1 . . /N A S ) percentage at ˜ T = 5 K, and (32 . /N A S ) per-centage at ˜ T = 15 K.Further results are shown in Fig. 6, where we plotthe dependence of the RMSD on the skyrmion size λ .We note that there is a critical radius λ cr ( ˜ T ) which sig-nals the interplay between long-time renormalization andshort-time dynamical effects. For λ < λ cr , the RMSD isinversely proportional to the skyrmion size, as expectedfor a massive particle with a mass proportional to thearea λ . Indeed, the time-dependent damping kernel γ ij ( t ) of Eq. (B4) is renormalized to the effective massof Eq. (2) in the long-time scale approximation. On thecontrary, for λ > λ cr , shorter time scale dynamical infor-mation becomes dominant and the RMSD scales linearlywith λ . Analogous results are obtained for very low tem-peratures below 2 K, illustrated in the inset of Fig. 6.Several conclusions can be drawn also from the timedependence of S ii (¯ t ) as illustrated in Fig. 7. At shorttimes ¯ t (cid:28)
1, we find a quadratic dependence, S ii (¯ t ) (cid:39) S ¯ t , which resembles the ballistic regime of the Brown-ian motion of a particle [68]. The constant S is foundfrom Eq. 43 under the replacement sin[( ε ν (cid:48) − ε ν )¯ t/ → ( ε ν (cid:48) − ε ν )¯ t/
2, while for the specific parameters plottedin Fig. 7 we find S = 4 . × . Such a ballistic mo-0tion is a direct consequence of the memory effects whichdominate the dynamics at short time scales. At longertimes, the memory effects become negligible and S ii (¯ t )saturates at a value which can be estimated from replac-ing sin [( ε ν (cid:48) − ε ν )¯ t/ → / − kg has been experimentally observed for shorttime scales of the inertia-dominated regime of µ s [69, 70].Here, the ballistic motion we predict for the quantum dy-manics of a magnetic skyrmion, with an inertial mass of0 . × − kg at T = 580 mK, is restricted to the im-measurably small femtosecond regime, which, however,is comparable to the duration of ultrafast light-inducedheat pulses needed to write and erase magnetic skyrmions[71]. We anticipate that the ballistic motion for a con-fined skyrmion with an inertial mass of about 10 − kg[40], could possibly take place within the experimentallyaccessible nanosecond regime. It suffices to mention thatthe classical dissipation is dominated by the contributionof some low-lying localized modes with energy ε in theGHz regime [72]. Thus, the quadratic short-time expan-sion is valid up to times ε − , i.e. the ballistic regimeextends in the nanosecond regime. We also note that ourpredictions significantly deviate from the classical resultsfor the mean squared displacement which, in the lattercase, increases linearly with time [37], a result that di-rectly follows from the assumption of a phenomenologi-cal thermal white noise which scales proportional to theGilbert damping parameter. VI. CONCLUSIONS
In this work, we consider the stochastic dynamics of amagnetic skyrmion in contact with a dissipative bath ofmagnons in the presence of a time-periodic external field,which directly couples to the magnon bath. We developa microscopic derivation of the Langevin equation of mo-tion based on a quantum field theory approach whichcombines the functional Keldysh and the collective coor-dinate formalism. The non-Markovian damping kernel isexplicitly related to the colored autocorrelation functionof the stochastic fluctuating fields, through the quantummechanical version of the fluctuation-dissipation theo-rem. Emphasis is given to the nontrivial temperaturedependence of the dynamical properties of the system, interms of the fundamental response and correlation func-tions. Contrary to the prediction of the classical theory,the damping kernel and the mass remain finite at van-ishingly small temperatures, due to the quantum natureof the bath considered in this work. This will give riseto a finite mean squared displacement at T →
0, whichincreases with temperature as T , a result that deviatesfrom the phenomenological prediction of a linear increase.We rigorously treat the effects of an external driveon the bath, and therefore on the skyrmion-bath cou-pling, and we generalize the theory of quantum dissi- pative response. The bath is dynamically engineeredout-of-equilibrium and through its interaction with theskyrmion gives rise to dissipation and random forces thatincorporate the bath’s dynamical activity. The magni-tude of these effects is illustrated in the diagonal andoff-diagonal response functions, which acquire an addi-tional time-periodicity inherited by the external drive.In addition, a super-Ohmic to Ohmic crossover behavioris signalled by new friction and topological charge renor-malization terms, similar to the effects predicted withina microscopic theory of classical dissipation with measur-able consequences for the skyrmion path [53]. We note,however, that, in contrast to Ref. 53, where the externaldrive couples to a well-pronounced bath mode, here wedo not consider resonance effects.Within our path integral formulation, we are able toestablish a generalization of the fluctuation-dissipationtheorem to the nonequilibrium case for weakly drivenmagnetic excitations. The spectral characteristics of thebath modes of the damping kernel are related to theones of the stochastic correlation function, irrespectivelyof the form of the external drive. Noteworthy, our re-sults apply to similar mesoscopic systems embedded inan driven bath. Advances in the theoretical understand-ing of skyrmion dynamics out of equilibrium is expectedto have an impact on similar particle-like objects suchas solitonic textures in quantum superfluids and domainwalls in ferromagnets. Our nonequilibrium formalism ofskyrmion dynamics can serve as a basis for future exper-imental investigations as well as theoretical studies thatgo beyond first order perturbation theory and beyond theslow dynamics of the GHz regime. VII. ACKNOWLEDGMENTS
This work was supported by the Swiss National ScienceFoundation (Switzerland) and the NCCR QSIT.
Appendix A: Autocorrelation Function
Here, we present in more detail the autocorrelationfunction of the stochastic fields, C ij ( t, t (cid:48) ), defined inEq. (16) of Sec. II. To evaluate the trace we use the func-tions Ψ ν ( r , t ), eigenfunctions of the magnon Hamiltonian H , which are presented in detail in Appendix D. Aftersome algebra, C ij ( t, t (cid:48) ) is expressed as, C ij ( t, t (cid:48) ) = C K,Kij ( t, t (cid:48) ) + C R,Aij ( t, t (cid:48) ) + C A,Rij ( t, t (cid:48) ) , (A1)where C a,bij ( t, t (cid:48) ) = iS (cid:88) ν (cid:48) (cid:90) ¯ t (cid:90) ¯ r , r , r (cid:48) ∂ t ∂ t (cid:48) (cid:2) Ψ † ν (¯ r , ¯ t ) G a (¯ r , r (cid:48) , ¯ t, t (cid:48) ) × Γ i ( r (cid:48) ) σ z G b ( r (cid:48) , r , t (cid:48) , t ) σ z Γ j ( r )Ψ ν ( r , t ) (cid:3) , (A2)1with a, b = K, R, A . A more transparent form is obtainedfor a bath of magnetic excitations at equilibrium, i.e. b ( t ) = 0, C ij ( t − t (cid:48) ) = i ∂ t ∂ t (cid:48) (cid:88) ν,ν (cid:48) (cid:48) B νν (cid:48) ij e − i ( ε ν (cid:48) − ε ν )( t − t (cid:48) ) × [Θ( t − t (cid:48) ) + Θ( t (cid:48) − t ) − F ( ε ν (cid:48) ) F ( ε ν )] , (A3)which is further simplified in Fourier space, C ij ( ω ) = iπω βω (cid:88) ν,ν (cid:48) (cid:48) B νν (cid:48) ij ¯ F ν (cid:48) ν δ ( ω − ε ν (cid:48) + ε ν ) , (A4)where, again, ¯ F νν (cid:48) = F ( ε ν ) − F ( ε ν (cid:48) ) and F ( ε ν ) =coth( βε ν / Appendix B: Equilibrium Damping Kernel
Our current task is to analyze the damping kernel ofEq. (20) by considering first the special case b ( t ) = 0.By a simple inspection of Eq. (14) we notice that cor-relations in equilibrium are time translation invariantand the Green functions depend on time differences, G R,A ( t, t (cid:48) ) = G R,A ( t − t (cid:48) ) and as a result the the diagonalpart of the damping kernel is found equal to γ ii ( t ) = Θ( t ) ∂ t (cid:88) νν (cid:48) (cid:48) (cid:60) ( B νν (cid:48) ii ) ¯ F νν (cid:48) sin[( ε ν (cid:48) − ε ν ) t ] , (B1)while the off-diagonal part can be cast into the form γ yx ( t ) = Θ( t ) ∂ t (cid:88) ν,ν (cid:48) (cid:48) (cid:61) ( B νν (cid:48) yx ) ¯ F νν (cid:48) cos[( ε ν (cid:48) − ε ν ) t ] . (B2)Here we sum over the quantum number ν = { q = ± , n } ,where the index q distinguishes between particle states( q = 1), solutions of the eigenvalue problem H Ψ n = ε qn σ z Ψ n , with positive eigenfrequency ε n = + ε n , and an-tiparticle states ( q = −
1) with negative eigenfrequency ε − n = − ε n [40]. The matrix elements are given by B νν (cid:48) ij = B n,q ; n (cid:48) ,q (cid:48) ij = ( qq (cid:48) / (cid:82) r Ψ † ν Γ i σ z Ψ ν (cid:48) (cid:82) r (cid:48) Ψ † ν (cid:48) Γ j σ z Ψ ν .From the structure of the matrix elements we concludethat B νν (cid:48) ii = (cid:60) ( B νν (cid:48) ii ) and B νν (cid:48) xy = i (cid:61) ( B νν (cid:48) ij ). We also notethat γ xy ( t ) = − γ yx ( t ) and that (cid:60) ( B ν (cid:48) νii ) = (cid:60) ( B νν (cid:48) ii ), while (cid:61) ( B ν (cid:48) νyx ) = −(cid:61) ( B νν (cid:48) yx ). Thus, both Eqs. (B1) and (B2) aresymmetric under the interchange of ν and ν (cid:48) .It appears convenient to derive the Langevin equationof Eq. (19) in the Laplace-frequency z space,˜ Q (cid:15) ij zR jc ( z ) + zR jc ( z ) γ ji ( z ) = ξ i ( z ) , (B3)where the frequency dependent kernel of Eq. (B1) equals γ ii ( z ) = z (cid:88) ν,ν (cid:48) (cid:48) (cid:60) ( B ν ; ν (cid:48) ii )( ε ν (cid:48) − ε ν ) ¯ F νν (cid:48) ( ε ν (cid:48) − ε ν ) + z , (B4) and the off-diagonal kernel of Eq. (B2) is found to be γ xy ( z ) = z (cid:88) ν,ν (cid:48) (cid:48) (cid:61) [ B ν ; ν (cid:48) xy ] ¯ F νν (cid:48) ( ε ν (cid:48) − ε ν ) + z , (B5)and it also holds that γ yx ( z ) = − γ xy ( z ). Note that agree-ment of the damping kernel γ ij ( z ) with earlier results de-rived in Matsubara space using the imaginary time pathintegral approach [40] can be established by simple ana-lytic continuation. Appendix C: Nonequilibrium Damping Kernel
In this section we provide explicit formulas for the re-duced expressions of the nonequilibrium damping kernelsappearing in Eq. (28). First we note that the Laplacetransform W ij ( z ) of the function W ij ( t ) given in Eq. (24)is expressed as W ji ( z ) = (cid:88) ν ,ν ,ν (cid:48) C ν ν ν ji [ w ν ν ( z ) − w ν ν ( z )]( ε ν − ε ν ) − ω ext , (C1)where w ν ν ( z ) = ¯ F ν ν ( ε ν − ε ν ) / [( ε ν − ε ν ) + z ] andwhere the matrix elements C ν ν ν ji are given in Eq. (25).Starting from Eq. (27) we define the temperature depen-dent dissipation constants through a Taylor expansion ofthe kernel ∆ γ ji ( t, z ) around z = 0 as∆ γ xx ( t,
0) = − ω ext [ W xx (0) + W xx ( iω ext )] sin( ω ext t ) (C2)for the linear friction-like terms and the next order isgiven through the relation ∂ z ∆ γ xx ( t, z ) | z =0 = cos( ω ext t )[ W xx (0) + W xx ( iω ext )+ iω ext W (cid:48) xx ( iω ext )] . (C3)Analogously, we find∆ γ yx ( t,
0) = ω ext [ W yx (0) + W yx ( iω ext )] cos( ω ext t ) (C4)for the linear friction-like terms and ∂ z ∆ γ yx ( t, z ) | z =0 = sin( ω ext t )[ W yx (0) + W yx ( iω ext )+ iω ext W (cid:48) yx ( iω ext )] . (C5)for the next order term. First we note that in the staticlimit ω ext →
0, all the terms vanish besides a mass renor-malization term W xx (0). At this point we emphasizethat our approach is valid only for slow dynamics andconsequently the frequency of the external drive shouldbe restricted to the GHz range, i.e. ω ext (cid:28) ε gap . Atthe same time we recall that the external potential V in-duces a finite but small overlap 0 < | V ν ν | (cid:28) ε d (cid:28) ω ext (cid:28) ε gap , with ε d = | ε ν − ε ν | , the resulting expressions are summarizedin Eqs. (28)–(29), with D ( T ) = − ω ext ¯ W ii , δM ( T ) = ¯ W ii , δQ ( T ) = ω ext ¯ W yx and G ( T ) = ¯ W yx . The ¯ W ji coefficientcan be found in Eq. (30).2 Appendix D: Magnon Spectrum
Here we briefly discuss the structure of the magnonexcitations, while a more detailed discussion can be foundin Refs. 40 and 53. The magnon Hamiltonian H for themodel of Eq. (3) in dimensionless units is given by H = 2[ −∇ + U ( ρ )] + 2 U ( ρ ) σ x − iU ( ρ ) ∂∂φ σ z , (D1)where U ( ρ ) = ρ − sin Θ ρ , and U ( ρ ) = sin 2Θ ρ − (Θ (cid:48) ) κ + 1 ρ ) sin Θ − Θ (cid:48) , (D2)and U ( ρ ) = h cos Θ − ρ − (Θ (cid:48) ) κ ρ )(1 + 3 cos 2Θ ) − Θ (cid:48) . (D3)The goal is to solve the eigenvalue problem of theform H Ψ n = ε n σ z Ψ n . Using the wave expansionsΨ n = e imφ ψ n,m ( ρ ) / √ π , the eigenvalue problem takesthe form H m ψ n,m ( ρ ) = ε n,m σ z ψ n,m ( ρ ), with H m = 2( −∇ ρ + U ( ρ ) + m ρ ) + 2 U ( ρ ) σ x + 2 U ( ρ ) mσ z , (D4)and ∇ ρ = ∂ ∂ ρ + ρ ∂∂ ρ . Scattering states Ψ m,k ( r ), classifiedby m as well as the radial momentum k (cid:62)
0, carry energy ε ( k ) = ε gap + k , with ε gap = 2 κ + h , and are of the form ψ m,k ( ρ ) = d m [cos( δ m ) J m +1 ( kρ ) − sin( δ m ) Y m +1 ( kρ )] (cid:18) (cid:19) , (D5)where J m ( Y m ) are the Bessel functions of the first (sec-ond) kind, d m ( k ) is a normalization constant and δ m ( k )is a scattering phase shift that determines the inten-sity of magnon scattering due to the presence of theskyrmion. The phase shifts are calculated within the WKB approximation discussed in detail in Refs. 40 and73. In the presence of an oscillating field b ( t ) = b Θ( t − t ) cos( ω ext t )(sin ϕ ext , , cos ϕ ext ), the magnons experiencea potential V ( r , t ) = b ( t ) · D = b Θ( t − t ) cos( ω ext t ) V ( r ),with D = δ χ † δ χ m | χ = χ † =0 and V ( r ) = V ( r ) + V ( r ) σ x + V ( r ) σ y . (D6)The potentials are V , ( r ) = csc[Θ ( ρ )] B ( r ) ± B ( r ), V ( r ) = (1 /
2) csc[Θ ( ρ )] B ( r ) + B ( r ), and B ( r ) = (1 /
2) sin( ϕ ext ) cos[Φ ( φ )] sin[Θ ( ρ )], B ( r ) = B ( r ) + (1 /
2) cos( ϕ ext ) cos[Φ ( φ )] sin[Θ( ρ )] and B ( r ) = − (1 /
2) sin( ϕ ext ) sin[Φ ( φ )] cos[Θ ( ρ )].We note that since the Hamiltonian H is invariant un-der the conjugation transformation C , where C = Kσ x with K the complex conjugation operator, there exists anadditional class of solutions Ψ − n = C σ x Ψ n with negativeeigenfrequency. To distinguish these two classes of solu-tions we use an additional index Ψ q = ± n , where the statesΨ n have positive eigenfrequencies ε n ≥
0, while Ψ − n havenegative eigenfrequencies ε − n ≤
0. The bi-orthogonalityconditions for the solutions are of the form (cid:104) Ψ qn | σ z | Ψ q (cid:48) m (cid:105) = qδ q,q (cid:48) δ n,m . Similarly, the resolution of the unity operatoris given by = (cid:80) q = ± (cid:80) n q | Ψ qn (cid:105)(cid:104) Ψ qn | σ z and the traceof an operator is Tr( A ) = (cid:80) q = ± (cid:80) n q (cid:104) Ψ qn | σ z A | Ψ qn (cid:105) . Tocalculate the mass M ( T ) of Eq. (22), as well as the drive-induced dissipation of Eq. (30), the sum over the quan-tum number ν is replaced in the following way: (cid:88) ν Ψ ν = (cid:88) q = ± ,n Ψ qn → (cid:88) q = ± (cid:88) m (cid:88) k Ψ qm,k . (D7)To render our results finite in the thermodynamiclimit, we subtract the background fluctuations [39] as (cid:80) k Ψ m,k → (cid:80) k (cid:16) Ψ free m,k − Ψ m,k (cid:17) , where Ψ free m,k are givenby Eq. (D5) for δ m ( k ) = 0. We also note that in additionto scattering states, a few localized modes which cor-respond to deformations of the skyrmion into polygonsexist in the range 0 < ε n < ε gap , but do not contributesignificantly compared to the continuum of modes Ψ m,k .For detailed formulas of the explicit calculation of themass M ( T ) we refer the reader to Appendix C of Ref. 40and in particular to Eq. (C8). [1] U. Weiss, Quantum Dissipative Systems , 4th Edition,World Scientific, Singapore, (2012).[2] J. E. Keizer,
Statistical Thermodynamics of Nonequilib-rium Processes , Springer-Verlag, New York, (1987).[3] L. E. Reichl,
A Modern Course in Statistical Physics , 2ndEdition, John Wiley & Sons, New York, (1998).[4] P. H¨anggi and G.-L. Ingold, Chaos The Nonequilibrium Sta-tistical Mechanics of Open and Closed Systems , VCH,New York, (1990).[6] W. F. Brown, Jr., Phys. Rev. , 1677 (1963). [7] C. Tannoudji, C. Dupont-Roc, and G. Grynberg,
AtomPhoton Interaction: Basic Processes and Application ,Wiley, New York, (1992).[8] Y. Makhlin, G. Schon, and A. Shnirman, Rev. Mod.Phys. , 357 (2001).[9] A. O. Caldeira, An introduction to macroscopic quantumphenomena and quantum dissipation , Cambridge Univer-sity Press, Cambridge UK (2014).[10] D. K. Efimkin, J. Hofmann, and V. Galitski, Phys. Rev.Lett. , 225301 (2016).[11] H. M. Hurst, D. K. Efimkin, I. B. Spielman, and V. Gal- itski, Phys. Rev. A , 178 (1989).[13] U. K. R¨ossler, A. A. Leonov, and A. N. Bogdanov, J.Phys. Conf. Ser. , 012105 (2011).[14] F. Wilczek and A. Zee, Phys. Rev. Lett. , 2250 (1983).[15] N. Papanicolaou and T.N. Tomaras, Nucl. Phys. B360 ,425 (1991).[16] S. Muehlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A.Rosch, A. Neubauer, R. Georgii, and P. Boeni, Science , 915 (2009).[17] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,Y. Matsui, N. Nagaosa, and Y. Tokura, Nature , 901(2010).[18] H. S. Park, X. Yu, S. Aizawa, T. Tanigaki, T. Akashi,Y. Takahashi, T. Matsuda, N. Kanazawa, Y. Onose, D.Shindo, A. Tonomura, and Y. Tokura, Nat. Nanotechnol. , 337 (2014).[19] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science , 198 (2012).[20] J. S. White, I. Levati´c, A. A. Omrani, N. Egetenmeyer,K. Prˇsa, I. ˇZivkovi´c, J. L. Gavilano, J. Kohlbrecher, M.Berger, and H. M. Rønnow, J. Phys.: Condens. Matter , 432201 (2012).[21] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Ku-betzka, R. Wiesendanger, G. Bihlmayer, and S. Bl¨ugel,Nat. Phys. , 713 (2011).[22] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B.Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesen-danger, Science , 636 (2013).[23] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. ,152 (2013).[24] F. Jonietz, S. Muehlbauer, C. Pfleiderer, A. Neubauer,W. Muenzer, A. Bauer, T. Adams, R. Georgii, P. Boeni,R. A. Duine, K. Everschor, M. Garst, and A. Rosch, Sci-ence , 1648 (2010).[25] X. Z. Yu, N. Kanazawa, W. Z. Zhang, T. Nagai, T. Hara,K. Kimoto, Y. Matsui, Y. Onose, and Y. Tokura, Nat.Commun. , 988 (2012).[26] J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert,Nat. Nanotechnol. , 839 (2013).[27] N. Nagaosa and Y. Tokura, Nat. Nanotechnol. , 899(2013).[28] R. Wiesendanger, Nat. Reviews Materials , 16044(2016).[29] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics,Part 2 , 3rd ed., Course of Theoretical Physics, Vol. 9(Pergamon, Oxford, 1980).[30] T. L. Gilbert, IEEE Trans. Magn., , 3443 (2004).[31] J. L. Garc´ıa-Palacios and F. J. L´azaro, Phys. Rev. B ,224403 (2014).[33] R. E. Troncoso and A. S. N´u˜nez, Annals of Physics ,850 (2014).[34] C. Sch¨utte, J. Iwasaki, A. Rosch, and N. Nagaosa, Phys.Rev. B , 174434 (2014).[35] J. Barker and O. A. Tretiakov, Phys. Rev. Lett. ,147203 (2016).[36] S. A. D´ıaz, C. J. O. Reichhardt, D. P. Arovas, A. Saxena,and C. Reichhardt, Phys. Rev. B , 085106 (2017).[37] J. Miltat, S. Rohart, and A. Thiaville, Phys. Rev. B ,214426 (2018). [38] T. Nozaki, Y. Jibiki, M. Goto, E. Tamura, T. Nozaki, H.Kubota, A. Fukushima, S. Yuasa, and Y. Suzuki, Appl.Phys. Lett. , 012402 (2019).[39] H.-B. Braun and D. Loss, Phys. Rev. B , 3237 (1996).[40] C. Psaroudaki, S. Hoffman, J. Klinovaja, and D. Loss,Phys. Rev. X , 041045 (2017).[41] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z.Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Nat.Mater. , 106 (2011).[42] J. Grenz, A. K¨ohler, A. Schwarz, and R. Wiesendanger,Phys. Rev. Lett. , 047205 (2017).[43] H. Grabert, P. Schramm, and G.-L. Ingold, Phys. Rep. , 115 (1988).[44] L. Keldysh, Zh. Eksp. Teor. Fiz. , 1515 (1964).[45] A. Kamenev and A. Levchenko, Advances in Physics ,197 (2009).[46] B. Sakita, Quantum Theory of Many Variable Systemsand Fields , World Scientific, Singapore (1985).[47] H. Grabert, Phys. Rev. B , 245433 (2015).[48] M. Frey and H. Grabert, Phys. Rev. B , 045429 (2016).[49] J. Reichert, P. Nalbach, and M. Thorwart, Phys. Rev.A , 032127 (2016).[50] H. Grabert, P. Nalbach, J. Reichert, and M. Thorwart,J. Phys. Chem. Lett. , 2015 (2016).[51] W. Wang, M. Beg, B. Zhang, W. Kuch, and H. Fangohr,Phys. Rev. B , 020403(R) (2015).[52] K.-W. Moon, D.-H. Kim, S.-G. Je, B. S. Chun, W. Kim,Z.Q. Qiu, S.-B. Choe, and C. Hwang, Scientific Reports , 20360 (2016).[53] C. Psaroudaki and D. Loss, Phys. Rev. Lett. , 237203(2018).[54] D. Pinna, F. Abreu Araujo, J.-V. Kim, V. Cros, D. Quer-lioz, P. Bessiere, J. Droulez, and J. Grollier, Phys. Rev.App. , 241 (1958).[57] T. Moriya, Phys. Rev. , 91 (1960).[58] N. Bogdanov and A. Hubert, J. Magn. Magn. Mater ,255 (1994).[59] A. B. Butenko, A. A. Leonov, U. K. R¨oßler, and A. N.Bogdanov, Phys. Rev. B , 052403 (2010).[60] S. M. Alamoudi, D. Boyanovsky, and F. I. Takakura,Phys. Rev. D , 105003 (1998).[61] A. A. Thiele, Phys. Rev. Lett. , 230 (1973).[62] M. Stone, Phys. Rev. B , 16573 (1996).[63] A. Altland and B. D. Simons, Condensed Matter FieldTheory , 2nd Edition, Cambridge University Press (2010).[64] C. Wang, D. Xiao, X. Chen, Y. Zhou, and Y. Liu, NewJournal of Physics , 083008 (2017).[65] S. Komineas and N. Papanicolaou, Phys. Rev. B ,281 (2013).[69] T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, Science , 1673 (2010).[70] R. Huang, I. Chavez, K. M. Taute, B. Luki´c, S. Jeney,M. G. Raizen, and E.-L. Florin, Nature Physics , 576 (2011).[71] G. Berruto, I. Madan, Y. Murooka, G. M. Vanacore,E. Pomarico, J. Rajeswari, R. Lamb, P. Huang, A. J.Kruchkov, Y. Togawa, T. LaGrange, D. McGrouther,H. M. Rønnow, and F. Carbone, Phys. Rev. Lett. , 064403 (2018).[73] M. V. Berry and K. E. Mount, Rep. Prog. Phys.35