Superfluid spin transport in magnetically ordered solids
SSuperfluid spin transport in magnetically ordered solids
E. B. Sonin
Racah Institute of Physics, Hebrew University of Jerusalem,Givat Ram, Jerusalem 91904, Israel (Dated: June 1, 2020)
Abstract
The paper reviews the theory of the long-distance spin superfluid transport in solid ferro- andantiferromagnets based on the analysis of the topology, the Landau criterion, and phase slips.Experiments reporting evidence of the existence of spin superfluidity are also overviewed. a r X i v : . [ c ond - m a t . o t h e r] M a y . INTRODUCTION The phenomenon of spin superfluidity is based on the analogy of special cases of theLandau–Lifshitz–Gilbert (LLG) theory in magnetism and superfluid hydrodynamics. Thisanalogy was clearly formulated long ago by Halperin and Hohenberg in their hydrody-namic theory of spin waves. While in a superfluid mass (charge in superconductors) can betransported by a current proportional to the gradient of the phase of the macroscopic wavefunction, in a magnetically ordered medium there are spin currents, which are proportionalto the gradient of the spin phase. The latter is defined as the angle of rotation aroundsome axis in the spin space. Strictly speaking this analogy is complete only if this axis is asymmetry axis in the spin space. Then according to Noether’s theorem the spin componentalong this axis is conserved. But possible violation of the spin conservation law usually israther weak because it is related with relativistically small (inversely proportional to thespeed of light) processes of spin-orbit interaction. In fact, the LLG theory itself is based onthe assumption of weak spin-orbit interaction .The analogy of the LLG theory with the theory of superfluidity suggests a new usefullanguage for description of phenomena in magnetism, but not a new phenomenon. Duringthe whole period of spin superfluidity investigations and up to now there have been disputesabout definition what is spin superfluidity. There is a school of thinking that the existence ofany spin current proportional to the spin phase (rotation angle) means spin superfluidity .This definition transforms spin superfluidity into a trivial ubiquitous phenomenon existingin any magnetically ordered medium. A spin current proportional to the spin phase emergesin any domain wall and in any spin wave. Under this broad definition spin superfluidity wasalready experimentally detected beyond reasonable doubt in old experiments of the middleof the 20th century detecting domain walls and spin waves. We use the term superfluidity inits original meaning known from the times of Kamerlingh Onnes and Kapitza: transport ofsome physical quantity (mass, charge, or spin) over macroscopic distances without essentialdissipation. This requires a constant or slowly varying phase gradient at macroscopic scalewith the total phase variation along the macroscopic sample equal to 2 π multiplied by avery large number. In examples of domain walls and spin waves this definitely does not takeplace. Gradients oscillate in space or time, or in both. The total phase variation is on theorder of π or much less. Currents transport spin on distances not more than the domain2all thickness, or the spin wavelength. Although such currents are also sometimes calledsupercurrents, we use the term supercurrent only in the case of macroscopic supercurrentpersistent at large spatial and temporal scales.The possibility of supercurrents is conditioned by the special topology of the magneticorder parameter space (vacuum manifold). Namely, this space must have topology of cir-cumference on the plane. In magnetically ordered systems this requires the presence ofeasy-plane uniaxial anisotropy. It is possible also in non-equilibrium coherent precessionstates, when spin pumping supports spin precession with fixed spin component along themagnetic field (the axis z ). Such non-equilibrium coherent precession states, which are callednowadays magnon BEC, were experimentally investigated in the B phase of superfluid Heand in YIG films.
Spin superfluid transport (in our definition of this phenomenon) is possible as long as thespin phase gradient does not exceeds the critical value determined by the Landau criterion.The Landau criterion checks stability of supercurrent states with respect to elementaryexcitations of all collective modes. The Landau criterion determines a threshold for thecurrent state instability, but it tells nothing about how the instability develops. The decayof the supercurrent is possible only via phase slips. In a phase slip event a vortex crossescurrent streamlines decreasing the phase difference along streamlines. Below the criticalvalue of supercurrent phase slips are suppressed by energetic barriers. The critical value ofthe supercurrent at which barriers vanish is of the same order as that estimated from theLandau criterion. This leads to a conclusion that the instability predicted by the Landaucriterion is a precursor of the avalanche of phase slips not suppressed by any activationbarrier.The present paper reviews the three essentials of the spin superfluidity concept: topology,Landau criterion, and phase slips. The paper focuses on the qualitative analysis avoidingdetails of calculations, which can be found in original papers. After the theoretical analysisthe experiments supporting the existence of spin superfluidity are discussed.
II. CONCEPT OF SUPERFLUIDITY
Since the idea of spin superfluidity emerged from the analogy of magnetodynamics andsuperfluid hydrodynamics let us remind the concept of the mass superfluidity in the theory3f superfluidity. In superfluid hydrodynamics there are the Hamilton equations for the pairof the canonically conjugate variables “phase – density”: (cid:126) dϕdt = − δ H δn , dndt = δ H (cid:126) δϕ . (1)Here δ H δn = ∂ H ∂n − ∇ · ∂ H ∂ ∇ n , δ H δϕ = ∂ H ∂ϕ − ∇ · ∂ H ∂ ∇ ϕ (2)are functional derivatives of the Hamiltonian H = (cid:126) n m ∇ ϕ + E ( n ) , (3)where E ( n ) is the energy of the superfluid at rest, which depends only on the particledensity n . Taking into account the gauge invariance (the energy does not depend on thephase directly, ∂ H /∂ϕ = 0, but only on its gradient) the Hamilton equations are reducedto the equations of hydrodynamics for an ideal liquid: m d v dt = − ∇ µ, (4) dndt = − ∇ · j . (5)In these expressions µ = ∂E ∂n + (cid:126) m ∇ ϕ (6)is the chemical potential, and j = n v = ∂ H (cid:126) ∂ ∇ ϕ (7) a) b) FIG. 1. Phase (in-plane rotation angle) variation at the presence of mass (spin) currents. a)Oscillating currents in a sound (spin) wave). b) Stationary mass (spin) supercurrent.
4s the particle current. We consider the zero-temperature limit, when the superfluid velocitycoincides with the center-of-mass velocity v = (cid:126) m ∇ ϕ. (8)A collective mode of the ideal liquid is a sound wave. In the sound wave the phasevaries in space, i.e., the wave is accompanied by mass currents [Fig. 1(a)]. An amplitudeof the phase variation is small, and currents transport mass on distances of the order ofthe wavelength. A real superfluid transport on macroscopic distances is possible in currentstates, which are stationary solutions of the hydrodynamic equations with finite constantcurrents, i.e., with constant nonzero phase gradients. In the current state the phase rotatesthrough a large number of full 2 π -rotations along streamlines of the current [Fig. 1(b)].These are supercurrents or persistent currents.The crucial point of the superfluidity concept is the question why the supercurrent is apersistent current, which does not decay despite it is not the ground state of the system. Theanswer to this question follows from the analysis of the topology of the order parameter space(vacuum manifold). At the equilibrium the order parameter of a superfluid is a complexwave function ψ = ψ e iϕ , where the modulus ψ of the wave function is a positive constantdetermined by minimization of the energy and the phase ϕ is a degeneracy parameter sincethe energy does not depend on ϕ . Any from the degenerate ground states in a closed annularchannel (torus) maps on some point at the circumference | ψ | = ψ in the complex plane ψ , while a current state with the phase change 2 πn around the torus maps onto a path[Fig. 2(a)] winding around the circumference n times. It is impossible to change the windingnumber n keeping the path on the circumference | ψ | = ψ all the time. In the languageof topology states with different n belong to different classes, and n is a topological charge .Only a vortex moving across the torus channel can change n to n −
1. This process is aphase slip. The phase slip costs energy, which is spent on creation of the vortex and itsmotion across current streamlines. The state with the vortex in the channel maps on thefull circle | ψ | ≤ ψ [Fig. 2(b)]. Thus, phase slips are impeded by potential barriers, whichmake the current state metastable.According to the Landau criterion, the current state is metastable as long as any quasi-particle of the superfluid in the laboratory frame has a positive energy and therefore itscreation requires an energy input. The Landau criterion checks the stability only with5 e ψ Im ψψ Re ψ Im ψψ a)b) FIG. 2. Topology of the uniform mass current and the vortex states. a) The current state in atorus maps onto the circumference | ψ | = | ψ | = const in the complex ψ - plane, where ψ is theequilibrium order parameter wave function of the uniform state. b) The vortex state maps ontothe circle | ψ | ≤ | ψ | . respect to weak elementary perturbations of the current state, while a vortex is a strongmacroscopic perturbation. However, the Landau critical gradients are of the same order asthe gradients at which barriers for phase slips disappear. The both are on the order of theinverse vortex core radius. III. SPIN SUPERFLUIDITYA. Ferromagnets
The phenomenological description of magnetically ordered media is given by the LLGtheory. For a ferromagnet with magnetization density M the LLG equation is ∂ M ∂t = γ [ H eff × M ] , (9)where γ is the gyromagnetic ratio between the magnetic and mechanical moment. Theeffective magnetic field is determined by the functional derivative of the total energy: H eff = − δ H δ M = − ∂ H ∂ M + ∇ i ∂ H ∂ ∇ i M . (10)According to the LLG equation, the absolute value M of the magnetization cannot vary.The evolution of M is a precession around the effective magnetic field H eff .6e shall consider the case when spin-rotational invariance is partially broken, and thereis uniaxial crystal magnetic anisotropy. The phenomenological Hamiltonian is H = A ∇ i M · ∇ i M + GM z M − H · M . (11)If the anisotropy energy G is positive, it is the “easy plane” anisotropy, which keeps themagnetization in the xy plane. If the external magnetic field H is directed along the z axis,the z component of spin is conserved because of invariance with respect to rotations aroundthe z axis. Since the absolute value M of magnetization is fixed, the magnetization vector M is fully determined by the z magnetization component M z and the angle ϕ showing thedirection of M in the easy plane xy : M x = M ⊥ cos ϕ, M y = M ⊥ sin ϕ, M ⊥ = (cid:112) M − M z . (12)In the new variables the Hamiltonian is H = AM ⊥ ( ∇ ϕ ) M z χ − HM z . (13)Here we neglected gradients of M z . The parameter A is stiffness of the spin system de-termined by exchange interaction, and the magnetic susceptibility χ = M /G along the z axis is determined by the uniaxial anisotropy energy G keeping the magnetization in theeasy plane. The LLG equation reduces to the Hamilton equations for a pair of canonicallyconjugate continuous variables “angle–angular momentum”:1 γ dϕdt = − δ H δM z = − ∂ H ∂M z , (14)1 γ dM z dt = δ H δϕ = − ∇ · ∂ H ∂ ∇ ϕ , (15)where functional derivatives on the right-hand sides are taken from the Hamiltonian givenby Eq. (13). Using the expressions for functional derivatives one can write the Hamiltonequations as 1 γ dϕdt = AM z ( ∇ ϕ ) − M z − χHχ , (16)1 γ dM z dt + ∇ · J = 0 , (17)7here J = − ∂ H ∂ ∇ ϕ = − AM ⊥ ∇ ϕ (18)is the spin current. Although our equations contain not the spin density but the magneti-zation, the vector J is defined as a current of spin with the spin density M z /γ .There is an evident analogy of Eqs. (16) and (17) with the hydrodynamic equations (4)and (5) for the superfluid. One of solutions of these equations describes the spin-wave mode.However, as well as the mass current in a sound wave, the small oscillating spin current inthe spin wave does not lead to long-distance superfluid spin transport, which this review Re ψ Im ψψ a)b) c) d) M z z M z z xy H xy H xy H Re ψ Im ψψ a)b) c) d) M z z M z z xy H xy H xy H Re ψ Im ψψ a)b) c) d) M z z M z z xy H xy H xy H a)c) b) a) b) c)d) FIG. 3. Mapping of spin current states on the order parameter space (vacuum manifold).a) Spin currents in an isotropic ferromagnet. The current state in torus maps on an equatorialcircumference on the sphere of radius M (top). Continuous shift of mapping on the surface ofthe sphere (middle) reduces it to a point at the northern pole (bottom), which corresponds to theground state without currents.b) Spin currents in an easy-plane ferromagnet. The easy-plane anisotropy reduces the order pa-rameter space to an equatorial circumference in the xy plane topologically equivalent to the orderparameter space in superfluids.c) Spin currents in an easy-plane ferromagnet in a magnetic field parallel to the axis z . Spin isconfined in the plane parallel to the xy plane but shifted closer to the northern pole.d) The vortex state maps on the surface of the upper or the lower semisphere in the vacuummanifold. π -rotations as shown in Fig. 1(b). Inthe current state with a constant gradient of the spin phase K = ∇ ϕ , there is a constantmagnetization component along the magnetic field (the axis z ): M z = χH − χAK . (19)Like in superfluids, the stability of current states is connected to the topology of the orderparameter space. In isotropic ferromagnets ( G = 0) the order parameter space is a sphericalsurface of radius equal to the absolute value of the magnetization vector M [Fig. 3(a)].All points on this surface correspond to the same energy of the ground state. Suppose wecreated the spin current state with monotonously varying phase ϕ in a torus. This statemaps on the equatorial circumference in the order parameter space. The topology allows tocontinuously shift the circumference and to reduce it to a point (the northern or the southernpole). During this process shown in Fig. 3(a) the path remains in the order parameter spaceall the time, and therefore, no energetic barrier resists to the transformation. Thus, themetastability of the current state is not expected in isotropic ferromagnets.In a ferromagnet with easy-plane anisotropy ( G >
0) the order parameter space reducesfrom the spherical surface to the equatorial circumference in the xy plane [Fig. 3(b)]. Thismakes the order parameter space topologically equivalent to that in superfluids. Now thetransformation of the circumference to the point costs the anisotropy energy. This allows toexpect metastable spin currents (supercurrents). The magnetic field along the anisotropyaxis z shifts the easy plane either up [Fig. 3(c)] or down away from the equator.The current states in easy-plane ferromagnets relax to the ground state via phase slipsevents, in which magnetic vortices cross spin current streamlines. States with vortices mapon a surface of a hemisphere of radius M either above or below the equator as shown inFig. 3(d).Up to now we considered states close to the equilibrium (ground) state. In a ferromagnetin a magnetic field the equilibrium magnetization is parallel to the field. However, bypumping magnons into the sample it is possible to tilt the magnetization with respect tothe magnetic field. This creates the state with the coherent spin precession around themagnetic field (the magnon BEC ). Although the state is far from the true equilibrium,but it, nevertheless, is a state of minimal energy at fixed magnetization M z . Because of9nevitable spin relaxation the state of uniform precession requires permanent pumping ofspin and energy. However, if processes violating the spin conservation law are weak, one canignore them and treat the state as a quasi-equilibrium state. The state of uniform precessionmaps on a circumference parallel to the xy plane. One can consider also a current state, inwhich the phase (the rotation angle in the xy plane) varies not only in time but also in spacewith a constant gradient. In this case the easy plane for the magnetization is not related tothe crystal anisotropy but created dynamically. However, in the quasi-equilibrium coherentprecession state demonstration of the long-distance superfluid spin transport is problematic(see Sec. VIII). B. Antiferromagnets
Long time ago it was widely accepted to describe the dynamics of a bipartite antiferro-magnet by the LLG equations for two spin sublattices coupled via exchange interaction: d M i dt = γ [ H i × M i ] . (20)Here the subscript i = 1 , M i belongs, and H i = − δ H δ M i = − ∂ H ∂ M i + ∇ j ∂ H ∂ ∇ j M i (21)is the effective field for the i th sublattice determined by the functional derivative of theHamiltonian H . For an isotropic antiferromagnet the Hamiltonian is H = M · M χ + A ( ∇ i M · ∇ i M + ∇ i M · ∇ i M )2 + A ∇ j M · ∇ j M − H · m . (22)In the uniform ground state the total magnetization m = M + M (23)is equal to m = χ H , while the staggered magnetization L = M − M (24)is normal to m . Without the magnetic field the two sublattice magnetizations are antiparal-lel, and the total magnetization m vanishes. The first term in the Hamiltonian (22), whichdetermines the susceptibility χ , originates from the exchange interaction between spins of10he two sublattices. This is the susceptibility normal to the staggered magnetization L .Since in the LLG theory absolute values of sublattice magnetizations M and M are equalto M and do not vary in space and time, the susceptibility parallel to L vanishes.Let us consider a uniform state but not necessary the ground state. There are no currentsin this state, and the gradient-dependent terms in the Hamiltonian Eq. (22) can be ignored.Rewriting the Hamiltonian in terms of m and L one obtains H = − L − m χ − H · m = − M χ + m χ − H · m . (25)Minimizing the Hamiltonian with respect to the absolute value of m (at it fixed direction)one obtains H = − M χ − χH m − M χ − χH χH L , (26)where H m = ( H · m ) /m and H L = ( H · L ) /L are the projections of the magnetic field onthe total magnetization m and on the staggered magnetization L . The first two terms areconstant, while the last term plays the role of the easy-plane anisotropy energy confining L in the plane normal to H . For H parallel to the axis z the anisotropy energy [the last termon the right-hand side of Eq. (26)] is E a = χH L z L . (27)In the analogy to the ferromagnetic case, one can describe the vectors of sublattice magne-tizations M i with the constant absolute value M by the two pairs of the conjugate variables( M iz , ϕ i ), which are determined by the two pairs of the Hamilton equations:1 γ dϕ i dt = − δ H δM iz = − ∂ H ∂M iz , (28)1 γ dM iz dt = δ H δϕ i = ∂ H ∂ϕ i − ∇ · ∂ H ∂ ∇ ϕ i . (29)Let us consider the axisymmetric solutions of these equations with ϕ = ϕ = π − ϕ and M z = M z = m z . Then there is only one pair of the Hamilton equations for the pair of theconjugate variables ( m z , ϕ ): 1 γ dϕdt = A − m z ( ∇ ϕ ) − m z − χHχ , (30)1 γ dm z dt + ∇ · J = 0 . (31)11ere J = − ∂ H ∂ ∇ ϕ = − A − L ∇ ϕ (32)is the the spin current and A − = A − A . These equations are identical to Eqs. (16) and(17) for the ferromagnetic after replacing the spontaneous magnetization component M z by the total magnetization component m z , A by A − /
2, and M ⊥ by L . In the stationarycurrent state there is a constant gradient K = ∇ ϕ of the spin phase and a constant totalmagnetization m z = χH − χA − K / . (33)While in ferromagnets the current state is a spiral spin structure with the spatial precessionof the in-plane spontaneous magnetization along current streamlines, in antiferromagnetsthe current states are related to the spatial precession of the staggered magnetization.The order parameter space for the isotropic antiferromagnet in the absence of the externalmagnetic field is a surface of a sphere. However, the order parameter is not the totalmagnetization but the unit N´eel vector l = L /L . While in the ferromagnet the magneticfield produces an easy axis for the total magnetization, in the antiferromagnet the magneticfield produces the easy plane for the order parameter vector l with the anisotropy energygiven by Eq. (27). Thus, the topology necessary for the spin superfluidity in antiferromagnetsdoes not require the crystal easy-plane anisotropy. IV. SPIN CURRENTS WITHOUT SPIN CONSERVATION LAW
Though processes violating the spin conservation law are relativistically weak, their effectis of principal importance and cannot be ignored in general. The attention to superfluidtransport in the absence of conservation law was attracted first in discussions of superflu-idity of electron-hole pairs. The number of electron-hole pairs can vary due to interbandtransitions, and the degeneracy with respect to the phase of the pair condensate is lifted.On the basis of it Guseinov and Keldysh concluded that the existence of spatially homoge-neous stationary current states is impossible and there is no analogy with superfluidity. Thisphenomenon was called “fixation of phase”. However some time later it was demonstrated that phase fixation does not rule out the existence of weakly inhomogeneous stationary12urrent states analogous to superfluid current states. This analysis was extended on spinsuperfluidity.
One can take into account processes violating the spin conservation law by adding the n -fold in-plane anisotropy energy ∝ G in to the Hamiltonian (13): H = M z χ − γM z H + AM ⊥ ( ∇ ϕ ) G in [1 − cos( nϕ )] . (34)Then the spin continuity equation (17) becomes dM z dt = − ∇ · J + nG in sin( nϕ ) = AM ⊥ (cid:20) ∇ ϕ − sin( nϕ ) l (cid:21) , (35)where l = (cid:115) AM ⊥ nG in (36)is the thickness of the wall separating domains with n equivalent easiest directions in theeasy plane. We focus on stationary states when dM z /dt = 0. The phase ϕ is a periodicalsolution of the sine-Gordon equation parametrized by the average phase gradients (cid:104)∇ ϕ (cid:105) .At small (cid:104)∇ ϕ (cid:105) (cid:28) /l the spin structure constitutes the chain of domains with the period2 π/n (cid:104)∇ ϕ (cid:105) . Any domain corresponds to some of the n equivalent easiest directions in theeasy plane. Spin currents (gradients) inside domains are negligible but there are essentialspin currents inside domain walls where ∇ ϕ ∼ /l . This hardly reminds the superfluidtransport on macroscopic scales: spin is transported over distances on the order of thedomain-wall thickness l . With increasing (cid:104)∇ ϕ (cid:105) the density of domain walls grows, and at (cid:104)∇ ϕ (cid:105) (cid:29) /l the domains coalesce. Deviations of the gradient ∇ ϕ from the constant average rp חon-unlform current states וn the exזtonlc condensate: Lozovik & Yudson ( 1977) Sonin ( 1977) Shevchenko (1977) X rp Extenslon of the analysls on spln cuחents:
Sonin (1978) X FIG. 4. The nonuniform spin-current states with (cid:104)∇ ϕ (cid:105) (cid:28) /l and (cid:104)∇ ϕ (cid:105) (cid:29) /l . (cid:104)∇ ϕ (cid:105) become negligible. This restores the analogy with the superfluid transportin superfluids. The transformation of the domain wall chain into a weakly inhomogeneouscurrent state at growing (cid:104)∇ ϕ (cid:105) is illustrated in Fig. 4.An important difference with conventional mass superfluidity is that the existence ofconventional superfluidity is restricted only from above by the Landau critical gradients,while the existence of spin superfluidity is restricted also from below: gradients should not beless than the value 1 /l . Since the upper Landau critical value is determined by the easy-planeuniaxial anisotropy G and the lower critical value is determined by the in-plane anisotropyenergy G in , spin superfluidity is possible only if G (cid:29) G in . The existence of the lower criticalgradient for spin superfluidity is important for spin superfluidity observation discussed inSec. IX. However, in the further theoretical analysis we ignore processes violating the spinconservation law assuming that the phase gradients essentially exceed the lower thresholdfor spin superfluidity. V. COLLECTIVE SPIN MODES AND THE LANDAU CRITERIONA. Ferromagnets
In order to check the Landau criterion, one should know the spectrum of collective modesin the current state with the constant value of the spin phase gradient K = ∇ ϕ andwith the longitudinal (along the magnetic field) magnetization given by Eq. (19). It isnecessary to solve the Hamilton equations Eqs. (16) and (17) linearized with respect toweak perturbations of the current state. We skip the standard algebra given elsewhere .Finally one obtains the spectrum of plane spin waves ∝ e i k · r − iωt : ω + w · k = ˜ c sw k. (37)Here ˜ c sw = (cid:112) − χAK c sw (38)is the spin-wave velocity in the current state and c sw = γM ⊥ (cid:115) Aχ (39)14s the spin velocity in the state without spin currents. The velocity w = 2 γM z A K (40)can be called Doppler velocity because its effect on the mode frequency is similar to theeffect of the mass velocity on the mode frequency in a Galilean invariant fluid (Dopplereffect). However, our system is not Galilean invariant, and the gradient K is present alsoon the right-hand side of the dispersion relation (37).We obtained the gapless Goldstone mode with the sound-like linear in k spectrum. Thecurrent state becomes unstable when at k parallel to w the frequency ω becomes negative.This happens at the gradient K equal to the Landau critical gradient K c = M ⊥ √ M − M ⊥ √ χA . (41)In the limit of weak magnetic fields when M z (cid:28) M the Landau critical gradient is K c = 1 √ χA = γMχc sw . (42)In this limit the pseudo-Doppler effect is not important, and the Landau critical gradient K c is determined by the condition that the spin-wave velocity ˜ c sw in the current state vanishes.In the opposite limit M z → M ( M ⊥ →
0) the Landau critical gradient, K c = M ⊥ M √ χA , (43)decreases, and the spin superfluidity becomes impossible at the phase transition to theeasy-axis anisotropy ( M ⊥ = 0).Deriving the sound-like spectrum of the spin wave we neglected in the Hamiltonian termsdependent on gradients ∇ M z . Taking these terms into account one obtains quadratic in k corrections to the spectrum. These corrections become important at k ∼ /ξ , where ξ = MM ⊥ (cid:112) χA (44)can be called the coherence length. The coherence length ξ determines the core radius ofvortices just because the gradients ∇ M z are important in the vortex core. On the otherhand, the calculation of the energy of the vortex in the current state (Sec. VI) indicates thatpotential barriers for phase slips disappear at the gradients of the order 1 /ξ . Since the 1 /ξ is of the same order of magnitude as the Landau critical gradient Eq. (41), the instability with15espect to elementary excitations (the Landau instability) and the instability with respectto macroscopic excitations (vortices participating in phase slips) start at approximately thesame gradients. B. Antiferromagnets
Directions of the sublattice magnetizations in a bipartite antiferromagnet are determinedby the two pairs of polar angles θ i , ϕ i ( i = 1 , M ix = M cos θ i cos ϕ i , M iy = M cos θ i sin ϕ i , M iz = M sin θ i . (45)In the further analysis it is convenient to use other angle variables: θ = π + θ − θ , Θ = π − θ − θ ,ϕ = ϕ + ϕ , Φ = ϕ − ϕ . (46)The polar angle Θ for the staggered magnetization L and the canting angle θ are shown inFig. 5 for the case when the both magnetizations are in the plane xz ( ϕ = Φ = 0). In theseangle variables the equations for two collective modes in antiferromagnets are decoupled.In the stationary current state the total magnetization m z = 2 M sin θ is given by Eq. (33)and ∇ ϕ = K , while Θ = Φ = 0. In a weakly perturbed current state small but nonzeroΘ and Φ appear. Also the angles θ and ϕ differ from their values in the stationary current M2 X M 1
FIG. 5. Angle variables θ and Θ for the case when the both magnetizations are in the plane xz ( ϕ = Φ = 0). θ → θ + θ (cid:48) , ϕ → ϕ + ϕ (cid:48) . As in the ferromagnetic case, we skip the algebra of thelinearization and the solution of linearized equations (see Ref. 24 for a detailed calculation)and give the resulting spectra of two spin-wave modes.The equations for the pair of perturbations ( θ (cid:48) , ϕ (cid:48) ) describe the Goldstone mode with thespectrum of the plane spin waves ω + w · k = ˜ c sw . (47)Here the spin-wave velocity in the ground state without spin currents, the spin-wave velocityin the current state, and the Doppler velocity are given by c sw = γL ⊥ (cid:115) A − χ , ˜ c sw = c sw (cid:114) − χA − K , w = γm z A − K , (48)where L ⊥ = L cos θ . The gapless Goldstone mode in an antiferromagnet does not differ fromthat in a ferromagnet, if one replaces in the expressions for the ferromagnet A by A − / M by 2 M .The equations for the pair of perturbations (Θ , Φ) describe the gapped mode with thespectrum ω + w · k = (cid:113) ω + c sw k , (49)where the gap is given by ω = (cid:115) γ m z χ − c sw K . (50)For better understanding of the physical nature of the two modes, let us consider varia-tions of the Cartesian components of the total and the staggered magnetizations producedby these perturbations in the uniform ground state without current and with L parallel tothe axis x ( ϕ = 0): m (cid:48) x = 2 M sin θ Θ , m (cid:48) y = 2 M cos θ Φ , m (cid:48) z = 2 M cos θθ (cid:48) ,L (cid:48) x = − θθ (cid:48) , L (cid:48) y = 2 M cos θϕ (cid:48) , L (cid:48) z = − M cos θ Θ , (51)From these expressions one can see that the pair of perturbations ( θ (cid:48) , ϕ (cid:48) ) is related to theGoldstone mode and produces rotation of the staggered magnetization L by the angle ϕ (cid:48) around the axis z and the oscillation of the total spin component m (cid:48) z along the same axis.On the other hand, the pair of perturbations (Θ , Φ) produces rotation of all spins by the17 ) b)
FIG. 6. The schematic picture of the two spin wave modes in the bipartite antiferromagnet in theplane xz . (a) The gapless Goldstone mode. There are oscillations of the canting angle and of thetotal magnetization component m z and rotational oscillations around the axis z . (b) The gappedmode. There are oscillations of the total magnetization component m y and rotational oscillationsaround the axis y . angle Θ around the axis y and the oscillation m (cid:48) y of the total spin component along the sameaxis. This is connected to the degree of freedom described by the pair of conjugate variables( m y , Θ). In the presence of the magnetic field the rotational invariance for the axis y isbroken, and the mode must have a gap. In the current state with the gradient of the angle ϕ the gapped mode is connected with the rotation of L around the axis which itself rotatesin the easy-plane xy along the current streamlines. Two modes are illustrated in Fig. 6.In the past decades there were numerous calculations of the spin wave spectrum bothin ferro- and antiferromagnets. However, the spin wave spectra discussed in the presentpaper were calculated not for the ground state, but for the metastable current states. Inthe magnetodynamics of antiferromagnets it was usually assumed that the spin polarizationis weak and the canting angle is small. Then the magnetodynamics can be reduced tothe single equation for the N´eel vector equivalent to that in the sigma model (see therecent review by Galkina and Ivanov and references therein). The derivation of spectrapresented in this paper did not use the assumption of small canting angles. Therefore,the obtained dispersion relations are valid up to the magnetic field at which the sublatticemagnetizations become equal, and the staggered magnetization L vanishes. This makes thespin superfluidity impossible. However, this magnetic field is on the order of the exchangefield, which is usually very strong. 18pplying the Landau criterion to the gapless mode at small canting angles θ , one obtainsthe critical gradient K c = (cid:115) χA − , (52)similar to the value Eq. (42) obtained for a ferromagnet. However, in contrast to a ferro-magnet where the susceptibility χ is connected with weak anisotropy energy, in an antifer-romagnet the susceptibility χ is determined by a much larger exchange energy and is rathersmall. As a result, in an antiferromagnet the gapless Goldstone mode becomes unstable atthe very high value of K . At much lower values of K the gapped mode loses its stabilitywhen the gap becomes negative and the mode frequency becomes complex. According tothe spectrum (49), the gap in the spectrum vanishes at the critical gradient K c = 1 ξ = γHc s = γm z χc s . (53)Here we introduced a new correlation length ξ = c s γH , (54)which is connected to the effective easy-plane anisotropy energy (27). The instability of thegapped mode is a precursor of the instability with respect to phase slips with vortices, whichhave the core radius of the order of ξ . VI. PHASE SLIPS AND BARRIERS FOR VORTEX MOTION ACROSS STREAM-LINES
For the estimation of barriers for phase slips one must consider the interaction of vorticeswith spin currents. The total energy of the vortex is mostly determined by the area outsidethe core (the London region) where one must take into account the interaction of vorticeswith spin currents. In the London region the main contribution to the energy is the first termin the Hamiltonian Eq. (13) proportional to ∇ ϕ (we consider now a ferromagnet). Thisterm plays the role of the kinetic energy of spin currents. The other terms are constants.The spin phase gradient in the current state with a straight vortex parallel to the axis z is ∇ ϕ = [ˆ z × r ] r + K , (55)19here the first term is the spin phase gradient introduced by the vortex, ˆ z is the unit vectoralong the z axis, r is a 2D position vector with the origin at the vortex axis, and the gradient K is related to the spin current. Substituting this into the kinetic energy and integratingthe energy over the whole space occupied by the ferromagnet one obtains a logarithmicallydivergent integral, which depends on the sample geometry. We consider a 2D problem ofthe straight vortex at the distance R from the plane border. The gradient K is parallel tothe border. Then the the energy of the straight vortex per unit length in the presence ofcurrents is E v = πAM ⊥ (cid:18) ln Rr c − KR (cid:19) . (56) a)b)c) FIG. 7. Skyrmion cores of vortices. Variation of magnetization vectors ( M in a ferromagnet, M and M in an antiferromagnet) in the vortex core as a function of the distance r from the vortexaxis is shown schematically. (a) The vortex in the ferromagnet corresponding to the single spinwave mode with the coherence length ξ given by Eq. (44). (b) The vortex in the antiferromagnetcorresponding to the Goldstone mode spin wave mode with the coherence length ξ given byEq. (44), where the order parameter stiffness A in the ferromagnet is replaced by the stiffness A − / ξ given by Eq. (54). r c . The vortex energy has a maximumat R = 1 / K . The energy at the maximum is a barrier preventing phase slips: E b = πAM ⊥ ln 12 Kr c . (57)The barrier vanishes if K becomes of the order of the inverse vortex core radius. Thisconclusion is applicable also to antiferromagnets.The core radius r c is of order of the coherence length determined from the spin wave spec-trum as was indicated earlier. However, different modes have different coherence lengths,and it is necessary to understand which kind of a vortex corresponds to which spin wavemode. In vortex cores spins form skyrmions. Variation of magnetization vectors in theskyrmion vortex core as a function of the distance r from the vortex axis is shown schemat-ically in Fig. 7. In the ferromagnet [Fig. 7(a)] there is only one spin wave mode, and theradius of the core is determined by the coherence length ξ for this mode [see Eq. (44)]. Inthe antiferromagnet there are two spin wave modes and, correspondingly, there are two typesof vortices. The skyrmion core connected to the Goldstone mode is illustrated in Fig. 7(b).Only the pair of the angle variables ( θ , ϕ ) vary inside the core, while Θ = Φ = 0. Figure 7(c)illustrates the skyrmion core connected to the gapped mode. The magnetization vectors M and M rotate around the axis normal to the magnetic field, and there are nonzero Θ andΦ. The core radius is determined by the coherence length ξ given by Eq. (54). VII. LONG-DISTANCE SUPERFLUID SPIN TRANSPORT
From the time when the concept of spin superfluidity was suggested , it was debatedabout whether the superfluid spin current is a “real” transport current. As a response tothis concern, in Ref. 1 a Gedanken (at that time) experiment for demonstration of reality ofsuperfluid spin transport was proposed (see also more recent Refs. 4, 6, and 7).The spin is injected to one side of a magnetically ordered layer of thickness d and spinaccumulation is checked at another side (Fig. 8). For the analysis of spin transport in this setup we must modified the continuity equation (17) for the ferromagnet adding two dissipationterms: 1 γ dM z dt = − ∇ · J − ∇ · J d − M (cid:48) z γT , (58)21 zx J L zx Spin injectionSpin injection Spin injection
Medium withoutspin super fl uidityMedium withspin super fl uidity m x z m z a)b) Spin detection dy xz
Pt PtCr O c) H c) Pt PtCr O H z dx y FIG. 8. Long distance spin transport. (a) Spin injection to a spin-nonsuperfluid medium. (b)Spin injection to a spin-superfluid medium.
Here M (cid:48) z = M z − χH is the non-equilibrium magnetization and the superfluid spin current J is given by Eq. (18). The first dissipation term is the spin diffusion current J d = − Dγ ∇ M z . (59)Spin diffusion does not violates the spin conservation law. The second dissipation term isconnected with the longitudinal spin relaxation. It is characterized by the Bloch time T and does violate the spin conservation law.In the absence of spin superfluidity ( J = 0) Eq. (16) for the spin phase is not relevant, andEq. (58) describes pure spin diffusion [Fig. 8(a)]. Its solution, with the boundary conditionthat the spin current J is injected at the interface x = 0, is J d = J e − x/L d , M (cid:48) z = γJ (cid:114) T D e − x/L d , (60)where L d = (cid:112) DT (61)is the spin-diffusion length. Thus, the effect of spin injection exponentially decays at thescale of the spin-diffusion length, and the density of spin accumulated at the other side ofthe layer decreases exponentially with growing distance d .22owever, if spin superfluidity is possible, the spin precession equation (16) becomes rel-evant. According to this equation, in a stationary state the magnetization M (cid:48) z cannot varyin space [Fig. 8(b)] since according to Eq. (16) the gradient ∇ M (cid:48) z is accompanied by thelinear in time growth of the gradient ∇ ϕ . The right-hand side of Eq. (16) is an analog of thechemical potential, and the requirement of constant in space magnetization M z is similar tothe requirement of constant in space chemical potential in superfluids, or the electrochemicalpotential in superconductors. As a consequence of this requirement, spin diffusion currentis impossible in the bulk since it is simply “short-circuited” by the superfluid spin current.The bulk spin diffusion current can appear only in AC processes.If the spin superfluidity is possible, the spin current can reach the spin detector at theplane x = d opposite to the border where spin is injected. As a boundary condition at x = d ,one can use a phenomenological relation connecting the spin current with the magnetiza-tion: J ( d ) = M (cid:48) z ( d ) v d , where v d is a phenomenological constant. This boundary conditionwas confirmed by the microscopic theory of Takei and Tserkovnyak . Together with theboundary condition J (0) = J at x = 0 this yields the solution of Eqs. (16) and (58): M (cid:48) z = T d + v d T γJ , J ( x ) = J (cid:18) − xd + v d T (cid:19) . (62)Thus, the spin accumulated at large distance d from the spin injector slowly decreases with d as 1 / ( d + C ) [Fig. 8(b)], in contrast to the exponential decay ∝ e − d/L d in the spin diffusiontransport [Fig. 8(a)]. The constant C is determined by the boundary condition at x = d . VIII. EXPERIMENTAL DETECTION OF SPIN SUPERFLUIDITY
A smoking gun of the possibility of spin supercurrents in the B -phase of superfluid Hewas an experiment with a spin current through a long channel connecting two cells filledby the superfluid He. The quasi-equilibrium state of the coherent spin precession (laterrebranded as magnon BEC ) was supported by spin pumping. The magnetic fields appliedto the two cells were slightly different, and therefore, the spins in the two cells precessedwith different frequencies. A small difference in the frequencies leads to a linear growth ofdifference of the precession phases in the cells and a phase gradient in the channel. Whenthe gradient reached the critical value, 2 π phase slips were detected in the experiment. Thesharp 2 π phase slip was reliable evidence of non-trivial spin supercurrents at phase gradients23 z β z ββ FIG. 9. Spin transport through a channel connecting two cells filled by the B phase of superfluid He. The horizontal arrow shows the direction of the spin current in the channel. restricted by finite critical values.It was important evidence that persistent spin currents are possible. However, real long-distance transportation of spin by these currents was not demonstrated. Moreover, it isimpossible to do in the non-equilibrium magnon BEC, which was realized in the B phase of He superfluid and in yttrium-iron-garnet magnetic films. The non-equilibrium magnonBEC requires pumping of spin in the whole bulk for its existence. In the geometry of theaforementioned spin transport experiment this would mean that spin is permanently pumpednot only by a distant injector but also all the way up the place where its accumulation isprobed. Thus, the spin detector measures not only spin coming from the distant injectorbut also spin pumped close to the detector. Therefore, the experiment does not prove theexistence of long-distance spin superfluid transport.The experiment suggested for detection of long-distance superfluid spin transport wasrecently done by Yuan et al. in antiferromagnetic Cr O . In the experiment of Yuan et al. the spin is created in the Pt injector by heating (the Seebeck effect) on one sideof the Cr O film and spin accumulation is probed on another side of the film by the Ptdetector via the inverse spin Hall effect (Fig. 10). In agreement with theoretical prediction,they observed spin accumulation inversely proportional to the distance from the interfacewhere spin was injected into Cr O .In Fig. 8 the spin flows along the axis x , while the spin and the magnetic field are directedalong the axis z . In the geometry of the experiment of Yuan et al. the spin flows along24 zx J L zx Spin injectionSpin injection Spin injection
Medium withoutspin super fl uidityMedium withspin super fl uidity m x z m z a)b) Spin detection dy xz
Pt PtCr O c) H c) Pt PtCr O H z dx y FIG. 10. Long distance spin transport in the geometry of the experiment by Yuan et al. . Spinis injected from the left Pt wire and flows along the Cr O film to the right Pt wire, which servesas a detector. The arrowed dashed line shows a spin-current streamline. the spin axis z parallel to the magnetic field. This geometry is shown in Fig. 10. Thedifference between two geometries is not essential if spin-orbit coupling is ignored. In ourtheoretical analysis we chose the geometry with different directions of the spin current andthe spin in order to stress the possibility of the independent choice of axes in the spin andthe configurational spaces.There were other reports on experimental detection of spin superfluidity in magneticallyordered solids. Bozhko et al. declared detection of spin superfluidity at high temperaturesin a decaying magnon condensate in a YIG film. In their experiment the phase gradientemerged because of spin precession difference produced by a temperature gradient. However,the estimate made in Ref. showed that the total phase difference across the magnon cloudin the experiment did not exceed about 1/3 of the full 2 π rotation. Thus, Bozhko et al. could detect only microscopic spin currents emerging in any spin wave. As explained above,“superfluidity” connected with such currents was well proved by numerous half-a-centuryold experiments on spin waves at all temperatures and does not need new experimentalconfirmations.Observation of the long-distance superfluid spin transport was also reported by Stepanov et al. in a graphene quantum Hall antiferromagnet. However, the discussion of this reportrequires an extensive theoretical analysis of the ν = 0 quantum Hall state of graphene, whichgoes beyond the scope of the present review. A reader can find this analysis in Ref. 29.25 X. DISCUSSION AND CONCLUSIONS
The paper addressed the basics of the spin superfluidity concept: topology, Landau cri-terion, and phase slips. Metastable (persistent) superfluid current states are possible if theorder parameter space (vacuum manifold) has the topology of a circumference on a planelike in conventional superfluids. In ferromagnets it is the circumference on the sphericalsurface in the space of spontaneous magnetizations M . In antiferromagnets it is the cir-cumference on the unit sphere in the space of the unit N´eel vector L /L , where L is thestaggered magnetization. The topology necessary for spin superfluidity requires the uniaxialeasy-plane anisotropy in ferromagnets, while in antiferromagnets this anisotropy is providedby the Zeeman energy, which confines the N´eel vector in the plane normal to the magneticfield.The Landau criterion was checked for the spectrum of elementary excitations, which arespin waves in our case. In ferromagnets there is only one Goldstone spin wave mode. Inbipartite antiferromagnets there are two modes: the Goldstone mode in which spins per-form rotational oscillations around the symmetry axis and the gapped mode with rotationaloscillations around the axis normal to the symmetry axis. At weak magnetic fields the Lan-dau instability starts not in the Goldstone mode, but in the gapped mode. In contrast tosuperfluid mass currents in conventional superfluids, metastable spin superfluid currents arerestricted not only by the Landau criterion from above but also from below. The restrictionfrom below is related to the absence of the strict conservation law for spin.The Landau instability with respect to elementary excitations is a precursor for theinstability with respect to phase slips. The latter instability starts when the spin phasegradient reaches the value of the inverse vortex core radius. This value is on the same orderof magnitude as the Landau critical gradient. Vortices participating in phase slips haveskyrmion cores, which map on the upper or lower part of the spherical surface in the spaceof spontaneous magnetizations in ferromagnets, or in the space of the unit N´eel vectors inantiferromagnets.It is worthwhile to note that in reality it is not easy to reach the critical gradients dis-cussed in the present paper experimentally. The decay of superfluid spin currents is possiblealso at subcritical spin phase gradients since the barriers for phase slips can be overcomeby thermal activation or macroscopic quantum tunneling. This makes the very definition26f the real critical gradient rather ambiguous and dependent on duration of observationof persistent currents. Calculation of real critical gradients requires a detailed dynamicalanalysis of processes of thermal activation or macroscopic quantum tunneling through phaseslip barriers, which is beyond the scope of the present paper. One can find examples of suchanalysis for conventional superfluids with mass supercurrents in Ref. 30.Although evidence of the existence of metastable superfluid spin currents in the B phase ofsuperfluid He were reported long ago the first experiment demonstrating the long-distancetransport of spin by these currents in the solid antiferromagnet was done only recently. This is not the end but the beginning of the experimental verification of the long-distancesuperfluid spin transport in magnetically ordered solids. In the experiment of Yuan et al. spin injection required heating of the Pt injector, and the spin current to the detector isinevitably accompanied by a heat flow. Lebrun et al. argued that probably Yuan et al. detected a signal not from spin coming from the injector but from spin generated by theSeebeck effect at the interface between the heated antiferromagnet and the Pt detector.Such effect has already been observed for antiferromagnet Cr O . If true, Yuan et al. observed not long-distance spin transport but long-distance heat transport. However, it isnot supported by the fact that Yuan et al. observed a threshold for superfluid spin transportat low intensity of injection, when according to the theory (see Sec. IV) the absence of thestrict spin conservation law becomes important. With all that said, the heat-transportinterpretation cannot be ruled out and deserves further investigation. According to thisinterpretation, one can see the signal observed by Yuan et al. at the detector even if the Ptinjector is replaced by a heater, which produces the same heat but no spin. An experimentalcheck of this prediction would confirm or reject the heat-transport interpretation.The present paper focused on spin superfluidity in magnetically ordered solids. Recentlyinvestigations of spin superfluidity were extended to spin-1 BEC, where spin and masssuperfluidity coexist and interplay. This interplay leads to a number of new nontrivialfeatures of the phenomenon of superfluidity. The both types of superfluidity are restrictedby the Landau criterion for the softer collective modes, which are the spin wave modes.As a result, the presence of spin superfluidity diminishes the possibility of the conventionalmass superfluidity. Another consequence of the coexistence of spin and mass superfluidityis phase slips with bicirculation vortices characterized by two topological charges (winding27umbers).
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