Interplay of spin and mass superfluidity in antiferromagnetic spin-1 BEC and bicirculation vortices
IInterplay of spin and mass superfluidity in antiferromagnetic spin-1 BECand bicirculation vortices
E. B. Sonin ∗ Racah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel (Dated: October 29, 2019)The paper investigates the coexistence and interplay of spin and mass superfluidity in the anti-ferromagnetic spin-1 BEC. The hydrodynamical theory describes the spin degree of freedom by theequations similar to the Landau–Lifshitz–Gilbert theory for bipartite antiferromagnetic insulator.The variables in the spin space are two subspins with absolute value (cid:126) /
2, which play the role of twosublattice spins in the antiferromagnetic insulators.As well as in bipartite antiferromagnetic insulators, in the antiferromagnetic spin-1 BEC thereare two spin-wave modes, one is a gapless Goldstone mode, another is gapped. The Landau cri-terion shows that in limit of small total spin (two subspins are nearly antiparallel) instability ofsupercurrents starts from the gapped mode. In the opposite limit of large total spin (two subspinsare nearly parallel) the gapless modes become unstable earlier than the gapped one. Mass and spinsupercurrents decay via phase slips, when vortices cross streamlines of supercurrent. The vorticesparticipating in phase slips are nonsingular bicirculation vortices. They are characterized by twotopological charges, which are winding numbers describing circulations of two angles around thevortex axis. The winding numbers can be half-integer. A particular example of a half-integer vortexis a half-quantum vortex with the superfluid velocity circulation h/ m . But the superfluid veloc-ity circulation is not a topological charge, and in general the quantum of this circulation can becontinuously tuned from 0 to h/ m . I. INTRODUCTION
Spin superfluidity in magnetically ordered systems isdiscussed a number of decades [1–12]. The phenomenonis based on the analogy of special cases of the Landau–Lifshitz–Gilbert (LLG) theory in magnetism and super-fluid hydrodynamics. While in a superfluid mass (chargein superconductors) can be transported by a current pro-portional to the gradient of the phase of the macroscop-ical wave function, in a magnetically ordered mediumthere are spin currents, which are proportional to thegradient of the spin phase. The latter is defined as theangle of rotation around some axis in the spin space.Strictly speaking this analogy is complete only if thisaxis is a symmetry axis in the spin space. Then accord-ing to Noether’s theorem the spin component along thisaxis is conserved. But possible violation of the spin con-servation law usually is rather weak because it is relatedwith relativistically small (inversely proportional to thespeed of light) processes of spin-orbit interaction. In fact,the LLG theory itself is based on the assumption of weakspin-orbit interaction [13].The analogy of the LLG theory with the theory of su-perfluidity suggests a new useful language for descriptionof phenomena in magnetism, but not a new phenomenon.During the whole period of spin superfluidity investiga-tions and up to now there have been disputes about def-inition what is spin superfluidity. There is a school ofthinking that the existence of spin current proportionalto the spin phase (rotation angle) means spin superfluid-ity [5]. There is no law in the book, which forbids the use ∗ [email protected] of this definition. But then spin superfluidity becomes atrivial ubiquitous phenomenon existing in any magneti-cally ordered medium. A spin current proportional to thespin phase emerges in any domain wall and in any spinwave. Under this broad definition of spin superfluidityspin superfluidity was experimentally detected beyondreasonable doubt in old experiments of the middle of the20th century detecting domain walls and spin waves.We prefer define the term superfluidity in its origi-nal meaning known from the times of Kamerlingh Onnesand Kapitza: transport of some physical quantity (mass,charge, or spin) over macroscopical distances without es-sential dissipation. This requires a constant or slowlyvarying phase gradient at macroscopical scale with thetotal phase variation along the macroscopic sample equalto 2 π multiplied by a very large number. In examples ofdomain walls and spin waves this definitely does not takeplace. Gradients oscillate in space or time, or in both.The total phase variation is on the order of π or much less.Currents transport spin on distances not more than thedomain wall thickness, or the spin wavelength. Althoughsuch currents are also sometimes called supercurrents, weuse the term supercurrent only in the case of macroscop-ical supercurrent persistent at large spatial and temporalscales.The possibility of supercurrents is conditioned by spe-cial topology of the magnetic order parameter space (vac-uum manifold). Namely, this space must have topologyof circumference on the plane. In magnetically orderedsystems this requires the presence of easy-plane uniaxialanisotropy. It is possible also in non-equilibrium coher-ent precession states, when spin pumping supports coher-ent spin precession with fixed spin component along themagnetic field (the axis z ). Such non-equilibrium coher- a r X i v : . [ c ond - m a t . o t h e r] O c t ent precession states were considered as manifestation ofmagnon BEC [14, 15]. These states were experimentallyinvestigated in the B phase of superfluid He [3] and inYIG films [16]. Resemblance and distinction of coherentprecession states with BEC and lasers was discussed inRef. 4.For assessment of possibility of observation of long-distance spin transport by spin supercurrents one shouldconsider the Landau criterion, which checks stability ofsupercurrent states with respect to weak excitations of allcollective modes. Although the Landau criterion pointsout a threshold for the current state instability, it tellsnothing about how the instability develops. The decayof the supercurrent is possible only via phase slips. Ina phase slip event a vortex crosses current streamlinesdecreasing the phase difference along streamlines. Belowsome critical value of supercurrent phase slips are sup-pressed by energetic barriers. The critical value of thesupercurrent at which barriers vanish is of the same or-der as those estimated from the Landau criterion. Thisleads to a conclusion that the instability predicted by theLandau criterion is a precursor of the avalanche of phaseslips not suppressed by any activation barrier.Recently investigations of spin superfluidity were ex-tended to spin-1 BEC, where spin and mass superfluid-ity coexist and interplay. Investigations focused on theferromagnetic BEC [17–19]. The present paper extendsthe analysis on all states of spin-1 BEC, either ferro- orantiferromagnetic. The interplay of spin and mass su-perfluidity is responsible for a number of new nontrivialfeatures.The first step of the analysis was reformulation of hy-drodynamics of spin-1 BEC presenting it in the formmore suitable for the goals of this paper. It was alreadyknown that hydrodynamics of ferromagnetic spin-1 BECis described by equations of spin motion similar to thosein the LLG theory in magnetism but taking into accountthe possibility of superfluid motion as a whole [17–19].This allowed to use some results known from investiga-tions of spin superfluidity in magnetically ordered solids.In the present paper we demonstrate that the hydrody-namics of the antiferromagnetic spin-1 BEC is similarto the LLG theory for a bipartite antiferromagnet withtwo sublattices each of which is characterized by a vectorof magnetization (spin). Despite translational invarianceis not broken and there are no sublattices in the spin-1 BEC, one can introduce two spins of absolute value (cid:126) /
2, which can vary their direction in space and time,but not their absolute values similarly to two sublatticemagnetizations in the LLG theory for a bipartite antifer-romagnet. We shall call them subspins. Thus, results ofthe recent analysis of spin superfluidity in solid antifer-romagnets with localized spins [20] become relevant forantiferromagnetic spin-1 BEC. In particular, like in solidantiferromagnets, in the antiferromagnetic spin-1 BECthere are two spin-wave modes: one is a Goldstone gap-less mode similar to that in the ferromagnetic BEC andanother has a gap depending on the magnetic field. At weak magnetic fields (the Zeeman energy is less than thespin-dependent interaction energy) the Landau criticalvalues are reached in the gapped mode earlier than inthe gapless one. At strong magnetic fields close to thefield at which spin polarization is completely saturatedalong the magnetic field, the situation is opposite: thegapless mode becomes unstable earlier than the gappedone.While in scalar superfluids the Landau critical veloc-ities for mass currents are scaled by the sound velocityand in magnetically ordered media with localized spinscritical velocities for spin currents are scaled by the spin-wave velocity, in the spin-1 BEC, where spin and masssuperfluidity coexist, critical velocities for both currentsare determined by the lesser from the sound and spin-wave velocity. Usually this is the spin-wave velocity, andthe analysis of the paper focuses on the spin degree offreedom. Another remarkable outcome of the interplay ofmass and spin superfluidity is properties of vortices par-ticipating in phase slips. In the multicomponent spin-1BEC vortices are determined not by one but by two wind-ing numbers (topological charges). The two charges arerelated with two gauge invariances: with respect to theglobal phase of the wave function and with respect to thespin phase. We call these vortices bicirculation vortices.In the spin-1 BEC, like in other multicomponent su-perfluids, the circulation of superfluid velocity is not atopological charge anymore, and the superfluid velocityis not curl-free. In single-component scalar superfluids,where the velocity circulation is a topological charge, thevelocity field around the vortex is singular and divergesas 1 /r , when the distance r from the vortex axis goes to0. The divergence in the energy can be avoided only ifthe superfluid density vanishes at the vortex axis. Butin the spin-1 BEC the singularity 1 /r can be compen-sated without suppression of the superfluid density in thevortex core by a proper choice of the ratio between twotopological charges (winding numbers). Such vortices arecalled nonsingular or continuous [21]. Normally the Lan-dau critical gradients and the critical gradients for theinstability with respect to phase slip are of the order ofthe inverse core radius. Therefore, the instability with re-spect to phase slips starts earlier for nonsingular vorticesbecause of their larger core radius compared to singularvortices. Existence of vortices with different ratios of twowinding numbers makes the decay of supercurrents morecomplicated. There are vortices, which are effective forrelaxation of mass supercurrents, and which are effectivefor relaxation of spin supercurrents. For complete relax-ation of all supercurrents to the ground state at least twodifferent types of vortices must participate in phase slips.In spin-1 BEC winding numbers of vortices can be notonly integer, but also half-integer. The known exampleof half-integer vortices is the half-quantum vortex, whichattracted a lot of attention in the literature [22, 23]. How-ever, in the spin-1 BEC the “quantum” of velocity circu-lation can be equal not only to the fundamental quantum h/m or its half, but in fact can be continuously tuned bya magnetic field, or by the intensity of spin pumping,which supports the non-equilibrium coherent precessionstate with fixed z -component of spin. II. HYDRODYNAMICS FROM THEGROSS–PITAEVSKII THEORY OF SPIN-1 BEC
The wave function of bosons with spin 1 is a 3D vectorin the spin space, In the Cartesian basis [23–25] the wavefunction vector is ψ = ψ x ψ y ψ z , (1)where ψ x = ψ + − ψ − √ , ψ y = i ( ψ + + ψ − ) √ , ψ z = − ψ , (2)and ψ ± , ψ are coefficients of the expansion of the wavefunction in eigenfunctions of the spin projection on thequantization axis (the axis z ) with eigenvalues ± , ψ , i (cid:126) ∂ ψ ∂t = δ H δ ψ ∗ , (3)is obtained by variation of the Lagrangian L = i (cid:126) (cid:18) ψ ∗ ∂ ψ ∂t − ψ ∂ ψ ∗ ∂t (cid:19) − H ( ψ , ψ ∗ ) (4)with respect to ψ ∗ . The complex-conjugate equation fol-lows from variation with respect to ψ . Here δ H δ ψ ∗ = ∂ H ∂ ψ ∗ − ∇ i ∂ H∇ i ∂ ψ ∗ (5)is a functional derivative of the Hamiltonian H = (cid:126) m ∇ i ψ ∗ ∇ i ψ + V | ψ | V s ( | ψ | − | ψ | )2 − γ H · S | ψ | , (6)where γ is the gyromagnetic ratio, H is the magneticfield, and V and V s are amplitudes of spin-independentand spin-dependent interaction of bosons respectively,Spin-1 BEC is ferromagnetic if V s is negative and an-tiferromagnetic if V s is positive. The complex vector ψ determines the particle density n = | ψ | of bosons withspin per particle S = − i (cid:126) [ ψ ∗ × ψ ] | ψ | , (7)and the superfluid velocity v i = − i (cid:126) m | ψ | ( ψ ∗ ∇ i ψ − ψ ∇ i ψ ∗ ) . (8) The equation of the spin balance is ∂ ( nS i ) ∂t + ∇ k J ik = G i , (9)where the tensor J ik = (cid:15) ist (cid:18) ψ ∗ s ∂ H ∂ ∇ k ψ t + ψ s ∂ H ∂ ∇ k ψ ∗ t (cid:19) (10)is the current of the i th spin component along the axis k , and G i = (cid:15) ist (cid:18) ψ ∗ s ∂ H ∂ψ t + ∇ k ψ ∗ s ∂ H ∂ ∇ k ψ t + ψ s ∂ H ∂ψ ∗ t + ∇ k ψ s ∂ H ∂ ∇ k ψ ∗ t (cid:19) (11)is the torque on the i th spin component, which vanishesif the Hamiltonian is spherically symmetric in the spinspace.Now we perform the generalized Madelung transfor-mation presenting the wave function vector as ψ = ψ e i Φ (cid:18) cos λ d + i sin λ f (cid:19) , (12)where the real scalar ψ = √ n , the two real unit mutuallyorthogonal vectors d and f , and the two phase (angle)variables Φ and λ are 6 parameters fully determining thecomplex 3D vector ψ . In new hydrodynamical variablesthe spin is S = (cid:126) sin λ [ d × f ] , (13)and the superfluid velocity is v i = (cid:126) m (cid:20) ∇ i Φ + sin λ d ∇ i f − f ∇ i d ) (cid:21) . (14)The two unit vectors d and f together with the thirdunit vector, s = S S = [ d × f ] , (15)fully determine the quantum state for the ferromagneticspin-1 BEC, when λ = π/
2. States with 0 ≤ λ < π/ S of the total spin less than its maximal value (cid:126) in theferromagnetic state. The pure antiferromagnetic statewith zero total spin ( λ = 0) is called polar phase [23].The Hamiltonian (6) transforms to H = mnv H , (16)where H = n (cid:126) m (cid:20) cos λ ∇ d + sin λ ∇ f − sin λ ( f ∇ d ) + ∇ λ (cid:21) + V n V s n sin λ − γn (cid:126) sin λ H · [ d × f ](17)is the Hamiltonian in the coordinate frame moving withthe superfluid velocity v .The dynamical equations for our hydrodynamical vari-ables follow from the nonlinear Schr¨odinger equation (3): (cid:126) [ ˙Φ + ( v · ∇ )Φ] = − δ H δn , ˙ n = 1 (cid:126) δ H δ Φ = − (cid:126) ∇ · ∂ H ∂ ∇ Φ = −∇ · ( n v ) , (18) n (cid:126) cos λ [ ˙ λ + ( v · ∇ ) λ ] = − (cid:18) f δ H δ d − d δ H δ f (cid:19) , (19) n (cid:126) [ ˙ d + ( v · ∇ ) d ] = f cos λ δ H δλ + s sin λ (cid:18) s δ H δ f (cid:19) , (20) n (cid:126) [ ˙ f + ( v · ∇ ) f ] = − d cos λ δ H δλ − s sin λ (cid:18) s δ H δ d (cid:19) . (21)The continuity equation [the second line of Eq. (18)] takesinto account that because of gauge invariance the Hamil-tonian depends on the gradient of Φ but not on the phaseΦ itself. The superfluid velocity is not curl-free and thegeneralized Mermin–Ho relation is ∇ × v = (cid:126) m { cos λ [ ∇ λ × ( d i ∇ f i − f i ∇ d i )]+ sin λ(cid:15) ijk s i [ ∇ s j × ∇ s k ] } . (22) In the ferromagnetic state ( λ = π/
2) this reduces to theoriginal Mermin–Ho relation [26].We want to demonstrate the analogy of the spin-1 BEChydrodynamics with LLG theory for bipartite antiferro-magnet. The vector L = (cid:126) cos λ d is an analogue of theantiferromagnetic vector (staggering magnetization) inthe LLG theory of a bipartite antiferromagnet. Continu-ing this analogy, we may introduce the spins S and S similar to spins of two sublattices of a bipartite antifer-romagnet, which determine the antiferromagnetic vector L = S − S and the total spin S = S + S . The vectors L and S are orthogonal one to another, and the absolutevalues of vectors S and S are equal to S = (cid:126) / S and S may replace λ , d , and f as hy-drodynamical variables. Then the canonical equationsfor the spin degree of freedom become n [ ˙ S i + ( v · ∇ ) S i ] = − (cid:20) S i × δ H δ S i (cid:21) , (23)where i = 1 ,
2. Apart from the term ∝ v in the left-handside taking into account the superfluid motion as a whole,Eq. (23) is exactly the LLG equations for a bipartite an-tiferromagnet. The Hamiltonian H in Eq. (23) directlyfollows from the Hamiltonian in the Gross–Pitaevskii the-ory: H = n m (cid:20) ( ∇ S + ∇ S ) λ ) + ( ∇ S − ∇ S ) + (cid:18) − λ λ − λ (cid:19) ∇ ( S · S ) sin λ + ( S ∇ S − S ∇ S ) (1 + cos λ ) cos λ (cid:21) + V s n ( S + S ) (cid:126) − nγ H · ( S + S ) . (24)In the LLG theory for localized spins they usually usethe Hamiltonian, which is a general quadratic form ofgradients ∇ S and ∇ S with constant coefficients. InEq. (24) the coefficients depend on the angle λ , whichdepends on S and S :cos 2 λ = − ( S · S ) S = − S · S ) (cid:126) . (25)In polar angles determining directions of S and S , S ix = S cos θ i cos ϕ i , S iy = S cos θ i sin ϕ i ,S iz = S sin θ i , (26)Eq. (23) transforms tocos θ i [ ˙ θ i + ( v · ∇ ) θ i ] = − nS δ H δϕ i , cos θ i [ ˙ ϕ i + ( v · ∇ ) ϕ i ] = 1 nS δ H δθ i . (27) Of course, there are no lattices or sublattices in thespin-1 BEC. But this strong analogy with an antiferro-magnet with two spin sublattices means that the LLGtheory remains valid even if spins are delocalized andsublattices melt down. In a bipartite antiferromagnetthe angle λ is a canting angle measuring deviation ofsublattice spins from the strictly antiparallel orientationin a pure antiferromagnetic state with zero total spin.We shall call spin vectors S and S subspins.In cold atoms BEC the spin-independent interaction ∝ V is much stronger than the spin-dependent one ∝ V s .According to Ho [24], the ratio | V s | /V is 0.04 for Naand 0.01 for Ru. Correspondingly, the ratio of thespin-wave velocity to the sound velocity proportional to (cid:112) | V s | /V is also small. This allows in the further analysisto ignore Eq. (18) describing motion of the superfluid asa whole and to assume that the superfluid is incompress-ible. This does not rule out possibility of superfluid masscurrents with v (cid:54) = 0, but the velocity must be divergence-feee: ∇ · v = 0. Stability of mass currents is also can beinvestigated considering only the spin degree of freedomand ignoring the mechanical degree of freedom. At wasshown in Ref. 20 for ferromagnetic spin-1 BEC and willbe shown below for antiferromagnetic spin-1 BEC, insta-bility starts in the softest mode, spin mode in our case.So our further analysis focuses on Eqs. (19)–(21) withthe Hamiltonian Eq. (24). III. COLLECTIVE MODES AND THE LANDAUCRITERION
For the further analysis it is convenient to transformthe angle variables in Eq. (26) to angles θ = π + θ − θ , θ = θ + θ − π ,ϕ = ϕ + ϕ , ϕ = ϕ − ϕ , (28)which have already been used in the analysis of spindynamics in antiferromagnetic insulators [20]. A ben-efit of these variables is that the dynamical equationsare reduced to decoupled equations for two noninteract-ing modes. In these variables the equations of motionEq. (27) transform to(cos 2 θ + cos 2 θ )[ ˙ θ + ( v · ∇ ) θ ]= − nS (cid:18) cos θ cos θ δ H δϕ + sin θ sin θ δ H δϕ (cid:19) , (cos 2 θ + cos 2 θ )[ ˙ ϕ + ( v · ∇ ) ϕ ]= 1 nS (cid:18) cos θ cos θ δ H δθ − sin θ sin θ δ H δθ (cid:19) , (29)(cos 2 θ + cos 2 θ )[ ˙ θ + ( v · ∇ ) θ ]= 1 nS (cid:18) cos θ cos θ δ H δϕ + sin θ sin θ δ H δϕ (cid:19) , (cos 2 θ + cos 2 θ )[ ˙ ϕ + ( v · ∇ ) ϕ ]= − nS (cid:18) cos θ cos θ δ H δθ − sin θ sin θ δ H δθ (cid:19) . (30)The canting angle λ is given bycos 2 λ = cos 2 θ (1 + cos 2 ϕ ) − cos 2 θ (1 − cos 2 ϕ )2 . (31)The meaning of the angles θ and θ for the simple case ϕ = ϕ = 0 is illustrated in Fig. 1.The Hamiltonian H in the angle variables Eq. (28) isgauge-invariant with respect to phases Φ and ϕ . Thegeneral expression for it rather clumsy, and further weshall present the Hamiltonian only for particular cases.We consider states with ϕ = 0 when the canting angle λ ✓
1. The gappedmode becomes unstable earlier than the gapless mode. The dotted line shows states, to which the current state relaxes afterphase slips with antiferromagnetic vortices (0 , ± θ = π/ − θ = 0 .
2. The gapless modebecomes unstable earlier than the gaped mode. The dotted line shows states, to which the current state relaxes after phaseslips with ferromagnetic vortices ( ± / , ± / trum ω − w · k = c s ξ (cid:115)(cid:20) − ξ ( K − k )2 (cid:21) − cos θ . (49)The Landau criterion imposes two inequalities on gradi-ents (velocities) in the current state. The first one, K < − cos θ ) ξ , (50)provides that at no k the frequency has an imaginarypart, i.e., the gap in the spectrum is positive. The secondinequality, w < m c s (cid:126) k (cid:40)(cid:20) − ξ ( K − k )2 (cid:21) − cos θ (cid:41) , (51) guarantees that the frequency is positive at any k . Theright-hand side of the inequality has a minimum at k = 2 ξ (cid:115)(cid:18) − ξ K (cid:19) − cos θ . (52)Using this value in Eq. (51), the latter becomes | w | < c s (cid:32)(cid:114) sin θ − ξ K (cid:114) cos θ − ξ K (cid:33) . (53)Figure 2 shows the stability areas for two modes inthe plane of two dimensionless parameters Kξ and v/c s .The current state is stable in the area where the bothmodes are stable. At weak spin polarization (small cant-ing angle θ ) the gapped mode destabilizes the currentstate earlier than the gapless one, like in antiferromag-netic insulators [20]. In the opposite limit θ → π/ IV. NONSINGULAR VORTICES INANTIFERROMAGNETIC SPIN-1 BEC
In scalar (single-component) superfluids only singularvortices are possible, in which the superfluid density mustvanish at the vortex axis in order to compensate the 1 /r divergence in the velocity field. In multicomponent su-perfluids it is possible to compensate the divergence atthe axis without suppression of the superfluid density inthe vortex core. Since we consider here an incompress-ible superfluid, further we shall focus only on phase slipswith nonsingular vortices.Studying nonsingular vortices we can use the Hamilto-nian Eq. (32) derived under assumption that ϕ = 0. Thevortex is characterized by two topological charges (wind-ing numbers), determined by circulations of the angles Φand ϕ , N Φ = 12 π (cid:73) ∇ Φ · d l , N ϕ = 12 π (cid:73) ∇ ϕ · d l , (54)where integration is along the closed path (loop) aroundthe vortex axis. We shall call these vortices bicirculationvortices and label them as ( N Φ , N ϕ )-vortices. The super-fluid velocity circulation around the path surrounding thevortex at large distances from its axis isΓ = (cid:73) v · d l = hm ( N Φ − sin θ ∞ N ϕ ) . (55)Here we introduced the angle θ ∞ equal to the value of θ far from the vortex axis where θ = 0 and the gradient-dependent energy is negligible. The angle θ ∞ dependson the magnetic field and is determined by minimizationof the energy E m given by Eq. (34):sin θ ∞ = (cid:126) γH cos θV s n . (56)Single-valuedness of the wave function Eq. (12) re-quires that N Φ is integer if the unit vectors d and f return back to their original values after going around thepath encircling the vortex. But the wave function alsoremains single-valued if d and f rotate by 180 ◦ aroundthe axis normal to both of them, i.e., parallel to the to-tal spin, while the charge N Φ is half-integer. The angleof rotation around the spin coincides with the angle ϕ only if the spin is strictly parallel or antiparallel to themagnetic field H (axis z ). In this case the topologicalcharges N ϕ and N Φ can be either both integer, or both half-integer. Correspondingly, we call vortices integer, orhalf-integer.We shall consider axisymmetric vortices. Then gra-dient of the angles Φ and ϕ equal to ∇ Φ v and ∇ ϕ v respectively have only azimuthal components, ∇ Φ v = N Φ [ˆ z × r ] r , ∇ ϕ v = N ϕ [ˆ z × r ] r , (57)while the angles θ and θ depend only on the distance r from the vortex axis. The angle gradients diverge as 1 /r .According to the Hamiltonian Eq. (32) and the expres-sion Eq. (33) for the superfluid velocity, this divergencecan be compensated only for the two types of vortices:(i) Any integer N ϕ , but N Φ = 0. At the vortex axis θ = 0 and θ = ± π . These are vortices (0 , N ). Thestructure of the skyrmion core of the vortex is illustratedschematically in Fig. 3(a) showing variation of two sub-spins with the distance r from the vortex axis. The vortexcan be called antiferromagnetic because the angle θ show-ing the direction of the antiferromagnetic vector varies inthe vortex core and there is no spin polarization at thevortex axis. A similar vortex was investigated in the bi-partite antiferromagnet in the LLG theory of localizedspins [20].The velocity circulation far from the (0 , N ) vortex axisis Γ = − N hm sin θ ∞ . (58)The result Eq. (58) is remarkable. The velocity circula-tion is quantized but with the circulation quantum de- ---•�H (a)Antiferromagnetic / vortex (O,N) � ��-------�-----ו=----+-----t---י--(b) Ferromagneticvortex (N,N) (c) Ferromagneticvortex (-N,N) (d) Vortex (-N,N) in ferromagnetic spin-1 BEC (e) Vortex (-N,N) at saturated spin polarization far from the vortex axis I \ \ I � ............ / \ I "' \ A t / �� "' == ==
00 == - r FIG. 3. Variation of two subspins S and S with the dis-tance r from the vortex axis in skyrmion cores of varioustypes of vortices. The rows (a)–(c) show vortices in the anti-ferromagnetic spin-1 BEC. The row (d) shows the vortex inthe ferromagnetic spin-1 BEC. The row (e) shows the vortex( − N, N ) at saturated spin polarization far from the vortexaxis ( θ → θ ∞ = π/ pendent on the magnetic field, since the canting angle θ ∞ depends on the magnetic field. The quantum varies fromzero (i.e., no quantization) to the fundamental quantum h/m .(ii) Two charges (winding numbers) satisfy the con-dition N Φ = ± N ϕ , At the axis θ = ± π and θ = 0.These are vortices ( ± N, N ), which we call ferromagneticbecause the spin is fully polarized at the vortex axis.The skyrmion structures of both ferromagnetic vorticesare shown in Figs. 3(b) and (c). Ferromagnetic vortices( ± N, N ) have the velocity circulationΓ =
N hm ( ± − sin θ ∞ ) . (59) The circulation quanta for these vortices can also betuned by the magnetic field. Since N can be half-integerthe circulation quantum varies from 0 to the fundamentalquantum h/m . The half-quantum vortices [22, 23] withΓ = ± h/ m are possible in the limit of very weak spinpolarization (weak magnetic field), when θ ∞ → θ and θ yields two coupled nonlinear Euler–Lagrange equations: − r ddr (cid:18) r dθ dr (cid:19) − (cid:18) dθdr (cid:19) sin θ − N Φ N ϕ cos θ cos θr + N ϕ sin θ sin θ r + (sin θ − sin θ ∞ cos θ ) cos θ ξ = 0 , (60) − r ddr (cid:20) r dθdr (1 + cos θ ) (cid:21) + N Φ N ϕ sin θ sin θr + N ϕ sin 2 θ (1 − cos θ − θ )4 r + sin θ ∞ sin θ sin θξ = 0 . (61)Let us start from vortices (0 , N ) when there is no cir-culation of the phase Φ. The (0 , N = 1 is energetically more favorablefor phase slips. The natural scale in Eq. (61) is the spincoherence length ξ , which determined the vortex-coreradius r c ∼ ξ excepting the limit of weak spin polariza-tion θ ∞ (cid:28)
1. In this limit the size of the vortex coreis much larger than the coherence length, as we shallsee soon. Then all gradient terms and terms ∝ /r inEq. (60) can be neglected and the small θ is determinedby a simple expression θ ( r ) = θ ∞ cos θ ( r ) . (62)Inserting it into Eq. (61) transforms it into − r ddr (cid:18) r dθdr (cid:19) − sin 2 θ r + θ ∞ sin 2 θ ξ = 0 . (63)The boundary conditions for this equation are θ = π/ r = 0) and θ = 0 at r → ∞ . Thespatial scale of this equation determines the vortex coreradius r c ∼ ξ θ ∞ , (64)which essentially exceeds the spin correlation length ξ .This justifies ignoring of gradient terms in Eq. (60).Switching to vortices ( ± N, N ) we also consider the vor-tices with minimal circulations. These are the (cid:0) ± , (cid:1) vortices with half-integer winding numbers. In these vor-tices θ = 0 everywhere in space, and Eq. (60) becomes − r ddr (cid:18) r dθ dr (cid:19) − cos θ r + (sin θ − sin θ ∞ ) cos θ ξ = 0 . (65) In the limit of strong spin polarization when two sub-spins are nearly parallel, the angle ˜ θ = π/ − θ is smalleverywhere varying from 0 at the axis of the vortex (cid:0) , (cid:1) to small ˜ θ ∞ far from the axis. The equation for ˜ θ ,14 r ddr (cid:32) r d ˜ θdr (cid:33) − ˜ θ r + (˜ θ ∞ − ˜ θ )˜ θ ξ = 0 , (66)similar to the equation for the amplitude of the wavefunction in the Gross–Pitaevskii theory for scalar super-fluids. The vortex core radius is r c ∼ ξ ˜ θ ∞ , (67)which essentially exceeds the coherence length ξ atsmall ˜ θ ∞ . If the Zeeman energy (cid:126) γH exceeds the spin-dependent interaction energy V s n [see Eq. (56)] spin po-larization is saturated and the both subspins become par-allel to the magnetic field. Then the core radius r c of thevortex (cid:0) , (cid:1) given by Eq. (67) becomes infinite. Thisvortex should be ignored because its energy and circu-lation vanish. However, the half-integer vortex (cid:0) − , (cid:1) has a finite core radius of the order ξ , and according toEq. (59) its circulation quantum h/m is the same as inthe scalar superfluids.It is interesting to compare the vortices in antiferro-magnetic spin-1 BEC with vortices in ferromagnetic spin-1 BEC, which were considered in Ref. 19. An exampleof the vortex in ferromagnetic spin-1 BEC is illustratedin Fig. 3(d). In antiferromagnetic spin-1 BEC in thevortex core the canting angle θ varies, while θ = 0 ev-erywhere [Fig. 3(c)]. In contrast, in ferromagnetic spin-1 BEC in the vortex core the angle θ varies, while the0canting angle is θ = π/ V s is negative and the structure with parallel subspins haslesser energy than the structure with antiparallel sub-spins. In this structure the total spin deviates from thedirection of the magnetic field, and this is incompatiblewith the existence of half-integer vortices. But at mag-netic fields sufficient for saturated spin polarization alongthe magnetic field there is a possibility of a half-integervortex also in ferromagnetic BEC, if the wave functionin its core is antiferromagnetic, i.e., θ < π/
2. The cor-responding structure of the skyrmion core is shown inFig. 3(e). Thus, a half-integer vortex is possible both inanti- and ferromagnetic phase. But there is a differencein the energy and the size of the core in these two phases.In the ferromagnetic spin-1 BEC the spin-dependentinteraction energy V s n is a negative constant, which doesnot affect the vortex structure. At weak magnetic fieldsthe core radius is scaled not by the coherence length ξ given by Eq. (44) but a much longer length determined bythe easy-plane anisotropy energy, which is usually smallerthan | V s | n . However, if inside the core the wave functionis antiferromagnetic ( θ < π/
2) the energy of the coredoes depend on | V s | n . As a result, the core radius of theinteger vortex (-1,1) is larger than the core radius of thehalf-integer vortex (cid:0) − , (cid:1) . This difference vanishes inthe limit of very strong magnetic fields when the Zee-man energy essentially exceeds | V s | n . In this limit thecore radius of all vortices is determined by the Zeemanenergy: r c ∼ (cid:126) / √ γHS . The integer vortex (-1,1) withdouble-quantum velocity circulation 2 h/m is an analogof the Anderson–Toulouse vortex existing in the A phaseof superfluid He [21, 27].
V. BICIRCULATION VORTICES AND PHASESLIPS
For the analysis of participation of nonsingular vor-tices in phase slips one must consider interaction of vor-tices with mass and spin currents. The total energy ofthe vortex is mostly determined by the area outside thecore (the London region) where one must take into ac-count interaction of vortices with mass and spin currents.In the London region the angles θ and θ are close totheir asymptotic values θ ∞ and 0 respectively. Then onlyterms quadratic in gradients ∇ Φ and ∇ ϕ are kept in theHamiltonian Eq. (32). The phase (angle) gradients in thepresence of a vortex and currents are ∇ Φ = ∇ Φ v + K Φ , ∇ ϕ = ∇ ϕ v + K , (68)where ∇ Φ v and ∇ ϕ v are given by Eq. (57). Substitut-ing these expressions into the Hamiltonian and integrat-ing over the plane normal to the vortex axis one obtainsthe energy of the straight vortex per unit length in the presence of currents: E v = πn (cid:126) m ( N − θ ∞ N Φ N ϕ + N ϕ ) ln Rr c − π (cid:126) n t · [ R × ˜ v ] , (69)where t is the unit vector along the vortex axis, R isthe position vector for the vortex axis with its origin be-ing either at a wall, or the position of the other vortex(antivortex), and ˜ v = (cid:126) m [( N Φ − sin θ ∞ N ϕ ) K Φ +( N ϕ − sin θ ∞ N Φ ) K ] = m Γ h v + N ϕ (cid:126) cos θ ∞ m K (70)is the effective velocity.The vortex energy E v has a maximum at R = (cid:126) ( N − θ ∞ N Φ N ϕ + N ϕ )2 m ˜ v , (71)and the energy at the maximum is a barrier preventingphase slips. The barrier vanishes if R becomes of theorder of the vortex core radius. Thus, phase slips aresuppressed by energetic barriers as far as˜ v < (cid:126) ( N − θ ∞ N Φ N ϕ + N ϕ ) mr c . (72)In single-component scalar superfluids the effective ve-locity ˜ v coincides with the superfluid velocity v . Whenthe latter vanishes the barrier is infinite, and phase slipsare impossible. This reflects the trivial fact that the cur-rentless state is the ground state, and phase slips cannotdecrease its energy. In our case of a multicomponent su-perfluid phase slips also cannot decrease the energy if theeffective velocity ˜ v vanishes. But the latter vanishes notonly in the currentless ground state but also in stateswith nonzero gradients satisfying the condition ˜ v = 0.Thus, the only one type of vortices with fixed ratio oftwo winding numbers is not sufficient for complete de-cay of a current state with arbitrary values of two phasegradients. Complete relaxation to the ground state re-quires at least two types of vortices with different ratiosof winding numbers.For the vortex (0,1) with the core radius Eq. (64) atweak spin polarization θ ∞ (cid:28) K < r c = θ ∞ ξ . (73)This condition imposes the same restriction on stabilityof the current state as the Landau criterion Eq. (50) onstability of the gapped mode at small θ ∞ . Thus, insta-bility of the gapped mode is a precursor of phase slipswith this type of vortices. Phase slips only with vortices(0 , ±
1) cannot result in complete relaxation of arbitrarycurrent states to the ground state. A final state after1these phase slips is a state with zero effective velocity ˜ v .The latter states lie on the dotted line in Fig. 2(a).For the vortex (cid:0) , (cid:1) close to saturated spin polar-ization ( θ ∞ ∼ π/
2) with the core radius Eq. (67) theinequality Eq. (72) yields the inequality mv (cid:126) + 2 K < r c = ˜ θ ∞ ξ , (74)which at small ˜ θ ∞ agrees with the Landau criterionEq. (45) for the gapless mode. Thus, instability of thismode is a precursor of phase slips with ferromagneticvortices (cid:0) , (cid:1) . These phase slips result in relaxation ofan initial arbitrary current state to current states with˜ v = 0, which lie on the dotted line in Fig. 2(b).At complete saturation of spin polarization ( θ ∞ = π/
2) no stable spin current is possible. The vortex (cid:0) , (cid:1) has zero energy and zero velocity circulation and is irrel-evant. But mass persistent currents are possible as far asthey are stable with respect to phase slips with vortices (cid:0) − , (cid:1) . The core radius of this vortex is of the orderof the coherence length ξ , and the condition of stabil-ity with respect to phase slips is similar to that obtainedfrom the Landau criterion.In our analysis the criterion of instability was disap-pearance of energetic barriers suppressing the decay ofmass or spin supercurrents. The critical velocities (gra-dients) are inversely proportional to the core radius ofvortices participating in phase slips. In reality phaseslips may occur even in the presence of barriers due tothermal activation or quantum tunneling, although theirprobability is low. At low velocities (gradients) the loga-rithmic factor in the vortex energy Eq. (69) is very largeand weakly depends on the value of the core radius r c .Then the chance of the vortex to participate in the phaseslip is determined mostly by the pre-logarithmic factor inEq. (69). The vortex with the smallest pre-factor is themost probable actor in the phase slip. This makes vor-tices with smaller circulations better candidates for phaseslips. In particular, in the ferromagnetic spin-1 BEC atmagnetic fields sufficient for spin orientation along mag-netic fields the half-integer vortex (cid:0) − , (cid:1) with single-quantum circulation h/m is a more probable actor inphase slips rather than the Anderson–Toulouse integervortex (-1,1) with double-quantum velocity circulation2 h/m despite the latter has a larger core radius. VI. CONCLUSIONS
The hydrodynamics of the antiferromagnetic spin-1BEC was derived from the Gross–Pitaevskii theory show- ing its analogy with the LLG theory of bipartite solidantiferromagnets. In the hydrodynamics of spin-1 BECtwo subspins with the absolute value (cid:126) / π/ h/ m . However, in gen-eral one can tune continuously the velocity circulationquantum of a vortex between 0 and h/ m . This musthave important consequences for properties of the spinorBEC, especially at its rotation. [1] E. B. Sonin, Zh. Eksp. Teor. Fiz. , 2097 (1978), [Sov.Phys.–JETP, , 1091 (1978)]. [2] E. B. Sonin, Usp. Fiz. Nauk , 267 (1982), [Sov. Phys.–Usp., , 409 (1982)]. [3] Yu. Bunkov, in Progress of Low Temperature Physics ,Vol. 14, edited by W. P. Halperin (Elsevier, 1995) p. 68.[4] E. B. Sonin, Adv. Phys. , 181 (2010).[5] Yu. M. Bunkov and G. E. Volovik, “Novel superfluids,”(Oxford University Press, 2013) Chap. IV, pp. 253–311.[6] S. Takei and Y. Tserkovnyak, Phys. Rev. Lett. ,227201 (2014).[7] S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak,Phys. Rev. B , 094408 (2014).[8] H. Chen and A. H. MacDonald, in Universal themesof Bose–Einstein condensation , edited by N. Proukakis,D. Snoke, and P. Littlewood (Cambridge UniversityPress, 2017) Chap. 27, pp. 525–548, arXiv:1604.02429.[9] C. Sun, T. Nattermann, and V. L. Pokrovsky, Phys. Rev.Lett. , 257205 (2016).[10] E. B. Sonin, Phys. Rev. B , 144432 (2017).[11] E. Iacocca, T. J. Silva, and M. A. Hoefer, Phys. Rev.Lett. , 017203 (2017).[12] A. Qaiumzadeh, H. Skarsv˚ag, C. Holmqvist, andA. Brataas, Phys. Rev. Lett. , 137201 (2017).[13] L. D. Landau and E. M. Lifshitz, Statistical physics. PartII (Pergamon Press, 1980).[14] Yu. D. Kalafati and V. L. Safonov, Pis’ma Zh. Eksp. Teor. Fiz. , 135 (1989), [JETP Lett.
149 (1989)].[15] E. B. Fel’dman and A. K. Khitrin, Zh. Eksp. Teor. Fiz. , 967 (1990), [Sov. Phys.–JETP , 538 (1990)].[16] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A.Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin,Nature , 430 (2006).[17] A. Lamacraft, Phys. Rev. B , 224512 (2017).[18] J. Armaitis and R. A. Duine, Phys. Rev. A , 053607(2017).[19] E. B. Sonin, Phys. Rev. B , 224517 (2018).[20] E. B. Sonin, Phys. Rev. B , 104423 (2019).[21] M. M. Salomaa and G. E. Volovik, Rev. Mod. Phys. ,533 (1985), erratum: , 573 (1988).[22] U. Leonhardt and G. E. Volovik, Pis’ma Zh. Eksp. Teor.Fiz. , 66 (2000), [JETP Lett. , 46–48 (2000)].[23] Y. Kawaguchi and M. Ueda, Phys. Rep. , 253 (2012).[24] T.-L. Ho, Phys. Rev. Lett. , 742 (1998).[25] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. , 1822(1998).[26] N. D. Mermin and T. L. Ho, Phys. Rev. Lett. , 594(1976).[27] P. W. Anderson and G. Toulouse, Phys. Rev. Lett.38