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Superfluid spin transport in magnetically ordered solids

The paper reviews the theory of the long-distance spin superfluid transport in solid ferro- and antiferromagnets 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.


I. INTRODUCTION
The phenomenon of spin superfluidity [1][2][3][4][5][6][7][8][9][10][11][12] is based on the analogy of special cases of the Landau-Lifshitz-Gilbert (LLG) theory in magnetism and superfluid hydrodynamics. This analogy was clearly formulated long ago by Halperin and Hohenberg 13 in their hydrodynamic theory of spin waves. While in a superfluid mass (charge in superconductors) can be transported by a current proportional to the gradient of the phase of the macroscopic wave function, in a magnetically ordered medium there are spin currents, which are proportional to the gradient of the spin phase. The latter is defined as the angle of rotation around some axis in the spin space. Strictly speaking this analogy is complete only if this axis is a symmetry axis in the spin space. Then according to Noether's theorem the spin component along this axis is conserved. But possible violation of the spin conservation law usually is rather weak because it is related with relativistically small (inversely proportional to the speed of light) processes of spin-orbit interaction. In fact, the LLG theory itself is based on the assumption of weak spin-orbit interaction 14 .
The analogy of the LLG theory with the theory of superfluidity suggests a new useful language for description of phenomena in magnetism, but not a new phenomenon. During the whole period of spin superfluidity investigations and up to now there have been disputes about definition what is spin superfluidity. There is a school of thinking that the existence of any spin current proportional to the spin phase (rotation angle) means spin superfluidity 5,15 .
This definition transforms spin superfluidity into a trivial ubiquitous phenomenon existing in any magnetically ordered medium. A spin current proportional to the spin phase emerges in any domain wall and in any spin wave. Under this broad definition spin superfluidity was already experimentally detected beyond reasonable doubt in old experiments of the middle of the 20th century detecting domain walls and spin waves. We use the term superfluidity in its original meaning known from the times of Kamerlingh Onnes and Kapitza: transport of some physical quantity (mass, charge, or spin) over macroscopic distances without essential dissipation. This requires a constant or slowly varying phase gradient at macroscopic scale with the total phase variation along the macroscopic sample equal to 2π multiplied by a very large number. In examples of domain walls and spin waves this definitely does not take place. 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 the domain 2 wall thickness, or the spin wavelength. Although such currents are also sometimes called supercurrents, we use the term supercurrent only in the case of macroscopic supercurrent persistent at large spatial and temporal scales.
The possibility of supercurrents is conditioned by the special topology of the magnetic order parameter space (vacuum manifold). Namely, this space must have topology of circumference on the plane. In magnetically ordered systems this requires the presence of easy-plane uniaxial anisotropy. It is possible also in non-equilibrium coherent precession states, when spin pumping supports spin precession with fixed spin component along the magnetic field (the axis z). Such non-equilibrium coherent precession states, which are called nowadays magnon BEC, were experimentally investigated in the B phase of superfluid 3 He and in YIG films. 5,16 Spin superfluid transport (in our definition of this phenomenon) is possible as long as the spin phase gradient does not exceeds the critical value determined by the Landau criterion.
The Landau criterion checks stability of supercurrent states with respect to elementary excitations of all collective modes. The Landau criterion determines a threshold for the current state instability, but it tells nothing about how the instability develops. The decay of the supercurrent is possible only via phase slips. In a phase slip event a vortex crosses current streamlines decreasing the phase difference along streamlines. Below the critical value of supercurrent phase slips are suppressed by energetic barriers. The critical value of the supercurrent at which barriers vanish is of the same order as that estimated from the Landau criterion. This leads to a conclusion that the instability predicted by the Landau criterion is a precursor of the avalanche of phase slips not suppressed by any activation barrier.
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 avoiding details of calculations, which can be found in original papers. After the theoretical analysis the 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 and superfluid hydrodynamics let us remind the concept of the mass superfluidity in the theory 3 of superfluidity. In superfluid hydrodynamics there are the Hamilton equations for the pair of the canonically conjugate variables "phase -density": Here are functional derivatives of the Hamiltonian where E 0 (n) is the energy of the superfluid at rest, which depends only on the particle density n. Taking into account the gauge invariance (the energy does not depend on the phase directly, ∂H/∂ϕ = 0, but only on its gradient) the Hamilton equations are reduced to the equations of hydrodynamics for an ideal liquid: In these expressions is the chemical potential, and 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.
4 is the particle current. We consider the zero-temperature limit, when the superfluid velocity coincides with the center-of-mass velocity v = m ∇ϕ.
A collective mode of the ideal liquid is a sound wave. In the sound wave the phase varies in space, i.e., the wave is accompanied by mass currents [ Fig. 1(a)]. An amplitude of the phase variation is small, and currents transport mass on distances of the order of the wavelength. A real superfluid transport on macroscopic distances is possible in current states, which are stationary solutions of the hydrodynamic equations with finite constant currents, i.e., with constant nonzero phase gradients. In the current state the phase rotates through a large number of full 2π-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 a persistent current, which does not decay despite it is not the ground state of the system. The answer 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 complex wave function ψ = ψ 0 e iϕ , where the modulus ψ 0 of the wave function is a positive constant determined by minimization of the energy and the phase ϕ is a degeneracy parameter since the energy does not depend on ϕ. Any from the degenerate ground states in a closed annular channel (torus) maps on some point at the circumference |ψ| = ψ 0 in the complex plane ψ, while a current state with the phase change 2πn around the torus maps onto a path [ Fig. 2(a)] winding around the circumference n times. It is impossible to change the winding number n keeping the path on the circumference |ψ| = ψ 0 all the time. In the language of 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 a phase slip. The phase slip costs energy, which is spent on creation of the vortex and its motion across current streamlines. The state with the vortex in the channel maps on the full circle |ψ| ≤ ψ 0 [ Fig. 2(b)]. Thus, phase slips are impeded by potential barriers, which make the current state metastable.
According to the Landau criterion, the current state is metastable as long as any quasiparticle of the superfluid in the laboratory frame has a positive energy and therefore its creation requires an energy input. The Landau criterion checks the stability only with

A. Ferromagnets
The phenomenological description of magnetically ordered media is given by the LLG theory. For a ferromagnet with magnetization density M the LLG equation is 17 ∂M where γ is the gyromagnetic ratio between the magnetic and mechanical moment. The effective magnetic field is determined by the functional derivative of the total energy: 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 ef f .
We shall consider the case when spin-rotational invariance is partially broken, and there is uniaxial crystal magnetic anisotropy. The phenomenological Hamiltonian is If the anisotropy energy G is positive, it is the "easy plane" anisotropy, which keeps the magnetization 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 around the z axis. Since the absolute value M of magnetization is fixed, the magnetization vector In the new variables the Hamiltonian is Here we neglected gradients of M z . The parameter A is stiffness of the spin system determined by exchange interaction, and the magnetic susceptibility χ = M 2 /G along the z axis is determined by the uniaxial anisotropy energy G keeping the magnetization in the easy plane. The LLG equation reduces to the Hamilton equations for a pair of canonically conjugate continuous variables "angle-angular momentum": 1 γ where functional derivatives on the right-hand sides are taken from the Hamiltonian given by Eq. (13). Using the expressions for functional derivatives one can write the Hamilton equations as where is the spin current. Although our equations contain not the spin density but the magnetization, 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 in the spin wave does not lead to long-distance superfluid spin transport, which this review 8 addresses. Spin superfluid transport on long distances is realized in current states with spin rotating in the plane through a large number of full 2π-rotations as shown in Fig. 1(b). In the current state with a constant gradient of the spin phase K = ∇ϕ, there is a constant magnetization component along the magnetic field (the axis z): Like in superfluids, the stability of current states is connected to the topology of the order parameter space. In isotropic ferromagnets (G = 0) the order parameter space is a spherical Up to now we considered states close to the equilibrium (ground) state. In a ferromagnet in a magnetic field the equilibrium magnetization is parallel to the field. However, by pumping magnons into the sample it is possible to tilt the magnetization with respect to the magnetic field. This creates the state with the coherent spin precession around the magnetic field (the magnon BEC 5,16 ). Although the state is far from the true equilibrium, but it, nevertheless, is a state of minimal energy at fixed magnetization M z . Because of inevitable spin relaxation the state of uniform precession requires permanent pumping of spin and energy. However, if processes violating the spin conservation law are weak, one can ignore them and treat the state as a quasi-equilibrium state. The state of uniform precession maps on a circumference parallel to the xy plane. One can consider also a current state, in which the phase (the rotation angle in the xy plane) varies not only in time but also in space with a constant gradient. In this case the easy plane for the magnetization is not related to the crystal anisotropy but created dynamically. However, in the quasi-equilibrium coherent precession 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 antiferromagnet by the LLG equations for two spin sublattices coupled via exchange interaction: 19 Here the subscript i = 1, 2 indicates to which sublattice the magnetization M i belongs, and is the effective field for the ith sublattice determined by the functional derivative of the Hamiltonian H. For an isotropic antiferromagnet the Hamiltonian is In the uniform ground state the total magnetization is equal to m = χH, while the staggered magnetization is normal to m. Without the magnetic field the two sublattice magnetizations are antiparallel, and the total magnetization m vanishes. The first term in the Hamiltonian (22), which determines the susceptibility χ, originates from the exchange interaction between spins of the two sublattices. This is the susceptibility normal to the staggered magnetization L.
Since in the LLG theory absolute values of sublattice magnetizations M 1 and M 2 are equal to 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 currents in 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 Minimizing the Hamiltonian with respect to the absolute value of m (at it fixed direction) one obtains In the analogy to the ferromagnetic case, one can describe the vectors of sublattice magnetizations 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: Let us consider the axisymmetric solutions of these equations with ϕ = ϕ 1 = π − ϕ 2 and Then there is only one pair of the Hamilton equations for the pair of the conjugate variables (m z , ϕ): Here is the the spin current and A − = A − A 12 . These equations are identical to Eqs. (16) and (17) While in ferromagnets the current state is a spiral spin structure with the spatial precession of the in-plane spontaneous magnetization along current streamlines, in antiferromagnets the 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 external magnetic field is a surface of a sphere. However, the order parameter is not the total magnetization but the unit Néel vector l = L/L. While in the ferromagnet the magnetic field produces an easy axis for the total magnetization, in the antiferromagnet the magnetic field produces the easy plane for the order parameter vector l with the anisotropy energy given by Eq. (27). Thus, the topology necessary for the spin superfluidity in antiferromagnets does not require the crystal easy-plane anisotropy. 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): Then the spin continuity equation (17) becomes where with the longitudinal (along the magnetic field) magnetization given by Eq. (19). It is necessary to solve the Hamilton equations Eqs. (16) and (17) linearized with respect to weak perturbations of the current state. We skip the standard algebra given elsewhere 24 .
Finally one obtains 11,24 the spectrum of plane spin waves ∝ e ik·r−iωt : Herec is the spin-wave velocity in the current state and is the spin velocity in the state without spin currents. The velocity can be called Doppler velocity because its effect on the mode frequency is similar to the effect of the mass velocity on the mode frequency in a Galilean invariant fluid (Doppler effect). However, our system is not Galilean invariant, 11 and the gradient K is present also on the right-hand side of the dispersion relation (37).
We obtained the gapless Goldstone mode with the sound-like linear in k spectrum. The current 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 In the limit of weak magnetic fields when M z M the Landau critical gradient is 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 velocityc sw in the current state vanishes.
In the opposite limit M z → M (M ⊥ → 0) the Landau critical gradient, In the further analysis it is convenient to use other angle variables: The polar angle Θ for the staggered magnetization L and the canting angle θ are shown in In the stationary current state the total magnetization m z = 2M sin θ is given by Eq. (33) and ∇ϕ = K, while Θ = Φ = 0. In a weakly perturbed current state small but nonzero Θ and Φ appear. Also the angles θ and ϕ differ from their values in the stationary current and give the resulting spectra of two spin-wave modes.
The equations for the pair of perturbations (θ , ϕ ) describe the Goldstone mode with the spectrum of the plane spin waves Here the spin-wave velocity in the ground state without spin currents, the spin-wave velocity in the current state, and the Doppler velocity are given by where L ⊥ = L cos θ. The gapless Goldstone mode in an antiferromagnet does not differ from that in a ferromagnet, if one replaces in the expressions for the ferromagnet A by A − /2 and M by 2M .
The equations for the pair of perturbations (Θ, Φ) describe the gapped mode with the spectrum ω + w · k = ω 2 0 + c sw k 2 , where the gap is given by angle Θ around the axis y and the oscillation m y of the total spin component along the same axis. 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 is broken, 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 rotates in 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 both in ferro-and antiferromagnets. However, the spin wave spectra discussed in the present paper were calculated not for the ground state, but for the metastable current states. In the magnetodynamics of antiferromagnets it was usually assumed that the spin polarization is weak and the canting angle is small. Then the magnetodynamics can be reduced to the single equation for the Néel vector equivalent to that in the sigma model (see the recent review by Galkina and Ivanov 25 and references therein). The derivation of spectra presented 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 sublattice magnetizations become equal, and the staggered magnetization L vanishes. This makes the spin superfluidity impossible. However, this magnetic field is on the order of the exchange field, which is usually very strong.
Applying the Landau criterion to the gapless mode at small canting angles θ, one obtains the critical gradient similar to the value Eq. (42) obtained for a ferromagnet. However, in contrast to a ferromagnet where the susceptibility χ is connected with weak anisotropy energy, in an antiferromagnet the susceptibility χ is determined by a much larger exchange energy and is rather small. As a result, in an antiferromagnet the gapless Goldstone mode becomes unstable at the very high value of K. At much lower values of K the gapped mode loses its stability when the gap becomes negative and the mode frequency becomes complex. According to the spectrum (49), the gap in the spectrum vanishes at the critical gradient Here we introduced a new correlation length which is connected to the effective easy-plane anisotropy energy (27). The instability of the gapped mode is a precursor of the instability with respect to phase slips with vortices, which have the core radius of the order of ξ.

LINES
For the estimation of barriers for phase slips one must consider the interaction of vortices with spin currents. The total energy of the vortex is mostly determined by the area outside the core (the London region) where one must take into account the interaction of vortices with spin currents. In the London region the main contribution to the energy is the first term in the Hamiltonian Eq. (13) proportional to ∇ϕ 2 (we consider now a ferromagnet). This term 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 where the first term is the spin phase gradient introduced by the vortex,ẑ is the unit vector along 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 integrating the energy over the whole space occupied by the ferromagnet one obtains a logarithmically divergent integral, which depends on the sample geometry. We consider a 2D problem of the straight vortex at the distance R from the plane border. The gradient K is parallel to the border. Then the the energy of the straight vortex per unit length in the presence of currents is

20
The lower cutoff of the logarithm is the core radius r c . The vortex energy has a maximum at R = 1/2K. The energy at the maximum is a barrier preventing phase slips: The barrier vanishes if K becomes of the order of the inverse vortex core radius. This conclusion is applicable also to antiferromagnets.
The core radius r c is of order of the coherence length determined from the spin wave spec-  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 1 and M 2 rotate around the axis normal to the magnetic field, and there are nonzero Θ and Φ. 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 1 , it was debated about whether the superfluid spin current is a "real" transport current. As a response to this concern, in Ref. 1 a Gedanken (at that time) experiment for demonstration of reality of superfluid 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 spin accumulation is checked at another side (Fig. 8). For the analysis of spin transport in this set up we must modified the continuity equation (17) for the ferromagnet adding two dissipation terms: Here M 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 Spin diffusion does not violates the spin conservation law. The second dissipation term is connected with the longitudinal spin relaxation. It is characterized by the Bloch time T 1 and does violate the spin conservation law.
In the absence of spin superfluidity (J = 0) Eq. (16) for the spin phase is not relevant, and Eq. (58) describes pure spin diffusion [ Fig. 8(a)]. Its solution, with the boundary condition that the spin current J 0 is injected at the interface x = 0, is where is the spin-diffusion length. Thus, the effect of spin injection exponentially decays at the scale of the spin-diffusion length, and the density of spin accumulated at the other side of the layer decreases exponentially with growing distance d.
However, if spin superfluidity is possible, the spin precession equation (16) 16) and (58): Thus, the spin accumulated at large distance d from the spin injector slowly decreases with It was important evidence that persistent spin currents are possible. However, real longdistance transportation of spin by these currents was not demonstrated. Moreover, it is impossible to do in the non-equilibrium magnon BEC, which was realized in the B phase of 3 He superfluid 5 and in yttrium-iron-garnet magnetic films. 16 The non-equilibrium magnon BEC requires pumping of spin in the whole bulk for its existence. In the geometry of the aforementioned spin transport experiment this would mean that spin is permanently pumped not only by a distant injector but also all the way up the place where its accumulation is probed. Thus, the spin detector measures not only spin coming from the distant injector but also spin pumped close to the detector. Therefore, the experiment does not prove the existence of long-distance spin superfluid transport.
The experiment suggested for detection of long-distance superfluid spin transport 1 was recently done by Yuan et al. 27 in antiferromagnetic Cr 2 O 3 . In the experiment of Yuan et al. 27 the spin is created in the Pt injector by heating (the Seebeck effect) on one side of the Cr 2 O 3 film and spin accumulation is probed on another side of the film by the Pt detector via the inverse spin Hall effect (Fig. 10). In agreement with theoretical prediction, they observed spin accumulation inversely proportional to the distance from the interface where spin was injected into Cr 2 O 3 .
In Fig. 8  the spin axis z parallel to the magnetic field. This geometry is shown in Fig. 10. The difference between two geometries is not essential if spin-orbit coupling is ignored. In our theoretical analysis we chose the geometry with different directions of the spin current and the spin in order to stress the possibility of the independent choice of axes in the spin and the configurational spaces.
There were other reports on experimental detection of spin superfluidity in magnetically ordered solids. Bozhko et al. 15 declared detection of spin superfluidity at high temperatures in a decaying magnon condensate in a YIG film. In their experiment the phase gradient emerged because of spin precession difference produced by a temperature gradient. However, the estimate made in Ref. 10 showed that the total phase difference across the magnon cloud in the experiment did not exceed about 1/3 of the full 2π rotation. Thus, Bozhko et al. 15 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-century old experiments on spin waves at all temperatures and does not need new experimental confirmations.
Observation of the long-distance superfluid spin transport was also reported by Stepanov et al. 28 in a graphene quantum Hall antiferromagnet. However, the discussion of this report requires an extensive theoretical analysis of the ν = 0 quantum Hall state of graphene, which goes beyond the scope of the present review. A reader can find this analysis in Ref. 29. It is worthwhile to note that in reality it is not easy to reach the critical gradients discussed in the present paper experimentally. The decay of superfluid spin currents is possible also at subcritical spin phase gradients since the barriers for phase slips can be overcome by thermal activation or macroscopic quantum tunneling. This makes the very definition of the real critical gradient rather ambiguous and dependent on duration of observation of persistent currents. Calculation of real critical gradients requires a detailed dynamical analysis of processes of thermal activation or macroscopic quantum tunneling through phase slip barriers, which is beyond the scope of the present paper. One can find examples of such analysis for conventional superfluids with mass supercurrents in Ref. 30.
Although evidence of the existence of metastable superfluid spin currents in the B phase of superfluid 3 He were reported long ago 26 the first experiment demonstrating the long-distance transport of spin by these currents in the solid antiferromagnet was done only recently. 27 This is not the end but the beginning of the experimental verification of the long-distance superfluid spin transport in magnetically ordered solids. In the experiment of Yuan et al. 27 spin injection required heating of the Pt injector, and the spin current to the detector is inevitably accompanied by a heat flow. Lebrun et al. 31 argued that probably Yuan et al. 27 detected a signal not from spin coming from the injector but from spin generated by the Seebeck effect at the interface between the heated antiferromagnet and the Pt detector.
Such effect has already been observed for antiferromagnet Cr 2 O 3 . 32 If true, Yuan et al. 27 observed not long-distance spin transport but long-distance heat transport. However, it is not supported by the fact that Yuan et al. observed a threshold for superfluid spin transport at low intensity of injection, when according to the theory (see Sec. IV) the absence of the strict spin conservation law becomes important. With all that said, the heat-transport interpretation cannot be ruled out and deserves further investigation. According to this interpretation, one can see the signal observed by Yuan et al. 27 at the detector even if the Pt injector is replaced by a heater, which produces the same heat but no spin. An experimental check of this prediction would confirm or reject the heat-transport interpretation.
The present paper focused on spin superfluidity in magnetically ordered solids. Recently investigations of spin superfluidity were extended to spin-1 BEC, where spin and mass superfluidity coexist and interplay. [33][34][35][36] This interplay leads to a number of new nontrivial features of the phenomenon of superfluidity. The both types of superfluidity are restricted by 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 conventional mass superfluidity. Another consequence of the coexistence of spin and mass superfluidity is phase slips with bicirculation vortices characterized by two topological charges (winding