Spin currents and spin superfluidity
Abstract
The present review analyzes and compares various types of dissipationless spin transport: (1) Superfluid transport, when the spincurrent state is a metastable state (a local but not the absolute minimum in the parameter space). (2) Ballistic spin transport, when spin is transported without losses simply because sources of dissipation are very weak. (3) Equilibrium spin currents, i.e., genuine persistent currents. (4) Spin currents in the spin Hall effect. Since superfluidity is frequently connected with Bose condensation, recent debates about magnon Bose condensation are also reviewed.
For any type of spin currents simplest models were chosen for discussion in order to concentrate on concepts rather than details of numerous models. The various hurdles on the way of using the concept of spin current (absence of the spinconservation law, ambiguity of spin current definition, etc.) were analyzed. The final conclusion is that the spincurrent concept can be developed in a fully consistent manner, and is a useful language for description of various phenomena in spin dynamics.
pin current; spin superfluidity; easyplane (anti)ferromagnet; Landau criterion; spinorbit coupling; spin Hall effect
10.1080/0001873YYxxxxxxxx \issn14606976 \issnp00018732 \jvol00 \jnum00 \jyear2008 \jmonthJune
REVIEW
1 Introduction
The problem of spin transport occupies minds of condensed matter physicists for decades. A simple example of spin transport is spin diffusion, which is a process accompanied with dissipation. Conceptually more complicated is “dissipationless” spin transport, which was also discussed long time but was in the past and remains now to be a matter of controversy. The main source of controversy is that spin is not a conserved quantity. This leads to many complications and ambiguities in defining such concepts as spin flow, current, or transport. Sometimes these complications are purely semantic. However, this does not make them simpler for discussion. “Semantic traps” very often are a serious obstacle for understanding physics and for deriving proper conclusions concerning observation and practical application of the phenomenon. The best strategy in these cases is to focus not on names but on concepts hidden under these names. Only after this one may “take sides” in semantic disputes not forgetting, however, that choosing names is to considerable extent a matter of convention and taste.
During long history of studying the problem of “dissipationless” spin transport one can notice three periods, when studies in this field were especially intensive. The first period started from theoretical suggestions on possible “superfluidity of electronhole pairs” [1], which were later extended on possible spin superfluidity [2, 3]. At the same period the concept of spin superfluidity was exploited [4, 5, 6] for interpretation of experiments demonstrating unusually fast spin relaxation in HeA [7]. The second period was marked by intensive theoretical and experimental work on spin superfluidity in HeB starting from interpretation of experiments on the socalled Homogeneously Precessing Domain (HPD) [8] in terms of spin supercurrents [9]. Finally in these days (the third period) we observe a growing interest to dissipationless spin currents in connection with work on spintronics [10]. The final goal of spintronics is to create devices based on spin manipulation, and transport of spin with minimal losses is crucial for this goal. Now one can find reviews summarizing the investigations done during the first [11] and the second [12, 13] periods of works on dissipationless spin transport. On the other hand, the work on spin transport in spintronics is a developing story, and probably it is still premature to write summarizing reviews. Nevertheless, some reviews mostly addressing the spin Hall effect have already appeared [14, 15]. It looks also useful to have a glance on the current status of the field from a broader viewpoint and to find bridges between current investigations and those done in the “last millennium”. The present review aims at this goal. The intention is to discuss mostly concepts without unnecessary deepening in details, and simplest models were chosen for this.
The term “superfluidity” is used in the literature to cover a broad range of phenomena, which have been observed in superfluid He and He, BoseEinstein condensates of cold atoms, and, in the broader sense of this term, in superconductors. In the present review superfluidity means only a possibility to transport a physical quantity (mass, charge, spin, …) without dissipation. Exactly this phenomenon gave a rise to the terms “superconductivity” and “superfluidity”, discovered nearly 100 years and 70 years ago respectively. It is worthwhile to stress that one should not understand the adjective “dissipationless” too literally. In reality we deal with an essential suppression of dissipation due to the presence of energetic barriers of the topological origin. How essential suppression could be, is a matter of a special analysis. In the present review we restrict discussion with the question whether activation barriers, which suppresses dissipation, can appear.
But superfluidity is not the only reason for suppression of dissipation in the transport process, and it is important to understand the difference between various types of dissipationless transport. In the present review we shall discuss four types of them:

Superfluid transport: The spincurrent state is a metastable state (a local but not the absolute minimum in the parameter space).

Ballistic transport. Here spin is transported without losses simply because sources of dissipation are very weak.

Equilibrium currents. Sometimes symmetry allows currents even at the equilibrium. A superconductor in a magnetic field is a simple example. Equilibrium spin currents are also possible, though there is a dispute on whether they have something to do with spin transport. Equilibrium spin currents are genuine persistent currents, since no dissipation is possible at the equilibrium by definition.

Spin currents in the spin Hall effect. These currents are also called dissipationless since they are normal to the driving force (electric field) and therefore do not produce any work. However, in the spin Hall effect there is dissipation connected with a longitudinal charge current through a conducting medium. On the other hand, it was recently revealed that the spin Hall effect is possible also in insulators where a charge current is absent. Then spin currents are not accompanied by any dissipation becoming similar to equilibrium spin currents.
The second type (ballistic) looks mostly trivial: dissipation is absent because sources of dissipation are absent. Still it is worth of short discussion since sometimes they confuse ballistic transport with superfluid transport (an example of it is discussed in section 7.2). The superfluid transport does not require the absence of dissipation mechanisms. One may expect that in an ideally clean metal at zero temperature resistance would be absent. But this would not be superconductivity. Superconductivity is the absence of resistance in a dirty metal at .
The first two types of spin currents are discussed in Part I of the review, which is devoted to magnetically ordered systems. The third and the fourth types are discussed mostly in Part II, which addresses timereversalinvariant systems without magnetic order, though in magnetically ordered media equilibrium spin currents are also possible (section 9). Since from the very beginning of the theory of superfluidity the relation between superfluidity and Bose condensation was permanently in the focus of attention, discussing spin superfluidity one cannot avoid to consider the concept of magnon Bose condensation, which is vividly debated nowadays. Section 10 addresses this issue.
Part I: Spin currents in magnetically ordered systems
2 Mass supercurrents
Since the idea of spin superfluidity originated from the analogy with the more common concept of mass superfluidity let us shortly summarize the latter. The essence of the transition to the superfluid or superconducting state is that below the critical temperature the complex order parameter , which has a meaning of the wave function of the bosons or the fermion Cooper pairs, emerges as an additional macroscopical variable of the liquid. For the sake of simplicity, we restrict ourselves to the case of a neutral superfluid at zero temperature putting aside the twofluid theory for finite temperatures. Then the theory of superfluidity tells that the order parameter determines the particle density and the velocity of the liquid is given by the standard quantummechanical expression
(1) 
Thus the velocity is a gradient of a scalar, and any flow is potential. Since the phase and the particle number are a pair of canonically conjugate variables, one can write down the Hamilton equations for the pair of the canonically conjugated variables “phase – density”:
(2) 
Here is the total liquid energy, whereas is the energy density, and and are functional derivatives of the total energy:
(3) 
(4) 
In these expressions is the chemical potential,
(5) 
is the particle current, and the dependence of the energy on the density gradient was ignored. Eventually the Hamilton equations are reduced to the equations of hydrodynamics for an ideal liquid:
(6) 
(7) 
A crucial property of the system is the gauge invariance: the energy does not depend on the phase directly () but only on its gradient. According to Noether’s theorem this must lead to the conservation law for a conjugate variable, the total number of particles. The conservation law manifests itself in the continuity equation (7), which contains the particle supercurrent. The prefix “super” stresses that this current is not connected with dissipation. It is derived from the Hamiltonian or the Lagrangian but not from the dissipation function. In contrast to the diffusion current proportional to the density gradient, the supercurrent is proportional to the phase gradient. Therefore it appears only in a coherent state with broken gauge invariance. The equations of superfluid hydrodynamics can be derived from the Gross–Pitaevskii equation for a weakly nonideal Bosegas. However, they are much more general than this model. They can be formulated from the most general principles of symmetry and conservation laws. Indeed, deriving the twofluid theory of superfluidity Landau did not use the concept of Bosecondensation.
An elementary collective mode of the ideal liquid is a sound wave. In a sound wave the phase varies in space, i.e., the wave is accompanied by mass supercurrents (figure 1a). An amplitude of the time and space dependent phase variation is small, and currents transport mass on distances of the order of the wavelength. A really superfluid transport on macroscopic distances is related with stationary solutions of the hydrodynamic equations corresponding to finite constant currents with constant nonzero phase gradients (current states). In the current state the phase rotates through a large number of full 2rotations along streamlines of the current (figure 1b).
The crucial point of the superfluidity concept is why the supercurrent is a persistent current, which does not decay despite it is not the ground state of the system and has a larger energy. The first explanation of the supercurrent stability was given on the basis of the well known Landau criterion [16]. According to this criterion, the current state is stable as far as any quasiparticle of the Boseliquid in the laboratory frame has a positive energy and therefore its creation requires an energy input. Let us suppose that elementary quasiparticles of the Boseliquid at rest have an energy spectrum where is the quasiparticle momentum. If the Boseliquid moves with the velocity the quasiparticle energy in the laboratory frame is . The energy cannot be negative (which would mean instability) if
(8) 
In superfluid He the Landau critical velocity is determined by the roton part of the spectrum. But in this review we focus on the longwavelength collective excitations, which are phonons with the spectrum . Then according to equation (8) the supercurrent cannot be stable if the velocity exceeds the sound velocity .
As far as one wants to check the Landau criterion for longwavelength collective modes like sound waves, it is not necessary to solve a dynamical problem looking for the spectrum of phonons. It is enough to estimate the energy of possible static fluctuations in the stationary current state with particle density and velocity . Let us write down the energy of the current state taking into account possible local fluctuations of the particle density, , and of the velocity, , up to the terms of the second order:
Here and are the chemical potential and the energy density of the liquid at rest. One may neglect terms of the first order with respect to the density fluctuation and the velocity since we look for an energy extremum at fixed averaged density and velocity and the firstorder term must vanish after integration. Using the thermodynamic relation and omitting the subscript 0 in and , one obtains
(9) 
The quadratic form under the integral is positive definite, i.e., the current state corresponds to the energy minimum, if the condition is satisfied. This condition is identical to the Landau criterion equation (8) for the phonon spectrum .
The theory of superfluidity tells that the Landau criterion is a necessary but not sufficient condition for current metastability. The Landau criterion checks only small deviations from the current state. Meanwhile the current state can be destroyed via large perturbations of the current state. In superfluids these large perturbations are vortices. In the current state the phase rotates along the current direction. The current can relax if one can remove one 2turn of the phase. This requires that a singular vortex line crossed or “cut” the channel crosssection. The process is called “phase slip”.
If the vortex axis (vortex line) coincides with the axis, the phase gradient around the vortex line is given by
(10) 
where is the position vector in the plane. The phase changes by 2 around the vortex line (figure 2a). Creation of the vortex requires some energy. The vortex energy per unit length (line tension) is determined by the kinetic (gradient) energy:
(11) 
where the upper cutoff is determined by geometry. For example, for the vortex shown in figure 2a it is the distance of the vortex line from a sample border. The lower cutoff is the vortexcore radius. It determines the distance at which the phase gradient is so high that the hydrodynamic expression for the energy becomes invalid. A good estimation for is , where is the circulation quantum of the velocity. Inside the core the modulus of the order parameter goes down to zero eliminating the singularity in the kinetic energy at the vortex axis. For the weakly nonideal Bosegas this estimation yields the coherence length.
Now suppose that a vortex appears in the current state with the constant gradient : The phase gradients induced by the vortex are superimposed on the constant phase gradient related to the current: . The total gradient energy includes that of the current, the vortex energy given by equation (11), and the energy from the cross terms of the two gradient fields. Only the last two contributions are connected with the vortex, and their sum determines the energy of the vortex in the current state:
(12) 
where is the length of the vortex line and is the area of the cut, at which the phase jumps by . For the 2D case shown in figure 2a (a straight vortex in a slab of thickness normal to the picture plane) . One can see that vortex motion across the channel (growth of ) is impeded by the barrier, which is determined by variation of the energy with respect to . The peak of the barrier corresponds to . The height of the barrier is
(13) 
Thus the barrier disappears at gradients , which are of the same order as the critical gradient determined from the Landau criterion. In the 3D geometry the phase slip is realized with expansion of vortex rings. For the ring of radius the vortexlength and the area of the cut are and respectively, and the barrier disappears at the same critical gradient as in the 2D case.
The barriers stabilizing metastable current states are connected with topology of the order parameter space. In a superfluid the order parameter is a complex wave function . At the equilibrium , where the modulus is a constant determined by minimization of the energy and the phase is a degeneration parameter since the energy does not depend on . Any current state in a closed annular channel (torus) with the phase change around the channel maps onto a circumference in the complex plane (figure 3a) winding the circumference times. It is evident that it is impossible to change keeping the path on the circumference all the time. Thus is a topological charge. One can change it (removing, e.g., one winding around the circumference) only by leaving the circumference (the equilibrium order parameter space in the case). This should cost energy, which is spent on creation of a vortex. Figure 3b shows mapping of the vortex state onto the circle .
Without such topological barriers superfluidity is ruled out. However, barriers do not automatically provide the lifetime of currents long enough. In practice, dissipation via phase slips is possible even in the presence of barriers due to thermal fluctuations or quantum tunneling. Here and later on we address only “ideal” critical currents (the upper bound for critical currents) at which barriers disappear leaving “practical” critical currents beyond the scope of the present review.
3 Phenomenology of magnetically ordered systems and spin currents
The main interaction responsible for magnetic order is exchange interaction, which is invariant with respect to rotations of the whole spin system. Then according to Noether’s theorem the total spin must be conserved. For ferromagnets where the order parameter is the spontaneous magnetization , this means that the exchange energy can depend on the absolute value of but not on its direction. Other contributions to the free energy (anisotropy energy or dipoledipole interaction) are related with spinorbit interaction, which does not conserve the total spin. But these interactions are relativistically small, i.e., governed by the small relativistic parameter , where is a typical electron velocity and is the speed of light. The spinorbit interaction does depend on direction, but because of its weakness cannot affect the absolute value in slow dynamics. This is a crucial point in the phenomenological theory of magnetism of Landau and Lifshitz [17], which determines the form of the equation of motion for ferromagnet magnetization known as the LandauLifshitz equation [18]:
(14) 
where is the gyromagnetic ratio between the magnetic and mechanical moment (). The effective magnetic field is determined by the functional derivative of the total free energy with density :
(15) 
According to the LandauLifshitz equation, the absolute value of the magnetization cannot vary. The evolution of is a precession around the effective magnetic field .
At first let us discuss exchange approximation, in which relativistic effects are ignored and the conservation of total spin is not violated. In this approximation the free energy density is
(16) 
The first exchangeenergy term , being the largest term, is crucial for determination of the equilibrium value of . But after determination of it can be ignored as an inessential large constant. Indeed, its contribution to the effective field in LandauLifshitz equation (14) does not produce any effect: the contribution is parallel to and vanishes in the vector product. In the absence of external fields, which break invariance with respect to rotations in the spin space, the LandauLifshitz equation reduces to the continuity equations for components of the spin density :
(17) 
where
(18) 
is the th component of the spin current transporting the th component of spin. Thus in an isotropic ferromagnet all three components of spin are conserved.
The LandauLifshitz equation has planewave solutions describing spatially nonuniform precession of the magnetization around the groundstate magnetization : . Here the magnetization deviation is small and normal to . Linearizing with respect to , one obtains spin waves with the spectrum
(19) 
In an isotropic ferromagnet spin waves at are accompanied by spin currents, but superfluid spin transport is impossible as will be clear from section 4.
Next we shall consider the case when spinrotational invariance is partially broken, and there is uniaxial crystal magnetic anisotropy given by the third term in the phenomenological free energy:
(20) 
If the anisotropy energy is positive, it is the “easy plane” anisotropy, which keeps the spontaneous magnetization in the plane (the continuous limit of the model). In this model the component of spin is conserved, because invariance with respect to rotations in the easy plane remains unbroken. Since the absolute value of magnetization is fixed, the vector of the magnetization is fully determined by the angle showing the direction of in the easy plane (, ) and by the component of the magnetization . We use the notation instead of in order to emphasize that is a small dynamic correction to the magnetization, which is absent at the equilibrium. In the new variables the free energy is
(21) 
The constant is stiffness of the spin system determined by exchange interaction, and the magnetic susceptibility along the axis is determined by the uniaxial anisotropy energy keeping the magnetization in the plane. The LandauLifshitz equation reduces to the Hamilton equations for a pair of canonically conjugate continuous variables “angle–angular momentum” (analogous to the canonically conjugate pair “coordinate–momentum”):
(22) 
(23) 
where functional derivatives on the righthand sides are taken from the free energy given by equation (21). Using the expressions for functional derivatives one can write the Hamilton equations as
(24) 
(25) 
where
(26) 
is the spin current.
There is an evident analogy of equations (24) and (25) with the hydrodynamic equations (6) and (7) for an ideal liquid, equation (25) being the continuity equation for spin. This analogy was exploited by Halperin and Hohenberg [19] in their hydrodynamic theory of spin waves. In contrast to the isotropic ferromagnet with the quadratic spinwave spectrum, the spin wave in the easyplane ferromagnet has a soundlike spectrum as in a superfluid: , where the spinwave velocity is . Halperin and Hohenberg introduced the concept of spin current, which appears in a propagating spin wave like a mass supercurrent appears in a sound wave (figure 1a). This current transports the component of spin on distances of the order of the wavelength. But as well as the mass supercurrent in a sound wave, this small oscillating spin current does not lead to superfluid spin transport, which this review addresses. Spin superfluid transport on long distances is realized in current states with magnetization rotating in the plane through a large number of full 2rotations as shown in figure 1b.
Let us consider now the case of antiferromagnetic order. The simplest model of an antiferromagnet is two sublattices with magnetizations and . In the absence of weak ferromagnetism and external magnetic fields two magnetizations completely compensate each other without producing any total magnetization . However, a small magnetization does appear due to external magnetic fields or dynamical effects. The amplitudes of and and their mutual orientation are mostly determined by strong exchange interaction, but the latter does not fix the direction of the staggered magnetization , which is the order parameter of a twosublattice antiferromagnet.The equations of motion for two vectors and can be derived from the two LandauLifshitz equations for and taking into account the exchange interaction between two sublattices. But it would be useful to present a more general version of the macroscopic phenomenological theory, which is able to describe an antiferromagnetic structure of any complexity [21, 20]. The theory of spin dynamics in superfluid phases of He developed by Leggett and Takagi [22] also belongs to this class. Following the same principle “exchange is the strongest interaction” as in the LandauLifshitz theory, macroscopic theories of this type deal with phenomena at scales essentially exceeding microscopic scales (the coherence length in the case of He), at which the exchange energy establishes the tensor structure of the order parameter. This permits to assume that the entire dynamic evolution of the order parameter reduces to rotations in the 3D spin state, which cannot change the exchange energy. Then the dynamics of the system is described by three independent pairs of canonically conjugated variables “angle–moment” – ():
(27) 
Here are the angles of spin rotations around three Cartesian axes (). Apart from spatial dependence of the variables, these equations are similar to the equations of motion of a 3D rigid top. In our case the top is an antiferromagnetic spin order parameter rigidly fixed by exchange interaction. As in the case of a twosublattice antiferromagnet, magnetization results from deformation of the equilibrium spin structure. The approach is valid as far as this deformation is weak, i.e., is smaller than the characteristic moments of the antiferromagnetic structure (staggered magnetization in the case of a twosublattice antiferromagnet). Since rotation around the vector has no effect on the state of the system the latter has only two degrees of freedom corresponding to two pairs “angle–moment”. Then the equations become the equations of motion of a rotator. In contrast to the spontaneous magnetization in the LandauLifshitz equation (14), the small absolute value of the magnetization is not kept constant.
Because the group of 3D rotations is noncommutative, the state of the system depends on the order, in which rotations around different axes are performed. In practice they frequently use the Euler angles (they are introduced in section 7). For the most content of Part I (except for section 7), one can choose one degree of freedom connected with the conjugate pair –, and the problem of noncommutativity is absent (further we shall omit the subscript of the angle ). If the energy of the ground state does not depend (or depends weakly as discussed in section 5) on the angle , the equations of motions for and are the same Hamilton equations (22) and(23), which were formulated for an easy–plane ferromagnet. In the case of a twosublattice antiferromagnet the angle is the angle of the staggered magnetization in the easy plane.
The discussion of this section has not made any reference to a concrete microscopic model of magnetism. Indeed, the approach is general enough and is valid for models of magnetism based on the concepts of either localized or itinerant electrons. In particular, ferromagnetism of localized electrons is described by the Heisenberg model with the Hamiltonian:
(28) 
where , are spins at the sites , and the summation over includes only the nearest neighbors to the site . In the continuum limit, when the spin rotates very slowly at scales of the intersite distance , the Hamiltonian (28) reduces to the free energy (16) in the LandauLifshitz theory with the magnetization and the stiffness constant .
The debates on reliability of the general phenomenological approach to magnetism are as old as the approach itself. Nearly sixty years ago Herring and Kittel [23] argued with their opponents that their phenomenological theory of spin waves “is not contingent upon the choice of any particular approximate model for the ferromagnetic electrons”. Interestingly these discussions are still continuing in connection with the concept of the spin current, which originates from the general phenomenological approach. Originally they connected spin supercurrents with counterflows of particles with opposite spins, for example, of He atoms in the Aphase of He [4, 5]. Bunkov [13] insisted that only a counterflow of particles with opposite spins would lead to superfluid spin transport, thus ruling out spin superfluidity in materials with magnetic order resulting from exchange interaction between localized spins (see p. 93 in his review). However, the spin current does not require itinerant electrons for its existence [2]. The presumption that spin transport in insulators is impossible is still alive nowadays. According to Shi et al. [24], it is a critical flaw of spincurrent definition if it predicts spin currents in insulators.
4 Stability of spincurrent states
For the sake of simplicity further we focus on current states in an easyplane ferromagnet, though the analysis can be easily generalized to other magnetically ordered systems discussed in the previous section. In the current state the spontaneous magnetization rotates in the easy plane through a large number of full rotations when the position vector is varying along the direction of spin current (figure 1b). The spincurrent state is metastable if it corresponds to a local minimum of the free energy, i.e., any transition to nearby states would require an increase of energy. This condition is an analog of the Landau criterion for mass supercurrents discussed in section 2. In order to check current metastability, one should estimate the energy of possible small static fluctuations around the stationary current state. For this estimation, one should take into account that the stiffness constant is proportional to the squared inplane component of the spontaneous magnetization , and in the presence of large angle gradients must be replaced with . So the free energy is
(29) 
One can see that if exceeds the current state is unstable with respect to the exit of from the easy plane. This is the Landau criterion for the stability of the spin current.
Like in superfluids, stability of current states is connected with topology of the order parameter space. For ferromagnets the order parameter is the magnetization vector . For isotropic ferromagnets the space of degenerated equilibrium states is a sphere , whereas for an easyplane ferromagnet this space reduced to an equatorial circumference on this sphere (figure 4a). Thus the order parameter space for an easyplane ferromagnet is topologically equivalent to that space for superfluids (the circumference on the complex plane shown in figure 3). Spincurrent states are stable because they belong to the topological classes different from the class of the uniform ground state and cannot be reduced to the latter by continuous deformation of the path. In contrast, for an isotropic ferromagnet the path around the equatorial circumference can be continuously transformed to a point on the sphere as shown in figure 4b. In this process the energy monotonously decreases, and topological barriers are absent. Topology of an easyaxis (anti)ferromagnet also does not allow stable spincurrent states.
The connection of superfluiditylike phenomena with topology of the order parameter space is universal and not restricted with the examples of mass and spin superfluidity considered here. The same arguments support possibility of exciton superfluidity, which was discussed even earlier than spin superfluidity (see the introductory section 1). Though the whole problem of superfluid exciton transport is far from its resolution, in some special case experimental evidences of this transport has already been reported. Kellogg et al. [25] observed vanishing resistance in double quantum Hall layers, which was interpreted as a consequence of Bose condensation of interlayer excitons (or pseudospin ferromagnetism).
As well as in the theory of mass superfluidity, the Landau criterion is a necessary but not sufficient condition for current metastability. One should also to check stability with respect to large perturbations, which are magnetic vortices. The magnetic vortices were well known in magnetism. Bloch lines in ferromagnetic domain walls are an example of them [26]. In the spincurrent state the magnetization traces a spiral at moving along the current direction. The spin current can relax if one can remove one turn of the spiral. This requires that a singular line (magnetic vortex) crossed or “cut” the channel crosssection [2, 3] as shown in figure 2b.
The structure of the magnetic vortex outside the vortex core is the same as of the mass superfluid vortex given by equation (10). Correspondingly, the magnetic vortex energy is determined by the expression similar to equation (11):
(30) 
where the upper cutoff depends on geometry. However, the radius and the structure of the magnetic vortex core are determined differently from the mass vortex. In a magnetic system the order parameter must not vanish at the vortex axis since there is a more effective way to eliminate the singularity in the gradient energy: an excursion of the spontaneous magnetization out of the easy plane . This would require an increase of the uniaxial anisotropy energy, which keeps in the plane, but normally this energy is much less than the exchange energy, which keeps the orderparameter amplitude constant. Finally the core size is determined as a distance at which the uniaxial anisotropy energy density is in balance with the gradient energy . This yields . Figure 4c shows mapping of the spin vortex state onto the order parameter space. In contrast to superfluid vortices mapping onto a plane circle, the spin vortex state can map onto one of two halves of the sphere . Thus a magnetic (spin) vortex has an additional topological charge having two values [27].
The energy of the spincurrent state with a vortex and the energy of the barrier, which blocks the phase slip, i.e., the decay of the current, are determined similarly to the case of mass superfluidity [see equations (12) and (13)]:
(31) 
(32) 
where is the length of the vortex line and is the area of the cut, at which the angle jumps by . Thus the barrier disappears at gradients , which are of the same order as the critical gradient determined from the Landau criterion. This is a typical situation in the superfluidity theory. But sometimes the situation is more complicated as we shall see in section 7.2.
5 Spin currents without spin conservation law
Though processes violating the conservation law for the total spin are relativistically weak, their effect is of principal importance and in no case can be ignored. The attention to superfluid transport in the absence of conservation law was attracted first in connection with discussions of superfluidity of electronhole pairs. The number of electronhole pairs can vary due to interband transitions. As was shown by Guseinov and Keldysh [28], interband transitions lift the degeneracy with respect to the phase of the “pair Bosecondensate” and make the existence of spatially homogeneous stationary current states impossible. On the basis of it Guseinov and Keldysh concluded that there is no analogy with superfluidity. This phenomenon was called “fixation of phase”. However some time later it was demonstrated [29] that phase fixation does not rule out existence inhomogeneous stationary current states, which admit some analogy with superfluid current states^{1}^{1}1Similar conclusions have been done with respect to possibility of supercurrents in systems with spatially separated electrons and holes [30, 31].. This analysis was extended on spin currents [2, 3].
In the spin system the role of the phase is played by the angle of the magnetization in the easy plane, and the degeneracy with respect to the angle is lifted by magnetic anisotropy in the plane. Adding the fold inplane anisotropy energy to the total free energy (21) the latter can be written as
(33) 
Then the spin continuity equation (25) becomes
(34) 
where
(35) 
Excluding from equations (24) and (34) one obtains the sine Gordon equation for the angle :
(36) 
where is the spinwave velocity. According to this equation, the inplane anisotropy leads to a gap in the spinwave spectrum:
(37) 
There are onedimensional solutions of the sine Gordon equation with nonzero average , which correspond to a periodic lattice of solitons (domain walls) of the width with the period moving with the velocity . The function inverse to is
(38) 
where the constant is determined by the equation
(39) 
The free energy of the soliton lattice is given by
(40) 
It is possible to develop the hydrodynamic theory of the soliton lattice in the terms of local density and velocity of solitons [32], which is able to describe deformations of the lattice slow in space and time. Here we focus on stationary current states when (). At small average twisting of the spontaneous magnetization the structure constitutes domains that correspond to the equivalent easiest directions in the easy plane. In this limit () the free energy density is the product of the energy of an isolated domain wall and the density of domain walls :
(41) 
Spin currents (gradients) inside domains are negligible but there are essential spin currents inside domain walls where . This hardly reminds genuine superfluid transport on macroscopical scales: spin is transported over distances on the order of the domainwall width . With increasing the density of domain walls grows, and at they coalesce while for a displacement along the direction of the gradient , the end point of the vector describes, a line close to a helix. The nonuniform states with and are shown in figure 5. Thus the processes violating spin conservation law are not important for large deformations (gradients) of the spin structure. This means that the analogy of these deformed states with current states in superfluids makes sense.
Studying stability of nonuniform current states it is possible to ignore the inplane anisotropy only for large spin currents when . Let us consider the opposite limit of when the spin structure reduces to a chain of domain walls. The relaxation of the spin current, which is proportional to the wall density, requires that some domain walls vanish from the channel. This process is illustrated in figure 2b for the fourfold inplane symmetry (). When a magnetic vortex appears, domain walls finish not at the wall but at the vortex line, around which the angle changes by 2. The angle jump occurs at the cut restricted by the vortex line. The domain walls disappear via motion of the vortex line across the channel cross section. In the course of this process, the change of the energy consists of the vortexline energy, which is proportional to the line length , and of a decrease of the surface energy of the walls themselves proportional to the cut area . The latter contribution is determined by the product of the free energy density (41) and the volume . Taking these two contributions into account, the energy during the process of annihilation of walls is
(42) 
Comparing it with the energy given by equation (31) one sees that the gradient is replaced by the maximum gradient inside the domain wall. Correspondingly for the 2D case shown in figure 2b the expression (32) for the barrier energy must be replaced by
(43) 
where the two lengths and are determined by the uniaxial and the inplane anisotropy energies and . Thus large barriers stabilizing spincurrent states are possible only if the condition is satisfied. This conclusion [2, 3, 11] was recently confirmed by the analysis of König et al. [33].
An important difference with conventional mass superfluidity is that in conventional superfluidity the barrier, which suppresses supercurrent relaxation, grows unrestrictedly when the gradient decreases. In contrast, in spin superfluidity the barrier growth stops when the gradient reaches the values of the order (inverse width of the domain wall). Since the current relaxation time exponentially depends on the barrier (whether the barrier is overcome due to thermal fluctuations or via quantum tunneling) the life time of the current state in conventional superfluidity diverges when the velocity (phase gradient) decreases. In contrast, the life time of the spin current can be exponentially large but always finite. This provides an ammunition for rigorists, who are not ready to accept the concept “spin superfluidity” (or superfluidity of any nonconserved quantity) in principle. In principle, one could agree with them. But in practice, whatever we call it, “nonideal superfluidity” or “quasisuperfluidity”, some consequences should outcome from the fact of the existence of topological barriers suppressing relaxation of spincurrent states. A key point is whether these consequences are observable. This is the topic of the next section.
6 Is superfluid spin transport “real”?
From early days of discussions on spin supercurrents and up to now there are arguments on whether the spin supercurrent can result in “real” transport of spin. Partially this is a semantic problem: One must carefully define what “real” transport really means. Let us suppose that one has a usual superfluid mass persistent current in a ring geometry. Nobody doubts that real mass transport occurs in this case, but how can one notice it in the experiment? In any part of the ring channel there is no accumulation (increase or decrease) of the mass. Of course, one can detect gyroscopic effects related with persistent currents, but it is an indirect evidence. What may be a direct evidence? One could suggest a Gedanken Experiment in which the ring channel is suddenly closed in some place. In the wake of it one can observe that the mass increases on one side from the closure and decreases on the other side. This would be a real transport if one required a demonstration of mass accumulation as a proof of it. Accepting this definition of transport reality one can notice real transport only in a nonequilibrium process, when the transported quantity decreases in some place and increases in another . Naturally one can discard these semantic exercises as irrelevant for practice, but only as far as they refer to mass currents. In the case of spin currents in the past and nowadays spin accumulation sometimes is considered as a necessary proof of real spin transport. Therefore, in old publications on spin superfluidity [2, 3, 11] much attention was paid to possible experimental demonstration of spin transport from one place to another.
Before starting discussion of possible spintransport demonstration it is useful to consider a mechanical analogue of superfluid mass or spin supercurrent [11]. Let us twist a long elastic rod so that a twisting angle at one end of the rod with respect to an opposite end reaches values many times . Bending the rod into a ring and connecting the ends rigidly, one obtains a ring with a circulating persistent angularmomentum flux (figure 6). The intensity of the flux is proportional to the gradient of twisting angle, which plays the role of the phase gradient in the mass supercurrent or the spinrotationangle gradient in the spin supercurrent. The analogy with spin current is especially close because spin is also a part of the angular momentum. The deformed state of the ring is not the ground state of the ring, but it cannot relax to the ground state via any elastic process, because it is topologically stable. The only way to relieve the strain inside the rod is plastic displacements. This means that dislocations must move across rod crosssections. The role of dislocations in the twisted rod is the same as the role of vortices in the mass or spin current states: In both of the cases some critical deformation (gradient) is required to switch the process on. There are various ways to detect deformations or strains in an elastically deformed body. Similarly, it is certainly possible, at least in principle, to notice deformation (angle gradient) of the spin structure in the spincurrent state. It would be a legitimate evidence of the spin current, not less legitimate than a magnetic field measured around the ring as an evidence of the persistent charge current in the ring.
Of course, it is not obligatory to discuss the twisted rod in terms of angularmomentum flux. One can describe it only in terms of deformations, stresses, and elastic stiffness. So we must have in mind that there are two languages, or descriptions of the same physical phenomenon. A choice of one of them is a matter of taste and tradition. For example, in order to describe the transfer of momentum they use the momentumflux tensor (“flux”, or “current” language) in hydrodynamics, while in the elasticity theory they prefer to call the same tensor as stress tensor. In principle one can avoid the term “superfluidity” and speak only about the “phase stiffness” even in the case of mass supercurrents.
Let us return to possible demonstration of “real” spin transport. Suppose that spin is injected into a sample at the sample boundary (figure 7). The injection can be realized practically either with an injection of a spinpolarized current (for the sake of simplicity we put aside the problem what happens with charge in this case), or with pumping the spin with a circularly polarized microwave irradiation. If the medium at cannot support superfluid spin transport, the only way of spin propagation is spin diffusion described by the equations
(44) 
where is the spindiffusion coefficient and is the time characterizing the Bloch longitudinal relaxation, which violates the spinconservation law. In the stationary case , and both the spin current and the nonequilibrium magnetization exponentially decay inside the sample: , where is the spindiffusion length. So no spin can reach the other boundary of the sample provided .
Now let us suppose that the medium at is magnetically ordered and can support superfluid spin transport. If the injection is so weak that the angle gradient is much less than , the perturbation of the medium by the injection can penetrate at the length not longer than the domainwall width . So the injection pushes a piece of a domain wall into the sample. The spin current exponentially decays like in the medium without superfluidity. If the injection is so strong that the angle gradient exceeds its maximum value in a center of an isolated domain wall, continuity of the spin current on the boundary requires appearance of a soliton lattice with a period of the order or smaller than . This means that the spin current can reach the other boundary . In the thermodynamic limit one may expect a stationary soliton lattice with . However, at the boundary the spin current must be injected into the medium without spin superfluidity. This is impossible without a finite , and the finite means that the soliton lattice is moving. In the limit the soliton lattice is rather dense and one may neglect periodical spincurrent modulation caused by inplane anisotropy. Then spin transport is described by the equations
(45) 
(46) 
with the boundary conditions for the supercurrent at and at . The current in the first condition is the spininjection current, while the second boundary condition takes into account that the medium at is not spinsuperfluid and spin injection there is possible only if some nonequilibrium magnetization is present. The coefficient can be found by solving the spindiffusion equations in the medium at [3]. It also depends on properties of the contact at . While the inplane anisotropy violating the spin conservation (phase fixation) was neglected, one cannot neglect irreversible dissipative processes, which also violate the spinconservation law. The simplest example of such a process is the longitudinal spin relaxation characterized by time .
The stationary solution of equations (45) and (46) is
(47) 
Though the solution is stationary in the sense that , but . We consider a nonequilibrium process (otherwise spin accumulation is impossible), which is accompanied by the precession of in the easy plane. But the process is stationary only if the precession angular velocity is constant in space. The condition const, which results from it, is similar to the condition of constant chemical potential in superfluids or electrochemical potential in superconductors in stationary processes. If this condition were not satisfied, there would be steady growth of the angle twist as is evident from equation (45). As already mentioned above, the nonzero means that the soliton lattice is moving. In our case the soliton velocity is rather slow since it is inversely proportional to : .
One can see that irreversible loss of spin is a more serious obstacle for superfluid spin transport than coherent phase fixation, to which most of attention was attracted in the literature. Because of spin relaxation, the spin current inevitably decreases while moving away from the injection point, in contrast to constant superfluid mass currents. However, in a spinsuperfluid medium this decrease is linear and therefore less destructive than exponential decay of currents in nonsuperfluid media. So one have a good chance to notice spin accumulation in the medium at rather distant from the place of original spin injection. This justifies using the term “superfluid”.
In the presented analysis we assumed that at the boundary the entire spininjection current is immediately transformed into a supercurrent. Actually spin injection can also generate the diffusion current close to the boundary. However, at some distance from the boundary the diffusion current inevitably transforms into a supercurrent [3]. If this distance (healing length) is much shorter than the size of the sample our boundary condition at is fully justified. Similar effects take place at contacts “normal metal  superconductor”: The current from the normal metal to the superconductor is completely transformed into the supercurrent at some finite distance from the contact.
Spin injection is not the only method of generation of spin currents. One can generate spin currents by a rotating inplane magnetic field, which is applied to one end of the sample and is strong enough to orient the magnetization parallel to it. Because of the stiffness of the spin system, the spin rotation at one end is transmitted to the other end of the sample, which is not subject to the direct effect of the rotating magnetic field. Transmission of the torque through the sample is spin current. Since the rotating field is acting on the phase (angle) of the order parameter, it can be called coherent method, in contrast to the incoherent method of spin injection. The coherent method of spincurrent generation has no analog in superfluids and superconductors, since in the latter cases there is no field linked to the phase of the order parameter. Referring to the set up shown in figure 7 with spin injection replaced by rotating magnetic field, in the coherent method the magnetization is fixed by the frequency of the rotating field. In this case there is no threshold for spincurrent generation, and the spin current appears whatever small the frequency could be. But if the frequency (and proportional to it) is low, the spin transmission is realized via generation of a chain of well separated solitons (domain walls), which propagate to the other end of the sample. Thus, a “moving soliton lattice” is another synonym for spin superfluid transport. Longdistance propagation of solitons through a slab of the phase of superfluid He generated by a pulse of a radiofrequency magnetic field has already experimentally realized by Bartolac et al. [34]. This experiment was discussed in terms of spin transport in reference [32].
7 Spinprecession superfluidity in superfluid HeB
7.1 Stationary uniform precession in HeB
Now we focus on the experimental and theoretical investigations of superfluid spin transport in the phase of superfluid He. The spin superfluidity in the phase has several important features, which distinguish it from the spin superfluidity discussed previously in this review. First, in contrast to what was considered earlier, observed spincurrent states in the phase are dynamical nonlinear states very far from the equilibrium, which require for their support permanent pumping of energy. Thus dissipation is always present, and speaking about “superfluidity”, i.e., “dissipationless” spin transport, we have in mind the absence of additional dissipation connected with the spin current itself. Second, while the previous discussion dealt with the transport of a single spin component (component), in the phase spin vector performs a more complicated 3D rotation and the spin current refers to the transport of some combination of spin components. This combination may be called “precession moment” because it is a canonical conjugate of the precession rotation angle (precession phase) rather the rotation angle of genuine spin in the spin space. So one should discern two types of spin superfluid transport: transport of spin precession and transport of spin [35]. In the experiment [8] they used slightly nonuniform magnetic fields, and precession took place only inside the homogeneously precessing domain (HPD). But for discussion of superfluid spin transport it is not so important, and in the following we consider only processes inside the HPD ignoring gradients of the magnetic field.
The spin dynamics of superfluid phases of He is described by the theory of Leggett and Takagi [22], which is an example of the general phenomenological theory of magnetically ordered systems in terms of conjugate canonical variables “angle–moment”, which was shortly discussed in section 3. As well as in studies of rotating solid tops, sometimes it is more convenient to describe spin rotations via the Euler angles , , and (figure 8). In He superfluid spin dynamics these angles were used by Fomin [9, 12], who, however, replaced the angle by the angle . The angle is the precession tipping angle, and is the precession phase determining the direction of the line of nodes . The angle characterizes the resultant rotation of the order parameter in the laboratory frame, and in the limit (no precession) becomes the angle of rotation around the axis. The magnetic moments canonically conjugate to the angles , , and are , , and respectively, where is the component of the magnetization in the laboratory coordinate frame, is the projection of on the axis of the rotating coordinate frame (see figure 8), and is the projection of on the line of nodes , which is perpendicular to the axes and . The free energy density consists of three terms, , where
(48) 
includes the magnetization and the Zeeman energies, and the gradient energy and the orderparameter dependent energy (dipole energy in the case of He) will be determined later on. Here is the magnetic susceptibility.
For the phenomena observed experimentally only one degree of freedom is essential, which is connected with precession, i.e., with the conjugate pair “precession phase –precession moment ”. In contrast to the mode connected to the longitudinal magnetic resonance (oscillations of the longitudinal spin component), which was discussed in previous sections, the precession mode is connected with the transverse magnetic resonance (nuclear magnetic resonance in the case of He), in which does not oscillate essentially.
Neglecting nutation, the directions of the axis and of the moment coincide. Then is constant, , , , and the free energy density becomes
(49) 
where is a strong constant magnetic field parallel to the axis. The Hamilton equations for the precession mode are:
(50) 
where is the Larmor frequency and is the total free energy. Since this section deals with nuclear spins, here is the nuclear gyromagnetic ratio. In the experiment the magnetization amplitude is determined by the magnetic field: .
Equation (50) describes free precession without dissipation. One of the most important mechanisms of dissipation is the Leggett–Takagi mechanism [22] related with the process of equilibration of the magnetization of the normal component with the precessing magnetization of the superfluid component (for details see the reviews by Bunkov [13] and Fomin [12] and references therein). This mechanism becomes ineffective at low temperatures, and this leads to the Suhl instability of the uniform precession, which is discussed below. So one cannot observe uniform precession in HeB at very low temperatures. The dissipation leads to a precession decay, and in order to support the state of uniform precession in the experiment the energy dissipation must be compensated by the energy pumped by the rotating transverse magnetic field. Assuming that the balance of the pumped energy and the dissipated energy eventually leads to stationary precession, we may further ignore the both. The stationary precession state corresponds to the extremum of the Gibbs thermodynamic potential, which is obtained from the free energy with the Legendre transformation: , where the precession frequency plays the role of the “chemical potential” conjugate to the precession moment density .
Up to now the theory was rather general and valid for precession in any magnetically ordered system. Referring to HeB particularly, the dipole energy in HeB is [13, 12]
(51) 
where is the longitudinal NMR frequency and . At the stationary precession the angle does not vary in time and can be found by minimization of the Gibbs potential. In the state of uniform precession without spatial gradients only the dipole energy depends on , and the equation for is
(52) 
Solution of this equation yields
(53) 
Thus at () and at () one must choose two different branches of the solution of equation (52). The critical angle is called the Leggett magic angle.
It has already been known about 50 years from studies of nonlinear ferromagnetic [36] and antiferromagnetic [37] resonance that the state of uniform spin precession with finite precession angle can be unstable with respect to excitation of spin waves (Suhl parametric instability). Though Suhl instability is a phenomenon of the nonlinear classical wave theory [38] it is easier to qualitatively explain it in terms of spinwave quanta (magnons). The processes leading to instability are transformations of quanta of uniform precession into two spinwave quanta with wave vectors :
(54) 
The precession is unstable if at least one of these processes is allowed by the laws of energy and momentum conservation. The threemagnon process () corresponds to the first order Suhl instability. The process is possible if a quantum of uniform precession of frequency can dissociate into two quanta of the lower spectral branch with frequency . Another possibility to destabilize uniform precession is a fourmagnon process of transformation of two quanta of uniform precession into two quanta of the same spectral branch with finite wave vectors (the secondorder Suhl instability, ). The process becomes possible if a nonlinear correction to the frequency of uniform precession has an opposite sign with respect to the frequency dispersion . In the theory of nonlinear waves the latter condition is called Lighthill’s condition, which is necessary for modulation instability [39]. The secondorder Suhl instability is an example of it. In all known examples of uniform precessions in magnetically ordered systems there are conditions for at least one type of Suhl instability. In superfluid He Suhl instability is possible in the A [40, 41] and B [42, 43] phases. But as any parametric instability, the Suhl instability can be suppressed by dissipation, which leads to a critical precession angle below which the state of uniform precession remains stable. In the B phase stable uniform precession is possible at temperatures . At lower temperatures dissipation is weak and cannot block the Suhl instability. This explains a sudden transition to the regime of “catastrophic relaxation” observed by Bunkov et al. [44].^{2}^{2}2In contrast to Surovtsev and Fomin [42, 43], who explained catastrophic relaxation with the bulk Suhl instability, Bunkov et al. [45] suggested another mechanism of Suhl instability, which exists near the surface (see arguments over the two mechanisms by Fomin [46] and Bunkov et al. [47])
7.2 Stability of spinprecession supercurrents (Landau criterion)
Let us consider the state with uniform spinprecession current proportional to the gradient . The total free energy should now include the gradient energy of HeB [48]:
(55) 
where
(56) 
, and and are longitudinal and transverse spin wave velocities. The expression for assumes that all gradients are normal to the axis parallel to the dc magnetic field.
An important feature of the dipole energy in HeB is its independence of the precession angle . In accordance with Noether’s theorem this means that the precession moment is strictly conserved. Thus the equations describing stationary precession with frequency are
(57) 
(58) 
where
(59) 
is the spinprecession current.
Apart from , other gradients are absent: . Then equations (53) and (55)–(57) yield the following equation for :
The proper way to check stability of the current state given by equation (LABEL:curSt) is to use the Landau criterion [35]. Since we check stability of the relative minimum at fixed averaged gradient , we must do a new Legendre transformation choosing a new Gibbs thermodynamic potential
(61) 
which has a minimum at the specified values of the precession and the gradient . Now we must find the energy increase due to fluctuations. It is easy to check that fluctuations of always increase , so it suffices to retain in the fluctuation energy only terms quadratic in small deviations and from the stationary values and (we omit hereafter the subscript 0):
(62) 
The fluctuation energy is positive definite, i.e., the current state is stable, so long as does not exceed the critical value
(63) 
This expression yields the critical gradient on the order of the inverse dipole length , but it is valid only at . At () the dipole energy vanishes, , and the superfluid precession transport is impossible.
Another definition of the critical gradient was suggested by Fomin [49]: He believed that the spincurrent state can be stable as far as the gradient does not exceed the value , which is the maximum gradient at which equation (LABEL:curSt) for has a solution. Fomin’s theory allows stable supercurrents for when the dipole energy and the Landau critical gradient vanish. Arguing in favor of his critical gradient, Fomin [50] stated that the Landau criterion is not necessary for the superfluid spin transport since emission of spin waves, which comes into play after exceeding the Landau critical gradient, is not essential in the experimental conditions (see also the similar conclusion after equation (2.39) in the review by Bunkov [13]). This argument is conceptually inconsistent. If the experimentalists observed “dissipationless” spin transport simply because dissipation were weak, they would deal with ballistic rather than superfluid transport. An example of ballistic spin transport will be considered in section 8. As was stressed in section 1, the essence of the phenomenon of superfluidity is not the absence of sources of dissipation, but ineffectiveness of these sources due to energetic and topological reasons. The Landau criterion is an absolutely necessary condition for superfluidity. Fortunately for the superfluidity scenario in the HeB, Fomin’s estimation of the role of dissipation by spinwave emission triggered by violation of the Landau criterion is not conclusive. He found that this dissipation is weak compared to dissipation by spin diffusion. But this is an argument in favor of importance rather than unimportance of the Landau criterion. Indeed, spindiffusion, whatever high the diffusion coefficient could be, is ineffective in the subcritical regime, in which the gradient of the “chemical potential” is absent. On the other hand, in the supercritical regime the “chemical potential” is not constant anymore and this triggers the strong spindiffusion mechanism of dissipation.
7.3 Experimental evidence of the superfluid spinprecession transport
As was discussed above, the appearance of spin current itself is not yet a manifestation of spin superfluid transport. Supercurrents appear in spin waves or domain walls where they transport spin on distances of the order of wavelength or width of domain walls, but hardly it would be reasonable to call it superfluid transport. Similarly spin currents in the domain wall separating HPD from the bulk without precession cannot be a manifestation of superfluid transport. A convincing evidence of spin superfluidity would be spin transport on long distances. This evidence was presented by BorovikRomanov et al. [51] studying spin current through a long channel connecting two cells filled by HPD. The schematic set up of their experiment is shown in figure 9. There is a dc magnetic field parallel to the vertical axis . The HPDs in the two cells were supported with independently rotating rf magnetic fields and with different precession phases as a result of it. A small difference in the frequencies of the two rf fields leads to a linear growth of difference of the precession phases in the cells. This creates a phase gradient in the channel accompanied by a spinprecession supercurrent. The rf coils can monitor precession phases in different parts of the set up. Due to a linear growth of in time eventually it reaches the critical value at which a 2 phase slip occurs. It is possible to register this event via its effect on NMR absorption [13]. Thus the critical gradient can be measured as a function of the precession frequency.
Despite aforementioned conceptual flaw of Fomin’s theory, BorovikRomanov et al. [51, 52] found an agreement of this theory with the experiment. Let us compare now the experiment with the theory based on the Landau criterion. Equation (63) for the Landau critical argument contains the value inside the channel, which is different from in the cells where there is no precession phase gradients (see figure 9). This is an analogy with the Bernoulli law in hydrodynamics (liquid density is less in areas with higher currents). The value of in the channel grows with according to equation (LABEL:curSt) at fixed precession frequency . The latter is controlled in the experiment and in stationary states does not vary in space, exactly like the chemical potential in stationary states of superfluids. If
(64) 
where is the critical gradient from equation (63) at , reaches the value 1/4 earlier than reaches the Landau critical gradient. Since no stable supercurrent is possible at the critical argument is determined as a solution of equation (LABEL:curSt) at :
(65) 
The experiment was done at small , and its results must be compared with equation (65). For the ratio [50] the latter gives the value of by the numerical factor smaller than Fomin’s result, which was in about 1.5 times larger than the critical gradient in the experiment. Thus the theory based on the Landau criterion even better agrees with the experiment [51, 52], and an approximate agreement of Fomin’s result with the experiment cannot be used as an argument in its favor. At larger values of , when the condition (64) is not satisfied, the critical argument is determined by (63) and not proportional to . So the difference with Fomin’s theory becomes more essential (for more details see reference [53]).
Further important development in experimental studies of spinprecession superfluidity was observation of a spincurrent analog of the Josephson effect [54]. The weak link was formed by making a constriction of the channel (orifice) connecting the two cells.
7.4 Spinprecession vortex and its nucleation
As was already discussed in section 4, at gradients less than the Landau critical gradient, the barrier, which impedes the current decay, is related to vortex motion across the flow streamlines (phase slips). Similarly, vortices called spinprecession vortices appear in the spinprecession flow, and the vortex core radius was estimated to be on the order of the dipole length [35]. The barrier for vortex growth in the phaseslip process vanishes at phase gradients of the order of the inverse core radius. So the threshold for vortex instability agrees with the critical gradient from the Landau criterion [equation (63)]. This is usual in the conventional superfluidity theory [11].
Later Fomin [50] showed that the vortex core must be determined by another scale , where and are the precession and the Larmor frequencies. This was supported by Misirpashaev and Volovik [55] on the basis of the topological analysis. According to equation (LABEL:curSt) in the ground state without spin currents . So if is not too close to 1/4 and are of the same order of magnitude. But if , i.e. the precession angle approaches to the critical value rad (or ) the core radius becomes , i.e., by the large factor differs from the earlier estimation [35]. So the latter is valid only far from the critical angle, where . Since no barrier impedes vortex expansion across a channel if the gradient is on the order of , the large core at leads to the strange (from the point of view of the conventional superfluidity theory) conclusion: The instability with respect to vortex expansion occurs at the phase gradients essentially less than the Landau critical gradient , obtained for any . Recently a resolution of this paradox was suggested [56]: At precession angles close to 104 at phase gradients less than the Landau critical gradient but larger than the inverse core radius, no barrier impedes phase slips at the stage of vortex motion across streamlines, but there is a barrier, which blocks phase slips on the very early stage of nucleation of the vortex core. So for these gradients stability of current states is determined not by vortices but by vortexcore nuclei.
It should be stressed that, in contrast to the previous subsection, where the growth of was accompanied by the growth of at fixed , the present analysis is performed at fixed in the channel excepting an area a vortex core or its nucleus. It never exactly equal to though could be whatever small. Thus grows with . Vortex nucleation starts from a ”protonucleus”, which is a slight localized depression of the superfluid density [determined by in our case]. The nucleus, which is related with a peak of a barrier, corresponds to an extremum (saddle point) of the Gibbs potential given by equation (61). Therefore, the nucleus structure should be found from solution of the EulerLagrange equations for this Gibbs potential. The first step is to vary the Gibbs potential with respect to . Let us restrict ourselves with a 1D problem, when the distribution in the nucleus depends only on one coordinate . Then the distribution of is given by