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Soviet Journal of Low Temperature Physics

Hydrodynamics of superfluid liquid crystals

A. É. Meǐerovich

Published Jul 1, 1983• 0 citations
Publisher

Abstract

A system of hydrodynamic equations has been obtained for superfluid Fermi systems with singlet pairing and nonzero momentum which are formed, e.g., in the case of small polarization of the spin system. As a result, the energy gap is a periodic function of the coordinates and a spin-density wave is formed. For such systems the complete set of dynamic variables includes both purely hydrodynamic (superfluid and normal) velocities and the displacement vector corresponding to the violations of the translational invariance. All of the tensors of normal and superfluid densities that appear in the system as well as the elastic moduli have been calculated. A dispersion relation has been obtained for thermal vibrations (second sound) and elastic vibrations, and this relation has turned out to be substantially anisotropic. Observation of the predicted effects is most probable in 3He–He II solutions in magnetic fields of the order of several kilooersteds at low temperatures.

DOI
10.1063/10.0030928

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A system of hydrodynamic equations has been obtained for superfluid Fermi systems with singlet pairing and nonzero momentum which are formed, e.g., in the case of small polarization of the spin system. As a result, the energy gap is a periodic function of the coordinates and a spin-density wave is formed. For such systems the complete set of dynamic variables includes both purely hydrodynamic (superfluid and normal) velocities and the displacement vector corresponding to the violations of the translational invariance. All of the tensors of normal and superfluid densities that appear in the system as well as the elastic moduli have been calculated. A dispersion relation has been obtained for thermal vibrations (second sound) and elastic vibrations, and this relation has turned out to be substantially anisotropic. Observation of the predicted effects is most probable in 3He–He II solutions in magnetic fields of the order of several kilooersteds at low temperatures.
The order parameters are constructed for a Fermi liquid with conformational degrees of freedom. Based on them, additional thermodynamic parameters were introduced: the spin unit vector dα (determining the anisotropy in the spin subspace), the unit spatial vectors mi and ni (determining the anisotropy in space), and also three scalar parameters determining the shape of the Cooper pair u, v, q ( first two items are half-axes of ellipsoid of Cooper pair and last item is mutual orientation in space of these half axes). The symmetry properties of the order parameter operator are considered. The equations of ideal hydrodynamics of a Fermi liquid are derived taking into account the influence of conformational degrees of freedom. By conformational degrees of freedom should be understood the parameters associated with the shape and size of the Cooper pair. Expressions are obtained for the flows of thermodynamic quantities of such a Fermi liquid in terms of the density of the energy functional. The energy functional depends both on the additive integrals of motion (classical fluid parameters) and on conformational parameters. The dispersion equation of such a liquid is obtained for a model representation of the energy functional (the work was performed as part of the Fermi-liquid approach). The dispersion equation includes spin modes, first, second, and third sounds. The dispersion equation for the spatial subsystem, including the first, second, and third sounds characteristic of superfluid systems, is analyzed. Particular solutions of the dispersion equation are simulated using the Maple software package (several 3D figures are given for the angular dependence of the speeds of 1 and 2 sounds in a spherical coordinate system). All of the above allows us to conclude that such a Fermi liquid can be considered as a superfluid liquid crystal of a nematic type. The presence of conformational parameters distinguishes the considered phase from the F phase of a superfluid Fermi liquid.
Abstract In this paper, we develop an appropriate set of hydrodynamic equations for a U(N) invariant superfluid that couple the dynamics of superflow and magnetization. In the special case when both the superfluid and normal velocities are zero, the hydrodynamic equations reduce to a generalized version of Landau–Lifshitz equation for ferromagnetism with U(N) symmetry. When both velocities are non-zero, there appears couplings between the superflow and magnetization dynamics, and the superfluid velocity no longer satisfies the irrotational condition. On the other hand, the magnitude of magnetization is no longer a constant of motion as was the case for the standard Landau–Lifshitz theory. In comparison with the simple superfluid, the first and second sounds are modified by a non-zero magnetization through various thermodynamic functions. For U(2) invariant superfluid, we get both (zero-) sound wave and a spin wave at zero temperature. It is found that the dispersion of spin wave is always quadratic, which is consistent with microscopic analysis. In the appendix, we show that the hydrodynamic equation for a U(N) invariant superfluid can be obtained from the general hydrodynamic equation with arbitrary internal symmetries.
Abstract We determine the hydrodynamic modes of the superfluid analog of a smectic-A liquid crystal phase, i.e., a state in which both gauge invariance and translational invariance along a single direction are spontaneously broken. Such a superfluid smectic provides an idealized description of the incommensurate supersolid state realized in Bose–Einstein condensates with strong dipolar interactions as well as of the stripe phase in Bose gases with spin–orbit coupling. We show that the presence of a finite normal fluid density in the ground state of these systems gives rise to a well-defined second-sound type mode even at zero temperature. It replaces the diffusive permeation mode of a normal smectic phase and is directly connected with the classic description of supersolids by Andreev and Lifshitz in terms of a propagating defect mode. An analytic expression is derived for the two sound velocities that appear in the longitudinal excitation spectrum. It only depends on the low-energy parameters associated with the two independent broken symmetries, which are the effective layer compression modulus and the superfluid fraction.
The Fermi-liquid approach generalized to superfluid systems is used to derive general formulas for the normal and abnormal distribution functions (DF) for quasiparticles of a superfluid Fermi liquid (SFL) consisting of electrically neutral fermions with spin-triplet pairing (the spin of a pair s=1, the orbital angular momentum l of the pair being an arbitrary odd number) in a static uniform magnetic field, taking into account the Landau spin-exchange Fermi-liquid interaction at temperatures 0<T<Tc (Tc is the temperature of phase transition from the normal to the superfluid state). In deriving the DF, the explicit form of the energy functional (EF) for SFL is not used. A system of coupled equations is obtained for the order parameter, energy of quasiparticles, and effective magnetic field for the SFL under investigation in the case of an EF quadratic in DF. The obtained DF and the system of equations are applicable for describing both unitary and nonunitary phases of superfluid He3 in a magnetic field, e.g., for determining the magnetization and magnetic susceptibility of these phases.
The Bose-Einstein condensation mechanism is shown to be capable of accounting for the existence of separate hydrodynamic velocity fields for the normal and the superfluid, provided suitable assumptions are made with regard to the single-particle energy spectrum. We consider the effect on the microcanonical gas distribution of imposing a nonzero value of the total momentum P. For an ordinary gas the effect is trivial, the whole distribution being merely shifted in momentum space. However, in the case of a degenerate Bose gas whose single-particle energy spectrum has a sharp minimum (gap or cusp), only the excited part of the gas (normal fluid) participates in the imposed motion, the condensate (superfluid) remaining "frozen" in momentum space. This rigidity of the condensate in momentum space plays the same role as the rigidity of the superelectrons on imposition of a magnetic field in the London theory of superconductivity. P acts as an additional thermodynamic variable, states with P \ensuremath{\ne} 0 being macroscopically metastable and corresponding to the existence of a relative velocity between normal and superfluid. The basic hydrodynamic assumption of the two-fluid model is thus reduced to an assumption concerning the form of an effective single-particle energy spectrum, and the parallelism between the theories of superconductivity and superfluidity is clearly exhibited. The present theory permits, in particular, the introduction of the mathematical form of Landau's phonon and "roton" spectra within the framework of the Bose-Einstein condensation picture. The statistical-thermodynamic formulas are derived and are shown to lead to characteristic two-fluid equations derived previously from a variational principle.
A Fermi-liquid approach to the theory of superfluidity is developed, based on a generalization of the combinatoric expression for the entropy to the case in which the state of the Fermi liquid is described not only by a normal but also by an anomalous distribution function. The energy of the superfluid Fermi liquid is given in the form of a functional of these distribution functions. In the weak-coupling approximation the Fermi-liquid approach is equivalent to the BCS theory. The interaction between the Fermi liquid and an electromagnetic field is gauge invariant. In addition to the singlet-pairing theory the triplet-pairing theory is also developed. A unified variational principle that gives rise to the theory of superfluidity (superconductivity) is used to obtain the London equations and the Ginzburg–Landau equation. Applications of this theory are made to the derivation of an equation for ideal two-component hydrodynamics.