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Telecommunications and Radio Engineering

Symmetry of Equilibrium State and Hydrodynamics Equation of a Superfluid Fermi-liquid

Mikhail Yurievich Kovalevsky, Alexander Alimovich Rozhkov

Published Jan 1, 2001• 0 citations
Publisher

Abstract

Three kinds of quantum liquids are considered. Order parameter operators are written for two quantum liquids (where spontaneous breaking of symmetry takes place), and its commutation properties are considered. Symmetry properties of the equilibrium state are noted and expressions of order parameters of operator means in the equilibrium state are obtained. Expressions of basic thermodynamical relations in terms of the local equilibrium of thermodynamical potential are obtained. Nondissipative hydrodynamics equations are derived for all three liquids. The relationship connecting the densities of fluxes of additive motion integrals is obtained in the local equilibrium state.

DOI
10.1615/telecomradeng.v56.i2.50

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Three kinds of quantum liquids are considered. Order parameter operators are written for two quantum liquids (where spontaneous breaking of symmetry takes place), and its commutation properties are considered. Symmetry properties of the equilibrium state are noted and expressions of order parameters of operator means in the equilibrium state are obtained. Expressions of basic thermodynamical relations in terms of the local equilibrium of thermodynamical potential are obtained. Nondissipative hydrodynamics equations are derived for all three liquids. The relationship connecting the densities of fluxes of additive motion integrals is obtained in the local equilibrium state.
An approach based on the method of quasiaverages and the reduced description method is developed as a microscopic tool for the derivation of hydrodynamic equations for super uid liquids with triplet pairing of quasiparticles.Additional thermodynamic parameters connected with different spontaneously broken symmetries of the system are determined with the use of the order parameter operator as functionals of the nonequilibrium statistical operator.The ux densities of additive motion integrals are obtained in terms of the locally equilibrium thermodynamic potential.Selfconsistent equations for the energy gap and quasiparticle energy are found which are analyzed for the unitary and nonunitary states of super uid Fermi{liquid.
This review presents a microscopic construction of the thermodynamics and hydrodynamics of superfluid bosons and fermions with singlet pairing, based on the concept of quasiaverages and the hypothesis of reduced description. Here we do not assume that the Hamiltonian possesses any dynamical symmetry. This has permitted obtaining results pertaining to both Galilean-invariant and relativistic systems. Account is taken of dissipative processes. The kinetic coefficients are presented in terms of the correlation functions of the flux operators. The approach is extended to solutions of quantum liquids. The influence of an external ac field on superfluid systems is studied, and the low-frequency asymptotic behavior of the Green's function is found in the hydrodynamic approximation. The symmetry properties of the equilibrium state are formulated, and the thermodynamics is constructed for superfluid Fermi systems with triplet pairing (the superfluid phases 3 He-B and 3 He-A). For the latter the flux densities of the additive integrals of motion are found in a state of equilibrium and the equations of "ideal" hydrodynamics are derived.
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.
A classification of the equilibrium states of a quantum liquid with d pairing is given on the basis of the concept of quasi-averages. The order-parameter operator is represented in terms of Fermi operators. It is shown that the set of such equilibrium states can be classified in terms of the quantum number corresponding to the projection of the orbital angular momentum of a Cooper pair on the anisotropy direction. The explicit form of the three admissible generators of the unbroken symmetry is found and the corresponding equilibrium values of the order parameter are obtained. A generalization to possible nonuniform equilibrium structures is carried out, and the corresponding form of the order parameter of such structures is found.
This paper presents the results of investigation of the superfluid states in a two-component Fermi liquid within the framework of the Fermi liquid approach. Particular attention is paid to the superfluid states in nuclear matter which are characterized by the superposition of singlet and triplet types of pairing in spin and isospin spaces. The fundamental principles of the Fermi liquid approach are formulated and further used in the study of superfluidity in nuclear matter with the superposition of singlet and triplet types of pairing. The system of self-consistency equations is derived and its solutions are presented. The conditions of existence of the considered states are determined. These conditions impose certain constraints on the interaction potential and particle density in the system. It is shown that the states with a complete set of nonzero order parameters are realized only in a narrow density range, the width and position of which on the density scale depend on the actual parametrization of the Skyrme potential employed. We considered 18 different parameterizations and indicated for which of them the studied types of superfluid states may appear. The problem of stability of the states with the superposition of singlet and triplet pairing is studied. It is shown that the lowest value of the thermodynamic potential corresponds to purely triplet states. Then, in the order of increasing, follow the thermodynamic potentials of purely singlet states and mixed singlet-triplet states. Separately, we consider the case of unitary states, for which the solutions of the self-consistency equations are also analyzed. It is shown that the density range in which such states can be realized is different from that corresponding to the nonunitary states. The problem of existence of the unitary superfluid states with the superposition of singlet and triplet superfluidity is considered in the case of asymmetrical nuclear matter. It is shown that the appearance of asymmetry in nuclear matter breaks the unitarity of superfluid states in nuclear matter with regard to isospin.