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Low Temperature Physics

Classification of the equilibrium states of a superfluid liquid with d pairing

A. P. Ivashin, Mikhail Yurievich Kovalevsky, Наталья Николаевна Чеканова

Published Sep 1, 2004• 4 citations
Publisher

Abstract

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.

DOI
10.1063/1.1802951

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The Journal of V N Karazin Kharkiv National University Series Physics•Jan 1, 2020•0 citations
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.
The classification of equilibrium states of superfluid with scalar, vector and tensor order parameters is carried out on the basis of the quasiaveragues concept. The generalization of a requirement of the residual symmetry for nonuniform equilibrium states is given. The admissible requirements of a spatial symmetry in the terms of integrals of motion are found. The connection of these requirements with helicoidal structure of vectors of a spin and spatial anisotropy is established. At some restrictions is shown, that the equilibrium structure of an order parameter can be represented as product of a nonuniform part, depending on spatial coordinates, and homogeneous part of an order parameter.
The problem of classification of degenerate equilibrium states in spin s = 1 systems with the phase and magnetic symmetries simultaneously broken is considered. An assumption about the residual symmetry of degenerate equilibrium states and the transformation properties of the spinor order-parameter operator under transformations generated by additive integrals of motion yields equations that classify its equilibrium values. The anisotropic structure of the equilibrium values of the order parameter is presented in terms of the parameters of the residual symmetry generator. Based on the model with a separated condensate, the structure of the densities of the magnetic additive integrals of motion in the equilibrium state is revealed. A comparison is made with the results of existing studies on superfluid spin s = 1 systems.
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.
General expressions are obtained in explicit form for the anomalous and normal distribution functions of quasiparticles in nonunitary phases of a superfluid paramagnetic Fermi liquid consisting of electrically neutral fermions with triplet pairing (the pairs have spin s=1 and an arbitrary odd value of the orbital angular momentum l) in a uniform static magnetic field. A generalized Fermi-liquid approach is used which takes into account the exchange Fermi-liquid interaction but does not specify the form of the energy functional of the superfluid Fermi liquid. The results are valid at any temperature in the interval 0⩽T⩽Tc, where Tc is the temperature of the phase transition from the normal to the superfluid state. When an explicit form of the energy functional is specified, the general formulas for the distribution function can be used to describe the various nonunitary phases of a superfluid Fermi liquid of the He3 type in a high magnetic field. In particular, for He3-A1, He3-A2, and the nonunitary two-dimensional phase of He3 in a high magnetic field at 0⩽T⩽Tc, an energy functional quadratic in the distribution function is used to find a system of coupled equations for the order parameter and the effective magnetic field and to obtain an expression is obtained for the nonlinear magnetic susceptibility.
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.
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.