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Low Temperature Physics

Two-gap superfluidity in the theory of a Fermi liquid

Alexander I. Akhiezer, A. A. Isaev, S. V. Peletminsky, A. A. Yatsenko

Published Jan 1, 1998• 1 citation
Publisher

Abstract

The possibility of a phase transition from the one-gap superfluid state with the singlet or triplet pairing of fermions to the two-gap superfluid state corresponding to the superposition of states with the singlet and triplet pairing in a Fermi liquid is considered. The thermodynamic stability of new two-gap solutions is analyzed.
DOI
10.1063/1.593536

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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.
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.
The hierarchy of equations for reduced density matrices relevant to thermodynamic equilibrium with account taken of the spin obtained earlier is modified in order to describe the state of a Fermi system with a condensate. Although the procedure is to some extent analogous with the one carried out by the author earlier for a Bose liquid peculiarities relevant to Fermi statistics complicate considerably the treatment. As in the case of the Bose liquid the condensate phase can be superfluid as well as nonsuperfluid, the physical causes of superfluidity being identical. A new mechanism of fermion pairing that acts even in the case of a purely repulsive Hamiltonian is pointed out. Special attention is given to the thermodynamics of a superfluid Fermi system. The example of a hard-sphere system is used to find out the form of phase diagrams, the character of the phase transition to a condensate phase and the properties of the last. Noticeable dissimilarities from a Bose system with the same Hamiltonian are revealed. Application of the present approach to superconductivity is discussed as well.
We investigated possible superfluid phases at finite temperature in a two-component Fermi gas with density imbalance. In the frame of a general four-fermion interaction theory, we solved in the BCS region the gap equations for the pairing gap and pairing momentum under the restriction of fixed number densities and analyzed the stability of different phases by calculating the superfluid density and number susceptibilities. The homogeneous superfluid is stable only at high temperature and low number asymmetry, the inhomogeneous Larkin-Ovchinnikov-Fudde-Ferrell state survives at low temperature and high number asymmetry, and in between them there exists another possible inhomogeneous phase, that of phase separation. The critical temperatures and the orders of the phase transitions among the superfluid phases and normal phase are calculated analytically and numerically. The phase diagram we obtained in the temperature and number asymmetry plane is quite different from the one in the temperature and chemical potential difference plane for a system with fixed chemical potentials.
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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.