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Journal of Mathematical Physics

Microscopic theory of superfluid Fermi systems. I. Binary expansion

L. E. Reichl

Published Mar 1, 1979• 1 citation
Publisher

Abstract

This is the first of two papers in which the binary expansion is used to study the properties of hard core fermi systems with broken gauge symmetry. In this paper, which is primarily formal, symmetry-breaking source terms are introduced into the generating function for thermodynamic quantities, and an exact expression is obtained for the generating function in terms of single-particle and two-particle sources and the reaction matrices for the two-body problem. In a subsequent paper, the theory is applied to a dilute hard sphere fluid.

DOI
10.1063/1.524083

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This is the first of two papers in which the binary expansion is used to study the properties of hard core fermi systems with broken gauge symmetry. In this paper, which is primarily formal, symmetry-breaking source terms are introduced into the generating function for thermodynamic quantities, and an exact expression is obtained for the generating function in terms of single-particle and two-particle sources and the reaction matrices for the two-body problem. In a subsequent paper, the theory is applied to a dilute hard sphere fluid.
This is the second of two papers in which the binary expansion is used to study properties of hard core Fermi systems with broken gauge symmetry. In this paper, we make contact with the phenomenology of superfluid Fermi systems commonly used in field theory. This requires, however, that each scattering event is approximated by its forward scattering value, an approximation which, at best is valid at very low densities. Within this approximation we show that a hard sphere Fermi fluid can exhibit superfluidity.
A theory of a Fermi superfluid is formulated based on the condensate wave function of the configuration space. First, two basic ingredients are extracted from the wave function, i.e. the effective condensate wave function v ( x 1 , x 2 ) and the quasiparticle field γ( x ) with x denoting the space and spin coordinates r α. After transforming a Hamiltonian into a normal-ordered product of γ, a perturbation expansion with respect to the quasiparticle interaction is formulated for both the thermodynamic potential Ω and the one-particle density matrix ρ. The thermodynamic potential thus obtained includes the normal-state expression of Luttinger and Ward as the limit of v → 0. The density matrix is written as a sum of the coherent part ρ ( c) and the quasiparticle part ρ ( q) , where ρ ( c) is expressed only with respect to v ( x 1 , x 2 ). It is shown that v and γ are responsible for the coherent and thermodynamic properties respectively. Thus the present formalism gives a microscopic foundation for the two-fluid model. Connections with the Gor'kov equations and the Bogoliubov-de Gennes equations are also clarified. It is found that the one-particle density matrices of those formalisms are not equivalent to ρ of the present formalism. The origin of the angular momentum paradox of 3 He - A is traced to this difference.
A gauge-symmetry-breaking parameter associated with the fluctuation of the zero-mode particles renders a source term to the continuity equation of interacting Bose systems. As a consequence, additional terms appear in the generalized Ward identities, which modify in an essential way the relation between the superfluid and condensate densities. An energy gap results as a function of the symmetry-breaking parameter and interatomic potential. The general results are worked out in the shielded potential approximation, and an equation for the condensate fraction n0{n}_{0}n0​(T) is obtained in closed form. At absolute zero the condensate fractions in dilute Bose systems and superfluid 4He^{4}\mathrm{He}4He are found to be n0{n}_{0}n0​=0.438 and n0{n}_{0}n0​=0.126, respectively.
The singlet and pair distribution functions of a Fermi gas are evaluated for low temperatures. It is assumed that the particles interact with a hard-sphere potential and have an arbitrary spin, and the evaluation is made to first order in the hard-sphere diameter. The binary-kernel method developed by Siegert, Lee, and Yang is used and the cluster-expansion theory given recently by Isihara and Yee for the distribution functions is applied. The pair distribution function is determined by a dimensionless variable $\frac{r}{\ensuremath{\lambda}}$, with $\ensuremath{\lambda}$ being the thermal de Broglie wavelength. Thus, changing the distance corresponds to varying the temperature. Moreover, the pair distribution function depends on the density. Therefore, its asymptotic expression for $T\ensuremath{\rightarrow}0$ depends on the ways to approach the limit. If we keep the density finite and reduce the temperature, the gas will be degenerate when $\ensuremath{\lambda}$ is of order of ${n}^{\ensuremath{-}\frac{1}{3}}$, where nnn is the number density. The pair distribution function for this case is an oscillating function of the Fermi momentum. On the other hand, if we reduce the density first and then decrease the temperature, the pair distribution function decays as ${r}^{\ensuremath{-}4}$, and the free-particle diagrams give a contribution of the same decay.
A number of problems involving the spectrum of elementary excitations of superfluid Fermi systems were studied by the methods of quantum field theory. A system with direct binary interaction is considered at zero and non-zero temperature, as well as an electron-phonon system at absolute zero. On finding Schwinger's equations for oneparticle Greene functions, use was made of the ground state approximation proposed by N. N. Bogoliubov, i.e., a formalism was applied which does not preserve the full number of particles. (auth)
We develop a general variational theory for inhomogeneous Fermi systems such as the electron gas in a metal surface, the surface of liquid 3He^{3}\mathrm{He}3He, or simple models of heavy nuclei. The ground-state wave function is expressed in terms of two-body correlations, a one-body attenuation factor, and a model-system Slater determinant. Massive partial summations of cluster expansions are performed by means of Born-Green-Yvon and hypernetted-chain techniques. An optimal single-particle basis is generated by a generalized Hartree-Fock equation in which the two-body correlations screen the bare interparticle interaction. The optimization of the pair correlations leads to a state-averaged random-phase-approximation equation and a strictly microscopic determination of the particle-hole interaction.