Green's functions for fermion systems with pairing correlations

M. K. Weigel

J M Irvine, L. D. Skouras

The Green function method is used to calculate an analytic expression for the excitation spectrum of a finite fermion system interacting through a pairing Hamiltonian. The number of particles is rigorously conserved and use is made of an approximation introduced by Flowers, Irvine and Johnstone in 1964 to simplify the equations and obtain results very similar to those of the Bardeen-Cooper-Schrieffer theory.

Yi‐Cai Zhang

In this work, a Josephson relation is generalized to a multi-component fermion superfluid. Superfluid density is expressed through a two-particle Green function for pairing states. When the system has only one gapless collective excitation mode, the Josephson relation is simplified, which is given in terms of the superfluid order parameters and the trace of two-particle normal Green function. In addition, it is found that the matrix elements of two-particle Green function is directly related to the matrix elements of the pairing fluctuations of superfluid order parameters. Furthermore, in the presence of inversion symmetry, the superfluid density is given in terms of the pairing fluctuation matrix. The results of the superfluid density in Haldane model show that the generalized Josephson relation can be also applied to a multi-band fermion superfluid in lattice.

H. Flocard

Joseph L. Birman, A. I. Solomon

The Thermal Matsubara Green Functions, and their zero temperature analogs, in many-fermion systems are shown to be built up from products, or sums of products, of factors each of which is determined by a Lie Algebra. These factors are: the structure constants of a mapping φ of the B2N+1 Fermion Algebra (consisting of singles and pairs of fermion operators) onto itself; and the structure constants of a mapping φ′ of the D2N+1 Dynamical Algebra of the Hamiltonian (consisting of pairs of fermion operators) onto itself; and eigenvalues of the generators of the Abelian Cartan subalgebra. Eigenstates of the many-fermion Hamiltonian typically relate to different types or numbers of elementary excitations (quasi-particles); these states are shwon to be bases for irreducible representations of the Dynamical Algebra. The contribution of each such irreducible set of states to the Green Functions is exhibited. These results are exact for mean-field Hamiltonians; generalization to include effects of fluctuations may be possible. The Green Functions for a model BCS singlet superconductor are determined (Dynamical Algebra SU (2)) to illustrate the various steps in the analysis.