Exact expectation values within Richardson's approach for the pairing Hamiltonian in a macroscopic system

TL;DR

This paper derives exact analytical expectation values for the pairing Hamiltonian in macroscopic systems using Richardson's approach, extending beyond BCS theory to cases with complex excitation spectra.

Contribution

It provides the first analytical expressions for expectation values within Richardson's solution for macroscopic superconductors, generalizing BCS results to more complex spectra.

Findings

Exact expectation values derived for Richardson's pairing Hamiltonian

Results applicable to systems with multiple energy gaps

Generalization of BCS expressions to non-mean-field regimes

Abstract

BCS superconductivity is explained by a simple Hamiltonian describing an attractive pairing interaction between pairs of electrons. The Hamiltonian may be treated using a mean-field method, which is adequate to study equilibrium properties and a variety of nonequilibrium effects. Nevertheless, in certain nonequilibrium situations, even in a macroscopic rather than a microscopic superconductor, the application of mean-field theory may not be valid. In such cases, one may resort to the full solution of the Hamiltonian, as given by Richardson in the 1960s. The relevance of Richardson's solution to macroscopic nonequilibrium superconductors was pointed out recently based on the existence of quantum instabilities out of equilibrium. It is then of interest to obtain analytical expressions for expectation values between exact eigenvalues of the pairing Hamiltonian within the Richardson approach for macroscopic systems. We undertake this task in the current paper. It should be noted that Richardson's approach yields the full set of eigenvalues of the Hamiltonian, while BCS theory yields only a subset. The results obtained here, then, generalize the familiar BCS expressions (e.g., for the electron occupation or pairing correlations) to cases where the spectrum of excitations diverges from BCS theory (e.g., in cases where the spectrum exhibits multiple gaps).