Free Energy of an Inhomogeneous Superconductor: a Wave Function Approach

TL;DR

This paper introduces a wave function-based method to calculate the free energy of inhomogeneous superconductors, applicable to various spatial configurations, and establishes connections between different theoretical equations for improved analysis.

Contribution

It presents a novel approach using the Andreev approximation and Gelfand-Dikii equation to compute free energy in inhomogeneous superconductors, extending previous methods.

Findings

Expresses local density of states and free energy density via the Andreev Hamiltonian resolvent.

Establishes the link between Eilenberger and Gelfand-Dikii equations.

Provides an algorithm for gauge-invariant gradient expansion at arbitrary temperatures.

Abstract

A new method for calculating the free energy of an inhomogeneous superconductor is presented. This method is based on the quasiclassical limit (or Andreev approximation) of the Bogoliubov-de Gennes (or wave function) formulation of the theory of weakly coupled superconductors. The method is applicable to any pure bulk superconductor described by a pair potential with arbitrary spatial dependence, in the presence of supercurrents and external magnetic field. We find that both the local density of states and the free energy density of an inhomogeneous superconductor can be expressed in terms of the diagonal resolvent of the corresponding Andreev Hamiltonian, resolvent which obeys the so-called Gelfand-Dikii equation. Also, the connection between the well known Eilenberger equation for the quasiclassical Green's function and the less known Gelfand-Dikii equation for the diagonal resolvent of the Andreev Hamiltonian is established. These results are used to construct a general algorithm for calculating the (gauge invariant) gradient expansion of the free energy density of an inhomogeneous superconductor at arbitrary temperatures.