Non-local equation for the superconducting gap parameter

TL;DR

This paper derives and analyzes a non-local integral equation for the superconducting gap parameter, revealing that the coarse-graining size corresponds to the Cooper pair size across the BCS-BEC crossover, and introduces an efficient numerical solution method.

Contribution

It introduces a non-local integral equation for the superconducting gap and a novel Fourier transform algorithm, providing insights into vortex structures across the BCS-BEC crossover.

Findings

Coarse-graining size matches Cooper pair size across the phase diagram.

The numerical method efficiently solves the integral equation for vortex analysis.

Insights into length scales and details relevant to vortex structures.

Abstract

The properties are considered in detail of a non-local (integral) equation for the superconducting gap parameter, which is obtained by a coarse-graining procedure applied to the Bogoliubov-deGennes (BdG) equations over the whole coupling-vs-temperature phase diagram associated with the superfluid phase. It is found that the limiting size of the coarse-graining procedure, which is dictated by the range of the kernel of this integral equation, corresponds to the size of the Cooper pairs over the whole coupling-vs-temperature phase diagram up to the critical temperature, even when Cooper pairs turn into composite bosons on the BEC side of the BCS-BEC crossover. A practical method is further implemented to solve numerically this integral equation in an efficient way, which is based on a novel algorithm for calculating the Fourier transforms. Application of this method to the case of an isolated vortex, throughout the BCS-BEC crossover and for all temperatures in the superfluid phase, helps clarifying the nature of the length scales associated with a single vortex and the kinds of details that are in practice disposed off by the coarse-graining procedure on the BdG equations.