Numerical Solution of the Bardeen-Cooper-Schrieffer Equation for Unconventional Superconductors

TL;DR

This paper develops an efficient numerical method to solve the Bardeen-Cooper-Schrieffer equation for unconventional superconductors with long-range interactions, providing insights into the superconducting gap structure on a lattice.

Contribution

It introduces a Galerkin method with B-splines for solving the BCS equation incorporating long-range power-law interactions on a lattice, advancing computational techniques in superconductivity.

Findings

Numerical solutions for a nodal superconductor on a 2D lattice.

Efficient computation of Epstein zeta function in the context of superconductivity.

Analysis of the equation's analytical properties and their implications.

Abstract

In this work, we consider the analytical properties and the efficient numerical solution of the Bardeen-Cooper-Schrieffer equation for unconventional superconductivity incorporating long-range power-law electron-electron interactions within a tight-binding model on a $d$-dimensional lattice. It is a nonlinear convolution equation for the complex matrix-valued superconducting gap under symmetry constraints imposed by the fermionic anticommutation rules. The long-range interaction enters in momentum space in the form of the now efficiently computable Epstein zeta function, which exhibits a power-law singularity at zero momentum. This needs to be accounted when evaluating the convolution. After a brief overview of some of the equation's analytical properties, we discuss its efficient numerical solution using a Galerkin method with B-splines. We present numerical results for a nodal superconductor on a two-dimensional square lattice.