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

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.

Figures (2)

• Figure 1: Equation $s/C_1=\varphi(s),\ C_1=1,\ C_1=1/2$ for $d=1$.
• Figure 2: Nodal d-wave solution of the gap equation for $C_1=0.75,\ C_2=0.7$, and $\nu=2.01$

Theorems & Definitions (1)

• theorem 1