Method of studying the Bogoliubov-de Gennes equations for the superconducting vortex lattice state

TL;DR

This paper introduces a novel method to solve the Bogoliubov-de Gennes equations for superconducting vortex lattices by constructing an eigenspace for electrons in a magnetic field, enabling analysis of large systems.

Contribution

The paper presents an exact unitary transformation approach to relate electron eigenstates to Harper's equation, facilitating the study of large superconducting vortex lattice systems.

Findings

Successfully applied to s-wave superconductor vortex lattice

Enables analysis of large system sizes

Provides a new computational framework for BdG equations

Abstract

In this paper, we present a method to construct the eigenspace of the normal-state electrons moving in a 2D square lattice in presence of a perpendicular uniform magnetic field which imposes (quasi)-periodic boundary conditions for the wave functions in the magnetic unit cell. An exact unitary transformations are put forward to correlate the discrete eigenvectors of the 2D electrons with those of the Harper's equation. The cyclic-tridiagonal matrix associated with the Harper's equation is then tridiagonalized by another unitary transformation. The obtained eigenbasis is utilized to expand the Bogoliubov-de Gennes equations for the superconducting vortex lattice state, which showing the merit of our method in studying the large-sized system. To test our method, we have applied our results to study the vortex lattice state of an s-wave superconductor.