Numerical construction of a low-energy effective Hamiltonian in a self-consistent Bogoliubov-de Gennes approach of superconductivity

TL;DR

This paper introduces a fast numerical method to construct low-energy effective Hamiltonians for superconductors using a matrix reduction technique, enabling efficient self-consistent calculations of physical properties.

Contribution

It presents a novel combination of matrix-size reduction and polynomial expansion methods for solving BdG equations in superconductivity.

Findings

Efficient calculation of quasi-particle excitations in vortex lattices

Accurate evaluation of thermal conductivity and relaxation rates

Applicable to large and nano-scale superconducting systems

Abstract

We propose a fast and efficient approach for solving the Bogoliubov-de Gennes (BdG) equations in superconductivity, with a numerical matrix-size reduction procedure proposed by Sakurai and Sugiura [J. Comput. Appl. Math. 159, 119 (2003)]. The resultant small-size Hamiltonian contains the information of the original BdG Hamiltonian in a given energy domain. In other words, the present approach leads to a numerical construction of a low-energy effective theory in superconductivity. The combination with the polynomial expansion method allows a self-consistent calculation of the BdG equations. Through numerical calculations of quasi-particle excitations in a vortex lattice, thermal conductivity, and nuclear magnetic relaxation rate, we show that our approach is suitable for evaluating physical quantities in a large-size superconductor and a nano-scale superconducting device, with the mean-field superconducting theory.