Reduced-Shifted Conjugate-Gradient Method for a Green's Function: Efficient Numerical Approach in a Nano-structured Superconductor

TL;DR

The paper introduces the reduced-shifted Conjugate-Gradient (RSCG) method, an efficient numerical technique for calculating Green's function matrix elements directly, enabling analysis of large nano-structured superconductors with many frequencies.

Contribution

The RSCG method provides a direct, efficient way to compute Green's function matrix elements without solving linear equations for each frequency, especially useful for large systems.

Findings

Successfully applied to a d-wave nano-island superconductor

Handled approximately 50,000 Matsubara frequencies simultaneously

Demonstrated efficiency over kernel-polynomial-based methods

Abstract

We propose the reduced-shifted Conjugate-Gradient (RSCG) method, which is numerically efficient to calculate a matrix element of a Green's function defined as a resolvent of a Hamiltonian operator, by solving linear equations with a desired accuracy. This method does not calculate solution vectors of linear equations but does directly calculate a matrix element of the resolvent. The matrix elements with different frequencies are simultaneously obtained. Thus, it is easy to calculate the exception value expressed as a Matsubara summation of these elements. To illustrate a power of our method, we choose a nano-structured superconducting system with a mean-field Bogoliubov-de Gennes (BdG) approach. This method allows us to treat with the system with the fabrication potential, where one can not effectively use the kernel-polynomial-based method. We consider the d-wave nano-island superconductor by simultaneously solving the linear equations with a large number (~ 50000) of Matsubara frequencies.