Acceleration of the Arnoldi method and real eigenvalues of the non-Hermitian Wilson-Dirac operator

TL;DR

This paper introduces an improved polynomial transformation-based Arnoldi method for efficiently computing low-lying real eigenvalues of the Wilson-Dirac operator, aiding in calculations relevant to QCD and super Yang-Mills theory.

Contribution

The paper presents a novel polynomial transformation approach and iterative procedure that significantly enhances the efficiency of eigenvalue computations for the Wilson-Dirac operator.

Findings

Improved eigenvalue computation efficiency demonstrated.

Method applicable to operators with symmetric, bounded spectra.

Facilitates calculations of fermion determinant signs and Pfaffians.

Abstract

In this paper, we present a method for the computation of the low-lying real eigenvalues of the Wilson-Dirac operator based on the Arnoldi algorithm. These eigenvalues contain information about several observables. We used them to calculate the sign of the fermion determinant in one-flavor QCD and the sign of the Pfaffian in N=1 super Yang-Mills theory. The method is based on polynomial transformations of the Wilson-Dirac operator, leading to considerable improvements of the computation of eigenvalues. We introduce an iterative procedure for the construction of the polynomials and demonstrate the improvement in the efficiency of the computation. In general, the method can be applied to operators with a symmetric and bounded eigenspectrum.