An iterative method to compute the sign function of a non-Hermitian matrix and its application to the overlap Dirac operator at nonzero chemical potential

TL;DR

This paper introduces an efficient iterative method using Krylov subspace and deflation schemes to compute the sign function of non-Hermitian matrices, crucial for lattice QCD at nonzero chemical potential.

Contribution

It develops a modified Arnoldi method with deflation to accurately and efficiently compute the sign function of non-Hermitian matrices, improving performance in lattice QCD applications.

Findings

The method effectively handles eigenvalues near discontinuities.

Deflation schemes improve convergence and efficiency.

Numerical results demonstrate significant performance gains.

Abstract

The overlap Dirac operator in lattice QCD requires the computation of the sign function of a matrix. While this matrix is usually Hermitian, it becomes non-Hermitian in the presence of a quark chemical potential. We show how the action of the sign function of a non-Hermitian matrix on an arbitrary vector can be computed efficiently on large lattices by an iterative method. A Krylov subspace approximation based on the Arnoldi algorithm is described for the evaluation of a generic matrix function. The efficiency of the method is spoiled when the matrix has eigenvalues close to a function discontinuity. This is cured by adding a small number of critical eigenvectors to the Krylov subspace, for which we propose two different deflation schemes. The ensuing modified Arnoldi method is then applied to the sign function, which has a discontinuity along the imaginary axis. The numerical results clearly show the improved efficiency of the method. Our modification is particularly effective when the action of the sign function of the same matrix has to be computed many times on different vectors, e.g., if the overlap Dirac operator is inverted using an iterative method.