An iterative method to compute the overlap Dirac operator at nonzero chemical potential

TL;DR

This paper introduces an improved iterative method using Krylov subspace and deflation techniques to efficiently compute the overlap Dirac operator at nonzero chemical potential, addressing challenges posed by eigenvalues near discontinuities.

Contribution

The authors develop a novel deflation-enhanced Krylov subspace method for computing the sign function of non-Hermitian matrices in lattice QCD, improving efficiency at nonzero chemical potential.

Findings

Deflation significantly speeds up convergence.

Method effective on large lattice sizes.

Improved accuracy near eigenvalue discontinuities.

Abstract

The overlap Dirac operator at nonzero quark chemical potential involves the computation of the sign function of a non-Hermitian matrix. In this talk we present an iterative method, first proposed by us in Ref. [1], which allows for an efficient computation of the operator, even on large lattices. The starting point is a Krylov subspace approximation, based on the Arnoldi algorithm, for the evaluation of a generic matrix function. The efficiency of this method is spoiled when the matrix has eigenvalues close to a function discontinuity. To cure this, a small number of critical eigenvectors are added to the Krylov subspace, and two different deflation schemes are proposed in this augmented subspace. The ensuing method is then applied to the sign function of the overlap Dirac operator, for two different lattice sizes. The sign function has a discontinuity along the imaginary axis, and the numerical results show how deflation dramatically improves the efficiency of the method.