Short-recurrence Krylov subspace methods for the overlap Dirac operator at nonzero chemical potential

TL;DR

This paper introduces new short-recurrence Krylov subspace methods for efficiently computing the sign function of non-Hermitian matrices in lattice QCD with nonzero chemical potential, improving over previous long-recurrence approaches.

Contribution

The paper proposes two novel short-recurrence Krylov methods based on restarted Arnoldi and two-sided Lanczos processes, enhancing computational efficiency for the overlap Dirac operator.

Findings

New methods outperform previous approaches on various lattice sizes.

Deflation significantly improves the efficiency of the proposed methods.

Direct evaluation via two-sided Lanczos is feasible and effective.

Abstract

The overlap operator in lattice QCD requires the computation of the sign function of a matrix, which is non-Hermitian in the presence of a quark chemical potential. In previous work we introduced an Arnoldi-based Krylov subspace approximation, which uses long recurrences. Even after the deflation of critical eigenvalues, the low efficiency of the method restricts its application to small lattices. Here we propose new short-recurrence methods which strongly enhance the efficiency of the computational method. Using rational approximations to the sign function we introduce two variants, based on the restarted Arnoldi process and on the two-sided Lanczos method, respectively, which become very efficient when combined with multishift solvers. Alternatively, in the variant based on the two-sided Lanczos method the sign function can be evaluated directly. We present numerical results which compare the efficiencies of a restarted Arnoldi-based method and the direct two-sided Lanczos approximation for various lattice sizes. We also show that our new methods gain substantially when combined with deflation.