A numerical method to compute derivatives of functions of large complex matrices and its application to the overlap Dirac operator at finite chemical potential

TL;DR

This paper introduces a numerical method for computing derivatives of functions of large complex matrices, enabling efficient calculations in lattice QCD at finite chemical potential, especially for the overlap Dirac operator.

Contribution

The paper presents a novel method compatible with implicit Krylov-Ritz approximations for derivatives of matrix functions, applicable to non-Hermitian matrices in lattice QCD.

Findings

Efficient derivative computation for the sign function of non-Hermitian matrices.

Application demonstrated on realistic SU(3) gauge configurations.

Method supports calculation of conserved currents and fermionic forces.

Abstract

We present a method for the numerical calculation of derivatives of functions of general complex matrices. The method can be used in combination with any algorithm that evaluates or approximates the desired matrix function, in particular with implicit Krylov-Ritz-type approximations. An important use case for the method is the evaluation of the overlap Dirac operator in lattice Quantum Chromodynamics (QCD) at finite chemical potential, which requires the application of the sign function of a non-Hermitian matrix to some source vector. While the sign function of non-Hermitian matrices in practice cannot be efficiently approximated with source-independent polynomials or rational functions, sufficiently good approximating polynomials can still be constructed for each particular source vector. Our method allows for an efficient calculation of the derivatives of such implicit approximations with respect to the gauge field or other external parameters, which is necessary for the calculation of conserved lattice currents or the fermionic force in Hybrid Monte-Carlo or Langevin simulations. We also give an explicit deflation prescription for the case when one knows several eigenvalues and eigenvectors of the matrix being the argument of the differentiated function. We test the method for the two-sided Lanczos approximation of the finite-density overlap Dirac operator on realistic $SU(3)$ gauge field configurations on lattices with sizes as large as $14\times14^3$ and $6\times18^3$.