A note on Zolotarev optimal rational approximation for the overlap Dirac operator

TL;DR

This paper explores the use of Zolotarev optimal rational approximation for the inverse square root function in lattice QCD, demonstrating high-precision preservation of chiral symmetry in the overlap Dirac operator across various gauge configurations.

Contribution

It provides a theoretical error bound for the approximation and empirical formulas with numerical data, enhancing the practical application of the overlap Dirac operator in lattice QCD.

Findings

Error bound is always satisfied in tested configurations

Empirical error formulas are established and numerically validated

High-precision chiral symmetry preservation is achievable

Abstract

We discuss the salient features of Zolotarev optimal rational approximation for the inverse square root function, in particular, for its applications in lattice QCD with overlap Dirac quark. The theoretical error bound for the matrix-vector multiplication $ H_w (H_w^2)^{-1/2}Y $ is derived. We check that the error bound is always satisfied amply, for any QCD gauge configurations we have tested. An empirical formula for the error bound is determined, together with its numerical values (by evaluating elliptic functions) listed in Table 2 as well as plotted in Figure 3. Our results suggest that with Zolotarev approximation to $ (H_w^2)^{-1/2} $, one can practically preserve the exact chiral symmetry of the overlap Dirac operator to very high precision, for any gauge configurations on a finite lattice.