Domain-Wall Fermion with $ R_5 $ Symmetry

TL;DR

This paper introduces a reflection-symmetric domain-wall fermion operator in five dimensions, utilizing Zolotarev optimal rational approximations to improve the approximation of the sign function in lattice QCD simulations.

Contribution

The paper develops a new domain-wall fermion operator with R_5 symmetry that employs Zolotarev rational functions for better approximation of the sign function.

Findings

Bound on the approximation error |1 - S(λ)| in terms of maximum deviation d_Z.

Explicit construction of the operator with degrees (n-1,n) or (n,n) depending on N_s.

Enhanced symmetry properties potentially lead to improved chiral symmetry in lattice simulations.

Abstract

We present the domain-wall fermion operator which is reflection symmetric in the fifth dimension, with the approximate sign function $ S(H) $ of the effective 4-dimensional Dirac operator satisfying the bound $ |1-S(\lambda)| \le 2 d_Z $ for $ \lambda^2 \in [\lambda_{min}^2, \lambda_{max}^2] $, where $ d_Z $ is the maximum deviation $ | 1- \sqrt{x} R_Z(x) |_{\rm max} $ of the Zolotarev optimal rational polynomial $ R_Z(x) $ of $ 1/\sqrt{x} $ for $ x \in [1, \lambda_{max}^2/\lambda_{min}^2] $, with degrees $ (n-1,n) $ for $ N_s = 2n $, and $ (n,n) $ for $ N_s = 2 n + 1 $.