Some remarks on the Ginsparg-Wilson fermion

TL;DR

This paper investigates the properties of generalized Ginsparg-Wilson fermions, comparing higher-order Overlap operators to the Neuberger-Dirac operator, and finds that the original operator exhibits superior chiral properties.

Contribution

It provides an explicit realization of generalized Ginsparg-Wilson relations and compares their chiral properties to the standard Neuberger-Dirac operator.

Findings

Neuberger-Dirac operator has better chiral properties than higher-order versions.

Higher-order operators show less favorable fermion propagator and anomaly behavior.

Results are based on lattice simulations up to size 16 x 16.

Abstract

We note that Fujikawa's proposal of generalization of the Ginsparg-Wilson relation is equivalent to setting $R = (a \gamma_5 D)^{2k}$ in the original Ginsparg-Wilson relation $D \gamma_5 + \gamma_5 D = 2 a D R \gamma_5 D$. An explicit realization of D follows from the Overlap construction. The general properties of D are derived. The chiral properties of these higher-order (k > 0) realizations of Overlap Dirac operator are compared to those of the Neuberger-Dirac operator (k = 0), in terms of the fermion propagator, the axial anomaly and the fermion determinant in a background gauge field. Our present results (up to lattice size 16 x 16) indicate that the chiral properties of the Neuberger-Dirac operator are better than those of higher-order ones.