What happens to Lattice Fermion near Continuum Limit?

TL;DR

This paper explores the behavior of lattice fermions near the continuum limit, deriving a Ginsparg-Wilson relation with chiral symmetry breaking, and constructs explicit operators in two dimensions.

Contribution

It introduces a Ginsparg-Wilson relation accommodating chiral symmetry breaking and provides a geometric interpretation and explicit construction of the Dirac operator in two dimensions.

Findings

Deviation from GWR diminishes with iterative block-spin transformations

Derived PCAC and anomaly relations on the lattice

Constructed explicit GW Dirac operator in two dimensions

Abstract

A Ginsparg-Wilson Relation (GWR) is obtained in the presence of chiral symmetry breaking terms. It leads to the PCAC relation as well as an anomaly relation on the lattice. For general fermions, the deviation from the exact GWR is getting small when the block-spin transformations are performed iteratively. Based on a simple geometrical interpretation of the Dirac operator satisfying the GWR, we find some physical properties shared by the lattice fermions near the continuum limit. In two-dimensions, we explicitly construct the GW Dirac operator by using a conformal mapping.