Form and index of Ginsparg-Wilson fermions

TL;DR

This paper analyzes a class of lattice Dirac operators related to Ginsparg-Wilson fermions, clarifying their index properties and chiral anomaly behavior, especially in the continuum limit, with implications for lattice gauge theory.

Contribution

It demonstrates that certain lattice Dirac operators based on Cayley transformations generally lack a nonvanishing index on finite lattices but recover it in the continuum limit.

Findings

Operators in this class do not admit a nonvanishing index on finite lattices.

The index defect disappears in the continuum limit due to Hilbert space properties.

The paper discusses the sum rule for the index and its implications.

Abstract

We clarify the questions rised by a recent example of a lattice Dirac operator found by Chiu. We show that this operator belongs to a class based on the Cayley transformation and that this class on the finite lattice generally does not admit a nonvanishing index, while in the continuum limit, due to operator properties in Hilbert space, this defect is no longer there. Analogous observations are made for the chiral anomaly. We also elaborate on various aspects of the underlying sum rule for the index.