Ginsparg-Wilson relation with R=(a \gamma_5 D)^{2k}

TL;DR

This paper explores a family of lattice Dirac operators satisfying a generalized Ginsparg-Wilson relation, revealing that their locality diminishes as a parameter increases, which impacts their topological properties.

Contribution

The authors construct explicit lattice Dirac operators for the Ginsparg-Wilson relation with a specific R and analyze their locality and topological features.

Findings

Operators tend to become nonlocal as parameter k increases

Locality of Dirac operators is not essential for their topological index

Explicit realization of the Dirac operator satisfying the relation

Abstract

The Ginsparg-Wilson relation $D \gamma_5 + \gamma_5 D = 2 a D R \gamma_5 D$ with $R = (a \gamma_5 D)^{2k}$ is discussed. An explicit realization of D is constructed. It is shown that this sequence of topologically-proper lattice Dirac operators tend to a nonlocal operator in the limit $k \to \infty$. This suggests that the locality of a lattice Dirac operator is irrelevant to its index.