Theoretical foundation for the Index Theorem on the lattice with staggered fermions

TL;DR

This paper develops a theoretical framework to identify topological zero-modes of staggered lattice fermions, linking their index to gauge field topology, and proposes a computationally efficient method for topological charge assignment.

Contribution

It introduces a spectral flow approach for staggered fermions, connecting their index to the Index Theorem and providing a new way to determine topological charge on the lattice.

Findings

The method accurately identifies zero-modes and their chiralities.

It performs comparably to Wilson index in 2D U(1) backgrounds.

The approach is more computationally efficient than existing methods.

Abstract

A way to identify the would-be zero-modes of staggered lattice fermions away from the continuum limit is presented. Our approach also identifies the chiralities of these modes, and their index is seen to be determined by gauge field topology in accordance with the Index Theorem. The key idea is to consider the spectral flow of a certain hermitian version of the staggered Dirac operator. The staggered fermion index thus obtained can be used as a new way to assign the topological charge of lattice gauge fields. In a numerical study in U(1) backgrounds in 2 dimensions it is found to perform as well as the Wilson index while being computationally more efficient. It can also be expressed as the index of an overlap Dirac operator with a new staggered fermion kernel.