Lattice fermions as spectral graphs

TL;DR

This paper applies spectral graph theory to analyze lattice fermions, providing a novel framework to determine the number of fermion species on various lattice topologies, including irregular and non-regular lattices.

Contribution

It introduces a spectral graph theory approach to study lattice fermions, enabling analysis of fermion species on arbitrary lattice topologies.

Findings

Reproduces known fermion species counts for Naive, Wilson, and Domain-wall fermions.

Extends analysis to fermions on a discretized four-dimensional hyperball.

Proposes potential for a new theorem on fermion species on arbitrary lattices.

Abstract

We study lattice fermions from the viewpoint of spectral graph theory (SGT). We find that a fermion defined on a certain lattice is identified as a spectral graph. SGT helps us investigate the number of zero eigenvalues of lattice Dirac operators even on the non-torus and non-regular lattice, leading to understanding of the number of fermion species (doublers) on lattices with arbitrary topologies. The procedure of application of SGT to lattice fermions is summarized as follows: (1) One investigates a spectral graph corresponding to a lattice fermion. (2) One obtains a matrix corresponding to the graph. (3) One finds zero eigenvalues of the matrix by use of the discrete Fourier transformation (DFT). (4) By taking an infinite-volume and continuum limits, one finds the number of species. We apply this procedure to the known lattice fermion formulations including Naive fermions, Wilson fermions and Domain-wall fermions, and reproduce the known fact on the number of species. We also apply it to the lattice fermion on the discretized four-dimensional hyperball and discuss the number of fermion species on the bulk. In the end of the paper, we discuss the application of the analysis to lattice fermions on generic lattices with arbitrary topologies, which could lead to constructing a new theorem regarding the number of species.