The Euclidean Spectrum of Kaplan's Lattice Chiral Fermions

TL;DR

This paper analyzes the spectral properties of Kaplan's lattice chiral fermions, revealing continuum and lattice differences, and discusses implications for the fermionic propagator and non-perturbative effects.

Contribution

It provides a detailed spectral analysis of Kaplan's domain wall fermions in both continuum and lattice settings, highlighting key differences and potential issues.

Findings

No bound states with non-zero momentum in continuum

Lattice bound states without energy gap unless fine-tuned

Fermionic propagator exhibits expected pole despite spectral peculiarities

Abstract

We consider the (2n+1)-dimensional euclidean Dirac operator with a mass term that looks like a domain wall, recently proposed by Kaplan to describe chiral fermions in $2n$ dimensions. In the continuum case we show that the euclidean spectrum contains {\it no} bound states with non-zero momentum. On the lattice, a bound state spectrum without energy gap exists only if $m$ is fine tuned to some special values, and the dispersion relation does not describe a relativistic fermion. In spite of these peculiarities, the fermionic propagator {\it has} the expected (1/p-slash) pole on the domain wall. But there may be a problem with the phase of the fermionic determinant at the non-perturbative level.