A Single Right-Moving Free Fermion Mode on an Ultra-Local $1+1$d Spacetime Lattice

TL;DR

This paper introduces a non-hermitian, ultra-local lattice model for a chiral free-fermion theory that maintains unitarity and Lorentz invariance, providing a new approach to simulating chiral fermions in lattice systems.

Contribution

It demonstrates that a chiral free-fermion theory can be realized on an ultra-local lattice with a non-hermitian Lagrangian, circumventing previous no-go theorems.

Findings

Model is formulated in Minkowskian time with Lorentz invariance.

The model is free from gauge anomalies despite theoretical expectations.

It describes a single chiral edge mode of Floquet systems, enabling potential physical realizations.

Abstract

Defining a Chiral Fermion Theory on a lattice has presented an ongoing challenge both in Condensed Matter physics and in Lattice Gauge Theory. In this paper, we demonstrate that a chiral free-fermion theory can live on an ultra-local spacetime lattice if we allow the Lagrangian to be non-hermitian. Rather than a violation of unitarity, the non-hermitian structure of our Lagrangian arises because time is discrete, and we show that our model is obeys an elementary unitarity condition: namely, that the norm of the two-point functions conserves probability. Beyond unitarity, our model displays several surprising properties: it is formulated directly in Minkowskian time; it has exactly Lorentz invariant dynamics for all frequencies and momenta (in the large volume limit); and it is free from all gauge anomalies, despite the prediction from field theory that it should suffer one. We show that our model is a discrete time description of a single chiral edge mode of several recently proposed $2+1$d Floquet models. That the chiral edge can be treated without the rest of the $2+1$d system, even when coupled to a gauge field, implies that the Floquet models are radically different from Integer Quantum Hall models, which also support chiral edge modes. Furthermore, the Floquet results imply that our model can be physically realized, which presents an opportunity for gauge theories to be simulated in a condensed matter or cold atom context. Our results present a solution to the `Chiral-fermion problem:' a chiral field theory can indeed be defined on an ultra-local spacetime lattice, and we address how our model avoids several no-go arguments.