Fate of Quantum Anomalies for 1d lattice chiral fermion with a simple non-Hermitian Hamiltonian

TL;DR

This paper introduces a 1+1D lattice chiral fermion model with a non-Hermitian Hamiltonian that reproduces quantum anomalies without requiring a 2D bulk, challenging conventional beliefs about lattice chiral fermions.

Contribution

It presents a novel 1D lattice chiral fermion model with a non-Hermitian Hamiltonian that captures quantum anomalies independently, bypassing the need for a 2D topological bulk.

Findings

Model exhibits the same chiral and gravitational anomalies as continuum theory.

Low energy behavior remains a unitary chiral fermion field theory.

Non-Hermitian damping suppresses opposite chirality modes.

Abstract

It is generally believed that the 1+1D model for a single chiral fermion does not exist by itself alone on lattice. The obstruction to such a lattice realization is the failure to reproduce the quantum anomalies of a chiral fermion in continuum. The conventional way to escape is to associate the anomalous 1d system with a 2d bulk, which is in a topologically non-trivial state, as the boundary of the latter. In this paper, we propose a 1+1D chiral fermion model on 1d spatial lattice, {standing alone} -- without being associated with a 2d bulk -- with a simple {non-Hermitian} hopping Hamiltonian. We demonstrate, using various methods, that the model possesses the same chiral anomaly and gravitational anomaly as in continuum theory. Furthermore, with appropriate parameters, the low energy effective theory of the model remains a field theory for unitary chiral fermions. The essential reason for the success is that the usual "doubled" fermion mode with opposite chirality is rapidly damped out because of non-Hermicity of the Hamiltonian.