Generalized Hall current on a finite lattice

TL;DR

This paper extends the concept of quantum Hall current to finite lattice systems with domain wall fermions, linking the number of gapless modes to the divergence of a generalized Hall current.

Contribution

It introduces a lattice-regulated finite volume framework for generalized Hall currents in 1+1D Wilson-Dirac fermions with domain walls, connecting gapless modes to current divergence.

Findings

Number of gapless fermions equals the integral of the divergence of the generalized Hall current.

Extension of quantum Hall current concept to odd-dimensional boundary theories on a lattice.

Provides a finite volume lattice model for analyzing boundary fermion phenomena.

Abstract

Gapped fermion theories with gapless boundary fermions can exist in any number of dimensions. When the boundary has even space-time dimensions and hosts chiral fermions, a quantum Hall current flows from the bulk to the boundary in a background electric field. This current compensate for the boundary chiral anomaly. Such a current inflow picture is absent when the boundary theory is odd dimensional. However, in recent work, the idea of quantum Hall current has been generalized to describe odd dimensional boundary theories in continuous Euclidean space-time dimension of infinite volume. In this paper we extend this idea to a lattice regulated finite volume theory of 1+1 dimensional Wilson-Dirac fermions. This fermion theory with a domain wall in fermion mass can host gapless modes on the wall. The number of gapless fermions is equal to the integral of the divergence of the lattice generalized Hall current.