Flavoured Lattice Schwinger Model with Chiral Anomaly

TL;DR

This paper introduces a flavoured lattice Schwinger model that preserves an exact axial symmetry at finite lattice spacing, resolves fermion doubling via flavor staggering, and reproduces the chiral anomaly in the continuum limit.

Contribution

It presents a novel flavoured lattice construction that maintains chiral symmetry and anomaly, linking lattice gauge theory with topological insulator boundary states.

Findings

The model reduces to two massless Schwinger models in the continuum limit.

It defines a gauge-invariant lattice axial charge with a correct anomaly equation.

Spatial separation of flavors corresponds to topological edge states in a 2D insulator.

Abstract

We introduce the \emph{flavoured lattice Schwinger model}, a $(1{+}1)$-dimensional $U(1)$ lattice gauge theory in which the fermion doubling problem is resolved by staggering a $\mathbb{Z}_{2}$ flavour degree of freedom rather than staggering chirality. Unlike all standard approaches, the flavoured construction preserves an exact axial $U(1)$ symmetry at finite lattice spacing. We derive the continuum limit, showing the model reduces to two copies of the massless Schwinger model labelled by $\alpha\in\{0,1\}$. The central result is that the flavoured construction admits a well-defined, regularized, gauge-invariant lattice axial charge $Q_{G}^{A}$ with chiral anomaly equation $\langle dQ_{G}^{A}/dt\rangle = -(2g/\pi)\int dx\,\langle E(x)\rangle$ in the continuum limit, derived as a direct dynamical consequence of minimal gauge coupling at finite lattice spacing. Restricting to the $\alpha=0$ sector recovers the standard single-flavour result. We further show that spatial separation of the flavour sectors can be realised as a helical edge states living on the boundaries of a ribbon shaped $(2{+}1)$-dimensional Bernevig--Hughes--Zhang topological insulator. This provides a bulk-boundary picture solution to fermion doubling and allows the chiral anomaly to be put on the lattice for a single flavour.