Flavoured Lattice Schwinger Model with Chiral Anomaly

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 the standard approaches, this construction preserves an exact axial $U(1)$ symmetry at finite lattice spacing. We derive the continuum limit, showing that the model reduces to the \emph{two-flavour} massless Schwinger model, with flavours $\alpha \in \{0,1\}$ sharing one dynamical $U(1)$ gauge field. The central result is a well-defined, regularised, gauge-invariant lattice axial charge $Q_G^A$ whose continuum non-conservation $⟨d{Q}_G^A/dt⟩=-(2g/\pi )\int dx⟨E(x)⟩$ arises as a direct dynamical consequence of minimal gauge coupling. A particle-hole transformation on the $\chi$ flavour exposes a hidden $U_L(2)\times U_R(2)$ chiral symmetry; non-Abelian bosonisation then identifies the model with a massive abelian Schwinger sector tensored with the level-$1$ $SU(2)$ Wess--Zumino--Witten model. Finally, we show that embedding the flavoured fermions in a ribbon-shaped $(2+1)$D Bernevig--Hughes--Zhang topological insulator and gauging the bulk in a constant background field factorises the boundary theory into \emph{two decoupled} single-flavour Schwinger models, one on each edge, identifying the lattice factor of $2$ as one quantum of Schwinger anomaly per edge.