$\mathbb{Z}_N$ gauge theories coupled to topological fermions: QED$_2$ with a quantum-mechanical $θ$ angle

Abstract

We present a detailed study of the topological Schwinger model [Phys. Rev. D 99, 014503 (2019)], which describes (1+1) quantum electrodynamics of an Abelian $U(1)$ gauge field coupled to a symmetry-protected topological matter sector, by means of a class of $\mathbb{Z}_N$ lattice gauge theories. Employing density-matrix renormalization group techniques that exactly implement Gauss' law, we show that these models host a correlated topological phase for different values of $N$, where fermion correlations arise through inter-particle interactions mediated by the gauge field. Moreover, by a careful finite-size scaling, we show that this phase is stable in the large-$N$ limit, and that the phase boundaries are in accordance to bosonization predictions of the $U(1)$ topological Schwinger model. Our results demonstrate that $\mathbb{Z}_N$ finite-dimensional gauge groups offer a practical route for an efficient classical simulation of equilibrium properties of electromagnetism with topological fermions. Additionally, we describe a scheme for the quantum simulation of a topological Schwinger model exploiting spin-changing collisions in boson-fermion mixtures of ultra-cold atoms in optical lattices. Although technically challenging, this quantum simulation would provide an alternative to classical density-matrix renormalization group techniques, providing also an efficient route to explore real-time non-equilibrium phenomena.