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

## TL;DR

This paper investigates a (1+1)D topological Schwinger model with $ ext{Z}_N$ gauge theories, revealing stable topological phases and proposing a quantum simulation scheme using ultra-cold atoms, bridging classical and quantum computational approaches.

## Contribution

It introduces a $ ext{Z}_N$ lattice gauge theory framework for the topological Schwinger model and demonstrates its stability and simulation potential.

## Key findings

- Identification of a stable topological phase across different N values.
- Finite-size scaling confirms phase stability in the large-N limit.
- Proposal of a quantum simulation scheme using ultra-cold atom mixtures.

## 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.