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We present a detailed study of the topological Schwinger model [Phys. Rev. D99, 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. Employingdensity-matrix renormalization group techniques that exactly implement Gauss'law, we show that these models host a correlated topological phase fordifferent values of $N$, where fermion correlations arise throughinter-particle interactions mediated by the gauge field. Moreover, by a carefulfinite-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 ofthe $U(1)$ topological Schwinger model. Our results demonstrate that$\mathbb{Z}_N$ finite-dimensional gauge groups offer a practical route for anefficient classical simulation of equilibrium properties of electromagnetismwith topological fermions. Additionally, we describe a scheme for the quantumsimulation of a topological Schwinger model exploiting spin-changing collisionsin boson-fermion mixtures of ultra-cold atoms in optical lattices. Althoughtechnically challenging, this quantum simulation would provide an alternativeto classical density-matrix renormalization group techniques, providing also anefficient route to explore real-time non-equilibrium phenomena.

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