  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.
