We study the ground-state properties of a class of ${\mathbb{Z}}_n$ lattice gauge theories in $1+1$ dimensions, in which the gauge fields are coupled to spinless fermionic matter. These models, stemming from discrete representations of the Weyl commutator for the U(1) group, preserve the unitary character of the minimal coupling and have, therefore, the property of formally approximating lattice quantum electrodynamics in one spatial dimension in the large-$n$ limit. The numerical study of such approximated theories is important to determine their effectiveness in reproducing the main features and phenomenology of the target theory, in view of implementations of cold-atom quantum simulators of QED. In this paper, we study the cases $n=2÷8$ by means of a DMRG code that exactly implements Gauss’s law. We perform a careful scaling analysis and show that, in absence of a background field, all ${\mathbb{Z}}_n$ models exhibit a phase transition which falls in the Ising universality class, with spontaneous symmetry breaking of the $CP$ symmetry. We then perform the large-$n$ limit and find that the asymptotic values of the critical parameters approach the ones obtained for the known phase transition of the zero-charge sector of the massive Schwinger model, which occurs at negative mass.

• Received 3 October 2017

DOI:https://doi.org/10.1103/PhysRevD.98.074503