# Phase Transitions in $Z_{n}$ Gauge Models: Towards Quantum Simulations of the Schwinger-Weyl QED

## TL;DR

This paper investigates $ ext{Z}_n$ lattice gauge theories coupled to fermions in 1+1 dimensions, analyzing phase transitions and their relation to quantum electrodynamics, with implications for quantum simulation of gauge theories.

## Contribution

It provides a detailed numerical analysis of $ ext{Z}_n$ models for $n=2$ to 8, demonstrating their phase transition behavior and connection to the Schwinger model in the large-$n$ limit.

## Key findings

- All $ ext{Z}_n$ models exhibit Ising universality class phase transitions.
- Phase transition parameters approach those of the Schwinger model as $n$ increases.
- Models show spontaneous $CP$ symmetry breaking in the absence of background fields.

## Abstract

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 $\mathrm{U}(1)$ 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 $n = 2 \div 8$ by means of a DMRG code that exactly implements Gauss' 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 the zero-charge sector of the massive Schwinger model, which occurs at negative mass.