Discrete Abelian lattice gauge theories on a ladder and their dualities with quantum clock models

TL;DR

This paper explores a duality between $ ext{Z}_N$ lattice gauge theories on a ladder and $N$-clock models, revealing sector-dependent phase diagrams and analyzing confined phases through numerical methods.

Contribution

It introduces a duality mapping from gauge-invariant states of a ladder gauge theory to a one-dimensional clock model with sector-dependent longitudinal fields.

Findings

Different superselection sectors exhibit distinct phase diagrams.

Numerical analysis suggests possible confined phases for certain sectors.

The duality mapping provides new insights into gauge theory phase structures.

Abstract

We study a duality transformation from the gauge-invariant subspace of a $\mathbb{Z}_N$ lattice gauge theory on a two-leg ladder geometry to an $N$-clock model on a single chain. The main feature of this mapping is the emergence of a longitudinal field in the clock model, whose value depends on the superselection sector of the gauge model, implying that the different sectors of the gauge theory can show quite different phase diagrams. In order to investigate this and see if confined phases might emerge, we perform a numerical analysis for $N = 2, 3, 4$, using exact diagonalization and DMRG.