$Z_N$ symmetry in $SU(N)$ gauge theories

TL;DR

This paper investigates the behavior of $Z_N$ symmetry in $SU(N)$ gauge theories with fundamental matter fields, demonstrating that symmetry breaking diminishes at large temporal extent and identifying factors influencing symmetry realization.

Contribution

It introduces an analytical approach to study $Z_N$ symmetry in gauge theories with matter, revealing conditions under which symmetry breaking vanishes and how spatial modes affect phase boundaries.

Findings

$Z_N$ symmetry breaking reduces at large temporal sites.

Spatial links and modes influence phase boundaries.

Analytical reduction to a 1D gauged chain facilitates calculations.

Abstract

We study $Z_N$ symmetry in $SU(N)$ gauge theories in the presence of matter fields in the fundamental representation, by restricting the lattice partition function integration to matter fields which are uniform in spatial directions and gauge fields with vanishing spatial components. In this approximation the gauge matter field interaction effectively reduces to a 1-dimensional gauged chain. This makes analytical calculations of the matter field contribution to the Polyakov loop free energy possible. We show that in the limit of large number of temporal sites the explicit breaking of $Z_N$ symmetry in this free energy vanishes, driven by dominance of the density of states. We argue that the spatial links as well as the spatial modes of the matter fields determine the boundaries separating regions where $Z_N$ symmetry is realised from rest of the phase diagram.