Confinement-Deconfinement transition and $Z_2$ symmetry in $Z_2+$Higgs theory

TL;DR

This paper investigates the $Z_2$ symmetry and phase transitions in 4D lattice $Z_2+$Higgs theory using Monte Carlo simulations and analytical models, revealing symmetry realization in the Higgs symmetric phase for large temporal lattice sizes.

Contribution

It demonstrates the realization of $Z_2$ symmetry in the Higgs symmetric phase and provides an analytical understanding of its dependence on the number of temporal sites.

Findings

$Z_2$ symmetry is realized in the Higgs symmetric phase for large temporal sites.

The analytical model confirms the symmetry in the large temporal site limit.

The symmetry is linked to $Z_2$ invariance and the density of states dominance.

Abstract

We study the Polyakov loop and the $Z_2$ symmetry in the lattice $Z_2+$Higgs theory in 4D Euclidean space using Monte Carlo simulations. The results show that this symmetry is realised in the Higgs symmetric phase for large number of temporal lattice sites. To understand the dependence on the number of temporal sites, we consider a one dimensional model by keeping terms of the original action corresponding to a single spatial site. In this approximation the partition function can be calculated exactly as a function of the Polyakov loop. The resulting free energy is found to have the $Z_2$ symmetry in the limit of large temporal sites. We argue that this is due to $Z_2$ invariance as well as dominance of the distribution or density of states corresponding to the action.