Dynamics of a lattice 2-group gauge theory model

TL;DR

This paper investigates a four-dimensional lattice gauge theory model with local symmetry based on a crossed module of finite groups, analyzing its phases, topological charges, and symmetry breaking using both analytical methods and Monte Carlo simulations.

Contribution

It introduces a new lattice model based on crossed modules, proves a factorization theorem for correlation functions, and maps out its phase diagram with numerical confirmation.

Findings

The model reduces to known topological quantum field theories in special limits.

A factorization theorem simplifies correlation function computations.

The Polyakov surface serves as an order parameter sensitive to phase transitions.

Abstract

We study a simple lattice model with local symmetry, whose construction is based on a crossed module of finite groups. Its dynamical degrees of freedom are associated both to links and faces of a four-dimensional lattice. In special limits the discussed model reduces to certain known topological quantum field theories. In this work we focus on its dynamics, which we study both analytically and using Monte Carlo simulations. We prove a factorization theorem which reduces computation of correlation functions of local observables to known, simpler models. This, combined with standard Krammers-Wannier type dualities, allows us to propose a detailed phase diagram, which form is then confirmed in numerical simulations. We describe also topological charges present in the model, its symmetries and symmetry breaking patterns. The corresponding order parameters are the Polyakov loop and its generalization, which we call a Polyakov surface. The latter is particularly interesting, as it is beyond the scope of the factorization theorem. As shown by the numerical results, expectation value of Polyakov surface may serve to detects all phase transitions and is sensitive to a value of the topological charge.