A New Type of Lattice Gauge Theory through Self-adjoint Extensions

TL;DR

This paper introduces a generalized lattice gauge theory framework using self-adjoint extensions of the electric field operator, revealing a phase with broken $Z_2$ symmetry at $ heta=\pi$, distinct from the standard Wilson theory.

Contribution

It proposes a novel generalization of lattice gauge theory via self-adjoint extensions, specifically analyzing the $ heta=\pi$ case and its unique phase structure.

Findings

Evidence of a broken $Z_2$ symmetry in the continuum limit at $ heta=\pi$

Dualization leads to a theory of staggered height variables

The phase diagram differs from the ordinary Wilson theory

Abstract

A generalization of Wilsonian lattice gauge theory may be obtained by considering the possible self-adjoint extensions of the electric field operator in the Hamiltonian formalism. In the special case of 3D $\mathrm{U}(1)$ gauge theory these are parametrised by a phase $\theta$, and the ordinary Wilson theory is recovered for $\theta=0$. We consider the case $\theta=\pi$, which, upon dualization, turns into a theory of staggered integer and half-integer height variables. We investigate order parameters for the breaking of the relevant symmetries, and thus study the phase diagram of the theory, which shows evidence of a broken $\mathbb{Z}_2$ symmetry in the continuum limit, in contrast to the ordinary theory.