# Comments on Abelian Higgs Models and Persistent Order

## TL;DR

This paper investigates anomalies and persistent order in Abelian Higgs models across 1+1 and 2+1 dimensions, revealing constraints on phase transitions and dualities, and demonstrating models with non-trivial ground states unaffected by fluctuations.

## Contribution

It uncovers discrete anomalies related to charge conjugation symmetry and shows that certain Abelian Higgs models exhibit persistent order, with a novel duality to the Ising model in 1+1 dimensions.

## Key findings

- Anomalies constrain domain wall degrees of freedom in 2+1D Abelian Higgs models.
- In 1+1D, the Abelian Higgs model is dual to the Ising model.
- Models without a dynamical unit charge particle have non-trivial ground states unaffected by fluctuations.

## Abstract

A natural question about Quantum Field Theory is whether there is a deformation to a trivial gapped phase. If the underlying theory has an anomaly, then symmetric deformations can never lead to a trivial phase. We discuss such discrete anomalies in Abelian Higgs models in 1+1 and 2+1 dimensions. We emphasize the role of charge conjugation symmetry in these anomalies; for example, we obtain nontrivial constraints on the degrees of freedom that live on a domain wall in the VBS phase of the Abelian Higgs model in 2+1 dimensions. In addition, as a byproduct of our analysis, we show that in 1+1 dimensions the Abelian Higgs model is dual to the Ising model. We also study variations of the Abelian Higgs model in 1+1 and 2+1 dimensions where there is no dynamical particle of unit charge. These models have a center symmetry and additional discrete anomalies. In the absence of a dynamical unit charge particle, the Ising transition in the 1+1 dimensional Abelian Higgs model is removed. These models without a unit charge particle exhibit a remarkably persistent order: we prove that the system cannot be disordered by either quantum or thermal fluctuations. Equivalently, when these theories are studied on a circle, no matter how small or large the circle is, the ground state is non-trivial.