Comments on Abelian Higgs Models and Persistent Order

TL;DR

The paper investigates how 't Hooft anomalies, including discrete and center-related anomalies, constrain the phase structure of Abelian Higgs models in $1+1$ and $2+1$ dimensions. By analyzing anomaly inflow, circle reductions, and dualities (notably the $1+1$ Ising duality for the $N=1$ case), it shows that symmetric, gapped trivial phases are generally forbidden when charge-conjugation symmetry or 1-form center symmetries participate nontrivially. A key finding is that for $p>1$ in both 1+1 and 2+1 dimensions, the presence of center symmetry yields persistent order even at finite temperature, as the anomalies constrain disordered phases on circles. The work also links domain-wall theories to nontrivial anomaly structures and identifies enhanced symmetry scenarios (e.g., $SO(5)$ at the CP$^1$ fixed point) and connections to condensed-matter phenomena such as Néel-VBS transitions. These results illuminate how discrete and higher-form symmetries shape IR dynamics, offering robust constraints across dimensions and potential applications to lattice models and beyond.

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.

Figures (5)

• Figure 1: The potential of $\tilde{\varphi}$ as the strength of instanton effect is varied at $\theta=\pi$ in the $p=1$, $N=1$ model. As one moves away from the deep Higgs regime the number of minima changes from one to two, resembling an Ising transition.
• Figure 2: The phase diagram of 1+1 scalar QED with one charge 1 scalar. The semi-infinite line at $\theta=\pi$ represents a first order transition which ends at some critical $m^2_*$ with a second order Ising transition.
• Figure 3: Potential energies of the $p$ superselection sectors in the Higgs phase. Here the action $S_0$ of a vortex instanton is taken to be infinity and the instanton effect is turned off. We have $p$ vacua ($p=4$) degenerate exactly in energy.
• Figure 4: Potential energies of the $p$ superselection sectors in the Higgs phase at $\theta=0$. Here $S_0$ is taken large but finite. The instanton vortices lift some of the degeneracies of the $p$ sectors, leaving only one sector with the lowest energy density.
• Figure 5: Potential energies of the $p$ superselection sectors in the Higgs phase at $\theta=\pi$. Here $S_0$ is taken large but finite. The instanton vortices lift some of the degeneracies of the $p$ sectors but there is a two-fold degeneracy in sectors with lowest energy densities.