# Walls, Anomalies, and (De)Confinement in Quantum Anti-Ferromagnets

## TL;DR

This paper explores the confinement and deconfinement phenomena in quantum anti-ferromagnets through the Abelian-Higgs model, revealing how domain walls can host deconfined spinons and exhibit anomalies, with implications for lattice simulations and phase transitions.

## Contribution

It demonstrates the presence of 't Hooft anomalies on domain walls in quantum anti-ferromagnets and shows how these walls can host deconfined particles, connecting field theory with lattice models and phase transitions.

## Key findings

- Domain walls carry 't Hooft anomalies.
- Deconfined spinons appear on domain walls.
- Wall theories can become gapless or break charge conjugation symmetry.

## Abstract

We consider the Abelian-Higgs model in 2+1 dimensions with instanton-monopole defects. This model is closely related to the phases of quantum anti-ferromagnets. In the presence of $\mathbb{Z}_2$ preserving monopole operators, there are two confining ground states in the monopole phase, corresponding to the Valence Bond Solid (VBS) phase of quantum magnets. We show that the domain-wall carries a 't Hooft anomaly in this case. The anomaly can be saturated by, e.g., charge-conjugation breaking on the wall or by the domain wall theory becoming gapless (a gapless model that saturates the anomaly is $SU(2)_1$ WZW). Either way the fundamental scalar particles (i.e. spinons) which are confined in the bulk are deconfined on the domain-wall. This $\mathbb{Z}_2$ phase can be realized either with spin-1/2 on a rectangular lattice, or spin-1 on a square lattice. In both cases the domain wall contains spin-1/2 particles (which are absent in the bulk). We discuss the possible relation to recent lattice simulations of domain walls in VBS. We further generalize the discussion to Abrikosov-Nielsen-Olsen (ANO) vortices in a dual superconductor of the Abelian-Higgs model in 3+1 dimensions, and to the easy-plane limit of anti-ferromagnets. In the latter case the wall can undergo a variant of the BKT transition (consistent with the anomalies) while the bulk is still gapped. The same is true for the easy-axis limit of anti-ferromagnets. We also touch upon some analogies to Yang-Mills theory.