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

TL;DR

The paper studies domain walls in the 2+1D Abelian-Higgs model with even monopoles, showing that a mixed 't Hooft anomaly between charge-conjugation and spin-rotation symmetries is saturated on the wall. This forces deconfinement of bulk-confined spinons on the wall, realized either by spontaneous C-symmetry breaking or by a gapless wall theory such as the $SU(2)_1$ WZW model, with concrete lattice and semiclassical pictures supporting the scenario. The analysis extends to easy-plane/axis deformations and to 3+1D Abelian-Higgs theories with ANO vortices, where the wall can undergo BKT-like transitions consistent with anomalies. The results connect condensed-matter realizations of VBS physics to anomaly inflow and to broader Yang–Mills analogies, offering predictions testable in lattice simulations and guiding the interpretation of domain-wall excitations.

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.

Figures (8)

• Figure 1: The schematic depiction of domain-walls in $\mathbb{Z}_2$ broken VBS. The domain-walls restore the topological symmetry, but break charge conjugation because they carry electric flux (depicted by the arrows). The domain-walls inside the domain-walls are elementary excitations of spinons which do not exist in the bulk as they are confined. In $a)$ a single spinon on the domain-wall is depicted, as well as terms which enter \ref{['eq:meff']}. In $b)$ a pair created from the vacuum of the domain-wall is shown, changing the vacuum in between them by the charge-conjugation (i.e. changing the flux on the domain wall).
• Figure 2: A lattice depiction of $\mathbb{Z}_2$ VBS domain walls which break the ${\cal C}$-symmetry.
• Figure 3: A lattice depiction of spin-$1$ system. a) The product state: circles label spin-$1$ states on sites, labeled by two fundamental indices. b) The formation of the valence bond between two sites is a contraction of one of the fundamental indices on each site. c) A formation of a long valence bond with fundamental indices at the end.
• Figure 4: a,b) Two vacua of the spin-$1$ VBS phase. c) A domain wall hosts free spin-$1/2$ particles (labeled by a blue circle), which are confined in the bulk.
• Figure 5: A depiction of the space-time of the quantum magnet. By setting the $\mathbb{Z}_2$-topological gauge field $A$ to a constant along the $x$ direction, the VBS domain wall is induced.