Walls, anomalies, and deconfinement in quantum antiferromagnets

We consider the Abelian-Higgs model in $2+1$ dimensions with instanton-monopole defects. This model is closely related to the phases of quantum antiferromagnets. 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 $SU{\left(2\right)}_1$ 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 antiferromagnets. 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 antiferromagnets. We also touch upon some analogies to Yang-Mills theory.

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 (6). 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 $\mathcal{C}$ symmetry.

Figure 3

A lattice depiction of a spin-one system. (a) The product state: circles label spin-one 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-one 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.

Figure 6

A representation of the domain-wall surface on a torus (a) and the two possible scenarios consistent with the 't Hooft anomaly inflow when the charge-conjugation gauge field is turned on along the cycle ${\gamma }_x$. This can be thought of as implementing a $C$ transformation along the disk $\mathrm{\Sigma }$ on which an integral ${\int }_{\mathrm{\Sigma }}B$ is not gauge invariant and requires a particle world line in the fundamental $SU\left(N_f\right)$ representation wrapping along the minor cycle of the torus to cancel the gauge variation.

Figure 7

(a) The preferred orientation of the $O\left(3\right)$ order parameter $\vec{n}={\sum }_{i,j=1,2}{\varphi }_i^{*}{\left(\vec{\sigma }\right)}_{ij}{\varphi }_j$ (red arrow), when $\lambda$ in (14) is positive. (b),(c) The preferred orientation of $\vec{n}$ when $\lambda$ is negative. The ${\mathbb{Z}}_2$ exchange symmetry corresponds to the exchange of north and south poles.

Figure 8

A representation of the domain wall of the ${\mathbb{Z}}_2$-exchange symmetry broken phase. On one side of the domain wall we must have the $O\left(3\right)$ order parameter pointing north (the red arrow) and on the other south (the blue arrow). The monopole-antimonopole is linearly confined in the bulk, but it is only logarithmically confined on the domain wall as a monopole becomes a vortex on the domain wall (pictured). The system can therefore undergo the BKT transition on the domain wall by percolating vortex monopoles. If only even monopoles are allowed in the bulk, correspondingly even vortices (winding modes) will be allowed on the domain wall.