Field theory of symmetry-protected valence bond solid states in (2+1) dimensions

This paper describes a semiclassical field-theory approach to the topological properties of spatially featureless Affleck-Kennedy-Lieb-Tasaki type valence bond solid ground states of antiferromagnets in spatial dimensions one to three. Using nonlinear sigma models set in the appropriate target manifold and augmented with topological terms, we argue that the path integral representation of the ground-state wave functional can correctly distinguish symmetry-protected topological ground states from topologically trivial ones. The symmetry-protection feature is demonstrated explicitly in terms of a dual field theory, where we take into account the nontrivial spatial structure of topological excitations, which are caused by competition among the relevant ordering tendencies. A temporal surface contribution to the action originating from the bulk topological term plays a central role in our study. We discuss how the same term governs the behavior of the so-called strange correlator. In particular, we find that the path integral expression for the strange correlator in two dimensions reduces to the well-known Haldane expression for the two point spin correlator of antiferromagnetic spin chains.

Figure 1

(a) The rewriting, following Ref. [40], of the staggered summation over planar spin Berry phases (3) into the net space-time vorticity (4). The directions of the vertical arrows on the temporal links account for the staggering of the Berry phases. Each arrow contributes to the action the piece $iS{(-1)}^j{\mathrm{\Delta }}_{\tau }\varphi$. The auxiliary horizontal arrows on the spatial links stand for $\pm iS{\mathrm{\Delta }}_x\varphi$. (b) An illustration depicting how space-time vortices induce a discontinuous jump (i.e., a phase slip) in the winding number (9) associated with the snapshot configuration.

Figure 2

The spatial distribution of the weight $Y_{\overline{j}}$ entering [via Eq. (13)] the Berry phases associated with monopoles residing on the dual sites $\overline{j}$ (i.e., the center of the plaquettes of the direct lattice). Each plaquette is assigned a sublattice index (A, B, C, and D) coinciding with that of the vertex to the southwest of the center (dashed lines).

Figure 3

The modified topological charge $Q_{\tau x}$ for the case $n_4={\delta }_4$. The sphere shown schematically is $S^3$. The topological charge is given by the shaded area $(={\int }_0^{arccos\delta }4\pi {sin}^2\theta d\theta )$ divided by $\mathrm{vol}\left(S^3\right)=2{\pi }^2$.

Figure 4

A top view (from the $+z$ direction) of a staggered array of (1+1)D O(4) $\text{NL}\sigma$ models with a WZ term, each running in the $z$ direction. In a precise analogy with the 2D counterpart discussed in Sec. 3, the staggered summation over the WZ terms is converted into a collection of monopole Berry phases. As in the 2D case, the weights ${\zeta }_{\overline{j}}=0,1/4,1/2,3/4$ are assigned to the monopoles occurring at the dual sites $\left\{\overline{j}\right\}$.

Figure 5

Thick orange lines depict an array of (1+1)D O(4) $\text{NL}\sigma$ models with WZ terms at level $k_{\mathrm{eff}}=4S/3$, which are placed in the background of a prefixed ${\text{VBS}}_z$ order (green lines)—see text for details. Condensation of monopoles results in a 3D AKLT-like state on the cubic lattice.