Instanton effects in lattice models of bosonic symmetry-protected topological states

Bosonic symmetry-protected topological (SPT) states are gapped disordered phases of matter possessing symmetry-preserving boundary excitations. It has been proposed that, at long wavelengths, the universal properties of an SPT system are captured by an effective nonlinear sigma model field theory in the presence of a quantized topological $\theta$ term. By studying lattice models of bosonic SPT states, we are able to identify, in their Euclidean path integral formulation, (discrete) Berry phases that hold relevant physical information on the nature of the SPT ground states. These discrete Berry phases are given intuitive physical interpretation in terms of instanton effects that capture the presence of a $\theta$ term on the microscopic scale.

Figure 1

Figure 1 depicts the slicing of the partition function into $M$ intervals, as described by Eq. (2.7a). At each time slice, we have an instantaneous representation of the $D$-dimensional SPT system, which, without boundaries, is generically represented by a circle. Figure 1 depicts the slicing of $Z_{\mathrm{SPT}}\left[\sigma \left({\tau }_k\right),\sigma \left({\tau }_{k+1}\right)\right]$ into $N$ subintervals, as described by Eq. (2.10).

Figure 2

Figures 2 and 2 capture the fluctuation of the middle spin giving a phase $e^{\mathrm{i}{\mathcal{S}}_j}=-1$ [Eq. (3.22)], as the configuration of nearest neighbor spins has $D_j=0$. Figures 2 and 2 capture the fluctuation of the middle spin giving a phase $e^{\mathrm{i}{\mathcal{S}}_j}=+1$ [Eq. (3.22)], as the configuration of nearest neighbor spins has $D_j=2$.