Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge

TL;DR

This paper extends the classification of symmetry-protected topological (SPT) phases to higher dimensions by analyzing the anomalous symmetry actions on edges, linking obstructions to cohomology groups and enabling calculations for physical models.

Contribution

It generalizes the edge-based classification of SPT phases to higher dimensions using local symmetry representations and cohomology, applicable to both bosonic and fermionic systems.

Findings

Obstructions to local symmetry action classify SPT phases in higher dimensions.

The cohomology group H^{d+1}(G, U(1)) characterizes these obstructions.

Framework applies to models like non-linear sigma models with theta terms.

Abstract

It is well known that (1+1)-D bosonic symmetry-protected topological (SPT) phases with symmetry group $G$ can be identified by the projective representation of the symmetry at the edge. Here, we generalize this result to higher dimensions. We assume that the representation of the symmetry on the spatial edge of a ($d+1$)-D SPT is /local/ but not necessarily /on-site/, such that there is an obstruction to its implementation on a region with boundary. We show that such obstructions are classified by the cohomology group $H^{d+1}(G, U(1))$, in agreement with the classification of bosonic SPT phases proposed in [Chen et al, Science 338, 1604 (2012)]. Our analysis allows for a straightforward calculation of the element of $H^{d+1}(G, U(1))$ corresponding to physically meaningful models such as non-linear sigma models with a theta term in the action. SPT phases outside the classification of Chen et al are those in which the symmetry cannot be represented locally on the edge. With some modifications, our framework can also be applied to fermionic systems in (2+1)-D.