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

It is well known that $\left(1+1\right)$-dimensional $\left[\right(1+1\left)\text{-D}\right]$ 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 $\left(d+1\right)$-D SPT is local but not necessarily on site, such that there is an obstruction to its implementation on a region with a boundary. We show that such obstructions are classified by the cohomology group $H^{d+1}(G,\text{U}\left(1\right))$, 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 $H^{d+1}(G,\text{U}\left(1\right))$ corresponding to physically meaningful models such as nonlinear $\sigma$ 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 $\left(2+1\right)$-D.

Figure 1

Obtaining a 2-cocycle on the $\left(0+1\right)$-D edge of a $\left(1+1\right)$-D SPT.

Figure 2

Obtaining a 3-cocycle on the $\left(1+1\right)$-D edge of a $\left(2+1\right)$-D system.

Figure 3

The reduction process to obtain a 4-cocycle $\omega$ on the $\left(2+1\right)$-D edge of a $\left(3+1\right)$-D system, assuming a symmetry representation on the edge of the form Eq. (8).

Figure 4

The regions $A$ and $A^{\prime }$ on which we can prove that the anomalous symmetry on the boundary is classified by the same element of $H^3(G,\mathrm{U}\left(1\right))$. The orientations of the boundaries $\partial A$ and $\partial A^{\prime }$ are depicted with arrows.