Classification of symmetry-protected phases for interacting fermions in two dimensions

Recently, it has been established that two-dimensional bosonic symmetry-protected topological (SPT) phases with on-site unitary symmetry $G$ can be completely classified by the group cohomology $H^3(G,\mathrm{U}\left(1\right))$. Later, group supercohomology was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the algebraic theory of symmetry and defects in two-dimensional topological phases. We reproduce the partial classifications given by group supercohomology, and we also show that with an additional $H^1(G,{\mathbb{Z}}_2)$ structure, a complete classification of SPT phases for two-dimensional interacting fermion systems with a total symmetry group $G\times {\mathbb{Z}}_2^f$ is obtained. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.

Figure 1

Heptagon equations which ensure consistency between $G$-crossed braiding and fusion of defects.

Figure 2

The Pentagon equation enforces the consistency between different sequences of $F$ moves starting and ending with the same fusion trees.