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Symmetry protected topological (SPT) phases are well understood in thecontext of free fermions and in the context of interacting but essentiallybosonic models. Recently it has been realized that intrinsically fermionic SPTsexist which only appear in interacting models. Here we show that the 3+1dimensional realizations of these phases have surface states characterized by anew 't Hooft anomaly, captured by a $H^3(G, Z_2)$ class. This is encoded in theanomalous action of symmetry on the surface states with topological order,which must necessarily permute the anyons. We discuss in detail an example withsymmetry group $G = Z_2 \times Z_4$. Using a network model of the surface wederive a candidate surface topological order given by a $Z_4$ gauge theory. Werelate our findings to anomalies valued in $H^3$ with various coefficientsintroduced previously in both bosonic and fermionic settings, and describe ageneral framework that unifies these various anomalies.

Surface Topological Order and a new 't Hooft Anomaly of Interaction Enabled 3+1D Fermion SPTs