Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly

The interplay between symmetry and topological properties plays a very important role in modern physics. In the past decade, the concept of symmetry-enriched topological (SET) phases was proposed and their classifications have been systematically studied for bosonic systems. Very recently, the concept of SET phases has been generalized into fermionic systems and their corresponding classification schemes are also proposed. Nevertheless, how to realize all these fermionic SET (fSET) phases in lattice models remains to be a difficult open problem. In this paper, we first construct exactly solvable models for non-anomalous non-chiral 2+1D fSET phases, namely, the symmetry-enriched fermionic string-net models, which are described by commuting-projector Hamiltonians whose ground states are the fixed-point wavefunctions of each fSET phase. Mathematically, we provide a partial definition to $G$-graded super fusion category, which is the input data of a symmetry-enriched fermionic string-net model. Next, we construct exactly solvable models for non-chiral 2+1D fSET phases with 't Hooft anomaly, especially the $H^3(G,\mathbb{Z}_2)$ fermionic 't Hooft anomaly which is different from the well known bosonic $H^4(G,U(1)_T)$ anomaly. In our construction, this $H^3(G,\mathbb{Z}_2)$ fermionic 't Hooft anomaly is characterized by a violation of fermion-parity conservation in some of the surface ${F}$-moves (a kind of renormalization moves for the ground state wavefunctions of surface SET phases), and also by a new fermionic obstruction $\Theta$ in the surface pentagon equation. We demonstrate this construction in a concrete example that the surface topological order is a $\mathbb{Z}_4$ gauge theory embedded into a fermion system and the total symmetry $G^f=\mathbb{Z}_2^f\times\mathbb{Z}_2\times\mathbb{Z}_4$.