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Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $σ$ models and a special group supercohomology theory

Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of super cocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (ie, non-on-site) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric} boundary must be gapless for 1+1D boundary, and must be gapless or topologically ordered beyond 1+1D. As an application of this general result, we construct a new SPT phase in 3D, for interacting fermionic superconductors with coplanar spin order (which have $T^2=1$ time-reversal $Z_2^T$ and fermion-number parity $Z_2^f$ symmetries described by a full symmetry group $Z_2^T\times Z_2^f$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion-pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in 2D with a full symmetry group $Z_2\times Z_2^f$. Those 2D fermionic SPT phases all have central-charge $c=1$ gapless edge excitations, if the symmetry is not broken.