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Symmetry-protected topological (SPT) phases are gapped short-range-entangledquantum phases with a symmetry $G$, which can all be smoothly connected to thetrivial product states if we break the symmetry. It has been shown that a largeclass of interacting bosonic SPT phases can be systematically described bygroup cohomology theory. In this paper, we introduce a (special) groupsupercohomology theory which is a generalization of the standard groupcohomology theory. We show that a large class of short-range interactingfermionic SPT phases can be described by the group supercohomology theory.Using the data of super cocycles, we can obtain the ideal ground state wavefunction for the corresponding fermionic SPT phase. We can also obtain the bulkHamiltonian that realizes the SPT phase, as well as the anomalous (ie,non-on-site) symmetry for the boundary effective Hamiltonian. The anomaloussymmetry on the boundary implies that the symmetric} boundary must be gaplessfor 1+1D boundary, and must be gapless or topologically ordered beyond 1+1D. Asan application of this general result, we construct a new SPT phase in 3D, forinteracting fermionic superconductors with coplanar spin order (which have$T^2=1$ time-reversal $Z_2^T$ and fermion-number parity $Z_2^f$ symmetriesdescribed by a full symmetry group $Z_2^T\times Z_2^f$). Such a fermionic SPTstate can neither be realized by free fermions nor by interacting bosons(formed by fermion-pairs), and thus are not included in the K-theoryclassification for free fermions or group cohomology description forinteracting bosons. We also construct three interacting fermionic SPT phases in2D with a full symmetry group $Z_2\times Z_2^f$. Those 2D fermionic SPT phasesall have central-charge $c=1$ gapless edge excitations, if the symmetry is notbroken.

physical review. b, condensed matter and materials physics

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