Zheng-Cheng Gu, Xiao-Gang Wen
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. In this paper, we introduce a (special) group super-cohomology theory which is a generalization of the standard group cohomology theory. We show that interacting fermionic SPT phases can be described by the group super-cohomology theory. Or more precisely, in any dimensions and where the fermions form 1D representations of the symmetry group in the fixed point wavefunctions, we can describe/construct interacting fermionic SPT phases using the cocycles in the (special) group super-cohomology theory. We further prove that the (special) group super-cohomology classes also has an Abelian group structure, which describes the stacking operation of the fermionic SPT phases. 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 realize the SPT phase, as well as the low energy e?ective Hamiltonian for the boundary excitations. As an application of this general result, we construct three new SPT phases in 3D, for interacting fermionic superconductors with coplanar spin order (which have T^2 = 1 time-reversal Z2T and fermion-number-parity Z2f symmetries described by a full symmetry group Z2T*Z2f). Those three Z2T*Z2f fermionic SPT states cannot be realized by free fermions, and thus are not included in the K-theory classification. We also construct three interacting fermionic SPT phases in 2D with a full symmetry group Z2*Z2f . Those 2D fermionic SPT phases all have central-charge c = 1 gapless edge excitations, if the symmetry is not broken.
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http://arxiv.org/abs/1201.2648
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