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

TLDR: In this paper, the authors introduced a group super-cohomology theory for symmetry-protected topological (SPT) phases, which is a generalization of the standard group cohomology theory.

Abstract: Symmetry-protected topological (SPT) phases are gapped quantum phases with a symmetry, which can be smoothly connected to the trivial product states only if we break the symmetry. For a given symmetry, we can have many different SPT phases. But how to describe/construct those different SPT phases that can not be distinguished by their symmetry? It has been shown that different bosonic SPT phases in any dimensions and for any symmetry groups can be described/constructed using group cohomology theory of the symmetry group. In this paper, we introduce a group super-cohomology theory which is a generalization of the standard group cohomology theory. Using the group super-cohomology theory, we can describe/construct different interacting fermionic SPT phases, in any dimensions and for symmetry groups where the fermions form 1D representations of the symmetry group. Just like the boson case, our systematic construction is based on constructing discrete fermionic topological non-linear σ-models from the group super-cohomology theory. Our discrete fermionic topological non-linear σ-model, when defined on a space-time with boundary, can be viewed as a “non-local” boundary effective Lagrangian, which is a fermionic and discrete generalization of the bosonic continuous Wess-Zumino-Witten term. Thus we believe that the boundary excitations of a non-trivial SPT phase are gapless if the symmetry is not broken. As an application of this general result, we construct three non-trivial SPT phases in 3D, for interacting fermionic superconductors with coplanar spin order (which have T 2 = 1 time-reversal Z 2 and fermion-number-parity Z 2 symmetries described by a full symmetry group Z T 2 × Z 2 ). We also construct three interacting fermionic SPT phases in 2D with a full symmetry group Z2×Z 2 . Those 2D fermionic SPT phases all have central-charge c = 1 gapless edge excitations, if the symmetry is not broken.