Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory

Abstract

The classification and construction of symmetry-protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that (generalized) group cohomology theory or cobordism theory gives rise to a complete classification of SPT phases in interacting boson or spin systems. The construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in three dimensions. In this work, we revisit this problem based on an equivalence class of fermionic symmetric local unitary transformations. We construct very general fixed-point SPT wave functions for interacting fermion systems. We naturally reproduce the partial classifications given by special group supercohomology theory, and we show that with an additional B[over ˜]H^{2}(G_{b},Z_{2}) structure [the so-called obstruction-free subgroup of H^{2}(G_{b},Z_{2})], a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group G_{f}=G_{b}×Z_{2}^{f} can be obtained for unitary symmetry group G_{b}. We also discuss the procedure for deriving a general group supercohomology theory in arbitrary dimensions.