Supercohomology symmetry protected topological (SPT) phases and bosonic 2-group SPT phases in (3+1)D

Symmetry-protected topological (SPT) phases of matter are described by short-range entangled states. For each bosonic SPT phase described by the group cohomology, there is a fixed-point state that can be prepared by a finite depth quantum circuit (FDQC) built from the corresponding cohomology data. In this talk, a generalization for (3+1)D intrinsically interacting fermionic SPT phases, known as the supercohomology phases, will be introduced. The derivation of the FDQC utilizes a series of exact lattice dualities that relate bosonic SPT phases with a certain 2-group symmetry to supercohomology phases. A short overview is that gauging the 1-form symmetry of a bosonic model gives a Z2 lattice gauge theory, and bosonizing a fermionic model also gives a Z2 lattice gauge theory. These Z2 lattice gauge theories can match, and this constructs a duality between certain bosonic and fermionic models. The concepts of "gauging 1-form symmetry" and "bosonization (gauging fermion parity)" will be clearly explained. A primary result of this approach is that the “symmetry fractionalization” on fermion parity flux loops is immediate, which is the characteristic of supercohomology phases.