Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions

Abstract

We build exactly solvable lattice Hamiltonians for fermionic symmetry-protected topological (SPT) phases in (3+1)D classified by group supercohomology. A central benefit of our construction is that it produces an explicit finite-depth quantum circuit (FDQC) that prepares the ground state from an unentangled symmetric state. The FDQC allows us to clearly demonstrate the characteristic properties of supercohomology phases—namely, symmetry fractionalization on fermion parity flux loops—predicted by continuum formulations. By composing the corresponding FDQCs, we also recover the stacking relations of supercohomology phases. Furthermore, we derive topologically ordered gapped boundaries for the supercohomology models by extending the protecting symmetries, analogous to the construction of topologically ordered boundaries for bosonic SPT phases. Our approach relies heavily on dualities that relate certain bosonic 2-group SPT phases with supercohomology SPT phases. We develop physical motivation for the dualities in terms of explicit lattice prescriptions for gauging a 1-form symmetry and for condensing emergent fermions. We also comment on generalizations to supercohomology phases in higher dimensions and to fermionic SPT phases outside of the supercohomology framework.