Condensed Matter Physics and Topological Field Theory
Symmetry protected topological(SPT) phase is a generalization of topological insulator(TI). Different from the intrinsic topological phase, e.g., the fractional quantum hall(FQH) phase, SPT phase is only distinguishable from a trivial disordered phase when certain symmetry is preserved. Indeed, SPT phase has a long history in 1D, and it has been shown that the well known Haldane phase of S=1 Heisenberg chain belongs to this class. However, in higher dimensions, most of the previous studies focus on free electron systems.
Unitary fusion categories are the algebraic input for the Turaev-Viro (TV) type TQFTs in (2+1)-dimensions and their Hamiltonian realization for the Levin-Wen model. We are interested in a generalization to unitary 2-fusion category for (3+1)-dimensions. Mackaay's spherical 2-categories are not general enough to include interesting examples such as the G-crossed braided fusion categories and general homotopy 2-types. We will discussion new (3+1)-TQFTs with G-crossed braided fusion categories as input, and their lattice realization based on the thesis work of Shawn X. Cui.