Hamiltonian models for topological phases of matter in three spatial dimensions

TL;DR

This paper develops Hamiltonian models for (3+1)D topological phases of matter using unitary G-crossed braided fusion categories, unifying various known topological quantum field theories and introducing new realizations.

Contribution

It provides commuting projector Hamiltonian constructions for a broad class of 3D topological models based on advanced mathematical structures, connecting and extending existing theories.

Findings

Hamiltonian models for 3D topological phases based on G-crossed categories

Unification of known topological quantum field theories within a Hamiltonian framework

New Hamiltonian realization of Kashaev's state sum TQFT

Abstract

We present commuting projector Hamiltonian realizations of a large class of (3+1)D topological models based on mathematical objects called unitary G-crossed braided fusion categories. This construction comes with a wealth of examples from the literature of symmetry-enriched topological phases. The spacetime counterparts to our Hamiltonians are unitary state sum topological quantum fields theories (TQFTs) that appear to capture all known constructions in the literature, including the Crane-Yetter-Walker-Wang and 2-Group gauge theory models. We also present Hamiltonian realizations of a state sum TQFT recently constructed by Kashaev whose relation to existing models was previously unknown. We argue that this TQFT is captured as a special case of the Crane-Yetter-Walker-Wang model, with a premodular input category in some instances.