Abstract
We show that -dimensional topological phases of matter generically support loop excitations with topological degeneracy. The loops carry “Cheshire charge”: topological charge that is not the integral of a locally defined topological charge density. Cheshire charge has previously been discussed in non-Abelian gauge theories, but we show that it is a generic feature of all (3+1)-D topological phases (even those constructed from an Abelian gauge group). Indeed, Cheshire charge is closely related to nontrivial three-loop braiding. We use a dimensional reduction argument to compute the topological degeneracy of loop excitations in the -dimensional topological phases associated with Dijkgraaf-Witten gauge theories. We explicitly construct membrane operators associated with such excitations in soluble microscopic lattice models in Dijkgraaf-Witten phases and generalize this construction to arbitrary membrane-net models. We explain why these loop excitations are the objects in the braided fusion 2-category , thereby supporting the hypothesis that 2-categories are the correct mathematical framework for -dimensional topological phases.
- Received 21 April 2017
DOI:https://doi.org/10.1103/PhysRevB.96.045136
©2017 American Physical Society