Cheshire charge in (3+1)-dimensional topological phases

We show that $(3+1)$-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 $(3+1)$-dimensional topological phases associated with Dijkgraaf-Witten gauge theories. We explicitly construct membrane operators associated with such excitations in soluble microscopic lattice models in ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ 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 $Z\left(\mathbf{2}{\mathbf{Vect}}_G^{\omega }\right)$, thereby supporting the hypothesis that 2-categories are the correct mathematical framework for $(3+1)$-dimensional topological phases.

Figure 1

Processes A, B, and C must have the same amplitude. Here the red loop is a flux loop and the black loop is a Cheshire charge loop.

Figure 2

Dimensional reduction of a loop excitation carrying flux $x$ in (3+1)-D. (a) Side view. (b) Dimensionally reduced picture. The small dimension in (a) is compactified (opposite surfaces are identified). The flux threaded through this compact dimension can be detected by moving a particle along a cycle (red lines). The two red lines in (a) differ by a braid around the $x$ flux, so they measure a flux differing by $x$.

Figure 3

The product $g_1{g}_2{g}_3$ around a threefold intersection between membranes must be 1.

Figure 4

The pentagon move relates two different membrane-net configurations (shown with black and red connecting arrows respectively).

Figure 5

The consistency condition for the membrane operator, in the two different graphical representations described in the text.

Figure 6

The hexagon move.

Figure 7

In panels (a) and (b), each of the two membrane-net configurations equated by the consistency condition Fig. 5 (shown with red and green arrows) is related to the same membrane-net configuration shown with gray arrows by a sequence of pentagon and hexagon moves.