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

Dominic V. Else, Chetan Nayak

TL;DR

This work demonstrates that Cheshire charge is a generic feature of (3+1)-D topological phases, intimately tied to nontrivial three-loop braiding. By combining dimensional reduction, exactly solvable lattice constructions, and membrane-net generalizations, the authors classify and realize loop-like excitations carrying Cheshire charge in Dijkgraaf-Witten theories, deriving their topological degeneracies from gapped boundaries of (2+1)-D phases and SPT data. The results connect loop degeneracy and three-loop braiding to a higher-categorical framework, showing that excitations in (3+1)-D phases live as objects in braided fusion 2-categories, specifically $Z(\mathbf{2Vect}_G^\omega)$. This establishes a concrete bridge between physical loop excitations, their fusion/braiding structure, and an abstract category-theoretical language, with implications for classification and potential quantum information applications. Overall, the paper provides both explicit constructions and a unifying mathematical viewpoint for Cheshire charge and higher-dimensional topological order.

Abstract

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 non-trivial 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 $Z(\mathbf{2Vect}_G^ω)$, thereby supporting the hypothesis that 2-categories are the correct mathematical framework for (3+1)-dimensional topological phases.

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

TL;DR

This work demonstrates that Cheshire charge is a generic feature of (3+1)-D topological phases, intimately tied to nontrivial three-loop braiding. By combining dimensional reduction, exactly solvable lattice constructions, and membrane-net generalizations, the authors classify and realize loop-like excitations carrying Cheshire charge in Dijkgraaf-Witten theories, deriving their topological degeneracies from gapped boundaries of (2+1)-D phases and SPT data. The results connect loop degeneracy and three-loop braiding to a higher-categorical framework, showing that excitations in (3+1)-D phases live as objects in braided fusion 2-categories, specifically . This establishes a concrete bridge between physical loop excitations, their fusion/braiding structure, and an abstract category-theoretical language, with implications for classification and potential quantum information applications. Overall, the paper provides both explicit constructions and a unifying mathematical viewpoint for Cheshire charge and higher-dimensional topological order.

Abstract

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 non-trivial 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 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 (3+1)-dimensional topological phases.

Paper Structure

This paper contains 17 sections, 2 theorems, 31 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Cheshire charge as a "non-Abelian" fusion rule
  3. Cheshire charge in the 3D toric code
  4. Cheshire charge in more general models and Three-Loop Braiding
  5. Cheshire charge in Dijkgraaf-Witten theories
  6. Enumerating the types of loop-like excitations
  7. General Result
  8. Example: $G = Z_2 \times Z_2$
  9. Constructing the loop excitations explicitly in a (3+1)-D lattice model
  10. More general constructions: membrane-net models
  11. Category-theoretical viewpoint
  12. The flux-(SPT boundary) correspondence in a (3+1)-D Dijkgraaf-Witten theory
  13. Dimensional reduction: understanding loop excitations in terms of gapped boundaries between (2+1)-D topological phases
  14. Equivalence of two forms of data characterizing loop excitations
  15. Membrane-net models
  16. ...and 2 more sections

Key Result

Lemma 1

(a) Suppose that $g \in G$ satisfies $\chi(g) = 1$ for all $\chi \in G^{*}$. Then $g = 1$. (b) Suppose that $g \in G$ satisfies $\chi(g) = 1$ for all $\chi \in C$. Then $g \in H$.

Figures (7)

  • 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. 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 three-fold 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.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof