Composing topological domain walls and anyon mobility

TL;DR

The paper develops an enriched fusion-category framework ${ ext{UFC}}^{ ext{A}}$ to describe (2+1)D TOs with anomalies, and introduces domain-wall tunneling operators as a concrete tool to analyze anyon mobility and the decomposition of parallel walls into indecomposable sectors. By situating domain walls as ${ ext{A}}$-enriched bimodules and employing relative Deligne products, condensation algebras, and their centers, the authors connect Witt equivalences, wall excitations, and lattice-model realizations (Walker–Wang) in a unified 3-category picture. They prove that composite walls generically decompose into multiple sectors, each described by minimal central projections in a convolution algebra, and demonstrate how tunneling channels identify the sector-specific wall type via explicit XYX examples (toric code, doubled Ising, TY3-, and dihedral groups). The results illuminate how anyon condensation, wall condensation, and domain-wall composition interplay to determine ground-state degeneracies and defect structures, with potential applications to quantum information and higher-categorical TOs. The work offers a scalable path toward analyzing more general boundaries and extended defects using fusion 2-categories and beyond, linking bulk, wall, and boundary data through enriched centers and Witt equivalences.

Abstract

Topological domain walls separating 2+1 dimensional topologically ordered phases can be understood in terms of Witt equivalences between the UMTCs describing anyons in the bulk topological orders. However, this picture does not provide a framework for decomposing stacks of multiple domain walls into superselection sectors - i.e., into fundamental domain wall types that cannot be mixed by any local operators. Such a decomposition can be understood using an alternate framework in the case that the topological order is anomaly-free, in the sense that it can be realized by a commuting projector lattice model. By placing these Witt equivalences in the context of a 3-category of potentially anomalous (2+1)D topological orders, we develop a framework for computing the decomposition of parallel topological domain walls into indecomposable superselection sectors, extending the previous understanding to topological orders with non-trivial anomaly. We characterize the superselection sectors in terms of domain wall particle mobility, which we formalize in terms of tunnelling operators. The mathematical model for the 3-category of topological orders is the 3-category of fusion categories enriched over a fixed unitary modular tensor category.

Figures (8)

• Figure 1: Standard description of topological order in terms of localized excitations, cf. MR2942952MR3246855; contrast with Figures \ref{['fig:standardDescAnomalyFree']} and \ref{['fig:enrichedDesc']} below. A Witt equivalence MR3039775${\mathcal{C}}\to{\mathcal{D}}$ is a unitary multifusion category ${\mathcal{X}}$ with a choice of braided equivalence $Z({\mathcal{X}})\cong{\mathcal{C}}\boxtimes\overline{{\mathcal{D}}}$; see § \ref{['ssec:EnrichedBimodules']}.
• Figure 2: Description of anomaly free topological order in terms of ingredients for commuting projector model, cf. MR29429521912.01760; compare with Figure \ref{['fig:enrichedDesc']} below.
• Figure 3: Description of topological order in terms of ingredients for commuting projector model afforded by ${\mathcal{A}}$-enriched fusion categories.
• Figure 4: Glossary for details of boundary for Walker-Wang models and algebraic higher categorical structure from $\mathsf{UFC}^{\mathcal{A}}$, expanding on MR2942952
• Figure 5: Since $Z({\mathcal{X}})\cong Z^{\mathcal{A}}({\mathcal{X}})\boxtimes {\mathcal{A}}$, attaching an ${\mathcal{A}}$-Walker-Wang bulk to ${\mathcal{X}}$ trivializes the ${\mathcal{A}}$-layer of topological order, leaving only $Z^{\mathcal{A}}({\mathcal{X}})$.

Theorems & Definitions (47)

• Example 2.2
• Example 2.3
• Remark 2.4
• Remark 2.5
• Definition 2.7
• Example 2.9
• Remark 2.10
• Lemma 2.12
• proof
• Proposition 2.13