Bulk-to-boundary anyon fusion from microscopic models

TL;DR

This work develops a constructive framework for bulk-to-boundary fusion of anyons in 2+1D non-chiral topological orders, formulating a bimodule approach based on tube and semi-tube algebras to classify bulk and boundary excitations. By diagonalizing twisted group algebras with action and employing the folding trick, the authors derive a closed formula for bulk-to-boundary fusion multiplicities in twisted gauge theories and compute Lagrangian algebra objects through condensation data in several gauge-theory models, including Vec$^oldsymbol{ω}(G)$ with $G= ext{Z}_N, ext{Z}_N imes ext{Z}_M$, and $S_3$. The paper also treats domain walls and islands, providing explicit tunneling tables and detailing how Abelian twists can host non-Abelian islands that enable non-Clifford operations within Abelian codes. The results bridge microscopic fixed-point lattice models with macroscopic anyon condensation and Lagrangian algebras, with direct implications for topological quantum error correction, lattice surgery, and potential generalizations to higher dimensions. Overall, the work offers a practical, algebraically explicit toolkit for analyzing defects, boundaries, and domain walls in twisted gauge theories and their computational applications.

Abstract

Topological quantum error correction based on the manipulation of the anyonic defects constitutes one of the most promising frameworks towards realizing fault-tolerant quantum devices. Hence, it is crucial to understand how these defects interact with external defects such as boundaries or domain walls. Motivated by this line of thought, in this work, we study the fusion events between anyons in the bulk and at the boundary in fixed-point models of 2+1-dimensional non-chiral topological order defined by arbitrary fusion categories. Our construction uses generalized tube algebra techniques to construct a bi-representation of bulk and boundary defects. We explicitly derive a formula to calculate the fusion multiplicities of a bulk-to-boundary fusion event for twisted quantum double models and calculate some exemplary fusion events for Abelian models and the (twisted) quantum double model of S3, the simplest non-Abelian group-theoretical model. Moreover, we use the folding trick to study the anyonic behavior at non-trivial domain walls between twisted S3 and twisted Z2 as well as Z3 models. A recurring theme in our construction is an isomorphism relating twisted cohomology groups to untwisted ones. The results of this work can directly be applied to study logical operators in two-dimensional topological error correcting codes with boundaries described by a twisted gauge theory of a finite group.

Figures (6)

• Figure 1: Different sequences of $F$-moves evaluating to the same transformation on of the graph have to compose to the same map on the associated vector spaces. In particular, the above diagram has to commute, i.e., the composite map corresponding to the top path has to evaluate to the same as composing the maps corresponding to the bottom path. The resulting condition on the $F$-symbols is called pentagon equation.
• Figure 2: Different sequences of $L$- and $F$-moves evaluating to the same transformation on of the graph have to compose to the same map on the associated vector spaces. In particular, the above diagram has to commute, i.e., the composite map corresponding to the top path has to evaluate to the same as composing the maps corresponding to the bottom path. The resulting condition on the $L$- and $F$-symbols is called boundary pentagon equation.
• Figure 3: Via the folding trick, a $\mathcal{C}$-$\mathcal{C}'$ domain wall, classified by a $\mathcal{C}$-$\mathcal{C}'$bimodule, is equivalent to a $\mathcal{C}\otimes \mathcal{C}'$ module, defined by $L$ symbols of the above form. In Eq. \ref{['eq:folding_Lsymbol']} we give the equation relating the data defining the bimodule to the boundary data after the fold.
• Figure 4: The multiplication of two tube algebra basis elements $(a,b,c,d)_T$ and $(a',b',c',d')_T$ is defined via gluing the two associated string diagrams together (left) and using $F$-moves to reduce it to the cellulation on the right. The phase acquired by the sequence of moves can be derived by evaluating the space-time complex that maps the two dual triangulations to each other, which is composed of three tetrahedra (middle). Note that the front- and the back-side edges of the space-time complex above are identified.
• Figure 5: The multiplication of two basis elements in $S$ is defined via gluing the two associated semi-tube string diagrams together (left) and using $L$ (and $F$) moves to reduce it to the cellulation on the right. The associated phase can be derived by considering the space-time complex that maps from the initial to the final cellulation.

Theorems & Definitions (15)

• Definition 1: (left) $G$-module
• Remark
• Definition 2: $n$-cochain
• Definition 3: (twisted) $n$-coboundary
• Remark
• Definition 4: (twisted) cohomology groups
• Remark
• Definition 5: slant product
• Remark
• Example B.1: $\mathbb{Z}_N$