Gapped Boundaries of Kitaev's Quantum Double Models: A Lattice Realization of Anyon Condensation from Lagrangian Algebras

TL;DR

This work addresses how to realize all gapped boundaries of Kitaev's quantum double models directly from Lagrangian algebras in $Z_1(\mathrm{Vec}_G)$, providing a microscopic lattice construction aligned with the macroscopic anyon-condensation framework. By developing two complementary pictures for bulk anyons (anyon-creating and anyon-probing) and deriving boundary consistency conditions, the authors produce explicit boundary terms and classify gapped boundaries into three constructive families, including Abelian and spontaneous-symmetry-breaking types, with a conjectural extension to non-Abelian cases. They establish a robust operator-state dictionary via $S$ and $T$ transformations, connect boundary terms to bulk-to-boundary condensation, and demonstrate the framework on $\mathbb{Z}_2$, $\mathbb{Z}_2\times\mathbb{Z}_2$, and $S_3$ quantum doubles, recovering known boundary structures and providing new lattice realizations. The results have broad implications for topological quantum codes, domain walls, and coarse-grained understanding of boundary phase transitions, and they offer a path to generalizations to extended string-net models and higher dimensions through Lagrangian algebra data.

Abstract

The macroscopic theory of anyon condensation, rooted in the categorical structure of topological excitations, provides a complete classification of gapped boundaries in topologically ordered systems, where distinct boundaries correspond to the condensation of different Lagrangian algebras. However, an intrinsic and direct understanding of anyon condensation in lattice models, grounded in the framework of Lagrangian algebras, remains undeveloped. In this paper, we propose a systematic framework for constructing all gapped boundaries of Kitaev's quantum double models directly from the data of Lagrangian algebras. Central to our approach is the observation that bulk interactions in the quantum double models admit two complementary interpretations: the anyon-creating picture and the anyon-probing picture. Generalizing this insight to the boundary, we derive the consistency condition for boundary ribbon operators that respect the mathematical axiomatic structure of Lagrangian algebras. Solving these conditions yields explicit expressions for the local boundary interactions required to realize gapped boundaries. We also provide three families of solutions that cover a broad range of cases. Our construction provides a microscopic characterization of the bulk-to-boundary anyon condensation dynamics via the action of ribbon operators. Moreover, all these boundary terms are supported within a common effective Hilbert space, making further studies on pure boundary phase transitions natural and convenient. Given the broad applicability of anyon condensation theory, we believe that our approach can be generalized to planar topological codes, extended string-net models, or higher-dimensional topologically ordered systems.

Figures (19)

• Figure 1: Kitaev's quantum double models defined on a honeycomb lattice.
• Figure 2: In the quantum double model, a site in the bulk is defined as a combination of a plaquette and an adjacent vertex. The illustrated site consists of the green plaquette and the blue vertex.
• Figure 3: Two distinct sites may share a common vertex or plaquette: (a) Two sites sharing the common green plaquette; (b) Two sites sharing the common blue vertex.
• Figure 4: The action of the flux ribbon operator must terminate at the vertex part of the respective sites at the endpoints of the path. In these two illustrated examples, the endpoint of the operator $\hat{Z}^h$ acts exclusively on the edges parallel to the orange line. The number of edges at the endpoint of $\hat{Z}^h$ varies depending on the position of the vertex component of the end site.
• Figure 5: A non-local operator $\hat{M}^{[C,R]}_{nq,\infty}$ creates a topological excitation $\ket{[C,R];nq}$ at its ending site. The purple vertices and plaquettes represent local eigenstates of the vertex operators and plaquette operators with eigenvalues equal to $1$, respectively.

Theorems & Definitions (14)

• Theorem 5.1: Boundary theory of 2+1D topological order
• Definition 5.1
• Theorem 5.2: Anyon condensation in 2+1D
• Theorem 5.3: Classification of Lagrangian algebras in $\text{Z}_1(\mathrm{Vec}_G)$
• Remark 5.1
• Remark 5.2
• Definition 5.2: Alternating Bicharacters
• Theorem 5.6: Isomorphism Between Bicharacters and 2-Cohomology
• Conjecture 5.1
• Remark 5.3