Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter

TL;DR

The paper presents an exactly solvable lattice Hamiltonian framework for gapped boundaries in Kitaev's quantum double/Dijkgraaf-Witten theories and develops a parallel algebraic model based on modular tensor categories and Lagrangian algebras. It provides a detailed boundary construction via subgroup data, boundary ribbon operators, and a bulk-to-boundary condensation formalism, including a Fourier-style basis that classifies boundary excitations as (T,R) pairs with multiplicities. The authors introduce M-3j and M-6j symbols to encode associativity of bulk-to-boundary condensation and demonstrate these ideas through the explicit example of D(S3), deriving multiple gapped boundary types and condensation channels. The algebraic side recasts gapped boundaries as indecomposable module categories and relates boundary excitations to Fun_C(M,M), establishing Morita-equivalence with Rep(G) and connecting to the Drinfeld center framework. Overall, the work unifies solvable Hamiltonian constructions with categorical data to fully describe gapped boundaries and their excitations in DW theories, with potential implications for topological quantum computation and domain-wall physics.

Abstract

We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries.

Figures (14)

• Figure 1.1: Pictorial summary of the Hamiltonian realization and algebraic model of gapped boundaries. In this picture, the hole $\mathfrak{h}$ is the inner boldfaced and blue rectangle.
• Figure 2.1: Lattice for the Kitaev model. For simplicity of illustration and calculation, we use a square lattice, but in general, one can use an arbitrary lattice. If the group $G$ is non-abelian, it is necessary to define orientations on edges, as we have shown here. The edges $j$ and $j_1,...j_m$, used to obtain $A^g(v)$ and $B^h(p)$, are illustrated for this example of $v,p$.
• Figure 2.2: Illustration of Definitions \ref{['cilium-def']}-\ref{['ribbon-def']}. The direct lattice is shown as before, and the dual lattice is shown in dotted lines. $s=(v,p)$ is a cilium. $\tau$ is a dual triangle, and $\tau'$ is a direct triangle. $\rho = \rho_1 \rho_2$ is a composite ribbon, formed by gluing the last site of $\rho_1$ to the first site of $\rho_2$. $\rho$ is an open ribbon, and $\sigma$ is a closed ribbon.
• Figure 2.3: Lattice for the Kitaev model with boundary. For any fixed group $G$, there can be multiple ways to define projection operators at the boundary such that all terms in the new Hamiltonian still commute. These are studied in Section \ref{['sec:bd-hamiltonian']}.
• Figure 2.4: Example: defining the Hamiltonian (\ref{['eq:gapped-bds-hamiltonian']}), in the case of two holes on an infinite lattice. The new Hamiltonians $H^{(K_1,1)}_{(G,1)}$ (resp., $H^{(K_2,1)}_{(G,1)}$) are applied to all edges within and along the blue (red) shaded rectangle.

Theorems & Definitions (36)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• proof
• Remark 2.7
• Remark 2.8
• Remark 2.9