The Quantum Double Model with Boundary: Condensations and Symmetries

TL;DR

This work extends Kitaev's quantum double model by introducing boundaries and domain walls, and it provides a complete algebraic framework to classify boundary condensations through subgroups and 2-cocycles. By folding domain walls to boundaries, it derives conditions under which two quantum doubles $\mathcal{Z}(G)$ and $\mathcal{Z}(G')$ are equivalent and demonstrates a nontrivial auto-equivalence for $\mathcal{Z}(\mathbf{F}_{q}^{+}\rtimes \mathbf{F}_q^{\times})$, including the $S_3$ case. The approach ties the representation theory of the Drinfeld double $D(G)$ to edge physics, showing how ribbon operators and boundary algebras govern tunneling, condensation, and symmetries of anyons. It also situates condensations within Davydov’s classification of algebras in modular categories and discusses near-field structures as a source of nontrivial symmetries. The results open paths for extending boundary constructions to more general settings and for deeper exploration of domain walls in topological quantum computation contexts.

Abstract

Associated to every finite group, Kitaev has defined the quantum double model for every orientable surface without boundary. In this paper, we define boundaries for this model and characterize condensations; that is, we find all quasi-particle excitations (anyons) which disappear when they move to the boundary. We then consider two phases of the quantum double model corresponding to two groups with a domain wall between them, and study the tunneling of anyons from one phase to the other. Using this framework we discuss the necessary and sufficient conditions when two different groups give the same anyon types. As an application we show that in the quantum double model for S_3 (the permutation group over three letters) there is a chargeon and a fluxion which are not distinguishable. This group is indeed a special case of groups of the form of the semidirect product of the additive and multiplicative groups of a finite field, for all of which we prove a similar symmetry.

Figures (7)

• Figure 1: A planer square lattice with directed edges. The direction of an edge can be reversed by changing the corresponding state according to the right figure. The pair of adjacent vertex $v_0$ and face $f_0$ consist a site. This site $s_0=(v_0, f_0)$ is depicted as a dotted line. A ribbon connecting two sites $s_0$ and $s_1$ is also shown. The corresponding ribbon operator acts on bold edges and is defined in Figure \ref{['fig:ribbon']}.
• Figure 2: Definition of operators $A_s^{g}$ and $B_s^{h}$. Note that if $h$ is in the center of $G$, $B_s^h$ depends only on the face $f$ and not vertex $v$.
• Figure 3: Definition of the ribbon operator $F_{\xi}^{h,g}$. The ribbon $\xi$ connects the starting site $s_0$ to the ending site $s_1$.
• Figure 4: A planar lattice with boundary. A ribbon connects the boundary site $s_0=(v_0, f_0)$ to the internal site $s_1=(v_1, f_1)$. Here, operators $A_{s_0}^{k}$ and $B_{s_0}^k$ are defined the same as before: $A_{s_0}^k \vert x, y, z\rangle = \vert kx, ky, kz\rangle$ and $B_{s_0}^k \vert x\rangle = \delta_{k, x}\vert x\rangle$.
• Figure 5: A lattice with boundary in which every other boundary edge is marked by a dotted line. Here we fix a boundary site $s_0=(v_0, f_0)$ where $f_0$ is adjacent to a solid boundary edge.

Theorems & Definitions (17)

• Lemma 3.1
• Remark 3.1
• Proposition 4.1
• Remark 4.1
• Lemma 5.1
• Remark 5.1
• Remark 5.2
• Proposition 5.1
• Proposition 5.2
• Lemma 5.2