The Quantum Double Model with Boundary: Condensations and Symmetries

TL;DR

This paper extends the quantum double model to include boundaries, characterizes particle condensations at these boundaries, and explores how different group-based models can produce equivalent anyon types, revealing new symmetries.

Contribution

It introduces a boundary framework for the quantum double model, characterizes condensations, and analyzes symmetries between different group-based models, including the S_3 case.

Findings

Boundaries lead to specific particle condensations.

Conditions for different groups to produce identical anyons.

Identification of symmetries in models based on finite groups.

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.