Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras

TL;DR

This paper rigorously defines and analyzes ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras, revealing subtleties and establishing their properties for understanding topological quantum phases.

Contribution

It introduces a rigorous framework for ribbon operators in the Hopf algebra generalization of the Kitaev model, clarifying distinctions and properties of these operators.

Findings

Distinction between locally clockwise and counterclockwise ribbons is crucial.

Ribbon operators create quasi-particles at ribbon ends, linked to irreducible representations of the Drinfeld double.

Properties hold in the generalized model but require complex proofs due to Hopf algebra complexities.

Abstract

Kitaev's quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. Originally it is based on finite groups, and is later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized Kitaev quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, this issue already exists in the original model based on finite non-Abelian groups. We show how certain properties would fail even in the original model if we do not distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than just group algebras.