Defects in Kitaev models and bicomodule algebras

TL;DR

This paper extends the Kitaev model to surfaces with boundaries and defects by using semisimple Hopf algebras and bicomodule algebras, introducing new algebraic structures for labeling cells and generalizing the underlying algebraic framework.

Contribution

It constructs a generalized Kitaev model incorporating boundaries and defects via semisimple bicomodule algebras, extending the algebraic structure of the standard model.

Findings

Generalized algebraic structure for Kitaev models with defects and boundaries

Introduction of an algebra with representations labeling 0-cells

Identification of a symmetric separability idempotent as a generalization of the Haar integral

Abstract

We construct a Kitaev model, consisting of a Hamiltonian which is the sum of commuting local projectors, for surfaces with boundaries and defects of dimension 0 and 1. More specifically, we show that one can consider cell decompositions of surfaces whose 2-cells are labeled by semisimple Hopf algebras and 1-cells are labeled by semisimple bicomodule algebras. We introduce an algebra whose representations label the 0-cells and which reduces to the Drinfeld double of a Hopf algebra in the absence of defects. In this way we generalize the algebraic structure underlying the standard Kitaev model without defects or boundaries, where all 1-cells and 2-cells are labeled by a single Hopf algebra and where point defects are labeled by representations of its Drinfeld double. In the standard case, commuting local projectors are constructed using the Haar integral for semisimple Hopf algebras. A central insight we gain in this paper is that in the presence of defects and boundaries, the suitable generalization of the Haar integral is given by the unique symmetric separability idempotent for a semisimple (bi-)comodule algebra.