Boundary and domain wall theories of 2d generalized quantum double model

TL;DR

This paper develops a comprehensive framework for 2d topological orders based on Hopf algebras, including boundary and domain wall theories, and classifies excitations using algebraic structures, with explicit constructions and solutions.

Contribution

It introduces systematic constructions of boundary and domain wall Hamiltonians for 2d quantum double models based on Hopf algebras, extending the understanding of topological excitations and dualities.

Findings

Classified boundary and domain wall excitations via bimodules.

Constructed boundary and domain wall Hamiltonians systematically.

Solved ground states with boundaries and defects using Hopf tensor networks.

Abstract

The generalized quantum double lattice realization of 2d topological orders based on Hopf algebras is discussed in this work. Both left-module and right-module constructions are investigated. The ribbon operators and the classification of topological excitations based on the representations of the quantum double of Hopf algebras are discussed. To generalize the model to a 2d surface with boundaries and surface defects, we present a systematic construction of the boundary Hamiltonian and domain wall Hamiltonian. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. The ribbon operator realization of boundary-bulk duality is also discussed. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.