Extend The Levin-Wen Model To Two-dimensional Topological Orders With Gapped Boundary Junctions

TL;DR

This paper extends the Levin-Wen model to include gapped boundary junctions in two-dimensional topological orders, providing an exactly solvable Hamiltonian that captures boundary defects and their quantum properties.

Contribution

It introduces a new exactly solvable Hamiltonian model for topological orders with gapped boundary junctions, incorporating boundary defects and their algebraic characterization.

Findings

Derived a formula for ground state degeneracy.

Constructed an explicit ground-state basis.

Identified boundary charges as quantum observables.

Abstract

A realistic material may possess defects, which often bring the material new properties that have practical applications. The boundary defects of a two-dimensional topologically ordered system are thought of as an alternative way of realizing topological quantum computation. To facilitate the study of such boundary defects, in this paper, we construct an exactly solvable Hamiltonian model of topological orders with gapped boundary junctions, where the boundary defects reside, by placing the Levin-Wen model on a disk, whose gapped boundary is separated into multiple segments by junctions. We find that the Hamiltonian of a gapped boundary junction is characterized by either a morphism between or a common Frobenius subalgebra of the two Frobenius algebras (in the input fusion category) characetrizing the two boundary segments joint by the junction. We derive a formula of the ground state degeneracy and an explicit ground-state basis of our model. We propose the notion of mobile and immobile charges on the boundary and find that they are quantum observables and label the ground-state basis. Our model is computation friendly.