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A realistic material may possess defects, which often bring the material newproperties that have practical applications. The boundary defects of atwo-dimensional topologically ordered system are thought of as an alternativeway of realizing topological quantum computation. To facilitate the study ofsuch boundary defects, in this paper, we construct an exactly solvableHamiltonian model of topological orders with gapped boundary junctions, wherethe boundary defects reside, by placing the Levin-Wen model on a disk, whosegapped boundary is separated into multiple segments by junctions. We find thatthe Hamiltonian of a gapped boundary junction is characterized by either amorphism between or a common Frobenius subalgebra of the two Frobenius algebras(in the input fusion category) characetrizing the two boundary segments jointby the junction. We derive a formula of the ground state degeneracy and anexplicit ground-state basis of our model. We propose the notion of mobile andimmobile charges on the boundary and find that they are quantum observables andlabel the ground-state basis. Our model is computation friendly.

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