Boundary Hamiltonian theory for gapped topological phases on an open surface

TL;DR

This paper develops a Hamiltonian framework for gapped topological phases on open surfaces, extending the Levin-Wen model to include boundary conditions, and classifies boundary types using Frobenius algebras.

Contribution

It introduces explicit boundary Hamiltonians compatible with the Levin-Wen model, classifies boundary types via Morita-equivalent Frobenius algebras, and constructs boundary quasiparticle operators.

Findings

Gapped boundary conditions are classified by Frobenius algebras.

Explicit ground-state wavefunctions are derived.

Boundary quasiparticles are characterized by bimodules of Frobenius algebras.

Abstract

In this paper we propose a Hamiltonian approach to gapped topological phases on an open surface with boundary. Our setting is an extension of the Levin-Wen model to a 2d graph on the open surface, whose boundary is part of the graph. We systematically construct a series of boundary Hamiltonians such that each of them, when combined with the usual Levin-Wen bulk Hamiltonian, gives rise to a gapped energy spectrum which is topologically protected; and the corresponding wave functions are robust under changes of the underlying graph that maintain the spatial topology of the system. We derive explicit ground-state wavefunctions of the system and show that the boundary types are classified by Morita-equivalent Frobenius algebras. We also construct boundary quasiparticle creation, measuring and hopping operators. These operators allow us to characterize the boundary quasiparticles by bimodules of Frobenius algebras. Our approach also offers a concrete set of tools for computations. We illustrate our approach by a few examples.