Boundary Hamiltonian theory for gapped topological phases on an open surface

Abstract

In this paper we propose a Hamiltonian approach to gapped topological phases on open surfaces. Our setting is an extension of the Levin-Wen model to a 2d graph on an 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. It is shown that 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 on a disk as well as on a cylinder. For boundary quasiparticle excitations, we are able to construct their creation, annihilation, measuring and hopping operators etc. Given a bulk string-net theory, our approach provides a classification scheme of possible types of gapped boundary conditions by Frobenius algebras (modulo Morita equivalence) of the bulk fusion category; the boundary quasiparticles are characterized by bimodules of the pertinent Frobenius algebras. Our approach also offers a set of concrete tools for computations. We illustrate our approach by a few examples.

Entanglement entropy, quantum fluctuations, and thermal entropy in topological phases

“…In this paper, we always choose a ground state in which there are no non-contractible anyon loops. In the sphere case, certainly there are no such loops anyway but on a cylinder, there may be because a topological order on a cylinder may have ground state degeneracy [30,39,40]. In general, we can choose an arbitrary ground state, with or without non-contractible anyon loops.…”

Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries

“…By comparing to the boundary vertex and plaquette operators in the extended LW model reviewed in Appendix B or in Ref. [11,12] and the bulk operators in the enlarged LW model in Ref. [8], we can actually identify the model H QD,Γ L G ,A G,K defined onΓ as the extended LW model H LW,Γ Rep G ,A with A = A G,K defined on the same latticeΓ.…”

“…Inequivalent Frobenius algebra objects in the UFC Rep S3 obtained by solving the defining conditions (4.1). Frobenius algebra objects A 1 = 0 and A 1 = 0 ⊕ 1 ⊕ 2 are Morita equivalent[12]. One thus can forget about A 1 .…”

“…But to comply with the definition of boundary vertex operators in Refs. [11,12], where the extended LW model was constructed. We rewrite this canonical Frobenius algebra as…”

“…As in the main text, here d s counts the multiplicity of s. Note that Refs. [11,12] actually did not consider the cases with multiplicities in a Frobenius algebra and thus did not require an extra input data on the boundary. In this review, we offer a generalization to such cases.…”