Boundary Hamiltonian theory for gapped topological phases on an open surface

TL;DR

The paper develops a concrete boundary Hamiltonian framework for gapped topological phases on open surfaces by adjoining a boundary term derived from Frobenius algebras inside the Levin-Wen unitary fusion category. It shows boundary degrees of freedom live in Frobenius algebras and boundary excitations are classified by A–A bimodules, with ground states and quasiparticles described explicitly via modules and bimodules; equivalence of boundary conditions is captured by Morita equivalence. Through disk and cylinder geometries and models like Z2, Fibonacci, and Ising, the authors derive ground-state wavefunctions, boundary operators, and excitation algebras, illustrating topological invariance under Pachner moves and the dependence of GSD on boundary data. The work connects to Kitaev–Kong’s module-category perspective, providing a practical, computable bridge between category theory and explicit Hamiltonians for open topological orders.

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.

Figures (11)

• Figure 1: Trivalent graph with boundary
• Figure 2: A configuration of string types on a directed trivalent graph. The configuration (b) and hence the associated Hilbert space is regarded the same as (a), with some of the directions of some edges reversed and the corresponding labels $j$ conjugated $j^*$.
• Figure 3: Two ways of composing $T_{2\rightarrow 2}$ moves to transform from the left most graph to the right most graph.
• Figure 4: (a) Initial and final graphs with marked plaquettes. (b) A possible sequence of Pachner moves taking the initial graph to the final.
• Figure 5: Boundary is a wall carrying tails. $j$'s are bulk labels and $a$ are tail labels.