# Boundary Hamiltonian theory for gapped topological orders

## TL;DR

This paper develops a systematic lattice Hamiltonian model for topological orders with explicit boundary terms, providing a robust framework for understanding gapped boundaries and boundary excitations in topological phases.

## Contribution

It introduces a comprehensive construction of boundary Hamiltonians for Levin-Wen models, classifies gapped boundary conditions via Frobenius algebras, and characterizes boundary excitations using module theory.

## Key findings

- Gapped boundary conditions are classified by Frobenius algebras.
- Explicit wavefunctions for ground states and boundary excitations are constructed.
- Boundary quasi-particle creation and hopping operators are explicitly formulated.

## Abstract

In this letter, we report our systematic construction of the lattice Hamiltonian model of topological orders on open surfaces, with explicit boundary terms. We do this mainly for the Levin-Wen stringnet model. The full Hamiltonian in our approach yields a topologically protected, gapped energy spectrum, with the corresponding wave functions robust under topology-preserving transformations of the lattice of the system. We explicitly present the wavefunctions of the ground states and boundary elementary excitations. We construct the creation and hopping operators of boundary quasi-particles. We find that given a bulk topological order, the gapped boundary conditions are classified by Frobenius algebras in its input data. Emergent topological properties of the ground states and boundary excitations are characterized by (bi-) modules over Frobenius algebras.