Twisted Quantum Double Model of Topological Orders with Boundaries

Abstract

We generalize the twisted quantum double model of topological orders in two dimensions to the case with boundaries by systematically constructing the boundary Hamiltonians. Given the bulk Hamiltonian defined by a gauge group $G$ and a three-cocycle in the third cohomology group of $G$ over $U(1)$, a boundary Hamiltonian can be defined by a subgroup $K$ of $G$ and a two-cochain in the second cochain group of $K$ over $U(1)$. The consistency between the bulk and boundary Hamiltonians is dictated by what we call the Frobenius condition that constrains the two-cochain given the three-cocyle. We offer a closed-form formula computing the ground state degeneracy of the model on a cylinder in terms of the input data only, which can be naturally generalized to surfaces with more boundaries. We also explicitly write down the ground-state wavefunction of the model on a disk also in terms of the input data only.

11 pages

References in corpus (7)

• Non-Abelian Anyons and Topological Quantum Computation
• A classification of symmetry enriched topological phases with exactly solvable models
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Twisted Gauge Theory Model of Topological Phases in Three Dimensions
• Ground State Degeneracy of Topological Phases on Open Surfaces
• String-Net Models with $Z_N$ Fusion Algebra
• Boundary Hamiltonian theory for gapped topological orders

Cited by in corpus (4)

• Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries
• Entanglement Entropy, Quantum Fluctuations, and Thermal Entropy in Topological Phases
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy