From effective Hamiltonian to anomaly inflow in topological orders with boundaries

Abstract

Whether two boundary conditions of a two-dimensional topological order can be continuously connected without a phase transition in between remains a challenging question. We tackle this challenge by constructing an effective Hamiltonian, describing anyon interaction, that realizes such a continuous deformation. At any point along the deformation, the model remains a fixed point model describing a gapped topological order with gapped boundaries. That the deformation retains the gap is due to the anomaly cancelation between the boundary and bulk. Such anomaly inflow is quantitatively studied using our effective Hamiltonian. We apply our method of effective Hamiltonian to the extended twisted quantum double model with boundaries (constructed by two of us in Ref.[1]). We show that for a given gauge group $G$ and a three-cocycle in $H^3[G,U(1)]$ in the bulk, any two gapped boundaries for a fixed subgroup $K\subseteq G$ on the boundary can be continuously connected via an effective Hamiltonian. Our results can be straightforwardly generalized to the extended Levin-Wen model with boundaries (constructed by two of us in Ref.[2].

5 pages

References in corpus (8)

• 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
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• String-Net Models with $Z_N$ Fusion Algebra
• On Defects Between Gapped Boundaries in Two-Dimensional Topological Phases of Matter
• Twisted Quantum Double Model of Topological Orders with Boundaries
• 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
• Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders
• Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy II -- Cutting through the boundary
• Topological Phase Transitions Induced by Varying Topology and Boundaries in the Toric Code