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.

arXiv comment11 pages

References in corpus (23)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• Topological quantum memory
• A "missing" family of classical orthogonal polynomials
• String-net condensation: A physical mechanism for topological phases
• An RVB phase in the triangular lattice quantum dimer model
• Braiding statistics approach to symmetry-protected topological phases
• Models for gapped boundaries and domain walls
• Proposed experiments to probe the non-abelian ν=5/2 quantum Hall state
• A classification of symmetry enriched topological phases with exactly solvable models
• The Quantum Double Model with Boundary: Condensations and Symmetries
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Twisted Gauge Theory Model of Topological Phases in Three Dimensions
• Generalizations and limitations of string-net models
• Boundary Degeneracy of Topological Order
• Ground State Degeneracy in the Levin-Wen Model for Topological Phases
• Ground State Degeneracy of Topological Phases on Open Surfaces
• A geometric approach to boundaries and surface defects in Dijkgraaf-Witten theories
• String-Net Models with $Z_N$ Fusion Algebra
• Boundary Hamiltonian theory for gapped topological orders
• Superconductivity with intrinsic topological order induced by pure Coulomb interaction and time-reversal symmetry breaking

Cited by in corpus (28)

• Dualities in one-dimensional quantum lattice models: topological sectors
• Tube algebras, excitations statistics and compactification in gauge models of topological phases
• Non-invertible Symmetries and Higher Representation Theory II
• Excitation basis for (3+1)d topological phases
• Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries
• Entanglement Entropy, Quantum Fluctuations, and Thermal Entropy in Topological Phases
• Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases
• Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy
• Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders
• Characterizing topological order by the information convex
• Seeing topological entanglement through the information convex
• Gapped boundaries of (3+1)d topological orders
• Boundary and domain wall theories of 2d generalized quantum double model
• On weak Hopf symmetry and weak Hopf quantum double model
• Electric-Magnetic duality in twisted quantum double model of topological orders
• Dual loop quantizations of 3d gravity
• Extend The Levin-Wen Model To Two-dimensional Topological Orders With Gapped Boundary Junctions
• Galois Orbits of TQFTs: Symmetries and Unitarity
• Higher Gauge Theory and Integrability
• From effective Hamiltonian to anomaly inflow in topological orders with boundaries
• Entanglement Entropy of Topological Orders with Boundaries
• Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization
• Finite-group gauge theories on lattices as Hamiltonian systems with constraints
• Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries
• Universal fault tolerant quantum computation in 2D without getting tied in knots
• Planar fault-tolerant circuits for non-Clifford gates on the 2D color code