Microscopic description of 2d topological phases, duality and 3d state sums

Abstract

Doubled topological phases introduced by Kitaev, Levin and Wen supported on two dimensional lattices are Hamiltonian versions of three dimensional topological quantum field theories described by the Turaev-Viro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state are shown in case of the honeycomb lattice and the gauge group being a finite group by means of the well-known duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three dimensional geometrical interpretation are given.

19 pages, typos corrected, style improved, a final paragraph added

References in corpus (7)

• Non-Abelian Anyons and Topological Quantum Computation
• Detecting topological order in a ground state wave function
• Bipartite entanglement and entropic boundary law in lattice spin systems
• Entanglement renormalization and topological order
• Explicit tensor network representation for the ground states of string-net models
• Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models
• Exact entanglement renormalization for string-net models

Cited by in corpus (19)

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• Quantum computation with Turaev-Viro codes
• Generalized string-nets for unitary fusion categories without tetrahedral symmetry
• Methods for simulating string-net states and anyons on a digital quantum computer
• String-net models for non-spherical pivotal fusion categories
• Electric-magnetic duality of lattice systems with topological order
• Topological phases from higher gauge symmetry in 3+1D
• Partition function of the Levin-Wen model
• Kitaev lattice models as a Hopf algebra gauge theory
• Mapping between Morita equivalent string-net states with a constant depth quantum circuit
• Kitaev models based on unitary quantum groupoids
• Fusing Binary Interface Defects in Topological Phases: The $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$ case
• Towards Non-Invertible Anomalies from Generalized Ising Models
• BF-theory in graphene: a route toward topological quantum computing?
• 2D Quantum Double Models From a 3D Perspective
• Wegner-Wilson loops in string nets
• A Hermitian TQFT from a non-semisimple category of quantum sl(2)-modules
• Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups
• Categorical Quantum Volume Operator