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 is 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.
Highlights
Topological quantum field theories TQFTs in three dimensions describe a variety of physical and toy models in many areas of modern physics
Since the idea of fault-tolerant quantum computation appeared in the literature 4, TQFTs are important in quantum information theory
On one hand we identified the ground states and the constraint operators of these models in case the underlying lattice is the honeycomb and the gauge group is a finite group
Summary
Topological quantum field theories TQFTs in three dimensions describe a variety of physical and toy models in many areas of modern physics. The emergence of topological phases from a description of microscopic degrees of freedom is modeled by the lattice models of Kitaev 4 and Levin and Wen 7 Since they generically have degenerate ground states and quasi-particle excitations insensitive to local disturbances, they are investigated in the theory of quantum computation 9 , their continuum limit being closely related to the spin network simulator 10, 11. A summary is given with a list of questions for future research
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