Deformed quantum double realization of the toric code and beyond

Abstract

Quantum double models, such as the toric code, can be constructed from transfer matrices of lattice gauge theories with discrete gauge groups and parametrized by the center of the gauge group algebra and its dual. For general choices of these parameters the transfer matrix contains operators acting on links which can also be thought of as perturbations to the quantum double model driving it out of its topological phase and destroying the exact solvability of the quantum double model. We modify these transfer matrices with perturbations and extract exactly solvable models which remain in a quantum phase, thus nullifying the effect of the perturbation. The algebra of the modified vertex and plaquette operators now obey a deformed version of the quantum double algebra. The Abelian cases are shown to be in the quantum double phase whereas the non-Abelian phases are shown to be in a modified phase of the corresponding quantum double phase. These are illustrated with the groups ${\mathbb{Z}}_n$ and $S_3$. The quantum phases are determined by studying the excitations of these systems namely their fusion rules and the statistics. We then go further to construct a transfer matrix which contains the other ${\mathbb{Z}}_2$ phase namely the double semion phase. More generally for other discrete groups these transfer matrices contain the twisted quantum double models. These transfer matrices can be thought of as being obtained by introducing extra parameters into the transfer matrix of lattice gauge theories. These parameters are central elements belonging to the tensor products of the algebra and its dual and are associated to vertices and volumes of the three dimensional lattice. As in the case of the lattice gauge theories we construct the operators creating the excitations in this case and study their braiding and fusion properties.

21 pages, 5 figures, A new section has been added which shows how the double semion model can be obtained from the transfer matrix formalism. This section also shows how the twisted quantum double model of arXiv:1211.3695 [cond-mat.str-el] can be obtained from the transfer matrix formalism. Algebra of operators added, main results unchanged

References in corpus (14)

• Non-Abelian Anyons and Topological Quantum Computation
• Topological Quantum Distillation
• Topological quantum order: stability under local perturbations
• Topological Order and Conformal Quantum Critical Points
• A classification of symmetry enriched topological phases with exactly solvable models
• Topological Computation without Braiding
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• A short proof of stability of topological order under local perturbations
• Topological order in a 3D toric code at finite temperature
• Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
• Self-duality and bound states of the toric code model in a transverse field
• A hierarchy of topological tensor network states
• Toric-boson model: Toward a topological quantum memory at finite temperature
• Topological Order Following a Quantum Quench

Cited by in corpus (1)

• Finite-group gauge theories on lattices as Hamiltonian systems with constraints