2D Quantum Double Models From a 3D Perspective

Abstract

In this paper we look at 3D lattice models that are generalizations of the state sum model used to define the Kuperberg invariant of 3-manifolds. The partition function is a scalar constructed as a tensor network where the building blocks are tensors given by the structure constants of an involutary Hopf algebra $\mathcal{A}$. These models are very general and are hard to solve in its entire parameter space. One can obtain familiar models, such as ordinary gauge theories, by letting $\mathcal{A}$ be the group algebra $\mathbb{C}(G)$ of a discrete group $G$ and staying on a certain region of the parameter space. We consider the transfer matrix of the model and show that Quantum double Hamiltonians are derived from a particular choice of the parameters. Such a construction naturally leads to the star and plaquette operators of the quantum double Hamiltonians, of which the toric code is a special case when $\mathcal{A}=\mathbb{C}({\mathbb{Z}}_2)$. This formulation is convenient to study ground states of these generalized quantum double models where they can naturally be interpreted as tensor network states. For a surface $\Sigma$, the ground state degeneracy is determined by the Kuperberg 3-manifold invariant of $\Sigma \times S^1$. It is also possible to obtain extra models by simple enlarging the allowed parameter space but keeping the solubility of the model. While some of these extra models have appeared before in the literature, our 3D perspective allows for an uniform description of them.

43 pages, 74 figures. A new section has been included which describes new topological and quasi-topological phases which can be obtained from the transfer matrix of lattice gauge theories. Apart from this the Introduction has been significantly improved

References in corpus (14)

• Non-Abelian Anyons and Topological Quantum Computation
• Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Topological Order and Conformal Quantum Critical Points
• Valence Bond Solids for Quantum Computation
• Tensor-entanglement renormalization group approach to 2D quantum systems
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models
• A quantum topological phase transition at the microscopic level
• Space-Time Geometry of Topological phases
• Wilson Line Picture of Levin-Wen Partition Functions
• Exactly soluble lattice models for non-abelian states of matter in 2 dimensions
• Comment on: Competing Interactions, the Renormalization Group, and the Isotropic-Nematic Phase Transition, by D. Barci and D. Stariolo, Phys. Rev. Lett. 98, 200604 (2007)
• Topological low-temperature limit of Z(2) spin-gauge theory in three dimensions

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