Miguel Jorge Bernabe Ferreira

2D quantum double models from a 3D perspective

In this paper we look at three dimensional (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 involutory Hopf algebra . 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 be the group algebra 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 . 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 ?, the ground state degeneracy is determined by the Kuperberg 3-manifold invariant of . It is also possible to obtain extra models by simply 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.

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 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 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.

Quasi-Topological Quantum Field Theories and Z 2 Lattice Gauge Theories

We consider a two-parameter family of ℤ2 gauge theories on a lattice discretization of a three-manifold and its relation to topological field theories. Familiar models such as the spin-gauge model are curves on a parameter space Γ. We show that there is a region Γ0 ⊂ Γ where the partition function and the expectation value 〈WR(γ)〉 of the Wilson loop can be exactly computed. Depending on the point of Γ0, the model behaves as topological or quasi-topological. The partition function is, up to a scaling factor, a topological number of . The Wilson loop on the other hand, does not depend on the topology of γ. However, for a subset of Γ0, 〈WR(γ)〉 depends on the size of γ and follows a discrete version of an area law. At the zero temperature limit, the spin-gauge model approaches the topological and the quasi-topological regions depending on the sign of the coupling constant.