2D quantum double models from a 3D perspective

Abstract

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 Σ, 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.

Campos de Gauge e matéria na rede - generalizando o Toric Code

“…In [41] the vector space A is set to be the group algebra C(G) and the QDM Hamiltonian is obtained from the partition function Z(C(G), z, z * ) of a three dimensional Lattice Gauge Theory. The procedure of obtaining such Hamiltionans relies on the fact that we can associate the partition function to a transfer matrix T [21], this is,…”

“…Also in [41] the particular case where G = Z 2 is obtained which reproduces the Toric Code, the formalism is also useful to explicitly write the ground states of the model as tensor networks. More general choices of parameters of the theory can be interpreted as adding perturbations that induce quantum phase transitions [42] making unsolvable the otherwise solvable models (as the QDM models); in [43] are considered more general choices of parameters at the level of the transfer matrix, the Hamiltionans obtained from such transfer matrices are explored for the abelian C(Z n ) and the non-abelian C(S 3 ) where we show that the inclusion of these parameters result on modified vertex operators although the models remain solvable and in the case of C(Z n ) the model is still in the QDM phase.…”

“…Also in [43] a more intricate deformation has been made, by associating weights to the volumes and the vertices of a 3D lattice it is possible to get transfer matrices that are not those of lattice gauge theories anymore. Likewise, following the procedure in [41] of splitting the transfer matrix one obtains a Hamiltonian defining quantum models that reproduce more general class of models known as Double Semion [44,45] and Twisted Quantum Double models [46]. Summarizing, this formalism has shown to be very versatile and useful for the study of topological order as different topological phases can be realized via the systematical obtainment of exactly solvable models from certain choices of the parameters z and z * .…”

“…By following the same procedure of [41] we obtain Hamiltonians from this transfer matrix that for certain parameters are made of mutually commuting projectors operators similar to the ones of the QDM with the inclusion of a new operator that plays the role of a parallel transporter acting on an edge and its two adjacent vertices. Such models are exactly solvable and they can describe new phases, topological or not.…”

“…In the second half of Introduction Chapter 2 we describe the algebraic structure of the Quantum Double Models for any finite and discrete group G, starting from the most elementary operators that act over on-site Hilbert spaces we construct the vertex and plaquette operators that define the Hamiltonian and develop on their algebraic properties, we end the chapter showing how the Toric Code is an special case of these more general models. As mentioned above these Hamiltonians can be obtained from transfer matrices of lattice gauge theories, the details of such procedure can be found in [41] and we do not include them here as it is not the main focus of this work.…”