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arXiv:1904.02153 · International Journal of Quantum Information

Quantum double models coupled to matter fields: A detailed review for a dualization procedure

M. F. Araujo de Resende, J. P. Ibieta Jimenez, J. Lorca Espiro

Published Feb 24, 2023• 0 citations
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Abstract

In this paper, we investigate how it is possible to define a new class of lattice gauge models based on a dualization procedure of a previous generalization of the Kitaev Quantum Double Models. In the case of this previous generalization that will be used as a basis, it was defined by adding new qudits (which were denoted as matter fields in reference to some works) to the lattice vertices with the intention of, for instance, interpreting its models as Kitaev Quantum Double Models coupled with Potts ones. Now, with regard to the generalization that we investigate here, which we want to define as the dual of this previous one, these new qudits were added to the lattice faces. And as the coupling between gauge and matter qudits of the previous generalization was performed by a gauge group action, we show that the dual behavior of these two generalizations was achieved by coupling these same qudits in the second one through a gauge group co-action homomorphism. One of the most striking dual aspects of these two generalizations is that, in both, part of the quasiparticles that were inherited from the Kitaev Quantum Double Models become confined when these action and co-action are nontrivial. But the big news here is that, in addition to the group homomorphism (that defines this gauge group co-action) allows us to classify all the different models of this second generalization, this same group homomorphism also suggests that all these models can be interpreted as two-dimensional restrictions of the 2-lattice gauge theories.

DOI
10.1142/s0219749923500181

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•Aug 11, 2014•2 citations
In this paper, we investigate how it is possible to define a new class of lattice gauge models based on a dualization procedure of a previous generalization of the Kitaev Quantum Double Models. In the case of this previous generalization that will be used as a basis, it was defined by adding new qudits (which were denoted as matter fields in reference to some works) to the lattice vertices with the intention of, for instance, interpreting its models as Kitaev Quantum Double Models coupled with Potts ones. Now, with regard to the generalization that we investigate here, which we want to define as the dual of this previous one, these new qudits were added to the lattice faces. And as the coupling between gauge and matter qudits of the previous generalization was performed by a gauge group action, we show that the dual behavior of these two generalizations was achieved by coupling these same qudits in the second one through a gauge group co-action homomorphism. One of the most striking dual aspects of these two generalizations is that, in both, part of the quasiparticles that were inherited from the Kitaev Quantum Double Models become confined when these action and co-action are nontrivial. But the big news here is that, in addition to the group homomorphism (that defines this gauge group co-action) allows us to classify all the different models of this second generalization, this same group homomorphism also suggests that all these models can be interpreted as two-dimensional restrictions of the 2-lattice gauge theories.
In this paper, we constructed a new generalization of a class of discrete bidimensional models, the so called Quantum Double Models, by introduce matter qunits to the faces of the lattice that supports these models. This new generalization can be interpreted as the algebraic dual of a first, where we introduce matter qunits to the vertices of this same lattice. By evaluating the algebraic and topological orders of these new models, we prove that, as in the first generalization, a new phenomenon of quasiparticle confinement may appear again: this happens when the co-action homomorphism between matter and gauge groups is non-trivial. Consequently, this homomorphism not only classifies the different models that belong to this new class, but also suggests that they can be interpreted as a 2-dimensional restriction of the 2-lattice gauge theories.
In this work, we evaluate one generalization of a class of discrete two-dimensional models, the so-called Quantum Double Models, by introduce matter fields to the vertices of the lattice that supports these models. Beside the basic model properties, we studied its topological order behaviour under the hypothesis that the basic states be indexed by cyclic Abelian groups. In this generalization, appears a new phenomenon of quasiparticle confinement due to the action of the gauge group. As a consequence, the ground state degeneracy becomes independent of the fundamental group of the manifold on which the model is defined, depending on this algebraic action and on the second group of homology. Another feature of this generalization is the presence of the quasiparticles with non-Abelian fusion rules. These latter quasiparticles are always required so that this lattice system can be excited and return to any of its vacuum states.
We evaluate one generalisation of a class of discrete bidimensional models, the so called Quantum Double Models, by introduce matter fields to the vertices of the lattice that supports these models. Beside the basic model properties, we studied its topological order behaviour under the hypothesis that the basic states be indexed by cyclic Abelian groups. In this generalisation, appears a new phenomenon of quasiparticle confinement due to the action of the gauge group. As a consequence, the ground state degeneracy becomes independent of the fundamental group of the manifold on which the model is defined, depending on this action and on the second group of homology. Another feature of this generalisation is the presence of the quasiparticles with non-Abelian fusion rules. These latter quasiparticles are always required so that this lattice system can be excited and return to any of its vacuum states.
In this work, we present a brief but insightful overview of the gauge theories, which are defined on n-dimensional lattices by using finite gauge groups, in order to show how they can be interpreted as a Hamiltonian system with constraints, analogous to what happens with the classical (continuous) gauge (field) theories. As this interpretation is not usually explored in the literature that discusses/introduces the concept of lattice gauge theory, but some recent works have been exploring Hamiltonian models in order to support some kind of quantum computation, we use this interpretation to, for example, present a brief geometric view of one class of these models: the Kitaev Quantum Double Models.
Abstract One generalization of the toric code is the so-called Kitaev Quantum Double model which realizes the Drinfeld double of a group. In this model edges of a lattice are directed and labeled with elements of a group G which then give us a planar diagram algebra. We consider mainly the “untwisted” case where the F-matrices (or equivalently 3-cocyles) are trivial. This generalizes the toric code where the group is Z2. Again we have vertex and plaquette operators, and a code space which satisfies both. As with the toric code, the ground state is topological although here (depending on the group G) the quasiparticles may be nonabelian. We turn to a continuum model for a more detailed analysis of the ground state space and the quasiparticles using the tube algebra, which, using some amount of group theory, is easily tractable. We discuss the so-called quasiparticle “ribbon operators” which generalize the error strings of the toric code. We relate the Kitaev quantum double model to lattice gauge theory and briefly discuss its generalization to higher dimensions.
We construct two dimensional lattice models from transfer matrices of lattice gauge theories with discrete gauge groups. These transfer matrices are built out of local operators acting on links, vertices and plaquettes and are parametrized by the center of the gauge group algebra and its dual. They contain the familiar 2D quantum double models for a particular choice of parameters, which includes the well studied example of the toric code. However for more general choices of 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 towards a paramagnetic phase. These perturbations can be thought of as magnetic fields added to the system which destroy the exact solvability of the quantum double model. Nevertheless from the same transfer matrix with perturbations we exhibit exactly solvable models which remain in a quantum phase, thus nullifying the effect of the perturbation. 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. This is shown by working with the groups Zn and S3 for the Abelian and non-Abelian cases respectively. The quantum phases are found by studying the excitations of these systems. The fusion rules and the statistics of these anyons indicate the quantum phases of these models. The implementation of these models can possibly improve the use of quantum double models for fault tolerant quantum computation. We then construct theories which arise from transfer matrices that are not the transfer matrices of lattice gauge theories. In particular we show that for the Z2 case this contains the double semion model. 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.