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A note on "Non-Abelian fusion rules from an Abelian system" by Pramod Padmanabhan and Paulo Teotonio-Sobrinho.

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

Published Nov 28, 2021• 0 citations
OpenAlex

Abstract

Since Ref. [1] shows the emergence of non-Abelian fusion rules in some examples of a class of Abelian models (which can be interpreted as a subclass of the Abelian Quantum Double Models coupled to matter fields on the lattice vertices), but does not prove whether these rules also exist in other cases, the purpose of this note is to present such proof emphasizing the importance of the existence of these rules. By the way, as the ground state of these models can be degenerate as a function of the group action, we prove that these non-Abelian fusion rules are always necessary for phase transitions to occur in this energy regime via a condensation mechanism.

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•Apr 3, 2019•1 citation
Since Ref. [1] shows the emergence of non-Abelian fusion rules in some examples of a class of Abelian models (which can be interpreted as a subclass of the Abelian Quantum Double Models coupled to matter fields on the lattice vertices), but does not prove whether these rules also exist in other cases, the purpose of this note is to present such proof emphasizing the importance of the existence of these rules. By the way, as the ground state of these models can be degenerate as a function of the group action, we prove that these non-Abelian fusion rules are always necessary for phase transitions to occur in this energy regime via a condensation mechanism.
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 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 have studied topology and dynamics of quantum vortices in spin-2 Bose-Einstein condensates. By computationally modeling controllable braiding and fusion of these vortices, we have demonstrated that certain vortices in such spinor condensates behave as non-Abelian anyons. We identify these anyons as fluxon, chargeon, and dyon quasiparticles. The pertinent anyon models are defined by the quantum double of the underlying discrete non-Abelian symmetry group of the condensate ground state order parameter.
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.
Higher-form symmetries generalize conventional global symmetries and act on lower-dimensional submanifolds of a quantum system. While Abelian topological phases can be organized by 1-form symmetries that form a group, non-Abelian topological phases based on finite groups require 1-form symmetry operators governed by non-invertible fusion algebras. In this work, we make this statement precise in quantum double lattice models D(G)\mathcal D(G)D(G) for finite non-Abelian groups GGG. We construct the electric, magnetic, and dyonic 1-form operators directly at the lattice fixed point and show that together they form a complete nonlocal diagnostic algebra for the topological Hilbert space. Using these operators, we explicitly determine the cylinder and torus ground-state subspaces for arbitrary finite GGG. Furthermore, we calculate the microscopic fusion and gluing of the 1-form symmetries and show that their topological deformation properties emerge after projection to the defect-free topological subspace. Our results establish ground states of non-Abelian quantum double models as a concrete microscopic realization of spontaneous non-invertible 1-form symmetry breaking and provide an operator language that may be useful for characterizing such states in quantum processors.
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.