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Kitaev model for mirror bicrossproduct Hopf algebras and its tensor network representation

Florian Girelli, Prince K. Osei, Abdulmajid Osumanu

Published Sep 2, 2017• 0 citations
OpenAlex

Abstract

Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double. We propose a new version based on Majid's bicrossproduct quantum group. Given a Hopf algebra $H$, we show that a triangulated oriented surface defines a representation of the bicrossproduct quantum group $H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. We obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.

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Kitaev lattice model for bicrossproduct Hopf algebras and tensor network representation

Florian Girelli, Prince K. Osei, Abdulmajid Osumanu

Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double. We propose a new version based on Majid's bicrossproduct quantum group. Given a Hopf algebra $H$, we show that a triangulated oriented surface defines a representation of the bicrossproduct quantum group $H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. We obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.

•Sep 1, 2017•0 citations

Semidual Kitaev lattice model and tensor network representation

Florian Girelli, Prince K. Osei, Abdulmajid Osumanu

Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double $D(H)=H\bowtie H^{*\text{op}} $, as an example of a double cross product quantum group. We propose a new version based instead on $M(H)=H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$ as an example of Majid's bicrossproduct quantum group, related by semidualisation or `quantum Born reciprocity' to $D(H)$. Given a finite-dimensional Hopf algebra $H$, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group $H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.

arXiv:1709.00522•Sep 2, 2017•0 citations

Semidual Kitaev lattice model and tensor network representation

Florian Girelli, Prince K. Osei, Abdulmajid Osumanu

Abstract Kitaev’s lattice models are usually defined as representations of the Drinfeld quantum double D(H) = H ⋈ H*op, as an example of a double cross product quantum group. We propose a new version based instead on M(H) = Hcop ⧑ H as an example of Majid’s bicrossproduct quantum group, related by semidualisation or ‘quantum Born reciprocity’ to D(H). Given a finite-dimensional Hopf algebra H, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop ⧑ H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.

Journal of High Energy Physics•Sep 1, 2021•7 citations

Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras

Bowen Yan, Penghua Chen, Shawn Xingshan Cui

Abstract Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.

arXiv:2105.08202•Journal of Physics A Mathematical and Theoretical•Feb 15, 2022•15 citations

Two-dimensional bialgebras and quantum groups: algebraic structures and tensor network realizations

José Garre-Rubio, András Molnár, Germán Sierra

Abstract We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is based on defining 2D coproducts through horizontal and vertical maps that satisfy compatibility and associativity conditions, enabling the consistent growth of vector spaces over lattice sites. We present several examples of 2D bialgebras, including group-like and Lie algebra-inspired constructions and a quasi-1D coproduct instance that is applicable to Taft-Hopf algebras and to quantum groups. The approach is further applied to the quantum group U q [ s u ( 2 ) ] , for which we construct 2D generalizations of its generators, analyze q -deformed singlet states, and derive a 2D R-matrix satisfying an intertwining relation in the semiclassical limit. Additionally, we show how tensor network states, particularly projected entangled pair states, naturally induce 2D coalgebra structures when supplemented with appropriate boundary conditions. Our results establish a local and algebraically consistent method to embed quantum group symmetries into higher-dimensional lattice systems, potentially connecting to the emerging theory of fusion 2-categories and categorical symmetries in quantum many-body physics.

arXiv:2507.22541•Journal of Physics A Mathematical and Theoretical•Feb 3, 2026•0 citations

2D Quantum Double Models From a 3D Perspective

Miguel Jorge, Bernabe Ferreiraa, Pramod Padmanabhana +4 more

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.

•Jan 1, 2014•22 citations
Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double. We propose a new version based on Majid's bicrossproduct quantum group. Given a Hopf algebra HHH, we show that a triangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop▶ ⁣ ⁣ ⁣◃HH^{\text{cop}}\blacktriangleright\!\!\!\triangleleft HHcop▶◃H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. We obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.
Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double. We propose a new version based on Majid's bicrossproduct quantum group. Given a Hopf algebra HHH, we show that a triangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop▶ ⁣ ⁣ ⁣◃HH^{\text{cop}}\blacktriangleright\!\!\!\triangleleft HHcop▶◃H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. We obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.
Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double $D(H)=H\bowtie H^{*\text{op}} $, as an example of a double cross product quantum group. We propose a new version based instead on M(H)=Hcop▶ ⁣ ⁣ ⁣◃HM(H)=H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft HM(H)=Hcop▶◃H as an example of Majid's bicrossproduct quantum group, related by semidualisation or `quantum Born reciprocity' to D(H)D(H)D(H). Given a finite-dimensional Hopf algebra HHH, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop▶ ⁣ ⁣ ⁣◃HH^{\text{cop}}\blacktriangleright\!\!\!\triangleleft HHcop▶◃H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.
Abstract Kitaev’s lattice models are usually defined as representations of the Drinfeld quantum double D(H) = H ⋈ H*op, as an example of a double cross product quantum group. We propose a new version based instead on M(H) = Hcop ⧑ H as an example of Majid’s bicrossproduct quantum group, related by semidualisation or ‘quantum Born reciprocity’ to D(H). Given a finite-dimensional Hopf algebra H, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop ⧑ H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.
Abstract Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.
Abstract We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is based on defining 2D coproducts through horizontal and vertical maps that satisfy compatibility and associativity conditions, enabling the consistent growth of vector spaces over lattice sites. We present several examples of 2D bialgebras, including group-like and Lie algebra-inspired constructions and a quasi-1D coproduct instance that is applicable to Taft-Hopf algebras and to quantum groups. The approach is further applied to the quantum group U q [ s u ( 2 ) ] , for which we construct 2D generalizations of its generators, analyze q -deformed singlet states, and derive a 2D R-matrix satisfying an intertwining relation in the semiclassical limit. Additionally, we show how tensor network states, particularly projected entangled pair states, naturally induce 2D coalgebra structures when supplemented with appropriate boundary conditions. Our results establish a local and algebraically consistent method to embed quantum group symmetries into higher-dimensional lattice systems, potentially connecting to the emerging theory of fusion 2-categories and categorical symmetries in quantum many-body physics.
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 A\mathcal{A}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 A\mathcal{A}A be the group algebra C(G)\mathbb{C}(G)C(G) of a discrete group GGG 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 A=C(Z2)\mathcal{A}=\mathbb{C}(\mathbb{Z}_2)A=C(Z2​). 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\SigmaΣ, the ground state degeneracy is determined by the Kuperberg 3-manifold invariant of Σ×S1\Sigma\times S^1Σ×S1. 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.