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arXiv:2208.06317 · Journal of Mathematical Physics

Algebraic aspects of boundaries in the Kitaev quantum double model

Alexander Cowtan, Shahn Majid

Published Oct 1, 2023• 2 citations
arXivHTMLPDF

Abstract

We provide a systematic treatment of boundaries based on subgroups K ⊆ G for the Kitaev quantum double D(G) model in the bulk. The boundary sites are representations of a *-subalgebra Ξ ⊆ D(G) and we explicate its structure as a quasi-Hopf *-algebra dependent on a choice of transversal R. We provide decomposition formulae for irreducible representations of D(G) pulled back to Ξ. As an application of our treatment, we study patches with boundaries based on K = G horizontally and K = {e} vertically and show how these could be used in a quantum computer using the technique of lattice surgery. More abstractly, we also provide explicitly the monoidal equivalence of the category of Ξ-modules and the category of G-graded K-bimodules and use this to prove that different choices of R are related by Drinfeld cochain twists. Examples include Sn−1 ⊂ Sn and an example related to the octonions where Ξ is also a Hopf quasigroup.

DOI
10.1063/5.0127285

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We provide a systematic treatment of boundaries based on subgroups K ⊆ G for the Kitaev quantum double D(G) model in the bulk. The boundary sites are representations of a *-subalgebra Ξ ⊆ D(G) and we explicate its structure as a quasi-Hopf *-algebra dependent on a choice of transversal R. We provide decomposition formulae for irreducible representations of D(G) pulled back to Ξ. As an application of our treatment, we study patches with boundaries based on K = G horizontally and K = {e} vertically and show how these could be used in a quantum computer using the technique of lattice surgery. More abstractly, we also provide explicitly the monoidal equivalence of the category of Ξ-modules and the category of G-graded K-bimodules and use this to prove that different choices of R are related by Drinfeld cochain twists. Examples include Sn−1 ⊂ Sn and an example related to the octonions where Ξ is also a Hopf quasigroup.
We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double D( G) symmetry, where G is a finite group. We provide projection operators for its quasiparticles content as irreducible representations of D( G) and combine this with D( G)-bimodule properties of open ribbon excitation spaces [Formula: see text] to show how open ribbons can be used to teleport information between their endpoints s0, s1. We give a self-contained account that builds on earlier work but emphasizes applications to quantum computing as surface code theory, including gates on D( S3). We show how the theory reduces to a simpler theory for toric codes in the case of [Formula: see text], including toric ribbon operators and their braiding. In the other direction, we show how our constructions generalize to D( H) models based on a finite-dimensional Hopf algebra H, including site actions of D( H) and partial results on ribbon equivariance even when the Hopf algebra is not semisimple.
Abstract The generalized quantum double lattice realization of 2d topological orders based on Hopf algebras is discussed in this work. Both left-module and right-module constructions are investigated. The ribbon operators and the classification of topological excitations based on the representations of the quantum double of Hopf algebras are discussed. To generalize the model to a 2d surface with boundaries and surface defects, we present a systematic construction of the boundary Hamiltonian and domain wall Hamiltonian. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. The ribbon operator realization of boundary-bulk duality is also discussed. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.
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.
A bstract In this work, we employ the Tannaka-Krein reconstruction to compute the quantum double 𝒟(𝒢) of a finite 2-group 𝒢 as a Hopf monoidal category. We also construct a 3+1D lattice model from the Dijkgraaf-Witten TQFT functor for the 2-group 𝒢, generalizing Kitaev’s 2+1D quantum double model. Notably, the string-like local operators in this lattice model are shown to form 𝒟(𝒢). Specializing to 𝒢 = ℤ 2 , we demonstrate that the topological defects in the 3+1D toric code model are modules over 𝒟(ℤ 2 ).
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.
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.