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arXiv:2401.13774

Ribbon operators in the Semidual lattice code model

Fred Soglohu, Prince K. Osei, Abdulmajid Osumanu

Published Jan 24, 2024• 0 citations
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Abstract

In this work, we provide a rigorous definition of ribbon operators in the Semidual Kitaev lattice model and study their properties. These operators are essential for understanding quasi-particle excitations within topologically ordered systems. We show that the ribbon operators generate quasi-particle excitations at the ends of the ribbon and reveal themselves as irreducible representations of the Bicrossproduct quantum group $M(H)=H^{\text{cop}}\lrbicross H$ or $M(H)^{\text{op}}$ depending on their chirality or local orientation.

DOI
10.48550/arxiv.2401.13774

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In this work, we provide a rigorous definition of ribbon operators in the Semidual Kitaev lattice model and study their properties. These operators are essential for understanding quasi-particle excitations within topologically ordered systems. We show that the ribbon operators generate quasi-particle excitations at the ends of the ribbon and reveal themselves as irreducible representations of the Bicrossproduct quantum group M(H)=Hcop\lrbicrossHM(H)=H^{\text{cop}}\lrbicross HM(H)=Hcop\lrbicrossH or M(H)opM(H)^{\text{op}}M(H)op depending on their chirality or local orientation.
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 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 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.
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.
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.