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Kitaev Quantum Double Model

Steven H. Simon

Published Sep 29, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1093/oso/9780198886723.003.0031

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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.
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.
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 Doubled-Semion model is another simple example of generalizing the toric code. Here we start with a planar diagram algebra that is the d=-1, Z2 loop gas. I.e, here a loop is given the value of -1. This is also known as the nontrivial cocyle of Z2. This model can be thought of as the simplest case of a twisted Kitaev Quantum Double, and it is also the simplest example of the Drinfeld double of a modular anyon theory (the double of the semion theory). In this case we construct a Hamiltonian such again we have a loop gas, but now flipping over a plaqutte incurs a sign in the wavefunction if we change the parity of the number of loops. We again turn to the tube algebra to identify the excitations of the model, which are both right- and left-handed semions. We run into a problem related to the nontrivial Frobenius-Schur indicator of the input diagram algebra. However, here we can resolve the issue by choosing a convenient gauge.
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.
In this paper we look at three dimensional (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 involutory Hopf algebra . 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 be the group algebra 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 . 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 , the ground state degeneracy is determined by the Kuperberg 3-manifold invariant of . It is also possible to obtain extra models by simply 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.
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 ).