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arXiv:1012.3857 · Reviews in Mathematical Physics

LOCALIZED ENDOMORPHISMS IN KITAEV'S TORIC CODE ON THE PLANE

Pieter Naaijkens

Published May 1, 2011• 45 citations
arXivHTMLPDF

Abstract

We consider various aspects of Kitaev's toric code model on a plane in the C*-algebraic approach to quantum spin systems on a lattice. In particular, we show that elementary excitations of the ground state can be described by localized endomorphisms of the observable algebra. The structure of these endomorphisms is analyzed in the spirit of the Doplicher–Haag–Roberts program (specifically, through its generalization to infinite regions as considered by Buchholz and Fredenhagen). Most notably, the statistics of excitations can be calculated in this way. The excitations can equivalently be described by the representation theory of [Formula: see text], i.e. Drinfel'd's quantum double of the group algebra of ℤ2.

DOI
10.1142/s0129055x1100431x

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We consider various aspects of Kitaev's toric code model on a plane in the C*-algebraic approach to quantum spin systems on a lattice. In particular, we show that elementary excitations of the ground state can be described by localized endomorphisms of the observable algebra. The structure of these endomorphisms is analyzed in the spirit of the Doplicher–Haag–Roberts program (specifically, through its generalization to infinite regions as considered by Buchholz and Fredenhagen). Most notably, the statistics of excitations can be calculated in this way. The excitations can equivalently be described by the representation theory of [Formula: see text], i.e. Drinfel'd's quantum double of the group algebra of ℤ2.
We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.
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 We study Kitaev’s quantum double model for arbitrary finite gauge group in infinite volume, using an operator-algebraic approach. The quantum double model hosts anyonic excitations which can be identified with equivalence classes of ‘localized and transportable endomorphisms’, which produce anyonic excitations from the ground state. Following the Doplicher–Haag–Roberts (DHR) sector theory from AQFT, we organize these endomorphisms into a braided monoidal category capturing the fusion and braiding properties of the anyons. We show that this category is equivalent to RepfD(G)\textbf{Rep}_f {{\mathcal {D}}}(G)Repf​D(G) Rep f D ( G ) , the representation category of the quantum double of G . This establishes for the first time the full DHR structure for a class of 2d quantum lattice models with non-abelian anyons.
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.
The color code model is a crucial instance of a Calderbank-Shor-Steane (CSS)-type topological quantum error-correcting code, which notably supports transversal implementation of the full Clifford group. Its robustness against local noise is rooted in the structure of its topological excitations. From the perspective of quantum phases of matter, it is essential to understand these excitations in the thermodynamic limit. In this work, we analyze the color code model on an infinite lattice within the quasi-local C∗C^{*}C∗-algebra framework, using a cone-localized Doplicher-Haag-Roberts (DHR) analysis. We classify its irreducible anyon superselection sectors and construct explicit string operators that generate anyonic excitations from the ground state. We further examine the fusion and braiding properties of these excitations. Our results show that the topological order of the color code is described by Rep(D(Z2×Z2))≃Rep(D(Z2))⊠Rep(D(Z2))\mathsf{Rep}(D(\mathbb{Z}_2 \times \mathbb{Z}_2)) \simeq \mathsf{Rep}(D(\mathbb{Z}_2)) \boxtimes \mathsf{Rep}(D(\mathbb{Z}_2))Rep(D(Z2​×Z2​))≃Rep(D(Z2​))⊠Rep(D(Z2​)), which is equivalent to a double layer of the toric code and consistent with established analyses on finite lattices. We also establish the existence and uniqueness of the KMS state at every finite inverse temperature and derive an explicit description of the corresponding thermal distribution of stabilizer excitations.
Abstract Following the general theory of categorified quantum groups developed by the author previously, we construct the Drinfel’d double 2-bialgebra associated to a finite group N = G0. For N = ℤ2, we explicitly compute the braided 2-categories of 2-representations of certain version of this Drinfel’d double 2-bialgebra, and prove that they characterize precisely the 4d toric code and its spin-ℤ2 variant. This result relates the two descriptions (categorical vs. field theoretical) of 4d gapped topological phases in existing literature and displays an instance of higher Tannakian duality for braided 2-categories. In particular, we show that particular twists of the underlying Drinfel’d double 2-bialgebra is responsible for much of the higher-structural properties that arise in 4d topological orders.