Localized endomorphisms in Kitaev's toric code on the plane

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 D(Z_2), i.e., Drinfel'd's quantum double of the group algebra of Z_2.

26 pages, 5 figures. v2: proof of Prop. 2.2 fixed, other minor corrections

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• Microscopic definitions of anyon data
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• Haag duality for Kitaev's quantum double model for abelian groups
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• Haag duality and the distal split property for cones in the toric code
• A derivation of braided $C^{\ast }$-tensor categories from gapped ground states satisfying the approximate Haag duality
• On the stability of charges in infinite quantum spin systems
• The complete set of infinite volume ground states for Kitaev's abelian quantum double models
• Kitaev's quantum double model from a local quantum physics point of view
• Kosaki-Longo index and classification of charges in 2D quantum spin models
• The split and approximate split property in 2D systems: stability and absence of superselection sectors
• Jones index, secret sharing and total quantum dimension
• Relativistic Quantum Fields Are Universal Entanglement Embezzlers
• A dynamical Toric Code model with fusion and de-fusion
• On the extension of stringlike localised sectors in 2+1 dimensions
• Dynamical abelian anyons with bound states and scattering states
• Subfactors and quantum information theory
• Local topological order and boundary algebras
• An algebraic quantum field theoretic approach to toric code with gapped boundary
• Homological codes and abelian anyons
• Lieb-Robinson bounds, Arveson spectrum and Haag-Ruelle scattering theory for gapped quantum spin systems
• Topologically ordered states in infinite quantum spin systems
• The category of anyon sectors for non-abelian quantum double models
• Classification of the anyon sectors of Kitaev's quantum double model
• $C^{\ast }$-categorical prefactorization algebras for superselection sectors and topological order
• Continuous Dependence on the Initial Data in the Kadison Transitivity Theorem and GNS Construction
• Uniqueness of purifications is equivalent to Haag duality
• On a C*-Diagonal Generated by the Toric Code