On the stability of charges in infinite quantum spin systems

Abstract

We consider a theory of superselection sectors for infinite quantum spin systems, describing charges that can be approximately localized in cone-like regions. The primary examples we have in mind are the anyons (or charges) in topologically ordered models such as Kitaev's quantum double models and perturbations of such models. In order to cover the case of perturbed quantum double models, the Doplicher-Haag-Roberts approach, in which strict localization is assumed, has to be amended. To this end we consider endomorphisms of the observable algebra that are almost localized in cones. Under natural conditions on the reference ground state (which plays a role analogous to the vacuum state in relativistic theories), we obtain a braided tensor $C^{\ast }$-category describing the sectors. We also introduce a superselection criterion selecting excitations with energy below a threshold. When the threshold energy falls in a gap of the spectrum of the ground state, we prove stability of the entire superselection structure under perturbations that do not close the gap. We apply our results to prove that all essential properties of the anyons in Kitaev's abelian quantum double models are stable against perturbations.

arXiv commentv2: 40 pages. Improved presentation, some corrections. v1: 37 pages. Some results were first reported in the dissertation of MC, arXiv:1708.05035

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• Fusion rules from entanglement
• A derivation of braided $C^{\ast }$-tensor categories from gapped ground states satisfying the approximate Haag duality
• Spectral Gaps and Incompressibility in a $\nu =1/3$ Fractional Quantum Hall System
• The split and approximate split property in 2D systems: stability and absence of superselection sectors
• On ${\mathbb{Z}}_2$-indices for ground states of fermionic chains
• An entropic invariant for 2D gapped quantum phases
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• A dynamical Toric Code model with fusion and de-fusion
• Quantum teleportation in the commuting operator framework
• Dynamical abelian anyons with bound states and scattering states
• Stability of the spectral gap and ground state indistinguishability for a decorated AKLT model
• Classification of the anyon sectors of Kitaev's quantum double model
• Continuous Dependence on the Initial Data in the Kadison Transitivity Theorem and GNS Construction