Topologically ordered states in infinite quantum spin systems
arXiv:1708.05035
Abstract
This dissertation discusses some properties of topologically ordered states as they appear in the setting of infinite quantum spin systems. We begin by studying the set of infinite volume ground states for Kitaev's abelian quantum double models. We show that states describing a single excitation in the bulk are infinite volume ground states, that is, local perturbations cannot remove the charge. The single excitations states, which are inequivalent for distinct charges, give a complete characterization of the sector theory for the set of ground states. Furthermore, any pure ground state is equivalent to some single excitation ground state. We proceed to study the stability of charges that are classified by certain representations of the algebra of observables. We introduce a new superselection criterion selecting almost localized and transportable -endomorphisms with respect to a vacuum state. Equivalence classes of representations satisfying the superselection criterion form different charged superselection sectors of the system. We show that if the vacuum state satisfies certain locality conditions then the superselection structure will be a braided tensor -category. Further, this superselection structure is stable up to deformations by a quasi-local dynamics. This result is then applied to show that the anyon structure of the abelian quantum double models is stable under local perturbations.
arXiv comment104 pages, Ph.D. dissertation, University of California, Davis. Chapter 4 is an extended version of arXiv:1608.04449
References in corpus (28)
- Fault-tolerant quantum computation by anyons
- Anyons in an exactly solved model and beyond
- Topological entanglement entropy
- An Area Law for One Dimensional Quantum Systems
- Lieb-Schultz-Mattis in Higher Dimensions
- Lieb-Robinson Bounds and the Exponential Clustering Theorem
- Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance
- Quantum codes on a lattice with boundary
- Ground state entanglement and geometric entropy in the Kitaev's model
- Propagation of Correlations in Quantum Lattice Systems
- A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
- Stability of Frustration-Free Hamiltonians
- Exponential Decay of Correlations Implies Area Law
- A statistical mechanics view on Kitaev's proposal for quantum memories
- Localized endomorphisms in Kitaev's toric code on the plane
- Anyons in discrete gauge theories with Chern-Simons terms
- Ground states of the infinite q-deformed Heisenberg ferromagnet
- Haag duality for Kitaev's quantum double model for abelian groups
- A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization I
- The complete set of infinite volume ground states for Kitaev's abelian quantum double models
- Product Vacua and Boundary State Models in d Dimensions
- Spectral Gap and Edge Excitations of -dimensional PVBS models on half-spaces
- Kosaki-Longo index and classification of charges in 2D quantum spin models
- On thermal stability of topological qubit in Kitaev's 4D model
- A model for charges of electromagnetic type
- Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons
- Asymptotic Abelianness and Braided Tensor C*-Categories
- C^1-Classification of gapped parent Hamiltonians of quantum spin chains with local symmetry