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arXiv:1708.05035

Topologically ordered states in infinite quantum spin systems

Matthew Cha

Published Aug 16, 2017• 1 citation
arXivHTMLPDF

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 $C^*$-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.

DOI
10.48550/arxiv.1708.05035

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Abstract We describe in a simple setting how to extract a braided tensor category from a collection of superselection sectors of a two-dimensional quantum spin system, corresponding to abelian anyons. We extract from this category its fusion ring as well as its F and R-symbols. We then construct the double semion state in infinite volume and extract the braided tensor category describing its semion, anti-semion, and bound state excitations. We verify that this category is equivalent to the representation category of the twisted quantum double Dϕ(Z2)\mathcal {D}^{\phi }(\mathbb {Z}_2)Dϕ(Z2​) D ϕ ( Z 2 ) .
In this dissertation, we detail an operator algebraic approach to studying topological order in the infinite volume setting. We give a thorough and self-contained review of the DHR-style approach on quantum spin systems, which builds a category DHR\mathrm{\textbf{DHR}}DHR of anyon sectors starting from microscopic lattice spin systems. In general, this category has the structure of a braided C∗\mathrm{C}^*C∗-tensor category. We will verify in full detail that DHR\mathrm{\textbf{DHR}}DHR is the expected category in Kitaev's Quantum Double model, a paradigmatic model for studying topological order on the lattice. We will then extend the DHR-style analysis to systems in the presence of a global on-site symmetry, and introduce a category of symmetry defects, GSecG\mathsf{Sec}GSec, and show that it has the structure of a GGG-crossed braided C∗\mathrm{C}^*C∗-tensor category.
We study a new kind of ordering — topological order — in rigid states (the states with no local gapless excitations). We concentrate on characterization of the different topological orders. As an example we discuss in detail chiral spin states of 2 + 1 dimensional spin systems. Chiral spin states are described by the topological Chern-Simons theories in the continuum limit. We show that the topological orders can be characterized by a non-Abelian gauge structure over the moduli space which parametrizes a family of the model Hamiltonians supporting topologically ordered ground states. In 2 + 1 dimensions, the non-Abelian gauge structure determines possible fractional statistics of the quasi-particle excitations over the topologically ordered ground states. The dynamics of the low lying global excitations is shown to be independent of random spatial dependent perturbations. The ground state degeneracy and the non-Abelian gauge structures discussed in this paper are very robust, even against those perturbations that break translation symmetry. We also discuss the symmetry properties of the degenerate ground states of chiral spin states. We find that some degenerate ground states of chiral spin states on torus carry non-trivial quantum numbers of the 90° rotation.
Abstract We introduce a set of axioms for locally topologically ordered quantum spin systems in terms of nets of local ground state projections, and we show they are satisfied by Kitaev’s Toric Code and Levin-Wen type models. For a locally topologically ordered spin system on Zk\mathbb {Z}^{k}Zk , we define a local net of boundary algebras on Zk−1\mathbb {Z}^{k-1}Zk−1 , which provides a mathematically precise algebraic description of the holographic dual of the bulk topological order. We construct a canonical quantum channel so that states on the boundary quasi-local algebra parameterize bulk-boundary states without reference to a boundary Hamiltonian. As a corollary, we obtain a new proof of a recent result of Ogata [Oga24] that the bulk cone von Neumann algebra in the Toric Code is of type II\mathrm {II}II , and we show that Levin-Wen models can have cone algebras of type III\mathrm {III}III . Finally, we argue that the braided tensor category of DHR bimodules for the net of boundary algebras characterizes the bulk topological order in (2+1)D, and can also be used to characterize the topological order of boundary states.
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An important class of models studied in quantum statistical mechanics consists of frustration-free interactions whose Hamiltonians exhibit a gap in the spectrum above their ground state energies. This paper discusses recent techniques which dramatically extend the class of models to which stability applies when the perturbations exhibit stretched exponential decay at long distances. We consider a class of initial models satisfying an assumption that prevents splitting of the ground state space. This assumption on the structure on the ground state space is referred to as local topological quantum order. A key tool in our analysis is the quasi-local unitary spectral flow automorphism which allows us to decompose the perturbed Hamiltonian into a perturbation model satisfying a relative form bound. We will present a result for the stability of frustration-free, one-dimensional models of this type in the form of an estimate uniform in the system size. We also present additional conditions ensuring the existence of a pure perturbed ground state whose GNS Hamiltonian remains gapped. As an application, we evaluate the stability of the Heisenberg XXZ spin-1/2 chain. Our work shows a new result on the stability of the XXZ model for perturbations respecting a local gauge symmetry and the stability above the translation invariant ground states in the thermodynamic limit. We also discuss challenges to additional conjectured stability results for the XXZ model.