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) D ϕ ( Z 2 ) .
This is the continuation of our study of the Levin-Wen model based on an arbitrary unitary fusion category
C on the infinite plane. The ground state of the Levin-Wen model hosts anyonic excitations whose fusion and braiding properties are captured by the associated braided
C∗-tensor category of superselection sectors
SSS. By constructing explicit isomorphisms between the fusion spaces of
SSS and those of the Drinfeld center
Z(C), we show that these two categories have isomorphic
F- and
R-symbols. It follows that the full subcategory of finite sectors is unitarily braided monoidally equivalent to the Drinfeld center,
SSSf≃Z(C). This provides the first complete characterisation of the category of superselection sectors for a class of two-dimensional lattice models supporting anyons with non-integer quantum 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) 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.
Anyons in
2+1 dimensions are not only characterized by exotic braiding statistics but also by intricate fusion properties. Two anyons may fuse into multiple topological charge sectors, and associativity of fusing three anyons to produce a fixed charge sector is governed by
F-symbols. While braiding invariants, such as the modular data, are typically complex valued, a complete description of general anyon models requires understanding the arithmetic properties of its fusion associativity data as well. The
F-symbols for many of the most common
2+1d topological orders, including all Abelian anyon models as well as Fibonacci and Ising anyons, can be made real valued. We show this phenomenon is not universal by exhibiting braided fusion categories whose
F-symbols cannot be made real. We call such
F-symbols "inherently complex." The examples we study lack a charge-conjugation symmetry and our results are therefore consistent with the converse of a statement proved in a companion work linking real
F-symbols in braided fusion categories with the existence of a suitable charge-conjugation symmetry. We analyse the smallest-rank braided fusion categories we know of with inherently complex
F-symbols:
Rep(Z7⋊Z3) and
Rep(Z5⋊Z4). Consequently, the corresponding
Z(Rep(Z7⋊Z3)) and
Z(Rep(Z5⋊Z4)) anyon models also have inherently complex
F-symbols. Our presentation connects these examples with recent results on classifying anyons beyond modular data.
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.
Until recently, a careful derivation of the fusion structure of anyons from some underlying physical principles has been lacking. In Shi et al. (2020), the authors achieved this goal by starting from a conjectured form of entanglement area law for 2D gapped systems. In this work, we instead start with the principle of exchange symmetry, and determine the minimal prescription of additional postulates needed to make contact with unitary ribbon fusion categories as the appropriate algebraic framework for modelling anyons. Assuming that 2D quasiparticles are spatially localised, we build a functor from the coloured braid groupoid to the category of finite-dimensional Hilbert spaces. Using this functor, we construct a precise notion of exchange symmetry, allowing us to recover the core fusion properties of anyons. In particular, given a system of n quasiparticles, we show that the action of a certain n-braid βn uniquely specifies its superselection sectors. We then provide an overview of the braiding and fusion structure of anyons in the usual setting of braided 6j fusion systems. By positing the duality axiom of Kitaev (2006) and assuming that there are finitely many distinct topological charges, we arrive at the framework of ribbon categories.