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 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.
We introduce and study a class of anyon models that are a natural generalization of Ising anyons and Majorana fermion zero modes. These models combine an Ising anyon sector with a sector associated with
SO(m)2 Chern-Simons theory. We show how they can arise in a simple scenario for electron fractionalization and give a complete account of their quasiparticle types, fusion rules, and braiding. We show that the image of the braid group is finite for a collection of
2n fundamental quasiparticles and is a proper subgroup of the metaplectic representation of $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})\ensuremath{\ltimes}H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$, where $Sp(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the symplectic group over the finite field
Fm and $H(2n\ensuremath{-}2,{\mathbb{F}}_{m})$ is the extra special group (also called the $(2n\ensuremath{-}1)$-dimensional Heisenberg group) over
Fm. Moreover, the braiding of fundamental quasiparticles combined with a restricted set of measurements can be efficiently simulated classically. However, computing the result of braiding a certain type of composite quasiparticle followed by fusion into the identity is $#P$-hard. It is not universal for quantum computation because it has a finite braid group image. This is a rare example of a topological phase that is not universal for quantum computation through braiding but nevertheless has $#P$-hard link invariants. We argue that our models are closely related to recent analyses finding non-Abelian anyonic properties for defects in quantum Hall systems, generalizing Majorana zero modes in quasi-1D systems.
This thesis aims at concluding the classification results for topological phases with symmetry in 2+1 dimensions. The main result is that topological phases are classified by a triple of unitary braided fusion categories
E⊂C⊂M plus the chiral central charge
c. Here
E is a symmetric fusion category,
E=Rep(G) for boson systems or
E=sRep(Gf) for fermion systems, consisting of the representations of the symmetry group and describing the local excitations with symmetry;
C is the category of all the quasiparticle excitations in the bulk, containing
E as its Müger center;
M is a minimal modular extension of
C, that also includes the gauged symmetry defects. We also study the stacking of topological phases with symmetry and two types of anyon condensations based on such classification.