Abstract Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.
We use a network of chiral junctions to construct a family of topological chiral spin liquids in two spatial dimensions. The chiral spin liquid phase harbors SU(2) _k k anyons, which stem from the underlying SU(2) _k k WZW models that describe the constituent spin chains of the network. The network exhibits quantized spin and thermal Hall conductances. We illustrate our construction by inspecting the topological properties of the SU(2) _2 2 model. We find that this model has emergent Ising anyons, with spinons acting as vortex excitations that bind Majorana zero modes. We also show that the ground state of this network is threefold degenerate on the torus, asserting its non-Abelian character. Our results shed new light on the stability of non-Abelian topological phases in artificial quantum materials.
In this thesis we investigate topological phases of matter that have a global, unbroken symmetry group -- also known as symmetry enriched topological (SET) phases. We address three questions about these phases: (1) how can we build exactly solvable models that realize them? (2) how can we determine if their edge theories can be gapped without breaking the symmetry? and (3) how do we understand the phenomenon of decoupled charge and neutral modes which occurs in certain fractional quantum Hall states? ,More specifically, we address the first question by constructing exactly solvable models for a wide class of symmetry enriched topological (SET) phases, which we call symmetry-enriched string nets. The construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group G, and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2 symmetry which exchanges the 'e' and 'm' type anyons. We further illustrate our construction with a number of additional examples. ,For the second question, we focus on the edge theories of 2D SET phases with Z2 symmetry. The central problem we seek to solve is to determine which edge theories can be gapped without breaking the symmetry. Previous attempts to answer this question in special cases relied on constructing perturbations of a particular type to gap the edge. This method proves the edge can be gapped when the appropriate perturbations can be found, but is inconclusive if they cannot be found. We build on this previous work by deriving a necessary and sufficient algebraic condition for when the edge can be gapped. Our results apply to Z2 symmetry protected topological phases as well as Abelian Z2 SET phases,Finally, in the fourth chapter, we describe solvable models that capture how impurity scattering in certain fractional quantum Hall edges can give rise to a neutral mode --- i.e. an edge mode that does not carry electric charge. These models consist of two counter-propagating chiral Luttinger liquids together with a collection of discrete impurity scatterers. Our main result is an exact solution of these models in the limit of infinitely strong impurity scattering. From this solution, we explicitly derive the existence of a neutral mode and we determine all of its microscopic properties including its velocity. We also study the stability of the neutral mode and show that it survives at finite but sufficiently strong scattering. Our results are applicable to a family of Abelian fractional quantum Hall states of which the nu = 2/3 state is the most prominent example.
We study the topological order that arises from chiral states with SU(N) or SO(N) edge-state symmetry. This extends our previous study of topological orders that descend from the bosonic
E8 quantum Hall state. We use exactly solvable models of coupled electron wires to construct states with
SU(m)n,SO(m)n, or
Sp(m)n topological order for various levels
n. We use our constructions to write down string operators for various non-Abelian anyons. We thereby provide a systematic, model derivation of quantum Hall states, topological superconductors, and spin liquids with emergent non-Abelian quasiparticle excitations, including those of Ising, metaplectic, and Fibonacci type.
In a two-dimensional topological superconductor hosting vortices with Majorana zero modes (MZMs), the exchange of vortices yields a projective representation of the braid group. We investigate whether the full anyonic exchange matrix is uniquely determined by the MZM parity operators and whether consistency with the hexagon equation of the Ising modular tensor category necessitates chiral edge modes. We show that the MZM algebra
γiγj+γjγi=2δij fixes the non-Abelian structure of the exchange operator up to an overall Abelian phase. Completing the modular data and imposing the Ising hexagon equation requires a topological spin
θσ=eiπ/8 for the vortices, which via the Gauss-Milgram sum implies a chiral central charge
c=1/2(mod8). Non-chiral systems with
c=0 yield incompatible topological spins, demonstrating that the MZM parity operators alone do not uniquely fix the full anyonic exchange matrix. We discuss implications for extrinsic defects and anomaly inflow. ## 1. Introduction Non-Abelian anyons in two-dimensional topological phases offer a route to topologically protected quantum computation. In a class of topological superconductors—exemplified by the chiral
px+ipy state—vortices trap Majorana zero modes (MZMs) whose algebraic exchange realizes a projective representation of the braid group (Ivanov, 2001). The resulting anyon theory is the Ising theory, with anyon types
{1,σ,ψ}, where
σ labels the vortex and
ψ labels the fermion. A central question is whether the MZM parity operators—the only local algebraic data available from the zero-mode subspace—uniquely determine the full anyonic exchange matrix, including all phases. Moreover, the consistency of braiding with the hexagon equation, a defining axiom of modular tensor categories (MTCs), imposes constraints relating the topological spin to the chiral central charge. This raises the question of whether non-chira
The non-Abelian topological phase with Fibonacci anyons minimally supports universal quantum computation. In order to investigate the possible phase transitions out of the Fibonacci topological phase, we propose a generic quantum-net wave function with two tuning parameters dual with each other, and the norm of the wave function can be exactly mapped into a partition function of the two-coupled ϕ^{2}-state Potts models, where ϕ=(sqrt[5]+1)/2 is the golden ratio. By developing the tensor network representation of this wave function on a square lattice, we can accurately calculate the full phase diagram with the numerical methods of tensor networks. More importantly, it is found that the non-Abelian Fibonacci topological phase is enclosed by three distinct nontopological phases and their dual phases of a single ϕ^{2}-state Potts model: the gapped dilute net phase, critical dense net phase, and spontaneous translation symmetry breaking gapped phase. We also determine the critical properties of the phase transitions among the Fibonacci topological phase and those nontopological phases.