Realizing non-Abelian statistics

Abstract

We construct a series of 2+1-dimensional models whose quasiparticles obey non-Abelian statistics. The adiabatic transport of quasiparticles is described by using a correspondence between the braid matrix of the particles and the scattering matrix of 1+1-dimensional field theories. We discuss in depth lattice and continuum models whose braiding is that of SO(3) Chern-Simons gauge theory, including the simplest type of non-Abelian statistics, involving just one type of quasiparticle. The ground-state wave function of an SO(3) model is related to a loop description of the classical two-dimensional Potts model. We discuss the transition from a topological phase to a conventionally-ordered phase, showing in some cases there is a quantum critical point.

20 pages in two-column format. v2: fixed typos and added references

References in corpus (30)

• Fault-tolerant quantum computation by anyons
• Non-abelian statistics of half-quantum vortices in p-wave superconductors
• String-net condensation: A physical mechanism for topological phases
• "Deconfined" quantum critical points
• Topological orders and Edge excitations in FQH states
• Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level
• Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm
• An RVB phase in the triangular lattice quantum dimer model
• Z_2 Gauge Theory of Electron Fractionalization in Strongly Correlated Systems
• $2n$ Quasihole States Realize $2^{n-1}$-Dimensional Spinor Braiding Statistics in Paired Quantum Hall States
• Topological Order and Conformal Quantum Critical Points
• Spiders for rank 2 Lie algebras
• Large N expansion for frustrated and doped quantum antiferromagnets
• A Class of $P,T$-Invariant Topological Phases of Interacting Electrons
• Fermions, strings, and gauge fields in lattice spin models
• New Class of Non-Abelian Spin-Singlet Quantum Hall States
• Quantum groups and nonabelian braiding in quantum Hall systems
• Quantum Criticality and Deconfinement in Phase Transitions Between Valence Bond Solids
• Quantum Field Theory of Nonabelian Strings and Vortices
• Non-abelian vortices and non-abelian statistics
• Quantum symmetries in discrete gauge theories
• On bipartite Rokhsar-Kivelson points and Cantor deconfinement
• Landau-Ginzburg Theories for Non-Abelian Quantum Hall States
• N=2 Supersymmetry, Painleve III and Exact Scaling Functions in 2D Polymers
• An extended Hubbard model with ring exchange: a route to a non-Abelian topological phase
• A Line of Critical Points in 2+1 Dimensions: Quantum Critical Loop Gases and Non-Abelian Gauge Theory
• Anyons in discrete gauge theories with Chern-Simons terms
• G/G models as the strong coupling limit of topologically massive gauge theory
• Non-Abelian topological phases in an extended Hubbard model
• Quasiparticle operators with non-Abelian braiding statistics

Cited by in corpus (54)

• Non-Abelian Anyons and Topological Quantum Computation
• Stability of the quantum spin Hall effect: effects of interactions, disorder, and Z_2 topology
• Universal topological quantum computation from a superconductor/Abelian quantum Hall heterostructure
• Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum
• Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension
• Topological order in an exactly solvable 3D spin model
• Braid Topologies for Quantum Computation
• Universal entanglement entropy in 2D conformal quantum critical points
• Creation, manipulation, and detection of Abelian and non-Abelian anyons in optical lattices
• Interferometry of non-Abelian Anyons
• Topological Quantum Compiling
• Topological Quantum Compiling with Reinforcement Learning
• Exact entanglement renormalization for string-net models
• Homological Error Correction: Classical and Quantum Codes
• Topological Quantum Computing with Only One Mobile Quasiparticle
• Topology driven quantum phase transitions in time-reversal invariant anyonic quantum liquids
• Why should anyone care about computing with anyons?
• Topological Quantum Computing with Read-Rezayi States
• Topological order from quantum loops and nets
• Dynamics at and near conformal quantum critical points
• Topological phases and topological entropy of two-dimensional systems with finite correlation length
• Anyons in Quantum Hall Interferometry
• Scaling of Entanglement Entropy at 2D quantum Lifshitz fixed points and topological fluids
• Multiparticle Entanglement in the Lipkin-Meshkov-Glick Model
• Linear independence of nearest neighbor valence bond states on the kagome lattice and construction of SU(2)-invariant spin-1/2-Hamiltonian with a Sutherland-Rokhsar-Kivelson quantum liquid ground state
• The N-eigenvalue Problem and Two Applications
• Creating and manipulating non-Abelian anyons in cold atom systems using auxiliary bosons
• Loop models and their critical points
• Condensation-Driven Phase Transitions in Perturbed String Nets
• Proposal for reading out anyon qubits in non-abelian $\nu =12/5$ quantum Hall state
• The breaking of quantum double symmetries by defect condensation
• Critical points in coupled Potts models and critical phases in coupled loop models
• Test of Einstein-Podolsky-Rosen Steering Based on the All-Versus-Nothing Proof
• Ashkin-Teller universality in a quantum double model of Ising anyons
• BF-theory in graphene: a route toward topological quantum computing?
• Exploiting Geometric Degrees of Freedom in Topological Quantum Computing
• Fibonacci topological order from quantum nets
• Genetic braid optimization: A heuristic approach to compute quasiparticle braids
• Unified approach to topological quantum computation with anyons: From qubit encoding to Toffoli gate
• Loop condensation in the triangular lattice quantum dimer model
• Quantum criticality of loops with topologically constrained dynamics
• Realizing string-net condensation: Fibonacci anyon braiding for universal gates and sampling chromatic polynomials
• Non-Abelian Anyon Superconductivity
• Projected Entangled Pair States with continuous virtual symmetries
• Doubled Lattice Chern-Simons-Yang-Mills Theories with Discrete Gauge Group
• A composite particle construction of the Fibonacci fractional quantum Hall state
• Braiding Majorana Fermions
• String-nets, single and double-stranded quantum loop gases for non-Abelian anyons
• Critical behavior of loops and biconnected clusters on fractals of dimension d < 2
• Geometric properties of loop condensed phases on the square lattice
• Merging Quantum Loop Gases: a Route to Non-Abelian Topological Phases
• Topological phases of the compass ladder model
• A chain of strongly correlated SU(2)_4 anyons: Hamiltonian and Hilbert space of states
• Boson-Lattice Construction for Anyon Models