Anyons in a weakly interacting system

1991

We obtain a hierarchy of effective Hamiltonians which allow for a unified treatment of the fractional quantum Hall effect and a gas of fractional-statistics particles (anyons) in two dimensions. Anyon superconductivity is the analog of the fractional quantum Hall effect. For a rational statistics parameter a, P/Q with PQ even, Q anyons bind forming a charge-Qe superfluid.

Reviews of Modern Physics, 2008

Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as Non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding quasiparticles, and then measuring the multi-quasiparticle states. The fault-tolerance of a topological quantum computer arises from the non-local encoding of the states of the quasiparticles, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the ν = 5/2 state, although several other prospective candidates have been proposed in systems as disparate as ultra-cold atoms in optical lattices and thin film superconductors. In this review article, we describe current research in this field, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. We address both the mathematical underpinnings of topological quantum computation and the physics of the subject using the ν = 5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

Physical Review Letters, 1990

In two space dimensions we generalize the boson-vortex duality picture for spinless particles to bosonvortex-Skyrmion duality for spin-2 particles. The spin-singlet fractional quantum Hall eff'ect and spinsinglet anyon superconductivity can be understood as the condensation of vortices and Skyrmions into various fractional quantum Hall states.

2016

The aim of this text is to provide an introduction to the theory of topological quantum computation. We give an introduction to the theory of anyons (two-dimensional quasi-particle excitations that have exotic statistics) and how we can use these to perform fault-tolerant quantum computation. Additionally, we give a complete description of an exactly solvable spin lattice model whose local low-energy excitations of the Hamiltonian behave as anyons. We conclude by indicating how this model can be generalized so as to perform universal quantum computation.

Physical Review B, 2000

Journal of Physics: Condensed Matter, 2005

An anyon wave function (characterized by the statistical factor n) projected onto the lowest Landau level is derived for the fractional quantum Hall effect states at filling factor ν = n/(2pn + 1) (p and n are integers). We study the properties of the anyon wave function by using detailed Monte Carlo simulations in disk geometry and show that the anyon ground-state energy is a lower bound to the composite fermion one. Our results suggest that the composite fermions can be viewed as a combination of anyons and a fluid of charge-neutral dipoles.

Annals of Physics, 2008

The dichotomy between fermions and bosons is at the root of many physical phenomena, from metallic conduction of electricity to super-fluidity, and from the periodic table to coherent propagation of light. The dichotomy originates from the symmetry of the quantum mechanical wave function to the interchange of two identical particles. In systems that are confined to two spatial dimensions particles that are neither fermions nor bosons, coined ''anyons'', may exist. The fractional quantum Hall effect offers an experimental system where this possibility is realized. In this paper we present the concept of anyons, we explain why the observation of the fractional quantum Hall effect almost forces the notion of anyons upon us, and we review several possible ways for a direct observation of the physics of anyons. Furthermore, we devote a large part of the paper to non-abelian anyons, motivating their existence from the point of view of trial wave functions, giving a simple exposition of their relation to conformal field theories, and reviewing several proposals for their direct observation.

Physical Review A, 2004

We show that bosonic fields may present anyonic behavior when interacting with a fermion in a Jaynes-Cummings-like model. The proposal is accomplished via the interaction of a two-level system with two quantized modes of a harmonic oscillator; under suitable conditions, the system acquires a fractional geometric phase. A crucial role is played by the entanglement of the system eigenstates, which provides a twodimensional confinement in the effective evolution of the system, leading to the anyonic behavior. For a particular choice of parameters, we show that it is possible to transmute the statistics of the system continually from fermions to bosons. We also present an experimental proposal, in an ion-trap setup, in which fractional statistical features can be generated, controlled, and measured.

An exotic feature of the fractional quantum Hall effect is the emergence of anyons, which are quasiparticle excitations with fractional statistics. In the presence of a symmetry, such as U (1) charge conservation, it is well known that anyons can carry fractional symmetry quantum numbers. In this work we reveal a different class of symmetry realizations: i.e. anyons can " breed " in multiples under symmetry operation. We focus on the global Ising (Z2) symmetry and show examples of these unconventional symmetry realizations in Laughlin-type fractional quantum Hall states. One remarkable consequence of such an Ising symmetry is the emergence of anyons on the Ising symmetry domain walls. We also provide a mathematical framework which generalizes this phenomenon to any Abelian topological orders.

Physical Review Letters, 2008

Anyons are particle-like excitations of strongly correlated phases of matter with fractional statistics, characterized by nontrivial changes in the wavefunction, generalizing Bose and Fermi statistics, when two of them are interchanged. This can be used to perform quantum computations . We show how to simulate the creation and manipulation of Abelian and non-Abelian anyons in topological lattice models using trapped atoms in optical lattices. Our proposal, feasible with present technology, requires an ancilla particle which can undergo single particle gates, be moved close to each constituent of the lattice and undergo a simple quantum gate, and be detected.