A topological argument similar to that of Leinaas and Myrheim implies that a non-trivial statistical phase factor can also arise from exchanging twice a pair of distinguishable particles in two dimensions. Some general properties of this phase factor are deduced. Wilczek's model for anyons and the Laughlin theory for the quasiparticles in the fractional quantum Hall ground states are examined in light of these properties, and the former is generalized for systems containing many species of anyons. The statistical properties of holons and spinons relative to each other are briefly discussed as an example.
A topological argument similar to that of Leinaas and Myrheim implies that a non-trivial statistical phase factor can also arise from exchanging twice a pair of distinguishable particles in two dimensions. Some general properties of this phase factor are deduced. Wilczek's model for anyons and the Laughlin theory for the quasiparticles in the fractional quantum Hall ground states are examined in light of these properties, and the former is generalized for systems containing many species of anyons. The statistical properties of holons and spinons relative to each other are briefly discussed as an example.
The quantum Hall effect has been the prototypical example for many emerging fields of Physics, including the study of topological phases of matter, topological order, and topological insulators. The objects of interest in this thesis are anyons, which are elementary excitations in fractional quantum Hall (FQH) fluids. Anyons are quasiparticles with fractional charge and fractional statistics. The latter prop- erty makes it a subject of immense interest due to application in fault-tolerant quantum computing. Here, we employ a perspective that anyons are the appro- priate degree of freedom to describe the relevant physics of FQH systems. This is because when a topological phase is assumed and the energy gap goes to infinity, anyons (“quasiholes”) are the only possible elementary excitations, since they are not energetically punished by the correspondng model Hamiltonian. With respect to a realistic Hamiltonian, however, this choice of degree of freedom depends on the exact effective interaction in the system. For example, while the Laughlin state is the model state at filling factor n + 1/3 (where n is some integer), when the effective electron-electron interaction is more long-ranged, the quasiholes of a Gaffnian state may be excited and become the better degree of freedom to describe physics in the same filling factor. This perspective results in a surprising observa- tion that a Laughlin quasihole may fractionalize under certain conditions, leading to a quasiparticle of charge e/6 that can potentially be detected in experiments. Another important aspect of anyons is their intrinsic spin and the corresponding spin-statistics relation, which must account for the fact that FQH anyons are not point particles, but objects with finite sizes and intricate internal structures. Our investigation leads to a rigorous derivation of the general spin-statistics relation for anyons in Abelian FQH phases. Furthermore, we show it is possible to fully capture the important features of the statistics by a conformal Hilbert space (CHS) description. Our analysis provides the necessary tools to study such geometric effects, which we use to investigate how disorder in a realistic system may affect different experimental schemes for measuring anyon statistics.
We have studied the ground-state (GS) properties of fractional-statistics particles (anyons) by exact numerical solution for small systems. We have seen in our results (depending on the choice of parameters for the Hamiltonian) the fractional-statistics analogs of a number of well-known ground states for conventional particles, including particularly the quantum Hall state (which we relate to the observed ‘hierarchy’ states), and a superconducting GS, the latter being formed (for half-statistics particles or semions) by pairing. We suggest possible experimental signatures for distinguishing semion from fermion superconductors. We also directly test the ‘mean-field theory’ of statistics against our exact results, and find it to treat GS properties well.
The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter θ. θ=0,π correspond to bosons and fermions respectively; at intermediate values we say that we have fractional statistics. In two dimensions, θ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve non-abelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of non-abelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern--Simons terms. The crossings of one-dimensional anyons on a ring are uni-directional, such that a fractional phase θ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently the anyon behavior predicted for quasiparticles in the ν=1/3 fractional quantum Hall state has been observed both in scattering and in interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.
Particles possessing fractional statistics in two dimension have been dubbed anyons. We want to show how it is possible to extend this concept in the case of conformally invariant systems in one dimension, and reproduce the flux-tube attachment construction due to Wilczek. We will introduce a class of models describing 1D anyons, which has the great advantage over its 2D counterparts to be exactly solvable. The properties of these 1D anyons generalize those of a Luttinger liquid and yield a complete classification of 1D fixed points when time-reversal symmetry is broken. I HOW ANYONS MAY ARISE IN ONE DIMENSION Fractional statistics arises in two dimensions because the exchange of particle is encoded by the braid group BN rather than by the permutation group S N. This fractional exchange statistics is often thought to be possible only in two dimensions in contrast with for instance fractional exclusion statistics which is defined independently of the dimensionality. However already in their seminal paper Leinaas and hfyrheim investigated the possibility to define intermediate statistics in one dimension by considering generalized boundary conditions for wavefunctions
We consider a thought experiment where two distinct species of 2D particles in a perpendicular magnetic field interact via repulsive potentials. If the magnetic field and the interactions are strong enough, one type of particles forms a Laughlin state and the other type couples to Laughlin quasiholes. We show that, in this situation, the motion of the second type of particles is described by an effective Hamiltonian, corresponding to the magnetic gauge picture for noninteracting anyons. The argument is in accord with, but distinct from, the Berry phase calculation of Arovas, Schrieffer, and Wilczek. It suggests possibilities to observe the influence of effective anyon statistics in fractional quantum Hall systems.
General kinematical restrictions on quantum-theories of N identical anyons have been established by using braid groups. For an orientable compact surface (without boundary) of genus g, the statistical parameter θ is a rational multiple of π, θ=πp/v (p and v mutually prime integers), and the number of components of the wavefunction is (a multiple of) vg. The particle-number is N=rv+1−g (r integer) for spinless (S=0) and N=rv for spinning (S=θ/2π) anyons. The restrictions for spinning anyons are consistent with results for fractional quantum Hall systems, nonlinear O(3) field theory, and Chern-Simons theory. The multi-component structure, which appears for g≥1, reflects an internal collective degree of freedom. A comparison with the multi-component wavefunctions needed to describe systems with fractionally charged quasi-particles yields consistency relations between charge and statistics. Non-orientable surfaces do not allow fractional statistics.