Mashkevich S. Quantum and statistical mechanics of systems with fractional statistics

Fractional statistics in low-dimensional systems is considered. The exact multiplicities of three-anyon states are obtained, and the spectrum of three anyons in a harmonic potential is calculated numerically. The notion of anyon trajectories, i.e., continuous dependences of energy levels on angular momentum, is introduced. It is shown how the concept of trajectories simplifies the understanding of spectra of an arbitrary number of anyons. The third and fourth virial coefficients of anyons are calculated numerically, and conclusions drawn concerning general thermodynamic properties of the anyon gas. The cluster and virial expansions for multispecies anyons are built. The mixed virial coefficients of third order are computed numerically, and an approximation of two-particle correlations, in which the virial expansion has a very simple form, is worked out. The (1+2) and (2+1)-body problems for a system of particles with magnetic impurities are solved numerically. Expressions for virial coefficients of finite-size anyons in two limiting cases are obtained. The statistical form factor in a specific field-theoretical model where finite-size anyons arise is calculated. Anyon perturbation theory is worked out consistently. It is shown how to "correct" perturbation theory, to avoid singularities inherent for pointlike anyons, and make it work both for pointlike and finite-size anyons. A three-dimensional model in which fractional statistics effectively arises is worked out. A quantum-mechanical model of exclusion statistics is constructed. The thermodynamics of exclusion statistics is worked out. The generic cluster and virial expansions both for one and many species of particles are obtained, and the connection of such statistics with the exactly solvable model of anyons on the lowest Landau level in the multispecies case established.