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Ukrainian Journal of Physics

Two-Parameter Modifications of Anyonic Statistics

M. Hornetska, Andrij A. Rovenchak

Published Feb 1, 2016• 2 citations
Publisher

Abstract

Two-parameter models of fractional statistics aimed at finding an expression for the occupation numbers of free anyons have been considered. Virial coefficients are found for statistics of several types: к-deformed Polychronakos and Haldane–Wu statistics, Polychronakos and Haldane–Wu statistics modified with the q-exponential in the bosonic limit, and incomplete and nonadditive Gentile statistics for various level-filling maxima. A relation between the anyonic statistics and various statistics of fractional types is found and analyzed.

DOI
10.15407/ujpe61.02.0168

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Two-parameter models of fractional statistics aimed at finding an expression for the occupation numbers of free anyons have been considered. Virial coefficients are found for statistics of several types: к-deformed Polychronakos and Haldane–Wu statistics, Polychronakos and Haldane–Wu statistics modified with the q-exponential in the bosonic limit, and incomplete and nonadditive Gentile statistics for various level-filling maxima. A relation between the anyonic statistics and various statistics of fractional types is found and analyzed.
The statistics of q-oscillators, quons, and, to some extent, of anyons are studied and the basic differences among these objects are pointed out In particular, the statistical distributions for different bosonic and fermionic q-oscillators are found for their corresponding Fock space representations in the case when the Hamiltonian is identified with the number operator. In this case and for non-relativistic particles, the single-particle temperature Green function is defined with q-deformed periodicity conditions. The equations of state for nonrelativistic and ultrarelativistic bosonic q-gases in an arbitrary space dimension are found near Bose statistics, as well as that for an anyonic gas near Bose and Fermi statistics. The first corrections to the second virial coefficients are also evaluated.
Do anyons, dynamically realized by the field theoretic Chern-Simons construction, obey fractional exclusion statistics? We find that they do if the statistical interaction between anyons and antianyons is taken into account. For this anyon model, we show perturbatively that the exchange statistical parameter of anyons is equal to the exclusion statistical parameter. We obtain the same result by applying the relation between the exclusion statistical parameter and the second virial coefficient in the nonrelativistic limit.
We obtain for an anyon gas in the high temperature limit a relation between the exclusion statistics parameter ggg and the Hausdorff dimension hhh, given by g=h(2−h)g=h(2-h)g=h(2−h). The anyonic excitations are classified into equivalence classes labeled by Hausdorff dimension, hhh, and in that limit, the parameter ggg give us the second virial coefficient for any statistics, ννν. The anyonic excitations into the same class hhh get the same value of this virial coefficient.
We discuss the connection between anyons (particles with fractional statistics) and deformed Lie algebras (quantum groups). After a brief review of the main properties of anyons, we present the details of the anyonic realization of all deformed classical Lie algebras in terms of anyonic oscillators. The deformation parameter of the quantum groups is directly related to the statistics parameter of the anyons. Such a realization is a direct generalization of the Schwinger construction in terms of fermions and is based on a sort of bosonization formula which yields the generators of the deformed algebra in terms of the undeformed ones. The entire procedure is well defined on two-dimensional lattices, but it can be consistently reduced also to one-dimensional chains.