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Physical Review Letters

Exclusion Statistics: Low-Temperature Properties, Fluctuations, Duality, and Applications

Chetan Nayak, Frank A. Wilczek

Published Nov 14, 1994• 132 citations
Publisher

Abstract

We derive some physical properties of ideal assemblies of identical particles obeying generalized exclusion statistics. We discuss fluctuations and in this connection point out a fundamental contrast to conventional quantum statistics. We demonstrate a duality relating the distribution of particles at statistics $g$ to the distribution of holes at statistics $\frac{1}{g}$. We suggest an application to Mott insulators.

DOI
10.1103/physrevlett.73.2740

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We derive some physical properties of ideal assemblies of identical particles obeying generalized exclusion statistics. We discuss fluctuations and in this connection point out a fundamental contrast to conventional quantum statistics. We demonstrate a duality relating the distribution of particles at statistics ggg to the distribution of holes at statistics 1g\frac{1}{g}g1​. We suggest an application to Mott insulators.
We derive some physical properties of ideal assemblies of identical particles obeying generalized exclusion statistics. We discuss fluctuations, and in this connection point out a fundamental contrast to conventional quantum statistics. We demonstrate a duality relating the distribution of particles at statistics ggg to the distribution of holes at statistics 1/g1/g1/g. We suggest applications to Mott insulators.
We analytically calculate some thermodynamic quantities of an ideal ggg-on gas obeying generalized exclusion statistics. We show that the specific heat of a ggg-on gas (g≠0g \neq 0g=0) vanishes linearly in any dimension as T→0T \to 0T→0 when the particle number is conserved and exhibits an interesting dual symmetry that relates the particle-statistics at ggg to the hole-statistics at 1/g1/g1/g at low temperatures. We derive the complete solution for the cluster coefficients bl(g)b_l(g)bl​(g) as a function of Haldane's statistical interaction ggg in DDD dimensions. We also find that the cluster coefficients bl(g)b_l(g)bl​(g) and the virial coefficients al(g)a_l(g)al​(g) are exactly mirror symmetric (lll=odd) or antisymmetric (lll=even) about g=1/2g=1/2g=1/2. In two dimensions, we completely determine the closed forms about the cluster and the virial coefficients of the generalized exclusion statistics, which exactly agree with the virial coefficients of an anyon gas of linear energies. We show that the ggg-on gas with zero chemical potential shows thermodynamic properties similar to the photon statistics. We discuss some physical implications of our results.
After a very brief introduction to the concept of fractional exclusion statistics (FES), introduced by Haldane in Phys. Rev. Lett. 67, 937 (1991), I shall present some recent results regarding the basic properties of this statistics. These properties have been overlooked for almost two decades and because of that, the application of FES to concrete physical systems was, to a certain degree, arbitrary. Essentially, the FES is a generalization of the Pauli Exclusion Principle which allows one to describe systems of interacting particles (bosons or fermions) as systems of ideal particles, but which obey a more peculiar type of statistics. In the second part of the paper I shall present some application of FES by analyzing concrete systems of interacting particles.
We consider the generalized exclusion statistics in the Kondo problem. The thermodynamic Bethe ansatz equations have been used for a multicomponent system of particles obeying the generalized exclusion principle. We have found a relation between the derivative of the phase shift of the scattering matrix for Fermi particles and for particles characterized by generalized exclusion statistics. We show that the statistical matrix in the Kondo problem has a universal form in the high and low temperature limits.