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Journal of the Physical Society of Japan

Exclusion Statistics Interpretation of the Quantum Calogero Model

Hideaki Ujino, Miki Wadati

Published Nov 15, 1995• 7 citations
Publisher

Abstract

Following the definition of Haldane's exclusion statistics, we propose a new and simple representation of the spectrum and the state counting of the quantum Calogero model. We show that N non-interacting particles with statistical interactions confined in an external harmonic well give exactly the same energy spectrum and the same degeneracy as those of the N -particle quantum Calogero model. We also show that this picture exactly corresponds to the generalized ideal gas, which was introduced by Bernard and Wu.

DOI
10.1143/jpsj.64.4064

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Following the definition of Haldane's exclusion statistics, we propose a new and simple representation of the spectrum and the state counting of the quantum Calogero model. We show that N non-interacting particles with statistical interactions confined in an external harmonic well give exactly the same energy spectrum and the same degeneracy as those of the N -particle quantum Calogero model. We also show that this picture exactly corresponds to the generalized ideal gas, which was introduced by Bernard and Wu.
One-dimensional model of nonrelativistic particles with inverse-square interaction potential known as the Calogero-Sutherland model is shown to possess fractional statistics. Using the theory of Jack polynomials the exact dynamical density-density correlation function and the hole propagator part of one-particle Green's function at any rational interaction coupling constant are obtained and used to show clear evidences of the fractional exclusion statistics in the sense of Haldane's "generalized Pauli exclusion principle." This model is also endowed with the corresponding natural exchange statistics.
We use the Thomas-Fermi method to examine the thermodynamics of particles obeying Haldane exclusion statistics. Specifically, we study Calogero-Sutherland particles placed in a given external potential in one dimension. For the case of a simple harmonic potential (constant density of states), we obtain the exact one-particle spatial density and a closed form for the equation of state at finite temperature, which are both new results. We then solve the problem of particles in a x2/3{x}^{2/3}x2/3 potential (linear density of states) and show that Bose-Einstein condensation does not occur for any statistics other than bosons.
We give two formulations of exclusion statistics (ES) using a variable number of bosonic or fermionic single-particle states which depends on the number of particles in the system. Associated bosonic and fermionic ES parameters are introduced and are discussed for FQHE quasiparticles, anyons in the lowest Landau level and for the Calogero-Sutherland model. In the latter case, only one family of solutions is emphasized to be sufficient to recover ES; appropriate families are specified for a number of formulations of the Calogero-Sutherland model. We extend the picture of a variable number of single-particle states to generalized ideal gases with statistical interaction between particles of different momenta. Integral equations are derived which determine the momentum distribution for single-particle states and distribution of particles over the single-particle states in the thermal equilibrium. \textcopyright{} 1996 the American Physical Society.