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arXiv:0801.1413 · Low Temperature Physics

The relation between fractional statistics and finite bosonic systems in the one-dimensional case

Andrij A. Rovenchak

Published May 1, 2009• 16 citations
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Abstract

The equivalence is established between a one-dimensional (1D) Bose system with a finite number of particles and a system obeying fractional (intermediate) Gentile statistics, in which the maximum occupation of single-particle energy levels is limited. A system of 1D harmonic oscillators, providing a model for a harmonically trapped Bose gas, is considered. The results are generalized for a system with a power-law energy spectrum.

DOI
10.1063/1.3132748

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The equivalence is established between a one-dimensional (1D) Bose system with a finite number of particles and a system obeying fractional (intermediate) Gentile statistics, in which the maximum occupation of single-particle energy levels is limited. A system of 1D harmonic oscillators, providing a model for a harmonically trapped Bose gas, is considered. The results are generalized for a system with a power-law energy spectrum.
We discuss recent results on the relation between the strongly interacting one-dimensional Bose gas and a gas of ideal particles obeying nonmutual generalized exclusion statistics (GES). The thermodynamic properties considered include the statistical profiles, the specific heat and local pair correlations. In the strong coupling limit γ → ∞, the Tonks-Girardeau gas, the equivalence is with Fermi statistics. The deviation from Fermi statistics during boson fermionization for finite but large interaction strength γ is described by the relation α ≈ 1 - 2/γ, where α is a measure of the GES. This gives a quantitative description of the fermionization process. In this sense the recent experimental measurement of local pair correlations in a 1D Bose gas of87Rb atoms also provides a measure of the deviation of the GES parameter α away from the pure Fermi statistics value α=1. Other thermodynamic properties, such as the distribution profiles and the specific heat, are also sensitive to the statistics. They also thus provide a way of exploring fractional statistics in the strongly interacting 1D Bose gas.
Abstract We study the unitary dynamics of a 1D gas of hard-core bosons trapped into a harmonic potential which varies periodically in time with frequency . Such periodic systems can be classified into orbits of different monodromies corresponding to two different physical situations, namely the case in which the bosonic cloud remains stable during the time-evolution and the case where it turns out to be unstable. In the present work we derive in the large particle number limit exact results for the stroboscopic evolution of the energy and particle densities in both physical situations.
We study the thermodynamic behavior of an ideal gas of bosons trapped in a three-dimensional anisotropic harmonic-oscillator potential. The condensate fraction as well as the specific heat is calculated using the Euler-Maclaurin approximation. For a finite number of particles there is no phase transition, but there is a well-defined temperature at which the condensation starts. We also consider condensation in lower dimensions, and, for one-dimensional systems, we discuss the dependence of the condensate fraction and heat capacity on the ensemble used.
Closed form, analytical results for the finite-temperature one-body density matrix, and Wigner function of a d -dimensional, harmonically trapped gas of particles obeying exclusion statistics are presented. As an application of our general expressions, we consider the intermediate particle statistics arising from the Gentile statistics, and compare its thermodynamic properties to the Haldane fractional exclusion statistics. At low temperatures, the thermodynamic quantities derived from both distributions are shown to be in excellent agreement. As the temperature is increased, the Gentile distribution continues to provide a good description of the system, with deviations only arising well outside of the degenerate regime. Our results illustrate that the exceedingly simple functional form of the Gentile distribution is an excellent alternative to the generally only implicit form of the Haldane distribution at low temperatures.
Bosons in the form of ultracold alkali-metal atoms can be confined to a one-dimensional (1D) domain by the use of harmonic traps. This motivates the study of the ground-state occupations ${\ensuremath{\lambda}}_{i}$ of effective single-particle states ${\ensuremath{\varphi}}_{i},$ in the theoretical 1D impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The ${\ensuremath{\lambda}}_{i}$ and ${\ensuremath{\varphi}}_{i}$ are the eigenvalues and eigenfunctions, respectively, of the one-body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed i the occupations ${\ensuremath{\lambda}}_{i}$ are asymptotically proportional to N\sqrt{N}N​ in both the circular and harmonically trapped cases.
The equivalence of a quantum system of particles interacting with a two-body inverse square potential to a system of noninteracting particles obeying 1D fractional statistics (1D anyons), stated by Polychronakos for particles on a line, is studied for the cases where the interacting system is placed (i) into a harmonic potential on a line, and (ii) on a ring, with imposing periodic boundary conditions. In the first case, reducibility of the interacting system to the Calogero system is used to explore the statistical distribution for free 1D anyons. On a ring, the thermodynamic limit is discussed in terms of the thermodynamic (asymptotic) Bethe ansatz. Yang and Yang’s integral equation is treated in this case as describing the statistical mechanics of free 1D anyons. It gives a functional equation for the statistical distribution of 1D anyons consistent with the harmonic potential approach. We show that on a ring of a finite circumference, a system of two free 1D anyons is equivalent to a system of two particles interacting with an inverse sine square potential (the Sutherland system). We also discuss the relation to the statistical mechanics of free anyons in 2+1 dimensions.