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arXiv:cond-mat/9503026 · Physical Review Letters

Comment on “Thermodynamics of a One-Dimensional Ideal Gas with Fractional Exclusion Statistics”

Alain Dasnières de Veigy, Stéphane Ouvry

Published Jul 10, 1995• 17 citations
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Abstract

In a recent letter -Phys. Rev. Lett. 73, 3331 (1995)-, the conclusion was reached that, in the one-dimensional Calogero model, only the second virial coefficient is affected by the statistical parameter $α$, where $α$ is related to the coupling constant $κ/ x_{ij}^2$ of the Calogero interaction by $κ=α(α+1)$. We argue that it is not so, i.e. all virial coefficients are affected, if the thermodynamic limit is properly taken.

DOI
10.1103/physrevlett.75.352

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In a recent letter -Phys. Rev. Lett. 73, 3331 (1995)-, the conclusion was reached that, in the one-dimensional Calogero model, only the second virial coefficient is affected by the statistical parameter ααα, where ααα is related to the coupling constant κ/xij2κ/ x_{ij}^2κ/xij2​ of the Calogero interaction by κ=α(α+1)κ=α(α+1)κ=α(α+1). We argue that it is not so, i.e. all virial coefficients are affected, if the thermodynamic limit is properly taken.
In Phys. Rev. Lett. 67, 937 (1991) [1], Haldane introduced the fruitful concept of fractional exclusion statistics (FES). One of the most influential papers in which the thermodynamics of FES systems was deduced is Y.-S. Wu, Phys. Rev. Lett. 73, 922 (1994). Unfortunately, some important, but eventually subtle, properties of the exclusion statistics parameters were overlooked in the original paper [1] and in all the papers after that, including [2]. This omission makes the thermodynamics of FES systems inconsistent when mutual exclusion statistics is manifesting in the system. By this Comment I want to point-out this error--an error that persisted for such a long time--and to give the correct statistical mechanics interpretation of FES.
It is mentioned that anyon thermodynamic potential Q(α,N)Q(\alpha, N)Q(α,N) could not be factorized in terms characteristic of the ideal boson α=0\alpha =0α=0 and fermion α=1\alpha =1α=1 gases by the relation Q(α,N)=(1−α)Q(0,Nb)+αQ(1,Nf)Q(\alpha, N) = (1-\alpha) Q(0, N_b)+ \alpha Q(1, N_f)Q(α,N)=(1−α)Q(0,Nb​)+αQ(1,Nf​) in which N=Nf+NbN=N_f +N_bN=Nf​+Nb​, that claimed in Phys. Rev. Lett. 78, 3233 (1997). Our analyses indicate that the thermodynamic quantities of anyon gas may be factorized as Q(α)=αQ(1)+(1−α)Q(0)Q(\alpha) = \alpha Q(1) + (1-\alpha) Q(0)Q(α)=αQ(1)+(1−α)Q(0) only in the two-dimension system.
One-dimensional fractional statistics is studied using the Calogero-Sutherland model (CSM) which describes a system of non-relativistic quantum particles interacting with inverse-square two-body potential on a ring.The inversesquare exchange can be regarded as a pure statistical interaction and this system can be mapped to an ideal gas obeying the fractional exclusion and exchange statistics.The details of the exact calculations of the dynamical correlation functions for this ideal system is presented in this paper.An effective low-energy one-dimensional "anyon" model is constructed; and its correlation functions are found to be in agreement with those in the CSM; and this agreement provides an evidence for the equivalence of the first-and the second-quantized construction of the 1D anyon model at least in the long wave-length limit.Furthermore, the finite-size scaling applicable to the conformally invariant systems is used to obtain the complete set of correlation exponents for the CSM.
We show that the thermodynamic Bethe ansatz equations for one-dimensional integrable many-body systems can be reinterpreted in such a way that they only code the statistical interactions, in the sense of Haldane, between particles of identical or different momenta. Thus, the thermodynamic properties of these systems can be characterized by the generalized ideal gases recently proposed by one of us. For example, the Yang-Yang δδδ-function gas is a gas with specific statistical interactions between particles of different momenta, while the Calogero-Sutherland system provides a model for an ideal gas of particles with a fractional statistics.
We report on a study of virial expansions for interacting electrons in the lowest Landau level of a two-dimensional electron gas. For hard-core-model interactions, we derive analytic results valid at low temperatures and filling factors smaller than 1/31/31/3, and comment on their relationship with virial expansions for exclusion statistics models. In the high-temperature limit the virial coefficients reduce to those for a noninteracting electron system. For the first five virial coefficients, the crossover between low- and high-temperature limits has been studied numerically by using partition functions obtained from small system exact diagonalization calculations. Our results show that the exclusion statistics description of fractional Hall thermodynamics breaks down when the temperature exceeds a small fraction of the gap temperature.