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arXiv:cond-mat/9401071 · Physical Review Letters

Haldane exclusion statistics and second virial coefficient

M. V. N. Murthy, Ravi Shankar

Published Jun 6, 1994• 80 citations
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Abstract

We show that Haldane's new definition of statistics, when generalized to infinite dimensional Hilbert spaces, is determined by the high temperature limit of the second virial coefficient. We thus show that this exclusion statistics parameter g of anyone is nontrivial and is completely determined by its exchange statistics parameter \ensuremath{\alpha}. We also compute g for quasiparticles in the Luttinger model and show that it is equal to \ensuremath{\alpha}.

DOI
10.1103/physrevlett.72.3629

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We show that Haldane's new definition of statistics, when generalized to infinite dimensional Hilbert spaces, is determined by the high temperature limit of the second virial coefficient. We thus show that this exclusion statistics parameter g of anyone is nontrivial and is completely determined by its exchange statistics parameter \ensuremath{\alpha}. We also compute g for quasiparticles in the Luttinger model and show that it is equal to \ensuremath{\alpha}.
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 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.
The thermodynamics of two-species gases described by the Haldane exclusion statistics is investigated. I use the high-temperature perturbation to find the third Virial coefficient for the gas in an arbitrary dimension. The perturbative result indicates that the two-dimensional thermodynamic potential becomes the linear function of the element of the Haldane statistics matrix, if the matrix is symmetric and particle mass is species independent. I also present a nonperturbative method to prove this factorizable property and an exact expression of the thermodynamic potential is suggested.
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.
I discuss Haldane's concept of generalized exclusion statistics (1991 Phys. Rev. Lett. 67 937) and show that it leads to inconsistencies in the calculation of the particle distribution that maximizes the partition function. These inconsistencies appear when mutual exclusion statistics is manifested between different subspecies of particles in the system. In order to eliminate these inconsistencies, I introduce new mutual exclusion statistics parameters, which are proportional to the dimension of the Hilbert subspace on which they act. These new definitions lead to properly defined particle distributions and thermodynamic properties. In another paper ( Preprint 0710.0728), I show that the fractional exclusion statistics manifested in general systems with interaction have these, physically consistent, statistics parameters.