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arXiv:cond-mat/9505053 · Physical review. B, Condensed matter

Bosonic and fermionic single-particle states in the Haldane approach to statistics for identical particles

Serguei B. Isakov

Published Mar 1, 1996• 3 citations

Abstract

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.

DOI: 10.1103/physrevb.53.6585

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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.

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

δ

-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.

The thermodynamic of particles with intermediate statistics interpolating between Bose and Fermi statistics is addressed in the simple case where there is one quantum number per particle. Such systems are essentially one-dimensional. As an illustration, one considers the anyon model restricted to the lowest Landau level of a strong magnetic field at low temperature, the generalization of this model to several particles species, and the one-dimensional Calogero model. One reviews a unified algorithm to compute the statistical mechanics of these systems. It is pointed out that Haldane's generalization of the Pauli principle can be deduced from the anyon model in a strong magnetic field at low temperature.

The thermodynamic of particles with intermediate statistics interpolating between Bose and Fermi statistics is addressed in the simple case where there is one quantum number per particle. Such systems are essentially one-dimensional. As an illustration, one considers the anyon model restricted to the lowest Landau level of a strong magnetic field at low temperature, the generalization of this model to several particles species, and the one-dimensional Calogero model. One reviews a unified algorithm to compute the statistical mechanics of these systems. It is pointed out that Haldane's generalization of the Pauli's principle can be deduced from the anyon model in a strong magnetic field at low temperature.

While elementary particles obey either bosonic or fermionic exchange statistics, generalized exchange statistics that interpolate between bosons and fermions---applicable to quasiparticles---constitute an intriguing topic from both the fundamental and practical points of view. This work develops a scattering framework for two identical one-dimensional (1D) bosonic anyons and two identical 1D fermionic anyons with zero-range contact interactions. The two-body system with zero-range interactions, in both free space and under external confinement, is used to illustrate the recently proposed bosonic-anyon---fermionic-anyon mapping (R. Hidalgo-Sacoto et al., Phys. Rev. A 112, 063309 (2025)), which connects the eigenstates of bosonic anyons to those of fermionic anyons and vice versa. Performing explicit calculations for two-particle systems, the momentum distributions and the off-diagonal correlations of the single-particle density matrix for bosonic anyons and fermionic anyons are confirmed to be distinct. We also confirm the previously derived asymptotic coefficients of the momentum distribution tail at orders ${k}^{\ensuremath{-}2}$ and ${k}^{\ensuremath{-}3}$ for two harmonically confined anyons. Nonuniversal contributions at order ${k}^{\ensuremath{-}4}$ are discussed.

The quantum statistical mechanics of an ideal gas with fractional exclusion (i.e., Haldane-Wu) statistics in arbitrary dimensions is discussed. The general formulation for pressure and density of the system is obtained in a closed form in terms of the

D

-dimensional momentum representation, which can be regarded as a natural generalization of the classic results for Fermi and Bose gases. Using this, it is shown that ideal gases with fractional exclusion statistics can be regarded as composites of fermions and bosons, and that no condensation occurs at low temperature except for the pure boson case.