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arXiv:cond-mat/9811415 · Physical Review Letters

Realization of Haldane's Exclusion Statistics in a Model of Electron-Phonon Interactions

C. Pépin, Alexei M. Tsvelik

Published May 10, 1999• 5 citations
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Abstract

We discuss an integrable model describing one-dimensional relativistic fermions interacting with the Toda array of $N$ scalar fields. In the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and at low temperatures it is possible to decouple scalar bosonic modes and consider fermions separately. They have a trivial two-body scattering matrix and obey fractional statistics. As far as we know the original model presents the first example of a model with local bare interactions generating purely statistical interactions between renormalized particles. As a by-product we obtain nontrivial thermodynamic equations for the Toda array in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$.

DOI
10.1103/physrevlett.82.3859

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We construct thermodynamics of the one-dimensional supersymmetric {\it t-J} model with the $ 1/\sin^2$ interaction and hopping. The thermodynamics is described exactly in terms of free spinons and holons obeying Haldane's fractional exclusion statistics at all temperatures. Moreover, at low temperatures the semionic spinons and holons decouple resulting in the spin-charge separation in thermodynamic properties. We obtain explicit results for the spin and charge susceptibilities and specific heat, and interpret them in terms of the fractional exclusion statistics. Extension to the multi-component {\it t-J} model shows that the excitations obey either fractional statistics for g-ons with partial polarization of components, or the parafermionic one without polarization.
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We construct thermodynamics of the one-dimensional supersymmetric t-J model with the 1/sin 2 interaction and hopping. The thermodynamics is described exactly in terms of free spinons and holons obeying Haldane's fractional exclusion statistics at all temperatures. Moreover, at low temperatures the semionic spinons and holons decouple resulting in the spin-charge separation in thermodynamic properties. We obtain explicit results for the spin and charge susceptibilities and specific heat, and interpret them in terms of the fractional exclusion statistics. Extension to the multi-component t-J model shows that the excitations obey either fractional statistics for g-ons with partial polarization of components, or the parafermionic one without polarization.
We consider N fermions in a two-dimensional harmonic oscillator potential interacting with a very short-range repulsive pair-wise potential. The ground-state energy of this system is obtained by performing a Thomas-Fermi as well as a self-consistent Hartree-Fock calculation. The two results are shown to agree even for a small number of particles. We next use the finite temperature Thomas-Fermi method to demonstrate that in the local density approximation, these interacting fermions are equivalent to a system of noninteracting particles obeying the Haldane-Wu fractional exclusion statistics. It is also shown that mapping onto to a system of N noninteracting quasi-particles enables us to predict the energies of the excited states of the N-body system.
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 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.