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arXiv:hep-th/9407107 · Modern Physics Letters B

CALOGERO MODELS FOR DISTINGUISHABLE PARTICLES

Cyril Furtlehner, Stéphane Ouvry

Published Apr 10, 1995• 15 citations
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Abstract

Motivated by topological bidimensional quantum models for distinguishable particles, and by Haldane's definition of mutual statistics for different species of particles, we propose a new class of one-dimensional [Formula: see text] Calogero model with coupling constants gij depending on the labels of the particles. We solve the ground state problem, and show how to build some classes of excited states.

DOI
10.1142/s0217984995000450

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Motivated by topological bidimensional quantum models for distinguishable particles, and by Haldane's definition of mutual statistics for different species of particles, we propose a new class of one-dimensional [Formula: see text] Calogero model with coupling constants gij depending on the labels of the particles. We solve the ground state problem, and show how to build some classes of excited states.
The 3-body Calogero problem is solved by separation of variables for arbitrary exchange statistics. A numerical computation of the 4-body spectrum is also presented. The results display new features in comparison with the standard case of bosons and fermions, for instance the energies are not linear with the interaction parameter ννν and Bethe ansatz as well as Haldane's statistics are not verified.
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.
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.
Following the definition of Haldane's exclusion statistics, we propose a new and simple representation of the spectrum and the state counting of the quantum Calogero model. We show that N non-interacting particles with statistical interactions confined in an external harmonic well give exactly the same energy spectrum and the same degeneracy as those of the N -particle quantum Calogero model. We also show that this picture exactly corresponds to the generalized ideal gas, which was introduced by Bernard and Wu.
We construct a new many-body Hamiltonian with two- and three-body interactions in two space dimensions and obtain its exact many-body ground state for an arbitrary number of particles. This ground state has a novel pairwise correlation. A class of exact solutions for the excited states is also found. These excited states display an energy spectrum similar to the Calogero-Sutherland model in one dimension. The model reduces to an analog of the well-known trigonometric Sutherland model when projected on to a circular ring.