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Journal of the Physical Society of Japan

Exclusion Statistical TBA and the Quantum Calogero Model

Hideaki Ujino, Miki Wadati

Published Dec 15, 1996• 2 citations
Publisher

Abstract

Thermodynamics of the quantum Calogero model is studied by the microcanonical ensemble approach. We formulate a thermodynamic Bethe ansatz (TBA), whose reference state is the generalized ideal gas with fractional statistics (free a -on). Using the a -onic TBA, the free-quasiparticle picture for the Calogero model is identified as the free a -on. We calculate the thermodynamic potential and the distribution function for the Calogero model in the thermodynamic limit. The thermodynamic potential at thermal equilibrium can be understood by the Bernoulli equation. We also show that the thermodynamics of the Calogero model and the Sutherland model are the same except for the dispersion relations.

DOI
10.1143/jpsj.65.3819

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Thermodynamics of the quantum Calogero model is studied by the microcanonical ensemble approach. We formulate a thermodynamic Bethe ansatz (TBA), whose reference state is the generalized ideal gas with fractional statistics (free a -on). Using the a -onic TBA, the free-quasiparticle picture for the Calogero model is identified as the free a -on. We calculate the thermodynamic potential and the distribution function for the Calogero model in the thermodynamic limit. The thermodynamic potential at thermal equilibrium can be understood by the Bernoulli equation. We also show that the thermodynamics of the Calogero model and the Sutherland model are the same except for the dispersion relations.
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.
Abstract The Thermodynamic Bethe Ansatz (TBA) allows us to study finite size and finite temperature effects of an integrable model. This chapter investigates the integral equations that determine the free energy and gives their physical interpretation. It discusses Casimir energy, Bethe relativistic wave function, the derivation of thermodynamics, the meaning of pseudo-energy (dressed energy and momentum), infrared and ultraviolet limits, the coefficient of bulk energy, the general form of the TBA equations, the thermodynamics of the free field theories, L-channel quantization and the LeClair–Mussardo formula. It also covers the application of the Yang–Lee S-matrix, the magnetic field Ising model, and the tricritical Ising model.
The thermodynamic Bethe ansatz (TBA) and the excited state TBA equations for an integrable spin chain related to the Lie superalgebra osp (1|2) are proposed by the quantum transfer matrix (QTM) method. We introduce the fusion hierarchy of the QTM and derive the functional relations among them (T-system) and their certain combinations (Y-system). Their analytical property leads to the nonlinear integral equations which describe the free energy and the correlation length at any finite temperatures. With regard to the free energy, they coincide with the TBA equation based on the string hypothesis.
The equivalence of a quantum system of particles interacting with a two-body inverse square potential to a system of noninteracting particles obeying 1D fractional statistics (1D anyons), stated by Polychronakos for particles on a line, is studied for the cases where the interacting system is placed (i) into a harmonic potential on a line, and (ii) on a ring, with imposing periodic boundary conditions. In the first case, reducibility of the interacting system to the Calogero system is used to explore the statistical distribution for free 1D anyons. On a ring, the thermodynamic limit is discussed in terms of the thermodynamic (asymptotic) Bethe ansatz. Yang and Yang’s integral equation is treated in this case as describing the statistical mechanics of free 1D anyons. It gives a functional equation for the statistical distribution of 1D anyons consistent with the harmonic potential approach. We show that on a ring of a finite circumference, a system of two free 1D anyons is equivalent to a system of two particles interacting with an inverse sine square potential (the Sutherland system). We also discuss the relation to the statistical mechanics of free anyons in 2+1 dimensions.