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Journal of the Physical Society of Japan

Exact Virial Expansion for the Quantum Calogero Model

Hideaki Ujino, Takayuki Tsuchida, Miki Wadati

Published Jan 15, 1998• 0 citations
Publisher

Abstract

Exact treatment of the microcanonical ensemble for the quantum Calogero model is presented through the exclusion statistical thermodynamic Bethe ansatz (TBA) method. All the virial coefficients for the model in the thermodynamic limit are rigorously calculated, which are in agreement with the conjectured results.
DOI
10.1143/jpsj.67.119

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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.
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We propose discrete TBA equations for models with discrete spectrum. We illustrate our construction on the Calogero-Moser model and determine the discrete 2-body TBA function which yields the exact N-body Calogero-Moser thermodynamics. We apply this algorithm to the Lieb-Liniger model in a harmonic well, a model which is relevant for the microscopic description of harmonically trapped Bose-Einstein condensates in one dimension. We find that the discrete TBA reproduces correctly the N-body groundstate energy of the Lieb-Liniger model in a harmonic well at first order in perturbation theory, but corrections do appear at second order. Typeset using REVTEX 1 I.
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.
We develop a regularization of the quantum microcanonical ensemble, called a Gaussian ensemble, which can be used for derivation of the canonical ensemble from microcanonical principles. The derivation differs from the usual methods by giving an explanation for the, at the first sight unreasonable, effectiveness of the canonical ensemble when applied to certain small, isolated, systems. This method also allows a direct identification between the parameters of the microcanonical and the canonical ensemble and it yields simple indicators and rigorous bounds for the effectiveness of the approximation. Finally, we derive an asymptotic expansion of the microcanonical corrections to the canonical ensemble for those systems, which are near, but not quite, at the thermodynamical limit and show how and why the canonical ensemble can be applied also for systems with an exponentially increasing density of states. The aim throughout the paper is to keep mathematical rigour intact while attempting to produce results which are both physically and practically interesting.