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Progress of Theoretical Physics

Classical Thermodynamics of the Toda Lattice: --As a Classical Limit of the Two-Component Bethe Ansatz Scheme--

Hajime Takayama, M. Ishikawa

Published Oct 1, 1986• 17 citations
Publisher

Abstract

In our previous work thermodynamics of two-componet (soliton and phonon) gas is constructed from the Yang-Yang equation for one-component integrable quantum gas. We call this the two-component Bethe ansatz (TCBA) scheme. The method is applied to the Toda lattice and its classical limit is analyzed. The key parameter kF which distinguishes solitons from phonons within the TCBA scheme is chosen such that the resultant equations for effective energies of the modes coincide with those expected from the simple-minded ideal-gas phenomenology. The soliton specific heat exhibits a peculiar temperature dependence at low temperatures. It is however compensated with the anharmonic part of phonons.

DOI
10.1143/ptp.76.820

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In our previous work thermodynamics of two-componet (soliton and phonon) gas is constructed from the Yang-Yang equation for one-component integrable quantum gas. We call this the two-component Bethe ansatz (TCBA) scheme. The method is applied to the Toda lattice and its classical limit is analyzed. The key parameter kF which distinguishes solitons from phonons within the TCBA scheme is chosen such that the resultant equations for effective energies of the modes coincide with those expected from the simple-minded ideal-gas phenomenology. The soliton specific heat exhibits a peculiar temperature dependence at low temperatures. It is however compensated with the anharmonic part of phonons.
A linear chain with exponential plus linear interactions (Toda lattice) can be mapped by the Bethe ansatz to a Fermi gas. The thermodynamics is treated by analogy to the delta-function Bose gas (Yang and Yang). Analytical solutions are given for the strongly anharmonic regime and low temperatures; the general case is treated numerically. For the case of weak anharmonicity the specific heats are compared with earlier approximate calculations. The anharmonic part of the specific heat is shown to be not very different from the corresponding classical result which can be interpreted as the specific heat of an ideal gas of solitons. For the quantum case the soliton-gas picture needs further investigation.
The Bethe-ansatz (BA) equation for the classical sine-Gordon (SG) thermodynamics is shown to reduce to the one for the classical Toda lattice within two approximations; the phonon-phonon interactions are neglected (the harmonic phonon) and solitons are treated nonrelativistically. Within these approximations the solitons in the SG model correspond to the particles in the Toda lattice. Making use of an analytical solution of the BA equation for the classical Toda lattice due to Opper, we obtain an analytical solution of the approximate SG equation. We propose also a similar approximate BA equation for the ${\ensuremath{\varphi}}^{4}$ model, although the ${\ensuremath{\varphi}}^{4}$ model does not have the BA equation for the classical thermodynamics.
Thermodynamics of the quantum Toda lattice is studied based on Gutzwiller's quantization condition and Yang - Yang's thermodynamic formulation. It is shown that Gutzwiller's quantization condition becomes identical to the Bethe ansatz equation in the thermodynamic limit. We have calculated the thermodynamic averages of the energy, higher-order conserved quantities and the specific heat. The average energy and the specific heat become close to those of the classical Toda lattice in the high temperature limit, while they are similar to those of a harmonic chain in the low temperature limit.
The finite-temperature excitations of a large Toda lattice in the semiclassical regime are analysed using the Bethe ansatz, and a physical interpretation is presented, showing clearly that the lattice phonon excitation corresponds to the Bethe ansatz hole, even in the classical limit at finite temperature. The authors' method gives a simple derivation of the formulae for excitation energies and momenta, which were first written down by Gruner-Bauer and Mertens (1988) from a consistency argument, and it suggests why solitons do not apparently contribute to the Toda lattice free energy.
The aim of the explicit evaluation of the equilibrium thermodynamic properties of the quantum Toda lattice is reached here by a method that reduces the evaluation of quantum-statistical averages to a classical configurational integral. This is done by constructing an effective potential of the same form as the original one, whose parameters are renormalized by the quantum fluctuations. In addition to quantum specific heat (in agreement with Bethe-ansatz calculations), explicit results for meaningful static correlation functions are reported.