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Physical review. B, Condensed matter

Thermal solitons in the Toda chain

Nikos Theodorakopoulos, N. C. Bacalis

Published Nov 1, 1992• 14 citations
Publisher

Abstract

The fraction of degrees of freedom occupied by solitons in the classical Toda chain is evaluated as a function of temperature. Our results, derived within the framework of the Bethe ansatz (BA), are in complete agreement with recent numerical simulations that employ a soliton counter and reveal a ${\mathit{T}}^{1/3}$ dependence in the limit of low temperatures. The agreement confirms the existence of a T-independent soliton-phonon boundary in momentum space within the BA formalism, which apparently shares a common physical basis with the soliton counter's effective distinction between bound and scattering states.

DOI
10.1103/physrevb.46.10706

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The fraction of degrees of freedom occupied by solitons in the classical Toda chain is evaluated as a function of temperature. Our results, derived within the framework of the Bethe ansatz (BA), are in complete agreement with recent numerical simulations that employ a soliton counter and reveal a T1/3{\mathit{T}}^{1/3}T1/3 dependence in the limit of low temperatures. The agreement confirms the existence of a T-independent soliton-phonon boundary in momentum space within the BA formalism, which apparently shares a common physical basis with the soliton counter's effective distinction between bound and scattering states.
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.
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 dynamical correlation spectra of the Toda chain are obtained by a molecular dynamics simulation and analyzed in terms of Bethe-ansatz-based soliton phenomenology. The spectrum of local temperature fluctuations exhibits a feature which can be unambiguously assigned to solitons. Detailed linewidth analysis demonstrates the phenomenon of nondissipative soliton diffusion, i.e., the stochastic sequence of spatial shifts experienced by the soliton as it moves through the lattice and interacts with other excitations without momentum exchange. Our results provide strong support for the dynamical predictions of the finite-temperature Bethe ansatz excitation spectrum.
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.
A review of the classical and quantum thermodynamic properties of the Toda chain is provided, together with a survey of the techniques that have been used to work them out, i.e., the bilateral Laplace transform method for the classical system, the Bethe–Ansatz and the variational path-integral approach for the quantum one. For the classical Toda chain we also recall some of the main dynamical features, i.e. the integrability of the model and the soliton-like solutions of the equations of motion. In the quantum case a comparison between the Bethe–Ansatz and the variational method is made. In particular it is shown as the latter offers the possibility of also evaluating static correlation functions.