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Physical Review Letters

Solitons and Nondissipative Diffusion

Nikos Theodorakopoulos, Michel Peyrard

Published Sep 20, 1999• 27 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.83.2293

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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 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.
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.
In this work we study the quantum Toda lattice, developing the asymptotic Bethe ansatz method first used by Sutherland. Despite its known limitations we find, on comparing with Gutzwiller's exact method, that it works well in this particular problem and in fact becomes exact as \ensuremath{\hbar} grows large. We calculate ground state and excitation energies for finite-sized lattices, identify excitations as phonons and solitons on the basis of their quantum numbers, and find their dispersions. These are similar to the classical dispersions for small \ensuremath{\hbar}, and remain similar all the way up to \ensuremath{\hbar}=1, but then deviate substantially as we go farther into the quantum regime. On comparing the sound velocities for various \ensuremath{\hbar} obtained thus with that predicted by conformal theory we conclude that the Bethe ansatz gives the energies per particle accurate to O(1/N2{\mathrm{N}}^{2}N2). On that assumption we can find correlation functions. Thus the Bethe ansatz method can be used to yield much more than the thermodynamic properties which previous authors have calculated.
In this work we study the quantum Toda lattice, developing the asymptotic Bethe ansatz method first used by Sutherland. Despite its known limitations we find, on comparing with Gutzwiller's exact method, that it works well in this particular problem and in fact becomes exact as ℏ\hbarℏ grows large. We calculate ground state and excitation energies for finite sized lattices, identify excitations as phonons and solitons on the basis of their quantum numbers, and find their dispersions. These are similar to the classical dispersions for small ℏ\hbarℏ, and remain similar all the way up to ℏ=1\hbar=1ℏ=1, but then deviate substantially as we go further into the quantum regime. On comparing the sound velocities for various ℏ\hbarℏ obtained thus with that predicted by conformal theory we conclude that the Bethe ansatz gives the energies per particle accurate to O(1/N21/N^21/N2). On that assumption we can find correlation functions. Thus the Bethe ansatz method can be used to yield much more than the thermodynamic properties which previous authors have calculated.
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.