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Physical Review Letters

Finite-Temperature Excitations of the Classical Toda Chain

Nikos Theodorakopoulos

Published Aug 27, 1984• 47 citations
Publisher

Abstract

The finite-temperature excitation spectrum of the classical Toda lattice is obtained by numerical solution of the Bethe Ansatz. Bethe-Ansatz thermodynamics agrees exactly with known transfer-integral results. A soliton-phonon phenomenology is developed and its validity assessed with the Bethe-Ansatz spectrum as an exact standard. The phenomenology's intrinsic limitations become evident in a transition region, where a distinction between phonon and (nontopological) soliton is impossible.

DOI
10.1103/physrevlett.53.871

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The finite-temperature excitation spectrum of the classical Toda lattice is obtained by numerical solution of the Bethe Ansatz. Bethe-Ansatz thermodynamics agrees exactly with known transfer-integral results. A soliton-phonon phenomenology is developed and its validity assessed with the Bethe-Ansatz spectrum as an exact standard. The phenomenology's intrinsic limitations become evident in a transition region, where a distinction between phonon and (nontopological) soliton is impossible.
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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 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.
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.
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.