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Progress of Theoretical Physics

Exact Solution of the Soliton-Phonon Theory for the Classical Sine-Gordon Thermodynamics

S. G. Chung

Published Jul 1, 1992• 0 citations
Publisher

Abstract

The soliton-phonon theory of Johnson et al. and Timonen et al. for the classical sine-Gordon thermodynamics is exactly solved numerically to find its prediction for the free energy. The obtained free energy agrees with the exact transfer matrix method result only in the low reduced temperature or small effective coupling regime. It is thus concluded that the soliton-phonon theory is not exact.

DOI
10.1143/ptp.88.29

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The soliton-phonon theory of Johnson et al. and Timonen et al. for the classical sine-Gordon thermodynamics is exactly solved numerically to find its prediction for the free energy. The obtained free energy agrees with the exact transfer matrix method result only in the low reduced temperature or small effective coupling regime. It is thus concluded that the soliton-phonon theory is not exact.
This paper reviews soliton theories for classical statistical mechanics of the massive-Thirring-sine-Gordon model. These theories attempt not only to reproduce the classical transfer matrix method result but also to provide thermodynamic information about exact nonlinear excitations of the model. Three theories are discussed. Firstly, the soliton–gas phenomenology which is successful in the Toda lattice is shown to be not even approximately correct for the sine-Gordon model. This breakdown is due to the incorrect counting of the number of degrees of freedom. Secondly, the soliton–phonon theory which reproduces the WKB result is also shown to be not exact. This non-exactness is ascribed to the fact that the nonlinear phonons are not exact elementary excitations in the model. Thirdly, a non-gas phenomenological soliton–breather theory is derived as a classical limit of quantum statistical mechanics of the model. The exactness of the last theory is demonstrated through an accurate numerical analysis of its basic equations.
We use the quantum Bethe ansatz method to compute the free energy of the sine-Gordon model in the classical limit. In this limit the number of breathers, and hence the number of coupled integral equations to be solved, diverges. By linearizing the breather mass spectrum and phase shifts, extending a method of Maki, we can reduce the breather ladder to anharmonic phonons. The divergent set of integral equations is reduced to only two, for interacting phonons and solitons. We solve these equations iteratively to give the free energy in a double series in the temperature t and the soliton density ${e}^{\mathrm{\ensuremath{-}}1/t}$ which agrees to high order with classical transfer integral results.
We use the quantum Bethe-Ansatz method to compute the free energy of the classical sine-Gordon system. Previous attempts to do this have failed because the number of coupled integral equations to be solved to find the free-energy diverges in the classical limit. We present a transformation, extending a method of Maki, which reduces this divergent set to only two coupled equations, for the densities of solitons and anharmonic phonons. These equations can be solved iteratively in the temperature ttt and the soliton density ${e}^{\frac{\ensuremath{-}1}{t}}$ to give a double series for the free energy. This series coincides to high order with classical transfer-matrix results.
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 Boltzmann gas consisting of solitons, antisolitons, and breathers of the classical sine-Gordon system is studied by taking their interactions into account. A compact expression for the thermodynamic potential is obtained, which agrees with that obtained from the classical limit of the Bethe-ansatz formulation of the quantum sine-Gordon thermodynamics. We further find that the average number of breathers excited at low temperatures (m\ensuremath{\ll}T\ensuremath{\ll}M, where m and M are the phonon and the soliton masses, respectively) is half the total number of degrees of freedom of the system.
We give two fundamental methods for evaluation of classical free energies of all the integrable models admitting soliton solutions; the sine-Gordon equation is one example. Periodic boundary conditions impose integral equations for allowed phonon and soliton momenta. From these, generalized Bethe-Ansatz and functional-integration methods using action-angle variables follow. Results for free energies coincide, and coincide with those that we find by transfer-integral methods. Extension to the quantum case, and quantum Bethe Ansatz, on the lines to be reported elsewhere for the sinh-Gordon equation, is indicated.