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Physical Review Letters

Thermodynamics of the Classical Massive-Thirring-Sine-Gordon Model

S. G. Chung

Published Feb 13, 1989• 11 citations
Publisher

Abstract

The proper classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ of the quantum Bethe-Ansatz thermodynamics gives a new set of integral equations for the thermodynamics of the classical massive-Thirring-sine-Gordon model. The theory contains the thermodynamic information about breathers and solitons, and reproduces the transfer-matrix-method result for arbitrary coupling constant.

DOI
10.1103/physrevlett.62.708

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Physical Review Letters•Mar 3, 1986•21 citations
The proper classical limit \ensuremathℏ\ensuremath→0\ensuremath{\hbar}\ensuremath{\rightarrow}0\ensuremathℏ\ensuremath→0 of the quantum Bethe-Ansatz thermodynamics gives a new set of integral equations for the thermodynamics of the classical massive-Thirring-sine-Gordon model. The theory contains the thermodynamic information about breathers and solitons, and reproduces the transfer-matrix-method result for arbitrary coupling constant.
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.
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 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.