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International Journal of Modern Physics B

SOLITON THEORY FOR CLASSICAL STATISTICAL MECHANICS OF THE MASSIVE-THIRRING-SINE-GORDON MODEL

S. G. Chung

Published Aug 15, 1994• 7 citations

Abstract

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.

DOI: 10.1142/s0217979294000981

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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.

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.

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.

We revisit the exact thermodynamic description of the classical sine-Gordon field theory, a well-known integrable model. We found that existing results in the literature based on the soliton-gas picture did not correctly take into account light, but extended, solitons and thus led to incorrect results. This issue is regularized upon requantization: we derive the correct thermodynamics by taking the semiclassical limit of the quantum model. Our results are then extended to transport settings by means of Generalized Hydrodynamics.