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Progress of Theoretical Physics Supplement

Statistical Mechanics of Soliton Systems

Toshio Tsuzuki, Kazuo Sasaki

Published Jan 1, 1988• 23 citations
Publisher

Abstract

The statistical mechanics of the soliton-bearing nonlinear Klein-Gordon systems in one dimension is reviewed. Several different theoretical approaches are discussed: the transfer-integral method, the soliton-gas approach, the Bethe-ansatz method, and a few methods of computing the quantum correction. The recent development in the low-temperature double-series expansion for the free energy is the main subject.

DOI
10.1143/ptps.94.73

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The statistical mechanics of the soliton-bearing nonlinear Klein-Gordon systems in one dimension is reviewed. Several different theoretical approaches are discussed: the transfer-integral method, the soliton-gas approach, the Bethe-ansatz method, and a few methods of computing the quantum correction. The recent development in the low-temperature double-series expansion for the free energy is the main subject.
Quantum statistical mechanics of soliton systems is developed in the path integral formalizm by making use of the non-canonical collective coordinate method. The free energy is calculated in the ideal-gas approximation up to the order of two-loop. Explicit calculations are given for the sine-Gordon model. The classical limit of the present quantum calculation agrees with the exact result of the low-temperature expansion of the transfer-integral method. Numerical calculation reveals the quantum effect on the specific heat.
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.
Recent developments in statistical mechanics of quantum integrable systems are reviewed. Those studies are fundamental and have a renewed interest related to newly developing fields such as atomic Bose-Einstein condensations, photonic crystals and quantum computations. After a brief summary of the basic concepts and methods, the following three topics are discussed. First, by the thermal Bethe ansatz (TBA), a hard-core Bose gas is exactly solved. The model includes fully the effect of excluded volume and is identified to be a c=1 conformal field theory. Second, the cluster expansion method based on the periodic boundary condition for the Bethe wave function, which we call the Bethe ansatz cluster expansion (BACE) method, is developed for a δ-function gas and the XXX Heisenberg chain. This directly proves the TBA and reveals intrinsic properties of quantum integrable systems. Third, for a δ-function gas, the integral equations for the distribution functions of the quasi-momentum and the quasi-particle energy are solved in the form of power series. In the weak coupling case, the results reproduce those of Bogoliubov theory.
The thermodynamic properties of soliton-bearing systems can be obtained from the statistics of their soliton components, which reflect restrictions on phase space due to interactions. Going beyond the established agreement of the free energy with transfer-integral results, we have successfully tested first derivatives (energy, number density) and second derivatives (specific heat, compressibility) for a one-component soliton gas (corresponding to the case of the low-temperature sine-Gordon chain). A complete description of the equilibrium statistical properties of the soliton gas is given in terms of the averages of occupation numbers and their fluctuations. The latter are characterized by the absence of off-diagonal correlations.
The collective coordinate method is applied to develop the statistical mechanics of soliton systems. The present perturbation expansion is a direct extention of that done in the quantum field theory and is applicable to the quantum case. A new type of divergence which has never met so far in the many body theory appears in the present treatment but it can be removed by an appropriate treatment giving reasonable results. As the first paper of this series the classical case is investigated and explicit calculation is done in the classical sine-Gordon model. The results agree with the exact low temperature corrections obtained by the transfer-integral method and with those obtained by Miyashita and Maki who applied the collective coordinate approach in a different way.
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.