Quantum Three Wave Interaction Models : Bethe Anstz and Statistical Mechanics(Development of Soliton Theory)

Kenji Ohkuma

Kenji Ohkuma

R. Bullough, D. J. Pilling, JUSSI TIMONEN

Miki Wadati, Go Kato, Toshiaki Iida

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.

Alan H. Luther

Nikos Theodorakopoulos, N. C. Bacalis

We examine the statistical mechanics of the classically integrable Ishimori-Faddeev-Haldane lattice spin model. Thermodynamic properties can be exactly described in terms of an interacting soliton gas. Our semiclassical scheme can be used to calculate the low-temperature thermodynamics of the S=1/2 Heisenberg chain, in very good agreement with recent numerical calculations based on the quantum-mechanical Bethe ansatz.