The subject of this book is on the borders of condensed matter physics, mathematics and field theory. It is believed that the N body problem is soluble only for N = 2, but is not soluble for systems with N ≥ 3. But, considering one dimensional systems, there are many solvable models, like the XXZ model. In condensed matter theory, the essential problem is that of solving many-body interacting systems and it is rare that we encounter solvable cases. But in these rare cases we can compare the theoretical results and experimental fact in detail. This kind of work is very valuable. Actually, orbits of planets and the energy spectrum of a hydrogen atom are treated in the regime of the exactly solvable case of two-body systems. Thus, the knowledge of exactly solvable systems is important not only for theorists but also for experimentalists. This book is planned for readers who have taken an elementary course of statistical mechanics and quantum mechanics.

There are also solvable two-dimensional classical systems like the six-vertex model. The Hamiltonians of 1D quantum systems and the transfer matrices of 2D classical systems sometimes have common eigenstates. In many cases we can write down many-body eigenfunctions of these matrices by the method of the Bethe-ansatz. The N-body wave function is represented as a linear combination of N! plane waves with N quasi-momenta. The energy eigenvalue of the lowest energy state in the thermodynamic limit is reduced to a distribution function of the quasi-momenta.