McCoy B.M., Wu T.T. The two-dimensional Ising model

New York: Dover Publications, Inc., 2014. — 528 p.Of all the systems in statistical mechanics on which exact calculations have been performed, the two-dimensional Ising model is not only the most thoroughly investigated; it is also the richest and most profound. In 1925, Ising introduced the statistical system which now bears his name and studied some of its properties in one dimension. Although the generalization of Ising’s system to higher dimensions was immediately obvious, it was not until 1941 that a quantitative statement about the phase transition in the two-dimensional case was made when Kramers and Wannier and also Montroll computed the Curie (or critical) temperature.
However, the most remarkable development was made in 1944 when Onsager was able to compute the thermodynamic properties of the two-dimensional lattice in the absence of a magnetic field. Onsager’s approach was greatly simplified by Kaufman in 1949, and in a companion paper Kaufman and Onsager studied spin correlation functions. The spontaneous magnetization was first published, without derivation, by Onsager in 1949, and the first derivation was given by Yang in 1952. For the next decade no new result of fundamental significance was derived, but a great deal was accomplished in simplifying the mathematics of these pioneering papers. The work of Kac, Kasteleyn, Montroll, Potts, Szegö, and Ward, among others, has been especially significant.
When the first edition of this book was published 40 years ago in 1973 the results presented were up-to-date and complete. However, very shortly after the book was published major new results were obtained which demonstrated that in the scaling limit near the critical temperature the two point function for a suitably large separation between the spins satisfied a Painlevé equation of the third kind. Since that time there have been many further major advances in our understanding of the Ising model and the book has become seriously out of date.
Over the years we have been asked by many people to rectify this problem by writing a new book which would derive the many significant results which have been obtained since 1973. This is a monumental task which we have been quite reluctant to undertake. However, it is, in our opinion, important to make these results available to the community of mathematical physicists and to mathematicians who work on solvable models in statistical mechanics.
When Dover proposed to issue a reprint of the first edition it seemed to be a unique opportunity to present the results of the last 40 years by adding a new final chapter to the original text of 1973. This was the inspiration for what is now chapter 17. In the space of one chapter it is quite impractical to present the derivations of these results and instead we give references to the original papers where the results are first derived.
However different authors use different notations, conventions, and normalizations, and, consequently, the process of presenting these results in one common notation results at times in the results quoted not being literally the same as what is in the original papers.
We have no illusions, however, that this final chapter will be the last word in the development of the Ising model because there are many open questions presented there for which, as was similarly said in the epilogue of the 1973 edition, “progress can certainly be made”. It is our hope that this book will stimulate that progress.