ISING MODEL - CALCULATIONS WITHOUT APPROXIMATIONS

(Personal webpage)

In this lecture, I wish to emphasize that the model examined by Ernst Ising about 75 years ago, though highly nontrivial, admits exact (i.e., without making any approximation) calculations of many statistical-mechanical quantities. Such exact calculations are very valuable providing an intuition that is worthwhile for more realistic (and more complicated) models. Furthermore, exact results also permit to test approximate approaches and often give hints for new approximations.

More specifically, in my lecture I deal with the initial (zero-field) static susceptibilities of a nonuniform spin-1/2 Ising chain [1]. If the chain is uniform (i.e., all exchange couplings are the same), the longitudinal susceptibility can be obtained with the help of the transfer-matrix method, whereas the transverse susceptibility can be obtained by the Jordan-Wigner fermionization method. Interestingly, the initial static susceptibilities can be calculated rigorously for the case of regularly varying exchange couplings or the case of random exchange couplings too [1]. These findings demonstrate how regular alternation or random disorder may lead either to quantitative or to qualitative changes in the temperature dependence of the initial static susceptibilities.

From the pedagogical side, I discuss a number of concepts such as transfer matrix, Jordan-Wigner fermionization, Bogolyubov transformation etc. which play important role in finding the solution of a square-lattice Ising model [2]. We do not know how to solve a three-dimensional Ising model. The search for other exact calculations is encouraged.