Ising Model

Abstract

From the Ising model, which can be considered to be a certain specialization of the general Heisenberg model, first, an overview of the really wide range of applications of this model and of the possibilities of solving the model will be given. The great interest this model provokes is based on the fact that it is the only genuine many-body model which shows a phase transition and is solved mathematically rigorously.

We calculate the partition function of the one-dimensional Ising model without external field and show that spontaneous magnetization can be unequal to zero only for T=0. The partition function for the case where the system is in the presence of external magnetic field is calculated using the transfer matrix method. From this, certain important thermodynamic quantities such as the free energy, the temperature, field-dependent magnetization, the isothermal magnetic susceptibility, the entropy and the specific heat will be derived and discussed.

In contrast to the one-dimnsional case, the two-dimensional model displays a phase transition between the paramagnetic and ferromagnetic spin ordering at a finite temperature T _c. First the existence of spontaneous magnetization will be proved with the help of the so-called “Peierls argument”. The proof is accomplished on the basis of some estimates with a minimum mathematical complexity. The exact evaluation of the free energy (without field) leads to a concrete result for the transition temperature _Tc. The phase transition manifests itself through a logarithmic singularity in the specific heat.