Introduction to Mathematical Physics/N body problems and statistical equilibrium/Ising Model

In this section, an example of the calculation of a partition function is presented. The Ising model , \index{Ising} is a model describing ferromagnetism\index{ferromagnetic}. A ferromagnetic material is constituted by small microscopic domains having a small magnetic moment. The orientation of those moments being random, the total magnetic moment is zero. However, below a certain critical temperature $$T_c$$, magnetic moments orient themselves along a certain direction, and a non zero total magnetic moment is observed . Ising model has been proposed to describe this phenomenom. It consists in describing each microscopic domain by a moment $$S_i$$ (that can be considered as a spin)\index{spin}, the interaction between spins being described by the following hamiltonian (in the one dimensional case):

partition function of the system is:

which can be written as:

It is assumed that $$S_l$$ can take only two values. Even if the one dimensional Ising model does not exhibit a phase transition, we present here the calculation of the partition function in two ways. $$\sum_{(S_l)}$$ represents the sum over all possible values of $$S_l$$, it is thus, in the same way an integral over a volume is the successive integral over each variable, the successive sum over the $$S_l$$'s. Partition function $$Z$$ can be written as:

with

We have:

Indeed:

Thus, integrating successively over each variable, one obtains:

This result can be obtained a powerful calculation method: the renormalization group method, \index{renormalisation group} proposed by K. Wilson. Consider again the partition function:

where

Grouping terms by two yields to:

where

This grouping is illustrated in figure figrenorm. Calculation of sum over all possible values of $$S_{i+1}$$ yields to:

Function $$\sum_{S_{i+1}}g(S_i,S_{i+1},S_{i+2})$$ can thus be written as a second function $$f_{K'}(S_i,S_{i+2})$$ with

Iterating the process, one obtains a sequence converging towards the partition function $$Z$$ defined by equation eqZisi.