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 [ma:equad:Schuster88], [ph:physt:Diu89] \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):

$H=-K\sum S_{l}S_{l+1}$

partition function of the system is:

$Z=\sum _{(S_{l})}\Pi _{l=0}^{N-1}e^{-KS_{l}S_{l+1}},$

which can be written as:

$Z=\sum _{(S_{l})}\Pi _{l=0}^{N-1}{\mbox{ ch }}K+S_{l}S_{l+1}{\mbox{ sh }}K.$

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:

$Z=\sum _{S_{1}}\dots \sum _{S_{n}}f(S_{1},S_{2})f(S_{2},S_{3})\dots$

with

$f_{K}(S_{i},S_{i+1})={\mbox{ ch }}K+S_{i}S_{i+1}{\mbox{ sh }}K$

We have:

$\sum _{S_{1}}f(S_{1},S_{2})=2{\mbox{ ch }}K.$

Indeed:

$\sum _{S_{1}}f(S_{1},S_{2})={\mbox{ ch }}K+S_{2}(+1){\mbox{ sh }}K+{\mbox{ ch }}K+S_{2}(-1){\mbox{ sh }}K.$

Thus, integrating successively over each variable, one obtains:

eqZisi

$Z=2^{n-1}({\mbox{ ch }}K)^{n-1}$

This result can be obtained a powerful calculation method: the renormalization group method[ph:physt:Diu89], [ma:equad:Schuster88]\index{renormalisation group} proposed by K. Wilson. Consider again the partition function:

$Z=\sum _{S_{1}}\dots \sum _{S_{n}}f_{K}(S_{1},S_{2})f_{K}(S_{2},S_{3})\dots$

where

$f_{K}(S_{i},S_{i+1})={\mbox{ ch }}K+S_{i}S_{i+1}{\mbox{ sh }}K$

Grouping terms by two yields to:

$Z=\sum _{S_{1}}\dots \sum _{S_{n}}g(S_{1},S_{2},S_{3}).g(S_{3},S_{4},S_{5})\dots$

where

$g(S_{i},S_{i+1},S_{i+2})=({\mbox{ ch }}K+S_{i}S_{i+1}{\mbox{ sh }}K)({\mbox{ ch }}K+S_{i+1}S_{i+2}{\mbox{ sh }}K)$

This grouping is illustrated in figure figrenorm.

Sum over all possible spin values $S_{i},S_{i+1},S_{i+2}$. The product$f_{K}(S_{i},S_{i+1})f_{K}(S_{i+1},S_{i+2})$ is the sum over all possible values of spins $S_{i}$ and $S_{i+2}$ of a function $f_{K'}(S_{i},S_{i+2})$ deduced from $f_{K}$ by a simple change of the value of the parameter $K$ associated to function $f_{K}$.}

figrenorm

Calculation of sum over all possible values of $S_{i+1}$ yields to:

$\sum _{S_{i+1}}g(S_{i},S_{i+1},S_{i+2})=2({\mbox{ ch }}^{2}K+S_{i}S_{i+2}{\mbox{ sh }}^{2}K)$

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

$K'={\mbox{ Arcth }}({\mbox{ th }}^{2}K).$

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