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):

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

partition function of the system is:

$Z=\sum_{(S_l)}\Pi_{l=0}^{N-1}e^{-KS_lS_{l+1}},$

which can be written as:

$Z=\sum_{(S_l)}\Pi_{l=0}^{N-1}\mbox{ ch } K +S_lS_{l+1}\mbox{ sh } K.$

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

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

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

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

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

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

This grouping is illustrated in figure figrenorm.

Sum over all possible spin values math>S_{i},S_{i+1},S_{i+2}</math>. 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:

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

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

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

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