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 ${\displaystyle T_{c}}$, magnetic moments orient themselves along a certain direction, and a non zero total magnetic moment is observed[1] . Ising model has been proposed to describe this phenomenom. It consists in describing each microscopic domain by a moment ${\displaystyle S_{i}}$ (that can be considered as a spin)\index{spin}, the interaction between spins being described by the following hamiltonian (in the one dimensional case):

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

partition function of the system is:

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

which can be written as:

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

It is assumed that ${\displaystyle S_{l}}$ 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. ${\displaystyle \sum _{(S_{l})}}$ represents the sum over all possible values of ${\displaystyle S_{l}}$, it is thus, in the same way an integral over a volume is the successive integral over each variable, the successive sum over the ${\displaystyle S_{l}}$'s. Partition function ${\displaystyle Z}$ can be written as:

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

with

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

We have:

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

Indeed:

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

Thus, integrating successively over each variable, one obtains:

eqZisi

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

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[2]. Consider again the partition function:

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

where

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

Grouping terms by two yields to:

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

where

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

This grouping is illustrated in figure figrenorm.

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

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

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

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

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

1. Ones says that a phase transition occurs.\index{phase transition} Historically, two sorts of phase transitions are distinguished [ph:physt:Diu89]
1. phase transition of first order (like liquid--vapor transition) whose characteristics are:
• Coexistence of the various phases.
• Transition corresponds to a variation of entropy.
• existence of metastable states.
2. second order phase transition (for instance the ferromagnetic--paramagnetic transition) whose characteristics are:
• symmetry breaking
• the entropy S is a continuous function of temperature and of the order parameter.
2. Kenneth Geddes ilson received the physics Nobel prize in 1982 for the method of analysis introduced here.