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

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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[1] . 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[2]. 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 productf_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.

  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 price in 1982 for the method of analysis introduced here.