Landau and Landau-Ginzburg Theory

Abstract

As an introduction to the methods of Landau theory we consider the simplest version of the Ising model ferromagnet described in Sect. 2.3 with Hamiltonian given by (1.49). The initial step in the development of Landau theory is essentially the same as in any other version of mean-field theory (see, for example, Volume 1, Sect. 3.1). This leads (Example 3.1) to the replacement of the Hamiltonian (1.49) by

$$ \hat H(H,J,\hat m) = - N(\frac{1}{2}zJ{\hat m^2} + H\hat m) $$

((3.1))

where m̂ = 𝓜̂/ N, with 𝓜̂ given by (1.46), and z is the coordination number of the lattice. The partition function Z, which is a function of L = 𝓗/T and K = J/T, is

$$ Z(L,K) = \sum\limits_{\{ \sigma (r)\} } {\exp [ - \hat H(H,J;\hat m)/T]} $$

((3.2))