Editing Landau theory (section)

==Including long-range correlations==
Consider the Ising model free energy above. Assume that the order parameter <math>\Psi </math> and external magnetic field, <math>h</math>, may have spatial variations. Now, the free energy of the system can be assumed to take the following modified form:

:<math> F :=  \int d^D x \ \left( a(T) + r(T) \psi^2(x) + s(T) \psi^4(x) \ + f(T) (\nabla \psi(x))^2 \ +h(x) \psi(x)\ \ + \mathcal{O}(\psi^6 ; (\nabla \psi)^4) \right) </math>

where <math>D</math> is the total ''spatial'' dimensionality. So,

:<math> \langle \psi(x) \rangle := \frac{\text{Tr}\ \psi(x) {\rm e}^{-\beta H}}{Z} </math>

Assume that, for a ''localized'' external magnetic perturbation <math> h(x) \rightarrow 0 + h_0 \delta(x) </math>, the order parameter takes the form <math> \psi(x) \rightarrow \psi_0 + \phi(x) </math>. Then,

:<math> \frac{\delta \langle \psi(x) \rangle}{\delta h(0)} = \frac{\phi(x)}{h_0} = \beta \left ( \langle \psi(x) \psi(0) \rangle - \langle \psi(x) \rangle \langle \psi(0) \rangle \right ) </math>

That is, the fluctuation <math>\phi(x)</math> in the order parameter corresponds to the order-order correlation. Hence, neglecting this fluctuation (like in the earlier mean-field approach) corresponds to neglecting the order-order correlation, which diverges near the critical point.

One can also solve <ref>"Equilibrium Statistical Physics" by Michael Plischke, Birger Bergersen, Section 3.10, 3rd ed</ref> for <math> \phi(x)</math>, from which the scaling exponent, <math> \nu </math>, for correlation length <math> \xi \sim (T-T_c)^{-\nu} </math> can deduced. From these, the [[Ginzburg criterion]] for the [[upper critical dimension]] for the validity of the Ising mean-field Landau theory (the one without long-range correlation) can be calculated as:
:<math> D \ge 2 + 2 \frac{\beta}{\nu}</math>

In our current Ising model, mean-field Landau theory gives <math>\beta = 1/2 = \nu </math> and so, it (the Ising mean-field Landau theory) is valid only for spatial dimensionality greater than or equal to 4 (at the marginal values of <math>D=4</math>, there are small corrections to the exponents). This modified version of mean-field Landau theory is sometimes also referred to as the Landau–Ginzburg theory of Ising phase transitions. As a clarification, there is also a [[Ginzburg–Landau theory]] specific to superconductivity phase transition, which also includes fluctuations.