Mean-field theory and spatial correlations in statistical physics

Question

In statistical physics, mean-field theory (MFT) is often introduced by working out the Ising model and it's properties. From a spin model point of view, the mean-field approximation is given by requiring that :

Eq.(1)$\hspace{75pt}$$\langle S_i S_j \rangle = \langle S_i \rangle \langle S_j \rangle $ for $i\neq j$

Where $S_i$ is the local spin observable supported at site $i$ of a given lattice (in the classical Ising case, it is just $\pm1$).

I divide my questions/comments into two parts :

Part 1 (a): I know that there are more sophisticated ways of formulating much more rigorously mean-field theory in statistical physics, but is the above relation an equivalent definition for the particular case of a spin model ?

Part 1 (b) : Given that the above relation is an equivalent definition of MFT for a spin model, is it true to say that : "Mean-field theory is equivalent to taking out any spatial spin correlations of our system." ? I think this follows from the Eq.(1).

Part 2 : However, and here is what is confusing me : Why can we define a correlation length $\xi$ and a corresponding critical exponent $\nu$ (c.g. $\nu=1/2$ for MFT applied to the Ising model) from the connected two-point correlation function ?

Eq.(2) $\hspace{75pt}\langle S_iS_j \rangle - \langle S_i \rangle \langle S_j\rangle\sim e^{-|i-j|/\xi}$

To me, Eq. (1) and Eq. (2) look contradictory for distances $|i-j|$ smaller than the correlation length, yet there are both derived from MFT...

Answer 1

In short, I think the answers are:

1) yes, the approximation $ \langle S_i S_j \rangle \approx \langle S_i \rangle \langle S_j \rangle $ gives you the correct behavior for a spin system with homogenous spin values, but

2) there is more to mean field theory than this level of calculation

The approximation $$ \langle S_i S_j \rangle \approx \langle S_i \rangle \langle S_j \rangle $$ provides an approximation for determining what the mean spin value is within the Ising model, but is insufficient to actually calculate $ \langle S_i S_j \rangle$, as you've noted. The result of this approximation is a free energy in terms of the mean field $\langle S \rangle = m$. To get the two-point correlation function, we have to determine the energetic cost for having non-uniform $m({\bf r})$. One natural way of doing this is to use the Landau expansion, i.e. we write (see, e.g. Chaikin and Lubensky chapter 4) $$ F = \int d^d x f + \int d^d x \frac{c}{2} |\nabla m |^2 $$ where $f(x) = \frac{1}{2} r m^2 + u m^4 + \cdots $.

The free energy from the first term is something that you can get from making the approximation $\langle S_i S_j \rangle \approx \langle S_i \rangle \langle S_j \rangle$. However, this does not get you the value $c$, which is essentially a phenomenological term (you can relate it to the effective line tension between domains in the Ising model). Whenever you see a description of the correlation function in MFT, some term like this has been included. There is also an equivalent MFT scheme in field theory where the MFT can be derived by a saddle-point approximation (see Kardar's Statistical Physics of Fields, for instance). However, I don't remember offhand how to get from an Ising model to the appropriate field theory... I think this is done with a Hubbard-Stratonovich transformation, but I don't remember the details.