condensed matter - Validity of mean-field approximation

In mean-field approximation we replace the interaction term of the Hamiltonian by a term, which is quadratic in creation and annihilation operators. For example, in the case of the BCS theory, where

$$ \sum_{kk^{\prime}}V_{kk^{\prime}}c_{k\uparrow}^{\dagger}c_{-k\downarrow}^{\dagger}c_{-k^{\prime}\downarrow}c_{k^{\prime}\uparrow}\to\sum_{k}\Delta_{k}c_{k\uparrow}^{\dagger}c_{-k\downarrow}^{\dagger} + \Delta_{k}^{\star}c_{-k\downarrow}c_{k\uparrow}\text{,} $$

with $\Delta_{k}=\sum_{k^{\prime}}V_{kk^{\prime}}\langle c_{-k^{\prime}\downarrow}c_{k^{\prime}\uparrow}\rangle\in\mathbb{C}$. Then, in books, like this by Bruss & Flensberg, there is always a sentence like "the fluactuations around $\Delta_{k}$ are very small", such that the mean-field approximation is a good approximation. But we known for example in the case of the 1D Ising model the mean-field approximation is very bad.

My question: Is there a inequality or some mathematical conditions which says something about the validity of the mean-field approach? Further, is there a mathematical rigoros derivation of the mean-field approximation and the validity of it?

Answer

You can introduce the ``would-be'' bosonic mean field exactly, using the Hubbard-Stratonich (a.k.a partial bosonization) method, see wikipedia and Interacting fermions on a lattice and Hubbard-Stratonovich transformation and mean-field approximation .

The mean field approximation correspond to performing the integral over the bosonic field using the stationary phase approximation. The fermion action is bilinear and can be performed exactly. Corrections to the stationary phase approximation correspond to fluctuations around the mean field. These are small if the four-fermion coupling in the BCS theory is small. If the coupling is strong it may be possible to justify the mean field approximation by a large N (N component vector field) or large d (number of dimensions) approximation.

Schmatically, the effective action of the bosonized theory is $$ S = {\rm Tr}\{\log[G_0^{-1}G(\phi)]\} -\frac{\phi^2}{g} $$ where $g$ is the coupling, $\phi$ is the bosonic field, and $$ G(\phi) = \left( \begin{matrix} p_0-\epsilon_p & \phi \\ \phi & p_0+\epsilon_p \end{matrix} \right) $$ is the propagator. I also define $G_0=G(0)$. Now $$ \left. \frac{\delta S}{\delta\phi}\right|_{\phi_0} = 0 $$ is the MFA gap equation $$ \phi_0 = g\int \frac{d^3p}{(2\pi)^3} \frac{\phi_0}{\sqrt{\epsilon_p^2+\phi_0^2}} . $$ Corrections can be found by expanding $S$ around $\phi=\phi_0+\delta \phi$. This will give higher loops containing $G(\phi_0)$. The expansion parameter is g. In physical units $1/g$ is the logarithm of the Fermi energy over the gap, $g\sim [\log(E_F/\phi_0)]^{-1}$.

Near $T_c$ corrections to mean field (=Landau-Ginsburg) are controlled by the Ginsburg criterion, as explained in Valrio92's answer. In weak coupling BCS the Ginsburg window is small, and mean field is accurate, except very close to $T_c$.