<PERSON> criterion and superconductivity
The <PERSON> criterion tells us quantitatively when mean field theory is valid. If $\phi$ is the order parameter of the system, then mean field theory requires that the fluctuations in the order parameter are much smaller than the actual value of the order parameter near the critical Point:
$$ \langle\left(\delta\phi\right)^{2}\rangle << \langle\phi^{2}\rangle\text{.} $$
For example in the case of a <PERSON> model. The order parameter is given by the magnetization and we can make the fluctuations Expansion
$$ m = m_{0} + \delta m $$
where $m_{0}$ is the order parameter and $\delta m$ describes the fluctuations. 346
Then, it can be shown that the Ginzburg criterion maps to
$$ \langle\delta m\left(x\right)\delta m\left(x^{\prime}\right)\rangle << m_{0}^{2} $$
where $\langle\delta m\left(x\right)\delta m\left(x^{\prime}\right)\rangle$ is the correlations function of the fluctuations.
I am interesting how is this working for BCS theory? Here the BCS action functional reads
$$ S_{\text{BCS}} = \sum_{Q}\Delta_{Q}^{\dagger}\left(\frac{g}{\beta V}\right)^{-1}\Delta_{Q} - \text{tr}\ln\left(\left(G_{\text{BCS}}^{-1}\right)\right) $$
with $\left(G_{\text{BCS}}^{-1}\right){k,q} = \begin{pmatrix} \left(-i\omega + \epsilon{k}\right)\delta_{k,q} & \Delta_{k-q} \ \Delta_{q-k}^{\dagger} & \left(-i\omega - \epsilon_{k}\right)\delta_{k,q} \end{pmatrix}$. Then with the fluctuations expansion
$$ \Delta_{Q} = \sqrt{\beta V}\Delta\delta_{Q,0} + \Phi_{Q} $$
where $\Delta$ is the mean-field order Parameter and $\Phi_{Q}$ is the fluctuations field. Then, because $\left(G_{\text{BCS}}^{-1}\right){k,q}$ is linear in $\Delta{Q}$ we can write $\left(G_{\text{BCS}}^{-1}\right){k,q} = \left(G{\text{MF}}^{-1}\right){k,q} + \left(\sum{\text{Fluc}}\right)_{k,q}$.
After a little bit of algebra it can be shown that the partition function factorized
$$ \mathcal{Z}{\text{BCS}} = \mathcal{Z}{\text{MF}}\mathcal{Z}_{\text{Fluc}} $$
with $\mathcal{Z}{\text{MF}}$ is the partition function for the mean-field order parameter and $\mathcal{Z}{\text{Fluc}}$ for the fluctuations. The Partition function for the fluctuations has the following form
$$ \mathcal{Z}{\text{Fluc}} = \int D\left[\Phi^{\dagger},\Phi\right]e^{-S{\text{Fluc}}} $$
with the Action functional of the fluctuations
$$ S_{\text{Fluc}} = \frac{1}{2}\sum_{Q}\left(\Phi_{Q}^{\dagger},\Phi_{-Q}\right)\begin{pmatrix} \Gamma_{11} & \Gamma_{12} \ \Gamma_{21} & \Gamma_{22} \end{pmatrix}\begin{pmatrix} \Phi_{Q}\ \Phi_{-Q}^{\dagger} \end{pmatrix} $$
Here we have a correlation matrix $\Gamma = \begin{pmatrix} \Gamma_{11} & \Gamma_{12} \ \Gamma_{21} & \Gamma_{22} \end{pmatrix}$.
My question is how I can relate the correlation matrix $\Gamma$ to the mean-field order parameter $\Delta$, which is a scalar, for the Ginzburg criterion?
Edit: For references, I used the following article Path-Integral Description of Cooper Pairing. 804