Ginzburg criterion and superconductivity

Question

The Ginzburg 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 Ising 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. 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. In this article there is a definition of the correlation matrix $\Gamma$.

Answer 1

The action has two classical (mean-field) solutions,one is zero and another one is the non-zero BCS order parameter. When you expand the action around $\Delta=0$ what you get is an infinite series. If you neglect the quantum fluctuations (put $\omega_{n\ne 0}=0$) the series becomes the Landau-Ginzburg statistical field Hamiltonian(So the LG theory works for the normal phase and for the superconducting phase near the transition point where the order parameter is small). For the Landau-Ginzburg Hamiltonian the Ginzburg criterion states that the effect of fluctuations is dominant in the behavior of the system in dimensions lesser than $d=4$ (You can see all of standard textbooks for more details and the derivation). So the mean-field theory can't describe superconductivity since $d<4$ in real world (actually the experimental results had agreement with the predictions of the LG theory and it was due to the lack of enough precision in the experiments in those days, for more info see standard references).

But if you insist on expanding the action in the "deep" BCS phase (where the order parameter is large), to find the $\Gamma$ matrix you should expand the trace term : $$ \mathrm{Tr}\ln(G^{-1}_{MF}+\Sigma)=\mathrm{Tr}\ln G^{-1}_{MF}+\mathrm{Tr}\ln (1+ G_{MF} \Sigma)=\\ \mathrm{Tr}\ln G^{-1}_{MF}+\mathrm{Tr}(G_{MF}\Sigma)-\frac12 \mathrm{Tr}(G_{MF}\Sigma G_{MF}\Sigma)+...$$ The first order terms is canceled for the mean-field value of $\Delta$. If you calculate the second order term you can derive the $\Gamma$ matrix(There can be a little algebra, though).