I would like to understand how can I differ 1-type and 2-type superconductors from microscopic theory.
To start, I write down the BCS hamiltonian in terms of fermionic fields, $$ S[\bar{\psi},\,\psi]=\int_{0}^{\beta}d\tau\int d^dr\left[\psi_{\sigma}\left(\partial_{\tau}+ie\phi+\frac{(-i\partial-e{\bf A})^2}{2m}-\mu\right)\psi_{\sigma}-g\bar{\psi}_{\uparrow}\bar{\psi}_{\downarrow}\psi_{\downarrow}\psi_{\uparrow}\right]$$ Then, I perform Hubbard-Stratonovich transformation in order to decouple 4-fermion interactions, $$ \exp\left[g\int d\tau\int d^dr\bar{\psi}_{\uparrow}\bar{\psi}_{\downarrow}\psi_{\downarrow}\psi_{\uparrow}\right]=\\=\int\mathcal{D}[\bar{\Delta},\,\Delta]\exp\left(-\int d\tau\int d^dr\left[\frac{|\Delta|^2}{g}-(\bar{\Delta}\psi_{\downarrow}\psi_{\uparrow}+\Delta\bar{\psi}_{\uparrow}\bar{\psi}_{\downarrow})\right]\right)$$ Next, I introduce Nambu spinors and rewrite effective action as follows, $$S_{\text{eff}}=\int d\tau d^3r\left(\frac{|\Delta|^2}{g}+\ln\det\mathcal{G}^{-1}\right),\\ \mathcal{G}^{-1}=\begin{pmatrix}-\partial_{\tau}+\partial^2/(2m)+\mu & \Delta \\ \bar{\Delta} & -\partial_{\tau}-\partial^2/(2m)-\mu\end{pmatrix}.$$ In this effective action I can expand $\ln\det$ with help of $\ln\det=\text{tr}\ln$, which gives $$\text{tr}\ln\mathcal{G}^{-1}=\text{tr}\ln\left[\mathcal{G}_0^{-1}(1+\mathcal{G}_0^{-1}\Delta)\right]=\text{tr}\ln\mathcal{G}_0^{-1}-\sum_{n=0}^{\infty}\frac{1}{2n}\text{tr}\left(\mathcal{G}_0\Delta\right)^{2n}.$$ Substituting this expansion into $S_{\text{eff}}$, I obtain exatcly the Ginzburg-Landau expansion. From this expansion, I can easily see Higgs mechanism, derive Meissner effect and etc.
But how can I differ 2-type superconductor from the 1-type superconductor in terms of such calculations?
