BCS pairing-field effective action ($S_{\mathrm{eff}}[\Delta ]=\int |\Delta {|}^2/\lambda -\mathrm{Tr}\ln {\mathcal{G}}^{-1}[\Delta ]$)

An attractive local pairing interaction can be decoupled with a complex Hubbard–Stratonovich transformation. Integrating the fermionic Grassmann variables gives an effective action for the pair field. Its uniform saddle is the BCS gap equation, and its slow-field expansion near the transition gives the Ginzburg-Landau theory of superconductivity. A trace over a fully doubled particle-hole basis requires the corresponding factor of one half; the displayed formula uses one independent pairing block.

Subtracting the normal-state grand potential fixes the reference constant in the uniform BCS pairing-field effective action. Here ${\omega }_n=(2n+1)\pi /\beta$ has energy units and $E_k^2={\xi }_k^2+|\Delta {|}^2$. The ratio cancels a field-independent determinant phase and produces a convergent high-frequency sum under the same momentum regulator. Differentiation gives the BCS gap equation. For a fixed attractive pairing shell, this is a finite regulated model; an unrestricted continuum contact interaction additionally needs ultraviolet renormalization. Its weak-coupling condensation density $-{\nu }_F{\Delta }^2/2$ uses a single-spin density of states per volume.

A two-component Nambu spinor $({\psi }_{k\uparrow },{\bar{\psi }}_{-k\downarrow })$ for every k accounts for both spin species once after relabelling. Its Grassmann integral contributes one functional determinant. A fully doubled spin-particle-hole basis instead has redundancy and requires one half in the trace logarithm or a corresponding Pfaffian convention. A reliable check is the zero-gap limit: the grand potential must become $-2{\beta }^{-1}{\sum }_k\ln (1+e^{-\beta {\xi }_k})$, with exactly two spin states and no invented vacuum offset.