= Pair-field effective action of a superconductor
{title2=$S_{\rm eff}[\Delta]=\int |\Delta|^2/g-\operatorname{Tr}\ln G^{-1}[\Delta]$}
A pairing-channel <Hubbard–Stratonovich transformation> replaces an attractive fermion interaction by a complex <Cooper pair> field coupled to fermion pairs. A <Grassmann Gaussian integral> then gives a trace-log effective <action>, with its trace convention fixed to avoid doubling independent spin-pair blocks. The uniform stationary equation is the <BCS gap equation>. A small-field, slow-gradient expansion near the transition yields <Ginzburg-Landau theory of superconductivity>; the normalization of the phenomenological field need not equal that of the microscopic gap.
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