 BCS gap equation

The BCS gap equation makes the superconducting gap function self-consistent with the BCS anomalous average that creates it. For a translation-invariant spin-singlet mean-field approximation, define ${\xi }_k={ϵ}_k-\mu$ and $E_k=\sqrt{{\xi }_k^2+|{\Delta }_k{|}^2}$. ThenThe interaction $V_{k{k}^{\prime }}$ and its normalization/cutoff are part of the model. A nonzero solution describes pairing; ${\Delta }_k=0$ is always a solution, so finding any solution alone does not establish which state minimizes the free energy.

For constant attraction $-V$ in a symmetric shell $|\xi |<\Lambda$ with constant single-spin density of states ${\nu }_s$, the zero-temperature equation reduces toAt weak coupling, this gives $\Delta \simeq 2\Lambda e^{-1/({\nu }_{s}V)}$. The exponential explains how a small attraction produces an extremely small pairing scale, matching the Cooper instability. The finite-temperature equation can be linearized as the gap tends to zero to determine the mean-field transition. ^[ref]