Matsubara derivation of the BCS saddle (${\beta }^{-1}{\sum }_n({\omega }_n^2+E^2{)}^{-1}=\tanh (\beta E/2)/(2E)$)

Differentiate the subtracted grand potential of the reduced BCS model with respect to the squared pairing amplitude. The displayed fermionic Matsubara sum converts the derivative of the trace logarithm into ${\sum }_k\tanh (\beta E_k/2)/(2E_k)$. Setting it equal to $V/\lambda$ gives the uniform nonzero-gap saddle. The limit E→0 is finite at positive temperature, $\beta /4$. The sum relies on antiperiodic temporal boundary conditions from the fermionic coherent-state trace.