 Matsubara derivation of the BCS saddle

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.