Fermionic pairing-block determinant normalization

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.