We will thus factorize the full determinant \( \vert\hat{D}\vert \) into two smaller ones, where each can be identified with \( \uparrow \) and \( \downarrow \) respectively:

$$ \vert\hat{D}\vert = \vert\hat{D}\vert_\uparrow\cdot \vert\hat{D}\vert_\downarrow $$