Fermionic one-level thermal determinant (${\det }_{\mathrm{AP}}({\partial }_{\tau }+\xi )=1+e^{-\beta \xi }$)

An antiperiodic free fermion field with inverse propagator ${\partial }_{\tau }+\xi$ has a thermally normalized determinant $1+e^{-\beta \xi }$. The factor counts empty and occupied states of one level. Its logarithmic Matsubara sum is meaningful only with a regularization fixing the determinant normalization; an unqualified infinite product is not convergent.