Antiperiodic boundary condition ($f(x+L)=-f(x)$)

An antiperiodic boundary condition is a phase twist of $\pi$ across the boundary. A Fourier mode then has $k=2\pi (n+1/2)/L$. It occurs in the even-fermion sector of a periodic Jordan–Wigner transformation and in imaginary-time fermionic thermal fields, for different underlying reasons.