= Parity-dependent boundary condition of the Jordan-Wigner chain
{title2=$c_{N+1}=-(-1)^{N_f}c_1$}
For $S_n^+=\prod_{m<n}(1-2n_m)c_n$ and periodic <spins>, the end exchange term carries the global <fermion parity> $P=(-1)^{N_f}$. Replacing the end bond by a uniform hopping expression requires $c_{N+1}=-Pc_1$: even parity gives antiperiodic <fermions>, odd parity periodic <fermions>. Each resulting quadratic spectrum must be restricted to states of the parity used in its <boundary condition>.
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