 Parity-dependent boundary condition of the Jordan-Wigner chain

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 $c_{N+1}=-P{c}_1$: 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.