= Jordan–Wigner transformation
{c}
{title2=$S_n^+=\prod_{m<n}(1-2c_m^\dagger c_m)c_n$}
{wiki}
The Jordan–Wigner transformation represents a one-dimensional <spin one-half> chain by fermions with a parity string preceding each site. With $P_n=\prod_{m<n}(1-2n_m)$, one has $S_n^+=P_nc_n$ and $S_n^z=1/2-n_n$. The string converts the off-site <canonical anticommutation relations> into commuting spin operators. Adjacent exchange terms become quadratic fermion expressions, while a periodic end bond depends on global <fermion parity>.
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