 Jordan–Wigner transformation

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 $S_n^{+}=P_n{c}_n$ 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.

The Jordan-Wigner transformation is a mathematical technique used in quantum mechanics and condensed matter physics to map spin systems to fermionic systems. It provides a way to express operators of spin-1/2 systems (like those found in quantum spin chains) in terms of fermionic creation and annihilation operators.