Jordan–Wigner transformation

Summary

The Jordan–Wigner transformation turns spin-$1/2$ operators on a chain of sites into fermion creation and annihilation operators. It attaches to the spin-lowering operator at site $j$ a string of sign operators on all the sites to the left of $j$. The string turns the commutation of operators on different sites into the anticommutation that fermions need. It maps some spin chains, such as the transverse-field Ising chain and the XY chain, to free fermions, which can then be solved exactly.