Jordan Wigner Transformation
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The Jordan-Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators. It originally was created for one-dimensional lattice models, but now two-dimensional analogues of the transformation have been created.
Analogy between Spins and Fermions
Take spin-1/2 operators acting on a site of a lattice, . Taking the anticommutator of and , we find , as would be expected from fermionic operators. We might be then tempted to set
However, on different sites, we have the relation , where , and so spins on different sites commute while fermions anti-commute. We cannot take the analogy as presented very seriously.
A transformation which recovers the true fermion commutation relations from spin-operators was performed in 1928 by Jordan and Wigner. We take a chain of fermions, and define a new set of operators
- .
They differ from the above only by a phase factor , where measures the number of up-spins to the right of site