Jordan Wigner Transformation

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 $j$ of a lattice, $S_{j}^{+},S_{j}^{-},S_{j}^{z}$. Taking the anticommutator of $S_{j}^{+}$ and $S_{j}^{-}$, we find $\{S_{j}^{+},S_{j}^{-}\}=1$, as would be expected from fermionic operators. We might be then tempted to set

$S_{j}^{+}=f^{\dagger }$

$S_{j}^{-}=f$

$S_{j}^{z}=f^{\dagger }f-{\frac {1}{2}}$

However, on different sites, we have the relation $[S_{j}^{+},S_{k}^{-}]=0$, where $j\neq k$, 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

$S_{j}^{+}=f^{\dagger }\exp {(-i\pi \phi _{j})}$

$S_{j}^{-}=\exp {(i\pi \phi _{j})}f$

$S_{j}^{z}=f^{\dagger }f-{\frac {1}{2}}$.

They differ from the above only by a phase factor $\exp {(i\pi \phi _{j})}$, where $\phi _{j}$ measures the number of up-spins to the right of site $j$

$\phi _{j}=\sum _{k=1}^{j-1}\left({\frac {1}{2}}+S_{k}^{z}\right)=\sum _{k=1}^{j-1}f_{k}^{\dagger }f_{k}$