Jordan–Wigner transformation: Difference between revisions

inverse transformation
(inverse transformation)
A transformation which recovers the true fermion commutation relations from spin-operators was performed in 1928 by Jordan and Wigner. This is a special example of a [[Klein transformation]]. We take a chain of fermions, and define a new set of operators
:<math>S_{j}^{+} = f^{\dagger}_j(-1)^{\phi_j} </math>
:<math>S_{j}^{-} = (-1)^{\phi_j}ff_j</math>
:<math>S_{j}^{z} = f^{\dagger}f_j f_j - \frac{1}{2}</math>.
They differ from the above only by a sign factor <math>(-1)^{\phi_j}</math>, where <math>\phi_j</math> measures the number of up-spins to the right of site <math>j</math>
:<math>\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</math>
<math>\phi_j</math> is has integral eigenvalues.
The inverse transformation is given by
:<math>f^\dagger_j = S_j^+ (-1)^{\phi_j}</math>
:<math>f_j = (-1)^{\phi_j} S_j^-</math>
:<math>f^\dagger_j f_j = S_j^z+\frac{1}{2}</math>
Note that the definition of the fermionic operators is nonlocal with respect to the bosonic operators because we have deal with an entire chain of operators to the right of the site the fermionic operators are defined with respect to. This is an example of a [['t Hooft operator]], which is a [[disorder operator]] instead of an [[order operator]]. This is also an example of an [[S-duality]].