Jordan–Wigner transformation: Difference between revisions

(In 1D, bosons=fermions)
However, on different sites, we have the relation <math>[S_{j}^{+},S_{k}^{-}] = 0</math>, where <math>j \neq k</math>, 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. 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}\exp{(-i\pi\phi_j)} </math>
:<math>S_{j}^{-} = \exp{(i\pi\phi_j)}f</math>
They differ from the above only by a phase factor <math>\exp{(i\pi\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>
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]].
== See also ==