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.

This transformation actually shows that at least in some cases with one spatial dimension, the distinction between bosons and fermions is nonexistent.

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 $j\; eq\; k$, 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. This is a special example of a Klein transformation. We take a chain of fermions, and define a new set of operators:$S\_\{j\}^\{+\}\; =\; f^\{dagger\}\_j(-1)^\{phi\_j\}$:$S\_\{j\}^\{-\}\; =\; (-1)^\{phi\_j\}f\_j$:$S\_\{j\}^\{z\}\; =\; f^\{dagger\}\_j\; f\_j\; -\; frac\{1\}\{2\}$.They differ from the above only by a sign factor $(-1)^\{phi\_j\}$, 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\}\; ight)\; =\; sum\_\{k=1\}^\{j-1\}\; f\_k^\{dagger\}f\_k$$phi\_j$ is has integral eigenvalues.

The inverse transformation is given by:$f^dagger\_j\; =\; S\_j^+\; (-1)^\{phi\_j\}$:$f\_j\; =\; (-1)^\{phi\_j\}\; S\_j^-$:$f^dagger\_j\; f\_j\; =\; S\_j^z+frac\{1\}\{2\}$

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 also true the other way around. 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

* Klein transformation
* S-duality