Jordan–Wigner transformation

The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators. It was proposed by Pascual Jordan and Eugene Wigner for one-dimensional lattice models, but now two-dimensional analogues of the transformation have also been created. The Jordan–Wigner transformation is often used to exactly solve 1D spin-chains such as the Ising and XY models by transforming the spin operators to fermionic operators and then diagonalizing in the fermionic basis.

This transformation actually shows that the distinction between spin-1/2 particles and fermions is nonexistent. It can be applied to systems with an arbitrary dimension.

In what follows we will show how to map a 1D spin chain of spin-1/2 particles to fermions.

Take spin-1/2 Pauli operators acting on a site ${\displaystyle j}$ of a 1D chain, ${\displaystyle \sigma _{j}^{+},\sigma _{j}^{-},\sigma _{j}^{z}}$. Taking the anticommutator of ${\displaystyle \sigma _{j}^{+}}$ and ${\displaystyle \sigma _{j}^{-}}$, we find ${\displaystyle \{\sigma _{j}^{+},\sigma _{j}^{-}\}=1}$, as would be expected from fermionic creation and annihilation operators. We might then be tempted to set

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