Jordan–Wigner transformation

Remarks

Check of the relations: on one site $a a^\dagger + a^\dagger a$ is the identity, and $Z a = a = -a Z$, so $Z$ and $a$ anticommute. For $i < j$, the products $c_i c_j$ and $c_j c_i$ differ only at site $i$, where they contain $a_i Z_i$ and $Z_i a_i = -a_i Z_i$, so $c_i c_j = -c_j c_i$. The occupation number is $n_j = c_j^\dagger c_j = (1 - Z_j)/2$, and the inverse map is $a_j = (1 - 2n_1) \cdots (1 - 2n_{j-1}) \, c_j$.

Example: with the Pauli matrices $X = a + a^\dagger$ and $Y = i(a^\dagger - a)$, one finds $X_j X_{j+1} + Y_j Y_{j+1} = 2(c_j^\dagger c_{j+1} + c_{j+1}^\dagger c_j)$. So the XX chain $\sum_j (X_j X_{j+1} + Y_j Y_{j+1})$ with open ends is a quadratic form in the fermion operators. The same computation applies to the transverse-field Ising model and to the XXZ model, where the $Z_j Z_{j+1}$ term becomes a fermion interaction.

Non-example: the operators $a_j$ without the string do not anticommute on different sites. For instance $a_1 a_2 = a_2 a_1 \ne 0$, so $\set{a_1, a_2} = 2 a_1 a_2 \ne 0$.

Conventions differ between sources: some use $\sigma^+$ or $\sigma^-$ in place of $a$, or the string $\prod_{k<j}(-Z_k)$.