JordanWigner

The Jordan-Wigner transformation is a nonlocal mapping between fermionic operators and spin-1/2 operators in one dimension. It was introduced by Pascual Jordan and Eugene Wigner in the late 1920s to enable exact solutions of certain quantum many-body problems by recasting fermionic degrees of freedom in terms of spins.

Concretely, for a chain of N sites with fermionic annihilation operators c_j and creation operators c_j†, and Pauli operators σ_j^x, σ_j^y, σ_j^z acting on site j, the transformation is defined by c_j = (∏_{k=1}^{j-1} σ_k^z) σ_j^− and c_j† = (∏_{k=1}^{j-1} σ_k^z) σ_j^+, where σ_j^± = (σ_j^x ± i σ_j^y)/2. The string of Z operators (the product over k < j) enforces the correct fermionic anticommutation relations: {c_i, c_j†} = δ_{ij} and {c_i, c_j} = 0 for i ≠ j.

The transformation maps local fermionic Hamiltonians with nearest-neighbor hopping and density interactions into spin Hamiltonians with nearest-neighbor couplings, at the cost of introducing nonlocal string operators in the spin representation. It is particularly powerful in one dimension and underpins exact analyses of models such as the tight-binding chain and the transverse-field Ising model when reformulated as free or interacting fermions.

Limitations include its nonlocality in the spin language and its restricted practicality to one dimension; periodic boundary conditions introduce subtleties related to fermion parity. Generalizations exist, notably the Bravyi–Kitaev mapping, and higher-dimensional mappings become increasingly complex, often sacrificing locality. The Jordan-Wigner transformation remains a foundational tool in quantum many-body theory.