Jordan-Wigner Transformation
What It Is
The Jordan-Wigner (JW) transformation is an exact mapping between spin-1/2 operators on a 1D chain and spinless fermion operators. It converts certain spin Hamiltonians into quadratic (free) fermion problems that can be solved exactly.
Definition
\[c_i = \left(\prod_{j<i}\sigma^z_j\right)\sigma^-_i, \qquad c^\dagger_i = \left(\prod_{j<i}\sigma^z_j\right)\sigma^+_i\]
The product $\prod_{j<i}\sigma^z_j$ (the JW string) ensures fermionic anti-commutation: $\{c_i, c^\dagger_j\} = \delta_{ij}$.
When It Applies
| Condition | Satisfied? | Consequence |
|---|---|---|
| 1D geometry | Required | JW string is well-defined only in 1D |
| Nearest-neighbor interactions | Helpful | JW string cancels for NN terms |
| No $\sigma^x\sigma^x$ with $\sigma^z\sigma^z$ mixing | Helpful | Avoids quartic fermion terms |
Limitations
- Strictly 1D: In 2D+, the JW string creates non-local interactions. Extensions exist (2D JW, Kitaev) but are more complex.
- Convention dependence: $\sigma^z\sigma^z$ and $\sigma^x\sigma^x$ conventions produce different fermion Hamiltonians — one may be free while the other is interacting. See TFIM calculation for how the Kramers-Wannier duality resolves this.
Applications in QAtlas
| Model | Calculation note | Result |
|---|---|---|
| TFIM | JW-TFIM-BdG | BdG quasiparticle spectrum |
| Heisenberg | (interacting after JW — not directly solvable) | Bethe ansatz instead |
| Kitaev chain | (planned) | Topological phase boundary |
References
- P. Jordan, E. Wigner, Z. Physik 47, 631 (1928).
- E. Lieb, T. Schultz, D. Mattis, Ann. Phys. 16, 407 (1961).