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 Incoming links: Two-site Jordan-Wigner normalization and signs

There is a normalization mismatch in the printed definitions:

σ^{x} \pm i σ^{y}

have nonzero entries of magnitude two, whereas a single fermionic creation or annihilation operator has magnitude one. To obtain the intended Jordan–Wigner transformation, use normalized spin ladders

s_{j}^{\pm} = (σ_{j}^{x} \pm i σ_{j}^{y}) /2

. The given fermion expressions represent these

s_{j}^{\pm}

Put

n_{j} = f_{j}^{†} f_{j}

and

P_{1} = e^{iπ n_{1}} = 1 - 2 n_{1}

P_{1}^{2} = I

P_{1} f_{1} = - f_{1} P_{1}

, and

[P_{1}, f_{2}] = [P_{1}, f_{2}^{†}] = 0

. Hence

[s_{1}^{-}, s_{1}^{+}] = [f_{1}, f_{1}^{†}] = 1 - 2 n_{1} = - σ_{1}^{z}, [σ_{1}^{z}, σ_{2}^{z}] = 4 [n_{1}, n_{2}] = 0,

[s_{1}^{-}, s_{2}^{-}] = f_{1} P_{1} f_{2} - P_{1} f_{2} f_{1} = - P_{1} {f_{1}, f_{2}} = 0,

and

[s_{2}^{-}, s_{2}^{+}] = [P_{1} f_{2}, f_{2}^{†} P_{1}] = f_{2} f_{2}^{†} - f_{2}^{†} f_{2} = 1 - 2 n_{2} = - σ_{2}^{z} .

These are the spin commutation relations, with distinct sites commuting. If one retains the unscaled symbols

σ_{j}^{\pm} = 2 s_{j}^{\pm}

, the first and last displayed commutators instead equal

- 4 σ_{j}^{z}

; the off-site commutators remain zero. Setting the unscaled

σ_{j}^{-}

equal to

f_{j}

would contradict the original Pauli matrices.