 Two-site Jordan-Wigner normalization and signs

ID: two-site-jordan-wigner-normalization-and-signs

With occupied fermion sites representing up-spins, use

s_{1}^{-} = f_{1}

s_{2}^{-} = P_{1} f_{2}

and

P_{1} = (- 1)^{n_{1}} = 1 - 2 n_{1}

P_{1}^{2} = I

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

and

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

. Consequently the two sites commute as spin operators, while

[s_{j}^{-}, s_{j}^{+}] = 1 - 2 n_{j} = - σ_{j}^{z}

. This normalization is essential: the unscaled combinations

σ_{j}^{x} \pm i σ_{j}^{y}

are twice

s_{j}^{\pm}

H = 2 h - 2 h (n_{1} + n_{2}) - 2 J (f_{1}^{†} f_{2} + f_{2}^{†} f_{1}) .

In particular,

f_{1}^{†} P_{1} = f_{1}^{†}

and

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

fix the hopping signs. The modes

b = (f_{1} + f_{2}) / 2

and

c = (f_{1} - f_{2}) / 2

H = 2 h + (- 2 h - 2 J) b^{†} b + (- 2 h + 2 J) c^{†} c

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