c_{2 j - 1} = a_{j} + a_{j}^{†}, c_{2 j} = - i (a_{j} - a_{j}^{†}) .

They obey

c_{r}^{†} = c_{r}

{c_{r}, c_{s}} = 2 δ_{rs}

, and

a_{j} = (c_{2 j - 1} + i c_{2 j}) /2

. Substituting these inverse relations expands every hopping and pairing monomial as a bilinear in the

c_{r}

. Diagonal terms

c_{r}^{2} = 1

contribute only a constant.

For

r \neq = s

c_{r} c_{s}

is anti-Hermitian. Hermiticity of the original quadratic fermion Hamiltonian therefore makes its coefficient purely imaginary, so it can be written

i A_{rs}

with

A_{rs}

real. Since

\sum_{rs} A_{rs} c_{r} c_{s} = \frac{1}{2} \sum_{rs} (A_{rs} - A_{sr}) c_{r} c_{s} + \frac{1}{2} tr A,

the symmetric part again changes only the constant. Absorbing conventional factors into

A

gives

H = i \sum_{rs} A_{rs} c_{r} c_{s} + constant, A^{T} = - A, A \in M_{2 n} (R) .

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