= Bosonic quadratic Nambu normalization
{c}
{title2=$H=C-\tfrac12\sum_k A_k+\tfrac12\sum_k\Psi_k^\dagger\begin{pmatrix}A_k&B_k\\B_k&A_k\end{pmatrix}\Psi_k$}
For real even coefficients, $\Psi_k=(a_k,a_{-k}^\dagger)^{\mathsf T}$ and a full-zone sum, this expression equals $C+\sum_k[A_ka_k^\dagger a_k+B_k(a_ka_{-k}+a_k^\dagger a_{-k}^\dagger)/2]$. The constant subtraction follows from $a_{-k}a_{-k}^\dagger=1+a_{-k}^\dagger a_{-k}$. Pairing redundancy and <normal ordering> determine both factors together. Switching to half a zone requires grouping both members of each pair; changing only a diagonal factor is inconsistent.
Back to article page
