 $$\begin{aligned}   \label{00}   \tilde{\cal H} &=&   \sum_{{\bf k},\sigma} \varepsilon_{\bf k} \,     c_{{\bf k},\sigma}^{\dagger}c_{{\bf k},\sigma}  +   \sum_{\bf q} \omega_{\bf q} \,     b_{\bf q}^{\dagger}b_{\bf q} -    \sum_{\bf k}   \left(     \