potential term we have an ideal Bose lattice gas, the Hamiltonian of which can be diagonalized by the canonical transformation $$\begin{aligned}
b(\vec{k}) &=& \frac{1}{\sqrt{N}} \sum_j b_j e^{i\vec{k}\cdot
\vec{r}_j},\nonumber \\
&& \\
b_j &=& \frac{1}{\sqrt{N}} \sum_{\vec{k}} b(\vec{k})
e^{-i \vec{k} \cdot \vec{r}_j},\nonumber
\label{trafo}\end{aligned}$$ where $N$ is the number of sites in the system. We use periodic boundary conditions, and the wave vectors $\vec{k}$ are restricted to the first Brillouin zone of the lattice.

Inserting (\[trafo\]) into the first two terms of (\[ham\]) we obtain the unperturbed Hamiltonian $${\cal H}_0 = \sum_k [\Delta + \varepsilon(\vec{k}) ] b^{\dagger}
(\vec{k}) b(\vec{k}),
\label{h0}$$ where $\varepsilon(\vec{k}) $ is given by $$\varepsilon(\vec{k}) = t \sum_{\delta} (1-\cos \vec{k}\cdot\vec{r}_
{\delta}),
\label{eps}$$ and $\vec{r}_{\delta}$ is the set of the $c$ nearest neighbour positions with respect to a centrally chosen site. $\Delta$ is the gap of the energy band, since we have $\varepsilon(\vec{0})
=0$ as lowest energy in the center of the Brillouin zone. One sees that $t$ is a measure for the bandwidth, which for a $d$-dimensional hypercubic lattice is $w=4dt$.

The interaction is written in terms of $b^{(\dagger)}(\vec{k})$ as $$V=\frac{U}{2N} \sum_{\vec{k}_i,\vec{G}} \delta_{\vec{k}_1+\vec{k}_2,
\vec{k}_3+\vec{k}_4+\vec{G}} b^{\dagger}(\vec{k}_1)b^{\dagger}
(\vec{k}_2)
b(\vec{k}_3) b(\vec{k}_4),
\label{vk}$$ where $\vec{G}$ is a vector of the reciprocal lattice. The matrix elements 