a common basis can be chosen. Thus the full solution of (\[eq:JCHMomentum\]) for a fixed total number of excitations $N=nL$ is given by the distribution $ \vec {n}=\{n_{   k_1},n_{   k_2},\dots\}$ of $N$ excitations on $L$ momentum modes with minimal energy $E_N[  \vec n]=\sum_{   k} E^{n_{   k}}_{   k}$ together with the constraint $\sum_{   k} n_{   k}\equiv N$. Note that the number of momentum modes $L$ is equal to the number of sites.\
When constructing the phase diagram, the energy of $N=nL-1$, $N=nL$ and $N=nL+1$ excitations needs to be calculated. In the limit of vanishing hopping ($t=0$) and for commensurate filling, i.e. $N=nL$, the distribution of occupation numbers which has the lowest energy is again $\vec n =\{n,n,\dots,n\}$. This corresponds to a MI state with an integer number of excitations on every lattice sites. The phase is gapped with a particle-hole gap as described in section \[sec:intro\]. When $t$ is increased the ground state remains the same, but the gap closes and a quantum phase transition occurs from the MI to the SF phase at some critical value of $t$. The only remaining thing in order to calculate the chemical potentials is to find the momentum mode where the addition (removal) of 