distribution, $n(k=0)$, are shown in Fig. \[fig:staticphasediagram\] as a function of the filling ratio and the lattice depth. One can immediately see that the coherence decreases quickly for unit filling, $F=1$, eventually reaching the value of $\lambda_0=1$ corresponding to the pinned phase. For double filling, $F=2$, a finite lattice depth is required before this transition can happen, as the lattice needs to be deep enough to overcome the repulsive two-body interactions at each site. For incommensurate fillings a slow decrease of the coherence with increasing lattice depth can be seen. This behavior is qualitatively mirrored in the magnitude of the momentum distribution at $k=0$ and in the deep lattice limit ($V_0=20 E_r$), both quantities display an oscillating dependence on the filling ratio $F$, as can be seen in the insets of Fig. \[fig:staticphasediagram\]. This is similar to the behaviour known for the superfluid fraction in the hard-core BHM [@Krutitsky2016].

The distinct characteristics of the pinned phase and the superfluid phase also manifest themselves in the full momentum distribution. In order to characterize the incommensurate phase we show the momentum distribution for $F=\frac{2}{5}$ as a function of the lattice depth in Fig. \[fig:momentumdistributionasfunctionofdepth\](a) and with a comparison to the commensurate case 