exponential decay is much faster than for a macroscopic number of defects ($F=\frac{2}{5}$). In fact it is similar to the one found in the pinned phase, but unlike the pinned phase it shows small revivals at subsequent lattice sites. The magnitude of these revivals decays slightly, but it is difficult to ascertain the precise nature of this small decay, as we are restricted to a periodic 50-site lattice.

Since the modulations in the correlation-function stem from the off-diagonal terms in the RSPDM, they relate directly to the peaks that appear in the momentum distribution and therefore to the superfluid properties of the system. The external lattice breaks the continuious spatial symmetry and imposes discrete spatial symmetry, that is LRO (crystaline order), while the incommensurability between the number of lattice sites and particles allows for a superfluid flow giving rise to QODLRO. This results in numerous peaks at multiples of $ 2 k_l$ with a structure that is dependent on the lattice depth and particle-to-site ratio $F$. Contrarily, in the hard-core BHM for incommensurate lattice-sites and particle numbers, the momentum distribution shows only a single peak at $k=0$ associated with superfluidity. This is due to the spatial symmetry not being broken 