the Gaussian contribution of the first $N=M$ particles remains, while the additional particles lead to a multi-peak structure on top of it. This means that the first $N=M$ particles are effectively filling the lowest energy-band, while the superfluid physics happens in the second band.

![ (a) Normalized momentum distribution as a function of the filling factor. (b) Momentum distribution for filling factors $F=1$ (magenta line), $F=\frac{2}{5}$ (dashed red line) and $F=\frac{7}{5}$ (black line). (c) Momentum distribution for filling factors $F=1$ (magenta line) and $F=\frac{49}{50}$ (dashed black line). All plots in this figure are for a lattice depth of $V_0 = 20 E_r$.[]{data-label="fig:momentumdistributionasfunctionofratio"}](momentumdistasfuncofratio.pdf){width="\linewidth"}

Spatial auto-correlations
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The multi-peak structure observed in the momentum distribution of the incommensurate TG gas is a clear indicator that off-diagonal long-range order (ODLRO) and density long-range order (LRO) exist in the gas. To confirm this we calculate the spatial auto-correlations, which we define at a distance $r$ as the average of the RSPDM-values at $(x,y)$, i.e. $$\rho_1(r) = \frac{1}{L^2}\int dx dy \rho_1(x,y) \delta(|x-y|-r).$$

This function then explicitly quantifies the long-range order of the system and we display it for different regimes in Fig. \[fig:correlationfunctions\]. One can see that for no external lattice (red curve in panel 