and ${N_\downarrow}$. A summary of the 18 sets of results obtained is shown in [Fig. \[nvsnu\]]{}.

The relationship between the ratio of commmunal state indices ${N_\uparrow}/{N_\downarrow}$ and Fermi surface density of states ratios ${\nu_{\uparrow}}/{\nu_{\downarrow}}$ is well described by the line ${N_\uparrow}/{N_\downarrow}={\nu_{\uparrow}}/{\nu_{\downarrow}}$, providing strong evidence for the communal pairing [@multipartsupconfew; @commupair] over FFLO. The relationship is particularly strong when the ratio can be written containing small integers [@multipartsupconfew] to minimise the product ${N_\uparrow}{N_\downarrow}$, mitigating the energy penalty for states with high ${N_\uparrow}{N_\downarrow}$ [@commupair]. The correlation coefficient between the gathered data and the line ${N_\uparrow}/{N_\downarrow}={\nu_{\uparrow}}/{\nu_{\downarrow}}$ is $R^2=0.95$.

Conclusions {#sec:conclusion}
===========

We have observed a spatially modulated superconducting state using DMC. Furthermore, the state is qualitatively different from an FFLO state [@ffloff; @fflolo], having condensate fraction peaks at multiple momenta, as opposed to the single peak expected for FFLO. Exact diagonalisation studies provide corroborating evidence that the distribution of condensate fraction with momenta is more indicative of communal superconductivity [@multipartsupconfew; @commupair] than traditional FFLO or crystalline FFLO superconductivity [@fflorevision1; @fflorevision2]. We have confirmed convergence of the state with respect to choices of system size, scattering length, pseudopotential cutoff length, DMC timestep, and DMC walker population.

This numerical evidence that builds on previous analytical 