a square lattice. The condensate fraction reflects the rotational symmetry of the underlying momentum space lattice, a feature shared with crystalline FFLO theories [@fflorevision1] and is nonzero beyond that of the optimal pairing momenta predicted by FFLO theory. While nonzero pairing at nonoptimal $q$ is not present in FFLO theory or any of its derivatives, it is however consistent with communal pairing [@commupair].

### Characterisation of the communal state

The condensate fraction indicates that the superconducting correlations are consistent with communal pairing. To probe the nature and number of fermions in the communal pairing state, we follow the prescription of Ref. [@multipartsupconfew] and perform exact diagonalisation focusing on $({N_\uparrow},{N_\downarrow})=(2,1)$ or $(3,1)$ fermions in a subset of the momentum states used in the DMC study, specifically those at the Fermi surfaces of the respective spin-species, and calculate the condensate fraction averaged across pair momenta of fixed $q$ as a function of $q$. The strength of the contact interaction for the exact diagonalisation study was chosen to match that used in the DMC study. Results are shown in Fig. \[cfractan\] in the azimuthal direction and Fig. \[cfraccomp\] radially, along with the averaged results obtained using DMC.

![Plot of normalised condensate fraction $f^{}_{\mathrm{norm}}$ 