Abstract
We explore in this paper a harmonically trapped asymmetric fermionic superfluid in a two-dimensional optical lattice with pairing interactions between the neighboring sites, by self-consistently solving Bogliubov--de Gennes equations. We find the superfluid generally phase separates into a $d$-wave pairing component and an extended $s$-wave pairing one, mixed with the unpaired normal fermions. For sufficient large number asymmetries, the ground state is found to be a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state without fourfold rotational symmetry. Both the $d$-wave and $s$-wave order parameters show spatially varying novel patterns with nodal points or nodal lines. These features are expected to be checked by the experiments on the repulsive ultracold fermionic atoms.
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