  Leveraging cutting-edge numerical methodologies, we study the ground state of
the two-dimensional spin-polarized Fermi gas in an optical lattice. We focus on
systems at high density and small spin polarization, corresponding to the
parameter regime believed to be most favorable to the formation of the elusive
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase. Our systematic study
of large lattice sizes, hosting nearly $500$ atoms, provides strong evidence of
the stability of the FFLO state in this regime, as well as a high-accuracy
characterization of its properties. Our results for the density correlation
function reveal the existence of density order in the system, suggesting the
possibility of an intricate coexistence of long-range orders in the ground
state. The ground-state properties are seen to differ significantly from the
standard mean-field description, providing a compelling avenue for future
theoretical and experimental explorations of the interplay between interaction
and superfluidity in an exotic phase of matter.
