  We study attractively interacting spin-1/2 fermions on the square lattice
subject to a spin population imbalance. Using unbiased diagrammatic Monte Carlo
simulations we find an extended region in the parameter space where the Fermi
liquid is unstable towards formation of Cooper pairs with non-zero
center-of-mass momentum, known as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)
state. In contrast to earlier mean-field and quasi-classical studies we provide
quantitative and well-controlled predictions on the existence and location of
the relevant Fermi-liquid instabilities. The highest temperature where the FFLO
instability can be observed is about half of the superfluid transition
temperature in the unpolarized system.
