  The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, characterized by Cooper
pairs condensed at finite-momentum are, at the same time, exotic and elusive.
It is partially due to the fact that the FFLO states allow superconductivity to
survive even in strong magnetic fields at the mean-field level. The effects of
induced interactions at zero temperature are calculated in both clean and dirty
cases, and it is found that the critical field at which the quantum phase
transition to an FFLO state occurs at the mean-field level is strongly
suppressed in imbalanced Fermi gases. This strongly shrinks the phase space
region where the FFLO state is unstable and more exotic ground state is to be
found. In the presence of high level impurities, this shrinkage may destroy the
FFLO state completely.
