  The evolution of an attractive polarized two-component Fermi gas at zero
temperature is analyzed as its polarization is progressively decreased, from
full polarization (corresponding to the polaronic limit) down to a critical
polarization when superfluidity sets in. This critical polarization and the
nature of the associated superfluid instability are determined within a fully
self-consistent $t$-matrix approach implemented exactly at zero temperature. In
this way, the polarization-vs-coupling phase diagram at zero temperature is
constructed throughout the whole BCS-BEC crossover. Depending on the coupling
strength of the inter-particle interaction between the two components, the
superfluid instability can be either toward a Fulde-Ferrel-Larkin-Ovchinnikov
(FFLO) phase or toward a standard polarized BCS phase. The evolution with
polarization of the quasi-particle parameters in the normal Fermi gas turns out
to be notably different in the two cases. When the instability is toward a
polarized BCS superfluid, quasi-particles in the proximity of the two Fermi
surfaces remain well defined for all polarizations. When the instability is
instead toward an FFLO superfluid, precursor effects become apparent upon
approaching the FFLO quantum critical point (QCP), where the quasi-particle
residues vanish and the effective masses diverge. This behavior leads to a
complete breakdown of the quasi-particle picture characteristic of a Fermi
liquid, similarly to what occurs in heavy-fermion materials at an
antiferromagnetic QCP. At unitarity, the system is further investigated at
finite temperature, making it possible to identify a non-Fermi liquid region in
the temperature-vs-polarization phase diagram associated with the underlying
FFLO QCP.
