## The lowest order particle–hole diagram which generates the induced interaction *g*_{ind}(*p*_{1}, *p*_{4})

**Figure 1.** The lowest order particle–hole diagram which generates the induced interaction *g*_{ind}(*p*_{1}, *p*_{4}). Arrowed and dashed lines describe fermionic propagators and the coupling *g* between the atoms, respectively.

**Abstract**

Motivated by the recent experimental realization of a candidate to the Fulde–Ferrell (FF) and the Larkin–Ovchinnikov (LO) states in one-dimensional (1D) atomic Fermi gases, we study the quantum phase transitions in these enigmatic, finite-momentum-paired superfluids. We focus on the FF state and investigate the effects of the induced interaction on the stability of the FFLO phase in homogeneous spin-imbalanced quasi-1D Fermi gases at zero temperature. When this is taken into account, we find a direct transition from the fully polarized to the FFLO state in agreement with exact solutions. Also, we consider the effect of a finite lifetime of the quasi-particle states in the normal-superfluid instability. In the limit of long lifetimes, the lifetime effect is irrelevant and the transition is directly from the fully polarized to the FFLO state. We show, however, that for sufficiently short lifetimes, there is a quantum critical point, at a finite value of the mismatch of the Fermi wave-vectors of the different quasi-particles, that we fully characterize. In this case, the transition is from the FFLO phase to a normal partially polarized state with increasing mismatch.