  Motivated by a realization of imbalanced Feshbach-resonant atomic Fermi
gases, we formulate a low-energy theory of the Fulde-Ferrell and the
Larkin-Ovchinnikov (LO) states and use it to analyze fluctuations, stability,
and phase transitions in these enigmatic finite momentum-paired superfluids.
Focusing on the unidirectional LO pair-density wave state, that spontaneously
breaks the continuous rotational and translational symmetries, we show that it
is characterized by two Goldstone modes, corresponding to a superfluid phase
and a smectic phonon. Because of the liquid-crystalline "softness" of the
latter, at finite temperature the 3d state is characterized by a vanishing LO
order parameter, quasi-Bragg peaks in the structure and momentum distribution
functions, and a "charge"-4, paired Cooper-pairs, off-diagonal-long-range
order, with a superfluid-stiffness anisotropy that diverges near a transition
into a nonsuperfluid state. In addition to conventional integer vortices and
dislocations the LO superfluid smectic exhibits composite half-integer
vortex-dislocation defects. A proliferation of defects leads to a rich variety
of descendant states, such as the "charge"-4 superfluid and Fermi-liquid
nematics and topologically ordered nonsuperfluid states, that generically
intervene between the LO state and the conventional superfluid and the
polarized Fermi-liquid at low and high imbalance, respectively. The fermionic
sector of the LO gapless superconductor is also quite unique, exhibiting a
Fermi surface of Bogoliubov quasiparticles associated with the Andreev band of
states, localized on the array of the LO domain-walls.
