  Increasing the spin imbalance in superconductors can spatially modulate the
gap by forming Cooper pairs with finite momentum. For large imbalances compared
to the Fermi energy, the inhomogeneous FFLO superconductor ultimately becomes a
normal metal. There is mounting experimental evidence for this scenario in 2D
organic superconductors in large in-plane magnetic fields; this is complemented
by ongoing efforts to realize this scenario in coupled tubes of atomic Fermi
gases with spin imbalance. Yet, a theory for the phase transition from a metal
to an FFLO superconductor has not been developed so far and the universality
class has remained unknown. Here we propose and analyze a spin imbalance driven
quantum critical point between a 2D metal and an FFLO phase in anisotropic
electron systems. We derive the effective action for electrons and bosonic FFLO
pairs at this quantum phase transition. Using this action, we predict non-Fermi
liquid behavior and the absence of quasi-particles at a discrete set of hot
spots on the Fermi surfaces. This results in strange power-laws in
thermodynamics and response functions, which are testable with existing
experimental set-ups on 2D organic superconductors and may also serve as
signatures of the elusive FFLO phase itself. The proposed universality class is
distinct from previously known quantum critical metals and, because its
critical fluctuations appear already in the pairing channel, a promising
candidate for naked metallic quantum criticality over extended temperature
ranges.
