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We present a theory for a lattice array of weakly coupled one-dimensionalultracold attractive Fermi gases (1D `tubes') with spin imbalance, where strongintratube quantum fluctuations invalidate mean field theory. We first constructan effective field theory, which treats spin-charge mixing exactly, based onthe Bethe ansatz solution of the 1D single tube problem. We show that the 1DFulde-Ferrel-Larkin-Ovchinnikov (FFLO) state is a two-component Luttingerliquid, and its elementary excitations are fractional states carrying bothcharge and spin. We analyze the instability of the 1D FFLO state againstinter-tube tunneling by renormalization group analysis, and find that it flowsinto either a polarized Fermi liquid or a FFLO superfluid, depending on themagnitude of interaction strength and spin imbalance. We obtain the phasediagram of the quasi-1D system and further determine the scaling of thesuperfluid transition temperature with intertube coupling.

Theory of quasi-one dimensional imbalanced Fermi gases