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Non-Fermi liquid fixed point for an imbalanced gas of fermions in $1+ε$ dimensions

We consider a gas of two species of fermions with population imbalance. Using the renormalization group in $d=1+ε$ dimensions, we show that for spinless fermions and $ε> 0$ a fixed point appears at finite attractive coupling where the quasiparticle residue vanishes, and identify this with the transition to Larkin--Ovchinnikov--Fulde--Ferrell order (inhomogeneous superconductivity). When the two species of fermions also carry spin degrees of freedom we find a fixed point indicating a transition to spin density wave order.