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

We consider a gas of two species of fermions with population imbalance. Using the renormalisation group in $d=1+\epsilon$ dimensions, we show that for spinless fermions and $\epsilon > 0$ a fixed point appears at finite attractive coupling where the quasiparticle residue $Z$ vanishes, and identify this fixed point 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 repulsive fixed point, indicating a transition to spin density wave order.