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 $d=1+ϵ$ spatial dimensions, we show that for spinless fermions and $2>ϵ>0$ 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.

Figure 1

Schematic phase diagram showing the continuous quantum phase transition between the normal Fermi liquid and LOFF states as a function of interaction strength and imbalance. The LOFF-BCS transition is first order, leading to phase separation. Inset: vanishing of the quasiparticle residue in the Fermi liquid as the transition is approached along the bold horizontal arrow. Note that the exponent varies continuously with $\mathcal{P}$.

Figure 2

Pairing of majority (blue [dark gray]) and minority (red [medium gray]) fermions. Left: $|\mathbf{Q}|=K_{F,a}-K_{F,b}$ leads to allowed values of fermion momentum indicated by the dash bounded region. Right: $|\mathbf{Q}|\ne K_{F,a}-K_{F,b}$ yields smaller allowed regions (cf. the $\mathcal{P}=0$ case [22]).

Figure 3

The two diagrams contributing to the vertex at one loop order. Dashed and solid lines indicate majority (a) and minority (b) propagators, respectively. The notation is explained in the text.

Figure 4

Flow diagram for the spinless case, indicating Fermi liquid (FL) and LOFF phases.