Exotic Superfluid States of Lattice Fermions in Elongated Traps

We present real-space dynamical mean-field theory calculations for attractively interacting fermions in three-dimensional lattices with elongated traps. The critical polarization is found to be 0.8, regardless of the trap elongation. Below the critical polarization, we find unconventional superfluid structures where the polarized superfluid and Fulde-Ferrell-Larkin-Ovchinnikov-type states emerge across the entire core region.

Figure 1

Cross-sectional visualization of the particle density and pair potential structure in the elongated trap with the aspect ratio $\alpha =7.5$ for selected polarizations $P\le 0.8$. (a) The density difference $\delta n\equiv n_{\uparrow }-n_{\downarrow }$ is plotted in the $y=0$ plane. The lighter (darker) gray indicates the larger (smaller) value of $\delta n$. Black defines $\delta n=0$. (b) The pair potential $\Delta (x,y=0,z)$ shows the evolution of the superfluid core structure as a function of $P$. The FFLO-like oscillations are observed at high $P$’s but below $P=0.8$ where $\Delta$ vanishes.

Figure 2

Transition to the normal phase with increasing polarization $P$ examined for various trap aspect ratios $\alpha$. (a) Superfluid order parameter $\sum _i|{\Delta }_i{|}^2/N_{\downarrow }$ vanishes above $P_c\simeq 0.8$, regardless of $\alpha$. (b) Central polarization $(n_{\uparrow }-n_{\downarrow })/(n_{\uparrow }+n_{\downarrow })$ of the site at the cloud center becomes finite far below $P_c$, implying the presence of a polarized superfluid core with finite $\delta n$ and $\Delta$ as indicated in the inset. The conventional DMFT data with LDA are given for comparison.

Figure 3

Evolution of the cloud structure with increasing polarization $P$. Axial density profiles $n$ and pair potential $\Delta$ are shown along the $z$ axis. The plots are selected for the trap aspect ratio $\alpha =7.5$. The coexistence of the finite dip in the density difference $\delta n\equiv n_{\uparrow }-n_{\downarrow }$ and a finite $\Delta$ at the central region characterizes the polarized superfluid core. The central dip in $\delta n$ disappears at high $P=0.74$, replaced by the FFLO-type oscillations in $\Delta (z)$ spreading across the trap center.

Figure 4

Trap aspect ratio $\alpha$ and polarization $P$ dependence of the phase of the central region of the trap. Fully paired superfluid (SF) is found at low $P$’s and continuously evolves into the polarized superfluid (pSF). The pSF is specified with the central polarization larger than 0.005. At higher $P$’s, the FFLO-type phase is identified for $\alpha \ge 5.0$. The dashed lines between the phases are drawn for guidance.

Figure 5

BdG calculations at unitarity in the elongated trap with $\alpha =10$. (a) The condensate fraction is plotted as a function of polarization $P$. (b) The density difference $\delta n_{\mathrm{BdG}}$ and (c) the pair potential ${\Delta }_{\mathrm{BdG}}$ (normalized by the Fermi energy $E_F$) in the $y=0$ plane are shown for selected $P$’s. The majority population $N_{\uparrow }$ is fixed at 200 for a given $P$.