Exotic Superfluid Phases in Spin-Polarized Fermi Gases in Optical Lattices

Leveraging cutting-edge numerical methodologies, we study the ground state of the two-dimensional spin-polarized Fermi gas in an optical lattice. We focus on systems at high density and small spin polarization, corresponding to the parameter regime believed to be most favorable to the formation of the elusive Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase. Our systematic study of large lattice sizes, hosting nearly 500 atoms, provides strong evidence of the stability of the FFLO state in this regime, as well as a high-accuracy characterization of its properties. Our results for the density correlation function reveal the existence of density order in the system, suggesting the possibility of an intricate coexistence of long-range orders in the ground state. The ground-state properties are seen to differ significantly from the standard mean-field description, providing a compelling avenue for future theoretical and experimental explorations of the interplay between spin imbalance, strong interactions, and superfluidity in an exotic phase of matter.

Figure 1

Structure factor of the pairing correlation for a $4\times 32$ lattice with $N_{\uparrow }=57$, $N_{\downarrow }=33$, and $U=-4$. We compare the AFQMC results (dashed lines) obtained from three different trial wave functions, the free-particle wave function (black squares), and two HFB wave functions with an effective interaction strength $U_{\mathrm{eff}}=-2.5$ (blue open circles) and $U_{\mathrm{eff}}=-3.5$ respectively (red filled circles). The wave function with $U_{\mathrm{eff}}=-2.5$ is obtained from the incommensurate HFB solution, and the wave function with $U_{\mathrm{eff}}=-3.5$ from the commensurate solution. The solid lines show the corresponding variational results. Inset: corresponding modulated long-range behavior of the AFQMC results for the pairing correlation in real space.

Figure 2

Structure factor of the pairing correlation for a $4\times 36$ lattice with $N_{\uparrow }=39$, $N_{\downarrow }=29$, and $U=-4$. The AFQMC results (red circles) display a shallow maximum, while the mean-field results (green dashed line) display a sharp peak. The vertical line indicates the leading pairing wave vector from [19]. Inset: pairing correlations in real space, as computed with AFQMC and mean field. Note the factor of 10 difference in the $y$ axis scales (with tick labels corresponding to AFQMC results on the left and mean-field on the right).

Figure 3

Pairing and density correlations for a $22\times 22$ system with $N_{\uparrow }=217$, $N_{\downarrow }=181$, and $U=-4$. Upper left: color plot of sign $[{\mathcal{C}}_{\mathrm{\Delta }}(\mathbf{r})]$, to highlight the nodal structure. Upper right: ${\mathcal{C}}_{\mathrm{\Delta }}(\mathbf{r})$ plotted along the horizontal line cuts indicated in the left panel. Lower left: color plot of sign $[{\mathcal{C}}_n(\mathbf{r})]$. Lower right: ${\mathcal{C}}_n(\mathbf{r})$ plotted along the horizontal line cuts indicated in the left panel.

Figure 4

Momentum distributions. Left: majority (top) and minority (bottom) momentum distributions are shown by the color maps. The circles indicate the occupied momenta in the noninteracting state. Right: values of $n_{\uparrow }(\mathbf{k})$ and $n_{\downarrow }(\mathbf{k})$ along two cuts, $k_x=0$ (solid lines) and the diagonal $k_x=k_y$ (dashed lines), as illustrated on the left. The vertical lines indicate the noninteracting Fermi surfaces. The system is a $22\times 22$ site periodic supercell with $N_{\uparrow }=217$, $N_{\downarrow }=181$, and $U=-4$.