Fulde-Ferrell-Larkin-Ovchinnikov pairing in one-dimensional optical lattices

Spin-polarized attractive Fermi gases in one-dimensional (1D) optical lattices are expected to be remarkably good candidates for the observation of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We model these systems with an attractive Hubbard model with population imbalance. By means of the density-matrix renormalization-group method, we compute the pairing correlations as well as the static spin and charge structure factors in the whole range from weak to strong coupling. We demonstrate that pairing correlations exhibit quasi-long-range order and oscillations at the wave number expected from the FFLO theory. However, we also show by numerically computing the mixed spin-charge static structure factor that charge and spin degrees of freedom appear to be coupled already for a small imbalance. We discuss the consequences of this coupling for the observation of the FFLO phase, as well as for the stabilization of the quasi-long-range order into long-range order by coupling many identical 1D systems, such as in quasi-1D optical lattices.

Figure 1

Top panel: DMRG results for the total site occupation $n_{\ell \uparrow }+n_{\ell \downarrow }$ as a function of site position $\ell$ for a system with $N_{\uparrow }=20$ fermions in $L=60$ lattice sites and $U∕t=5$. The numbers of down-spin fermions are $N_{\downarrow }=20$, 16, 12, 6, and 2 (the corresponding spin polarization is $\delta =0\%$,11%, 25%, 54%, and 82%). Bottom panel: the local spin polarization $n_{\ell \uparrow }-n_{\ell \downarrow }$ as a function of $\ell$ for the same system parameters as in the top panel. The thin solid lines are just guides to the eye.

Figure 2

Top panel: DMRG results for the pairing correlator ${\mathcal{C}}_{\ell {\ell }^{\prime }=L∕2}$ as a function of site position $\ell$ for a system with $N_{\uparrow }=20$ fermions in $L=60$ lattice sites and $U∕t=5$. The numbers of down-spin fermions are $N_{\downarrow }=20$, 16, 12, 6, and 2 (the corresponding spin polarization is $\delta =0\%$,11%, 25%, 54%, and 82%). Bottom panel: Interaction contribution to the momentum-space pairing correlator $\Delta \mathcal{C}\left(k_m\right)$ as a function of wave number $k_m=\pi m∕(L+1)$. The symbol coding is the same as in the top panel. Note the well-defined peaks at $q_{\mathrm{FFLO}}$. In the inset, we show $\Delta \mathcal{C}\left(k_m\right)$ for the unpolarized system with $N_{\uparrow }=N_{\downarrow }=20$, which shows a narrow peak at $k_1$. The thin solid lines are just guides to the eye.

Figure 3

Top panel: Interaction contribution to the momentum-space pairing correlator $\Delta \mathcal{C}\left(k_m\right)$ as a function of $k_m$ for $N_{\uparrow }=20$, $N_{\downarrow }=12$ $(\delta =25\%)$, and $L=60$. The interaction strength $U∕t$ is decreased from 50 to 0.2. In the inset, we show $\mathcal{C}\left(k_m\right)$ for $U∕t=5$, 1, and 0. Bottom panel: The FFLO anomaly $\Gamma$ in the pair correlator [see Eq. (3)] as a function of $U∕t$ for $N_{\uparrow }=20$ and $N_{\downarrow }=18$ (squares, $\delta =5\%$), $N_{\downarrow }=12$ (circles, $\delta =25\%$), and $N_{\downarrow }=6$ (triangles, $\delta =54\%$). Inset: peak visibility $\nu$ [see Eq. (4)] as a function of $2⩽U∕t⩽5$ for the same system parameters as in the main body of the figure ($\nu$ is about 30% at $U∕t=50$). The thin solid lines are just guides to the eye.

Figure 4

Top panel: the density-density structure factor $S_{nn}\left(k_m\right)$ as a function of $k_m$ for $N_{\uparrow }=20$ and $N_{\downarrow }=18$ and different values of $U∕t=0.4$, 1, 2, 3, 4, and 5 (from bottom to top). Bottom panel: the spin-spin structure factor $S_{mm}\left(k_m\right)$. The symbol coding is as in the top panel.

Figure 5

The mixed density-spin structure factor $S_{nm}\left(k_m\right)$ as a function of $k_m$ for $N_{\uparrow }=20$ and $N_{\downarrow }=18$ and different values of $U∕t=0.4$, 1, 2, 3, 4, and 5. The symbol coding is as in Fig. 4. Whereas the results at small $U∕t$ (see, for example, filled squares corresponding to $U∕t=0.4$) are likely to be somewhat affected by finite-size effects, which prevent the full development of a pairing gap, the results at larger $U$ clearly show a nonvanishing weight of $S_{nm}\left(k_m\right)$ at small $k_m$, thus indicating that spin and charge degrees of freedom are coupled at long wavelengths.