Unconventional spin-density waves in dipolar Fermi gases

We show that unconventional spin order arises naturally in two-component dipolar Fermi gases of atoms or molecules, which recently became accessible experimentally, in optical lattices. Using an unbiased functional renormalization-group analysis, we find that dipolar interactions lead to an instability of the gas toward an $\ell =1$ spin-density wave state. This phase is the particle-hole analog of spin-triplet, $p$-wave Cooper pairs. The order parameter for such spin-density waves of $p$-wave orbital symmetry is a vector in spin space and, moreover, is defined on lattice bonds rather than on lattice sites. We determine the rich quantum phase diagram of dipolar fermions at half filling on the square lattice as a function of the dipolar orientation and discuss how these exotic spin-density waves emerge amidst competition with superfluid and charge-density wave phases.

Figure 1

Schematic spin and charge order for pseudo-spin-1/2 dipolar fermions in two dimensions. The central image illustrates the on-site interaction $U$ between opposite spins (red arrows) and the dipolar interaction $V_{\mathbf{r}}$, here assumed to be spin independent. The characteristic scale for $V_{\mathbf{r}}$ is $V_d=d^2/a^3$, with $a$ being the lattice spacing. $V_{\mathbf{r}}$ depends also sensitively on the orientation of the dipoles labeled by angles $(\theta ,\varphi )$. (a) The conventional antiferromagnetic spin-density wave (SDW${}_s$). (b) Checkerboard charge-density wave (CDW${}_s$), where “charge” is defined as the total density. (c) An example of $p$-wave spin-density waves (SDW${}_p$) with modulation of $y$ bond variables. (d) An example of mixed (extended) $s$- and $d$-wave spin-density waves (SDW${}_{s+d}$). Red arrows in (c) and (d) indicate the direction of the spin vector $\mathcal{S}$ defined on the bonds (yellow ellipsoids).

Figure 2

Phase diagram of dipolar fermions on a square lattice at half filling obtained from FRG. It is shown on the surface of a sphere as a function of the dipole orientation angle $\theta$ and $\varphi$ for fixed interactions (a) $V_d=0.5,U=0.1$ and (b) $V_d=0.5,U=0.5$ in units of $t$. Unconventional SDWs (SDW${}_p$ and SDW${}_{s+d}$) are sandwiched between the CDW and BCS superfluid phase. The FRG eigen-wave-function corresponding to a representative point (marked by “$*$”) in each phase is shown in the $k_x-k_y$ polar plots with matching colors. Note that FRG predicts a mixed $s+d$-wave (rather than pure $d$-wave) SDW. As the on-site interaction is increased from $U=0.1$ in (a) to $U=0.5$ in (b), the SDW${}_{s+d}$ phase expands and squeezes out the neighboring phases. The $s$-wave component of SDW${}_{s+d}$ increases with $U$, while the $d$-wave component diminishes.

Figure 3

Mean-field results for the SDW${}_p$ phase. (a) Preferred direction of the spin-polarization vector $\mathcal{S}$ as a function of the ratio $V_d^{\parallel }/V_d^{\perp }$. It is along the $z$ direction for $V_d^{\perp }<V_d^{\parallel }$ and lies in the $x$-$y$ plane for $V_d^{\perp }>V_d^{\parallel }$. (b) The energy difference between the mean-field states with ${\mathcal{S}}^z$ and ${\mathcal{S}}^x$ order. Inset: Magnitude of the corresponding order parameter. The parameters are $U=0.1$, $V_d^{\parallel }=1.0$, $\theta ={47}^{\circ }$, and $\varphi =0$. All energies are in units of $t$. (c) The mean-field energy gap of the SDW${}_p$ phase, in units of the Fermi energy, as a function of the interspin long-range dipolar interaction $V_d^{\perp }$ for $\theta ={47}^{\circ }$ and $\varphi =0$. For the SU(2) symmetric case plotted in (c), the energy gaps for different vector polarizations are degenerate.