Liquid crystal phases of ultracold dipolar fermions on a lattice

Motivated by the search for quantum liquid crystal phases in a gas of ultracold atoms and molecules, we study the density-wave and nematic instabilities of dipolar fermions on the two-dimensional square lattice (in the $x\text{−}y$ plane) with dipoles pointing to the $z$ direction. We determine the phase diagram using two complementary methods, the Hartree-Fock mean-field theory and the linear-response analysis of compressibility. Both give consistent results. In addition to the staggered $\left(\pi ,\pi \right)$ density wave, over a finite range of densities and hopping parameters, the ground state of the system first becomes nematic and then smectic, when the dipolar interaction strength is increased. Both phases are characterized by the same broken fourfold $\left(C_4\right)$ rotational symmetry. The difference is that the nematic phase has a closed Fermi surface but the smectic does not. The transition from the nematic to the smectic phase is associated with a jump in the nematic order parameter. This jump is closely related to the Van Hove singularities. We derive the kinetic equation for collective excitations in the normal isotropic phase and find that the zero sound mode is strongly Landau damped and thus is not a well-defined excitation. Experimental implications of our results are discussed.

Figure 1

(Color online) (a) The experimental setup: a square lattice in the $x\text{−}y$ plane and the applied field (either electric or magnetic) along the $z$ direction. The dipole moments of fermions are aligned along the $z$ direction which leads to large intersite density-density interaction. (b) The staggered density wave: the density is more concentrated on one sublattice than the other. The size of circle indicates the density at the given site. (c) The nematic phase: the effective hoppings along $x$ and $y$ are different. (d) The first Brillouin zone of the square lattice. Four circles $\left(A,B,\alpha ,\beta \right)$ mark the regions near the Van Hove points. The interaction between the opposite Van Hove points is $U_1$ and for neighboring Van Hove points is $U_2$.

Figure 2

(Color online) The phase boundaries of staggered density wave (solid) and smectic phase (dashed) for $t^{\prime }=0$, $n=0.44–0.5$. The ordered quantum phase is above the respective transition line.

Figure 3

(Color online) (a) The nematic order parameter as a function of $U$. (b) The Fermi surface in the first quadrant of the Brillouin zone for $U=5.8$ (isotropic), 6 (nematic), and 6.2 (smectic). These results are computed at $t^{\prime }=-0.2$ and $n=0.4$.

Figure 4

The eigenvector of the softest mode of Eq. (12) computed for $U=5.8$, $t^{\prime }=-0.2$, and $n=0.4$.