Quantum Phases of Quadrupolar Fermi Gases in Optical Lattices

We introduce a new platform for quantum simulation of many-body systems based on nonspherical atoms or molecules with zero dipole moments but possessing a significant value of electric quadrupole moments. We consider a quadrupolar Fermi gas trapped in a 2D square optical lattice, and show that the peculiar symmetry and broad tunability of the quadrupole-quadrupole interaction results in a rich phase diagram encompassing unconventional BCS and charge density wave phases, and opens up a perspective to create a topological superfluid. Quadrupolar species, such as metastable alkaline-earth atoms and homonuclear molecules, are stable against chemical reactions and collapse and are readily available in experiment at high densities.

Figure 1

Recipe for realization of quadrupolar particles: (a) with alkaline-earth atoms in long-living ${}^{3}P_2$ levels; and (b) with homonuclear molecules in rotational states with $J>1/2$. (c) Angular “shape” of quadrupolar particles exemplified by $|2,M⟩$ states.

Figure 2

(a) Schematic representation of quadrupolar fermions on a square lattice. Alignment of the quadrupoles is given by the quantization axis of the external field $\mathbf{F}$, pointing along $\widehat{F}=({\theta }_F,{\varphi }_F)$. The nearest-neighbor interaction is represented by green and red solid lines, while the next-nearest-neighbor interaction is shown in blue. (b) 3D plot showing the interactions $V_{\widehat{x}}$ (red), $V_{\widehat{y}}$ (green), and $V_{\widehat{x}+\widehat{y}}$ (blue) as a function of the angles (${\theta }_F$, ${\varphi }_F$); “*” marks the point in the vicinity of which both $V_{\widehat{x},\widehat{y}}$ and $V_{\widehat{x}+\widehat{y}}$ change the sign.

Figure 3

Quantum phase diagram for quadrupolar fermions on a square lattice. (a) FRG phase diagram in the weak coupling limit, $V/t=0.2$, and at half filling shown as a function of the magnetic field direction $\widehat{F}=({\theta }_F,{\varphi }_F)$. The point marked by “*”, where 5 different phases seem to meet, corresponds to $V_{\widehat{x}}$, $V_{\widehat{y}}$, $V_{\widehat{x}+\widehat{y}}\approx 0$ as shown in Fig. 2b. This suggests the likelihood of at most three distinct tricritical points in close proximity to each other, which are hard to resolve due to the smallness of the couplings. (b) The orbital symmetry of the CDW and BCS phases shown in (a) is plotted in red. This symmetry corresponds to the symmetry of the most divergent eigenvector of the BCS and CDW vertices combined. The $p_x$ ($p_y$) wave CDW has the same orbital structure as the $p_x$ ($p_y$) wave BCS.