Ultracold Atoms in a Square Lattice with Spin-Orbit Coupling: Charge Order, Superfluidity, and Topological Signatures

We present an ab initio, numerically exact study of attractive fermions in square lattices with Rashba spin-orbit coupling. The ground state of this system is a supersolid, with coexisting charge and superfluid order. The superfluid is composed of both singlet and triplet pairs induced by spin-orbit coupling. We perform large-scale calculations using the auxiliary-field quantum Monte Carlo method to provide the first full, quantitative description of the charge, spin, and pairing properties of the system. In addition to characterizing the exotic physics, our results will serve as essential high-accuracy benchmarks for the intense theoretical and especially experimental efforts in ultracold atoms to realize and understand an expanding variety of quantum Hall and topological superconductor systems.

Figure 1

Density-density (upper panel) and pair-pair (lower panel) correlation functions. The left column is for one SOC strength $\lambda =0.25$, while the right column shows the evolution versus $\lambda$. The density plots on the left show the correlations for $i$ running through the entire periodic supercell ($14\times 14$). The upper right panel shows $D(i,0)$ for a slice of sites $i$ along $i_y=-3$, while the lower right panel shows $P_{1s}(i,0)$ for a slice along the diagonal, $i_x=i_y$.

Figure 2

Spin distribution and momentum-space pair wave functions. In (a), the arrow length and direction give the magnitude and orientation of the spin expectation, respectively, at each lattice momentum, and the dot size is proportional to the total occupation. The amplitude of the triplet and singlet parts of the pairing wave function are plotted in (b) and (c), respectively, with the noninteracting Fermi surfaces indicated by the dashed curves. The system is a $20\times 20$ periodic supercell.

Figure 3

Triplet (left) and singlet (right) pair wave functions in real space. From top to bottom, $\lambda =0.25$, 1.0, 1.5. The interaction strength is $U/t=-4$, and the system is a $14\times 14$ periodic supercell.

Figure 4

Edge current in the open system. The top panel shows the edge current at half filling for spin $\uparrow$, with the direction and magnitude of the current indicated by the arrow direction and its color. The bottom panel illustrates the evolution of the current with doping, plotting the current in the $y$ direction versus $y$ at $x=1$.