Topological Varma Superfluid in Optical Lattices

Topological states of matter are peculiar quantum phases showing different edge and bulk transport properties connected by the bulk-boundary correspondence. While noninteracting fermionic topological insulators are well established by now and have been classified according to a tenfold scheme, the possible realization of topological states for bosons has not been explored much yet. Furthermore, the role of interactions is far from being understood. Here, we show that a topological state of matter exclusively driven by interactions may occur in the $p$ band of a Lieb optical lattice filled with ultracold bosons. The single-particle spectrum of the system displays a remarkable parabolic band-touching point, with both bands exhibiting non-negative curvature. Although the system is neither topological at the single-particle level nor for the interacting ground state, on-site interactions induce an anomalous Hall effect for the excitations, carrying a nonzero Chern number. Our work introduces an experimentally realistic strategy for the formation of interaction-driven topological states of bosons.

Figure 1

(a) A face-centered square lattice (left-hand image) is obtained by superimposing the two square lattices shown in the center and the right-hand images (with ${\lambda }_1\approx 2{\lambda }_2$). Colored disks show potential minima, gray disks or lines show maxima. (b) By varying the relative depth of the superimposed square lattices, the relative depth of the $\mathcal{A}$ and the $\mathcal{B}$ wells of the resulting Lieb lattice can be tuned. (c) Experimental realization of a Lieb-lattice potential with built-in band swapping functionality (see text for details).

Figure 2

(a) Tight-binding lattice model. (b) Comparison between the tight-binding band structure (dashed blue line) and the exact band structure (continuous red line) for $V_{1,0}=10.6E_{\text{rec},2}$ and $V_{2,0}=9.4E_{\text{rec},2}$. The fitting parameters $t=0.166$, $t^{\prime }=0.011$, ${\xi }_p=0.209$ and overall energy shift $E_0=-16.367$ are given in units of the recoil energy, $E_{\text{rec},2}\equiv {\hslash }^2{k}_2^2/(2m)$, with $m$ the atomic mass.

Figure 3

(a) $X-X^{\prime }$ condensate: A pattern of staggered angular momenta and currents arises. (b) $M$ condensate: Angular momenta are rectified and currents form loops in the elementary plaquettes (dashed line).

Figure 4

(a) Bogolyubov spectrum. Parameters are chosen in units where $t=1$: $t^{\prime }=0.2$, ${\xi }_p=1.5$, $U=0.005$. Numerical minimization of the mean-field free energy under the constraint ${\rho }_s+{\rho }_p=100$ yields $\mu =-1.01882$, ${\rho }_{p}U=0.1730$, and ${\rho }_{s}U=0.3269$. (b) Berry curvature $F_{xy}$ of the lowest branch of the Bogolyubov spectrum. The white spot at $M=(\pi ,\pi )$ indicates that the Berry curvature is not calculated at this point, since this is the momentum at which condensation occurs.

Figure 5

(a) Bogolyubov spectrum in a cylindrical geometry with periodic boundary conditions along the $y$ direction and 36 unit cells along the $x$ direction. The red continuous line indicates the edge-state dispersion. Parameters are chosen as in Fig. 4. (b) Expanded view on the region marked by the dashed box in (a).