Topological bands with a Chern number $C=2$ by dipolar exchange interactions

We demonstrate the realization of topological band structures by exploiting the intrinsic spin-orbit coupling of dipolar interactions in combination with broken time-reversal symmetry. The system is based on polar molecules trapped in a deep optical lattice, where the dynamics of rotational excitations follows a hopping Hamiltonian which is determined by the dipolar exchange interactions. We find topological bands with Chern number $C=2$ on the square lattice, while a very rich structure of different topological bands appears on the honeycomb lattice. We show that the system is robust against missing molecules. For certain parameters we obtain flat bands, providing a promising candidate for the realization of hard-core bosonic fractional Chern insulators.

Figure 1

(a) Setup: Each lattice site of a two-dimensional optical lattice is occupied by a single polar molecule. The molecules can be excited into two different rotational states. Dipole-dipole interactions induce long-range tunneling links for the excitations. (b) Rotational level structure of each molecule with applied electric field and additional microwave field with Rabi frequency $\mathrm{\Omega }$ and detuning $\mathrm{\Delta }$.

Figure 2

(a) Dispersion relation for the $|+\rangle$ and $|-\rangle$ states on the square lattice. The dashed line shows the time-reversal invariant point $t=\mu =0$ with band touching at the $\mathrm{\Gamma }$ and $\text{M}$ point. The solid line shows the gapped topological bands in the time-reversal-broken system for $w/\overline{t}=3,\mu =0$, and $t/\overline{t}=0.4$. (b) Dispersion relation for the $|+\rangle$ and $|1\rangle$ states for electric field angles ${\mathrm{\Theta }}_0=0$ (dashed) and ${\mathrm{\Theta }}_0=\pi /4$ (solid), respectively. The latter has a lower band with flatness $f\approx 1$.

Figure 3

(a) Sample-averaged Chern number $\langle C\rangle$ in the disordered system for increasing density $\rho$ of defects. A single realization either yields $C=2$ or $C=0$. Bars indicate two standard errors. The results are shown for square lattices of size $L\times L$ with $L=10,20,40$. The long-range tunneling stabilizes the topological phase for defect densities $\rho \lesssim 0.45$. (b) Two-dimensional projection of the dispersion relation in the honeycomb lattice for $t/\overline{t}=0.54,w/\overline{t}=1.97$ and $\mu /\overline{t}=-4.54$. The lowest band has a flatness ratio of $f\approx 6.4$ and a Chern number of $C=-1$. (c) Topological phase diagram in the honeycomb lattice for $t/\overline{t}=0.54$. The labels give the Chern numbers of the four bands (bar indicates negative number) from bottom to top while the solid lines correspond to touching points between two bands. The color indicates the flatness $f$ of the lowest band. The arrow shows the parameters of the flat-band model in panel (b).

Figure 4

(a) Dispersion relation for the $|+\rangle$ and $|1\rangle$ states on a cylindrical square lattice geometry with infinite extent in $x$ direction and 31 sites in the $y$ direction. Four edge states cross the bandgap in the $C=2$ phase (two for each edge). (b) Exponentially decaying amplitude of the two edge states in logarithmic scale, corresponding to the points shown in the spectrum. (c) Edge state amplitude $|{\psi }_{+}{(x,y)|}^2+{\left|{\psi }_{-}(x,y)\right|}^2$ on a finite $50\times 50$ square lattice with a filling fraction of $0.8$ (missing sites are indicated by crosses).

Figure 5

Hopping strengths and flux pattern of a single layer in different lattices. Tunneling elements without arrow are real numbers. Complex hoppings have the indicated strength along the arrow and the complex conjugate in the opposite direction. (a) Square lattice: A single layer can be constructed by stripes of one component along one of the primitive vectors, effectively doubling the unit cell. The second layer is given by a translation along the second primitive vector. (b) Honeycomb lattice: By distributing the $|+\rangle ,|-\rangle$ orbitals to the two distinct sublattices it is possible to retain the symmetry of the lattice. The second layer is given by a ${60}^{\circ }$ rotation.