Tunable Chern insulator with optimally shaken lattices

Driven optical lattices permit the engineering of effective dynamics with well-controllable tunneling properties. By specifically designing polychromatic driving forces, we describe how an optimal realization of a tunable Chern insulator can be achieved with a system of interacting particles on a shaken hexagonal lattice. Its implementation does not require shallow lattices and favors the study of strongly correlated phases with nontrivial topology.

Figure 1

Sketch of the hexagonal lattice constructed with a triangular Bravais lattice spanned by two of the three vectors ${\mathbf{b}}_1, {\mathbf{b}}_2$, and ${\mathbf{b}}_3$ and a two-point basis comprising the sites $A$ (red) and $B$ (green). The vectors ${\mathbf{a}}_1, {\mathbf{a}}_2$, and ${\mathbf{a}}_3$ connect the different neighboring lattice sites. The relative sign between the rates of the effective next-nearest-neighbor tunneling along the direction ${\mathbf{b}}_3$ is exemplified in blue.

Figure 2

Phase diagram of the isotropic effective Hamiltonian in Eq. (21) (black lines and dark colors) overlapped with the phase diagram of the Haldane model [39] (gray lines and lighter colors), giving the Chern number $\mathcal{C}$ of the lowest-energy band as a function of the phase $\varphi$ and ratio $\mathrm{\Delta }/j_2$. Orange represents $\mathcal{C}=-1$, blue $\mathcal{C}=1$, and white $\mathcal{C}=0$. For $\varphi =\pm \pi /2$ the Chern number of the two Hamiltonians coincide independently of $\mathrm{\Delta }/j_2$.

Figure 3

Phase $\varphi$ of the complex effective next-nearest-neighbor tunneling rates as a function of $A_1/\omega$ and $A_2/\omega$ for a force ${\mathbf{F}}_{+}\left(t\right)$ with $N=2$ and ${\delta }_2=\pi /2$. All phases can be explored with a suitable choice of $A_1$ and $A_2$. The dashed and solid white lines indicate the contour lines for $j_1/j_0=0.5$ and $j_1/j_0=0.25$ and limit the region of accessible phases given a constraint with $r_{\mathrm{th}}=0.5$ or $r_{\mathrm{th}}=0.25$, respectively.

Figure 4

Plot of the real and imaginary part of $R({\varphi }_{\mathrm{tg}},r_{\mathrm{th}})e^{i{\varphi }_{\mathrm{tg}}}$ calculated numerically as a function of a discrete set of target phases ${\varphi }_{\mathrm{tg}}$ and for two different values of $r_{\mathrm{th}}$. In polar coordinates, the radius and argument of each data point coincide with $R({\varphi }_{\mathrm{tg}},r_{\mathrm{th}})$ and ${\varphi }_{\mathrm{tg}}$, respectively. Dots correspond to $r_{\mathrm{th}}=0.25$, and crosses to $r_{\mathrm{th}}=0.5$. The results in blue have been obtained with a driving force ${\mathbf{F}}_{+}\left(t\right)$ and the results in orange with ${\mathbf{F}}_{-}\left(t\right)$.

Figure 5

Plot of $R({\varphi }_{\mathrm{tg}},r_{\mathrm{th}}=0.5)$ calculated numerically as a function of a discrete set of target phases ${\varphi }_{\mathrm{tg}}={\varphi }_n=n\pi /40$ indexed by $n=0,1,\cdots ,20$. Thus, the labels $n=0$ and $n=20$ refer to ${\varphi }_{\mathrm{tg}}=0$ and ${\varphi }_{\mathrm{tg}}=\pi /2$, respectively. The results in green correspond to a driving force ${\mathbf{F}}_{+}\left(t\right)$ with $N=3$, while in blue crosses we display for comparison the results for $N=2$.