${\mathbb{Z}}_2$ Topological Insulators in Ultracold Atomic Gases

We describe how optical dressing can be used to generate band structures for ultracold atoms with nontrivial ${\mathbb{Z}}_2$ topological order. Time-reversal symmetry is preserved by simple conditions on the optical fields. We first show how to construct optical lattices that give rise to ${\mathbb{Z}}_2$ topological insulators in two dimensions. We then describe a general method for the construction of three-dimensional ${\mathbb{Z}}_2$ topological insulators. A central feature of our approach is a new way to understand ${\mathbb{Z}}_2$ topological insulators starting from the nearly free electron limit.

Figure 1

Lowest energy bands for the 2D topological insulator with $V=0.5E_R$ and $\delta =0.25$ for wave vectors on the perimeter of the parallelogram (shown in the inset) with corners ${\mathbf{\Gamma }}_{nm}=(n{\mathit{\kappa }}_1+m{\mathit{\kappa }}_2)/2$ ($n,m=0,1$). Each band is twofold degenerate, with fourfold degeneracies at ${\mathbf{\Gamma }}_{nm}\ne 0$. The four bands with $E/E_R\lesssim -0.1$ have nontrivial ${\mathbb{Z}}_2$ invariant, as follows from the eigenstate parities ($\pm$) at ${\mathbf{\Gamma }}_{nm}$.

Figure 2

Lowest energy bands for the model Eq. (9), for a set of wave vectors $\mathit{k}=\sum _i{k}_i{\mathit{\kappa }}_i$ with $k_2$ and $k_3$ uniformly spaced across the Brillouin zone. The parameters are $V=0.9E_R$, $\delta =1.0$, and $\mu =-0.2$. The four bands with $E/E_R\lesssim -0.9$ have nontrivial ${\mathbb{Z}}_2$ invariant which, together with the clear band gap, demonstrates the topological insulator phase. The right-hand panel shows the density of states.