Topology-induced phase transitions in quantum spin Hall lattices

Physical phenomena driven by topological properties, such as the quantum Hall effect, have the appealing feature that they are robust with respect to external perturbations. Lately, a new class of materials has emerged that manifests topological properties at room temperature and without the need of external magnetic fields. These topological insulators are band insulators with large spin-orbit interactions and exhibit the quantum spin-Hall (QSH) effect. Here we investigate the transition between QSH and normal insulating phases under topological deformations of a two-dimensional lattice. We demonstrate that the QSH phase present in the honeycomb lattice loses its robustness as the occupancy of extra lattice sites is allowed. Furthermore, we propose a method for verifying our predictions with fermionic cold atoms in optical lattices. In this context, the spin-orbit interaction is engineered via an appropriate synthetic gauge field.

Figure 1

(a) HCL with lattice vectors ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$ and two atoms in the unit cell. (b) ${\mathcal{T}}_3$ lattice with three atoms in the unit cell. Rims A and B (open circle and square) have a lower connectivity of 3 compared to 6 for the hub site (filled circle). The ${\mathcal{T}}_3$ lattice can be viewed as two nested HCLs, emphasized by the left (red) and right (blue) hexagon.

Figure 2

(a) Energy spectrum as a function of the in-plane momentum $(k_x,k_y)$ and for a fixed value of the spin-orbit interaction $t_{\mathrm{SO}}=0.25t$. The first Brillouin zone and its six corners are indicated. (b) Cut of the energy spectrum at $k_x=0$ [shadowed plane in (a)].

Figure 3

One-dimensional energy bands for a strip of (a) HCL and (b) ${\mathcal{T}}_3$ lattice with $t_{\mathrm{SO}}=0.05\hspace{0.5em}t$. The bands crossing the gap (blue lines) are spin-filtered edge states.

Figure 4

(a) Phase diagrams in the Fermi energy-phase ($E_{\mathrm{F}}-\varphi$)-plane for the honeycomb lattice. The dark regions represent the energy spectrum. The gaps are colored according to the ${\mathrm{Z}}_2$ index: orange (green) gaps correspond to $\nu =1$ ($\nu =0$). (b) Phase diagrams in the ($E_{\mathrm{F}}-\varphi$) plane for the ${\mathcal{T}}_3$ lattice. (c) Cut of the phase diagram for the HCL in (a) for $\varphi =1/21$. The orange bars denote the energy intervals for which $\nu =1$. (d) Cut of the phase diagram for the ${\mathcal{T}}_3$ in (b) for $\varphi =1/11$. Within the relativistic regime limited by $E_{\mathrm{VHS}}=1$ ($E_{\mathrm{VHS}}\approx 1.4$), the gauge field (4) opens nontrivial (trivial) ${\mathrm{Z}}_2$ bulk gaps for the HCL (${\mathcal{T}}_3$ lattice).

Figure 5

Peierls phases generating the SU(2) gauge field (a) for the honeycomb lattice and (b) in the ${\mathcal{T}}_3$ lattice.