Topological insulators and fractional quantum Hall effect on the ruby lattice

We study a tight-binding model on the two-dimensional ruby lattice. This lattice supports several types of first- and second-neighbor spin-dependent hopping parameters in an $s$-band model that preserves time-reversal symmetry. We discuss the phase diagram of this model for various values of the hopping parameters and filling fractions and note an interesting competition between spin-orbit terms that individually would drive the system to a $Z_2$ topological insulating phase. We also discuss a closely related spin-polarized model with only first- and second-neighbor hoppings and show that extremely flat bands with finite Chern numbers result, with a ratio of the band gap to the band width of approximately 70. Such flat bands are an ideal platform to realize a fractional quantum Hall effect at appropriate filling fractions. The ruby lattice can be possibly engineered in optical lattices and may open the door to studies of transitions among quantum spin liquids, topological insulators, and integer and fractional quantum Hall states.

Figure 1

Schematic of the ruby lattice and illustration of the nearest-neighbor hopping, $t,t_1$ (real) and $t^{\prime },t_1^{\prime }$ (complex), and the three types of “second-neighbor” spin-orbit coupling or hopping indicated by the dashed or dot dashed lines, $t_2,t_3$, and $t_{4r}$. (a) The spin-orbit coupling strength $t_2$ within a hexagon. (b) The spin-orbit coupling strength $t_3$ within a pentagon composed of one triangle and one square. The hoppings $t_2$ and $t_3$ are present on all such bonds of the type shown that are consistent with the symmetry of the lattice. (c) The unit cell of the ruby lattice. (d) Schematic of the hopping parameters used to obtain a flat band with a finite Chern number and $W/E_g\approx 70$. (e) The first Brillouin zone of the ruby lattice, with the high symmetry points $\Gamma$, $M$, and $K$.

Figure 2

The energy bands without any spin-orbit coupling and with finite spin-orbit coupling, given by Eq. (2). (a) Bulk energy bands along high symmetry directions in the absence of spin-orbit coupling. Note that there is a Dirac point at $K$ for 1/6 and 2/3 filling and a quadratic band touching point at $\Gamma$ for 1/2 and 5/6 filling. (The underlying lattice is triangular, as it is for the honeycomb lattice.) Note also the flat bands along the $\Gamma –M$ direction at 1/2 and 5/6 filling. (b) The energy bands for finite spin-orbit coupling, with $t_2=t_3=0.1t$. It is clearly seen that the Dirac points at the $K$ points and quadratic band touching point at the $\Gamma$ point are removed.

Figure 3

The energy bands on a strip geometry for the case that $t_1=t$, $t_2=t_3=0.1t$. At filling fractions 1/6 and 2/3, edge modes cross the band gap an odd number of times, which clearly reveals the topological insulator phases. Their identification is also confirmed with a direct evaluation of the $Z_2$ invariant.

Figure 4

Phase diagrams for different filling fractions. Figures (a)–(e) are for the filling fraction from 1/6 to 5/6, with filling fraction increasing in units of 1/6. For each filling, the vertical and horizontal axes measure $t_1$ and $t_2=t_3={\lambda }_{\mathrm{SO}}$, in units of $t$, respectively. The gray-scale coding is as follows: black $=$ conductor, gray $=$ insulator, and white $=$ topological insulator.

Figure 5

Phase diagrams for $t_1=0$. The horizontal axis is $t_2/t$, and the vertical axis is $t_3/t$. Figures (a)–(e) are for the filling fraction from 1/6 to 5/6, increasing in units of 1/6. For all fillings the origin $t_2=t_3=0$ is a trivial insulator.

Figure 6

Phase diagrams for $t_1=t$. The horizontal axis is $t_2/t$, and the vertical axis is $t_3/t$. Figures (a)–(e) are for the filling fraction from 1/6 to 5/6, increasing in units of 1/6.

Figure 7

Phase diagrams for $t_1=2t$. The horizontal axis is $t_2/t$, and the vertical axis is $t_3/t$. Figures (a)–(e) are for the filling fraction from 1/6 to 5/6, increasing in units of 1/6.

Figure 8

The energy bands with $t_i=1.2t,t_{1r}=-1.2t,t_{1i}=2.6t,\text{and}{t}_{4r}=-1.2t$.