Topological states in multi-orbital HgTe honeycomb lattices

Research on graphene has revealed remarkable phenomena arising in the honeycomb lattice. However, the quantum spin Hall effect predicted at the K point could not be observed in graphene and other honeycomb structures of light elements due to an insufficiently strong spin–orbit coupling. Here we show theoretically that 2D honeycomb lattices of HgTe can combine the effects of the honeycomb geometry and strong spin–orbit coupling. The conduction bands, experimentally accessible via doping, can be described by a tight-binding lattice model as in graphene, but including multi-orbital degrees of freedom and spin–orbit coupling. This results in very large topological gaps (up to 35 meV) and a flattened band detached from the others. Owing to this flat band and the sizable Coulomb interaction, honeycomb structures of HgTe constitute a promising platform for the observation of a fractional Chern insulator or a fractional quantum spin Hall phase.


Supplementary Figure 2: Ribbon with broken inversion symmetry.
Atomistic tight-binding calculations for a honeycomb ribbon of HgTe nanocrystals, with broken inversion symmetry. The nanocrystals have a truncated nanocube shape (truncation factor q=0.5, honeycomb lattice spacing a=5.0 nm). A single plane of atoms is removed from the right side of the ribbon in order to break the inversion symmetry (not shown). f, schematic view of the ribbon. The unit cell of 16 nanocrystals, shown in magenta, is reproduced periodically along the direction indicated by the arrow. a, dispersion of the p bands. The position of the bulk bands is indicated by pink vertical bars along the left axis. b,c,d,e, 2D plots of the wave functions of four states calculated at = 0.3×2 / where l is the length of the unit cell. The labels 1 (b), 2 (c), 3 (d) and 4 (e) refer to the states indicated in a. The plots are restricted to the unit cell of the ribbon. The white dots indicate the atoms.    Supplementary Fig. 5a. ! , ! ! , ! ! , and ! ! are the onsite energies on the s, p x , p y , and p z orbitals, respectively. !!" , !!" , !!" , and !"# are the hopping parameters, following the notations of Ref.

Supplementary
[1]. !!" , !!" , and !!" are the terms describing the Rashba SOC, following the same notations. The intrinsic SOC is defined by ISO ! and ISO ! on s and p orbitals, respectively.

Supplementary Note 1 Influence of size on lattices of HgTe nanocrystals.
When we vary the lattice spacing a (3−8 nm) and the nanocube truncation (q between 0.25 and 0.5) in lattices of HgTe nanocrystals, all band structures have similar behaviour. Examples of band structure are shown in Supplementary Fig. 1. The presence of well-separated s and p bands with helical gaps is quite general, even if the width of the gaps may vary substantially depending on the geometry of the superlattices. The overall shape of the s bands is always the same, whereas for the p bands the variations are more important because not only the nearestneighbour hopping, but also the respective positions of the p x , p y and p z states depend on nanocrystal size and truncation. However, the lowest p band is always detached from the next higher one. The width of the lowest gap in the p bands is given in Supplementary Table 1 for different values of a and q.

Supplementary Note 2 Edge localization in ribbons of HgTe nanocrystal superlattices
In this section, we present additional results on ribbons made from lattices of HgTe nanocrystals. We consider the same nanocrystals as in Figs. 1 and 2 (q=0.5, a=5.0 nm) but we investigate a ribbon in which the two edges are not symmetric by inversion. First, we discuss the effect of this asymmetry on the band structure. Second, we present plots of the wave functions. When the two edges of the ribbon are equivalent by inversion symmetry, the topological edge states on the opposite sides of the ribbon are quasi-degenerate for each value of k, the coupling between opposite edge states being negligible (Fig. 2). In order to observe the effect of geometry on the results, we have also considered a ribbon in which the inversion symmetry has been broken. For that purpose, we have removed all atoms of the last atomic plane on the right side of the ribbon and we have saturated the broken bonds with pseudo-hydrogen atoms. Supplementary Fig. 2a shows that the edge states are preserved thanks to their topological protection but their degeneracy at a given k is lifted due to the asymmetry between the two sides of the ribbon.
The 2D plots of the wavefunctions of the four states denoted 1−4 in Supplementary Fig. 2a for the asymmetric ribbon are shown in Supplementary Figs. 2b−e. These states calculated at = 0.3×2 / are strongly localized on the edges of the ribbon. State 3 is more delocalized than the other three states because it lies very close to the bulk band edge.

Supplementary Note 3 Band structures of honeycomb lattices of vertical cylinders
We have investigated a third type of honeycomb structure composed of HgTe cylinders. The axes of the cylinders are parallel to each other and are organized on a honeycomb lattice ( Supplementary Fig. 4a). Such structures could be fabricated from a HgTe layer, grown for example by gas-phase approaches. The honeycomb nanogeometry is then defined using nanoscale lithography.
Quite similar band structures are obtained for these lattices ( Supplementary Fig. 4b). Once again, s and p-like bands can be easily identified. The Rashba SOC is in general much larger than for lattices of nanocrystals due to a stronger coupling between neighbouring sites. Interestingly, a very similar behaviour was found for superlattices of spheres connected by cylinders when the coupling between neighbouring spheres is strong, i.e., for large values of d/D (Fig. 3). As a consequence of the large Rashba SOC, the gap in the s sector is closed. Nontrivial gaps remain in the p bands, for many configurations that we have investigated, in spite of larger spin splitting. However, the larger Rashba SOC tends to increase the dispersion of the lowest p band.
Once again, the results of the atomistic TB calculations are well described by the effective model (Supplementary Table 2 and Supplementary Fig. 5a). Only the highest bands of Supplementary Fig. 4b are not reproduced by the effective model because they involve higherenergy orbitals which are not considered in the model. The topological properties of the bands are demonstrated by edge-state analyses in ribbons, using the atomistic TB calculations ( Supplementary Fig. 4c) or the effective TB model ( Supplementary Fig. 5b) which give results in excellent agreement. The non-trivial topology of the bands is confirmed by calculations of the ! topological invariants using the effective-model Hamiltonian.