Real-space obstruction in quantum spin Hall insulators

The recently introduced classification of two-dimensional insulators in terms of topological crystalline invariants has been applied so far to “obstructed” atomic insulators characterized by a mismatch between the centers of the electronic Wannier functions and the ionic positions. We extend this notion to quantum spin Hall insulators in which the ground state cannot be described in terms of time-reversal symmetric localized Wannier functions. A system equivalent to graphene in all its relevant electronic and topological properties except for a real-space obstruction is identified and studied via symmetry analysis as well as with density functional theory. The low-energy model comprises a local spin-orbit coupling and a nonlocal symmetry breaking potential, which turn out to be the essential ingredients for an obstructed quantum spin Hall insulator. An experimental fingerprint of the obstruction is then measured in a large-gap triangular quantum spin Hall material.

Figure 1

(a) Wyckoff positions in the hexagonal unit cell giving rise to the triangular and honeycomb lattices. The former is made of the filled black circle at $1a$, while the latter by the orange “A” and green “B” open circles at $2b$. A different Bloch phase $exp\left(i\mathbf{k}·\mathbf{R}\right)$ (purple, yellow, and cyan colors) calculated at $\mathbf{k}$=K or ${\mathrm{K}}^{\prime }$ is assigned to each unit cell (diamond shapes). The filled-black circle in the center corresponds to $\mathbf{R}=(0,0)$ and represents the atom assigned to the purple unit cell). (b) Interference between orbital and Bloch phases. The blue arrow shows how the phase of the chiral orbitals sitting at the $\mathbf{R}$ positions of the triangular lattice winds going around the A/B points. In red we denote instead the contribution from the Bloch phases along the same path. For this specific example, we have chosen an orbital with $m\text{mod}3=1$ at K.

Figure 2

Band structure of the $\{p_{\pm },p_z\}$ model on the freestanding (a) triangular lattice and in the absence of a mirror symmetry with respect to the horizontal plane (b). The color code of the solid lines denotes the orbital character for the model without SOC while the bands with SOC are plotted with thick-dashed lines. The thin-dashed vertical lines indicate the position of the $p_{\pm }\text{−}{p}_z$ degeneracy (in the absence of SOC and mirror symmetry breaking) and the inset to (a) shows the path in the BZ. The insulating phase with SOC for the freestanding case, see thick-dashed line in (a), is topologically trivial ($\nu =0)$ whereas, the concomitant presence of SOC and of the mirror symmetry breaking, thick-dashed lines in (b), turns it into a QSHI ($\nu =1)$.

Figure 3

$\det \left[S\right(\mathbf{k}\left)\right]$ along the diagonal of the BZ for a trial basis of $|j,j_z\rangle =\{|1/2,\pm 1/2\rangle$ orbitals on the triangular site (a) and $|s{p}_z^A,+\rangle ,|s{p}_z^B,-\rangle$ orbitals on the interstitial sites (b). The dashed-vertical lines indicate the position of the crossing of the $p_{\pm }$ and the $p_z$ band in the absence of SOC and mirror symmetry breaking.

Figure 4

Band structure and sublattice projected DOS of the triangular and Kane-Mele model in the nontrivial phase. The solid lines/filled curve correspond to the triangular lattice, whereas the dashed lines to the Kane-Mele model. The blue-red color code denotes the OAM character (inset to the band structure), while the green-orange one encodes the A/B localization. The dashed box indicates the region shown in the inset to the sublattice projected DOS.

Figure 5

Side (a) and top view (b) on the unit cell: Gray and cyan spheres represent indium and silicon atoms, respectively. Carbon atoms are denoted in red and the pink spheres correspond to the hydrogen atoms used for passivation [49]; just for the clarity of the illustration, we show here the topmost SiC layer only. (c) Comparison of ARPES measurements of indenene and $G_0{W}_0$ band structure along the $\mathrm{\Gamma }\text{−}\mathrm{K}\text{−}\mathrm{M}\text{−}\mathrm{\Gamma }$ path. (d) STS measurements stabilized at tip-sample distance ${\mathrm{z}}_0$ and ${\mathrm{z}}_1={\mathrm{z}}_0\text{−}7.8\text{Å}$ (inset). At ${\mathrm{z}}_0$ STS is less sensitive to in-plane states effectively probing only $p_z$-like indenene states. High $n$-type doping of the substrate shifts the valence band onset (VBO) to approximately –$300\mathrm{mV}$ aligning with our ARPES data. (e) Renormalized $dI/dV$ signal recorded at the A (orange), B (green), and T1 (black) site of the indenene unit cell.

Figure 6

(a) Conventional unit cell (gray) and edge geometry. (b) 2D BZ and backfolding of high-symmetry momenta to 1D slab BZ. Edge localization (c) and OAM polarization (d) in the slab model.

Figure 7

Sketch of the ISB $p_x\text{−}{p}_y$ interaction on the triangular lattice activated by a staggered potential peaking at the A/B voids of the triangular lattice. First-neighbor hoppings can be decomposed into inequivalent second order processes via the A/B points as indicated by the gray arrows ${\widehat{\mathbf{d}}}_1$ and ${\widehat{\mathbf{d}}}_2$.

Figure 8

Local orbital angular momentum polarization of the two valence bands in the $\nu =0$ and $\nu =1$ phases. To lift the degeneracy of the valence bands at the valley momenta, a small values of ${\lambda }_{\text{ISB}}=0.1$ has been chosen and in the $\nu =0$ the compensating in-plane OAM is visualized by setting ${\lambda }_{\text{MIR}}=0.01$ (otherwise the in-plane OAM vanishes). The dashed lines indicate the first BZ.

Figure 9

Logarithmic plot of $\det \left[S\right(\mathbf{k}\left)\right]$ of the vertical-reflection symmetric model ${\lambda }_{\text{ISB}}=0$ for a time-reversal symmetric $J=1/2$ and time-reversal asymmetric A/B trial basis sets. The dashed lines indicate the first BZ.

Figure 10

Theoretical graph analysis of the band structures in layer group $\mathrm{p}6mm$ without (a) and with (b) SOC. Similarly to the color code of Fig. 2, the green (red) color denotes schematically the dominating $p_{\pm }\left(p_z\right)$ orbital character. The dashed line illustrates the position of the Fermi level for a filling of two electrons. The irreducible band representation notation follows Ref. [61].

Figure 11

$GW$ scaling of the energy gap (${\mathrm{E}}_{\mathrm{g}}$), Rashba splitting in the conduction band (${\mathrm{R}}_{\mathrm{c}}$) and Rashba splitting in the valence band (${\mathrm{R}}_{\mathrm{v}}$), as a function of the $k$ points, in the energy range [100: 300] meV. The dots correspond to the computed values and the solid lines to the interpolation of the results.