Quantum Spin Hall Effect and Spin Bott Index in a Quasicrystal Lattice

Despite the rapid progress in the field of the quantum spin Hall (QSH) effect, most of the QSH systems studied up to now are based on crystalline materials. Here we propose that the QSH effect can be realized in quasicrystal lattices (QLs). We show that the electronic topology of aperiodic and amorphous insulators can be characterized by a spin Bott index $B_s$. The nontrivial QSH state in a QL is identified by a nonzero spin Bott index $B_s=1$, associated with robust edge states and quantized conductance. We also map out a topological phase diagram in which the QSH state lies in between a normal insulator and a weak metal phase due to the unique wave functions of QLs. Our findings not only provide a better understanding of electronic properties of quasicrystals but also extend the search of the QSH phase to aperiodic and amorphous materials that are experimentally feasible.

Figure 1

(a) Penrose tiling containing 521 vertices. The red dashed line defines a unit cell under periodic approximation. The inset shows that the atomic orbitals are placed on the vertices of rhombuses and indicates the nearest-neighbor hopping between them. (b) Atomic model of a QSH state in a surface-based 2D QL. The red and blue arrows represent helical edge states with opposite spin polarizations.

Figure 2

Calculation of a quasicrystal sample with 1364 atoms (three orbitals and two spins on each atom). The parameters used here are ${\epsilon }_s=1.8$, ${\epsilon }_p=-6.5$, $\lambda =0.8$, $V_{ss\sigma }=-0.4$, $V_{sp\sigma }=0.9$, $V_{pp\sigma }=1.8$, and $V_{pp\pi }=0.05\mathrm{eV}$. (a) Energy eigenvalues $E_n$ versus the state index $n$. The system with a periodic boundary condition (PBC) shows a gap, while that with an open boundary condition (OBC) shows midgap states. (b) The wave function $|\psi (\mathbf{r})⟩=\rho (\mathbf{r})e^{i\varphi (\mathbf{r})}$ of the midgap state [marked as the yellow star in (a)] is localized on the edge of the system. The size and the color of the blob indicate the norm $|\rho (\mathbf{r}){|}^2$ and phase $\varphi (\mathbf{r})$ of the wave function, respectively. (c) Two-terminal conductance $G$ as a function of the Fermi energy $E$, showing a quantized plateau in the energy gap. (d) Local density of state $D_n(E)$ at $E=0\mathrm{eV}$ for the central quasicrystal in the transport simulation. The size of the blue dot represents the relative value of the local density of state.

Figure 3

(a) Topological phase diagram in the parameter space of energy difference $\mathrm{\Delta }={\epsilon }_s-{\epsilon }_p$ and SOC strength $\lambda$. (b) Energy gap $E_g$ and spin Bott index $B_s$ as a function of the interaction strength scale $d_0$ calculated in a quasicrystal approximant containing 3571 atoms. Topological phase transitions between a normal insulator (NI), a QSH insulator, and weak metal (WM) are clearly visible. The dark blue line in the QSH region represents the defect mode in the energy gap. (c)–(e) Orbital-resolved spectrum of QL in the (c) NI, (d) QSH insulator, and (e) WM states. An $s\text{−}p$ band inversion occurs in the QSH state. The colors of the dots represent the relative weights of $p$ or $s$ orbitals.