Minimal models for Wannier-type higher-order topological insulators and phosphorene

A higher-order topological insulator (HOTI) is an extended notion of the conventional topological insulator. It belongs to a special class of topological insulators to which the conventional bulk-boundary correspondence is not applicable. Provided the mirror symmetries are present, the bulk topological number is described by the quantized Wannier center located at a high-symmetry point of the crystal. The emergence of corner states is a manifestation of nontrivial topology in the bulk. In this paper we propose minimal models for the Wannier-type second-order topological insulator in two dimensions and the third-order topological insulator in three dimensions. They are anisotropic chiral-symmetric two-band models. It is explicitly shown that the Wannier center is identical to the winding number in the present model, demonstrating that it is indeed a topological quantum number. Finally we point out that the essential physics of phosphorene near the Fermi energy is described by making a perturbation of the Wannier-type HOTI. We predict that these corner states will be observed in the rhombus structure of phosphorene near the Fermi energy around $-0.16\mathrm{eV}$.

Figure 1

Illustration of rhombi with (a) $t_a=0$, (b) $\left|t_a\right|<t_b$, and (c) $t_b=0$. The size $L$ is defined by the number of benzene rings on one side in (b). Here, $L=5$. There are two isolated atoms at the top and bottom corners of the rhombus for $t_a=0$ in (a). (d) The square root of the local density of states $\sqrt{{\rho }_i}$ for the rhombus. The amplitude is represented by the radius of the spheres. (e) Energy spectrum of the rhombus made of the anisotropic honeycomb lattice. There emerge topological corner states (marked in red) at the Fermi energy for $|t_a|<t_b/2$. (f) Energy spectrum of the rhombus made of phosphorene. The horizontal axis is $t_a$. We have chosen the parameters as $t_b=3.665\mathrm{eV},t_c=-0.105\mathrm{eV}$, and $t_a^{*}=-1.220\mathrm{eV}$ for phosphorene.

Figure 2

Illustration and band structure of (a1),(a2) a normal-zigzag nanoribbon and (b1),(b2) a skew-zigzag nanoribbon, where $t_a=-1.220\mathrm{eV},t_b=3.665\mathrm{eV}$, and $t_c=-0.105\mathrm{eV}$. Star symbols represent WCs in (a1) and (b1). The horizontal axis is $k$ in (a2) and (b2). (a2) When the edge cuts through WCs as in (a1), perfect flatbands emerge at the Fermi energy (red line). These perfect flatbands do not represent topological edge states. (b2) When the edge cuts through no WCs as in (b1), no edge states emerge at the Fermi energy.

Figure 3

The Fermi surface of the bulk anisotropic diamond lattice with (a1) $t_a=t_b$, (b1) $t_a=t_b/2$, (c1) $t_a=t_b/3$, and (d1) $t_a=t_b/4$. (a2)–(d2) The corresponding band structure of a thin film. (a3)–(d3) The corresponding band structure of a diamond prism. The size of the diamond is $L=3$.

Figure 4

Illustration of a rhombohedron made of the diamond lattice with (a) $t_a=0$, (b) $t_a=t_b$, and (c) $t_b=0$. (d) Energy spectrum of the rhombohedron made of the anisotropic diamond lattice with $L=3$. The horizontal axis is $t_a/t_b$. There emerge zero-energy states (marked in red) for $|t_a/t_b|<1/3$. They are topological boundary states. (e) The square root of the local density of states $\sqrt{{\rho }_i}$ for the rhombohedron with $L=3$ and $t_a/t_b=1/4$. The amplitude is represented by the radius of the spheres.