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 eV.