Multiorbital model reveals a second-order topological insulator in $1H$ transition metal dichalcogenides

Recently, a new class of second-order topological insulators (SOTIs) characterized by an electronic dipole has been theoretically introduced and proposed to host topological corner states. As a novel topological state, it has been attracting great interest and experimentally realized in artificial systems of various fields of physics based on multisublattice models, e.g., breathing kagome lattice. In order to realize such kind of SOTI in natural materials, we proposed a symmetry-faithful multiorbital model. Then, we reveal several familiar transition metal dichalcogenide (TMD) monolayers as a material family of two-dimensional SOTI with large bulk gaps. The topologically protected corner state with fractional charge is pinned at Fermi level due to the charge neutrality and filling anomaly. Additionally, we propose that the zero-energy corner state is preserved in the heterostructure composed of a topological nontrivial flake embedded in a trivial material. The novel second-order corner states in familiar TMD materials hold promise for revealing unexpected quantum properties and applications.

Figure 1

Atomic structure and electronic structure of a $1H\text{−}M{X}_2$ monolayer. $M$ stands for (W, Mo) and $X$ stands for (Te, Se, S). (a) Atomic structure of a $1H\text{−}M{X}_2$ monolayer. The arrows ${\mathit{a}}_1$ and ${\mathit{a}}_2$ (${\mathit{b}}_1$ and ${\mathit{b}}_2$) are the two (reciprocal) lattice vectors. One rhombic unit cell is colored in cyan. The star marks the Wannier charge located at the hollow site in the unit cell, corresponding to an electronic dipole $\mathit{P}=(\frac{1}{3},\frac{2}{3})$. The black hexagon in the lower right corner is the first Brillouin zone. (b) Orbital projected band structure of a $1H\text{−}\mathrm{Mo}{\mathrm{S}}_2$ monolayer. Colored circles represent contributions from different $M\text{−}d$ orbitals. The Fermi energy $E_{\text{F}}$ is set to be zero as a reference. (c) Wilson loop of WCC of the highest valence band showing the calculated electronic dipole $\mathit{P}=(\frac{1}{3},\frac{2}{3})$.

Figure 2

Schematic lattice diagram and phase diagram of the simplified multiorbital model Eq. (1). In (a), the three ellipses in different colors represent three orbitals on the same site. The thicker and thinner lines present two different kinds of nearest-neighbor hoppings. The dashed lines present the crystal field effects. The star marks the Wannier charge located at the center of the thicker triangle, corresponding to an electronic dipole $\mathit{P}=(\frac{1}{3},\frac{2}{3})$. Topological phase diagrams of the multiorbital model for the (b) $t_{\text{E}}=0$ and (c) $t_{23}=0$ cases.

Figure 3

Energy spectrum of a triangular $\mathrm{Mo}{\mathrm{S}}_2$ flake with armchair edges. The red dots on the Fermi level represent the six in-gap corner states. Other bulk and edge states are colored in cyan. The atomic structure and the charge distribution of the corner state $n=808$, bulk state $n=760$, and edge states $n=814$ and 869 are shown in the inset. For the corner state $n=808$, the electron is localized and equally distributed on the three corners with $-\frac{1}{3}\left|e\right|$ charge on each corner.

Figure 4

Energy spectrum of a heterostructure composed of a triangular $\mathrm{Mo}{\mathrm{S}}_2$ flake in a $\mathrm{Ti}{\mathrm{S}}_2$ monolayer. In (a), the red dots on the Fermi level represent the six in-gap corner states. (b) The charge distribution of corner state $n=3160$. The electron is localized and equally distributed on the three corners with $-\frac{1}{3}\left|e\right|$ charges on each corner.