In higher-order topological insulators, the ingap states can be found in (d-n)-dimensional edges (n > 1), such as the corner states of two-dimensional (2D) systems or the hinge states of three-dimensional systems. Higher-order topological insulators have been introduced in the precursory Benalcazar-Bernevig-Hughes quadrupole model [1], but no electronic compound has been proposed to be a quadrupole topological insulator (QTI) yet. In recent years, van der Waals layered materials of A₂M_1,3Te₅ (A=Ta, Nb; M=Pd, Ni; X=Se, Te) family have attracted attentions because of their special properties, such as quantum spin Hall effect in Ta₂Pd₃Te₅ monolayer [2], excitonic state in Ta₂NiSe₅, and superconductivity in Nb₂Pd₃Te₅. In particular, the monolayers of A₂M_1,3Te₅ family can be exfoliated easily, serving as a good platform for studying topology and interactions in lower dimensions.

Recently, the research group of Prof. Zhijun Wang, Drs. Zhaopeng Guo, Junze Deng, and Yue Xie from Institute of Physics of the Chinese Academy of Sciences reported that Ta₂Ni₃Te₅ monolayer can be 2D QTI with second-order topology due to the double-band inversion [3]. A time-reversal-invariant system with two mirror reflections (M_x and M_y) can be classified by Stiefel-Whitney numbers (w₁, w₂) due to the combined symmetry TC_2z. Using the Wilson loop method (Figure 1), we compute w₁ = 0 and w₂ = 1 for Ta₂Ni₃Te₅, indicating a QTI with q^xy = e/2. Thus, gapped edge states and localized corner states are obtained. By analyzing atomic band representations, we demonstrate that its unconventional nature with an essential band representation at an empty site, i.e., Ag@4e, is due to the remarkable double-band inversion on Y-Γ with elementary band representations of A'@4e (Ta-d_z²) and A''@4e (Te-p_x). Then, we construct an eight-band quadrupole model with M_x and M_y successfully for electronic materials, and confirm the higher-order topology (Figure 2). These transition-metal compounds of A₂M_1,3Te₅ (A=Ta, Nb; M=Pd, Ni; X=Se, Te) family provide a good platform for realizing the QTI and exploring the interplay between topology and interactions.