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Two-dimensional Obstructed Atomic Insulators with Fractional Corner Charge in MA$_2$Z$_4$ Family

According to topological quantum chemistry, a class of electronic materials have been called obstructed atomic insulators (OAIs), in which a portion of valence electrons necessarily have their centers located on some empty $\textit{Wyckoff}$ positions without atoms occupation in the lattice. The obstruction of centering these electrons coinciding with their host atoms is nontrivial and results in metallic boundary states when the boundary is properly cut. Here, on basis of first-principles calculations in combination with topological quantum chemistry analysis, we propose two dimensional MA$_2$Z$_4$ (M = Cr, Mo and W; A = Si and Ge, Z = N, P and As) monolayer family are all OAIs. A typical case is the recently synthesized MoSi$_2$N$_4$. Although it is a topological trivial insulator with the occupied electronic states being integer combination of elementary band representations, it has valence electrons centering empty $\textit{Wyckoff}$ positions. It exhibits unique OAI-induced metallic edge states along the (1$\bar{1}$0) edge of MoSi$_2$N$_4$ monolayer and the in-gap corner states at three vertices of certain hexagonal nanodisk samples respecting C$_3$ rotation symmetry. The readily synthesized MoSi$_2$N$_4$ is quite stable and has a large bulk band gap of 1.94 eV, which makes the identification of these edge and corner states most possible for experimental clarification.