Title: Atomically thin obstructed atomic insulators with robust edge modes and quantized spin Hall effect

Abstract: Symmetry-protected edge states serve as direct evidence of nontrivial electronic topology in atomically thin materials. Finding these states in experimentally realizable single-phase materials presents a substantial challenge for their use for both fundamental studies and developing functional nanoscale devices. Here, we show the presence of robust edge states in phosphorene and group-Va monolayers with puckered lattice structures. By carefully analyzing the symmetry of the atomic sites and edge modes properties, we demonstrate that these atomically thin two-dimensional (2D) materials realize recently introduced obstructed atomic insulator states with partially occupied edge modes. The obstructed edge modes attain a Rashba-type spin splitting with Rashba parameter ($\alpha$) of 1.52 eV \r{A} for arsenene. Under strain or doping effects, these obstructed insulators transition to a phase with substantial spin-Berry curvature, yielding a double quantum spin Hall state with a spin Hall conductivity of 4$\frac{e^2}{h}$. The experimental availability of phosphorene and other group-Va materials featuring puckered lattice structures could enable verification of obstructed atomic states and enhanced spin-Berry curvature effects discussed in this study, offering the potential for applications in topological electronic and spintronic devices.