Hidden higher-order topology in nonsymmorphic group IV and V tetragonal monolayers

In recent years, two-dimensional (2D) second-order topological insulators (SOTIs) have garnered significant interest, with indications of their potential realization in various symmorphic 2D electronic materials. However, up to this point, no nonsymmorphic 2D electronic SOTIs have been identified, probably due to the inability of nonsymmorphic operations to maintain the invariance of nanoflakes. In this paper, we investigate the existence of nonsymmorphic 2D SOTIs, unveiling hidden higher-order topology within 2D nonsymmorphic electronic systems. Our findings are substantiated by symmetry analyses, tight-binding (TB) models, and first-principles calculations. The emergence of topological corner states in these nonsymmorphic 2D SOTIs is attributed to the filling anomaly within a set of symmorphic Wannier orbitals, which exhibit a symmorphic distribution. We identify square-octagon monolayers (so-MLs) of group IV and V elements, including 2D tetragonal P, and 2D hydrogenated tetragonal Si and Ge, as promising material candidates. The corner states in these nonsymmorphic so-MLs are protected by a point symmetry (${C}_{4}$ rotation). The TB model of so-MLs behaves similarly to the Su-Schrieffer-Heeger model, with higher-order topological insulating phases having greater intersquare hoppings compared to intrasquare hoppings, while the reverse is considered trivial. These discoveries not only enrich our theoretical comprehension of higher-order topology but also introduce potential material candidates for experimental exploration, thus advancing the field of topological crystalline materials.