Topological Corner States in Bilayer and Trilayer Systems with Vertically Stacked Topologically Distinct Layers

We investigate bilayer and trilayer systems composed of topologically distinct, vertically stacked layers based on the Benalcazar-Bernevig-Hughes model. We have identified a topological phase transition that significantly alters the number of the topological corner states in these systems. Additionally, we find that traditional nested Wilson loop analysis inaccurately classifies certain phases, leading us to evaluate multipole chiral numbers (MCNs) as a more appropriate topological invariant. This approach accurately identifies topological phases and the number of MCNs coincides with the number of corner modes, enhancing our understanding of topological insulators and opening avenues for further applications.