Topological corner states in bilayer and trilayer systems with vertically stacked topological heterostructures

We investigate bilayer and trilayer systems composed of topologically distinct, vertically stacked layers, forming topological heterostructures based on the Benalcazar-Bernevig-Hughes model. We find that a topological phase transition induced by interlayer coupling significantly alters the number of corner states in these topological structures. Furthermore, 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 for this scenario. The MCNs not only enable accurate classification of topological phases but also directly correspond to the number of zero-energy corner states, effectively characterizing $\mathbb{Z}$-class higher-order topological insulator phases. Our study proposes the concept of topological heterostructures, providing critical insights into the control of localized corner states within multilayer systems and expanding potential research directions avenues.