  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.
