Effect of coordination on topological phases on self-similar structures

Topologically nontrivial phases have recently been reported on self-similar structures. Here we investigate the effect of local structure, specifically the role of the coordination number, on the topological phases on self-similar structures embedded in two dimensions. We study a geometry dependent model on two self-similar structures having different coordination numbers, constructed from the Sierpinski gasket. For different nonspatial symmetries present in the system, we numerically study and compare the phases on both structures. We characterize these phases by the localization properties of the single-particle states, their robustness to disorder, and by using a real-space topological index. We find that both structures host topologically nontrivial phases and the phase diagrams are different on the two structures. This suggests that, in order to extend the present classification scheme of topological phases to nonperiodic structures, one should use a framework which explicitly takes the coordination of sites into account.