Topologically nontrivial phases are linked to the appearance of localized modes in the boundaries of an open insulator. On the other hand, the existence of geometric frustration gives rise to degenerate localized bulk states. The interplay of these two phenomena may, in principle, result in an enhanced protection/localization of edge states. In this paper, we study a two-dimensional Lieb-based topological insulator with staggered hopping parameters and diagonal open boundary conditions. This system belongs to the $C_{2v}$ class and sustains one-dimensional (1D) boundary modes except at the topological transition point, where the $C_{4v}$ symmetry allows for the existence of localized (0D) corner states. Our analysis reveals that, while a large set of boundary states have a common well-defined topological phase transition, other edge states reflect a topological nontrivial phase for any finite value of the hopping parameters, are completely localized (compact) due to destructive interference, and evolve into corner states when reaching the higher symmetry point. We consider the robustness of these compact edge states with respect to time-dependent perturbations and indicate ways that these states could be prepared and measured in experiments with ultracold atoms.

• Received 26 March 2019

DOI:https://doi.org/10.1103/PhysRevB.100.125123