Robustness of chiral edge modes in fractal-like lattices below two dimensions: A case study

One of the most prominent characteristics of two-dimensional quantum Hall systems are chiral edge modes. Their existence is a consequence of the bulk-boundary correspondence and their stability guarantees the quantization of the transverse conductance. In this paper, we study two microscopic models, the Hofstadter lattice model and an extended version of Haldane's Chern insulator. Both models host quantum Hall phases in two dimensions. We transfer them to lattice implementations of fractals with a dimension between one and two and study the existence and robustness of their edge states. Our main observation is that, contrary to their two-dimensional counterpart, there is no universal behavior of the edge modes in fractals. Instead, their presence and stability critically depends on details of the models and the lattice realization of the fractal.