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

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