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

TL;DR

This paper investigates the stability of chiral edge modes in fractal-like lattices with dimensions between one and two, revealing that their existence is highly model-dependent and not universally guaranteed as in traditional two-dimensional systems.

Contribution

It provides a comparative analysis of quantum Hall models on fractal lattices, highlighting the non-universality and model-specific nature of edge mode robustness in lower-dimensional fractal structures.

Findings

Edge modes are not universally stable in fractal lattices.

The presence of edge states depends on specific model details.

Robustness of edge modes varies with lattice realization.

Abstract

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 work, 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.