# The existence of robust edge currents in Sierpinsky Fractals

## TL;DR

This paper demonstrates the existence of robust, quantized edge currents in a Sierpinski carpet fractal, showing that edge modes persist across various magnetic field strengths and are stable against disorder.

## Contribution

It provides the first detailed analysis of Hall conductivity and edge modes in a Sierpinski fractal, revealing their robustness and establishing a counting rule for edge modes.

## Key findings

- Edge modes with Hall conductivity up to ±e^2/h are generically present.
- Quantized edge conductance is stable under disorder.
- A simple counting rule determines the maximum number of edge modes.

## Abstract

We investigate the Hall conductivity in a Sierpinski carpet, a fractal of Hausdorff dimension $d_f=\ln(8)/\ln(3) \approx 1.893$, subject to a perpendicular magnetic field. We compute the Hall conductivity using linear response and the recursive Green function method. Our main finding is that edge modes, corresponding to a maximum Hall conductivity of at least $\sigma_{xy}=\pm \frac{e^2}{h}$, seems to be generically present for arbitrary finite field strength, no mater how one approaches the thermodynamic limit of the fractal. We discuss a simple counting rule to determine the maximal number of edge modes in terms of paths through the system with a fixed width. This quantized edge conductance, as in the case of the conventional Hofstadter problem, is stable with respect to disorder and thus a robust feature of the system.