# Hall conductivity of Sierpinski carpet

## TL;DR

This paper investigates how the Hall conductivity behaves in a Sierpinski carpet fractal, revealing the loss of quantization and topological features as the fractal's complexity increases.

## Contribution

It provides a numerical analysis of Hall conductivity and edge states in a fractal system, highlighting the breakdown of topological quantization with increasing fractal depth.

## Key findings

- Hall conductivity quantization disappears with increased fractal depth
- Hall conductivity is no longer proportional to Chern number at higher depths
- Edge states become less prominent as the fractal becomes more complex

## Abstract

We calculate the Hall conductivity of a Sierpinski carpet using Kubo-Bastin formula. The quantization of Hall conductivity disappears when we increase the depth of the fractal. The Hall conductivity is no more proportional to the Chern number. Nevertheless, these quantities behave in a similar way showing some reminiscence of a topological nature of the Hall conductivity. We also study numerically the bulk-edge correspondence and find that the edge states become less manifested when the depth of Sierpinski carpet is increased.