Hall conductivity of a Sierpiński carpet

Abstract

We calculate the Hall conductivity of a Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpet using Kubo-Bastin formula. The quantization of Hall conductivity disappears when we increase the depth of the fractal, and 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 Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpet is increased.