Distribution of Chern number by Landau level broadening in Hofstadter's butterfly

TL;DR

This paper explores how Landau level broadening affects the distribution of Chern numbers within Hofstadter's butterfly energy spectrum, linking fractal band structure to quantum Hall conductance.

Contribution

It introduces a classification method for energy bands using continued fractions to relate conductance to Landau level broadening in Hofstadter's butterfly.

Findings

Conductance at band gaps matches denominators of continued fraction expansions.

Landau level broadening explains the fractal distribution of Chern numbers.

The classification scheme reveals underlying structure in quantum Hall conductance.

Abstract

We discuss the relationship between the quantum Hall conductance and a fractal energy band structure, Hofstadter's butterfly, on a square lattice under a magnetic field. At first, we calculate the Hall conductance of Hofstadter's butterfly on the basis of the linear responce theory. By classifying the bands into some groups with a help of continued fraction expansion, we find that the conductance at the band gaps between the groups accord with the denominators of fractions obtained by aborting the expansion halfway. The broadening of Landau levels is given as an account of this correspondance.