The Colored Hofstadter Butterfly for the Honeycomb Lattice

Abstract: We rely on a recent method for determining edge spectra and we use it to compute the Chern numbers for Hofstadter models on the honeycomb lattice having rational magnetic flux per unit cell. Based on the bulk-edge correspondence, the Chern number $\sigma_H$ is given as the winding number of an eigenvector of a $2 \times 2$ transfer matrix, as a function of the quasi-momentum $k \in (0,2 \pi)$. This method is computationally efficient (of order $O(n^4)$ in the resolution of the desired image). It also shows that for the honeycomb lattice the solution for $\sigma_H $ for flux $p/q$ in the $r$-th gap conforms with the Diophantine equation $r=\sigma_H\cdot p+ s\cdot q$, which determines $\sigma_H \mod q$. A window such as $\sigma_H \in(-q/2,q/2)$, or possibly shifted, provides a natural further condition for $\sigma_H$, which however turns out not to be met. Based on extensive numerical calculations, we conjecture that the solution conforms with the relaxed condition $\sigma_H\in(-q,q)$.