Disorder broadening of Hofstadter bands in the honeycomb lattice

For a large range of twist angles, above the magic angle, the Hofstadter bands of twisted bilayer and trilayer graphene are dual to the Hofstadter bands of the honeycomb lattice. Here, we calculate the broadening of the Hofstadter bands of a honeycomb lattice in the presence of disorder within the self-consistent Born approximation (SCBA). In this study, the disorder is treated by the method of impurity averaging with the disorder fluctuation having white noise correlations. We calculate the local density of states which can be compared with scanning tunneling microscopy (STM) measurements of Hofstadter bands in twisted bilayer graphene.