Local density of states of the Harper-Hofstadter model with random disorder.

Anderson localization plays a crucial role in the integer quantum hall systems. The degenerated Landau level is split by random disorder and the increase of disorder strength leads to the localization of the states in the tails of the Landau level. The recently developed Cluster-Typical Medium Theory (Cluster-TMT) can be modified to study lattice models for the integer quantum effect, in particular the Harper-Hofstadter model with local random disorder. The cluster-TMT method is successful in reproducing the full phase diagram of the three dimensional Anderson model. Its success is hinted by the observation that the local density of states of the localized phase follows a log-normal distribution. We investigate the statistics of the local density of states of the Harper-Hofstadter model with local random potential. We find that the local density of states changes very closely from normal to log-normal distribution as the disorder strength increases. This provides support on the validity of the cluster-TMT when applied to the integer quantum hall systems.