Numerical study of Integer quantum Hall transition in Harper-Hofstadter model with local random disorder

Anderson localization plays a crucial role in the integer quantum hall(IQH) 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 estimate the exponent corresponding to localization length method by performing the finite size scaling analysis for the smallest Lyapunov exponent obtained from Transfer Matrix method. In light of the recent studies in critical scaling at the IQH transitions, we carefully investigate the role of irrelevant exponent in our finite size scaling analysis. We use these results to perform the finite size scaling of the typical local density states. We find that the proposed order parameter, typical local density of states, fits well to the scaling hypothesis from which we obtain the corresponding critical exponent. We also find that the local density of states qualitatively changes very closely from normal to log-normal distribution as the disorder strength increases. This opens a possibility to apply cluster-TMT to the integer quantum hall systems.