Study of Correlated Disorders and interaction in the Hofstadter Butterfly
We investigate the impact of several quasiperiodic disorders and their continuous interpolation with the Aubry-Andre (AA) potential on the Hofstadter butterfly using mean field approximation at zero temperature for a two-dimensional square lattice. Weak disorder mildly smears the fractal spectrum, while strong quasiperiodic potentials destroy the butterfly and generate multiple energy gaps. The AA potential produces the strongest spectral restructuring, creating prominent gaps near half-filling. Interpolating AA with other quasiperiodic potentials reveals competing gap-opening mechanisms, ranging from AA-dominated gaps at small interpolation parameters to a robust half-filling gap generated by the competing disorders at large parameters. Entanglement entropy follows the area law at low and high magnetic fields but shows pronounced deviations at intermediate fields, with opposite trends for strong AA versus other quasiperiodic potentials. Localization analysis using IPR and NPR confirms enhanced localization with increasing disorder; the AA potential yields the largest IPR, with notable field dependence. Interpolation produces smooth crossovers between distinct localization regimes.