Fractal defect states in the Hofstadter butterfly

We investigate the electronic properties in Bloch electron on a square lattice with vacancies uniform magnetic field. show that single vacancy site introduced to system creates defect energy level every innumerable fractal gap Hofstadter butterfly. The wavefunctions of different levels have all localization lengths depending their generations, and they can be described by universal function after an appropriate scaling. also each state has its own characteristic orbital moment, which is exactly correlated gradient diagram. Probing spatial nature defect-localized states provides powerful way elucidate