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