Hofstadter butterflies in a modulated magnetic field: Superconducting wire network with magnetic decoration

Hofstadter butterfly spectra of tight-binding electron systems under spacially modulated magnetic fields are calculated. The dependence of the spectrum on the symmetry and strength of the spatially varying magnetic field is elucidated. The Little-Parks oscillation of the superconducting network under a spatially modulated magnetic field produced by decoration with mesoscopic magnetic structure exhibits behavior reproducing the edge of the corresponding Hofstadter spectra.