Hofstadter-Hubbard model with opposite magnetic fields: Bardeen-Cooper-Schrieffer pairing and superfluidity in the nearly flat butterfly bands

Despite the multiband spectrum of the widely known Hofstadter butterfly, it turns out that the pairing correlations of the time-reversal-symmetric Hofstadter-Hubbard model are well described by a single order parameter that is uniform in real space. By exploiting a BCS mean-field theory for the nearly flat butterfly-bands regime of lowmagnetic-flux limit, here we reveal a number of unusual superfluid properties both in the ground state and at finite temperatures. Our thorough analysis includes but is not limited to the order parameter, condensate, and superfluid fractions, and the critical BCS and Berezinskii-Kosterlitz-Thouless transition temperatures.