  Despite the multi-band 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 low magnetic-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 BKT
transition temperatures.
