  In a lattice model subject to a perpendicular magnetic field, when the
lattice constant is comparable to the magnetic length, one enters the
"Hofstadter regime," where continuum Landau levels become fractal magnetic
Bloch bands. Strong mixing between bands alters the nature of the resulting
quantum phases compared to the continuum limit; lattice potential, magnetic
field, and Coulomb interaction must be treated on equal footing. Using
determinant quantum Monte Carlo (DQMC) and density matrix renormalization group
(DMRG) techniques, we study this regime numerically in the context of the
Hubbard-Hofstadter model on a triangular lattice. In the field-filling phase
diagram, we find a broad wedge-shaped region of ferromagnetic ground states for
filling factor $\nu \leq 1$, bounded below by filling factor $\nu = 1$ and
bounded above by half-filling the lowest Hofstadter subband. We observe
signatures of SU(2) quantum Hall ferromagnetism at filling factors $\nu=1$ and
$\nu=3$. The phases near $\nu=1$ are particle-hole asymmetric, and we observe a
rapid decrease in ground state spin polarization consistent with the formation
of skyrmions only on the electron doped side. At large fields, above the
ferromagnetic wedge, we observe a low-spin metallic region with spin
correlations peaked at small momenta. We argue that the phenomenology of this
region likely results from exchange interaction mixing fractal Hofstadter
subbands. The phase diagram derived beyond the continuum limit points to a rich
landscape to explore interaction effects in magnetic Bloch bands.
