  In contrast to the fractional quantum Hall (FQH) effect, where electron
density fixes the applied magnetic field, fractional Chern insulators (FCIs)
can realize FQH states in comparatively weak or even zero magnetic fields.
Previous theoretical work highlighted magic angle graphene as a promising FCI
platform, satisfying the twin requirements of flat bands and
lowest-Landau-level-like quantum geometry. Indeed, recent experiments have
demonstrated FCIs in magic angle graphene with weak magnetic fields. Here we
conduct a detailed theoretical study of the most prominent FCI state observed,
and clarify the role of the magnetic field in stabilizing this state. We
introduce two new technical tools: first, we generalize the notion of ideal
quantum geometry to Hofstadter minibands and, second, we extend the
Hartree-Fock theory of magic-angle graphene to finite field, to account for the
interaction generated bandwidth. We show that magnetic field both dramatically
reduces the effective bandwidth and improves the quantum geometry for hosting
FCIs. Using density matrix renormalization group (DMRG) simulations of a
microscopic model of magic angle graphene, we establish the regime of bandwidth
and quantum geometry indicators where FCIs are stabilized. Further
characterizing the finite-field bands by the same quantities we show how a
zero-field charge density wave state gives way to an FCI state at a magnetic
flux consistent with experiment. We also speculate on the other FCIs seen in
the same experiments, including anomalous incompressible states and
even-denominator fractions which may host non-Abelian states. Finally, when
bandwidth is the limiting factor, we propose a range of experimental parameters
where FCIs should appear at zero magnetic field.
