  ABC-stacked multilayer graphene (ABC-MLG) exhibits topological surface flat
bands with a divergent density of states, leading to many-body instabilities at
charge neutrality. Here, we explore electronic ordering within a mean-field
approach with full generic treatment of all spin-isotropic, two-site charge
density and spin interactions up to next-nearest neighbor (NNN) sites. We find
that surface superconductivity and magnetism are significantly enhanced over
bulk values. We find spin-singlet $s$ wave and unconventional NNN bond
spin-triplet $f$ wave to be the dominant superconducting pairing symmetries,
both with a full energy gap. By establishing the existence of ferromagnetic
intra-sublattice interaction, $(J_2<0)$ we conclude that the $f$-wave state is
favored in ABC-MLG, in sharp contrast to bulk ABC-graphite where chiral $d$- or
$p$-wave states, together with s-wave states, display stronger ordering
tendencies albeit not achievable at charge neutrality. We trace this
distinctive surface behavior to the strong sublattice polarization of the
surface flat bands. We also find competing ferrimagnetic order, fully
consistent with density functional theory (DFT) calculations. The magnetic
order interpolates between sublattice ferromagnetism and antiferromagnetism,
but only with the ratio of the sublattice magnetic moments ($R$) being
insensitive to the DFT exchange correlation functional. We finally establish
the full phase diagram by constraining the interactions to the $R$-value
identified by DFT. We find $f$-wave superconductivity being favored for all
weak to moderately strong couplings $J_2$ and as long as $J_2$ is a
sufficiently large part of the full interaction mix. Gating ABC-MLG away from
charge neutrality further enhances the $f$-wave state over the ferrimagnetic
state, establishing ABC-MLG as a strong candidate for $f$-wave
superconductivity.

Superconductivity;Mesoscale and Nanoscale Physics;Atomic and Molecular Clusters