  Multipolar orderings in degenerate orbital systems offer unique opportunities
for emergent topological phases. The phase diagram of interacting spinless
fermions in a $p$-band diamond lattice at unit filling is first studied to
elucidate the essential role of orbital multipolar orderings in the evolution
of multifold degenerate band nodes. The free band structure around the
Brillouin zone center is described by two quadratic band nodes each with a
threefold degeneracy, which are spanned by the bonding and anti-bonding
$p$-orbital multiplets, respectively. Upon switching on interactions, the
triply degenerate band node is split into a pair of Weyl fermions with opposite
chirality due to the onset of orbital multipolar orderings. Further raising
interactions ultimately drives the system into an insulating phase with the
orbital quadrupolar ordering. Our study is then generalized to spin-$1/2$
fermions, which has direct relevance with solid-state materials. The system
develops full spin polarization through a ferromagnetic transition at tiny
interactions, leaving the remaining orbital sector activated. The ensuing
transitions take place in the orbital sector as a natural consequence,
qualitatively recovering the phase diagram of spinless fermions. Our findings
shed new light on the realization of emergent novel fermions with a prospect
being a frontier at the confluence of topology, orbital physics and strong
correlation.
