  Inspired by our recent works[1, 2] of SU(2) and SU(4) Dirac fermions
subjected to plaquette interactions on square lattice, here we extend the
large-scale quantum Monte Carlo investigations to the phase digram of
correlated Dirac fermions with SU(6) and SU(8) symmetries subjected to the
plaquette interaction on the same lattice. From SU(2) to SU(8), the rich phase
diagram exhibits a plethora of emerging quantum phases such as the Dirac
semimetal, the antiferromagnetic Mott insulator, valence bond solid (VBS) and
the Dirac spin liquid and phase transitions including the Gross-Neveu chiral
transitions with emergent continuous symmetry, the deconfined quantum
criticality and the first order transition between interaction-driven columnar
VBS and plaquette VBS. These rich phenomena coming from the simple-looking
lattice models, firmly convey the message that the interplay between the
$SU(N)$ Dirac fermions -- with enhanced internal symmetries -- and extended
plaquette interactions -- beyond the on-site Hubbard type -- is the new
playground to synthesise novel highly entangled quantum matter both at the
model level and with experimental feasibilities.
