  In the past decades, tremendous efforts have been made towards understanding
the exotic physics emerging from competition between various ordering
tendencies in strongly correlated systems. Employing state-of-the-art quantum
Monte-Carlo simulation, we investigate an interacting SU($N$) fermionic model
with varying interaction strength and value of $N$, and unveil the ground-state
phase diagram of the model exhibiting a plethora of exotic phases. For small
value of $N$, namely $N=2,3$, the ground state is antiferromagnetic (AFM)
phase, whereas in the large-$N$ limit, valence bound solid (VBS) order is
dominant. For the intermediate value of $N$ such as $N=4$, remarkably, our
study reveals distinct VBS orders appear in the weak and strong coupling
regimes. More fantastically, the competition between staggered and columnar VBS
ordering tendencies gives rise to a Mott insulating phase without spontaneously
symmetry breaking (SSB), existing in a large interacting parameter regime,
which is consistent with a gapped quantum spin liquid. Our study not only
provides a platform to investigate the fundamental physics of quantum many-body
systems, but also offers a novel route towards searching for exotic states of
matter such as quantum spin liquid in realistic quantum materials.
