  We employ the large-scale quantum Monte-Carlo simulations to investigate the
Mott-insulating states of the half-filled SU(4) Hubbard model on the square
lattice with a staggered-flux pattern. The noninteracting band structure that
evolves from a nested Fermi surface at zero flux to isotropic Dirac cones at
$\pi$ flux, exhibits anisotropic Dirac cones as the flux varies in between. Our
simulations show transitions between the three phases of Dirac semimetal,
antiferromagnet and valence-bond solid. A direct continuous transition between
the antiferromagnetic phase and the valence-bond-solid phase is realized via
varying the flux in the Mott regime. The simulated critical exponents
remarkably agree with those of SU(4) $J$-$Q$ model. Inside the
valence-bond-solid phase induced by the flux, the plaquette valence-bond state
with vanishing single-particle gap is identified. At strong coupling, the
valence-bond-solid phase disappears and the Mott-insulating state is always
accompanied by antiferromagnetic ordering, regardless of the magnitude of the
flux.
