  We employ the sign-problem-free projector determinant quantum Monte Carlo
method to study a microscopic model of SU($N$) fermions with singlet-bond and
triplet-current interactions on the square lattice. We find the gapped singlet
$p_x$ and gapless triplet $d_{x^2-y^2}$ density wave states in the half-filled
$N=4$ model. Specifically, the triplet $d_{x^2-y^2}$ density wave order is
observed in the weak triplet-current interaction regime. As the triplet-current
interaction strength is further increased, our simulations demonstrate a
transition to the singlet $p_x$ density wave state, accompanied by a gapped
mixed-ordered area where the two orders coexist. With increasing singlet-bond
interaction strength, the triplet $d_{x^2-y^2}$-wave order persists up to a
critical point after which the singlet $p_x$ density wave state is stabilized,
while the ground state is disordered in between the two ordered phases. The
analytical continuation is then performed to derive the single-particle
spectrum. In the spectra of triplet $d_{x^2-y^2}$ and singlet $p_x$ density
waves, the anisotropic Dirac cone and the parabolic shape around the Dirac
point are observed, respectively. As for the mixed-ordered area, a
single-particle gap opens and the velocities remain anisotropic at the Dirac
point.
