Quantum phase transition (QPT) has long been among the most fascinating topics in condensed matter physics. QPT occurs at zero temperature and is driven by quantum fluctuations that are tuned by varying certain parameters in a quantum many-body system. In recent years, with the rapid development of experimental approaches, optical traps and lattices loaded with ultracold atoms have become an excellent lab for studying strong correlation physics in a quantum many-body system. Interestingly, ultracold alkali and alkaline-earth fermions carry large hyperfine spins, which provides the opportunities to study SU(N) symmetries that are rare in solids, and to realize the quantum many-body states that are unstable in electron systems. In order to study the strongly correlated ultracold fermionic atoms in optical lattices, this work performs the sign-problem-free quantum Monte Carlo (QMC) simulations on various models of interacting SU(N) fermions.

For SU(N) fermions with singlet-bond and triplet-current interactions, the QMC study of the half-filled SU(4) model finds the gapped singlet p_x and gapless triplet d_x2-y2 density-wave orders. Specifically, the triplet d_x2-y2 density wave order is observed in the weak triplet-current interaction regime. As the triplet-current interaction strength is 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 the singlet-bond interaction strength, the triplet d_x2-y2-wave order persists up to a critical point after which the singlet p_x density wave state is stabilized, but 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_x2-y2 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.

On the square lattice with a staggered pattern of flux, the half-filled SU(4) Hubbard model is investigated. The noninteracting band structure that evolves from a nested Fermi surface at zero flux to isotropic Dirac cones at π-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 numerically 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.

On the honeycomb lattice, the half-filled attractive SU(3) Hubbard model is investigated. At strong coupling, we show that on-site and off-site trions coexist and the local off-site trion forms a bond state. With increasing attractive Hubbard interaction, our simulations demonstrate a continuous quantum phase transition from the semimetal to charge density wave at the critical coupling U_c/t=-1.52(2). The critical exponents ν=0.82(3) and η=0.58(4) determined by the QMC simulation remarkably disagree with those of the N=3 chiral Ising universality class suggested by the effective Gross-Neveu-Yukawa (GNY) theory, but coincide with the N=1 chiral Ising universality class. Our work not only illustrates the existence of off-site trions in two-dimensional Hubbard model, but also raise doubts about the extent of applicability of GNY model on the attractive SU(3) Dirac fermions.