Metal-insulator transition and magnetism of SU(3) fermions in the square lattice

We study the SU(3) symmetric Fermi-Hubbard model (FHM) in the square lattice at $1/3$-filling using numerically exact determinant quantum Monte Carlo and numerical linked-cluster expansion techniques. We present the different regimes of the model in the $T\text{\ensuremath{-}}U$ plane, which are characterized by local and short-range correlations, and capture signatures of the metal-insulator transition and magnetic crossovers. These signatures are detected as the temperature scales characterizing the rise of the compressibility, and an interaction-dependent change in the sign of the diagonal spin-spin correlation function. The analysis of the compressibility estimates the location of the metal-insulator quantum critical point at ${U}_{c}/t\ensuremath{\sim}6$, and provides a temperature scale for observing Mott physics at finite $T$. Furthermore, from the analysis of the spin-spin correlation function we observe that for $U/t\ensuremath{\gtrsim}6$ and $T\ensuremath{\sim}J=4{t}^{2}/U$ there is a development of a short-range two-sublattice (2SL) antiferromagnetic structure, as well as an emerging three-sublattice (3SL) antiferromagnetic structure as the temperature is lowered below $T/J\ensuremath{\lesssim}0.57$. This crossover from 2SL to 3SL magnetic ordering agrees with Heisenberg limit predictions, and has observable effects on the density of on-site pairs. Finally, we describe how the features of the regimes in the $T\ensuremath{-}U$ plane can be explored with alkaline-earth-like atoms in optical lattices with currently achieved experimental techniques and temperatures. The results discussed in this paper provide a starting point for the exploration of the SU(3) FHM upon doping.