The $\mathrm{SU}\left(N\right)$ Hubbard model is a natural extension of the $\mathrm{SU}\left(2\right)$ model. However, even the $N=3$ case remains poorly understood. We report a ground-state phase diagram of the square lattice $\mathrm{SU}\left(3\right)$ Fermi-Hubbard model. Using a backflow Ansatz, we identify strong signatures of a Mott transition and a subsequent magnetic transition, and the suppression of a previously predicted magnetic phase. We study the hole-doped model, identifying a transition from magnetic to paramagnetic behavior in the strong-coupling regime. Our findings offer a different ground-state picture. More broadly, our work suggests a path to study general $\mathrm{SU}\left(N\right)$ Hubbard models with arbitrary filling and geometry.

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• Cold and ultracold molecules
• Magnetic order
• Magnetic phase transitions
• Strongly correlated systems
• Hubbard model
• SU(N) symmetries