Ultracold multicomponent fermions (atoms or molecules) loaded in optical lattices provide an ideal platform for simulating $\mathrm{SU}\left(N\right)$ Hubbard models that host unconventional many-body quantum states beyond SU(2). A prime example is the attractive three-color Hubbard model, in which trion states emerge at strong coupling. Nevertheless, much of its trion ordering on two-dimensional lattices remains uncertain. Here, we employ the determinant quantum Monte Carlo method to simulate the attractive three-color Hubbard model on a $\pi$-flux square lattice at half filling. We show that color-dependent attractive interaction can induce the coexisting charge density wave (CDW) and Néel-ordered states in the three-color $\pi$-flux Hubbard model. In particular, enhanced charge fluctuations (cf. honeycomb lattice) cause much stronger Néel ordering on the $\pi$-flux square lattice. The coexisting charge and Néel orders survive up to a melting temperature, at which they vanish simultaneously. The Ginzburg-Landau (GL) analysis on the coexistence of CDW and Néel orders demonstrates how color-dependent Hubbard interactions stabilize coexisting orders from the perspective of GL free energy principle.

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• Charge order
• Magnetic order
• Optical lattices & traps
• Phase transitions
• Thermal properties
• Strongly correlated systems
• Magnetic moment
• Hubbard model
• Quantum Monte Carlo