Trion ordering in the attractive three-color Hubbard model on a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> -flux square lattice

Ultracold multicomponent fermions (atoms or molecules) loaded in optical lattices provide an ideal platform for simulating $\mathrm{SU}(N)$ 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 $\ensuremath{\pi}$-flux square lattice at half filling. We show that color-dependent attractive interaction can induce the coexisting charge density wave (CDW) and N\'eel-ordered states in the three-color $\ensuremath{\pi}$-flux Hubbard model. In particular, enhanced charge fluctuations (cf. honeycomb lattice) cause much stronger N\'eel ordering on the $\ensuremath{\pi}$-flux square lattice. The coexisting charge and N\'eel orders survive up to a melting temperature, at which they vanish simultaneously. The Ginzburg-Landau (GL) analysis on the coexistence of CDW and N\'eel orders demonstrates how color-dependent Hubbard interactions stabilize coexisting orders from the perspective of GL free energy principle.