Trion ordering in the attractive three-color Hubbard model on a $π$-flux square lattice

Ultracold multicomponent fermions (atoms/molecules) loaded in optical lattices provide an ideal platform for simulating 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 (DQMC) method to simulate the attractive three-color Hubbard model on a $π$-flux square lattice at half filling. We show that color-dependent attractive interaction can induce coexisting charge density wave (CDW) and Néel ordered states in the three-color $π$-flux Hubbard model. In particular, enhanced charge fluctuations (cf. honeycomb lattice) cause much stronger Néel ordering on the $π$-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.

💡 Research Summary

The authors investigate the attractive three‑color (SU(3)) Hubbard model on a π‑flux square lattice at half filling using sign‑problem‑free determinant quantum Monte Carlo (DQMC). By introducing color‑dependent on‑site attractions (U₁₂ = U, U₁₃ = U₂₃ = U′ < 0), they explore how the lattice coordination number (z = 4) and the Dirac‑like band structure affect the formation of bound states (trions) and long‑range order. Two types of trions are identified: on‑site trions, where all three colors occupy the same site, quantified by the triple occupancy P₃, and off‑site trions, where a pair of colors resides on one site and the third color on a neighboring site, quantified by P₃^{off;1} and P₃^{off;3}. Compared with the honey‑comb lattice (z = 3), the π‑flux square lattice exhibits markedly lower P₃ and substantially higher off‑site trion densities, indicating that the increased coordination enhances charge fluctuations and promotes the conversion of on‑site trions into off‑site ones.

The off‑site trions carry a net color moment and give rise to Néel (antiferromagnetic) order. The Néel order parameter m_Q, defined via staggered correlations of the local color magnetization m_i = ¼(n_{i1}+n_{i2}−2n_{i3}), is extracted through finite‑size scaling. The authors find a nearly linear relationship between m_Q and the density of off‑site trions P₃^{off;3}, demonstrating that the strength of Néel ordering is directly tied to the population of off‑site trions. For a given interaction anisotropy |U′| < 6, the Néel order on the π‑flux lattice is considerably stronger than on the honey‑comb lattice, confirming that a single increase in coordination number can dramatically boost magnetic ordering in this multicomponent system.

To probe whether off‑site trions develop any directional preference due to the alternating hopping signs along y, the authors compute bond‑bond correlation functions B_{ab}(i,j) for all combinations of bond directions a,b ∈ {x,y}. All correlations decay rapidly to zero with distance, and no discernible difference exists between B_{xx} and B_{yy} or between cross‑terms B_{xy}, B_{yx}. This indicates that off‑site trions are randomly oriented along the four lattice directions, and the Néel order emerges despite the lack of any spatial anisotropy in the trion distribution.

Both charge‑density‑wave (CDW) order, characterized by a staggered modulation of the on‑site triple occupancy, and Néel order vanish simultaneously at a finite melting temperature T_c ≈ 0.6 t. The authors perform a Ginzburg‑Landau (GL) analysis, writing the free energy as F = α_Δ|Δ|² + β_Δ|Δ|⁴ + α_M|M|² + β_M|M|⁴ + γ|Δ|²|M|², where Δ and M denote the CDW and Néel order parameters, respectively. The DQMC data imply γ < 0, meaning that the coupling term lowers the free energy when both orders coexist. Consequently, the color‑dependent Hubbard interactions stabilize a phase where CDW and Néel orders are intertwined, rather than competing.

Overall, the study demonstrates that the π‑flux square lattice provides a unique platform to explore trion‑driven intertwined orders in multicomponent fermionic systems. By tuning the ratio U′/U, one can control the balance between on‑site and off‑site trions, thereby adjusting both charge and magnetic ordering. The findings are directly relevant to ongoing ultracold‑atom experiments with three hyperfine states of ⁶Li or with microwave‑shielded polar molecules, where synthetic π‑flux lattices and color‑selective interactions can be engineered. The work opens avenues for realizing and probing exotic composite orders, such as trion‑mediated supersolids or quantum liquids, in highly controllable quantum simulators.