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

TL;DR

This work tackles trion ordering in the half-filled attractive three-color Hubbard model realized on a $\pi$-flux square lattice. It employs sign-problem-free determinant quantum Monte Carlo (DQMC) to reveal that color-dependent inter-color attractions drive coexisting CDW and Néel orders, with enhanced Néel order arising from increased charge fluctuations on the $\pi$-flux lattice and a common melting temperature. The coexistence is interpreted through a Ginzburg-Landau framework showing that coupling between CDW and Néel channels, under color anisotropy, stabilizes the ordered state and dictates their simultaneous thermal dissolution. The findings provide a quantitative benchmark for ultracold-atom simulations of SU($N$) physics and suggest experimentally accessible signatures via fluorescence imaging of charge and spin/color distributions.

Abstract

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.

Figures (8)

• Figure 1: (a) Schematic illustration of the $\pi$-flux square lattice. Black and red lines denote hopping integrals $t$ and $-t$, respectively, resulting in a $\pi$ flux per plaquette. (b) Band structure of the $\pi$-flux square lattice with Dirac points at momentum-space locations $(k_x,k_y)=(\pm\frac{\pi}{2},\pm\frac{\pi}{2})$. (c) Schematic illustration of the CDW order and on-site trions. (d) Schematic illustration of the Néel order and off-site trions.
• Figure 2: The on-site triple occupancy $P_3$ and off-site triple occupancies $P_{3\mathrm{off};1}$, $P_{3\mathrm{off};3}$ are plotted as functions of $|U^{\prime}|$. Solid and dashed curves correspond to the data on the $L=12$$\pi$-flux square lattice and $L=9$ honeycomb lattice (extracted from Ref. li2022), respectively. A zoom-in view of $P_{3\mathrm{off};1}$ and $P_{3\mathrm{off};3}$ curves is shown in the inset.
• Figure 3: (a) The finite-size extrapolations of the Néel order parameter $m_Q$ for various $|U^{\prime}|$. The quadratic polynomial fitting is used. (b) The Néel order parameter $m_Q$ is plotted as a function of $|U^{\prime}|$ on the $\pi$-flux square lattice (red squares) and honeycomb lattice (extracted from Ref. li2022) (purple hexagons). The inset shows the nearly linear relationship between $m_Q$ and $P_{3\text{off};3}$ on the $\pi$-flux square lattice, indicating enhanced Néel order with increased off-site trion density.
• Figure 4: The bond-bond correlations (a) $B_{xx}(i,j)$, (b) $B_{yy}(i,j)$, (c) $B_{xy}(i,j)$, and (d) $B_{yx}(i,j)$ are plotted as functions of the distance $r_{ij}$ between sites $i$ and $j$ for various $|U^{\prime}|$ on a $L=14$ lattice.
• Figure 5: The finite-size extrapolations of order parameters (a) $D$, (b) $M_1$, (c) $M_3$, and (d) $m_Q$ at $|U^{\prime}| = 3$ for various temperatures $T$. Order parameters $D$, $M_1$, and $M_3$ characterize charge spatial modulation, while $m_Q$ quantifies Néel ordering. The quadratic polynomial fitting is used.