arXiv:2506.12300 · Physical Review Letters

Trion Formation and Ordering in the Attractive SU(3) Fermi-Hubbard Model

Jonathan Stepp, Eduardo Ibarra-García-Padilla, Richard T. Scalettar, Kaden R. A. Hazzard

Published Apr 30, 2026• 1 citation

Abstract

Recent advances in microwave shielding have increased the stability and control of large numbers of polar molecules, allowing for the first realization of a molecular Bose-Einstein condensate. Remarkably, it was also recently realized that shielded polar molecules exhibit an SU(N) symmetry among their hyperfine states, opening the door to SU(N) systems with larger N, bosonic particle statistics, and tunable interactions-both repulsive and attractive. Motivated by these results, we have studied the SU(3) attractive Fermi-Hubbard model (FHM) on a square lattice. Using the determinant quantum Monte Carlo (DQMC) method, we explore the finite temperature phase diagram and provide evidence for three distinct regions-a three-component Fermi liquid (FL) region, a "trion" liquid (TL) region, and an ordered charge density wave (CDW) phase. The CDW phase is stable at finite temperature [in contrast to the SU(2) CDW], while the FL to TL crossover appears to point to a quantum phase transition at zero temperature. Our method extends straightforwardly to larger N and is sign problem free for even values of N. With these results, we demonstrate the potential physics enabled by using polar molecules as a quantum simulation platform for the attractive SU(N) FHM.

DOI: 10.1103/vnbs-r69s

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arXiv:2506.12300 · Physical Review Letters

Trion Formation and Ordering in the Attractive SU(3) Fermi-Hubbard Model

Jonathan Stepp, Eduardo Ibarra-García-Padilla, Richard T. Scalettar, Kaden R. A. Hazzard

Published Apr 30, 2026• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.1103/vnbs-r69s

Related papers

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

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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.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.

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Rodrigo A. Fontenele, Natanael C. Costa, Raimundo R. dos Santos +1 more

Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak (Bardeen-Cooper-Schrieffer, BCS) and strong (Bose-Einstein condensation, BEC) couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice, from which several different quantities have been calculated and should be useful as a roadmap to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature, $T_c$ , in terms of the band filling, $\ave{n}$, and interaction strength, $U$ , from which we pinpoint a somewhat wide region $|U|/t \approx 5 \pm 1$ ( $t$ is the hopping amplitude) and $\ave{n} \approx 0.79 \pm 0.09$ leading to a maximum $T_c \approx 0.16 t$ . Two additional temperature scales, namely pairing, $T_p$ , and degeneracy, $T_d$ , have been highlighted: the former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), while the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites, for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak (Bardeen-Cooper-Schrieffer, BCS) and strong (Bose-Einstein condensation, BEC) couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice, from which several different quantities have been calculated and should be useful as a roadmap to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature, $T_c$, in terms of the band filling, $\ave{n}$, and interaction strength, $U$, from which we pinpoint a somewhat wide region $|U|/t \approx 5 \pm 1$ ($t$ is the hopping amplitude) and $\ave{n} \approx 0.79 \pm 0.09$ leading to a maximum $T_c \approx 0.16 t$. Two additional temperature scales, namely pairing, $T_p$, and degeneracy, $T_d$, have been highlighted: the former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), while the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites, for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.

arXiv:2201.02156Jan 6, 20221 citation

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.

We employ the determinant quantum Monte Carlo method to study the finite-temperature properties of the half-filled attractive

\mathrm{SU}(3)

Hubbard model on a honeycomb lattice. We calculate the phase diagram in which the phase boundary separates the disordered phase and the charge density wave (CDW) phase and the transition temperature

{T}_{\text{tr}}(|U|)

varies nonmonotonically with attractive Hubbard interaction

|U|

. As the Hubbard

|U|

increases at constant temperature

T<\text{max}({T}_{\text{tr}}(|U|))

, the system first undergoes a transition from thermal Dirac semimetal phase to CDW phase, and eventually the CDW state is thermally melted at a strong Hubbard

|U|

where the system enters a trion liquid phase. In between the two transition points the nonmonotonic

|U|

dependence of CDW order strength is strikingly different from the zero-temperature monotonic behavior. In the trion CDW state where off-site trions arise from quantum fluctuations (a fermion inside an on-site trion hops to a nearest-neighbor site), the simulated triple occupancy at constant Hubbard

|U|

surprisingly increases with temperature, implying that the formation of off-site trions is suppressed by the thermal delocalization of on-site trions. We have also calculated the entropy-temperature relations for various attractive Hubbrad interactions, which exhibit the prominent characteristic of the Pomeranchuk effect. Our work has revealed that the formation of on-site and off-site trions has significant consequences for thermodynamic properties of SU(3) Dirac fermions.

U_c=-1.52(2)

. We demonstrate that this quantum phase transition belongs to the chiral Ising universality class according to the numerically determined critical exponents

\nu=0.82(3)

and

\eta=0.58(4)

. With the increase of coupling strength, the trion formation is investigated, and the change in probability of the on-site trion occupancy infers the coexistence of on-site trionic and off-site trionic CDW states at half-filling.

N=3

chiral Ising universality class suggested by the effective Gross-Neveu-Yukawa (GNY) theory, but coincide with the

N=1

chiral Ising universality class. In the CDW phase, we show that on-site and off-site trions coexist and the off-site trion forms a local bond state. Our work not only illustrates the formation of off-site trions in two-dimensional Hubbard model but also raises doubts about the extent of applicability of GNY model on attractive SU(3) Dirac fermions.

T_c

, in terms of the band filling, $\ave{n}$, and interaction strength,

U

, from which we pinpoint a somewhat wide region

|U|/t \approx 5 \pm 1

(

t

is the hopping amplitude) and $\ave{n} \approx 0.79 \pm 0.09$ leading to a maximum

T_c \approx 0.16 t

. Two additional temperature scales, namely pairing,

T_p

, and degeneracy,

T_d

, have been highlighted: the former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), while the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites, for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.