Finite-Dimensional Fate of the Color-Superfluid–Trion Transition in the Attractive Three-Component (SU(3)) Hubbard Model

Rapp, Zaránd, Honerkamp, and Hofstetter [Phys. Rev. Lett. 98, 160405 (2007)] predicted, in a variational treatment of the $d\to\infty$ attractive three-component ("color") Hubbard model, a second-order quantum phase transition at a critical attraction $U_C$ between a color superfluid (CSF) of Cooper pairs of two of the three fermion species — with a secondary ferromagnetic polarization driving spontaneous domain formation — and a phase of immobile on-site three-fermion bound states (trions), the optical-lattice analogues of two-flavor color superconductivity and baryons in QCD. They explicitly left open how this $d\to\infty$, $T=0$ picture is modified in finite dimensions: the itinerant trionic Fermi liquid and its scales ($t_{\mathrm{eff}}$, $T^*$), competing orders near half filling, finite-temperature and low-dimensional fluctuation regimes, domain-wall energetics, and quantitative detection signatures. The subsequent literature shows that the half-filling charge-order competition and the Ginzburg–Landau analysis of the superfluid–magnetism interplay are now substantially mapped, while a precise open core survives — the order, universality, and quantitative finite-dimensional characterization of the CSF–trion transition at generic fillings (where methods currently disagree and the fermion sign problem blocks unbiased simulation), its fluctuation-dominated finite-temperature boundaries, and its experimental realization under realistic three-body losses.

Background

Ultracold fermionic atoms in optical lattices — crystals of light formed by interfering laser beams, in which atoms hop between potential minima like electrons in a solid — can be trapped in three internal hyperfine states, called "colors". When the dominant interaction is an attractive on-site attraction $U<0$ between any two different colors, the system realizes the three-component (SU(3)-symmetric) Hubbard model, the cold-atom analogue of two-flavor quantum chromodynamics (QCD) at high density. Rapp, Zaránd, Honerkamp, and Hofstetter [Phys. Rev. Lett. 98, 160405 (2007)] studied this model with a Gutzwiller-type variational wave function (a trial ground state that optimizes the local on-site fermion configurations) in the limit of infinite spatial dimension $d\to\infty$ (where $t^*=t\sqrt{d}$ is the scaled kinetic energy), at equal color densities and zero temperature. They found two phases. At weak attraction $|U|<|U_C|$ (with $U_C\approx-1.774t^*$ at filling $\varrho=1/3$, i.e. one atom per lattice site on average, since $\varrho\equiv n/3$ with $n$ the total atom number per lattice site), the ground state is a color superfluid (CSF): atoms of two colors form Cooper pairs, breaking the SU(3) symmetry to SU(2), analogously to color superconductivity in dense QCD. At $|U|>|U_C|$ the variational parameter diverges and the ground state becomes a trion phase: on-site singlet bound triples of three different colors, the analogues of baryons. The transition was found to be second order, with the CSF developing a secondary "ferromagnetic" polarization (nonzero expectation value of spin operators built from the Gell-Mann matrices, the SU(3) generalization of Pauli matrices), which at equal total color populations implies spontaneous phase separation into domains. In $d\to\infty$ the trion hopping vanishes as $1/d^{3/2}$, so trions are immobile and carry a finite residual entropy. The authors explicitly acknowledged what their analysis could not decide: (i) in finite $d$ the trions should hop with an effective amplitude $t_{\mathrm{eff}}$ and form an itinerant trionic Fermi liquid with a small Fermi temperature $T^*$ whose scaling as $|U|\to|U_C|^+$ they could not compute; (ii) near half filling ($\varrho\approx1/2$) Fermi-surface nesting (matching of occupied and empty states on the Brillouin-zone boundary, a common source of density-wave instabilities) and charge ordering could modify the phase diagram; (iii) at $T>0$ thermal fluctuations destroy CSF order at finite $T$ in $d=3$, and for $d<3$ phase fluctuations preclude any long-range CSF order, so the finite-temperature boundaries in low-dimensional geometries were beyond the $d\to\infty$ treatment; (iv) the energetics and length scales of the domain formation were deferred to a Ginzburg–Landau (GL) analysis announced as unpublished; (v) detection proposals — Bragg scattering of the soft Anderson–Bogoliubov collective mode (the superfluid's gapless, sound-like density oscillation) and lattice-modulation ("shaking") spectroscopy of trion breakup at $\omega\approx2|U|$ — were qualitative, without line shapes, widths, or temperature dependence.

Problem Statement

For the attractive three-component Hubbard model (three fermion colors with equal attractive on-site interactions $U<0$) in finite spatial dimensions $d=2$ and $3$, including quasi-one-dimensional and trapped geometries, determine how the $d\to\infty$ color-superfluid–trion picture of Rapp et al. is modified, and determine the conditions under which it can be realized and observed experimentally. Concretely, all four of the following are required: (a) establish the nature — order (second versus first), universality class, and critical behavior — of the zero-temperature color-superfluid–to–trion transition at generic fillings away from particle-hole symmetry, reconciling the conflicting existing determinations (second order in the $d\to\infty$ variational treatment; first order at low temperature in self-energy-functional studies; a Fermi-liquid–trion-liquid crossover pointing to a quantum phase transition in sign-problem-free determinant quantum Monte Carlo (DQMC) at half filling on the square lattice); (b) characterize the itinerant trionic Fermi liquid quantitatively: the trion effective hopping $t_{\mathrm{eff}}$ as a function of coupling and dimension, and the trionic Fermi temperature $T^*$ and its scaling as $|U|\to|U_C|^+$ (approaching the critical attraction of the $d\to\infty$ variational study from the trion side) and with $d$, replacing the $d\to\infty$ picture of immobile trions with residual entropy; (c) determine the finite-temperature color-superfluid phase boundaries and crossovers in fluctuation-dominated regimes — two-dimensional and quasi-one-dimensional or trapped geometries, where no long-range CSF order survives at $T>0$ — beyond $d\to\infty$ and mean-field treatments; and (d) specify the conditions (namely, interaction anisotropy, three-body-loss constraints, temperature and detection protocols) under which the color superfluid and the trion liquid can be realized and observed in three-flavor optical-lattice gases, including quantitative observable predictions such as structure-factor and lattice-modulation spectra near the transition. Consolidating the already-mapped half-filling charge-order competition and the Ginzburg–Landau superfluid–magnetism analysis into this finite-dimensional phase diagram is part of the objective; establishing those restricted results anew is not — in subgoal (a), the cited half-filling DQMC determination enters as existing evidence to be reconciled with the generic-filling picture, not as a filling at which the charge-order competition must be re-established.

References

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