Robustness of the color-selective Mott transition and the repulsion-driven color superfluid of three-component fermionic atoms under harmonic trapping and finite-dimensional lattice geometry

Three-component (color) fermions in optical lattices with purely repulsive, color-dependent interactions undergo a color-selective Mott transition (paired or color-selective Mott states) near half filling and, on the doped side, an s-wave color superfluid in which the two weakly repelling colors form Cooper pairs through an effective attraction generated by density fluctuations of the third color. These predictions were established only within homogeneous dynamical mean-field theory (DMFT) on the infinite-dimensional Bethe lattice with two-site or perturbative impurity solvers. Later literature has settled the finite-temperature phase diagrams of the homogeneous model and independently confirmed the superfluid with continuous-time quantum Monte Carlo impurity solvers, and has partially filled in finite-dimensional and one-dimensional aspects. What remains open is the experimentally decisive robustness question: whether and where the color-selective/paired Mott physics and the repulsion-driven color superfluid survive a harmonic trapping potential (spatially varying local filling, Mott/superfluid shell structure) and finite-dimensional lattice geometries with realistic non-semicircular densities of states, including how the known finite-temperature phase boundaries deform under these conditions.

Background

Ultracold fermionic atoms can be loaded into optical lattices (standing-wave light crystals) whose low-energy physics is captured by a Hubbard model, with the atomic hyperfine states playing the role of a flavor index called color. When three colors are present, the on-site inter-color repulsions $U_{12}$, $U_{23}$, and $U_{31}$ can be tuned independently via Feshbach resonances, realizing a three-component Hubbard model with color-dependent (SU(3)-breaking) repulsion; three-component gases of $^6$Li and alkaline-earth-like atoms such as $^{173}$Yb are the experimental platforms. Miyatake, Inaba, and Suga studied this model near half filling (3/2 atoms per site) with dynamical mean-field theory (DMFT), which maps the lattice problem onto a self-consistent quantum impurity problem and becomes exact in infinite dimensions. They used the homogeneous Bethe lattice, an infinite-coordination lattice whose non-interacting density of states is semi-circular with bandwidth $W=4t$ ($t$ the hopping integral), and solved the impurity problem with the two-site DMFT approximation of Potthoff, an analytically tractable solver, later supplemented by modified iterative perturbation theory and the self-energy functional approach. Their central findings were: (i) for strongly anisotropic repulsions, one pair of colors can localize while the third remains metallic — a color-selective Mott transition (CSMT), the cold-atom analog of the orbital-selective Mott transition in the degenerate Hubbard model of Koga et al. — together with a paired Mott insulator (PMI) in which preformed pairs of the two weakly repelling colors and atoms of the third color are both localized; and (ii) on the doped side, close to the PMI transition, an exotic color superfluid (CSF): the two weakly repelling colors form $s$-wave Cooper pairs while the third color stays a Fermi liquid. The pairing mechanism is purely repulsive: in a random phase approximation (RPA) treatment, the effective interaction between colors 1 and 2 becomes $\tilde{U}_{12}\approx U_{12}-U_{23}^{2}\chi_{3}$, where $\chi_{3}$ is the density susceptibility of the third color, so that density fluctuations of the third color generate an attraction that can overcome the bare repulsion $U_{12}$. This is distinct from the attractive-interaction color superfluidity of Rapp et al. and from weak-coupling RG pairing in the SU(3)-symmetric model of Honerkamp and Hofstetter. At exactly half filling the low-temperature ground states are in fact staggered ordered — a color density wave (CDW, pairs and third-color atoms alternating between sublattices) or a color-selective antiferromagnet (CSAF, two colors alternating while the third is itinerant) — and the superfluid dome sits at nearby incommensurate fillings. The 2010 source paper and its companion Rapid Communications posed as an open outlook the extension of these results beyond the modeling approximations: the harmonic trapping potential that confines any real gas (neglected in the homogeneous calculation), finite-temperature stability ranges, finite-dimensional lattice geometries with non-semicircular densities of states, and impurity solvers more accurate than the two-site approximation. A trapped gas is spatially inhomogeneous: within a local-density approximation or real-space DMFT (in which a separate impurity problem is solved at every lattice site under the local chemical potential), the local filling varies across the cloud, so commensurate Mott cores, staggered-ordered regions, and doped superfluid shells can coexist, and the question of survival becomes a question about this shell structure. Finite-dimensional lattices such as the square or cubic optical lattice bring perfect nesting at half filling (which strengthens the competing CDW/CSAF orders), non-semicircular densities of states, and spatial fluctuations beyond single-site DMFT, addressable with cluster methods such as the variational cluster approximation (VCA) or with density matrix renormalization group (DMRG) in one dimension. Modern site- and flavor-resolved quantum gas microscopy can measure flavor-resolved occupation statistics, double occupancies, and pairing correlations in three-flavor lattice gases, so a quantitative answer would connect directly to ongoing experiments.

Problem Statement

Determine, by controlled many-body calculation, whether the characteristic phases of the anisotropic repulsive three-component Hubbard model near half filling — the color-selective Mott transition with its paired and color-selective Mott insulating states, and the repulsion-driven s-wave color superfluid in which the two weakly repelling colors pair via third-color density fluctuations — survive under experimentally realistic conditions, and map how their phase boundaries shift. Concretely: (1) include the harmonic trapping potential, computing the spatially resolved phase structure in which the local filling varies across the cloud and commensurate Mott cores, staggered-ordered regions, and doped superfluid shells can coexist, and establish the trap parameters and interaction anisotropies for which the color superfluid and the color-selective Mott physics persist; (2) treat finite-dimensional lattice geometries with realistic non-semicircular densities of states (square and cubic optical lattices), quantifying how the superfluid dome, the Mott transitions, and the competing color-density-wave and color-selective antiferromagnetic orders rearrange relative to the Bethe-lattice results; and (3) determine how the known finite-temperature phase boundaries of the homogeneous model deform under (1) and (2), since the existing temperature phase diagrams were computed only for the homogeneous Bethe lattice. The finite-temperature stability of the homogeneous model and the adequacy of impurity solvers beyond the two-site approximation have already been settled and are not re-asked; the common objective of the remaining components is the experimentally relevant phase diagram of the repulsion-driven pairing mechanism under spatial confinement and finite-dimensional lattice geometry.

References

1. Shin-ya Miyatake, Kensuke Inaba, Sei-ichiro Suga, "Color-selective Mott transition and color superfluid of three-component fermionic atoms with repulsive interaction in optical lattices", Physica C: Superconductivity 470, S916-S918 (2010). DOI: 10.1016/j.physc.2009.11.162, https://doi.org/10.1016/j.physc.2009.11.162
2. Shin-ya Miyatake, Kensuke Inaba, Sei-ichiro Suga, "Three-component fermionic atoms with repulsive interaction in optical lattices", Physical Review A 81, 021603(R) (2010). DOI: 10.1103/PhysRevA.81.021603, arXiv:1002.2444, https://arxiv.org/abs/1002.2444
3. Kensuke Inaba, Shin-ya Miyatake, Sei-ichiro Suga, "Mott transitions of three-component fermionic atoms with repulsive interaction in optical lattices", Physical Review A 82, 051602(R) (2010). DOI: 10.1103/PhysRevA.82.051602, arXiv:1006.1685, https://arxiv.org/abs/1006.1685
4. Kensuke Inaba, Sei-ichiro Suga, "Superfluid State of Repulsively Interacting Three-Component Fermionic Atoms in Optical Lattices", Physical Review Letters 108, 255301 (2012). DOI: 10.1103/PhysRevLett.108.255301, arXiv:1205.5964, https://arxiv.org/abs/1205.5964
5. Kensuke Inaba, Sei-ichiro Suga, "Superfluid, staggered state, and Mott insulator of repulsively interacting three-component fermionic atoms in optical lattices", Modern Physics Letters B 27, 1330008 (2013). DOI: 10.1142/S0217984913300081, arXiv:1305.1972, https://arxiv.org/abs/1305.1972
6. S. Suga, K. Inaba, "Three-component repulsive fermionic atoms in optical lattices at finite temperatures", Journal of Physics: Conference Series 273, 012016 (2011). DOI: 10.1088/1742-6596/273/1/012016, https://doi.org/10.1088/1742-6596/273/1/012016
7. S. Suga, K. Inaba, "Perturbative approach to three-component fermionic atoms in optical lattices", Journal of Physics: Conference Series 391, 012166 (2012). DOI: 10.1088/1742-6596/391/1/012166, https://doi.org/10.1088/1742-6596/391/1/012166
8. Yuki Okanami, Nayuta Takemori, Akihisa Koga, "Stability of the superfluid state in three-component fermionic optical lattice systems", Physical Review A 89, 053622 (2014). DOI: 10.1103/PhysRevA.89.053622, arXiv:1401.5610, https://arxiv.org/abs/1401.5610
9. Sei-ichiro Suga, "Controlled pairing symmetry of the superfluid state in systems of three-component repulsive fermionic atoms in optical lattices", Physical Review A 92, 023617 (2015). DOI: 10.1103/PhysRevA.92.023617, arXiv:1408.6582, https://arxiv.org/abs/1408.6582
10. Andrii Sotnikov, Walter Hofstetter, "Magnetic ordering of three-component ultracold fermionic mixtures in optical lattices", Physical Review A 89, 063601 (2014). DOI: 10.1103/PhysRevA.89.063601, arXiv:1402.3397, https://arxiv.org/abs/1402.3397
11. Takumi Hasunuma, Tatsuya Kaneko, Shohei Miyakoshi, Yukinori Ohta, "Mott Transitions and Staggered Orders in the Three-component Fermionic System: Variational Cluster Approach", Journal of the Physical Society of Japan 85, 074704 (2016). DOI: 10.7566/JPSJ.85.074704, arXiv:1605.06648, https://arxiv.org/abs/1605.06648
12. Hiromasa Yanatori, Akihisa Koga, "Finite Temperature Properties of Three-Component Fermion Systems in Optical Lattice", Journal of the Physical Society of Japan 85, 014002 (2016). DOI: 10.7566/JPSJ.85.014002, arXiv:1510.01471, https://arxiv.org/abs/1510.01471
13. Duong-Bo Nguyen, Duy-Khuong Phung, Van-Nham Phan, Minh-Tien Tran, "Metal-insulator transition induced by mass imbalance in a three-component Hubbard model", Physical Review B 91, 115140 (2015). DOI: 10.1103/PhysRevB.91.115140, arXiv:1503.08972, https://arxiv.org/abs/1503.08972
14. S. Suga, K. Inaba, "Three-body losses of repulsively interacting three-component fermionic atoms in optical lattices", Journal of Physics: Conference Series 529, 012024 (2014). DOI: 10.1088/1742-6596/529/1/012024, arXiv:1404.5994, https://arxiv.org/abs/1404.5994
15. Jirayu Mongkolkiattichai, Liyu Liu, Sohail Dasgupta, Kaden R. A. Hazzard, Peter Schauss, "Quantum gas microscopy of three-flavor Hubbard systems" (2025). arXiv:2503.05687, https://arxiv.org/abs/2503.05687
16. Sheng Chen, Qiao Yang, Wéi Wú, Fadi Sun, "Dominant superconductivity in a three-component Hubbard chain", Physical Review B 114, 045105 (2026). DOI: 10.1103/kpth-wy8h, arXiv:2512.08784, https://arxiv.org/abs/2512.08784
17. Fei-Jie Huang, Qi-Hui Chen, Yong-Ping Fu, Rong-Li Wang, "A continuous line of Berezinskii-Kosterlitz-Thouless phase transition points and the golden ratio point in the three-component fermion ladder", Journal of Physics: Condensed Matter 38, 125602 (2026). DOI: 10.1088/1361-648X/ae5438, https://doi.org/10.1088/1361-648X/ae5438
18. Ákos Rapp, Gergely Zaránd, Carsten Honerkamp, Walter Hofstetter, "Color Superfluidity and "Baryon" Formation in Ultracold Fermions", Physical Review Letters 98, 160405 (2007). DOI: 10.1103/PhysRevLett.98.160405, https://doi.org/10.1103/PhysRevLett.98.160405
19. Carsten Honerkamp, Walter Hofstetter, "Ultracold Fermions and the SU(N) Hubbard Model", Physical Review Letters 92, 170403 (2004). DOI: 10.1103/PhysRevLett.92.170403, https://doi.org/10.1103/PhysRevLett.92.170403
20. Michael Potthoff, "Two-site dynamical mean-field theory", Physical Review B 64, 165114 (2001). DOI: 10.1103/PhysRevB.64.165114, https://doi.org/10.1103/PhysRevB.64.165114
21. Akihisa Koga, Norio Kawakami, T. M. Rice, Manfred Sigrist, "Orbital-Selective Mott Transitions in the Degenerate Hubbard Model", Physical Review Letters 92, 216402 (2004). DOI: 10.1103/PhysRevLett.92.216402, https://doi.org/10.1103/PhysRevLett.92.216402
22. E. V. Gorelik, N. Blümer, "Mott transitions in ternary flavor mixtures of ultracold fermions on optical lattices", Physical Review A 80, 051602(R) (2009). DOI: 10.1103/PhysRevA.80.051602, https://doi.org/10.1103/PhysRevA.80.051602
23. O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.-S. Lühmann, B. A. Malomed, T. Sowiński, J. Zakrzewski, "Non-standard Hubbard models in optical lattices: a review", Reports on Progress in Physics 78, 066001 (2015). DOI: 10.1088/0034-4885/78/6/066001, arXiv:1406.0181, https://arxiv.org/abs/1406.0181
24. Shin-ya Miyatake, Kensuke Inaba, Sei-ichiro Suga, "Color superfluidity and trionic state of three-component lattice fermionic atoms", Journal of Physics: Conference Series 273, 012008 (2011). DOI: 10.1088/1742-6596/273/1/012008, https://doi.org/10.1088/1742-6596/273/1/012008