Unresolved extensions of color superfluidity in three-component ultracold fermions with color-orbit and color-flip fields: Efimov trimers, nonuniform (LOFF) phases, fluctuation corrections to the critical temperature, trap effects, and unequal interactions with finite detuning

For a three-component (three-color) ultracold Fermi gas subjected to synthetic color-orbit and color-flip fields, Kurkcuoglu and Sá de Melo (Phys. Rev. A 97, 023632 (2018)) classified the low-temperature color-superfluid phases within saddle-point mean-field theory for equal inter-color s-wave interactions, zero detuning, and a homogeneous gas, and in their Section V ("Final Remarks") explicitly listed four unresolved directions that complete the physics of this model: (i) Efimov trimer formation and the low-density trimer-versus-pair phase diagram under the fields; (ii) whether narrow Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) regions exist in the BCS regime at very low temperature; (iii) the fluctuation-corrected critical temperature across the BCS-BEC crossover; and (iv) the spatial coexistence of phases in harmonic traps within the local density approximation. A fifth direction goes beyond the restriction the source explicitly imposes — equal inter-color interactions and zero detuning, with the paper noting that nonzero detuning breaks parity and enriches the topology of the phases: (v) the generalization to unequal inter-color interactions and nonzero detuning. None of these five directions has been calculated for this model. The experimental realization of SU(3) color-orbit coupling in 2025 makes the problem timely. The problem is open.

Background

Ultracold fermionic atoms with three internal hyperfine states — realizable in $^{6}\mathrm{Li}$, $^{40}\mathrm{K}$, and $^{173}\mathrm{Yb}$ — can be labeled by a color index $c\in\{R,G,B\}$ by analogy with the three quark colors of quantum chromodynamics (QCD). When only inter-color s-wave contact interactions are present and the three coupling constants are equal ($g_{RG}=g_{RB}=g_{GB}=g$), the system is approximately SU(3) symmetric and, at low temperature, forms a color superfluid, the cold-atom analogue of color superconductivity in dense quark matter; its pairing is characterized by the interaction parameter $1/(k_F a_s)$, which interpolates from weak-coupling BCS pairing ($1/(k_F a_s)\ll -1$) to a Bose-Einstein condensate (BEC) of dimers ($1/(k_F a_s)\gg 1$), with $k_F$ the Fermi momentum and $a_s$ the s-wave scattering length.

Laser or radio-frequency fields can additionally couple the internal states. A resonant field that coherently converts one color into another is a color-flip field (a Rabi coupling with frequency $\Omega$, the SU(3) generalization of a spin-flip Zeeman field), while a configuration in which the momentum transfer of the absorbed photon depends on the color produces color-orbit coupling — a momentum-dependent non-Abelian synthetic gauge field of strength set by the recoil momentum $k_T$, generalizing spin-orbit coupling from SU(2) to SU(3). A mismatch of the photon frequencies from two-photon resonance gives a detuning $\delta$ (a color-shift field analogous to a Zeeman detuning). The single-particle Hamiltonian of the simplest experimentally relevant configuration ($\Omega_{RB}=0$, $\delta=0$) is diagonalized in a mixed-color basis, in which the pairing order parameter becomes a tensor with six independent, explicitly momentum-dependent components.

Kurkcuoglu and Sá de Melo [Phys. Rev. A 97, 023632 (2018); arXiv:1707.09923] solved this model at the saddle-point (mean-field) level for equal interactions, zero detuning, and a homogeneous gas: they found five nodal color-superfluid phases (phases whose quasiparticle excitation spectrum has zeros at points, lines, or surfaces in momentum space) plus one fully gapped phase, a pentacritical quintuple point where the five nodal phases meet at $k_T=0.35\,k_F$, Lifshitz-type topological transitions between phases, and spectroscopic signatures (quasiparticle spectra, momentum distributions, density of states) in the singlet, triplet, and quintet total-pseudo-spin pairing sectors.

The authors' Section V ("Final Remarks") lists four topics that remain unresolved — Efimov states, non-uniform superfluidity, the critical temperature of color superfluids, and trap effects — and a fifth direction goes beyond the restriction the source explicitly imposes (equal couplings $g_{RG}=g_{RB}=g_{GB}$ and zero detuning $\delta=0$, parity being a good quantum number only at $\delta=0$); these five items define the present problem. First, at very low densities ($n\lesssim 10^{12}\,\mathrm{cm}^{-3}$), Efimov trimers — universal three-body bound states of three resonantly interacting particles, occurring here because the three colors are distinguishable — can form; a conjectured zero-temperature trimer-versus-pair phase diagram exists for three-component fermions in narrow Feshbach resonances without synthetic fields [Nishida, Phys. Rev. Lett. 109, 240401 (2012)], but how color-orbit and color-flip fields modify Efimov physics (including the Efimov three-body parameter) and the resulting low-density phase diagram was left open. Second, in the BCS corner of the phase diagram ($\Omega/E_F\lesssim 0.2$, $1/(k_F a_s)\lesssim -1$), nonuniform superfluid phases of the Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) type — pairing at finite center-of-mass momentum, possible when nesting of the dressed Fermi surfaces is strong — may exist in a narrow region at very low temperatures ($T/T_F\ll 10^{-3}$); their stability and extent under the fields is unknown. Third, the critical temperature $T_c$ of the color superfluid beyond mean field — via Gaussian fluctuations or a $T$-matrix approach (a self-consistent summation of pair-propagation diagrams), as done for three-component gases without fields [Ozawa and Baym, Phys. Rev. A 82, 063615 (2010); Tajima and Naidon, New J. Phys. 21, 073051 (2019)] and for two-component spin-orbit- and Rabi-coupled fermions [Powell, Baym, and Sá de Melo, Phys. Rev. A 105, 063304 (2022)] — has not been calculated for the present three-color model with its six-component momentum-dependent order-parameter tensor. Fourth, the effect of a harmonic trapping potential — treated within the local density approximation (LDA), in which the chemical potential becomes position-dependent, $\tilde\mu(\mathbf{r})=\mu-V_{\mathrm{trap}}(\mathbf{r})$, so that different phases of the homogeneous phase diagram coexist as spatial shells — has not been mapped. Fifth, the generalization to non-identical inter-color interactions ($g_{RG}\neq g_{RB}\neq g_{GB}$) and nonzero detuning ($\delta\neq 0$, which breaks parity of the single-particle spectrum and hence changes the topology of the phases) remains unexplored.

Problem Statement

For a homogeneous three-component ultracold Fermi gas ($^{6}\mathrm{Li}$, $^{40}\mathrm{K}$, or $^{173}\mathrm{Yb}$ with three internal states labeled $R$, $G$, $B$) with equal inter-color s-wave contact interactions, subjected to a color-orbit coupling of strength $k_T$ and a color-flip (Rabi) field of frequency $\Omega$, the low-temperature phase diagram and the robustness of the color-superfluid phases found by Kurkcuoglu and Sá de Melo at the saddle-point level remain to be completed along five directions: the four listed in the source's Section V ("Final Remarks") and the generalization to unequal inter-color interactions and nonzero detuning indicated by the source's explicit restrictions. The common objective is to determine the true equilibrium phase behavior of this model beyond the idealized mean-field, equal-interaction, zero-detuning, homogeneous, two-body-only treatment. Specifically: (1) Low-density three-body sector — compute the Efimov spectrum and three-body parameter of the three distinguishable fermions in the presence of the full momentum-dependent color-orbit and color-flip single-particle Hamiltonian, including the effect of the fields and of medium (Pauli-blocking) corrections at $n\lesssim 10^{12}\,\mathrm{cm}^{-3}$, and construct the resulting low-density trimer-versus-color-superfluid phase diagram in the plane of $k_F/\kappa_*^{(0)}$ versus $1/(k_F a_s)$ (with $\kappa_*^{(0)}$ the three-body parameter), generalizing the field-free conjecture of Nishida. (2) Nonuniform phases — determine whether LOFF-type finite-momentum pairing is stable in the BCS corner ($\Omega/E_F\lesssim 0.2$, $1/(k_F a_s)\lesssim -1$) at $T/T_F\ll 10^{-3}$, and delimit its extent, allowing all six order-parameter components to acquire finite-momentum structure. (3) Critical temperature and fluctuations — calculate the fluctuation-corrected $T_c$ across the BCS-BEC crossover via Gaussian fluctuations or a $T$-matrix (or comparable) approach for the six-component momentum-dependent order-parameter tensor, and establish how the color-orbit and color-flip fields modify the transition, including its order (continuous versus discontinuous). (4) Trap effects — within the local density approximation for a harmonic trap, map the spatial coexistence of the normal phase, all five nodal color-superfluid phases, and the fully gapped phase (for example at unitarity) as a function of trap parameters for fixed $k_T$ and $\Omega$. (5) Broken symmetry of couplings — extend the phase diagram to non-identical inter-color interactions $g_{RG}\neq g_{RB}\neq g_{GB}$ and to nonzero detuning $\delta\neq0$ (where parity is no longer conserved), establishing which of the nodal topological phases survive and what new phases appear. Each of the five directions is an open problem in its own right, and the five are independent: a complete, self-contained treatment of any one of them resolves the task, and none is a prerequisite for another. Results for the direction(s) addressed should use the single-particle Hamiltonian and conventions of the source paper (or justify modifications thereof) and reduce to known results in the limits $k_T,\Omega,\delta\to 0$ and $g_{RG}=g_{RB}=g_{GB}$.

References

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