Universal Borromean Binding in Spin-Orbit Coupled Ultracold Fermi Gases

Borromean rings and Borromean binding, a class of intriguing phenomena as three objects are linked (bound) together while any two of them are unlinked (unbound), widely exist in nature and have been found in systems of biology, chemistry and physics. Previous studies have suggested that the occurrence of such a binding in physical systems typically relies on the microscopic details of pairwise interaction potentials at short-range, and is therefore non-universal. Here, we report a new type of Borromean binding in ultracold Fermi gases with Rashba spin-orbit coupling, which is {\it universal} against short-range interaction details, with its binding energy only dependent on the s-wave scattering length and the spin-orbit coupling strength. We show that the occurrence of this universal Borromean binding is facilitated by the symmetry of the single-particle dispersion under spin-orbit coupling, and is therefore {\it symmetry-selective} rather than interaction-selective. The state is robust over a wide range of mass ratio between composing fermions, which are accessible by Li-Li, K-K and K-Li mixtures in cold atoms experiments. Our results reveal the importance of symmetry factor in few-body physics, and shed light on the emergence of new quantum phases in a many-body system with exotic few-body correlations.

The fascinating topological structure of Borromean rings has attracted much attention in biology [1] and chemistry [2]; while in physics, their quantum mechanical analog, the Borromean binding, has been reported in halo nuclei 6 He and 11 Li [3,4] and in ultracold atomic gases [5][6][7][8][9][10][11][12][13][14] manifested as the Efimov effect [15,16]. Despite its wide existence in nature, the Borromean phenomenon seems quite intricate and peculiar, as it especially requires three bodies being more favorably bound than two bodies. Previous studies have shown that such a requirement can be fulfilled by fine-tuning the pairwise short-range interaction potentials. For instance, in three dimensions (3D), the coupling constant should vary with the specific shape of the short-range potential [17,18], while in two dimensions (2D), it is necessary for the potential to include a repulsive barrier outside an attractive core [19,20]. Meanwhile, for Efimov-type Borromean states, a short-range (three-body) parameter is essential to uniquely determine the binding energies as well as the locations of their emergence [16]. In all these studies, the Borromean binding appears to be a non-universal phenomenon, which inevitably relies on the short-range details of interaction potentials. This non-universality makes a unified understanding of the Borromean binding conceptually difficult, and renders its experimental detection inconveniently system-dependent.
To overcome these difficulties, we aim at engineering a universal Borromean binding, where the shortrange interaction details are completely irrelevant and its occurrence is physically transparent. Motivated by a simple fact that few-body physics also crucially depend on single-particle properties, we realize that a po-tential route toward our goal is through the modification of single-particle physics. In ultracold atomic gases, an outstanding candidate to achieve this is the synthetic spin-orbit coupling (SOC) [21][22][23][24][25][26][27][28][29][30][31], with the form of SOC highly tunable according to a number of proposals [32][33][34][35][36][37][38][39]. Indeed, the significant change of single-particle dispersion by SOC has been shown to result in rich and exciting physics in few-and many-body systems [40]. In particular, it has been found that an isotropic SOC can support dimer for arbitrarily weak interactions [38,[41][42][43], and can induce universal trimer in a wide parameter regime of interaction strength and mass ratio [44]. These are in distinct contrast to the dimer and the Kartavtsev-Malykh trimer [45] in the absence of SOC. So far, however, no universal Borromean binding has yet been identified.
In this work we report the discovery of universal Borromean bindings in ultracold Fermi-Fermi mixtures with Rashba SOC. The three-body system can be denoted as a −ã − b, whereã is a two-component fermion subject to Rashba SOC, with one of its components tuned close to a wide Feshbach resonance with the b atom [46]. The mechanism for the Borromean binding in this system is schematically shown in Fig. 1. Under Rashba SOC, the single-particle ground state ofã possesses a U(1) degeneracy (see Fig. 1(a)). With such a spectral symmetry, the two-body (ã − b) scattering within the lowest energy subspace is blocked due to total momentum conservation ( Fig. 1(b)), which effectively suppresses the dimer formation. In contrast, the three-body scattering can take full advantage of this U(1) degeneracy, where an initial state ofã −ã − b atoms at {k, −k, 0} can be scattered to a different state at {k , −k , 0} with a conserved total momentum ( Fig. 1(c)). Here, k and k both lie on the circle of the U(1) degenerate manifold ofã. This enhanced low-energy scattering phase space strongly suggests the trimer formation be much easier than the dimer formation, which, as we will show, would give rise to the Borromean binding. As the emergence of this Borromean binding is symmetry-selective rather than interactionselective, its universality is naturally guaranteed: the binding energy only relies on the s-wave scattering length and the SOC strength. We identify the existence of such bindings in a wide range of mass ratio between composing fermions, which are readily accessible by Li-Li, K-K and K-Li mixtures in current cold atoms experiments. The robustness of this Borromean binding suggests the importance of the symmetry factor in few-body physics, which has rarely been discussed before.
Model. The Hamiltonian of our system is written as: where λ is the strength of Rashba SOC between two spin species (α =↑, ↓) ofã-atom; U is the bare interaction between a ↑ and b, and is related to the s-wave scattering length a s via 1/U = µ/(2πa s )−(1/V ) k 1/(2µk 2 ), with V the quantization volume and µ = m a m b /(m a + m b ) the reduced mass. As Feshbach resonances are statedependent and have a finite width, it is reasonable to assume negligible interactions in other two-body subsystems [46]. Note we have taken = 1 for brevity. Under SOC, the single-particle eigen-state ofã in the helicity basis is created by The ground state has U (1) degeneracy in k-space with k ⊥ = λ and a threshold energy E th = −λ 2 /(2m a ). Given the single-particle spectrum b k = k 2 /(2m b ) for b-atom, the two-bodyã − b and the three-bodyã −ã − b systems respectively have threshold energies E th and 2E th .
Dimer state. We start by addressing the dimer state of theã − b system. The dimer wave function with a center-of-mass momentum Q can be written as: The coefficient Ψ (2) can be solved in a standard way based on the Lippman-Schwinger equation [47]: where E 2 is the two-body binding energy determined by Among all Q sectors, the lowest bound state (E 2 < 0) is found with Q = 0. Different from previous two-body solutions with Rashba SOC [41], to support a bound state here, a finite critical interaction strength, 1/(λa s ) c , is required, which can be solved analytically as a function of mass ratio η = m a /m b : The function of 1/(λa s ) c in terms of η is plotted in Fig. 2(a). As η is increased from zero, 1/(λa s ) c first increases from −∞ to a positive maximum value around η ∼ 1, then decreases and finally approaches 0 + as η → ∞. This behavior can be understood from the analysis of the two-body scattering energy E (2) k,σ = b −k + a k,σ − E th , whose low-energy property is crucial for the formation of a shallow bound state. It is easy to see that the minimum of E (2) k,σ , denoted as E min , lies on a ring with radius k ⊥ = λ/(1 + η) in the (k x , k y ) plane. As η increases from 0 to ∞, the radius evolves from λ to 0, indicating a dimensional crossover from effectively 2D to 3D. This is also manifested in the density of state ρ at E min , which approaches zero from a finite value as η increases (see Fig. 2(a) insets). Consequently, the critical 1/(λa s ) c changes from −∞ to 0, corresponding to a crossover of the two-body bound state threshold from 2D to 3D. An important feature in Fig. 2(a) is that the twobody threshold 1/(a s ) c is pushed from resonance to positive values for a considerable range of mass ratio η ∈ [0.44, ∞), indicating the suppression of dimer formation by Rashba SOC. This is consistent with the schematic picture in Fig. 1(b). For an initialã−b state in the lowest energy subspace (|Q| = λ), it cannot be scattered into a different state among the U(1) degenerate ground states due to the conservation of total momentum. Given the blocked threshold scattering with |Q| = λ, the ground state dimer with E 2 < 0 is found to be at Q = 0, where the U(1) symmetry is restored at the cost of higher threshold energy (E min > 0). In Fig. 2(b1,b2), we plot the momentum distribution of such dimers for two different mass ratios η = 1 and 40/6, corresponding to the cases of Li-Li (or K-K) and K-Li mixtures. For both cases, the largest weight of the wave function lies on a ring with radius k ⊥ < λ and with E min > 0. Borromean binding. We are now in position to examine the three-body problem. According to the analysis in Fig. 1(c), the ground state trimer is expected to have zero center-of-mass momentum, for which the wave function can be written as Following similar procedures as in solving the two-body problem, we obtain the integral equations for the threebody bound state solution [47]: where F σ (k) = U qξ Ψ (3) (−k − q; kσ; qξ)γ ↑ kσ γ ↑ qξ , and the trimer binding energy E 3 can be obtained by requiring non-zero solution of F σ (k). Under Rashba SOC, the F −function can be decoupled into sectors with different magnetic angular momentum: where θ m is an arbitrary phase shift that turns out to be irrelevant to the final solution of E 3 . Note that due to Fermi statistics, the ground state is in the m = 1 sector. Given F σ (k), the wave function Ψ (3) can be obtained accordingly [47]. In Fig. 3, we plot the ground state trimer energy E 3 for the 40 K(ã)-40 K(ã)-6 Li(b) case as a function of interaction strength 1/(λa s ). As expected, when 1/(λa s ) increases, the trimer is found to emerge well before the dimer, which leads to the occurrence of the Borromean binding. For the 40 K-40 K- 6 Li system, the Borromean state is stable within the range of 1/(λa s ) ∈ [0.2, 0.31), while the most tightly bound Borromean occurs at the phase boundary against the ordinary trimer, i.e., when the dimer starts to develop at 1/(λa s ) = 0.31. At this point, the Borromean binding energy can be as large as nearly 30% of the SOC energy λ 2 /(2m a ). The ordinary trimer finally merges into the atom-dimer threshold at a larger 1/(λa s ) = 0.76.
To gain further understanding of the binding mechanism, we plot in Fig. 4(a) the momentum distribution of the Borromean state at 1/(λa s ) = 0.3. In contrast to that of dimers shown in Fig. 2(b1,b2), here the most weight of the probability distribution, |Ψ (3) (0; k, −; −k, −)| 2 , spreads along the U(1) circle in the lowest energy subspace forã atoms. Thus, scattering among these lowenergy states contributes the most to the bound state formation, consistent with the schematics in Fig. 1(c).
An outstanding feature of the Borromean binding in the current system is its universality, i.e. the binding energy does not rely on the short-range interaction details. This can be shown by imposing different high-momentum cutoffs Λ for the argument of F σ -function in Eq.(7), (k c ⊥ , |k z | c ) = ( √ 2Λ, Λ). In Fig. 4(b), we plot E 3 as a function of λ/Λ for the Borromean binding at 1/(λa s ) = 0.3. If the binding is universal, E 3 should be independent of the actual cutoff Λ, and all the points should fall onto a straight line in the (λ/Λ, E 3 /(Λ 2 /(2m a ))) plane. This is exactly the case in Fig. 4(b). The only relevant length scales are then a s and 1/λ. The universality of the Borromean binding here distinguishes itself from those in the previous studies where the short-range (or high-energy) details of the interaction potential play essential roles.  Fig. 2(a)), which also marks the boundary between "B" and "T" for η 0.39.
Furthermore, we find that the Borromean binding in the current system is remarkably robust. As shown in the ground state phase diagram for theã −ã − b system in Fig. 5, the Borromean binding can be stabilized over a wide range of mass ratio with η 0.39, thus covering all Li-Li-Li, K-K-K, and K-K-Li systems. We have checked that the momentum distributions of these Borromean states for different η all exhibit similar structures as shown in Fig. 4(a). Therefore, these Borromean states all share the same binding mechanism, which is closely related to the spectral symmetry in the low-energy manifold due to Rashba SOC (see Fig. 1).
Final remark.
The universal Borromean bindings demonstrated in our work are expected to have dramatic effects on the many-body system. With the Borromean binding energy on the same order of the SOC energy, a dilute gas with strong SOC is anticipated to be comprised of self-bound Borromean clusters, which function as composite fermions. Moreover, as the emergence of such a binding is associated with the three-body scattering resonance, a scattering system near this resonance will exhibit strong three-body correlations which dominate over the two-body ones. These prominent three-body correlations would potentially lead to intriguing collective phenomena in both the attractive and the scattering branches of the underlying fermion system.
Finally, we remark that the mechanism of universal Borromean bindings established in this work can be generalized to a vast class of systems, where the singleparticle symmetry is modified by intrinsic or external potentials. Our work thus paves the way for the study of systems where the symmetry factor, instead of interaction details, plays the dominant role in generating exotic few-body correlations which should lead to new quantum phases in many-body systems.

SUPPLEMENTARY MATERIAL
In this supplementary material we provide some details for solving two-body and three-body problems considered in the main text.

I. Two-body problem
According to the Lippman-Schwinger equation, a two-body bound state satisfies |Ψ (2) is the non-interacting Green's function for two particles. We therefore obtain: By introducing a quantity f Q = U k ,σ Ψ (2) (Q − k ; k σ )γ ↑ k σ , one can get the self-consistent equation (4), as well as the expression of wave function in equation (3).

II. Three-body problem
Given the ansatz wave function (6), the Lippman-Schwinger equation gives rise to: By introducing an auxiliary function F σ (k) = U qξ Ψ (3) (−k − q; kσ; qξ)γ ↑ kσ γ ↑ qξ , we obtain the self-consistent equation (7) for F σ (k), and the expression of wave function: For the Rashba SOC, |γ ↑ kσ | 2 is a constant 1/2 for any k and σ. This leads to the decomposition of F σ (k) in terms of the magnetic angular momentum m (see equation (8)). Physically, this is because the Rashba SOC does not take effect within the b † a † ↑ a † ↑ sector. The decomposition generally applies to other types of SOC as long as the SOC does not include k z σ z term.
For each angular momentum sector m, the integral equation for the trimer binding energy E 3 reads: where G (m) σξ (k ⊥ , k z ; q ⊥ , q z ) =