Three-Boson Spectrum in the Presence of 1D Spin-Orbit Coupling: Efimov’s Generalized Radial Scaling Law

Abstract

Spin-orbit coupled cold atom systems, governed by Hamiltonians that contain quadratic kinetic energy terms typical for a particle's motion in the usual Schrödinger equation and linear kinetic energy terms typical for a particle's motion in the usual Dirac equation, have attracted a great deal of attention recently since they provide an alternative route for realizing fractional quantum Hall physics, topological insulators, and spintronics physics. The present work focuses on the threeboson system in the presence of 1D spin-orbit coupling, which is most relevant to ongoing cold atom experiments. In the absence of spin-orbit coupling terms, the three-boson system exibits the Efimov effect: the entire energy spectrum is uniquely determined by the s-wave scattering length and a single three-body parameter, i.e., using one of the energy levels as input, the other energy levels can be obtained via Efimov's radial scaling law, which is intimately tied to a discrete scaling symmetry. It is demonstrated that the discrete scaling symmetry persists in the presence of 1D spinorbit coupling, implying the validity of a generalized radial scaling law in five-dimensional space. The dependence of the energy levels on the scattering length, spin-orbit coupling parameters, and center-of-mass momentum is discussed. It is conjectured that three-body systems with other types of spin-orbit coupling terms are also governed by generalized radial scaling laws, provided the system exhibits the Efimov effect in the absence of spin-orbit coupling.PACS numbers:

Energetics and structural properties of two- and three-boson systems in the presence of one-dimensional spin-orbit coupling

“…The term "universal three-boson state" in this context refers to a state described nearly perfectly by Efimov's zero-range theory [5,23]. For the parameters considered in this work, the most strongly bound threeboson state is the lowest universal three-boson state [41].…”

“…For example, theδ = 0 system with infinitesimally small Ω supports a bound state for (a s k so ) −1 ≥ −0.504 while that with Ω = 2E so supports a bound state for (a s k so ) −1 ≥ −1.304. Applying the generalized radial scaling law [41], these energy plots also describe higher-lying Efimov manifolds. Figure 5(a) shows that the binding energy for Ω = 2E so is enhanced the most forδ ≈ 0.…”

“…The transformation from the single-particle position vectors #» r j to the Jacobi vectors #» ρ j (to be used below) and from the single spin operators #» σ j to the Jacobi spin operators #» Σ j proceeds analogously (Appendix A of Ref. [41] provides explicit expressions for Σ 1,z and Σ 2,z ).…”

“…The binding energy is directly related to the dissociation properties of few-body systems. For example, weakly-bound states tell one the critical scattering lengths at which enhanced three-body losses are expected to occur in the presence of 1D spin-orbit coupling [41]. Once the critical generalized detuning δcr at which the binding energy E (n) N,binding ( δ) is the largest has been determined, the z-component of the generalized total momentum of the corresponding state is uniquely determined via Eq.…”

“…In a recent work [41], it was demonstrated that Efimov's discrete scaling law persists in the presence of 1D spin-orbit coupling in an enlarged parameter space that includes not only the two-body s-wave scattering length, but also the parameters that characterize the 1D spinorbit coupling terms. The discrete scaling law tells us that once we know the shape of four energy surfaces in a five-dimensional parameter space, we can predict all other energy surfaces in the five-dimensional parameter space.…”