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 Schr\"odinger 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 three-boson 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 spin-orbit 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.
Highlights
Under which conditions do two, three, or more particles form weakly bound states, i.e., bound states that are larger than the range of the two, three, and higher-body forces that bind the particles together? And under which conditions are the characteristics of these few-body bound states governed by underlying symmetries? These questions are of utmost importance across physics
The triton has played an important role in the context of the Thomas collapse [5] and the Efimov effect [6,7], which is intimately tied to a discrete scaling symmetry of the three-body Schrödinger equation
The key objective of the present work is to show that the three-boson system in the presence of 1D spin-orbit coupling obeys a generalized radial scaling law, which reflects the existence of a discrete scaling symmetry in the limit of zero-range interactions
Summary
Under which conditions do two, three, or more particles form weakly bound states, i.e., bound states that are larger than the range of the two-, three-, and higher-body forces that bind the particles together? And under which conditions are the characteristics of these few-body bound states governed by underlying symmetries? These questions are of utmost importance across physics. If two-body short-range interactions are added, the modified single-particle dispersion curves can significantly alter the properties of weakly bound two- and three-body states This has been demonstrated extensively for two identical fermions for 1D, 2D, and 3D spin-orbit coupling [32,33,34,35,36,37,38,39,40,41] and for two identical bosons for 2D and 3D spinorbit coupling [41,42,43,44,45] but not for the 1D spin-orbit coupling considered in this work. The key objective of the present work is to show that the three-boson system in the presence of 1D spin-orbit coupling obeys a generalized radial scaling law, which reflects the existence of a discrete scaling symmetry in the limit of zero-range interactions.
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