Universal low-energy properties of three two-dimensional bosons

Abstract

Universal low-energy properties are studied for three identical bosons confined in two dimensions. The short-range pairwise interaction in the low-energy limit is described by means of the boundary condition model. The wave function is expanded in a set of eigenfunctions on the hypersphere, and the system of hyperradial equations is used to obtain analytical and numerical results. Within the framework of this method, exact analytical expressions are derived for the eigenpotentials and the coupling terms of hyperradial equations. The derivation of the coupling terms is generally applicable to a variety of three-body problems provided the interaction is described by the boundary condition model. The asymptotic form of the total wave function at a small and a large hyperradius is studied, and the universal logarithmic dependence ϳln 3 in the vicinity of the triple-collision point is derived. Precise three-body binding energies and the ͑2+1͒-scattering length are calculated.

Three-boson bound states in two dimensions

“…53 All these approaches involve several numerical integrations over unbounded spaces or kernel inversion, some of them are limited to s-wave resonant scattering. The hyperspherical method has been used for fewbody problems in two dimensions, [36][37][38][39][40][41] with the usual approach of representing states as a sum of many hyperspherical harmonics. Our paper therefore provides an alternative approach to the quantum three-body problems, simple and efficient, involving only direct root finding and evolving of a first-order ordinary differential equation to an intermediate length scale (for example, the divergence behavior showing the existence of bound state is already clear at a relatively small length scale r ∼ 20 in Fig.…”

“…Usually, the angular part of four dimensional vector is represented in terms of hyperspherical coordinates in the literature, [35][36][37][38][39][40][41][42][43] but the resulting algorithms have slow convergence and the number of states scales as the square of the number of levels included. Here we adopt the Hopf coordinates, which gives faster convergence and number of states proportional to the number of levels included (see Appendix A ):…”

Bound states and scattering lengths of three two-component particles with zero-range interactions under one-dimensional confinement

“…Experiments with ultra-cold gases in the one-dimensional (1D) and quasi-1D traps have been recently performed [1,11,12,13], amid the rapidly growing interest to the investigation of mixtures of ultra-cold gases [14,15,16,17,18,19,20]. Different aspects of the three-body dynamics in 1D have been analyzed in a number of recent papers, e. g., the bound-state spectrum of two-component compound in [21], low-energy three-body recombination in [22], application of the integral equations in [23], and variants of the hyperradial expansion in [24,25,26].…”

“…In the low-energy limit, the contact potential is a good approximation for any short-range interaction and its usage provides a universal, i. e., independent of the potential form, description of the dynamics [23,26,34,35,36,37]. More specifically, it is assumed that one particle interacts with the other two via an attractive contact interac-tion of strength λ < 0 while the sign of the interaction strength λ 1 for the identical particles is arbitrary.…”

Quantum correlations and degeneracy of identical bosons in a two-dimensional harmonic trap

“…[20]. Other 2D works include two and three-body exact solutions for fermions and bosons with contact interaction [21], fast-converging numerical methods for computing the energy spectrum for a few bosons [22], the study of finite-range effects [23,24] and universality [25,26], condensation in trapped fewboson systems [27], and interacting few-fermions systems [28,29].…”