I consider nonrelativistic bosons interacting via pairwise potentials with infinite scattering length and supporting no two-body bound states. To lowest order in effective field theory, these conditions lead to noninteracting bosons, since the coupling constant of the Lieb-Liniger model vanishes identically in this limit. Since any realistic pairwise interaction is not a mere delta function, the noninteracting picture is an idealization indicating that the effect of interactions is weaker than in the case of off-resonant potentials. I show that the leading-order correction to the ground-state energy for more than two bosons is accurately described by the lowest-order three-body force in effective field theory that arises due to the off-shell structure of the two-body interaction. For natural two-body interactions with a short-distance repulsive core and an attractive tail, the emergent three-body interaction is repulsive and, therefore, three bosons do not form any bound states. This situation is analogous to the two-dimensional repulsive Bose gas, when treated using the lowest-order contact interaction, where the scattering amplitude exhibits an unphysical Landau pole. The avoidance of this state in the three-boson problem proceeds in a way that parallels the two-dimensional case. These results pave the way for the experimental realization of one-dimensional Bose gases with pure three-body interactions using ultracold atoms.
I consider nonrelativistic bosons interacting via pairwise potentials with infinite scattering length and supporting no two-body bound states. To lowest order in effective field theory, these conditions lead to noninteracting bosons, since the coupling constant of the Lieb-Liniger model vanishes identically in this limit. Since any realistic pairwise interaction is not a mere delta function, the noninteracting picture is an idealization indicating that the effect of interactions is weaker than in the case of off-resonant potentials. I show that the leading-order correction to the ground-state energy for more than two bosons is accurately described by the lowest-order three-body force in effective field theory that arises due to the off-shell structure of the two-body interaction. For natural two-body interactions with a short-distance repulsive core and an attractive tail, the emergent three-body interaction is repulsive and, therefore, three bosons do not form any bound states. This situation is analogous to the two-dimensional repulsive Bose gas, when treated using the lowest-order contact interaction, where the scattering amplitude exhibits an unphysical Landau pole. The avoidance of this state in the three-boson problem proceeds in a way that parallels the two-dimensional case. These results pave the way for the experimental realization of one-dimensional Bose gases with pure three-body interactions using ultracold atoms.
We study the wave function φ(3) of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length a. We derive asymptotic expansions of φ(3) in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as ln3(B/a) at large hyperradius B=(s12+s22+s32)/2. At order B−2ln−3(B/a), we identify a three-body parameter D with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, D acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of D in terms of φ(3) and the changes of the two-body and three-body potentials. We also study the effects of D and φ(3) on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.
We solve the three-boson problem with contact two-and three-body interactions in one dimension and analytically calculate the ground and excited trimer-state energies.Then, by using the diffusion Monte Carlo technique we calculate the binding energy of three dimers formed in a one-dimensional Bose-Bose or Fermi-Bose mixture with attractive interspecies and repulsive intraspecies interactions.Combining these results with our three-body analytics, we extract the three-dimer scattering length close to the dimer-dimer zero crossing.In both considered cases the three-dimer interaction turns out to be repulsive.Our results constitute a concrete proposal for obtaining a one-dimensional gas with a pure three-body repulsion.
A one-dimensional system of a finite number of bosons with a δ-like interaction potential is treated. A three-body boson repulsion is taken into account, along with the pairwise attraction. The bound state of N bosons is found in the self-consistent-field approximation. In such a system there is no collapse of the Bose gas, and the ground state in the thermodynamic limit is the bound state of N particles with a practically constant density. Such a state can be considered as a condensate droplet in the coordinate space.
It is shown that the effective interaction strength of three bosons at small collision energies can be extracted from their wave function at zero energy. Asymptotic expansions of this wave function at large interparticle distances are derived, from which is defined a quantity D named three-body scattering hypervolume, which is an analog of the two-body scattering length. Given any finite-range interactions, one can thus predict the effective three-body force from a numerical solution of the Schr\"odinger equation. In this way, the constant D for hard-sphere bosons is computed, leading to the first complete result for the ground-state energy per particle of a dilute Bose-Einstein condensate (BEC) of hard spheres to order ${\ensuremath{\rho}}^{2}$, where $\ensuremath{\rho}$ is the number density. Effects of D are also demonstrated in the three-body energy in a finite box of size L, which is expanded to the order ${L}^{\ensuremath{-}7}$, and in the three-body scattering amplitude in vacuum. The three-body scattering amplitude calculated in this paper disagrees with an earlier calculation in the literature, because of the omission of the two-body effective range in that earlier work. Another key prediction is the condensate fractions of dilute BECs, which also disagree with an earlier work in the literature based on the effective field theory (EFT), as a result of short-range physics. An EFT prediction of the BEC ground-state energy, however, is corroborated.
Abstract In 1963, Lieb and Liniger solved exactly a one-dimensional model of bosons interacting by a repulsive δ -potential and calculated the ground state in the thermodynamic limit. In the present work, we extend this model to a potential of three δ -functions—one of them is repulsive and the other two are attractive—modeling some aspects of the interaction between atoms, and we present an approximate solution for a dilute gas. In this limit, for low energy states, the results are found to be reduced to the ones of an effective Lieb–Liniger model with an effective δ -function of strength c eff and the regime of stability is identified. This may shed light on some aspects of interacting bosons.
We show that the Lieb-Liniger model for one-dimensional bosons with repulsive δ-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length a and the radius r of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant g∼a/r2 is derived. Our bounds are uniform in g in the whole parameter range 0≤g≤∞, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size ∼r−2 above the ground state energy.