Three-boson problem at low energy and Implications for dilute Bose-Einstein condensates

TL;DR

This paper introduces the three-body scattering hypervolume D, enabling prediction of effective three-body interactions in dilute Bose-Einstein condensates from wave functions, and explores its effects on ground state energy and scattering properties.

Contribution

It defines the three-body scattering hypervolume D and demonstrates its use in predicting three-body forces and energies in dilute BECs, revealing limitations of effective field theory.

Findings

Computed D for hard-sphere bosons.

Derived ground state energy to order ρ^2.

Identified EFT violations in condensate fraction.

Abstract

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. An asymptotic expansion of this wave function at large interparticle distances is 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 interaction potentials, one can thus predict the effective three-body force from a numerical solution of the Schr\"{o}dinger equation. In this way the constant $D$ for hard-sphere bosons is computed, leading to the complete result for the ground state energy per particle of a dilute Bose-Einstein condensate (BEC) of hard spheres to order $\rho^2$, where $\rho$ 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 $L^{-7}$, and in the three-body scattering amplitude in vacuum. Another key prediction is that there is a violation of the effective field theory (EFT) in the condensate fraction in dilute BECs, caused by short-range physics. EFT predictions for the ground state energy and few-body scattering amplitudes, however, are corroborated.