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arXiv:cond-mat/0405206 · Physical Review Letters

Universal Properties of Two-Dimensional Boson Droplets

H.‐W. Hammer, Dam Thanh Son

Published Dec 16, 2004• 81 citations
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Abstract

We consider a system of N nonrelativistic bosons in two dimensions, interacting weakly via a short-range attractive potential. We show that for N large, but below some critical value, the properties of the N-boson bound state are universal. In particular, the ratio of the binding energies of (N+1)- and N-boson systems, B(N+1)/B(N), approaches a finite limit, approximately 8.567, at large N. We also confirm previous results that the three-body system has exactly two bound states. We find for the ground state B(0)(3)=16.522 688(1)B2 and for the excited state B(1)(3)=1.270 409 1(1)B2.

DOI
10.1103/physrevlett.93.250408

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Using lattice effective field theory, we study the ground state binding energy of NNN distinct particles in two dimensions with equal mass interacting weakly via an attractive SU(N)\mathrm{SU}(N)SU(N)-symmetric short range potential. We find that in the limit of zero range and large NNN, the ratio of binding energies ${B}_{N}∕{B}_{N\ensuremath{-}1}$ approaches the value 8.3(6).
When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions, predict the existence of two three-body bound states when both interactions are attractive, and determine their binding energies as universal functions of the two-body and three-body scattering lengths. We also show that an infinitesimal three-body attraction induces an excited bound state only for 3, 39, or more bosons. Our findings herein have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.
When quantum particles are confined into lower dimensions, an effective three-body interaction inevitably arises and may cause significant consequences. Here we study bosons in one dimension with weak two-body and three-body interactions and predict the existence of two three-body bound states when both interactions are attractive. Their binding energies are universal functions of the two-body and three-body scattering lengths and have direct relevance to a broad range of quasi-one-dimensional systems realized with ultracold atoms.
Ultracold atoms offer valuable opportunities where interparticle interactions can be controlled at will. In particular, by extinguishing the two-body interaction, one can realize unique systems governed by the three-body interaction, which is otherwise hidden behind the two-body interaction. Here we study one-dimensional bosons with a weak three-body attraction and show that they form few-body bound states as well as a many-body droplet stabilized by the quantum mechanical effect. Their binding energies relative to that of three bosons are all universal and the ground-state energy of the dilute droplet is found to grow exponentially as ${E}_{N}/{E}_{3}\ensuremath{\rightarrow}exp(8{N}^{2}/\sqrt{3}\ensuremath{\pi})$ with increasing particle number $N\ensuremath{\gg}1$. The realization of our system with coupled two-component bosons in an optical lattice is also discussed.
We investigate a system of bosons in a two-dimensional harmonic trap. In the limit of strong attractive interactions, the bosons make a droplet insensitive to external confinement. For weak interactions, in contrast, the ground state is given by the harmonic trap. In this work, we conduct a variational study of the transition between these two limits. We find that this transition occurs at the critical interaction strength whose value is universal if scaled appropriately with the number of particles. To connect the change in the properties of the system to the classical description of phase transitions, we analyze the static response of the Bose gas related to the isothermal compressibility. Finally, we perform numerically exact calculations for a few particles to demonstrate the effects of finite-range interactions on this transition. We conclude that finite-range effects wash out the point of transition.