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arXiv:physics/0512085 · Physical Review A

Large- N droplets in two dimensions

Dean Lee

Published Jun 19, 2006• 30 citations
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Abstract

Using lattice effective field theory, we study the ground state binding energy of $N$ distinct particles in two dimensions with equal mass interacting weakly via an attractive $\mathrm{SU}(N)$-symmetric short range potential. We find that in the limit of zero range and large $N$, the ratio of binding energies ${B}_{N}∕{B}_{N\ensuremath{-}1}$ approaches the value 8.3(6).

DOI
10.1103/physreva.73.063204

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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 BN∕BN\ensuremath−1{B}_{N}∕{B}_{N\ensuremath{-}1}BN​∕BN\ensuremath−1​ approaches the value 8.3(6).
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.
We consider the nonrelativistic four-boson system in two dimensions interacting via a short-range attractive potential. For a weakly attractive potential with one shallow two-body bound state with binding energy B-2, the binding energies B-N of shallow N-body bound states are universal and thus do not depend on the details of the interaction potential. We compute the four-body binding energies in an effective quantum mechanics approach. There are exactly two bound states: the ground state with B-4((0)) = 197.33(1)B-2 and one excited state with B-4((1)) = 25.5 (1)B-2. We compare our results to recent predictions for N-body bound states with large N much greater than 1.
Close upper and lowei bounds are found on the binding energy of a system of N identical bosons interacting in one dimension by an attractive square-well pair potential. In the limit in which the potential has the shape of a δ function the bounds are found to be consistent with McGuire's exact solution. In three dimensions the limit of vanishingly small range, in which the two-particle binding energy is kept constant, gives infinite binding to the N -particle system, in agreement with Thomas's theorem Consideration of other limiting Proc.esses eliminates an apparent contradiction between Thomas and Svartholm.
In this article, we illustrate the scaling properties of a family of solutions for NNN attractive bosonic atoms in the limit of large NNN. These solutions represent the quantized dynamics of solitonic degrees of freedom in atomic droplets. In dimensions lower than two, or $d=2\ensuremath{-}\ensuremath{\epsilon}$, we demonstrate that the number of isotropic droplet states scales as ${N}^{3/2}/{\ensuremath{\epsilon}}^{1/2}$, and for $\ensuremath{\epsilon}=0$, or d=2d=2d=2, scales as N2{N}^{2}N2. The ground-state energies scale as ${N}^{2/\ensuremath{\epsilon}+1}$ in $d=2\ensuremath{-}\ensuremath{\epsilon}$, and when d=2d=2d=2, scale as an exponential function of NNN. We obtain the universal energy spectra and the generalized Tjon relation; their scaling properties are uniquely determined by the asymptotic freedom of quantum bosonic fields at short distances, a distinct feature in low dimensions. We also investigate the effect of quantum loop corrections that arise from various virtual processes and show that the resultant lifetime for a wide range of excited states scales as ${N}^{\ensuremath{\epsilon}/2}{E}^{1\ensuremath{-}\ensuremath{\epsilon}/2}$.
We consider bosons interacting through a narrow sss-wave resonance. Such a resonance is characterized by an infinite scattering length and a large and negative effective range r0r_0r0​. We argue that any number N≥3N\ge3N≥3 of bosons can form a self-bound cluster with the binding energy per particle increasing as N2N^2N2 for 1≪N≪(−r0/abg)1/21\ll N\ll (-r_0/a_\text{bg})^{1/2}1≪N≪(−r0​/abg​)1/2, where abga_\text{bg}abg​ is the background scattering length (between atoms and molecules). In the opposite limit N≫(−r0/abg)1/2N\gg (-r_0/a_\text{bg})^{1/2}N≫(−r0​/abg​)1/2, bosons form droplets with binding energy per particle saturating to a constant value independent of the particle number. The stability of clusters and droplets when the interaction is detuned from the resonance is also studied.
Ultradilute quantum droplets are intriguing new state of matter, in which the attractive mean-field force can be balanced by the repulsive force from quantum fluctuations to avoid collapse. Here, we present a microscopic theory of ultradilute quantum droplets in low-dimensional two-component Bose-Bose mixtures, by generalizing the conventional Bogoliubov theory to include the bosonic pairing arising from the inter-species attraction. Our pairing theory is fully equivalent to a variational approach and hence gives an upper bound for the energy of quantum droplets. In one dimension, we find that the energy calculated by the pairing theory is in an excellent agreement with the latest diffusion Monte Carlo simulation {[}Phys. Rev. Lett. \textbf{122}, 105302 (2019){]}, for nearly all the interaction strengths at which quantum droplets exist. In two dimensions, we show that quantum droplets disappear and may turn into a soliton-like many-body bound state, when the inter-species attraction exceeds a critical value. Below the threshold, the pairing theory predicts more or less the same results as the Bogoliubov theory derived by Petrov and Astrakharchik {[}Phys. Rev. Lett. \textbf{117}, 100401 (2016){]}. The predicted energies from both theories are higher than the diffusion Monte Carlo results, due to the weak inter-species attraction and the increasingly important role played by the beyond-Bogoliubov-approximation effect in two dimensions. Our pairing theory provides an ideal starting point to understand interesting ground-state properties of quantum droplets, including their shape and collective oscillations.