SciPost Submission Page

Beyond Lee-Huang-Yang description of self-bound Bose mixtures

by Miki Ota, Grigori E. Astrakharchik

Submission summary

As Contributors: Miki Ota
Arxiv Link: (pdf)
Date submitted: 2020-05-21
Submitted by: Ota, Miki
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Atomic, Molecular and Optical Physics - Theory
Approach: Theoretical


We investigate the properties of self-bound ultradilute Bose-Bose mixtures, beyond the Lee-Huang-Yang description. Our approach is based on the determination of the beyond mean-field corrections to the phonon modes of the mixture in a self-consistent way and calculation of the associated equation of state. The newly obtained ground state energies show excellent agreement with recent quantum Monte Carlo calculations, providing a simple and accurate description of the self-bound mixtures with contact type interaction. We further show numerical results for the equilibrium properties of the finite size droplet, by adjusting the Gross-Pitaevskii equation. Finally, we extend our analysis to the one-dimensional mixtures where an excellent agreement with quantum Monte Carlo predictions is found for the equilibrium densities.

Current status:
Editor-in-charge assigned

Reports on this Submission

Anonymous Report 2 on 2020-6-20 Invited Report


1 - A much needed analysis finally addressing a "loophole" in the original model for self-bound quantum droplets arising in binary Bose mixtures.

2 - The key idea is very transparent and relies upon a very simple argument, avoiding unnecessary complications.

3 - The comparison between the analytical predictions and QMC numerical outcomes reinforces the strength of the analysis and the assumptions on which it is built on.


- Despite repeatedly mentioning the availability of experimental data, the paper somewhat lacks a satisfactory comparison between theory and experiments.

- Connected to the point above, since the approach is any case perturbative, it is not clear if the detachments from Petrov's original proposal predicted here may actually be revealed.


The manuscript by Ota and Astrakharchik is focused on the solution of a known issue in Petrov's seminal proposal about quantum mechanical stabilization of binary Bose mixtures. Indeed, it is a known fact that one branch of the collective excitation spectrum (the density one) becomes purely imaginary beyond the mean-field instability threshold.

By computing beyond-Gaussian corrections to the Bogoliubov sound, the authors manage to extract analytical meaningful result, hopefully better fitting the increasing amount of data coming from current experimental setups.

In my opinion, this investigation for sure deserves the publication in SciPost, but I'd like the authors to address some minor points I am listing below:

1) The authors notice that both the density and spin compressibility in Eqs. (8)-(9) display a non-zero imaginary part. They then mention the fact that this is an expected outcome for a perturbative approach and that it can be safely neglected within the current experimental regimes. I'd like the authors to expand a little bit on why it is a safe procedure, since the reader may have the impression that the pitfall of Petrov's theory (the appearance of imaginary quantities signalling some sort of instability) is simply shifted towards the compressibility.

2) At page 6, just below the caption of Fig. 2, the authors refer to the spinodal point pointing out the instability of the liquid phase towards the nucleation of multiple droplets. Has this process ever been observed? It seems to me a very relevant experimental point and a comment should be in order.

3) In the caption of Fig. 2 the authors mention five different lines, but I do not see the black dotted one, neither in the figure or in the legend.

4) Again about Fig. 2 and its discussion in page 7, I certainly agree with the claim that the main correction to the energy density comes from the spin channel, but I do not understand what is the strong suppression of energy one should see comparing the green and the blue line. I think the authors should clarify this point.

5) Concerning Eq. (13), it seems important to mention that such an effective description with only a single field $\Phi$ actually neglects the eventual internal dynamics between the mixture components.

6) In page 10-11, while discussing the results of their GPE calculations based on the effective energy functional in Eq. (10), the author often mention relevant experimental investigations about the self-bound droplets such as Refs. (8-10), but I do not see a clear comparison between their theory and such investigations. Is it because the experimental setups are still lacking the sensitivity to probe this detachment from Petrov's theory? If not, I think this work would greatly benefit from a more extended discussion (maybe a figure, if data are available) about this point.

Related to the point above, at the end of Section 4 (page 11), the authors writes about "trap confined measurements" but do not cite any reference.

7) In the conclusion, the authors mention the possible extension of their theory, I would also add the possibility to deal with a Rabi-coupled mixtures and also a spin-orbit coupling. In both cases, theoretical investigations base on Petrov's theory are already available.

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2020-6-19 Invited Report


The manuscript by M. Ota and G. Astrakharchik analyzes the properties of liquid droplets in 1D quantum Bose-Bose mixtures. The authors sensibly extend the available formalism by including corrections to the "standard" LHY treatment. These corrections are shown to have a number of very beneficial effects, and yield energies and equilibrium densities which are in excellent agreement with available QMC calculations. Moreover, the authors compute improved density profiles, droplet sizes and critical numbers of atoms for droplet formation which display a better agreement with existing experiments.

The manuscript is excellently written, the formalism is clearly outlined, and the results are thoroughly discussed. Moreover, this physics is certainly for ongoing experiments. As such, I can certainly recommend publication of this manuscript in SciPost.

Here below I collected a series of minor typos, remarks and comments (listed in chronological order) which the Authors may want to address upon resubmission.

Sec. 1
# "The liquid droplets ... since THEY ARISE ..."

Sec. 2
# appropriated --> appropriate

# The authors mention here that corrections to the LHY theory may be obtained in two ways, following Beliaev theory, or through thermodynamics relations. Are the two results expected to match? or may one find different answers?
I see that this point is partially touched later, towards the end of Sec. III, but the authors may want to comment something here already.
By the way, there may be a typo in Sec. III, because the authors write "a systematic calculation of the THIRD-order terms of the perturbative theory", but earlier Beliaev theory is introduced as being a SECOND order one. Please check?

# atoms density --> atom density

# disbalance --> imbalance

# I couldn't find an explicit definition of the scattering length "a". It is obviously defined through "g", but the authors may want to add it, for completeness.

Sec. 3
# Fig. 2: a black-dotted line is mentioned in the caption, and also in the main text, but I couldn't find it in the Figure.

# Fig. 1 shows that the beyond-LHY correction to the spin mode is tiny in the liquid region. On the other hand, when discussing Fig. 2 the authors write that this has a strong effect on the energy. Is there a simple explanation for this? Or am I missing something? Please clarify.
Moreover, for easier reference in the caption of this figure and of Fig.7, I suggest to include the "equal density version" of Eq.(4) as a stand-alone equation right before Eq.(7).

# Fig.3: the discrepancy between the improved theory and QMC seems to be maximal at $\delta g/g=-0.1$ (green), while it is much smaller at the next value shown, -0.05 (orange). Is there a simple reason for this?

# caption of Fig.3 (And also text): "color dots" --> "colored dots"

# "hints that that" --> "suggests that"

Sec. 4
# the authors write "the main contributions to the LHY terms in Eq. (3) arise from “short” distances $\propto 1/c_+$". However, $c_+ \rightarrow 0$ in the region of interest. Could the authors clarify?

# Eq.(14): is a $\Phi$ missing at the end of the equation? else, what is the Laplacian operator acting upon?

# Eq.(16): if $\Phi$ is a 3D wavefunction, it should have dimensions of length^{-3/2}, but it seems to have length^{-3/4}; check the prefactor?

# Fig.6: I'm confused: can one relate the values for $\tilde{N}$ given in the text with the data shown in this figure? For which value of $n_0\xi^3$ is the Figure obtained? (or is this not relevant?) Please clarify.

# do the corrections to the speed of sound yield a "real part"? or rather a "real result"?

# could the results found here be used to evaluate finite-temperature thermodynamics of such binary mixtures?

  • validity: top
  • significance: high
  • originality: top
  • clarity: top
  • formatting: perfect
  • grammar: excellent

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