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Heavy-light $N+1$ clusters of two-dimensional fermions

by Jules Givois, Andrea Tononi, Dmitry S. Petrov

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Submission summary

Authors (as registered SciPost users): Jules Givois · Andrea Tononi
Submission information
Preprint Link: scipost_202403_00030v1  (pdf)
Date submitted: 2024-03-21 13:51
Submitted by: Givois, Jules
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
Approach: Theoretical


We study binding of $N$ identical heavy fermions by a light atom in two dimensions assuming zero-range attractive heavy-light interactions. By using the mean-field theory valid for large $N$ we show that the $N+1$ cluster is bound when the mass ratio exceeds $1.074N^2$. The mean-field theory, being scale invariant in two dimensions, predicts only the shapes of the clusters leaving their sizes and energies undefined. By taking into account beyond-mean-field effects we find closed-form expressions for these quantities. We also discuss differences between the Thomas-Fermi and Hartree-Fock approaches for treating the heavy fermions.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2024-5-16 (Invited Report)


1-Interesting subject for a broad audience not yet considered in the litterature;
2-Relevant analysis with comparisons between different approaches;
3-Can be used for further studies in this domain


Even if of good quality,
1- the presentation can be improved
2- the question concerning the numerics can be more detailed

(see report for the details of these two issues)


The physics of one impurity interacting resonantly with N identical fermions is a subject of general interest in the many body problem. With this study in two spatial dimensions, the authors explore a configuration which was not yet solved. After an introduction with an overview of known results, the authors derive the mean field approach (TF) which gives the shape of the density profile but does not fix the spatial scale. This first approach permits one to exhibit the two important dimensionless parameters : the parameter alpha proportional to N^2 over the mass ratio and the second, gamma proportional to the coupling contant g times N where g<0. The system can bind only when alpha is less than a critical value and for a given value of alpha in this interval the mean field equation gives a value of gamma. Next, they use a beyond mean field correction in the local density approximation to determine this unkown scale by showing that in a second order perturbation theory, the coupling constant is a function of the Fermi momentum and of the dimer energy. They thus obtain the cluster energy as a function of the dimer energy. To get more accuracy they use a Hartree-Fock (HF) approach and compare the results obtained numerically with the analytical results of the TF approach. Finally, they compare the HF results with few-body exact results obtained for small N.

The manuscript gives interesting results in the large N limit where exact few-body techniques cannot give any answer. Nevertheless, from my point of view, it can be improved to have more impact.

1) In the introduction, I suggest to explicitly write that the issue studied here concerns the binding of N+1 particles in presence of a shallow dimer (whatever the dimension D) corresponding to the limit of a large and positive scattering length. This is more precise an clear that saying that the interaction is attractive (cf the renormalization or regularization of a delta interaction which means that g delta itself is ill-defined)

2) Part 2: the introduction of the formula of g function of E(1+1) and the UV cut-off kappa is not useful and can bring confusion to the reader: for each alpha there is a gamma and this suggests that kappa is fixed by alpha... However this is not a good reasoning and shows the limitation of the use of a first order perturbation theory.Instead, I suggest to say that writting g delta in the functionnal where g is a given negative constant is a first guess for treating the interaction. Except that, all the analysis is interesting. Perhaps it is valuable to recall that the scale invariance is expected to be broken similarly to what happens in 2D Bosonic systems with a contact force (cf Pitaevskii Rosch collective modes) and it is thus necessary to treat the interaction at the second order of the perturbation theory.

3) Part 3: It is important to emphasize that the shallow dimer energy is the relevant scale: why not express E_{N+1} in terms of E_{1+1} ?

4) Part 4: The dispersion of the points in the right panel of Fig 2 is puzzling:

a) Why not plotting the relative dispersion Delta gamma/gamma (also more relevant) ? the dispersion will be reduced

b) Even with (a) I guess that there will be a larger dispersion for alpha=8 than for alpha=2. Nevertheless, the explanations given are not clear: there are two possibilities as suggested in the text. If the dispersion is due to the vicinity of the threshold, this a very interesting effect which deserves further future studies. If this is a numerical effect, this is less interesting but this can be tested by changing the mesh size and/or the interval of integration. Thus, more infiormations are needed in the text for the numerical analysis (grid used : logarithmic, linear ? , mesh size, interval) and at least vary these parameters would give indications if this is a purely numerical effect.

5) Part 5: In the figure, I suggest to replace E^{exact}_{1+1} which appears no where else in the manuscript by E_{1+1}

Except these remarks/suggestions of possible improvments, I think that the paper meets the criteria to be published in Scipost

Requested changes

see report


Ask for minor revision

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

Anonymous Report 1 on 2024-5-12 (Invited Report)


This paper studies the large-N limit of systems of N heavy identical fermions bound (with zero-range interactions) by a light atom in two dimensions. Such N+1 clusters only bind above a critical mass ratio where the interspecies attraction overcomes the Fermi pressure. By using a mean-field theory, together with a Thomas-Fermi approximation for the kinetic energy of the heavy atoms, the authors determine this critical ratio. Since the mean-field theory is scale invariant, it can additionally predict the shapes of the clusters (up to a rescaling), but not their absolute sizes and energies. In order to obtain the latter, they execute a beyond-mean-field analysis based on the local-density approximation — first by treating the heavy fermions with a Thomas-Fermi approach, and then to improve the accuracy, with a Hartree-Fock approach. They discuss the relative merits of both methods. Last, they apply the (many-body) Hartree-Fock technique to small clusters for which exact solutions are known and find that it performs quite well. The literature is nicely reviewed in the introduction. It is hence evident that this work probes an unexplored region of the parameter space of fermionic (N+1)-body systems with zero-range interactions. The results are also relevant to experiments on ultracold Fermi gases with large spin and mass imbalances (e.g., mixtures of Li-6 and Yb-173). The writing is clear and the methods are detailed. While there is scope for future investigation, this paper tells a complete story. Therefore, I do not believe that anything further needs to be done to this submission, and in my opinion, it is suitable for publication in SciPost Physics.


Publish (easily meets expectations and criteria for this Journal; among top 50%)

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
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