SciPost Submission Page
Heavylight $N+1$ clusters of twodimensional fermions
by Jules Givois, Andrea Tononi, Dmitry S. Petrov
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Jules Givois · Andrea Tononi 
Submission information  

Preprint Link:  scipost_202403_00030v1 (pdf) 
Date submitted:  20240321 13:51 
Submitted by:  Givois, Jules 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study binding of $N$ identical heavy fermions by a light atom in two dimensions assuming zerorange attractive heavylight interactions. By using the meanfield theory valid for large $N$ we show that the $N+1$ cluster is bound when the mass ratio exceeds $1.074N^2$. The meanfield theory, being scale invariant in two dimensions, predicts only the shapes of the clusters leaving their sizes and energies undefined. By taking into account beyondmeanfield effects we find closedform expressions for these quantities. We also discuss differences between the ThomasFermi and HartreeFock approaches for treating the heavy fermions.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024516 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202403_00030v1, delivered 20240516, doi: 10.21468/SciPost.Report.9068
Strengths
1Interesting subject for a broad audience not yet considered in the litterature;
2Relevant analysis with comparisons between different approaches;
3Can be used for further studies in this domain
Weaknesses
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)
Report
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 HartreeFock (HF) approach and compare the results obtained numerically with the analytical results of the TF approach. Finally, they compare the HF results with fewbody exact results obtained for small N.
The manuscript gives interesting results in the large N limit where exact fewbody 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 illdefined)
2) Part 2: the introduction of the formula of g function of E(1+1) and the UV cutoff 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
Recommendation
Ask for minor revision
Report #1 by Anonymous (Referee 1) on 2024512 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202403_00030v1, delivered 20240511, doi: 10.21468/SciPost.Report.9038
Report
This paper studies the largeN limit of systems of N heavy identical fermions bound (with zerorange 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 meanfield theory, together with a ThomasFermi approximation for the kinetic energy of the heavy atoms, the authors determine this critical ratio. Since the meanfield 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 beyondmeanfield analysis based on the localdensity approximation — first by treating the heavy fermions with a ThomasFermi approach, and then to improve the accuracy, with a HartreeFock approach. They discuss the relative merits of both methods. Last, they apply the (manybody) HartreeFock 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 zerorange interactions. The results are also relevant to experiments on ultracold Fermi gases with large spin and mass imbalances (e.g., mixtures of Li6 and Yb173). 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.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)