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

Dear Editors,

We thank the reviewers for their careful reading of our manuscript, for their positive opinion, and for constructive comments. The first referee has no points to clarify. Below is our response to the second reviewer and the summary of changes. We hope that our manuscript is now suitable for publication.

Sincerely, Jules Givois, Andrea Tononi, and Dmitry Petrov

Referee: 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)

Response: In the first paragraph of the revised version we now explicitly mention that the scattering length is positive and that it determines the size of the 1+1 cluster.

It is also specified in the abstract and in the first phrase of the introduction that we are dealing with zero-range interactions. Therefore, mentioning that the dimer is shallow may be confusing. The paper is already quite technical and we would not like to touch effective-range effects. When we introduce the cutoff kappa we say that it should be much larger than any other momentum scale in the problem. In particular, it is larger than the inverse size of the dimer.

Referee: 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.

Response: The Referee does not point to any error and the question is semantic. In our formulation g is an auxiliary quantity, which drops out of the final answer. However, it has a well-defined physical meaning. g is the depth of the interaction potential in momentum space and kappa is the corresponding cutoff. We follow the standard procedure of replacing the zero-range boundary condition (governed by the scattering length) by an effective short-range potential (governed by g and kappa). The formula relating these quantities is well known. In contrast to what the Referee says, this formula is useful for us. By developing the perturbation theory for small g up to the second order we obtain results in the g-independent and kappa-independent form, i.e., only in terms of a. This is a very standard way of perturbatively treating two-dimensional systems with weak short-range interactions. The condition of weak interaction allows us to choose a sufficiently large kappa such that one can claim the validity of the result in the zero-range limit.

Thinking of the Referee's comment we guess that the confusion may be due to the difference between gamma_c and gamma which we do not sufficiently emphasize. gamma is an external parameter, whereas gamma_c is not. These two quantities are allowed to deviate from each other. In the new version we add a paragraph in Sec.2 where we explain this. We hope that this solves the issue.

Referee: 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.

Response: In Sec.3 we mention that the scaling invariance is broken by beyond-MF effects citing the work of Hammer and Son Ref.[14]. As far as we know, Pitaevskii and Rosch in Ref.[17] did not mention this mechanism of symmetry breaking, but we refer to their work in Sec.2 in another context.

Referee: 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} ?

Response: We added this expression to Eq.(13) and also note that the result is independent of the cutoff, consistent with the fact that the dimer size a is the only length scale in the problem.

Referee: 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

Response: gamma is a dimensionless quantity. It is not obvious for us that dividing by gamma brings in more information. In any case, on the log scale this would just shift all the data points in the vertical direction without changing their dispersion. We also note that the values of gamma are given in the caption.

Referee: 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.

Response: The numerical procedure is described after Eq.(15). It amounts to solving two coupled one-dimensional (we assume cylindrical symmetry) differential equations, which is a rather easy task by current standards. The grid in the radial direction contains a few thousands of points in the region of interest (inside and around the cluster). We have varied the grid parameters and we are confident that the numerical uncertainty is negligible on the scale of Fig.2. The dispersion is not a numerical artifact. We agree with the Referee that this is an interesting phenomenon for future studies.

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

Response: Done.

1) On page 1 in the Introduction we mention that a is positive and it determines the size of the 1+1 cluster.

2) On page 4 we add a paragraph explaining the difference between gamma_c and gamma. We also mention that the final result of the calculation (after taking into account the beyond-MF effects) should not depend on the choice of the pair g and kappa.

3) In Eq.(13) we added the expression for the energy of the N+1 cluster in terms of the energy of the 1+1 cluster. Right after this equation we point out that g and kappa drop out of the final answer.

4) We removed the superscript "exact" from the vertical label in Fig. 3.