Polaron spectroscopy of interacting Fermi systems: insights from exact diagonalization

We thank the two Referees for their constructive remarks. Apart from a series of changes in the main text and related plots (that we discussed below), we have performed extensive additional studies, which are now presented in four new Appendices. We believe that this additional material, which nicely supports the main results of our work, constitute a sizable improvement of our manuscript.

Report 1

We thank the Referee for their appreciation of our manuscript and for their useful remarks. We have now added a new Appendix A, which addresses the questions of the Referee about convergence. The bottom line of this analysis is that by comparing different sizes, fillings and lattices it is indeed possible to distinguish the features that would also hold in a thermodynamically large system, from those that are artifacts of finite-size effects.

Regarding the study of Rabi oscillations in this framework: we agree that it may be a very interesting application of our method, and we comment on it in the outlines. However, since this would still require a (conceptually straightforward) rewriting of some parts of our code, and since this does not completely fit the main topic of “interacting fermions”, we have decided to leave it as a future direction.

Report 2

We thank the Referee for their reading of our manuscript and for their useful remarks. In the revised manuscript we adopt a more critical attitude towards the “double line of the attractive polaron branch in a superfluid”. Indeed, while we are inclined to believe that this feature could likely be a finite-size effect, it might still suggest a strong contribution of bosonic, Goldstone-like excitations at energies very close to the AP line (as motivated by the comparison with Appendix D).
We reply point by point to the their questions 1-11:

1) We now explicitly mention the finite-size effect issue: “While
our calculations are performed using an exact-diagonalization approach, hence limiting
our analysis to systems of few interacting Fermi particles, we identify robust spectral
features supported by theoretical results.”.

2) We now discuss this in Appendix A, Fig. 7.c.

3) We now address these issues in Appendix A (finite size corrections) and Appendix B (umklapp peak).

4) First of all, most of the results do not rely on whether the microscopic lattice is chosen to be a square or a triangular one: Indeed, as also pointed out by the Referee, only the band minimum matters.
Following their suggestion, we now mainly display triangular-lattice results in the main text, and moved the square-lattice results to the new Appendix C (showing excellent agreement).

However, we point out that there is a genuine difference between triangular and square lattices in the case of the charge density wave because they give rise to different charge order, with different symmetries. For this reason, we have decided to plot both results in the main text.

Concerning the extra feature in the triangular-lattice results shown in Fig. 3a: We are currently investigating the spectra obtained from a Hartree-Fock approach combined with the Chevy ansatz; while these results will be presented in a future publication, we hereby summarize our current findings: while this Hartree-Fock-Chevy approach can reproduce the main avoided crossing due to the increase of the quasi-particle gap, we find that it does not capture this extra feature. Since larger system sizes are prohibitive, we cannot confirm whether this is a ‘beyond HF+Chevy’ effect or a finite size effect. This limitation is commented in the text.

5) We now explicitly define the +/-K points and use \kappa for the interaction parameter.

6) To reassure the reader, we now added the binding energies in Fig. 4.a, which perfectly match the ED lines. However, as expected, this agreement is only found at small E_F/E_B, since there is a strong redshift due to the Fermi sea at large E_F (this is why we had decided not to plot the binding energy in vacuum in the original version). Notice that this is different from Fig. 5, since in that case it is E_pair that is varied.

7) We now improved the presentation and explicitly provide the relation between the two quantities, U and E_B. The binding energy of the impurity-spin up molecule is still given by Eq. (5), and analogously for the spin down one.
To strengthen and clarify the comparison with the Chevy results, we have now added a new Appendix D, devoted to the BCS+Chevy calculation with the exact same setting as in ED.

8) Now we briefly discuss the main advantages and drawbacks of the cold atom and TMD platforms. Following the Referee’s suggestion, we also point out that our approach is interesting in addressing the question of how the many-body limit is reached starting from few particles and increasing their number.

9) This has now been fixed.

10) We are now more careful when drawing the colorbars of the plots, with proper units 1/E_F.

11) We have fixed all these presentation issues.

We thank once again the Refeere for all their constructive and detailed comments.