Correlation functions and characteristic lengthscales in flat band superconductors

1- In the new version of the manuscript the authors consider an alternative definition of the coherence length, namely the one used in Ref. 32 [J.-X. Hu, S. A. Chen, and K. T. Law, arXiv:2308.05686]. In their analysis the authors show that this definition is not without problems and fails to give sensible results in the case of some lattice models.

1- The authors claim in the abstract that the size of the Cooper pair is disconnected from the quantum metric. On the other hand, in the conclusion it is written "Nevertheless, ⟨g⟩ [the quantum metric] provides the correct order of magnitude in the sawtooth chain and for the stub lattice." I agree with the other reference that the abstract and conclusion should provide a more balanced description of the results obtained in the paper and they should also be consistent with each other.
2- Whereas the authors have used the definition of coherence length provided in Ref. 32 in the new version of the manuscript, this is only one of the possible options. As I pointed out in my previous report, the best definition of coherence length is in my opinion the one of Ref. 39, properly extended to multiorbital lattice models. This reference is cited in the new version but its results are mentioned only briefly. On the other hand, I think that this paper is rather important and the authors should analyze what happens when the definition of coherence length of Ref. 39 is used.

I appreciate the fact the authors have extended their analysis and used the same definition of coherence length of Ref. 32, namely the reference claiming that there is a relation between quantum metric (integrated over the Brillouin zone) and coherence length. As shown in the new Eq. 11, this definition of coherence length amounts to the spread of the normal correlation function $G_{\lambda\lambda}(\mathbb{r})$. Interestingly, they highlight the fact that with this definition, the coherence length is zero in the stub lattice, the Lieb lattice and the $\chi$-lattice. Moreover, with the same definition it is found that the coherence length scales as $1/\sqrt{\Delta}$ and not $1/\Delta$, which is the usual result of BCS theory. In my opinion,this is a rather strong evidence that the results of Ref. 32 should be revisited.

On the other hand, these arguments are not sufficient to claim that the coherence length is completely disconnected from the quantum metric, as written in the abstract and in the conclusion. Indeed, the problems encountered with the definition of Ref. 32 might be easily resolved by using instead the definition of Ref. 39, which in the notation of the manuscript would read
\[
\xi^2 = \frac{\sum_{\lambda\eta} |\mathrm{r}|^2|K_{\lambda\eta}(r)|^2}{\sum_{\lambda\eta} |K_{\lambda\eta}(r)|^2}\,.
\]
Note that instead of the normal correlation function $G_{\lambda\eta}(\mathrm{r})$, the anomalous correlation function $K_{\lambda\eta}(\mathrm{r})$ appears. Moreover, the sum is carried out over all possible orbital pairs $\lambda\eta$ and not just for $\lambda=\eta$. Thus, the exact results in Eqs. 5 and 9 do not apply and the coherence length is finite in the case the stub lattice, the Lieb lattice and the $\chi$-lattice. Moreover, it has been shown in Ref. 39 that this definition reproduces the result of BCS theory for a dispersive band
\[
\xi = \frac{\hbar v_{\rm F}}{2\sqrt{2}\Delta}\,.
\]
Note the prefactor $1/(2\sqrt{2})$ which is ignored in the manuscript, for instance in Figure 7. The fact that one should sum over all possible orbital pairs probably modifies the numerical values of the coherence length shown in Fig. 7, maybe leading to a better agreement with the QM result.
From a conceptual point of view, it also seems more sensible that the coherence length should be calculated from the anomalous correlation function, which is the order parameter of the superconducting state.

Concluding, it seems to me that the definition of coherence length of Ref. 39 is the most appropriate one and does lead to consistent results, whereas the one in Ref. 32 has some issues. This leaves the question of the relation between the coherence length and the quantum metric very much open at present since the analytical results of Ref. 32 should be revisited, in particular one should check whether the approximations employed in this latter reference are indeed justified.

1- (Optional) Study what happens when the definition of coherence length of Ref. 39 is applied to the various lattice models considered in the manuscript.
2- Revisit abstract and conclusion to provide a balanced picture of the results. The claim that the coherence length is disconnected from the quantum metric is very strong and does not reflect accurately the results. In particular, in order to make such a claim it is necessary to consider all possible reasonable definitions of coherence length found in the literature, but at present this is not done in the current version of the manuscript since the definition provided in Ref. 39 is not taken into account.
3- Minor change: in the introduction the sentence “its square root provides a measure of the typical spread of the FB Bloch eigenstates.” should be modified. It is more accurate to say that the QM measures the minimal spread of the Wannier functions.
4- Minor change: after equation 4 "Note that a similar quantity have been used in Ref. [39] to extract the the Cooper pair size in conventional superconductors within the exact two-body problem." To my understanding the results of Ref. 39 are obtained withint the usual mean-field BCS approximation not by the solution of the two-body problem. Modify the sentence.
5- Minor change: Above Eq. 8 “Surprisingly, it is found that there are only two non-vanishing values corresponding respectively to |r| = 0 and a.” This is not surprising in view of the results of Ref. 43. Modify the sentence.

Ask for major revision

The authors of the manuscript provided satisfactory answers to the previous referee reports. Only a few minor presentational issues (summarized below) that the authors should clarify hinder me from recommending the publication of the manuscript.

1- The current abstract does not reflect the findings adjusted during the review process. I encourage the authors to highlight their analysis more precisely and extend it beyond the last sentence in its current form.
2- I thank the authors for adding Table 1 as a summary of the models. Unfortunately, the current position is not optimal, and a few further explanatory words are needed for the mentioned concepts (What does tunable QM precisely mean? What is uniform pairing?). Furthermore, a few more words at the end of the introduction would help to give a motivation why the five properties are relevant for the following discussion.
3- I thank the authors for clarifying the BCS-BEC crossover in Fig 2. Unfortunately, the presentation of Fig 2. is still misleading. It suggests that the crossover scale is at 0.04 (gray area), although the scale is around 1. I suggest removing the red coloring or extending the x-axis of this plot such that, indeed, only the BEC regime is indicated by the red color.
4- The authors connect the property of being bipartite to the existence of an FB at zero energy above Eq. (5). However, the chi-lattice does not show a zero mode although being bipartite. The authors may clarify this confusion.
5- The authors should clarify what they mean by "without changing the nature of the compact localized eigenstates."
6- The authors should clarify that "one would expect Delta_avg^{-1} from a standard BCS analysis" for \tilde \xi (see paragraph below Eq. (13)), where the authors potentially refer to the 1/Delta dependence of xi^(K).

Ask for minor revision