Boson-fermion pairing and condensation in two-dimensional Bose-Fermi mixtures

Dear Editor,

Thank you for your Editorial Recommendation of July 5th, 2024, asking for a minor revision of our manuscript. We hereby submit a revised version that takes into due account all recommendations by the Referees. Together with this resubmission, we have also sent a point-by-point response to both Referees.

We are really grateful to the Referees for their valuable comments and suggestions, which have helped us to considerably improve our manuscript.

Yours sincerely,
Leonardo Pisani, Pietro Bovini, Fabrizio Pavan, Pierbiagio Pieri

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Summary of the changes made
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We have changed the layout of our text to adopt the SciPost latex template.
We refer to the new version when pointing to page, equation, and figure numbers.
Our manuscript has been modified to take into account all the suggestions by the Referees. We have also made some changes to improve the presentation and quoted a few additional references. All of these changes (except minor stylistic/typos corrections) are listed below.

a) Pag. 2, changed “micro-cavities” with “nanostructures” in the 2nd paragraph, added four new sentences to the 2rd paragraph (“The so-called … on more solid ground.”)
b) Pag. 3, re-written the final part of the 5th paragraph (from “For this reason …”)
c) Pag. 5, added a sentence at the end of the first paragraph, re-worded the first two sentences of the last paragraph of Sec. 2.1, re-worded the end of the first paragraph in Sec. 2.2 (“There, the authors … to the boson concentration.”), added a new paragraph at the end of the page (starting with “A warning is however in order.”)
d) Pag. 6, added two new paragraphs (“Let us see now the rationale … controlled approximations are not available”, “The same strategy is then adopted … with (one-component) fermions [76].”), inserted Eq.5 and corresponding discussion
e) Pag. 7, re-worded the paragraph above Eq.7 and the paragraph above Eq.9, added a comment below Eq.9, introduced more specific names for the T-matrices (“many-body” and ”two-body”) below Eq.9
f) Pag. 8, re-written Eq.11 and the paragraph below it
g) Pag. 9, changed the titles of Sec.3 and Sec.3.1, re-worded the last two sentences of the first paragraph of Sec.3 and the first sentence of Sec. 3.1, added the explicit definition of n^0_{\mu_F} below Eq.21, changed \Sigma_{CF} to \Sigma^0_{CF} below Eq.22 and everywhere else
h) Pag. 10, added Eq.24 and its explanation below it, added the sentence “in the many-body T-matrix in the condensed phase” above Eq.25, added a new paragraph (“Here, G_{CF}(P,\Omega), … condensate.”) below Eq.27, re-worded the first sentence below Eq.29, added a whole new paragraph (“One sees from Eq. (29) … value for all intermediate concentrations.”) above Eq.30, changed “pair” with “composite-fermion” before Eq.30
i) Pag. 11, added Eq.34 and comments above and below it, changed “two-body correlator” with “many-body T-matrix” at the end of Sec.3.1, re-worded the paragraph containing Eqs. 35 and 36
j) Pag. 12, re-organized the end of Sec.3.2, in particular moving text and equations from former Sec.IIIB (from former Eq.33 to 46b) to the new Appendix D2 while re-organizing and re-wording the rest; re-worded the sentence after Eq.44
k) Pag. 13, added a comment at the end of the paragraph after Eq.47, re-written the last two paragraphs of the page
l) Pag. 14, re-written the 1st paragraph, added a second inset in Fig.3(b) updating the caption, re-worded the last sentence of the page
m) Pag. 15, added a sentence “So, in this regime, …”, modifies the first sentence of the 2nd paragraph, re-written the last two sentences of the 2nd paragraph, inserted new paragraph “The Luttinger theorem states …” at the end of the page
n) Pag. 16, inserted new paragraph (2nd paragraph: “In a BF mixture with a condensate …”), updated the caption of Fig.5, completely re-written the last paragraph of the page with several new sentences
o) Pag. 17, completely re-written all paragraphs before Section 4 with several new sentences, added a new sentence (last sentence of the page)
p) Pag. 19, added Eq.49
q) Pag. 20, added insets in Fig.9 and their explanations in the caption
r) Pag. 21, re-worded a sentence in the caption of Fig.11, added two new sentences after Eq. 51
s) Pag. 22, substituted “asymptotic limit” with “large-momentum behavior” in the caption of Fig.12, added two new sentences in the 2nd paragraph: “to check to what extent …” and “The use of the same thermodynamic…”,, added “internal part of the” before “molecular wave-function” in the first sentence of the 3rd paragraph
t) Pag. 23, replaced “asymptotics” with “approximation” in the caption of Fig.13, added a new sentence immediately after Eq.55
u) Pag. 24, added a new curve and a new set of points in Fig.14(a) updating the corresponding caption, re-written a sentence in the 1st paragraph, inserted a new paragraph (2nd paragraph), re-worded the last sentence of Sec.4.3
v) Pag. 25, added two new sentences (“As an independent check…to possible numerical errors”) near the end of the 3rd paragraph, added a new 4th paragraph (“As a matter of fact, … followed by the limit x -> 0.”), added four new sentences at the end of the last paragraph
w) Pag. 26, changed the notation of Eqs.56-57
x) Pag. 27, reworded the last sentence of the last paragraph at the end of the page
y) Pag. 28, re-written the first two sentences of the 4th paragraph in Sec.5 “A further difference with …”
z) Pag. 29, added words “in particular at small values of x …” at the end of the second sentence of the 2nd paragraph, added a new paragraph at the end of the page (”It is worth also discussing the relevance … sufficient to guarantee stability.“)
aa) Pag. 30, added two new paragraphs (“As a final remark …are available online [131].”) before the Acknowledgements
bb) Pag. 30-31, renamed, re-organized, and re-written Appendix A
cc) Pag. 32, added Fig.17 and its caption
dd) Pag. 34-35, added Appendix D1 containing the whole content of former Appendix D
ee) Pag. 35-37, added Appendix D2 containing material from former Sec.IIIB (from former Eq.33 to 46b)
ff) Bibliography, added the following references 42, 44, 48, 49, 95, 96, 100, 101, 102, 111, 112, 113, 114, 115, 128, 129, 130, 131
gg) Bibliography, updated the following references (due to publication or change of version): former 34 now 39, former 45 now 46, former 53 now 56
hh) Bibliography, removed former reference 40