Two-particle self-consistent approach for broken symmetry phases

Interesting method development.

The manuscript is clearly addressed to a specialist audience. This is also reflected by a selective choice of references.

The present work develops a two-particle self-consistent (TPSC) approach for broken symmetry phases and applies it to the three-dimensional Hubbard model. A benchmark against the Quantum Monte Carlo (QMC) data of Ref. [13] looks promising.

However, the manuscript clearly addresses a specialist audience. For example, there is a big leap from Eq. (1) to Eq. (2), i.e., as of the second page of the text, the author is talking to specialists only. Moving parts of Appendix A into the main text may help, but I expect that the manuscript will remain very technical nevertheless.

Another point if the exponent $\beta=1/2$ mentioned at the beginning of section 4. For the three-dimensional Heisenberg ($O(3)$) universality class, it should rather be $\beta \approx 0.369$, see for example M. Campostrini et al., Phys. Rev. B 63, 214503 (2001) (also mentioned in Ref. [13]). To me, this is a strong indication that TPSC remains a mean-field theory after all, and I would find a related discussion plus references appropriate.

Furthermore, there are other (lattice) QMC investigation of the Néel transition in the half-filled three-dimensional Hubbard model. P. R. C. Kent et al., Phys. Rev. B 72, 060411(R) (2005) and S. Fuchs et al., Phys. Rev. Lett. 106, 030401 (2011) may be a good starting point to access further literature (actually, both references are also cited in Ref. [13]).

There are a number of further relatively minor points that I mention in "Requested changes".

To conclude, I believe that there is some interest in this work such that it ultimately merits publication in some form. However, my impression at least of the present manuscript is a minor technical progress that is of interest mainly to specialists. Therefore, I recommend transfer of a suitably revised version to SciPost Physics Core.

1- Move part of Appendix A to the beginning of section 3 in order to make the discussion more self-contained.
2- Add discussion to the result $\beta=1/2$ at the beginning of section 4 and cite relevant references, such as Phys. Rev. B 63, 214503 (2001) .
3- Mention further investigations of the half-filled three-dimensional Hubbard model such as P. R. C. Kent et al., Phys. Rev. B 72, 060411(R) (2005) and S. Fuchs et al., Phys. Rev. Lett. 106, 030401 (2011) and compare, e.g., the Néel temperature $T_N$ with these.
4- I liked the last three sentences of the Introduction (section 1) and hope that I have passed the test.
5- I believe that many equations would fit on one line, which would render the manuscript more readable. Examples: Eqs. (4), (10), (16), (18), (19), (21), (28), (37), (40), and (42). Same for a reduction of Eqs. (29) and (41) from three to two lines.
6- Between Eqs. (5) and (6) there is an abbreviation "BSE" that has not been introduced.
7- Appendix C is quite short such that the cross-references generate unnecessary overhead. Why not move the content of the appendix to the appropriate place in section 4?
8- In panels (c) and (d) of Fig. 3, clarity would be improved of the labels "TPSC" and "RPA" had the same colors as the corresponding lines.
9- I can guess what the bar means, e.g., in the $\delta_{a\bar{b}}$ below Eq. (17), but I believe that an explanation would be helpful.
10- If the meaning of the bar in the $\overline{y}$ in Eq. (18) and below was specified, I have missed this description.
11- Below Eq. (41), there is a reference to "The last equation", but Eq. (41) actually is not an equation.
12- Some preprint references are actually published. Maybe some appeared only after submission of the manuscript, but I still recommend an update. Specifically:
[13] is published in Phys. Rev. Lett. 132, 246505 (2024).
[19] is published in Phys. Rev. B 109, 045155 (2024).
[28] is published in Phys. Rev. B 108, 075144 (2023).
[34] is published in Phys. Rev. B 109, 075143 (2024).
13- The URLs in Refs. [29,30] are redundant and could be omitted.
14-There are issues with the English text, such as:
a) The manuscript mixes British and American English. For example, there are occurrences of "magnetisation" (British English) and "magnetization" (American English). I recommend that the author settles on one version and runs the manuscript through an appropriate spellchecker.
b) There are several instances of "as following" which in my opinion should read "as follows".
c) There is a duplicate "for for" in the caption of Fig. 3.
Overall, I believe that the manuscript would benefit from careful proofreading, preferably also from somebody else than the author, and ideally a native English speaker.

Accept in alternative Journal (see Report)