Multi-orbital two-particle self-consistent approach -- strengths and limitations

1. This paper shows how to solve a multi-orbital Hubbard system with SU(2) symmetry with TPSC. This is a great addition to the domain as TPSC has been
known to show physical features unaccessible to Dynamical mean-field calculations or Monte-Carlo ones.
2. The derivations are thorough and well explained, and the author added many details in Appendices, which are all really pertinent.
3. The analysis of the limits of the Hartree-Fock decoupling is really interesting and gives insights to the limitations of TPSC in its pure form.
4. The method is benchmarked against a previous formulation of multi-orbital TPSC, DMFT and D-TRILEX

1. Some of the claims still need further explanation and/or references.
2. There are still some mathematical errors and/or typos with small inconsistencies, but it is almost all resolved from the first submission.

I have followed the redprint for my comments. It was discussed with André-Marie Tremblay.
The two-particle self-consistent approach (TPSC) is an appealing method to study strongly correlated systems. It satisfies many exact results and even though multi-orbital formulations were already done before, this generalization in the SU(2) symmetry which is presented in this paper is non trivial and will be a great addition to the field.
Unfortunately again, in the present form, this paper cannot be published. It needs small corrections of mathematical issues. Furthermore, there is a section in which the discussion needs either more results shown, a physical explanation of some references to back up some statements.

After the authors respond to the constructive criticism below and after further review by me or some other referee, I hope publication can be recommended.

1. After the Eq. (16), the following definition is shown : $n^{s_1 s_2}_{o_1 o_2} = c^{\dagger}_{o_2 s_2} (\tau, \mathbf{r}) c_{o_1 s_1} (\tau, \mathbf{r})$. First of all, I think the previous introduced notation should be kept here $\boldsymbol{\tau}=(\mathbf {r},\tau)$ , for consistency. Second, shouldn’t there also be a $\boldsymbol{\tau}$ dependance to $n^{s_1 s_2}_{o_1 o_2}$? Like such : $n^{s_1 s_2}_{o_1 o_2}(\boldsymbol{\tau}) = c^{\dagger}_{o_2 s_2} (\boldsymbol{\tau}) c_{o_1 s_1} (\boldsymbol{\tau})$ ?

2. For equation 8, but also for all of the other equations where the sum are redundant. I have to stress that equations written as is, with two sums over
one index, is not mathematically valid: at least not for the equation that is written. There are many ways to keep the equation mathematically valid and explicitly show the sum:
either to remove the prime over the indices that are linked with the summation operator. eg. $\sum_{s_4} c_{s_4} c_{s_1}$. To make an even greater distinction, one could use a different notation for the explicitly summed indices: e.g. $\sum_{\alpha} c_{\alpha} c_{s_1}$.

3. In the same vein as the previous point, the cases of sums shown explicitly to take into account excluded orbitals, e.g. eq. (47), the $1-\delta$ could be used: keeping the indices primed and adding $1-\delta$ would mean exactly what is written in the text. e.g. $ (1-\delta_{o_1 o_2}) c_{o_1} $.

4. In Section 3.2, there is a discussion about how the approximation of local and static vertices break down as one lowers temperature. Why? First of
all, the results shown in the paper are almost all at the same temperature , which is mostly a high temperature. There are no results comparing to
DMFT vertices as the temperature goes down. In the papers mentioned, I have not seen any mention of locality of vertices breaking down. I want to
stress here that I am not fully against that possibility, but it seems to me that this is a claim that needs backup or a physical explanation which I have
not found here. Secondly, single-orbital TPSC does not fail at low temperature, except in the renormalized classical regime. And even in the
renormalized classical regime with static and local vertices for doubly self-consistent TPSC, there was great correspondance with benchmarks. [Vilk, Y. M., et al. PRB 110, no. 12 (2024): 125154. https://doi.org/10.1103/PhysRevB.110.125154.]

Ask for minor revision

The new version of the manuscript is an improvement when compared to its previous one however I still have some doubts about one of the main aspect of the paper and therefore I am not convinced to recommend this paper for publication yet.

1.) From what I understand, the authors argue that the source of ambiguity arises from the structure of the equation of motion, specifically whether the sum over the internal spin index is incorporated into the ansatz definition or not. However, in TPSC, the ansatz is typically chosen to reproduce the correct static and local limits of the two-particle Green’s functions. This can be achieved by defining G(2) without directly relying on the equation of motion. Once this quantity is fixed at the level of G(2), it is then used in the equation of motion.
My question is: would this ambiguity persist if one were to adopt this protocol for fixing the ansatz?

2.) I would honestly suggest the authors to move Figure 2 into the appendix.

3.) The authors considered my advice to perform additional calculation partially. While I appreciate their effort in doing so I think that it would be informative to compare the self-energy computed using the different methods fixing two or more k-points as a function of frequency.

Ask for major revision