Non-perturbative intertwining between spin and charge correlations: A "smoking gun" single-boson-exchange result

We thank the Referee for reviewing our manuscript as well as for her/his overall positive assessment on our work, consistent to the positive evaluation of Referee 1. The Referee has asked us to consider specific points to be addressed prior to publications. We have thoroughly considered all of them (see the detailed reply below) and included the corresponding changes into the manuscript text and the appendix.

In what follows I mention some minor suggestions that I invite the authors to consider: 1. Some physical aspects, although quite basic could be not obvious for not experts. For example what is the physical meaning of diagonal and not-diagonal component in the Matsubara space in terms of static and dynamic fluctuations, the physical meaning of the matric eigenvalues, as well as the need to invert the generalized susceptibility. The authors could spend a few words in the introductory sections to make the reader able to better appreciate the physics behind the quantities considered.

In the view of our study, some formal aspects, such as the distinction between diagonal and off-diagonal entries of the generalized susceptibilities, are important mostly for their effects in driving the sign-change of the eigenvalues and the associated irreducible vertex divergences (whose link with the structure of the generalized susceptibility have been now more precisely highlighted in the last paragraph of p. 12 of the revised manuscript). Further, as the Referee correctly notes our analysis, based on the SBE decomposition of the generalized susceptibility, also allows to ascribe specific contributions to its diagonal, e.g. those associated to the exchange of a spin collective mode, to the static spin response of the system, which becomes the dominating damping factor of the on-site charge fluctuation in the local moment regime (cf. detailed discussion in Sec. 3.2 and in previous replies). At the same time, one cannot associate, in general, the whole diagonal/off-diagonal entries to one specific static/dynamic response of the systems, because more than one SBE/multiboson term, as well as the bubble-term for the diagonal entries, might contribute to the different matrix diagonal/offdiagonal elements. In this context, in order to highlight the general properties, which characterize the frequency diagonal/off diagonal entries of the generalized charge susceptibility and, in particular, their relation with the bubble and the vertex correction terms, we have slightly extended the discussion right after Eq. (8) in Sec. 2 of the revised manuscript. As for the physical interpretation of the eigenvalues, apart from the suppression effects of on-site fluctuations we mention in our work, which has been extensively discussed, at the level of a significant ``observation'' in the recent literature, see Refs. [3, 25] in the revised manuscript, one should also recall the crucial effect that negative eigenvalues can have in lattice model, when lifting the condition of perfect particle-hole symmetry: The lowest (negative) eigenvalue can trigger the onset of a phase-separation instability, such that observed in the proximity of the Mott metal-insulator transition in the DMFT solution of the Hubbard model. Motivated by the Referee's question, we have inserted an extra sentence at p. 6 before Eq. (8), mentioning this important eigenvalue property and including the specific reference to the papers discussing this aspect (e.g. Ref. [21] and [24]). Eventually, we agree with the Referee that it is also important to explain to the readership, why one might actually need to invert the Bethe-Salpeter equation (and hence the generalized susceptibility matrix). In fact, as we now emphasize in Sec. 2.3 at the end of p. 5 of the revised manuscript, computing such inversions is necessary, e.g., to provide the input for diagrammatic approaches based on parquet equations, such as D$\Gamma$A and QUADRILEX. In fact, at intermediate-to-strong coupling, these approaches might be directly affected by the divergences of the two-particle irreducible vertex functions, triggered by vanishing eigenvalues. In this respect, we also mention in the revised manuscript, that for related reasons, other algorithmic schemes, such as bold diagrammatic Monte Carlo and the nested cluster approaches can face problems in the same parameter regime, as it is discussed in Refs. [2,9].

1. To help the reader I would use the extensive names of the models in Sec. 2.2 and define here the acronym used in the following part of the paper.

In the light of this Referee suggestion, we now use the full names also in Sec. 2.2 of the revised manuscript and define the acronyms there for a second time.

1. The paper is already very long and dense. I suggest the authors to have a last revision and try to rephrase/shorten a bit the text whenever is possible.

We thank to Referee for this comment, which has also been mentioned by Referee 1. We have worked throughout the whole manuscript and, whenever possible, we have simplified the text, by shortening and/or rewording too long and complex sentences, without compromising the completeness of information (s. enclosed .pdf file, with the highlighted changes in the text)

1. While the results of the approximations can be briefly mentioned within the discussion to further support the findings, I would suggest to remove “3.4 Insights from Approximations” from the main text and create an appendix.

We agree with the Referee that the manuscript is long, and can thus benefit from the proposed rearrangement. Hence, we have followed her/his advise by moving Sec. 3.4 into a newly created appendix. In order to keep reference of most important information in the main part of the manuscript we have added a concise paragraph at the end of Sec. 3.3, which summarizes the results obtained via different approximations, with reference to the appendix.

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manuscript_changes_5BFxpca.pdf

We thank the Referee for the careful review of our manuscript, for her/his overall positive evaluation, as well as for her/his constructive observations. Below, we detail our Reply to all specific points raised in her/his report:

a) My main concern is about the convoluted and often hard-to-understand way to discuss the results. I urge you to go over the text again, try to sharpen the message and shorten sentences whereever possible. Here is an example from the bottom of page 12: "At the same time, the low-frequency bubble contribution is, as expected, almost vanishing in the HA, reflecting the ground-state, while it is sizable (though, generally smaller than in the perturbative regime) in the AIM or the DMFT solution of the Hubbard model, consistent with the Fermi-liquid nature of their ground states." Here is another example from page 5: "The strong suppression of the diagonal entries in $\chi$, even down to negative values, drives the breakdown of the self-consistent perturbative description, as it is responsible for several sign flips (from positive to negative) of the eigenvalues of the generalized susceptibility and, hence, for corresponding divergences of irreducible vertex functions in the corresponding channel [1]. " Similar constructions appear throughout the manuscript. You need to avoid this type of convoluted sentences if you want to give non-experts a chance to understand your paper. In the following I list a number of minor points that should be addressed to further improve the readability of the manuscript.

We thank the Referee for this comment. Following her/his suggestion we have now simplified long and convoluted sentences throughout the whole manuscript, improving the readability of our discussions. (s. enclosed .pdf file, with the highlighted changes in the text)

b) On page 14, please explain why you qualify a Green's function with a suppressed low frequency part as "insulating".

For $\text{i}\nu\rightarrow0$ the imaginary Green's function is directly connected to the spectral function at the Fermi level (namely: $-\frac{1}{\pi} G(i\nu \rightarrow 0) = A(\omega=0)$). Therefore, if its value tends to zero, the spectral function at the Fermi level is correspondingly suppressed, featuring an insulating behavior.

c) Please make it very clear where in the manuscript you talk about ONE-line and TWO-line irreducible vertices.

We thank the Referee for his observation: The adjective ``irreducible" was indeed used in two different contexts, though always at the level of two-particle quantities: (i) two-particle vertex functions irreducible w.r.t. a cut of two fermionic lines (mostly in a given channel), which define the typical kernel of parquet/Bethe-Salpeter equations, and (ii) two-particle vertex functions irreducible w.r.t. a cut of a single interaction line ($U$-irreducibility) which naturally appear in the SBE decomposition. Following the observation of the Referee, in the revised manuscript we now always explicitly specify whether we are referring to two-particle irreducible quantities (i.e. to two-particle diagrams that do not fall apart if two fermionic lines are cut) or to U-irreducible quantities (i.e. to two-particle diagrams that do not fall apart when a single interaction line is cut). At the same time, note that we have not used in the paper the word one-particle irreducible for the one-particle quantities, such as the self-energy, as this diagrammatic specification was not needed for our discussions.

d) You talk about the sign change of the eigenvalues of $\chi$ many times. Why not showing these eigenvalues as a function of T for the models considered if it is so important. Also the need to invert the generalized susceptibility could be explained for completeness. Yes, you may use formulas here!

We did not show the results for the sign-changes of the eigenvalues of the generalized charge susceptibility as extensive studies of this specific issue has already been performed in the literature for these models, e.g. in Ref. [1,3,7,9,20,21,23-25,28] mentioned in the Introduction and in Sec. II of the revised manuscript. On the other hand, we agree with the Referee that showing the mathematical expression linking the vanishing eigenvalues of the generalized susceptibility and the divergences of the two-particle vertex functions could definitely help the clarity of our discussions. Hence, in the revised manuscript, namely at p. 14, we have now reported the explicit definition of the two-particle irreducible vertex function, given by the inversion of the corresponding Bethe-Salpeter equation. From that expression, it becomes fully transparent why a zero eigenvalue of a generalized susceptibility triggers a divergence in the two-particle irreducible vertex of the corresponding sector.

e) You put yourself in the comfortable position that you only study models for which you can obtain exact correlation functions. But what are the implications of your findings for method development aiming at other more realistic models?

While the numerical/analytical results of our manuscript rigorously hold for the specific models considered, some general considerations can be plausibly drawn for more general cases, where no (numerically) exact solution is available, and may be used as a guidance for future studies beyond the framework of our work. Indeed, the SBE decomposition is quite general and strong spin fluctuations could manifest themselves in a nonlocal generalized charge susceptibility similarly as in the (purely local) cases studied here. Notice, however, that in the most general case, not only the physical susceptibilities, but also the Hedin vertices are nonlocal quantities. Therefore, it is not easy to predict how local or nonlocal spin fluctuations may be related to corresponding local and/or nonlocal divergences of the two-particle irreducible vertices. These questions are certainly relevant for future investigations, as similar two-particle irreducible vertex divergences in the charge channel have been reported to occur in the cluster DMFT (approximated) solution of the two-dimensional Hubbard model even at lower U-values than those found in the purely local DMFT solution, cf. Refs. [9,16]. Interestingly, these divergences, beyond affecting the lowest frequency structure of the vertex functions, are additionally associated to the specific momentum structure of (commensurate) AF fluctuations (Ref. [16]). One could speculate, thus, that the highly nonperturbative physics of the two-dimensional Hubbard model could be somewhat encoded in a ``generalization" of the strong link between different scattering sectors, which we discuss in our work, beyond the purely on-site physics. In that case, the role of the static local response might be replaced by strong antiferromagnetic and/or RVB fluctuations and could affect, depending on the parameters, the CDW and/or $d-$wave superconduncting response of the systems. Future studies could shed light on these (at the moment purely speculative) ideas.

f) There is a typo in Fig. 11 (label "perTurbative") and in the end of the caption of Fig. 14

We thank the referee for noticing it and have fixed the typo in the revised manuscript.

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