diff --git a/DeltaTbeta.pdf b/DeltaTbeta.pdf
deleted file mode 100644
index 388ce90abe23b48c6158b7ea66c848860471239c..0000000000000000000000000000000000000000
Binary files a/DeltaTbeta.pdf and /dev/null differ
diff --git a/Fig3_1_old.pdf b/Fig3_1_old.pdf
deleted file mode 100644
index a3e4ce4262097ac5150e5d9f94eebd1748dae1a6..0000000000000000000000000000000000000000
Binary files a/Fig3_1_old.pdf and /dev/null differ
diff --git a/Fig3_2_old.pdf b/Fig3_2_old.pdf
deleted file mode 100644
index 6546b930fea5fb968a0db593274ca28eed3e8762..0000000000000000000000000000000000000000
Binary files a/Fig3_2_old.pdf and /dev/null differ
diff --git a/Fig4_1_old.pdf b/Fig4_1_old.pdf
deleted file mode 100644
index 05e14b51520b9bac1a178c03ad0b2768959118c4..0000000000000000000000000000000000000000
Binary files a/Fig4_1_old.pdf and /dev/null differ
diff --git a/Fig4_2_old.pdf b/Fig4_2_old.pdf
deleted file mode 100644
index fc170cf452bed0d9bedb84741d80cfe1a6348090..0000000000000000000000000000000000000000
Binary files a/Fig4_2_old.pdf and /dev/null differ
diff --git a/Nexample.pdf b/Nexample.pdf
deleted file mode 100644
index 072a528943311f95a6e5d725422ace76c65dabcb..0000000000000000000000000000000000000000
Binary files a/Nexample.pdf and /dev/null differ
diff --git a/reply.tex b/reply.tex
deleted file mode 100644
index c04a69af293c2996edb7befea4a9253ac406b0ab..0000000000000000000000000000000000000000
--- a/reply.tex
+++ /dev/null
@@ -1,322 +0,0 @@
-\documentclass[a4paper,11pt]{report}
-
-\usepackage{graphicx}
-\usepackage{enumitem}
-\usepackage[export]{adjustbox}
-\usepackage{float}
-\usepackage{color}
-\usepackage{amsmath}
-%\usepackage{xcolor}
-\usepackage[dvipsnames]{xcolor}
-%\usepackage[font=small,labelfont=bf,tableposition=top]{caption}
-%\DeclareCaptionLabelFormat{andtable}{#1~#2 \& \tablename~\thetable}
-\usepackage{ulem}
-
-\begin{document}
-\pagestyle{empty}
-
-\newcommand{\SA}[1]{{\color{purple} #1}}
-\newcommand{\SAc}[1]{{\color{purple} SA: #1}}
-\newcommand{\AT}[1]{{\color{teal} #1}}
-\newcommand{\ATc}[1]{{\color{teal}AT: #1}}
-\newcommand{\old}[1]{{\color{gray}{\sout{#1}}}}
-\newcommand{\MP}[1]{{\color{orange} #1}}
-\newcommand{\MR}[1]{{\color{Green} #1}}
-\newcommand{\MPc}[1]{{\color{orange}{ MP: #1}}}
-\newcommand{\TD}[1]{{\color{red}{ TODO: #1}}}
-
-\phantom{a}
-\vskip -5mm
-
-\noindent
-Dear Dr. Kancharla,
-
-\vskip 8mm
-
-\noindent
-Thank you very much for forwarding the two Referee reports on our manuscript:
-
-\begin{center}
-{\sl The highly nonperturbative nature of the Mott metal-insulator transition:\\Two-particle vertex divergences in the coexistence region}
-\end{center}
-
-
-\noindent
-We are glad that our work has been positively evaluated by the two Referees. In particular, Referee 1 finds {\sl ``the topic relevant to the current interest in the
-many-electrons theory community"} and acknowledges {\sl ``the endeavor of heavy computations performed to obtain the results presented"}, while Referee 2 believes that {\sl ``the results are both sound and important and thus
-warrant publication as a regular paper in PRB".}
-
-
-\noindent
-Both Referees appear in favor of publication of our manuscript, after we consider their specific
-observations to improve the strength of our presentation.\\
-
-\noindent
-Herewith, we resubmit our manuscript carefully revised, after addressing all the questions raised by the two Referees, as we illustrate in our detailed replies. \\
-Hence, we hope that our revised manuscript can be accepted for publication in Physical Review B. \\
-
-\noindent
-Thank you very much for your assistance,
-
-\vskip 7mm
-
-\noindent
-Sincerely yours, \\
-
-\vskip 4mm
-
-\noindent
-M.~Pelz, S.~Adler, M.~Reitner, A.~Toschi
-
-\vskip 10mm
-
-\newpage
-
-\newpage
-
-\pagestyle{plain}
-\pagenumbering{arabic}
-
-
-\noindent
-{\bf REPLY TO REFEREE A}\\
-
-\noindent
-We thank the Referee for carefully reading our manuscript, for the positive evaluation of our work and her/his constructive observations. \\
-
-
-\noindent
-The Referee has asked us to consider specific points to be addressed prior to publications.
-We have regarded all of them very thoroughly and included the corresponding changes into the manuscript text and figures.
-Specifically, we report below a detailed reply to all Referee's observations.
-
-\noindent
-The main questions posed by the Referee are the following:\\
-
-\noindent
-{\color{blue}
-Appendix A reads that the frequency grid used is 100 by 100. But
-the number of negative eigenvalues for some data points exceeds 100.
-How is it possible? }\\
-
-We thank the Referee for noticing this missing information in the appendix and the resulting inconsistency with the main text. Indeed, while for the higher temperatures we used the mentioned 100$\times$100 frequency grids, for the lowest temperature ($\beta=300$) the number of Matsubara frequencies we computed was larger, i.e., 200$\times$200 grids have been considered. The corrected information has been included in the corresponding section of the appendix.\\
-
-{\color{blue} Also, aren’t there any artifacts arising from the
-finite size of the frequency window?
-}\\
-
-The Referee is right in noticing that, in principle, restricting the study of the on-site generalized susceptibility (i.e., of a matrix with an infinite number of Matsubara frequencies) to a finite size frequency window may originate artifacts in the eigenvalue spectrum. However, the intrinsic frequency structure of the generalized susceptibility itself neutralizes these possible drawbacks, provided that the finite size matrix is not taken too small.
-In fact, for frequencies larger than $|\nu| \sim U$, the structure of the generalized charge susceptibility matrix gets dominated by its asymptotics behavior [J. Kuneš
-PRB 83, 085102 (2011); G. Rohringer et al., PRB 86, 125114 (2012); A. Tagliavini PRB 97, 235140 (2018); N. Wentzell et al., PRB 102, 085106 (2020)], which is of purely perturbative nature (and, hence, associated to positive eigenvalues, whose eigenvectors have mostly high-frequency components). Conversely, all relevant information on non-perturbative eigenvalues is encoded in the low-frequency part,
-i.e., in the part of the frequency matrix we explicitly consider.
-
-To make this argument more precise, we express Eq.~(4) of the main text in the form
-\begin{equation}
- \label{eq:chi_vertex}
- \boldsymbol{\chi}_{ph,\sigma\sigma^\prime}^{\nu\nu^\prime\Omega = 0}= -\beta \delta_{\sigma \sigma'} \delta_{\nu \nu'} G^2(\nu) - G^2(\nu) \boldsymbol{F}_{ph,\sigma\sigma^\prime}^{\nu\nu^\prime\Omega=0} G^2(\nu'),
-\end{equation}
-where $\boldsymbol{F}_{ph,\sigma\sigma^\prime}^{\nu\nu^\prime\Omega=0}$ is the full vertex. Since the first (bubble-like) term in Eq.~(\ref{eq:chi_vertex}) is diagonal and at half-filling also positive, negative eigenvalues can only emerge due to the influence of the second (vertex-correction) term. The latter, which has also off-diagonal entries, decays fast for high frequencies ($\propto \cfrac{1}{\nu^4}$ along the diagonal). Hence, a finite frequency grid can be expected to capture the negative eigenvalues, stemming from the vertex $\boldsymbol{F}_{ph,\sigma\sigma^\prime}^{\nu\nu^\prime\Omega=0}$, quite well, provided that the frequency grid is chosen to be ``large enough'' to contain all the relevant structure of $\boldsymbol{F}_{ph,\sigma\sigma^\prime}^{\nu\nu^\prime\Omega=0}$. In practice, to make sure that a big enough frequency grid for the parameter sets of our calculations has been used, we performed check-up calculations for exemplary parameter sets, for which we selected an increasing number of frequencies, e.g. by starting from a 150$\times$150 frequency grid. There, the test calculations showed no change in the number of negative eigenvalues, meaning that our frequency grid was large enough for our analysis. Accordingly, we have added a sentence to the appendix in Sec.~A to give more details about our choice of the frequency box sizes for the evaluation of $N_{\lambda<0}$.\\
-
-\noindent
-{\color{blue}
-Because of the limitation of CT-QMC, the lowest temperature the
-authors could reach is D/300 or D/200, where D is the half-bandwidth
-of the non-interacting band. Then they extrapolate the results to the
-infinite inverse temperature $\beta$ to make some argument on $T = 0$. But
-I am not sure whether such extrapolation makes sense. Within the
-temperature range of their data points, $U_{c2} – U_c$ increases as $T$
-decreases. However, at lower $T$ (for which CT-QMC cannot work), $U_{c2} –
-U_c$ becomes decreasing; finally $U_c$ becomes equal to $U_{c2}$ at $T = 0$.
-From the solid lines and the dash-dotted lines shown in Fig. 6, it is
-clear that the number of the vertex divergent lines between the U and
-$U_c$ lines increases in a similar trend as the dependence of $U_{c2} – U_c$
-vs $\beta$. In this regard, I don’t think the authors have reached the
-proper scaling regime that connects to $T = 0$. I anticipate that the
-number of negative eigenvalues along the $U_c$ and $U_{c2}$ lines should
-meet or at least scale similarly, near $T = 0$. But for the current data
-shown, it’s hard to see that the two scaling curves, $\beta$ and ln
-$\beta$, will reconcile at last.
-}\\
-\begin{figure*}[t!]
- \centering
- \includegraphics[width=\linewidth]{DeltaTbeta.pdf}
- \caption{The difference of the inverse temperature $\beta$ along $U_c(\beta)$ and $U_{c2}(\beta)$, namely $\Delta\beta = \beta_c(U) - \beta_{c2}(U)$ for $U \to U_{c0}$, shows a monotonic behavior (left panel), in contrast to the difference in temperature $\Delta T = 1/\beta_c(U) - 1/\beta_{c2}(U)$ (right panel).}
- \label{fig:DeltaTbeta}
-\end{figure*}
-
-Here the Referee mentions a quite delicate aspect of our analysis. Indeed, it is certainly true that we cannot reach much lower temperatures than those showed (or even $T \!=\! 0$!) with our CT-QMC solver, and therefore that we cannot exclude, in principle, that the difference in the number of the vertex divergence lines between $U_{c2}(\beta)$ and $U_c(\beta)$ might indeed become smaller again for lower temperatures in contrary to our extrapolations, as the Referee argued.
-
-However, we want to point out that our extrapolations appear, at least, consistent with the overall shape of $U_{c2}(T)$ and $U_c(T)$ and, hence, they might indeed remain valid down to $T=0$.
-More specifically, our data for $\beta_c(U)$ and $\beta_{c2}(U)$ can be quite accurately fitted to the following form
-\begin{equation}
-\label{eq:beta}
- \beta \approx -a + \frac{b}{(U_{c0}-U)^{\gamma}}
-\end{equation}
-where the corresponding fitted parameters we get, having set $U_{c0}=U_{c}(T=0)=2.92$, are reported in Tab.~\ref{tab:beta_fits}.
-
-One can then observe, how the difference between the inverse temperatures of the two lines, i.e. $\Delta\beta = \beta_c(U) - \beta_{c2}(U)$ for $U \to U_{c0}$ monotonically increases by increasing $\beta$ (decreasing $T$), although both lines converge to the same value of $U_{c0}$ for $T=0$ (the difference in temperature $\Delta T = \frac{1}{\beta_c(U)} - \frac{1}{\beta_{c2}(U)}$ shows, instead, the expected non-monotonic behavior, see Fig.~\ref{fig:DeltaTbeta}.
-
-In this situation, the existence of a unique function $N_{\lambda<0}(\beta, U)$ encoding the number of negative eigenvalues in the two-dimensional parameter plane $\{ \beta, U\}$, analytical in the whole correlated metallic regime except for a single pole (divergence) at $U_c(T\!=\! 0) \! = \! U_{c2}(T\!=\! 0)$, would naturally allow for different diverging behaviors along the different paths $U_{c}(\beta)$ and $U_{c2}(\beta)$.
-
-To show this, we can resort to the following (\underline{merely illustrative}!) example:
-\begin{equation}
-\label{eq:testN}
-N^{\mathrm{example}}_{\lambda<0}(\beta,U) = \frac{1}{\frac{1}{\beta^2}+(U-U_{c0})^2}.
-\end{equation}
-
-\begin{table}[]
- \centering
- \begin{tabular}{c|c|c|c}
- &a&b&$\gamma$ \\
- \hline
- $\beta_c$ & 28.66 & 19.46 & 2.266 \\
- \hline
- $\beta_{c2}$ & 25.94 & 27.02 & 1.631 \\
- \end{tabular}
- \caption{Parameters for $\beta_c(U)$ and $\beta_{c2}(U)$ fitted to the functional form in Eq.~(\ref{eq:beta}).}
- \label{tab:beta_fits}
-\end{table}
-It is easy to verify that the illustrative function $N^{\mathrm{example}}_{\lambda<0}(\beta,U)$ is indeed an analytic function of $U$ and $\beta$ except for the pole at $(U=U_{c0}, \beta \to \infty)$.
-
-The respective evaluation along $U_c(\beta)$ and $U_{c2}(\beta)$ then gives
-\begin{equation}
- N^{\mathrm{c, example}}_{\lambda<0}(\beta,U(\beta)) \approx \frac{1}{\frac{1}{\beta^2}+(\frac{b}{a+\beta})^{\frac{2}{\gamma}}},
-\end{equation}
- for which one can readily see how $N^{\mathrm{c2, example}}_{\lambda<0} - N^{\mathrm{c, example}}_{\lambda<0}$ monotonically increases for $\beta \to \infty$. Nevertheless, both $U_c(\beta)$ and $U_{c2}(\beta)$ converge to $U_{c0}$ for $\beta \to \infty$, for which $N^{\mathrm{example}}_{\lambda<0}(U_{c0})$ has one (but an infinite) value (see Fig.~\ref{fig:Nexample}).
-This generic example reconciles the results presented in our paper, with the smooth evolution of several physical quantities, such as
-quasi-particle spectral weight, double occupancy, and energy as a function of $U$ reported in the recent DMFT literature (see e.g., M. Karski et al., PRB 72, 113110 (2005)) for the $T=0$ limit.\\
-
-\begin{figure*}[t!]
- \centering
- \includegraphics[width=\linewidth]{Nexample.pdf}
- \caption{Illustration of the monotonic increasing difference between $N_{\lambda<0}$ along $U_{c}(\beta)$ and $U_{c2}(\beta)$ by using a simple exemplary test function $N^{\mathrm{example}}_{\lambda<0}(\beta,U)$, given in Eq.~(\ref{eq:testN}). Left panel: $N^{\mathrm{example}}_{\lambda<0}(T,U)$, indicated by the background color scale, for $U_{c}(T)$ and $U_{c2}(T)$. Right panel: $N^{\mathrm{example}}_{\lambda<0}(\beta,U_c(\beta))$ and $N^{\mathrm{example}}_{\lambda<0}(\beta,U_{c2}(\beta))$.}
- \label{fig:Nexample}
-\end{figure*}
-\noindent
-
-{\color{blue}
-Moreover, at the end of Sec. III C, the authors claim that the
-number of negative eigenvalues will change continuously at $T = 0$.
-However, Fig. 5 shows the contrary; the jump across the $U_c$ line gets
-more abrupt as T decreases.
-}\\
-
-
-Indeed, we expect the difference in the number of the vertex divergence lines $\Delta N_{\lambda<0}(\beta)$ between the paramagnetic metal (PM) and the paramagnetic insulator (PI) as function of inverse temperature $\beta$ along the $U_{c}(\beta)$ line to monotonically increase for $\beta \to \infty$, similar to the difference between $N_{\lambda<0}(\beta)$ at $U_{c2}(\beta)$ and $U_{c}(\beta)$ in the PM, as discussed above. Nevertheless, for $T=0$, the PM and PI extrapolations yield both an infinite value, similar to $U_{c2}(\beta)$ at $U_{c}(\beta)$ in the PM solution, suggesting $\Delta N_{\lambda<0}(T=0)=0$ at the critical point $U_c$ for zero temperature. To clarify the according discussion in the main text, we changed the last two sentences in Sec.~IIIC and added a paragraph at the end of Sec.~IIID and an additional footnote.\\
-
-\noindent
-{\color{blue}
-In Figs. 3 and 4, there are labels of type “$n_{HA} =$ …â€. But except
-for the “$n_{HM} = 28$†label in Fig. 4 which accompanies a small arrow
-pointing to the corresponding line, it’s hard to associate those
-labels with specific lines.
-}\\
-
-We thank the Referee for noticing that the $n_{HA}$ labels in Figs.~3 and 4 have been hard to associate to the corresponding divergence lines. We have added small arrows to all labels of $n_{HA}$ and $n_{HM}$ to improve these figures and make the labels clearer for the reader.\\
-
-\noindent
-{\color{blue}
-As far as I see, the authors used the same colormap for the
-intensity plots throughout this paper, which is good. This point
-should be emphasized somewhere in the paper.
-} \\
-
-Yes, the same color map range is used for all plots for a better comparability. We thank the Referee for noticing this (positive) aspect of our data representation and included a corresponding sentence in the text of Sec.~IIIA to emphasize this, ass suggested by the Referee.\\
-
-\newpage
-
-\noindent
-{\bf REPLY TO REFEREE B} \\
-
-We thank the Referee for reviewing our manuscript, for the high appreciation of our work, which she/he considers of interest for being published.
-
-The Referee has listed some very helpful and constructive suggestions to be carefully considered prior to publication.
-We did so, by carefully revising our manuscript and supplemental material.
-
-Below, we detail our Reply to all specific points raised by the Referee: \\
-
-\noindent
-{\color{blue}
-In the Conclusion and Outlook section the authors propose to extend
-the current studies by including spatial correlations and check if the
-expected shift towards lower U values of the Mott MIT would be
-accompanied by the corresponding shift of the accumulation point of
-the irreducible vertex divergence lines.
-
-While the precise parameter values at which the transition to
-insulating behavior occurs shall indeed depend on the cluster size,
-the authors seem to overlook a more serious consequence driven by
-spatial correlations - a well-known artifact of DMFT is that the
-thermodynamic transition line $U_c(T)$ is increasing with decreasing $T$.
-In contrast in the cluster extension of DMFT, the critical line $U_c(T)$
-bends back, and $U_c$ decreases with decreasing $T$ , see H. Park, K.
-Haule, and G. Kotliar, PRL 101, 186403 (2008) and M. Balzer et al 2009
-EPL 85 17002.
-
-That shall have a big impact on the nature of the coexistence region
-in Figs. 3 and 4.
-} \\
-
-The observation of the Referees is fully correct, and let us also emphasize here, makes any future development of our DMFT analysis of the MIT-behavior of vertex divergences to the case of cluster extensions of DMFT (C-DMFT) particularly interesting.
-Indeed, first (though non-systematic) numerical indications have been reported [see, e.g. Gunnarsson et al., PRB 93, 245102 (2016); Vucicevic et al., PRB 97, 125141 (2018)] that a progressive reduction of the $U$ values marking the location of vertex divergences is driven by a progressive inclusion of nonlocal correlation in Cluster DMFT calculations (both Cellular-DMFT and DCA) for the 2D Hubbard model. This trend matches the corresponding reduction of the interaction values at which the MIT occurs when increasing the cluster size.
-At the same time, preliminary numerical DCA calculations (not yet published) performed in the past (in collaboration with Prof.~Le~Blanc and Prof.~E.~Gull) also outlined a qualitative change in the low-temperature curvature of the first vertex divergence line encountered (from weak-coupling), w.r.t. DMFT. This seems to reflect the corresponding change in the bending of the $U_c(T)$ lines in C-DMFT (and, thus, the different entropy balance between the correlated metallic and the insulating phase).
-Evidently, this calls for a systematic, thorough study of the site/momentum dependence vertex functions in the proximity of the MIT computed in C-DMFT, possibly exploiting different cluster sizes to identify emerging trends.
-While the effort of such challenging studies, which are actually already programmed in the future months, goes well beyond the work presented in our manuscript, this will provide a clear-cut starting point for the upcoming investigations, allowing to better disentangle the effects of non-local and local correlation in driving the breakdown of the perturbative skeleton expansion.
-As a final remark on this point, let us also mention, that we may expect, purely on a speculative level, that the accumulation point of vertex divergences discussed in our DMFT work could be dragged in C-DMFT together with the MIT-location towards lower values of $U$ (somewhat in the spirit of Fig.~8 of the main text), whereas the curvature of the corresponding vertex divergence lines would also change to match the modified bending of $U_c(T)$. Eventually, it would be intriguing to consider what might happen in the thermodynamic limit (infinite cluster size), where diagrammatic extensions of DMFT as well as determinant Quantum Monte Carlo suggests the MIT (or MI-crossover) of the unfrustrated 2D Hubbard model to occur for vanishingly small values of $U$.
-
-In the revised version of the manuscript, we have hence included this aspect of the probable change in the shape of the vertex divergence lines upon introducing non-local correlations, as mentioned by the Referee.\\
-
-
-\noindent
-{\color{blue}
-There is an ongoing controversy concerning the nature (continuous
-or discontinuous) of the zero-temperature metal-insulator transition
-in the infinite-dimensional Hubbard model, see R. Bulla PRL 83, 136
-(1999) versus J. Schlipf et al., PRL 82, 4890 (1999).
-
-In this limit all the vertex corrections vanish which is consistent
-with the DMFT assumption of a momentum-independent self-energy.
-
-I wonder if the authors' findings establishing a connection between
-the irreducible-vertex divergences and the occurrence of the
-first-order Mott transition can help to solve the above controversy?
-}\\
-
-
-To the best of our knowledge the controversy on the nature of the Mott MIT at $T\!=\!0$ has been eventually clarified, at least on the numerical level, see e.g., M. Karski et al., PRB 72, 113110 (2005): The $T\!=\!0$ Mott MIT of DMFT appears characterized by a continuous vanishing of the spectral weight of the central quasi-particle peak at $U=U_{c2}(T\!=\!0)=U_{c0}$, which for $U\geq U_{c0}$, thus, leaves the two Hubbard bands only, yielding the perfectly gapped spectral function of the Mott insulating phase. At the same time, the non-equivalence of $U_{c1}$ and $U_{c0}$ at $T\! = \! 0$ reflects the existence of a (metastable) insulating solution in the whole coexistence region between $U_{c1}(T\!=\!0) \! < \! U_{c0}$.
-Remaining on the numerical level, we note that some of authors of J. Schlipf et al., PRL 82, 4890 (1999) have subsequently confirmed the presence of (a finite $T$) hysteresis by refined Hirsh-Fye QMC calculations (s. PhD Thesis by N. Bl\"umer) down to the lowest temperature accessible at that time.
-
-Arguably more relevant w.r.t.~the interpretation of our results, however, is the analytical derivation of S. Kehrein, PRL 81, 3912 (1998). There, on the basis of the assumption of a pointwise convergence of the perturbative skeleton expansion in the whole metallic regime up to $U_{c0}$, the possibility of observing the specific kind of Mott MIT described above (i.e., the one numerically reported by R. Bulla PRL 83, 136 (1999) and later by M. Karski et al., PRB 72, 113110 (2005)) was explicitly ruled out. Indeed, the clear numerical evidence presented in our manuscript for the occurrence of a large number of vertex divergences in the correlated metallic region for $U \! < \! U_{c0}$ would allow to reconcile the analytical results by S. Kehrein with the more recent numerical studies on the nature of the MIT. In fact, after crossing the first vertex divergence line, the skeleton expansion does no longer converge to the correct result, but to an unphysical solution [see E. Kozik et al., PRL 114, 156402 (2015); O. Gunnarsson et al., PRL 119, 056402 (2017)]. This indicates that the assumption of S. Kehrein, PRL 81, 3912 (1998), gets violated in the correlated metallic regime well before $U_{c0}$, paving the path to the manifestation of the Mott MIT in the specific way we (numerically) observe it.
-In this context, it is interesting to note, how the presence of vertex divergences (and the associated breakdown of the self-consistent perturbation expansion), already before $U_{c0}$, matches quite well, a posteriori, the following comment made in S. Kehrein, PRL 81, 3912 (1998), "Therefore we obtain a contradiction already for some $U \!<\! U_{c}$ in the metallic regime.", whereas the origin of the contradiction was precisely the assumption of a pointwise convergence of the skeleton expansion in the whole metallic region at $T\!=\!0$ till the Mott MIT.
-We thank the Referee for her/his comment, which motivates us to briefly mention this interesting link in the revised version of our manuscript.
-
-\newpage
-
-\noindent
-{\bf LIST OF CHANGES}
-
-\vskip 5mm
-
-\begin{enumerate}
-
-\item An additional sentence in Sec.~IIIA about the consistent use of the color map for the different figures has been added.
-\item In Figs. 3 and 4 additional arrows to better associate the labels to the corresponding divergence lines have been included.
-\item In Sec.~IIIC the last two sentences discussing the difference in the number of divergence lines between the PM and PI phases at the MIT at $T=0$ have been changed to clarify the discussion.
-\item For the same reason, a paragraph and a footnote have been added to the end of Sec.~IIID.
-\item In Sec.~IV an additional paragraph has been included: Here, we discuss the implications of the numerically extrapolated infinite number of vertex divergences at the MIT at $T=0$ w.r.t. the derivation of S.Kehrein, PRL 81, 3912 (1998) and how this connects to the numerically observed smooth transition at $T=0$ of M. Karski et al. PRB 72, 113110 (2005) .
-\item Two sentences have been slightly modified in the last paragraph of Sec.~V to discuss the change of slope in the transition line of the MIT and its possible effect on the overall shape of the vertex divergence lines upon including non-local correlations.
-\item In Sec.~A of the Appendix, we added the missing information about the sizes of the frequency grids for the low temperature calculations.
-\item An additional sentence providing more details about our choice of the frequency
-box sizes for the evaluation of $N_{\lambda<0}$, has been added to Sec.~A of the Appendix.
-\item At the end of Sec.~A of the Appendix, we adjusted the number of CPU hours.
-
-
-
-
-
-
-\end{enumerate}
-
-
-
-\end{document}
\ No newline at end of file
diff --git a/second_reply.tex b/second_reply.tex
deleted file mode 100644
index d716d40e75fbe383c188da0dd6d1d1f1075a1d12..0000000000000000000000000000000000000000
--- a/second_reply.tex
+++ /dev/null
@@ -1,227 +0,0 @@
-\documentclass[a4paper,11pt]{report}
-
-\usepackage{graphicx}
-\usepackage{enumitem}
-\usepackage[export]{adjustbox}
-\usepackage{float}
-\usepackage{color}
-\usepackage{amsmath}
-%\usepackage{xcolor}
-\usepackage[dvipsnames]{xcolor}
-%\usepackage[font=small,labelfont=bf,tableposition=top]{caption}
-%\DeclareCaptionLabelFormat{andtable}{#1~#2 \& \tablename~\thetable}
-\usepackage{ulem}
-
-\begin{document}
-\pagestyle{empty}
-
-\newcommand{\SA}[1]{{\color{purple} #1}}
-\newcommand{\SAc}[1]{{\color{purple} SA: #1}}
-\newcommand{\AT}[1]{{\color{teascalingl} #1}}
-\newcommand{\ATc}[1]{{\color{teal}AT: #1}}
-\newcommand{\old}[1]{{\color{gray}{\sout{#1}}}}
-\newcommand{\MP}[1]{{\color{orange} #1}}
-\newcommand{\MR}[1]{{\color{Green} #1}}
-\newcommand{\MPc}[1]{{\color{orange}{ MP: #1}}}
-\newcommand{\TD}[1]{{\color{red}{ TODO: #1}}}
-
-
-
-
-\noindent
-Dear Dr. Kancharla,
-
-\vskip 8mm
-
-\noindent
-Thank you for forwarding the two Referee reports on our manuscript:
-
-\begin{center}
-{\sl The highly nonperturbative nature of the Mott metal-insulator transition:\\Two-particle vertex divergences in the coexistence region}
-\end{center}
-
-
-\noindent
-We are glad that our work has been very positively evaluated by Referee 2. In particular, she/he found that our {\sl “results are both sound and important and thus
-warrant publication as a regular paper in PRBâ€}. Further, she/he acknowledges that her/his comments have been {\sl ``fully addressed}, and {\sl ``led to a fresh insight''} on {\sl ``the analytical results by S. Kehrein,
-PRL 81, 3912 (1998)''}.
-
-\noindent
-In the second report of Referee 1 an observation on a rather specific aspect of our findings has been made, related to the interpretation of the $T\rightarrow 0$ extrapolated data at the Mott transition.
-While we have explicitly addressed her/his concern in our reply, explaining why they are not grounded, and also slightly modified some wording in the manuscript to avoid possible misunderstandings, we note that Referee 2, after explicitly examining Referee 1 observation, fully supports the validity of our analysis. In particular,
-she/he finds our previous response to Referee 1 {\sl ``consistent with a number of negative eigenvalues along the $U_c$ and
-$U_{c2}$ lines converging to that of $U_{c0}$ at $T=0$''} and that the {\sl ``other referee's doubt concerning the soundness of the paper seems not
-to be valid.''}, underlying that her/his {\sl ``previous recommendation [of publishing the manuscript] still holds, definitely''}.
-
-
-\noindent \\
-Hence, we hope that after the inconsistency of the new point raised by Referee 1 has been clarified, and in consideration of the validity and the importance of our finding highlighted by Referee 2, our (slightly revised) manuscript can be now accepted for publication in Physical Review B. \\
-
-\noindent
-Thank you very much for your assistance,
-
-\vskip 7mm
-
-\noindent
-Sincerely yours, \\
-
-\vskip 4mm
-
-\noindent
-M.~Pelz, S.~Adler, M.~Reitner, and A.~Toschi
-
-\vskip 10mm
-
-\newpage
-
-\newpage
-
-\pagestyle{plain}
-\pagenumbering{arabic}
-
-
-\noindent
-{\bf REPLY TO THE FIRST REFEREE}\\
-
-\noindent
-%We thank the Referee for carefully reading our manuscript, for the positive evaluation of our work and her/his constructive observations. \\
-
-
-\noindent
-%he Referee has asked us to consider specific points to be addressed prior to publications.
-%We have regarded all of them very thoroughly and included the corresponding changes into the manuscript text and figures.
-%pecifically, we report below a detailed reply to all Referee's observations.
-
-\noindent
-The new critique raised by the Referee 1 in her/his second report is the following:\\
-
-\noindent
-{\color{blue}
- Unfortunately, I cannot understand the authors' logic that the number
-of the vertex divergence lines, $N_{\lambda < 0}$, should change
-continuously across the critical $U$lines ($U_c$ and $U_{c2}$) at zero
-temperature ($T = 0$). In my view, all the results shown in the
-manuscript and the reply indicate the opposite: the number changes
-discontinuously across the lines, and the discrete change becomes
-divergent at $T = 0$.
-
-In their manuscript and reply, the authors mix up the argument based
-on inverse temperature $\beta$ and that on temperature T, resulting in
-the wrong claim. They should not be mixed; one should stick to a
-single choice to be consistent.
-
-If we use T, then it's clear that the critical U lines meet at a
-single point, $U = U_{c0}$ and $T = 0$. However, if we use $\beta$ (which is
-preferred by the authors), then the lines do not meet; as depicted in
-the left panel of Fig. 1 of the reply, the two lines, $U_c$ vs. $\beta$
-and $U_{c2}$ vs. $\beta$, are divergently separated along the $\beta$ axis, in
-the limit of $U_c, U_{c2} \to U_{c0}$. So the authors' logic following their
-Eqs. (3) and (4) in their reply do not make sethense, since those
-equations cannot capture the fact that $N_{\lambda < 0}$ is divergent
-over the infinite interval of $\beta$ at $U = U_{c0}$. Especially, the
-phrase above Eq. (4), "the pole at $(U = U_c0, \beta \to \infty)$", is
-wrong!
-
-The same problem appears in the authors' argument on the continuity
-across the phase transition between the metallic and insulating
-solutions. In the rightmost panels of Figs. 6 and 7 in the manuscript,
-the authors perform the scaling analysis of the $N_{\lambda < 0}$ at
-critical U's, as functions of $\beta$. As $\beta$ increases, the lines
-further separate. Given this, how can one conclude that the lines will
-meet in the infinite $\beta $limit?
-
-To have $N_{\lambda < 0}$ evolve continuously across the phase
-transition at T = 0, the only resort is that $N_{\lambda < 0}$ changes
-its scaling behavior at temperatures lower than the values treated by
-the authors.
-
-I find this issue more critical than I thought before. When I wrote my
-first report, I assumed that the authors' continuity claim was a typo
-or I missed some logic. But now, I suspect that the authors first fix
-their conclusion ('$N_{\lambda < 0}$ should change continuously at T$ =
-0$') and then invent some reasoning, which contradicts their own data.
-This part harms the soundness of the whole paper, even though the rest
-of the manuscript is fine to me.}\\
-
-Eventually, the Referee appears to disagree only on one very specific aspect of our work, namely on the interpretation of the extrapolation of our finite temperature results to zero temperature at the MIT, in terms of the MIT properties. Below we clarify, why the the Referee's doubt is not founded.
-
-In particular, her/his doubt can be essentially summarized in two points:\\
-
-\textbf{From the viewpoint of the Referee, we would have unreasonably inferred from our extrapolations that the number of the vertex divergence lines, $N_{\lambda < 0}$, should change continuously across the critical $U$-lines ($U_c$ and $U_{c2}$) at zero temperature ($T = 0$). Thereby, the Referee does not agree that a single-valued function $N_{\lambda<0}(T,U)$ depending on two variables $U,T$, being analytic except for a pole at $T=0, U=U_c$, can show different divergent behaviors along two different paths to the pole, as we observe. Therefore, she/he concludes that at lower temperatures, there {\sl must} be a different scaling behavior than the one we reported.}\\
-
-While in general every extrapolation procedure might overlook the onset of a different behavior in the not-accessible parameter regime, this specific criticism of the Referee appears unfounded. In fact, the possibility of observing different diverging behaviors along differing paths in the variable-space is a \underline{general property} of the functions of more than one variable, and not a consequence of our choice of parametrizing the path down to $T \rightarrow 0$, e.g., in terms of $\beta$ instead of $T$ (Note: $\beta$ has been chosen, since $N_{\lambda < 0}$ shows a clear functional dependence on it).
-
-This can be readily seen by looking at the following (merely illustrative) simple example: Let us consider the two-variable, single-valued function $f(x,y)=\frac{1}{x^4+y^2}$, having a pole at the origin, and the two paths $(x(t),y(t))=(0,t)$ and $(x(t),y(t))=(t,0)$ for $t\to 0$. Along both paths the observed divergent behavior is different ($t^{-4}$ and $t^{-2}$, respectively), however the limit of $f(x\to 0,y\to 0) = \infty$ is uniquely defined.
-
-For our extrapolations of $N_{\lambda<0}(T,U)$ to $U_{c}$ at $T=0$ in the metallic phase, the singularity of $N_{\lambda<0}$ is reached along all paths to $T=0$ at $U_{c2}$, always yielding $N_{\lambda <0}(T=0,U=U_{c2}) \to \infty$. This is evidently consistent with the possible existence of a single-valued function $N_{\lambda <0}$ in the whole metallic region characterized by a pole-singularity at $(T, U)=(0, U_{c2})$. In such case, considering the systematic divergence of $N_{\lambda <0}(T \! =\! 0,U \! \geq \! U_{c2})$ in the whole Mott insulating regime, our results would not contradict
-\footnote{Note that we have intentionally used this cautious formulation (here, as well as in the manuscript) because in any $T \!= \!0$-extrapolation problem one cannot a priori exclude that a different scaling behavior may emerge at temperatures lower than the values which are explicitly calculated. This general consideration is, however, not related to the Referee criticism. In any case, let us stress here that we consider the emergence of different scaling for $N_{\lambda<0}$ at $U_{c}(T)$ at lower temperatures than those accessible to our DMFT calculations \underline{unlikely}, since our lowest temperature values already lie \underline{below} the ``effective Kondo temperature'' $T_K$ [P. Chalupa {\sl et al.}, Phys. Rev. Lett. {\bf 126}, 056403 (2021)] of the auxiliary AIMs of the DMFT solutions and no other relevant temperature scale is expected to emerge below $T_K$.} the possibility of observing a ``continuous behavior'' of some related physical quantities, such as, e.g., the on-site charge/spin static response across the Mott MIT.
-
- We note that the Referee criticism could have been valid, only if our extrapolations had yielded \underline{different values} along the different paths, since, in such a situation, the results would have indeed suggested the presence of a first-order-like discontinuous behavior at the MIT. Since this feature is evidently not present in our data, the cautious statements made in our manuscript fully stand.\\
-
-
-\textbf{The Referee states that we unreasonably conclude that the number of the vertex divergence lines should change continuously across the phase transition at zero temperature ($T = 0$), in contrast to our own extrapolations.}\\
-
-From our extrapolations, we see the metallic solutions along $U_c(T)$ and $U_{c2}(T)$ share a \underline{distinctive feature} with the insulating solution, which is the divergent value of $N_{\lambda <0}$ for $T\to 0$. This is in contrast to a finite number of $N_{\lambda<0}$ at $T=0$ for any path of metallic solutions ending at $U<U_c$ for $T\rightarrow 0$.
-These findings would be \underline{consistent}, as we state in the manuscript, with the existence of a \underline{unique} function describing the increase of $N_{\lambda<0}(U)$ along the $T=0$-axis up to its divergence at $U \geq U_{c2}(T\!=\!0)$, and, hence, \underline{do not rule out} a smooth $T=0$-evolution of the corresponding $\chi_c^{\nu,\nu'}$ across $U = U_{c2}(T\!=\!0)$, as well as of the related physical charge response of the system.
-
-
-We note that in the manuscript, we clearly state that the difference in the number of negative eigenvalues between the metallic and insulating solution $\Delta N_{\lambda<0}$ is growing for $\beta \to \infty$ along $U_c(\beta)$ ``\textit{On a more quantitative level, we note that $\Delta N_{\lambda<0}$ between the PM and the PI along $U_c$ increases with decreasing temperature})''. And, having carefully considered the first Referee report, in the first revised version of the manuscript we have also removed the potentially misleading sentence on the continuous change of the number of $N_{\lambda<0}$ at the transition at $T=0$ (as the word ''continuous'' is mathematically not appropriate for a singular behavior).
-As we explained before, however, the different singular behavior observed along different path does not affect any other statements/consideration made in the manuscript. \\
-
-In conclusion, we think that the revision of the manuscript we carefully made on the line of the comments of both Referees has clearly improved the quality and the potential impact of our work (as also explicitly acknowledged by the second Referee). Hence, having illustrated why the last doubt of the first Referee does not pose any problem for the conclusions (cautiously) made in the revised manuscript, we believe this should be considered now ready to be published in Physical Review B as a regular article.
-
-
-%However, we do recognize that the statement in the reply (not the manuscript) about $\Delta N_{\lambda<0}$ at $T=0$ was not correct, confusing and should have been avoided (we wanted to emphasize the similarity between the two solutions at $T=0$ and $U_c$):``\textit{Nevertheless, for $T=0$, the PM and PI extrapolations yield both an infinite value, similar to $U_{c2}(\beta)$ at $U_{c}(\beta)$ in the PM solution, suggesting $\Delta N_{\lambda<0}(T=0)=0$ at the critical point $U_c$ for zero temperature.}'' However, we firmly repudiate the following statement by the referee ``the authors first fix
-%their conclusion and then invent some reasoning, which contradicts their own data''.
-
-
-
-\newpage
-
-\noindent
-{\bf REPLY TO THE SECOND REFEREE } \\
-
-We thank the Referee for reviewing our modified manuscript and for her/his overall appreciation of our work.
-We are also very grateful to the Referee for carefully examining the new criticism made by the first Referee in her/his second report and for explicitly confirming the validity of our results and interpretations. \\
-
-We believe that, the revision of our manuscript that we specifically made to answer to the comments of the Referee has significantly improved the potential impact of our work, as these changes have allowed to directly link our new findings on the two-particle level with fundamental open questions about the theoretical interpretation of the Mott-Hubbard MIT in DMFT. \\
-
-
-Hence, we hope that our manuscript can be soon accepted for publication as a regular article in Physical Review B.
-
-
-\noindent
-{\color{blue}
-} \\
-
-
-\newpage
-
-\noindent
-{\bf LIST OF CHANGES}
-
-\vskip 5mm
-
-\begin{enumerate}
-
-\item On page 7 in the second paragraph of section III.D we fixed a typo by adding a missing ``the'' in the sentence ``In the right panel, then, we report our ...''
-
-\item On page 8 in section III.D we changed the following sentence ``From our extrapolations of $N_{\lambda<0}(\beta)$ we can now determine the overall behavior of $N_{\lambda<0}$ at $T\! =\! 0$ across
-the MIT.'' to ``From our extrapolations of $N_{\lambda<0}(\beta)$ we can now try to estimate the overall behavior of $N_{\lambda<0}$ at $T\! =\! 0$ across
-the MIT.''
-to make the phrasing more clear.
-
-\item On page 8 at the end of section III.D we have modified the last sentence and added a footnote, to further clarify the discussion about our extrapolations of $N_{\lambda<0}$ to $T=0$.:\\
-
-However, as the asymptotic value of $N_{\lambda<0}(\beta \rightarrow \infty)$ always diverges when approaching $U_c(T\!=\!0) \!= \! U_{c2}(T\!=\!0)$ independently of the path, our extrapolated results for $N_{\lambda<0}(\beta \rightarrow \infty)$ appear consistent {\sl{(footnote: In particular, the $T\!=\!0$-extrapolated behavior of $N_{\lambda<0}$ could be consistent with the existence of a unique function describing the increase of $N_{\lambda<0}$ for increasing $U$ along the $T\!=\!0$-axis up to its divergence at $U=U_{c2}(T\!=\!0)$, and, hence, does not rule out the possibility of a smooth evolution of the generalized susceptibility $\boldsymbol{\chi}^{\nu \nu'}_c$ and of the corresponding physical response function.)}} with the smooth evolution of several physical quantities (such as the quasiparticle spectral weight, the double-occupancy, etc.) numerically reported [citations: Karski et al., PRB 72, 113110 – (2005); Raas et al.,PRB 79, 115136 (2009)] at the $T=0$-Mott MIT.
-
-
-
-
-
-
-\end{enumerate}
-
-
-
-\end{document}
\ No newline at end of file
diff --git a/soley_foreditor.tex b/soley_foreditor.tex
deleted file mode 100644
index 0f5bdbce2a06d1baf8e9d0eb4cd14d84f8f60d6f..0000000000000000000000000000000000000000
--- a/soley_foreditor.tex
+++ /dev/null
@@ -1,54 +0,0 @@
-\documentclass[a4paper,11pt]{report}
-
-\usepackage{graphicx}
-\usepackage{enumitem}
-\usepackage[export]{adjustbox}
-\usepackage{float}
-\usepackage{color}
-\usepackage{amsmath}
-%\usepackage{xcolor}
-\usepackage[dvipsnames]{xcolor}
-%\usepackage[font=small,labelfont=bf,tableposition=top]{caption}
-%\DeclareCaptionLabelFormat{andtable}{#1~#2 \& \tablename~\thetable}
-\usepackage{ulem}
-
-\begin{document}
-\pagestyle{empty}
-
-\newcommand{\SA}[1]{{\color{purple} #1}}
-\newcommand{\SAc}[1]{{\color{purple} SA: #1}}
-\newcommand{\AT}[1]{{\color{teascalingl} #1}}
-\newcommand{\ATc}[1]{{\color{teal}AT: #1}}
-\newcommand{\old}[1]{{\color{gray}{\sout{#1}}}}
-\newcommand{\MP}[1]{{\color{orange} #1}}
-\newcommand{\MR}[1]{{\color{Green} #1}}
-\newcommand{\MPc}[1]{{\color{orange}{ MP: #1}}}
-\newcommand{\TD}[1]{{\color{red}{ TODO: #1}}}
-
-
-
-\noindent
-Dear Dr. Kancharla, \\
-
-\noindent
-We were quite surprised by the second report of the first Referee.
-While it is always possible to push forward a new criticism, the new observation of the Referee (indeed regarding a side-aspect of our work) appears quite weak, and formulated in a way we do not consider as fully professional (see, e.g., {\sl ``I suspect that the authors first fix
-their conclusion and then invent some reasoning, which contradicts their own data.''}).
-We found this (unscientific) comment somewhat offending, since by revising our manuscript we have carefully considered all comments of both Referees, and worked seriously to address them in our first Reply, as explicitly acknowledged by the second Referee.
-
-In any case, in agreement with the Referee 2, we think that the manuscript impact has relevantly improved after the first revision based on the comments of both Referees, since it provides now the solution of apparent contradictions in the fundamental understanding of the Mott-Hubbard metal-insulator transition in DMFT, which have remained debated for more than two decades!
-
-Hence, since the new criticism (on a quite marginal aspect of the interpretation) made by the first Referee is *not* valid, as the second Referee explicitly double-checked and we also clearly illustrate in our second reply, and considering the significance of our findings, we hope that our manuscript can be now considered ready to be published on Physical Review B as a regular article.\\
-
-\noindent
-Thank you very much for your assistance,\\
-
-
-\vskip 4mm
-
-\noindent
-M.~Pelz, S.~Adler, M.~Reitner, and A.~Toschi
-
-
-
-\end{document}
\ No newline at end of file