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few typos in Pade section

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17
Manuscript/EPAWTFT.bbl
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@ -6,7 +6,7 @@
%Control: page (0) single %Control: page (0) single
%Control: year (1) truncated %Control: year (1) truncated
%Control: production of eprint (0) enabled %Control: production of eprint (0) enabled
\begin{thebibliography}{139}% \begin{thebibliography}{141}%
\makeatletter \makeatletter
\providecommand \@ifxundefined [1]{% \providecommand \@ifxundefined [1]{%
\@ifx{#1\undefined} \@ifx{#1\undefined}
@ -1311,6 +1311,21 @@
{Loos}},\ }\href {\doibase 10.1063/1.4790613} {\bibfield {journal} {\bibinfo {Loos}},\ }\href {\doibase 10.1063/1.4790613} {\bibfield {journal} {\bibinfo
{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {138}},\ \bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {138}},\ \bibinfo
{pages} {064108} (\bibinfo {year} {2013})}\BibitemShut {NoStop}% {pages} {064108} (\bibinfo {year} {2013})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Pavlyukh}(2017)}]{Pavlyukh_2017}%
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
{Pavlyukh}},\ }\href {\doibase 10.1038/s41598-017-00355-w} {\bibfield
{journal} {\bibinfo {journal} {Sci. Rep.}\ }\textbf {\bibinfo {volume}
{7}},\ \bibinfo {pages} {504} (\bibinfo {year} {2017})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Tarantino}\ and\ \citenamefont
{Di~Sabatino}(2019)}]{Tarantino_2019}%
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
{Tarantino}}\ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
{Di~Sabatino}},\ }\href {\doibase 10.1103/PhysRevB.99.075149} {\bibfield
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\bibitem [{\citenamefont {Gluzman}(2020)}]{Gluzman_2020}% \bibitem [{\citenamefont {Gluzman}(2020)}]{Gluzman_2020}%
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Manuscript/EPAWTFT.bib
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@ -1,13 +1,38 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-11-25 15:13:39 +0100 %% Created for Pierre-Francois Loos at 2020-11-25 22:26:55 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Pavlyukh_2017,
author = {Y. Pavlyukh},
date-added = {2020-11-25 22:26:26 +0100},
date-modified = {2020-11-25 22:26:52 +0100},
doi = {10.1038/s41598-017-00355-w},
journal = {Sci. Rep.},
pages = {504},
title = {Pade resummation of many-body perturbation theory},
volume = {7},
year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1038/s41598-017-00355-w}}
@article{Tarantino_2019,
author = {Tarantino, Walter and Di Sabatino, Stefano},
date-added = {2020-11-25 22:23:50 +0100},
date-modified = {2020-11-25 22:24:04 +0100},
doi = {10.1103/PhysRevB.99.075149},
journal = {Phys. Rev. B},
pages = {075149},
title = {Diagonal Pad\'e approximant of the one-body Green's function: A study on Hubbard rings},
volume = {99},
year = {2019},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.99.075149},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.99.075149}}
@article{Goodson_2000, @article{Goodson_2000,
author = {Goodson,David Z.}, author = {Goodson,David Z.},
date-added = {2020-11-25 10:05:02 +0100}, date-added = {2020-11-25 10:05:02 +0100},

15
Manuscript/EPAWTFT.tex
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@ -1238,16 +1238,17 @@ We refer the interested reader to more specialised reviews for additional inform
\subsection{Pad\'e approximant} \subsection{Pad\'e approximant}
%==========================================% %==========================================%
The inability of Taylor series to model properly the energy function $E(\lambda$) can be simply understood by the fact that one aims at modelling a complicated function with potentially poles and singularities by a simple polynomial of finite order. The inability of Taylor series to model properly the energy function $E(\lambda$) can be simply understood by the fact that one aims at modelling a complicated function with potentially poles and singularities by a simple polynomial of finite order.
This description can be significantly improved thanks to Pad\'e approximant. \cite{Pade_1892,BakerBook} A Taylor series just does not have enough flexibility for this job.
Nonetheless, the description of complex energy functions can be significantly improved thanks to Pad\'e approximant, \cite{Pade_1892} and related techniques. \cite{BakerBook,BenderBook}
According to Wikipedia, \textit{``a Pad\'e approximant is the best approximation of a function by a rational function of given order''}. According to Wikipedia, \textit{``a Pad\'e approximant is the best approximation of a function by a rational function of given order''}.
A $[d_A/d_B]$ Pad\'e approximant is defined as More specifically, a $[d_A/d_B]$ Pad\'e approximant is defined as
\begin{equation} \begin{equation}
\label{eq:PadeApp} \label{eq:PadeApp}
E_{[d_A/d_B]}(\lambda) = \frac{A(\lambda)}{B(\lambda)} = \frac{\sum_{k=0}^{d_A} a_k \lambda^k}{\sum_{k=0}^{d_B} b_k \lambda^k} E_{[d_A/d_B]}(\lambda) = \frac{A(\lambda)}{B(\lambda)} = \frac{\sum_{k=0}^{d_A} a_k \lambda^k}{\sum_{k=0}^{d_B} b_k \lambda^k}
\end{equation} \end{equation}
(with $b_0 = 1$), where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms according to power of $\lambda$. (with $b_0 = 1$), where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms according to power of $\lambda$.
Pad\'e approximants are extremely useful in many areas of physics and chemistry \cite{Loos_2013,Gluzman_2020} as they can model poles, which appears at the roots of the polynomial $B(\lambda)$. Pad\'e approximants are extremely useful in many areas of physics and chemistry \cite{Loos_2013,Pavlyukh_2017,Tarantino_2019,Gluzman_2020} as they can model poles, which appears at the roots of the polynomial $B(\lambda)$.
However, they are unable to model functions with square-root branch points, which are ubiquitous in the singularity structure of a typical perturbative treatment. However, they are unable to model functions with square-root branch points, which are ubiquitous in the singularity structure of a typical perturbative treatment.
Figure \ref{fig:PadeRMP} illustrates the improvement brought by Pad\'e approximants as compared to the usual Taylor expansion in the case of the RMP series of the Hubbard dimer for $U/t = 4.5$. Figure \ref{fig:PadeRMP} illustrates the improvement brought by Pad\'e approximants as compared to the usual Taylor expansion in the case of the RMP series of the Hubbard dimer for $U/t = 4.5$.
@ -1262,7 +1263,7 @@ Figure \ref{fig:PadeRMP} illustrates the improvement brought by Pad\'e approxima
%==========================================% %==========================================%
\subsection{Quadratic approximant} \subsection{Quadratic approximant}
%==========================================% %==========================================%
In a nutshell, the idea behind quadratic approximant is to model the singularity structure of the energy function $E(\lambda)$ via a generalized version of the square-root singularity expression \cite{Mayer_1985,Goodson_2011,Goodson_2019} In a nutshell, the idea behind quadratic approximant is to model the singularity structure of the energy function $E(\lambda)$ via a generalised version of the square-root singularity expression \cite{Mayer_1985,Goodson_2011,Goodson_2019}
\begin{equation} \begin{equation}
\label{eq:QuadApp} \label{eq:QuadApp}
E(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ] E(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ]
@ -1276,7 +1277,7 @@ where
& &
R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k
\end{align} \end{align}
are polynomials, such that $d_P + d_Q + d_R = n - 1$, where $n$ is the truncation order of the Taylor of $E(\lambda)$. are polynomials, such that $d_P + d_Q + d_R = n - 1$, and $n$ is the truncation order of the Taylor series of $E(\lambda)$.
Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ie, Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ie,
\begin{equation} \begin{equation}
Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}} Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}}
@ -1291,8 +1292,8 @@ As shown in Ref.~\onlinecite{Goodson_2000}, quadratic approximants provide conve
For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant are quite poor approximation, but its $[1/0,1]$ version already model perfectly the RMP energy function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm i 4t/U$. For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant are quite poor approximation, but its $[1/0,1]$ version already model perfectly the RMP energy function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm i 4t/U$.
This is expected knowing the form of the RMP energy [see Eq.~\eqref{eq:E0MP}] which perfectly suits the purpose of quadratic approximants. This is expected knowing the form of the RMP energy [see Eq.~\eqref{eq:E0MP}] which perfectly suits the purpose of quadratic approximants.
We can anticipate that the singularity structure of the UMP energy function is going to be much more challenging to model properly, and this is indeed the case as the UMP energy function which contains three branches. We can anticipate that the singularity structure of the UMP energy function is going to be much more challenging to model properly, and this is indeed the case as the UMP energy function contains three branches.
However, by ramping up high enough the degree of the polynomials, one is able to get an accurate estimates of the radius of convergence of the UMP series as shown in Fig.~\ref{fig:QuadUMP}. However, by ramping up high enough the degree of the polynomials, one is able to get an accurate estimates of the radius of convergence of the UMP series as shown in Fig.~\ref{fig:QuadUMP} and Table \ref{tab:QuadUMP}.
\titou{Here comes a discussion of Fig.~\ref{fig:QuadUMP} and Table \ref{tab:QuadUMP}.} \titou{Here comes a discussion of Fig.~\ref{fig:QuadUMP} and Table \ref{tab:QuadUMP}.}

4
hugh_notes.txt
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@ -217,7 +217,7 @@ Class A) Monotonic convergence expected for systems in which the electron pairs
Class B) Initial oscillatory convergence with strong pair and three-electron correlation effects. Class B) Initial oscillatory convergence with strong pair and three-electron correlation effects.
eg. Ne, F, F^-, FH eg. Ne, F, F^-, FH
In these systems, there are closely spaced electron pairs that cluster in a small region of space. In these systems, there are closely spaced electron pairs that cluster in a small region of space.
One might imagine that this requires greater orbital relaxation, perhaps ``breating'' relaxation, One might imagine that this requires greater orbital relaxation, perhaps ``breathing'' relaxation,
to allow the electron pairs to become separated? Or maybe that it generally introduces stronger to allow the electron pairs to become separated? Or maybe that it generally introduces stronger
dynamic correlation effects? Orbital relaxation plus pair correlation comes through T1 T2 terms. dynamic correlation effects? Orbital relaxation plus pair correlation comes through T1 T2 terms.
@ -225,7 +225,7 @@ They observe both E(SDQ) and E(T) terms negative in Class A systems, but E_MP5(S
in Class B. Oscillations in the T correlation terms drive the oscillatory convergence behaviour. This behaviour in Class B. Oscillations in the T correlation terms drive the oscillatory convergence behaviour. This behaviour
does not appear to be caused by multiconfigurational effects, but may be amplified by them. does not appear to be caused by multiconfigurational effects, but may be amplified by them.
Class B has more improtant orbital relaxation effects and three-electron correlation than Class A. Class B has more important orbital relaxation effects and three-electron correlation than Class A.