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Manuscript/EPAWTFT.tex

@@ -1622,6 +1622,7 @@ energy using low-order perturbation expansions.
 
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 \subsection{Shanks Transformation}
+\label{sec:Shanks}
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 While the Pad\'e and quadratic approximants can yield a convergent series representation
@@ -1729,19 +1730,20 @@ Seminal contributions from various research groups around the world have evidenc
 
 Later, these erratic behaviours were investigated and rationalised in terms of avoided crossings and singularities in the complex plane. 
 In that regard, it is worth highlighting the key contribution of Cremer and He who proposed a classification of the types of convergence: \cite{Cremer_1996} ``class A'' systems that exhibit monotonic convergence, and ``class B'' systems for which convergence is erratic after initial oscillations. 
-Further insights were brought thanks to a series of papers by Olsen and coworkers \cite{Christiansen_1996,Olsen_1996,Olsen_2000,Olsen_2019} where they employ a two-state model to dissect the various convergence behaviours of Hermitian and non-Hermitian perturbation series.
+Further insights were brought thanks to a series of papers by Olsen and coworkers \cite{Christiansen_1996,Olsen_1996,Olsen_2000,Olsen_2019} where they employed a two-state model to dissect the various convergence behaviours of Hermitian and non-Hermitian perturbation series.
 Building on the careful mathematical analysis of Stillinger who showed that the mathematical origin behind the divergent series with odd-even sign alternation is due to a dominant singularity on the negative real $\lambda$ axis, \cite{Stillinger_2000} Sergeev and Goodson proposed a more refined singularity classification: $\alpha$ singularities which have ``large'' imaginary parts, and $\beta$ singularities which have very small imaginary parts. \cite{Goodson_2000a,Goodson_2000b,Goodson_2004,Sergeev_2005,Sergeev_2006}
 We have further highlighted that these so-called $\beta$ singularities are connected to quantum phase transitions and symmetry breaking.
 
 Finally, we have discussed several resummation techniques, such as Pad\'e and quadratic approximants, that can be used to improve energy estimates for both convergent and divergent series.
 However, it is worth mentioning that the construction of these approximants requires high-order MP coefficients which are quite expensive to compute in practice.
+The Shanks transformation presented in Sec.~\ref{sec:Shanks} can, in some cases, alleviate this issue.
 
-Most of the concepts reviewed in the present manuscript has been illustrated on the symmetric (or asymmetric in one occasion) Hubbard dimer at half-filling.
+Most of the physical concepts and mathematical tools reviewed in the present manuscript has been illustrated on the symmetric (or asymmetric in one occasion) Hubbard dimer at half-filling.
 Although extremely simple, this clearly illustrates the obvious versatility of the Hubbard model to understand perturbation theory as well as other concepts such as Kohn-Sham density-functional theory (DFT), \cite{Carrascal_2015} linear-response theory, \cite{Carrascal_2018} many-body perturbation theory, \cite{Romaniello_2009,Romaniello_2012,DiSabatino_2015,Tarantino_2017}, DFT for ensembles, \cite{Deur_2017,Deur_2018,Senjean_2018,Sagredo_2018,Fromager_2020} and many others.
-We believe that the Hubbard dimer could then be used for further developments around perturbation theory.
+We believe that the Hubbard dimer could then be used for further developments and comprehension around perturbation theory.
 
 As a concluding remark and from a broader point of view, the present work shows that our understanding of the singularity structure of the energy is still incomplete.
-Yet, we hope that the present contribution will open new perspectives for the understanding of the physics of exceptional points in electronic structure theory.
+Yet, we hope that the present contribution will open new perspectives on the physics of exceptional points in electronic structure theory.
 
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 \begin{acknowledgements}