intro

Manuscript/EPAWTFT.tex

@@ -1830,16 +1830,17 @@ 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 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}
+Further insights were provided by 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 axis, \cite{Stillinger_2000} Sergeev and Goodson proposed a 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.
+As mentioned earlier in this manuscript, turning low-order truncated MP series into convergent and systematically improvable series would be a highly desirable feature that could drastically improve the general applicability of such methods.
 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 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} thermal DFT, \cite{Smith_2016,Smith_2018} and many others.
+Although extremely simple, this clearly illustrates the obvious versatility of the Hubbard model to understand the subtle notions linked to the extension of perturbation theory into the complex plane, 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} thermal DFT, \cite{Smith_2016,Smith_2018}, correlated methods, \cite{} and many others.
 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.