comment on confocal map

Manuscript/EPAWTFT.bbl

@@ -6,7 +6,7 @@
 %Control: page (0) single
 %Control: year (1) truncated
 %Control: production of eprint (0) enabled
-\begin{thebibliography}{132}%
+\begin{thebibliography}{133}%
 \makeatletter
 \providecommand \@ifxundefined [1]{%
  \@ifx{#1\undefined}
@@ -1261,6 +1261,13 @@
   {\bibinfo  {journal} {J. Phys. C.: Solid State Phys.}\ }\textbf {\bibinfo
   {volume} {18}},\ \bibinfo {pages} {3297} (\bibinfo {year}
   {1985})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Feenberg}(1956)}]{Feenberg_1956}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
+  {Feenberg}},\ }\href {\doibase 10.1103/PhysRev.103.1116} {\bibfield
+  {journal} {\bibinfo  {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume}
+  {103}},\ \bibinfo {pages} {1116} (\bibinfo {year} {1956})}\BibitemShut
+  {NoStop}%
 \bibitem [{\citenamefont {Surj{\'a}n}\ and\ \citenamefont
   {Szabados}(2000)}]{Surjan_2000}%
   \BibitemOpen

Manuscript/EPAWTFT.bib

@@ -1,13 +1,31 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-11-27 20:55:11 +0100 
+%% Created for Pierre-Francois Loos at 2020-12-01 13:28:00 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Feenberg_1956,
+	author = {Feenberg, Eugene},
+	date-added = {2020-12-01 13:27:51 +0100},
+	date-modified = {2020-12-01 13:27:57 +0100},
+	doi = {10.1103/PhysRev.103.1116},
+	issue = {4},
+	journal = {Phys. Rev.},
+	month = {Aug},
+	numpages = {0},
+	pages = {1116--1119},
+	publisher = {American Physical Society},
+	title = {Invariance Property of the Brillouin-Wigner Perturbation Series},
+	url = {https://link.aps.org/doi/10.1103/PhysRev.103.1116},
+	volume = {103},
+	year = {1956},
+	Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRev.103.1116},
+	Bdsk-Url-2 = {https://doi.org/10.1103/PhysRev.103.1116}}
+
 @article{Kais_2006,
 	abstract = {Finite size scaling for calculations of the critical parameters of the few-body Schr{\"o}dinger equation is based on taking the number of elements in a complete basis set as the size of the system. We show in an analogy with Yang and Lee theorem, which states that singularities of the free energy at phase transitions occur only in the thermodynamic limit, that singularities in the ground state energy occur only in the infinite complete basis set limit. To illustrate this analogy in the complex-parameter space, we present calculations for Yukawa type potential, and a Coulomb type potential for two-electron atoms.},
 	author = {Sabre Kais and Craig Wenger and Qi Wei},

Manuscript/EPAWTFT.tex

@@ -1238,7 +1238,7 @@ eigenstates as a function of $\lambda$ indicate the presence of a zero-temperatu
 Meanwhile, as an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis, eventually forming a critical point.
 The existence of an EP \emph{on} the real axis is therefore diagnostic of a QPT.\cite{Cejnar_2005, Cejnar_2007a}
 Since the MP critical point corresponds to a singularity on the real $\lambda$ axis, it can immediately be
-recognised as a QPT \hugh{with respect to varying the perturbation parameter $\lambda$}.
+recognised as a QPT with respect to varying the perturbation parameter $\lambda$.
 However, a conventional QPT can only occur in the thermodynamic limit, which here is analogous to the complete 
 basis set limit.\cite{Kais_2006}
 The MP critical point and corresponding $\beta$ singularities in a finite basis must therefore be modelled by pairs of EPs
@@ -1293,11 +1293,11 @@ which corresponds to strictly localising the two electrons on the left site.
 %\begin{equation}
 %   E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon).
 %\end{equation}
-With this representation, the parametrised \hugh{asymmetric} RMP Hamiltonian becomes
+With this representation, the parametrised asymmetric RMP Hamiltonian becomes
 \begin{widetext}
 \begin{equation}
-\label{eq:H_RMP}
-\hugh{\bH_\text{asym}\qty(\lambda)} = 
+\label{eq:H_asym}
+\bH_\text{asym}\qty(\lambda) = 
 \begin{pmatrix}
     2(U-\epsilon) - \lambda U &	-\lambda t				 &	-\lambda t	            &	0	        \\
     -\lambda t				  &	(U-\epsilon) - \lambda U &	0		                &	-\lambda t	\\
@@ -1421,7 +1421,7 @@ occurs exactly at $\lambda = 1$.
 In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided 
 crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}).
 We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes
-a new type of MP critical point and \hugh{represents a QPT as the perturbation parameter $\lambda$ is varied.}
+a new type of MP critical point and represents a QPT as the perturbation parameter $\lambda$ is varied.
 Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the 
 radius of convergence (see Fig.~\ref{fig:RadConv}).
 
@@ -1607,6 +1607,10 @@ Besides, they provide accurate estimates of the ground-state energy at $\lambda
 	\end{ruledtabular}
 \end{table}
 
+An interesting point raised in Ref.~\onlinecite{Goodson_2019} suggests that low-order quadratic approximants might struggle to model the correct singularity structure when the energy function has poles in both the positive and negative half-planes. 
+In such a scenario, the quadratic approximant will have the tendency to place its branch points in-between, potentially introducing singularities quite close to the origin.
+A simple potential cure for this consists in applying a judicious transformation (like a bilinear conformal mapping) which does not affect the points at $\lambda = 0$ and $\lambda = 1$. \cite{Feenberg_1956}
+
 %==========================================%
 \subsection{Analytic continuation}
 %==========================================%
@@ -1614,7 +1618,7 @@ Besides, they provide accurate estimates of the ground-state energy at $\lambda
 Recently, Mih\'alka \textit{et al.} studied the partitioning effect on the convergence properties of Rayleigh-Schr\"odinger perturbation theory by considering the MP and the EN partitioning as well as an alternative partitioning \cite{Mihalka_2017a} (see also Ref.~\onlinecite{Surjan_2000}).
 Taking as an example (in particular) the water molecule at equilibrium and at stretched geometries, they could estimate the radius of convergence via a quadratic Pad\'e approximant and convert divergent perturbation expansions to convergent ones in some cases thanks to a judicious choice of the level shift parameter.
 In a subsequent study by the same group, \cite{Mihalka_2017b} they use analytic continuation techniques to resum divergent MP series \cite{Goodson_2011} taking again as an example the water molecule in a stretched geometry.
-In a nutshell, their idea consists in calculating the energy of the system for several values of $\lambda$ for which the MP series is rapidly convergent (\ie, for $\lambda < r_c$), and to extrapolate the final energy to the physical system at $\lambda = 1$ via a polynomial- or Pad\'e-based fit. 
+In a nutshell, their idea consists in calculating the energy of the system for several values of $\lambda$ for which the MP series is rapidly convergent (\ie, for $\abs{\lambda} < r_c$), and to extrapolate the final energy to the physical system at $\lambda = 1$ via a polynomial- or Pad\'e-based fit. 
 However, the choice of the functional form of the fit remains a subtle task.
 This technique was first generalised by using complex scaling parameters and applying analytic continuation by solving the Laplace equation, \cite{Surjan_2018} and then further improved thanks to Cauchy's integral formula \cite{Mihalka_2019}
 \begin{equation}
@@ -1626,8 +1630,6 @@ Their method consists in refining self-consistently the values of $E(\lambda)$ c
 When the values of $E(\lambda)$ on the so-called contour are converged, Cauchy's integrals formula \eqref{eq:Cauchy} is invoked to compute the values at $E(\lambda=1)$ which corresponds to the final estimate of the FCI energy.
 The authors illustrate this protocol on the dissociation curve of \ce{LiH} and the stretched water molecule showing encouraging results. \cite{Mihalka_2019} 
 
-\titou{T2 will add a comment about Goodson's remark on the failure of low-order approximants when $\alpha$ and $\beta$ singularities are present.}
-
 %%%%%%%%%%%%%%%%%%%%
 \section{Conclusion}
 %%%%%%%%%%%%%%%%%%%%