review Hugh changes

Manuscript/EPAWTFT.bbl

@@ -6,7 +6,7 @@
 %Control: page (0) single
 %Control: year (1) truncated
 %Control: production of eprint (0) enabled
-\begin{thebibliography}{131}%
+\begin{thebibliography}{132}%
 \makeatletter
 \providecommand \@ifxundefined [1]{%
  \@ifx{#1\undefined}
@@ -1188,6 +1188,15 @@
   {journal} {\bibinfo  {journal} {J. Phys. A: Math. Theor.}\ }\textbf {\bibinfo
   {volume} {40}},\ \bibinfo {pages} {581} (\bibinfo {year} {2007})}\BibitemShut
   {NoStop}%
+\bibitem [{\citenamefont {Kais}\ \emph {et~al.}(2006)\citenamefont {Kais},
+  \citenamefont {Wenger},\ and\ \citenamefont {Wei}}]{Kais_2006}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
+  {Kais}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Wenger}}, \ and\
+  \bibinfo {author} {\bibfnamefont {Q.}~\bibnamefont {Wei}},\ }\href {\doibase
+  https://doi.org/10.1016/j.cplett.2006.03.035} {\bibfield  {journal} {\bibinfo
+   {journal} {Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume} {423}},\ \bibinfo
+  {pages} {45 } (\bibinfo {year} {2006})}\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Goodson}(2019)}]{Goodson_2019}%
   \BibitemOpen
   \bibfield  {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont

Manuscript/EPAWTFT.tex

@@ -1142,12 +1142,11 @@ regardless of whether this point is inside or outside the complex unit circle. \
 %=======================================
 
 % STILLINGER INTRODUCES THE CRITICAL POINT
-\hugh{%
 In the early 2000's, Stillinger reconsidered the mathematical origin behind the divergent series with odd-even
 sign alternation.\cite{Stillinger_2000} 
 This type of convergence behaviour corresponds to Cremer and He's class B systems with closely spaced
 electron pairs and includes \ce{Ne}, \ce{HF}, \ce{F-}, and \ce{H2O}.\cite{Cremer_1996}
-Stillinger proposed that the divergence of these series occurs arise from a dominant singularity
+Stillinger proposed that the divergence of these series arise from a dominant singularity
 on the negative real $\lambda$ axis, corresponding to a multielectron autoionisation threshold.\cite{Stillinger_2000}
 To understand Stillinger's argument, consider the paramterised MP Hamiltonian in the form
 \begin{multline}
@@ -1171,12 +1170,10 @@ This autoionisation effect is closely related to the critial point for electron
 atoms (see Ref.~\onlinecite{Baker_1971}).
 Furthermore, a similar set of critical points exists along the positive real axis, corresponding to single-electron ionisation
 processes.\cite{Sergeev_2005}
-}
 
 % CLASSIFICATIONS BY GOODSOON AND SERGEEV
-\hugh{%
 To further develop the link between the critical point and types of MP convergence, Sergeev and Goodson investigated
-the relationship with the location of the dominant singularity that controls the radius of convergence.\cite{Goodson_2004}.
+the relationship with the location of the dominant singularity that controls the radius of convergence.\cite{Goodson_2004}
 They demonstrated that the dominant singularity in class A corresponds to a dominant EP with a positive real component, 
 with the magnitude of the imaginary component controlling the oscillations in the signs of the MP 
 term.\cite{Goodson_2000a,Goodson_2000b}
@@ -1187,10 +1184,8 @@ HF potential $v^{\text{HF}}$ is relatively localised and the autoionization is f
 $\lambda$ values closer to the origin.
 With these insights, they regrouped the systems into new classes: i) $\alpha$ singularities which have ``large'' imaginary parts, 
 and ii) $\beta$ singularities which have very small imaginary parts.\cite{Goodson_2004,Sergeev_2006} 
-}
 
 % RELATIONSHIP TO BASIS SET SIZE
-\hugh{%
 The existence of the MP critical point can also explain why the divergence observed by Olsen \etal\ in the \ce{Ne} atom 
 and \ce{HF} molecule occured when diffuse basis functions were included.\cite{Olsen_1996}
 Clearly diffuse basis functions are required for the electrons to dissociate from the nuclei, and indeed using
@@ -1200,10 +1195,9 @@ by a cluster of sharp avoided crossings between the ground state and high-lying
 Alternatively, Sergeev \etal\ demonstrated that the inclusion of a ``ghost'' atom also
 allows the formation of the critical point as the electrons form a bound cluster occuping the ghost atom orbitals.\cite{Sergeev_2005}
 This effect explains the origin of the divergence in the \ce{HF} molecule as the \ce{F} valence electrons jump to \ce{H} 
-for a sufficiently negative $\lambda$.\cite{Sergeev_2005}
+for a sufficiently negative $\lambda$ value.\cite{Sergeev_2005}
 Furthermore, the two-state model of Olsen \etal{}\cite{Olsen_2000} was simply too minimal to understand the complexity of 
 divergences caused by the MP critical point.
-}
 
 % BASIS SET DEPENDENCE (INCLUDE?)
 %Finally, it was shown that $\beta$ singularities are very sensitive to changes in the basis set but not to the bond stretching. 
@@ -1223,21 +1217,19 @@ divergences caused by the MP critical point.
 %This explains that Olsen \textit{et al.}, because they used a simple two-state model, only observed the first singularity of this cluster of singularities causing the divergence. \cite{Olsen_2000}
 
 % RELATIONSHIP TO QUANTUM PHASE TRANSITION
-\hugh{%
 When a Hamiltonian is parametrised by a variable such as $\lambda$, the existence of abrupt changes in the 
 eigenstates as a function of $\lambda$ indicate the presence of a zero-temperature quantum phase transition (QPT).%
 \cite{Heiss_1988,Heiss_2002,Borisov_2015,Sindelka_2017,CarrBook,Vojta_2003,SachdevBook,GilmoreBook} 
-As an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis.
+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 a QPT within the perturbation theory approximation.
 However, a conventional QPT can only occur in the thermodynamic limit, which here is analogous to the complete 
-basis set limit.\cite{Kais_2006}
+basis set limit in our case.\cite{Kais_2006}
 The MP critical point $\beta$ singularity in a finite basis must therefore be modelled by pairs of EPs
-that tend towards the real axis, exactly as described by Sergeev \etal.\cite{Sergeev_2005}
+that tend towards the real axis, exactly as described by Sergeev \etal\cite{Sergeev_2005}
 In contrast, $\alpha$ singularities correspond to large avoided crossings that are indicative of low-lying excited
 states which share the symmetry of the ground state,\cite{Goodson_2004} and are thus not manifestations of a QPT.
-}
 
 %=======================================
 \subsection{Critical Point in the Hubbard Dimer}