Altered fig1b. Added comment about EPs converging on real axis.

Manuscript/EPAWTFT.bbl

@@ -6,7 +6,7 @@
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-\begin{thebibliography}{180}%
+\begin{thebibliography}{179}%
 \makeatletter
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@@ -50,15 +50,6 @@
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-\bibitem [{\citenamefont {Dirac}\ and\ \citenamefont
-  {Fowler}(1929)}]{Dirac_1929}%
-  \BibitemOpen
-  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {P.~A.~M.}\
-  \bibnamefont {Dirac}}\ and\ \bibinfo {author} {\bibfnamefont {R.~H.}\
-  \bibnamefont {Fowler}},\ }\href {\doibase 10.1098/rspa.1929.0094} {\bibfield
-  {journal} {\bibinfo  {journal} {Proc. R. Soc. Lond. A}\ }\textbf {\bibinfo
-  {volume} {123}},\ \bibinfo {pages} {714} (\bibinfo {year}
-  {1929})}\BibitemShut {NoStop}%
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   \BibitemOpen
   \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.~W.~S.}\
@@ -1441,9 +1432,9 @@
   {G.~Rauhut}}\ and\ \bibinfo {author} {\bibfnamefont {H.-J.}\ \bibnamefont
   {Werner}},\ }\href {\doibase
   10.1002/(SICI)1096-987X(199808)19:11<1241::AID-JCC4>3.0.CO;2-K} {\bibfield
-  {journal} {\bibinfo  {journal} {J. Comp. Chem.}\ }\textbf {\bibinfo {volume}
+  {journal} {\bibinfo  {journal} {J. Comput. Chem.}\ }\textbf {\bibinfo
-  {19}},\ \bibinfo {pages} {1241} (\bibinfo {year} {1998})}\BibitemShut
+  {volume} {19}},\ \bibinfo {pages} {1241} (\bibinfo {year}
-  {NoStop}%
+  {1998})}\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Sch{\"u}tz}\ \emph {et~al.}(1999)\citenamefont
   {Sch{\"u}tz}, \citenamefont {Hetzer},\ and\ \citenamefont
   {Werner}}]{Schutz_1999}%

Manuscript/EPAWTFT.tex

@@ -297,8 +297,9 @@ unless otherwise stated, atomic units will be used throughout.
     \end{subfigure}
 	\caption{%
 	Exact energies for the Hubbard dimer ($U=4t$) as functions of $\lambda$ on the real axis (\subref{subfig:FCI_real}) and in the complex plane (\subref{subfig:FCI_cplx}).
-    Only the interacting closed-shell singlets are plotted in the complex plane, becoming degenerate at the EP (black dot).
+    Only the interacting closed-shell singlets are shown in the complex plane, becoming degenerate at the EP (black dot).
-    The contour followed around the EP in order to interchange states is also represented.
+    Following a contour around the EP (black solid) interchanges the states, while a second rotation (black dashed)
+    returns the states to their original energies.
 	\label{fig:FCI}}
 \end{figure*}
 
@@ -357,6 +358,8 @@ These $\lambda$ values correspond to so-called EPs and connect the ground and ex
 Crucially, the energy surface becomes non-analytic at $\lambda_{\text{EP}}$ and a square-root singularity forms with two branch cuts running along the imaginary axis from $\lambda_{\text{EP}}$  to $\pm \i \infty$ (see Fig.~\ref{subfig:FCI_cplx}).
 On the real $\lambda$ axis, these EPs lead to the singlet avoided crossing at $\lambda = \Re(\lambda_{\text{EP}})$.
 The ``shape'' of this avoided crossing is related to the magnitude of $\Im(\lambda_{\text{EP}})$, with smaller values giving a ``sharper'' interaction.
+\hugh{In the limit $U/t \to 0$, the two EPs converge at $\lep \to 0$ to create a conical intersection with 
+a gradient discontinuity on the real axis.}
 
 Remarkably, the existence of these square-root singularities means that following a complex contour around an EP in the complex $\lambda$ plane will interconvert the closed-shell ground and excited states (see Fig.~\ref{subfig:FCI_cplx}).
 This behaviour can be seen by expanding the radicand in Eq.~\eqref{eq:singletE} as a Taylor series around $\lambda_{\text{EP}}$ to give
@@ -1174,16 +1177,21 @@ divergences caused by the MP critical point.
 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} 
-Meanwhile, as an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis, eventually forming a critical point.
+Meanwhile, as an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis.
-\hugh{The existence of an EP \emph{on} the real axis is therefore diagnostic of a QPT.\cite{Cejnar_2005, Cejnar_2007a}}
+\hugh{When these points converge on the real axis, they effectively ``annihilate'' each other and no longer behave as EPs.
+Instead, they form a ``critical point'' singularity that resembles a conical intersection, and 
+the convergence of a pair of complex-conjugate EPs 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 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
+The MP critical point and corresponding $\beta$ singularities in a finite basis must therefore be modelled by pairs of
-that tend towards the real axis, exactly as described by Sergeev \etal\cite{Sergeev_2005}
+complex-conjugate EPs 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.
+\hugh{Notably, since the exact MP critical point corresponds to the interaction between a bound state
+and the continuum, its functional form is more complicated than a conical intersection and remains an open question.}
 
 %=======================================
 \subsection{Critical Points in the Hubbard Dimer}