comment on complex adiabatic connection, added figure legend

Manuscript/EPAWTFT.tex

@@ -499,6 +499,28 @@ and the ground-state RHF energy (Fig.~\ref{fig:HF_real})
 However, in the strongly correlated regime (large $U$), the closed-shell restriction on the orbitals prevents RHF from 
 correctly modelling the physics of the system with the two electrons on opposing sites.
 
+%%% FIG 3 (?) %%%
+% Analytic Continuation of HF
+%%%%%%%%%%%%%%%%%
+\begin{figure*}[t]
+	\begin{subfigure}{0.49\textwidth}
+    \includegraphics[height=0.65\textwidth,trim={0pt 0pt 0pt -35pt},clip]{HF_cplx_angle}
+	\subcaption{\label{subfig:UHF_cplx_angle}}
+    \end{subfigure}
+	\begin{subfigure}{0.49\textwidth}
+	\includegraphics[height=0.65\textwidth]{HF_cplx_energy}
+	\subcaption{\label{subfig:UHF_cplx_energy}}
+    \end{subfigure}
+	\caption{%
+    \hugh{(\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$.
+    Symmetry-broken solutions correspond to an individual sheets and become equivalent at the quasi-EP (black dot).
+    The RHF solution is independent of $\lambda$, giving constant plane at $\pi/2$.
+    (\subref{subfig:UHF_cplx_energy}) The corresponding HF energy surfaces show a non-analytic 
+    point at the quasi-exceptional point.}
+	\label{fig:HF_cplx}}
+\end{figure*}
+%%%%%%%%%%%%%%%%%
+
 As the on-site repulsion is increased from 0, the HF approximation reaches a critical value at $U=2t$ where a symmetry-broken 
 UHF solution appears with a lower energy than the RHF one.
 This critical point is analogous to the infamous Coulson--Fischer point identified in the hydrogen dimer.\cite{Coulson_1949}
@@ -523,24 +545,6 @@ of the HF energy rather than a minimum.
 \subsection{Self-consistency as a perturbation} %OR {Complex adiabatic connection}
 %============================================================%
 
-%%% FIG 3 (?) %%%
-% Analytic Continuation of HF
-%%%%%%%%%%%%%%%%%
-\begin{figure*}[t]
-	\begin{subfigure}{0.49\textwidth}
-    \includegraphics[height=0.65\textwidth,trim={0pt 0pt 0pt -30pt},clip]{HF_cplx_angle}
-	\subcaption{\label{subfig:UHF_cplx_angle}}
-    \end{subfigure}
-	\begin{subfigure}{0.49\textwidth}
-	\includegraphics[height=0.65\textwidth]{HF_cplx_energy}
-	\subcaption{\label{subfig:UHF_cplx_energy}}
-    \end{subfigure}
-	\caption{%
-    Analytic continuation of HF into the complex $\lambda$ plane.
-	\label{fig:HF_cplx}}
-\end{figure*}
-%%%%%%%%%%%%%%%%%
-
 % INTRODUCE PARAMETRISED FOCK HAMILTONIAN
 \hugh{The inherent non-linearity in the Fock eigenvalue problem arises from self-consistency 
 in the HF approximation, and is usually solved through an iterative approach.\cite{SzaboBook}
@@ -559,6 +563,7 @@ in the Hubbard dimer directly mirror the energies shown in Fig.~\ref{fig:HF_real
 with coalesence points at 
 \begin{equation}
     \lambda_{\text{c}} = \pm \frac{2t}{U}.
+    \label{eq:scaled_fock}
 \end{equation}
 In contrast, when $\lambda$ becomes complex, the HF equations become non-Hermitian and 
 each HF solutions can be analytically continued for all $\lambda$ values using
@@ -573,16 +578,27 @@ accurate representation for the true HF ground state at $\lambda = 1$.
 For example, in the Hubbard dimer with $U > 2t$, one finds $\lambda_{\text{c}} < 1$ and the symmetry-pure orbitals
 do not provide a good representation of the HF ground state.
 In contrast, $U < 2t$ yields $\lambda_{\text{c}} > 1$ and corresponds to
-the regime where the HF ground state is correctly represented by symmetry-pure orbitals.}
+the regime where the HF ground state is correctly represented by symmetry-pure orbitals.
+}
 
 % COMPLEX ADIABATIC CONNECTION
-\titou{In a recent paper, \cite{Burton_2019} using holomorphic Hartree-Fock (h-HF) \cite{Hiscock_2014,Burton_2018,Burton_2016} as an analytic continuation of conventional HF theory, we have demonstrated, on a simple model, that one can interconvert states of different symmetries and natures by following well-defined contours in the complex $\lambda$-plane, where $\lambda$ is the strength of the electron-electron interaction (see Fig.~\ref{fig:iAC}).
-In particular, by slowly varying $\lambda$ in a similar (yet different) manner to an adiabatic connection in density-functional theory, \cite{Langreth_1975,Gunnarsson_1976,Zhang_2004} one can ``morph'' a ground-state wave function into an excited-state wave function via a stationary path of HF solutions. \cite{Seidl_2018}
-In such a way, we could obtain a doubly-excited state wave function starting from the ground state wave function, a process which is not as easy as one might think. \cite{Gilbert_2008,Thom_2008,Shea_2018}
-One of the fundamental discovery we made was that Coulson-Fisher points (where multiple symmetry-broken solutions coalesce) play a central role and can be classified as \textit{quasi}-exceptional points, as the wave functions do not become self-orthogonal.
-The findings reported in Ref.~\onlinecite{Burton_2019} represent the very first study of non-Hermitian quantum mechanics for the exploration of multiple solutions at the HF level. 
-It perfectly illustrates the deeper topology of electronic states revealed using a complex-scaled electron-electron interaction.
-Through the introduction of non-Hermiticity, we have provided a more general framework in which the complex and diverse characteristics of multiple solutions can be explored and understood.}
+\hugh{We have recently shown that the complex scaled Fock operator Eq.~\eqref{eq:scaled_fock}
+also allows states of different symmetries to be interconverted by following a well-defined
+contour in the complex $\lambda$-plane.\cite{Burton_2019}
+In particular, by slowly varying $\lambda$ in a similar (yet different) manner
+to an adiabatic connection in density-functional theory,\cite{Langreth_1975,Gunnarsson_1976,Zhang_2004} 
+a ground-state wave function can be ``morphed'' into an excited-state wave function 
+via a stationary path of HF solutions.
+This novel approach to identifying excited-state wave functions demonstrates the fundamental 
+role of quasi-EPs in determining the behaviour of the HF approximation.} 
+
+%\titou{In a recent paper, \cite{Burton_2019} using holomorphic Hartree-Fock (h-HF) \cite{Hiscock_2014,Burton_2018,Burton_2016} as an analytic continuation of conventional HF theory, we have demonstrated, on a simple model, that one can interconvert states of different symmetries and natures by following well-defined contours in the complex $\lambda$-plane, where $\lambda$ is the strength of the electron-electron interaction (see Fig.~\ref{fig:iAC}).
+%In particular, by slowly varying $\lambda$ in a similar (yet different) manner to an adiabatic connection in density-functional theory, \cite{Langreth_1975,Gunnarsson_1976,Zhang_2004} one can ``morph'' a ground-state wave function into an excited-state wave function via a stationary path of HF solutions. \cite{Seidl_2018}
+%In such a way, we could obtain a doubly-excited state wave function starting from the ground state wave function, a process which is not as easy as one might think. \cite{Gilbert_2008,Thom_2008,Shea_2018}
+%One of the fundamental discovery we made was that Coulson-Fisher points (where multiple symmetry-broken solutions coalesce) play a central role and can be classified as \textit{quasi}-exceptional points, as the wave functions do not become self-orthogonal.
+%The findings reported in Ref.~\onlinecite{Burton_2019} represent the very first study of non-Hermitian quantum mechanics for the exploration of multiple solutions at the HF level. 
+%It perfectly illustrates the deeper topology of electronic states revealed using a complex-scaled electron-electron interaction.
+%Through the introduction of non-Hermiticity, we have provided a more general framework in which the complex and diverse characteristics of multiple solutions can be explored and understood.}
 
 %=====================================================%
 \subsection{M{\o}ller-Plesset perturbation theory}