Transferred RMP critical point discussion from notebook as skeleton.

Manuscript/EPAWTFT.tex

@@ -9,6 +9,7 @@
 \definecolor{hughgreen}{RGB}{0, 128, 0}
 \newcommand{\titou}[1]{\textcolor{red}{#1}}
 \newcommand{\hugh}[1]{\textcolor{hughgreen}{#1}}
+\newcommand{\hughDraft}[1]{\textcolor{orange}{#1}}
 \newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}}
 \newcommand{\trashHB}[1]{\textcolor{orange}{\sout{#1}}}
 
@@ -1239,10 +1240,95 @@ In contrast, $\alpha$ singularities correspond to large avoided crossings that a
 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}
-%\label{sec:critical_point_hubbard}
+\subsection{Critical Points in the Hubbard Dimer}
+\label{sec:critical_point_hubbard}
 %=======================================
 
+%------------------------------------------------------------------%
+% Figure on the RMP critical point
+%------------------------------------------------------------------%
+\begin{figure*}[t]
+	\begin{subfigure}{0.49\textwidth}
+	\includegraphics[height=0.65\textwidth]{rmp_critical_point}
+	\subcaption{\label{subfig:rmp_cp}}
+    \end{subfigure}
+	\begin{subfigure}{0.49\textwidth}
+	\includegraphics[height=0.65\textwidth]{rmp_critical_point_surf}
+	\subcaption{\label{subfig:rmp_cp_surf}}
+    \end{subfigure}
+	\caption{%
+    \hugh{RMP critical point in the Hubbard dimer.}
+	\label{fig:RMP_cp}}
+\end{figure*}
+%------------------------------------------------------------------%
+
+\hughDraft{%
+The simplified site basis of the Hubbard dimer makes explicitly modelling the ionisation continuum 
+impossible.
+To model the auto-ionisation in the Hubbard dimer, we need to turn one of the sites into a ghost atom that will act as
+a destination for ionised electrons. We do this by considering an asymmetric Hubbard dimer where the "atomic" site
+has a negative diagonal term in the one-electron Hamiltonian, representing the nuclear attraction. This term is already
+encoded in the Initialisation section such that the core Hamiltonian is
+To model the doubly-occupied atom, we can define our reference HF state as the configuration with $\theta = 0$ and energy
+\begin{equation}
+    \frac{1}{2} (2 U - 4 \epsilon)
+\end{equation}
+The RMP Hamiltonian then becomes
+...
+}
+
+\hughDraft{%
+Now let's think about the physics of the problem...
+For the ghost site to truly represent ionised electrons, we need the hopping term to vanish (or become very small).
+$U$ controls the strength of the HF repulsive potential. A stronger repulsion will encourage the electrons to be forced
+away from the "atomic" site at a less negative value of $\lambda$.
+$\epsilon$ controls the strength of attraction to the atom. A stronger attraction to the nucleus will mean that the electrons
+are more tightly bound to the atom and a more negative $\lambda$ will be needed for auto-ionisation.
+We therefore expect that the position of the RMP critical point will be controlled by the ratio $\epsilon / U$, with smaller ratios
+making ionisation occur closer to $\lambda$ origin, and making the divergence in these cases more likely.
+}
+
+\hughDraft{%
+Taking the exact case with $t=0$, the RMP energies becomes
+\begin{subequations}
+\begin{align}
+    E_{-} &= 2U - 2 \epsilon - U \lambda
+    \\
+    E_{\text{S}} &= U - \epsilon - U \lambda
+    \\ 
+    E_{+} &= U \lambda 
+\end{align}
+\end{subequations}
+By comparison with Fig.~\ref{fig:RMP_cp}, the critical point can be identified as by solving $E_{-} = E_{+}$,
+giving 
+\begin{equation}
+\lc = 1 - \epsilon / U.
+\end{equation}
+Thus, as expected, the critical point lies on the real axis and moves closer to the origin for larger 
+$U / \epsilon$. 
+}
+
+\hughDraft{%
+The position of the critical point is controlled by the ratio $\epsilon / U$.
+If the magnitude of the ratio becomes greater than one, then the series diverges.
+We can interpret large $U$ as strong electron repulsion effects in electron dense molecules, 
+such as \ce{F-}. A small $\epsilon$ is also likely to correspond to 
+strong nuclear screening by the core and valence electrons. Both of these factors are 
+common in atoms on the right-hand-side of periodic table, 
+\eg\ \ce{F}, \ce{O}, \ce{Ne}, etc, as well as negatively-charged species, and so we recover the
+class $\beta$ system classification.
+}
+
+\hughDraft{%
+In practical calculations, one considers the perturbation energy in a finite basis and the critical point is modelled
+as a cluster of branch points close to the real axis. While we cannot change the size of our basis, we can adjust
+the extent to which the second site behaves as a ghost by making the hopping term larger.
+When $t$ is (slightly) non-zero, our modelled critical point becomes an EP point close to the real axis with a sharp
+associated avoided crossing for real $\lambda$. If $t$ becomes larger, this avoided crossing becomes less sharp and the EPs
+move away from the real axis. This mirrors the discussion of EPs approaching the real axis in the exact basis.
+This effect is shown by the dashed lines in Fig.~\ref{fig:RMP_cp}.
+}
+
 %%====================================================
 %\subsection{The physics of quantum phase transitions}
 %%====================================================