updated some figures

Manuscript/EPAWTFT.tex

@@ -773,6 +773,8 @@ and identified that the slow UMP convergence arises from its failure to correctl
 low-lying double excitation.
 This erroneous description of the double excitation amplitude has the same origin as the spin-contamination in the reference
 UHF wave function, creating the first direct link between spin-contamination and slow UMP convergence.\cite{Gill_1988}
+
+% LEPETIT CHAT
 Lepetit \etal\ later analysed the difference between perturbation convergence using the unrestricted MP 
 and EN partitionings. \cite{Lepetit_1988}
 They argued that the slow UMP convergence for stretched molecules arises from 
@@ -1195,28 +1197,6 @@ $\lambda$ 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} 
 
-%------------------------------------------------------------------%
-% 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{Modelling the RMP critical point using the asymmetric Hubbard dimer.
-        (\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed).
-        (\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the
-        real axis, giving a sharp avoided crossing on the real axis (solid).
-    }
-	\label{fig:RMP_cp}}
-\end{figure*}
-%------------------------------------------------------------------%
-
 % RELATIONSHIP TO BASIS SET SIZE
 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 occurred when diffuse basis functions were included.\cite{Olsen_1996}
@@ -1268,6 +1248,28 @@ states which share the symmetry of the ground state,\cite{Goodson_2004} and are
 \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{Modelling the RMP critical point using the asymmetric Hubbard dimer.
+        (\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed).
+        (\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the
+        real axis, giving a sharp avoided crossing on the real axis (solid).
+    }
+	\label{fig:RMP_cp}}
+\end{figure*}
+%------------------------------------------------------------------%
+
 % INTRODUCING THE MODEL
 \hugh{%
 The simplified site basis of the Hubbard dimer makes explicilty modelling the ionisation continuum impossible.
@@ -1356,6 +1358,87 @@ In the limit $t \rightarrow 0$, these EPs approach the real axis, mirroring Serg
 set representations of the MP critical point.\cite{Sergeev_2006}
 }
 
+%------------------------------------------------------------------%
+% Figure on the UMP critical point
+%------------------------------------------------------------------%
+\begin{figure*}[t]
+	\begin{subfigure}{0.49\textwidth}
+	\includegraphics[height=0.65\textwidth]{ump_critical_point}
+	\subcaption{\label{subfig:ump_cp}}
+    \end{subfigure}
+	\begin{subfigure}{0.49\textwidth}
+	\includegraphics[height=0.65\textwidth]{ump_ep_to_cp}
+	\subcaption{\label{subfig:ump_ep_to_cp}}
+    \end{subfigure}
+	\caption{%
+    \hugh{Modelling the RMP critical point using the asymmetric Hubbard dimer.
+        (\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed).
+        (\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the
+        real axis, giving a sharp avoided crossing on the real axis (solid).
+    }
+	\label{fig:RMP_cp}}
+\end{figure*}
+%------------------------------------------------------------------%
+
+% RELATIONSHIP BETWEEN QPT AND UMP
+\hughDraft{%
+The ground-state EP observed in the UMP series also has a small imaginary part and falls close to the real axis. 
+As the correlation strength increases, this EP moves closer to the real axis and closer to the radius of convergence at 
+$\lambda = 1$. So can we understand this using the arguments related to the critical point?
+Closed-shell case:
+The work by Stillinger builds an argument around the HF potential $v_{\text{HF}}$ which, by itself, 
+is repulsive and concentrated around the 
+occupied orbitals. As $\lambda$ becomes increasingly positive along the real axis, this HF potential 
+becomes attractive for $\lambda > 1$.
+However, the explicit two-electron becomes increasingly repulsive for larger positive $\lambda$, 
+until eventually single electrons are successively expelled from the molecule.
+Effect of symmetry-breaking:
+Things become more subtle for the symmetry-broken case because we must now consider the $\alpha$ and $\beta$ HF potentials.
+When UHF symmetry breaking occurs in the Hubbard dimer, with the $\alpha$ electron localising on the left site, 
+the $\alpha$ HF potential
+will then be a repulsive interaction localised around the $\beta$ electron, so on the right site. 
+The same is true for the $\beta$ HF potential.
+Therefore, as $\lambda$ becomes greater than 1 and the HF potentials become attractive, 
+there is a driving force for the $\alpha$ and $\beta$.
+electrons to swap sites. If both electrons swap at the same time, then we get a double excitation. Therefore, we expect an 
+avoided crossing as $\lambda$ is increased beyond 1. 
+The "sharpness" of this avoided crossing is controlled by the strength of the electron-electron repulsion... see below.
+}
+
+\hughDraft{%
+For small $U/t$, the HF potentials will be weak and the electrons will slowly delocalise over 
+both sites as $\lambda$ increases. The UHF reference itself already has some degree of delocalisation over 
+both sites as we are only just beyond the CFP. This leads to a
+"shallow" avoided crossing, and the corresponding EPs corresponding with have non-zero imaginary parts. 
+At larger $U/t$, the HF potentials will becomes increasingly repulsive and the "swapping" driving force will become stronger. 
+Furthermore, the strong on-site repulsion will make delocalisation over both sites increasingly unfavourable. 
+We therefore see that the electrons swap sites almost instantaneously as $\lambda$ passes through 1, as this is the threshold point 
+where the HF potential becomes attractive. This very rapid change in the nature of the state corresponds to a "sharp" avoided
+crossing, with EPs close to the real axis.
+Note that, although this appears to be an avoided crossing with the first-excited state, 
+by the time we have reached $\lambda \approx 1$,
+we are beyond the excited-state avoided crossing and thus the first-excited state qualitatively corresponds to a double
+excitation from the reference. This matches our expectation of both electrons swapping sites.
+Okay we can't actually do $U/t = \infty$, but we can make it very large. With such a strong attractive HF potential and such strong
+on-site repulsion, the electrons swapping process will occur exactly at $\lambda=1$. In this limit, the change is so sudden that it
+now corresponds to a QPT in the perturbation approximation, and again the EPs fall on the real axis.
+}
+
+\hughDraft{%
+By applying the separation of the Hamiltionian proposed by Stillinger, we have been able to rationalise why the UMP ground-state
+occurs so close to radius of convergence and increasingly close to the real axis for large $U/t$. The key feature here is the fact that the
+HF potential is different for the $\alpha$ and $\beta$ electrons, and by extension the potential felt by 
+an electron is not strictly localised around
+that electrons orbitals. This is completely different to the symmetric structure in the closed-shell case. Because of these different 
+potentials, there can be a large driving force for spatial rearrangement occurring as soon as the HF potential becomes positive 
+(at $\lambda=1$). 
+This sudden change in electron distribution is made more extreme if the HF potential and electron repulsion dominate over the 
+one-electron terms, which also corresponds to the regime of strong wave function symmetry breaking. 
+We have therefore provided a physical motivation behind why this EP can behave like a QPT for strong symmetry breaking. 
+This supports the conclusions in Antoine's report that the symmetry-broken EP behaves as a class $\beta$ singularity.
+}
+
+
 %%====================================================
 %\subsection{The physics of quantum phase transitions}
 %%====================================================