resub version

Manuscript/EPAWTFT.tex

@@ -560,7 +560,7 @@ the total spin operator $\hat{\mathcal{S}}^2$, leading to ``spin-contamination''
 \label{sec:HF_hubbard}
 %================================================================%
 
-%%% FIG 2 (?) %%%
+%%% FIG 2 %%%
 % HF energies as a function of U/t
 %%%%%%%%%%%%%%%%%
 \begin{figure}
@@ -611,7 +611,7 @@ We can therefore consider these as symmetry-pure molecular orbitals.}
 However, in the strongly correlated regime $U>2t$, the closed-shell orbital restriction prevents RHF from 
 modelling the correct physics with the two electrons on opposite sites.
 
-%%% FIG 3 (?) %%%
+%%% FIG 3 %%%
 % Analytic Continuation of HF
 %%%%%%%%%%%%%%%%%
 \begin{figure*}[t]
@@ -865,7 +865,7 @@ gradient discontinuities or spurious minima.
 \label{sec:spin_cont}
 %==========================================%
 
-%%% FIG 2 %%%
+%%% FIG 4 %%%
 \begin{figure*}
 	\begin{subfigure}{0.32\textwidth}
 	\includegraphics[height=0.75\textwidth]{fig4a}	
@@ -939,6 +939,7 @@ whether the perturbation series will converge or not.}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % RADIUS OF CONVERGENCE PLOTS
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% FIG 5 %%%
 \begin{figure}[htb]
 	\includegraphics[width=\linewidth]{fig5}
 	\caption{
@@ -949,7 +950,7 @@ whether the perturbation series will converge or not.}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
-%%% FIG 3 %%%
+%%% FIG 6 %%%
 \begin{figure*}
 	\begin{subfigure}{0.32\textwidth}
 	\includegraphics[height=0.75\textwidth]{fig6a}	
@@ -1240,6 +1241,7 @@ and the continuum, its functional form is more complicated than a conical inters
 %------------------------------------------------------------------%
 % Figure on the RMP critical point
 %------------------------------------------------------------------%
+%%% FIG 7 %%%
 \begin{figure*}[t]
 	\begin{subfigure}{0.32\textwidth}
 	\includegraphics[height=0.75\textwidth]{fig7a}	
@@ -1327,8 +1329,9 @@ a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % RMP critical point density
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% FIG 8 %%%
 \begin{figure}[b]
-	\includegraphics[width=\linewidth]{rmp_crit_density}
+	\includegraphics[width=\linewidth]{fig8}
 	\caption{
 \titou{Electron density $\rho_\text{atom}$ on the ``atomic'' site of the asymmetric Hubbard dimer with 
 $\epsilon = 2.5 U$. 
@@ -1360,19 +1363,20 @@ set representations of the MP critical point.\cite{Sergeev_2006}
 %------------------------------------------------------------------%
 % Figure on the UMP critical point
 %------------------------------------------------------------------%
+%%% FIG 9 %%%
 \begin{figure*}[t]
 	\begin{subfigure}{0.32\textwidth}
-    \includegraphics[height=0.75\textwidth,trim={0pt 5pt -10pt 15pt},clip]{fig8a}
+    \includegraphics[height=0.75\textwidth,trim={0pt 5pt -10pt 15pt},clip]{fig9a}
 		\subcaption{\label{subfig:ump_cp}}
     \end{subfigure}
     %
     \begin{subfigure}{0.32\textwidth}
-	\includegraphics[height=0.75\textwidth]{fig8b}
+	\includegraphics[height=0.75\textwidth]{fig9b}
 		\subcaption{\label{subfig:ump_cp_surf}}
     \end{subfigure}
     %
     \begin{subfigure}{0.32\textwidth}
-	\includegraphics[height=0.75\textwidth]{fig8c}	
+	\includegraphics[height=0.75\textwidth]{fig9c}	
 		\subcaption{\label{subfig:ump_ep_to_cp}}
     \end{subfigure}
 %    \includegraphics[height=0.65\textwidth,trim={0pt 5pt 0pt 15pt}, clip]{ump_critical_point}
@@ -1405,8 +1409,9 @@ and a single electron dissociates from the molecule (see Ref.~\onlinecite{Sergee
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % UMP critical point density
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% FIG 10 %%%
 \begin{figure}[b]
-	\includegraphics[width=\linewidth]{ump_crit_density}
+	\includegraphics[width=\linewidth]{fig10}
 	\caption{
 \titou{Difference in the electron densities on the left and right sites for the UMP ground state in the symmetric Hubbard dimer
 [see Eq.~\eqref{eq:ump_dens}]. 
@@ -1465,9 +1470,10 @@ radius of convergence (see Fig.~\ref{fig:RadConv}).
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%
+%%% FIG 11 %%%
 \begin{figure*}
-    \includegraphics[height=0.23\textheight]{fig9a}
+    \includegraphics[height=0.23\textheight]{fig11a}
-    \includegraphics[height=0.23\textheight]{fig9b}
+    \includegraphics[height=0.23\textheight]{fig11b}
     \caption{\label{fig:PadeRMP}
     RMP ground-state energy as a function of $\lambda$ in the Hubbard dimer obtained using various truncated Taylor series and approximants
     at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).}
@@ -1566,8 +1572,9 @@ while the Pad\'e approximants still offer relatively accurate energies and recov
 a convergent series.
 
 %%%%%%%%%%%%%%%%%
+%%% FIG 12 %%%
 \begin{figure}[t]
-    \includegraphics[width=\linewidth]{fig10}
+    \includegraphics[width=\linewidth]{fig12}
     \caption{\label{fig:QuadUMP}
     UMP energies in the Hubbard dimer as a function of $\lambda$ obtained using various approximants at $U/t = 3$.}
 \end{figure}
@@ -1676,19 +1683,20 @@ The remedy for this problem involves applying a suitable transformation of the c
 	\end{ruledtabular}
 \end{table}
 
+%%% FIG 13 %%%
 \begin{figure*}
     \begin{subfigure}{0.32\textwidth}
-	\includegraphics[height=0.85\textwidth]{fig11a}	
+	\includegraphics[height=0.85\textwidth]{fig13a}	
     \subcaption{\label{subfig:322quad} [3/2,2] Quadratic}
     \end{subfigure}
     %
     \begin{subfigure}{0.32\textwidth}
-	\includegraphics[height=0.85\textwidth]{fig11b}
+	\includegraphics[height=0.85\textwidth]{fig13b}
 		\subcaption{\label{subfig:exact} Exact}
     \end{subfigure}
     %
 	\begin{subfigure}{0.32\textwidth}
-        \includegraphics[height=0.85\textwidth]{fig11c}	
+        \includegraphics[height=0.85\textwidth]{fig13c}	
     \subcaption{\label{subfig:304quad} [3/0,4] Quadratic}
     \end{subfigure}
 \caption{%
@@ -1849,8 +1857,9 @@ divergent series into a convergent one by increasing the magnitude of these deno
 However, like the UMP series in stretched \ce{H2},\cite{Lepetit_1988} 
 the cost of larger denominators is an overall slower rate of convergence.
 
+%%% FIG 14 %%%
 \begin{figure}
-    \includegraphics[width=\linewidth]{fig12}
+    \includegraphics[width=\linewidth]{fig14}
     \caption{%
         Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function
         of $\lambda$ for the Hubbard dimer at $U/t = 4.5$.