From 3d903b9c5d91269f8ccde86997a233e12c90fe81 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 2 Feb 2021 13:16:55 +0100 Subject: [PATCH] resub version --- Manuscript/EPAWTFT.tex | 41 +++++++++++------- Manuscript/fig10.pdf | Bin 24804 -> 29284 bytes Manuscript/fig11a.pdf | Bin 428579 -> 32612 bytes Manuscript/fig11b.pdf | Bin 289610 -> 18329 bytes Manuscript/fig12.pdf | Bin 19924 -> 24804 bytes Manuscript/fig13a.pdf | Bin 0 -> 428579 bytes Manuscript/fig13b.pdf | Bin 0 -> 289610 bytes Manuscript/{fig11c.pdf => fig13c.pdf} | Bin Manuscript/fig14.pdf | Bin 0 -> 19924 bytes Manuscript/{rmp_crit_density.pdf => fig8.pdf} | Bin Manuscript/fig8a.pdf | Bin 21741 -> 0 bytes Manuscript/fig8b.pdf | Bin 410695 -> 0 bytes Manuscript/fig9a.pdf | Bin 32612 -> 21741 bytes Manuscript/fig9b.pdf | Bin 18329 -> 410695 bytes Manuscript/{fig8c.pdf => fig9c.pdf} | Bin Manuscript/ump_crit_density.pdf | Bin 29284 -> 0 bytes 16 files changed, 25 insertions(+), 16 deletions(-) create mode 100644 Manuscript/fig13a.pdf create mode 100644 Manuscript/fig13b.pdf rename Manuscript/{fig11c.pdf => fig13c.pdf} (100%) create mode 100644 Manuscript/fig14.pdf rename Manuscript/{rmp_crit_density.pdf => fig8.pdf} (100%) delete mode 100644 Manuscript/fig8a.pdf delete mode 100644 Manuscript/fig8b.pdf rename Manuscript/{fig8c.pdf => fig9c.pdf} (100%) delete mode 100644 Manuscript/ump_crit_density.pdf diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 7bbf0e6..7f57ddf 100644 --- a/Manuscript/EPAWTFT.tex +++ b/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]{fig9b} + \includegraphics[height=0.23\textheight]{fig11a} + \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 $U/t = 4.5$. diff --git a/Manuscript/fig10.pdf b/Manuscript/fig10.pdf index da8c123626f286ac70b824498c621fe730100dca..7eeb3d821cf9c70a57faa36d5a086dbddae775ed 100644 GIT binary patch delta 27523 zcmV)1K+V78!2#st0gz08m3_;uB|CE6JilTOZ3BIj$yX);!2<(^0RbM^>Y1Q{v295Z zss^?P{(W;jg5*9`t?rg^``%caCzHuXFd|lvyk6({|36&ge^Py3@1Mv2KF+T`{_m-; 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