From 4a711e256f0d3e2a7d22254cb515b4851721b515 Mon Sep 17 00:00:00 2001 From: Hugh Burton Date: Wed, 2 Dec 2020 17:23:02 +0000 Subject: [PATCH] Added more discussion about quadratics... --- Manuscript/EPAWTFT.tex | 238 +++++++++++++++++++++++++++------------ Manuscript/ump_exact.pdf | Bin 0 -> 289610 bytes Manuscript/ump_qa304.pdf | Bin 0 -> 238584 bytes Manuscript/ump_qa322.pdf | Bin 0 -> 428579 bytes 4 files changed, 166 insertions(+), 72 deletions(-) create mode 100644 Manuscript/ump_exact.pdf create mode 100644 Manuscript/ump_qa304.pdf create mode 100644 Manuscript/ump_qa322.pdf diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 797ba79..429acc4 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -1327,6 +1327,17 @@ radius of convergence (see Fig.~\ref{fig:RadConv}). \label{sec:Resummation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%% +\begin{figure*} + \includegraphics[height=0.23\textheight]{PadeRMP35} + \includegraphics[height=0.23\textheight]{PadeRMP45} + \caption{\label{fig:PadeRMP} + RMP ground-state energy as a function of $\lambda$ obtained using various resummation + techniques at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).} +\end{figure*} +%%%%%%%%%%%%%%%%% + + %As frequently claimed by Carl Bender, \hugh{It is frequently stated that} \textit{``the most stupid thing that one can do with a series is to sum it.''} @@ -1345,10 +1356,12 @@ We refer the interested reader to more specialised reviews for additional inform %==========================================% \subsection{Pad\'e Approximant} %==========================================% + \hugh{The failure of a Taylor series for correctly modelling the MP energy function $E(\lambda)$ -arises because one is trying to model a complicated function containing branch points and +arises because one is trying to model a complicated function containing multiple branches, branch points and singularities} using a simple polynomial of finite order. -A truncated Taylor series just does not have enough flexibility to do the job properly. +A truncated Taylor series \hugh{can only predict a single sheet and} does not have enough +flexibility to adequately describe the MP energy. Alternatively, the description of complex energy functions can be significantly improved by introducing Pad\'e approximants, \cite{Pade_1892} and related techniques. \cite{BakerBook,BenderBook} @@ -1371,6 +1384,34 @@ where the nature of the solution undergoes a sudden transition). \hugh{Despite this limitation, the successive diagonal Pad\'e approximants (\ie, $d_A = d_B $) often define a convergent perturbation series in cases where the Taylor series expansion diverges.} +\begin{table}[b] + \caption{RMP ground-state energy estimate at $\lambda = 1$ provided by various truncated Taylor + series and Pad\'e approximants at $U/t = 3.5$ and $4.5$. + We also report the \hugh{closest pole to the origin $\lc$} provided by the diagonal Pad\'e approximants. + \label{tab:PadeRMP}} + \begin{ruledtabular} + \begin{tabular}{lccccc} + & & \mc{2}{c}{$\abs{\lc}$} & \mc{2}{c}{$E_{-}(\lambda = 1)$} \\ + \cline{3-4} \cline{5-6} + Method & degree & $U/t = 3.5$ & $U/t = 4.5$ & $U/t = 3.5$ & $U/t = 4.5$ \\ + \hline + Taylor & 2 & & & $-1.01563$ & $-1.01563$ \\ + & 3 & & & $-1.01563$ & $-1.01563$ \\ + & 4 & & & $-0.86908$ & $-0.61517$ \\ + & 5 & & & $-0.86908$ & $-0.61517$ \\ + & 6 & & & $-0.92518$ & $-0.86858$ \\ + \hline + Pad\'e & [1/1] & $2.29$ & $1.78$ & $-1.61111$ & $-2.64286$ \\ + & [2/2] & $2.29$ & $1.78$ & $-0.82124$ & $-0.48446$ \\ + & [3/3] & $1.73$ & $1.34$ & $-0.91995$ & $-0.81929$ \\ + & [4/4] & $1.47$ & $1.14$ & $-0.90579$ & $-0.74866$ \\ + & [5/5] & $1.35$ & $1.05$ & $-0.90778$ & $-0.76277$ \\ + \hline + Exact & & $1.14$ & $0.89$ & $-0.90754$ & $-0.76040$ \\ + \end{tabular} + \end{ruledtabular} +\end{table} + Fig.~\ref{fig:PadeRMP} illustrates the improvement provided by diagonal Pad\'e approximants compared to the usual Taylor expansion in cases where the RMP series of the Hubbard dimer converges ($U/t = 3.5$) and diverges ($U/t = 4.5$). @@ -1379,15 +1420,24 @@ energy at $\lambda = 1$ provided by various truncated Taylor series and Pad\'e approximants for these two values of the ratio $U/t$. While the truncated Taylor series converges laboriously to the exact energy as the truncation degree increases at $U/t = 3.5$, the Pad\'e approximants yield much more accurate results. -\hugh{Furthermore, the position of the closest pole to origin $\lc$ in the Pad\'e approximants -indicate that they a relatively good approximation to the true branch point singularity in the RMP energy. +\hugh{Furthermore, the distance of the closest pole to origin $\abs{\lc}$ in the Pad\'e approximants +indicate that they a relatively good approximation to the position of the +true branch point singularity in the RMP energy. For $U/t = 4.5$, the Taylor series expansion performs worse and eventually diverges, while the Pad\'e approximants still offer relaitively accurate energies and recovers a convergent series.} +%%%%%%%%%%%%%%%%% +\begin{figure}[t] + \includegraphics[width=\linewidth]{QuadUMP} + \caption{\label{fig:QuadUMP} + UMP energies as a function of $\lambda$ obtained using various resummation techniques at $U/t = 3$.} +\end{figure} +%%%%%%%%%%%%%%%%% + \hugh{% -We can expect that the singularity structure of the UMP energy will be much more challenging -to model properly as the UMP energy function contains three connected branches +We can expect the UMP energy function to be much more challenging +to model properly as it contains three connected branches (see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}). Fig.~\ref{fig:QuadUMP} and Table~\ref{tab:QuadUMP} indicate that this is indeed the case. In particular, Fig.~\ref{fig:QuadUMP} illustrates that the Pad\'e approximants are trying to model @@ -1402,43 +1452,6 @@ even in cases where the convergence of the UMP series is incredibly slow (see Fig.~\ref{subfig:UMP_cvg}). } -\begin{table} - \caption{RMP ground-state energy estimate at $\lambda = 1$ provided by various truncated Taylor - series and Pad\'e approximants at $U/t = 3.5$ and $4.5$. - We also report the \hugh{closest pole to the origin $\lc$} provided by the diagonal Pad\'e approximants. - \label{tab:PadeRMP}} - \begin{ruledtabular} - \begin{tabular}{lccccc} - & & \mc{2}{c}{$\lc$} & \mc{2}{c}{$E_{-}(\lambda = 1)$} \\ - \cline{3-4} \cline{5-6} - Method & degree & $U/t = 3.5$ & $U/t = 4.5$ & $U/t = 3.5$ & $U/t = 4.5$ \\ - \hline - Taylor & 2 & & & $-1.01563$ & $-1.01563$ \\ - & 3 & & & $-1.01563$ & $-1.01563$ \\ - & 4 & & & $-0.86908$ & $-0.61517$ \\ - & 5 & & & $-0.86908$ & $-0.61517$ \\ - & 6 & & & $-0.92518$ & $-0.86858$ \\ - Pad\'e & [1/1] & $2.29$ & $1.78$ & $-1.61111$ & $-2.64286$ \\ - & [2/2] & $2.29$ & $1.78$ & $-0.82124$ & $-0.48446$ \\ - & [3/3] & $1.73$ & $1.34$ & $-0.91995$ & $-0.81929$ \\ - & [4/4] & $1.47$ & $1.14$ & $-0.90579$ & $-0.74866$ \\ - & [5/5] & $1.35$ & $1.05$ & $-0.90778$ & $-0.76277$ \\ - \hline - Exact & & $1.14$ & $0.89$ & $-0.90754$ & $-0.76040$ \\ - \end{tabular} - \end{ruledtabular} -\end{table} - -%%%%%%%%%%%%%%%%% -\begin{figure*} - \includegraphics[height=0.23\textheight]{PadeRMP35} - \includegraphics[height=0.23\textheight]{PadeRMP45} - \caption{\label{fig:PadeRMP} - RMP ground-state energy as a function of $\lambda$ obtained using various resummation - techniques at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).} -\end{figure*} -%%%%%%%%%%%%%%%%% - %==========================================% \subsection{Quadratic Approximant} %==========================================% @@ -1468,6 +1481,7 @@ their respective expressions \eqref{eq:PQR} yields $n+1$ linear equations for th $p_k$, $q_k$, and $r_k$ (where we are free to assume that $q_0 = 1$). A quadratic approximant, characterised by the label $[d_P/d_Q,d_R]$, generates, by construction, $n_\text{bp} = \max(2d_p,d_q+d_r)$ branch points at the roots of the polynomial $P^2(\lambda) - 4 Q(\lambda) R(\lambda)$ \hugh{and $d_q$ poles at the roots of $Q(\lambda)$}. + Generally, the diagonal sequence of quadratic approximant, \ie, $[0/0,0]$, $[1/0,0]$, $[1/0,1]$, $[1/1,1]$, $[2/1,1]$, is of particular interest \hugh{as the order of the corresponding Taylor series increases on each step. @@ -1480,60 +1494,140 @@ provide convergent results in the most divergent cases considered by Olsen and collaborators\cite{Christiansen_1996,Olsen_1996} and Leininger \etal \cite{Leininger_2000} -For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant -are quite poor approximations, but the $[1/0,1]$ version perfectly models the RMP energy -function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm i 4t/U$. -This is expected from the form of the RMP energy [see Eq.~\eqref{eq:E0MP}], which matches -the ideal target for quadratic approximants. +As a note of caution, Ref.~\onlinecite{Goodson_2019} suggests that low-order +quadratic approximants can struggle to correctly model the singularity structure when +the energy function has poles in both the positive and negative half-planes. +In such a scenario, the quadratic approximant will tend to place its branch points in-between, potentially introducing singularities quite close to the origin. +The remedy for this problem involves applying a suitable transformation of the complex plane (such as a bilinear conformal mapping) which leaves the points at $\lambda = 0$ and $\lambda = 1$ unchanged. \cite{Feenberg_1956} -%%%%%%%%%%%%%%%%% -\begin{figure} - \includegraphics[width=\linewidth]{QuadUMP} - \caption{\label{fig:QuadUMP} - UMP energies as a function of $\lambda$ obtained using various resummation techniques at $U/t = 3$.} -\end{figure} -%%%%%%%%%%%%%%%%% - -\begin{table} - \caption{Estimate of the radius of convergence $r_c$ of the UMP energy function provided +\begin{table}[b] + \caption{Estimate \hugh{for the distance of the closest singularity (pole or branch point) to the origin $\abs{\lc}$} + in the UMP energy function provided by various resummation techniques at $U/t = 3$ and $7$. The truncation degree of the Taylor expansion $n$ of $E(\lambda)$ and the number of branch points $n_\text{bp} = \max(2d_p,d_q+d_r)$ generated by the quadratic approximants are also reported. \label{tab:QuadUMP}} \begin{ruledtabular} \begin{tabular}{lccccccc} - & & & & \mc{2}{c}{$\lp$} & \mc{2}{c}{$E_{-}(\lambda)$} \\ + & & & & \mc{2}{c}{$\abs{\lc}$} & \mc{2}{c}{$E_{-}(\lambda)$} \\ \cline{5-6}\cline{7-8} \mc{2}{c}{Method} & $n$ & $n_\text{bp}$ & $U/t = 3$ & $U/t = 7$ & $U/t = 3$ & $U/t = 7$ \\ \hline + Taylor & & 2 & & & & $-0.74074$ & $-0.29155$ \\ + & & 3 & & & & $-0.78189$ & $-0.29690$ \\ + & & 4 & & & & $-0.82213$ & $-0.30225$ \\ + & & 5 & & & & $-0.85769$ & $-0.30758$ \\ + & & 6 & & & & $-0.88882$ & $-0.31289$ \\ + \hline Pad\'e & [1/1] & 2 & & $9.000$ & $49.00$ & $-0.75000$ & $-0.29167$ \\ & [2/2] & 4 & & $0.974$ & $1.003$ & $\hphantom{-}0.75000$ & $-17.9375$ \\ & [3/3] & 6 & & $1.141$ & $1.004$ & $-1.10896$ & $-1.49856$ \\ & [4/4] & 8 & & $1.068$ & $1.003$ & $-0.85396$ & $-0.33596$ \\ - & [5/5] & 10 & & $1.122$ & $1.004$ & $-0.97254$ & $-0.35513$ \\ + & [5/5] & 10 & & $1.122$ & $1.004$ & $-0.97254$ & $-0.35513$ \\ + \hline Quadratic & [2/1,2] & 6 & 4 & $1.086$ & $1.003$ & $-1.01009$ & $-0.53472$ \\ & [2/2,2] & 7 & 4 & $1.082$ & $1.003$ & $-1.00553$ & $-0.53463$ \\ & [3/2,2] & 8 & 6 & $1.082$ & $1.001$ & $-1.00568$ & $-0.52473$ \\ & [3/2,3] & 9 & 6 & $1.071$ & $1.002$ & $-0.99973$ & $-0.53102$ \\ - & [3/3,3] & 10 & 6 & $1.071$ & $1.002$ & $-0.99966$ & $-0.53103$ \\ + & [3/3,3] & 10 & 6 & $1.071$ & $1.002$ & $-0.99966$ & $-0.53103$ \\[0.5ex] + (pole-free) & [3/0,2] & 6 & 6 & $1.059$ & $1.003$ & $-1.13712$ & $-0.57199$ \\ + & [3/0,3] & 7 & 6 & $1.073$ & $1.002$ & $-1.00335$ & $-0.53113$ \\ + & [3/0,4] & 8 & 6 & $1.071$ & $1.002$ & $-1.00074$ & $-0.53116$ \\ + & [3/0,5] & 9 & 6 & $1.070$ & $1.002$ & $-1.00042$ & $-0.53114$ \\ + & [3/0,6] & 10 & 6 & $1.070$ & $1.002$ & $-1.00039$ & $-0.53113$ \\ \hline Exact & & & & $1.069$ & $1.002$ & $-1.00000$ & $-0.53113$ \\ \end{tabular} \end{ruledtabular} \end{table} -\hugh{On the other hand, the greater flexibility of the diagonal quadratic approximants provides a significantly -improved model of the UMP energy in comparison to the Pad\'e approximants or Taylor series. -In particular, these quadratic approximants provide an effect model for the avoided crossings -(Fig.~\ref{fig:QuadUMP}) and a far better estimate for the location of the branch point singularities. -Furthermore, they provide remarkably accurate estimates of the ground-state energy at $\lambda = 1$, -as shown in Table~\ref{tab:QuadUMP}} +\begin{figure*} + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.85\textwidth]{ump_qa322} + \subcaption{\label{subfig:ump_ep_to_cp} [3/2,2] Quadratic} + \end{subfigure} + % + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.85\textwidth]{ump_exact} + \subcaption{\label{subfig:ump_cp_surf} Exact} + \end{subfigure} + % + \begin{subfigure}{0.32\textwidth} + \includegraphics[height=0.85\textwidth]{ump_qa304} + \subcaption{\label{subfig:ump_cp} [3/0,4] Quadratic} + \end{subfigure} +\caption{% +\hugh{% +Comparison of the [3/2,2] and [3/0,4] quadratic approximants with the exact UMP energy surface in the complex $\lambda$ +plane with $U/t = 3$. +Both quadratic approximants correspond to the same truncation degree of the Taylor series and model the branch points +using a radicand polynomial of the same order. +However, the [3/2,2] approximant introduces poles into the surface that limits it accuracy, while the [3/0,4] approximant +is free of poles.} +\label{fig:nopole_quad}} +\end{figure*} -However, as a note of caution, Ref.~\onlinecite{Goodson_2019} suggests that low-order -quadratic approximants can struggle to correctly model the singularity structure when -the energy function has poles in both the positive and negative half-planes. -In such a scenario, the quadratic approximant will tend to place its branch points in-between, potentially introducing singularities quite close to the origin. -The remedy for this problem involves applying a suitable transformation of the complex plane (such as a bilinear conformal mapping) which leaves the points at $\lambda = 0$ and $\lambda = 1$ unchanged. \cite{Feenberg_1956} +For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant +are quite poor approximations, but the $[1/0,1]$ version perfectly models the RMP energy +function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm \i 4t/U$. +This is expected from the form of the RMP energy [see Eq.~\eqref{eq:E0MP}], which matches +the ideal target for quadratic approximants. +\hugh{% +Furthermore, the greater flexibility of the diagonal quadratic approximants provides a significantly +improved model of the UMP energy in comparison to the Pad\'e approximants or Taylor series. +In particular, these quadratic approximants provide an effective model for the avoided crossings +(Fig.~\ref{fig:QuadUMP}) and an improved estimate for the distance of the +closest branch point to the origin. +Table~\ref{tab:QuadUMP} shows that they provide remarkably accurate +estimates of the ground-state energy at $\lambda = 1$.} + +\hugh{% +While the diagonal quadratic approximants provide significanty improved estimates of the +ground-state energy, we can use our knowledge of the UMP singularity structure to develop +even more accurate results. +We have seen in previous sections that the UMP energy surface +contains only square-root branch cuts that approach the real axis in the limit $U/t \rightarrow \infty$. +Since there are no true poles on this surface, we can obtain more accurate quadratic approximants by +taking $d_q = 0$ and increasing $d_r$ to retain equivalent accuracy in the square-root term. +Fig.~\ref{fig:nopole_quad} illustrates this improvement for the pole-free [3/0,4] quadratic +approximant compared to the [3/2,2] approximant with the same truncation degree in the Taylor +expansion. +Clearly modelling the square-root branch point using $d_q = 2$ has the negative effect of +introducing spurious poles in the energy, while focussing purely on the branch point with $d_q = 0$ +leads to a significantly improved model. +Table~\ref{tab:QuadUMP} shows that these pole-free quadratic approximants +provide a rapidly convergent series with essentially exact energies at low order. +} + + +\hugh{Finally, to emphasise the improvement that can be gained by using either Pad\'e, diagonal quadratic, +or pole-free approximants, we consider the energy and error obtained using only the first 10 terms in the UMP +in Table~\ref{tab:UMP_order10}. +The accuracy of these approximants reinforces how our understanding of the MP +energy surface in the complex plane can be leveraged to significantly improve estimates of the exact +energy using low-order perturbation expansions. +} + +\begin{table}[h] + \caption{ + \hugh{% + Estimate and associated error of the exact UMP energy at $U/t = 7$ for + various approximants using up to ten terms in the Taylor expansion. + } + \label{tab:UMP_order10}} + \begin{ruledtabular} + \begin{tabular}{lccc} + \mc{2}{c}{Method} & $E_{-}(\lambda)$ & Abs.\ Error \\ + \hline + Taylor & 10 & $-0.33338$ & $0.197290$ \\ + Pad\'e & [5/5] & $-0.35513$ & $0.176000$ \\ + Quadratic (diagonal) & [3/3,3] & $-0.53104$ & $0.000103$ \\ + Quadratic (pole-free)& [3/0,6] & $-0.53113$ & $0.000003$ \\ + \hline + Exact & & $-0.53113$ & \\ + \end{tabular} + \end{ruledtabular} +\end{table} %==========================================% diff --git a/Manuscript/ump_exact.pdf b/Manuscript/ump_exact.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e8f55f6cf5865d63e3a4d4728757fbfebe2e62cc GIT binary patch literal 289610 zcmeFZcT|&E_dhx_&WsI%qap$U480?vhB|_PNE49~>L3Y7htLT{M+K#a9*T^jfP^AJ zN=N`l1QJ3~Az)|`2%S)ZbOQIm_xro+w{BVY-}}d#wUUrL=Q(G$v(Mh2y@UAe8%FX< z3aV$sKMi(#J)?5<+SP!2-efUi(S^6t0Y}AFwR&`eTIB8O7Uo4eb>GI*=A_ 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