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@ -1327,6 +1327,17 @@ radius of convergence (see Fig.~\ref{fig:RadConv}).
\label{sec:Resummation} \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, %As frequently claimed by Carl Bender,
\hugh{It is frequently stated that} \hugh{It is frequently stated that}
\textit{``the most stupid thing that one can do with a series is to sum it.''} \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} \subsection{Pad\'e Approximant}
%==========================================% %==========================================%
\hugh{The failure of a Taylor series for correctly modelling the MP energy function $E(\lambda)$ \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. 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 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} 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 $) \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.} 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 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 approximants compared to the usual Taylor expansion in cases where the RMP series of
the Hubbard dimer converges ($U/t = 3.5$) and diverges ($U/t = 4.5$). 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 $U/t$. approximants for these two values of the ratio $U/t$.
While the truncated Taylor series converges laboriously to the exact energy as the truncation 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. 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 \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 true branch point singularity in the RMP energy. 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, 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 while the Pad\'e approximants still offer relaitively accurate energies and recovers
a convergent series.} 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{% \hugh{%
We can expect that the singularity structure of the UMP energy will be much more challenging We can expect the UMP energy function to be much more challenging
to model properly as the UMP energy function contains three connected branches to model properly as it contains three connected branches
(see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}). (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. 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 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}). (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} \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 $q_0 = 1$). $p_k$, $q_k$, 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, 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 $Q(\lambda)$}. $n_\text{bp} = \max(2d_p,d_q+d_r)$ 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, Generally, the diagonal sequence of quadratic approximant,
\ie, $[0/0,0]$, $[1/0,0]$, $[1/0,1]$, $[1/1,1]$, $[2/1,1]$, \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. 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} collaborators\cite{Christiansen_1996,Olsen_1996}
and Leininger \etal \cite{Leininger_2000} and Leininger \etal \cite{Leininger_2000}
For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant As a note of caution, Ref.~\onlinecite{Goodson_2019} suggests that low-order
are quite poor approximations, but the $[1/0,1]$ version perfectly models the RMP energy quadratic approximants can struggle to correctly model the singularity structure when
function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm i 4t/U$. the energy function has poles in both the positive and negative half-planes.
This is expected from the form of the RMP energy [see Eq.~\eqref{eq:E0MP}], which matches 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 ideal target for quadratic approximants. 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{table}[b]
\begin{figure} \caption{Estimate \hugh{for the distance of the closest singularity (pole or branch point) to the origin $\abs{\lc}$}
\includegraphics[width=\linewidth]{QuadUMP} in the UMP energy function provided
\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
by various resummation techniques at $U/t = 3$ and $7$. 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 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. points $n_\text{bp} = \max(2d_p,d_q+d_r)$ generated by the quadratic approximants are also reported.
\label{tab:QuadUMP}} \label{tab:QuadUMP}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lccccccc} \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} \cline{5-6}\cline{7-8}
\mc{2}{c}{Method} & $n$ & $n_\text{bp}$ & $U/t = 3$ & $U/t = 7$ & $U/t = 3$ & $U/t = 7$ \\ \mc{2}{c}{Method} & $n$ & $n_\text{bp}$ & $U/t = 3$ & $U/t = 7$ & $U/t = 3$ & $U/t = 7$ \\
\hline \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 & & $9.000$ & $49.00$ & $-0.75000$ & $-0.29167$ \\ Pad\'e & [1/1] & 2 & & $9.000$ & $49.00$ & $-0.75000$ & $-0.29167$ \\
& [2/2] & 4 & & $0.974$ & $1.003$ & $\hphantom{-}0.75000$ & $-17.9375$ \\ & [2/2] & 4 & & $0.974$ & $1.003$ & $\hphantom{-}0.75000$ & $-17.9375$ \\
& [3/3] & 6 & & $1.141$ & $1.004$ & $-1.10896$ & $-1.49856$ \\ & [3/3] & 6 & & $1.141$ & $1.004$ & $-1.10896$ & $-1.49856$ \\
& [4/4] & 8 & & $1.068$ & $1.003$ & $-0.85396$ & $-0.33596$ \\ & [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 & $1.086$ & $1.003$ & $-1.01009$ & $-0.53472$ \\ Quadratic & [2/1,2] & 6 & 4 & $1.086$ & $1.003$ & $-1.01009$ & $-0.53472$ \\
& [2/2,2] & 7 & 4 & $1.082$ & $1.003$ & $-1.00553$ & $-0.53463$ \\ & [2/2,2] & 7 & 4 & $1.082$ & $1.003$ & $-1.00553$ & $-0.53463$ \\
& [3/2,2] & 8 & 6 & $1.082$ & $1.001$ & $-1.00568$ & $-0.52473$ \\ & [3/2,2] & 8 & 6 & $1.082$ & $1.001$ & $-1.00568$ & $-0.52473$ \\
& [3/2,3] & 9 & 6 & $1.071$ & $1.002$ & $-0.99973$ & $-0.53102$ \\ & [3/2,3] & 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 \hline
Exact & & & & $1.069$ & $1.002$ & $-1.00000$ & $-0.53113$ \\ Exact & & & & $1.069$ & $1.002$ & $-1.00000$ & $-0.53113$ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table}
\hugh{On the other hand, the greater flexibility of the diagonal quadratic approximants provides a significantly \begin{figure*}
improved model of the UMP energy in comparison to the Pad\'e approximants or Taylor series. \begin{subfigure}{0.32\textwidth}
In particular, these quadratic approximants provide an effect model for the avoided crossings \includegraphics[height=0.85\textwidth]{ump_qa322}
(Fig.~\ref{fig:QuadUMP}) and a far better estimate for the location of the branch point singularities. \subcaption{\label{subfig:ump_ep_to_cp} [3/2,2] Quadratic}
Furthermore, they provide remarkably accurate estimates of the ground-state energy at $\lambda = 1$, \end{subfigure}
as shown in Table~\ref{tab:QuadUMP}} %
\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 For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant
quadratic approximants can struggle to correctly model the singularity structure when are quite poor approximations, but the $[1/0,1]$ version perfectly models the RMP energy
the energy function has poles in both the positive and negative half-planes. function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm \i 4t/U$.
In such a scenario, the quadratic approximant will tend to place its branch points in-between, potentially introducing singularities quite close to the origin. This is expected from the form of the RMP energy [see Eq.~\eqref{eq:E0MP}], which matches
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} 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}
%==========================================% %==========================================%

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