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Manuscript/EPAWTFT.tex
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@ -108,6 +108,7 @@
\newcommand{\e}{\mathrm{e}} % Euler number \newcommand{\e}{\mathrm{e}} % Euler number
\newcommand{\rc}{r_{\text{c}}} \newcommand{\rc}{r_{\text{c}}}
\newcommand{\lc}{\lambda_{\text{c}}} \newcommand{\lc}{\lambda_{\text{c}}}
\newcommand{\lp}{\lambda_{\text{p}}}
\newcommand{\lep}{\lambda_{\text{EP}}} \newcommand{\lep}{\lambda_{\text{EP}}}
% Some energies % Some energies
@ -997,7 +998,8 @@ analysing the relation between the dominant singularity (\ie, the closest singul
and the convergence behaviour of the series.\cite{Olsen_2000} and the convergence behaviour of the series.\cite{Olsen_2000}
Their analysis is based on Darboux's theorem: \cite{Goodson_2011} Their analysis is based on Darboux's theorem: \cite{Goodson_2011}
\begin{quote} \begin{quote}
\textit{``In the limit of large order, the series coefficients become equivalent to the Taylor series coefficients of the singularity closest to the origin. ''} \textit{``In the limit of large order, the series coefficients become equivalent to
the Taylor series coefficients of the singularity closest to the origin. ''}
\end{quote} \end{quote}
Following this theory, a singularity in the unit circle is designated as an intruder state, Following this theory, a singularity in the unit circle is designated as an intruder state,
with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative). with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative).
@ -1401,39 +1403,55 @@ More specifically, a $[d_A/d_B]$ Pad\'e approximant is defined as
= \frac{\sum_{k=0}^{d_A} a_k\, \lambda^k}{1 + \sum_{k=1}^{d_B} b_k\, \lambda^k}, = \frac{\sum_{k=0}^{d_A} a_k\, \lambda^k}{1 + \sum_{k=1}^{d_B} b_k\, \lambda^k},
\end{equation} \end{equation}
where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms for each power of $\lambda$. where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms for each power of $\lambda$.
Pad\'e approximants are extremely useful in many areas of physics and chemistry \cite{Loos_2013,Pavlyukh_2017,Tarantino_2019,Gluzman_2020} as they can model poles, which appears at the locations of the roots of $B(\lambda)$. Pad\'e approximants are extremely useful in many areas of physics and
However, they are unable to model functions with square-root branch points (which are ubiquitous in the singularity structure of a typical perturbative treatment) and more complicated functional forms appearing at critical points (where the nature of the solution undergoes a sudden transition). chemistry\cite{Loos_2013,Pavlyukh_2017,Tarantino_2019,Gluzman_2020} as they can model poles,
which appears at the locations of the roots of $B(\lambda)$.
However, they are unable to model functions with square-root branch points
(which are ubiquitous in the singularity structure of a typical perturbative treatment)
and more complicated functional forms appearing at critical points (
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 $)
often define a convergent perturbation series in cases where the Taylor series expansion diverges.} often define a convergent perturbation series in cases where the Taylor series expansion diverges.}
Figure \ref{fig:PadeRMP} illustrates the improvement provided by diagonal Pad\'e approximants compared to the usual Taylor expansion in cases where the RMP series of the Hubbard dimer converges ($U/t = 3.5$) and diverges ($U/t = 4.5$). Fig.~\ref{fig:PadeRMP} illustrates the improvement provided by diagonal Pad\'e
More quantitatively, Table \ref{tab:PadeRMP} gathers estimates of the RMP ground-state energy at $\lambda = 1$ provided by various truncated Taylor series and Pad\'e approximants for these two values of the ratio $U/t$. approximants compared to the usual Taylor expansion in cases where the RMP series of
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. the Hubbard dimer converges ($U/t = 3.5$) and diverges ($U/t = 4.5$).
\hugh{Furthermore, the Pad\'e approximants provide a rather good estimate of the radius of convergence of the RMP series.} More quantitatively, Table \ref{tab:PadeRMP} gathers estimates of the RMP ground-state
For $U/t = 4.5$, the Taylor series expansion performs worse (and eventually diverges), energy at $\lambda = 1$ provided by various truncated Taylor series and Pad\'e
while the Pad\'e approximants still offer relaitively accurate energies even outside the radius of convergence of the RMP series. 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.
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.}
\hugh{% \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 (see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}). We can expect that the singularity structure of the UMP energy will be much more challenging
Figure~\ref{fig:QuadUMP} and Table~\ref{tab:QuadUMP} indicate that this is indeed the case. to model properly as the UMP energy function 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
the square root branch point that lies close to $\lambda = 1$ by placing a pole on the real axis
(\eg, [3/3]) or with a very small imaginary component (\eg, [4/4]).
The proximity of these poles to the physical point $\lambda = 1$ means that any error in the Pad\'e
functional form becomes magnified in the estimate of exact energy, as seen for the low-order
approximants in Table~\ref{tab:QuadUMP}.
However, with sufficiently high degree polynomials, one obtains However, with sufficiently high degree polynomials, one obtains
accurate estimates of both the radius of convergence and the ground-state energy at $\lambda = 1$, accurate estimates for the position of the closest singularity and the ground-state energy at $\lambda = 1$,
even in cases where the convergence of the UMP series is incredibly slow 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}).
In Figure \ref{fig:QuadUMP}, it becomes clear that the Pad\'e approximants are trying to model
the square root branch point that lies close to $\lambda = 1$ by placing a pole on the real axis
(for [3/3]) or with a very small imaginary component (for [4/4]).
The proximity of these poles to the radius of convergence means that any error in the Pad\'e
functional form becomes magnified in the estimate of energy at $\lambda = 1$.
} }
\begin{table} \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 $4.5$. \caption{RMP ground-state energy estimate at $\lambda = 1$ provided by various truncated Taylor
We also report the estimate of the radius of convergence $r_c$ provided by the diagonal Pad\'e approximants. 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}} \label{tab:PadeRMP}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lccccc} \begin{tabular}{lccccc}
& & \mc{2}{c}{$r_c$} & \mc{2}{c}{$E_{-}(\lambda = 1)$} \\ & & \mc{2}{c}{$\lc$} & \mc{2}{c}{$E_{-}(\lambda = 1)$} \\
\cline{3-4} \cline{5-6} \cline{3-4} \cline{5-6}
Method & degree & $U/t = 3.5$ & $U/t = 4.5$ & $U/t = 3.5$ & $U/t = 4.5$ \\ Method & degree & $U/t = 3.5$ & $U/t = 4.5$ & $U/t = 3.5$ & $U/t = 4.5$ \\
\hline \hline
@ -1458,14 +1476,17 @@ functional form becomes magnified in the estimate of energy at $\lambda = 1$.
\includegraphics[height=0.23\textheight]{PadeRMP35} \includegraphics[height=0.23\textheight]{PadeRMP35}
\includegraphics[height=0.23\textheight]{PadeRMP45} \includegraphics[height=0.23\textheight]{PadeRMP45}
\caption{\label{fig:PadeRMP} \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).} 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*} \end{figure*}
%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%
%==========================================% %==========================================%
\subsection{Quadratic Approximant} \subsection{Quadratic Approximant}
%==========================================% %==========================================%
Quadratic approximants \hugh{are designed} to model the singularity structure of the energy function $E(\lambda)$ via a generalised version of the square-root singularity expression \cite{Mayer_1985,Goodson_2011,Goodson_2019} Quadratic approximants \hugh{are designed} to model the singularity structure of the energy
function $E(\lambda)$ via a generalised version of the square-root singularity
expression \cite{Mayer_1985,Goodson_2011,Goodson_2019}
\begin{equation} \begin{equation}
\label{eq:QuadApp} \label{eq:QuadApp}
E(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ] E(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ]
@ -1484,14 +1505,28 @@ Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \
\begin{equation} \begin{equation}
Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}} Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}}
\end{equation} \end{equation}
and substituting $E(\lambda$) by its $n$th-order expansion and the polynomials by their respective expressions \eqref{eq:PQR} yields $n+1$ linear equations for the coefficients $p_k$, $q_k$, and $r_k$ (where we are free to assume that $q_0 = 1$). and substituting $E(\lambda$) by its $n$th-order expansion and the polynomials by
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 $P^2(\lambda) - 4 Q(\lambda) R(\lambda)$. their respective expressions \eqref{eq:PQR} yields $n+1$ linear equations for the coefficients
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. $p_k$, $q_k$, and $r_k$ (where we are free to assume that $q_0 = 1$).
However, by construction, a quadratic approximant has only two branches, which hampering the faithful description of more complicated singularity structures. A quadratic approximant, characterised by the label $[d_P/d_Q,d_R]$, generates, by construction,
As shown in Ref.~\onlinecite{Goodson_2000a}, quadratic approximants provide convergent results in the most divergent cases considered by Olsen and collaborators \cite{Christiansen_1996,Olsen_1996} and Leininger \etal \cite{Leininger_2000} $n_\text{bp} = \max(2d_p,d_q+d_r)$ 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.
However, while a quadratic approximant can reproduce multiple branch points, it can only describe
a total of two branches.
Since every branch point must therefore correspond to a degeneracy of the same two branches, this constraint
can hamper the faithful description of more complicated singularity structures such as the MP energy surface.
Despite this limitiation,} Ref.~\onlinecite{Goodson_2000a} demonstrates that quadratic approximants
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 $\lambda_\text{EP} = \pm i 4t/U$. For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant
This is expected from the form of the RMP energy [see Eq.~\eqref{eq:E0MP}], which matches the ideal target for quadratic approximants. 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.
%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%
\begin{figure} \begin{figure}
@ -1502,12 +1537,14 @@ This is expected from the form of the RMP energy [see Eq.~\eqref{eq:E0MP}], whic
%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%
\begin{table} \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 $7$. \caption{Estimate of the radius of convergence $r_c$ of the UMP energy function provided
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. 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}} \label{tab:QuadUMP}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lccccccc} \begin{tabular}{lccccccc}
& & & & \mc{2}{c}{$r_c$} & \mc{2}{c}{$E_{-}(\lambda)$} \\ & & & & \mc{2}{c}{$\lp$} & \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
@ -1526,9 +1563,10 @@ This is expected from the form of the RMP energy [see Eq.~\eqref{eq:E0MP}], whic
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table}
\hugh{On the other hand, the greater flexibility of the quadratic approximants provides a significantly
improved model of the UMP energy in comparison to the Pad\' approximants or Taylor series. \hugh{On the other hand, the greater flexibility of the diagonal quadratic approximants provides a significantly
In particular, the quadratic approximants provide an effect model for the avoided crossings 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. (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 $\lambda = 1$, Furthermore, they provide remarkably accurate estimates of the ground-state energy at $\lambda = 1$,
as shown in Table~\ref{tab:QuadUMP}} as shown in Table~\ref{tab:QuadUMP}}

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