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Hugh Burton 2020-12-02 20:35:20 +00:00
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@ -1581,7 +1581,7 @@ While the diagonal quadratic approximants provide significanty improved estimate
ground-state energy, we can use our knowledge of the UMP singularity structure to develop ground-state energy, we can use our knowledge of the UMP singularity structure to develop
even more accurate results. even more accurate results.
We have seen in previous sections that the UMP energy surface We have seen in previous sections that the UMP energy surface
contains only \titou{square-root branch cuts} that approach the real axis in the limit $U/t \to \infty$. contains only square-root branch cuts that approach the real axis in the limit $U/t \to \infty$.
Since there are no true poles on this surface, we can obtain more accurate quadratic approximants by 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 [see Eq.\eqref{eq:QuadApp}]. taking $d_q = 0$ and increasing $d_r$ to retain equivalent accuracy in the square-root term [see Eq.\eqref{eq:QuadApp}].
Figure~\ref{fig:nopole_quad} illustrates this improvement for the pole-free [3/0,4] quadratic Figure~\ref{fig:nopole_quad} illustrates this improvement for the pole-free [3/0,4] quadratic
@ -1620,7 +1620,6 @@ energy using low-order perturbation expansions.
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table}
%==========================================% %==========================================%
\subsection{Shanks Transformation} \subsection{Shanks Transformation}
%==========================================% %==========================================%
@ -1633,31 +1632,31 @@ that can be used to extract further information about the exact result.
The Shanks transformation presents one approach for extracting this information The Shanks transformation presents one approach for extracting this information
and accelerating the rate of convergence of a sequence.\cite{Shanks_1955,BenderBook} and accelerating the rate of convergence of a sequence.\cite{Shanks_1955,BenderBook}
\hugh{Consider the partial sums Consider the partial sums
$S_N = \sum_{k=0}^{N} a_k$ $S_n = \sum_{k=0}^{n} s_k$
defined from the truncated summation of an infinite series defined from the truncated summation of an infinite series
$ S = \sum_{k=0}^{\infty} a_k$. $S = \sum_{k=0}^{\infty} s_k$.
If the series converges, then the partial sums will tend to the exact result If the series converges, then the partial sums will tend to the exact result
\begin{equation} \begin{equation}
\lim_{N\rightarrow \infty} S_N = S. \lim_{n \to \infty} S_n = S.
\end{equation} \end{equation}
The Shanks transformation attempts to generate increasingly accurate estimates of this The Shanks transformation attempts to generate increasingly accurate estimates of this
limt by defining a new series as limit by defining a new series as
\begin{equation} \begin{equation}
T(S_N) = \frac{S_{N+1} S_{N-1} - S_{N}^2}{S_{N+1} + S_{N-1} - 2 S_{N}}. T(S_n) = \frac{S_{n+1} S_{n-1} - S_{n}^2}{S_{n+1} + S_{n-1} - 2 S_{n}}.
\end{equation} \end{equation}
This series can converge faster than the original partial sums and can thus provide greater This series can converge faster than the original partial sums and can thus provide greater
accuracy using only the first few terms in the series. accuracy using only the first few terms in the series.
However, it is designed to accelerate exponentially converging partial sums with However, it is designed to accelerate \titou{exponentially?} converging partial sums with
the approximate form the approximate form
\begin{equation} \begin{equation}
S_N \approx S + a\,b^N. S_n \approx S + \alpha\,\beta^n.
\end{equation} \end{equation}
Furthermore, while this transformation can accelerate the convergence of a series, Furthermore, while this transformation can accelerate the convergence of a series,
there is no guarantee that this acceleration will be fast enough to significantly there is no guarantee that this acceleration will be fast enough to significantly
improve the accuracy of low-order approximations.} improve the accuracy of low-order approximations.
\hugh{To the best of our knowledge, the Shanks transformation has never previously been applied To the best of our knowledge, the Shanks transformation has never previously been applied
to the acceleration of the MP series. to the acceleration of the MP series.
We have therefore applied it to the convergent Taylor series, Pad\'e approximants, and quadratic We have therefore applied it to the convergent Taylor series, Pad\'e approximants, and quadratic
approximants for RMP and UMP in the symmetric Hubbard dimer. approximants for RMP and UMP in the symmetric Hubbard dimer.
@ -1669,17 +1668,14 @@ can significantly improve the estimate of the energy using low-order perturbatio
as shown in Table~\ref{tab:RMP_shank}. as shown in Table~\ref{tab:RMP_shank}.
Even though the RMP series diverges at $U/t = 4.5$, the combination Even though the RMP series diverges at $U/t = 4.5$, the combination
of diagonal Pad\'e approximants with the Shanks transformation reduces the absolute error of of diagonal Pad\'e approximants with the Shanks transformation reduces the absolute error of
the best energy estimate to 0.002\,\% using only the lowest 10 terms in the corresponding the best energy estimate to 0.002\,\% using only the lowest 10 terms in the Taylor series.
Taylor series.}
This remarkable result indicates just how much information is contained in the first few This remarkable result indicates just how much information is contained in the first few
terms of a perturbation series, even if it diverges. terms of a perturbation series, even if it diverges.
\begin{table}[th] \begin{table}[th]
\caption{ \caption{
\hugh{%
Acceleration of the diagonal Pad\'e approximant sequence for the RMP energy Acceleration of the diagonal Pad\'e approximant sequence for the RMP energy
using the Shanks transformation. using the Shanks transformation.
}
\label{tab:RMP_shank}} \label{tab:RMP_shank}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lcccc} \begin{tabular}{lcccc}
@ -1702,8 +1698,6 @@ terms of a perturbation series, even if it diverges.
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table}
%==========================================% %==========================================%
\subsection{Analytic continuation} \subsection{Analytic continuation}
%==========================================% %==========================================%