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Hugh Burton 2020-11-25 11:55:23 +00:00
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@ -1216,6 +1216,15 @@ A $[d_A/d_B]$ Pad\'e approximant is defined as
(with $b_0 = 1$), where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms according to power of $\lambda$. (with $b_0 = 1$), where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms according to power of $\lambda$.
Pad\'e approximants are extremely useful in many areas of physics and chemistry \cite{Loos_2013,Gluzman_2020} as they can model poles, which appears at the roots of the polynomial $B(\lambda)$. Pad\'e approximants are extremely useful in many areas of physics and chemistry \cite{Loos_2013,Gluzman_2020} as they can model poles, which appears at the roots of the polynomial $B(\lambda)$.
However, they are unable to model functions with square-root branch points, which are ubiquitous in the singularity structure of a typical perturbative treatment. However, they are unable to model functions with square-root branch points, which are ubiquitous in the singularity structure of a typical perturbative treatment.
Figure \ref{fig:PadeRMP} illustrates the improvement brought by Pad\'e approximants as compared to the usual Taylor expansion in the case of the RMP series of the Hubbard dimer for $U/t = 4.5$.
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\begin{figure}
\includegraphics[width=\linewidth]{PadeRMP}
\caption{\label{fig:PadeRMP}
RMP ground-state energy as a function of $\lambda$ obtained with various approximations for $U/t = 4.5$.}
\end{figure}
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\subsection{Quadratic approximant} \subsection{Quadratic approximant}

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