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starting working on pade and stuff

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Pierre-Francois Loos 2020-11-24 23:07:27 +01:00
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40
Manuscript/EPAWTFT.tex
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@ -1167,7 +1167,46 @@ We believe that $\alpha$ singularities are connected to states with non-negligib
%==========================================% %==========================================%
\subsection{Pad\'e approximant} \subsection{Pad\'e approximant}
%==========================================% %==========================================%
According to Wikipedia, \textit{``a Pad\'e approximant is the best approximation of a function by a rational function of given order''}.
A $[d_A/d_B]$ Pad\'e approximant is defined as
\begin{equation}
\label{eq:PadeApp}
E(\lambda) = \frac{A(\lambda)}{B(\lambda)} = \frac{\sum_{k=0}^{d_A} a_k \lambda^k}{\sum_{k=0}^{d_B} b_k \lambda^k}
\end{equation}
with $b_0 = 1$.
Pad\'e approximants are nice and they can model poles, but they cannot model functions with square-root branch points, and that sucks.
%==========================================%
\subsection{Quadratic approximant}
%==========================================%
In a nutshell, the idea behind quadratic approximant is to model the singularity structure of the energy function $E(\lambda)$ via a generalized version of the square-root singularity expression
\begin{equation}
\label{eq:QuadApp}
E(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ]
\end{equation}
where
\begin{align}
\label{eq:PQR}
P(\lambda) & = \sum_{k=0}^{d_P} p_k \lambda^k,
&
Q(\lambda) & = \sum_{k=0}^{d_Q} q_k \lambda^k,
&
R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k
\end{align}
are polynomials, such that $d_P + d_Q + d_R = n - 1$ where $n$ is the highest-order series coefficient known from the Taylor expansion of $E(\lambda)$.
Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ie,
\begin{equation}
Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}}
\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 $q_0 = 1$).
A quadratic approximant, characterised by the label $[d_P/d_Q,d_R]$, generates, by construction, $\max(2d_p,d_q+d_r)$ branch points.
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.
For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant are quite poor approximation, but its $[1/0,1]$ version already model perfectly the RMP energy function by predicting a single pair of EPs at $\lambda_{EP} = \pm i 4t/U$.
This is expected knowing the form of the RMP energy [see Eq.~\eqref{eq:E0MP}] which perfectly suits the purpose of quadratic approximants.
We can anticipate that the singularity structure of the UMP energy function is going to be much more challenging to model properly, and this is indeed the case.
However, by ramping up high enough the degree of the polynomials, one is able to get an accurate estimates of the radius of convergence of the UMP series as shown in Fig.~??.
%==========================================% %==========================================%
\subsection{Analytic continuation} \subsection{Analytic continuation}
@ -1188,7 +1227,6 @@ Their method consists in refining self-consistently the values of $E(\lambda)$ c
When the values of $E(\lambda)$ on the so-called contour are converged, Cauchy's integrals formula \eqref{eq:Cauchy} is invoked to compute the values at $E(\lambda=1)$ which corresponds to the final estimate of the FCI energy. When the values of $E(\lambda)$ on the so-called contour are converged, Cauchy's integrals formula \eqref{eq:Cauchy} is invoked to compute the values at $E(\lambda=1)$ which corresponds to the final estimate of the FCI energy.
The authors illustrate this protocol on the dissociation curve of \ce{LiH} and the stretched water molecule showing encouraging results. \cite{Mihalka_2019} The authors illustrate this protocol on the dissociation curve of \ce{LiH} and the stretched water molecule showing encouraging results. \cite{Mihalka_2019}
%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}
%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%

95
Notebooks/UMP_rc.nb
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