starting working on pade and stuff
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@ -1167,7 +1167,46 @@ We believe that $\alpha$ singularities are connected to states with non-negligib
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%==========================================%
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\subsection{Pad\'e approximant}
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%==========================================%
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According to Wikipedia, \textit{``a Pad\'e approximant is the best approximation of a function by a rational function of given order''}.
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A $[d_A/d_B]$ Pad\'e approximant is defined as
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\begin{equation}
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\label{eq:PadeApp}
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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}
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\end{equation}
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with $b_0 = 1$.
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Pad\'e approximants are nice and they can model poles, but they cannot model functions with square-root branch points, and that sucks.
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%==========================================%
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\subsection{Quadratic approximant}
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%==========================================%
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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
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\begin{equation}
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\label{eq:QuadApp}
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E(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ]
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\end{equation}
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where
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\begin{align}
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\label{eq:PQR}
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P(\lambda) & = \sum_{k=0}^{d_P} p_k \lambda^k,
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&
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Q(\lambda) & = \sum_{k=0}^{d_Q} q_k \lambda^k,
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&
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R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k
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\end{align}
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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 $E(\lambda)$.
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Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ie,
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\begin{equation}
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Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}}
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\end{equation}
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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 $q_0 = 1$).
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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.
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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.
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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 $\lambda_{EP} = \pm i 4t/U$.
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This is expected knowing the form of the RMP energy [see Eq.~\eqref{eq:E0MP}] which perfectly suits the purpose of quadratic approximants.
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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.
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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.~??.
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%==========================================%
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\subsection{Analytic continuation}
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@ -1188,7 +1227,6 @@ Their method consists in refining self-consistently the values of $E(\lambda)$ c
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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.
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The authors illustrate this protocol on the dissociation curve of \ce{LiH} and the stretched water molecule showing encouraging results. \cite{Mihalka_2019}
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%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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%%%%%%%%%%%%%%%%%%%%
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