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@ -1453,12 +1453,12 @@ even in cases where the convergence of the UMP series is incredibly slow
%==========================================% %==========================================%
\subsection{Quadratic Approximant} \subsection{Quadratic Approximant}
%==========================================% %==========================================%
Quadratic approximants \hugh{are designed} to model the singularity structure of the energy Quadratic approximants are designed to model the singularity structure of the energy
function $E(\lambda)$ via a generalised version of the square-root singularity function $E(\lambda)$ via a generalised version of the square-root singularity
expression \cite{Mayer_1985,Goodson_2011,Goodson_2019} 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)} ],
\end{equation} \end{equation}
with the polynomials with the polynomials
\begin{align} \begin{align}
@ -1467,27 +1467,28 @@ with the polynomials
& &
Q(\lambda) & = \sum_{k=0}^{d_Q} q_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 R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k,
\end{align} \end{align}
defined such that $d_P + d_Q + d_R = n - 1$, and $n$ is the truncation order of the Taylor series of $E(\lambda)$. defined such that $d_P + d_Q + d_R = n - 1$, and $n$ is the truncation order of the Taylor series of $E(\lambda)$.
Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ie, Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ie,
\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 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 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 $q_0 = 1$). $p_k$, $q_k$, 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, 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 $P^2(\lambda) - 4 Q(\lambda) R(\lambda)$ \hugh{and $d_q$ poles at the roots of $Q(\lambda)$}. $n_\text{bp} = \max(2d_p,d_q+d_r)$ branch points at the roots of the polynomial
$P^2(\lambda) - 4 Q(\lambda) R(\lambda)$ and $d_q$ poles at the roots of $Q(\lambda)$.
Generally, the diagonal sequence of quadratic approximant, Generally, the diagonal sequence of quadratic approximant,
\ie, $[0/0,0]$, $[1/0,0]$, $[1/0,1]$, $[1/1,1]$, $[2/1,1]$, \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. is of particular interest 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 However, while a quadratic approximant can reproduce multiple branch points, it can only describe
a total of two branches. a total of two branches.
Since every branch point must therefore correspond to a degeneracy of the same two branches, this constraint \titou{Since every branch points 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. 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 Despite this limitation, Ref.~\onlinecite{Goodson_2000a} demonstrates that quadratic approximants
provide convergent results in the most divergent cases considered by Olsen and provide convergent results in the most divergent cases considered by Olsen and
collaborators\cite{Christiansen_1996,Olsen_1996} collaborators\cite{Christiansen_1996,Olsen_1996}
and Leininger \etal \cite{Leininger_2000} and Leininger \etal \cite{Leininger_2000}
@ -1499,9 +1500,8 @@ In such a scenario, the quadratic approximant will tend to place its branch poin
The remedy for this problem involves applying a suitable transformation of the complex plane (such as a bilinear conformal mapping) which leaves the points at $\lambda = 0$ and $\lambda = 1$ unchanged. \cite{Feenberg_1956} The remedy for this problem involves applying a suitable transformation of the complex plane (such as a bilinear conformal mapping) which leaves the points at $\lambda = 0$ and $\lambda = 1$ unchanged. \cite{Feenberg_1956}
\begin{table}[b] \begin{table}[b]
\caption{Estimate \hugh{for the distance of the closest singularity (pole or branch point) to the origin $\abs{\lc}$} \caption{Estimate for the distance of the closest singularity (pole or branch point) to the origin $\abs{\lc}$
in the UMP energy function provided in the UMP energy function provided by various resummation techniques at $U/t = 3$ and $7$.
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 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. 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}}
@ -1555,14 +1555,13 @@ The remedy for this problem involves applying a suitable transformation of the c
\subcaption{\label{subfig:ump_cp} [3/0,4] Quadratic} \subcaption{\label{subfig:ump_cp} [3/0,4] Quadratic}
\end{subfigure} \end{subfigure}
\caption{% \caption{%
\hugh{%
Comparison of the [3/2,2] and [3/0,4] quadratic approximants with the exact UMP energy surface in the complex $\lambda$ Comparison of the [3/2,2] and [3/0,4] quadratic approximants with the exact UMP energy surface in the complex $\lambda$
plane with $U/t = 3$. plane with $U/t = 3$.
Both quadratic approximants correspond to the same truncation degree of the Taylor series and model the branch points Both quadratic approximants correspond to the same truncation degree of the Taylor series and model the branch points
using a radicand polynomial of the same order. using a radicand polynomial of the same order.
However, the [3/2,2] approximant introduces poles into the surface that limits it accuracy, while the [3/0,4] approximant However, the [3/2,2] approximant introduces poles into the surface that limits it accuracy, while the [3/0,4] approximant
is free of poles.} is free of poles.}
\label{fig:nopole_quad}} \label{fig:nopole_quad}
\end{figure*} \end{figure*}
For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant
@ -1570,24 +1569,22 @@ are quite poor approximations, but the $[1/0,1]$ version perfectly models the RM
function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm \i 4t/U$. 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 This is expected from the form of the RMP energy [see Eq.~\eqref{eq:E0MP}], which matches
the ideal target for quadratic approximants. the ideal target for quadratic approximants.
\hugh{%
Furthermore, the greater flexibility of the diagonal quadratic approximants provides a significantly Furthermore, the greater flexibility of the diagonal quadratic approximants provides a significantly
improved model of the UMP energy in comparison to the Pad\'e approximants or Taylor series. improved model of the UMP energy in comparison to the Pad\'e approximants or Taylor series.
In particular, these quadratic approximants provide an effective model for the avoided crossings In particular, these quadratic approximants provide an effective model for the avoided crossings
(Fig.~\ref{fig:QuadUMP}) and an improved estimate for the distance of the (Fig.~\ref{fig:QuadUMP}) and an improved estimate for the distance of the
closest branch point to the origin. closest branch point to the origin.
Table~\ref{tab:QuadUMP} shows that they provide remarkably accurate Table~\ref{tab:QuadUMP} shows that they provide remarkably accurate
estimates of the ground-state energy at $\lambda = 1$.} estimates of the ground-state energy at $\lambda = 1$.
\hugh{%
While the diagonal quadratic approximants provide significanty improved estimates of the While the diagonal quadratic approximants provide significanty improved estimates of the
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 square-root branch cuts that approach the real axis in the limit $U/t \rightarrow \infty$. contains only \titou{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. taking $d_q = 0$ and increasing $d_r$ to retain equivalent accuracy in the square-root term [see Eq.\eqref{eq:QuadApp}].
Fig.~\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
approximant compared to the [3/2,2] approximant with the same truncation degree in the Taylor approximant compared to the [3/2,2] approximant with the same truncation degree in the Taylor
expansion. expansion.
Clearly modelling the square-root branch point using $d_q = 2$ has the negative effect of Clearly modelling the square-root branch point using $d_q = 2$ has the negative effect of
@ -1595,23 +1592,18 @@ introducing spurious poles in the energy, while focussing purely on the branch p
leads to a significantly improved model. leads to a significantly improved model.
Table~\ref{tab:QuadUMP} shows that these pole-free quadratic approximants Table~\ref{tab:QuadUMP} shows that these pole-free quadratic approximants
provide a rapidly convergent series with essentially exact energies at low order. provide a rapidly convergent series with essentially exact energies at low order.
}
Finally, to emphasise the improvement that can be gained by using either Pad\'e, diagonal quadratic,
\hugh{Finally, to emphasise the improvement that can be gained by using either Pad\'e, diagonal quadratic, or pole-free approximants, we consider the energy and error obtained using only the first 10 terms of the UMP series
or pole-free approximants, we consider the energy and error obtained using only the first 10 terms in the UMP
in Table~\ref{tab:UMP_order10}. in Table~\ref{tab:UMP_order10}.
The accuracy of these approximants reinforces how our understanding of the MP The accuracy of these approximants reinforces how our understanding of the MP
energy surface in the complex plane can be leveraged to significantly improve estimates of the exact energy surface in the complex plane can be leveraged to significantly improve estimates of the exact
energy using low-order perturbation expansions. energy using low-order perturbation expansions.
}
\begin{table}[h] \begin{table}[h]
\caption{ \caption{
\hugh{%
Estimate and associated error of the exact UMP energy at $U/t = 7$ for Estimate and associated error of the exact UMP energy at $U/t = 7$ for
various approximants using up to ten terms in the Taylor expansion. various approximants using up to ten terms in the Taylor expansion.
}
\label{tab:UMP_order10}} \label{tab:UMP_order10}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lccc} \begin{tabular}{lccc}
@ -1638,7 +1630,7 @@ will be fast enough for low-order approximations to be useful.
However, these low-order partial sums or approximants often contain a remarkable amount of information However, these low-order partial sums or approximants often contain a remarkable amount of information
that can be used to extract further information about the exact result. 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} and accelerating the rate of convergence of a sequence.\cite{Shanks_1955,BenderBook}
Consider the partial sums $S_N$ defined from the truncated summation of an infinite series Consider the partial sums $S_N$ defined from the truncated summation of an infinite series
\begin{equation} \begin{equation}