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OK up to Sec IIIF

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Pierre-Francois Loos 2020-12-05 14:49:23 +01:00
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Manuscript/EPAWTFT.tex
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@ -1139,7 +1139,7 @@ to identify and remains an open question.
% CLASSIFICATIONS BY GOODSOON AND SERGEEV % CLASSIFICATIONS BY GOODSOON AND SERGEEV
To further develop the link between the critical point and types of MP convergence, Sergeev and Goodson investigated To further develop the link between the critical point and types of MP convergence, Sergeev and Goodson investigated
the relationship with the location of the dominant singularity that controls the radius of convergence.\cite{Goodson_2004} the relationship with the location of the dominant singularity that controls the radius of convergence.\cite{Goodson_2004}
They demonstrated that the dominant singularity in class A systems corresponds to a EP with a positive real component, They demonstrated that the dominant singularity in class A systems corresponds to an EP with a positive real component,
where the magnitude of the imaginary component controls the oscillations in the signs of successive MP where the magnitude of the imaginary component controls the oscillations in the signs of successive MP
terms.\cite{Goodson_2000a,Goodson_2000b} terms.\cite{Goodson_2000a,Goodson_2000b}
In contrast, class B systems correspond to a dominant singularity on the negative real $\lambda$ axis representing In contrast, class B systems correspond to a dominant singularity on the negative real $\lambda$ axis representing
@ -1159,7 +1159,7 @@ While a finite basis can only predict complex-conjugate branch point singulariti
by a cluster of sharp avoided crossings between the ground state and high-lying excited states.\cite{Sergeev_2005} by a cluster of sharp avoided crossings between the ground state and high-lying excited states.\cite{Sergeev_2005}
Alternatively, Sergeev \etal\ demonstrated that the inclusion of a ``ghost'' atom also Alternatively, Sergeev \etal\ demonstrated that the inclusion of a ``ghost'' atom also
allows the formation of the critical point as the electrons form a bound cluster occupying the ghost atom orbitals.\cite{Sergeev_2005} allows the formation of the critical point as the electrons form a bound cluster occupying the ghost atom orbitals.\cite{Sergeev_2005}
This effect explains the origin of the divergence in the \ce{HF} molecule as the fluorine valence electrons jump to hydrogen at This effect explains the origin of the divergence in the \ce{HF} molecule as the fluorine valence electrons jump to \titou{the} hydrogen at
a sufficiently negative $\lambda$ value.\cite{Sergeev_2005} a sufficiently negative $\lambda$ value.\cite{Sergeev_2005}
Furthermore, the two-state model of Olsen and collaborators \cite{Olsen_2000} was simply too minimal to understand the complexity of Furthermore, the two-state model of Olsen and collaborators \cite{Olsen_2000} was simply too minimal to understand the complexity of
divergences caused by the MP critical point. divergences caused by the MP critical point.
@ -1368,8 +1368,8 @@ radius of convergence (see Fig.~\ref{fig:RadConv}).
\includegraphics[height=0.23\textheight]{fig9a} \includegraphics[height=0.23\textheight]{fig9a}
\includegraphics[height=0.23\textheight]{fig9b} \includegraphics[height=0.23\textheight]{fig9b}
\caption{\label{fig:PadeRMP} \caption{\label{fig:PadeRMP}
RMP ground-state energy as a function of $\lambda$ obtained using various resummation RMP ground-state energy as a function of $\lambda$ obtained using various \titou{truncated Taylor series and approximants}
techniques at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).} at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).}
\end{figure*} \end{figure*}
%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%
@ -1467,7 +1467,7 @@ a convergent series.
\begin{figure}[t] \begin{figure}[t]
\includegraphics[width=\linewidth]{fig10} \includegraphics[width=\linewidth]{fig10}
\caption{\label{fig:QuadUMP} \caption{\label{fig:QuadUMP}
UMP energies as a function of $\lambda$ obtained using various resummation techniques at $U/t = 3$.} UMP energies as a function of $\lambda$ obtained using various \titou{approximants} at $U/t = 3$.}
\end{figure} \end{figure}
%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%
@ -1538,7 +1538,7 @@ The remedy for this problem involves applying a suitable transformation of the c
\begin{table}[b] \begin{table}[b]
\caption{Estimate 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 by various resummation techniques at $U/t = 3$ and $7$. in the UMP energy function provided by various \titou{truncated Taylor series and approximants} 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}}
@ -1758,6 +1758,7 @@ the cost of larger denominators is an overall slower rate of convergence.
Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function
of $\lambda$ for the symmetric Hubbard dimer at $U/t = 4.5$. of $\lambda$ for the symmetric Hubbard dimer at $U/t = 4.5$.
The two functions correspond closely within the radius of convergence. The two functions correspond closely within the radius of convergence.
\titou{T2: are we keeping this?}
} }
\label{fig:rmp_anal_cont} \label{fig:rmp_anal_cont}
\end{figure} \end{figure}