add fig RadConv

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Pierre-Francois Loos 2020-11-19 17:50:01 +01:00
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@ -727,7 +727,14 @@ The UMP partitioning yield the following $\lambda$-dependent Hamiltonian:
\end{equation}
\end{widetext}
A closed-form expression for the ground-state energy can be obtained but it is cumbersome, so we eschew reporting it.
The radius of convergence of the UMP series can obtained numerically as a function of $U/t$ and is depicted in \titou{Fig.~\ref{eq:UMP_rc}}.
The radius of convergence of the UMP series can obtained numerically as a function of $U/t$ and is depicted in Fig.~\ref{fig:RadConv}.
\begin{figure}
\includegraphics[width=\linewidth]{RadConv}
\caption{
Evolution of the radius of convergence $r_c$ associated with the RMP ground state (red), the UMP ground state (blue), and the UMP excited state (orange) as functions of the ration $U/t$.
\label{fig:RadConv}}
\end{figure}
The convergence of the UMP as a function of the ratio $U/t$ is shown in Fig.~\ref{subfig:UMP_cvg} for two specific values: the first ($U = 3t$) is well within the RMP convergence region, while the second ($U = 7t$) falls outside.
Note that in the case of UMP, there are now two pairs of EPs as the open-shell singlet now couples strongly with both the ground and doubly-excited states.

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