Merge branch 'master' of github.com:pfloos/EPAWTFT

Manuscript/EPAWTFT.tex

@@ -753,7 +753,15 @@ 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}.
+From it, we clearly see that the UMP series has \titou{always?} a larger radius of convergence than the RMP series \titou{(except maybe at $U/t = 2^+$)}, and that the UMP ground-state series is always convergent as $r_c > 1$ for all $U/t$.
+ 
+\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 ratio $U/t$.
+	\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.