Stuff on quadratic approximant

Manuscript/EPAWTFT.bib

@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-11-25 10:05:17 +0100 
+%% Created for Pierre-Francois Loos at 2020-11-25 15:13:39 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 

Manuscript/EPAWTFT.tex

@@ -1191,7 +1191,7 @@ Figure \ref{fig:PadeRMP} illustrates the improvement brought by Pad\'e approxima
 \begin{figure}
     \includegraphics[width=\linewidth]{PadeRMP}
     \caption{\label{fig:PadeRMP}
-    RMP ground-state energy as a function of $\lambda$ obtained with various approximations for $U/t = 4.5$.}
+    RMP ground-state energy as a function of $\lambda$ obtained using various resummation techniques at $U/t = 4.5$.}
 \end{figure}
 %%%%%%%%%%%%%%%%%
 
@@ -1212,13 +1212,13 @@ where
 	&
 	R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k 
 \end{align}
-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 $E(\lambda)$.
+are polynomials, such that $d_P + d_Q + d_R = n - 1$, where $n$ is the truncation order of the Taylor of $E(\lambda)$.
 Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ie,
 \begin{equation}
 	Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}}
 \end{equation}
 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 $q_0 = 1$).
-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 at the roots of the polynomial $P^2(\lambda) - 4 Q(\lambda) R(\lambda)$.
+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 $P^2(\lambda) - 4 Q(\lambda) R(\lambda)$.
 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.
 Note that, by construction, a quadratic approximant has only two branches which hampers the faithful description of more complicated singularity structures.
 
@@ -1228,7 +1228,39 @@ For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic a
 This is expected knowing the form of the RMP energy [see Eq.~\eqref{eq:E0MP}] which perfectly suits the purpose of quadratic approximants.
 
 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 as the UMP energy function which contains three branches.
-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.~??.
+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.~\ref{fig:QuadUMP}.
+
+\titou{Here comes a discussion of Fig.~\ref{fig:QuadUMP} and Table \ref{tab:QuadUMP}.}
+
+%%%%%%%%%%%%%%%%%
+\begin{figure}
+    \includegraphics[width=\linewidth]{QuadUMP}
+    \caption{\label{fig:QuadUMP}
+    UMP energies as a function of $\lambda$ obtained using various resummation techniques at $U/t = 3$.}
+\end{figure}
+%%%%%%%%%%%%%%%%%
+
+\begin{table}
+	\caption{Radius of convergence $r_c$ for 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 points $n_\text{bp} = \max(2d_p,d_q+d_r)$ generated by the quadratic approximant are also reported.
+	\label{tab:QuadUMP}}
+	\begin{ruledtabular}
+		\begin{tabular}{llcccc}
+						&			&			&					&	\mc{2}{c}{$r_c$}			\\
+																		\cline{5-6}
+			\mc{2}{c}{Method}		&	$n$		&	$n_\text{bp}$	&	$U/t = 3$	&	$U/t = 7$	\\
+			\hline
+			Pad\'e		&	[2/2]	&	4	&		&	0.97448		&	1.00030	\\
+						&	[3/3]	&	6	&		&	1.14138		&	1.00448	\\
+			Quadratic	&	[2/1,2]	&	6	&	4	&	1.08640		&	1.00310	\\
+						&	[2/2,2]	&	7	&	4	&	1.08193		&	1.00310	\\
+						&	[3/2,2]	&	8	&	6	&	1.08247		&	1.00106	\\
+						&	[3/2,3]	&	9	&	6	&	1.07069		&	1.00239	\\
+						&	[3/3,3]	&	10	&	6	&	1.07064		&	1.00239	\\
+			Exact		&			&		&		&	1.06917		&	1.00239\\
+		\end{tabular}
+	\end{ruledtabular}
+\end{table}
 
 %==========================================%
 \subsection{Analytic continuation}