saving work on RMP pade

Manuscript/EPAWTFT.tex

@@ -1250,7 +1250,33 @@ More specifically, a $[d_A/d_B]$ Pad\'e approximant is defined as
 (with $b_0 = 1$), where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms according to power of $\lambda$.
 Pad\'e approximants are extremely useful in many areas of physics and chemistry \cite{Loos_2013,Pavlyukh_2017,Tarantino_2019,Gluzman_2020} as they can model poles, which appears at the roots of the polynomial $B(\lambda)$. 
 However, they are unable to model functions with square-root branch points, which are ubiquitous in the singularity structure of a typical perturbative treatment.
-Figure \ref{fig:PadeRMP} illustrates the improvement brought by diagonal (\ie, $d_A = d_B$) Pad\'e approximants as compared to the usual Taylor expansion in the case of the RMP series of the Hubbard dimer for $U/t = 4.5$.
+Figure \ref{fig:PadeRMP} illustrates the improvement brought by diagonal (\ie, $d_A = d_B$) Pad\'e approximants as compared to the usual Taylor expansion in cases where the RMP series of the Hubbard dimer converges ($U/t = 3.5$) and diverges ($U/t = 4.5$).
+
+\begin{table}
+	\caption{RMP ground-state energy estimate at $\lambda = 1$ provided by various Taylor expansions and Pad\'e approximants at $U/t = 3.5$ and $4.5$.
+	We also report the estimate of the radius of convergence $r_c$ provided by the diagonal Pad\'e approximants.
+	\label{tab:PadeRMP}}
+	\begin{ruledtabular}
+		\begin{tabular}{lccccc}
+						&			&	\mc{2}{c}{$r_c$}			&	\mc{2}{c}{$E_{-}(\lambda = 1)$}			\\
+																		\cline{3-4} \cline{5-6}
+			Method		&	degree	&	$U/t = 3.5$	&	$U/t = 4.5$	&	$U/t = 3.5$	&	$U/t = 4.5$	\\
+			\hline
+			Taylor		&	2		&		&		&	$-1.01563$	&	$-1.01563$	\\
+						&	3		&		&		&	$-1.01563$	&	$-1.01563$	\\
+						&	4		&		&		&	$-0.86908$	&	$-0.61517$	\\
+						&	5		&		&		&	$-0.86908$	&	$-0.61517$	\\
+						&	6		&		&		&	$-0.92518$	&	$-0.86858$	\\
+			Pad\'e		&	[1/1]	&	$2.29$	&	$1.78$	&	$-1.61111$	&	$-2.64286$	\\
+						&	[2/2]	&	$2.29$	&	$1.78$	&	$-0.82124$	&	$-0.48446$	\\
+						&	[3/3]	&	$1.73$	&	$1.34$	&	$-0.91995$	&	$-0.81929$	\\
+						&	[4/4]	&	$1.47$	&	$1.14$	&	$-0.90579$	&	$-0.74866$	\\
+						&	[5/5]	&	$1.35$	&	$1.05$	&	$-0.90778$	&	$-0.76277$	\\
+			\hline
+			Exact		&			&	$1.14$	&	$0.89$	&	$-0.90754$	&	$-0.76040$	\\
+		\end{tabular}
+	\end{ruledtabular}
+\end{table}
 
 %%%%%%%%%%%%%%%%%
 \begin{figure*}
@@ -1306,23 +1332,24 @@ However, by ramping up high enough the degree of the polynomials, one is able to
 %%%%%%%%%%%%%%%%%
 
 \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.
+	\caption{Estimate of the radius of convergence $r_c$ of the UMP energy function provided 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 points $n_\text{bp} = \max(2d_p,d_q+d_r)$ generated by the quadratic approximants are also reported.
 	\label{tab:QuadUMP}}
 	\begin{ruledtabular}
-		\begin{tabular}{llcccc}
-						&			&			&					&	\mc{2}{c}{$r_c$}			\\
-																		\cline{5-6}
-			\mc{2}{c}{Method}		&	$n$		&	$n_\text{bp}$	&	$U/t = 3$	&	$U/t = 7$	\\
+		\begin{tabular}{lccccccc}
+						&			&			&					&	\mc{2}{c}{$r_c$}	&	\mc{2}{c}{$E_{-}(\lambda)$}			\\
+																		\cline{5-6}\cline{7-8}
+			\mc{2}{c}{Method}		&	$n$		&	$n_\text{bp}$	&	$U/t = 3$	&	$U/t = 7$	&	$U/t = 3$	&	$U/t = 7$	\\
 			\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\\
+			Pad\'e		&	[2/2]	&	4	&		&	0.974		&	1.000	\\
+						&	[3/3]	&	6	&		&	1.141		&	1.004	\\
+						&	[4/4]	&	8	&		&	1.068		&	1.003	\\
+			Quadratic	&	[2/1,2]	&	6	&	4	&	1.086		&	1.003	\\
+						&	[2/2,2]	&	7	&	4	&	1.082		&	1.003	\\
+						&	[3/2,2]	&	8	&	6	&	1.082		&	1.001	\\
+						&	[3/2,3]	&	9	&	6	&	1.071		&	1.002	\\
+						&	[3/3,3]	&	10	&	6	&	1.071		&	1.002	\\
+			Exact		&			&		&		&	1.069		&	1.002	\\
 		\end{tabular}
 	\end{ruledtabular}
 \end{table}