From 694f69532375a4d317fc30434398e5fa02f4a603 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 25 Oct 2021 16:07:45 +0200 Subject: New data in table. --- ising_scaling.tex | 157 ++++++++++++++++++++++++++---------------------------- 1 file changed, 77 insertions(+), 80 deletions(-) diff --git a/ising_scaling.tex b/ising_scaling.tex index 2fa2528..7b5da7b 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -576,7 +576,7 @@ that the codimension of the fit is constant. We performed this procedure starting at $n=2$, or matching the scaling function at the low and high temperature zero field points to quadratic order, through $n=7$. The resulting fit coefficients can be found in Table -\ref{tab:fits} without any sort of uncertainty, which is difficult to quantify +\ref{tab:data} without any sort of uncertainty, which is difficult to quantify directly due to the truncation of series. However, precise results exist for the value of the scaling function at the critical isotherm, or equivalently for the series coefficients of the scaling function $\mathcal F_0$. Since we do not @@ -610,110 +610,107 @@ to their known values at the critical isotherm, or $\theta=1$. \cite{Mangazeev_2008_Variational}. Those without are taken from Fonseca \textit{et al.}, and are assumed to be accurate to within their last digit \cite{Fonseca_2003_Ising}. - } \label{tab:fits} + } \label{tab:data} \end{table} \begin{table} \singlespacing - \begin{tabular}{c|llllllll} + \begin{tabular}{c|lllllll} \multicolumn{1}{c|}{$n$} & \multicolumn{1}{c}{$\theta_\mathrm{YL}$} & - \multicolumn{1}{c}{$A_\mathrm{YL}$} & + \multicolumn{1}{c}{$C_\mathrm{YL}$} & \multicolumn{1}{c}{$G_1$} & \multicolumn{1}{c}{$G_2$} & \multicolumn{1}{c}{$G_3$} & \multicolumn{1}{c}{$G_4$} & - \multicolumn{1}{c}{$G_5$} & - \multicolumn{1}{c}{$G_6$} \\ + \multicolumn{1}{c}{$G_5$} \\ \hline 2 & - 0.18041 & - 2.1295 & - 1.2447 & - 0.49975 \\ + 0.198152 & + 2.39338 & + 1.52182 & + 0.455868 \\ 3 & - 0.19613 & - 2.1999 & - 1.2402 & - 0.39028 \\ + 0.199048 & + 2.25700 & + 1.30767 & + 0.396157 \\ 4 & - 0.19563 & - 2.2433 & - 1.2964 & - 0.37080 & - $-0.028926$ \\ + 0.198918 & + 2.36150 & + 1.46058 & + 0.307198 & + \hphantom{$-$}0.0323334 \\ 5 & - 0.19553 & - 2.2321 & - 1.2804 & - 0.36318 & - $-0.028924$ & & \\ + 0.198100 & + 2.43224 & + 1.55298 & + 0.362791 & + $-0.108393$ & + 0.0228573 \\ 6 & - 0.19737 & - 2.3981 & - 1.5091 & - 0.39200 & - $-0.090023$ & - 0.017233 & \\ + 0.197027 & + 2.45864 & + 1.59639 & + 0.484865 & + $-0.119149$ & + 0.0150607 \\ 7 & - 0.19730 & - 2.4229 & - 1.5454 & - 0.41320 & - $-0.12161$ & - 0.026346 & \\ - 8 & - 0.19655 & - 2.5513 & - 1.7323 & - 0.59677 & - $-0.29521$ & - 0.078509 & - $-0.0072514$ \\ - 9 & - 1.3754 & - 0.19652 & - 2.5482 & - 1.7278 & - 0.60278 & - $-0.28737$ & - 0.072411 & - $-0.0072455$ \\ + 0.196809 & + 2.49636 & + 1.65176 & + 0.54349 & + $-0.141089$ & + 0.00877823 & + $-0.00110698$ \\ \hline \end{tabular} - \begin{tabular}{c|llllllll} - \hline + \begin{tabular}{c|llllll} + \hline $n$ & \multicolumn{1}{c}{$\theta_0$} & - \multicolumn{1}{c}{$h_1$} & - \multicolumn{1}{c}{$h_2$} & - \multicolumn{1}{c}{$h_3$} & - \multicolumn{1}{c}{$h_4$} & - \multicolumn{1}{c}{$h_5$} & - \multicolumn{1}{c}{$h_6$} & - \multicolumn{1}{c}{$h_7$} \\ + \multicolumn{1}{c}{$g_1$} & + \multicolumn{1}{c}{$g_2$} & + \multicolumn{1}{c}{$g_3$} & + \multicolumn{1}{c}{$g_4$} & + \multicolumn{1}{c}{$g_5$} \\ \hline - 2 & - 1.2114 \\ - 3 & - 1.3498 & - $-0.014909$ \\ + 2 & + 1.22182 & + $-0.0140696$ \\ + 3 & + 1.33128 & + $-0.0144135$ & + $-0.000233014$ \\ 4 & - 1.4490 & - $-0.10871$ & $-0.0031747$ \\ + 1.33106 & + $-0.0268822$ & + \hphantom{$-$}$0.0332121$ \\ 5 & - 1.4719 & - $-0.11399$ & $-0.0031669$ & $8.8574\times10^{-7}$ \\ + 1.51661 & + $-0.248646$ & + \hphantom{$-$}$0.0513219$ & + $0.00433870$ \\ 6 & - 1.4358 & - $-0.19533$ & 0.029301 & 0.0039906 & $-0.00011913$ \\ + 1.37654 & + $-0.236244$ & + $-0.00860939$ & + 0.00204312 & + $-0.000954046$ \\ 7 & - 1.4324 & - $-0.22077$ & 0.036245 & 0.010120 & $-0.0011434$ & 0.00010095 \\ - 8 & - 1.3710 & - $-0.35150$ & 0.0050232 & 0.053659 & $-0.019806$ & 0.0033531 & $-0.00026034$ \\ + 1.33161 & + $-0.252714$ & + $-0.0308462$ & + 0.00491578 & + \hphantom{$-$}0.000264989 & + $-0.000210698$ \end{tabular} + \caption{ + Free parameters in the fit of the parametric coordinate transformation and + scaling form to known values of the scaling function series coefficients + for $\mathcal F_\pm$. The fit at stage $n$ matches those coefficients up to + and including order $n$. + } \label{tab:fits} \end{table} \begin{figure} @@ -778,7 +775,7 @@ accurate to within $2\times10^{-3}$. This approximation for the scaling function \caption{ The series coefficients for the scaling function $\mathcal F_-$ as a function of polynomial order $m$. The numeric values are from Table - \ref{tab:fits}, and those of Caselle \textit{et al.} are from the most + \ref{tab:data}, and those of Caselle \textit{et al.} are from the most accurate scaling function listed in \cite{Caselle_2001_The}. } \label{fig:glow.series} \end{figure} @@ -835,7 +832,7 @@ accurate to within $2\times10^{-3}$. This approximation for the scaling function \caption{ The series coefficients for the scaling function $\mathcal F_+$ as a function of polynomial order $m$. The numeric values are from Table - \ref{tab:fits}, and those of Caselle \textit{et al.} are from the most + \ref{tab:data}, and those of Caselle \textit{et al.} are from the most accurate scaling function listed in \cite{Caselle_2001_The}. } \end{figure} -- cgit v1.2.3-70-g09d2