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-rw-r--r--ising_scaling.tex179
1 files changed, 176 insertions, 3 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex
index e1b8589..bed9db3 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -1,7 +1,7 @@
\documentclass[
aps,
pre,
- reprint,
+ preprint,
longbibliography,
floatfix
]{revtex4-2}
@@ -474,10 +474,183 @@ When analytically continued to complex $\theta$, \eqref{eq:2d.real.Fc} has branc
\end{equation}
analytic for all $\theta\in\mathbb C$ outside the Langer branch cuts.
-
-
\subsection{Fitting}
+The scaling function has a number of free parameters: the position $\theta_c$ of the abrupt transition, prefactors in front of singular functions from the abrupt transition and the Yang--Lee point, the coefficients in the analytic part of $\mathcal F$, and the coefficients in the undetermined function $h$.
+
+\begin{table}
+ \begin{tabular}{c|ccccccccc}
+ & \multicolumn{9}{c}{$n$} \\
+ & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 \\
+ \hline
+ $\theta_c$ &
+ 1.19022 &
+ 1.24954 &
+ 1.24954 &
+ 1.27419 &
+ 1.28016 &
+ 1.31927 &
+ 1.32612 &
+ 1.33347 &
+ \\
+ $A_c$ &
+ 0.100228 &
+ 0.0720104 &
+ 0.0809725 &
+ 0.0561657 &
+ 0.0530319 &
+ 0.0377424 &
+ 0.0353443 &
+ 0.0328499 &
+ \\
+ $A_{\mathrm{YL}}$ &
+ 2.3876 &
+ 2.37518 &
+ 2.46094 &
+ 2.45696 &
+ 2.47971 &
+ 2.4754 &
+ 2.47666 &
+ 2.4769 &
+ \\
+ $F_0$ &
+ 1.07383 &
+ 1.14375 &
+ 1.16003 &
+ 1.23109 &
+ 1.25496 &
+ 1.31044 &
+ 1.32179 &
+ 1.33327 &
+ \\
+ $F_1$ &
+ 1.99663 &
+ 2.14753 &
+ 2.15397 &
+ 2.30314 &
+ 2.34538 &
+ 2.46209 &
+ 2.48527 &
+ 2.50906 &
+ \\
+ $F_2$ &
+ 0.228235 &
+ 0.226794 &
+ 0.240421 &
+ 0.218875 &
+ 0.228138 &
+ 0.224167 &
+ 0.231472 &
+ 0.232317 &
+ \\
+ $h_0$ &
+ 1.19078 &
+ 1.2259 &
+ 1.18183 &
+ 1.20578 &
+ 1.19878 &
+ 1.21439 &
+ 1.21605 &
+ 1.21825 &
+ \\
+ $h_1/10^{-2}$ &
+ $0.0377284$ &
+ $-1.32208$ &
+ $-1.98552$ &
+ $-3.4617$ &
+ $-4.05948$ &
+ $-5.27306$ &
+ $-5.53426$ &
+ $-5.79592$ &
+ \\
+ $h_2/10^{-3}$ &
+ &
+ 1.68015 &
+ 3.72445 &
+ 5.89419 &
+ 7.03625 &
+ 8.88373 &
+ 9.31123 &
+ 9.7244 &
+ \\
+ $h_3/10^{-3}$ & &
+ $-0.300336$ &
+ $-1.11915$ &
+ $-1.68817$ &
+ $-2.06784$ &
+ $-2.55373$ &
+ $-2.6766$ &
+ $-2.79303$ &
+ \\
+ $h_4/10^{-4}$ & & &
+ $2.59026$ &
+ $4.92004$ &
+ $6.6988$ &
+ $8.39954$ &
+ $8.83288$ &
+ $9.25784$ &
+ \\
+ $h_5/10^{-4}$ & & &
+ $-0.578821$ &
+ $-1.50074$ &
+ $-2.55766$ &
+ $-3.15364$ &
+ $-3.30063$ &
+ $-3.4621$ &
+ \\
+ $h_6/10^{-4}$ & & & &
+ $0.350651$ &
+ $0.965437$ &
+ $1.24302$ &
+ $1.29961$ &
+ $1.36707$ &
+ \\
+ $h_7/10^{-5}$ & & & &
+ $-0.617004$ &
+ $-3.5398$ &
+ $-5.06226$ &
+ $-5.40426$ &
+ $-5.70884$ &
+ \\
+ $h_8/10^{-5}$ & & & & &
+ $0.978241$ &
+ $1.90968$ &
+ $2.22565$ &
+ $2.40644$ &
+ \\
+ $h_9/10^{-6}$ & & & & &
+ $-1.76567$ &
+ $-6.25434$ &
+ $-8.84571$ &
+ $-10.1167$ &
+ \\
+ $h_{10}/10^{-6}$ & & & & & &
+ $1.51279$ &
+ $3.12976$ &
+ $4.0249$ &
+ \\
+ $h_{11}/10^{-7}$ & & & & & &
+ $-2.17053$ &
+ $-9.37655$ &
+ $-14.7593$ &
+ \\
+ $h_{12}/10^{-7}$ & & & & & & &
+ $2.0809$ &
+ $4.687$ &
+ \\
+ $h_{13}/10^{-8}$ & & & & & & &
+ $-2.83978$ &
+ $-12.363$ &
+ \\
+ $h_{14}/10^{-8}$ & & & & & & & &
+ $2.39528$ &
+ \\
+ $h_{15}/10^{-9}$ & & & & & & & &
+ $-2.99667$ &
+ \\
+ \end{tabular}
+\end{table}
+
\subsection{Comparison}
\section{The three-dimensional Ising model}