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-rw-r--r--data/glow_series_numeric.dat27
-rw-r--r--ising_scaling.tex52
2 files changed, 67 insertions, 12 deletions
diff --git a/data/glow_series_numeric.dat b/data/glow_series_numeric.dat
index 27927c1..e943522 100644
--- a/data/glow_series_numeric.dat
+++ b/data/glow_series_numeric.dat
@@ -1,12 +1,15 @@
-0 0 0
-1 -1.3578383417065955 1.e-14
-2 -0.04895328972 2.e-12
-3 0.038863929 1.e-10
-4 -0.068362121 1.e-9
-5 0.18388371 1.e-8
-6 -0.65917 1.e-6
-7 2.937665 3.e-6
-8 -15.61 0.01
-9 96.76 0
-10 -679 0
-11 5340 0
+0 0
+1 -1.3578383417065955
+2 -0.04895328972
+3 0.038863929
+4 -0.068362121
+5 0.18388371
+6 -0.65917
+7 2.937665
+8 -15.61
+9 96.76
+10 -679
+11 5340
+12 -4.66e4
+13 4.46e5
+14 -4.66e6
diff --git a/ising_scaling.tex b/ising_scaling.tex
index 54bada0..c6dc29a 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -480,6 +480,27 @@ analytic for all $\theta\in\mathbb C$ outside the Langer branch cuts.
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|ccc}
+ $n$ & $\mathcal F_-^{(n)}$ & $\mathcal F_0^{(n)}$ & $\mathcal F_+^{(n)}$ \\\hline
+ 0 & 0 & $-1.197733383797993$ & 0 \\
+ 1 & $-1.35783834$ & $-0.318810124891$ & 0 \\
+ 2 & $-0.048953289720$ & $0.110886196683$ & $-1.84522807823$ \\
+ 3 & 0.0388639290 & $0.01642689465$ & 0 \\
+ 4 & $-0.068362121$ & $-2.639978\times10^{-4}$ & 8.3337117508 \\
+ 5 & 0.18388371 & $-5.140526\times10^{-4}$ & 0 \\
+ 6 & $-0.659170$ & $2.08856\times 10^{-4}$ & $-95.16897$ \\
+ 7 & 2.937665 & $-4.4819\times10^{-5}$ & 0 \\
+ 8 & $-15.61$ & $3.16\times10^{-7}$ & 1457.62 \\
+ 9 & 96.76 & $4.31\times10^{-6}$ & 0 \\
+ 10 & $-679$ & $-1.99\times10^{-6}$ & -25891 \\
+ 11 & $5.34\times10^3$ & & 0 \\
+ 12 & $-4.66\times10^4$ & & $5.02\times10^5$ \\
+ 13 & $4.46\times10^5$ & & 0 \\
+ 14 & $-4.66\times10^6$ & & $-1.04\times10^7$
+ \end{tabular}
+\end{table}
+
+\begin{table}
\begin{tabular}{c|cccccccccc}
& \multicolumn{9}{c}{$n$} \\
& 0 & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 \\
@@ -705,6 +726,7 @@ The scaling function has a number of free parameters: the position $\theta_c$ of
set logscale y
set xlabel '$n$'
set ylabel '$\mathcal F_n$'
+
plot \
dat1 using 1:(abs($2)) title 'Numeric', \
dat2 using 1:(abs($2)) title 'Ours ($n=0$)', \
@@ -764,6 +786,36 @@ The three-dimensional Ising model is easier in some ways, since its hyperbolic c
\mathcal F_c(\theta)=F_c(\theta_c^2-\theta^2)^{-7/3}e^{-1/[B(\theta_c^2-\theta^2)]^2}
\end{equation}
+For $\theta\in\mathbb R$,
+\begin{equation}
+ \begin{aligned}
+ \operatorname{Im}\mathcal F(\theta+0i)&=F_c[\Theta(\theta-\theta_c)\mathcal I(\theta)-\Theta(-\theta-\theta_c)\mathcal I(-\theta)]
+ \end{aligned}
+\end{equation}
+\begin{equation}
+ \mathcal I(\theta)=(\theta-\theta_c)^{-7/3}e^{-1/B(\theta-\theta_c)^2}
+\end{equation}
+The dispersion integral \eqref{} can be used to find the real part of $\mathcal F_c$ for $\theta\in\mathbb R$, or
+\begin{equation} \label{eq:2d.real.Fc}
+ \operatorname{Re}\mathcal F_c(\theta)=F_c[\mathcal R(\theta)+\mathcal R(-\theta)]
+\end{equation}
+where $\mathcal R$ is given by the function
+\begin{equation}
+ \begin{aligned}
+ \mathcal R(\theta)
+ &=
+ -\frac1{12\pi}\left\{
+ 4B\Gamma(2/3)\left[\frac{e^{-1/B^2\theta_c^2}}{\theta_c}\operatorname{Ei}_{\frac53}(-1/B^2\theta_c^2)
+ +\frac{e^{-1/B^2(\theta-\theta_c)^2}}{\theta-\theta_c}\operatorname{Ei}_{\frac53}(-1/B^2(\theta-\theta_c)^2)\right]\right. \\
+ &\left.-\Gamma(1/6)\left[\frac{e^{-1/B^2\theta_c^2}}{\theta_c^2}\operatorname{Ei}_{\frac76}(-1/B^2\theta_c^2)
+ +\frac{e^{-1/B^2(\theta-\theta_c)^2}}{(\theta-\theta_c)^2}\operatorname{Ei}_{\frac76}(-1/B^2(\theta-\theta_c)^2)
+ \right]
+ \right\}
+ \end{aligned}
+\end{equation}
+
+\cite{Connelly_2020_Universal} report the location of the Yang--Lee singularity.
+
\section{Outlook}
The successful smooth description of the Ising free energy produced in part by analytically continuing the singular imaginary part of the metastable free energy inspires an extension of this work: a smooth function that captures the universal scaling \emph{through the coexistence line and into the metastable phase}. Indeed, the tools exist to produce this: by writing $t(\theta)=(1-\theta^2)(1-(\theta/\theta_m)^2)$ for some $\theta_m>\theta_c$, the invariant scaling combination