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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 |