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| -rw-r--r-- | ising_scaling.tex | 44 |
1 files changed, 23 insertions, 21 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index 5630fe4..ca1bd15 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -328,13 +328,15 @@ argument for all real $\theta$ by \end{equation} For small $\theta$, $\mathcal F(\theta)$ will resemble $\mathcal F_+$, for $\theta$ near one it will resemble $\mathcal F_0$, -and for $\theta$ near $\theta_0$ it will resemble $\mathcal F_-$. This can be seen explicitly using the definitions \eqref{eq:schofield} to relate the above form to the original scaling functions, giving +and for $\theta$ near $\theta_0$ it will resemble $\mathcal F_-$. This can be +seen explicitly using the definitions \eqref{eq:schofield} to relate the above +form to the original scaling functions, giving \begin{equation} \label{eq:scaling.function.equivalences.2d} \begin{aligned} \mathcal F(\theta) - &=|t(\theta)|^{D\nu}\mathcal F_\pm\left[g(\theta)|1-\theta^2|^{-\Delta}\right] - +\frac{(1-\theta^2)^2}{8\pi}\log t(\theta)^2\\ - &=|h(\theta)|^{D\nu/\Delta}\mathcal F_0\left[(1-\theta^2)|g(\theta)|^{-1/\Delta}\right] + &=|1-\theta^2|^{D\nu}\mathcal F_\pm\left[g(\theta)|1-\theta^2|^{-\Delta}\right] + +\frac{(1-\theta^2)^2}{8\pi}\log(1-\theta^2)^2\\ + &=|g(\theta)|^{D\nu/\Delta}\mathcal F_0\left[(1-\theta^2)|g(\theta)|^{-1/\Delta}\right] +\frac{(1-\theta^2)^2}{8\pi}\log g(\theta)^{2/\Delta} \end{aligned} \end{equation} @@ -578,22 +580,22 @@ the series coefficients of the scaling function $\mathcal F_0$, and the accuracy of the fit results can be checked against the known values here. \begin{table}\label{tab:fits} - \begin{tabular}{c|ccc} - $m$ & $\mathcal F_-^{(m)}$ & $\mathcal F_0^{(m)}$ & $\mathcal F_+^{(m)}$ \\\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 \\ + \begin{tabular}{c|lll} + $m$ & \multicolumn{1}{c}{$\mathcal F_-^{(m)}$} & $\mathcal F_0^{(m)}$ & $\mathcal F_+^{(m)}$ \\\hline + 0 & \hphantom{$-$}0 & $-1.197\,733\,383\,797\ldots$ & \hphantom{$-$}0 \\ + 1 & $-1.357\,838\,341\,707\ldots$ & \hphantom{$-$}$0.318\,810\,124\,891\ldots$ & \hphantom{$-$}0 \\ + 2 & $-0.048\,953\,289\,720\ldots$ & \hphantom{$-$}$0.110\,886\,196\,683(2)$ & $-1.845\,228\,078\,233\ldots$ \\ + 3 & \hphantom{$-$}$0.038\,863\,932(3)$ & $-0.016\,426\,894\,65(2)$ & \hphantom{$-$}0 \\ + 4 & $-0.068\,362\,119(2)$ & $-2.639\,978(1)\times10^{-4}$ & \hphantom{$-$}$8.333\,711\,750(5)$ \\ + 5 & \hphantom{$-$}$0.183\,883\,70(1)$ & \hphantom{$-$}$5.140\,526(1)\times10^{-4}$ & \hphantom{$-$}0 \\ + 6 & $-0.659\,171\,4(1)$ & \hphantom{$-$}$2.088\,65(1)\times 10^{-4}$ & $-95.168\,96(1)$ \\ + 7 & \hphantom{$-$}$2.937\,665(3)$ & \hphantom{$-$}$4.481\,9(1)\times10^{-5}$ & \hphantom{$-$}0 \\ + 8 & $-15.61(1)$ & \hphantom{$-$}$3.16\times10^{-7}$ & \hphantom{$-$}1457.62(3) \\ + 9 & \hphantom{$-$}96.76 & $-4.31\times10^{-6}$ & \hphantom{$-$}0 \\ + 10 & $-679$ & $-1.99\times10^{-6}$ & $-25\,891(2)$ \\ + 11 & \hphantom{$-$}$5.34\times10^3$ & & \hphantom{$-$}0 \\ + 12 & $-4.66\times10^4$ & & \hphantom{$-$}$5.02\times10^5$ \\ + 13 & \hphantom{$-$}$4.46\times10^5$ & & \hphantom{$-$}0 \\ 14 & $-4.66\times10^6$ & & $-1.04\times10^7$ \end{tabular} \end{table} @@ -823,7 +825,7 @@ accuracy of the fit results can be checked against the known values here. \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 $\theta_m>\theta_0$, the invariant scaling combination +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 $t=R(1-\theta^2)(1-(\theta/\theta_m)^2)$ for some $\theta_m>\theta_0$, the invariant scaling combination \begin{acknowledgments} The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien |