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| -rw-r--r-- | ising_scaling.tex | 91 |
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diff --git a/ising_scaling.tex b/ising_scaling.tex index b7b76b2..e229b33 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -1,4 +1,10 @@ -\documentclass[aps,prb,reprint,longbibliography,floatfix]{revtex4-2} +\documentclass[ + aps, + prb, + reprint, + longbibliography, + floatfix +]{revtex4-2} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} @@ -9,14 +15,18 @@ citecolor=purple, filecolor=purple, linkcolor=purple -]{hyperref} % ref and cite links with pretty colors - -\usepackage{amsmath, graphicx, xcolor} +]{hyperref} +\usepackage{amsmath} +\usepackage{graphicx} +\usepackage{xcolor} \begin{document} -\title{Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line} +\title{Smooth Ising universal scaling functions} + \author{Jaron Kent-Dobias} +\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} + \author{James P.~Sethna} \affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} @@ -29,6 +39,77 @@ \cite{Campostrini_2000_Critical} + +\section{The 2D Ising model} + +\subsection{Definition of functions} + +\begin{equation} \label{eq:free.energy.2d.low} + F(u_t, u_h) + = |u_t|^2\mathcal F_{\pm}(u_h|u_t|^{-\beta\delta}) + +\frac{u_t^2}{8\pi}\log u_t^2 +\end{equation} +where the functions $\mathcal F_\pm$ have expansions in nonnegative integer powers of their arguments. +\begin{equation} \label{eq:free.energy.2d.mid} + F(u_t, u_h) + = |u_h|^{2/\beta\delta}\mathcal F_0(u_t|u_h|^{-1/\beta\delta}) + +\frac{u_t^2}{8\pi}\log u_h^{2/\beta\delta} +\end{equation} +where the function $\mathcal F_0$ has a convergent expansion in nonnegative integer powers of its argument \footnote{ + To connect with Mangazeev and Fonseca, $\mathcal F_0(x)=\tilde\Phi(-x)=\Phi(-x)+(x^2/8\pi) \log x^2$ and $\mathcal F_\pm(x)=G_{\mathrm{high}/\mathrm{low}}(x)$. +}. + +\begin{align} + \label{eq:schofield.free.energy} + F(R, \theta) &= R^2\mathcal F(\theta) + t(\theta)^2\frac{R^2}{8\pi}\log R^2 \\ + \label{eq:schofield.temperature} + u_t(R, \theta) &= Rt(\theta) \\ + \label{eq:schofield.field} + u_h(R, \theta) &= R^{\beta\delta}h(\theta) +\end{align} +The scaling function $\mathcal F$ can be defined in terms of the more conventional ones above by substituting \eqref{eq:schofield.temperature} and \eqref{eq:schofield.field} into \eqref{eq:free.energy.2d.low} and \eqref{eq:free.energy.2d.mid}, yielding +\begin{equation} + \begin{aligned} + &\mathcal F(\theta) + =t(\theta)^2\mathcal F_\pm\left[h(\theta)|t(\theta)|^{-\beta\delta}\right] + +\frac{t(\theta)^2}{8\pi}\log t(\theta)^2 \\ + &=|h(\theta)|^{2/\beta\delta}\mathcal F_0\left[t(\theta)|h(\theta)|^{-1/\beta\delta}\right] + +\frac{t(\theta)^2}{8\pi}\log h(\theta)^{2/\beta\delta} + \end{aligned} +\end{equation} +We choose the functions $t$ and $h$ so as to ensure that $F$ has an integer power series in \emph{all} regimes. $t$ is an even function of $\theta$ with $t(0)=1$ and $t(1)=0$. $h$ is an odd function with $h(0)=h(\theta_c)=0$ for some $\theta_c>1$. + +\begin{align} + t(\theta)&=1-\theta^2 \\ + h^{(n)}(\theta)&=\left(1-\frac{\theta^2}{\theta_c^2}\right)\sum_{i=0}^nh_i\theta^{2i+1} +\end{align} + + +\begin{equation} + f(x)=\Theta(-x) |x| e^{-1/|x|} +\end{equation} +where $\Theta$ is the Heaviside function. + +\begin{equation} + \operatorname{Im}\mathcal F(\theta)=A\left\{f\left[b(\theta_c-\theta)\right]+f\left[b(\theta_c+\theta)\right]\right\} +\end{equation} + +\begin{equation} + \begin{aligned} + \operatorname{Re}\mathcal F(\theta) + &=G(\theta^2)-\frac{\theta^2}\pi\int d\vartheta\, \frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} \\ + &=G(\theta^2)+\frac A\pi\left\{f[b(\theta_c-\theta)]+f[b(\theta_c+\theta)]\right\} + \end{aligned} +\end{equation} +for arbitrary analytic function $G$ and +\begin{equation} + f(x)=xe^{1/x}\operatorname{Ei}(-1/x) +\end{equation} + +\section{The 3D Ising model} + +\section{Outlook} + \begin{acknowledgments} The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien for helpful conversations. The authors would also like to think Jacques Perk |