From 8c7ba65464178104b3bb68e1a15b1e6fbf3e3231 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 22 Dec 2020 19:45:59 +0100
Subject: Started new paper.

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-
-%
-%  Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017.
-%  Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved.
-%
-\documentclass[aps,prl,reprint]{revtex4-1}
-
-\usepackage[utf8]{inputenc}
-\usepackage{amsmath,amssymb,latexsym,mathtools,xifthen}
-
-% uncomment to label only equations that are referenced in the text
-%\mathtoolsset{showonlyrefs=true}
-
-% I want labels but don't want to type out ``equation''
-\def\[{\begin{equation}}
-\def\]{\end{equation}}
-
-% math not built-in
-\def\arcsinh{\mathop{\mathrm{arcsinh}}\nolimits}
-\def\arccosh{\mathop{\mathrm{arccosh}}\nolimits}
-\def\ei{\mathop{\mathrm{Ei}}\nolimits} % exponential integral Ei
-\def\re{\mathop{\mathrm{Re}}\nolimits}
-\def\im{\mathop{\mathrm{Im}}\nolimits}
-\def\sgn{\mathop{\mathrm{sgn}}\nolimits}
-\def\dd{d} % differential
-\def\O{O}          % big O
-\def\o{o}          % little O
-
-% subscript for ``critical'' values, e.g., T_\c
-\def\c{\mathrm c}
-
-% scaling functions
-\def\fM{\mathcal M}  % magnetization
-\def\fX{\mathcal Y}  % susceptibility
-\def\fXt{\check{\mathcal Y}}  % susceptibility
-\def\fF{\mathcal F}  % free energy
-\def\fiF{\mathcal H} % imaginary free energy
-\def\fS{\mathcal S}  % surface tension
-\def\fG{\mathcal G}  % exponential factor
-
-% lattice types
-\def\sq{\mathrm{sq}}
-\def\tri{\mathrm{tri}}
-\def\hex{\mathrm{hex}}
-
-% dimensions
-\def\dim{d}
-\def\twodee{\textsc{2d}}
-\def\threedee{\textsc{3d}}
-\def\fourdee{\textsc{4d}}
-
-% fancy partial derivative
-\newcommand\pd[3][]{
-  \ifthenelse{\isempty{#1}}
-    {\def\tmp{}}
-    {\def\tmp{^#1}}
-  \frac{\partial\tmp#2}{\partial#3\tmp}
-}
-
-% used to reformat display math to fit in two-column ``reprint'' mode
-\makeatletter
-\newif\ifreprint
-\@ifclasswith{revtex4-1}{reprint}{\reprinttrue}{\reprintfalse}
-\makeatother
-
-\begin{document}
-
-\title{Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line}
-\author{Jaron Kent-Dobias}
-\author{James P.~Sethna}
-\affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA}
-
-\date\today
-
-\begin{abstract}
-  Renormalization group ideas and results from critical droplet theory are
-  used to construct a scaling ansatz for the imaginary component of the free
-  energy of an Ising model in its metastable state close to the critical
-  point. The analytic properties of the free energy are used to determine
-  scaling functions for the free energy in the vicinity of the critical point
-  and the abrupt transition line. These functions have essential singularities
-  at the abrupt transition. Analogous forms for the magnetization and susceptibility in
-  two dimensions are fit to numeric data and show good agreement, especially
-  when their nonsingular behavior is modified to match existing numeric results.
-\end{abstract}
-
-\maketitle
-
-The Ising model is the canonical example of a system with a continuous phase
-transition, and the study of its singular properties marked the first success
-of the renormalization group (\textsc{rg}) method in statistical physics
-\cite{wilson.1971.renormalization}. Its status makes sense: it's a simple
-model whose continuous phase transition contains all the essential features of
-more complex ones, but admits \textsc{rg} methods in a straightforward way and
-has exact solutions in certain dimensions and for certain parameter
-restrictions. However, the Ising critical point is not simply a continuous
-transition: it also ends a line of abrupt phase transitions extending from it
-at zero field below the critical temperature. Though typically neglected in
-\textsc{rg} scaling analyses of the critical point, we demonstrate that there
-are numerically measurable contributions to scaling due to the abrupt
-transition line that cannot be accounted for by analytic changes of control or
-thermodynamic variables.
-
-\textsc{Rg} analysis predicts that the singular part of the free energy per
-site $F$ as a function of reduced temperature $t=1-T_\c/T$ and field $h=H/T$
-in the vicinity of the critical point takes the scaling form
-$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})$ for the low temperature phase
-$t<0$ \cite{cardy.1996.scaling}. When studying the properties of the Ising
-critical point, it is nearly always assumed that the universal scaling
-function $\fF$ is analytic, i.e., has a convergent Taylor series. However, it
-has long been known that there exists an essential singularity in $\fF$ at
-zero argument, though its effects have long been believed to be unobservable
-\cite{fisher.1967.condensation}, or simply just neglected
-\cite{guida.1997.3dising,schofield.1969.parametric,schofield.1969.correlation,caselle.2001.critical,josephson.1969.equation,fisher.1999.trigonometric}.
-With careful analysis, we have found that assuming the presence of the
-essential singularity is predictive of the scaling form of, for instance, the
-susceptibility and magnetization.
-
-The provenance of the essential singularity can be understood using the
-methods of critical droplet theory for the decay of an Ising system in a
-metastable state, i.e., an equilibrium Ising state for $T<T_\c$, $H>0$
-subjected to a small negative external field $H<0$. The existence of an
-essential singularity has also been suggested by transfer matrix
-\cite{mccraw.1978.metastability,enting.1980.investigation,mangazeev.2008.variational,mangazeev.2010.scaling} and \textsc{rg}
-methods \cite{klein.1976.essential}, and a different kind of essential
-singularity is known to exist in the zero-temperature susceptibility
-\cite{orrick.2001.susceptibility,chan.2011.ising,guttmann.1996.solvability,nickel.1999.singularity,nickel.2000.addendum,assis.2017.analyticity}.  It has long been known that the decay
-rate $\Gamma$ of metastable states in statistical mechanics is often related
-to the metastable free energy $F$ by $\Gamma\propto\im F$
-\cite{langer.1969.metastable,penrose.1987.rigorous,gaveau.1989.analytic,privman.1982.analytic}.
-`Metastable free energy' can be thought of as either an analytic continuation
-of the free energy through the abrupt phase transition, or restriction of the
-partition function trace to states in the vicinity of the local free energy
-minimum that characterizes the metastable state. In any case, the free energy
-develops a nonzero imaginary part in the metastable region. Heuristically,
-this can be thought of as similar to what happens in quantum mechanics with a
-non-unitary Hamiltonian: the imaginary part describes loss rate of probability
-that the system occupies any `accessible' state, which corresponds to decay.
-
-In critical droplet theory, the metastable state decays when a domain of the
-equilibrium state forms whose surface-energy cost for growth is outweighed by
-bulk-energy gains. There is numerical evidence that, near the critical point,
-these droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy
-cost of the surface of a droplet of radius $R$ is $\Sigma S_\dim R^{\dim-1}$
-and that of its bulk is $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are
-the surface area and volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$
-is the surface tension of the equilibrium--metastable interface. The critical
-droplet size is then $R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the
-critical droplet is $\Delta F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$.
-Assuming the typical singular scaling forms
-$\Sigma/T=|t|^\mu\fS(h|t|^{-\beta\delta})$ and $M=|t|^\beta\mathcal
-M(h|t|^{-\beta\delta})$ and using known hyperscaling relations
-\cite{widom.1981.interface}, this implies a scaling form
-\def\eqcritformone{
-  T\frac{\pi^{\dim/2}(\dim-1)^{\dim-1}}{\Gamma(1+\dim/2)}
-    \frac{\fS^\dim(h|t|^{-\beta\delta})}{(-h|t|^{-\beta\delta}
-    \fM(h|t|^{-\beta\delta}))^{\dim-1}}
-}
-\def\eqcritformtwo{
-  T\fG^{-(\dim-1)}(h|t|^{-\beta\delta})
-}
-\ifreprint
-\[
-  \begin{aligned}
-    \Delta F_\c
-    &=\eqcritformone\\
-    &\sim\eqcritformtwo
-  \end{aligned}
-\]
-\else
-\[
-  \Delta F_\c=\eqcritformone\sim\eqcritformtwo
-\]
-\fi
-for the free energy change due to the critical droplet.
-Since both surface tension and magnetization are finite and nonzero for $H=0$
-at $T<T_\c$, $\fG(X)=-BX+\O(X^2)$ for small negative $X$ with
-\[
-  B=\frac{\fM(0)}{\dim-1}\bigg(\frac{\Gamma(1+\dim/2)}
-    {\pi^{\dim/2}\fS(0)^\dim}\bigg)^{1/(\dim-1)}.
-\]
-This first term in the scaling function $\fG$ is related to the ratio between
-the correlation length $\xi$ and the critical domain radius $R_\c$, with
-\[
-  Bh|t|^{-\beta\delta}
-  =\frac\xi{R_\c}\bigg(\frac{\Gamma(1+\dim/2)}
-    {\pi^{\dim/2}\fS(0)(\xi_0^-)^{\dim-1}}\bigg)^{1/(\dim-1)}
-\]
-where the critical amplitude for the correlation length $\xi_0^-$ is defined
-by $\xi=\xi_0^-|t|^{-\nu}$ for $t<T_\c$ and $H=0$. Since $\fS(0)(\xi_0^-)^{\dim-1}$ is a
-universal amplitude ratio \cite{zinn.1996.universal},
-$(Bh|t|^{-\beta\delta})/(\xi/R_\c)$ is a universal quantity.  The decay rate
-of the metastable state is proportional to the Boltzmann factor corresponding
-to the
-creation of a critical droplet, yielding
-\[
-  \im F\sim\Gamma\propto e^{-\Delta F_\c/T}
-    =e^{-\fG(h|t|^{-\beta\delta})^{-(\dim-1)}}.
-\]
-For $d>1$ this function has an essential singularity in the invariant
-combination $h|t|^{-\beta\delta}$.
-
-This form of $\im F$ for small $h$ is well known
-\cite{langer.1967.condensation,harris.1984.metastability}.  We make the
-scaling ansatz that the imaginary part of the metastable free energy has the
-same singular behavior as the real part of the equilibrium free energy, and
-that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$,
-where
-\[
-  \fiF(X)=A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}}
-  \label{eq:im.scaling}
-\]
-and $\Theta$ is the Heaviside function. Results from combining an analysis of
-fluctuations on the surface of critical droplets with \textsc{rg} recursion
-relations suggest that $b=-(d-3)d/2$ for $d=2,4$ and $b=-7/3$ for $d=3$
-\cite{houghton.1980.metastable,rudnick.1976.equations,gunther.1980.goldstone}.
-Assuming that $F$ is analytic in the upper complex-$h$ plane, the real part of
-$F$ in the equilibrium state can be extracted from this imaginary metastable
-free energy using the Kramers--Kronig relation
-\[
-  \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h'.
-  \label{eq:kram-kron}
-\]
-This relationship has been used to compute high-order moments of the free
-energy with $H$ in good agreement with transfer matrix expansions
-\cite{lowe.1980.instantons}. Here, we evaluate the integral explicitly to come
-to
-functional forms.  In \threedee\ and \fourdee\ this can be done directly
-given our scaling ansatz, yielding
-\def\eqthreedeeone{
-  \fF^\threedee(Y/B)&=
-  \frac{A}{12}\frac{e^{-1/Y^2}}{Y^2}
-  \bigg[\Gamma(\tfrac16)E_{7/6}(-Y^{-2})
-}
-\def\eqthreedeetwo{
-  -4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2})\bigg]
-}
-\def\eqfourdeeone{
-  \fF^\fourdee(Y/B)&=
-  -\frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2}
-  \Big[3\ei(-Y^{-3})
-}
-\def\eqfourdeetwo{
-  +3\Gamma(\tfrac23)\Gamma(\tfrac13,Y^{-3})
-  +\Gamma(\tfrac13)\Gamma(-\tfrac13,Y^{-3})\Big]
-}
-\ifreprint
-\begin{align}
-  &\begin{aligned}
-    \eqthreedeeone\\
-    &\hspace{7em}
-    \eqthreedeetwo
-  \end{aligned}
-  \\
-  &\begin{aligned}
-    \eqfourdeeone
-    \\
-    &\hspace{-0.5em}
-    \eqfourdeetwo,
-  \end{aligned}
-\end{align}
-\else
-\begin{align}
-  \eqthreedeeone\eqthreedeetwo
-  \\
-  \eqfourdeeone\eqfourdeetwo,
-\end{align}
-\fi
-where $E_n$ is the generalized exponential integral and $\Gamma(x,y)$ is the
-incomplete gamma function.
-At the level of truncation of \eqref{eq:im.scaling} at which we are working
-the Kramers--Kronig relation does not converge in \twodee. However, higher
-moments can still be extracted, e.g., the susceptibility, by taking
-\[
-  \chi=\pd MH=-\frac1{T}\pd[2]Fh
-  =-\frac2{\pi T}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'.
-\]
-With a scaling form defined by $T\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$,
-this yields
-\[
-  \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big]
-  \label{eq:sus_scaling}
-\]
-Scaling forms for the free energy can then be extracted by direct integration
-and their constants of integration fixed by known zero-field values, yielding
-\begin{align}
-  \label{eq:mag_scaling}
-  \fM^\twodee(Y/B)
-    &=\fM(0)+\frac{AB}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\
-  \fF^\twodee(Y/B)
-    &=-Y\bigg(\frac{\fM(0)}B-\frac{A}\pi e^{1/Y}\ei(-1/Y)\bigg)
-    \label{eq:2d_free_scaling}
-\end{align}
-with $F(t,h)=t^2\fF(h|t|^{-15/8})+t^2\log t^2$ in
-two dimensions, as $\alpha=0$ and $\beta\delta=\frac{15}8$.
-
-How are these functional forms to be interpreted? Though the scaling function
-\eqref{eq:im.scaling} for the imaginary free energy of the metastable state is
-asymptotically correct sufficiently close to the critical point, the results
-of the integral relation \eqref{eq:kram-kron} are not, since there is no limit
-of $t$ or $h$ in which it becomes arbitrarily correct for a given truncation
-of \eqref{eq:im.scaling}. It is well established that this method of using
-unphysical or metastable elements of a theory to extract properties of the
-stable or equilibrium theory is only accurate for high moments of those
-predictions \cite{parisi.1977.asymptotic,bogomolny.1977.dispersion}.  The
-functions above should be understood as possessing exactly the correct
-singularity  at the coexistence line, but requiring polynomial corrections,
-especially for smaller integer powers. Using these forms in conjunction with
-existing methods of describing the critical equation of state or critical
-properties with analytic functions in $h$ will incorporate these low-order
-corrections while preserving the correct singular structure. In other words,
-the scaling functions can be \emph{exactly} described by
-$\tilde\fF(X)=\fF(X)+f(X)$ for some analytic function $f$. Higher order terms
-in the expansion of $\tilde\fF$ become asymptotically equal to those of $\fF$
-because, as an analytic function, progressively higher order terms of $f$ must
-eventually become arbitrarily small \cite{flanigan.1972.complex}.
-
-How predictive are these scaling forms in the proximity of the critical point
-and the abrupt transition line? We simulated the \twodee\ Ising model on square
-lattice using a form of the Wolff algorithm modified to remain efficient in
-the presence of an external field. Briefly, the external field $H$ is applied
-by adding an extra spin $s_0$ with coupling $|H|$ to all others
-\cite{dimitrovic.1991.finite}. A quickly converging estimate for the
-magnetization in the finite-size system was then made by taking
-$M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin
-\cite{kent-dobias.2018.wolff}.
-Data was then taken for susceptibility and magnetization for
-$T_\c-T,H\leq0.1$. This data, rescaled as appropriate to collapse onto a
-single curve, is plotted in Fig.~\ref{fig:scaling_fits}.
-
-For the \twodee\ Ising model on a square lattice, exact results at zero
-temperature have $\fS(0)=4/T_\c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$
-\cite{onsager.1944.crystal}, and $\fX(0)=C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that
-$B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we
-assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of
-the susceptibility scaling function, then
-$T\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant
-$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$.  The
-resulting scaling functions $\fX$ and $\fM$ are plotted as solid blue lines in
-Fig.~\ref{fig:scaling_fits}. Though there is good agreement
-between our functional forms and what is measured, there
-are systematic differences that can be seen most clearly in the
-magnetization. This is to be expected based on our earlier discussion: these
-scaling forms should only be expected to well-describe the singularity at the
-abrupt transition. Our forms both exhibit incorrect low-order
-coefficients at the transition (Fig.~\ref{fig:series}) and incorrect
-asymptotics as $h|t|^{-\beta\delta}$ becomes very large.
-
-In forthcoming work,
-we develop a method to incorporate the essential singularity in the scaling
-functions into a form that also incorporates known properties of the scaling
-functions in the rest of the configuration space using a Schofield-like
-parameterization \cite{schofield.1969.parametric,caselle.2001.critical,kent-dobias.2018.parametric}. Briefly, we define
-parameters $R$ and $\theta$ by
-\begin{align}
-  t=R(1-\theta^2)
-  &&
-  h=h_0R^{\beta\delta}g(\theta)
-\end{align}
-where $h_0$ is a constant and $f$ is an arbitrary odd function whose first
-finite zero $\theta_\c>1$ corresponds to the abrupt transition. In these
-coordinates the invariant combination $h|t|^{-\beta\delta}$ is given by
-\[
-  h|t|^{-\beta\delta}=\frac{h_0g(\theta)}{|1-\theta^2|^{\beta\delta}}=\frac{h_0(-g'(\theta_\c))}{(\theta_c^2-1)^{\beta\delta}}(\theta_\c-\theta)
-  +\O\big( (\theta_\c-\theta)^2\big),
-\]
-an analytic function of $\theta$ about $\theta_\c$.
-The simplest
-function of the coordinate $\theta$ that exhibits the correct singularity at
-the abrupt transitions at $h=\pm0$, $t<0$ is
-\[
-  \fX(\theta)=\fX\bigg(\frac{h_0(-g'(\theta_\c))}{(\theta_\c^2-1)^{\beta\delta}}(\theta_\c-\theta)\bigg)+\fX\bigg(\frac{h_0(-g'(\theta_\c))}{(\theta_\c^2-1)^{\beta\delta}}(\theta_\c+\theta)\bigg)
-\]
-This function is analytic in the range $-\theta_\c<\theta<\theta_\c$.
-In order to correct its low-order behavior to match that expected, we
-make use of both the freedom of the coordinate transformation $g$ and an
-arbitrary analytic additive function $Y$,
-\begin{align}
-  g(\theta)=\bigg(1-\frac{\theta^2}{\theta_\c^2}\bigg)\sum_{n=0}^\infty
-  g_n\theta^{2n+1}
-  &&
-  Y(\theta)=\sum_{n=0}^\infty Y_n\theta^{2n}
-\end{align}
-so that $\tilde\fX(\theta)=\fX(\theta)+Y(\theta)$. By manipulating these
-coefficients, we can attempt to give the resulting scaling form a series
-expansion consistent with known values. One such prediction---made by fixing
-the first four terms in the low-temperature, critical isotherm, and
-high-temperature expansions 
-of $\tilde\fX$---is shown as a dashed yellow line in
-Fig.~\ref{fig:scaling_fits}.
-As shown in Fig.~\ref{fig:series}, the low-order free energy coefficients of this prediction match known values
-exactly up to $n=5$, and improve the agreement with higher-order coefficients.
-Unlike scaling forms which treat $\fX$ as analytic at the coexistence line,
-the series coefficients of the scaling form developed here increase without
-bound at high order.
-
-
-\begin{figure}
-  \input{fig-susmag}
-  \caption{
-    Scaling functions for (top) the susceptibility and (bottom) the
-    magnetization plotted in terms of the invariant combination
-    $h|t|^{-\beta\delta}$. Points with error bars show data with sampling
-    error taken from simulations of a $4096\times4096$ square-lattice Ising
-    model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$
-    and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid blue lines
-    show our analytic results \eqref{eq:sus_scaling} and
-    \eqref{eq:mag_scaling}, the dashed yellow lines show 
-    a scaling function modified to match known series expansions of the
-    susceptibility
-    to third order, and the
-    dotted green lines show the
-    polynomial resulting from truncating the known series expansion after the eight terms
-    reported by \cite{mangazeev.2008.variational,mangazeev.2010.scaling}.
-  }
-  \label{fig:scaling_fits}
-\end{figure}
-
-\begin{figure}
-  \input{fig-series}
-  \caption{
-    The series coefficients defined by $\tilde\fF(X)=\sum_nf_nX^n$. The blue
-    pluses correspond to the scaling form \eqref{eq:2d_free_scaling}, the
-    yellow saltires correspond to a scaling function modified to match known
-    series expansions of the susceptibility to third order---and therefore
-    the free energy to fifth order---and the green
-    stars
-    correspond to the first eight coefficients from
-    \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. The modified
-    scaling function and the known coefficients match exactly up to $n=5$.
-  }
-  \label{fig:series}
-\end{figure}
-
-Abrupt phase transitions, such as the jump in magnetization in the Ising
-model below $T_\c$, are known to imply essential singularities in the free
-energy that are usually thought to be unobservable in practice. We have
-argued that this essential singularity controls the universal scaling
-behavior near continuous phase transitions, and have derived an explicit
-analytical form for the singularity in the free energy, magnetization,
-and susceptibility for the Ising model. We have developed a Wolff algorithm 
-for the Ising model in a field, and showed that incorporating our singularity
-into the scaling function gives good convergence to the simulations in \twodee.
-
-Our results should allow improved high-precision functional forms for the free
-energy~\cite{caselle.2001.critical}, and should have implications for the scaling
-of correlation functions~\cite{chen.2013.universal,wu.1976.spin}. Our methods might be generalized
-to predict similar singularities in systems where nucleation and metastability
-are proximate to continuous phase transitions, such as 2D superfluid
-transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{dahm.1989.dynamics}, and
-freezing transitions in glasses, spin glasses, and other disordered systems.
-
-
-%We have used results from the properties of the metastable Ising ferromagnet
-%and the analytic nature of the free energy to derive universal scaling
-%functions for the free energy, and in \twodee the magnetization and
-%susceptibility, in the limit of small $t<0$ and $h$. Because of an essential
-%singularity in these functions at $h=0$---the abrupt transition line---their
-%form cannot be brought into that of any regular function by analytic
-%redefinition of control or thermodynamic variables. These predictions match
-%the results of simulations well. Having demonstrated that the essential
-%singularity in thermodynamic functions at the abrupt transition leads to
-%observable scaling effects, we hope that these functional forms will be used in
-%conjunction with traditional perturbation methods to better express the
-%equation of state of the Ising model in the whole of its parameter space.
-
-\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
-  for pointing us to several insightful studies. JPS thanks Jim Langer for past inspiration,
-  guidance, and encouragement. This work was supported by NSF grants
-  DMR-1312160 and DMR-1719490.
-\end{acknowledgments}
-
-\bibliography{essential-ising}
-
-\end{document}
-
-- 
cgit v1.2.3-70-g09d2

