From 91cf18f2f55706c7b764fdc5f048b0c6106c698e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 12 Jul 2017 12:26:38 -0400
Subject: many final changes

---
 essential-ising.tex | 288 ++++++++++++++++++++++++++--------------------------
 1 file changed, 144 insertions(+), 144 deletions(-)

(limited to 'essential-ising.tex')

diff --git a/essential-ising.tex b/essential-ising.tex
index 1aa5ea1..4673847 100644
--- a/essential-ising.tex
+++ b/essential-ising.tex
@@ -76,11 +76,10 @@
   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 zero field. Analogous forms for the magnetization
-  and susceptibility in two-dimensions are fit to numeric data and show good
-  agreement.
+  scaling functions for the free energy in the vicinity of the critical point
+  and the abrupt transition line. These functions have essential singularities
+  at zero field. Analogous forms for the magnetization and susceptibility in
+  two dimensions are fit to numeric data and show good agreement.
 \end{abstract}
 
 \maketitle
@@ -88,30 +87,32 @@
 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}. This status makes sense: it's a simple
-model whose phase transition admits \textsc{rg} methods in a straightforward way,
-and has exact solutions in certain dimensions and for certain parameter
-restrictions. However, in one respect the Ising critical point is not simply a
-continuous transition: it ends the line of abrupt phase transitions 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
+\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,aharony.1983.fields}. When studying the properties of
-the Ising critical point, it is nearly always assumed that $\fF(X)$,
-the universal scaling function, is an analytic function of $X$. However, it
-has long been known that there exists an essential singularity in $\fF$
-at $X=0$, though its effects have long been believed to be unobservable
+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 e.g. the
-susceptibility.
+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
@@ -136,19 +137,21 @@ 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,
 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 then is
-$R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the critical
-droplet is $\Delta
-F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$.  
+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 then is $R_\c=(\dim-1)\Sigma/M|H|$ 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}}
+  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})
@@ -169,70 +172,55 @@ M(h|t|^{-\beta\delta})$ and using known hyperscaling relations
 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
 \[
-  B=\frac{\fM(0)}{\dim-1}\bigg(\frac{\Gamma(1+\dim/2)}{\pi^{\dim/2}\fS(0)^\dim}\bigg)^{1/(\dim-1)}.
+  B=\frac{\fM(0)}{\dim-1}\bigg(\frac{\Gamma(1+\dim/2)}
+    {\pi^{\dim/2}\fS(0)^\dim}\bigg)^{1/(\dim-1)}.
 \]
-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
+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
 \[
-  Bh|t|^{-\beta\delta}=\bigg(\frac{\Gamma(1+\dim/2)}{\pi^{\dim/2}\fS(0)(\xi_0^-)^{\dim-1}}\bigg)^{1/(\dim-1)}\frac\xi{R_\c}
+  Bh|t|^{-\beta\delta}
+  =\frac\xi{R_\c}\bigg(\frac{\Gamma(1+\dim/2)}
+    {\pi^{\dim/2}\fS(0)(\xi_0^-)^{\dim-1}}\bigg)^{1/(\dim-1)}
 \]
-where $\xi=\xi_0^-|t|^{-\nu}$ for $t<T_c$. Since $\fS(0)(\xi_0^-)^{\dim-1}$ is a
-universal amplitude ratio \cite{zinn.1996.universal}, $\frac{Bh|t|^{-\beta\delta}}{\xi/R_c}$ is a
-universal quantity.
-% The constant $B$ should be universal near the critical point given careful
-% definition of the variable $X$.
-% \[
-%   \begin{aligned}
-%     \frac\xi{R_\c}
-%     &=\frac{\xi_0^-\fM(0)}{(d-1)\mathcal
-%     S(0)}h|t|^{-\beta\delta}
-%     =\frac{(\xi_0^-/\xi_0^+)R_\chi
-%     R_\xi^d}{(d-1)R_CR_\Sigma}\frac{h|t|^{-\beta\delta}}{\fM(0)^\delta D_\c}\\
-%     &=C\frac{h}{D_\c}|\fM(0)^{1/\beta}t|^{-\beta\delta}
-%   \end{aligned}
-% \]
-% \[
-%   \frac BC=\bigg(\frac{\Gamma(1+\frac d2)}{\pi^{d/2}\fS(0)(\xi_0^-)^{d-1}}\bigg)^{1/(d-1)}
-%   =\bigg(\frac{\Gamma(1+\frac d2)}{\pi^{d/2}R_\Sigma(\xi_0^-/\xi_0^+)^{d-1}}\bigg)^{1/(d-1)}
-% \]
-% $R_\Sigma=\fS(0)\xi_0^{d-1}$
-% These are $R^+_\xi=\frac1{\sqrt{2\pi}}$, $R_\Sigma^+=1$
-% $R_C=0.3185699$ $R_\chi=6.77828502$ $\xi_0^-/\xi_0^+=\frac12$
-The decay rate of the metastable state is proportional to the Boltzmann factor
-for the creation of a critical droplet, yielding
+where the critical amplitude for the correlation length $\xi_0^-$ is defined
+by $\xi=\xi_0^-|t|^{-\nu}$ for $t<T_\c$. 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 for the
+creation of a critical droplet, yielding
 \[
-  \im F\sim\Gamma\propto e^{-\Delta F_\c/T}=e^{-\fG(h|t|^{-\beta\delta})^{-(\dim-1)}}.
+  \im F\sim\Gamma\propto e^{-\Delta F_\c/T}
+    =e^{-\fG(h|t|^{-\beta\delta})^{-(\dim-1)}}.
 \]
 For $d>1$ this function has an essential singularity in the invariant
 combination $h|t|^{-\beta\delta}$.
 
-% $\Gamma/\Gamma_\sq=(D_\sq/D)(\fM(0)^{1/\beta}/\fM_\sq(0)^{1/\beta})^{-7/4}$
-
 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 $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$ for
+\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
 \[
-  \fiF(X)=-A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}},
+  \fiF(X)=-A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}}
   \label{eq:im.scaling}
 \]
-where $\Theta$ 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
-$d=3$
+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 in $H$ in good agreement with transfer matrix expansions
 \cite{lowe.1980.instantons}. Here, we compute the integral to come to explicit
-functional forms.  In \threedee and \fourdee this can be computed
-explicitly given our scaling ansatz, yielding
+functional forms.  In \threedee and \fourdee this can be computed explicitly
+given our scaling ansatz, yielding
 \def\eqthreedeeone{
   \fF^\threedee(Y/B)&=
   \frac{A}{12}\frac{e^{-1/Y^2}}{Y^2}
@@ -254,14 +242,14 @@ explicitly given our scaling ansatz, yielding
 \begin{align}
   &\begin{aligned}
     \eqthreedeeone\\
-    &\hspace{6em}
+    &\hspace{8em}
     \eqthreedeetwo
   \end{aligned}
   \\
   &\begin{aligned}
     \eqfourdeeone
     \\
-    &\hspace{2em}
+    &\hspace{-0.5em}
     \eqfourdeetwo.
   \end{aligned}
 \end{align}
@@ -272,119 +260,131 @@ explicitly given our scaling ansatz, yielding
   \eqfourdeeone\eqfourdeetwo.
 \end{align}
 \fi
-At the level of truncation we are working at, the Kramers--Kronig relation
-does not converge in \twodee. However, the higher moments can still be
-extracted, e.g., the susceptibility, by taking
+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
 \[
   \chi=\pd[2]Fh
   =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'.
 \]
-With $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields
+With a scaling form defined by $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$,
+this yields
 \[
   \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big]
   \label{eq:sus_scaling}
 \]
-The constant $A$ can be fixed by zero-field results, with $\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=\frac{2AB^2}\pi$.
 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{ABT_c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\
+    &=\fM(0)+\frac{ABT_\c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\
   \fF^\twodee(Y/B)
-    &=\fF(0)+T_cY\bigg(\frac{\fM(0)}B+\frac{AT_c}\pi e^{1/Y}\ei(-1/Y)\bigg)
+    &=\fF(0)+T_\c Y\bigg(\frac{\fM(0)}B+\frac{AT_\c}\pi e^{1/Y}\ei(-1/Y)\bigg)
 \end{align}
-with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in two dimensions.
+with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in
+two dimensions.
 
-How are these functional forms to be interpreted? They are not asymptotic
-forms in any sense, as there is no limit of $t$ or $h$ in which they become
-aribitrarily correct. It is well established that this method of using
+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 only works for high moments of those predictions
-\cite{parisi.1977.asymptotic,bogomolny.1977.dispersion}.
-These functions should be understood as possessing exactly the correct
+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.
+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. 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 $\fX(0)=C_{0-}/T_\c$ with $C_{0-}=0.025\,536\,971\,9$
-\cite{barouch.1973.susceptibility}, so that $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. 
-Data was then taken for susceptibility and
-magnetization for $T_\c-T,H\leq0.1$. The resulting scaling functions $\fX$ and
-$\fM$ are plotted using this data in
+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.
+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 $\fX(0)=C_0^-/T_\c$ with
+$C_0^-=0.025\,536\,971\,9$ \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
+$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant
+$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_0^-$.  The
+resulting scaling functions $\fX$ and $\fM$ are plotted as solid lines in
 Fig.~\ref{fig:scaling_fits}. As can be seen, there is very good agreement
-between our proposed functional forms and what is measured.
-However, there are systematic differences that can be seen most clearly in the
-magnetization. Since our method is known to only be accurate for high moments
-of the free energy, we should expect that low moments require corrections.
-Therefore, we also fit those corrections using 
+between our proposed functional forms and what is measured.  However, there
+are systematic differences that can be seen most clearly in the magnetization.
+Since our method is known to only be accurate for high moments of the free
+energy, we should expect that low moments require corrections.  Therefore, we
+also fit those corrections using 
 \begin{align}
-  \fX^{\twodee\prime}(X)&=\fX^\twodee(X)+\sum_{n=1}^Nf_n(BX)\label{eq:sus_scaling_poly}\\
-  \fM^{\twodee\prime}(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=1}^NF_n(BX)\label{eq:mag_scaling_poly}
+  \tilde\fX^\twodee(X)&=\fX^\twodee(X)+\sum_{n=0}^Nf_n(BX)\label{eq:sus_scaling_poly}\\
+  \tilde\fM^\twodee(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=0}^NF_n(BX)\label{eq:mag_scaling_poly}
 \end{align}
-where $F_n'(x)=f_n(x)$ and
+where $F_n'(Y)=f_n(Y)$ and
 \[
   \begin{aligned}
-    f_n(x)&=\frac{C_nx^n}{1+(\lambda x)^{n+1}}\\
-    F_n(x)&=\frac{C_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda x)^{n+1}).
+    f_n(Y)&=\frac{c_nx^n}{1+(\lambda Y)^{n+1}}\\
+    F_n(Y)&=\frac{c_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda Y)^{n+1}).
   \end{aligned}
   \label{eq:poly}
 \]
 The functions $f_n$ have been chosen to be pure integer power laws for small
-argument, but vanish appropriately at large argument.
-We fit these functions to our numeric data for $N=3$. The resulting curves are
-also plotted in Fig.~\ref{fig:scaling_fits} as dashed lines.
-
-%\begin{table}
-%  \centering
-%  \begin{tabular}{c|llc}
-%    Lattice    & $T_\c$             & $\fM(0)^{1/\beta}$        & $D/D_\sq$   \\
-%    \hline % -------------------------------------------------------------------
-%    Square     & $2/\log(1+\sqrt2)$ & $2^{5/2}\arcsinh1$        & 1           \\
-%    Triangular & $4/\log3$          & $4\log3$                  & $3^{3/2}/4$ \\
-%    Hexagonal  & $2/\log(2+\sqrt3)$ & $\frac8{\sqrt3}\arccosh2$ & $3^{3/2}/8$
-%  \end{tabular}
-%  \caption{
-%    The critical temperatures and amplitudes for the magetization along both the coexistence line and the critical isotherm, for three different lattices.
-%  }
-%  \label{tab:consts}
-%\end{table}
+argument, but vanish appropriately at large argument. This is necessary
+because the susceptibility vanishes with $h|t|^{-\beta\delta}$ while bare
+polynomial corrections would not.  We fit these functions to our numeric data
+for $N=0$ while requiring that $C_0^-/T_\c=\fX'(0)=c_0+2AB^2/\pi$. The
+resulting curves are also plotted as dashed lines in
+Fig.~\ref{fig:scaling_fits}. Our singular scaling function with one low-order
+correction appears to match data quite well.
 
 \begin{figure}
-  \input{figs/fig-susmag}
+  \input{fig-susmag}
   \caption{
     Scaling functions for (top) the susceptibility and (bottom) the
-    magnetization plotted in terms of the invarient 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
-    $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Color denotes the value of
-    $T$. The solid lines show our
-    analytic results \eqref{eq:sus_scaling} and \eqref{eq:mag_scaling}, while
-    the dashed lines show fits of \eqref{eq:sus_scaling_poly} and
-    \eqref{eq:mag_scaling_poly} to the data for $N=3$, with $C_1=-0.00368$,
-    $C_2=-0.0191$, $C_3=0.0350$, and $\lambda=2.42$.
+    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 $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Color denotes the value of
+    $T$, with red corresponding to $0.01$ and violet to $0.1$. The solid lines
+    show our analytic results \eqref{eq:sus_scaling} and
+    \eqref{eq:mag_scaling}, while the dashed lines show fits of
+    \eqref{eq:sus_scaling_poly} and \eqref{eq:mag_scaling_poly} to the data
+    for $N=0$, with $c_0=-0.0124$ and $\lambda=1.77$.
   }
   \label{fig:scaling_fits}
 \end{figure}
 
 We have used results from the properties of the metastable Ising ferromagnet
-and the analytic nature of the free energy to derive the universal scaling
+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 $t$ and $h$. Because of an essential
+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 regular functions 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 singularity leads to observable 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.
+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
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