From d1a57e6446970c2149add56c4e017bfa991c777a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 30 May 2017 10:11:26 -0400
Subject: a lot of writing work

---
 essential-ising.tex | 161 +++++++++++++++++++++++++++++++++++++---------------
 1 file changed, 116 insertions(+), 45 deletions(-)

(limited to 'essential-ising.tex')

diff --git a/essential-ising.tex b/essential-ising.tex
index 9e4d2c3..ef98476 100644
--- a/essential-ising.tex
+++ b/essential-ising.tex
@@ -32,7 +32,7 @@
 
 \begin{document}
 
-\title{Essential Singularity in the Ising Abrupt Transition}
+\title{Essential Singularities in the Ising Universal Scaling Functions}
 \author{Jaron Kent-Dobias}
 \author{James P.~Sethna}
 \affiliation{Cornell University}
@@ -40,7 +40,7 @@
 \date{April 20, 2017}
 
 \begin{abstract}
-  Test
+  This is an abstract!
 \end{abstract}
 
 \maketitle
@@ -61,14 +61,21 @@ thermodynamic variables.
 
 {\sc Rg} analysis predicts that the singular part of the free energy per site
 $F$ as a function of reduced temperature $t=1-\frac{T_c}T$ and field $h=H/T$ in the vicinity of the critical point takes the scaling form
-$F(t,h)=g_t^{2-\alpha}\mathcal F(g_hg_t^{-\Delta})$, where
+$F(t,h)=|g_t|^{2-\alpha}\mathcal F(g_h|g_t|^{-\Delta})$ \footnote{Technically we should write $\mathcal
+  F_{\pm}$ to indicate that the universal scaling function takes a different
+  form for $t<0$ and $t>0$, but we will restrict ourselves entirely to $t<0$
+  and hence $\mathcal F_-$
+for the purposes of this paper.}, where
 $\Delta=\beta\delta$ and $g_t$, $g_h$ are analytic functions of $t$, $h$ that
-transform exactly linearly under {\sc rg}. When studying the properties of the
+transform exactly linearly under {\sc rg}
+\cite{cardy.1996.scaling}. When studying the properties of the
 Ising critical point, it is nearly always assumed that $\mathcal F(X)$, the
 universal scaling function, is an analytic function of $X$. However, it has
 long been known that there exists an essential singularity in $\mathcal F$ at
 $X=0$, though its effects have long been believed to be unobservable
-\cite{fisher.1967.condensation}. With careful analysis, we have found that
+\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.
 
@@ -79,7 +86,7 @@ subjected to a small negative external field $H<0$.
 It's 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.1967.condensation,langer.1969.metastable,gaveau.1989.analytic}.
+\cite{langer.1967.condensation,langer.1969.metastable,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
@@ -90,22 +97,23 @@ Hamiltonian: the imaginary part describes loss of probability in the system
 that corresponds to decay. 
 
 In critical droplet theory, the metastable state decays when a domain of the
-equilibrium state forms that is sufficiently large to grow and envelop the
-whole system. Assuming the free energy cost of the surface of the droplet
+equilibrium state forms whose surface-energy cost for growth is outweighed by
+bulk-energy gains. Assuming the free energy cost of the surface of the droplet
 scales with the number of spins $N$ like $\Sigma N^\sigma$ and that of its
-bulk scales like $-MHN$, the critical droplet size scales like
-$N_\c\sim(MH/\Sigma)^{-1/(1-\sigma)}$ and the free energy of the critical
-droplet scales like $\Delta F_\c\sim\Sigma^{1/(1-\sigma)}(HM)^{-\sigma/(1-\sigma)}$.
-Assuming domains have minimal surfaces, $\sigma=1-\frac1d$ and
-$\Delta F_\c\sim\Sigma^d(HM)^{-(d-1)}$. Assuming the scaling forms
-$\Sigma=g_t^\mu\mathcal S(g_hg_t^{-\Delta})$ and $M=g_t^\beta\mathcal
-M(g_hg_t^{-\Delta})$ and using known hyperscaling relations
+bulk scales like $-M|H|N$, the critical droplet size scales like
+$N_\c\sim(M|H|/\Sigma)^{-1/(1-\sigma)}$ and the free energy of the critical
+droplet scales like $\Delta F_\c\sim\Sigma^{1/(1-\sigma)}(M|H|)^{-\sigma/(1-\sigma)}$.
+Assuming domains have minimal surfaces, as evidenced by numeric studies
+\cite{gunther.1993.transfer-matrix}, $\sigma=1-\frac1d$ and
+$\Delta F_\c\sim\Sigma^d(M|H|)^{-(d-1)}$. Assuming the scaling forms
+$\Sigma=|g_t|^\mu\mathcal S(g_h|g_t|^{-\Delta})$ and $M=|g_t|^\beta\mathcal
+M(g_h|g_t|^{-\Delta})$ and using known hyperscaling relations
 \cite{widom.1981.interface}, this implies a scaling form
 \begin{align}
   \Delta F_c&
-  \sim\mathcal S^d(g_hg_t^{-\Delta})(g_hg_t^{-\Delta}\mathcal
-    M(g_hg_t^{-\Delta}))^{-(d-1)}\\
-  &\sim\mathcal G^{-(d-1)}(g_h g_t^{-\Delta})
+  \sim\mathcal S^d(g_h|g_t|^{-\Delta})(-g_h|g_t|^{-\Delta}\mathcal
+    M(g_h|g_t|^{-\Delta}))^{-(d-1)}\notag\\
+  &\sim\mathcal G^{-(d-1)}(g_h|g_t|^{-\Delta})
 \end{align}
 Since both surface tension and magnetization are finite and nonzero for $H=0$
 at $T<T_c$, $\mathcal G(X)=\O(X)$ for small $X$.
@@ -113,19 +121,20 @@ The decay rate of the metastable state will be roughly given by the Boltzmann
 factor for the creation of a critical droplet, or $\Gamma\sim e^{-\beta\Delta
 F_c}$, so that
 \[
-  \im F\sim e^{-\mathcal G(g_hg_t^{-\Delta})^{-(d-1)}}
+  \im F\sim e^{-\mathcal G(g_h|g_t|^{-\Delta})^{-(d-1)}}
 \]
 For $d>1$ this function has an essential singularity in the invariant
-combination $g_hg_t^{-\Delta}$.
+combination $g_h|g_t|^{-\Delta}$.
 
 This form of $\im F$ for small $h$ is known. Henceforth we will assume $h$ and
 $t$ are sufficiently small that $g_t\simeq t$, $g_h\simeq h$, and all
 functions of both variables can be truncated at lowest order. 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
-$t$, $h$, $\im F(t,h)=t^{2-\alpha}\mathcal H(ht^{-\Delta})$ for
+$t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\mathcal H(h|t|^{-\Delta})$ for
 \[
-  \mathcal H(X)=A\Theta(-X)X^\zeta e^{-1/(BX)^{d-1}}
+  \mathcal H(X)=A\Theta(-X)(-X)^\zeta e^{-1/(-BX)^{d-1}}
+  \label{eq:im.scaling}
 \]
 where $\Theta$ is the Heaviside function and with $\zeta=-(d-3)d/2$ for $d=2,4$ and $\zeta=-7/3$ for $d=3$
 \cite{houghton.1980.metastable,gunther.1980.goldstone}. Assuming that $F$ is
@@ -137,25 +146,21 @@ using the Kramers--Kronig relation
 \]
 In {\sc 3d} and {\sc 4d} this can be computed explicitly given our scaling
 ansatz, yielding
-\[
-  \begin{aligned}
-    \mathcal F(X)&=
-    \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2}
-    \bigg[\Gamma(\tfrac16)E_{7/6}((BX)^{-2})\\
-    &\hspace{10em}-4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg]
-  \end{aligned}
+\begin{align}
+  \mathcal F^{\text{\sc 3d}}(X)&=
+  \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2}
+  \bigg[\Gamma(\tfrac16)E_{7/6}((BX)^{-2})\\
+  &\hspace{10em}-4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg]
   \notag
-\]
-\[
-  \begin{aligned}
-    \mathcal F(X)&=
-    \frac{A}{9\pi X^2}e^{1/(BX)^3}
-    \Big[3\Gamma(0,(BX)^{-3})\\
-    &\hspace{2em}-3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3})
-    -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big]
-  \end{aligned}
+\\
+  \mathcal F^{\text{\sc 4d}}(X)&=
+  \frac{A}{9\pi X^2}e^{1/(BX)^3}
+  \Big[3\Gamma(0,(BX)^{-3})\\
+  &\hspace{2em}-3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3})
+  -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big]
   \notag
-\]
+\end{align}
+for {\sc 4d}.
 At the level of truncation we are working at, the Kramers--Kronig relation
 does not converge in {\sc 2d}. However, the higher moments can still be
 extract, e.g., the susceptibility, by taking
@@ -163,27 +168,93 @@ extract, e.g., the susceptibility, by taking
   \chi\propto\pd[2]Fh
   =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'
 \]
-This yields
+With $\chi=|t|^{-\gamma}\mathcal Y(h|t|^{-\Delta})$, this yields
 \[
-  \chi=|t|^{-\gamma}\frac{C}{2(BX)^3}\big[BX(BX-1)-e^{1/BX}\ei(-1/BX)\big]
+  \mathcal Y^{\text{\sc 2d}}(X)=\frac{C}{2(BX)^3}\big[BX(BX-1)-e^{1/BX}\ei(-1/BX)\big]
+  \label{eq:sus_scaling}
 \]
+for some constant $C$. Previous work at zero field suggests that
+$C=C_{0-}/T_c$, with $C_{0-}=0.025\,536\,971\,9$
+\cite{barouch.1973.susceptibility}.
 Scaling forms for the free energy can then be extracted by integration and
 comparison with known exact results at zero field, yielding
 \[
-  \mathcal M(X)=\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX)-D+\mathcal M(0)
+  \mathcal M^{\text{\sc 2d}}(X)=\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX)-D+\mathcal M(0)
+  \label{eq:mag_scaling}
 \]
 with $\mathcal
-M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$, and
+M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$
+\cite{onsager.1944.crystal} and
+$D$ constant, and
 \[
-  \mathcal F(X)=\mathcal F(0)+EX(\mathcal M(0)+De^{1/BX}\ei(-1/BX))
+  \mathcal F^{\text{\sc 2d}}(X)=\mathcal F(0)+EX(\mathcal M(0)+De^{1/BX}\ei(-1/BX))
 \]
+for $\mathcal F(0)=?$ and $E$ constant.
 
 How predictive are these scaling forms in the proximity of the critical point
 and the abrupt transition line? We used a form of the Wolff algorithm modified
 to remain efficient in the presence of an external field by incorporating the
 field as another spin with coupling $|H|$ to all others
 \cite{dimitrovic.1991.finite}. Data was then taken for susceptibility and
-magnetization for $|t|,h\leq0.1$
+magnetization for $|t|,h\leq0.1$. This data is plotted in Figs.~\ref{fig:sus}
+and \ref{fig:mag}, along with collapses of data onto a single universal curve
+in the insets of those figures. As can be seen, there is very good agreement
+between our proposed functional forms and what is measured.
+
+\begin{figure}
+  \caption{Fit of scaling form \eqref{eq:sus_scaling} to numeric data.}
+  \label{fig:sus}
+\end{figure}
+
+\begin{figure}
+  \caption{Fit of scaling form \eqref{eq:mag_scaling} to numeric data.}
+  \label{fig:mag}
+\end{figure}
+
+The most accurate components of this analysis are its predictions for the
+higher-order terms in the expansion
+\[
+  \mathcal F(X)=-\sum_{n=0}^\infty f_n(-X)^n
+\]
+\cite{brezin.1976.perturbation,bogomolny.1977.dispersion,lipatov.1977.divergence,parisi.1977.asymptotic}
+which can be computed directly from the integral
+\[
+  f_n=\frac{(-1)^{n+1}}{n!}\pd{\mathcal F}X\bigg|_{X=0}
+  =\frac{(-1)^{n+1}}{\pi}\int_{-\infty}^\infty\frac{\mathcal H(X)}{X^{n+1}}\;\dd X
+\]
+Applied to our ansatz \eqref{eq:im.scaling}, we find
+\begin{align}
+  f_n^{\text{\textsc{2d}}}
+  &=\frac{AB^{n-1}\Gamma(n-1)}{\pi}
+  &&
+  n \geq 2
+  \label{eq:fn.2d}
+  \\
+  f_n^{\text{\textsc{3d}}}
+  &=\frac{AB^{n+\frac73}\Gamma(\frac n2+\frac76)}{2\pi}
+  \\
+  f_n^{\text{\textsc{4d}}}
+  &=\frac{AB^{n+2}\Gamma(\frac n3+\frac23)}{3\pi}
+\end{align}
+The form of \eqref{eq:fn.2d} is identical to that found in
+\cite{baker.1980.ising}, with our $f_n$ relating to their $a_n$ by
+$a_{n-1}=2^nf_n$.
+\[
+  r_n=\frac{a_n}{a_{n-1}}=\frac{2f_{n+1}}{f_{n}}=2B\frac{\Gamma(n)}{\Gamma(n-1)}
+\]
+
+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
+functions for the free energy, and in \textsc{2d} the magnetization and
+susceptibility, in the limit of small $t$ and $h$. Because of an essential
+singularity in these functions at $h=0$---the abrupt transition line---their
+form cannot be modified 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.
 
 
 \bibliographystyle{apsrev4-1}
-- 
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