From e539f537677e645a6804063a7bd3eac2d2e57113 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 5 Jun 2017 21:51:35 -0400
Subject: many changes, most notably removing our section on the high-order
 terms and a lot of rewriting

---
 essential-ising.tex | 238 +++++++++++++++++++++++++---------------------------
 1 file changed, 112 insertions(+), 126 deletions(-)

(limited to 'essential-ising.tex')

diff --git a/essential-ising.tex b/essential-ising.tex
index b9ee8a2..4bab30f 100644
--- a/essential-ising.tex
+++ b/essential-ising.tex
@@ -6,8 +6,7 @@
 \documentclass[aps,prl,reprint]{revtex4-1}
 
 \usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage{amsmath,amssymb,latexsym,concmath,mathtools,xifthen,mfpic}
+\usepackage{amsmath,amssymb,latexsym,mathtools,xifthen}
 
 \mathtoolsset{showonlyrefs=true}
 
@@ -17,7 +16,7 @@
 \def\re{\mathop{\mathrm{Re}}\nolimits}
 \def\im{\mathop{\mathrm{Im}}\nolimits}
 \def\dd{\mathrm d}
-\def\O{\mathcal O}
+\def\O{O}
 \def\o{\mathcal o}
 \def\ei{\mathop{\mathrm{Ei}}\nolimits}
 \def\b{\mathrm b}
@@ -37,64 +36,74 @@
 \author{James P.~Sethna}
 \affiliation{Cornell University}
 
-\date{April 20, 2017}
+\date\today
 
 \begin{abstract}
-  This is an 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
+  asymptotic 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
 
 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 ({\sc rg}) method in statistical physics
+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 {\sc rg} methods in a straightforward way,
+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 {\sc rg}
-scaling analysis of the critical point, we demonstrate that there are
+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.
 
-{\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
-$F(t,h)=|g_t|^{2-\alpha}\mathcal F(g_h|g_t|^{-\Delta})$ \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}
-\cite{cardy.1996.scaling,aharony.1983.fields}. 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}, 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.
+\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
+$F(t,h)=|g_t|^{2-\alpha}\mathcal F(g_h|g_t|^{-\Delta})$ \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 \textsc{rg}
+\cite{cardy.1996.scaling,aharony.1983.fields}. 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}, 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.
 
 The providence 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 $H<0$.
-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,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 of probability in the system
-that corresponds to decay. 
+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} and \textsc{rg}
+methods \cite{klein.1976.essential}.  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 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 whose surface-energy cost for growth is outweighed by
@@ -102,10 +111,11 @@ 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 $-M|H|N$, 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
+droplet scales like $\Delta
+F_\c\sim\Sigma^{1/(1-\sigma)}(M|H|)^{-\sigma/(1-\sigma)}$.  Assuming domains
+have minimal surfaces, as evidenced by transfer matrix studies
+\cite{gunther.1993.transfer-matrix}, $\sigma=1-\frac1d$ and $\Delta
+F_\c\sim\Sigma^d(M|H|)^{-(d-1)}$. Assuming the singular 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
@@ -113,81 +123,86 @@ M(g_h|g_t|^{-\Delta})$ and using known hyperscaling relations
   \Delta F_c&
   \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})
+  &\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 $X$.
-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
+at $T<T_c$, $\mathcal G(X)=\O(X)$ for small $X$.  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
 \[
-  \im F\sim e^{-\mathcal G(g_h|g_t|^{-\Delta})^{-(d-1)}}
+  \im F\sim e^{-\mathcal G(g_h|g_t|^{-\Delta})^{-(d-1)}}.
 \]
 For $d>1$ this function has an essential singularity in the invariant
 combination $g_h|g_t|^{-\Delta}$.
 
-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
-$t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\mathcal H(h|t|^{-\Delta})$ for
+This form of $\im F$ for small $h$ is well known
+\cite{langer.1967.condensation,harris.1984.metastability}. 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 $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\mathcal
+H(h|t|^{-\Delta})$ 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
-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
+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 $\zeta=-(d-3)d/2$ for $d=2,4$ and $\zeta=-7/3$ 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
 \[
-  \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h'
+  \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h'.
 \]
-In {\sc 3d} and {\sc 4d} this can be computed explicitly given our scaling
-ansatz, yielding
+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 \textsc{3d} and \textsc{4d} this can be computed
+explicitly given our scaling ansatz, yielding
 \begin{align}
-  \mathcal F^{\text{\sc 3d}}(X)&=
+  \mathcal F^{\text{\textsc{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
 \\
-  \mathcal F^{\text{\sc 4d}}(X)&=
+  \mathcal F^{\text{\textsc{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}.
+for \textsc{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
+does not converge in \textsc{2d}. However, the higher moments can still be
 extracted, e.g., the susceptibility, by taking
 \[
   \chi\propto\pd[2]Fh
-  =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'
+  =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'.
 \]
 With $\chi=|t|^{-\gamma}\mathcal Y(h|t|^{-\Delta})$, this yields
 \[
-  \mathcal Y^{\text{\sc 2d}}(X)=\frac{C}{2(BX)^3}\big[BX(BX-1)-e^{1/BX}\ei(-1/BX)\big]
+  \mathcal Y^{\text{\textsc{2d}}}(X)=\frac{C}{2(BX)^3}\big[BX(BX-1)-e^{1/BX}\ei(-1/BX)\big]
   \label{eq:sus_scaling}
 \]
 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
+\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
 \[
-  \mathcal M^{\text{\sc 2d}}(X)=\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX)-D+\mathcal M(0)
+  \mathcal M^{\text{\textsc{2d}}}(X)=\frac{D}{BX}(BX-1)e^{1/BX}\ei(-1/BX)-D+\mathcal M(0)
   \label{eq:mag_scaling}
 \]
-with $\mathcal
-M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$
-\cite{onsager.1944.crystal} and
-$D$ constant, and
+with $\mathcal M(0)=\big(2(\sqrt2-1)\big)^{1/4}\big((4+3\sqrt2)\sinh^{-1}1\big)^{1/8}$
+\cite{onsager.1944.crystal} and $D$ constant, and
 \[
-  \mathcal F^{\text{\sc 2d}}(X)=\mathcal F(0)+EX(\mathcal M(0)+De^{1/BX}\ei(-1/BX))
+  \mathcal F^{\text{\textsc{2d}}}(X)=\mathcal F(0)+EX(\mathcal M(0)+De^{1/BX}\ei(-1/BX))
 \]
 for $\mathcal F(0)=?$ and $E$ constant.
 
@@ -203,58 +218,29 @@ between our proposed functional forms and what is measured.
 
 \begin{figure}
   \input{figs/fig-sus}
-  %\includegraphics{figs/fig_sus-collapse}
-  \caption{Fit of scaling form \eqref{eq:sus_scaling} to numeric data from a
-  $L=1024$ square-lattice Ising model.
-Different colored points show different values of $t$, which vary from
-$0.01$, $0.02$, \dots, $0.1$. Solid black line shows fitted form, with
-$C=0.0111\pm0.0023$ and $B=0.173\pm0.072$.}
+  \caption{
+    Fit of scaling form \eqref{eq:sus_scaling} to numeric data.  Data with
+    sampling error taken from Monte Carlo simulations of an $L=2048$
+    square-lattice Ising model with $t=0.01,0.02,\ldots,0.1$ and
+    $h=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form,
+    with $C=0.0111\pm0.0023$ and $B=0.173\pm0.072$.
+  }
   \label{fig:sus}
 \end{figure}
 
 \begin{figure}
   \input{figs/fig-mag}
-  %\includegraphics{figs/fig_mag-collapse}
-  \caption{Fit of scaling form \eqref{eq:mag_scaling} to numeric data from a
-$L=1024$ square-lattice Ising model. Different colored points show different
-values of $t$, which vary from $0.01$, $0.02$, \dots, $0.1$. Solid black lines
-shows fitted form, with $\mathcal M(0)=1.21039\pm0.00031$,
-$D=0.09400\pm0.00035$, and $B=0.0861\pm0.0010$.}
+  \caption{
+    Fit of scaling form \eqref{eq:mag_scaling} to numeric data. Data with
+    sampling error taken from Monte Carlo simulations of an $L=2048$
+    square-lattice Ising model with $t=0.01,0.02,\ldots,0.1$ and
+    $h=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form,
+    with $\mathcal M(0)=1.21039\pm0.00031$,
+    $D=0.09400\pm0.00035$, and $B=0.0861\pm0.0010$.
+  }
   \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
@@ -263,7 +249,7 @@ 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
+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.
-- 
cgit v1.2.3-54-g00ecf

