From d16d1e6b3394f7cd604cb9c64630042c8c77d6dc Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 26 May 2017 10:43:44 -0400
Subject: moved files to new name

---
 essential-ising.bib |  99 +++++++++++++++++++++++++
 essential-ising.tex | 206 ++++++++++++++++++++++++++++++++++++++++++++++++++++
 essential_ising.bib |  99 -------------------------
 essential_ising.tex | 206 ----------------------------------------------------
 4 files changed, 305 insertions(+), 305 deletions(-)
 create mode 100644 essential-ising.bib
 create mode 100644 essential-ising.tex
 delete mode 100644 essential_ising.bib
 delete mode 100644 essential_ising.tex

diff --git a/essential-ising.bib b/essential-ising.bib
new file mode 100644
index 0000000..c84c1a7
--- /dev/null
+++ b/essential-ising.bib
@@ -0,0 +1,99 @@
+
+@article{houghton.1980.metastable,
+  title={The metastable Ising magnet in a negative field},
+  author={Hougton, Anthony and Lubensky, Tom C},
+  journal={Physics Letters A},
+  volume={77},
+  number={6},
+  pages={479--480},
+  year={1980},
+  publisher={North-Holland}
+}
+
+@article{langer.1967.condensation,
+  title={Theory of the condensation point},
+  author={Langer, James S},
+  journal={Annals of Physics},
+  volume={41},
+  number={1},
+  pages={108--157},
+  year={1967},
+  publisher={Elsevier}
+}
+
+@article{langer.1969.metastable,
+  title={Statistical theory of the decay of metastable states},
+  author={Langer, JS},
+  journal={Annals of Physics},
+  volume={54},
+  number={2},
+  pages={258--275},
+  year={1969},
+  publisher={Elsevier}
+}
+
+@article{gaveau.1989.analytic,
+  title={Metastable decay rates and analytic continuation},
+  author={Gaveau, B and Schulman, LS},
+  journal={Letters in Mathematical Physics},
+  volume={18},
+  number={3},
+  pages={201--208},
+  year={1989},
+  publisher={Springer}
+}
+
+@article{bogomolny.1977.dispersion,
+  title={Calculation of the green functions by the coupling constant dispersion relations},
+  author={Bogomolny, Evgeny B},
+  journal={Physics Letters B},
+  volume={67},
+  number={2},
+  pages={193--194},
+  year={1977},
+  publisher={Elsevier}
+}
+
+@article{gunther.1993.transfer-matrix,
+  title={Numerical transfer-matrix study of metastability in the d= 2 Ising model},
+  author={G{\"u}nther, Christoph CA and Rikvold, Per Arne and Novotny, MA},
+  journal={Physical review letters},
+  volume={71},
+  number={24},
+  pages={3898},
+  year={1993},
+  publisher={APS}
+}
+
+@inproceedings{widom.1981.interface,
+  title={Structure of the interface between fluid phases},
+  author={Widom, Benjamin},
+  booktitle={Faraday Symposia of the Chemical Society},
+  volume={16},
+  pages={7--21},
+  year={1981},
+  organization={Royal Society of Chemistry}
+}
+
+@article{gunther.1980.goldstone,
+  title={Goldstone modes in vacuum decay and first-order phase transitions},
+  author={G{\"u}nther, NJ and Wallace, DJ and Nicole, DA},
+  journal={Journal of Physics A: Mathematical and General},
+  volume={13},
+  number={5},
+  pages={1755},
+  year={1980},
+  publisher={IOP Publishing}
+}
+
+@article{barouch.1973.susceptibility,
+  title={Zero-field susceptibility of the two-dimensional Ising model near ${T}_c$},
+  author={Barouch, Eytan and McCoy, Barry M and Wu, Tai Tsun},
+  journal={Physical Review Letters},
+  volume={31},
+  number={23},
+  pages={1409},
+  year={1973},
+  publisher={APS}
+}
+
diff --git a/essential-ising.tex b/essential-ising.tex
new file mode 100644
index 0000000..d6a92cc
--- /dev/null
+++ b/essential-ising.tex
@@ -0,0 +1,206 @@
+%  Ising model abrupt transition.
+%
+%  Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017.
+%  Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved.
+%
+\documentclass[fleqn]{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath,amssymb,latexsym,concmath,mathtools,xifthen,mfpic}
+
+\mathtoolsset{showonlyrefs=true}
+
+\title{Essential Singularity in the Ising Abrupt Transition}
+\author{Jaron Kent-Dobias}
+
+\date{April 20, 2017}
+
+\begin{document}
+
+\def\[{\begin{equation}}
+\def\]{\end{equation}}
+
+\def\im{\mathop{\mathrm{Im}}\nolimits}
+\def\dd{\mathrm d}
+\def\O{\mathcal O}
+\def\ei{\mathop{\mathrm{Ei}}\nolimits}
+\def\b{\mathrm b}
+
+\newcommand\pd[3][]{
+  \ifthenelse{\isempty{#1}}
+    {\def\tmp{}}
+    {\def\tmp{^#1}}
+  \frac{\partial\tmp#2}{\partial#3\tmp}
+}
+
+\maketitle
+
+\begin{abstract}
+\end{abstract}
+
+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
+\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic}
+\[
+  \Gamma\propto\im F
+\]
+What exactly is meant by `metastable free energy' is important to establish,
+since formally the free energy relies on the existence of an equilibrium
+state. Here one can imagine either analytic continuation of the free energy
+through an 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. 
+
+One can estimate the scaling of the decay rate of the {\sc 2d} Ising model
+using ideas from nucleation theory. In this framework, the metastable state
+decays when a sufficiently large domain in the stable state forms to grow
+stably to fill out the whole system. The free energy of a domain of $N$ spins
+causes a free energy change
+\[
+  \Delta F=\Sigma N^\sigma-MHN
+\]
+where $\Sigma$ is the surface tension and $1-\frac1d\leq\sigma<1$. This is
+maximized by
+\[
+  N_c=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)}
+\]
+which corresponds to a free energy change
+\[
+  \Delta F_c\sim\bigg(\frac\Sigma{(MH)^\sigma}\bigg)^{1/(1-\sigma)}
+\]
+The rate of formation is proportional to the Boltzmann factor,
+\[
+  \Gamma\sim e^{-\beta \Delta
+  F_c}=e^{-\beta(\Sigma/(MH)^\sigma)^{1/(1-\sigma)}}
+\]
+For domains whose boundary is minimal, $\sigma=1-\frac1d$ and this becomes
+\[
+  \Gamma\sim e^{-\beta(\Sigma/(MH)^\sigma)^{d-1}}
+\]
+Since $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$
+\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$
+with $\mathcal S(0)=\O(1)$ and $\mathcal M(0)=\O(1)$,
+\[
+  \Gamma\sim e^{-1/\mathcal G(ht^{-\beta\delta})^{d-1}}
+\]
+with $\mathcal G(X)=\O(X)$. This establishes the form of $\im F$
+besides the prefactor. Results from field theory predict that, for small $H$
+and $1<d<5$, $d\neq 3$,
+\[
+  \im F\simeq\bigg(\frac h{t^\Delta}\bigg)^{-(d-3)d/2}(g^*)^{-d(d-1)/4}
+  \exp\bigg[-B\bigg(\frac h{|t|^\Delta}\bigg)^{-(d-1)}(g^*)^{-(d+1)/2}\bigg]
+\]
+\[
+  \im F\simeq\bigg(\frac
+  h{t^\Delta}\bigg)^{-7/3}(g^*)^{-8/3}\exp\bigg[-B\bigg(\frac
+  h{t^\Delta}\bigg)^{-2}(g^*)^{-2}\bigg]
+\]
+with $\Delta=3-\frac\epsilon2$, $g^*=2\pi^2\frac\epsilon{n+8}$
+\cite{houghton.1980.metastable,gunther.1980.goldstone}. This is consistent
+with our form above. We therefore predict that
+\[
+  \im F=t^{2-\alpha}\mathcal F(ht^{-\beta\delta})^{-(d-3)d/2}e^{-1/\mathcal
+    G(ht^{-\beta\delta})^{d-1}}
+\]
+In {\sc 2d} we have
+\[
+  \im F=t^2\mathcal F(ht^{-\Delta})e^{-1/\mathcal G(ht^{-\Delta})}
+\]
+with $\Delta=\beta\delta=\frac{15}8$. In terms of $X=ht^{-\Delta}$, this is
+\[
+  \im F=t^2\mathcal F(X)e^{-1/\mathcal G(X)}\simeq At^2|X|e^{-1/B|X|}
+\]
+
+\cite{langer.1967.condensation}
+
+\[
+  F(X)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(X')}{X'-X}\,\dd X'
+  =\frac{At^2}\pi\int_{-\infty}^0\frac{|X'|e^{-1/B|X'|}}{X'-X}\,\dd
+  X'
+  =-\frac{At^2}\pi\int_0^\infty\frac{X'e^{-1/BX'}}{X'+X}\,\dd
+  X'
+\]
+since $\im F=0$ for $X>0$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$.
+Unfortunately this integral doesn't converge, and it seems we cannot evaluate
+this result at the level of truncation we've chosen. However, 
+
+\[
+  F(H)=At^{2-\alpha}\sum_{n=0}^\infty f_nX^n
+\]
+\[
+  f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(X)}{X^{n+1}}\,\dd X
+  =\frac{(-1)^{n+1}}\pi\int_0^{\infty}\frac{Xe^{-1/BX}}{X^{n+1}}\,\dd X
+  =\frac1\pi(-1)^{n+1}B^{n-1}\Gamma(n-1)
+\]
+for $n>1$. 
+
+\begin{align}
+  \chi
+  &=\pd[2]Fh
+  =t^{-2\Delta}\pd[2]FX
+  =-\frac{2}\pi At^{2-2\Delta}\int_0^\infty\frac{X'e^{-1/BX'}}{(X+X')^3}\,\dd
+  X'\\
+  &=\frac2\pi
+  \frac{ABt^{-\gamma}}{(BX)^3}\big[BX(1-BX)+e^{1/BX}\ei(-1/BX)\big]
+\end{align}
+
+\[
+  \lim_{X\to0}\chi=-\frac4\pi ABt^{-\gamma}
+\]
+
+\[
+  \beta^{-1}\chi=C_{0\pm}|t|^{-7/4}+C_{1\pm}|t|^{-3/4}+\O(1)
+\]
+$C_{0-}=0.025\,536\,971\,9$ $C_{1-}=-0.001\,989\,410\,7$
+\cite{barouch.1973.susceptibility}
+
+CORRECTIONS TO SCALING, $u_t$ and $u_h$ instead of $t$ and $h$.
+
+\begin{align}
+  u_h
+  &=h[1+c_ht+dht^2+e_hh^2+f_ht^3+\O(t^4,th^2)]\\
+  u_t
+  &=t+b_th^2+c_t^2+d_t^3+e_tth^2+f_tt^4+\O(t^5,t^2h^2,h^4)
+\end{align}
+\begin{align}
+  c_h=\frac{\beta_c}{\sqrt2}
+  &&
+  d_h=\frac{23\beta_c^2}{16}
+  &&
+  f_h=\frac{191\beta_c^3}{48\sqrt2}\\
+  c_t=\frac{\beta_c}{\sqrt2}
+  &&
+  d_t=\frac{7\beta_c^2}6
+  &&
+  f_t=\frac{17\beta_c^3}{6\sqrt2}\\
+  e_t=b_t\beta_c\sqrt2
+  &&
+  b_t=-\frac{E_0\pi}{16\beta_c^2}
+\end{align}
+$E_0=0.040\,325\,5003$ $e_h=-0.007\,27(15)$
+\[
+  F(t,h)-F(t,0)=\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2n}(t)h^{2n}
+\]
+\[
+  \chi(t,h)=\pd[2]Fh=\chi_2(t)+\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2(n+1)}h^{2n}
+\]
+
+\begin{align}
+  \chi
+  &=\pd[2]Fh
+  =\pd[2]{F_\b}h
+  +\frac d{y_t}\bigg(\frac d{y_t}-1\bigg)|u_t|^{d/y_t-2}\bigg(\pd{u_t}h\bigg)^2
+\end{align}
+
+\input{figs/scaling_func.tex}
+
+\bibliographystyle{plain}
+\bibliography{essential_ising}
+
+\end{document}
+
diff --git a/essential_ising.bib b/essential_ising.bib
deleted file mode 100644
index c84c1a7..0000000
--- a/essential_ising.bib
+++ /dev/null
@@ -1,99 +0,0 @@
-
-@article{houghton.1980.metastable,
-  title={The metastable Ising magnet in a negative field},
-  author={Hougton, Anthony and Lubensky, Tom C},
-  journal={Physics Letters A},
-  volume={77},
-  number={6},
-  pages={479--480},
-  year={1980},
-  publisher={North-Holland}
-}
-
-@article{langer.1967.condensation,
-  title={Theory of the condensation point},
-  author={Langer, James S},
-  journal={Annals of Physics},
-  volume={41},
-  number={1},
-  pages={108--157},
-  year={1967},
-  publisher={Elsevier}
-}
-
-@article{langer.1969.metastable,
-  title={Statistical theory of the decay of metastable states},
-  author={Langer, JS},
-  journal={Annals of Physics},
-  volume={54},
-  number={2},
-  pages={258--275},
-  year={1969},
-  publisher={Elsevier}
-}
-
-@article{gaveau.1989.analytic,
-  title={Metastable decay rates and analytic continuation},
-  author={Gaveau, B and Schulman, LS},
-  journal={Letters in Mathematical Physics},
-  volume={18},
-  number={3},
-  pages={201--208},
-  year={1989},
-  publisher={Springer}
-}
-
-@article{bogomolny.1977.dispersion,
-  title={Calculation of the green functions by the coupling constant dispersion relations},
-  author={Bogomolny, Evgeny B},
-  journal={Physics Letters B},
-  volume={67},
-  number={2},
-  pages={193--194},
-  year={1977},
-  publisher={Elsevier}
-}
-
-@article{gunther.1993.transfer-matrix,
-  title={Numerical transfer-matrix study of metastability in the d= 2 Ising model},
-  author={G{\"u}nther, Christoph CA and Rikvold, Per Arne and Novotny, MA},
-  journal={Physical review letters},
-  volume={71},
-  number={24},
-  pages={3898},
-  year={1993},
-  publisher={APS}
-}
-
-@inproceedings{widom.1981.interface,
-  title={Structure of the interface between fluid phases},
-  author={Widom, Benjamin},
-  booktitle={Faraday Symposia of the Chemical Society},
-  volume={16},
-  pages={7--21},
-  year={1981},
-  organization={Royal Society of Chemistry}
-}
-
-@article{gunther.1980.goldstone,
-  title={Goldstone modes in vacuum decay and first-order phase transitions},
-  author={G{\"u}nther, NJ and Wallace, DJ and Nicole, DA},
-  journal={Journal of Physics A: Mathematical and General},
-  volume={13},
-  number={5},
-  pages={1755},
-  year={1980},
-  publisher={IOP Publishing}
-}
-
-@article{barouch.1973.susceptibility,
-  title={Zero-field susceptibility of the two-dimensional Ising model near ${T}_c$},
-  author={Barouch, Eytan and McCoy, Barry M and Wu, Tai Tsun},
-  journal={Physical Review Letters},
-  volume={31},
-  number={23},
-  pages={1409},
-  year={1973},
-  publisher={APS}
-}
-
diff --git a/essential_ising.tex b/essential_ising.tex
deleted file mode 100644
index d6a92cc..0000000
--- a/essential_ising.tex
+++ /dev/null
@@ -1,206 +0,0 @@
-%  Ising model abrupt transition.
-%
-%  Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017.
-%  Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved.
-%
-\documentclass[fleqn]{article}
-
-\usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage{amsmath,amssymb,latexsym,concmath,mathtools,xifthen,mfpic}
-
-\mathtoolsset{showonlyrefs=true}
-
-\title{Essential Singularity in the Ising Abrupt Transition}
-\author{Jaron Kent-Dobias}
-
-\date{April 20, 2017}
-
-\begin{document}
-
-\def\[{\begin{equation}}
-\def\]{\end{equation}}
-
-\def\im{\mathop{\mathrm{Im}}\nolimits}
-\def\dd{\mathrm d}
-\def\O{\mathcal O}
-\def\ei{\mathop{\mathrm{Ei}}\nolimits}
-\def\b{\mathrm b}
-
-\newcommand\pd[3][]{
-  \ifthenelse{\isempty{#1}}
-    {\def\tmp{}}
-    {\def\tmp{^#1}}
-  \frac{\partial\tmp#2}{\partial#3\tmp}
-}
-
-\maketitle
-
-\begin{abstract}
-\end{abstract}
-
-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
-\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic}
-\[
-  \Gamma\propto\im F
-\]
-What exactly is meant by `metastable free energy' is important to establish,
-since formally the free energy relies on the existence of an equilibrium
-state. Here one can imagine either analytic continuation of the free energy
-through an 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. 
-
-One can estimate the scaling of the decay rate of the {\sc 2d} Ising model
-using ideas from nucleation theory. In this framework, the metastable state
-decays when a sufficiently large domain in the stable state forms to grow
-stably to fill out the whole system. The free energy of a domain of $N$ spins
-causes a free energy change
-\[
-  \Delta F=\Sigma N^\sigma-MHN
-\]
-where $\Sigma$ is the surface tension and $1-\frac1d\leq\sigma<1$. This is
-maximized by
-\[
-  N_c=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)}
-\]
-which corresponds to a free energy change
-\[
-  \Delta F_c\sim\bigg(\frac\Sigma{(MH)^\sigma}\bigg)^{1/(1-\sigma)}
-\]
-The rate of formation is proportional to the Boltzmann factor,
-\[
-  \Gamma\sim e^{-\beta \Delta
-  F_c}=e^{-\beta(\Sigma/(MH)^\sigma)^{1/(1-\sigma)}}
-\]
-For domains whose boundary is minimal, $\sigma=1-\frac1d$ and this becomes
-\[
-  \Gamma\sim e^{-\beta(\Sigma/(MH)^\sigma)^{d-1}}
-\]
-Since $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$
-\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$
-with $\mathcal S(0)=\O(1)$ and $\mathcal M(0)=\O(1)$,
-\[
-  \Gamma\sim e^{-1/\mathcal G(ht^{-\beta\delta})^{d-1}}
-\]
-with $\mathcal G(X)=\O(X)$. This establishes the form of $\im F$
-besides the prefactor. Results from field theory predict that, for small $H$
-and $1<d<5$, $d\neq 3$,
-\[
-  \im F\simeq\bigg(\frac h{t^\Delta}\bigg)^{-(d-3)d/2}(g^*)^{-d(d-1)/4}
-  \exp\bigg[-B\bigg(\frac h{|t|^\Delta}\bigg)^{-(d-1)}(g^*)^{-(d+1)/2}\bigg]
-\]
-\[
-  \im F\simeq\bigg(\frac
-  h{t^\Delta}\bigg)^{-7/3}(g^*)^{-8/3}\exp\bigg[-B\bigg(\frac
-  h{t^\Delta}\bigg)^{-2}(g^*)^{-2}\bigg]
-\]
-with $\Delta=3-\frac\epsilon2$, $g^*=2\pi^2\frac\epsilon{n+8}$
-\cite{houghton.1980.metastable,gunther.1980.goldstone}. This is consistent
-with our form above. We therefore predict that
-\[
-  \im F=t^{2-\alpha}\mathcal F(ht^{-\beta\delta})^{-(d-3)d/2}e^{-1/\mathcal
-    G(ht^{-\beta\delta})^{d-1}}
-\]
-In {\sc 2d} we have
-\[
-  \im F=t^2\mathcal F(ht^{-\Delta})e^{-1/\mathcal G(ht^{-\Delta})}
-\]
-with $\Delta=\beta\delta=\frac{15}8$. In terms of $X=ht^{-\Delta}$, this is
-\[
-  \im F=t^2\mathcal F(X)e^{-1/\mathcal G(X)}\simeq At^2|X|e^{-1/B|X|}
-\]
-
-\cite{langer.1967.condensation}
-
-\[
-  F(X)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(X')}{X'-X}\,\dd X'
-  =\frac{At^2}\pi\int_{-\infty}^0\frac{|X'|e^{-1/B|X'|}}{X'-X}\,\dd
-  X'
-  =-\frac{At^2}\pi\int_0^\infty\frac{X'e^{-1/BX'}}{X'+X}\,\dd
-  X'
-\]
-since $\im F=0$ for $X>0$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$.
-Unfortunately this integral doesn't converge, and it seems we cannot evaluate
-this result at the level of truncation we've chosen. However, 
-
-\[
-  F(H)=At^{2-\alpha}\sum_{n=0}^\infty f_nX^n
-\]
-\[
-  f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(X)}{X^{n+1}}\,\dd X
-  =\frac{(-1)^{n+1}}\pi\int_0^{\infty}\frac{Xe^{-1/BX}}{X^{n+1}}\,\dd X
-  =\frac1\pi(-1)^{n+1}B^{n-1}\Gamma(n-1)
-\]
-for $n>1$. 
-
-\begin{align}
-  \chi
-  &=\pd[2]Fh
-  =t^{-2\Delta}\pd[2]FX
-  =-\frac{2}\pi At^{2-2\Delta}\int_0^\infty\frac{X'e^{-1/BX'}}{(X+X')^3}\,\dd
-  X'\\
-  &=\frac2\pi
-  \frac{ABt^{-\gamma}}{(BX)^3}\big[BX(1-BX)+e^{1/BX}\ei(-1/BX)\big]
-\end{align}
-
-\[
-  \lim_{X\to0}\chi=-\frac4\pi ABt^{-\gamma}
-\]
-
-\[
-  \beta^{-1}\chi=C_{0\pm}|t|^{-7/4}+C_{1\pm}|t|^{-3/4}+\O(1)
-\]
-$C_{0-}=0.025\,536\,971\,9$ $C_{1-}=-0.001\,989\,410\,7$
-\cite{barouch.1973.susceptibility}
-
-CORRECTIONS TO SCALING, $u_t$ and $u_h$ instead of $t$ and $h$.
-
-\begin{align}
-  u_h
-  &=h[1+c_ht+dht^2+e_hh^2+f_ht^3+\O(t^4,th^2)]\\
-  u_t
-  &=t+b_th^2+c_t^2+d_t^3+e_tth^2+f_tt^4+\O(t^5,t^2h^2,h^4)
-\end{align}
-\begin{align}
-  c_h=\frac{\beta_c}{\sqrt2}
-  &&
-  d_h=\frac{23\beta_c^2}{16}
-  &&
-  f_h=\frac{191\beta_c^3}{48\sqrt2}\\
-  c_t=\frac{\beta_c}{\sqrt2}
-  &&
-  d_t=\frac{7\beta_c^2}6
-  &&
-  f_t=\frac{17\beta_c^3}{6\sqrt2}\\
-  e_t=b_t\beta_c\sqrt2
-  &&
-  b_t=-\frac{E_0\pi}{16\beta_c^2}
-\end{align}
-$E_0=0.040\,325\,5003$ $e_h=-0.007\,27(15)$
-\[
-  F(t,h)-F(t,0)=\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2n}(t)h^{2n}
-\]
-\[
-  \chi(t,h)=\pd[2]Fh=\chi_2(t)+\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2(n+1)}h^{2n}
-\]
-
-\begin{align}
-  \chi
-  &=\pd[2]Fh
-  =\pd[2]{F_\b}h
-  +\frac d{y_t}\bigg(\frac d{y_t}-1\bigg)|u_t|^{d/y_t-2}\bigg(\pd{u_t}h\bigg)^2
-\end{align}
-
-\input{figs/scaling_func.tex}
-
-\bibliographystyle{plain}
-\bibliography{essential_ising}
-
-\end{document}
-
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