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%
% research_midsummer.tex - Research Presentation for the Topaz lab.
%
% Created by Jaron Kent-Dobias on Tue Mar 20 20:57:40 PDT 2012.
% Copyright (c) 2012 pants productions. All rights reserved.
%
\documentclass[fleqn]{beamer}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,latexsym,graphicx}
\usepackage{concmath}
%\usepackage{bera}
%\usepackage{merriweather}
\usepackage[T1]{fontenc}
\usecolortheme{beaver}
\usefonttheme{serif}
\setbeamertemplate{navigation symbols}{}
\title{Universal scaling and the essential singularity at the abrupt Ising transition}
\author{ Jaron~Kent-Dobias\inst{1} \and James~Sethna\inst{1}}
\institute{\inst{1}Cornell University}
\date{16 March 2016}
\begin{document}
\def\dd{\mathrm d}
\def\im{\mathop{\mathrm{Im}}}
\def\ei{\mathop{\mathrm{Ei}}}
\def\crit{\mathrm{crit}}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}
\frametitle{Renormalization and Universality}
\begin{columns}
\begin{column}{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/fig2}
\tiny
From \emph{Scaling and Renormalization in Statistical Physics} by John
Cardy
\end{column}
\begin{column}{0.6\textwidth}
Renormalization is an analytic scaling transformation that acts on
system space.
\vspace{1em}\pause
Fixed points are scale invariant, corresponding to systems representing
idealized phases or critical behavior.
\vspace{1em}\pause
Nonanalytic behavior---like power laws and logarithms---are preserved
under {\sc rg} and shared by \emph{any} system that flows to the same
point.
\vspace{1em}\pause
Not all nonanalytic behavior is singular!
\end{column}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{The Metastable Ising Model}
\begin{columns}
\begin{column}{0.3\textwidth}
\includegraphics[width=\textwidth]{figs/fig3}
\vspace{1em}
\includegraphics[width=\textwidth]{figs/fig4}
\end{column}
\begin{column}{0.7\textwidth}
Consider an Ising-class model brought into a metastable state.
\vspace{1em} \pause
A domain of $N$ spins entering the stable phase causes a free energy
change
\[
\Delta F=\Sigma N^\sigma-MHN
\]
\pause
The metastable phase is stable to domains smaller than
\[
N_\crit=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)}
\]
but those larger will grow to occupy the entire system.
\end{column}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{The Metastable Ising Model}
The formation of a critical domain has energy cost
\[
\Delta F_\crit\sim MH\bigg(\frac{MH}{\Sigma}\bigg)^{-1/(1-\sigma)}
\]
\pause
The decay rate of the metastable is proportional to the probability of
forming a critical domain $e^{-\beta\Delta F_\crit}$.
\pause \vspace{1em}
Decay of the equilibrium state implies existence of an imaginary part in the
free energy,
\[
\im F\sim e^{-\beta\Delta F_\crit}
\]
\end{frame}
\begin{frame}
\frametitle{The Metastable Ising Model}
Near the Ising critical point, $\sigma=1-\frac1d$ and
\begin{align*}
M=t^\beta\mathcal M(h/{t^{\beta\delta}})
&&
\Sigma=t^\mu\mathcal S(h/{t^{\beta\delta}})
\end{align*}
with $\mathcal M(0)$ and $\mathcal S(0)$ nonzero and finite.
\pause \vspace{1em}
Therefore,
\[
\Delta F_\crit\sim\Sigma\bigg(\frac{MH}{\Sigma}\bigg)^{-(d-1)}
=X^{-(d-1)}\mathcal F(X)
\]
for $X=h/t^{\beta\delta}$, and
\[
\im F=\mathcal I(X)e^{-\beta/X^{(d-1)}}
\]
\end{frame}
\begin{frame}
\frametitle{The Essential Singularity}
\begin{center}
\includegraphics[width=.7\textwidth]{figs/fig1}
\end{center}
Imaginary free energy is nonanalytic at $H=0$.
\pause\vspace{1em}
This and its implications are therefore a universal feature of the Ising class.
\end{frame}
\begin{frame}
\frametitle{The Essential Singularity}
Analytic properties of the partition function imply that
\[
F(X)=\frac1\pi\int_{-\infty}^0\frac{\im F(X')}{X'-X}\;\dd X'
\]
\pause
Only predictive for high moments of $F$, or
\[
f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(X')}{X^{\prime n+1}}\;\dd X'
\]
for $F=\sum f_nX^n$.
\end{frame}
\begin{frame}
\frametitle{The Essential Singularity}
Results from field theory indicate that $\mathcal I(X)\propto X$ for $d=2$
and small $X$, so that
\[
\im F=AXe^{-\beta/X^{(d-1)}}
\]
\[
f_n=\frac{A\Gamma(n-1)}{\pi(-B)^{n-1}}
\]
Not a convergent series---the real part of $F$ for $H>0$ is also
nonanalytic.
\end{frame}
\begin{frame}
\frametitle{The Essential Singularity}
In two dimensions, the Cauchy integral does not converge, normalize with
\[
F(X\,|\,\lambda)=\frac1\pi\int_{-\infty}^0\frac{\im
F(X')}{X'-X}\frac1{1+(\lambda X')^2}\;\dd X'
\]
\[
\begin{aligned}
\frac{A}\pi
\frac{Xe^{B/X}\ei(-B/X)+\frac1\lambda\im(e^{-i\lambda B}(i+\lambda
X)(\pi+i\ei(i\lambda B)))}{1+(\lambda X)^2}
\end{aligned}
\]
\[
\chi=t^{-\gamma}\mathcal X(h/t^{\beta\delta})
\]
\[
\mathcal X(X)=\frac A{\pi X^3}\big[(B-X)X+B^2e^{B/X}\ei(-B/X)\big]
\]
\end{frame}
\begin{frame}
\includegraphics[width=0.8\textwidth]{figs/fig5}
$A=-0.0939(8)$, $B=5.45(6)$.
\end{frame}
\end{document}
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