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%  research_midsummer.tex - Research Presentation for the Topaz lab.
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%  Created by Jaron Kent-Dobias on Tue Mar 20 20:57:40 PDT 2012.
%  Copyright (c) 2012 pants productions. All rights reserved.
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\title{Universal scaling and the essential singularity at the Ising first-order 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}}}

\begin{frame}
	\titlepage
\end{frame}

\begin{frame}
	\frametitle{Renormalization and free energy}
	Rescale a system by a factor $b$, with couplings $K\to K'$.
	From John Cardy's \emph{Scaling and Renormalization in Statistical Physics},
	free energy per site $f$
	\[
		f(\{K\})=g(\{K\})+b^{-d}f(\{K'\})
	\]
	\begin{quote}
		However, if we are interested in extracting only the \emph{singular}
		behavior of $f$, \dots we may obtain a \emph{homogeneous} transformation law for the
		\emph{singular part} of the free energy $f_s$
		\[
			f_s(\{K\})=b^{-d}f_s(\{K'\})
		\]
	\end{quote}
	Defense: $g(\{K\})$ is an analytic function of $\{K\}$, while the singular
	part is nonanalytic
\end{frame}

\begin{frame}
	Follow thermodynamic functions onto metastable branch.
\end{frame}

\begin{frame}
	\[\Delta f\sim\Sigma\gamma(N)-HMN\]
	Near the critical point, $\gamma(N)\sim N^{\frac{d-1}d}$ 
	\[
		M=|t|^\beta\mathcal M(h/{t^{\beta\delta}})
	\]
	\[
		N_{crit}\sim\bigg(\frac{\Sigma}{HM}\Big(1-\tfrac1d\Big)\bigg)^d
	\]
	\[
		\Delta f_{crit}\sim\Sigma\bigg(\frac{\Sigma}{HM}\bigg)^{d-1}
		\sim X^{-(d-1)}\frac{\mathcal
		S^d(X)}{\mathcal M^{d-1}(X)}
	\]
	$X=h/t^{\beta\delta}$
	The probability that such a domain forms and the metastable state decays is given by the Boltzmann factor,
	so that
	$\Sigma\sim|t|^\mu$, $\mu=-\nu+\gamma+2\beta$
	\[
		\im f\sim e^{-\beta\, \Delta f_{crit}}
		\sim\mathcal F(X)e^{-1/X^{d-1}}
	\]
\end{frame}

\begin{frame}
	$e^{-1/x}$ is nonanalytic at $x=0$: all derivatives vanish, means that free
	energy (which has no imaginary part in stable phase) is smooth

	\centering
	\includegraphics[width=0.7\textwidth]{figs/fig1}
\end{frame}
\begin{frame}
	Analyticity of $F$ means that the imaginary

	\[
		f(h)=\sum_n A_nh^N
	\]
	\[
		A_n=(-B)^{1-n}\Gamma(n-1)
	\]

	\[
		f(h)=\frac1\pi\int_{h'<0}\frac{\dd h'\,\im f(h')}{h'-h}
	\]
\end{frame}

\begin{frame}
	Field theorists (Lubensky, blah blah blah)
	\[
		\mathcal
	\]
\end{frame}

\end{document}

