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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 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}

