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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{Parametric Ising Models}
  \begin{columns}
    \begin{column}{0.4\textwidth}
      \includegraphics[width=\textwidth]{figs/fig7}

      \vspace{1em}

      \includegraphics[width=\textwidth,height=0.5\textwidth]{susceptibility.jpg}

      \tiny\texttt{a plot of susceptibility with precision parametric fit and it
        isn't
        very good at the abrupt transition}
    \end{column}
    \begin{column}{0.6\textwidth}
      Scaling forms of Ising variables that do well globally.

      \pause\vspace{1em}

      Incorporate the critical point in a natural way:
      \begin{itemize}
        \item singular scaling with the ``radial coordinate''
        \item analytic scaling with the ``angular coordinate''
      \end{itemize}

      \vspace{1em} \pause

      Typically do a very poor job near the abrupt transition at $H=0$.
    \end{column}
  \end{columns}
\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 are power laws!
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{The Metastable Ising Model}

  \begin{columns}
    \begin{column}{0.3\textwidth}
      \only<1-1>{\includegraphics[width=\textwidth]{figs/fig3}}
      \only<2->{\includegraphics[width=\textwidth]{figs/fig4}}

      \vspace{1em}

      \includegraphics[width=\textwidth]{figs/fig8}

    \end{column}
    \begin{column}{0.7\textwidth}
      Consider an Ising-class model brought into a metastable state.

      \vspace{1em} \pause\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 
  \[
    \im F=t^{2-\alpha}\mathcal I(X)e^{-\beta/X^{(d-1)}}
  \]
\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 $\mathcal I(X)\propto X+\mathcal
  O(X^2)$ for $d=2$, so that
  \[
    \im F=t^{2-\alpha}\big(AX+\mathcal O(X^2)\big)e^{-\beta/X^{(d-1)}}
  \]

  \pause

  The resulting moments for $n>1$ are
  \[
    f_n=At^{2-\alpha}\frac{\Gamma(n-1)}{\pi(-B)^{n-1}}
  \]

  \pause

  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
  $\lambda$,
  \[
    F(X\,|\,\lambda)=\frac1\pi\int_{-\infty}^0\frac{\im
    F(X')}{X'-X}\frac1{1+(\lambda X')^2}\;\dd X'
  \]

  \pause

  Exact result has form
  \[
    \begin{aligned}
      F(X\,|\,\lambda)&=\frac{A}\pi\frac1{1+(\lambda X)^2}\Big[
      Xe^{B/X}\ei(-B/X)\\
      &\qquad+\frac1\lambda\im(e^{-i\lambda B}(i+\lambda
      X)(\pi+i\ei(i\lambda B)))\Big]
    \end{aligned}
  \]

  \pause

  The Cauchy integral is only predictive for high moments.

\end{frame}
\begin{frame}
  \frametitle{The Essential Singularity}

  What about the susceptibility $\chi=\frac{\partial^2\!F}{\partial h^2}$?

  \pause \vspace{1em}

  Has a well-defined limit as $\lambda\to0$, simple functional form:
  \[
    \chi=t^{-\gamma}\mathcal X(h/t^{\beta\delta})
  \]
  where the scaling function is
  \[
    \mathcal X(X)=\frac A{\pi X^3}\big[(B-X)X+B^2e^{B/X}\ei(-B/X)\big]
  \]
  
  \centering
  \includegraphics[width=0.6\textwidth]{figs/fig9}
\end{frame}

\begin{frame}
  \frametitle{The Essential Singularity}

  Two parameter fit to simulations yields $A=-0.0939(8)$, $B=5.45(6)$, close
  agreement in limit of small $t$ and $H$!

  \vspace{1em}

  \only<1-1>{\includegraphics[width=\textwidth]{figs/fig6}}
  \only<2-2>{\includegraphics[width=\textwidth]{figs/fig5}}

  \vspace{1em}\pause
\end{frame}

\begin{frame}
  \frametitle{What's Next}

  We have an explicit form for a new component of the universal scaling forms
  near the Ising abrupt transition.

  \vspace{1em} \pause

  Hope to form a parametric scaling variables that include this, correct
  leading
  analytic corrections to scaling, and (maybe?) extend smoothly through the
  metastable region.

  \vspace{1em} \pause

  Remain on the lookout for other universal properties to incorporate.
\end{frame}

\begin{frame}
  \huge
  \centering
  {\sl Questions?}
\end{frame}

\end{document}