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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 \and James~Sethna}
\institute{Cornell University}
\date{16 March 2017}

\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{Outline}
  \begin{itemize}
    \item Renormalization and the Ising model
      \pause
    \item Metastability and complex free energy
      \pause
    \item Analytic constraints on the stable free energy
      \pause
    \item Closed-form results for {\sc 2d} Ising 
  \end{itemize}
  \vfill
\end{frame}

\begin{frame}
  \frametitle{Renormalization and the Ising Model}
  \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}
      {\sc Rg} methods typically used to study critical points.
      \vspace{1em}\pause\\
      {\sc Rg} analytically maps system space onto itself.
      \vspace{1em}\pause\\
      Nonanalytic behavior is preserved by {\sc rg}.
      \vspace{1em}\pause\\
      Critical points characterized by common nonanalyticities.
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Renormalization and the Ising Model}
  \begin{columns}
    \begin{column}{0.4\textwidth}
      \includegraphics{figs/fig3}
    \end{column}
    \begin{column}{0.6\textwidth}
      Ising critical point has power laws, logarithms in thermodynamic
      variables.
      \vspace{1em}\pause\\
      Connected to line of abrupt transitions.
      \vspace{1em}\pause\\
      We've identified predictive nonanalytic behavior along the abrupt transition line.
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Metastability \& Complex Free Energy}
  \begin{columns}
    \begin{column}{0.4\textwidth}
      \includegraphics{figs/fig11}
    \end{column}
    \begin{column}{0.6\textwidth}
      \includegraphics{figs/fig12}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Metastability \& Complex Free Energy}
  \begin{columns}
    \begin{column}{0.4\textwidth}
      \includegraphics{figs/fig13}
    \end{column}
    \begin{column}{0.6\textwidth}
      \includegraphics{figs/fig14}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Metastability \& Complex Free Energy}
  \begin{columns}
    \begin{column}{0.4\textwidth}
      \includegraphics{figs/fig4}
    \end{column}
    \begin{column}{0.6\textwidth}
      Thermodynamics can be continued into metastable phase.
      \vspace{1em}\pause\\
      Decay rate related to imaginary free energy,
      $\Gamma\propto\frac1{kT}\im F$ (Langer 1969).
      \vspace{1em}\pause\\
      Ising metastable decay somewhat well studied (G\"unther 1980, Houghton
      1980)
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Metastability \& Complex Free Energy}
  Decay of metastable phase occurs when domain of critical size forms.
  \vspace{1em}\pause\\
  Domain of $N$ spins entering the stable phase causes a free energy
  change
  \[
    \Delta F=\Sigma N^\sigma-MHN
  \]
  with $1-\frac1d\leq\sigma<1$.
  \pause\vspace{1em}\\
  Metastable phase is stable to domains smaller than
  \[
    N_\crit=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)}
  \]
  but larger will grow to occupy the entire system, decay to stable phase.
\end{frame}

\begin{frame}
  \frametitle{Metastability \& Complex Free Energy}
  Formation of critical domain has energy cost
  \[
    \Delta F_\crit=\Delta F\Big|_{N=N_\crit}\sim
    \bigg(\frac{\Sigma}{(MH)^\sigma}\bigg)^{1/(1-\sigma)}
  \]
  \pause
  Probability of such a domain forming is
  \[
    P_\crit\sim e^{-\beta\Delta F_\crit}
  \]
  \pause
  Imaginary free energy is therefore
  \[
    \im F\sim\Gamma\sim P_\crit\sim e^{-\beta\Delta F_\crit}
    =e^{-\beta(\Sigma/(MH)^\sigma)^{1/(1-\sigma)}}
  \]

\end{frame}

\begin{frame}
  \frametitle{Metastability \& Complex Free Energy}
  \begin{columns}
    \begin{column}{0.4\textwidth}
      \includegraphics{figs/fig1}
    \end{column}
    \begin{column}{0.6\textwidth}
      $\im F$ has an essential singularity of the form
      $e^{-1/H^{\sigma/(1-\sigma)}}$.
      \vspace{1em}\pause\\
      Near critical point, $\sigma=1-\frac1d$, and
      \[\im F\sim e^{-1/H^{d-1}}\]
      \pause
      Nonanalytic behavior is universal!
      \vspace{1em}\pause\\
      Can directly observe by measuring metastable decay rate, but what else?
      \vspace{1em}\pause\\
      Thought to be unobservable (Fisher 1980).
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Analytic Constraints on the Stable Free Energy}
  Analytic properties of $F(H)$ give Cauchy-style constraint
  \[
    F(H)=\frac1\pi\int_{-\infty}^0\frac{\im F(H')}{H'-H}\;\dd H'
  \]
  \pause
  Only know $\im F(H)$ for $|H|\ll 1$, so constraint only
  predictive for higher moments, for $F(H)=\sum_nf_nH^n$
  \[
    f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(H')}{H^{\prime n+1}}\;\dd H'
  \]
  \pause
  Approach well-established in statistical physics and field theory
  (Parisi 1977, Bogomolny 1977, others)
\end{frame}

\begin{frame}
  \frametitle{Closed-form results for {\sc 2d} Ising}
  Near the critical point with $X=h/t^{\beta\delta}$ and $h=H/T$,
  \begin{align}
    M=t^\beta\mathcal M(X)
    &&
    \Sigma=t^\mu\mathcal S(X)
    \notag
  \end{align}
  \pause
  Our analysis with some considerations of field theory (Houghton 1980) yields
  \[
    \im F=t^{2-\alpha}\big[AX+\mathcal O(X^2)\big]e^{-[B+\mathcal
    O(X)]/X}
  \]
  \pause
  Yields moments for $n\geq2$ which agree with others
  (Baker 1980),
  \[
    f_n=At^{2-\alpha}\frac{\Gamma(n-1)}{\pi(-B)^{n-1}}
  \]
  \pause
  Cauchy-style integral diverges for truncation, $f_0=f_1=\pm\infty$.
\end{frame}

\begin{frame}
  \frametitle{Closed-form results for {\sc 2d} Ising}
  We can use the constraint to compute the susceptibility
  \[
    \chi=\frac{\partial^2F}{\partial h^2}
  \]
  \pause
  Yields a scaling form
  \begin{align}
    \chi=t^{-\gamma}\Xi(h/t^{\beta\delta})
    &&
    \Xi(X)=-\frac1\pi\frac AX\Bigg[1-\frac BX-\bigg(\frac
      BX\bigg)^2e^{B/X}\ei\bigg(-\frac BX\bigg)
    \Bigg]
      \notag
  \end{align}
  \pause
  Prefactor fixed by known results for zero-field susceptibility
  \[
    A=-\frac{B\pi C_{0_-}}{2T_c}
  \]
  with $C_{0_-}=0.0255369719$ (Barouch 1973).
\end{frame}

\begin{frame}
  \frametitle{Closed-form results for {\sc 2d} Ising}

  \includegraphics{figs/fig6}
\end{frame}

\begin{frame}
  \frametitle{Closed-form results for {\sc 2d} Ising}
  \includegraphics{figs/fig5}

\end{frame}

\begin{frame}
  \frametitle{Closed-form results for {\sc 2d} Ising}
  \includegraphics{figs/fig20}
\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 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 novel universal properties to incorporate.
\end{frame}

\begin{frame}
  \frametitle{Questions?}
  \small
  \begin{align}
    \chi=t^{-\gamma}\Xi(h/t^{\beta\delta})
    &&
    \Xi(X)=-\frac1\pi\frac AX\Bigg[1-\frac BX-\bigg(\frac
      BX\bigg)^2e^{B/X}\ei\bigg(-\frac BX\bigg)
    \Bigg]
      \notag
  \end{align}
  \centering
  \includegraphics[width=0.7\textwidth]{figs/fig20}
\end{frame}

\end{document}