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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.
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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 the {\sc 2d} Ising susceptibility 
  \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} analytically maps system space onto itself.
      \vspace{1em}\pause\\
      Fixed points correspond to phases, criticality.
      \vspace{1em}\pause\\
      Nonanalytic behavior is preserved by {\sc rg}.
    \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 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?
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Analytic Constraints on the Stable Free Energy}
\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}

