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-rw-r--r--aps_mm_2017.aux52
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-rw-r--r--aps_mm_2017.nav52
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diff --git a/aps_mm_2017.pdf b/aps_mm_2017.pdf
index fd007e7..4940523 100644
--- a/aps_mm_2017.pdf
+++ b/aps_mm_2017.pdf
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diff --git a/aps_mm_2017.synctex.gz b/aps_mm_2017.synctex.gz
index d487693..b1b88d8 100644
--- a/aps_mm_2017.synctex.gz
+++ b/aps_mm_2017.synctex.gz
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diff --git a/aps_mm_2017.tex b/aps_mm_2017.tex
index d7b0a7f..e85f277 100644
--- a/aps_mm_2017.tex
+++ b/aps_mm_2017.tex
@@ -35,6 +35,38 @@
\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}
@@ -63,7 +95,7 @@
\vspace{1em}\pause
- Not all nonanalytic behavior is singular!
+ Not all nonanalytic behavior are power laws!
\end{column}
\end{columns}
\end{frame}
@@ -73,16 +105,18 @@
\begin{columns}
\begin{column}{0.3\textwidth}
- \includegraphics[width=\textwidth]{figs/fig3}
+ \only<1-1>{\includegraphics[width=\textwidth]{figs/fig3}}
+ \only<2->{\includegraphics[width=\textwidth]{figs/fig4}}
\vspace{1em}
- \includegraphics[width=\textwidth]{figs/fig4}
+ \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
+ \vspace{1em} \pause\pause
A domain of $N$ spins entering the stable phase causes a free energy
change
@@ -141,9 +175,9 @@
\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
+ for $X=h/t^{\beta\delta}$, and
\[
- \im F=\mathcal I(X)e^{-\beta/X^{(d-1)}}
+ \im F=t^{2-\alpha}\mathcal I(X)e^{-\beta/X^{(d-1)}}
\]
\end{frame}
@@ -181,49 +215,108 @@
\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
+ Results from field theory indicate that $\mathcal I(X)\propto X+\mathcal
+ O(X^2)$ for $d=2$, so that
\[
- \im F=AXe^{-\beta/X^{(d-1)}}
+ \im F=t^{2-\alpha}\big(AX+\mathcal O(X^2)\big)e^{-\beta/X^{(d-1)}}
\]
+ \pause
+
+ The resulting moments for $n>1$ are
\[
- f_n=\frac{A\Gamma(n-1)}{\pi(-B)^{n-1}}
+ f_n=At^{2-\alpha}\frac{\Gamma(n-1)}{\pi(-B)^{n-1}}
\]
+ \pause
+
Not a convergent series---the real part of $F$ for $H>0$ is also
- nonanalytic.
+ 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
\[
\begin{aligned}
- \frac{A}\pi
- \frac{Xe^{B/X}\ei(-B/X)+\frac1\lambda\im(e^{-i\lambda B}(i+\lambda
- X)(\pi+i\ei(i\lambda B)))}{1+(\lambda X)^2}
+ F(X\,|\,\lambda)&=\frac{A}\pi\frac1{1+(\lambda X)^2}\Big[
+ Xe^{B/X}\ei(-B/X)\\
+ &\qquad+\frac1\lambda\im(e^{-i\lambda B}(i+\lambda
+ X)(\pi+i\ei(i\lambda B)))\Big]
\end{aligned}
\]
+ \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}
- \includegraphics[width=0.8\textwidth]{figs/fig5}
+ \frametitle{What's Next}
+
+ We have an explicit form for a new component of the universal scaling forms
+ near the Ising abrupt transition.
- $A=-0.0939(8)$, $B=5.45(6)$.
+ \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}
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