march meeting pres changes

9 files changed, 396 insertions, 154 deletions

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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
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
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}
diff --git a/figs/fig7.pdf b/figs/fig7.pdf
new file mode 100644
index 0000000..45bbe58
--- /dev/null
+++ b/figs/fig7.pdf
diff --git a/figs/fig8.pdf b/figs/fig8.pdf
new file mode 100644
index 0000000..8ce799c
--- /dev/null
+++ b/figs/fig8.pdf
diff --git a/figs/fig9.pdf b/figs/fig9.pdf
new file mode 100644
index 0000000..4d94c8f
--- /dev/null
+++ b/figs/fig9.pdf