From 9fb0c1e6012c6fd04902ce71f87d6124bdb542d5 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 4 Sep 2019 21:59:58 -0400
Subject: final changes

---
 aps_mm_2017.tex | 188 ++++++++++++++++++++------------------------------------
 1 file changed, 67 insertions(+), 121 deletions(-)

(limited to 'aps_mm_2017.tex')

diff --git a/aps_mm_2017.tex b/aps_mm_2017.tex
index 9149ab9..89f4ef8 100644
--- a/aps_mm_2017.tex
+++ b/aps_mm_2017.tex
@@ -43,7 +43,7 @@
       \pause
     \item Analytic constraints on the stable free energy
       \pause
-    \item Closed-form results for the {\sc 2d} Ising susceptibility 
+    \item Closed-form results for {\sc 2d} Ising 
   \end{itemize}
   \vfill
 \end{frame}
@@ -59,11 +59,13 @@
       Cardy
     \end{column}
     \begin{column}{0.6\textwidth}
-      {\sc Rg} analytically maps system space onto itself.
+      {\sc Rg} methods typically used to study critical points.
       \vspace{1em}\pause\\
-      Fixed points correspond to phases, criticality.
+      {\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}
@@ -80,7 +82,7 @@
       \vspace{1em}\pause\\
       Connected to line of abrupt transitions.
       \vspace{1em}\pause\\
-      We've identified nonanalytic behavior along the abrupt transition line.
+      We've identified predictive nonanalytic behavior along the abrupt transition line.
     \end{column}
   \end{columns}
 \end{frame}
@@ -177,162 +179,97 @@
       $e^{-1/H^{\sigma/(1-\sigma)}}$.
       \vspace{1em}\pause\\
       Near critical point, $\sigma=1-\frac1d$, and
-      \[
-        \im F\sim e^{-1/H^{d-1}}
-      \]
+      \[\im F\sim e^{-1/H^{d-1}}\]
       \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}
-\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,
+  Analytic properties of $F(H)$ give Cauchy-style constraint
   \[
-    \Delta F_\crit\sim\Sigma\bigg(\frac{MH}{\Sigma}\bigg)^{-(d-1)}
-    =X^{-(d-1)}\mathcal F(X)
+    F(H)=\frac1\pi\int_{-\infty}^0\frac{\im F(H')}{H'-H}\;\dd H'
   \]
-  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
+  Only know $\im F(H)$ for $|H|\ll 1$, so constraint only
+  predictive for higher moments, for $F(H)=\sum_nf_nH^n$
   \[
-    f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(X')}{X^{\prime n+1}}\;\dd X'
+    f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(H')}{H^{\prime n+1}}\;\dd H'
   \]
-  for $F=\sum f_nX^n$.
+  \pause
+  Approach well-established in statistical physics and field theory
+  (Parisi 1977, Bogomolny 1977, others)
 \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
+  \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
   \[
-    \im F=t^{2-\alpha}\big(AX+\mathcal O(X^2)\big)e^{-\beta/X^{(d-1)}}
+    \im F=t^{2-\alpha}\big[AX+\mathcal O(X^2)\big]e^{-[B+\mathcal
+    O(X)]/X}
   \]
-
   \pause
-
-  The resulting moments for $n>1$ are
+  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
-
-  Not a convergent series---the real part of $F$ for $H>0$ is also
-  nonanalytic!
+  Cauchy-style integral diverges for truncation, $f_0=f_1=\pm\infty$.
 \end{frame}
 
 \begin{frame}
-  \frametitle{The Essential Singularity}
-
-  In two dimensions, the Cauchy integral does not converge, normalize with
-  $\lambda$,
+  \frametitle{Closed-form results for {\sc 2d} Ising}
+  We can use the constraint to compute the susceptibility
   \[
-    F(X\,|\,\lambda)=\frac1\pi\int_{-\infty}^0\frac{\im
-    F(X')}{X'-X}\frac1{1+(\lambda X')^2}\;\dd X'
+    \chi=\frac{\partial^2F}{\partial h^2}
   \]
-
   \pause
-
-  Exact result has form
-  \[
-    \begin{aligned}
-      F(X\,|\,\lambda)&=\frac{A}\pi\frac1{1+(\lambda X)^2}\Big[
-      Xe^{B/X}\ei(-B/X)\\
-      &\qquad+\frac1\lambda\im(e^{-i\lambda B}(i+\lambda
-      X)(\pi+i\ei(i\lambda B)))\Big]
-    \end{aligned}
-  \]
-
+  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
-
-  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
+  Prefactor fixed by known results for zero-field susceptibility
   \[
-    \mathcal X(X)=\frac A{\pi X^3}\big[(B-X)X+B^2e^{B/X}\ei(-B/X)\big]
+    A=-\frac{B\pi C_{0_-}}{2T_c}
   \]
-  
-  \centering
-  \includegraphics[width=0.6\textwidth]{figs/fig9}
+  with $C_{0_-}=0.0255369719$ (Barouch 1973).
 \end{frame}
 
 \begin{frame}
-  \frametitle{The Essential Singularity}
-
-  $A$ is fixed by prior calculations (Barouch 1973)
-
-  Two parameter fit to simulations yields $A=-0.0939(8)$, $B=5.45(6)$, close
-  agreement in limit of small $t$ and $H$!
+  \frametitle{Closed-form results for {\sc 2d} Ising}
 
-  \vspace{1em}
+  \includegraphics{figs/fig6}
+\end{frame}
 
-  \only<1-1>{\includegraphics{figs/fig6}}
-  \only<2-2>{\includegraphics{figs/fig5}}
+\begin{frame}
+  \frametitle{Closed-form results for {\sc 2d} Ising}
+  \includegraphics{figs/fig5}
 
-  \vspace{1em}\pause
 \end{frame}
 
 \begin{frame}
+  \frametitle{Closed-form results for {\sc 2d} Ising}
   \includegraphics{figs/fig20}
 \end{frame}
 
@@ -344,20 +281,29 @@
 
   \vspace{1em} \pause
 
-  Hope to form a parametric scaling variables that include this, correct
+  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 universal properties to incorporate.
+  Remain on the lookout for other novel universal properties to incorporate.
 \end{frame}
 
 \begin{frame}
-  \huge
+  \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
-  {\sl Questions?}
+  \includegraphics[width=0.7\textwidth]{figs/fig20}
 \end{frame}
 
 \end{document}
-- 
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