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-rw-r--r--aps_mm_2017.tex188
-rw-r--r--figs/fig20.pdfbin67080 -> 86306 bytes
2 files changed, 67 insertions, 121 deletions
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}
diff --git a/figs/fig20.pdf b/figs/fig20.pdf
index 9810e1e..a0ca1a2 100644
--- a/figs/fig20.pdf
+++ b/figs/fig20.pdf
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