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-rw-r--r--aps_mm_2017.aux38
-rw-r--r--aps_mm_2017.log282
-rw-r--r--aps_mm_2017.nav38
-rw-r--r--aps_mm_2017.pdfbin88369 -> 232269 bytes
-rw-r--r--aps_mm_2017.synctex.gzbin12244 -> 19629 bytes
-rw-r--r--aps_mm_2017.tex240
-rw-r--r--figs/fig1.pdfbin7095 -> 8212 bytes
-rw-r--r--figs/fig2.pngbin0 -> 86041 bytes
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-rw-r--r--figs/fig4.pdfbin0 -> 16885 bytes
-rw-r--r--figs/fig5.pdfbin0 -> 38153 bytes
-rw-r--r--figs/fig6.pdfbin0 -> 33985 bytes
12 files changed, 417 insertions, 181 deletions
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@@ -16,8 +16,9 @@
\usecolortheme{beaver}
\usefonttheme{serif}
+\setbeamertemplate{navigation symbols}{}
-\title{Universal scaling and the essential singularity at the Ising first-order transition}
+\title{Universal scaling and the essential singularity at the abrupt Ising transition}
\author{ Jaron~Kent-Dobias\inst{1} \and James~Sethna\inst{1}}
\institute{\inst{1}Cornell University}
\date{16 March 2016}
@@ -26,86 +27,203 @@
\def\dd{\mathrm d}
\def\im{\mathop{\mathrm{Im}}}
+\def\ei{\mathop{\mathrm{Ei}}}
+\def\crit{\mathrm{crit}}
\begin{frame}
- \titlepage
+ \titlepage
\end{frame}
\begin{frame}
- \frametitle{Renormalization and free energy}
- Rescale a system by a factor $b$, with couplings $K\to K'$.
- From John Cardy's \emph{Scaling and Renormalization in Statistical Physics},
- free energy per site $f$
- \[
- f(\{K\})=g(\{K\})+b^{-d}f(\{K'\})
- \]
- \begin{quote}
- However, if we are interested in extracting only the \emph{singular}
- behavior of $f$, \dots we may obtain a \emph{homogeneous} transformation law for the
- \emph{singular part} of the free energy $f_s$
- \[
- f_s(\{K\})=b^{-d}f_s(\{K'\})
- \]
- \end{quote}
- Defense: $g(\{K\})$ is an analytic function of $\{K\}$, while the singular
- part is nonanalytic
+ \frametitle{Renormalization and Universality}
+
+ \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}
+ Renormalization is an analytic scaling transformation that acts on
+ system space.
+
+ \vspace{1em}\pause
+
+ Fixed points are scale invariant, corresponding to systems representing
+ idealized phases or critical behavior.
+
+ \vspace{1em}\pause
+
+ Nonanalytic behavior---like power laws and logarithms---are preserved
+ under {\sc rg} and shared by \emph{any} system that flows to the same
+ point.
+
+ \vspace{1em}\pause
+
+ Not all nonanalytic behavior is singular!
+ \end{column}
+ \end{columns}
\end{frame}
\begin{frame}
- Follow thermodynamic functions onto metastable branch.
+ \frametitle{The Metastable Ising Model}
+
+ \begin{columns}
+ \begin{column}{0.3\textwidth}
+ \includegraphics[width=\textwidth]{figs/fig3}
+
+ \vspace{1em}
+
+ \includegraphics[width=\textwidth]{figs/fig4}
+ \end{column}
+ \begin{column}{0.7\textwidth}
+ Consider an Ising-class model brought into a metastable state.
+
+ \vspace{1em} \pause
+
+ A domain of $N$ spins entering the stable phase causes a free energy
+ change
+ \[
+ \Delta F=\Sigma N^\sigma-MHN
+ \]
+
+ \pause
+
+ The metastable phase is stable to domains smaller than
+ \[
+ N_\crit=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)}
+ \]
+ but those larger will grow to occupy the entire system.
+ \end{column}
+ \end{columns}
\end{frame}
\begin{frame}
- \[\Delta f\sim\Sigma\gamma(N)-HMN\]
- Near the critical point, $\gamma(N)\sim N^{\frac{d-1}d}$
- \[
- M=|t|^\beta\mathcal M(h/{t^{\beta\delta}})
- \]
- \[
- N_{crit}\sim\bigg(\frac{\Sigma}{HM}\Big(1-\tfrac1d\Big)\bigg)^d
- \]
- \[
- \Delta f_{crit}\sim\Sigma\bigg(\frac{\Sigma}{HM}\bigg)^{d-1}
- \sim X^{-(d-1)}\frac{\mathcal
- S^d(X)}{\mathcal M^{d-1}(X)}
- \]
- $X=h/t^{\beta\delta}$
- The probability that such a domain forms and the metastable state decays is given by the Boltzmann factor,
- so that
- $\Sigma\sim|t|^\mu$, $\mu=-\nu+\gamma+2\beta$
- \[
- \im f\sim e^{-\beta\, \Delta f_{crit}}
- \sim\mathcal F(X)e^{-1/X^{d-1}}
- \]
+ \frametitle{The Metastable Ising Model}
+
+ The formation of a critical domain has energy cost
+ \[
+ \Delta F_\crit\sim MH\bigg(\frac{MH}{\Sigma}\bigg)^{-1/(1-\sigma)}
+ \]
+
+ \pause
+
+ The decay rate of the metastable is proportional to the probability of
+ forming a critical domain $e^{-\beta\Delta F_\crit}$.
+
+ \pause \vspace{1em}
+
+ Decay of the equilibrium state implies existence of an imaginary part in the
+ free energy,
+ \[
+ \im F\sim e^{-\beta\Delta F_\crit}
+ \]
+
+\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
+ \[
+ \im F=\mathcal I(X)e^{-\beta/X^{(d-1)}}
+ \]
\end{frame}
\begin{frame}
- $e^{-1/x}$ is nonanalytic at $x=0$: all derivatives vanish, means that free
- energy (which has no imaginary part in stable phase) is smooth
+ \frametitle{The Essential Singularity}
- \centering
- \includegraphics[width=0.7\textwidth]{figs/fig1}
+ \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}
- Analyticity of $F$ means that the imaginary
-
- \[
- f(h)=\sum_n A_nh^N
- \]
- \[
- A_n=(-B)^{1-n}\Gamma(n-1)
- \]
-
- \[
- f(h)=\frac1\pi\int_{h'<0}\frac{\dd h'\,\im f(h')}{h'-h}
- \]
+ \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}
- Field theorists (Lubensky, blah blah blah)
- \[
- \mathcal
- \]
+ \frametitle{The Essential Singularity}
+
+ Results from field theory indicate that $\mathcal I(X)\propto X$ for $d=2$
+ and small $X$, so that
+ \[
+ \im F=AXe^{-\beta/X^{(d-1)}}
+ \]
+
+ \[
+ f_n=\frac{A\Gamma(n-1)}{\pi(-B)^{n-1}}
+ \]
+
+ 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
+ \[
+ F(X\,|\,\lambda)=\frac1\pi\int_{-\infty}^0\frac{\im
+ F(X')}{X'-X}\frac1{1+(\lambda X')^2}\;\dd X'
+ \]
+
+ \[
+ \begin{aligned}
+ \frac{A}\pi
+ \frac{Xe^{B/X}\ei(-B/X)+\frac1\lambda\im(e^{-i\lambda B}(i+\lambda
+ X)(\pi+i\ei(i\lambda B)))}{1+(\lambda X)^2}
+ \end{aligned}
+ \]
+
+ \[
+ \chi=t^{-\gamma}\mathcal X(h/t^{\beta\delta})
+ \]
+ \[
+ \mathcal X(X)=\frac A{\pi X^3}\big[(B-X)X+B^2e^{B/X}\ei(-B/X)\big]
+ \]
+\end{frame}
+
+\begin{frame}
+ \includegraphics[width=0.8\textwidth]{figs/fig5}
+
+ $A=-0.0939(8)$, $B=5.45(6)$.
\end{frame}
\end{document}
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