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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} |