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diff --git a/aps_mm_2017.tex b/aps_mm_2017.tex index eda12da..d7b0a7f 100644 --- a/aps_mm_2017.tex +++ b/aps_mm_2017.tex @@ -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} |