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| -rw-r--r-- | aps_mm_2017.tex | 188 | ||||
| -rw-r--r-- | figs/fig20.pdf | bin | 67080 -> 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 Binary files differindex 9810e1e..a0ca1a2 100644 --- a/figs/fig20.pdf +++ b/figs/fig20.pdf |