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diff --git a/ising_scaling.tex b/ising_scaling.tex index 83351fc..b148eec 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -188,9 +188,13 @@ of statistical mechanics, Langer demonstrated that the decay rate is asymptotically proportional to the imaginary part of the free energy in the metastable phase, with \begin{equation} - \operatorname{Im}F\propto\Gamma\sim e^{-\beta\Delta F_c}\simeq e^{-1/\tilde B|\xi|} + \operatorname{Im}F\propto\Gamma\sim e^{-\beta\Delta F_c}\simeq e^{-1/b|\xi|} \end{equation} -which can be more rigorously related in the context of quantum field theory [ref?]. +which can be more rigorously related in the context of quantum field theory +[ref?]. The constant $b=2M_0/\pi\sigma_0^2$ is predicted by known properties, +and for our conventions for $u_t$ and $u_h$, $M_0/\sigma_0^2=\bar +s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant. + \begin{figure} \includegraphics{figs/F_lower_singularities.pdf} @@ -207,12 +211,12 @@ alone. It is therefore suggestive that this should be considered a part of the singular free energy and moreover part of the scaling function that composes it. We will therefore make the ansatz that \begin{equation} \label{eq:essential.singularity} - \operatorname{Im}\mathcal F_-(\xi+i0)=A\Theta(-\xi)\xi e^{-1/\tilde B|\xi|}\left[1+O(\xi)\right] + \operatorname{Im}\mathcal F_-(\xi+i0)=A_0\Theta(-\xi)\xi e^{-1/b|\xi|}\left[1+O(\xi)\right] \end{equation} \cite{Houghton_1980_The} The linear prefactor can be found through a more careful accounting of the entropy of long-wavelength fluctuations in the droplet surface -\cite{Gunther_1980_Goldstone}. +\cite{Gunther_1980_Goldstone}. In conformal field theory, the prefactor is also known to be $A_0=\bar s/2\pi$. \subsection{Yang--Lee edge singularity} @@ -254,7 +258,7 @@ $\xi_\mathrm{YL}$ \cite{Cardy_1985_Conformal, Fonseca_2003_Ising}. This creates a branch cut stemming from the critical point along the imaginary-$\xi$ axis with a growing imaginary part \begin{equation} - \operatorname{Im}\mathcal F_+(i\xi\pm0)=\pm A\frac12\Theta(\xi^2-\xi_\mathrm{YL}^2)[(\xi/\xi_\mathrm{YL})^2-1]^{1+\sigma}[1+O[(\xi-\xi_\mathrm{YL})^2]] + \operatorname{Im}\mathcal F_+(i\xi\pm0)=\pm A_\mathrm{YL}\frac12\Theta(\xi^2-\xi_\mathrm{YL}^2)[(\xi/\xi_\mathrm{YL})^2-1]^{1+\sigma}[1+O[(\xi-\xi_\mathrm{YL})^2]] \end{equation} This results in analytic structure for $\mathcal F_+$ shown in Fig.~\ref{fig:higher.singularities}. @@ -384,11 +388,13 @@ where \begin{equation} \mathcal I(\theta)=(\theta-\theta_0)e^{-1/B(\theta-\theta_0)} \end{equation} -reproduces the essential singularity in \eqref{eq:essential.singularity}. Independently, we require for $\theta\in\mathbb R$ +reproduces the essential singularity in \eqref{eq:essential.singularity}. +Independently, we require for $\theta\in\mathbb R$ \begin{equation} \operatorname{Im}\mathcal F(i\theta+0)=\operatorname{Im}\mathcal F_\mathrm{YL}(i\theta+0)=\frac{C_\mathrm{YL}}2\Theta(\theta^2-\theta_\mathrm{YL}^2)[(\theta/\theta_\mathrm{YL})^2-1]^{1+\sigma} \end{equation} -Fixing these requirements for the imaginary part of $\mathcal F(\theta)$ fixes its real part up to an analytic even function. +Fixing these requirements for the imaginary part of $\mathcal F(\theta)$ fixes +its real part up to an analytic even function $G(\theta)$, real for real $\theta$. \begin{figure} \includegraphics{figs/contour_path.pdf} @@ -400,19 +406,17 @@ Fixing these requirements for the imaginary part of $\mathcal F(\theta)$ fixes i } \label{fig:contour} \end{figure} -As $\theta\to\infty$, $\mathcal -F(\theta)\sim\theta^{2/\beta\delta}=\theta^{16/15}$. In order that the -contribution from the arc of the contour vanish, we must have the integrand -vanish sufficiently fast at infinity. Since $2/\beta\delta<2$ in all -dimensions, we will simply use 2. +To find the real part of the nonanalytic part of the scaling function, we make +use of the identity \begin{equation} 0=\oint_{\mathcal C}d\vartheta\,\frac{\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} \end{equation} -where $\mathcal C$ is the contour in Figure \ref{fig:contour}. The only -nonvanishing contributions from this contour are along the real line and along -the branch cut in the upper half plane. For the latter contributions, the real -parts of the integration up and down cancel out, while the imaginary part -doubles. This gives +where $\mathcal C$ is the contour in Figure \ref{fig:contour}. The integral is +zero because there are no singularites enclosed by the contour. The only +nonvanishing contributions from this contour as the radius of the semicircle is +taken to infinity are along the real line and along the branch cut in the upper +half plane. For the latter contributions, the real parts of the integration up +and down cancel out, while the imaginary part doubles. This gives \begin{equation} \begin{aligned} 0&=\left[\int_{-\infty}^\infty+\lim_{\epsilon\to0}\left(\int_{i\infty-\epsilon}^{i\theta_{\mathrm{YL}}-\epsilon}+\int^{i\infty+\epsilon}_{i\theta_{\mathrm{YL}}+\epsilon}\right)\right] @@ -423,12 +427,15 @@ doubles. This gives +2i\int_{i\theta_{\mathrm{YL}}}^{i\infty}d\vartheta\,\frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} \end{aligned} \end{equation} -In principle one would need to account for the residue of the pole at zero, but since its order is less than two and $\mathcal F(0)=\mathcal F'(0)=0$, this evaluates to zero. +where $\mathcal P$ is the principle value. In principle one would need to +account for the residue of the pole at zero, but since its order is less than +two and $\mathcal F(0)=\mathcal F'(0)=0$, this evaluates to zero. Rearranging, this gives \begin{equation} \mathcal F(\theta) =\frac{\theta^2}{i\pi}\mathcal P\int_{-\infty}^\infty d\vartheta\,\frac{\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} +\frac{2\theta^2}\pi\int_{i\theta_{\mathrm{YL}}}^{i\infty}d\vartheta\,\frac{\operatorname{Im}\mathcal F(\theta')}{\vartheta^2(\vartheta-\theta)} \end{equation} +Taking the real part of both sides, we find \begin{equation} \operatorname{Re}\mathcal F(\theta) =\frac{\theta^2}{\pi}\mathcal P\int_{-\infty}^\infty d\vartheta\,\frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} @@ -441,32 +448,14 @@ Because the real part of $\mathcal F$ is even, the imaginary part must be odd. T \int_{\theta_0}^\infty d\vartheta\,\frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2}\left(\frac1{\vartheta-\theta}+\frac1{\vartheta+\theta}\right) -\frac{2\theta^2}\pi\int_{\theta_{\mathrm{YL}}}^{\infty}d\vartheta\,\frac{\operatorname{Im}\mathcal F(i\vartheta)}{\vartheta(\vartheta^2+\theta^2)} \end{equation} - -Now we must make our assertion of the form of the imaginary part of -$\operatorname{Im}\mathcal F(\theta)$. Since both of the limits we are -interested in---\eqref{eq:essential.singularity} along the real axis and -\eqref{eq:yang.lee.sing} along the imaginary axis---have symmetries which make -their imaginary contribution vanish in the domain of the other limit, we do not -need to construct a sophisticated combination to have the correct asymptotics: -a simple sum will do! - -For $\theta\in\mathbb C$, we take +Evaluating these ordinary integrals, we find for $\theta\in\mathbb R$ \begin{equation} - \mathcal F(\theta)=\mathcal F_c(\theta)+\mathcal F_{\mathrm{YL}}(\theta)+\sum_{i=1}^\infty F_{i}\theta^{2i}, + \operatorname{Re}\mathcal F(\theta)=\operatorname{Re}\mathcal F_0(\theta)+\mathcal F_\mathrm{YL}(\theta)+G(\theta) \end{equation} -where $\mathcal F_{\textrm{YL}}$ and $\mathcal F_c$ are functions that -contribute the appropriate singularities expected at the Yang--Lee point and -the first order transition. The first is simply -\begin{equation} - \mathcal F_{\mathrm{YL}}(\theta)=F_{\mathrm{YL}}\left[(\theta^2+\theta_{\mathrm{YL}}^2)^{1+\sigma}-\theta_{\mathrm{YL}}^{2(1+\sigma)}\right] -\end{equation} -The second must be determined using the relationship \eqref{eq:dispersion}. -The real part for $\theta\in\mathbb R$ is therefore +where \begin{equation} \label{eq:2d.real.Fc} - \operatorname{Re}\mathcal F_c(\theta+0i) - =\frac{\theta^2}{\pi} - \int_{\theta_0}^\infty d\vartheta\,\frac{\operatorname{Im}\mathcal F_c(\vartheta+0i)}{\vartheta^2}\left(\frac1{\vartheta-\theta}+\frac1{\vartheta+\theta}\right) - =F_c[\mathcal R(\theta)+\mathcal R(-\theta)] + \operatorname{Re}\mathcal F_0(\theta) + =C_0[\mathcal R(\theta)+\mathcal R(-\theta)] \end{equation} where $\mathcal R$ is given by the function \begin{equation} @@ -476,37 +465,51 @@ where $\mathcal R$ is given by the function +(\theta-\theta_0)e^{-1/B(\theta-\theta_0)}\operatorname{Ei}(1/B(\theta-\theta_0)) \right] \end{equation} -When analytically continued to complex $\theta$, \eqref{eq:2d.real.Fc} has branch cuts in the incorrect places. To produce a function with the correct analytic properties, these real and imaginary parts combine to yield +and +\begin{equation} + \mathcal F_{\mathrm{YL}}(\theta)=C_{\mathrm{YL}}\left[(\theta^2+\theta_{\mathrm{YL}}^2)^{1+\sigma}-\theta_{\mathrm{YL}}^{2(1+\sigma)}\right] +\end{equation} +We have also included the analytic part $G$, which we assume has a simple series expansion \begin{equation} - \mathcal F_c(\theta)=F_c\left\{ + G(\theta)=\sum_{i=1}^\infty G_i\theta^{2i} +\end{equation} +From the form of the real part, we can infer the form of $\mathcal F$ that is analytic for the whole complex plane except at the singularities and branch cuts previously discussed. +For $\theta\in\mathbb C$, we take +\begin{equation} + \mathcal F(\theta)=\mathcal F_0(\theta)+\mathcal F_{\mathrm{YL}}(\theta)+G(\theta), +\end{equation} +where +\begin{equation} + \mathcal F_0(\theta)=C_0\left\{ \mathcal R(\theta)+\mathcal R(-\theta)+i\operatorname{sgn}(\operatorname{Im}\theta)[\mathcal I(\theta)-\mathcal I(-\theta)] \right\} \end{equation} -analytic for all $\theta\in\mathbb C$ outside the Langer branch cuts. - - \section{Fitting} -The scaling function has a number of free parameters: the position $\theta_0$ of the abrupt transition, prefactors in front of singular functions from the abrupt transition and the Yang--Lee point, the coefficients in the analytic part of $\mathcal F$, and the coefficients in the undetermined function $h$. Other parameters are determined by known properties. +The scaling function has a number of free parameters: the position $\theta_0$ +of the abrupt transition, prefactors in front of singular functions from the +abrupt transition and the Yang--Lee point, the coefficients in the analytic +part $G$ of $\mathcal F$, and the coefficients in the undetermined function +$g$. Other parameters are determined by known properties. For $\theta>\theta_0$, \begin{equation} \begin{aligned} \operatorname{Im}u_f - &\simeq A u_t(\theta)^{D\nu}\xi(\theta)\exp\left\{\frac1{\tilde B\xi(\theta)}\right\} \\ - &=AR^{D\nu}t(\theta_0)^{D\nu}\xi'(\theta_0)(\theta-\theta_0) - \exp\left\{\frac1{\tilde B\xi'(\theta_0)}\left(\frac1{\theta-\theta_0} + &\simeq A_0 u_t(\theta)^{D\nu}\xi(\theta)\exp\left\{\frac1{b\xi(\theta)}\right\} \\ + &=A_0R^{D\nu}(\theta_0-1)^{D\nu}\xi'(\theta_0)(\theta-\theta_0) + \exp\left\{\frac1{b\xi'(\theta_0)}\left(\frac1{\theta-\theta_0} -\frac{\xi''(\theta_0)}{2\xi'(\theta_0)}\right) \right\}\left(1+O[(\theta-\theta_0)^2]\right) \end{aligned} \end{equation} \begin{equation} - B=-\tilde B\xi'(\theta_0)=-\tilde B\frac{h'(\theta_0)}{|t(\theta_0)|^{1/\beta\delta}} + B=-b\xi'(\theta_0)=-b\frac{h'(\theta_0)}{(\theta_0^2-1)^{1/\beta\delta}} \end{equation} \begin{equation} \begin{aligned} - F_c&=At(\theta_0)^{D\nu}\xi'(\theta_0)\exp\left\{ + C_0&=At(\theta_0)^{D\nu}\xi'(\theta_0)\exp\left\{ -\frac{\xi''(\theta_0)}{2\tilde B\xi'(\theta_0)^2} \right\} \\ &= |