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diff --git a/ising_scaling.tex b/ising_scaling.tex index 2b6dab4..83351fc 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -207,7 +207,7 @@ 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)=A\Theta(-\xi)|\xi|e^{-1/\tilde B|\xi|}\left[1+O(\xi)\right] + \operatorname{Im}\mathcal F_-(\xi+i0)=A\Theta(-\xi)\xi e^{-1/\tilde 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 @@ -254,7 +254,7 @@ $\xi_\mathrm{YL}$ \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)\sim\pm\frac12\Theta(\xi^2-\xi_\mathrm{YL}^2)[(\xi/\xi_\mathrm{YL})^2-1]^{1+\sigma}+\cdots + \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]] \end{equation} This results in analytic structure for $\mathcal F_+$ shown in Fig.~\ref{fig:higher.singularities}. @@ -378,7 +378,7 @@ simplest form of the imaginary part to be fixed later by the real part. \] We require that, for $\theta\in\mathbb R$ \begin{equation} - \operatorname{Im}\mathcal F(\theta+0i)=\operatorname{Im}\mathcal F_0(\theta+0i)=F_0[\Theta(\theta-\theta_0)\mathcal I(\theta)-\Theta(-\theta-\theta_0)\mathcal I(-\theta)] + \operatorname{Im}\mathcal F(\theta+0i)=\operatorname{Im}\mathcal F_0(\theta+0i)=C_0[\Theta(\theta-\theta_0)\mathcal I(\theta)-\Theta(-\theta-\theta_0)\mathcal I(-\theta)] \end{equation} where \begin{equation} @@ -386,8 +386,9 @@ where \end{equation} 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)=F_\mathrm{YL}[\Theta(\theta-\theta_\mathrm{YL})-\Theta(-\theta-\theta_\mathrm{YL})\mathcal I(-\theta)] + \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. \begin{figure} \includegraphics{figs/contour_path.pdf} |