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| -rw-r--r-- | ising_scaling.tex | 68 |
1 files changed, 45 insertions, 23 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index 98cd27f..2b6dab4 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -249,10 +249,15 @@ function $\mathcal F_+$ takes the form \mathcal F_+(\xi) =A(\xi) +B(\xi)[1+(\xi/\xi_{\mathrm{YL}})^2]^{1+\sigma}+\cdots \end{equation} -with edge exponent $\sigma=\frac16$ and $A$ and $B$ analytic functions at -$\xi_\mathrm{YL}$. - -\cite{Cardy_1985_Conformal} +with edge exponent $\sigma=-\frac16$ and $A$ and $B$ analytic functions at +$\xi_\mathrm{YL}$ \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 +\end{equation} +This results in analytic structure for $\mathcal F_+$ shown in +Fig.~\ref{fig:higher.singularities}. \section{Parametric coordinates} @@ -319,10 +324,12 @@ and for $\theta$ near $\theta_0$ it will resemble $\mathcal F_-$. This can be se \end{equation} This leads us to expect that the singularities present in these functions will likewise be -present in $\mathcal F(\theta)$. This is shown in Figure -\ref{fig:schofield.singularities}. Two copies of the Langer branch cut stretch -out from $\pm\theta_0$, where the equilibrium phase ends, and the Yang--Lee -edge singularities are present on the imaginary-$\theta$ line, where they must be since $\mathcal F$ has the same symmetry in $\theta$ as $\mathcal F_+$ has in $\xi$. +present in $\mathcal F(\theta)$. The analytic structure of this function is +shown in Fig.~\ref{fig:schofield.singularities}. Two copies of the Langer +branch cut stretch out from $\pm\theta_0$, where the equilibrium phase ends, +and the Yang--Lee edge singularities are present on the imaginary-$\theta$ +line, where they must be since $\mathcal F$ has the same symmetry in $\theta$ +as $\mathcal F_+$ has in $\xi$. The location of the Yang--Lee edge singularities can be calculated directly from the coordinate transformation \eqref{eq:schofield}. Since $g(\theta)$ is @@ -349,18 +356,38 @@ As we have seen in the previous sections, the unavoidable singularities in the scaling functions are readily expressed as singular functions in the imaginary part of the free energy. -Our strategy follows. First, we take the known singular expansions of the imaginary parts of the scaling functions $\mathcal F_{\pm}(\xi)$ and produce simplest form accessible under polynomial coordinate changes of $\xi$. Second, we assert that the imaginary part of $\mathcal F(\theta)$ must have this simplest form. Third, we perform a Kramers--Kronig type transformation to establish an explicit form for the real part of $\mathcal F(\theta)$. Finally, we make good on the assertion posited in the second step by fixing the Schofield coordinate transformation to produce the correct coefficients known for the real part of $\mathcal F_{\pm}$. +Our strategy follows. First, we take the singular imaginary parts of the +scaling functions $\mathcal F_{\pm}(\xi)$ and truncate them to the lowest order +accessible under polynomial coordinate changes of $\xi$. Then, we assert that +the imaginary part of $\mathcal F(\theta)$ must have this simplest form, +implicitly defining the parametric coordinate change. Third, we perform a +Kramers--Kronig type transformation to establish an explicit form for the real +part of $\mathcal F(\theta)$. Finally, we make good on the assertion made in +the second step by finding the coordinate transformation that produces the +correct series coefficients of $\mathcal F_{\pm}$. This success of this stems from the commutative diagram below. So long as the -application of Schofield coordinates and the dispersion relation can be said to -commute, we may assume we have found correct coordinates for the simplest form -of the imaginary part to be fixed in reality by the real part. +application of Schofield coordinates and the Kramers--Kronig relation can be +said to commute, we may assume we have found correct coordinates for the +simplest form of the imaginary part to be fixed later by the real part. \[ \begin{tikzcd}[row sep=large, column sep = 9em] \operatorname{Im}\mathcal F_\pm(\xi) \arrow{r}{\text{Kramers--Kronig in $\xi$}} \arrow[]{d}{\text{Schofield}} & \operatorname{Re}\mathcal F_{\pm}(\xi) \arrow{d}{\text{Schofield}} \\% \operatorname{Im}\mathcal F(\theta) \arrow{r}{\text{Kramers--Kronig in $\theta$}}& \operatorname{Re}\mathcal F(\theta) \end{tikzcd} \] +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)] +\end{equation} +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 $\theta\in\mathbb R$ +\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)] +\end{equation} \begin{figure} \includegraphics{figs/contour_path.pdf} @@ -372,7 +399,11 @@ of the imaginary part to be fixed in reality by the real part. } \label{fig:contour} \end{figure} -As $\theta\to\infty$, $\mathcal F(\theta)\sim\theta^{2/\beta\delta}$. 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. +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. \begin{equation} 0=\oint_{\mathcal C}d\vartheta\,\frac{\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} \end{equation} @@ -428,16 +459,7 @@ 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}. To -match the behavior we expect, we should have for $\theta\in\mathbb R$ -\begin{equation} - \operatorname{Im}\mathcal F_c(\theta+0i)=F_c[\Theta(\theta-\theta_0)\mathcal I(\theta)-\Theta(-\theta-\theta_0)\mathcal I(-\theta)] -\end{equation} -where -\begin{equation} - \mathcal I(\theta)=(\theta-\theta_0)e^{-1/B(\theta-\theta_0)} -\end{equation} -reproduces the singularity in \eqref{eq:essential.singularity}. +The second must be determined using the relationship \eqref{eq:dispersion}. The real part for $\theta\in\mathbb R$ is therefore \begin{equation} \label{eq:2d.real.Fc} \operatorname{Re}\mathcal F_c(\theta+0i) |