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| -rw-r--r-- | ising_scaling.tex | 90 |
1 files changed, 84 insertions, 6 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index 0bf301e..3f11729 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -246,7 +246,12 @@ 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, Houghton_1980_The}. In the Ising conformal field theory, the prefactor is known to be $A_0=\bar s/2\pi$ -\cite{Voloshin_1985_Decay, Fonseca_2003_Ising}. +\cite{Voloshin_1985_Decay, Fonseca_2003_Ising}. The signature of this +singularity in the scaling function is a superexponential divergence in the +series coefficients about $\xi=0$, which asymptotically take the form +\begin{equation} \label{eq:low.asymptotic} + \mathcal F_-^\infty(m)=\frac{A_0}\pi b^{m-1}\Gamma(m-1) +\end{equation} \subsection{Yang--Lee edge singularity} @@ -288,10 +293,15 @@ $\xi_\mathrm{YL}$ \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\tilde 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]] + \operatorname{Im}\mathcal F_+(i\xi\pm0)=\pm\tilde A_\mathrm{YL}\Theta(\xi-\xi_\mathrm{YL})(\xi-\xi_\mathrm{YL})^{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}. +Fig.~\ref{fig:higher.singularities}. The signature of this in the scaling +function is an asymptotic behavior of the coefficients which goes like +\begin{equation} \label{eq:high.asymptotic} + \mathcal F_+^\infty(m)=A_\mathrm{YL}2(-1)^{2m}\theta_\mathrm{YL}^{1-\sigma-m}\binom{1-\sigma}{m} +\end{equation} + \section{Parametric coordinates} @@ -425,7 +435,12 @@ where 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} + \operatorname{Im}\mathcal F(i\theta+0) + =\operatorname{Im}\mathcal F_\mathrm{YL}(i\theta+0) + =\frac12C_\mathrm{YL}\left[ + \Theta(\theta-\theta_\mathrm{YL})(\theta-\theta_\mathrm{YL})^{1+\sigma} + -\Theta(\theta+\theta_\mathrm{YL})(\theta+\theta_\mathrm{YL})^{1+\sigma} + \right] \end{equation} 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 $\theta$. @@ -501,7 +516,7 @@ where $\mathcal R$ is given by the function \end{equation} 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] + \mathcal F_{\mathrm{YL}}(\theta)=2C_\mathrm{YL}\left[2(\theta^2+\theta_\mathrm{YL}^2)^{(1+\sigma)/2}\cos\left((1+\sigma)\tan^{-1}\frac\theta{\theta_\mathrm{YL}}\right)-\theta_\mathrm{YL}^{1+\sigma}\right] \end{equation} We have also included the analytic part $G$, which we assume has a simple series expansion @@ -568,7 +583,7 @@ branch cut fixes the value of $C_\mathrm{YL}$ by &\simeq A_\mathrm{YL}|u_h(\theta)|^{D\nu/\Delta}(\eta_{\mathrm YL}-\eta(\theta))^{1+\sigma} \\ &=A_\mathrm{YL}R^{D\nu}|g(i\theta_\mathrm{YL})|^{D\nu/\Delta}[-\eta'(i\theta_\mathrm{YL})]^{1+\sigma}(\theta-i\theta_\mathrm{YL})^{1+\sigma}\left(1+O[(\theta-i\theta_\mathrm{YL})^2]\right)\\ &\simeq R^{D\nu}\mathcal F_\mathrm{YL}(\theta) - =C_\mathrm{YL}R^{D\nu}(2i\theta_{YL})^{1+\sigma}(\theta-i\theta_\mathrm{YL})^{1+\sigma}\left(1+O[(\theta-i\theta_\mathrm{YL})^2]\right) + =C_\mathrm{YL}R^{D\nu}(\theta-i\theta_\mathrm{YL})^{1+\sigma}\left(1+O[(\theta-i\theta_\mathrm{YL})^2]\right) \end{aligned} \end{equation} \begin{equation} @@ -857,6 +872,37 @@ Fig.~\ref{fig:phi.series}. \begin{figure} \begin{gnuplot}[terminal=epslatex] + dat1 = 'data/glow_numeric.dat' + dat2 = 'data/glow_series_ours_0.dat' + dat3 = 'data/glow_series_ours_6.dat' + dat4 = 'data/glow_series_caselle.dat' + + set xlabel '$m$' + set xrange [0:14.5] + + set key top left Left reverse + set ylabel '$\mathcal F_-^{(m)}/\mathcal F_-^\infty(m)$' + + os = 1.3578383417065954956 + asmp(n) = os / (2 * pi) * (2 * os / pi)**(n-1) * gamma(n - 1) / pi + + plot \ + dat1 using 1:(abs($2) / asmp($1)):($3 / asmp($1)) title 'Numeric' with yerrorbars, \ + dat2 using 1:(abs($2) / asmp($1)) title 'This work ($n=2$)', \ + dat3 using 1:(abs($2) / asmp($1)) title 'This work ($n=6$)', \ + dat4 using 1:(abs($2) / asmp($1)) title 'Caselle \textit{et al.}' + \end{gnuplot} + \caption{ + The series coefficients for the scaling function $\mathcal F_-$ as a + function of polynomial order $m$, rescaled by their asymptotic limit + $\mathcal F_-^\infty(m)$ from \eqref{eq:low.asymptotic}. The numeric values + are from Table \ref{tab:data}, and those of Caselle \textit{et al.} are + from the most accurate scaling function listed in \cite{Caselle_2001_The}. + } \label{fig:glow.series} +\end{figure} + +\begin{figure} + \begin{gnuplot}[terminal=epslatex] dat1 = 'data/ghigh_numeric.dat' dat2 = 'data/ghigh_series_ours_2.dat' dat3 = 'data/ghigh_series_ours_6.dat' @@ -885,6 +931,38 @@ Fig.~\ref{fig:phi.series}. \begin{figure} \begin{gnuplot}[terminal=epslatex] + dat1 = 'data/ghigh_numeric.dat' + dat2 = 'data/ghigh_series_ours_2.dat' + dat3 = 'data/ghigh_series_ours_6.dat' + dat4 = 'data/ghigh_caselle.dat' + + set xlabel '$m$' + set ylabel '$\mathcal F_+^{(m)}/\mathcal F_+^\infty(m)$' + set yrange [0.8:1.5] + set xrange [1.5:14.5] + + xYL = 0.18930 + AYL = 1.37 + sigma = 0.833333333333 + asmp(n) = -AYL * 2 * exp(log(xYL)*(sigma-n))*gamma(sigma+1)/gamma(n+1)/gamma(sigma-n+1) + + plot \ + dat1 using 1:(abs($2) / abs(asmp($1))):($3 / asmp($1)) title 'Numeric' with yerrorbars, \ + dat2 using 1:(abs($2) / asmp($1)) title 'This work ($n=2$)', \ + dat3 using 1:(abs($2) / asmp($1)) title 'This work ($n=6$)', \ + dat4 using 1:(abs($2) / asmp($1)) title 'Caselle \textit{et al.}' + \end{gnuplot} + \caption{ + The series coefficients for the scaling function $\mathcal F_+$ as a + function of polynomial order $m$, rescaled by their asymptotic limit + $\mathcal F_+^\infty(m)$ from \eqref{eq:high.asymptotic}. The numeric + values are from Table \ref{tab:data}, and those of Caselle \textit{et al.} + are from the most accurate scaling function listed in \cite{Caselle_2001_The}. + } \label{fig:ghigh.series} +\end{figure} + +\begin{figure} + \begin{gnuplot}[terminal=epslatex] dat1 = 'data/phi_numeric.dat' dat2 = 'data/phi_series_ours_2.dat' dat3 = 'data/phi_series_ours_6.dat' |