diff options
Diffstat (limited to 'ising_scaling.tex')
| -rw-r--r-- | ising_scaling.tex | 57 |
1 files changed, 29 insertions, 28 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index 21cc0fe..c7e7adb 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -259,7 +259,7 @@ $\xi_\mathrm{YL}$ \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_\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}\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}. @@ -340,7 +340,7 @@ The location of the Yang--Lee edge singularities can be calculated directly from the coordinate transformation \eqref{eq:schofield}. Since $g(\theta)$ is an odd real polynomial for real $\theta$, it is imaginary for imaginary $\theta$. Therefore, -\begin{equation} +\begin{equation} \label{eq:yang-lee.theta} i\xi_{\mathrm{YL}}=\frac{g(i\theta_{\mathrm{YL}})}{(1+\theta_{\mathrm{YL}}^2)^{-\Delta}} \end{equation} The location $\theta_0$ is not fixed by any principle. @@ -522,7 +522,23 @@ and \right)\right\} \end{aligned} \end{equation} -fixing $B$ and $C_0$. +fixing $B$ and $C_0$. Similarly, \eqref{eq:yang-lee.theta} puts a constraint on the value of $\theta_\mathrm{YL}$, while the known amplitude of the Yang--Lee branch cut fixes the value of $C_\mathrm{YL}$ by +\begin{equation} + \begin{aligned} + u_f + &\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) +\end{aligned} +\end{equation} +\begin{equation} + C_\mathrm{YL}=A_\mathrm{YL}|g(i\theta_\mathrm{YL})|^{D\nu/\Delta}\left[\frac{-\eta'(i\theta_\mathrm{YL})}{2i\theta_\mathrm{YL}}\right]^{1+\sigma} +\end{equation} +where $A_\mathrm{YL}=-1.37(2)$ and $\xi_\mathrm{YL}=0.18930(5)$ +\cite{Fonseca_2003_Ising}. Because these parameters are not known exactly, +these constraints are added to the weighted sum of squares rather than +substituted in. This leaves as unknown variables the positions $\theta_0$ and $\theta_{\mathrm{YL}}$ of the abrupt transition and Yang--Lee edge singularity, @@ -702,29 +718,6 @@ accuracy of the fit results can be checked against the known values here. \end{figure} \begin{figure} - \begin{gnuplot}[terminal=epslatex] - dat = 'data/yl_comparison.dat' - - set xlabel '$n$' - set ylabel '$|\Delta\xi_\mathrm{YL}|$' - set xrange [1.5:9.5] - set yrange [0.000005:0.05] - - set format y '$10^{%T}$' - set style data yerrorlines - set logscale y - unset key - - plot dat using 1:2:3 - \end{gnuplot} - \caption{ - The error in the location of the Yang--Lee edge singularity as a function - of the polynomial order $n$ at which the scaling function was fit. Error - bars denote the uncertainty in the known location of the singularity. - } -\end{figure} - -\begin{figure} \begin{gnuplot}[terminal=epslatex, terminaloptions={size 8.65cm,5.35cm}] dat9 = 'data/h_series_ours_9.dat' dat11 = 'data/h_series_ours_11.dat' @@ -810,6 +803,8 @@ accuracy of the fit results can be checked against the known values here. \begin{gnuplot}[terminal=epslatex] dat1 = 'data/ghigh_numeric.dat' dat2 = 'data/ghigh_caselle.dat' + dat3 = 'data/ghigh_series_ours_2.dat' + dat4 = 'data/ghigh_series_ours_9.dat' set key top left Left reverse set logscale y @@ -821,6 +816,8 @@ accuracy of the fit results can be checked against the known values here. plot \ dat1 using 1:(abs($2)):3 title 'Numeric' with yerrorbars, \ dat2 using 1:(abs($2)) title 'Caselle', \ + dat3 using 1:(abs($2)) title 'Caselle', \ + dat4 using 1:(abs($2)) title 'Caselle' \end{gnuplot} \caption{ } @@ -829,7 +826,9 @@ accuracy of the fit results can be checked against the known values here. \begin{figure} \begin{gnuplot}[terminal=epslatex] dat1 = 'data/phi_numeric.dat' - set key top left Left reverse + dat2 = 'data/phi_series_ours_2.dat' + dat3 = 'data/phi_series_ours_9.dat' + set key top right set logscale y set xlabel '$n$' set ylabel '$|\mathcal F_0^{(n)}|$' @@ -837,7 +836,9 @@ accuracy of the fit results can be checked against the known values here. set xrange [-0.5:10.5] plot \ - dat1 using 1:(abs($2)) title 'Numeric' with yerrorbars + dat1 using 1:(abs($2)):3 title 'Numeric' with yerrorbars, \ + dat2 using 1:(abs($2)) title 'Numeric', \ + dat3 using 1:(abs($2)) title 'Numeric' \end{gnuplot} \caption{ } |