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Diffstat (limited to 'ising_scaling.tex')
| -rw-r--r-- | ising_scaling.tex | 62 |
1 files changed, 47 insertions, 15 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index b148eec..e3461fa 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -381,7 +381,7 @@ simplest form of the imaginary part to be fixed later by the real part. \end{tikzcd} \] We require that, for $\theta\in\mathbb R$ -\begin{equation} +\begin{equation} \label{eq:imaginary.abrupt} \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 @@ -493,37 +493,40 @@ 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 $\theta>\theta_0$, +For $\theta>\theta_0$, the form \eqref{eq:essential.singularity} can be +expanded around $\theta=\theta_0$ to yield \begin{equation} \begin{aligned} \operatorname{Im}u_f &\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) + &=A_0R^{D\nu}(\theta_0^2-1)^{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} +Comparing this with the requirement \eqref{eq:imaginary.abrupt}, we find that \begin{equation} - B=-b\xi'(\theta_0)=-b\frac{h'(\theta_0)}{(\theta_0^2-1)^{1/\beta\delta}} + B=-b\xi'(\theta_0)=-b\frac{g'(\theta_0)}{(\theta_0^2-1)^{1/\beta\delta}} \end{equation} +and \begin{equation} \begin{aligned} - C_0&=At(\theta_0)^{D\nu}\xi'(\theta_0)\exp\left\{ - -\frac{\xi''(\theta_0)}{2\tilde B\xi'(\theta_0)^2} + C_0&=A_0t(\theta_0^2-1)^{D\nu}\xi'(\theta_0)\exp\left\{ + -\frac{\xi''(\theta_0)}{2b\xi'(\theta_0)^2} \right\} \\ &= - A|t(\theta_0)|^{D\nu-\Delta}h'(\theta_0) - \exp\left\{-\frac1{\tilde B}\left(\frac{|t(\theta_0)|^\Delta h''(\theta_0)}{2h'(\theta_0)^2}+\frac{\Delta|t(\theta_0)|^{\Delta - 1} t'(\theta_0)}{h'(\theta_0)} - \right)\right\} + A_0(\theta_0^2-1)^{D\nu-\Delta}g'(\theta_0) + \exp\left\{-\frac1b\left(\frac{(\theta_0^2-1)^\Delta g''(\theta_0)}{2g'(\theta_0)^2}-\frac{2\Delta(\theta_0^2-1)^{\Delta - 1}\theta_0}{g'(\theta_0)} + \right)\right\} \end{aligned} \end{equation} -fixing $B$ and $F_c$. Since $A$ and $\tilde B$ are known exactly, these forms can be substituted. +fixing $B$ and $C_0$. This leaves as unknown variables the positions $\theta_0$ and $\theta_{\mathrm{YL}}$ of the abrupt transition and Yang--Lee edge singularity, -the amplitude $A_\mathrm{YL}$ of the latter, and the unknown functions $F$ and -$h$. We determine these approximately by iteration in the polynomial order at +the amplitude $C_\mathrm{YL}$ of the latter, and the unknown functions $G$ and +$g$. We determine these approximately by iteration in the polynomial order at which the free energy and its derivative matches known results. We write as a cost function the difference between the known series coefficients of the scaling functions $\mathcal F_\pm$ and the series coefficients of our @@ -671,22 +674,51 @@ accuracy of the fit results can be checked against the known values here. \end{table} \begin{figure} - \begin{gnuplot}[terminal=epslatex, terminaloptions={size 8.65cm,5.35cm}] + \begin{gnuplot}[terminal=epslatex] dat = 'data/phi_comparison.dat' set xlabel '$n$' - set ylabel '$|\mathcal F_0^{(n)}-|$' + set ylabel '$|\Delta\mathcal F_0^{(m)}(0)|$' + set format y '$10^{%T}$' set style data linespoints set logscale y + set key title '\raisebox{0.5em}{$m$}' bottom left plot \ dat using 1:2 title '0', \ dat using 1:3 title '1', \ dat using 1:4 title '2', \ - dat using 1:5 title '3' + dat using 1:5 title '3', \ + dat using 1:6 title '4' + \end{gnuplot} + \caption{ + The error in the $m$th derivative of the scaling function $\mathcal F_0$ + with respect to $\eta$ evaluated at $\eta=0$, as a function of the + polynomial order $n$ at which the scaling function was fit. + } +\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} |