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Diffstat (limited to 'ising_scaling.tex')
-rw-r--r-- | ising_scaling.tex | 44 |
1 files changed, 29 insertions, 15 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index 909d99f..744f2d5 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -37,6 +37,17 @@ linkcolor=purple \date\today \begin{abstract} + We describe a method for approximating the universal scaling functions for + the Ising model in a field. By making use of parametric coordinates, the free + energy scaling function has a polynomial series everywhere. Its form is + taken to be a sum of the simplest functions that contain the singularities + which must be present: the Langer essential singularity and the Yang--Lee + edge singularity. Requiring that the function match series expansions in + the low- and high-temperature zero-field limits fixes the parametric + coordinate transformation. For the two-dimensional Ising model, we show that + this procedure converges exponentially with the order to which the series are + matched. We speculate that with appropriately modified parametric + coordinates, the method might also converge deep in the metastable phase. \end{abstract} \maketitle @@ -494,7 +505,7 @@ and \end{equation} We have also included the analytic part $G$, which we assume has a simple series expansion -\begin{equation} +\begin{equation} \label{eq:analytic.free.enery} G(\theta)=\sum_{i=1}^\infty G_i\theta^{2i} \end{equation} From the form of the real part, we can infer the form of $\mathcal F$ that is @@ -517,8 +528,8 @@ where The scaling function has a number of free parameters: the position $\theta_0$ of the abrupt transition, prefactors in front of singular functions from the 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. +part $G$ of $\mathcal F$, and the coefficients in the undetermined coordinate +function $g$. Other parameters are determined by known properties. For $\theta>\theta_0$, the form \eqref{eq:essential.singularity} can be expanded around $\theta=\theta_0$ to yield @@ -581,7 +592,8 @@ machine-precision cutoff, whichever is larger. We also add the difference between the predictions for $A_\mathrm{YL}$ and $\xi_\mathrm{YL}$ and their known numeric values, again weighted by their uncertainty. In order to encourage convergence, we also add to the cost the weighted coefficients -$j!g_j$ and $j!G_j$. +$j!g_j$ and $j!G_j$ defining the function $g$ and $G$ in +\eqref{eq:schofield.funcs} and \eqref{eq:analytic.free.enery}. A Levenberg--Marquardt algorithm is performed on the cost function to find a parameter combination which minimizes it. As larger polynomial order in the @@ -781,12 +793,12 @@ to their known values at the critical isotherm, or $\theta=1$. dat = 'data/phi_comparison.dat' set xlabel '$n$' - set xrange [1.5:7.5] + set xrange [1.5:6.5] set logscale y set format y '$10^{%T}$' set ylabel '$|\Delta\mathcal F_0^{(m)}|$' - set yrange [0.000002:0.003] + set yrange [0.000000005:0.0005] set style data linespoints set key title '\raisebox{0.5em}{$m$}' bottom left @@ -801,7 +813,9 @@ to their known values at the critical isotherm, or $\theta=1$. \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. + polynomial order $n$ at which the scaling function was fit. The point + $\eta=0$ corresponds to the critical isotherm at $T=T_c$ and $H>0$, roughly + midway between the limits used in the fit at $H=0$ and $T\neq T_c$. } \label{fig:error} \end{figure} @@ -824,7 +838,7 @@ Fig.~\ref{fig:phi.series}. \begin{gnuplot}[terminal=epslatex] dat1 = 'data/glow_numeric.dat' dat2 = 'data/glow_series_ours_0.dat' - dat3 = 'data/glow_series_ours_7.dat' + dat3 = 'data/glow_series_ours_6.dat' dat4 = 'data/glow_series_caselle.dat' set xlabel '$m$' @@ -838,7 +852,7 @@ Fig.~\ref{fig:phi.series}. plot \ dat1 using 1:(abs($2)):3 title 'Numeric' with yerrorbars, \ dat2 using 1:(abs($2)) title 'This work ($n=2$)', \ - dat3 using 1:(abs($2)) title 'This work ($n=7$)', \ + dat3 using 1:(abs($2)) title 'This work ($n=6$)', \ dat4 using 1:(abs($2)) title 'Caselle \textit{et al.}' \end{gnuplot} \caption{ @@ -853,7 +867,7 @@ Fig.~\ref{fig:phi.series}. \begin{gnuplot}[terminal=epslatex] dat1 = 'data/ghigh_numeric.dat' dat2 = 'data/ghigh_series_ours_2.dat' - dat3 = 'data/ghigh_series_ours_7.dat' + dat3 = 'data/ghigh_series_ours_6.dat' dat4 = 'data/ghigh_caselle.dat' set key top left Left reverse @@ -866,7 +880,7 @@ Fig.~\ref{fig:phi.series}. plot \ dat1 using 1:(abs($2)):3 title 'Numeric' with yerrorbars, \ dat2 using 1:(abs($2)) title 'This work ($n=2$)', \ - dat3 using 1:(abs($2)) title 'This work ($n=7$)', \ + dat3 using 1:(abs($2)) title 'This work ($n=6$)', \ dat4 using 1:(abs($2)) title 'Caselle \textit{et al.}' \end{gnuplot} \caption{ @@ -881,7 +895,7 @@ Fig.~\ref{fig:phi.series}. \begin{gnuplot}[terminal=epslatex] dat1 = 'data/phi_numeric.dat' dat2 = 'data/phi_series_ours_2.dat' - dat3 = 'data/phi_series_ours_9.dat' + dat3 = 'data/phi_series_ours_6.dat' set key top right set logscale y set xlabel '$m$' @@ -892,7 +906,7 @@ Fig.~\ref{fig:phi.series}. plot \ dat1 using 1:(abs($2)):3 title 'Numeric' with yerrorbars, \ dat2 using 1:(abs($2)) title 'This work ($n=2$)', \ - dat3 using 1:(abs($2)) title 'This work ($n=7$)' + dat3 using 1:(abs($2)) title 'This work ($n=6$)' \end{gnuplot} \caption{ The series coefficients for the scaling function $\mathcal F_0$ as a @@ -915,7 +929,7 @@ the ratio. \begin{gnuplot}[terminal=epslatex] dat1 = 'data/glow_numeric.dat' dat2 = 'data/glow_series_ours_0.dat' - dat3 = 'data/glow_series_ours_7.dat' + dat3 = 'data/glow_series_ours_6.dat' dat4 = 'data/glow_series_caselle.dat' ratLast(x) = (back2 = back1, back1 = x, back1 / back2) back1 = 0 @@ -929,7 +943,7 @@ the ratio. plot \ dat1 using (1/$1):(abs(ratLast($2))) title 'Numeric', \ dat2 using (1/$1):(abs(ratLast($2))) title 'This work ($n=2$)', \ - dat3 using (1/$1):(abs(ratLast($2))) title 'This work ($n=7$)', \ + dat3 using (1/$1):(abs(ratLast($2))) title 'This work ($n=6$)', \ dat4 using (1/$1):(abs(ratLast($2))) title 'Caselle \textit{et al.}' \end{gnuplot} \caption{ |