diff options
| -rw-r--r-- | essential-ising.tex | 20 | ||||
| -rw-r--r-- | figs/fig-series.gplot | 12 |
2 files changed, 19 insertions, 13 deletions
diff --git a/essential-ising.tex b/essential-ising.tex index dd2c083..f789eb4 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -350,7 +350,7 @@ In forthcoming work, we develop a method to incorporate the essential singularity in the scaling functions into a form that also incorporates known properties of the scaling functions in the rest of the configuration space using a Schofield-like -parameterization \cite{kent-dobias.2018.parametric}. Briefly, we define +parameterization \cite{schofield.1969.parametric,caselle.2001.critical,kent-dobias.2018.parametric}. Briefly, we define parameters $R$ and $\theta$ by \begin{align} t=R(1-\theta^2) @@ -381,7 +381,7 @@ arbitrary analytic additive function $Y$, && Y(\theta)=\sum_{n=0}^\infty Y_n\theta^{2n} \end{align} -so that $\tilde\fX(\theta)=\fX(\theta)+Y(\theta)$. By manipulating thees +so that $\tilde\fX(\theta)=\fX(\theta)+Y(\theta)$. By manipulating these coefficients, we can attempt to give the resulting scaling form a series expansion consistent with known values. One such prediction---made by fixing the first four terms in the low-temperature, critical isotherm, and @@ -390,6 +390,9 @@ of $\tilde\fX$---is shown as a dashed yellow line in Fig.~\ref{fig:scaling_fits}. As shown in Fig.~\ref{fig:series}, the low-order free energy coefficients of this prediction match known values exactly up to $n=5$, and improve the agreement with higher-order coefficients. +Unlike scaling forms which treat $\fX$ as analytic at the coexistence line, +the series coefficients of the scaling form developed here increase without +bound at high order. \begin{figure} @@ -403,10 +406,11 @@ exactly up to $n=5$, and improve the agreement with higher-order coefficients. and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid blue lines show our analytic results \eqref{eq:sus_scaling} and \eqref{eq:mag_scaling}, the dashed yellow lines show - a scaling function modified to match known series expansions - in several known limits, and the + a scaling function modified to match known series expansions of the + susceptibility + to third order, and the dotted green lines show the - polynomial resulting from truncating the series after the eight terms + polynomial resulting from truncating the known series expansion after the eight terms reported by \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. } \label{fig:scaling_fits} @@ -418,10 +422,12 @@ exactly up to $n=5$, and improve the agreement with higher-order coefficients. The series coefficients defined by $\tilde\fF(X)=\sum_nf_nX^n$. The blue pluses correspond to the scaling form \eqref{eq:2d_free_scaling}, the yellow saltires correspond to a scaling function modified to match known - series expansions in several known limits, and the green + series expansions of the susceptibility to third order---and therefore + the free energy to fifth order---and the green stars correspond to the first eight coefficients from - \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. + \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. The modified + scaling function and the known coefficients match exactly up to $n=5$. } \label{fig:series} \end{figure} diff --git a/figs/fig-series.gplot b/figs/fig-series.gplot index 729ff73..49aa4cc 100644 --- a/figs/fig-series.gplot +++ b/figs/fig-series.gplot @@ -1,5 +1,5 @@ -set terminal pslatex rotate size 3.417,2.111 +set terminal pslatex rotate size 3.6,2.111 cc1 = "#5e81b5" cc2 = "#e19c24" @@ -10,18 +10,18 @@ set logscale y data = "figs/fig-series-data.dat" -set xrange [0.5:8.5] +set xrange [0.5:15.5] set yrange [0.000005:5] set key off set xlabel '$n$' -set ylabel offset 1 '$|f_n|$' +set ylabel offset 2 '$|f_n|$' set ytics format '' set ytics add ('$\footnotesize10^{-5}$' 10**(-5),'$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3),'$\footnotesize10^{-2}$' 10**(-2), '$\footnotesize10^{-1}$' 10**(-1),'$\footnotesize10^{0}$' 10**(0)) plot \ - data using 1:2 with points lc rgb cc1, \ - data using 1:3 with points lc rgb cc2, \ - data using 1:4 with points lc rgb cc3 + data using 1:3 with points lc rgb cc1, \ + data using 1:2 with points lc rgb cc2, \ + data using 1:5 with points lc rgb cc3 |