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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} |