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| -rw-r--r-- | ising_scaling.tex | 6 |
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diff --git a/ising_scaling.tex b/ising_scaling.tex index 1933983..53f76ad 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -47,7 +47,7 @@ linkcolor=purple coordinate transformation. For the two-dimensional Ising model, we show that this procedure converges exponentially with the order to which the series are matched, up to seven digits of accuracy. - To facilitate use, we provide Python and Mathematica implementations of the code at both lowest order (three digit) and high accuracy. + To facilitate use, we provide a Mathematica implementation of the code at both lowest order (three digit) and high accuracy. %We speculate that with appropriately modified parametric coordinates, the method may converge even deep into the metastable phase. \end{abstract} @@ -100,7 +100,7 @@ With six derivatives, it is accurate to about $10^{-7}$. We hope that with some refinement, this idea might be used to establish accurate scaling functions for critical behavior in other universality classes, doing for scaling functions what advances in conformal bootstrap did for critical exponents -\cite{Gliozzi_2014_Critical}. Mathematica and Python implementations will be provided in the supplemental material. +\cite{Gliozzi_2014_Critical}. A Mathematica implementation will be provided in the supplemental material. \section{Universal scaling functions} \label{sec:UniversalScalingFunctions} @@ -1208,7 +1208,7 @@ the ratio. We have introduced explicit approximate functions forms for the two-dimensional Ising universal scaling function in the relevant variables. These functions are smooth to all orders, include the correct singularities, and appear to converge -exponentially to the function as they are fixed to larger polynomial order. The universal scaling function will be available in both Mathematica and Python in the supplemental material. It is implicitly defined by $\mathcal{F}_0$ and $\mathcal{F}_\pm$ in Eq.~\eqref{eq:scaling.function.equivalences.2d}, where $g(\theta)$ is defined in Eq.~\eqref{eq:schofield.funcs}, $\mathcal{F}$ in Eqs.~\eqref{eq:FfromFoFYLG}--\eqref{eq:FYL}, and the fit constants at various levels of approximation are given in Table~\ref{tab:fits}. +exponentially to the function as they are fixed to larger polynomial order. The universal scaling function will be available in Mathematica in the supplemental material. It is implicitly defined by $\mathcal{F}_0$ and $\mathcal{F}_\pm$ in Eq.~\eqref{eq:scaling.function.equivalences.2d}, where $g(\theta)$ is defined in Eq.~\eqref{eq:schofield.funcs}, $\mathcal{F}$ in Eqs.~\eqref{eq:FfromFoFYLG}--\eqref{eq:FYL}, and the fit constants at various levels of approximation are given in Table~\ref{tab:fits}. This method, although spectacularly successful, could be improved. It becomes difficult to fit the unknown functions at progressively higher order due to the complexity of the |