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diff --git a/ising_scaling.tex b/ising_scaling.tex index d91e5ca..1fdacc3 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -90,6 +90,16 @@ their simplest form. Then, the arbitrary analytic functions that compose those coordinates are approximated by truncated polynomials whose coefficients are fixed by matching the series expansions of the universal function. +For the two-dimensional Ising model, his method produces scaling functions +accurate to within $10^{-4}$ using just the values of the first three +derivatives of the function evaluated at two points, e.g., critical amplitudes +of the magnetization, susceptibility, and first generalized susceptibility. +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}. + \section{Universal scaling functions} A renormalization group analysis predicts that certain thermodynamic functions |