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diff --git a/essential-ising.tex b/essential-ising.tex index c98e938..b7f432c 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -267,10 +267,10 @@ moments can still be extracted, e.g., the susceptibility, by taking \chi=\pd MH=-\frac1{T_\c}\pd[2]Fh =-\frac2{\pi T_\c}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] -With a scaling form defined by $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, +With a scaling form defined by $T_\c\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields \[ - \fX^\twodee(Y/B)=\frac{AB^2}{\pi T_\c Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] + \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] \label{eq:sus_scaling} \] Scaling forms for the free energy can then be extracted by direct integration @@ -320,13 +320,12 @@ single curve, is plotted in Fig.~\ref{fig:scaling_fits}. For the \twodee Ising model on a square lattice, exact results at zero temperature have $\fS(0)=4/T_\c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ -\cite{onsager.1944.crystal}, and $\fX(0)=C_0^-/T_\c$ with -$C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that +\cite{onsager.1944.crystal}, and $\fX(0)=C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of the susceptibility scaling function, then -$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi T_\c$ and the constant -$A$ is fixed to $A=\pi T_\c\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The +$T_\c\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant +$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The resulting scaling functions $\fX$ and $\fM$ are plotted as solid lines in Fig.~\ref{fig:scaling_fits}. As can be seen, there is very good agreement between our proposed functional forms and what is measured. However, there |