From 75153282053738f549f2737f0fe42a18bb17b5a6 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 26 Sep 2017 11:19:54 -0400 Subject: changed paper body to incorporate new analysis --- essential-ising.tex | 67 +++++++++++++++++++++++------------------------------ 1 file changed, 29 insertions(+), 38 deletions(-) (limited to 'essential-ising.tex') diff --git a/essential-ising.tex b/essential-ising.tex index d641a57..7dd6686 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -316,7 +316,8 @@ the presence of an external field. Briefly, the external field $H$ is applied by adding an extra spin $s_0$ with coupling $|H|$ to all others \cite{dimitrovic.1991.finite}. A quickly converging estimate for the magnetization in the finite-size system was then made by taking -$M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin. +$M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin +\cite{kent-dobias.2018.wolff}. Data was then taken for susceptibility and magnetization for $T_\c-T,H\leq0.1$. This data, rescaled as appropriate to collapse onto a single curve, is plotted in Fig.~\ref{fig:scaling_fits}. @@ -330,34 +331,23 @@ the susceptibility scaling function, then $T\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ 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 -are systematic differences that can be seen most clearly in the magnetization. -Since our method is known to only be accurate for high moments of the free -energy, we should expect that low moments require corrections. Therefore, we -also fit those corrections using -\begin{align} - \tilde\fX^\twodee(X)&=\fX^\twodee(X)+\sum_{n=0}^Nf_n(BX)\label{eq:sus_scaling_poly}\\ - \tilde\fM^\twodee(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=0}^NF_n(BX)\label{eq:mag_scaling_poly} -\end{align} -where $F_n'(Y)=f_n(Y)$ and -\[ - \begin{aligned} - f_n(Y)&=\frac{c_nx^n}{1+(\lambda Y)^{n+1}}\\ - F_n(Y)&=\frac{c_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda Y)^{n+1}). - \end{aligned} - \label{eq:poly} -\] -The functions $f_n$ have been chosen to be pure integer power laws for small -argument, but vanish appropriately at large argument. This is necessary -because the susceptibility vanishes with $h|t|^{-\beta\delta}$ while bare -polynomial corrections would not. We fit these functions to known moments of -the free energy's scaling function -\cite{mangazeev.2008.variational,mangazeev.2010.scaling} our numeric data -for $N=0$. The -resulting curves are also plotted as dashed lines in -Fig.~\ref{fig:scaling_fits}. Our singular scaling function with one low-order -correction appears to match data quite well. +Fig.~\ref{fig:scaling_fits}. Though there is good agreement +between our functional forms and what is measured, there +are systematic differences that can be seen most clearly in the +magnetization. This is to be expected based on our earlier discussion: these +scaling forms should only be expected to well-describe the singularity at the +abrupt transition. Our forms both exhibit incorrect low-order +coefficients at the transition (Fig.~\ref{fig:series}) and incorrect +asymptotics as $h|t|^{-\beta\delta}$ becomes very large. 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}. Fig.~\ref{fig:scaling_fits} shows a result as a +dashed yellow line, which depicts the scaling form resulting from +incorporating our singularity and the known series expansions of the scaling +function at high temperature, low temperature, and at the critical isotherm to +quadratic order. The low-order series coefficients of this modified form are +also shown in Fig.~\ref{fig:series}. \begin{figure} \input{fig-susmag} @@ -370,10 +360,11 @@ correction appears to match data quite well. 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 - \eqref{eq:sus_scaling_poly} and \eqref{eq:mag_scaling_poly} for $N=0$, the - dotted green lines show the same for $N=4$, and the red line show the - polynomial resulting from truncating the series after the eight known - terms. + a scaling function modified to match known series expansions + in several known limits, and the + dotted green lines show the + polynomial resulting from truncating the series after the eight terms + reported by \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. } \label{fig:scaling_fits} \end{figure} @@ -383,11 +374,11 @@ correction appears to match data quite well. \caption{ 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 that form with the first four coefficients - fixed to known values (\eqref{eq:sus_scaling_poly} with $N=0$), the green - stars correspond to that form with the first eight coefficients fixed to - known values (\eqref{eq:sus_scaling_poly} with $N=4$), and the red squares - correspond to the first eight coefficients. + yellow saltires correspond to a scaling function modified to match known + series expansions in several known limits, and the green + stars + correspond to the first eight coefficients from + \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. } \label{fig:series} \end{figure} -- cgit v1.2.3-70-g09d2