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diff --git a/essential-ising.tex b/essential-ising.tex index 3241954..90c42b9 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -3,7 +3,7 @@ % Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. % Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. % -\documentclass[aps,prl,preprint]{revtex4-1} +\documentclass[aps,prl,reprint]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,latexsym,mathtools,xifthen} @@ -292,9 +292,9 @@ The constant $A$ can be fixed by zero-field results, with $\chi(t,0)|t|^\gamma=\ Scaling forms for the free energy can then be extracted by direct integration and their constants of integration fixed by known zero field values, yielding \begin{align} + \label{eq:mag_scaling} \fM^\twodee(Y/B) &=\fM(0)+\frac{ABT_c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ - \label{eq:mag_scaling} \fF^\twodee(Y/B) &=\fF(0)+T_cY\bigg(\frac{\fM(0)}B+\frac{AT_c}\pi e^{1/Y}\ei(-1/Y)\bigg) \end{align} @@ -308,54 +308,62 @@ model have $\fS(0)=4$ and $\fM(0)=(2^{5/2}\arcsinh1)^\beta$, so $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\sinh^{-1}1)^{15/8})^{-1}$. How predictive are these scaling forms in the proximity of the critical point -and the abrupt transition line? We simulated the \twodee Ising model on square, triangluar, and hexagonal lattices using a form of the Wolff algorithm modified +and the abrupt transition line? We simulated the \twodee Ising model on square lattice using a form of the Wolff algorithm modified to remain efficient in 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. 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 $B=\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. Data from other lattices can be mapped onto that of the square lattice using lattice--lattice scaling rules \cite{betts.1971.lattice}. If $D$ is the critical amplitude for the magnetization along the critical isotherm, i.e., $h=DM^\delta$ for $t=0$, then $t$ and $h$ for lattice $X$ are related to those of lattice $Y$ by $\frac{h_X}{D_X}=\frac{h_Y}{D_Y}$ and $t_X\fM_X(0)^{1/\beta}=t_Y\fM_Y(0)^{1/\beta}$. While $D$ is not known exactly, the ratios $D/D_\sq$ for any exactly solvable \twodee lattice are. These are tabulated, along with $\fM(0)$, in Table~\ref{tab:consts}. +\cite{barouch.1973.susceptibility}, so that $B=\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. Data was then taken for susceptibility and -magnetization for $T_\c-T,H\leq0.1$. This data is plotted in Figs.~\ref{fig:sus} -and \ref{fig:mag}, along with collapses of data onto a single universal curve +magnetization for $T_\c-T,H\leq0.1$. This data is plotted in +Fig.~\ref{fig:scaling_fits}, along with collapses of data onto a single universal curve in the insets of those figures. 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 of the form +\begin{align} + \fX^{\twodee\prime}(X)&=\fX^\twodee(X)+\sum_{n=1}^Nf_n(BX)\\ + \fM^{\twodee\prime}(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=1}^NF_n(BX) +\end{align} +where $F_n'(x)=f_n(x)$ and +\begin{align} + f_n(x)&=\frac{C_nx^n}{1+(\lambda x)^{n+1}}\\ + F_n(x)&=\frac{C_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda x)^{n+1}) +\end{align} +We fit these functions to our numeric data for $N=3$. The resulting curves are +also plotted in Fig.~\ref{fig:scaling_fits} as a dashed line. -\begin{table} - \centering - \begin{tabular}{c|llc} - Lattice & $T_\c$ & $\fM(0)^{1/\beta}$ & $D/D_\sq$ \\ - \hline % ------------------------------------------------------------------- - Square & $2/\log(1+\sqrt2)$ & $2^{5/2}\arcsinh1$ & 1 \\ - Triangular & $4/\log3$ & $4\log3$ & $3^{3/2}/4$ \\ - Hexagonal & $2/\log(2+\sqrt3)$ & $\frac8{\sqrt3}\arccosh2$ & $3^{3/2}/8$ - \end{tabular} - \caption{ - The critical temperatures and amplitudes for the magetization along both the coexistence line and the critical isotherm, for three different lattices. - } - \label{tab:consts} -\end{table} -\begin{figure} - \input{figs/fig-sus} - \caption{ - Fit of scaling form \eqref{eq:sus_scaling} to numeric data. Data with - sampling error taken from Monte Carlo simulations of an $L=2048$ - square-lattice Ising model with $T_\c-T=0.01,0.02,\ldots,0.1$ and - $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form, - with $C=0.0111\pm0.0023$ and $B=0.173\pm0.072$. - } - \label{fig:sus} -\end{figure} + + +%\begin{table} +% \centering +% \begin{tabular}{c|llc} +% Lattice & $T_\c$ & $\fM(0)^{1/\beta}$ & $D/D_\sq$ \\ +% \hline % ------------------------------------------------------------------- +% Square & $2/\log(1+\sqrt2)$ & $2^{5/2}\arcsinh1$ & 1 \\ +% Triangular & $4/\log3$ & $4\log3$ & $3^{3/2}/4$ \\ +% Hexagonal & $2/\log(2+\sqrt3)$ & $\frac8{\sqrt3}\arccosh2$ & $3^{3/2}/8$ +% \end{tabular} +% \caption{ +% The critical temperatures and amplitudes for the magetization along both the coexistence line and the critical isotherm, for three different lattices. +% } +% \label{tab:consts} +%\end{table} \begin{figure} - \input{figs/fig-mag} + \input{figs/fig-susmag} \caption{ - Fit of scaling form \eqref{eq:mag_scaling} to numeric data. Data with - sampling error taken from Monte Carlo simulations of an $L=2048$ - square-lattice Ising model with $T_\c-T=0.01,0.02,\ldots,0.1$ and - $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Solid line shows fitted form, - with $\fM(0)=1.21039\pm0.00031$, - $D=0.09400\pm0.00035$, and $B=0.0861\pm0.0010$. + Comparisons of scaling forms \eqref{eq:sus_scaling} and + \eqref{eq:mag_scaling} to numeric data. Data with + sampling error taken from Monte Carlo simulations of a $4096\times4096$ + square-lattice Ising model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ and + $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid lines show our + analytic results, while the dashed lines have polynomial corrections of + the form \eqref{eq:poly} fit to the data for $N=3$, with $C_1=-0.00368$, + $C_2=-0.0191$, $C_3=0.0350$, and $\lambda=2.42$. } - \label{fig:mag} + \label{fig:scaling_fits} \end{figure} We have used results from the properties of the metastable Ising ferromagnet |