diff options
| -rw-r--r-- | data_square-4096.dat | 486 | ||||
| -rw-r--r-- | essential-ising.tex | 86 | ||||
| -rw-r--r-- | figs/fig-mag.gplot | 44 | ||||
| -rw-r--r-- | figs/fig-mag_data-func.dat | 1001 | ||||
| -rw-r--r-- | figs/fig-mag_data-num.dat | 500 | ||||
| -rw-r--r-- | figs/fig-sus.gplot | 46 | ||||
| -rw-r--r-- | figs/fig-sus_data-func.dat | 1001 | ||||
| -rw-r--r-- | figs/fig-sus_data-num.dat | 500 | ||||
| -rw-r--r-- | figs/fig-susmag.gplot | 81 | ||||
| -rw-r--r-- | makefile | 2 |
10 files changed, 129 insertions, 3618 deletions
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+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 diff --git a/figs/fig-mag.gplot b/figs/fig-mag.gplot deleted file mode 100644 index c5c00db..0000000 --- a/figs/fig-mag.gplot +++ /dev/null @@ -1,44 +0,0 @@ - -set terminal pslatex rotate size 3.417,2.111 -set multiplot - -Tc = 2 / log(1 + sqrt(2)) -Delta = 15. / 8 -beta = 1. / 8 - -t(T) = abs((T - Tc) / T) -h(T, H) = H / T -X(T, H) = h(T, H) * t(T)**(-Delta) - -B = 1 / (2**(27./16) * pi * asinh(1)**(15./8)) -C0 = 0.0255369719 -#A = 2**(11./8) * pi**3 * asinh(1)**(19./4) * C0 -A = pi / 2 * C0 / (B**2 * Tc) -M0 = (2 * (sqrt(2) - 1))**(1./4) * ((4 + 3 * sqrt(2)) * asinh(1))**(1./8) - -func = "figs/fig-mag_scaling-func.dat" -num = "data/data_square-4096.dat" - -set samples 10000 -set key off - -set xrange [-20:1200] -set yrange [1.18:1.75] -set ylabel offset 1,0 '$M|t|^{-\beta}$' -set xlabel '$h|t|^{-\Delta}$' - -plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc rgb "black" - -set size 0.6,0.6 -set origin 0.38,0.2 -set xrange [-3:3] -set yrange [1.18:1.75] -set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' -set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' -set xtics format '\footnotesize$10^{%g}$' -2,2,3 -set ytics format '\footnotesize %g' - -plot num using (log10(X($2, $3))):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using ($1-log10(B)):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black - - - diff --git a/figs/fig-mag_data-func.dat b/figs/fig-mag_data-func.dat deleted file mode 100644 index 29653f0..0000000 --- a/figs/fig-mag_data-func.dat +++ /dev/null @@ -1,1001 +0,0 @@ --5. -4.698976620880227 --4.99 -4.688976675606296 --4.98 -4.678976738233927 --4.97 -4.668977103542201 --4.96 -4.658977175055251 --4.95 -4.64897725444246 --4.94 -4.6389776168638575 --4.93 -4.628977705544413 --4.92 -4.618977802145736 --4.91 -4.608977906594962 --4.9 -4.598978269856612 --4.89 -4.588978384162751 --4.88 -4.578978506414977 --4.87 -4.568978870802035 --4.86 -4.558979003497131 --4.85 -4.548979144167584 --4.84 -4.538979511480103 --4.83 -4.528979663220596 --4.82 -4.518979823098883 --4.81 -4.508980093110478 --4.8 -4.498980366682533 --4.79 -4.488980546555846 --4.78 -4.478980829968869 --4.77 -4.468981024367286 --4.76 -4.45898131817752 --4.75 -4.448981616365706 --4.74 -4.438981832304746 --4.73 -4.428982141813062 --4.72 -4.41898245622204 --4.71 -4.408982694750725 --4.7 -4.398983021480837 --4.69 -4.38898327631837 --4.68 -4.378983615975297 --4.67 -4.368983961528195 --4.66 -4.3589842409399875 --4.65 -4.3489846005174 --4.64 -4.338984897588365 --4.63 -4.328985271917661 --4.62 -4.318985653142753 --4.61 -4.30898597712615 --4.6 -4.298986374275912 --4.59 -4.288986717536211 --4.58 -4.278987131460024 --4.57 -4.268987553391627 --4.56 -4.258987926201059 --4.55 -4.248988366205847 --4.54 -4.238988814948243 --4.53 -4.228989219150657 --4.52 -4.218989687352525 --4.51 -4.208990139510039 --4.5 -4.198990602608149 --4.49 -4.188991076885322 --4.48 -4.178991562587806 --4.47 -4.168992083269395 --4.46 -4.158992592054487 --4.45 -4.14899311305065 --4.4399999999999995 -4.138993646517517 --4.43 -4.128994214017157 --4.42 -4.11899477281752 --4.41 -4.10899534496253 --4.4 -4.098995930752351 --4.39 -4.088996549873465 --4.38 -4.078997163435489 --4.37 -4.068997791591377 --4.36 -4.058998434672682 --4.35 -4.048999092999943 --4.34 -4.038999784226322 --4.33 -4.029000473731878 --4.32 -4.019001179564121 --4.31 -4.009001902084939 --4.3 -3.9990026574533086 --4.29 -3.9890034141655693 --4.28 -3.9790041887463223 --4.27 -3.969004981596725 --4.26 -3.9590057931477527 --4.25 -3.949006637866626 --4.24 -3.939007487773016 --4.23 -3.929008357692256 --4.22 -3.9190092480716765 --4.21 -3.909010165801212 --4.2 -3.899011104656105 --4.1899999999999995 -3.889012059012884 --4.18 -3.879013041771961 --4.17 -3.8690140413498035 --4.16 -3.8590150643886725 --4.15 -3.8490161170082184 --4.14 -3.8390171884668987 --4.13 -3.8290182903827445 --4.12 -3.8190194125471444 --4.11 -3.8090205660956147 --4.1 -3.799021741331879 --4.09 -3.789022948958459 --4.08 -3.779024179771146 --4.07 -3.769025444034952 --4.06 -3.7590267330222162 --4.05 -3.7490280521725925 --4.04 -3.7390294065078074 --4.03 -3.7290307879815963 --4.02 -3.719032205884891 --4.01 -3.7090336526107213 --4. -3.699035137097331 --3.99 -3.689036652139535 --3.98 -3.679038206348264 --3.9699999999999998 -3.669039792916126 --3.96 -3.659041420145048 --3.95 -3.6490430816057717 --3.94 -3.639044781858449 --3.9299999999999997 -3.6290465251671513 --3.92 -3.6190483056529694 --3.91 -3.6090501309051843 --3.9 -3.5990519969600037 --3.8899999999999997 -3.5890539049175336 --3.88 -3.579055858890956 --3.87 -3.5690578583445545 --3.86 -3.5590599043356437 --3.8499999999999996 -3.549061997949381 --3.84 -3.539064140294103 --3.83 -3.529066332504504 --3.8200000000000003 -3.5190685757400506 --3.81 -3.5090708711870926 --3.8 -3.4990732200614882 --3.79 -3.4890756223915274 --3.7800000000000002 -3.479078081909809 --3.77 -3.469080598671056 --3.76 -3.4590831740086907 --3.75 -3.449085809287373 --3.74 -3.439088505899216 --3.73 -3.4290912652708436 --3.7199999999999998 -3.419094088863227 --3.71 -3.409096978171543 --3.7 -3.399099934723846 --3.69 -3.3891029600820928 --3.6799999999999997 -3.379106054903357 --3.67 -3.3691092227381514 --3.66 -3.359112464293308 --3.65 -3.349115781283293 --3.6399999999999997 -3.339119175463116 --3.63 -3.3291226486284966 --3.62 -3.3191262026152755 --3.61 -3.3091298393019732 --3.5999999999999996 -3.299133560219415 --3.59 -3.2891373681310316 --3.58 -3.2791412646469493 --3.5700000000000003 -3.2691452518265933 --3.56 -3.2591493317778184 --3.55 -3.249153506306756 --3.54 -3.2391577783300276 --3.5300000000000002 -3.2291621497455276 --3.52 -3.2191666228630815 --3.51 -3.2091712000467383 --3.5 -3.1991758837159594 --3.49 -3.189180676039264 --3.48 -3.179185580164998 --3.4699999999999998 -3.169190598372289 --3.46 -3.15919573331219 --3.45 -3.149200987695975 --3.44 -3.139206364026145 --3.4299999999999997 -3.129211865692244 --3.42 -3.119217495321564 --3.41 -3.109223255885425 --3.4 -3.0992291504242426 --3.3899999999999997 -3.0892351818078763 --3.38 -3.0792413537100236 --3.37 -3.069247669139224 --3.36 -3.0592541314274238 --3.3499999999999996 -3.049260743984669 --3.34 -3.0392675102999243 --3.33 -3.029274433730568 --3.3200000000000003 -3.019281518355097 --3.31 -3.009288767592547 --3.3 -2.99929618556805 --3.29 -2.9893037757942587 --3.2800000000000002 -2.9793115425672396 --3.27 -2.9693194897899162 --3.26 -2.9593276218370663 --3.25 -2.9493359428995682 --3.24 -2.9393444572755207 --3.23 -2.929353169623009 --3.2199999999999998 -2.9193620844433785 --3.21 -2.909371206351528 --3.2 -2.8993805403120763 --3.19 -2.8893900910799566 --3.1799999999999997 -2.879399863837082 --3.17 -2.869409863648785 --3.16 -2.859420095704796 --3.15 -2.849430565532074 --3.1399999999999997 -2.8394412785648133 --3.13 -2.829452240370517 --3.12 -2.819463456847051 --3.11 -2.809474933758619 --3.0999999999999996 -2.7994866772662554 --3.09 -2.7894986934758235 --3.08 -2.779510988640467 --3.0700000000000003 -2.7695235693382174 --3.06 -2.759536442114671 --3.05 -2.7495496136729214 --3.04 -2.73956309103892 --3.0300000000000002 -2.7295768811752987 --3.02 -2.7195909914236975 --3.01 -2.7096054291312823 --3. -2.699620201844405 --2.9899999999999998 -2.689635317333078 --2.98 -2.6796507835196555 --2.9699999999999998 -2.6696666085307026 --2.96 -2.659682800609395 --2.95 -2.6496993682585663 --2.94 -2.639716320151525 --2.93 -2.6297336651607974 --2.92 -2.6197514123815897 --2.91 -2.609769571055506 --2.9 -2.59978815069819 --2.89 -2.5898071610409876 --2.88 -2.5798266119785587 --2.87 -2.5698465136915165 --2.86 -2.5598668765719665 --2.85 -2.5498877112671097 --2.84 -2.539909028613332 --2.83 -2.5299308397484213 --2.82 -2.519953156044999 --2.81 -2.5099759891510587 --2.8 -2.4999993509305676 --2.79 -2.4900232535671973 --2.78 -2.480047709504156 --2.77 -2.470072731482451 --2.76 -2.460098332487579 --2.75 -2.450124525845017 --2.7399999999999998 -2.440151325166729 --2.73 -2.430178744386789 --2.7199999999999998 -2.4202067977142354 --2.71 -2.4102354997207076 --2.6999999999999997 -2.4002648652996363 --2.69 -2.3902949096623476 --2.68 -2.3803256483886286 --2.67 -2.370357097385258 --2.66 -2.360389272942012 --2.65 -2.35042219169209 --2.64 -2.3404558706661645 --2.63 -2.3304903272556396 --2.62 -2.3205255792649755 --2.61 -2.310561644876933 --2.6 -2.3005985427034865 --2.59 -2.290636291753244 --2.58 -2.2806749114810807 --2.57 -2.2707144217574156 --2.56 -2.260754842916369 --2.55 -2.2507961957276423 --2.54 -2.2408385014428007 --2.53 -2.2308817817693334 --2.52 -2.2209260589159023 --2.51 -2.2109713555681405 --2.5 -2.2010176949328364 --2.4899999999999998 -2.191065100715562 --2.48 -2.181113597163801 --2.4699999999999998 -2.171163209046577 --2.46 -2.1612139616962747 --2.4499999999999997 -2.1512658809902976 --2.44 -2.1413189933918875 --2.43 -2.1313733259336227 --2.42 -2.121428906256971 --2.41 -2.111485762598099 --2.4 -2.1015439238261844 --2.39 -2.091603419430962 --2.38 -2.081664279558834 --2.37 -2.0717265350094807 --2.36 -2.0617902172565485 --2.35 -2.051855358460175 --2.34 -2.0419219914805016 --2.33 -2.0319901498921658 --2.32 -2.0220598679982764 --2.31 -2.0121311808453988 --2.3 -2.002204124238002 --2.29 -1.9922787347538824 --2.28 -1.9823550497591325 --2.27 -1.9724331074239951 --2.26 -1.962512946738382 --2.25 -1.952594607528142 --2.2399999999999998 -1.942678130471098 --2.23 -1.932763557113799 --2.2199999999999998 -1.9228509298880272 --2.21 -1.9129402921280458 --2.1999999999999997 -1.903031688087607 --2.19 -1.893125162957692 --2.18 -1.8832207628839481 --2.17 -1.873318534984801 --2.16 -1.8634185273697916 --2.15 -1.8535207891577528 --2.14 -1.8436253704956727 --2.13 -1.8337323225775013 --2.12 -1.8238416976634244 --2.11 -1.8139535490990388 --2.1 -1.8040679313354107 --2.09 -1.7941848999477779 --2.08 -1.7843045116571161 --2.07 -1.7744268243493577 --2.06 -1.7645518970961578 --2.05 -1.7546797901756386 --2.04 -1.7448105650930918 --2.03 -1.734944284602154 --2.02 -1.7250810127260756 --2.01 -1.7152208147790668 --2. -1.7053637573881035 --1.9899999999999998 -1.695509908514507 --1.98 -1.6856593374759956 --1.9699999999999998 -1.6758121149687653 --1.96 -1.6659683130897343 --1.9499999999999997 -1.6561280053589285 --1.94 -1.646291266741926 --1.9299999999999997 -1.6364581736725032 --1.92 -1.6266288040754073 --1.9100000000000001 -1.6168032373890027 --1.9 -1.6069815545882935 --1.8900000000000001 -1.597163838207777 --1.88 -1.5873501723644963 --1.87 -1.5775406427810115 --1.8599999999999999 -1.5677353368084173 --1.85 -1.5579343434495032 --1.8399999999999999 -1.5481377533816552 --1.83 -1.5383456589799542 --1.8199999999999998 -1.528558154340099 --1.81 -1.5187753353012445 --1.7999999999999998 -1.5089972994688996 --1.79 -1.4992241462375855 --1.7799999999999998 -1.4894559768133293 --1.77 -1.4796928942361862 --1.7599999999999998 -1.4699350034023968 --1.75 -1.4601824110865107 --1.7399999999999998 -1.4504352259631639 --1.73 -1.440693558628672 --1.7199999999999998 -1.4309575216223198 --1.71 -1.421227229447387 --1.6999999999999997 -1.411502798591876 --1.69 -1.4017843475487688 --1.6799999999999997 -1.3920719968360253 --1.67 -1.3823658690161156 --1.6600000000000001 -1.3726660887150783 --1.65 -1.3629727826411806 --1.6400000000000001 -1.3532860796030315 --1.63 -1.3436061105271015 --1.62 -1.333933008474815 --1.6099999999999999 -1.3242669086588303 --1.6 -1.3146079484589026 --1.5899999999999999 -1.3049562674368613 --1.58 -1.2953120073510491 --1.5699999999999998 -1.2856753121699165 --1.56 -1.2760463280847851 --1.5499999999999998 -1.266425203521926 --1.54 -1.2568120891536827 --1.5299999999999998 -1.247207137908661 --1.52 -1.2376105049810833 --1.5099999999999998 -1.2280223478390593 --1.5 -1.2184428262319036 --1.4899999999999998 -1.2088721021963773 --1.48 -1.1993103400618217 --1.4699999999999998 -1.1897577064541625 --1.46 -1.1802143702987218 --1.4499999999999997 -1.1706805028218277 --1.44 -1.1611562775511566 --1.4299999999999997 -1.15164187031478 --1.42 -1.1421374592388622 --1.4100000000000001 -1.1326432247439977 --1.4 -1.123159349540098 --1.3900000000000001 -1.1136860186198891 --1.38 -1.104223419250825 --1.37 -1.094771740965557 --1.3599999999999999 -1.0853311755507704 --1.35 -1.0759019170344577 --1.3399999999999999 -1.0664841616715452 --1.33 -1.0570781079278306 --1.3199999999999998 -1.047683956462245 --1.31 -1.0383019101073474 --1.2999999999999998 -1.028932173848078 --1.29 -1.019574954798705 --1.2799999999999998 -1.0102304621779452 --1.27 -1.0008989072822523 --1.2599999999999998 -0.9915805034572028 --1.25 -0.9822754660670188 --1.2399999999999998 -0.9729840124621528 --1.23 -0.9637063619449444 --1.2199999999999998 -0.9544427357333266 --1.21 -0.945193356922564 --1.1999999999999997 -0.9359584504450108 --1.19 -0.9267382430278873 --1.1799999999999997 -0.9175329631490576 --1.17 -0.908342840990786 --1.1600000000000001 -0.899168108391533 --1.15 -0.8900089987957054 --1.1400000000000001 -0.8808657472014276 --1.13 -0.8717385901063053 --1.12 -0.8626277654512194 --1.1099999999999999 -0.8535335125621172 --1.1 -0.8444560720898667 --1.0899999999999999 -0.8353956859481511 --1.08 -0.8263525972494354 --1.0699999999999998 -0.8173270502390506 --1.06 -0.8083192902273725 --1.0499999999999998 -0.7993295635201763 --1.04 -0.790358117347167 --1.0299999999999998 -0.7814051997887319 --1.02 -0.7724710597009488 --1.0099999999999998 -0.7635559466389006 --1. -0.7546601107783285 --0.9900000000000002 -0.745783802835678 --0.9799999999999995 -0.7369272739865957 --0.9699999999999998 -0.7280907757829148 --0.96 -0.71927456006819 --0.9500000000000002 -0.710478878891856 --0.9399999999999995 -0.7017039844220527 --0.9299999999999997 -0.692950128857191 --0.9199999999999999 -0.6842175643363237 --0.9100000000000001 -0.6755065428483992 --0.9000000000000004 -0.6668173161404544 --0.8899999999999997 -0.6581501356248398 --0.8799999999999999 -0.6495052522855466 --0.8700000000000001 -0.6408829165837034 --0.8600000000000003 -0.6322833783623495 --0.8499999999999996 -0.623706886750539 --0.8399999999999999 -0.6151536900668886 --0.8300000000000001 -0.6066240357226315 --0.8200000000000003 -0.5981181701242757 --0.8099999999999996 -0.5896363385759692 --0.7999999999999998 -0.5811787851816343 --0.79 -0.5727457527469868 --0.7800000000000002 -0.5643374826815244 --0.7699999999999996 -0.5559542149005716 --0.7599999999999998 -0.5475961877274909 --0.75 -0.53926363779613 --0.7400000000000002 -0.5309567999536288 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0.00004158329141608537 -7.383606715786582 1.2791108435331098 0.000029319131676017066 -19.55257415073327 1.3376071366101412 0.000030292064810427814 -17.3461628974399 1.3280960335537169 0.000031189876269939783 -84.30660745665303 1.4408037359643604 0.00003068334519370323 -574.1962089538827 1.624631608929271 0.00005435541210213813 -39.0762492247403 1.3831816213355879 0.00003044237299058765 -10.741717550626605 1.298236558077201 0.000027921277193101825 -220.42841371606121 1.5288903086509298 0.00004407832575303174 -102.64541615442957 1.460646965581937 0.00003569882228363849 -9.510739490224248 1.295217760381441 0.00003175656916650802 -8.78064417526713 1.286541441436241 0.000026700623415373695 -23.25207760064521 1.347592711754534 0.00003159967868888296 -20.628171367700244 1.337623810799078 0.000027808928460787825 -682.8387252576288 1.6416588258900962 0.00004789259177285275 -46.46971627785783 1.394554147576384 0.000029722313113898786 -11.310244655967265 1.3033915929920061 0.000033361006018547356 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\ No newline at end of file diff --git a/figs/fig-sus.gplot b/figs/fig-sus.gplot deleted file mode 100644 index 49eee6b..0000000 --- a/figs/fig-sus.gplot +++ /dev/null @@ -1,46 +0,0 @@ - -set terminal pslatex rotate size 3.417,2.111 -set multiplot - -Tc = 2 / log(1 + sqrt(2)) -Delta = 15. / 8 -gamma = 7. / 4 - -t(T) = abs((T - Tc) / T) -h(T, H) = H / T -X(T, H) = h(T, H) * t(T)**(-Delta) - -B = 1 / (2**(27./16) * pi * asinh(1)**(15./8)) -A = 0.0255369719 / Tc * pi / (2 * B**2) - -C1 = -0.0160968 -C2 = 0.116242 -C3 = -0.402641 -lambda = 4.39547 - -func = "figs/fig-sus_scaling-func.dat" -num = "data/data_square-4096.dat" - -set samples 10000 -set key off - -set xrange [-20:1200] -set yrange [-0.5:13] -set ylabel offset 1,0 '$\chi|t|^{\gamma}\times 10^3$' -set xlabel '$h|t|^{-\Delta}$' - -plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", func using (10**$1 / B):(10**(3+$2) * A * B**2 + 10**3 * (C1 * (10**$1) + C2 * (10**$1)**2 + C3 * (10**$1)**3) / (1 + (lambda * (10**$1))**4)) with lines lt 5 lw 1 lc rgb "red" - -set size 0.6,0.6 -set origin 0.38,0.37 -set xrange [-3:3] -set yrange [-4.5:-1.5] -set ylabel offset 2.5,0 '\footnotesize$\chi|t|^\gamma$' -set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' -set xtics format '\footnotesize$10^{%g}$' -2,2,3 -set ytics format '\footnotesize$10^{%g}$' -4,1,-2 - -plot num using (log10(X($2, $3))):(log10($10 * t($2)**gamma)):($11 / $10):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, func using ($1-log10(B)):($2+log10(A * B**2)) with linespoints pt 0 lw 2 lc rgb "black", func using ($1-log10(B)):(log10(10**$2 * A * B**2 + (C1 * (10**$1) + C2 * (10**$1)**2 + C3 * (10**$1)**3) / (1 + (lambda * (10**$1))**4))) with lines lt 5 lw 1 lc rgb "red" - - - diff --git a/figs/fig-sus_data-func.dat b/figs/fig-sus_data-func.dat deleted file mode 100644 index 972c77b..0000000 --- a/figs/fig-sus_data-func.dat +++ /dev/null @@ -1,1001 +0,0 @@ --5. 0.010094128168228536 --4.99 0.0004986394100467353 --4.98 -0.008441735254514486 --4.97 0.010954806238037133 --4.96 0.002527852467923256 --4.95 -0.005308016370920201 --4.94 0.0114293828525401 --4.93 0.004018232985220573 --4.92 -0.002858226194217986 --4.91 -0.009219894501557 --4.9 0.005081385417224557 --4.89 -0.0009618799060500723 --4.88 -0.006542816311444805 --4.87 0.005798926687913552 --4.86 0.0004888958038767844 --4.85 -0.004414469974308102 --4.84 0.006246050655940169 --4.83 0.0015752929121863565 --4.82 -0.0027318837493295303 --4.81 -0.00005661662281440823 --4.8 0.002370862728926167 --4.79 -0.0014169159468057889 --4.78 0.0008653476550368721 --4.77 -0.0025424410197080745 --4.76 -0.0003992250773220128 --4.75 0.0015469120819587867 --4.74 -0.001449908548519092 --4.73 0.00037816511803153026 --4.72 0.0020366851344616396 --4.71 -0.000600885400499167 --4.7 0.0009588492800455502 --4.69 -0.0014114510648439034 --4.68 0.000051240900424078435 --4.67 0.001381830333725127 --4.66 -0.0007050759037017436 --4.65 0.0005442342992692582 --4.64 -0.0013309799160835763 --4.63 -0.00015920310383799646 --4.62 0.00090685946257924 --4.61 -0.0007436599033119168 --4.6 0.00025588478516999245 --4.59 -0.0012259011969075148 --4.58 -0.0002881636867819425 --4.57 0.000565656154053291 --4.56 -0.0007402474141691176 --4.55 0.00006049475547055314 --4.54 0.0007892444246832854 --4.53 -0.00036070222068024964 --4.52 0.00032246997992554417 --4.51 0.00011803988377951132 --4.5 -0.00006888425539588968 --4.49 -0.00023948312541515455 --4.48 -0.0003949015873374922 --4.47 0.00015190608910263618 --4.46 -6.6898721768690905e-6 --4.45 -0.00015161214983287755 --4.4399999999999995 -0.00028403138338422624 --4.43 0.00016823072195665058 --4.42 0.00003363534826398597 --4.41 -0.00008944735150832387 --4.4 -0.00020179211112555123 --4.39 0.00017169958797518965 --4.38 0.000057425176276815505 --4.37 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-0.00013913297587281638 --4.04 -0.00011482127614658982 --4.03 -0.00013830914731367494 --4.02 -0.00011673800758250929 --4.01 -0.0001387102203417803 --4. -0.00011954970898984866 --3.99 -0.00014014783895817452 --3.98 -0.00012325673626363838 --3.9699999999999998 -0.00014262512687336033 --3.96 -0.00012776515321090274 --3.95 -0.00014597702279445089 --3.94 -0.00016298061823332674 --3.9299999999999997 -0.00015021536486419364 --3.92 -0.000166282243401589 --3.91 -0.00015521121853453868 --3.9 -0.00015801005761407916 --3.8899999999999997 -0.00017289193186674 --3.88 -0.0001754995870406425 --3.87 -0.00017830377526368898 --3.86 -0.00018131275010842317 --3.8499999999999996 -0.0001845069139014325 --3.84 -0.00018788018188933215 --3.83 -0.00019142848961655414 --3.8200000000000003 -0.00019515665850990488 --3.81 -0.00019906684890120174 --3.8 -0.00020315241152553515 --3.79 -0.00021490772480537348 --3.7800000000000002 -0.00021898372684717342 --3.77 -0.00022325030826297776 --3.76 -0.0002276981322952671 --3.75 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--3.45 -0.0004615682990100785 --3.44 -0.0004729405272929491 --3.4299999999999997 -0.0004837916543026933 --3.42 -0.0004949091077014542 --3.41 -0.0005063002798864301 --3.4 -0.0005179718353941347 --3.3899999999999997 -0.0005305228409217725 --3.38 -0.0005427403281909289 --3.37 -0.0005552548568725174 --3.36 -0.0005680734695650386 --3.3499999999999996 -0.0005812011839741446 --3.34 -0.000594643286150463 --3.33 -0.0006088590400995427 --3.3200000000000003 -0.0006229318141184147 --3.31 -0.00063754802542772 --3.3 -0.0006520959264674927 --3.29 -0.0006673881043398832 --3.2800000000000002 -0.0006828426314800055 --3.27 -0.0006988363931001364 --3.26 -0.0007150251360428137 --3.25 -0.0007315968499505627 --3.24 -0.0007487092605883688 --3.23 -0.0007660666668581646 --3.2199999999999998 -0.0007838333705097472 --3.21 -0.0008021477637153458 --3.2 -0.0008207538652094974 --3.19 -0.000839916450953925 --3.1799999999999997 -0.0008594023462756419 --3.17 -0.0008793449606781264 --3.16 -0.0008998585744393441 --3.15 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--2.84 -0.0018764744139451385 --2.83 -0.0019200259537334508 --2.82 -0.0019645841926706487 --2.81 -0.002010161750931191 --2.8 -0.002056803609383859 --2.79 -0.0021045228845014597 --2.78 -0.0021533445151011147 --2.77 -0.0022032850034109366 --2.76 -0.00225438790039576 --2.75 -0.002306670507446541 --2.7399999999999998 -0.0023601599793289787 --2.73 -0.0024148766790701138 --2.7199999999999998 -0.002470863533935367 --2.71 -0.0025281418872277355 --2.6999999999999997 -0.002586738190313637 --2.69 -0.002646688947676934 --2.68 -0.0027080189256341804 --2.67 -0.0027707655554273986 --2.66 -0.002834955427618004 --2.65 -0.00290062683703282 --2.64 -0.002967808411423396 --2.63 -0.0030365394677594796 --2.62 -0.0031068505252962075 --2.61 -0.0031787820724533014 --2.6 -0.003252366593645861 --2.59 -0.0033276458287398365 --2.58 -0.003404654266771937 --2.57 -0.0034834349420884464 --2.56 -0.003564024463496181 --2.55 -0.0036464672507797165 --2.54 -0.003730802012476773 --2.53 -0.0038170747276273656 --2.52 -0.0039053262310897803 --2.51 -0.003995604150857169 --2.5 -0.004087951501388745 --2.4899999999999998 -0.004182417646514152 --2.48 -0.004279047886384368 --2.4699999999999998 -0.004377893388078129 --2.46 -0.004479001793067613 --2.4499999999999997 -0.004582426212866536 --2.44 -0.0046882166769889105 --2.43 -0.004796428318102009 --2.42 -0.004907113670032493 --2.41 -0.005020329988918604 --2.4 -0.005136132341465379 --2.39 -0.005254580257299012 --2.38 -0.005375731779775511 --2.37 -0.005499647730983637 --2.36 -0.0056263898918032925 --2.35 -0.005756021310188049 --2.34 -0.005888606430569073 --2.33 -0.006024211016306962 --2.32 -0.006162902275618674 --2.31 -0.006304748798499813 --2.3 -0.006449820661072364 --2.29 -0.00659818938295447 --2.28 -0.006749928022982928 --2.27 -0.0069051111415831465 --2.26 -0.007063814887433234 --2.25 -0.007226116973457367 --2.2399999999999998 -0.007392096760524994 --2.23 -0.007561835230730505 --2.2199999999999998 -0.007735415069329403 --2.21 -0.007912920647854224 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\ No newline at end of file diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot new file mode 100644 index 0000000..dd002a4 --- /dev/null +++ b/figs/fig-susmag.gplot @@ -0,0 +1,81 @@ + +set terminal pslatex rotate size 3.417,4.222 +set multiplot + +Tc = 2 / log(1 + sqrt(2)) +Delta = 15. / 8 +gamma = 7. / 4 +beta = 1. / 8 + +t(T) = abs((T - Tc) / T) +h(T, H) = H / T +X(T, H) = h(T, H) * t(T)**(-Delta) + +poly(A, l, m, X) = A * X**m / (1 + (l * X)**(m + 1)) +polyint(A, l, m, X) = A * l**(-(m + 1)) * log(1 + (l * X)**(m + 1)) / (m + 1) + +M0 = (2**2.5 * asinh(1))**0.125 +B = Tc**2 * M0 / (16 * pi) +C0 = 0.0255369719 +A = pi / 2 * C0 / (B**2 * Tc) + +C1 = -0.00368218 +C2 = -0.0190753 +C3 = 0.0349601 +lamb = 2.41718 + +susfunc = "figs/fig-sus_scaling-func.dat" +magfunc = "figs/fig-mag_scaling-func.dat" + +num = "data/data_square-4096.dat" + +set samples 10000 +set key off + +set size 1,1 - 0.52 +set origin 0,0.52 +set xrange [-20:1200] +set yrange [-0.5:13] +set ylabel offset 1,0 '$\chi|t|^{\gamma}\times 10^3$' +set lmargin 6 +set xtics format '' +set bmargin 0.2 + +plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B):(10**(3+$2) * A * B**2 + 10**3 * (poly(C1, lamb, 1, 10**$1) + poly(C2, lamb, 2, 10**$1) + poly(C3, lamb, 3, 10**$1))) with lines dt 2 lw 2 lc black + +set bmargin -1 +set tmargin 0.2 +set size 1,0.52 +set origin 0,0 +set xrange [-20:1200] +set yrange [1.18:1.75] +set ylabel offset 1,0 '$M|t|^{-\beta}$' +set xlabel '$h|t|^{-\Delta}$' +set xtics format '%g' + +plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) smooth csplines with lines dt 2 lw 2 lc black + +set tmargin -1 +set lmargin -1 +set size 0.65,0.325 +set origin 0.31,0.5 + 0.29 / 2 +set xrange [-3:3] +set yrange [-4.5:-1.5] +set ylabel offset 2.5,0 '\footnotesize$\chi|t|^\gamma$' +set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' +set xtics format '\footnotesize$10^{%g}$' -2,2,3 +set ytics format '\footnotesize$10^{%g}$' -4,1,-2 + +plot num using (log10(X($2, $3))):(log10($10 * t($2)**gamma)):($11 / $10):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using ($1-log10(B)):($2+log10(A * B**2)) with linespoints pt 0 lw 2 lc rgb "black", susfunc using ($1-log10(B)):(log10(10**$2 * A * B**2 + poly(C1, lamb, 1, 10**$1) + poly(C2, lamb, 2, 10**$1) + poly(C3, lamb, 3, 10**$1))) with lines dt 2 lw 2 lc black + +set size 0.65,0.325 +set origin 0.31,0.23 / 2 +set xrange [-3:3] +set yrange [1.18:1.75] +set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' +set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' +set xtics format '\footnotesize$10^{%g}$' -2,2,3 +set ytics format '\footnotesize {%g}' 1.2,.2,1.6 + +plot num using (log10(X($2, $3))):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using ($1-log10(B)):(M0 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using ($1 - log10(B)):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) with lines dt 2 lw 2 lc black + @@ -1,6 +1,6 @@ DOC=essential-ising -FIGS=fig-sus fig-mag +FIGS=fig-susmag all: ${FIGS:%=figs/%.tex} rubber $(DOC).tex |