diff options
Diffstat (limited to 'ising_scaling.tex')
| -rw-r--r-- | ising_scaling.tex | 288 |
1 files changed, 20 insertions, 268 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index 7fa1ae6..5881786 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -20,7 +20,6 @@ linkcolor=purple \usepackage{graphicx} \usepackage{xcolor} \usepackage{tikz-cd} -\usepackage[subfolder]{gnuplottex} % need to compile separately for APS \usepackage{setspace} \usepackage{tabularx} @@ -246,7 +245,7 @@ s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant \begin{figure} - \includegraphics{figs/F_lower_singularities} + \includegraphics{F_lower_singularities} \caption{ Analytic structure of the low-temperature scaling function $\mathcal F_-$ in the complex $\xi=u_h|u_t|^{-\Delta}\propto H$ plane. The circle @@ -293,7 +292,7 @@ branch cuts beginning at $\pm i\xi_{\mathrm{YL}}$ for a universal constant $\xi_{\mathrm{YL}}$. \begin{figure} - \includegraphics{figs/F_higher_singularities} + \includegraphics{F_higher_singularities} \caption{ Analytic structure of the high-temperature scaling function $\mathcal F_+$ in the complex $\xi=u_h|u_t|^{-\Delta}\propto H$ plane. The squares @@ -355,81 +354,8 @@ In practice the infinite series in \eqref{eq:schofield.funcs} cannot be entirely fixed, and it will be truncated at finite order. \begin{figure} - \begin{gnuplot}[terminal=epslatex] - t0 = 1.36261 - g0 = 0.438453 - g1 = -0.0531270 - g2 = -0.00391478 - g3 = -0.000408016 - g4 = 0.0000262629 - g5 = -0.00000109745 - - f(t) = 1-t**2 - g(t) = (1-(t/t0)**2)*(g0*t + g1*t**3 + g2*t**5 + g3*t**7 + g4*t**9 + g5*t**11) - del = 15.0/8.0 - - set xlabel '$u_t$' - set ylabel '$u_h$' - unset key - - set xrange [-4.5:4.5] - set yrange [-3.5:3.5] - - set parametric - set style data lines - set trange [-t0:t0] - - plot sample \ - [-t0:t0:0.001] '+' u (0.5*f($1)):(0.5*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (f($1)):(g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (1.5*f($1)):(1.5**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (2*f($1)):(2**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (2.5*f($1)):(2.5**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (3*f($1)):(3**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (3.5*f($1)):(3.5**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (4*f($1)):(4**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (4.5*f($1)):(4.5**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (5*f($1)):(5**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (5.5*f($1)):(5.5**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (6*f($1)):(6**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (6.5*f($1)):(6.5**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (7*f($1)):(7**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (7.5*f($1)):(7.5**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (8*f($1)):(8**del*g($1)) lc black lw 2 , \ - [-t0:t0:0.001] '+' u (8.5*f($1)):(8.5**del*g($1)) lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(0)):($1**del*g(0)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(t0/16)):($1**del*g(t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(2*t0/16)):($1**del*g(2*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(3*t0/16)):($1**del*g(3*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(4*t0/16)):($1**del*g(4*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(5*t0/16)):($1**del*g(5*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(6*t0/16)):($1**del*g(6*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(7*t0/16)):($1**del*g(7*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(8*t0/16)):($1**del*g(8*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(9*t0/16)):($1**del*g(9*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(10*t0/16)):($1**del*g(10*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(11*t0/16)):($1**del*g(11*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(12*t0/16)):($1**del*g(12*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(13*t0/16)):($1**del*g(13*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(14*t0/16)):($1**del*g(14*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(15*t0/16)):($1**del*g(15*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(16*t0/16)):($1**del*g(16*t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(-t0/16)):($1**del*g(-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(2*-t0/16)):($1**del*g(2*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(3*-t0/16)):($1**del*g(3*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(4*-t0/16)):($1**del*g(4*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(5*-t0/16)):($1**del*g(5*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(6*-t0/16)):($1**del*g(6*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(7*-t0/16)):($1**del*g(7*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(8*-t0/16)):($1**del*g(8*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(9*-t0/16)):($1**del*g(9*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(10*-t0/16)):($1**del*g(10*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(11*-t0/16)):($1**del*g(11*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(12*-t0/16)):($1**del*g(12*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(13*-t0/16)):($1**del*g(13*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(14*-t0/16)):($1**del*g(14*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(15*-t0/16)):($1**del*g(15*-t0/16)) dt 2 lc black lw 2 - \end{gnuplot} + \centering + \includegraphics{ising_scaling-gnuplottex-fig1.pdf} \caption{ Example of the parametric coordinates. Solid lines are of constant $R=\frac12,1,\ldots,8\frac12$ and dashed lines are of constant @@ -495,7 +421,7 @@ $\theta$. Therefore, The location $\theta_0$ is not fixed by any principle. \begin{figure} - \includegraphics{figs/F_theta_singularities} + \includegraphics{F_theta_singularities} \caption{ Analytic structure of the global scaling function $\mathcal F$ in the complex $\theta$ plane. The circles depict essential singularities of the @@ -552,7 +478,7 @@ Fixing these requirements for the imaginary part of $\mathcal F(\theta)$ fixes its real part up to an analytic even function $G(\theta)$, real for real $\theta$. \begin{figure} - \includegraphics{figs/contour_path} + \includegraphics{contour_path} \caption{ Integration contour over the global scaling function $\mathcal F$ in the complex $\theta$ plane used to produce the dispersion relation. The @@ -914,34 +840,8 @@ values of both are plotted. \end{table} \begin{figure} - \begin{gnuplot}[terminal=epslatex] - set linetype 1 lc rgb "#5e81b5" lw 2 ps 2 - set linetype 2 lc rgb "#e19c24" lw 2 ps 2 - set linetype 3 lc rgb "#8fb032" lw 2 ps 2 - set linetype 4 lc rgb "#eb6235" lw 2 ps 2 - set linetype 5 lc rgb "#8778b3" lw 2 ps 2 - set linetype 6 lc rgb "#c56e1a" lw 2 ps 2 - - dat = 'data/phi_comparison.dat' - - set xlabel '$n$' - set xrange [1.5:6.5] - - set logscale y - set format y '$10^{%T}$' - set ylabel '$|\Delta\mathcal F_0^{(m)}|$' - set yrange [0.000000005:0.0005] - - set style data yerrorlines - set key title '\raisebox{0.5em}{$m$}' bottom left - - plot \ - dat using 1:2:3 title '0', \ - dat using 1:4:5 title '1', \ - dat using 1:6:7 title '2', \ - dat using 1:8:9 title '3', \ - dat using 1:10:11 title '4' - \end{gnuplot} + \centering + \includegraphics{ising_scaling-gnuplottex-fig2.pdf} \caption{ The error in the $m$th derivative of the scaling function $\mathcal F_0$ with respect to $\eta$ evaluated at $\eta=0$, as a function of the @@ -955,29 +855,8 @@ values of both are plotted. \begin{figure} - \begin{gnuplot}[terminal=epslatex] - set linetype 1 lc rgb "#5e81b5" lw 2 ps 2 - set linetype 2 lc rgb "#e19c24" lw 2 ps 2 - set linetype 3 lc rgb "#8fb032" lw 2 ps 2 - set linetype 4 lc rgb "#eb6235" lw 2 ps 2 - set linetype 5 lc rgb "#8778b3" lw 2 ps 2 - set linetype 6 lc rgb "#c56e1a" lw 2 ps 2 - - dat1 = 'data/phi_numeric.dat' - dat2 = 'data/phi_series_ours_2.dat' - dat3 = 'data/phi_series_ours_6.dat' - set key top right - set logscale y - set xlabel '$m$' - set ylabel '$|\mathcal F_0^{(m)}|$' - set format y '$10^{%T}$' - set xrange [-0.5:10.5] - - plot \ - dat1 using 1:(abs($2)):3 title 'Numeric' with yerrorbars, \ - dat2 using 1:(abs($2)) title 'This work ($n=2$)', \ - dat3 using 1:(abs($2)) title 'This work ($n=6$)' - \end{gnuplot} + \centering + \includegraphics{ising_scaling-gnuplottex-fig3.pdf} \caption{ The series coefficients for the scaling function $\mathcal F_0$ as a function of polynomial order $m$. The numeric values are from Table @@ -995,33 +874,8 @@ $\mathcal F_+$ in Fig.~\ref{fig:ghigh.series}, and for $\mathcal F_0$ in Fig.~\ref{fig:phi.series}. \begin{figure} - \begin{gnuplot}[terminal=epslatex] - set linetype 1 lc rgb "#5e81b5" lw 2 ps 2 - set linetype 2 lc rgb "#e19c24" lw 2 ps 2 - set linetype 3 lc rgb "#8fb032" lw 2 ps 2 - set linetype 4 lc rgb "#eb6235" lw 2 ps 2 - set linetype 5 lc rgb "#8778b3" lw 2 ps 2 - set linetype 6 lc rgb "#c56e1a" lw 2 ps 2 - - dat1 = 'data/glow_numeric.dat' - dat2 = 'data/glow_series_ours_0.dat' - dat3 = 'data/glow_series_ours_6.dat' - dat4 = 'data/glow_series_caselle.dat' - - set xlabel '$m$' - set xrange [0:14.5] - - set key top left Left reverse - set logscale y - set ylabel '$\mathcal F_-^{(m)}$' - set format y '$10^{%T}$' - - plot \ - dat1 using 1:(abs($2)):3 title 'Numeric' with yerrorbars, \ - dat2 using 1:(abs($2)) title 'This work ($n=2$)', \ - dat3 using 1:(abs($2)) title 'This work ($n=6$)', \ - dat4 using 1:(abs($2)) title 'Caselle \textit{et al.}' - \end{gnuplot} + \centering + \includegraphics{ising_scaling-gnuplottex-fig4.pdf} \caption{ The series coefficients for the scaling function $\mathcal F_-$ as a function of polynomial order $m$. The numeric values are from Table @@ -1031,32 +885,8 @@ Fig.~\ref{fig:phi.series}. \end{figure} \begin{figure} - \begin{gnuplot}[terminal=epslatex] - set linetype 1 lc rgb "#5e81b5" lw 2 ps 2 - set linetype 2 lc rgb "#e19c24" lw 2 ps 2 - set linetype 3 lc rgb "#8fb032" lw 2 ps 2 - set linetype 4 lc rgb "#eb6235" lw 2 ps 2 - set linetype 5 lc rgb "#8778b3" lw 2 ps 2 - set linetype 6 lc rgb "#c56e1a" lw 2 ps 2 - - dat1 = 'data/ghigh_numeric.dat' - dat2 = 'data/ghigh_series_ours_2.dat' - dat3 = 'data/ghigh_series_ours_6.dat' - dat4 = 'data/ghigh_caselle.dat' - - set key top left Left reverse - set logscale y - set xlabel '$m$' - set ylabel '$\mathcal F_+^{(m)}$' - set format y '$10^{%T}$' - set xrange [1.5:14.5] - - plot \ - dat1 using 1:(abs($2)):3 title 'Numeric' with yerrorbars, \ - dat2 using 1:(abs($2)) title 'This work ($n=2$)', \ - dat3 using 1:(abs($2)) title 'This work ($n=6$)', \ - dat4 using 1:(abs($2)) title 'Caselle \textit{et al.}' - \end{gnuplot} + \centering + \includegraphics{ising_scaling-gnuplottex-fig5.pdf} \caption{ The series coefficients for the scaling function $\mathcal F_+$ as a function of polynomial order $m$. The numeric values are from Table @@ -1081,34 +911,8 @@ would converge better. Notice that this infelicity does not appear to cause significant errors in the function $\mathcal F_-(\theta)$ or its low order derivatives, as evidenced by the convergence in Fig.~\ref{fig:error}. \begin{figure} - \begin{gnuplot}[terminal=epslatex] - set linetype 1 lc rgb "#5e81b5" lw 2 ps 2 - set linetype 2 lc rgb "#e19c24" lw 2 ps 2 - set linetype 3 lc rgb "#8fb032" lw 2 ps 2 - set linetype 4 lc rgb "#eb6235" lw 2 ps 2 - set linetype 5 lc rgb "#8778b3" lw 2 ps 2 - set linetype 6 lc rgb "#c56e1a" lw 2 ps 2 - - dat1 = 'data/glow_numeric.dat' - dat2 = 'data/glow_series_ours_0.dat' - dat3 = 'data/glow_series_ours_6.dat' - dat4 = 'data/glow_series_caselle.dat' - - set xlabel '$m$' - set xrange [0:14.5] - - set key top left Left reverse - set ylabel '$\mathcal F_-^{(m)}/\mathcal F_-^\infty(m)$' - - os = 1.3578383417065954956 - asmp(n) = os / (2 * pi) * (2 * os / pi)**(n-1) * gamma(n - 1) / pi - - plot \ - dat1 using 1:(abs($2) / asmp($1)):($3 / asmp($1)) title 'Numeric' with yerrorbars, \ - dat2 using 1:(abs($2) / asmp($1)) title 'This work ($n=2$)', \ - dat3 using 1:(abs($2) / asmp($1)) title 'This work ($n=6$)', \ - dat4 using 1:(abs($2) / asmp($1)) title 'Caselle \textit{et al.}' - \end{gnuplot} + \centering + \includegraphics{ising_scaling-gnuplottex-fig6.pdf} \caption{ The series coefficients for the scaling function $\mathcal F_-$ as a function of polynomial order $m$, rescaled by their asymptotic limit @@ -1119,35 +923,8 @@ Notice that this infelicity does not appear to cause significant errors in the f \end{figure} \begin{figure} - \begin{gnuplot}[terminal=epslatex] - set linetype 1 lc rgb "#5e81b5" lw 2 ps 2 - set linetype 2 lc rgb "#e19c24" lw 2 ps 2 - set linetype 3 lc rgb "#8fb032" lw 2 ps 2 - set linetype 4 lc rgb "#eb6235" lw 2 ps 2 - set linetype 5 lc rgb "#8778b3" lw 2 ps 2 - set linetype 6 lc rgb "#c56e1a" lw 2 ps 2 - - dat1 = 'data/ghigh_numeric.dat' - dat2 = 'data/ghigh_series_ours_2.dat' - dat3 = 'data/ghigh_series_ours_6.dat' - dat4 = 'data/ghigh_caselle.dat' - - set xlabel '$m$' - set ylabel '$\mathcal F_+^{(m)}/\mathcal F_+^\infty(m)$' - set yrange [0.8:1.5] - set xrange [1.5:14.5] - - xYL = 0.18930 - AYL = 1.37 - sigma = 0.833333333333 - asmp(n) = -AYL * 2 * exp(log(xYL)*(sigma-n))*gamma(sigma+1)/gamma(n+1)/gamma(sigma-n+1) - - plot \ - dat1 using 1:(abs($2) / abs(asmp($1))):($3 / asmp($1)) title 'Numeric' with yerrorbars, \ - dat2 using 1:(abs($2) / asmp($1)) title 'This work ($n=2$)', \ - dat3 using 1:(abs($2) / asmp($1)) title 'This work ($n=6$)', \ - dat4 using 1:(abs($2) / asmp($1)) title 'Caselle \textit{et al.}' - \end{gnuplot} + \centering + \includegraphics{ising_scaling-gnuplottex-fig7.pdf} \caption{ The series coefficients for the scaling function $\mathcal F_+$ as a function of polynomial order $m$, rescaled by their asymptotic limit @@ -1168,33 +945,8 @@ and the approximate functions defined here show the appropriate divergence in the ratio. \begin{figure} - \begin{gnuplot}[terminal=epslatex] - set linetype 1 lc rgb "#5e81b5" lw 2 ps 2 - set linetype 2 lc rgb "#e19c24" lw 2 ps 2 - set linetype 3 lc rgb "#8fb032" lw 2 ps 2 - set linetype 4 lc rgb "#eb6235" lw 2 ps 2 - set linetype 5 lc rgb "#8778b3" lw 2 ps 2 - set linetype 6 lc rgb "#c56e1a" lw 2 ps 2 - - dat1 = 'data/glow_numeric.dat' - dat2 = 'data/glow_series_ours_0.dat' - dat3 = 'data/glow_series_ours_6.dat' - dat4 = 'data/glow_series_caselle.dat' - ratLast(x) = (back2 = back1, back1 = x, back1 / back2) - back1 = 0 - back2 = 0 - - set xlabel '$1/m$' - set xrange [0:0.55] - set ylabel '$\mathcal F_-^{(m)}/\mathcal F_-^{(m-1)}$' - set yrange [0:15] - - plot \ - dat1 using (1/$1):(abs(ratLast($2))) title 'Numeric', \ - dat2 using (1/$1):(abs(ratLast($2))) title 'This work ($n=2$)', \ - dat3 using (1/$1):(abs(ratLast($2))) title 'This work ($n=6$)', \ - dat4 using (1/$1):(abs(ratLast($2))) title 'Caselle \textit{et al.}' - \end{gnuplot} + \centering + \includegraphics{ising_scaling-gnuplottex-fig8.pdf} \caption{ Sequential ratios of the series coefficients of the scaling function $\mathcal F_-$ as a function of inverse polynomial order $m$. The |