diff options
| -rw-r--r-- | essential-ising.bib | 1 | ||||
| -rw-r--r-- | essential-ising.tex | 20 | ||||
| -rw-r--r-- | figs/fig-series.gplot | 19 | ||||
| -rw-r--r-- | figs/fig-susmag.gplot | 36 |
4 files changed, 61 insertions, 15 deletions
diff --git a/essential-ising.bib b/essential-ising.bib index 16a0883..f490dd9 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -115,7 +115,6 @@ year={2013} } - @article{dimitrovic.1991.finite, title={Finite-size effects, goldstone bosons and critical exponents in the d= 3 Heisenberg model}, author={Dimitrovi{\'c}, I and Hasenfratz, P and Nager, J and Niedermayer, Ferenc}, diff --git a/essential-ising.tex b/essential-ising.tex index 9397aa0..ba8e452 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -281,6 +281,7 @@ and their constants of integration fixed by known zero field values, yielding &=\fM(0)+\frac{AB}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ \fF^\twodee(Y/B) &=-Y\bigg(\frac{\fM(0)}B-\frac{A}\pi e^{1/Y}\ei(-1/Y)\bigg) + \label{eq:2d_free_scaling} \end{align} with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in two dimensions. @@ -364,12 +365,13 @@ correction appears to match data quite well. $h|t|^{-\beta\delta}$. Points with error bars show data with sampling error taken from 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 $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Color denotes the value of - $T$, with red corresponding to $0.01$ and violet to $0.1$. The solid lines + 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}, while the dashed lines show fits of - \eqref{eq:sus_scaling_poly} and \eqref{eq:mag_scaling_poly} to the data - for $N=0$, with $c_0=-0.0124$ and $\lambda=1.77$. + \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. } \label{fig:scaling_fits} \end{figure} @@ -377,7 +379,13 @@ correction appears to match data quite well. \begin{figure} \input{fig-series} \caption{ - Something + 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. } \label{fig:series} \end{figure} diff --git a/figs/fig-series.gplot b/figs/fig-series.gplot index bde0cae..b9cfc57 100644 --- a/figs/fig-series.gplot +++ b/figs/fig-series.gplot @@ -1,13 +1,26 @@ set terminal pslatex rotate size 3.417,2.111 +cc1 = "#5e81b5" +cc2 = "#e19c24" +cc3 = "#8fb032" +cc4 = "#eb6235" + set logscale y data = "figs/fig-series-data.dat" -set xrange [0:20] -set yrange [0.000001:10] +set xrange [0:30] +set yrange [0.000004:600] set key off +set xlabel '$n$' +set ylabel offset 2 '$|f_n|$' + +set ytics format '\footnotesize$10^{%T}$' 0.00001,10,1000 -plot data using 1:2 with points, data using 1:3 with points, data using 1:4 with points, data using 1:5 with points +plot \ + data using 1:2 with points lc rgb cc1, \ + data using 1:3 with points lc rgb cc2, \ + data using 1:4 with points lc rgb cc3, \ + data using 1:5 with points lc rgb cc4 diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index 80679b6..c56c898 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -2,6 +2,11 @@ set terminal pslatex rotate size 3.417,4.222 set multiplot +cc1 = "#5e81b5" +cc2 = "#e19c24" +cc3 = "#8fb032" +cc4 = "#eb6235" + Tc = 2 / log(1 + sqrt(2)) Delta = 15. / 8 gamma = 7. / 4 @@ -52,7 +57,7 @@ magfunc = "figs/fig-mag_scaling-func.dat" num = "data/data_square-4096.dat" -set samples 10000 +set samples 20000 set key off set size 1,1 - 0.52 @@ -66,7 +71,12 @@ set mxtics 5 set mytics 5 set bmargin 0.2 -plot num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $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 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B3):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B3):(-10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (10**$1 / B3)**(i-1))) with lines dt 5 lw 2 lc black +plot \ + num using (X($2, $3)):(10**3 * Tc * $10 * t($2)**gamma):(10**3 * Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ + susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ + susfunc using (10**$1 / B2):(10**(3+$2) * A2 * B2**2 + 10**3 * (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines lw 2 lc rgb cc2 dt 2, \ + susfunc using (10**$1 / B3):(10**(3+$2) * A3 * B3**2 + 10**3 * (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines lw 2 lc rgb cc3 dt 3, \ + -10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (x)**(i-1)) with lines lw 2 lc rgb cc4 dt 4 set bmargin -1 set tmargin 0.2 @@ -77,7 +87,12 @@ set ylabel offset 1,0 '$M|t|^{-\beta}$' set xlabel '$h|t|^{-\beta\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) * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) smooth csplines with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) smooth csplines with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10**$1 / B)**(i-1)) with lines dt 5 lw 2 lc black +plot \ + num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)) with yerrorbars pt 0 lc black, \ + magfunc using (10**$1 / B):(M0 + 10**($2) * A * B) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ + magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) smooth csplines with lines lw 2 lc rgb cc2 dt 2, \ + magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) smooth csplines with lines lw 2 lc rgb cc3 dt 3, \ + -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 4 set logscale xy set tmargin -1 @@ -94,7 +109,13 @@ set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$' set mytics 5 set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 -plot num using (X($2, $3)):(Tc * $10 * t($2)**gamma):(Tc * $11 * t($2)**gamma):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb "black", susfunc using (10**$1 / B2):(10**$2 * A2 * B2**2 + (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines dt 2 lw 2 lc black, susfunc using (10**$1 / B3):(10**$2 * A3 * B3**2 + (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines dt 3 lw 2 lc black, susfunc using (10**$1 / B):(-sum[i=2:8] GC(i) * i * (i-1) * (10**$1 / B)**(i-2)) with lines dt 5 lw 2 lc black +plot \ + num using (X($2, $3)):(Tc * $10 * t($2)**gamma):(Tc * $11 * t($2)**gamma) with yerrorbars pt 0 lc black, \ + susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ + susfunc using (10**$1 / B2):(10**$2 * A2 * B2**2 + (sum[i=1:n] poly(c(i), lamb, i-1, 10**$1))) with lines lw 2 lc rgb cc2 dt 2, \ + susfunc using (10**$1 / B3):(10**$2 * A3 * B3**2 + (sum[i=1:n2] poly(c2(i), lamb2, i-1, 10**$1))) with lines lw 2 lc rgb cc3 dt 3, \ + -sum[i=2:8] GC(i) * i * (i-1) * x**(i-2) with lines dt 4 lw 2 lc rgb cc4, \ + 0.0705991 * x**(-14./15) unset logscale xy set logscale x @@ -105,5 +126,10 @@ set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' set ytics format '\footnotesize {%g}' 1.2,0.2,1.8 set mytics 5 -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 + A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) with lines dt 3 lw 2 lc black, magfunc using (10**$1 / B):(-sum[i=1:8] GC(i) * i * (10 **$1 / B)**(i-1)) with lines dt 5 lw 2 lc black +plot \ + num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)) with yerrorbars pt 0 lc black, \ + magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 2 lc rgb cc1 dt 1, \ + magfunc using (10**$1 / B2):(M0 + 10**($2) * A2 * B2 + (sum[i=1:n] polyint(c(i), lamb, i-1, 10**$1)) / B2) with lines lw 2 lc rgb cc2 dt 2, \ + magfunc using (10**$1 / B3):(M0 + 10**($2) * A3 * B3 + (sum[i=1:n2] polyint(c2(i), lamb2, i-1, 10**$1)) / B3) with lines lw 2 lc rgb cc3 dt 3, \ + -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 4 |