diff options
| -rw-r--r-- | essential-ising.bib | 13 | ||||
| -rw-r--r-- | essential-ising.tex | 67 | ||||
| -rw-r--r-- | figs/fig-series-data.dat | 16 | ||||
| -rw-r--r-- | figs/fig-series.gplot | 10 | ||||
| -rw-r--r-- | figs/fig-susmag.gplot | 35 |
5 files changed, 74 insertions, 67 deletions
diff --git a/essential-ising.bib b/essential-ising.bib index 1de2478..64dacb0 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -305,6 +305,18 @@ publisher={IOP Publishing} } +@article{kent-dobias.2018.parametric, + author={Kent-Dobias, Jaron and Sethna, James P}, + journal={(unpublished)}, + year={2018} +} + +@article{kent-dobias.2018.wolff, + author={Kent-Dobias, Jaron and Sethna, James P}, + journal={(unpublished)}, + year={2018} +} + @article{klein.1976.essential, title={Essential singularities at first-order phase transitions}, author={Klein, W and Wallace, DJ and Zia, RKP}, @@ -544,7 +556,6 @@ publisher={APS} } - @article{zinn.1996.universal, title={Universal surface-tension and critical-isotherm amplitude ratios in three dimensions}, author={Zinn, Shun-yong and Fisher, Michael E}, 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} diff --git a/figs/fig-series-data.dat b/figs/fig-series-data.dat index 8873b25..18e7599 100644 --- a/figs/fig-series-data.dat +++ b/figs/fig-series-data.dat @@ -1,7 +1,9 @@ -0 0.025536974521879204 0.025536974521879207 0.025536974521879207 -1 0.009593512562216971 0.01762155820793826 0.017621558207938266 -2 0.004805345190888446 0.01681987670730995 0.017961054818812806 -3 0.003008718424800099 0.018814565927230777 0.02332905039996879 -4 0.0022605793007909824 0.02318633276267007 0.03634395601353185 -5 0.0019815486405162243 0.0306005114760245 0.06569811399718999 -6 0.001985096734582422 0.04264499878327929 0.13485933959294732
\ No newline at end of file +0 0 0 0 +1 1.22241 1.22241 1.22241 +2 0.012768487260939602 0.012768487260939603 0.012768487260939603 +3 0.0015989187603694952 0.0029369263679897097 0.002936926367989711 +4 0.0004004454325740371 0.0014016563922758293 0.0014967545682344004 +5 0.00015043592124000494 0.0009407282963615389 0.0011664525199984396 +6 0.00007535264335969941 0.000772877758755669 0.0012114652004510617 +7 0.000047179729536100574 0.0007285836065720119 0.0015642408094569044 +8 0.0000354481559746861 0.0007615178354157016 0.002408202492731202
\ No newline at end of file diff --git a/figs/fig-series.gplot b/figs/fig-series.gplot index 171af71..729ff73 100644 --- a/figs/fig-series.gplot +++ b/figs/fig-series.gplot @@ -10,13 +10,15 @@ set logscale y data = "figs/fig-series-data.dat" -set xrange [-0.5:6.5] -set yrange [0.0006:0.4] +set xrange [0.5:8.5] +set yrange [0.000005:5] set key off set xlabel '$n$' -set ylabel offset 2 '$|f_n|$' +set ylabel offset 1 '$|f_n|$' -#set ytics format '\footnotesize$10^{%T}$' 0.001,10,1 +set ytics format '' + +set ytics add ('$\footnotesize10^{-5}$' 10**(-5),'$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3),'$\footnotesize10^{-2}$' 10**(-2), '$\footnotesize10^{-1}$' 10**(-1),'$\footnotesize10^{0}$' 10**(0)) plot \ data using 1:2 with points lc rgb cc1, \ diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index 3140df2..901576f 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -75,9 +75,9 @@ set bmargin 0.2 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, \ - susfuncm using 1:(10**3 * $2) with lines lw 2 lc rgb cc2 dt 2, \ - -10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (x)**(i-1)) with lines lw 2 lc rgb cc3 dt 3 + susfunc using (10**$1 / B):(10**(3+$2) * A * B**2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ + susfuncm using 1:(10**3 * $2) with lines lw 3 lc rgb cc2 dt 2, \ + -10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (x)**(i-1)) with lines lw 3 lc rgb cc3 dt 3 set bmargin -1 set tmargin 0.2 @@ -90,30 +90,31 @@ set xtics format '%g' 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, \ - magfuncm using 1:2 with lines lw 2 lc rgb cc2 dt 2, \ - -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 3 + magfunc using (10**$1 / B):(M0 + 10**($2) * A * B) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ + magfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ + -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 3 lc rgb cc3 dt 3 set logscale xy set tmargin -1 set lmargin -1 -set size 0.65,0.325 -set origin 0.31,0.5 + 0.29 / 2 +set size 0.7,0.325 +set origin 0.26,0.5 + 0.29 / 2 set xrange [0.0015:1900] set yrange [0.00002:0.08] -set ylabel offset 2.5,0 '\footnotesize$T\chi|t|^\gamma$' +set ylabel offset 4,0 '\footnotesize$T\chi|t|^\gamma$' set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' set mxtics 5 -set xtics format '' -2,10,1000 +#set xtics format '' -2,10,1000 set xtics add ('$\footnotesize10^{-2}$' 10**(-2), "" 0.1, '$\footnotesize10^0$' 1, "" 10, '$\footnotesize10^2$' 100, "" 1000) set mytics 5 -set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 +#set ytics format '\footnotesize$10^{%T}$' 0.00001,10,0.01 +set ytics add ('$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3), '$\footnotesize10^{-2}$' 10**(-2)) 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, \ - susfuncm using 1:2 with lines lw 2 lc rgb cc2 dt 2, \ - -sum[i=2:8] GC(i) * i * (i-1) * x**(i-2) with lines dt 3 lw 2 lc rgb cc4 + susfunc using (10**$1 / B):(10**$2 * A * B**2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ + susfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ + -sum[i=2:8] GC(i) * i * (i-1) * x**(i-2) with lines dt 3 lw 3 lc rgb cc3 unset logscale xy set logscale x @@ -126,7 +127,7 @@ set mytics 5 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, \ - magfuncm using 1:2 with lines lw 2 lc rgb cc2 dt 2, \ - -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 2 lc rgb cc4 dt 3 + magfunc using (10**$1 / B):(M0 + A * B * 10**$2) with linespoints pt 0 lw 3 lc rgb cc1 dt 1, \ + magfuncm using 1:2 with lines lw 3 lc rgb cc2 dt 2, \ + -sum[i=1:8] GC(i) * i * x**(i-1) with lines lw 3 lc rgb cc3 dt 3 |