changed paper body to incorporate new analysis

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+  journal={(unpublished)},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+  journal={(unpublished)},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