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-rw-r--r--data/ghigh_series_ours_2.dat14
-rw-r--r--data/ghigh_series_ours_6.dat15
-rw-r--r--data/glow_series_ours_2.dat28
-rw-r--r--data/glow_series_ours_6.dat15
-rw-r--r--data/phi_comparison.dat11
-rw-r--r--data/phi_series_ours_2.dat30
-rw-r--r--data/phi_series_ours_6.dat15
-rw-r--r--ising_scaling.tex44
8 files changed, 115 insertions, 57 deletions
diff --git a/data/ghigh_series_ours_2.dat b/data/ghigh_series_ours_2.dat
index 4f3dd27..47e089f 100644
--- a/data/ghigh_series_ours_2.dat
+++ b/data/ghigh_series_ours_2.dat
@@ -1,15 +1,15 @@
0 0
1 0
-2 -1.845228078232858
+2 -1.8452280782328416
3 0
-4 8.30011667399619
+4 8.377514122546064
5 0
-6 -93.91176560012791
+6 -96.52148911058097
7 0
-8 1429.3561030820913
+8 1493.9976737243942
9 0
-10 -25294.653782288846
+10 -26818.45130191777
11 0
-12 490304.7125789622
+12 526377.7118305506
13 0
-14 -1.0099250773153434e7 \ No newline at end of file
+14 -1.0966384870468803e7 \ No newline at end of file
diff --git a/data/ghigh_series_ours_6.dat b/data/ghigh_series_ours_6.dat
new file mode 100644
index 0000000..efa49ac
--- /dev/null
+++ b/data/ghigh_series_ours_6.dat
@@ -0,0 +1,15 @@
+0 0
+1 0
+2 -1.8452280782328523
+3 0
+4 8.333711749926277
+5 0
+6 -95.16896371253469
+7 0
+8 1457.5503638974826
+9 0
+10 -25869.797578475864
+11 0
+12 501951.0982257904
+13 0
+14 -1.033917515714893e7 \ No newline at end of file
diff --git a/data/glow_series_ours_2.dat b/data/glow_series_ours_2.dat
index f9927e8..3385257 100644
--- a/data/glow_series_ours_2.dat
+++ b/data/glow_series_ours_2.dat
@@ -1,15 +1,15 @@
0 0
-1 -1.3578383417064466
-2 -0.048953289719890325
-3 0.04145804952829153
-4 -0.07859966533925074
-5 0.2198470108765085
-6 -0.790971823869045
-7 3.4711729442114767
-8 -18.069245794661075
-9 109.29036827414234
-10 -754.7713051203647
-11 5864.4413713769845
-12 -50640.97000480466
-13 481133.8191124413
-14 -4.987584839224958e6 \ No newline at end of file
+1 -1.3578383417062967
+2 -0.04895328972041035
+3 0.040152590897806796
+4 -0.07634150977872403
+5 0.22553840411972031
+6 -0.8705719288881231
+7 4.0294426461487145
+8 -21.503792136560655
+9 130.1609902246785
+10 -886.8731778001119
+11 6762.939912929589
+12 -57330.54580849964
+13 536110.1451178609
+14 -5.486439916757801e6 \ No newline at end of file
diff --git a/data/glow_series_ours_6.dat b/data/glow_series_ours_6.dat
new file mode 100644
index 0000000..555f1c3
--- /dev/null
+++ b/data/glow_series_ours_6.dat
@@ -0,0 +1,15 @@
+0 0
+1 -1.357838341706558
+2 -0.04895328972070345
+3 0.038863930625903244
+4 -0.06836211971140452
+5 0.1838836984300239
+6 -0.6591714228830016
+7 2.938563888747659
+8 -15.659413828476081
+9 97.6039830158417
+10 -705.9286232118421
+11 5933.433358981995
+12 -58038.580008779725
+13 653987.4117822968
+14 -8.280938794522655e6 \ No newline at end of file
diff --git a/data/phi_comparison.dat b/data/phi_comparison.dat
index d111f53..7821568 100644
--- a/data/phi_comparison.dat
+++ b/data/phi_comparison.dat
@@ -1,6 +1,5 @@
-2 0.0015185988806263317 0.0016424512339846542 0.00010931409748554666 0.0014019918834638455 0.0009799159023033766
-3 0.0007353808953864949 0.0014547698806950948 0.0011618922642676333 0.0001926030528925822 0.0004716780917582721
-4 0.0006832169448554026 0.0015493096608610868 0.001202074914739018 0.00024563197897510894 0.0011902522778089069
-5 0.0001498262988313126 0.00034058740497194817 0.0003177652580080431 0.00008748091339953398 0.00012570023181056306
-6 9.052199169801156e-6 0.00003198782581631532 0.00005362053295303315 0.00005162642890723171 0.00002227019515451439
-7 5.551832441996751e-6 0.000013915434556921458 9.941126498061448e-6 0.000011350077847509937 0.00003260973372059311 \ No newline at end of file
+2 0.0002955829708120028 0.0002625212059513826 0.000028239602411814224 0.00008275586984924158 0.00005011431128301583
+3 0.00004906258666981245 0.00006482894953951934 0.000028531647414078987 6.68686237088878e-6 0.00001369543769757386
+4 2.148412799218846e-6 2.603799919176364e-6 6.527516478171957e-7 8.371111562578692e-7 7.601714303177897e-7
+5 7.595046134234451e-7 1.303033890176497e-6 8.262734071118327e-7 1.228725780497153e-8 3.3812614840688813e-7
+6 6.932139018367423e-8 1.2168572560700142e-7 7.317986536192755e-8 1.1355372860760715e-8 4.499344289949237e-8 \ No newline at end of file
diff --git a/data/phi_series_ours_2.dat b/data/phi_series_ours_2.dat
index f16c881..beedc59 100644
--- a/data/phi_series_ours_2.dat
+++ b/data/phi_series_ours_2.dat
@@ -1,15 +1,15 @@
-0 -1.1962147849173665
-1 -0.3204525761245943
-2 0.11077688258551482
-3 0.017828886533464094
-4 -0.0012439137023040104
-5 -0.0006436835676704981
-6 0.0008119273579228864
-7 -0.0003697032267769685
-8 -0.00006515301480336557
-9 0.0001753495754505909
-10 -0.00007628792820886294
-11 -0.000016198436209520737
-12 0.000032017927181286665
-13 -0.000012952206779860703
-14 -2.4795512728255315e-7 \ No newline at end of file
+0 -1.197437800827181
+1 -0.3190726460965614
+2 0.11091443628541181
+3 0.016509650519849282
+4 -0.0003141121112829187
+5 -0.0005146254923651025
+6 0.0002221366986115451
+7 -0.00005010818239482555
+8 -5.828936646513926e-7
+9 5.746698725714958e-6
+10 -2.317572453533117e-6
+11 2.8319049007218993e-7
+12 1.3282081081853632e-7
+13 -7.401803626641128e-8
+14 1.6095359608980834e-8 \ No newline at end of file
diff --git a/data/phi_series_ours_6.dat b/data/phi_series_ours_6.dat
new file mode 100644
index 0000000..4d46109
--- /dev/null
+++ b/data/phi_series_ours_6.dat
@@ -0,0 +1,15 @@
+0 -1.197733314476603
+1 -0.3188102465763356
+2 0.11088626986286532
+3 0.016426906005372888
+4 -0.00026404279344288684
+5 -0.0005140260159366009
+6 0.00020886534275493685
+7 -0.000044828373716923474
+8 3.2472263422728396e-7
+9 4.312795231984893e-6
+10 -1.989930723631579e-6
+11 4.3398967612728695e-7
+12 3.0343615844906865e-8
+13 -6.741502699179006e-8
+14 2.8695485833852726e-8 \ No newline at end of file
diff --git a/ising_scaling.tex b/ising_scaling.tex
index 909d99f..744f2d5 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -37,6 +37,17 @@ linkcolor=purple
\date\today
\begin{abstract}
+ We describe a method for approximating the universal scaling functions for
+ the Ising model in a field. By making use of parametric coordinates, the free
+ energy scaling function has a polynomial series everywhere. Its form is
+ taken to be a sum of the simplest functions that contain the singularities
+ which must be present: the Langer essential singularity and the Yang--Lee
+ edge singularity. Requiring that the function match series expansions in
+ the low- and high-temperature zero-field limits fixes the parametric
+ coordinate transformation. For the two-dimensional Ising model, we show that
+ this procedure converges exponentially with the order to which the series are
+ matched. We speculate that with appropriately modified parametric
+ coordinates, the method might also converge deep in the metastable phase.
\end{abstract}
\maketitle
@@ -494,7 +505,7 @@ and
\end{equation}
We have also included the analytic part $G$, which we assume has a simple
series expansion
-\begin{equation}
+\begin{equation} \label{eq:analytic.free.enery}
G(\theta)=\sum_{i=1}^\infty G_i\theta^{2i}
\end{equation}
From the form of the real part, we can infer the form of $\mathcal F$ that is
@@ -517,8 +528,8 @@ where
The scaling function has a number of free parameters: the position $\theta_0$
of the abrupt transition, prefactors in front of singular functions from the
abrupt transition and the Yang--Lee point, the coefficients in the analytic
-part $G$ of $\mathcal F$, and the coefficients in the undetermined function
-$g$. Other parameters are determined by known properties.
+part $G$ of $\mathcal F$, and the coefficients in the undetermined coordinate
+function $g$. Other parameters are determined by known properties.
For $\theta>\theta_0$, the form \eqref{eq:essential.singularity} can be
expanded around $\theta=\theta_0$ to yield
@@ -581,7 +592,8 @@ machine-precision cutoff, whichever is larger. We also add the difference
between the predictions for $A_\mathrm{YL}$ and $\xi_\mathrm{YL}$ and their
known numeric values, again weighted by their uncertainty. In order to
encourage convergence, we also add to the cost the weighted coefficients
-$j!g_j$ and $j!G_j$.
+$j!g_j$ and $j!G_j$ defining the function $g$ and $G$ in
+\eqref{eq:schofield.funcs} and \eqref{eq:analytic.free.enery}.
A Levenberg--Marquardt algorithm is performed on the cost function to find a
parameter combination which minimizes it. As larger polynomial order in the
@@ -781,12 +793,12 @@ to their known values at the critical isotherm, or $\theta=1$.
dat = 'data/phi_comparison.dat'
set xlabel '$n$'
- set xrange [1.5:7.5]
+ set xrange [1.5:6.5]
set logscale y
set format y '$10^{%T}$'
set ylabel '$|\Delta\mathcal F_0^{(m)}|$'
- set yrange [0.000002:0.003]
+ set yrange [0.000000005:0.0005]
set style data linespoints
set key title '\raisebox{0.5em}{$m$}' bottom left
@@ -801,7 +813,9 @@ to their known values at the critical isotherm, or $\theta=1$.
\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
- polynomial order $n$ at which the scaling function was fit.
+ polynomial order $n$ at which the scaling function was fit. The point
+ $\eta=0$ corresponds to the critical isotherm at $T=T_c$ and $H>0$, roughly
+ midway between the limits used in the fit at $H=0$ and $T\neq T_c$.
} \label{fig:error}
\end{figure}
@@ -824,7 +838,7 @@ Fig.~\ref{fig:phi.series}.
\begin{gnuplot}[terminal=epslatex]
dat1 = 'data/glow_numeric.dat'
dat2 = 'data/glow_series_ours_0.dat'
- dat3 = 'data/glow_series_ours_7.dat'
+ dat3 = 'data/glow_series_ours_6.dat'
dat4 = 'data/glow_series_caselle.dat'
set xlabel '$m$'
@@ -838,7 +852,7 @@ Fig.~\ref{fig:phi.series}.
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=7$)', \
+ dat3 using 1:(abs($2)) title 'This work ($n=6$)', \
dat4 using 1:(abs($2)) title 'Caselle \textit{et al.}'
\end{gnuplot}
\caption{
@@ -853,7 +867,7 @@ Fig.~\ref{fig:phi.series}.
\begin{gnuplot}[terminal=epslatex]
dat1 = 'data/ghigh_numeric.dat'
dat2 = 'data/ghigh_series_ours_2.dat'
- dat3 = 'data/ghigh_series_ours_7.dat'
+ dat3 = 'data/ghigh_series_ours_6.dat'
dat4 = 'data/ghigh_caselle.dat'
set key top left Left reverse
@@ -866,7 +880,7 @@ Fig.~\ref{fig:phi.series}.
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=7$)', \
+ dat3 using 1:(abs($2)) title 'This work ($n=6$)', \
dat4 using 1:(abs($2)) title 'Caselle \textit{et al.}'
\end{gnuplot}
\caption{
@@ -881,7 +895,7 @@ Fig.~\ref{fig:phi.series}.
\begin{gnuplot}[terminal=epslatex]
dat1 = 'data/phi_numeric.dat'
dat2 = 'data/phi_series_ours_2.dat'
- dat3 = 'data/phi_series_ours_9.dat'
+ dat3 = 'data/phi_series_ours_6.dat'
set key top right
set logscale y
set xlabel '$m$'
@@ -892,7 +906,7 @@ Fig.~\ref{fig:phi.series}.
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=7$)'
+ dat3 using 1:(abs($2)) title 'This work ($n=6$)'
\end{gnuplot}
\caption{
The series coefficients for the scaling function $\mathcal F_0$ as a
@@ -915,7 +929,7 @@ the ratio.
\begin{gnuplot}[terminal=epslatex]
dat1 = 'data/glow_numeric.dat'
dat2 = 'data/glow_series_ours_0.dat'
- dat3 = 'data/glow_series_ours_7.dat'
+ dat3 = 'data/glow_series_ours_6.dat'
dat4 = 'data/glow_series_caselle.dat'
ratLast(x) = (back2 = back1, back1 = x, back1 / back2)
back1 = 0
@@ -929,7 +943,7 @@ the ratio.
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=7$)', \
+ 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}
\caption{