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-rw-r--r--essential-ising.bib61
-rw-r--r--essential-ising.tex34
-rw-r--r--figs/fig-susmag.gplot71
3 files changed, 143 insertions, 23 deletions
diff --git a/essential-ising.bib b/essential-ising.bib
index df4a96b..16a0883 100644
--- a/essential-ising.bib
+++ b/essential-ising.bib
@@ -1,4 +1,22 @@
+@article{ambegaokar.1978.dissipation,
+ author = "V. Ambegaokar and B. I. Halperin and D. Nelson and E. Siggia",
+ year = "1978",
+ journal = "Physical Review Letters",
+ volume = "40",
+ pages = "783--6",
+ title = "Dissipation in two-dimensional superfluids",
+}
+
+@article{ambegaokar.1980.dynamics,
+ author = "V. Ambegaokar and B. I. Halperin and D. Nelson and E. Siggia",
+ year = "1980",
+ journal = "Physical Review B",
+ volume = "21",
+ pages = "1806--26",
+ title = "Dynamics of superfluid films",
+}
+
@article{aharony.1983.fields,
title={Nonlinear scaling fields and corrections to scaling near criticality},
author={Aharony, Amnon and Fisher, Michael E},
@@ -90,6 +108,14 @@
publisher={IOP Publishing}
}
+@article{chen.2013.universal,
+ title={Universal scaling function for the two-dimensional Ising model in an external field: A pragmatic approach},
+ author={Chen, Yan-Jiun and Paquette, Natalie M and Machta, Benjamin B and Sethna, James P},
+ journal={arXiv preprint arXiv:1307.6899},
+ 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},
@@ -294,6 +320,28 @@
publisher={IOP Publishing}
}
+@article{mangazeev.2008.variational,
+ title={Variational approach to the scaling function of the 2D Ising model in a magnetic field},
+ author={Mangazeev, Vladimir V and Batchelor, Murray T and Bazhanov, Vladimir V and Dudalev, Michael Yu},
+ journal={Journal of Physics A: Mathematical and Theoretical},
+ volume={42},
+ number={4},
+ pages={042005},
+ year={2008},
+ publisher={IOP Publishing}
+}
+
+@article{mangazeev.2010.scaling,
+ title={Scaling and universality in the two-dimensional Ising model with a magnetic field},
+ author={Mangazeev, Vladimir V and Dudalev, Michael Yu and Bazhanov, Vladimir V and Batchelor, Murray T},
+ journal={Physical Review E},
+ volume={81},
+ number={6},
+ pages={060103},
+ year={2010},
+ publisher={APS}
+}
+
@article{mccraw.1978.metastability,
title={Metastability in the two-dimensional Ising model},
author={McCraw, RJ and Schulman, LS},
@@ -413,6 +461,18 @@
publisher={APS}
}
+@article{wu.1976.spin,
+ title={Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region},
+ author={Wu, Tai Tsun and McCoy, Barry M and Tracy, Craig A and Barouch, Eytan},
+ journal={Physical Review B},
+ volume={13},
+ number={1},
+ pages={316},
+ year={1976},
+ 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},
@@ -424,3 +484,4 @@
publisher={Elsevier}
}
+
diff --git a/essential-ising.tex b/essential-ising.tex
index ea8e9eb..c98e938 100644
--- a/essential-ising.tex
+++ b/essential-ising.tex
@@ -202,7 +202,7 @@ same singular behavior as the real part of the equilibrium free energy, and
that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$,
where
\[
- \fiF(X)=-A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}}
+ \fiF(X)=A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}}
\label{eq:im.scaling}
\]
and $\Theta$ is the Heaviside function. Results from combining an analysis of
@@ -224,14 +224,14 @@ given our scaling ansatz, yielding
\def\eqthreedeeone{
\fF^\threedee(Y/B)&=
\frac{A}{12}\frac{e^{-1/Y^2}}{Y^2}
- \bigg[4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2})
+ \bigg[\Gamma(\tfrac16)E_{7/6}(-Y^{-2})
}
\def\eqthreedeetwo{
- -\Gamma(\tfrac16)E_{7/6}(-Y^{-2})\bigg]
+ -4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2})\bigg]
}
\def\eqfourdeeone{
\fF^\fourdee(Y/B)&=
- \frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2}
+ -\frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2}
\Big[3\ei(-Y^{-3})
}
\def\eqfourdeetwo{
@@ -242,7 +242,7 @@ given our scaling ansatz, yielding
\begin{align}
&\begin{aligned}
\eqthreedeeone\\
- &\hspace{8em}
+ &\hspace{7em}
\eqthreedeetwo
\end{aligned}
\\
@@ -264,13 +264,13 @@ At the level of truncation of \eqref{eq:im.scaling} at which we are working
the Kramers--Kronig relation does not converge in \twodee. However, higher
moments can still be extracted, e.g., the susceptibility, by taking
\[
- \chi=\pd[2]Fh
- =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'.
+ \chi=\pd MH=-\frac1{T_\c}\pd[2]Fh
+ =-\frac2{\pi T_\c}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'.
\]
With a scaling form defined by $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$,
this yields
\[
- \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big]
+ \fX^\twodee(Y/B)=\frac{AB^2}{\pi T_\c Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big]
\label{eq:sus_scaling}
\]
Scaling forms for the free energy can then be extracted by direct integration
@@ -278,9 +278,9 @@ and their constants of integration fixed by known zero field values, yielding
\begin{align}
\label{eq:mag_scaling}
\fM^\twodee(Y/B)
- &=\fM(0)+\frac{ABT_\c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\
+ &=\fM(0)+\frac{AB}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\
\fF^\twodee(Y/B)
- &=\fF(0)+T_\c Y\bigg(\frac{\fM(0)}B+\frac{AT_\c}\pi e^{1/Y}\ei(-1/Y)\bigg)
+ &=-Y\bigg(\frac{\fM(0)}B-\frac{A}\pi e^{1/Y}\ei(-1/Y)\bigg)
\end{align}
with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in
two dimensions.
@@ -325,8 +325,8 @@ $C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that
$B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we
assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of
the susceptibility scaling function, then
-$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant
-$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_0^-$. The
+$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi T_\c$ and the constant
+$A$ is fixed to $A=\pi T_\c\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. 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
@@ -349,8 +349,10 @@ where $F_n'(Y)=f_n(Y)$ and
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 our numeric data
-for $N=0$ while requiring that $C_0^-/T_\c=\fX'(0)=c_0+2AB^2/\pi$. The
+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.
@@ -385,10 +387,10 @@ into the scaling function gives good convergence to the simulations in \twodee.
Our results should allow improved high-precision functional forms for the free
energy~\cite{caselle.2001.critical}, and should have implications for the scaling
-of correlation functions~\cite{YJXXX,XXX}. Our methods might be generalized
+of correlation functions~\cite{chen.2013.universal,wu.1976.spin}. Our methods might be generalized
to predict similar singularities in systems where nucleation and metastability
are proximate to continuous phase transitions, such as 2D superfluid
-transitions~\cite{ALHN}, the melting of 2D crystals~\cite{XXX}, and
+transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{XXX}, and
freezing transitions in glasses, spin glasses, and other disordered systems.
diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot
index 9682953..a2f4584 100644
--- a/figs/fig-susmag.gplot
+++ b/figs/fig-susmag.gplot
@@ -7,6 +7,8 @@ Delta = 15. / 8
gamma = 7. / 4
beta = 1. / 8
+Ch = 0.838677624411
+
t(T) = abs((T - Tc) / T)
h(T, H) = H / T
X(T, H) = h(T, H) * t(T)**(-Delta)
@@ -14,14 +16,69 @@ X(T, H) = h(T, H) * t(T)**(-Delta)
poly(A, l, m, X) = A * X**m / (1 + (l * X)**(m + 1))
polyint(A, l, m, X) = A * l**(-(m + 1)) * log(1 + (l * X)**(m + 1)) / (m + 1)
+G(i) = i == 1 ? -1.3578383417066 : \
+ i == 2 ? -0.048953289720 : \
+ i == 3 ? 0.038863932 : \
+ i == 4 ? -0.068362119 : \
+ i == 5 ? 0.18388370 : \
+ i == 6 ? -0.6591714 : \
+ i == 7 ? 2.937665 : \
+ i == 8 ? -15.61 : 0
+GC(i) = G(i) * (2 * asinh(1))**2 * (Ch * (2 * asinh(1))**(-Delta))**i
+
M0 = (2**2.5 * asinh(1))**0.125
B = Tc**2 * M0 / (16 * pi)
C0 = 0.0255369719
A = pi / 2 * C0 / (B**2 * Tc)
-c0 = -0.012384
-lamb = 1.76962
-A2 = pi / 2 * (C0 / Tc - c0) / B**2
+#c0 = -0.012384
+#lamb = 1.76962
+#A2 = pi / 2 * (C0 / Tc - c0) / B**2
+
+#c0 = 0
+#lamb = 1
+#B2 = 0.521944
+#A2 = pi / 2 * C0 / (B2**2 * Tc)
+
+n = 1
+c(i) = i == 1 ? 0.0037735 : 0
+lamb = 10.487
+B2 = B
+A2 = 0.749317
+
+#n = 2
+#c(i) = i == 1 ? -0.177238 : \
+# i == 2 ? -0.0545988 : 0
+#lamb = 3.14865
+#B2 = B
+#A2 = 18.8816
+
+#n = 3
+#c(i) = i == 1 ? 0.00280714 : \
+# i == 2 ? -0.00243938 : \
+# i == 3 ? -0.0140978 : 0
+#lamb = 12.1954
+#B2 = B
+#A2 = 0.846118
+
+n2 = 4
+c2(i) = i == 1 ? 0.00245324 : \
+ i == 2 ? -0.00468448 : \
+ i == 3 ? 0.0092602 : \
+ i == 4 ? -0.536727 : 0
+lamb2 = 12.6068
+B3 = B
+A3 = 0.88157
+
+#n2 = 5
+#c2(i) = i == 1 ? 0.00281829 : \
+# i == 2 ? 0.000632215 : \
+# i == 3 ? -0.0911689 : \
+# i == 4 ? 1.93584 : \
+# i == 5 ? -24.7397 : 0
+#lamb2 = 13.2489
+#B3 = B
+#A3 = 0.845002
susfunc = "figs/fig-sus_scaling-func.dat"
magfunc = "figs/fig-mag_scaling-func.dat"
@@ -42,7 +99,7 @@ set mxtics 5
set mytics 5
set bmargin 0.2
-plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $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 / B):(10**(3+$2) * A2 * B**2 + 10**3 * poly(c0, lamb, 0, 10**$1)) with lines dt 2 lw 2 lc black
+plot num using (X($2, $3)):(10**3 * $10 * t($2)**gamma):(10**3 * $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 / B2):(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 / B2):(-10**3 * (sum[i=1:7] GC(i + 1) * i * (i + 1) * (10**$1 / B2)**(i-1) / Tc)) with lines dt 5 lw 2 lc black
set bmargin -1
set tmargin 0.2
@@ -53,7 +110,7 @@ set ylabel offset 1,0 '$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) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A2 * B + polyint(Tc * c0, lamb, 0, 10**$1) / B) smooth csplines with lines dt 2 lw 2 lc black
+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) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * Tc * A2 * B2 + (sum[i=1:n] polyint(Tc * 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) * Tc * A3 * B3 + (sum[i=1:n2] polyint(Tc * 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**$1 / B)**(i-1)) with lines dt 5 lw 2 lc black
set logscale xy
set tmargin -1
@@ -70,7 +127,7 @@ set xtics add ('$\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)):($10 * t($2)**gamma):($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 / B):(10**$2 * A2 * B**2 + poly(c0, lamb, 0, 10**$1)) with lines dt 2 lw 2 lc black
+plot num using (X($2, $3)):($10 * t($2)**gamma):($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**$1 / B)**(i-2) / Tc) with lines dt 5 lw 2 lc black
unset logscale xy
set logscale x
@@ -81,5 +138,5 @@ set ylabel offset 4,0 '\footnotesize$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 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A2 * B + polyint(Tc * c0, lamb, 0, 10**$1) / B) with lines dt 2 lw 2 lc black
+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 + Tc * A * B * 10**$2) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B2):(M0 + 10**($2) * Tc * A2 * B2 + (sum[i=1:n] polyint(Tc * c(i), lamb, i-1, 10**$1)) / B2) with lines dt 2 lw 2 lc black, magfunc using (10**$1 / B3):(M0 + 10**($2) * Tc * A3 * B3 + (sum[i=1:n2] polyint(Tc * c2(i), lamb2, i-1, 10**$1)) / B2) 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