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authorJaron Kent-Dobias <jaron@kent-dobias.com>2017-08-07 14:01:30 -0400
committerJaron Kent-Dobias <jaron@kent-dobias.com>2017-08-07 14:01:30 -0400
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1 files changed, 14 insertions, 6 deletions
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}