From f92bd7967eab5ed5fd62c74d9b876066804bf5e1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 16 Jun 2021 15:25:21 +0200
Subject: Added plot of two-spin Stokes line.

---
 stokes.tex | 27 +++++++++++++++++++++++----
 1 file changed, 23 insertions(+), 4 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 5a58265..fc73b03 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -11,6 +11,7 @@
   linkcolor=purple
 ]{hyperref} % ref and cite links with pretty colors
 \usepackage{amsmath, graphicx, xcolor} % standard packages
+\usepackage[subfolder]{gnuplottex} % need to compile separately for APS
 
 \begin{document}
 
@@ -336,10 +337,28 @@ critical points are at real $z$, we make this restriction, and find
     \frac d{dt}\frac{z_2}{z_1}=\Delta\frac{z_2}{z_1}
   \end{aligned}
 \end{equation}
-Therefore $z_2/z_1=e^{\Delta t}$, with $z_1^2+z_2^2=N$ as necessary.  Depending on the sign of $\Delta$, $z$ flows
-from one critical point to the other over infinite time. This is a Stokes line,
-and establishes that any two critical points in the 2-spin model with the same
-imaginary energy will possess one.
+Therefore $z_2/z_1=e^{\Delta t}$, with $z_1^2+z_2^2=N$ as necessary.  Depending
+on the sign of $\Delta$, $z$ flows from one critical point to the other over
+infinite time. This is a Stokes line, and establishes that any two critical
+points in the 2-spin model with the same imaginary energy will possess one.
+These trajectories are plotted in Fig.~\ref{fig:two-spin}.
+
+\begin{figure}
+  \begin{gnuplot}[terminal=epslatex, terminaloptions={size 8.65cm,5.35cm}]
+    set xlabel '$\Delta t$'
+    set ylabel '$z(t) / \sqrt{N}$'
+
+    plot 1 / sqrt(1 + exp(2 * x)) t '$z_1$', \
+         1 / sqrt(1 + exp(- 2 * x)) t '$z_2$'
+  \end{gnuplot}
+  \caption{
+    The Stokes line in the 2-spin model when the critical points associated
+    with the first and second cardinal directions are brought to the same
+    imaginary energy. $\Delta$ is proportional to the difference between the
+    real energies of the first and the second critical point; when $\Delta >0$
+    flow is from first to second, while when $\Delta < 0$ it is reversed.
+  } \label{fig:two-spin}
+\end{figure}
 
 Since they sit at the corners of a simplex, the critical points of the 2-spin
 model are all adjacent: no critical point is separated from another by the
-- 
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