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Diffstat (limited to 'stokes.tex')
| -rw-r--r-- | stokes.tex | 27 |
1 files changed, 23 insertions, 4 deletions
@@ -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 |