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@@ -643,72 +643,43 @@ constant complex factor. \subsection{The conditions for Stokes points} -As we have seen in the previous sections, gradient descent dynamics results in flow that enters and leaves each thimble - -What does the topology of the space of thimbles look like? Let us consider the -generic case, where the critical points of $\beta\mathcal S$ have distinct -energies. Consider initially the situation in the absence of any critical -point, e.g., as for a constant or linear function. Now, `place' a generic -(nondegenerate) critical point in the function at $z_0$. In the vicinity of the -critical point, the flow is locally -\begin{equation} - \dot z - \simeq-\frac{\beta^*}2(\operatorname{Hess}\mathcal S|_{z=z_0})^\dagger P(z-z_0)^* -\end{equation} -The matrix $(\operatorname{Hess}\mathcal S)^\dagger P$ has a spectrum identical to that of -$(\operatorname{Hess}\mathcal S)^\dagger$ save marginal directions corresponding to the normals to -manifold. Assuming we are working in a diagonal basis, this becomes -\begin{equation} - \dot z_i=-\frac12(\beta\lambda_i)^*\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2) -\end{equation} -Breaking into real and imaginary parts gives -\begin{eqnarray} - \frac{d\Delta x_i}{dt}&=-\frac12\left( - \operatorname{Re}\lambda_i\Delta x_i+\operatorname{Im}\beta\lambda_i\Delta y_i - \right) \\ - \frac{d\Delta y_i}{dt}&=-\frac12\left( - \operatorname{Im}\lambda_i\Delta x_i-\operatorname{Re}\beta\lambda_i\Delta y_i - \right) -\end{eqnarray} -Therefore, in the complex plane defined by each eigenvector of -$(\operatorname{Hess}\mathcal S)^\dagger P$ there is a separatrix flow. - -Continuing to `insert' critical points whose imaginary energy differs from $C$, -one repeatedly partitions the space this way with each insertion. Therefore, -for the generic case with $\mathcal N$ critical points, with $C$ differing in -value from all critical points, the level set $\operatorname{Im}\mathcal S=C$ has -$\mathcal N+1$ connected components, each of which is simply connected, -diffeomorphic to $\mathbb R^{2D-1}$ and connects two sectors of infinity to -each other. - -When $C$ is brought to the same value as the imaginary part of some critical -point, two of these disconnected surfaces pinch grow nearer and pinch together -at the critical point when $C=\operatorname{Im}\mathcal S$, as in the black lines of -Figure \ref{fig:local_flow}. The unstable manifold of the critical point, which -corresponds with the portion of this surface that flows away, produce the -Lefschetz thimble associated with that critical point. - -Stokes lines are trajectories that approach distinct critical points as time -goes to $\pm\infty$. From the perspective of dynamics, these correspond to -\emph{heteroclinic orbits}. What are the conditions under which Stokes lines -appear? Because the dynamics conserves imaginary energy, two critical points -must have the same imaginary energy if they are to be connected by a Stokes -line. This is not a generic phenomena, but will happen often as one model -parameter is continuously varied. When two critical points do have the same -imaginary energy and $C$ is brought to that value, the level set -$C=\operatorname{Im}\mathcal S$ sees formally disconnected surfaces pinch together in -two places. We shall argue that generically, a Stokes line will exist whenever -the two critical points in question lie on the same connected piece of this -surface. - -What are the ramifications of this for disordered Hamiltonians? When some -process brings two critical points to the same imaginary energy, whether a -Stokes line connects them depends on whether the points are separated from each -other by the separatrices of one or more intervening critical points. -Therefore, we expect that in regions where critical points with the same -energies tend to be nearby, Stokes lines will proliferate, while in regions -where critical points with the same energies tend to be distant compared to -those with different energies, Stokes lines will be rare. +As we have seen, gradient descent on the real part of the action results in a +flow which preserves the imaginary part of the action. Therefore, a necessary +condition for the existence of a Stokes line between two stationary points is +for those points to have the same imaginary action. However, this is not a +sufficient condition. This can be seen in Fig.~\ref{fig:4.spin}, which shows +the thimbles of the circular 6-spin model. The argument of $\beta$ has been +chosen such that the stationary points marked by $\clubsuit$ and +$\blacktriangle$ have exactly the same imaginary energy, and yet they do not +share a thimble. + +\begin{figure} + \includegraphics{figs/6_spin.pdf} + \caption{ + Some thimbles of the circular 6-spin model, where the argument of $\beta$ has + been chosen such that the imaginary parts of the action at the stationary + points $\clubsuit$ and $\blacktriangle$ are exactly the same (and, as a + result of conjugation symmetry, the points $\bigstar$ and $\blacksquare$). + } \label{fig:4.spin} +\end{figure} + +This is because these stationary points are not adjacent: they are separated +from each other by the thimbles of other stationary points. This is a +consistent story in one complex dimension, since the codimension of the +thimbles is the same as the codimension of the constant imaginary energy +surface is one, and such a surface can divide space into regions. However, in higher dimensions thimbles do not have codimension high enough to divide space into regions. + +\begin{figure} + \includegraphics{figs/2_spin_thimbles.pdf} + \caption{ + Thimbles of the $N=3$ spherical 2-spin model projected into the + $\operatorname{Re}\theta$, $\operatorname{Re}\phi$, + $\operatorname{Im}\theta$ space. The blue and green lines trace gradient + descent of the two minima, while the red and orange lines trace those of + the two saddles. The location of the maxima are marked as points, but their + thimbles are not shown. + } +\end{figure} \subsection{Evaluating thimble integrals} |