diff options
Diffstat (limited to 'stokes.tex')
| -rw-r--r-- | stokes.tex | 20 |
1 files changed, 18 insertions, 2 deletions
@@ -101,7 +101,7 @@ divergent partition function, and can only be properly defined by continuation from a parameter regime where everything is well-defined \cite{}. Others result in oscillatory phase space measures that spoil the use of Monte Carlo or saddle point techniques, but can be treated in a regime where the measure does not -oscillated and the results continued to the desired model \cite{}. +oscillated and the results continued to the desired model \cite{Alexandru_2022_Complex}. In any case, the nicest modern technique (which we will describe in some detail later) consists of deforming the phase space integral into a complex phase @@ -157,6 +157,7 @@ action potentially has more stationary points. We'll call $\Sigma$ the set of \emph{real} stationary points $\Sigma_0$. \begin{figure} + \hspace{5pc} \includegraphics{figs/action.pdf}\hfill \includegraphics{figs/stationaryPoints.pdf} @@ -194,9 +195,12 @@ without changing their value. This means that we are free to choose a nicer contour than our initial phase space $\Omega$. \begin{figure} + \hspace{5pc} \includegraphics{figs/hyperbola_1.pdf}\hfill \includegraphics{figs/hyperbola_2.pdf}\hfill - \includegraphics{figs/hyperbola_3.pdf}\\ + \includegraphics{figs/hyperbola_3.pdf} + + \hspace{5pc} \includegraphics{figs/anglepath_1.pdf}\hfill \includegraphics{figs/anglepath_2.pdf}\hfill \includegraphics{figs/anglepath_3.pdf} @@ -264,6 +268,7 @@ of the thimbles must represent the same element of this relative homology group. \begin{figure} + \hspace{5pc} \includegraphics{figs/thimble_homology.pdf} \hfill \includegraphics{figs/antithimble_homology.pdf} @@ -316,6 +321,7 @@ two stationary points are called \emph{Stokes lines}. An example of this behavior can be seen in Fig.~\ref{fig:1d.stokes}. \begin{figure} + \hspace{5pc} \includegraphics{figs/thimble_stokes_1.pdf}\hfill \includegraphics{figs/thimble_stokes_2.pdf}\hfill \includegraphics{figs/thimble_stokes_3.pdf} @@ -339,6 +345,7 @@ behavior can be seen in Fig.~\ref{fig:1d.stokes}. \end{figure} \begin{figure} + \hspace{5pc} \includegraphics{figs/thimble_orientation_1.pdf}\hfill \includegraphics{figs/thimble_orientation_2.pdf}\hfill \includegraphics{figs/thimble_orientation_3.pdf} @@ -431,6 +438,8 @@ to $z$, i.e., $z^\dagger u=0$, $Pu=u$, the identity. \begin{figure} + \hspace{5pc} + \hfill \includegraphics{figs/thimble_flow.pdf} \caption{Example of gradient descent flow on the action $\mathcal S$ featured @@ -677,6 +686,8 @@ $\blacktriangle$ have exactly the same imaginary energy, and yet they do not share a thimble. \begin{figure} + \hspace{5pc} + \hfill \includegraphics{figs/6_spin.pdf} \caption{ Some thimbles of the circular 6-spin model, where the argument of $\beta$ has @@ -693,6 +704,8 @@ 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} + \hspace{5pc} + \hfill \includegraphics{figs/2_spin_thimbles.pdf} \caption{ Thimbles of the $N=3$ spherical 2-spin model projected into the @@ -1474,7 +1487,9 @@ along with the inequality produce the required bound on $|\delta|$ as a function of $\Delta$ and $\arg\delta$. \begin{figure} + \hspace{5pc} \includegraphics{figs/bound.pdf} + \hfill \includegraphics{figs/example_bound.pdf} \caption{ @@ -1565,6 +1580,7 @@ in $\beta$, which would be equivalent to allowing non-constant terms in the Jacobian over the thimble. \begin{figure} + \hspace{5pc} \hfill\includegraphics{figs/obuchi_3-spin.pdf} \caption{ Phases of the 3-spin model in the complex-$\beta$, following Obuchi \& |