Writing on the p-spin section (especially 2-spin)

1 files changed, 38 insertions, 22 deletions

diff --git a/stokes.tex b/stokes.tex
index b98c46b..6081426 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -680,11 +680,11 @@ These eigenvalues associated with the Takagi vectors can be further related to p
 complex symmetric matrix $\beta\operatorname{Hess}\mathcal S$. Suppose that
 $u\in\mathbb R^N$ satisfies the eigenvalue equation
 \begin{equation}
-  (\beta\operatorname{Hess} S)^\dagger(\beta\operatorname{Hess} S)u
+  (\beta\operatorname{Hess}\mathcal S)^\dagger(\beta\operatorname{Hess}\mathcal S)u
   =\sigma u
 \end{equation}
 for some positive real $\sigma$ (because $(\beta\operatorname{Hess}
-S)^\dagger(\beta\operatorname{Hess} S)$ is self-adjoint). The square root of these
+S)^\dagger(\beta\operatorname{Hess}\mathcal S)$ is self-adjoint). The square root of these
 numbers, $\sqrt{\sigma}$, are the definition of the \emph{singular values} of
 $\beta\operatorname{Hess}\mathcal S$. A direct relationship between these singular
 values and the eigenvalues of the real hessian immediately follows by taking a
@@ -693,10 +693,10 @@ and writing
 \begin{equation}
   \eqalign{
     \sigma v^\dagger u
-    &=v^\dagger(\beta\operatorname{Hess} S)^\dagger(\beta\operatorname{Hess} S)u
-    =(\beta\operatorname{Hess} Sv)^\dagger(\beta\operatorname{Hess} S)u\\
-    &=(\lambda v^*)^\dagger(\beta\operatorname{Hess} S)u
-    =\lambda v^T(\beta\operatorname{Hess} S)u
+    &=v^\dagger(\beta\operatorname{Hess}\mathcal S)^\dagger(\beta\operatorname{Hess}\mathcal S)u
+    =(\beta\operatorname{Hess}\mathcal Sv)^\dagger(\beta\operatorname{Hess}\mathcal S)u\\
+    &=(\lambda v^*)^\dagger(\beta\operatorname{Hess}\mathcal S)u
+    =\lambda v^T(\beta\operatorname{Hess}\mathcal S)u
     =\lambda^2 v^\dagger u
   }
 \end{equation}
@@ -1080,8 +1080,8 @@ which is a sum of the `pure' actions
   \mathcal S_p(x)=\frac1{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}x_{i_1}\cdots x_{i_p}
 \end{equation}
 The variables $x\in\mathbb R^N$ are constrained to lie on the sphere $x^2=N$,
-making the model $D=N-1$ dimensional. The couplings $J$ form a totally symmetry
-$p$-tensor whose components are normally distributed with zero mean and
+making the model $D=N-1$ dimensional. The couplings $J$ form totally symmetric
+$p$-tensors whose components are normally distributed with zero mean and
 variance $\overline{J^2}=p!/2N^{p-1}$. The `pure' $p$-spin models have
 $a_i=\delta_{ip}$, while the mixed have some more complicated coefficients $a$.
 
@@ -1106,10 +1106,15 @@ The gradient of the constraint is simple with $\partial g=z$, and \eqref{eq:mult
   =\frac1Nz^T\partial\mathcal S
   =\sum_{p=2}^\infty a_pp\frac{\mathcal S_p(z)}N
 \end{equation}
-which for the pure $p$-spin in particular implies that $\mu=p\epsilon$ for specific energy $\epsilon$.
+which for the pure $p$-spin in particular implies that $\mu=p\epsilon$ for
+specific energy $\epsilon$.
 
 \subsection{2-spin}
 
+The pure 2-spin model is diagonalizable and therefore exactly solvable, and is
+not complex in the sense of having a superextensive number of stationary points
+in its action. However, it makes a good exercise of how the ideas of analytic
+continuation will apply in the literally more complex case of the $p$-spin for $p>2$.
 The Hamiltonian of the pure $2$-spin model is defined by
 \begin{equation}
   \mathcal S_2(z)=\frac12z^TJz.
@@ -1132,20 +1137,27 @@ zero. In the direction in question,
   \frac1N\frac12\lambda_iz_i^2=\epsilon=\frac12\lambda_i,
 \end{equation}
 whence $z_i=\pm\sqrt{N}$. Thus there are $2N$ stationary points, each
-corresponding to $\pm$ the cardinal directions in the diagonalized basis.
-
-Suppose that two stationary points have the same imaginary energy; without loss
-of generality, assume these are associated with the first and second
-cardinal directions. Since the gradient is proportional to $z$, any components that are
-zero at some time will be zero at all times. The dynamics for the components of
-interest assuming all others are zero are
+corresponding to $\pm$ the cardinal directions in the diagonalized basis. The
+energy at each stationary point is real if the couplings are real, and
+therefore there are no complex stationary points in the ordinary 2-spin model.
+
+Imagine for a moment that the coupling are allowed to be complex, giving the
+stationary points of the 2-spin complex energies and therefore potentially
+interesting thimble structure. Generically, the eigenvalues of the coupling
+matrix will have distinct imaginary parts, and there will be no Stokes lines.
+Suppose that two stationary points are brought to the same imaginary energy by
+some continuation; without loss of generality, assume these are associated with
+the first and second cardinal directions. Since the gradient is proportional to
+$z$, any components that are zero at some time will be zero at all times. The
+upward flow dynamics for the components of interest assuming all others are
+zero are
 \begin{equation}
   \dot z_1
-  =-z_1^*\left(d_1^*-\frac{d_1^*z_1^*z_1+d_2^*z_2^*z_2}{|z_1|^2+|z_2|^2}\right)
-  =-(d_1-d_2)^*z_1^*\frac{|z_2|^2}{|z_1|^2+|z_2|^2}
+  =-z_1^*\left(\lambda_1^*-\frac{\lambda_1^*z_1^*z_1+\lambda_2^*z_2^*z_2}{|z_1|^2+|z_2|^2}\right)
+  =-(\lambda_1-\lambda_2)^*z_1^*\frac{|z_2|^2}{|z_1|^2+|z_2|^2}
 \end{equation}
-and the same for $z_2$ with all indices swapped.  Since $\Delta=d_1-d_2$ is
-real, if $z_1$ begins real it remains real, with the same for $z_2$. Since the
+and the same for $z_2$ with all indices swapped.  Since $\Delta=\lambda_1-\lambda_2$ is
+real when the energies and therefore eigenvalues have the same imaginary part, if $z_1$ begins real it remains real, with the same for $z_2$. Since the
 stationary points are at real $z$, we make this restriction, and find
 \begin{equation}
   \frac d{dt}(z_1^2+z_2^2)=0 \qquad
@@ -1174,12 +1186,16 @@ These trajectories are plotted in Fig.~\ref{fig:two-spin}.
   } \label{fig:two-spin}
 \end{figure}
 
-Since they sit at the corners of a simplex, the stationary points of the 2-spin
+Since they sit at the corners of a simplex, the distinct stationary points of the 2-spin
 model are all adjacent: no stationary point is separated from another by the
 separatrix of a third. This means that when the imaginary energies of two
 stationary points are brought to the same value, their surfaces of constant
-imaginary energy join.
+imaginary energy join. However, this is not true for stationary points related
+by the symmetry $z\to-z$, as seen in Fig.~\ref{fig:3d.thimbles}.
 
+Since the 2-spin model with real couplings does not have any stationary points
+in the complex plane, analytic continuation can be made without any fear of
+running into Stokes points. Starting from real, large $\beta$, making an infinitesimal phase rotation into the complex plane results in a decomposition into thimbles where that of each stationary point is necessary, because all stationary points are real. The curvature of the action at the stationary points lying at $z_i=\delta_{ik}$ in the $j$th direction is given by $\lambda_k-\lambda_j=2(\epsilon_k-\epsilon_k)$. Therefore the generic case of $N$ distinct eigenvalues of the coupling matrix leads to $2N$ stationary points with $N$ distinct energies, two at each index from $0$ to $N-1$. Starting with the expression \eqref{eq:real.thimble.partition.function} valid for the partition function contribution from the thimble of a real stationary point, we have
 \begin{equation}
   \eqalign{
     Z(\beta)