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\begin{document}

\title{}

\author{Jaron Kent-Dobias}
\author{Jorge Kurchan}

\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}

\date\today

\begin{abstract}
\end{abstract}

\maketitle

\cite{Witten_2011_Analytic}

Consider a thermodynamic calculation involving the (real) $p$-spin model for a
particular instantiation of the coupling tensor $J$
\begin{equation} \label{eq:partition.function}
  Z_J(\beta)=\int_{S^{N-1}}ds\,e^{-\beta H_J(s)}
\end{equation}
where $S^{N-1}$ is the $N-1$-sphere of radius $N$. Quantities of interest are
usually related to the quenched free energy, produced by averaging over the
$J$s the sample free energy $F_J$
\begin{equation}
  \overline F(\beta,\kappa)=-\beta^{-1}\int dJ\,p_\kappa(J)\log Z_J
\end{equation}
which can depend in general on the inverse temperature $\beta$ and on some
parameter $\kappa$ which governs the distribution of $J$s. For most
applications, $\beta$ is taken to be real and positive, and the distribution
$p_\kappa$ is taken to be Gaussian or discrete on $\pm1$.

We are interested in analytically continuing expressions like $\overline F$
into the region of complex $\beta$ or distributions $p_\kappa$ involving
complex $J$. The former has been considered extensively for the Gaussian
$p$-spin in the past \cite{complex_energy}, while the latter is largely
unexplored.

When the argument of the exponential integrand in \eqref{eq:partition.function}
acquires an imaginary component, various numeric and perturbative schemes for
approximating its value can face immediate difficulties due to the emergence of
a sign problem, resulting from rapid oscillations coinciding with saddles.

\section{Geometry}

The surface $M\subset\mathbb C^N$ defined by $N=f(z)=z^2$ is an $N-1$ dimensional
\emph{Stein manifold}, a type of complex manifold defined by the level set of a
holomorphic function \cite{Forstneric_2017_Stein}. One can define a Hermitian
metric $h$ on $M$ by taking the restriction of the standard metric of $\mathbb
C^N$ to vectors tangent along $M$. For any smooth function $\phi:M\to\mathbb
R$, its gradient $\nabla\phi$ is a holomorphic vector field given by
\begin{equation}
  \nabla^\alpha\phi=h^{\bar\beta\alpha}\bar\partial_{\bar\beta}\phi
\end{equation}
Dynamics consists of gradient descent on $\operatorname{Re}H$, or
\begin{equation} \label{eq:flow}
  \dot u^\alpha=-\nabla^\alpha\operatorname{Re}H=-\tfrac12h^{\bar\beta\alpha}\bar\partial_{\bar\beta}\bar H
\end{equation}
Gradient descent on $\operatorname{Re}H$ is equivalent to Hamiltonian dynamics
with the Hamiltonian $\operatorname{Im}H$. This is because $(M, h)$ is Kähler
and therefore admits a symplectic structure, but that the flow conserves
$\operatorname{Im}H$ can be seen from the Cauchy--Riemann equations and
\eqref{eq:flow}:
\begin{equation}
  \begin{aligned}
    \frac d{dt}\operatorname{Im}H
    &=\dot u^\alpha\partial_\alpha\operatorname{Im}H+\dot{\bar u}^{\bar\alpha}\bar\partial_{\bar\alpha}\operatorname{Im}H \\
    &=\tfrac i4\left(\bar\partial_{\bar\beta}\bar Hh^{\bar\beta\alpha}\partial_\alpha H-\partial_\beta H\bar h^{\beta\bar\alpha}\bar\partial_{\bar\alpha}\bar H  \right)\\
    &=0
  \end{aligned}
\end{equation}
since $h$ is a Hermitian operator with $\bar h=h^T$.

Working with a particular map is inconvenient, and we would like to develop a
map-independent dynamics. Suppose that $z:\mathbb C^{N-1}\to M$ is a map. Using
the chain rule, one finds
\begin{equation}
  \begin{aligned}
    \dot z^i
    &=\dot u^\alpha\partial_\alpha z^i
    =-\tfrac12\bar\partial_{\bar\beta}\bar Hh^{\bar\beta\alpha}\partial_\alpha z^i
    =-\tfrac12\bar\partial_j\bar H\partial_{\bar\beta}\bar z^{\bar\jmath}h^{\bar\beta\alpha}\partial_\alpha z^i \\
    &=-\tfrac12(J^\dagger h^{-1}J)^{\bar\jmath i}\bar\partial_{\bar\jmath}\bar H\\
  \end{aligned}
\end{equation}
where $J$ is the Jacobian of the coordinate map and $h$ is the Hermitian
metric. In stereographic coordinates this can be worked out directly.
Consider the coordinates $z^i=u^i$ for $1\leq i\leq N-1$ and
\begin{equation}
  z^N=\sqrt{N-u^2}
\end{equation}
The Jacobian is
\begin{equation}
  J_\alpha^{\hphantom\alpha i}=\partial_\alpha z^i=\delta_\alpha^{\hphantom\alpha i}-\delta_N^{\hphantom Ni}\frac{u_\beta}{\sqrt{N-u^2}}
\end{equation}
and therefore the Hermitian metric induced by the map is
\begin{equation}
  h_{\alpha\bar\beta}=\bar J_{\alpha}^{\hphantom\alpha i}J_{\bar\beta}^{\hphantom\beta\bar\jmath}\delta_{i\bar\jmath}
  =\delta_{\bar\alpha\beta}+\frac{\bar u^{\alpha}u^{\bar\beta}}{|N-u^2|}
\end{equation}
The metric can be inverted explicitly:
\begin{equation}
  h^{\bar\beta\alpha}
  =\delta^{\bar\beta\alpha}-\frac{\bar u^{\bar\beta}u^\alpha}{|N-u^2|+|u|^2}.
\end{equation}
Putting these pieces together, we find
\begin{equation}
  (J^\dagger h^{-1}J)^{\bar\jmath i}
  =\delta^{\bar\jmath i}-\frac{z^{\bar\jmath}\bar z^i}{|z|^2}
\end{equation}
which is just the projector onto the constraint manifold \cite{Morrow_2006_Complex}.
Therefore, a map-independent dynamics for $z\in M$ is given by
\begin{equation}
  \dot z
  =-\frac12(\partial H)^\dagger\left(I-\frac{zz^\dagger}{|z|^2}\right)
\end{equation}

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.

The level sets defined by $\operatorname{Im}H=c$ for $c\in\mathbb R$ are surfaces of
$2(N-1)-1$ real dimensions. They must be simply connected, since gradient
descent in $\operatorname{Re}H$ cannot pass the same point twice.



\section{2-spin}

The Hamiltonian of the $2$-spin model is defined for $z\in\mathbb C^N$ by
\begin{equation}
  H_0=\frac12z^TJz.
\end{equation}
$J$ is generically diagonalizable by a complex orthogonal matrix $P$. With
$z\mapsto Pz$, $J\mapsto D$ for diagonal $D$. Then $\partial_i H=d_iz_i$. We will henceforth assume to be working in this basis. To find the critical points, we use the method of Lagrange multipliers. Introducing $\epsilon$, the constrained Hamiltonian is
\begin{equation}
  H=H_0+\epsilon(N-z^2)
\end{equation}
As usual, $\epsilon$ is equivalent to the energy per spin at any critical point.
Critical points must satisfy
\begin{equation}
  0=\partial_iH=(d_i-2\epsilon)z_i
\end{equation}
which is only possible for $z_i=0$ or $\epsilon=\frac12 d_i$. Generically the $d_i$ will all differ, so this can only be satisfied for one $d_i$ at a time, and to be a critical point all other $z_j$ must be zero. In the direction in question,
\begin{equation}
  \epsilon=\frac1N\frac12d_iz_i^2=\frac12 d_i,
\end{equation}
whence $z_i=\pm\sqrt{N}$. Thus there are $2N$ critical points, each corresponding to $\pm$ the cardinal directions in the diagonalized basis.

Suppose that two critical 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
\begin{equation}
  \begin{aligned}
    \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}
  \end{aligned}
\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
critical points are at real $z$, we make this restriction, and find
\begin{equation}
  \begin{aligned}
    \frac d{dt}(z_1^2+z_2^2)=0\\
    \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.

The critical points of the 2-spin model are all adjacent: no critical point is separated from another by the separatrix of a third. This means that when the imaginary energies of two critical points are brought to the same value, their surfaces of constant imaginary energy join.

\section{p-spin}

\section{(2 + 4)-spin}

\begin{acknowledgments}
  MIT mathematicians have been no help
\end{acknowledgments}

\bibliography{stokes}

\end{document}