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\begin{document}

\title{Complex complex landscapes}

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

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

\date\today

\begin{abstract}
  We study the saddle-points of the $p$-spin model -- the best understood
  example of a `complex' (rugged) landscape -- when its $N$ variables are
  complex. These points are the solutions to a system of $N$ random equations
  of degree $p-1$.  We solve for $\overline{\mathcal{N}}$, the number of
  solutions averaged over randomness in the $N\to\infty$ limit.  We find that
  it saturates the Bézout bound $\log\overline{\mathcal{N}}\sim N \log(p-1)$.
  The Hessian of each saddle is given by a random matrix of the form $M=a A^2+b
  B^2 +ic [A,B]_-+ d [A,B]_+$, where $A$ and $B$ are GOE matrices and $a-d$
  real.  Its spectrum has a transition from one-cut to two-cut that generalizes
  the notion  of `threshold level' that is well-known in the real problem.  The
  results from the real problem are recovered in the limit of real disorder. In
  this case, only the square-root of the total number solutions are real.  In
  terms of real and imaginary parts of the energy, the solutions are divided in
  sectors where the saddles have different topological properties.
\end{abstract}

\maketitle

Spin-glasses have long been considered the paradigm of `complex landscapes' of
many variables, a subject that includes Neural Networks and optimization
problems, most notably  Constraint Satisfaction ones.  The most tractable
family of these  are the mean-field spherical p-spin models
\cite{Crisanti_1992_The} (for a review see \cite{Castellani_2005_Spin-glass})
defined by the energy:
\begin{equation} \label{eq:bare.hamiltonian}
  H_0 = \sum_p \frac{c_p}{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
\end{equation}
where the $J_{i_1\cdots i_p}$ are real Gaussian variables and the $z_i$ are
real and constrained to a sphere $\sum_i z_i^2=N$. If there is a single  term
of a given $p$, this is known as the `pure $p$-spin' model, the case we shall
study here.  This problem has been studied also  in the algebra \cite{Cartwright_2013_The} and probability
literature \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}.
It  has been attacked from several angles: the replica trick to
compute the Boltzmann--Gibbs distribution \cite{Crisanti_1992_The}, a Kac--Rice
\cite{Kac_1943_On, Rice_1939_The, Fyodorov_2004_Complexity} procedure (similar
to the Fadeev--Popov integral)  to compute the number of saddle-points of the
energy function \cite{Crisanti_1995_Thouless-Anderson-Palmer}, and the
gradient-descent -- or more generally Langevin -- dynamics staring from a
high-energy configuration \cite{Cugliandolo_1993_Analytical}.  Thanks to the
relative simplicity of the energy, all these approaches are possible
analytically in the large $N$ limit.

In this paper we shall extend the study to the case  where the variables are complex
$z\in\mathbb C^N$ and $J$ is a symmetric tensor whose elements are complex normal with
$\overline{|J|^2}=p!/2N^{p-1}$ and $\overline{J^2}=\kappa\overline{|J|^2}$ for
complex parameter $|\kappa|<1$. The constraint becomes $z^2=N$.

The motivations for this paper are of two types. On the practical side, there
are indeed situations in which  complex variables  in a disorder problem appear
naturally: such is the case in which they are {\em phases}, as in random laser
problems \cite{Antenucci_2015_Complex}. Another problem where a Hamiltonian
very close to ours has been proposed is the Quiver Hamiltonians
\cite{Anninos_2016_Disordered} modeling Black Hole horizons in the
zero-temperature limit.  

There is however a more fundamental reason for this study: we know from
experience that extending a problem to the complex plane often uncovers an
underlying simplicity that is hidden in the purely real case. Consider, for
example, the procedure of starting from a simple, known Hamiltonian $H_{00}$
and studying $\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from
$\lambda=0$ to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$
is a polynomial of degree $p$  chosen to have simple, known saddles. Because we
are working in complex variables, and the saddles are simple all the way (we
shall confirm this), we may follow a single one from $\lambda=0$ to $\lambda=1$, while  with
real variables minima of functions appear and disappear, and this procedure is
not possible. The same idea may be implemented by performing diffusion in the
$J$'s, and following the roots, in complete analogy with Dyson's stochastic
dynamics.




  For our model the  constraint we  choose  $z^2=N$,
rather than $|z|^2=N$, in order to preserve the holomorphic nature of the
functions. In addition, the  nonholomorphic spherical constraint has a
disturbing lack of critical points nearly everywhere, since $0=\partial^*
H=-p\epsilon z$ is only satisfied for $\epsilon=0$, as $z=0$ is forbidden by the constraint.  It is enforced using the method of Lagrange multipliers:
introducing the $\epsilon\in\mathbb C$, this gives
\begin{equation} \label{eq:constrained.hamiltonian}
  H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
\end{equation}
It is easy to see that {\em for a pure $p$-spin}, at  any critical point
$\epsilon=H/N$, the average energy.

Critical points are given by the set of equations:

\begin{equation}
\frac{c_p}{(p-1)!}\sum_{ i, i_2\cdots i_p}^NJ_{i, i_2\cdots i_p}z_{i_2}\cdots z_{i_p} = \epsilon z_i
\end{equation}
which for given $\epsilon$ are a set of $N$ equations of degree $p-1$, to which one must add the constraint condition.
In this sense this study also provides a complement to the work on the distribution of zeroes
of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$ and $p \rightarrow \infty$.

Since $H$ is holomorphic, a critical point of $\Re H_0$ is also a critical  point of $\Im H_0$. The number of
critical points of $H$ is therefore the number of critical points of
$\mathop{\mathrm{Re}}H$. Writing $z=x+iy$, $\mathop{\mathrm{Re}}H$ can be
interpreted as a real function of $2N$ real variables. The number of critical
points it has is given by the usual Kac--Rice formula:
\begin{equation} \label{eq:real.kac-rice}
  \begin{aligned}
    \mathcal N_J(\kappa,\epsilon)
      &= \int dx\,dy\,\delta(\partial_x\mathop{\mathrm{Re}}H)\delta(\partial_y\mathop{\mathrm{Re}}H) \\
      &\qquad\times\left|\det\begin{bmatrix}
          \partial_x\partial_x\mathop{\mathrm{Re}}H & \partial_x\partial_y\mathop{\mathrm{Re}}H \\
          \partial_y\partial_x\mathop{\mathrm{Re}}H & \partial_y\partial_y\mathop{\mathrm{Re}}H
        \end{bmatrix}\right|.
  \end{aligned}
\end{equation}
The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in
a manifestly complex way.  Using the Wirtinger derivative
$\partial=\partial_x-i\partial_y$, one can write
$\partial_x\mathop{\mathrm{Re}}H=\mathop{\mathrm{Re}}\partial H$ and
$\partial_y\mathop{\mathrm{Re}}H=-\mathop{\mathrm{Im}}\partial H$. Carrying
these transformations through, we have
\begin{equation} \label{eq:complex.kac-rice}
  \begin{aligned}
    \mathcal N_J&(\kappa,\epsilon)
      = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H) \\
        &\qquad\qquad\qquad\times\left|\det\begin{bmatrix}
            \mathop{\mathrm{Re}}\partial\partial H & -\mathop{\mathrm{Im}}\partial\partial H \\
            -\mathop{\mathrm{Im}}\partial\partial H & -\mathop{\mathrm{Re}}\partial\partial H
          \end{bmatrix}\right| \\
      &= \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
        \left|\det[(\partial\partial H)^\dagger\partial\partial H]\right| \\
      &= \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
        |\det\partial\partial H|^2.
  \end{aligned}
\end{equation}
This gives three equivalent expressions for the determinant of the Hessian: as
that of a $2N\times 2N$ real matrix, that of an $N\times N$ Hermitian matrix,
or the norm squared of that of an $N\times N$ complex symmetric matrix.

These equivalences belie a deeper connection between the spectra of the
corresponding matrices: each eigenvalue of the real matrix has a negative
partner. For each pair $\pm\lambda$ of the real matrix, $\lambda^2$ is an
eigenvalue of the Hermitian matrix and $|\lambda|$ is a \emph{singular
value} of the complex symmetric matrix. The distribution of positive
eigenvalues of the Hessian is therefore the same as the distribution of
singular values of $\partial\partial H$, while both are the same as the
distribution of square-rooted eigenvalues of $(\partial\partial
H)^\dagger\partial\partial H$.

The expression \eqref{eq:complex.kac-rice} is to be averaged over the $J$'s as
$N \Sigma= \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation
that involves the replica trick. In most the parameter-space that we shall
study here, the {\em annealed approximation} $N \Sigma \sim \ln \overline{
\mathcal N_J} = \ln \int dJ \; N_J$ is exact. 

A useful property of the Gaussian distributions is that gradient and Hessian
may be seen to be independent \cite{Bray_2007_Statistics,
Fyodorov_2004_Complexity}, so that we may treat the $\delta$-functions and the
Hessians as independent. We compute each by taking the saddle point. The
$\delta$-functions are converted to exponentials by the introduction of
auxiliary fields $\hat z=\hat x+i\hat y$. The average over $J$ can then be
performed. A generalized Hubbard--Stratonovich then allows a change of
variables from the $4N$ original and auxiliary fields to eight bilinears
defined by
\begin{equation}
  \begin{aligned}
    Na=z^*\cdot z
    &&
    N\hat c=\hat z\cdot\hat z
    &&
    Nb=\hat z^*\cdot z \\
    N\hat a=\hat z^*\cdot\hat z
    &&
    Nd=\hat z\cdot z
  \end{aligned}
\end{equation}
and their conjugates. The result is, to leading order in $N$,
\begin{equation} \label{eq:saddle}
    \overline{\mathcal N_J}(\kappa,\epsilon)
        = \int da\,d\hat a\,db\,db^*d\hat c\,d\hat c^*dd\,dd^*e^{Nf(a,\hat a,b,\hat c,d)},
\end{equation}
where
\begin{widetext}
  \begin{equation}
    f=2+\frac12\log\det\frac12\begin{bmatrix}
      1 & a & d & b \\
      a & 1 & b^* & d^* \\
      d & b^* & \hat c & \hat a \\
      b & d^* & \hat a & \hat c^*
    \end{bmatrix}
    +\mathop{\mathrm{Re}}\left\{\frac18\left[\hat aa^{p-1}+(p-1)|d|^2a^{p-2}+\kappa(\hat c^*+(p-1)b^2)\right]-\epsilon b\right\}
    +\int d\lambda\,\rho(\lambda)\log|\lambda|^2
  \end{equation}
  where $\rho(\lambda)$, the distribution of eigenvalues $\lambda$ of
  $\partial\partial H$, is dependant on $a$ alone. This function has a maximum in
  $\hat a$, $b$, $\hat c$, and $d$ at which its value is (for simplicity, with
  $\kappa\in\mathbb R$)
  \begin{equation} \label{eq:free.energy.a}
    \begin{aligned}
      f(a)&=1+\frac12\log\left(\frac4{p^2}\frac{a^2-1}{a^{2(p-1)}-\kappa^2}\right)+\int d\lambda\,\rho(\lambda)\log|\lambda|^2 \\
          &\hspace{80pt}-\frac{a^p(1+p(a^2-1))-a^2\kappa}{a^{2p}+a^p(a^2-1)(p-1)-a^2\kappa^2}(\mathop{\mathrm{Re}}\epsilon)^2
          -\frac{a^p(1+p(a^2-1))+a^2\kappa}{a^{2p}-a^p(a^2-1)(p-1)-a^2\kappa^2}(\mathop{\mathrm{Im}}\epsilon)^2,
    \end{aligned}
  \end{equation}
\end{widetext}
This leaves a single parameter, $a$, which dictates the magnitude of $z^*\cdot
z$, or alternatively the magnitude $y^2$ of the imaginary part. The latter
vanishes as $a\to1$, where we should recover known results for the real
$p$-spin.

\begin{figure}[htpb]
  \centering

  \includegraphics{fig/spectra_0.0.pdf}
  \includegraphics{fig/spectra_0.5.pdf}\\
  \includegraphics{fig/spectra_1.0.pdf}
  \includegraphics{fig/spectra_1.5.pdf}

  \caption{
    Eigenvalue and singular value spectra of the matrix $\partial\partial H$
    for $p=3$, $a=\frac54$, and $\kappa=\frac34e^{-i3\pi/4}$ with (a)
    $\epsilon=0$, (b) $\epsilon=-\frac12|\epsilon_{\mathrm{th}}|$, (c)
    $\epsilon=-|\epsilon_{\mathrm{th}}|$, and (d)
    $\epsilon=-\frac32|\epsilon_{\mathrm{th}}|$. The shaded region of each inset
    shows the support of the eigenvalue distribution. The solid line on each
    plot shows the distribution of singular values, while the overlaid
    histogram shows the empirical distribution from $2^{10}\times2^{10}$ complex
    normal matrices with the same covariance and diagonal shift as
    $\partial\partial H$.
  } \label{fig:spectra}
\end{figure}


The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial\partial
H=\partial\partial H_0-p\epsilon I$, or the Hessian of
\eqref{eq:bare.hamiltonian} with a constant added to its diagonal. The
eigenvalue distribution $\rho$ of the constrained Hessian is therefore related
to the eigenvalue distribution $\rho_0$ of the unconstrained one by a similar
shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of
\eqref{eq:bare.hamiltonian} is
\begin{equation} \label{eq:bare.hessian}
  \partial_i\partial_jH_0
  =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}},
\end{equation}
which makes its ensemble that of Gaussian complex symmetric matrices. Given its variances
$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and
$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is
constant inside the ellipse
\begin{equation} \label{eq:ellipse}
  \left(\frac{\mathop{\mathrm{Re}}(\lambda e^{i\theta})}{a^{p-2}+|\kappa|}\right)^2+
  \left(\frac{\mathop{\mathrm{Im}}(\lambda e^{i\theta})}{a^{p-2}-|\kappa|}\right)^2
  <\frac{p(p-1)}{2a^{p-2}}
\end{equation}
where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue
spectrum of $\partial\partial H$ therefore is that of an ellipse in the complex
plane whose center lies at $-p\epsilon$. Examples of these distributions are
shown in the insets of Fig.~\ref{fig:spectra}.

The eigenvalue spectrum of the Hessian of the real part, or equivalently the
eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$, is the
singular value spectrum of $\partial\partial H$. When $\kappa=0$ and the
elements of $J$ are standard complex normal, this corresponds to a complex
Wishart distribution. For $\kappa\neq0$ the problem changes, and to our
knowledge a closed form is not known.  We have worked out an implicit form for
this spectrum using the saddle point of a replica symmetric calculation for the
Green function.

Introducing replicas to bring the partition function to
the numerator of the Green function gives
\begin{widetext}
  \begin{equation} \label{eq:green.replicas}
    G(\sigma)=\frac1N\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)}
      \exp\left\{
      \frac12\sum_\alpha^n\left[(\zeta_i^{(\alpha)})^*\zeta_i^{(\alpha)}\sigma
        -\Re\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right)
      \right]
    \right\}
  \end{equation}
  \textcolor{red}{\textbf{Not sure if the $N$ belongs here...}}
  with sums taken over repeated latin indices.
  The average can then be made over $J$ and Hubbard--Stratonovich used to change
  variables to replica matrices
  $N\alpha_{\alpha\beta}=(\zeta^{(\alpha)})^*\cdot\zeta^{(\beta)}$ and
  $N\chi_{\alpha\beta}=\zeta^{(\alpha)}\cdot\zeta^{(\beta)}$ and a series of replica
  vectors. Taking the replica-symmetric ansatz leaves all off-diagonal elements
  and vectors zero, and $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$,
  $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is
  \begin{equation}\label{eq:green.saddle}
    \overline G(\sigma)=\lim_{n\to0}\int d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0
    \exp\left\{nN\left[
      1+\frac{p(p-1)}{16}a^{p-2}\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2)
      +\frac p4\mathop{\mathrm{Re}}\left(\frac{(p-1)}8\kappa^*\chi_0^2-\epsilon^*\chi_0\right)
    \right]\right\}.
  \end{equation}
\end{widetext}
The argument of the exponential has several saddles. The solutions $\alpha_0$
are the roots of a sixth-order polynomial, but the root with the
smallest value of $\mathop{\mathrm{Re}}\alpha_0$ appears gives the correct
solution. A detailed analysis of the saddle point integration is needed to
understand why this is so. Given such $\alpha_0$, the density of singular
values follows from the jump across the cut, or
\begin{equation} \label{eq:spectral.density}
  \rho(\sigma)=\frac1{i\pi}\left(
    \lim_{\mathop{\mathrm{Im}}\sigma\to0^+}\overline G(\sigma)
    -\lim_{\mathop{\mathrm{Im}}\sigma\to0^-}\overline G(\sigma)
  \right)
\end{equation}
\textcolor{red}{\textbf{Missing a factor of two? Please check...}}

The transition from a one-cut to two-cut singular value spectrum naturally
corresponds to the origin leaving the support of the eigenvalue spectrum.
Weyl's theorem requires that the product over the norm of all eigenvalues must
not be greater than the product over all singular values \cite{Weyl_1912_Das}.
Therefore, the absence of zero eigenvalues implies the absence of zero singular
values. The determination of the threshold energy -- the energy at which the
distribution of singular values becomes gapped -- is therefore reduced to a
geometry problem, and yields
\begin{equation} \label{eq:threshold.energy}
  |\epsilon_{\mathrm{th}}|^2
  =\frac{p-1}{2p}\frac{(1-|\delta|^2)^2a^{p-2}}
  {1+|\delta|^2-2|\delta|\cos(\arg\kappa+2\arg\epsilon)}
\end{equation}
for $\delta=\kappa a^{-(p-2)}$.

With knowledge of this distribution, the integral in \eqref{eq:free.energy.a}
may be preformed for arbitrary $a$. For all values of $\kappa$ and $\epsilon$,
the resulting expression is always maximized for $a=\infty$. Taking this saddle
gives
\begin{equation} \label{eq:bezout}
  \overline{\mathcal N}(\kappa,\epsilon)
  =e^{N\log(p-1)}
  =(p-1)^N.
\end{equation}
This is precisely the Bézout bound, the maximum number of solutions that $N$
equations of degree $p-1$ may have \cite{Bezout_1779_Theorie}. More insight is
gained by looking at the count as a function of $a$, defined by
\begin{equation} \label{eq:count.def.marginal}
  \overline{\mathcal N}(\kappa,\epsilon)
  =\int da\,\overline{\mathcal N}(\kappa,\epsilon,a)
\end{equation}
and likewise the $a$-dependant complexity
$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}(\kappa,\epsilon,a)$. In
the large-$N$ limit, the $a$-dependant expression may be considered the
cumulative number of critical points up to the value $a$.

The integral in \eqref{eq:free.energy.a} can only be performed explicitly for
certain ellipse geometries. One of these is at $\epsilon=0$ any values of
$\kappa$ and $a$, which yields the $a$-dependent complexity
\begin{equation} \label{eq:complexity.zero.energy}
  \Sigma(\kappa,0,a)
  =\log(p-1)-\frac12\log\left(\frac{1-|\kappa|^2a^{-2(p-1)}}{1-a^{-2}}\right).
\end{equation}
Notice that the limit of this expression as $a\to\infty$ corresponds with
\eqref{eq:bezout}, as expected. Equation \eqref{eq:complexity.zero.energy} is 
plotted as a function of $a$ for several values of $\kappa$ in
Fig.~\ref{fig:complexity}. For any $|\kappa|<1$, the complexity goes to
negative infinity as $a\to1$, i.e., as the spins are restricted to the reals.
This is natural, given that the $y$ contribution to the volume shrinks to zero
as that of an $N$-dimensional sphere $\sim(a-1)^N$.  However, when the result
is analytically continued to $\kappa=1$ (which corresponds to real $J$)
something novel occurs: the complexity has a finite value at $a=1$.  Since the
$a$-dependence gives a cumulative count, this implies a $\delta$-function
density of critical points along the line $y=0$.  The number of critical points
contained within is
\begin{equation}
  \lim_{a\to1}\lim_{\kappa\to1}\overline{\mathcal N}(\kappa,0,a)
  =(p-1)^{N/2},
\end{equation}
the square root of \eqref{eq:bezout} and precisely the number of critical
points of the real pure spherical $p$-spin model. (note the role of conjugation
symmetry, already underlined in Ref\cite{Bogomolny_1992_Distribution}). In
fact, the full $\epsilon$-dependence of the real pure spherical $p$-spin is
recovered by this limit as $\epsilon$ is varied.

\begin{figure}[htpb]
  \centering
  \includegraphics{fig/complexity.pdf}
  \caption{
    The complexity of the pure 3-spin model at $\epsilon=0$ as a function of
    $a$ at several values of $\kappa$. The dashed line shows
    $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$.
  } \label{fig:complexity}
\end{figure}

These qualitative features carry over to nonzero $\epsilon$. In
Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $a$
close to one for which there are no solutions. For the case $\kappa=1$ -- the
analytic continuation to the usual real computation -- the situation is more
interesting. In the range of energies where there are real solutions this gap
closes, and this is only possible if the density of solutions diverges at
$a=1$.  Another remarkable feature of the limit $\kappa=1$ is that there is
still a gap without solutions around `deep' real energies where there is no
real solution. A moment's thought tells us that this is a necessity: otherwise
a small perturbation of the $J$'s could produce a real, unusually deep solution
for the real problem, in a region where we expect this not to happen.

\begin{figure}[htpb]
  \centering
  \includegraphics{fig/desert.pdf}
  \caption{
    The minimum value of $a$ for which the complexity is positive as a function
    of (real) energy $\epsilon$  for the pure 3-spin model at several values of
    $\kappa$.
  } \label{fig:desert}
\end{figure}

The relationship between the threshold and ground state energies is richer than
in the real case. In Fig.~\ref{fig:eggs} these are shown in the
complex-$\epsilon$ plane for several examples. Depending on the parameters, the
threshold always come at smaller magnitude than the ground state, or always
come at larger magnitude than the ground state, or change relative size as a
function of complex argument. For sufficiently large $a$ the threshold always
comes at larger magnitude than the ground state. If this were to happen in the
real case, it would likely imply our replica symmetric computation is unstable,
as having the ground state above the threshold would imply a ground state
Hessian with many negative eigenvalues, a contradiction with the notion of a
ground state. However, this is not a contradiction in the complex case, since
there cannot be minima of a complex function and the ground state therefore
takes on a different meaning. The relationship between the threshold, i.e.,
where the gap appears, and the dynamics of, e.g., a minimization algorithm or
physical dynamics, are a problem we hope to address in future work.

\begin{figure}[htpb]
  \centering

  \includegraphics{fig/threshold_2.000.pdf}
  \includegraphics{fig/threshold_1.325.pdf} \\
  \includegraphics{fig/threshold_1.125.pdf}
  \includegraphics{fig/threshold_1.000.pdf}

  \caption{
    Energies at which states exist (green shaded region) and threshold energies
    (black solid line) for the pure 3-spin model with
    $\kappa=\frac34e^{-i3\pi/4}$ and (a) $a=2$, (b) $a=1.325$, (c) $a=1.125$,
    and (d) $a=1$. No shaded region is shown in (d) because no states exist an
    any energy.
  } \label{fig:eggs}
\end{figure}

\begin{acknowledgments}
  JK-D and JK are supported by the Simons Foundation Grant No.~454943.
\end{acknowledgments}

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