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\signature{
  \vspace{-6\medskipamount}
  \smallskip
  Jaron Kent-Dobias \& Jorge Kurchan
}

\address{
  Laboratoire de Physique\\
  Ecole Normale Sup\'erieure\\
  24 rue Lhomond\\ 
  75005 Paris
}

\begin{document}
\begin{letter}{
  Agnese I.~Curatolo, Ph.D.\\
  Physical Review Letters\\
  1 Research Road\\
  Ridge, NY 11961
}

\opening{Dear Dr.~Curatolo,}

Enclosed please find a revised manuscript.
Neither referee  criticized  the scientific content of our paper,
nor substantively addressed its presentation. We have followed their comments
in the direction of  highlighting the importance of having a full solution. In particular
we have emphasized that going to the full replica treatment uncovers a phase-space structure that needs to be taken into account, and that is absent in the annealed treatment.


Among other changes, we have added the paragraph:

\begin{quote}
Having  a full, exact  (`quenched') solution of the generic problem  is not 
primarily a matter of {\em accuracy}.
Very basic structural questions are omitted in the approximate `annealed' solution. What is lost  is  the nature, at any given
energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with  an exponentially small number of saddles, or
-- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number? 
These questions need to be answered for the understanding of the relevance of more complex objects such as
barrier crossing (which barriers?) \footfullcite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate of long-time dynamics
(which end in what kind of  target states?). 
\end{quote}


Both referees find that our paper is clearly written but technical, and
that its topic of ``the different RSB schemes'' are not suitable for a
broad audience. This is surprising to the authors, since a quick
search on Google Scholar reveals several recent PRLs with heavy use of
RSB schemes.

We would also like to submit to the referees  that it is somewhat incongruous
that the solution to a problem that had remained open for 42 years -- during
which it was always present in articles in PRL
\footfullcite{Fyodorov_2004_Complexity, Bray_2007_Statistics,
Fyodorov_2012_Critical, Wainrib_2013_Topological, Dennis_2020_Jamming}-- is
rejected because  it demands of the readers a slightly longer attention span.
These previous works were often limited by the fact that general landscapes
(for which an annealed solution is not exact) were inaccessible. It is perhaps
true that the final solution of an open problem may often be more technical
than the previous ones.


Below, we respond to the referees' comments.
A comprehensive accounting of the changes to our manuscript can be found appended to this letter.

\begin{quote}
  \begin{center}
    Report of Referee A -- LY17256/Kent-Dobias
  \end{center}
  \it
     The authors consider spin glass models with mixed p-spin interactions
     on the N-Sphere and calculate the number of stationary points, the
     logarithm of which yields the complexity. The disorder average of this
     logarithm is computed with the replica trick, and for different model
     variants different replica symmetry breaking (RSB) solutions are
     obtained. A new feature of the solutions, in contrast to previous
     replica symmetric calculations, is that RSB must occur in parts of the
     energy-stability phase diagram.

     \hspace{2em}The paper is clearly written although the content is rather technical
     and probably only accessible to experts in mean field spin glass
     models and the different RSB schemes developed in this field. In
     connection with the well-studied p=3 spin glass model it is briefly
     mentioned that the complexity and its transitions as a function of
     energy and/or stability is relevant for the equilibrium and the
     dynamical behavior of this model – but such a connection has not been
     made here.

     \hspace{2em}Therefore, I feel that the results presented here are only interesting
     for group of experts and I do not assess the finding that the
     complexity of mixed p-spin glass models shows RSB as a major
     breakthrough in the field. Therefore, the manuscript is not suitable
     for publication in Phys.\ Rev.\ Lett., and the publication of the
     accompanying longer paper, submitted to PRE, is sufficient to
     disseminate the results summarized in this manuscript.
\end{quote}


Referee A correctly points out that one new feature of the solutions
outlined in our manuscript is that RSB must occur in parts of the
phase diagram for these models. However, they neglect another feature:
that they are the solutions of the \textit{quenched} complexity, which has
not been correctly calculated until now. We agree with the referee
that ``the complexity of the mixed p-spin glass models'' is not a major
breakthrough in and of itself, we just
chose to demonstrate  the problem in simplest toy model. But believe that the technique for
computing the quenched complexity is a major breakthrough  
\textit{because it brings in the features of organization of saddles of all
kinds that are invisible in the annealed scheme}. 


Referee A states that a connection between the complexity and the
equilibrium and dynamical behavior is not made in our paper. Until
recently, this connection was taken for granted, and the demonstration
that the standard correspondence does not hold in the mixed p-spin
spherical models was exciting enough news to be published in PRX 10,
031045 (2020). It is true that our work doesn't solve the problem that
paper opened, but it does deepen it by showing definitively that the
use of RSB and the quenched complexity are not sufficient to
reestablish a landscape–dynamics connection. 
{\bf One can hardly expect that the structure of saddles at a given energy may be connected
with dynamics (for example in Sherrington Kirkpatrick) if it is unknown}.

\begin{quote}
  \begin{center}
    Report of Referee B -- LY17256/Kent-Dobias
  \end{center}
  \textit{The paper presents a computation of the complexity in spherical
     spin-glass models. Neither the techniques nor the results are
     sufficiently new and relevant to justify publication on PRL. This is
     not surprising given that the topic has been studied extensively in
     the last thirty years and more, the only novelty with respect to
     previous work is that the results are obtained at zero temperature but
     this is definitively not enough. Essential open problems in the field
     involves dynamics and activated processes and some results have
     appeared recently, instead the analysis of the static landscape, to
     which the present paper is a variation, failed to deliver answers to
     these questions up to now.
   }
\end{quote}

Concerning the statement of Referee B that ``the only novelty
with respect to previous work is that the results are obtained at zero
temperature,'' we do not know what to make of the referee's statement.
The novelty of the paper is most definitely
not the fact of treating a zero temperature case. 
We have added the following phrase, that should clarify the situation:

  For simplicity we have concentrated here on the energy, rather
than  {\em free-energy} landscape, although the latter is sometimes
more appropriate. From the technical point of view, this makes no fundamental difference, it suffices 
to apply the same computation to the Thouless-Andreson-Palmer  (TAP) free energy, \footfullcite{Crisanti_1995_Thouless-Anderson-Palmer} instead of the energy. We do not expect new features or technical 
complications arise.

We agree with Referee B's assessment of ``essential open problems in
the field,'' and agree that our work does not deliver answers. However,
delivering answers for all essential open problems is not the acceptance
criterion of PRL. These are

\begin{itemize}
  \item Open a new research area, or a new avenue within an established area.
  \item Solve, or make essential steps towards solving, a critical problem.
  \item Introduce techniques or methods with significant impact.
  \item Be of unusual intrinsic interest to PRL's broad audience.
\end{itemize}

We believe our manuscript makes essential steps toward solving the
critical problem of connecting analysis of the static landscape to
dynamics. We believe that its essential step is through the
introduction of a new technique, calculation of the quenched
complexity, which we believe will have significant impact as it is
applied to more complicated models.

\closing{Sincerely,}

\vspace{1em}

\end{letter}

\end{document}
