Merge branch 'master' into aps

4 files changed, 326 insertions, 103 deletions

diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib
index 157c14f..b11ad6c 100644
--- a/frsb_kac-rice.bib
+++ b/frsb_kac-rice.bib
@@ -294,7 +294,7 @@
 
 @article{Crisanti_1995_Thouless-Anderson-Palmer,
  author = {Crisanti, A. and Sommers, H.-J.},
- title = {Thouless-Anderson-Palmer Approach to the Spherical $p$-Spin Spin Glass Model},
+ title = {{Thouless}-{Anderson}-{Palmer} Approach to the Spherical $p$-Spin Spin Glass Model},
  journal = {Journal de Physique I},
  publisher = {EDP Sciences},
  year = {1995},
@@ -926,4 +926,3 @@ complexity},
  url = {https://doi.org/10.1103%2Fphysrevlett.110.118101},
  doi = {10.1103/physrevlett.110.118101}
 }
-
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index bdcbe58..f21864f 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -36,8 +36,8 @@
 \begin{abstract}
   We derive the general solution for counting the stationary points of
   mean-field complex landscapes. It incorporates Parisi's solution
-  for the ground state, as it should. Using this solution, we count the
-  stationary points of two models: one with multi-step replica symmetry
+  for the ground state, as it should. Using this solution, we count 
+   {\color{red} and discuss the distribution of the stability indices } of  stationary points of two {\color{red} representative} models: one with multi-step replica symmetry
   breaking, and one with full replica symmetry breaking.
 \end{abstract}
 
@@ -52,33 +52,56 @@ Sherrington--Kirkpatrick model, in a paper remarkable for being one of the
 first applications of a replica symmetry breaking (RSB) scheme. As was clear
 when the actual ground-state of the model was computed by Parisi with a
 different scheme, the Bray--Moore result was not exact, and the problem has
-been open ever since \cite{Parisi_1979_Infinite}.  To date, the program of
-computing the number of stationary points---minima, saddle points, and
-maxima---of  mean-field complex landscapes has been only carried out for a small subset of
-models, including most notably the (pure) $p$-spin model ($p>2$)
-\cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1998_Stationary} and for similar
-energy functions inspired by molecular biology, evolution, and machine learning
-\cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}.  In
-a parallel development, it has evolved into an active field of probability
+been open ever since \cite{Parisi_1979_Infinite}.  
+Many other interesting aspects of the problem have been treated, and the subject has
+evolved into an active field of probability
 theory \cite{Auffinger_2012_Random, Auffinger_2013_Complexity,
-BenArous_2019_Geometry}.
+BenArous_2019_Geometry} and  has been applied to
+energy functions inspired by molecular biology, evolution, and machine learning
+\cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}.
+
+To date, however, the program of
+computing the statistics of stationary points---minima, saddle points, and
+maxima---of  mean-field complex landscapes has been only carried out in an exact form for a small subset of
+models, including most notably the (pure) $p$-spin model ($p>2$)
+\cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1998_Stationary}.
+
+{\color{red} Not having  a full, exact  (`quenched') solution of the generic problem  is not 
+primarily a matter of {\em accuracy} of the actual numbers  involved.
+In the same spirit (but in a geometrically distinct way) as in the case of the equilibrium properties of glasses, 
+much more 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?  Also, in an energy
+level where almost all saddle points are unstable, are there still a few  stable ones?
+
+These questions need to be answered for the understanding of the relevance of more complex objects such as
+barrier crossing (which barriers?) \cite{Ros_2021_Dynamical}, or the fate of long-time dynamics
+(which are the target states?). 
+
+
+
+
+
 
 In this paper we present what we argue is the general replica ansatz for the
 number of stationary points of generic mean-field models, which we expect to
 include the Sherrington--Kirkpatrick model. It reproduces the Parisi result in
-the limit of small temperature for the lowest states, as it should.
+the limit of small temperature for the lowest states, as it should. For this kind of situation
+it clarifies the structure of lowest saddles: there is a continuous distribution of them,
+with stability characterized by a continuous distribution of indices.
+}
 
-To understand the importance of this computation, consider the following
-situation. When one solves the problem of spheres in large dimensions, one
-finds that there is a transition at a given temperature to a one-step replica symmetry
+From the point of view of glassy systems, consider the following
+situation. Generically, we now know \cite{Charbonneau_2014_Fractal}
+ that there is a transition at a given temperature to a one-step replica symmetry
 breaking (1RSB) phase at a Kauzmann temperature, and, at a lower temperature,
 another  transition to  a full RSB (FRSB) phase (see \cite{Gross_1985_Mean-field,
-Gardner_1985_Spin}, the so-called `Gardner' phase
-\cite{Charbonneau_2014_Fractal}).  Now, this transition involves the lowest
-equilibrium states. Because they are obviously unreachable at any reasonable
-timescale, a common question is: what is the signature of the Gardner
-transition line for higher than equilibrium energy-densities? This is a
-question whose answers are significant to interpreting the results of myriad
+Gardner_1985_Spin}, the so-called `Gardner' phase.
+Now, this transition involves the lowest
+equilibrium states which are obviously unreachable at any reasonable
+timescale.  We should rather ask the question of what is the signature of the Gardner
+transition line for states with higher  energy-densities: the answer 
+will then be  significant to interpreting the results of myriad
 experiments and simulations \cite{Xiao_2022_Probing, Hicks_2018_Gardner,
 Liao_2019_Hierarchical, Dennis_2020_Jamming, Charbonneau_2015_Numerical,
 Li_2021_Determining, Seguin_2016_Experimental, Geirhos_2018_Johari-Goldstein,
@@ -89,11 +112,9 @@ transition are high energy (or low density) states reachable dynamically?' One
 approach to answering such questions makes use of `state following,'
 which tracks metastable thermodynamic configurations to their zero temperature
 limit \cite{Rainone_2015_Following, Biroli_2016_Breakdown,
-Rainone_2016_Following, Biroli_2018_Liu-Nagel, Urbani_2017_Shear}.  In the
-present paper we give a purely geometric appoarch: we consider the local energy
-minima at a given energy and study their number and other properties: the
-solution involves a replica-symmetry breaking scheme that is well-defined, and
-corresponds directly to the topological characteristics of those minima.
+Rainone_2016_Following, Biroli_2018_Liu-Nagel, Urbani_2017_Shear}.  
+The present paper we provide a purely geometric approach, since we shall address the local energy
+minima at any given energy and study their number and stability properties.
 
 
 Perhaps the most interesting application of this computation is in the context
@@ -126,6 +147,12 @@ $3+16$ model with a 2RSB ground state and a 1RSB complexity, and a $2+4$ with a
 FRSB ground state and a FRSB complexity. Finally \S\ref{sec:interpretation}
 provides some interpretation of our results.
 
+{\color{red}   A final remark is in order here: for simplicity we have concentrated on the energy, rather
+than the {\em free-energy} landscape. Clearly, in the presence of thermal fluctuations, the latter is
+more appropriate. However, from the technical point of view, this makes no fundamental difference, it suffices 
+to apply the same computation to the Thouless-Andreson-Palmer \cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the energy. No new 
+complications arise.}
+
 \section{The model}
 \label{sec:model}
 
diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index 2e4afa4..bdcec74 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -26,14 +26,17 @@
 \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}
 
 \begin{abstract}
-  Complexity is a measure of the number of stationary points in complex
-  landscapes. We derive a general solution for the complexity of mean-field
-  complex landscapes. It incorporates Parisi's solution for the ground state,
-  as it should. Using this solution, we count the stationary points of two
-  models: one with multi-step replica symmetry breaking, and one with full
-  replica symmetry breaking. These examples demonstrate the consistency of the
-  solution and reveal that the signature of replica symmetry breaking at high
-  energy densities is found in high-index saddles, not minima.
+  Complex landscapes are defined by their many saddle points. Determining their
+  number and organization is a long-standing problem, in particular for
+  tractable Gaussian mean-field potentials, which include glass and spin glass
+  models. The annealed approximation is well understood, but is generally not
+  exact. Here we describe the exact quenched solution for the general case,
+  which incorporates Parisi's solution for the ground state, as it should. More
+  importantly, the quenched solution correctly uncovers the full distribution
+  of saddles at a given energy, a structure that is lost in the annealed
+  approximation. This structure should be a guide for the accurate
+  identification of the relevant activated processes in relaxational or driven
+  dynamics.
 \end{abstract}
 
 \maketitle
@@ -45,83 +48,80 @@ size of the system \cite{Maillard_2020_Landscape, Ros_2019_Complex,
 Altieri_2021_Properties}. Though they are often called `rough landscapes' to
 evoke the intuitive image of many minima in something like a mountain range,
 the metaphor to topographical landscapes is strained by the reality that these
-complex landscapes also exist in very high dimensions: think of the dimensions
-of phase space for $N$ particles, or the number of parameters in a neural
-network.
-
-The \emph{complexity} of a function is the average of the logarithm of the
-number of its minima, maxima, and saddle points (collectively stationary
-points), under conditions fixing the value of the energy or the index of the
-stationary point 
-\cite{Bray_1980_Metastable}. 
-Since in complex landscapes this
-number grows exponentially with system size, their complexity is an extensive
-quantity. Understanding the complexity offers an understanding about the
-geometry and topology of the landscape, which can provide insight into
-dynamical behavior.
-
-When complex systems are fully connected, i.e., each degree of freedom
-interacts directly with every other, they are often described by a hierarchical
-structure of the type first proposed by Parisi, the \emph{replica symmetry
-breaking} (RSB) \cite{Parisi_1979_Infinite}. This family of structures is rich, spanning uniform
-\emph{replica symmetry} (RS), an integer $k$ levels of hierarchical nested
-structure ($k$RSB), a full continuum of nested structure (full RSB or FRSB),
-and arbitrary combinations thereof. Though these rich structures are understood
-in the equilibrium properties of fully connected models, the complexity has
-only been computed in RS cases.
-
-In this paper and its longer companion, we share the first results for the
-complexity with nontrivial hierarchy \cite{Kent-Dobias_2022_How}. Using a
-general form for the solution detailed in a companion article, we describe the
-structure of landscapes with a 1RSB complexity and a full RSB complexity
-\footnote{The Thouless--Anderson--Palmer (TAP) complexity is the complexity of
-  a kind of mean-field free energy. Because of some deep thermodynamic
-  relationships between the TAP complexity and the equilibrium free energy, the
-TAP complexity can be computed with extensions of the equilibrium method. As a
-result, the TAP complexity has been previously computed for nontrivial
-hierarchical structure.}.
-
-We study the mixed $p$-spin spherical models, with Hamiltonian
-\begin{equation} \label{eq:hamiltonian}
-  H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
+complex landscapes  exist in very high dimensions. Many  interesting versions
+of the problem have been treated, and the subject has evolved into an active
+field of probability theory \cite{Auffinger_2012_Random,
+  Auffinger_2013_Complexity, BenArous_2019_Geometry} and  has been applied to
+  energy functions inspired by molecular biology, evolution, and machine
+  learning \cite{Maillard_2020_Landscape, Ros_2019_Complex,
+  Altieri_2021_Properties}.
+
+The computation of the number of metastable states in such a landscape was
+pioneered forty years ago by Bray and Moore \cite{Bray_1980_Metastable} on the
+Sherrington--Kirkpatrick (SK) model in one of the first applications of any
+replica symmetry breaking (RSB) scheme. As was clear from the later results by
+Parisi \cite{Parisi_1979_Infinite}, their result was not exact, and the
+problem has been open ever since. To date the program of computing the
+statistics of stationary points---minima, saddle points, and maxima---of
+mean-field complex landscapes has been only carried out in an exact form for a
+relatively small subset of models, including most notably the (pure) $p$-spin spherical
+model ($p>2$) \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer,
+Cavagna_1997_An, Cavagna_1998_Stationary}.
+
+Having  a full, exact  (`quenched') solution of the generic problem  is not
+primarily a matter of {\em accuracy}. Basic structural questions are
+omitted in the approximate `annealed' solution. What is lost is the nature of
+the stationary points at a given energy level: 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 if one hopes to correctly describe more complex
+objects such as barrier crossing (which barriers?) \cite{Ros_2019_Complexity,
+Ros_2021_Dynamical} or the fate of long-time dynamics (that end in which kind
+of states?).
+
+In this paper we present what we argue is the general replica ansatz for the
+number of stationary points of generic mean-field models, which we expect to
+include the SK model. This allows us to clarify the rich structure of all the
+saddles, and in particular the lowest ones. Using a
+general form for the solution detailed in a companion article \cite{Kent-Dobias_2022_How}, we describe the
+structure of landscapes with a 1RSB complexity and a full RSB complexity. The interpretation of a Parisi
+ansatz itself must be recast to make sense of the new order parameters
+involved.
+
+For simplicity we concentrate 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 and it suffices
+to apply the same computation to the Thouless--Anderson--Palmer
+\cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the
+energy. We do not expect new features or technical complications to arise.
+
+For definiteness, we consider the standard example of the mixed $p$-spin
+spherical models, which exhibit a zoo of disordered phases. These models can be
+defined by drawing a random Hamiltonian $H$ from a distribution of isotropic
+Gaussian fields defined on the $N-1$ sphere. Isotropy implies that the
+covariance in energies between two configurations depends on only their dot
+product (or overlap), so for $\mathbf s_1,\mathbf s_2\in
+S^{N-1}$,
+\begin{equation} \label{eq:covariance}
+  \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right),
 \end{equation}
-is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the $N-1$ sphere
-$S^{N-1}=\{\mathbf s\mid\|\mathbf s\|^2=N\}$. The coupling coefficients $J$ are taken at random, with
-zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ chosen so that
-the energy is typically extensive. The overbar will always denote an average
-over the coefficients $J$. The factors $a_p$ in the variances are freely chosen
-constants that define the particular model. For instance, the so-called `pure'
-models have $a_p=1$ for some $p$ and all others zero.
+where $f$ is a function with positive coefficients. The overbar will always
+denote an average over the functions $H$. The choice of function $f$ uniquely
+fixes the model. For instance, the `pure' $p$-spin models have
+$f(q)=\frac12q^p$.
 
 The complexity of the $p$-spin models has been extensively studied by
 physicists and mathematicians. Among physicists, the bulk of work has been on
- the so-called  `TAP' complexity, 
-which counts minima in the mean-field Thouless--Anderson--Palmer () free energy \cite{Rieger_1992_The,
+ the so-called  `TAP' complexity of pure models \cite{Rieger_1992_The,
 Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An,
 Cavagna_1997_Structure, Cavagna_1998_Stationary, Cavagna_2005_Cavity,
-Giardina_2005_Supersymmetry}. The landscape complexity has been proven for pure
-and mixed models without RSB \cite{Auffinger_2012_Random,
-Auffinger_2013_Complexity, BenArous_2019_Geometry}. The mixed models been
-treated without RSB \cite{Folena_2020_Rethinking}. And the methods of
+Giardina_2005_Supersymmetry}, and more recently mixed models \cite{Folena_2020_Rethinking} without RSB \cite{Auffinger_2012_Random,
+Auffinger_2013_Complexity, BenArous_2019_Geometry}. And the methods of
 complexity have been used to study many geometric properties of the pure
 models, from the relative position of stationary points to one another to shape
 and prevalence of instantons \cite{Ros_2019_Complexity, Ros_2021_Dynamical}.
 
-The variance of the couplings implies that the covariance of the energy with
-itself depends on only the dot product (or overlap) between two configurations.
-In particular, one finds
-\begin{equation} \label{eq:covariance}
-  \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right),
-\end{equation}
-where $f$ is defined by the series
-\begin{equation}
-  f(q)=\frac12\sum_pa_pq^p.
-\end{equation}
-One needn't start with a Hamiltonian like
-\eqref{eq:hamiltonian}, defined as a series: instead, the covariance rule
-\eqref{eq:covariance} can be specified for arbitrary, non-polynomial $f$, as in
-the `toy model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}. In fact, defined this way the mixed spherical model encompasses all isotropic Gaussian fields on the sphere.
-
 The family of spherical models thus defined is quite rich, and by varying the
 covariance $f$ nearly any hierarchical structure can be found in
 equilibrium. Because of a correspondence between the ground state complexity
diff --git a/response.tex b/response.tex
new file mode 100644
index 0000000..4addddf
--- /dev/null
+++ b/response.tex
@@ -0,0 +1,197 @@
+\documentclass[a4paper]{letter}
+
+\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
+\usepackage[T1]{fontenc} % vector fonts plz
+\usepackage{newtxtext,newtxmath} % Times for PR
+\usepackage[
+  colorlinks=true,
+  urlcolor=purple,
+  linkcolor=black,
+  citecolor=black,
+  filecolor=black,
+]{hyperref} % ref and cite links with pretty colors
+\usepackage{xcolor}
+\usepackage[style=phys]{biblatex}
+
+\renewcommand{\thefootnote}{\fnsymbol{footnote}}
+
+\addbibresource{frsb_kac-rice.bib}
+
+\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.
+
+
+We have thus 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.
+
+\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}