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----------------------------------------------------------------------
Response to Referee A -- LZ16835/Kent-Dobias
----------------------------------------------------------------------
Referee A wrote:
> The authors consider the mean-field p-spin spherical model with
> *complex* variables and study the number of saddle points of the
> energy and the eigenvalue distribution of their Hessian matrix. The
> main result of the rather technical computation is that in a
> particular limit (concretely kappa->1) the known results for the real
> p-spin spherical model are reproduced, the (expected) Bézout bound for
> the number of solutions of the saddle point equations is reached and
> that the relationships between the “threshold” and extremal state
> energies is richer in the complex case than in the real case.
>
> I must admit that I was not able to grasp any far-reaching
> consequences of the computational tour de force only hinted at in the
> manuscript, and I fear that a nonexpert reader would also not be able
> to do so. Two arguments are pushed forward by the authors to justify
> the dissemination of their results to the broader readership of PRL:
> One is that there are indeed situations in which complex variables
> appear naturally in disordered system. The first example the authors
> mention is a Hamiltonian that could be relevant for with random Laser
> problems and was analyzed 2015 in PRA, which has up to now 30
> citations according to Google Scholar, and the second example is a
> Hamiltonian from sting theory that was analyzed in 2016 in JHEP, which
> has up to now 31 citations. I do not feel that these two examples
> prove that the enumeration of saddle points if the p-spin model is
> important or of broad interest.
>
> The second argument of the authors is that extending a real problem to
> the complex plane often uncovers underlying simplicity that is
> otherwise hidden, shedding light on the original real problem. Here I
> come back to what I already mentioned above: I do not see any
> simplicity emerging from the present calculation and I also do not see
> the original problem in a new light. Therefore, I do not think that
> one of the four PRL criteria is actually fulfilled and I recommend to
> transfer the manuscript to PRE.
We disagree with the referee's assessment here, as we have also explained in
our letter to the editors. Something in particular that goes unaddressed is
another motivation (which in the referee's defense we did not enumerate clearly
in our draft): that understanding the distribution of complex critical points
is necessary in the treatment of a large class of integrals involved both in
the definition of quantum mechanics with a complex action and in ameliorating
the sign problem in, e.g., lattice QCD.
If the criteria for publication is to be "first past the post" of cited
citations, one might examine our citations of that literature:
- Analytic continuation of Chern-Simons theory, E Witten (2011): 444 citations
- New approach to the sign problem in quantum field theories: High density
QCD on a Lefschetz thimble, M Cristoforetti et al (2012): 285 citations
Both works are concerned with the location and relative positions of critical
points of complex theories. In the resubmitted manuscript we have better
emphasized this motivation.
Referee A wrote:
> Although the first part of the manuscript is well written and well
> understandable (at least for me) from page 2 on it becomes very
> technical and unreadable for a non-expert. If the reader skips to the
> results and tries to understand the figures she/he is left with the
> ubiquitous parameter a, whose physical meaning is hidden deep in the
> saddle point calculation (“dictates the magnitude of |z|^2” – well,
> with respect to the solutions of (3): is “a” the average value of the
> modulus squared of the solution z’s or not?). Similar with epsilon:
> apparently it is the average energy of the saddle point solution – why
> not writing so also in the figure captions? The paper would profit a
> lot from a careful rewriting of at least the result section and to
> provide figure captions with the physical meaning of the quantities
> and parameters shown.
We thank the referee for their helpful suggestions with regards to the
readability of our manuscript. In the resubmitted version, much has been
rewritten for clarity. We would like to highlight several of the most
substantive changes:
- The ubiquitous parameter 'a' was replaced by the more descriptive 'r^2', as
it is a sort of radius, along with a new parameter 'R^2' which bounds it.
Descriptions in English of these were added to the figure captions.
- The technical portion of the paper was reordered to connect better with the
sections preceding and following it.
- The location of the results is now indicated before the beginning of the
technical portion for readers interested in skipping ahead.
Referee A wrote:
> A couple of minor, technical, quibbles:
>
> 1) If there is any real world application of a p-spin model with
> complex variables it will NOT have a spherical constraint. I would
> suggest to discuss the consequences of this constraint, which is
> introduced for computational simplicity.
>
> 2) After eq. (2): ”We choose to constrain our model by z^2=N.“ Then it
> is not a spherical constraint any more – does it have any physical
> relevance?
We have added a more detailed discussion of the constraint to address these
confusions, emphasizing its purpose. The new paragraphs are:
> One might balk at the constraint $z^Tz=N$---which could appropriately be
> called a \emph{hyperbolic} constraint---by comparison with $z^\dagger z=N$.
> The reasoning behind the choice is twofold.
>
> First, we seek draw conclusions from our model that are applicable to generic
> holomorphic functions without any symmetry. Samples of $H_0$ nearly provide
> this, save for a single anomaly: the value of the energy and its gradient at
> any point $z$ correlate along the $z$ direction, with $\overline{H_0\partial
> H_0}\propto \overline{H_0(\partial H_0)^*}\propto z$. This anomalous
> direction should thus be forbidden, and the constraint surface $z^Tz=N$
> accomplishes this.
>
> Second, taking the constraint to be the level set of a holomorphic function
> means the resulting configuration space is a \emph{bone fide} complex
> manifold, and therefore permits easy generalization of the integration
> techniques referenced above. The same cannot be said for the space defined by
> $z^\dagger z=N$, which is topologically the $(2N-1)$-sphere and cannot admit
> a complex structure.
>
> Imposing the constraint with a holomorphic function makes the resulting
> configuration space a \emph{bone fide} complex manifold, which is, as we
> mentioned, the situation we wish to model. The same cannot be said for the
> space defined by $z^\dagger z=N$, which is topologically the $(2N-1)$-sphere,
> does not admit a complex structure, and thus yields a trivial structure of
> saddles. However, we will introduce the bound $r^2\equiv z^\dagger z/N\leq
> R^2$ on the `radius' per spin as a device to classify saddles. We shall see
> that this `radius' $r$ and its upper bound $R$ are insightful knobs in our
> present problem, revealing structure as they are varied. Note that taking
> $R=1$ reduces the problem to that of the ordinary $p$-spin.
Referee A wrote:
> 3) On p.2: “…a, which dictates the magnitude of |z|^2, or
> alternatively the magnitude y^2 of the imaginary part. The last part
> is hard to understand, should be explained.
We thank the referee for pointing out this confusing statement, which was
unnecessary and removed.
> 4) On p.2: “In most the parameter space we shall study her, the
> annealed approximation is exact.” I think it is necessary to provide
> some evidence her, because the annealed approximation is usually a
> pretty severe approximation.
We have nuanced the statement in question and added a citation to a review
article which outlines the reasoning for analogous models. The amended sentence
reads:
> Based on the experience from similar problems \cite{Castellani_2005_Spin-glass},
> the \emph{annealed approximation} $N\Sigma\sim\log\overline{\mathcal N}$ is
> expected to be exact wherever the complexity is positive.
Sincerely,
Jaron Kent-Dobias & Jorge Kurchan
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