Generating dense packings of hard spheres by soft interaction design

1. The introduction is very pedagogical
2. The work has high originality

1. Outcome is not impressive

The authors study the random packing problem in the infinite dimensional limit. Practically, construction of hard sphere packings in higher dimensions is very difficult task because it is hampered by the dynamic transition. The authors attempt to bypass this difficulty by tuning biased potentials so that the dynamic transition is pushed toward higher density. This inverse problem is numerically performed thanks to the exact solution for the dynamic transition. This work shows high originality and provides some interesting physical insights such as connection between the number of different glass phases and relevant lengthscales in a given pair potential. The work would be useful for a wide range of fields, e.g., statistical physics, information science, and soft matter.

I recommend the draft for publication. However I would like to ask the authors the following questions and comments.

1) Is there a gain eventually?
I think the original goal is obtaining high density hard sphere packings (unbiased system) more effectively. The biased potential system have higher dynamic transition density than the unbiased system, which enables us to access higher density equilibrium configurations for the “biased” system. Starting from such a configuration, once the biased potential is turned off, the configuration is not in equilibrium anymore for the “unbiased” system. Therefore, subsequent further compression would not produce higher density jammed packings. In other words, it is not guaranteed that the biased potential system provides well-annealed configurations for the pure hard sphere system. I think this reasoning explains the fact that the authors did not find much denser jammed packing (inherent structure of the unbiased system) after the compression of the biased system.

2) Finite excess entropy
The authors mention that the obtained packings are thermodynamically stable because of finite excess entropy. Conventionally speaking, thermodynamic stability is related to the convexity of the entropy or its derivative, thus the absolute value is meaningless. Therefore the authors should add more words for the role of finite excess entropy in the context.

3) Another functional form for the biasing potential
This is just a suggestion. The authors found that the shape of the best-packing potentials consists of a sticky attraction and a positive tail. This kind of potential form can be more effectively explored by the Jagla potential which is a model of water. This model has 4 parameters (n=2), thus one could easily study by full sampling.

1) Typo
There is no overline on the label of Y-axis.

1) The paper is very interesting, sound and well written;
2) I also appreciate the appendix section, where more details on the algorithmic protocol can be found, and the final comparison with very recent rigorous results on the same subject;
3) This work can give important insights on finite-dimensional systems as well, as the authors themselves highlight in the perspectives.

No weak point to mention.

The authors study the challenging theoretical problem of sphere packings, which dates back to Kepler, in a statistical mechanics approach.
With respect to a purely hard-sphere system in high dimensions, they bias the uniform Gibbs measure by adding an attractive potential, which, in the simplest case, is also known as square-shoulder potential in the colloids’ literature.
After a very useful recap of the main results known thus far concerning the critical packing fraction bounds and the relations in the high-dimensional limit between the potential method and the dynamical density, they address the problem numerically, via both a “brute force” search and a modified gradient descent method.
In the presence of a short-range attractive potential, a reentrance transition between the liquid and the glass phases appears, as two of the authors already noticed in two previous related works.
The main aim of this paper is to achieve a better understanding of this reentrance transition and to extend it, by solving the inverse problem of maximizing the dynamical density over the parameter space of a trial family of potentials. The numerical analysis is performed using piecewise constant potentials with a given number of steps, from n=1 to n=6.
They describe in detail both the single-step case and its generalization to a higher number of steps.
However, going beyond the n=5 case does not provide any particular improvement since in any case the packing fraction saturates close to the value 7d2^{-d}.

As I wrote in the first section above, I find the paper very interesting and extremely clear, with numerous potential applications in finite-dimensional condensed matter physics.
I thus recommend the paper for immediate publication.

No requested changes.

I have only a personal suggestion that I leave to the authors' choice.
Since they mention possible connections with experiments and colloidal material setups in the conclusions, Sec. 5.3, I would suggest to include some references while mentioning the “square-shoulder potential” already in Sec. 3.3 (first paragraphs) to better remark why this potential can be of interest in related fields.

1-The basic idea of the paper is original and useful in practice.
2-The results are interesting and sound.
3-The presentation is quite pedagogical and easy to follow.

1-The paper could be streamlined and made shorter.

As said above, I deem the ideas proposed in the paper to be original and the results to be interesting and sound, certainly fit for publication on this platform. However, I have a concern. In the introduction, the authors write that "dressing" the hard-core repulsion with a potential tail is not supposed to influence the SAT-UNSAT transition, which is a reasonable statement. However, in paragraph 5.2 they mention that, once the optimized potential is added to the interaction, the jamming density of the configurations produced at its associated dynamical transition, $\varphi=\varphi_d$, increases (if only a little), when the optimized potential is added to the interaction. This means that all the out-of-equilibrium packings with a jamming density between the bare hard-sphere one ($\hat{\varphi_j,d} \simeq 7.4$) and the new one obtained from the optimized potential, are cut out of the J-line and supposedly unreachable by adiabatic compression from the liquid, despite the potential tail being supposedly (and reasonably) irrelevant on the J-line.
I'd like the authors the elaborate on this (both in their answer to this report and in the paper), before publication can be recommended.

I'd like the authors to address the concern described in the report, before I can recommend publication of the manuscript.