# Critical energy landscape of linear soft spheres

### Submission summary

 As Contributors: Antonio Sclocchi Arxiv Link: https://arxiv.org/abs/2002.04987v3 (pdf) Date accepted: 2020-06-26 Date submitted: 2020-06-22 Submitted by: Sclocchi, Antonio Submitted to: SciPost Physics Discipline: Physics Subject area: Statistical and Soft Matter Physics Approaches: Theoretical, Computational

### Abstract

We show that soft spheres interacting with a linear ramp potential when overcompressed beyond the jamming point fall in an amorphous solid phase which is critical, mechanically marginally stable and share many features with the jamming point itself. In the whole phase, the relevant local minima of the potential energy landscape display an isostatic contact network of perfectly touching spheres whose statistics is controlled by an infinite lengthscale. Excitations around such energy minima are non-linear, system spanning, and characterized by a set of non-trivial critical exponents. We perform numerical simulations to measure their values and show that, while they coincide, within numerical precision, with the critical exponents appearing at jamming, the nature of the corresponding excitations is richer. Therefore, linear soft spheres appear as a novel class of finite dimensional systems that self-organize into new, critical, marginally stable, states.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

We would like to warmly thank the referees for their positive feedback on our work.
The three reports agree on publishing our work with minor changes.
We tried to implement the suggestions from Referee 1 and 3 in both a revisited version of the manuscript
and with a point by point reply here below.
We also provide the resubmitted manuscript in the Scipost template.

Reply to the questions from Referee 1.

1) We added the numerical details the referee requested. In the caption of each figure (apart the first one)
we wrote the system size, number of samples and the dimension of the system.

2) We checked the captions and we fixed them.

3) We thank the referee for raising the interesting question about the critical behavior of bulk quantities
(such as the pressure) at the unjamming transition.
We are investigating this point extensively but we decided not to discuss it in detail in this work.
However, since the referee is asking, we added to Fig.2 a plot close to unjamming where we show
that we suspect logarithmic behavior for bulk physical quantities.
In order to asses this properly we need to consider a refined algorithm to perform
decompressions close to unjamming which allows to reduce sample to sample fluctuations.
This will be the subject of a forthcoming paper.

4) We changed the layout of fig.4 and we hope that now it is more readable.

Reply to the questions from Referee 3.

1) Jamming is encountered when the energy of local minima goes to zero.
Apart from this property, the scaling description of the unjamming transition induced by the linear ramp potential
is still open and we plan to investigate it in a forthcoming paper.

2) The spatial fluctuations of the local connectivity of the contact network are hyperuniform:
the variance of the number of contacts in a volume $V$ grows slower than $|V|$.
We have explicitly mentioned that in the second item in page 3.

3) We apologize for the confusion, fig. 2 shows an inflection rather than a plateau, we changed the sentence into:
"We conclude by noting that while increasing the packing fraction, the fraction of overlaps display an inflection around phi ~ 1.2".
For what concerns additional data, we provided in Fig.7 the corresponding figure for 3D packings which shows a milder inflection.

4) The structure factor is the Fourier transform of the space-dependent correlation function.
We study its small momentum behavior as a means to access large scale fluctuations.
At the large momentum, the structure factor tends to a constant that just describes the amplitude of local fluctuations.

5) Since the independence of the critical exponents from space dimension was first found in jamming
many people have been scratching their head in the attempt to understand this property.
Although there are proposals, based essentially on finite size scaling analysis,
that suggest that the upper critical dimension is 2, how to derive this result form phase transition theory
is an open problem. This is due to the fact that we lack a field theoretical description of the phase transition itself.
Linear spheres are not an exception and we have the same problem also in this case.
It is clear that for jamming phenomenology to take place, a prominent role is played by isostaticity
but it is unknown how it could be described at a field theoretical level.

### List of changes

- Added numerical details in the caption of every figure and modified them to make them more readable.
- Added right panel in figure 2 and discussed it in the main text at page 6-7.
- Added a sentence in the second item in page 3 to describe briefly the concept of hyperuniformity.

### Submission & Refereeing History

Resubmission 2002.04987v3 on 22 June 2020
Submission 2002.04987v2 on 16 April 2020