Critical energy landscape of linear soft spheres
SSciPost Physics Submission
Critical energy landscape of linear soft spheres
Silvio Franz , Antonio Sclocchi , Pierfrancesco Urbani Universit´e Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France Universit´e Paris-Saclay, CNRS, CEA, Institut de physique th´eorique, 91191,Gif-sur-Yvette, France.*[email protected]
Abstract
We show that soft spheres interacting with a linear ramp potential when over-compressed beyond the jamming point fall in an amorphous solid phase which iscritical, mechanically marginally stable and share many features with the jammingpoint itself. In the whole phase, the relevant local minima of the potential energylandscape display an isostatic contact network of perfectly touching spheres whosestatistics is controlled by an infinite lengthscale. Excitations around such energyminima are non-linear, system spanning, and characterized by a set of non-trivialcritical exponents. We perform numerical simulations to measure their valuesand show that, while they coincide, within numerical precision, with the criticalexponents appearing at jamming, the nature of the corresponding excitations isricher. Therefore, linear soft spheres appear as a novel class of finite dimensionalsystems that self-organize into new, critical, marginally stable, states.
Contents a r X i v : . [ c ond - m a t . d i s - nn ] J un ciPost Physics SubmissionReferences 15 Since more than twenty years, the ideal jamming points of systems of frictionless spheres haveshaped our thinking of low temperature glasses, suggested principles underlying amorphousrigidity, and provided mechanisms to rationalize low energy excitations in glasses [1, 2]. Topicfeature of packings at jamming is mechanical marginal stability. The number of contactsbetween the spheres is isostatic, in d dimensions each sphere has on average 2 d contacts, thatis the least one for which the system can sustain pressure [3–6]. As such, mechanical marginalstability brings about criticality and diverging lengthscales [7]. The jamming point is a criticalpoint characterized by a set of critical exponents describing both the behavior of bulk physicalquantities, such as pressure, energy and contacts [4, 8] as well as the microstructure of amor-phous packings [9–11]. In particular, a common characterization is provided by local statisticsof contact forces and interparticle distances. Marginal stability implies power law behaviorof the distribution of these quantities at small argument [9, 10] and predicts a non trivialrelation between the corresponding exponents [12]. These exponents have been computedexactly in [13, 14] and have been shown to agree -within numerical precision- with numericalsimulations of hard and soft spheres in various physical dimensions [15]. The universalityclass of marginally stable jamming points has been further shown to go beyond finite dimen-sional sphere systems and to encompass more generally a large class of continuous constraintsatisfaction problems in machine learning and computer science [16–19]. For soft constraints,jamming points are isolated critical points: in general, typical soft sphere systems (Harmonicor Hertzian spheres) compressed beyond the jamming point loose most of the salient criticalfeatures of jamming, becoming mechanically stable with a finite correlation length. In thispaper, we show that if we fine tune the soft sphere interaction potential - choosing it as alinear ramp - we can get a new amorphous solid phase which is mechanically marginally stableand critical for all densities beyond the jamming point. Furthermore, the emerging marginalstability is richer that the one appearing at the boundary jamming transition, with additionalsystem spanning non-linear excitations. We consider a set of N spheres in d dimensions whose centers are d -dimensional vectorsdenoted by { x i } i =1 ,...,N . We define a gap between two spheres, say i and j , as h ij = r ij − σ ij ,where we have denoted by σ ij the sum of the radii of the corresponding spheres and by r ij = | x i − x j | the distance between their centers. In the overcompressed phase, above jamming,spheres cannot be arranged without creating overlap between them. Therefore one typicallydefines a pure power interaction potential v α ( h ij ) = f c ( h ij ) α + where x + = | x | θ ( − x ). f c is aconstant that essentially sets the unit of forces. Common choices for the penalty exponent α are α = 2 or α = 5 /
2, corresponding respectively to Harmonic and Hertzian spheres. If α > r ij = σ ij , i.e. when spheres just2 ciPost Physics Submission Figure 1: A snapshot of a configuration of linear disks at packing fraction φ = 1. The contactnetwork is in red while the overlap network is in black. The thickness of the lines reflects theintensities of forces. While black lines carry all forces equal to one, red lines, associated tocontacts, carry a varying force in the interval [0 , f c = 1].touch. As a consequence, given a contact at jamming, an infinitesimal normal force is enoughto destabilize it and can cause an overlap between the corresponding particles. Therefore, for α > α = 1, abovethe jamming transition point. We show that in this case the jammed phase presents new andunexpected features: the linear ramp potential makes the overcompressed phase critical andmarginally stable, characterized by a set of non-linear excitations whose nature is richer thanthe ones appearing at jamming. The linear ramp potential, which is at the boundary betweenconvex and non-convex interparticle potentials, presents important qualitative differences fromthe case α >
1. First of all, it is non-differentiable: small forces applied to contacts do notnecessarily destabilize them. To induce an overlap, a total force greater than f c is necessary.In addition, the modulus of the force generated by an overlap does not depend on the extentof the overlap itself.We focus on systems of two and three dimensional polydisperse spheres and produce localminima by gradient descent minimization. Our main findings are: • Accessible local minima of the PEL are isostatic. Even if there is a finite fraction ofoverlapping spheres making the total energy finite, there is also an isostatic number ofpairs of spheres that just touch. We call interacting spheres couples of spheres thateither are in perfect contact ( contacts ) h ij = 0, or that overlap ( overlaps ) h ij < • Contacts play a crucial role in the stability of the system. Their number is fixed to beexactly equal to the number of degrees of freedom and its fluctuations are suppressed, asit happens at jamming [20]. We show that, as at jamming [21], the spatial fluctuationsof the local connectivity of the contact network are hyperuniform implying that thevariance of the number of contacts in a volume V grows slower than | V | . Conversely,the fluctuations of the number of overlaps follow central limit theorem and spatial fluc-tuations of the overlap network are only uniform. • If we look at gap variables and we focus on strictly positive and negative gaps, we find3 ciPost Physics Submission that both distributions have a power law divergence for small argument (in absolutevalue). The power law exponents controlling the divergence appear to be the same -within numerical precision - for both distributions and very close to the one of positivegaps at jamming. • Contacts can be associated with forces in the interval [0 , f c ]. We measure the forceempirical distribution and show that it displays two singular pseudogaps, close to zeroand close to f c . The pseudogap exponents appear to depend on the packing fractionclose to jamming. However, if we carefully separate the contribution of “bucklers”,namely spheres that have d + 1 interacting spheres [11], from the bulk statistics, thepseudogaps are universal and characterized by the same exponents in the whole jammedphase, far from jamming. The values of the critical exponents appear to be the same-within numerical precision - as the one of small force distribution at jamming.Isostaticity and critical behavior in the force and gap distributions have been shown to ap-pear in the unsatisfiable phase of the spherical perceptron optimization problem with linearcost function, which is a mean field model for linear spheres [22]. The main result of thepresent work is that these properties appear to survive in a robust way when we go to finitedimension. This implies that jammed packings of linear spheres are characterized by diverg-ing isostatic lengthscales and therefore are critical even far from jamming in the compressedphase. Therefore they provide a new, richer example of self-organized critical, marginallystable, finite dimensional systems. The linear ramp v ( h ) is a singular interparticle potential and therefore, both for the sake oftheoretical comprehension and to perform numerical simulations, it is very useful to smooththe singularity out and to define a differentiable (cid:15) -regularized potential between particles.From now on we set f c = 1 . We can define a regularized interparticle potential as v ( h ; (cid:15) ) = h > (cid:15) (cid:15) ( h − (cid:15) ) − (cid:15) < h < (cid:15) | h | h < − (cid:15) (1)which in the (cid:15) → N spheres is therefore defined as H (cid:15) ( x ) = (cid:80) i 2; it also allows to properly define theHessian controlling minima’s stability and to argue in favor of isostaticity.We consider systems of N up to 4096 disks in dimension d = 2 and N up to 1024 spheresin dimension d = 3, inside a d − dimensional box of side-length L with periodic boundaryconditions. The particles’ radii R { i =1 ...N } are random uniformly distributed between thevalues 1 − p and 1 + p , with polydispersity p = 0 . 2. The side-length L of the box is set Note that f c sets only the overall scale of the maximal force and therefore we do not loose generality insetting it to one. ciPost Physics Submission . . . . . . 84 1 1 . . . . . . . 84 1 . . c o n t a c t s , o v e r l a p s φ contactsoverlaps φ energypressure . . . . 01 0 . e n e r g y , o v e r l a p s , p r e ss u r e φ − φ J energyoverlapspressure . φ/ p | log(∆ φ/ | . φ . . / | log(∆ φ/ | Figure 2: Left Panel . Main plot: the isostaticity index defined as c = C/ ( N d − d ) andthe fraction of overlaps defined as n O = O/ ( N d − d ) as a function of the packing fraction(we find the jamming transition at φ J (cid:39) . φ > φ J . Energy, pressure and number of overlaps are increasing functions continuous atjamming. Data produced with system size N = 512, dimensions d = 2, averaged over ∼ Right Panel. The behavior of pressure, energy and overlaps close to theunjamming transition. We attempted some logarithmic fits of the form e ∼ | ∆ φ | / (cid:112) log(∆ φ/ p ∼ / (cid:112) log(∆ φ/ 2) and n O ∼ | ∆ φ | ν e . The unjamming packing fraction φ J is extracted fromthe fit of the energy.by the volume density φ = (cid:80) Ni =1 k d R di /L d , with k d = π d/ / Γ(1 + d/ 2) where Γ( x ) is Eulergamma function. Starting from a random configuration of particles’ positions, we minimizethe energy of the system H (cid:15) ( x ) = (cid:80) i 84 and φ dJ (cid:39) . 64. This procedure provides the set C of the C = |C| particles pairs µ = (cid:104) ij (cid:105) , with i < j , that are in contact (i.e. − (cid:15)/ < h ij < (cid:15)/ 2) and the set O of the O = |O| particles pairs µ = (cid:104) ij (cid:105) that are overlapping (i.e. that have negative gaps h ij < − (cid:15)/ f µ = f ij that form a C − dimensional vector (cid:126)f = { f ij } , while overlapping spheres exchange forces ofintensity 1, whose corresponding O − dimensional vector is simply (cid:126) , ..., f ij can be computed from the regularized potential of eq. 1 as f ij = | h ij − (cid:15) | /(cid:15) ,implying 0 < f ij < 1. Introducing the matrices S and T , with dimensions C × N d and O × N d respectively, defined as S kα (cid:104) ij (cid:105) = ( δ jk − δ ik ) n αij , with (cid:104) ij (cid:105) ∈ C , and T kα (cid:104) ij (cid:105) = ( δ jk − δ ik ) n αij , with (cid:104) ij (cid:105) ∈ O , where n αij is the α -component of the versor n ij = ( x j − x i ) / | x j − x i | , we can computethe contact forces f ij also in an algebraic manner using the force-balance condition S T (cid:126)f = −T T (cid:126) . (2)Notice that the system self-organizes in a way that the forces solving the linear system (2) liein the interval (0 , ciPost Physics Submission . . . 01 0 . s tr u c t u r e f a c t o r q contacts φ = 0 . contacts φ = 1 . contacts φ = 2 . ∼ x . overlaps φ = 1 . overlaps φ = 2 . Figure 3: Structure factor of the local connectivity of the network of contacts and overlaps.For small momentum, the structure factor of the contact network decreases down to zeroimplying hyperuniformity in the fluctuations of connectivity. The exponent controlling thebehavior of the structure factor appears to be close to ∼ . 53 which is the same found atjamming [21]. On the contrary, the connectivity of the overlap network is not hyperuniform.Data produced with system size N = 4096, dimensions d = 2, averaged over 44 samples for φ = 0 . 85, over 50 samples for φ = 1 . φ = 2.In Fig.1, we show an example of a configuration we obtain through the numerical procedurejust described. In red and black we draw respectively the contact and overlap networks. Inthe following, we present data for d = 2 and N = 512 (unless otherwise specified). The datawe got in d = 3 are qualitatively similar to the d = 2 case and therefore we report them inthe appendix. In the jammed phase, for φ > φ J , particles overlap and therefore the numbers of contacts C and of overlaps O , the energy E and the pressure p are different from zero. In two andin three dimensions, in all the minima we found, once removed the rattlers , C is equal tothe isostatic value ( N ∗ − d , where N ∗ is the number of spheres which are not rattlers. On the other hand, O , E and p are continuous at jamming, having defined the pressure p as p = V − (cid:80) i Left panel : the cumulative of the contact force distri-bution at φ = 0 . 85 in 2 d , close to the unjamming transition. We plot the cumulative bothstarting from the edge at f = 0 and at f = 1. While a blind statistics of forces is controlledby a hybrid power law exponent, once the effects of bucklers are removed we clearly observepower laws controlled by the mean field exponents, both close to f = 0 + and f = 1 − . In theinset we plot the empirical probability distribution function. Data produced with system size N = 512, dimensions d = 2, averaged over 30 samples. Right panel : Cumulative distributionof contact forces close to zero and one at φ = 2 in 2 d , far from jamming. We observe thatboth distributions follow the mean field exponent. Our statistics is not sufficient to detect anylocalized excitations at this packing fraction and therefore in this case we consider directlyall forces without separating the contribution of bucklers from the analysis. Data producedwith system size N = 512, dimensions d = 2, averaged over 35 samples.potential and logarithmic behavior has to be expected. In order to establish the precise formof the scalings close to unjamming one needs to consider proper decompression algorithmsthat allow to reduce sample to sample fluctuations close to the transition. This goes beyondthe scope of this paper and will be the subject of a forthcoming work [26]. To characterizethe networks of interaction, we study the fluctuations in the local contact number and overlapnumber. Following [21], we look at the local connectivity fluctuations of the networks ofinteractions by measuring the structure factors S c,o ( q ) = 1 N N (cid:88) i,j =1 (cid:104) δc i δc j e iq · r ij (cid:105) (3)where c i represents the number of contacts or overlaps of particle i for the contact or overlapstructure factors respectively, δc i = c i − (cid:104) c (cid:105) is its local fluctuations and the angular bracketsrepresent average over different minima. We plot both structure factors in Fig.3 for differentdensities in 2 d . The behavior at small q reveals a different behavior of fluctuations of contactand overlap numbers. The structure factor of the contact network decreases to zero at smallargument, while the one of overlaps tends to a positive value. This signals that the fluctuationsin contact number are hyperuniform in space, within a volume V , the square fluctuations of C scale subextensively in V , while the ones of the overlap number are normal and scale as V .This difference is a manifestation of the different role that contacts and overlaps play in thestability of the system. As the system is progressively compressed from the jamming pointto higher densities, the networks self-organize keeping the number of contacts fixed while7 ciPost Physics Submission − − − − − − − − − − ga p s C D F | h | / h| h |i φ = 0 . h > φ = 1 . h > φ = 2 . h > φ = 0 . h < φ = 1 . h < φ = 2 . h < ∼ x − γ J Figure 5: The positive and negative cumulative gap distribution. The y -axis of the negativegap is rescaled by an artificial factor 0.1 to improve the readability of the figure. We observethat both cumulative distribution appear to be described by the same power law exponentfor small argument. Data produced with system size N = 512, dimensions d = 2, averagedover 35 samples for densities φ = 0 . φ = 2 . φ = 1 . φ ∼ . 2. We empirically observe that at the same pointthe overlap network seems to undergo to a kind of percolation transition whose nature andproperties are left for future investigations. Having established that the system is isostatic, it is natural to turn the attention to the distri-bution of non-zero gap variables, which at jamming provides an important characterization ofcriticality. While at jamming all gaps are positive or zero, here we also have ’negative gaps’,quantifying the overlaps between particles. Both the distributions of positive and negativegaps appear to be singular at small argument. In Fig. 5 we plot the cumulative distribution ofboth positive and negative gaps for several packing fractions beyond the jamming transitionpoint. The small gap behavior is controlled in both cases by a power law. If we denote by g ± ( h ) the positive and negative gap distribution, we have that g ± ( h ) ∼ | h | − γ ± . (4)at small argument. The two exponents coincide within the errors, γ + ≈ γ − and their numericalvalue appear to be independent of density and equal to the one of positive gaps at jamming γ ± = γ J ≈ . . . . [13], as predicted by mean field theory [22]. Local minima contain an isostatic number of contacts to which we can associate contactforces and study their empirical distribution. Scalar contact forces are naturally defined in8 ciPost Physics Submission the interval [0 , f c = 1] and we observe that, as soon as we enter in the jammed phase, theirdistribution develops two pseudogaps close to the edges f ∼ , f ∼ + . However, following [11] we perform a statisticalanalysis in which we remove bucklers, namely spheres interacting with d + 1 spheres. Theresult of the analysis is plotted in Fig. 4 and we show that, independently from the packingfraction, the force distribution behaves as p ( f ) ∼ (cid:26) f θ − f ∼ + (1 − f ) θ + f ∼ − (5)with θ + ≈ θ − ≈ θ J , where θ J (cid:39) . . . . is the critical exponent controlling small forcesbetween hard spheres at jamming [13]. In Fig. 4, we also plot the cumulative distributionfunction of bucklers’ forces close to f ∼ 0. Again we see a power law behavior controlled -within numerical precision- by the same power law exponents controlling bucklers at jammingof hard spheres [9, 11]. Finally, we note that deep in the jammed phase, localized effects suchas bucklers (but also rattlers) disappear (with the statistics we have access to) and we do notneed to separate them from the statistics of forces to observe a critical power law with meanfield exponent θ J . The exact solution of the perceptron optimization problem with a linear cost function [22]provides a comprehensive mean-field framework that predicts the main features discussed inthis paper, namely isostaticity, identity of exponents θ + = θ − , γ + = γ − , their numerical valuesand so on. This theory can be adapted straightforwardly to soft linear spheres in infinitedimension [27]. In the next section we complement the mean field theory with marginalstability arguments. The configurations of minima of the PEL at finite energy density contain overlapping particles.It is easy to understand that there should also be pure contacts. The forces correspondingto overlapping particles are constant in modulus (equal to f c = 1) and, without contacts,only very symmetric configurations of particles would be mechanically stable. In fact, moregenerically, a number of contacts less than d on a particle would require a highly symmetricconfiguration to be stabilized by only overlaps. The minimal number of contacts necessary tostabilize a single particle is therefore d , with a number of overlaps larger or equal to one (orwith at least another contact). When we go from the jammed phase towards the jammingpoint, the number of overlaps vanishes and we recover that at jamming a number of contactslarger or equal to d + 1 is required to block a sphere. Particles with d + 1 interactions areprone to local excitations and are usually called bucklers. Local minima of the linear ramp potential are anharmonic due to the singularity in thepairwise interaction potential. However, one can consider the (cid:15) -regularized potential and look9 ciPost Physics Submission at the Hessian of local minima in this case. This is indeed well defined and reads H abij = (cid:40) − r ij v (cid:48) ( h ij ; (cid:15) )( δ ab − n aij n bij ) − v (cid:48)(cid:48) ( h ij ; (cid:15) ) n aij n bij i (cid:54) = j − (cid:80) k (cid:54) = i H abik i = j (6)with n aij = ( x ai − x bj ) / | x i − x j | , a, b = 1 , . . . , d . Focusing on i (cid:54) = j , we have the first term, oftencalled prestress, which vanishes at jamming, while we call the second term the elastic term.Because of the regularization, we have that v (cid:48) ( h ; (cid:15) ) = ( h − (cid:15) ) /(cid:15) I [ h ∈ [ − (cid:15)/ , (cid:15)/ − [ h < − (cid:15)/ v (cid:48)(cid:48) ( h ; (cid:15) ) = I [ h ∈ [ − (cid:15)/ , (cid:15)/ /(cid:15) , where we have defined I [ A ] the indicator function whichis equal to one if A is true and zero otherwise. Notice that the Hessian receives contributionsboth from overlaps and contacts. Overlaps contribute just to the prestress. Contacts insteadcontribute both to the prestress, with a finite term (notice that ( h − (cid:15) ) /(cid:15) is actually thecontact force), and to the elastic part with a term proportional to 1 /(cid:15) . This implies thatfor a variation of the position of the particles such that | δx i | (cid:46) (cid:15) , the energy stored in theelastic term is of order (cid:15) , and dominates the one stored in the prestress which is of order (cid:15) .This is a crucial property, which is at the basis of isostaticity and the criticality of non-linearexcitations in the compressed phase.Despite giving only a relatively small contribution, the contribution of the prestress termis important. In fact, as usual in repulsive sphere systems, this is a destabilizing term (itcorresponds to a negative definite matrix) that, though small, would imply unstable directionsif the elastic part is not full ranked. We conclude that the number of contacts should be atleast isostatic so that the total matrix is positive definite and the minimum is stable.The Hessian is therefore dominated by its isostatic random elastic part. Isostatic randommatrices are gapless [4, 17, 28–31] and characterized by an abundance of soft modes, theirspectrum should behave as λ − / at small argument, where λ represents the eigenvalues.We measure the spectrum of the elastic term matrix, namely the spectrum of lim (cid:15) → (cid:15) H abij .In Fig.6, we plot the corresponding density of states (DOS) with respect to the vibrationalfrequency ω = √ λ . Varying the density from φ = 0 . 85 to φ = 2 . 0, our numerical simulationsare compatible with having a constant DOS for ω → 0. In the appendix we develop a meanfield theory for such behavior supporting this numerical finding. Further information can be gained considering a non-linear stability analysis for the localminima of the PEL. The data we presented clearly shows that minima are isostatic configura-tions where the distributions of both positive and negative gaps display a power law behaviorat small argument. At the same time the isostatic delta peak of marginally satisfied gaps isaccompanied by a contact force distribution which has two pseudogaps close to forces equalto zero or one. The emergence of these power laws controls the non-linear excitations thatdominate the dynamics of the system when perturbing it away from such local minima. Onecan understand the nature of those excitations generalizing the lines of reasoning employedin [9, 12] for the jamming point. The simplest excitations are the ones in which isostaticity isoff by one contact. There are here two possibilities, either separating two spheres in contactand opening a positive gap, or on the contrary pushing two spheres in contact to make themoverlap and create a negative gap. The softest excitations are then the ones correspondingto either very week contacts in the former case, or to contacts carrying a force close to onein the latter case. When such contacts are removed, the system would become mechanically10 ciPost Physics Submission . . . . . . . . . − − − − − − D ( ω ) ωφ φ φ ω R ω D ( t ) d t ω/ h ω i φ = N = 512 N = 1024 N = 2048 N = 4096 Figure 6: Density of states (DOS) of the elastic part of the Hessian matrix of the regularizedpotential, see Eq. (6), for different packing fraction above jamming in d = 2 and with N =4096, averaged over 44 samples for φ = 0 . 85, over ... samples for φ = 1 . 0, over 48 samples for φ = 2 . 0. Inset: the finite size behavior of the left tail of the DOS is consistent with having afinite value for D ( ω = 0).unstable unless a new contact forms in the system and again we have two possibilities, eithera gap closes, or an overlap relaxes to become a contact. Assuming that both processes occurwith finite probability, we have θ + = θ − and γ + = γ − . Following [9, 12], one arrives at thescaling relation γ + = 1 / (2 + θ + ) controlling the critical exponents, which is verified by bothour numerics and the mean field theory of [22]. In this work we have described the emergence of a new critical phase obtained when linearspheres are compressed above the jamming point. The criticality of local minima of the PELof linear soft spheres is described by a set of power laws controlling the positive and negativegap distributions as well as contact forces. The critical exponents controlling such distribu-tions appear to be numerically indistinguishable from the corresponding ones at jamming.Furthermore, the critical behavior is again directly controlled by isostaticity of local minima.This is an interesting result that opens the way to study jamming criticality in a differentand complementary way. Indeed, typically, in order to look for the critical properties of thejamming transition, one needs to fine-tune the numerical simulations in order to be close tojamming. Linear soft spheres instead allow us to get to jamming-like critical configurationsjust by looking at local energy minima which can be obtained using standard numerical rou-tines to minimize the energy. In this case, no fine-tuning is needed. The rich physics that weobserve in linear spheres is due to isostaticity which we robustly find with descent dynamicsin local minima at finite N [20]. While the relevance of our work for materials as e.g. softcolloids or granulars is left for future investigations, the novelty of our results is directly man-ifested in the emergence of a new mechanism for marginal stability leading to criticality in afinite dimensional system.Our work opens a series of perspectives: on one hand, it would be extremely interestingto characterize the rheology of strained linear spheres [32, 33]. A possible way to look for11 ciPost Physics Submission that would be to perform similar experiments as in Ref. [34–36] and to analyze the statisticalproperties of avalanches. On the other hand, it would be interesting to investigate otherconcave penalty exponents α < 1, or more complex potentials, to see if different non-linearcriticality may arise. Moreover, by switching on temperature, one may investigate if marginalstability emerges at a critical point, the Gardner transition [37–39]. Finally, further work isrequired to understand the behavior of bulk quantities such as energy and pressure close tounjamming. Likely, this cannot be obtained from the scaling valid for α > α close to one. Whileour data hints at such phenomenology, further investigations are needed. A possible way toinvestigate this point would be to progressively compress a configuration from jamming. Thedynamics should be dominated by contacts carrying forces close to one becoming overlapswhile small gaps becoming contacts with a net flux of gaps from the positive to the negativeside of the distribution. How to describe such dynamics is left for future work. Funding information This work was supported by “Investissements d’Avenir” LabEx-PALM (ANR-10-LABX-0039-PALM) and by the Simons foundation (grants No. 454941,S. Franz). SF is a member of the Institut Universitaire de France. . . . . . . 64 0 . . . . . . . 64 1 1 . c o n t a c t s , o v e r l a p s φ contactsoverlaps φ energypressure . . s tr u c t u r e f a c t o r q contacts φ = 0 . contacts φ = 1 . contacts φ = 1 . overlaps φ = 1 . overlaps φ = 1 . ∼ x . Figure 7: Left panel : isostaticity index and fraction of overlaps. While at all densities theminima are isostatic, the number of overlaps increases monotonically. In the inset, the cor-responding energy and pressure. Right panel : structure factor of the local coordination ofthe contact and overlap network. While fluctuations of the local connectivity of the overlapnetwork follow the central limit theorem, the ones of the contact network are hyperuniform.Data produced with system size N = 1024, dimensions d = 3, averaged over ∼ 30 samples foreach density φ . A Properties of energy minima of linear spheres in three di-mensions Here we report the results of numerical simulations of three dimensional linear soft spheres.We consider N = 1024 spheres with varying packing fraction φ . With the polidispersity weare using, the jamming point is at φ (cid:39) . 64. We are interested in the properties of the PELof linear spheres above this packing fraction. As for d = 2, we find that our minimization12 ciPost Physics Submission I s o s t a t i c i t y i n d e x Figure 8: The behavior of the isostaticity index as a function of the regularizer parameter (cid:15) for a system of N = 2048 spheres in d = 2 at φ = 2, averaged over 29 samples. For (cid:15) → φ . In the right panel we plot also the structurefactor of the local connectivity of the overlap and contact network. We show that while theoverlap network obeys central limit theorem, the fluctuations of the number of contacts aresuppressed and the structure factor goes to zero for small momenta.Therefore both in d = 2 and d = 3 isostaticity is reached when minimizing the energy ofthe system. In order to see how this happens numerically, in Fig. 8 we plot the isostaticityindex as a function of the regularization parameter (cid:15) (we plot data for d = 2 for simplicity).We clearly see that as soon as the linear potential limit is reached, the system self organizesto sit on an isostatic minimum.Therefore the main conclusion of this analysis is that, as for the jamming transition, theproperties of soft spheres interacting with linear potential do not depend on the dimensionalityof the system (apart from local bucklers/rattlers effects).Finally in Fig.9 we plot the contact force and gap distribution. Contact forces displaytwo pseudogaps close to the edges f = 0 , B Mean field theory of the density of states of the contactnetwork matrix In the main text we have argued that the elastic part of the contact network matrix hasa density of states D ( ω ) which goes to a positive constant for ω → ciPost Physics Submission − − − − − − − . f o r ce s C D F f/ h f i , (1 − f ) / h − f i all forces f all forces − f ∼ x θ J P D F f − − − − − − − − − − ga p s C D F | h | / h| h |i φ = 0 . h > φ = 1 . h > φ = 1 . h > φ = 0 . h < φ = 1 . h < φ = 1 . h < ∼ x − γ J Figure 9: Left panel : Cumulative distribution of contact forces close to the two edges at φ = 1 . 5. The solid line corresponds to the mean field theory prediction. In the inset we plotthe corresponding empirical distribution function. Right panel : Cumulative distribution ofsmall positive and negative gaps for different packing fractions. The solid line represents themean field theory prediction. Data produced with system size N = 1024, dimensions d = 3,averaged over 45 samples for density φ = 0 . 7, 44 samples for φ = 1 . 5, 27 samples for φ = 2 . N variables x i arranged in a vector x = { x , . . . , x N } lying on the sphere | x | = N are sought to minimize the cost function H [ x ] = αN (cid:88) µ =1 | h µ | θ ( − h µ ) (7)where the gaps h µ are defined as h µ = 1 √ N ξ µ · x − σ (8)with ξ µ = { ξ µ , . . . ξ µN } a set of N dimensional vectors with components extracted from aNormal distribution and σ a constant control parameter. Local minima of the PEL areisostatic meaning that there is an isostatic number of gaps h µ = 0 and characterized by criticalpower laws in the gap and forces distribution. The Hessian of the non-analytic minima canbe defined by smoothing out the singularity of the linear potential close to h µ = 0 as we havedone with linear soft spheres. Calling (cid:15) the smoothing parameter, the Hessian becomes H ij = 1 (cid:15)N (cid:88) µ : h µ =0 ξ µi ξ µj + ζδ ij (9)being ζ a Lagrange multiplier needed to enforce the spherical constraint which plays here thesame role of the prestress in spheres. In the glassy phase, ζ < (cid:15) → ω given by D ( ω ) = 1 π (cid:112) − ω . (10)14 ciPost Physics Submission . . . . . . . . . − − − − D ( ω ) ω E = E = π √ − ω ω R ω D ( t ) d t ω/ h ω i E = N = 512 N = 1024 N = 2048 Figure 10: Density of states of the linear perceptron optimization problem for two valuesof the energy at fixed α = 5 (at energy E = 2 averaged over 30 samples with system size N = 2048, at energy E = 0 . 005 averaged over 50 samples with system size N = 1024). Theblack line is the theoretical prediction given by Eq. (10). In the Inset we plot the left tail ofthe DOS for different sizes which shows that the behavior is compatible with having a positiveDOS for ω → α = 5 and at two different values of the energy and we compare it with the theoreticalprediction. References [1] A. J. Liu and S. R. Nagel, Jamming is not just cool any more , Nature (6706), 21(1998), doi:https://doi.org/10.1038/23819.[2] A. J. Liu and S. R. Nagel, The jamming transition and the marginally jammed solid ,Annu. Rev. Condens. Matter Phys. (1), 347 (2010), doi:10.1146/annurev-conmatphys-070909-104045.[3] C. S. O’Hern, S. A. Langer, A. J. Liu and S. R. Nagel, Random packings of frictionlessparticles , Phys. Rev. Lett. (7), 075507 (2002), doi:10.1103/PhysRevLett.88.075507.[4] C. S. O’Hern, L. E. Silbert, A. J. Liu and S. R. Nagel, Jamming at zero temperatureand zero applied stress: The epitome of disorder , Phys. Rev. E (1), 011306 (2003),doi:10.1103/PhysRevE.68.011306.[5] M. E. Cates, J. Wittmer, J.-P. Bouchaud and P. Claudin, Jamming, force chains, andfragile matter , Phys. Rev. Lett. (9), 1841 (1998), doi:10.1103/PhysRevLett.81.1841.[6] A. V. Tkachenko and T. A. Witten, Stress propagation through frictionless granularmaterial , Phys. Rev. E , 687 (1999), doi:10.1103/PhysRevE.60.687.15 ciPost Physics Submission [7] A. J. Liu, S. R. Nagel, W. Van Saarloos and M. Wyart, The jamming scenario – anintroduction and outlook , In L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti andW. van Saarloos, eds., Dynamical Heterogeneities and Glasses . Oxford University Press,doi:10.1093/acprof:oso/9780199691470.003.0009 (2011).[8] C. P. Goodrich, A. J. Liu and J. P. Sethna, Scaling ansatz for thejamming transition , Proc. Natl. Acad. Sci. U.S.A. (35), 9745 (2016),doi:https://doi.org/10.1073/pnas.1601858113.[9] E. Lerner, G. During and M. Wyart, Low-energy non-linear excitations in sphere pack-ings , Soft Matter , 8252 (2013), doi:10.1039/C3SM50515D.[10] P. Charbonneau, E. I. Corwin, G. Parisi and F. Zamponi, Universal microstructureand mechanical stability of jammed packings , Phys. Rev. Lett. , 205501 (2012),doi:10.1103/PhysRevLett.109.205501.[11] P. Charbonneau, E. I. Corwin, G. Parisi and F. Zamponi, Jamming criticality re-vealed by removing localized buckling excitations , Phys. Rev. Lett. , 125504 (2015),doi:10.1103/PhysRevLett.114.125504.[12] M. Wyart, Marginal stability constrains force and pair distributions at random closepacking , Phys. Rev. Lett. , 125502 (2012), doi:10.1103/PhysRevLett.109.125502.[13] P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani and F. Zamponi, Fractal free energiesin structural glasses , Nat. Commun. , 3725 (2014), doi:10.1038/ncomms4725.[14] P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani and F. Zamponi, Exact theory ofdense amorphous hard spheres in high dimension. iii. the full replica symmetry break-ing solution , JSTAT (10), P10009 (2014), doi:https://doi.org/10.1088/1742-5468/2014/10/P10009.[15] P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani and F. Zamponi, Glass and jammingtransitions: From exact results to finite-dimensional descriptions , Annu. Rev. Condens.Matter Phys. , 265 (2017), doi:10.1146/annurev-conmatphys-031016-025334.[16] S. Franz and G. Parisi, The simplest model of jamming , Journal of Physics A:Mathematical and Theoretical (14), 145001 (2016), doi:https://doi.org/10.1088/1751-8113/49/14/145001.[17] S. Franz, G. Parisi, P. Urbani and F. Zamponi, Universal spectrum of normal modesin low-temperature glasses , Proc. Natl. Acad. Sci. U.S.A. (47), 14539 (2015),doi:https://doi.org/10.1073/pnas.1511134112.[18] S. Franz, G. Parisi, M. Sevelev, P. Urbani and F. Zamponi, Universality of the sat-unsat(jamming) threshold in non-convex continuous constraint satisfaction problems , SciPostPhysics (3), 019 (2017), doi:10.21468/SciPostPhys.2.3.019.[19] S. Franz, S. Hwang and P. Urbani, Jamming in multilayer supervised learning models ,Phys. Rev. Lett. (16), 160602 (2019), doi:10.1103/PhysRevLett.123.160602.[20] D. Hexner, P. Urbani and F. Zamponi, Can a large packing be assembled from smallerones? , Phys. Rev. Lett. , 068003 (2019), doi:10.1103/PhysRevLett.123.068003.16 ciPost Physics Submission [21] D. Hexner, A. J. Liu and S. R. Nagel, Two diverging length scales inthe structure of jammed packings , Phys. Rev. Lett. (11), 115501 (2018),doi:10.1103/PhysRevLett.121.115501.[22] S. Franz, A. Sclocchi and P. Urbani, Critical jammed phase of the linear perceptron ,Phys. Rev. Lett. (11), 115702 (2019), doi:10.1103/PhysRevLett.123.115702.[23] E. Bitzek, P. Koskinen, F. G¨ahler, M. Moseler and P. Gumbsch, Struc-tural relaxation made simple , Phys. Rev. Lett. (17), 170201 (2006),doi:10.1103/PhysRevLett.97.170201.[24] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, A limited memory algorithm for boundconstrained optimization , SIAM Journal on Scientific Computing (5), 1190 (1995),doi:https://doi.org/10.1145/279232.279236.[25] Wyart, M., On the rigidity of amorphous solids , Ann. Phys. Fr. (3), 1 (2005),doi:10.1051/anphys:2006003.[26] S. Franz, A. Sclocchi and P. Urbani, In preparation .[27] G. Parisi, P. Urbani and F. Zamponi, Theory of Simple Glasses: Exact Solutions inInfinite Dimensions , Cambridge University Press, doi:10.1017/9781108120494 (2020).[28] M. Wyart, S. R. Nagel and T. A. Witten, Geometric origin of excess low-frequencyvibrational modes in weakly connected amorphous solids , EPL (Europhysics Letters) (3), 486 (2005), doi:https://doi.org/10.1209/epl/i2005-10245-5.[29] L. Yan, E. DeGiuli and M. Wyart, On variational arguments for vibrational modes nearjamming , EPL (2), 26003 (2016), doi:https://doi.org/10.1209/0295-5075/114/26003.[30] G. Parisi, Soft modes in jammed hard spheres (i): Mean field theory of the isostatictransition , arXiv preprint (2014), doi:arXiv:1401.4413.[31] F. P. Benetti, G. Parisi, F. Pietracaprina and G. Sicuro, Mean-field model forthe density of states of jammed soft spheres , Phys. Rev. E (6), 062157 (2018),doi:10.1103/PhysRevE.97.062157.[32] G. Biroli and P. Urbani, Breakdown of elasticity in amorphous solids , Nature physics (12), 1130 (2016), doi:https://doi.org/10.1038/nphys3845.[33] C. Rainone, P. Urbani, H. Yoshino and F. Zamponi, Following the evolution of hard sphereglasses in infinite dimensions under external perturbations: Compression and shearstrain , Phys. Rev. Lett. (1), 015701 (2015), doi:10.1103/PhysRevLett.114.015701.[34] G. Combe and J.-N. Roux, Strain versus stress in a model granular ma-terial: a devil’s staircase , Physical Review Letters (17), 3628 (2000),doi:10.1103/PhysRevLett.85.3628.[35] S. Franz and S. Spigler, Mean-field avalanches in jammed spheres , Phys. Rev. E (2),022139 (2017), doi:10.1103/PhysRevE.95.022139.17 ciPost Physics Submission [36] B. Shang, P. Guan and J.-L. Barrat, Elastic avalanches reveal marginal be-havior in amorphous solids , Proc. Natl. Acad. Sci. U.S.A. (1), 86 (2020),doi:10.1073/pnas.1915070117.[37] J. Kurchan, G. Parisi, P. Urbani and F. Zamponi, Exact theory of dense amorphous hardspheres in high dimension. II. The high density regime and the gardner transition , J.Phys. Chem. B , 12979 (2013), doi:https://doi.org/10.1021/jp402235d.[38] L. Berthier, P. Charbonneau, Y. Jin, G. Parisi, B. Seoane and F. Zamponi, Growingtimescales and lengthscales characterizing vibrations of amorphous solids , Proc. Natl.Acad. Sci. U.S.A. (30), 8397 (2016), doi:10.1073/pnas.1607730113.[39] G. Biroli and P. Urbani, Liu-nagel phase diagrams in infinite dimension , SciPost Physics (4), 020 (2018), doi:10.21468/SciPostPhys.4.4.020.[40] D. J. Durian, Foam mechanics at the bubble scale , Phys. Rev. Lett. (26), 4780 (1995),doi:10.1103/PhysRevLett.75.4780.[41] M. Griniasty and H. Gutfreund, Learning and retrieval in attractor neural networksabove saturation , J. Phys. A (3), 715 (1991), doi:https://doi.org/10.1088/0305-4470/24/3/030.[42] P. Majer, A. Engel and A. Zippelius, Perceptrons above saturation , J. Phys. A (24),7405 (1993), doi:https://doi.org/10.1088/0305-4470/26/24/015.[43] G. Gy¨orgyi, Techniques of replica symmetry breaking and the storageproblem of the mcculloch–pitts neuron , Phys. Rep.342