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(6, ed ´ rl eLuan,C-05Luan,Switzerland Lausanne, CH-1015 Lausanne, de erale ´ l’ a ` nri tmqee aux et Atomique Energie ´ 1 the under Published Submission.y Direct PNAS a performed is article research, This interest.y of designed conflict paper.y no F.Z. the declare wrote authors and and The data, M.W., analyzed tools, P.U., reagents/analytic new H.I., contributed research, C.B., contributions: Author ffedmprpril 9 8 9.A osqec fhypo- of consequence a As 29). 28, (9, staticity, particle per freedom of t ls felpod,B,adohrnnpeia atce that universal- particles same nonspherical other the and in replica BP, is ellipsoids, the model of using this class analytically that ity solved confirm be that We can model method. and perceptron BP we random the Next, the argument. mimics of stability generalization marginal a a propose the crit- using same on the by have Based exponents indeed models 43. ical two ref. the that in show we introduced mapping, recently (BP), spher- “breathing” particles of model ical of a example into first ellipsoids a map As we universality, jamming. hypostatic of class universality the a,tepyia ehns htidcsasaigbhvo such behavior scaling particu- a In induces as 42). that (29, mechanism infancy physical its the in lar, still is spherical particles nonspherical with transition jamming of the Compared of understanding (40), (41). theoretical the polygons particles particles, superballs deformable convex-shaped even (33–37), other and spherocylinders (39), for superellipsoids by pack- obtained stabilized Hypostatic (38), also (29–32). are energy are they potential ings because the of modes” terms “quartic quartic as to referred predict would thus which expected condition, is is stability what system z Maxwell’s than naive The lower the is shape. by number spherical contact The perfectly hypostatic: the isostatic from the ation number from as contact spheres, increases the of those continuously Notably, value point of 27–32). jamming 9, the effects parti- the in 3, The each detail at (2, in position. ellipsoids of investigated of been particle direction have case the freedom the of to specify degrees extra addition should in one cle case, nonspherical. this general, in In are, particles constituent reality, in and, owo orsodnesol eadesd mi:[email protected] Email: addressed. be should correspondence whom To J nsadtervbain cl steprilseov toward pack- evolve of particles the structure internal as the with scale spheres. how vibrations particles their derive and for and ings class freedom of universality method. degrees replica a the unravel and combina- arguments We a stability using marginal observations of these tion struc- explain we both by Here affected that shape. fundamentally are indicates properties vibrational spherocylinders and tural work and theoretical com- ellipsoids and not in empirical are Previous properties understood. scaling pletely its particles, situation nonspherical generic the of In suspensions. dense materials, and granular of sand property including key a is transition jamming The Significance nti ok epooeatertclfaeokt describe to framework theoretical a propose we work, this In oee,asse fshrclprilsi nielzdmodel idealized an is particles spherical of system a However, z 2(d = J c atiuWyart Matthieu , − 2d + D ∝ d (ω ex ∆ b ) aoaor ePyiu The Physique de Laboratoire ) 1/2 NSlicense.y PNAS a nmlu eomdsat modes zero anomalous has where sunclear. is z nrisAtraie CA,F-91191 (CEA), Alternatives Energies J ´ − d d ex n rnec Zamponi Francesco and , 2d stenme frttoa degrees rotational of number the is ∝ ∆ 1/2 where , ,CR,705Prs France; Paris, 75005 CNRS, e, ´ rqe D orique, ´ NSLts Articles Latest PNAS ∆ preetde epartement ´ eoe h devi- the denotes ϕ J hc are which , b | f6 of 1

PHYSICS display hypostatic jamming. This analysis further predicts the where the gap function is defined as scaling behavior of g(h) and P(f ) near the jamming point. Inter- estingly, we find that these functions do not show a power-law |xi − xj | − σij hij = , behavior even at the jamming point, in marked contrast to the σ0 jamming of spherical particles. Also the simplicity of the model σij 1 allows us to derive an analytical expression of the density of = s . [5] σ0  2 2  D(ω) (ˆrij ·ˆui +ˆrij ·ˆuj ) (ˆrij ·ˆui −ˆrij ·ˆuj ) states , which exhibits the very same scaling behavior as 1 − χ + that of ellipsoids and BP. Finally, we confirm our predictions by 2 1+χˆui ·ˆuj 1−χˆui ·ˆuj numerical simulations of the BP model. ˆ BP Model Here, rij = (xi − xj )/ |xi − xj | is the unit vector connecting the ith and j th particles, εσ0 is the length of the principal axis, and The BP model (43) was originally introduced to understand the χ = (ε2 − 1)/(ε2 + 1), where ε denotes the aspect ratio. Because physics of the swap Monte Carlo algorithm (44), but here we we are interested in the nearly spherical case, we expand the pair focus on its relation with the jamming of ellipsoids. The model potential in small ∆ = ε − 1 as consists of N spherical particles with positions xi in d dimensions and radius Ri ≥ 0, interacting via the potential energy ∆ v(h ) = v(h(0)) − v 0(h(0)) (ˆr · uˆ )2 + (ˆr · uˆ )2 ij ij 2 ij ij i ij j VN ({x}, {R}) = UN ({x}, {R}) + µN ({R}), [1] 2 + ∆ wij , [6] θ(x) where, defining as the Heaviside theta function, (0) 2 2 where hij = rij /σ0 − 1 and ∆ wij denotes the O(∆ ) term that 2 we do not need to write explicitly. Substituting this in Eq. 4 and X hij 2 U = k θ(−h ), h = |x − x | − R − R , [2] keeping terms up to ∆ , we obtain VN ≈ UN + µN , where N 2 ij ij i j i j i

 0  ab −1 P 0 (0) a b kR X 0 2 Ri 2 The stiffness matrix is ki = −∆ j (=6 i) v (hij )ˆrij ˆrij , where µN = (Ri − Ri ) . [3] 2 Ri a, b = 1, ... , d. Note that near the jamming point, ki behaves as i 0 ki ∼ v (h)/∆ ∼ p/∆, which is the same scaling of the stiffness kR of the BP model, Eq. 3. Hence, if we identify ∆uˆi with Ri , Here, kR is determined by imposing that the dimensionless SD in the vicinity of jamming the potential for ellipsoids can be ana- pP 0 2 2 −1 P 0 ∆ ∝ i (Ri − Ri ) /(NR0 ) is constant, with R0 = N i Ri . lyzed essentially in the same way as in the BP model (43), as we Note that ∆ = 0 (corresponding to kR = ∞) gives back the usual discuss next. spherical particles (5) and that the full distribution of radii, P(R), can generically change even if ∆ is kept fixed. Upon Marginal Stability approaching jamming, where the adimensional pressure p (in The distinctive feature of both BP and ellipsoids is that the 2−d units of kR0 ) vanishes, it is found that kR = p/∆ and P(R) total potential, and thus the Hessian matrix, can be split into remains constant (43). two parts: one having finite stiffness and the second one hav- Because the BP model has Nd translational degrees of free- ing vanishing stiffness p/∆ by dimensional arguments. The dom and N radial degrees of freedom, the naive Maxwell stabil- zero modes of the first term are stabilized by the second one, ity condition requires z ≥ 2(d + 1) in the thermodynamic limit as recognized in refs. 29 and 32. We now provide additional (19, 45). However, a marginal stability argument and numeri- insight on this structure by generalizing a marginal stability cal simulations prove that the contact number at the jamming argument discussed for the BP in ref. 43. At jamming, p = 0 1/2 point zJ increases continuously as zJ − 2d ∝ ∆ (43) and the and VN = UN because µN ∝ p. The N3 ≡ Nz/2 constraints com- system is hypostatic for sufficiently small ∆; i.e., the number of ing from UN , one per mechanical contact, stabilize the same constraints is smaller than that required by Maxwell’s stability number of vibrational modes. Because the system is hypostatic, condition. This is very similar to ellipsoids and motivates us to there remain N0 ≡ N (d + dex) − Nz/2 = N (dex − δz/2) zero- conjecture that the two models could belong to the same uni- frequency modes, where δz = z − 2d and dex is the number of versality class. In the following, we show that this expectation extra degrees of freedom per particle; i.e., dex = 1 for the BP and is indeed true: Hypostatic packings of the BP and ellipsoids are dex = d − 1 for ellipsoids. Above jamming, where p > 0, the N0 stabilized by a common mechanism and have the same critical zero modes are stabilized by the “soft” constraint coming from exponents. µN whose characteristic stiffness is kR ∼ ki ∼ k(p/∆)  k, where k is the stiffness associated to UN . Hence, the energy scale of Mapping from Ellipsoids to BP these modes remains well separated from that of the N3 other We now construct a mapping from a system of ellipsoids to the modes, and we can restrict to the N0-dimensional subspace of the spherical BP model introduced above. Ellipsoids are described soft modes. In this space, we have N0 = N (dex − δz/2) degrees by their position xi and by unit vectors uˆi along their princi- of freedom, and µN provides Ndex constraints; hence the number pal axis, and for concreteness we model them by the Gay–Berne of degrees of freedom is N δz/2 less than the number of con- potential (31, 46) straints. When δz  1, a variational argument developed in refs. 17 and 47 describes the low-frequency spectrum. It shows that 2 2 the soft modes are shifted above a characteristic frequency ω∗ ∼ X h 2 2 −1 2 VN ({x}, {uˆ}) = v(hij ), v(h) = k θ(−h), [4] ki δz ∼ kRδz ∼ ∆ p δz , which is reduced by ∼ −p by the so- 2 2 −1 2 i

2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1812457115 Brito et al. Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 anyectstezr oe 3) pligteagmn in Eq. using argument which and as the modes deformation, behaves Applying zero shear the (30). to to modes 18 response zero ref. the the excites study mainly also can We where ellipsoids, of results numerical where tas it n that ing Eq. BP, Eq. recover γ the In holds: scaling following the and Eqs. ubrsol aetesm cln fshrclparticles: spherical of scaling same the have should number eas nteagmn easmdt ecoet jamming to close be to assumed Eq. we 1. argument Fig. (p the in illustrated in as because 43), 29, (9, els contact the of singularity number square-root universal the explains This aafrnnpeia atce r erdcdfo e.4,adfo the from and 40, ref. from reproduced are sphericity prediction particles theoretical nonspherical the for denotes line Data solid the while result, ical 1. Fig. range frequency the in found be can modes that suggests argument stability marginal The Spectrum Vibrational (30). result numerical the stable, marginally is system where rt tal. et Brito small for p ellipses of ratio aspect set and sphericity the between relation = 1 = ∼ c 10 = f 8 tfixed at 0) /2 −6 (x oclas l aa aafrteB orsodt pressure a to correspond BP the for Data data. all collapse to 1/6 and nvra cln ftecnatnme.Smosdnt h numer- the denote Symbols number. contact the of scaling Universal c ∆ . z = ) and f 1 edefined we A, 10 z J (x J → 2 = and 9 G scnre ynmrclsmltos(3.Assum- (43). simulations numerical by confirmed is c ) bevdi lisis P n eea te mod- other several and BP, ellipsoids, in observed f ml that imply hc requires which 8, 0 0 sarglrfnto around function regular a is (x ∼ d + ii,Eq. limit, c + ) δ ∆ 2 c zk → 1 c ntecnrr,when contrary, the On . r nnw osat.Asmn htthe that Assuming constants. unknown are x 0 R x ∆ + = ∆ ∼ 1/2 1/2 · · · p z δ δ z zk − and c for hsi optbewt previous with compatible is This . (A 10 ∆ = δ n obtains and i δ z ω z ∼ z J − x ∗ ∆ ∼ eue oEq. to reduces ∼ ∼ (p 1) p  γ f / ∆ aetesm cln dimension scaling same the have f 1/2 p ∆ (x eut n(43) in results 0, = ) √ nthe In 1. 1/2 (p 1/2 −1/2 hc eoestecretscaling correct the recovers which , ) ∆ /∆). → . . npretareetwith agreement perfect in , z p const − , h ha modulus shear the 8, z x p J 8 ∼ → ∼ ω ∆ od when holds for N ∗ n a expand can one 0, hc requires which 9, ∆ 0  0 . −0.35±1 ii,w should we limit, otvibrational soft x ω p  h contact the , . o the For 1. √ δ k z p R ∼ p We ∆. with ,  ∆ (48). [11] [10] [9] [8] 1/2 ∆ G , . osidpnetyanra itiuino eoma n unit and mean zero of distribution normal a independently lows The where “size” and ω atceadteosalsis obstacles the and particle the to corresponds band is frequency lowest The (i) N regions: three lowing where freedom ellipsoids, of of degrees one angular the the to onto scaling mapped is similar a has it that potential interaction the of with model, BP of the density of vibrational the detail more states in discuss to argument the ing ing xoet fhpsai amn.I ossso n rcrpar- radius tracer of one hypersphere coordinate of critical consists the with of It all ticle jamming. reproduces hypostatic solution of the analyt- show, exponents model, solved we be The as can perceptron, and, ellipsoids, particles. the ically of BP, nonspherical generalization of the a is models in which other jamming many hypostatic and of class universality all in jamming is dimension isostatic mod- of in Both exponents (24–26). particles dimensions critical model the by perceptron reproduce the analytically els understood: by well described equivalently, is be or, jamming can isostatic of It class universality The Model Mean-Field theoretical the follow of also function which a BP, as fraction the prediction the for band report each we in 3 modes Fig. in Finally, theory. the and fre- peak fixed characteristic we with theory, here bands our separated by 43; quencies predicted three ref. As of point. in consists state explained each about for are samples Details perceptron BP for 2. the data the Fig. for show of in results reported simulations analytical are the and below 43 introduced ref. model from BP the or (rotational ω freedom extra of the degrees to radial) corresponding band intermediate an ω particles, spherical band. of translational packings single their a isostatic displays that of which predicts that which from nificantly 47), argu- (17, variational is standard particles frequency typical the spherical apply of can ment one and modes, these ∆ the to sponding 1 1,2 ∗ 2 0 enwitoueama-edmdlwihdsrbsthe describes which model mean-field a introduce now We Then, ueia eut for results Numerical  ∼ ∼ = N N V ω ξ ontcag uhwith much change not do h diinldgeso reo ontsrnl affect strongly not do freedom of degrees additional the 1, ∂ 0 3 N p N 3 µ 2 D v 2 olw h hoeia predictions theoretical the follows , u omria tblt and stability marginal to due = oe aefiiefeunya amn.W o refine now We jamming. at frequency finite have modes ∝ = V (d (h r rznvrals n aho hi opnnsfol- components their of each and variables, frozen are (ω N − N D ω N ∆ U ex = ) ti ovnett en the define to convenient is It ). σ 0,1,2 /∂ (ω N 1/2 d ω − + d edn otecnetr htislwrcritical lower its that conjecture the to leading , 0 + (∆ ) h 2 δ eas n that find also We . 2 = hi cln with scaling Their . R N 2 ∼ z erjmigcnb eaae notefol- the into separated be can jamming near µ 0 θ N µ −1 /2) N (−h ∂ = 484 = h neato oeta ewe h tracer the between potential interaction The . 2 (49). 2 , ω R = Nz µ 2 2 U eomdssaiie by stabilized modes zero i h N )/2 ) Nd µ ∼ N x N 2 √ /2 /∂ = ∼ δ atce,aeae vra es 1,000 least at over averaged particles, D ntesraeo the of surface the on = N = n h a variable gap the and z (∆ oe a esltit w ad:(ii) bands: two into split be can modes rnltoa ere ffedm For freedom. of degrees translational ∆ ∆ x (ω 2 µ=1 √ X V N N M and · ∼ 2 −1 N N ξ and ) (d 1 h ihs adcorre- band highest the (iii) and , µ h resulting The ∆. v felpod rmrf 0adof and 30 ref. from ellipsoids of = p w.r.t. R + (h − hc saancnitn with consistent again is which , M i N p ) µ d σ . 2 ω lorpre nFg at 2 Fig. in reported also ∆, δ k ex ), btce fcoordinates of obstacles ∼ − 0 R z x stescn derivative second the as ), ihtpclfrequency typical with /2, i ∝ µ ∼ k R and R N NSLts Articles Latest PNAS p µ p N × N ∆ 1/2 . = hl h remain- the while /∆, 2 R ω k ∼ 2 i o small for R 0 /∆ h µ ∆ ∝ D µ u. ˆ µ=1 X N N M esa matrix Hessian p ∆ nsc way a such in , (ω sdfie as defined is hi typical Their . h remain- The . f -dimensional d i 1/2 R ) = ∞ → ifr sig- differs µ 2 , N , p ω i while , | 1 /N R D ∝ f6 of 3 [12] [13] (15) i (ω /∆ ∆, ξ of µ )

PHYSICS Fig. 2. Universality of the density of states. (Top) Density of states for ellipses, BP, and the perceptron. (Bottom) Evolution with ∆ of the characteristic −4 1/2 1/2 frequencies at p = 10 . Solid lines denote the theoretical predictions, ω0 ∝ ∆ , ω1 ∝ ∆, and ω2 ∝ ∆ , respectively. Data of ellipses are reproduced from ref. 32.

variance. The dynamical variables are x and the Rµ, whose vari- rations. Upon increasing α, q increases and eventually reaches ance is controlled by the chemical potential µN . We fix the value q = 1 at αJ , which is the jamming transition point (Fig. 4). Nat- PM 2 2 urally, due to the additional degrees of freedom when ∆ > 0, of kR so that µ=1 Rµ = M ∆ . In the ∆ → 0 limit, the system reduces to the standard perceptron model investigated in ref. 26, we have αJ (∆) > αJ (0) for equal σ. For σ > 0, the RS ansatz α while for ∆ > 0 the Rµ play the same role as the particle sizes in is stable for all values of and it describes the jamming transi- the BP model. tion. For σ < 0 instead, the jamming line is surrounded by a RSB Because the model can be solved by the same procedure region where the RS ansatz is unstable. The jamming transition as that of the standard perceptron model, here we give just a should thus be described by the RSB ansatz where q(x) is not brief sketch of our calculation. The free energy of the model at constant, corresponding to a rough energy landscape. The qual- temperature T = 1/β can be calculated by the replica method, itative behavior of the phase diagram is independent of ∆; in 1 n R N M −βVN −βf = limn→0 nN log Z , where Z = d xd Re and the overbar denotes the averaging over the quenched randomness ξµ. Here we are interested in the athermal limit T → 0. Using the saddle-point method, the free energy can be expressed as a b a b a function of the overlap qab = x · x /N , where x and x denote the positions of the tracer particles of the ath and bth replicas, respectively. In the n → 0 limit, qab is parameterized by a continuous variable x ∈ [0, 1], qab → q(x). The function q(x) plays the role of the order parameter and characterizes the hierarchical structure of the metastable states (50). We first cal- culate the phase diagram assuming a constant q(x) = q, which is the so-called replica symmetric (RS) ansatz that describes an energy landscape with a single minimum. The result for ∆ = 0.1 is shown in Fig. 4. The control parameters are the obstacle α = M /N σ α density and size . If is small, the tracer parti- Fig. 3. Weights of the density of states. Shown is the fraction of modes cle can easily find islands of configurations x that satisfy all of fi = Ni/N in the three bands of D(ω) given in Fig. 2, plotted as functions the constraints hµ > 0: The total potential energy UN and the of p at fixed ∆ = 10−1 (Left) and ∆ at fixed p = 10−4 (Right) for BP (with pressure vanish and the system is unjammed. The overlap q < 1 d = 2 and dex = 1). The theoretical predictions f0 = (1 − δz/2)/3, f1 = δz/6, measures the typical distance between two zero-energy configu- and f2 = 2/3 are plotted as solid lines, inferred from the measured δz.

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Fig. rt tal. et Brito distribution gap positive the introduce g we results, numerical (−h (h ∼ (f h iuto scmltl ifrn if different completely is situation The nipratosral ocaatrz amn sthe is jamming characterize to observable important An α 1,2–6.Near 24–26). (15, y ) α p ) J g χ ≡ 1/2 : J )ρ(h r nt n eua ucin tjmig n h square- the and jamming, at functions regular and finite are h (h h eua ouinsol onc otejamming the to connect should solution regular the , speoiatycnrle ytecnatnumber contact the by controlled predominantly is µ h hs iga ftepreto oe for model perceptron the of diagram phase The θ ) hc stesm cln eaira hto spherical of that as behavior scaling same the is which , (h htaeeatyeult eo o oprsnwith comparison For zero. to equal exactly are that and ) )ρ(h ∂ ∂ h f g / P P (h g 1 R )/ (f (f (h −∞ 0 = ∆ ) ρ(h − 0 α z κ R ) ∼ ) ) 0 q J ∼ = ∞ ' r eua n nt ucin.Teprefac- The functions. finite and regular are and  ∆ ) ρ(h (x h oe sdsrbdb h “jamming” the by described is model the ,  R ≡ 1.42, dh h ∆ α → ) −∞ and 0 f ∆ −γ ) J ∼ −µγ N P ρ 1 θ ∆ ∂ ∂ h eua ouinsol connect should solution regular the , θν 0 (h (f h x f P dh γ → < σ nrf 6and 26 ref. in p df −κ p ) ) 0 z (2 = µ=1 M 0 α ρ(h n h oc distribution force the and (f xii oe-a eair(24– behavior power-law a exhibit sn h omo h scaling the of form the Using 0. where , (h , 1 = & amn siottcwith isostatic is jamming 0, g ∆ < σ ∆ tjamming, At ). (h hδ α 1 − −ν α −µ 2)adterglrsolution regular the and (26) J − (h ) and κ)/κ, J h ytmi ecie ya by described is system the , 0 ∼ ( ) q µ Consequently, . ( ) f sawy uruddb a by surrounded always is (x h − z = (f J −γ (h f ) z h h ≥ −h ∆ ∼ ∼ ∼ )i − and , ∼ ∼ )i uruddb the by surrounded is 0) enn htthe that meaning 1, y > 1) ∆ z /p htas ie the gives also that 1) ∆ χ 2 θ J ν n a show can One 0. x (3κ = µ ∼ ) P −2 g (corresponding ) , = ∆ (h p , z z (f for hsmatch- this , − ) ) onsthe counts .Tered The 0.1. ∼ − and z g q J (h f 4)/(2 (x P = θ ) z with , ) (f z P c [15] [14] ∼ and and ∆ z 1 = ) (f p − − ≡ 1, ) . 2,tedniyo states of density density eigenvalue the the 52), for formula Edwards–Jones to leads also 1/2 exponents critical new with ensaohruieslt ls fhpsai amn.We jamming. which hypostatic (40), of criticality class jamming nonspheri- which universality same of identical, another the models are defines display other models particles Many two that cal numerically. the predicts confirmed of theory we Our behaviors scaling particles. scaling the breathing the derived and we ellipsoids of number argument, contact the stability of behavior marginal a Using Conclusions Eq. of finite For g 15). 14, (6, tion where for usual When as Here, behavior. systems, critical particle the observe for to enough 43) small ref. in as Eq. point, jamming ming hypostatic the ∆ of con- properties analyti- This critical reproduce (43). the hypostatic can BP transition. of displaying perceptron and all models generalized 32) the cally the (31, of that ellipsoids all firms as of such those jamming, as from same starts the band At the highest peak. that the sharp while implies (24), a phase perceptron at exhibits RSB standard terminates band from lowest the full The starts in the band As in lowest the 2). that stability from find (Fig. marginal derived We bands scaling functions equation. arate analytical arbitrary RSB the are used full with we of compatible here properties we are because scaling 2 simplicity, the Fig. for in of but only the function numerically interested equations solving a RSB by obtained as full be form should quantities closed These in interac- analytically the of derivatives second the potential as states tion matrix of density Hessian the the of form ical Eqs. argument, stability marginal iebhvo of behavior tive i.5. Fig. ybl eoetenmrclrsl,wietesldln eoe h the- the Eq. denotes to line according solid the prediction, while result, oretical numerical the denote Symbols (h safia hc fuieslt,w ettepeito o the for prediction the test we universality, of check final a As analyt- the derive to us allows model the of simplicity The eedneo h a itiuinfunction distribution gap the of dependence ) n nvra cln function scaling universal a and sctof(i.5,cnitn ihtetertclprediction theoretical the with consistent 5), (Fig. off cut is 0 = ∆ 14. a distribution Gap γ 0 = ossetwt rvosnmrclobserva- numerical previous with consistent .413, , z g J 14. (h nFg ,w hwnmrclrsls(obtained results numerical show we 5, Fig. In 14. V − g(h) ) N D g 1 g xiisapwrlwdivergence, power-law a exhibits (h Eq. , (ω ∼ (h fB ertejmigpoint, jamming the near BP of g(h) ∝ ) ∆ ) ) h snraie by normalized is 1/2 n h cln of scaling the and −0.413 fteB oe at model BP the of w.r.t. 12, ∆ ω and D µ ntecnrr,tedvrec of divergence the contrary, the on , 0 = ( Right . = (ω ω ω 1 c z κ/(4κ 2ωρ = ) 0 ∼ ∆ n o small for and 8 n h est fstates of density the and ∼ ∼ D and ∆ cln lto h aedata same the of plot Scaling ) ∆ x p D (ω ∆ i 0 − 1/2 ω et eki found, is peak delta a (ω (x −1/2 11. sbfr,w define we before, As ). (ω and NSLts Articles Latest PNAS 2 )=0 = 4) z g p ) h cln analysis scaling The ). ∼ , (h 2 1/2 ) ω ossetwt the with consistent , ipastresep- three displays ∆ k ) R 0 R p a eexpressed be can 1/2 → , erwhich near .851 µ D and , 10 = g ω /∆ (h 1 z (ω h qualita- The . ω 1 , , p and , o larger for ) sn the Using . oobtain to ), and g = k −5 D ttejam- the at (h R 10 ρ(λ (ω value a , and , p ) −6 ν ∼ ) ω | = ) which (Left . ∼ D D 2 h f6 of 5 (51, µ (ω (ω −γ ω are h p − 2 ) ) ) , . . .

PHYSICS introduced an analytically solvable model which allows us to quite different from that of spherical particles, in agreement with derive analytically the critical exponents associated to this recent experiments (57). universality class. One of the most surprising outputs of our theory is the uni- ACKNOWLEDGMENTS. We thank B. Chakraborty, A. Ikeda, J. Kurchan, D(ω) S. Nagel, and S. Franz for interesting discussions. We thank the authors versality of the density of states (Fig. 2). This might be of refs. 40 and 32 for sharing their data used in Figs. 1 and 2, respec- relevant for some colloidal experiments where the constituents tively. This project received funding from the European Research Council are nonspherical (53), in which the vibrational modes could be under the European Union’s Horizon 2020 Research and Innovation pro- experimentally extracted from the fluctuations of positions (54, gram (Grant 723955-GlassUniversality). This work was supported by Grants 55). Another relevant question is how nonspherical particles 689 454953 (to M.W.) and 454955 (to F.Z.) from the Simons Foundation and by a public grant from the “Laboratoire d’Excellence Physics Atoms Light would flow under shear (30). The divergence of the at Mater” (LabEx PALM) overseen by the French National Research Agency jamming is related to the low eigenvalues of D(ω) (56), which (ANR) as part of the “Investissements d’Avenir” program (reference no. suggests that the shear flow of nonspherical particles should be ANR-10-LABX-0039-PALM; to P.U.).

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