Shear bands as manifestation of a criticality in yielding amorphous solids
Giorgio Parisi, Itamar Procaccia, Corrado Rainone, Murari Singh
aa r X i v : . [ c ond - m a t . d i s - nn ] M a r D R A F T Shear bands as manifestation of a criticality inyielding amorphous solids
Giorgio Parisi a, , Itamar Procaccia b,1 , Corrado Rainone b , and Murari Singh b a Dipartimento di Fisica, Sapienza Universitá di Roma, INFN, Sezione di Roma I, IPFC – CNR, Piazzale Aldo Moro 2, I-00185 Roma, Italy; b Department of ChemicalPhysics, the Weizmann Institute of Science, Rehovot 76100, IsraelThis manuscript was compiled on October 16, 2018
Amorphous solids increase their stress as a function of an appliedstrain until a mechanical yield point whereupon the stress cannotincrease anymore, afterwards exhibiting a steady state with a con-stant mean stress. In stress controlled experiments the system sim-ply breaks when pushed beyond this mean stress. The ubiquity ofthis phenomenon over a huge variety of amorphous solids calls for ageneric theory that is free of microscopic details. Here we offer sucha theory: the mechanical yield is a thermodynamic phase transition,where yield occurs as a spinodal phenomenon. At the spinodal pointthere exists a divergent correlation length which is associated withthe system-spanning instabilities (known also as shear bands) whichare typical to the mechanical yield. The theory, the order parameterused and the correlation functions which exhibit the divergent corre-lation length are universal in nature and can be applied to any amor-phous solids that undergo mechanical yield. yielding | shear bands | criticality | correlation lenghts | replica method A solid, be it crystalline or amorphous, is operatively de-fined as any material capable to respond elastically toan externally applied shear deformation (1). However, anysolid material, when subject to a large enough shear-strain,finally undergoes a mechanical yield. Here we focus on themechanical yield of amorphous materials such as molecularand colloidal glasses, foams, and granular matter. The phe-nomenology exhibited by the yielding point within this vastclass of materials, as reported in countless strain-controlledsimulations (2–8) and experiments (9–11) shows a remarkabledegree of universality despite the highly varied nature of themodel systems involved. Among these universal features isthe presence, at the onset of flow at yielding, of system span-ning excitations referred to as shear-bands (12, 13), whereinthe shear strongly localizes, leaving the rest of the materialunperturbed. This phenomenon is of capital importance forengineering applications as it is responsible for the brittlenesstypical of glassy materials, in particular metallic glasses (14),whose potential for practical use is stymied by their tendencyto shear-band and fracture (13, 15, 16).In athermal amorphous solids the phenomenon has univer-sal features. For strains γ smaller than some critical valuedenoted as γ Y the stress in the material grows on the av-erage when the strain is increased. After yield the stresscannot grow on the average, no matter how much the strainis increased. The universality of the basic phenomenologyof yielding begs a picture of its characteristics in terms ofa universal theory, in the sense that such a theory shouldrely on a statistical-mechanical framework and be indepen-dent of details such as chemical composition and productionprocess of the material. This need was addressed in a recentwork (17), wherein building up from ideas first advanced in(18) there emerged a picture of mechanical yielding as a first- order phenomenon, i.e. as a discontinuous phase transitionin a suitable overlap order parameter Q ab (defined in Eq. (1)below) which jumps from a value of order 1 to a value oforder zero as strain is increased above the yielding thresh-old γ Y . The physical meaning of this observation is that be-fore yielding the amorphous system was limited to a smallpatch in the configuration space, very far from any kind ofergodicity. The yielding transition is an opening of a muchlarger available configuration space, whereupon the system isergodized subject to the constraint of constant mean stress.Within this framework, the yielding transition is essentiallyenvisioned as a spinodal point (19) i.e. the point where themetastable, high Q ab glassy patch of available configurationsbecomes unstable with respect to a new phase with low Q ab ,associated with an ergodized system in the presence of disor-der (20). A paradigmatic example of such a spinodal is theMode Coupling crossover (12), characterized by dynamicalslowing down and heterogeneities, whose behavior is charac-terized by a dynamical lengthscale which can be extractedfrom suitable multi-point correlators (12). According to ourpicture, this kind of critical behavior should also be found atthe yielding transition, conditional that one is able to derivethe expression of the right correlator to measure. This sugges-tion seems even more reasonable in light of a recent study (21)wherein the similarity of shear bands with dynamical hetero-geneities has been pointed out; also, some oscillatory shearsimulations seem to indicate that a slow-down of the dynam-ics on approaching yielding may indeed be present (22, 23).It is important to stress here that the reason that a spinodal.. Significance Statement
The art of making structural, polymeric, and metallic glassesis rapidly developing with many applications. A limitation isthat under increasing external strain all amorphous solids havea yield stress which, when exceeded, results in a plastic re-sponse leading to mechanical failure. Understanding this iscrucial for assessing the risk of failure of glassy materials un-der loads. The universality of the mechanical yield requiresa theory that is general enough to transcend the microscopicdetails of different glasses, which all show similar stress-straincurves with a yield point. We provide what appears to be thefirst general theory which is thermodynamic in nature, showingthat the mechanical yield is a spinodal criticality in an appro-priately constructed free energy landscape.
GP, IP and CR designed research. IP, CR and MS performed research and analyzed data, and IPand CR wrote the paper.The authors declare no conflict of interest. To whom correspondence should be addressed. E-mail: [email protected]
October 16, 2018 | vol. XXX | no. XX | R A F T point can be exposed and measured is that the glassy timescales and the athermal conditions stabilize the metastablesystem until the spinodal point is crossed and the system be-comes unstable against constrained ergodization.Within a generic statistical-mechanical theory, formulatedin terms of a suitable Gibbs free energy G [ φ ] (i.e. the free-energy for fixed order parameter φ ), stable phases are iden-tified with its points of minimum in φ , and phase transi-tions happen when the curvature of these minima goes tozero, inducing a critical behavior which manifests divergingsusceptibilities-fluctuations, critical slowing down of the dy-namics, and growing correlation lengths (24). At a spinodalpoint, for example, one such minimum becomes unstable andtransforms into a saddle. In the case of the order parameter Q ab the general form of the free energy s [ Q ab ] had been al-ready derived and studied (see (25) for a review) in the contextof the theory of replicas originally developed for the study ofspin-glasses, and its properties, at least at mean-field level, arewell known (we refer to (18, 26) for the derivation of s [ Q ab ] inthe specific case of mean-field hard spheres); the matrix of sec-ond derivatives (or, using a more field-theoretic terminology,the mass matrix) is not diagonal in the base of Q ab , and afterdiagonalization is found to have only three distinct modes, ormasses (25). Of these, the most relevant ones are the so called replicon mode λ R , which for example goes to zero at the newlyproposed Gardner transition (27), and the longitudinal mode λ L which is instead related to spinodal points (18, 19) suchas our yielding transition. In the Supplementary Informationto this paper we review briefly the background theory that isat the basis of the present approach.In this paper, we build up from the results of (17) and,following the line of reasoning formulated above, we employthe expression of the correlation function relative to the longi-tudinal mode λ L as it can be derived from the replicated fieldtheory (25) to reveal the critical features of the yielding tran-sition. We measure this correlator in numerical simulation,and use it to expose the critical properties of the yieldingtransition, showing how shear bands manifest the divergingcorrelation length encoded in this correlator. We show howthe order parameter Q ab and its associated replicated fieldtheory are thereby able to provide a unified and universal pic-ture of the yielding transition in terms of a spinodal point inpresence of disorder, with an associated criticality. The correlation functions
The relevant order parameter for the problem at hand is theoverlap function Q ab which measures the distance betweentwo configurations “ a " and “ b " of the same system. Denotingthe position of the i th particle as r ai in configuration “ a " and r bi in configuration “ b " we define Q ab ≡ N N X i =1 θ ( ℓ − | r ai − r bi | ) , [1]where θ ( x ) is the Heaviside step function and ℓ is a constantlength which is taken below to be 1/3 in Lennard-Jones units(see below for numerical details). Thus Q ab = 1 for two iden-tical configurations and Q ab = 0 when the distance betweenthe positions of all the particles i in the two configurations ex-ceed ℓ . Based on the introductory discussion, we now derive an expression for the correlator associated with the longitudi-nal mode, from whence one can extract the correlation lengthassociated with the onset of criticality at the yielding point,and define an associated susceptibility which will shoot up asthe yielding point is approached. The first step is to “localize"the overlap function and define the r -dependent quantity Q ab ( r ) ≡ N X i =1 θ ( ℓ − | r ai − r bi | ) δ ( r − r ai ) , [2]Next, as mentioned above, the expression for the longitudinalcorrelator in terms of four-replica correlation functions canbe found by diagonalization of the correlation matrix G ab ; cd ,which is defined as the inverse of the mass matrix M ab ; cd of thereplicated field theory of the overlap order parameter Q ab (25).The derivation is a matter of standard diagonalization algebra,so we shall not report it here and refer to the SI for the details.The expression, employed for example in (28, 29) in the caseof a model with spins on a lattice, reads for athermal systems G L ( r ) = 2 G R ( r ) − Γ ( r ) , [3]with the definitions G R ( r ) ≡ h Q ab ( r ) Q ab (0) i − h Q ab ( r ) Q ac (0) i [4]+ h Q ab ( r ) i h Q cd (0) i , Γ ( r ) ≡ h Q ab ( r ) Q ab (0) i − h Q ab ( r ) i h Q ab (0) i . [5]Here angular brackets denote a thermal average in the thermalcase and an evaluation in an inherent state in the athermalcase; an ( • ) indicates an average over different samples ofthe glass. The quantity G R ( r ) is the correlation function ofthe replicon mode (25) and Γ ( r ) is just the garden-varietyfour-point correlator.Using these definitions and taking Eq. Eq. (2) into account,the quantities we compute in numerical simulation, beforetaking the ensemble average, are (see the SI and Ref. (30)):˜Γ ( r ) = P i = j ( u abi − Q ab )( u abj − Q ab ) δ ( r − ( r ai − r aj )) P i = j δ ( r − ( r ai − r aj )) [6]and˜ G R ( r ) = P i = j [ u abi u abj − u abi u acj + Q ab Q cd ] δ ( r − ( r ai − r aj )) P i = j δ ( r − ( r ai − r aj )) . [7]with u abi ≡ θ ( ℓ − | r ai − r bi | ) . [8]These four-replica objects can be computed for any quadru-plet of distinct replicas. The ensemble averaged correlationfunctions are simply obtained as Γ ≡ ˜Γ ab and G R ≡ ˜ G abR ,and cf. the SI for a proof. We stress that one must keepthe full space dependence of the correlators in the definitionsabove, as the introduction of shear breaks the rotational sym-metry of the glass samples and so the correlators are not justfunctions of a distance r . et al. R A F T Numerics
To measure the quantities defined above, we performedmolecular dynamics simulations of a Kob-Andersen 65-35%Lennard Jones Binary Mixture in 2 d . We have three systemsizes, N = 1000, N = 4000 and N = 10000. We chose Q with ℓ = 0 . ℓ leavethe emerging picture invariant.Following the procedure reported in Ref. (17), as a firststep we prepared a glass by equilibrating the system at T = 0 .
4, and then quenching it (the rate is 10 − ) down to T = 1 · − into a glassy configuration. The sample is thenheated up again to T = 0 .
2, and a starting configuration ofparticle positions is chosen at this temperature. Note thatwhile at T = 0 . T = 0 . T = 0 .
2, and these different samples are then quencheddown to T = 0 at a rate of 0 .
1. This procedure can be re-peated any number of times (say 100 times), and it allowsus to get a sampling of the configurations, or replicas, insideone single “glassy patch". We then perform this procedureagain, using each time a different configuration from the par-ent melt at T = 0 .
4, and in doing so we get an ensembleof these glassy patches, each of them representing a distinctglass sample. For each of these patches, we measure the four-replica correlators defined above for any distinct quadrupletof replicas, averaging the result over any possible permuta-tions of the quadruplet to gain statistics (29). The ensembleaverage is then performed by averaging the result over all theglass samples. To perform these measurement, below we use100 patches for N = 1000, each with 100 configurations, 100patches for N = 4000 each with 50 configurations and 50patches for N = 10000 each with 50 configurations. A strain γ xy (denoted below as γ ) is then applied quasi-statically toall configurations in all patches. In this protocol after everystep of increased strain the system undergoes energy gradi-ent minimization to return to mechanical equilibrium. Thiscreates an ensemble of strained patches for every value of thestrain parameter γ , from whence we measure again the abovedefined correlators, which then become functions of the strain γ . This is simply a consequence of the response of the configu-rations, i.e. each position r i in the definitions above becomes r i ( γ ). Thus for example G R ( r ) becomes G R ( r ; γ ) etc. We areinterested in the behavior of the correlators as the yieldingpoint γ Y is approached. Results
We consider first the susceptibilities χ GL , χ GR and χ Γ2 thatcan be obtained from the correlators, for example χ GL ( γ ) ≡ Z d x G L ( x, y ; γ ) . [9]In figure 1 upper panel we show the susceptibility χ Γ2 as afunction of γ for the three system sizes at our disposal. Super-imposed are the stress vs. strain curves obtained by averagingthe individual curves over all the available configurations andglass samples. One sees very clearly the singularity that de-velops near the yield point as a function of the system size. Inthe lower panel of the same figure we show the susceptibility -1.4-1.2-1.0-0.8-0.6-0.4-0.2 0.0 0.00 0.05 0.10 0.15 0.20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 χ Γ σ γ (a)N=1000N=4000N=10000 0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 0.00 0.05 0.10 0.15 0.20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 χ G R σ γ (b)N=1000N=4000N=10000 Fig. 1.
The susceptibilities χ Γ2 (upper panel) and χ GR (lower panel) as a functionof γ for the three systems sizes available. Superimposed are the stress vs. straincurves for comparison. The color code is violet for N = 1000 , red for 4000 andgreen for 10000. χ GR as a function of the strain γ , again with the stress-straincurve superimposed for comparison. As we expected, the sus-ceptibilities show a distinct peak at the spinodal point γ Y wherein yielding occurs. Since χ Γ2 is much smaller in am-plitude than χ GR there is no much new information in χ GL which is approximately 2 χ GR .More detailed information is provided by the full depen-dence of the correlators on their arguments. To see mostclearly the change in the correlators as the spinodal pointis approached, it is best to consider for example the one-dimensional function G R ( x = 0 , y ; γ ), shown for N = 4000in Fig. 2. Similar results for the other systems sizes are avail-able in the SI. We note that the correlator changes both inamplitude and in extent when we approach the critical point.To quantify these changes we fit a 3 parameter function to G R ( x = 0 , y ) in the form G R ( x = 0 , y ; γ ) ≈ C + A exp( − y/ξ ) , [10]where all the fitting coefficients are functions of γ . In Fig. 3 wepresent the γ dependence of the amplitude A ( γ ), the constant C ( γ ) and the correlation length ξ ( γ ).It is interesting to notice that the constant C decreaseswith the system size, presumably becoming irrelevant in the Parisi et al.
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October 16, 2018 | vol. XXX | no. XX | R A F T G R ( x = , y ) y N=4000, γ =0.000050.040050.050050.055050.060050.064050.070050.074050.080050.09405 Fig. 2.
The function G R ( x = 0 , y ; γ ) for various values of γ from × − to 0.09405. Note the increase in the overall amplitude of the correlator as well asthe increase in the correlation length. The lines through the data are the fit functionEq. (10). thermodynamic limit. The amplitude A is still increasingwith the system size, and it is difficult to assert whether itconverges or not. On the other hand we can safely concludethat the data present a strong evidence for the increase in thecorrelation length; it is very likely that it should diverge inthe thermodynamic limit.A relevant question is whether one can define critical expo-nents that can be measured also in experimental situations,and whether such exponents can be computed from theory,even on the mean-field level. Clearly, the standard thermalmean-field approach cannot be employed, since averages hereare computed over replicas at T = 0, and fluctuations due toquenched disorder are expected to dominate the thermal fluc-tuations that stem only from the mother super-cooled liquidfrom which the replicas at T = 0 are created. Considerationsof the effect of such fluctuations are beyond the scope of thispaper and will be discussed elsewhere. Physical interpretation
To conclude this paper we present a physical interpretationto these new insights, connecting them to what is knownabout the mechanical yield in athermal amorphous solids.The most important characteristic of the mechanical yieldin athermal amorphous solids is the change from plastic re-sponses that are localized, typically in the form of Eshelbyquadrupoles, to subextensive plastic events that are systemspanning (31, 32). The energy drops associated with the local-ized Eshelby quadrupoles are system size independent, scalinglike N where N is the total number of particles in the system.Mechanical yield is associated with the spontaneous appear-ance of concatenated lines of quadrupoles (in 2 dimensions,or planes in 3 dimensions, (13, 15, 16)). The latter are asso-ciated with energy drops that are subextensive, scaling like N / in 2 dimensions. Importantly, the concatenated lines ofquadrupoles change drastically the displacement field associ-ated with the plastic events. Each quadrupole has an armwith a displacement field pointing outward and an arm withthe displacement field pointing inward. When the quadrupoleis isolated the displacement field decays algebraically to infin- ξ γ (a)N=1000N=4000N=10000 0.00 0.02 0.04 0.06 0.08 0.00 0.05 0.10 0.15 0.20 A γ (b)N=1000N=4000N=10000 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.00 0.05 0.10 0.15 0.20 C γ (c)N=1000N=4000N=10000 Fig. 3.
The γ dependence of the correlation length ξ ( γ ) , the amplitude A ( γ ) andthe constant C ( γ ) in the best fit to the function G R ( x = 0 , y ; γ ) , cf. Eq. 10. ity. In contrast, when the quadrupoles are organized in theline there is a global connection between the outgoing direc-tion of one quadrupole and the incoming direction of the next,making the displacement field strongly localized around theline of quadrupoles (or around a plane in 3 dimensions), andall the shear is there. This is a microscopic shear band. Anexample of the displacement field associated with such as sys-tem spanning event is shown in Fig. 4, and see Ref. (13) fordetails. The main point of this paper is that the highly corre- et al. R A F T lated phenomenon of such a shear band can only occur whenthere exists a correlation length that approaches the systemsize in magnitude. This is the correlation length ξ that isidentified in this paper, and cf. the upper panel of Fig. 3. Fig. 4.
Example of the spontaneous plastic event exhibiting a concatenation of aseries of Eshelby quadrupoles resulting in a correlated displacement field with shearlocalization over a thin region which is system spanning. Left panel: direct numericalsimulations. Right panel: inserted line of Eshelby quadrupoles.
To understand the relevance of the spinodal point forthis scenario, we provide two figures that were obtained inRef. (17). In the upper panel of Fig. 5 one sees the order 〈 Q 〉 〈 σ 〉 γ Yield N=4000Q - γσ - γ 〈 P γ ( Q ) 〉 Q N=4000 γ =0.080 γ =0.084 γ =0.088 γ =0.092 γ =0.097 Fig. 5.
Upper panel: the order parameter Q ab as a function of the strain γ , su-perimposed on the stress vs. strain curve. Lower panel: the probability distributionfunction P ( Q ab ) for different values of γ in the vicinity of the mechanical yield value γ Y . parameter Q ab as a function of γ , superimposed on the stressvs. strain curve of the system under study. The point “yield"was obtained with the help of the results shown in the lowerpanel, in which the probability of observing Q ab is plotted for values of γ around the mechanical yield point γ Y . The yielditself is identified when the probability distribution functionhas two peaks of the same height. The spinodal point is ata slightly higher value of γ , where the peak occurring aroundhigh values of Q ab is about to disappear, with a characteris-tic spinodal vanishing of the slope. This is occurring in thepresent system around γ = 0 .
1. Of course in the thermody-namic limit the whole range of γ values where the exchangeof stability is occurring is becoming very narrow.It is important to stress again that the ability to observethe divergence of the susceptibility and the correlation lengthdue to the spinodal phenomenon stems from the fact thatwe deal with an athermal glassy system whose typical relax-ation times are immense. In a liquid system the fluctuationswould have caused the system to make the transition beforethe spinodal point is reached. Conclusions
In conclusion, we have presented evidence that the yieldingtransition is a spinodal point with disorder, characterizedby a criticality whose features can be picked up by suitablemulti-point correlators whose expression can be obtained fromreplica theory. The treatment presented here pertains to anathermal setting, but an obvious direction for future researchwill be the application of these ideas to thermal glasses undershear (21); in that case, the system will generally be able to es-cape through thermal activation from the high- Q ab minimumbefore this has a chance to flatten and the relative suscep-tibility to diverge. However, since the nucleation time willanyway be fairly long, one should anyway be able to observe transient shear-bands/heterogeneities, as long as the temper-ature is low enough that nucleation does not take place untilthe system is close to the spinodal, which, interestingly, isprecisely the behavior of transient shear bands as reportedin (21). In this thermal setting, we expect that the study ofthe ideas presented in this paper will have to proceed muchas it does in the case of dynamical heterogeneities around theMCT crossover, entailing for example the definition and studyof time-dependent multi-point susceptibilities and correlators. ACKNOWLEDGMENTS.
IP acknowledges with thanks receivingthe “Premio Rita Levi-Montalcini" which facilitated the collabora-tion with GP. We thank George Hentschel and Francesco Zamponifor inspiring discussions. GP acknowldeges funding from the Euro-pean Research Council (ERC) under the European Union’s Hori-zon 2020 research and innovation programme (grant agreement No[694925]). IP was supported in part by the Minerva foundationwith funding from the Federal German Ministry for Education andResearch, and by the Israel Science Foundation (Israel SingaporeProgram).
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Giorgio Parisi , Itamar Procaccia , Corrado Rainone and Murari Singh Dipartimento di Fisica, Sapienza Universit´a di Roma, INFN,Sezione di Roma I, IPFC – CNR, Piazzale Aldo Moro 2, I-00185 Roma, Italy Department of Chemical Physics, the Weizmann Institute of Science, Rehovot 76100, Israel
I. THE LONGITUDINAL CORRELATION FUNCTION
Let us start from the expression of the free energy of a glass state, prepared by equilibrating a generic glass formerdown to a glass transition temperature T g where it can still be equilibrated, and then quenching it out of equilibriumto a given temperature T < T g . Such a free energy was first defined in [1] in the context of spin-glass physics.Its definition in the case of structural glasses, and its computation in the particular case of hard spheres were firstdiscussed in [2]. The definition, in the case of a generic glass former made of N particles is based on comparing twoconfigurations X a and X b of the same glass. Here X a ≡ { r ai } Ni =1 , X b ≡ { r bi } Ni =1 , (1)where the labeling r i refers to the position of the same particle i in the two different configurations. For a genericinteraction potential V ( X ) the definition of the free energy is f [ T, T g ] ≡ − βN Z dX e − β g V ( X ) Z g log (cid:20)Z dX e βV ( X ) δ ( q ∗ r − Q ) (cid:21) . (2)where β g = 1 / ( k B T g ), β = 1 / ( k B T ) and q ∗ r is the value of q r = 0 whereupon the free energy attains a local minimum [2].The overlap function Q for any two configuration, say a and b is [3] Q ab = 1 N N X i =1 θ ( ℓ − | r ai − r bi | ) . (3)Here ℓ is a coarse graining parameter (in [3], ℓ ≃ . T of the glass former, which is constrained to stay close to an amorphous configuration X which isselected from the equilibrium ensemble, using the canonical distribution when the glass is still at equilibrium at T g .The properties and computation of the free energy (2) are discussed extensively in [2, 4], so we refer the interestedreader to those works. The explicit analytic computation is accomplished in the mean field approximation. In ourpaper we use the results far from the mean field limit, but we ascertain that the relevant correlation functions that arefleshed out in the mean field calculation are the relevant ones also in the general case. Of course, critical exponentscan differ. In the sequel we sketch how from this mean-field theory in terms of an overlap order parameter Q ab onecan extract the definitions of the correlation functions that are expected to show critical behavior.The outermost integral in the Eq. (2) can be computed with the replica trick, f [ T, T g ] = lim s → ∂ s Φ[ T, T g ; s ] , (4)where Φ is defined asΦ[ T, T g ; s ] = − βN log Z dX dX · · · dX s e − β g V ( X ) e − βV ( X ) δ ( q − Q ) · · · e − βV ( X s ) δ ( q − Q s ) , (5)so we are considering s replicas of the X configuration. In infinite dimensions for the case of hard spheres it wasshown [5] that the functional defined above can be written asΦ = − βN Z D Q ab e − dS ( Q ab ) . (6)Here D Q ab denotes an integration measure over all the distinct Q ab s, D Q ab ≡ ,s Y a
In the paper we have exhibited in Fig. 1 the susceptibilities χ GR and χ Γ2 . For completeness we show here also thesusceptibility χ GL , see Fig. 1. As discussed in the paper, this susceptibility is very close to 2 χ GR . On the other handthe correlation functions themselves differ G R ( x, y ) and G L ( x, y ) differ significantly since the function Γ ( x, y ) is notpositive definite, becoming negative towards the edges of the available fields. This is seen clearly in Fig. 2 where weshow the three correlation function at the value of γ = 0 . χ G L σ γ N=1000N=4000N=10000
FIG. 1: The susceptibility χ GL as a function of γ for the three systems sizes available. Superimposed are the stress vs. straincurves for comparison. The color code is violet for N = 1000, red for 4000 and green for 10000. An interesting observation concerns the constant C used in the fit Eq. 10 in the paper. This constant is alsosensitive to the approach of the criticality, cf. the lower panel in Fig.3 in the paper. One could worry that integratingthis constant over y could contribute to the divergence of the susceptibilities. In fact the rise in C near the spinodalpoint goes down with the system size and its contribution to the integral is reduced as well, as can be seen in Fig. 3which presents the integral R dyG R ( x = 0 , y ) from which C × L is subtracted. The conclusion is that indeed thecontribution of C goes down also when integrated over the system size, showing that the main contribution to thedivergence of the susceptibility is from the divergence of the correlation length.Finally we need to discuss the fitting procedure for the correlation function G R ( x = 0 , y ). In Fig. 4 we showthe full results for this correlation function for all the available values of γ and for two larger systems sizes at our x γ =0.09405 -20-30-30-20-100 y Γ -0.0200.020.040.060.080.10.12 x γ =0.09405 -20-30-30-20-100 y G R x -10 γ =0.09405 -20-30-30-20-100 y G L FIG. 2: A 3-dimensional projection of the three correlation function as a function of x, y . ’ ∫ G R ( x = , y ) dy - C × L γ N=1000N=4000N=10000
FIG. 3: The difference between R dy G R ( x = 0 , y ) and C × L disposal. One sees that the exponents decay that is used for the fit is only reliable up to the minima of the functions.The reason for the upward trend is the periodic boundary condition that reflects the correlations. To eliminate thisspurious effect we presented in the paper the fit up to the minimum in the function. One should note however thatthe distance to the minimum increases with the system size, presumably diverging in the thermodynamic limit. Thusthe fit up to the minimum allows a faithful estimate of the correlation length ξ . G R ( x = , y ) y N=4000, γ =0.000050.040050.050050.055050.060050.064050.070050.074050.080050.09405 G R ( x = , y ) y N=10000, γ =0.000050.020050.040050.050050.055050.060050.065050.070050.075050.08505 FIG. 4: The full y dependence of G R ( x = 0 , y ). The region fitted by Eq. (10) in the paper is shown. Note that the minimumin the function resides in higher values of y for larger system sizes. [1] S. Franz and G. Parisi, J. Phys. I France , 1401 (1995), URL http://dx.doi.org/10.1051/jp1:1995201 .[2] C. Rainone, P. Urbani, H. Yoshino, and F. Zamponi, Phys. Rev. Lett. , 015701 (2015), URL http://link.aps.org/doi/10.1103/PhysRevLett.114.015701 .[3] P. K. Jaiswal, I. Procaccia, C. Rainone, and M. Singh, Phys. Rev. Lett. , 085501 (2016), URL http://link.aps.org/doi/10.1103/PhysRevLett.116.085501 .[4] C. Rainone and P. Urbani, Journal of Statistical Mechanics: Theory and Experiment , 053302 (2016), URL http://stacks.iop.org/1742-5468/2016/i=5/a=053302 .[5] J. Kurchan, G. Parisi, and F. Zamponi, Journal of Statistical Mechanics: Theory and Experiment , P10012 (2012),URL http://stacks.iop.org/1742-5468/2012/i=10/a=P10012 .[6] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I (Springer Science &Business Media, 1999).[7] C. Rainone, arXiv preprint arXiv:1411.3941 (2014).[8] A. Crisanti and H.-J. Sommers, Zeitschrift f¨ur Physik B , 341 (1992), ISSN 0722-3277, URL http://dx.doi.org/10.1007/BF01309287 .[9] A. Bray and M. Moore, Journal of Physics C: Solid State Physics , 79 (1979).[10] C. De Dominicis, I. Kondor, and T. Temesv´ari, in Spin glasses and random fields (World Scientific, Singapore, 1998), pp.119–160.[11] F. Zamponi, ArXiv e-prints (2010), 1008.4844.[12] P. Urbani and F. Zamponi, ArXiv e-prints (2016), 1610.06804.[13] L. Berthier, P. Charbonneau, Y. Jin, G. Parisi, B. Seoane, and F. Zamponi, Proceedings of the Na-tional Academy of Sciences