Critical behavior of the yielding spinodal of amorphous solids: computing the exponents, finite-size scaling, and the finite-temperature fate

The mechanical yield of an athermal amorphous solid under quasi-static shear was proposed (Parisi, Procaccia, Rainone, Singh, PNAS 2017) to be a spinodal point of the replicated free energy of the overlap order parameter $Q_{ab}$, approached through a correlation length $\xi(\gamma)$ — extracted from four-replica correlators — that grows as the yield strain $\gamma_Y$ is approached, with susceptibility peaks that sharpen with system size. The theory of this criticality is unfinished, and all of its load-bearing cores remain open. First, no critical exponents of the yielding spinodal have been computed from theory, even at the mean-field replica level: only unconstrained numerical estimates exist ($\nu \approx 2.4 \pm 0.35$, with the authors explicitly offering no theoretical basis), and the dominant physical ingredient — fluctuations of the quenched disorder over which replicas are averaged — is untreated; general disordered-spinodal theory (developed for the random-field Ising model) suggests quenched disorder may convert the singularity into a Griffiths-like, non-critical one, and a competing school describes brittle yield as rare-event nucleation. Second, the finite-size scaling of the measured correlator fit $G_R \approx C + A e^{-y/\xi}$ is undetermined: whether the amplitude $A(\gamma)$ converges or diverges in the thermodynamic limit, and whether $\xi$ truly diverges as $\gamma \to \gamma_Y$, are open. Third, the finite-temperature fate is unknown: thermal activation can nucleate escape from the high-$Q_{ab}$ glass minimum before the spinodal flattens, and the measurement program the source proposed for this regime — time-dependent multipoint susceptibilities of transient shear-banding heterogeneity — has never been implemented in this framework. The problem is to complete the theory of the yielding-spinodal criticality: compute its exponents or prove their absence, establish its finite-size scaling, and determine its finite-temperature fate.

Background

Amorphous solids — metallic, polymeric and colloidal glasses, foams, gels, and granular packings — resist small shear deformation elastically, but beyond a yield strain $\gamma_Y$ they can no longer sustain increasing stress and begin to flow plastically. In atomistic simulations this is studied in the athermal quasi-static (AQS) limit: shear is applied in small strain increments, each followed by energy minimization, so temperature plays no role and all fluctuations are those frozen into the disordered structure itself (quenched disorder). The approach to $\gamma_Y$ is punctuated by localized plastic rearrangements; at yield, rearrangements become system-spanning and concentrate in narrow regions called shear bands, which underlie the brittleness of e.g. metallic glasses. A long-standing question is whether this mechanical instability is a critical phenomenon — whether growing correlation lengths and susceptibilities precede yield and diverge at it.

The source paper placed this question in an equilibrium statistical-mechanics framework borrowed from glass theory. One considers replicas (independent copies) of the same glass: configurations prepared by quenching the same well-equilibrated starting point, particle labels preserved. The overlap between replicas $a$ and $b$, $Q_{ab} = (1/N) \sum_i \theta(\ell - |\mathbf{r}_i^a - \mathbf{r}_i^b|)$ with $\ell$ a small cutoff of order the cage size, serves as an order parameter for solidity. In the Franz–Parisi / state-following construction, the free energy constrained to remain close to the reference glass is a functional $V[Q_{ab}]$ depending parametrically on the imposed strain $\gamma$; quasi-static shear acts as an external field on this free energy, analogous to a magnetic field on a ferromagnet. A predecessor study (Jaiswal et al., PRL 2016) showed that yield appears as a first-order transition in $Q_{ab}$: as $\gamma$ grows, a metastable high-overlap minimum (the glass) coexists with a low-overlap phase (the flowed liquid), and the system falls out of the metastable minimum at $\gamma_Y$.

The source paper developed the field theory of this picture. Expanding the replicated free energy in fluctuations of the overlap field $Q_{ab}(\mathbf{r})$ around the metastable minimum gives an action whose Hessian (the matrix of second functional derivatives) organizes into replicon and longitudinal sectors — the replicon modes being the fluctuations that distinguish the replicas from one another (named after the corresponding sector of spin-glass theory), the longitudinal modes those that change the value of the overlap itself; at a spinodal (where a metastable minimum merges with a saddle and a curvature vanishes) it is the longitudinal mode that becomes soft. Diagonalization yields exact relations among disorder-averaged correlators, in particular $G_L(r) = 2 G_R(r) - \Gamma_2(r)$, where $\Gamma_2$ is a connected four-replica (overlap–overlap) correlation function and $G_R$ the replicon correlator. These are four-replica objects, measurable in simulation by preparing, for each glass sample ('patch'), many replicas at the same strain and averaging both over replicas and over patches; the corresponding susceptibilities (spatial integrals of the correlators) quantify the extent of regions poised to rearrange. In two-dimensional Kob–Andersen model glasses (a standard binary Lennard-Jones glass former) of $N = 1000$ to $10000$ particles the authors found that the longitudinal susceptibility and the four-point susceptibility $\chi_{\Gamma_2}$ develop peaks that sharpen with system size as $\gamma \to \gamma_Y$, consistent with a diverging correlation length whose growth coincides with the onset of shear banding. The length was extracted from the replicon correlator via the fit $G_R(x=0, y; \gamma) \approx C(\gamma) + A(\gamma) \exp(-y/\xi(\gamma))$, where $x$ and $y$ are the components of the pair separation along and transverse to the shear direction: $\xi(\gamma)$ grows on approach to $\gamma_Y$ and is expected to diverge in the thermodynamic limit; the constant $C$ decreases with $N$ (presumably vanishing), while the amplitude $A$ still grows with $N$ within accessible sizes, so whether it converges could not be decided.

What remains unfinished is the theory of this criticality, in a setting with two complicating features. (i) The averages are over replicas at zero temperature, so fluctuations of the quenched (sample-to-sample) disorder — not thermal fluctuations — dominate; the general theory of spinodals with quenched disorder, developed for the random-field Ising model (an Ising magnet with quenched random local fields; Nandi, Biroli, Tarjus, PRL 2016), predicts that in finite dimensions such spinodals need not show mean-field power-law criticality at all, but rather exponentially diverging lengths, Griffiths-like rounding (rare-region effects that replace sharp critical singularities by weak, non-universal ones), and escape controlled by rare-droplet nucleation. (ii) A competing school (Ozawa et al., PNAS 2018; Popović, de Geus, Wyart, PRE 2018) describes brittle yield of well-annealed glasses as nucleation at rare weak zones governed by extreme-value statistics, and reports strong sample-to-sample fluctuations and rare-event control of brittle yielding in 2D and 3D; mean-field hard-sphere calculations (Urbani–Zamponi; Altieri–Zamponi) confirm that the glass yields at a spinodal in $d = \infty$ but compute no critical behavior. Elastoplastic models do possess well-studied criticality, but at neighboring transitions — the steady flow curve and a finite-disorder critical point separating brittle from ductile yield (Rossi et al., PRL 2022) — not at the glass-basin spinodal approached from the elastic side. Finally, at any finite temperature the system can escape the metastable high-$Q_{ab}$ minimum by thermal activation before it flattens, so the fate of the spinodal criticality at finite $T$ — and its detection through time-dependent multipoint susceptibilities of the transient shear-banding heterogeneity, in analogy with the four-point susceptibilities of dynamical heterogeneities in supercooled liquids — is part of the same unfinished theory.

Problem Statement

The mechanical yield of an athermal amorphous solid under quasi-static shear was proposed (Parisi et al., PNAS 2017) to be a spinodal point of the replicated free energy of the overlap order parameter $Q_{ab}$, approached through a correlation length $\xi(\gamma)$ measured from four-replica correlators, with susceptibility peaks that sharpen with system size. Complete the theory of the criticality of this yielding spinodal: establish whether and how the yielding instability is critical, from theory, from finite-size scaling, and at finite temperature — three linked levels of one question.

1. Critical behavior from theory.

(a) At the mean-field (replica/state-following, $d = \infty$) level: determine from theory whether the approach to the yielding spinodal is governed by well-defined critical exponents (for the correlation length, for the longitudinal and four-point susceptibilities, and for the finite-size sharpening of the yield strain), and if so compute them; or establish instead at this level that the singularity is non-critical, and specify the resulting scaling forms.

(b) Beyond that level, and in addition to (a): treat the effect of the quenched-disorder fluctuations that dominate in this athermal, replica-averaged setting, in the finite-dimensional regime to which the Background assigns that effect — determine from theory whether quenched disorder rounds the spinodal into a non-critical (e.g., exponentially diverging, Griffiths-like) singularity, and specify the resulting scaling forms. For reference, unconstrained numerical fits give a correlation-length exponent $\nu \approx 2.4 \pm 0.35$ and an exponent $\theta \approx 0.41 \pm 0.09$ governing the finite-size sharpening of the yield strain; no theoretical basis for these numbers currently exists.

2. Finite-size scaling and the thermodynamic limit. Establish the finite-size scaling of the measured correlator fit $G_R(x=0,y;\gamma) \approx C + A e^{-y/\xi}$: the thermodynamic-limit behavior of the amplitude $A(\gamma)$ (convergent or system-size divergent) and of the constant $C(\gamma)$, and a decisive verification or refutation of the divergence of $\xi(\gamma)$ as $\gamma \to \gamma_Y$ — including whether any divergence is a power law, and how it coexists (or conflicts) with the rare-event/nucleation phenomenology of brittle yielding and with sample-to-sample disorder fluctuations.

3. Finite-temperature fate. Determine what remains of the spinodal criticality at finite temperature, where the system can escape the high-$Q_{ab}$ minimum by thermal activation before it flattens: construct a quantitative theory of the competition between nucleation and spinodal instability of the overlap free energy (nucleation rates versus distance to the spinodal, in the spirit of disordered-spinodal theory). Also construct and analyse the time-dependent multi-point susceptibilities and correlators the source proposed for this regime — analogous to those used for dynamical heterogeneities — to detect transient shear-banding heterogeneity as yield is approached at finite $T$.

Settle whether mechanical yield is a genuine critical phenomenon with universal, theory-computable exponents or a disorder-rounded, rare-event-controlled instability, and determine the observable precursors (growing lengths, sharpening susceptibilities) as functions of system size and temperature.

References

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