Marginal Stability in Structural, Spin, and Electron Glasses

Physical Review E

Journal of Statistical Mechanics: Theory and Experiment

Recently, we showed that optimization problems, both in infinite as well as in finite dimensions, for continuous variables and soft excluded volume constraints, can display entire isostatic phases where local minima of the cost function are marginally stable configurations endowed with nonlinear excitations [1, 2]. In this work we describe an athermal adiabatic algorithm to explore with continuity the corresponding rough high-dimensional landscape. We concentrate on a prototype problem of this kind, the spherical perceptron optimization problem with linear cost function (hinge loss). This algorithm allows to 'surf' between isostatic marginally stable configurations and to investigate some properties of such landscape. In particular we focus on the statistics of avalanches occurring when local minima are destabilized. We show that when perturbing such minima, the system undergoes plastic rearrangements whose size is power law distributed and we characterize the corresponding critical exponent. Finally we investigate the critical properties of the unjamming transition, showing that the linear interaction potential gives rise to logarithmic behavior in the scaling of energy and pressure as a function of the distance from the unjamming point. For some quantities, the logarithmic corrections can be gauged out. This is the case of the number of soft constraints that are violated as a function of the distance from jamming which follows a non-trivial power law behavior. Contents

Physical Review Letters, 2016

Physical Review E

We consider a spring-block model with both dry and viscous frictions, subjected to a periodic driving allowing mechanically stable configurations to be sampled. We show that under strong driving, the scaling of the correlation length with the energy density is incompatible with the prediction of Edwards statistical approach, which assumes a uniform sampling of mechanically stable configurations. A crossover between the Edwards scaling and the non-standard high energy scaling is observed at energy scales that depend on the viscous friction coefficient. Generalizing Edwards thermodynamics, we propose a statistical framework, based on a sampling of marginally stable states, that is able to describe the scaling of the correlation length in the highly viscous regime.

Proceedings of the National Academy of Sciences

We present a mechanism for the anomalous behavior of the specific heat in low-temperature amorphous solids. The analytic solution of a mean-field model belonging to the same universality class as high-dimensional glasses, the spherical perceptron, suggests that there exists a cross-over temperature above which the specific heat scales linearly with temperature, while below it, a cubic scaling is displayed. This relies on two crucial features of the phase diagram: (i) the marginal stability of the free-energy landscape, which induces a gapless phase responsible for the emergence of a power-law scaling; and (ii) the vicinity of the classical jamming critical point, as the cross-over temperature gets lowered when approaching it. This scenario arises from a direct study of the thermodynamics of the system in the quantum regime, where we show that, contrary to crystals, the Debye approximation does not hold.

Physical Review Letters, 2015

Marginal stability is the notion that stability is achieved, but only barely so. This property constrains the ensemble of configurations explored at low temperature in a variety of systems, including spin, electron and structural glasses. A key feature of marginal states is a (saturated) pseudo-gap in the distribution of soft excitations. We study how such a pseudo-gap appears dynamically in the case of the Sherrington-Kirkpatrick (SK) spin glass. After revisiting and correcting the multi-spinflip criterion for local stability, we show that stationarity along the hysteresis loop requires that soft spins are frustrated among each other, with a correlation that diverges as C(λ) ∼ 1/λ, where λ is the larger of two considered local fields. We explain how this arises spontaneously in a marginal system and develop an analogy between the spin dynamics in the SK model and random walks in two dimensions. We discuss the applicability of these findings to hard sphere packings.

Physical Review E, 2018

The response of amorphous materials to an applied strain can be continuous, or instead display a macroscopic stress drop when a shear band nucleates. Such discontinuous response can be observed if the initial configuration is very stable. We study theoretically how such brittleness emerges in athermal, quasi-statically driven, materials as their initial stability is increased. We show that this emergence is well reproduced by elasto-plastic models and is predicted by a mean field approximation, where it corresponds to a continuous transition. In mean field, failure can be forecasted from the avalanche statistics. We show that this is not the case for very brittle materials in finite dimensions due to rare weak regions where a shear band nucleates. Their critical radius is predicted to follow ac ∼ (Σ − Σ b) −2 , where Σ is the stress and Σ b the stress a shear band can carry.

Physical Review E, 2019

Quasi-localised modes appear in the vibrational spectrum of amorphous solids at low-frequency. Though never formalised, these modes are believed to have a close relationship with other important local excitations, including shear transformations and two-level systems. We provide a theory for their frequency density, DL(ω) ∼ ω α , that establishes this link for systems at zero temperature under quasi-static loading. It predicts two regimes depending on the density of shear transformations P (x) ∼ x θ (with x the additional stress needed to trigger a shear transformation). If θ > 1/4, α = 4 and a finite fraction of quasi-localised modes form shear transformations, whose amplitudes vanish at low frequencies. If θ < 1/4, α = 3 + 4θ and all quasi-localised modes form shear transformations with a finite amplitude at vanishing frequencies. We confirm our predictions numerically.

Proceedings of the National Academy of Sciences, 2019

Sliding at a quasi-statically loaded frictional interface can occur via macroscopic slip events, which nucleate locally before propagating as rupture fronts very similar to fracture. We introduce a microscopic model of a frictional interface that includes asperity-level disorder, elastic interaction between local slip events, and inertia. For a perfectly flat and homogeneously loaded interface, we find that slip is nucleated by avalanches of asperity detachments of extension larger than a critical radius A c governed by a Griffith criterion. We find that after slip, the density of asperities at a local distance to yielding x σ presents a pseudogap P ( x σ ) ∼ ( x σ ) θ , where θ is a nonuniversal exponent that depends on the statistics of the disorder. This result makes a link between friction and the plasticity of amorphous materials where a pseudogap is also present. For friction, we find that a consequence is that stick–slip is an extremely slowly decaying finite-size effect, whi...

Physical Review B, 2016

A salient feature of cyclically driven first-order phase transformations in crystals is their scale-free avalanche dynamics. This behavior has been linked to the presence of a classical critical point but the mechanism leading to criticality without extrinsic tuning remains unexplained. Here we show that the source of scaling in such systems is an annealed disorder associated with transformationinduced slip which co-evolves with the phase transformation, thus ensuring the crossing of a critical manifold. Our conclusions are based on a model where annealed disorder emerges in the form of a random field induced by the phase transition. Such disorder exhibits super-transient chaotic behavior under thermal loading, obeys a heavy-tailed distribution and exhibits long-range spatial correlations. We show that the universality class is affected by the long-range character of elastic interactions. In contrast, it is not influenced by the heavy-tailed distribution and spatial correlations of disorder.

Physical review. E, Statistical, nonlinear, and soft matter physics, 2015

Recent theoretical advances predict the existence, deep into the glass phase, of a novel phase transition, the so-called Gardner transition. This transition is associated with the emergence of a complex free energy landscape composed of many marginally stable sub-basins within a glass metabasin. In this study, we explore several methods to detect numerically the Gardner transition in a simple structural glass former, the infinite-range Mari-Kurchan model. The transition point is robustly located from three independent approaches: (i) the divergence of the characteristic relaxation time, (ii) the divergence of the caging susceptibility, and (iii) the abnormal tail in the probability distribution function of cage order parameters. We show that the numerical results are fully consistent with the theoretical expectation. The methods we propose may also be generalized to more realistic numerical models as well as to experimental systems.

Physical review. E, 2017

The mechanical and transport properties of jammed materials originate from an underlying percolating network of contact forces between the grains. Using extensive simulations we investigate the force-percolation transition of this network, where two particles are considered as linked if their interparticle force overcomes a threshold. We show that this transition belongs to the random percolation universality class, thus ruling out the existence of long-range correlations between the forces. Through a combined size and pressure scaling for the percolative quantities, we show that the continuous force percolation transition evolves into the discontinuous jamming transition in the zero pressure limit, as the size of the critical region scales with the pressure.

Physical Review E, 2019

We report on a non-equilibrium phase of matter, the minimally disordered crystal phase, which we find exists between the maximally amorphous glasses and the ideal crystal. Even though these near crystals appear highly ordered, they display glassy and jamming features akin to those observed in amorphous solids. Structurally, they exhibit a power-law scaling in their probability distribution of weak forces and small interparticle gaps as well as a flat density of vibrational states. Dynamically, they display anomalous aging above a characteristic pressure. Quantitatively this disordered crystal phase has much in common with the Gardner-like phase seen in maximally disordered solids. Near crystals should be amenable to experimental realizations in commercially-available particulate systems and are to be indispensable in verifying the theory of amorphous materials.

Physical Review E, 2016

It is known by now [1] that amorphous solids at zero temperature do not possess a nonlinear elasticity theory: besides the shear modulus which exists, all the higher order coefficients do not exist in the thermodynamic limit. Here we show that the same phenomenon persists up to temperatures comparable to the glass transition. The zero temperature mechanism due to the prevalence of dangerous plastic modes of the Hessian matrix is replaced by anomalous stress fluctuations that lead to the divergence of the variances of the higher order elastic coefficients. The conclusion is that in amorphous solids elasticity can never be decoupled from plasticity: the nonlinear response is very substantially plastic.

Communications physics, 2023

Much of the qualitative nature of physical systems can be predicted from the way it scales with system size. Contrary to the continuum expectation, we observe a profound deviation from logarithmic scaling in the impedance of a two-dimensional LC circuit network. We find this anomalous impedance contribution to sensitively depend on the number of nodes N in a curious erratic manner and experimentally demonstrate its robustness against perturbations from the contact and parasitic impedance of individual components. This impedance anomaly is traced back to a generalized resonance condition reminiscent of Harper's equation for electronic lattice transport in a magnetic field, even though our circuit network does not involve magnetic translation symmetry. It exhibits an emergent fractal parametric structure of anomalous impedance peaks for different N that cannot be reconciled with a continuum theory and does not correspond to regular waveguide resonant behavior. This anomalous fractal scaling extends to the transport properties of generic systems described by a network Laplacian whenever a resonance frequency scale is simultaneously present.

Physical Review E, 2020

We address the question of why larger, high-symmetry crystals are mostly weak, ductile, and statistically subcritical, while smaller crystals with the same symmetry are strong, brittle and supercritical. We link it to another question of why intermittent elasto-plastic deformation of submicron crystals features highly unusual size sensitivity of scaling exponents. We use a minimal integer-valued automaton model of crystal plasticity to show that with growing variance of quenched disorder, which can serve in this case as a proxy for increasing size, submicron crystals undergo a crossover from spin-glass marginality to criticality characterizing the second order brittle-to-ductile (BD) transition. We argue that this crossover is behind the nonuniversality of scaling exponents observed in physical and numerical experiments. The nonuniversality emerges only if the quenched disorder is elastically incompatible, and it disappears if the disorder is compatible.

Physical review, 2018

While perfect crystals may exhibit a purely elastic response to shear all the way to yielding, the response of amorphous solids is punctuated by plastic events. The prevalence of this plasticity depends on the number of particles N of the system, with the average strain interval before the first plastic event, ∆γ, scaling like N α with α negative: larger samples are more susceptible to plasticity due to more numerous disorder-induced soft spots. In this paper we examine this scaling relation in ultra-stable glasses prepared with the Swap Monte Carlo algorithm, with regard to the possibility of protocol-dependent scaling exponent, which would also imply a protocol dependence in the distribution of local yield stresses in the glass. We show that, while a superficial analysis seems to corroborate this hypothesis, this is only a pre-asymptotic effect and in fact our data can be well explained by a simple model wherein such protocol dependence is absent.