Corrado Rainone, Pierfrancesco Urbani, Francesco Zamponi, Edan Lerner, and Eran Bouchbinder
SciPost Phys. Core 4, 008 (2021) ·
published 16 April 2021

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Structural glasses feature quasilocalized excitations whose frequencies $\omega$ follow a universal density of states ${\cal D}(\omega)\!\sim\!\omega^4$. Yet, the underlying physics behind this universality is not yet fully understood. Here we study a meanfield model of quasilocalized excitations in glasses, viewed as groups of particles embedded inside an elastic medium and described collectively as anharmonic oscillators. The oscillators, whose harmonic stiffness is taken from a rather featureless probability distribution (of upper cutoff $\kappa_0$) in the absence of interactions, interact among themselves through random couplings (characterized by strength $J$) and with the surrounding elastic medium (an interaction characterized by a constant force $h$). We first show that the model gives rise to a gapless density of states ${\cal D}(\omega)\!=\!A_{\rm g}\,\omega^4$ for a broad range of model parameters, expressed in terms of the strength of stabilizing anharmonicity, which plays a decisive role in the model. Then  using scaling theory and numerical simulations  we provide a complete understanding of the nonuniversal prefactor $A_{\rm g}(h,J,\kappa_0)$, of the oscillators' interactioninduced mean square displacement and of an emerging characteristic frequency, all in terms of properly identified dimensionless quantities. In particular, we show that $A_{\rm g}(h,J,\kappa_0)$ is a nonmonotonic function of $J$ for a fixed $h$, varying predominantly exponentially with $(\kappa_0 h^{2/3}\!/J^2)$ in the weak interactions (small $J$) regime  reminiscent of recent observations in computer glasses  and predominantly decays as a powerlaw for larger $J$, in a regime where $h$ plays no role. We discuss the physical interpretation of the model and its possible relations to available observations in structural glasses, along with delineating some future research directions.
SciPost Phys. 10, 013 (2021) ·
published 21 January 2021

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We investigate the properties of local minima of the energy landscape of a continuous nonconvex optimization problem, the spherical perceptron with piecewise linear cost function and show that they are critical, marginally stable and displaying a set of pseudogaps, singularities and nonlinear excitations whose properties appear to be in the same universality class of jammed packings of hard spheres.
The piecewise linear perceptron problem appears as an evolution of the purely linear perceptron optimization problem that has been recently investigated in [1]. Its cost function contains two nonanalytic points where the derivative has a jump. Correspondingly, in the nonconvex/glassy phase, these two points give rise to four pseudogaps in the force distribution and this induces four power laws in the gap distribution as well. In addition one can define an extended notion of isostaticity and show that local minima appear again to be isostatic in this phase. We believe that our results generalize naturally to more complex cases with a proliferation of nonlinear excitations as the number of nonanalytic points in the cost function is increased.
Silvio Franz, Antonio Sclocchi, Pierfrancesco Urbani
SciPost Phys. 9, 012 (2020) ·
published 23 July 2020

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We show that soft spheres interacting with a linear ramp potential when
overcompressed beyond the jamming point fall in an amorphous solid phase which
is critical, mechanically marginally stable and share many features with the
jamming point itself. In the whole phase, the relevant local minima of the
potential energy landscape display an isostatic contact network of perfectly
touching spheres whose statistics is controlled by an infinite lengthscale.
Excitations around such energy minima are nonlinear, system spanning, and
characterized by a set of nontrivial critical exponents. We perform numerical
simulations to measure their values and show that, while they coincide, within
numerical precision, with the critical exponents appearing at jamming, the
nature of the corresponding excitations is richer. Therefore, linear soft
spheres appear as a novel class of finite dimensional systems that
selforganize into new, critical, marginally stable, states.
SciPost Phys. 4, 020 (2018) ·
published 27 April 2018

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We study Harmonic Soft Spheres as a model of thermal structural glasses in
the limit of infinite dimensions. We show that cooling, compressing and
shearing a glass lead to a Gardner transition and, hence, to a marginally
stable amorphous solid as found for Hard Spheres systems. A general outcome of
our results is that a reduced stability of the glass favors the appearance of
the Gardner transition. Therefore using strong perturbations, e.g. shear and
compression, on standard glasses or using weak perturbations on weakly stable
glasses, e.g. the ones prepared close to the jamming point, are the generic
ways to induce a Gardner transition. The formalism that we discuss allows to
study general perturbations, including strain deformations that are important
to study soft glassy rheology at the mean field level.
Silvio Franz, Giorgio Parisi, Maksim Sevelev, Pierfrancesco Urbani, Francesco Zamponi
SciPost Phys. 2, 019 (2017) ·
published 2 June 2017

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Random constraint satisfaction problems (CSP) have been studied extensively
using statistical physics techniques. They provide a benchmark to study average
case scenarios instead of the worst case one. The interplay between statistical
physics of disordered systems and computer science has brought new light into
the realm of computational complexity theory, by introducing the notion of
clustering of solutions, related to replica symmetry breaking. However, the
class of problems in which clustering has been studied often involve discrete
degrees of freedom: standard random CSPs are random KSAT (aka disordered Ising
models) or random coloring problems (aka disordered Potts models). In this work
we consider instead problems that involve continuous degrees of freedom. The
simplest prototype of these problems is the perceptron. Here we discuss in
detail the full phase diagram of the model. In the regions of parameter space
where the problem is nonconvex, leading to multiple disconnected clusters of
solutions, the solution is critical at the SAT/UNSAT threshold and lies in the
same universality class of the jamming transition of soft spheres. We show how
the critical behavior at the satisfiability threshold emerges, and we compute
the critical exponents associated to the approach to the transition from both
the SAT and UNSAT phase. We conjecture that there is a large universality class
of nonconvex continuous CSPs whose SATUNSAT threshold is described by the
same scaling solution.
Dr Urbani: "We warmly thank our referee: w..."
in Submission on Universality of the SATUNSAT (jamming) threshold in nonconvex continuous constraint satisfaction problems by Silvio Franz, Giorgio Parisi, Maksim Sevelev, Pierfrancesco Urbani, Francesco Zamponi