A recently formulated mean-field model of glasses — scalar anharmonic oscillators with quenched random stiffnesses coupled all-to-all by Gaussian interactions — develops, after an instantaneous quench, a non-equilibrium replica-symmetry-broken phase whose Hessian reproduces key glass phenomenology: a universal quartic nonphononic vibrational density of states populated by localized modes and a generic boson peak. Suda, Lerner and Bouchbinder (arXiv:2607.23603, 2026) equipped this model with a micromechanical framework based on the linear response to force monopoles, deriving an exact identity between monopole stiffnesses and global susceptibilities, reconstructing the vibrational spectrum from monopole statistics, and showing that a single stabilized-sector stiffness scale controls the boson-peak frequency. They leave open how far this framework extends beyond the fully-connected, harmonic setting in which it was derived: whether the results survive on sparse graphs, whether the framework yields new falsifiable predictions — notably quantitative relations between mesoscopic elasticity and the boson peak in computer and laboratory glasses — and how the stabilized sector enters anharmonic and strongly driven (elasto-plastic, yielding, memory-forming) dynamics.

Background

The low-temperature physics of glasses is often approached at the mean-field level, where a glass is represented by NN scalar displacements xix_i (each standing for a small group of particles oscillating near a local equilibrium) with the Hamiltonian

H=12∑i=1Nkixi2+14!∑i=1Nxi4+∑i<jJijxixj−f(t)∑i=1Nxi,H=\frac{1}{2}\sum_{i=1}^{N}k_i x_i^{2}+\frac{1}{4!}\sum_{i=1}^{N}x_i^{4}+\sum_{i<j}J_{ij}x_i x_j-f(t)\sum_{i=1}^{N}x_i,

where the bare stiffnesses kik_i are drawn uniformly from [km,k0][k_{\rm m},k_0], the couplings JijJ_{ij} are independent Gaussians of variance J2/NJ^{2}/N (all-to-all, mean-field connectivity), and the stabilizing quartic term ensures that minimizing HH — the analogue of an instantaneous quench of a liquid into one of its inherent structures — yields well-defined minima. In a parameter range the equilibrium model undergoes a replica-symmetry-breaking (RSB) transition: in spin-glass language, the assumption that independent copies (replicas) of the system are statistically equivalent fails, signaling a rugged energy landscape with many metastable minima. Numerical quenches place the model deep in a non-equilibrium RSB phase whose Hessian M\mathcal{M} (the matrix of second derivatives of HH, whose eigenvalues are squared vibrational frequencies) features a nonphononic vibrational density of states (VDoS) with the universal quartic tail DG(ω)∼ω4{\cal D}_{\rm G}(\omega)\sim\omega^{4} — as observed in computer glasses — populated by strongly localized modes, plus a generic excess peak at a frequency ωp\omega_{\rm p} analogous to the boson peak, the universal excess of terahertz vibrational states over the Debye expectation (the ω2\omega^{2} VDoS of a homogeneous elastic continuum) found in essentially all laboratory glasses. In finite-dimensional computer glasses, a central micromechanical probe is the linear response to a force dipole (equal and opposite forces applied to a pair of neighboring particles, sometimes called 'pinching' the glass): the response field is quasi-localized, and its stiffness κdipole\kappa_{\rm dipole} defines a mesoscopic elastic scale that varies much more strongly with glass preparation and disorder than the global elastic moduli and is tied to the population of soft quasi-localized modes (Lerner and Bouchbinder, J. Chem. Phys. 148, 214502 (2018); Rainone, Bouchbinder and Lerner, PNAS 117, 5228 (2020)). Suda, Lerner and Bouchbinder (arXiv:2607.23603) constructed the mean-field analogue: because the model lacks translational symmetry, the leading local perturbation is a force monopole applied to a single oscillator, with monopole stiffness κi=(M−1)ii/(M−2)ii\kappa_i=(\mathcal{M}^{-1})_{ii}/(\mathcal{M}^{-2})_{ii}. They showed numerically and analytically that (i) the monopole response is strongly localized, with mean participation number ⟨Ne⟩≃4\langle Ne\rangle\simeq 4 (the participation number counts how many oscillators carry the response; O(1)O(1) means strong localization despite all-to-all coupling); (ii) the stiffness distribution p(κ)p(\kappa) is emergently pseudogapped — it vanishes smoothly as κ→0\kappa\to 0, a signature of marginal stability (the system sits at the edge of mechanical instability, with no gap protecting it against arbitrarily soft modes), since the bare p(k)p(k) is nonzero at k=0k=0; (iii) an exact mean-field identity relates κi\kappa_i to purely local quantities (kik_i and the local displacement xix_i) and to the global displacement susceptibility χ\chi and spin-glass susceptibility χSG\chi_{\rm SG} (the latter measuring frozen-in replica-to-replica fluctuations); (iv) the entire VDoS can be reconstructed from the monopole statistics alone, with a single stiffness scale — the mean stiffness ⟨κ⟩st\langle\kappa\rangle_{\rm st} of the 'stabilized sector' (the oscillators whose bare stiffness is down-shifted below the collective level, ki<J2χk_i<J^{2}\chi) — controlling both the boson-peak frequency ωp\omega_{\rm p} and the upper cutoff ωg\omega_{\rm g} of the quartic tail, with ωp∼(Δf)1/3\omega_{\rm p}\sim(\Delta f)^{1/3}, where Δf\Delta f is the level of self-consistent force fluctuations; and (v) this mesoscopic stiffness varies more strongly with quenched disorder JJ than the model's global elasticity, mirroring computer glasses. These results build a concrete bridge between a solvable mean-field glass model (Rainone et al., SciPost Phys. Core 4, 008 (2021); Bouchbinder et al., Phys. Rev. B 103, 174202 (2021); analytic structure: Folena and Urbani, J. Stat. Mech. 2022, 053301) and the micromechanics of finite-dimensional glasses. They stand next to a parallel, contested program — heterogeneous-elasticity theory, which attributes the boson peak to spatially fluctuating elastic constants (Schirmacher et al., Phys. Rev. E 113, 065406 (2026); Lerner and Bouchbinder, Phys. Rev. E 111, L013402 (2025)) — and to the proposal that mesoscopic elasticity controls viscous slowing-down in glassforming liquids (Kapteijns et al., J. Chem. Phys. 155, 074502 (2021)). The mean-field model itself has also been pushed to the strongly driven regime, where it reproduces the oscillatory yielding transition, mechanical annealing and memory formation of cyclically sheared glasses (Suda, Lerner and Bouchbinder, Phys. Rev. Lett. 135, 258201 (2025)), but without the new monopole/stabilized-sector observables.

Problem Statement

As posed at the close of arXiv:2607.23603: determine how far the force-monopole micromechanical framework of the mean-field glass model extends beyond the fully-connected, harmonic setting in which it was derived. Concretely, with a common objective of establishing the robustness and predictive content of the monopole/stabilized-sector bridge between mean-field glass theory and amorphous solids: (1) Sparse connectivity — define the oscillator Hamiltonian displayed in the Background (Eq. (1) of the source paper) on a sparse (diluted) random graph in place of the all-to-all graph — since the source fixes neither the sparse-graph ensemble (e.g. its mean degree) nor the coupling-variance scaling that replaces J2/NJ^{2}/N on existing bonds, these choices are themselves part of the problem, and the determination must state how each verdict depends on them, within connectivity regimes in which the diluted model still admits the quenched interacting glassy (RSB) phase of the Background rather than decoupling into independent oscillators — and determine which of the framework's results remain valid: strong localization of the monopole response (⟨Ne⟩=O(1)\langle Ne\rangle=O(1)), the emergent pseudogap of p(κ)p(\kappa) as κ→0\kappa\to 0, the exact identity relating the monopole stiffness κi\kappa_i to the local quantities ki,xik_i, x_i and the global susceptibilities χ\chi and χSG\chi_{\rm SG}, the reconstruction of the vibrational density of states from monopole statistics, the control of the boson-peak frequency ωp\omega_{\rm p} and quartic-tail cutoff ωg\omega_{\rm g} by the stabilized-sector stiffness scale ⟨κ⟩st\langle\kappa\rangle_{\rm st}, and the stronger variation of this mesoscopic stiffness with quenched disorder JJ than the model's global elasticity; (2) Predictive content — establish whether the framework not only reproduces established results in computer glasses but offers new predictions, most notably quantitative relations between mesoscopic elasticity and the boson peak in computer and laboratory glasses; (3) Nonlinear regimes — extend the statistical-structural harmonic analysis and linear response to anharmonic properties of the underlying energy landscape and to strongly driven dynamics, testing the stated expectation that the stabilized-sector degrees of freedom play important roles in elasto-plastic response, yielding, and memory formation. The problem is a program of independently settleable sub-claims: within strand (1), each of the results named above; within strand (2), whether the framework merely reproduces established results in computer glasses and whether it yields new falsifiable quantitative relations between mesoscopic elasticity and the boson peak in computer and in laboratory glasses; and within strand (3), whether the stabilized-sector degrees of freedom govern the model's anharmonic landscape properties and its strongly driven elasto-plastic, yielding and memory phenomena. A complete resolution of the program requires a determination of all three strands, but each of these sub-claims may be settled independently, so a determination on any one of them is a self-contained partial result while the rest remain open.

References

  1. M. Suda, E. Lerner, and E. Bouchbinder, 'Local micromechanics in a mean-field model of glasses reveal key properties of its non-equilibrium RSB phase' (2026), arXiv:2607.23603, DOI 10.48550/arXiv.2607.23603, https://arxiv.org/abs/2607.23603 (source paper)
  2. E. Bouchbinder, E. Lerner, C. Rainone, P. Urbani, and F. Zamponi, 'Low-frequency vibrational spectrum of mean-field disordered systems', Physical Review B 103, 174202 (2021), DOI 10.1103/PhysRevB.103.174202, https://doi.org/10.1103/PhysRevB.103.174202
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