Low-frequency vibrational spectrum of mean-field disordered systems

Abstract

We study a recently introduced and exactly solvable mean-field model for the density of vibrational states D(ω) of a structurally disordered system. The model is formulated as a collection of disordered anharmonic oscillators, with random stiffness κ drawn from a distribution p(κ ), subjected to a constant field h and interacting bilinearly with a coupling of strength J. We investigate the vibrational properties of its ground state at zero temperature. When p(κ ) is gapped, the emergent D(ω) is also gapped, for small J. Upon increasing J, the gap vanishes on a critical line in the (h, J ) phase diagram, whereupon replica symmetry is broken. At small h, the form of this pseudogap is quadratic, D(ω) ∼ ω 2 , and its modes are delocalized, as expected from previously investigated mean-field spin glass models. However, we determine that for large enough h, a quartic pseudogap D(ω) ∼ ω 4 , populated by localized modes, emerges, the two regimes being separated by a special point on the critical line. We thus uncover that mean-field disordered systems can generically display both a quadraticdelocalized and a quartic-localized spectrum at the glass transition.

Delocalization transition in low energy excitation modes of vector spin glasses

“…In both cases, the corresponding eigenstates are delocalized, since they are related to the eigenvectors of simple random matrices. 1 Only very recently a mean-field spin glass model has been studied [18][19][20] displaying, in presence of an external field, D(ω) ∝ ω 4 with localized modes. However, this model has unbounded variables subject to a spherical constraint, very useful for performing computations, but rather unrealistic.…”

Effects of intraspecific cooperative interactions in large ecosystems

“…The techniques we are going to use in the following are standard and based on the computation of the first corrections to the ground state energy in a quantum mechanical system of anharmonic oscillators. Following the same hypothesis of [61], in the T → 0 limit we can write the solution as a problem of decoupled oscillators subject to a given effective potential.…”

Low energy excitations of mean-field glasses

“…From the replica free-energy one can also compute the 'replicon eigenvalue' Λ, whose positiveness is a necessary stability condition for the free-energy (16). Its expression is rather lengthy and we give it in Appendix A. Eq.…”

“…Putting everything together and using the saddle point equation q = β 2 f (q) we get (16). The physical overlap is found by extremizing G with respect to q and is given by eq.…”

“…It appears that deeper states in the landscape, corresponding to better optimized glasses, have less and less the low energy excitations, reflecting in smaller and smaller values of A 4 , and correspondingly, the excitations are more and more localized [12,13]. This spectrum of localized modes was first rationalized through phenomenological theories [14,15], while new predictions have recently enriched the picture [11][12][13][16][17][18][19]. In addition to typical ungapped minima, found by usual minimization protocols, it has been noticed in [20] that in some model glasses gapped mimima can be found through the use of smart minimization protocols that include particle swap [21,22].…”