Revision 1

Accepted · first revision · record

Title: Why quasilocalized modes in glasses scale as $\omega$ to the fourth
Plain: Computer glasses contain soft vibration patterns concentrated on a few dozen particles, and their number grows as the fourth power of frequency in every dimension tested. A finite-dimensional theory that derives this law is missing.
Precise: Simulations of quenched glasses show non-phononic quasilocalized modes with density $D_{\mathrm{loc}}(\omega) = A_g\,\omega^4$ at low $\omega$, with the same exponent in $d = 3\ \text{and}\ 4$ and, with stronger finite-size effects, in $d = 2$, and with the prefactor $A_g$ decreasing for better-annealed glasses. Derive the $\omega^4$ law and the dependence of $A_g$ on the preparation (parent) temperature from a controlled microscopic theory in finite d.
Settled by: A derivation of $D_{\mathrm{loc}}(\omega) \sim \omega^{4}$ and of $A_{g}(T_{\mathrm{parent}})$ in finite d that matches simulation data quantitatively.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: partially-resolved
Status note: The $\omega^{4}$ law is established numerically (2016-2020) and several mean-field derivations exist, without consensus on the finite-d theory (2026).
Posed since: 2016