Exotic Acoustic-Edge and Thermal Scaling in Disordered Hyperuniform Networks

Abstract

We develop a first-principles theory for the vibrational density of states (VDOS) and thermal properties of network materials built on stationary correlated disordered point configurations. For scalar (mass--spring) models whose dynamical matrix is a distance-weighted graph Laplacian, we prove that the limiting spectral measure is the pushforward of Lebesgue measure by a Fourier symbol that depends only on the edge kernel \(f\) and the two-point statistics \(g_2\) (equivalently the structure factor \(S\)). For hyperuniform systems with small-$k$ scaling \(S(k)\sim k^\alpha\) and compensated kernels, {the VDOS exhibits an algebraic \emph{pseudogap} at low frequency, \(g(\omega)\sim \omega^{\,2d/\beta-1}\) with \(\beta=\min\{4,\alpha+2\}\), which implies a low-temperature specific heat \(C(T)\sim T^{\,2d/\beta}\) and a heat-kernel decay \(Z(t)\sim t^{-d/\beta}\), defining a spectral dimension \(d_s=2d/\beta\).} This hyperuniformity-induced algebraic edge depletion could enable novel wave manipulation and low-temperature applications. Generalization to vector mechanical models and implications on material design are also discussed.