Exotic Acoustic-Edge and Thermal Scaling in Disordered Hyperuniform Networks

TL;DR

This paper develops a first-principles theory for vibrational and thermal properties of disordered hyperuniform networks, revealing algebraic pseudogaps and spectral dimensions that influence low-temperature behavior and wave manipulation.

Contribution

It introduces a novel theoretical framework linking hyperuniformity to vibrational spectra and thermal properties of disordered networks, with explicit formulas for the density of states.

Findings

Hyperuniform systems show algebraic pseudogaps at low frequencies.

The low-temperature specific heat scales as a power law determined by hyperuniformity.

Spectral dimension is derived as a key parameter influencing vibrational behavior.

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.