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 \(S(k)\sim k^α\) and compensated kernels, {the VDOS exhibits an algebraic \emph{pseudogap} at low frequency, \(g(ω)\sim ω^{\,2d/β-1}\) with \(β=\min\{4,α+2\}\), which implies a low-temperature specific heat \(C(T)\sim T^{\,2d/β}\) and a heat-kernel decay \(Z(t)\sim t^{-d/β}\), defining a spectral dimension \(d_s=2d/β\).} 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.

Figures (1)

• Figure 1: Comparisons of theoretical predictions for eigen value distribution $\varphi_L(\lambda)\sim \lambda^{d/\beta-1}$, VDOS $g(\omega) \sim \omega^{2d/\beta-1}$, heat capacity $C(T)\sim T^{2d/\beta}$ (per node, with unit $k_B$) and heat kernel $Z(t) \sim t^{-d/\beta}$ with numerical results of a variety of hyperuniform systems with $\alpha = 1/2$, 1, 2 and stealthy systems in 1D with $N=10,000$ points in a unitary periodic box and the compensated kernel (\ref{['eq_ck']}) with $a = 0.002$ and $b=0.004$. The scaling behaviors of the reported quantities are not sensitive to choice of $a$ and $b$ values. The results are obtained by averaging over 10 independent realizations for each case.