Disordered Hyperuniform Networks Exhibit a VDOS Pseudogap with Α = 1.5 and Spectral Dimension D = 2

Disordered materials, often dismissed for their lack of predictable behaviour, are now revealing surprising acoustic and thermal properties, thanks to research led by Yang Jiao from Arizona State University. This work establishes a fundamental theory explaining how the arrangement of particles within these materials dictates their vibrational characteristics and heat transfer. The team demonstrates that specific disordered arrangements, known as hyperuniform networks, exhibit an unusual depletion of low-frequency vibrations, creating a ‘pseudogap’ that dramatically alters how heat propagates through the material. This discovery opens exciting possibilities for designing materials with tailored thermal properties and manipulating acoustic waves in unconventional ways, potentially leading to innovations in diverse fields from energy storage to advanced sensing technologies.

The research establishes that for disordered systems modelled as distance-weighted graph Laplacians, the limiting spectral measure corresponds to the pushforward of Lebesgue measure by a Fourier symbol dependent solely on the edge kernel and the two-point statistics. For hyperuniform systems exhibiting small-k scaling of the structure factor, the vibrational density of states displays an algebraic pseudogap at low frequency, scaling as ω d/β-1, with β defined by the system’s characteristics. This result implies a low-temperature specific heat scaling as C(T) ~ T2d/β and a heat-kernel decay of Z(t) ~ t−d/β, thereby defining a spectral dimension for these materials. The observed hyperuniformity-induced depletion of vibrational modes suggests potential applications in novel wave manipulation techniques and low-temperature technologies.

Disorder, Networks, and Spectral Graph Theory

This body of work encompasses a broad range of research related to disordered systems, materials science, and the mathematical tools used to study them. It explores the theory of disordered systems and amorphous materials, connecting these concepts to random walks, network theory, and percolation. A strong emphasis is placed on the mathematical foundations, including spectral graph theory, operator theory, harmonic analysis, and probability theory, which are essential for analyzing these complex systems. The research also delves into solid state physics, examining the properties of materials such as phonons, thermal conductivity, and mechanical behaviour.

Wave propagation in disordered media and the phenomenon of Anderson localization are also key areas of investigation. Recent studies suggest the application of machine learning techniques for analyzing disordered systems, alongside a significant focus on phonon transport and thermal conductivity. This comprehensive collection of work highlights the interdisciplinary nature of the field, drawing from physics, mathematics, materials science, and computer science.

Disordered Networks Exhibit Predictable Vibrational Properties

Scientists have developed a first-principles theory to predict the vibrational properties of network materials built from disordered arrangements of points, revealing a direct connection between material structure and its vibrational density of states. The research demonstrates that the limiting spectral measure of these systems is determined by the edge kernel and the two-point statistics of the point configuration, offering a powerful method for predicting material behaviour. Experiments and calculations show that for hyperuniform systems, the vibrational density of states exhibits an algebraic pseudogap at low frequency, specifically scaling as ω d/β-1, where β is determined by the system’s characteristics. This algebraic depletion implies a unique low-temperature specific heat and a heat-kernel decay proportional to t -d/β, defining a spectral dimension for these materials.

The team discovered that the small-wavelength form of the material’s structure factor dictates the algebraic depletion, enabling precise control over the material’s vibrational properties. Calculations and simulations, performed on one-dimensional systems, demonstrate the scaling behaviours of the eigenvalue distribution, vibrational density of states, heat capacity, and heat kernel, validating the theoretical predictions. These findings establish a clear link between the arrangement of points within a material and its resulting vibrational characteristics, opening new avenues for the design of materials with tailored properties.

Disorder, Vibrations, and the Pseudogap Feature

This research establishes a fundamental connection between the arrangement of particles in disordered materials and their vibrational properties, with implications for thermal behaviour. Scientists developed a theoretical framework demonstrating that the vibrational density of states, which dictates how a material responds to mechanical stress and heat, can be predicted by knowing only the two-point structural statistics of the material. Specifically, they show that hyperuniform disordered systems exhibit an algebraic pseudogap at low frequencies, influencing both their specific heat and heat-kernel decay. This depletion of vibrational modes suggests potential for designing quieter materials and more precise supports with reduced softness.

The team’s theory successfully explains recent numerical observations, clarifying the role of hyperuniformity in controlling vibrational behaviour and distinguishing scenarios where genuine gaps require more complex structural arrangements. They demonstrate that the small-wavelength form of the material’s structure factor governs the low-frequency vibrational spectrum and low-temperature specific heat, enabling a form of “structure-factor engineering” for materials design. The work provides a powerful theoretical foundation for understanding and manipulating the thermal and mechanical properties of disordered materials.