Multi-Hyperuniformity in Complex Materials

• Multi-hyperuniformity is the phenomenon where multiple structural fluctuations—such as density, area, or perimeter—are concurrently suppressed over large length scales.
• Advanced methods like variance scaling, spectral analysis, and HUDLS reveal a hierarchy of disorder lengths critical for understanding phase transitions in complex systems.
• This framework enables the design of optimized materials with controlled photonic, mechanical, and transport properties by simultaneously managing diverse fluctuations.

Multi-hyperuniformity refers to the simultaneous presence of hyperuniformity in several distinct structural or statistical fields within a single system. In such systems, not only are number density fluctuations anomalously suppressed at large length scales, but additional degrees of freedom—such as shape, geometric, or even many-body property fluctuations—are hyperuniform over the same or disparate ranges of scale. This emergent phenomenon is of growing significance for the classification and design of complex materials and point patterns that require simultaneous control over multiple structural metrics at various scales.

1. Hyperuniformity: Single-Observable Framework

Hyperuniformity is defined by anomalous suppression of large-scale fluctuations, typically in the local number density or volume fraction. Formally, for a statistically homogeneous point pattern in $\mathbb{R}^d$, let $N(R)$ be the number of points inside a spherical window of radius $R$. The local number variance is

$$\sigma_N^2(R) = \langle N(R)^2 \rangle - \langle N(R) \rangle^2.$$

For Poisson (uncorrelated) distributions, $\sigma_N^2(R)\sim R^d$, while for hyperuniform systems, $\sigma_N^2(R)\sim R^{\epsilon}$ with $\epsilon<d$ (Maher et al., 2024). The structure factor $S(k)$ vanishes as $k \rightarrow 0$. Three classes of hyperuniformity exist, classified by the scaling exponents in real and reciprocal space (Torquato, 2018, Vanoni et al., 28 Jul 2025):

• Class I (Strong): $\sigma_N^2(R)\sim R^{d-1}$, $N(R)$0 with $N(R)$1.
• Class II: $N(R)$2, $N(R)$3.
• Class III (Weak): $N(R)$4, $N(R)$5, $N(R)$6.

These frameworks control density fluctuations, but do not capture the suppression of fluctuations in other (possibly coupled) observables.

2. Extension to Multiple Observables: Multi-Hyperuniformity

Multi-hyperuniform systems are characterized by simultaneous suppression of fluctuations in several independent or coupled observables across length scales. For a Voronoi-based tessellation, for example, in addition to local density $N(R)$7, one may define a geometrical field such as the sum of cell perimeters, $N(R)$8, or local area. The corresponding variance,

$N(R)$9

exhibits hyperuniform-like suppression for $R$0 up to a characteristic scale, with a crossover to normal (Poissonian) fluctuation growth at larger $R$1 (Zheng et al., 2018).

Different observables can possess distinct hyperuniformity crossover lengths and scaling regimes, resulting in a hierarchy of hyperuniform length scales. In jammed cellular models, density fluctuations are suppressed up to a length scale $R$2, while geometrical fluctuations (e.g., perimeters) are suppressed up to a generally larger scale $R$3, each characterized by critical scaling near phase transitions. Multi-hyperuniformity thus generalizes the hyperuniformity concept to a vector-valued or multi-indexed status.

3. Quantitative Metrics: Disorder Lengths and Spectra

The control and quantification of hyperuniformity (and multi-hyperuniformity) rely on disorder lengths. For number density, the "hyperuniformity disorder length" $R$4 is defined as the crossover scale between ordinary scaling and the anomalous hyperuniform regime, e.g.,

$R$5

where $R$6 and $R$7 are coefficients from the large-$R$8 expansion $R$9 (Maher et al., 2024).

For more general fields, distinct disorder lengths (e.g., $$\sigma_N^2(R) = \langle N(R)^2 \rangle - \langle N(R) \rangle^2.$$0 for perimeter fluctuations) are defined analogously, and often scale differently with system parameters. The concept is further extended by "Hyperuniformity Disorder Length Spectroscopy" (HUDLS), which provides a scale-dependent spectrum $$\sigma_N^2(R) = \langle N(R)^2 \rangle - \langle N(R) \rangle^2.$$1 of disorder lengths by inverting the real-space variance equations for various observables (area, edge length, weighted centroids), revealing the full scale hierarchy of hyperuniform suppression and crossovers (Chieco et al., 2021, Durian, 2017, Chieco et al., 2017).

In cellular foams and Voronoi patterns, for instance, area-weighted centroids exhibit strongly hyperuniform $$\sigma_N^2(R) = \langle N(R)^2 \rangle - \langle N(R) \rangle^2.$$2 at large $$\sigma_N^2(R) = \langle N(R)^2 \rangle - \langle N(R) \rangle^2.$$3, while edge-based fluctuations decay less rapidly, $$\sigma_N^2(R) = \langle N(R)^2 \rangle - \langle N(R) \rangle^2.$$4 with $$\sigma_N^2(R) = \langle N(R)^2 \rangle - \langle N(R) \rangle^2.$$5 (Chieco et al., 2021).

4. Criticality, Scaling, and Divergence in Multi-Hyperuniform Systems

Near structural critical points—such as the jamming or rigidity transition—multi-hyperuniform systems display critical scaling in their respective disorder lengths. In jamming models, the lengthscale for density hyperuniformity diverges as $$\sigma_N^2(R) = \langle N(R)^2 \rangle - \langle N(R) \rangle^2.$$6 (with $$\sigma_N^2(R) = \langle N(R)^2 \rangle - \langle N(R) \rangle^2.$$7), while the corresponding scale for geometrical observables (e.g., perimeter) diverges with a distinct exponent $$\sigma_N^2(R) = \langle N(R)^2 \rangle - \langle N(R) \rangle^2.$$8 (Zheng et al., 2018). In absorbing state transitions, similar but independent divergences occur for the hyperuniform length $$\sigma_N^2(R) = \langle N(R)^2 \rangle - \langle N(R) \rangle^2.$$9 (fixed by second moments) and a further length scale $\sigma_N^2(R)\sim R^d$0 governing hidden many-body correlations (Zheng et al., 2020). This suggests a proliferation of correlation lengths, each associated with a different underlying field, in the critical domain.

The co-existence of multiple diverging disorder lengths constitutes a hallmark of multi-hyperuniformity in the vicinity of critical points, with implications for hierarchical self-organization in disordered media.

5. Computational and Experimental Methodologies

Quantitative analysis of multi-hyperuniformity employs a combination of real-space windowed-variance measurements, spectral density analysis, and boundary layer inversion. Methodologies include:

• Variance extraction: $\sigma_N^2(R)\sim R^d$1 for each observable $\sigma_N^2(R)\sim R^d$2 (e.g., number, area, perimeter).
• Scaling fits: Fitting $\sigma_N^2(R)\sim R^d$3 vs. $\sigma_N^2(R)\sim R^d$4 to $\sigma_N^2(R)\sim R^d$5 to identify crossover lengths via $\sigma_N^2(R)\sim R^d$6 (Zheng et al., 2018).
• HUDLS: Direct inversion to obtain the disorder length $\sigma_N^2(R)\sim R^d$7 for multiple fields (Chieco et al., 2021, Durian, 2017, Chieco et al., 2017).
• Spectral-domain analysis: Computing structure factors or spectral densities for each observable and relating their small-$\sigma_N^2(R)\sim R^d$8 asymptotics to real-space disorder lengths.
• Critical scaling analysis: Extracting exponents from the divergence of disorder lengths as function of control parameters (e.g., $\sigma_N^2(R)\sim R^d$9 in rigidity, $\sigma_N^2(R)\sim R^{\epsilon}$0).

High-statistics sampling over large ensembles or images is typically required to resolve the hierarchical structure of disorder lengths in multi-hyperuniform systems.

6. Physical Implications and Applications

Multi-hyperuniform systems are of theoretical and practical interest for the design of advanced materials in which multiple fluctuating properties must be simultaneously controlled or suppressed. This includes photonic and phononic media, jammed and glassy states, and biological tissues exhibiting organized cellular patterns (Torquato, 2018, Maher et al., 2024). The ability to engineer multiple, possibly decoupled or hierarchically nested hyperuniform regimes enables optimization of mechanical, transport, or optical properties.

In addition, the identification of nested or mutually divergent disorder lengths reveals the hidden organization in complex systems—e.g., the presence of "hidden order beyond hyperuniformity" in absorbing-state systems (Zheng et al., 2020). Such findings underscore the necessity of going beyond single-observable analyses to fully characterize the order and disorder in real materials and stochastic processes.

7. Perspectives and Open Problems

The mathematical framework for multi-hyperuniformity is still developing. Key open questions include:

• How generic is multi-hyperuniformity in non-equilibrium and biological systems?
• What are the coupling mechanisms between different observables leading to simultaneous (or hierarchical) hyperuniformity?
• Can higher-order statistical moments and more complex fields (e.g., topological invariants) display hyperuniformity, and under what structural rules?
• How can multi-hyperuniformity be exploited in the inverse design of materials with prescribed combinations of suppressed fluctuations?

Promising directions involve the extension of local variance and integrated metrics to incorporate higher moments, as well as the theoretical classification of possible multi-hyperuniform scaling regimes (Maher et al., 2024). The continued development of real-space and spectral methodologies for arbitrary observables is likely to advance both the fundamental understanding and the functional exploitation of multi-hyperuniform systems across domains.