  Disordered hyperuniform many-body systems are distinguishable states of
matter that lie between a crystal and liquid: they are like perfect crystals in
the way they suppress large-scale density fluctuations and yet are like liquids
or glasses in that they are statistically isotropic with no Bragg peaks. These
systems play a vital role in a number of fundamental and applied problems:
glass formation, jamming, rigidity, photonic and electronic band structure,
localization of waves and excitations, self-organization, fluid dynamics,
quantum systems, and pure mathematics. systems. Here, we derive new rigorous
criteria that disordered hyperuniform two-phase heterogeneous materials must
obey and explore their consequences. Two-phase heterogeneous media are
ubiquitous, examples include composites and porous media, biological media,
foams, polymer blends, granular media, cellular solids, and colloids. We
rigorously establish the requirements for sphere packings to be
"multihyperuniform." We apply realizability conditions for an autocovariance
function and its associated spectral density of a two-phase medium, and then
incorporate hyperuniformity as a constraint in order to derive new conditions.
We show that some functional forms can immediately be eliminated from
consideration and identify other forms that are allowable. Specific examples
and counterexamples are described. Contact is made with well-known
microstructural models as well as irregular phase-separation and Turing-type
patterns. We also ascertain a family of spectral densities that are realizable
by disordered hyperuniform two-phase media in any space dimension, and present
explicit constructions. These studies provide insight into the nature of
disordered hyperuniformity in the context of heterogeneous materials and have
implications for the design of such novel amorphous materials.
