  Disordered many-particle hyperuniform systems are exotic amorphous states
characterized by anomalous suppression of large-scale density fluctuations.
Here we substantially broaden the hyperuniformity concept along four different
directions. This includes generalizations to treat fluctuations in the
interfacial area in heterogeneous media and surface-area driven evolving
microstructures, random scalar fields, divergence-free random vector fields, as
well as statistically anisotropic many-particle systems and two-phase media.
Interfacial-area fluctuations play a major role in characterizing the
microstructure of two-phase systems , physical properties that intimately
depend on the geometry of the interface, and evolving two-phase microstructures
that depend on interfacial energies (e.g., spinodal decomposition). In the
instances of divergence-free random vector fields and statistically anisotropic
structures, we show that the standard definition of hyperuniformity must be
generalized such that it accounts for the dependence of the relevant spectral
functions on the direction in which the origin in Fourier space
(nonanalyticities at the origin). Using this analysis, we place some well-known
energy spectra from the theory of isotropic turbulence in the context of this
generalization of hyperuniformity. We show that there exist many-particle
ground-state configurations in which directional hyperuniformity imparts exotic
anisotropic physical properties (e.g., elastic, optical and acoustic
characteristics) to these states of matter. Such tunablity could have
technological relevance for manipulating light and sound waves in ways
heretofore not thought possible. We show that disordered many-particle systems
that respond to external fields (e.g., magnetic and electric fields) are a
natural class of materials to look for directional hyperuniformity.
