  Disordered many-particle hyperuniform systems are exotic amorphous states of
matter that lie between crystals and liquids. Hyperuniform systems have
attracted recent attention because they are endowed with novel transport and
optical properties. Recently, the hyperuniformity concept has been generalized
to characterize scalar fields, two-phase media and random vector fields. In
this paper, we devise methods to explicitly construct hyperuniform scalar
fields. We investigate explicitly spatial patterns generated from Gaussian
random fields, which have been used to model the microwave background radiation
and heterogeneous materials, the Cahn-Hilliard equation for spinodal
decomposition, and Swift-Hohenberg equations that have been used to model
emergent pattern formation, including Rayleigh-B{\' e}nard convection. We show
that the Gaussian random scalar fields can be constructed to be hyperuniform.
We also numerically study the time evolution of spinodal decomposition patterns
and demonstrate that these patterns are hyperuniform in the scaling regime.
Moreover, we find that labyrinth-like patterns generated by the Swift-Hohenberg
equation are effectively hyperuniform. We show that thresholding a hyperuniform
Gaussian random field to produce a two-phase random medium tends to destroy the
hyperuniformity of the progenitor scalar field. We then propose guidelines to
achieve effectively hyperuniform two-phase media derived from thresholded
non-Gaussian fields. Our investigation paves the way for new research
directions to characterize the large-structure spatial patterns that arise in
physics, chemistry, biology and ecology. Moreover, our theoretical results are
expected to guide experimentalists to synthesize new classes of hyperuniform
materials with novel physical properties via coarsening processes and using
state-of-the-art techniques, such as stereolithography and 3D printing.
