  Positioned between crystalline solids and liquids, disordered many-particle
systems which are stealthy and hyperuniform represent new states of matter that
are endowed with novel physical and thermodynamic properties. Such stealthy and
hyperuniform states are unique in that they are transparent to radiation for a
range of wavenumbers around the origin. In this work, we employ recently
developed inverse statistical-mechanical methods, which seek to obtain the
optimal set of interactions that will spontaneously produce a targeted
structure or configuration as a unique ground state, to investigate the
spin-spin interaction potentials required to stabilize disordered stealthy
hyperuniform one-dimensional (1D) Ising-like spin chains. By performing an
exhaustive search over the spin configurations that can be enumerated on
periodic 1D integer lattices containing $N=2,3,\ldots,36$ sites, we were able
to identify and structurally characterize \textit{all} stealthy hyperuniform
spin chains in this range of system sizes. Within this pool of stealthy
hyperuniform spin configurations, we then utilized such inverse optimization
techniques to demonstrate that stealthy hyperuniform spin chains can be
realized as either unique or degenerate disordered ground states of radial
long-ranged (relative to the spin chain length) spin-spin interactions. Such
exotic ground states are distinctly different from spin glasses in both their
inherent structural properties and the nature of the spin-spin interactions
required to stabilize them. As such, the implications and significance of the
existence of such disordered stealthy hyperuniform ground state spin systems
warrants further study, including whether their bulk physical properties and
excited states, like their many-particle system counterparts, are singularly
remarkable, and can be experimentally realized.
