In this work, we develop and extend the class of inverse statistical-mechanical methods to the design of spin systems. For simplicity, we focus on the two-state Ising model (or equivalently speaking, the lattice gas model) with radial spin-spin interactions of finite range that extend beyond nearest-neighbor sites. Our interest is to find the optimal set of shortest-range pair interactions within this family of Hamiltonians, whose corresponding ground state is a targeted spin configuration. This is accomplished using a competitor-based 0K optimization algorithm which maximizes the gap between the energetically closest competitor and the target spin configuration for an exhaustive list of competitors. As a case study, we have investigated both striped-phase spin configurations, which are comprised of alternating parallel bands of up- and down-spins of varying thicknesses, and block checkerboard spin configurations, which are comprised of varying block sizes and are generalizations of the classic anti-ferromagnetic Ising model. Interestingly, our findings demonstrate that the structurally anisotropic striped phases, in which the thicknesses of the up- and down-spin bands are equal, are unique ground states for isotropic short-ranged interactions. By contrast, virtually all of the block checkerboard targets are either degenerate or cannot be stabilized by radial pair interactions. In this work, I will also discuss our recent investigation into the inverse design of "stealthy" and hyperuniform disordered two-state Ising (or lattice gas) systems--that is, configurations which possess novel scattering properties and are characterized by hidden long-range order.
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