Ensemble Theory for Stealthy Hyperuniform Disordered Ground States

It has been shown numerically that systems of particles interacting with isotropic "stealthy" bounded long-ranged pair potentials (similar to Friedel oscillations) have classical ground states that are (counterintuitively) disordered, hyperuniform, and highly degenerate. Disordered hyperuniform systems have received attention recently because they are distinguishable exotic states of matter poised between a crystal and liquid that are endowed with novel thermodynamic and physical properties. The task of formulating an ensemble theory that yields analytical predictions for the structural characteristics and other properties of stealthy degenerate ground states in d-dimensional Euclidean space R-d is highly nontrivial because the dimensionality of the configuration space depends on the number density rho and there is a multitude of ways of sampling the ground-state manifold, each with its own probability measure for finding a particular ground-state configuration. The purpose of this paper is to take some initial steps in this direction. Specifically, we derive general exact relations for thermodynamic properties (energy, pressure, and isothermal compressibility) that apply to any ground-state ensemble as a function of rho in any d, and we show how disordered degenerate ground states arise as part of the ground-state manifold. We also derive exact integral conditions that both the pair correlation function g(2)(r) and structure factor S(k) must obey for any d. We then specialize our results to the canonical ensemble (in the zero-temperature limit) by exploiting an ansatz that stealthy states behave remarkably like "pseudo"-equilibrium hard-sphere systems in Fourier space. Our theoretical predictions for g(2)(r) and S(k) are in excellent agreement with computer simulations across the first three space dimensions. These results are used to obtain order metrics, local number variance, and nearest-neighbor functions across dimensions. We also derive accurate analytical formulas for the structure factor and thermal expansion coefficient for the excited states at sufficiently small temperatures for any d. The development of this theory provides new insights regarding our fundamental understanding of the nature and formation of low-temperature states of amorphous matter. Our work also offers challenges to experimentalists to synthesize stealthy ground states at the molecular level.