Generating large disordered stealthy hyperuniform systems with ultrahigh accuracy to determine their physical properties
Peter K. Morse, Jaeuk Kim, Paul J. Steinhardt, Salvatore Torquato
Abstract
Hyperuniform many-particle systems are characterized by a structure factor $S(\mathbf{k})$ that is precisely zero as $|\mathbf{k}|\ensuremath{\rightarrow}0$; and stealthy hyperuniform systems have $S(\mathbf{k})=0$ for the finite range $0<|\mathbf{k}|\ensuremath{\le}K$, called the ``exclusion region.'' Through a process of collective-coordinate optimization, energy-minimizing disordered stealthy hyperuniform systems of moderate size have been made to high accuracy, and their novel physical properties have shown great promise. However, minimizing $S(\mathbf{k})$ in the exclusion region is computationally intensive as the system size becomes large. In this paper, we present an improved methodology to generate such states using double-double precision calculations on graphical processing units (GPUs) that reduces the deviations from zero within the exclusion region by a factor of approximately ${10}^{30}$ for system sizes more than an order of magnitude larger. We further show that this ultrahigh accuracy is required to draw conclusions about their corresponding characteristics, such as the nature of the associated energy landscape and the presence or absence of Anderson localization, which might be masked, even when deviations are relatively small.