May 28, 2019

High-dimensional generalizations of the kagomé and diamond crystals and the decorrelation principle for periodic sphere packings

In this paper, we introduce constructions of the high-dimensional generalizations of the kagomé and diamond crystals. The two-dimensional kagomé crystal and its three-dimensional counterpart, the pyrochlore crystal, have been extensively studied in the context of geometric frustration in antiferromagnetic materials. Similarly, the polymorphs of elemental carbon include the diamond crystal and the corresponding two-dimensional honeycomb structure, adopted by graphene. The kagomé crystal in d Euclidean dimensions consists of vertex-sharing d-dimensional simplices in which all of the points are topologically equivalent. The d-dimensional generalization of the diamond crystal can then be obtained from the centroids of each of the simplices, and we show that this natural construction of the diamond crystal is distinct from the Dd + family of crystals for all dimensions . We analyze the structural properties of these high-dimensional crystals, including the packing densities, coordination numbers, void exclusion probability functions, covering radii and quantizer errors. Our results demonstrate that the so-called decorrelation principle, which formally states that unconstrained correlations vanish in asymptotically high dimensions, remarkably applies to the case of periodic point patterns with inherent long-range order. We argue that the decorrelation principle is already exhibited in periodic crystals in low dimensions via a 'smoothed' pair correlation function obtained by convolution with a Gaussian kernel. These observations support the universality of the decorrelation principle for any point pattern in high dimensions, whether disordered or not. This universal property in turn suggests that the best conjectural lower bound on the maximal sphere-packing density in high Euclidean dimensions derived by Torquato and Stillinger (2006 Expt. Math. 15 307) is, in fact, optimal.


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