Density fluctuations in many-body systems are of fundamental importance throughout various scientific disciplines. Hyperuniform systems, which include crystals and quasicrystals, have density fluctuations that are anomalously suppressed at long wavelengths compared to the fluctuations in typical disordered point distributions such as in ideal gases and liquids. Such systems are characterized by a local number variance associated with points within a spherical observation window of radius R that grows more slowly than the window volume in the large-R limit. In this talk, we will provide the first rigorous hyperuniformity analysis of quasicrystals obtained by cut-and-projection method and related points sets derived from substitution tilings. Most importantly, we reveal that one-dimensional quasicrystals produced by projection from a two-dimensional lattice fall into two distinct classes determined by the width of the projection window. The number variance is either uniformly bounded in the one class for large R, or it scales logarithmically in R in the other class. This distinction provides a new classification of one-dimensional quasicrystalline systems and, as we show, the two classes exhibit distinct physical properties. Our analysis further suggests that measures of hyperuniformity may define new classes of quasicrystals in higher dimensions as well.

Erdal C. Oğuz obtained his PhD from University of Düsseldorf, Germany. His doctoral studies have addressed various aspects of statistical mechanics of soft condensed matter with an emphasis on structure formation and dynamics in colloidal systems. In his postdoctoral studies at Princeton University he investigated self-assembly of soft quasicrystals and hyperuniformity in several (non-) equilibrium systems. His current postdoctoral studies at Tel Aviv University concern the understanding of topological defects and their consequences in mechanical metamaterials from a condensed-matter perspective. Dr. Oğuz is a theoretical physicist with a strong affinity to numerical techniques. His research portfolio contains purely theoretical works of fundamental interest as well as practical research with potential real-life applications. In his career, Dr. Oğuz initiated and joined several collaborations with numerous experimental physicists and chemists.