Quasicrystals: What do we know? What do we want to know? What can we know?

Abstract

More than 35 years and 11 000 publications after the discovery of quasicrystals by Dan Shechtman, quite a bit is known about their occurrence, formation, stability, structures and physical properties. It has also been discovered that quasiperiodic self-assembly is not restricted to intermetallics, but can take place in systems on the meso-and macroscales. However, there are some blank areas, even in the centre of the big picture. For instance, it has still not been fully clarified whether quasicrystals are just entropy-stabilized high-temperature phases or whether they can be thermodynamically stable at 0 K as well. More studies are needed for developing a generally accepted model of quasicrystal growth. The state of the art of quasicrystal research is briefly reviewed and the main as-yet unanswered questions are addressed, as well as the experimental limitations to finding answers to them. The focus of this discussion is on quasicrystal structure analysis as well as on quasicrystal stability and growth mechanisms.

Mean-field phases of an ultracold gas in a quasicrystalline potential

“…Their order can be considered to arise from incommensurate projections of higher-dimensional periodic crystals [4,5], or due to a continous tiling of space with discrete unit cells [6]. Over the last few decades, quasicrystals have been studied in condensed matter systems [3,7,8], photonics [9,10], twisted bilayer graphene [11], the Gross-Pitaevskii equation with solitons [12], and as an emergent phase in ultracold dipolar gases [13]. There has also been a significant amount of research into the case of one-dimensional disordered, quasirandom and/or incommensurate systems that are related to quasicrystals [14][15][16][17][18][19][20][21][22][23][24].…”

Matter-Wave Diffraction from a Quasicrystalline Optical Lattice

“…Important manifestations of this non-trivial order on all length scales include the absence of universal power-law scaling near criticality [15] and its application to quantum complexity [16]. Moreover, quasicrystals exhibit fascinating phenomena such as phasonic degrees of freedom [6,17,18]. To date, quasicrystals have been extensively studied in condensed matter and material science [1,3,4,6,17], in photonic structures [9,13,[18][19][20], using laser-cooled atoms in the dissipative regime [21,22], and very recently in twisted bilayer graphene [23].…”

“…standing waves of light, have become a cornerstone in experimental research on quantum manybody physics [25]. They offer an ideal environment for examining quasicrystals since optical potentials are free of defects which greatly complicate measurements on quasicrystalline solids [6]. In addition, we are able to directly impose 'forbidden' rotational symmetries, thereby circumventing the elaborate synthesis of stable single crystals [26].…”

Quantum percolation in quasicrystals using continuous-time quantum walk

“…Quasicrystals do not show periodicity and have no long range translational symmetry. Yet they present longrange order and have a well-defined Bragg diffraction pattern [41][42][43][44][45][46][47]. The diffraction patterns from quasicrystal lattices reveal fivefold, eightfold and tenfold symmetries that cannot originate from a periodic arrangement of unit cells.…”