On the Formation of Solid States Beyond Perfect Crystals: Quasicrystals, Geometrically-Frustrated Crystals and Glasses

Abstract

There are three kinds of solid states of matter that can exist in physical space: quasicrystalline (quasiperiodic), crystalline (periodic) and amorphous (aperiodic). Herein, we consider the degree of orientational order that develops upon the formation of a solid state to be characterized by the application of quaternion numbers. The formation of icosahedral quasicrystalline solids is considered alongside the development of bulk superfluidity, characterized by a complex order parameter, that occurs by spontaneous symmetry breaking in three-dimensions. Crystalline solid states are viewed as higher-dimensional analogues to phase-coherent topologically-ordered superfluid states of matter that develop in restricted dimensions (Hohenberg-Mermin- Wagner theorem). Lastly, amorphous solid states are viewed as dual to crystalline solids, in analogy to Mott-insulating states of matter that are dual to topologically-ordered superfluids.