Crystallization in Three-Dimensions: Defect-Driven Topological Ordering and the Role of Geometrical Frustration

Abstract

Herein, fundamentals of topology and symmetry breaking are used to understand crystallization and geometrical frustration in topologically close-packed structures. This frames solidification from a new perspective that is unique from thermodynamic discussions. Crystallization is considered as developing from undercooled liquids, in which orientational order is characterized by a surface of a sphere in four-dimensions (quaternion) with the binary polyhedral representation of the preferred orientational order of atomic clustering inscribed on its surface. As a consequence of the dimensionality of the quaternion orientational order parameter, crystallization is seen as occurring in "restricted dimensions." Homotopy theory is used to classify all topologically stable defects, and third homotopy group defect elements are considered to be generalized vortices that are available in superfluid ordered systems. This topological perspective approaches the liquid-to-crystalline solid transition in three-dimensions from fundamental concepts of: Bose-Einstein condensation, the Mermin-Wagner theorem and Berezinskii-Kosterlitz-Thouless (BKT) topological-ordering transitions. In doing so, in this article, concepts that apply to superfluidity in "restricted dimensions" are generalized in order to consider the solidification of crystalline solid states.

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Cited by in corpus (6)

• Resonant spin amplification meets electron spin resonance in $n$-GaAs
• Anomalous magnetic suppression of spin relaxation in a two-dimensional electron gas in a GaAs/AlGaAs quantum well
• On the Formation of Solid States Beyond Perfect Crystals: Quasicrystals, Geometrically-Frustrated Crystals and Glasses
• Importance of the 'Higgs' Amplitude Mode in Understanding the "Ideal Glass Transition" and the Kauzmann Entropy Paradox at the Four-Dimensional Crystal/Glass Quantum Critical Point
• Solidification of Icosahedral Quasicrystals: Viewpoint Based on a Quaternion Orientational Order Parameter and the Formation of the E8 Lattice
• Quantized Hall Effect Phenomena and Topological-Order in 4D Josephson Junction Arrays in the Vicinity of a Quantum Phase Transition