Caroline S. Gorham and David E. Laughlin
Phys. Rev. B 99, 144106 (2019)
Publication year: 2019

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 a “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-crystal transition from the fundamental concepts of Bose-Einstein condensation, the Mermin-Wagner theorem and Berezinskii-Kosterlitz-Thouless (BKT) topological ordering transitions and thereby generalizes the concepts that apply to superfluidity in “restricted dimensions” to the formation of the solid state.

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