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

## TL;DR

This paper presents a topological framework for understanding three-dimensional crystallization and geometrical frustration, emphasizing defect classification and the role of topological order in phase transitions.

## Contribution

It introduces a novel topological approach to crystallization using quaternion order parameters and homotopy theory, linking concepts from superfluidity and BKT transitions to solidification.

## Key findings

- Classifies topologically stable defects in crystal structures.
- Reframes crystallization as a topological transition in restricted dimensions.
- Connects superfluidity concepts to solidification processes.

## 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.