Order and Transport Properties of Glassy Solids: Importance of Topological Defects*

The topological origins of the solidification of crystalline and non-crystalline states, and their inverse thermal transport properties are considered herein, making use of a quaternion orientational order parameter. Owing to the 4D nature of the quaternion numbers, quaternion ordered systems that exist in 4D/(3D+1t) must be considered to exist in ``restricted dimensions’’ (Mermin-Wagner). This is owing to the point-like nature of available third homotopy group topological defects in four-dimensions. Dual Berezinskii-Kosterlitz-Thouless (BKT) transitions are anticipated, defect-driven and particle-driven, to allow for the existence of crystalline and non-crystalline solid states. A full phase diagram for solidification is presented that includes frustration, in the range of finite kinetic and potential energy effects. The inverse thermal transport properties, above ~50K, are compared with the electrical transport properties of O(2) Josephson junction arrays (JJA) across the superconductor-to-insulator transition.