The Importance of "Topological Ordering" in Understanding the Structure and Thermal Conductivity of Crystalline and Glassy Solid States
Thermal transport properties of solid states are tied to the structure that forms from the undercooled atomic liquid. Above approximately 50 K, the thermal conductivity of glasses and crystalline materials show inverse behavior that has remained a matter of inquiry over the past century. This research elucidates the topological origins of solidification, of crystalline and non-crystalline solid states, and their thermal transport properties by adopting quaternion numbers to characterize orientational ordering. In doing so, this topological viewpoint generalizes the fundamental concepts of Bose-Einstein condensation, the Hohenberg-Mermin-Wagner theorem and Berezinskii-Kosterlitz-Thouless (BKT) topological ordering transitions from 2D/1D complex ordered systems to 4D/3D quaternion ordered systems.
The Kauzmann point ("ideal glass transition") is identified as a self-dual critical point that marks a quantum phase transition between crystalline and non-crystalline solids at a critical value of geometrical frustration. This transition is viewed as a higher-dimensional analogue to the superconductor-to-superinsulator quantum phase transition (or, superfluid-to-Mott insulator) in 2D/1D O(2) quantum rotor models. Using this topological viewpoint, the inverse thermal transport behavior of crystalline and non-crystalline solid states is considered alongside the dual electrical transport properties of Josephson junction arrays across the singularity at the superconductor-to-superinsulator transition.