Caroline S. Gorham
QGR visitor and friend Caroline Gorham delivered a guest lecture to the QGR team on what Hopf Fibrations can tell us about crystallization and glass formation.
This research elucidates the topological origins of the formation of crystalline and non-crystalline solid states, by adopting a quaternion orientational order parameter. Crystalline solids are considered to form in “restricted dimensions’’ 4D/(3D+1t), by a defect-binding topological transition of third homotopy group defects and disclinations, as a higher-dimensional analogue to the formation of topologically-ordered superfluid states in “restricted dimensions’’ 2D/(1D+1t) (e.g., Josephson junction arrays). Ultimately, it is suggested that, the notion of restricted dimensions may be interpreted by considering the dimensionality of the topological defects associated with the complex or quaternion order parameter (fiber space of the 1st and 2nd Hopf fibrations). This work thereby generalizes the fundamental concepts of: Bose-Einstein condensation, the Hohenberg-Mermin-Wagner theorem and Berezinskii-Kosterlitz-Thouless (BKT) topological ordering transitions from complex to quaternion Lie algebra domains. O(n) quantum rotor models mathematically model rotating Bose-Einstein condensates that exist in “restricted dimensions.” These models allow for the existence of frustrated ground states and, ultimately, a quantum phase transition between orientationally-ordered and orientationally-disordered ground states. These spectrum of ground states are ultimately realized in the laboratory, e.g., a superfluid-to-Mott insulator transition (complex) and a crystalline-to-glass transition (quaternion). The Kauzmann point (“ideal glass transition’’) is identified as a self-dual critical point between crystalline and non-crystalline solid states, achieved at a critical value of geometrical frustration (e.g., in topologically close-packed crystals) or for an infinitely slow cooling rate in glass-forming liquids. The transport properties of the ground states depend intimately on the ratio of potential and kinetic energies in the relevant O(n) quantum rotor model. Using this topological viewpoint, the inverse thermal transport properties of crystalline and non-crystalline solid states (above approximately 50 K) are considered alongside the electrical transport properties of JJAs across the singularity at the superconductor-to-superinsulator transition.
Caroline Gorham received her Ph.D. in Materials Science and Engineering at Carnegie Mellon University in August 2018. Her primary research interests have focused on the applications of topology to understand structure and thermal transport properties in crystalline and non-crystalline solid state forms of condensed matter.