Topological-Order Beyond 2D U(1) Systems: A Novel Perspective on Solidification and Curvature in Crystals and Glassy Solids

This work illuminates the importance of topology in providing a description of the formation and structure of solid states, with SO(3) symmetries, beyond perfect crystals. We view crystallization as a “flattening transition,” in which complementary curvature-carrying topological defects (disclinations and 3^rd homotopy group) bind in a 4D XYZW model – via a Kosterlitz-Thouless mechanism. This minimizes misorientational fluctuations of the relevant SU(2) quaternion orientational order parameter in the crystalline solid state. The ordered network of permanent defects (e.g., major skeleton) that stabilizes topologically-close-packed crystals, with icosahedral short-range orientational order that is geometrically-frustrated from filling space, is viewed as an uncharged analogue to the Abrikosov flux lattice that characterizes the ground state of Josephson junction arrays in the presence of a frustrating applied transverse magnetic field.


We suggest that the existence of glassy solids is an artifact of quaternion orientational order parameter undergoing a phase transition in the quaternion plane (4D/3D+1t). In these “restricted dimensions,” the phase-amplitude uncertainty principle that applies to the quaternion order parameter plays an interesting role in allowing for a rich spectra of ground states accessed by either the minimization of phase-angle uncertainty or amplitude uncertainty throughout the system. Whereas orientational-order in crystals is facilitated by minimization of phase-angle uncertainty, through the binding of topological defects, atomic components in glassy solids are frozen-in – minimizing amplitude uncertainty – and the resulting solid is orientationally-disordered as a consequence. Hence, within this topological framework, the “ideal glass transition" (that occurs at the finite Kauzmann temperature) is considered to be a higher-dimensional analogue to the 2D/1D superfluid-to-Mott insulator quantum phase transition.