Quaternionic unitary group symmetry breaking in condensed matter: A deeper look at solidification and resulting thermophysical properties

The nature of glass and the glass transition are widely thought to be two of the most challenging problems in condensed matter physics. Despite their ubiquity, a fundamental understanding of amorphous solids and the glass transition has yet to be obtained – in contrast to our extensive knowledge of crystalline solids. Making use of a four-dimensional quaternion orientational order parameter, theories of solidification of crystalline and glassy solid states may be unified about the “ideal glass transition” positioned as a quantum phase transition.

Owing to the dimensionality of the quaternion orientational order parameter, concepts of the Mermin-Wagner theorem and Berezinskii-Kosterlitz-Thouless (BKT) topological-ordering transitions may be extended to describe solidification in three-dimensions - where there is no possibility for conventional long-range orientational order to develop at the melting temperature.

It follows that the development of crystalline solids occurs by a defect-binding transition within a plasma of spontaneously-generated misorientational fluctuations that coexists with atomic clusters just below the melting temperature. In the presence of geometrical frustration, e.g., in the case of frustrated short-range icosahedral orientational order, excess signed topological defects exist that are unable to form bound pairs. These disclinations impart negative curvature, such that the overall space is flat, and persist to the ground state in a periodic manner (e.g., Frank-Kasper structures that express short-range icosahedral order) which satisfies the third law of thermodynamics.

With sufficient geometrical frustration, defect-driven crystallization is entirely suppressed at the “ideal glass transition,” which marks a self-dual critical point in-between crystalline and glassy ground states. Glass formation may be understood in dual terms, driven by thermal pinning of atomic clusters. Low-temperature thermal properties of glasses are a consequence of this dual-ordering, that is well-described within the framework of a quaternion orientational order parameter in “restricted dimensions.”