  In this article, a theoretical description of the "ideal glass transition" is
approached upon the adoption of a quaternion orientational order parameter.
Unlike first-order phase transitions of liquids into crystalline solid states,
glass transitions are entirely different phenomena that are non-equilibrium and
that are highly dependent on the applied cooling rate. Herein, the "ideal glass
transition" that occurs at the finite Kauzmann temperature at which point the
configurational entropy of an undercooled liquid matches that of its
crystalline counterpart is identified as a first-order quantum critical point.
We suggest that this quantum critical point belongs to quaternion ordered
systems that exist in four- and three-dimensions. The Kauzmann quantum critical
point is considered to be a higher-dimensional analogue to the
superfluid-to-Mott insulator quantum phase transition, in two- and
one-dimensional complex ordered systems. Such quantum critical points are
driven by tuning a non-thermal frustration parameter, and result due to
characteristic softening of the 'Higgs' type mode that corresponds to amplitude
fluctuations of the order parameter. The first-order nature of the finite
temperature Kauzmann quantum critical point is seen as a consequence of the
discrete change of the topology of the ground state manifold that applies to
crystalline and non-crystalline solid states.
