On the formation of crystalline and non-crystalline solid states and their thermal transport properties: A topological perspective via a quaternion orientational order parameter

The work presented in this thesis is a topological approach for understanding the formation of
structures from the liquid state. The strong dierence in the thermal transport properties of noncrystalline
solid states as compared to crystalline counterparts is considered within this topological
framework. Herein, orientational order in undercooled atomic liquids, and derivative solid states, is
identied with a quaternion order parameter.
In light of the four-dimensional nature of quaternion numbers, spontaneous symmetry breaking
from a symmetric high-temperature phase to a low-temperature phase that is globally orientationally
ordered by a quaternion order parameter is forbidden in three- and four-dimensions. This is a
higher-dimensional realization of the Mermin-Wagner theorem, which states that continuous symmetries
cannot be spontaneously broken at nite temperatures in two- and one-dimensions.
Understanding the possible low-temperature ordered states that may exist in these scenarios (of
restricted dimensions) has remained an important problem in condensed matter physics. In approaching
a topological description of solidication in three-dimensions, as characterized by a quaternion
orientational order parameter, it is instructive to rst consider the process of quaternion orientational
ordering in four-dimensions. This 4D system is a direct higher-dimensional analogue to planar models
of complex n􀀀vector (n = 2) ordered systems, known as Josephson junction arrays.
Just as Josephson junction arrays may be described mathematically using a lattice quantum rotor
model with O(2) symmetry, so too can 4D quaternion n􀀀vector (n = 4) ordered systems be modeled
using a lattice quantum rotor model with O(4) symmetry. O(n) quantum rotor models (that
apply to n􀀀vector ordered systems that exist in restricted dimensions) include kinetic and potential
energy terms. It is the inclusion of the kinetic energy term that leads to the possible realization
of two distinct ground states, because the potential and kinetic energy terms cannot be minimized
simultaneously.
The potential energy term is minimized by the total alignment of O(n) rotors in the ground state,
such that it is perfectly orientationally ordered and free of topological defects. On the other hand, minimization of the kinetic energy term favors a low-temperature state in which rotors throughout
the system are maximally orientationally disordered.
In four-dimensions, the O(4) quantum rotor model may be used to describe a 4D plastic crystal
that forms below the melting temperature. A plastic crystal is a mesomorphic state of matter between
the liquid and solid states. The realization of distinct low-temperature states in four-dimensions, that
are orientationally-ordered and orientationally-disordered, is compared with the realization of phasecoherent
and phase-incoherent low-temperature states of O(2) Josephson junction arrays. Such planar
arrays have been studied extensively as systems that demonstrate a topological ordering transition, of
the Berezinskii-Kosterlitz-Thouless (BKT) type, that allows for the development of a low-temperature
phase-coherent state.
In O(2) Josephson junction arrays, this topological ordering transition occurs within a gas of
misorientational
uctuations in the form of topological point defects that belong to the fundamental
homotopy group of the complex order parameter manifold (S1). In this thesis, the role that an analogous
topological ordering transition of third homotopy group point defects in a four-dimensional
O(4) quantum rotor model plays in solidication is investigated. Numerical Monte-Carlo simulations,
of the four-dimensional O(4) quantum rotor model, provide evidence for the existence of this novel
topological ordering transition of third homotopy group point defects.
A non-thermal transition between crystalline and non-crystalline solid ground states is considered
to exist as the ratio of importance of kinetic and potential energy terms of the O(4) Hamiltonian
is varied. In the range of dominant potential energy, with nite kinetic energy eects, topologically
close-packed crystalline phases develop for which geometrical frustration forces a periodic arrangement
of topological defects into the ground state (major skeleton network). In contrast, in the range
of dominant kinetic energy, orientational disorder is frozen in at the glass transition temperature
such that frustration induced topological defects are not well-ordered in the solid state.
Ultimately, the inverse temperature dependence of the thermal conductivity of crystalline and
non-crystalline solid states that form from the undercooled atomic liquid is considered to be a consequence
of the existence of a singularity at the point at which the potential and kinetic energy
terms become comparable. This material transport property is viewed in analogue to the electrical
transport properties of charged O(2) Josephson junction arrays, which likewise exhibit a singularity
at a non-thermal phase transition between phase-coherent and phase-incoherent ground states.