# 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

identied 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 solidication 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 nvector (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 nvector (n = 4) ordered systems be modeled

using a lattice quantum rotor model with O(4) symmetry. O(n) quantum rotor models (that

apply to nvector 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 solidication 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.

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

identied 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 solidication 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 nvector (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 nvector (n = 4) ordered systems be modeled

using a lattice quantum rotor model with O(4) symmetry. O(n) quantum rotor models (that

apply to nvector 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 solidication 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.

## History

## Date

2018-08-21## Degree Type

- Dissertation

## Department

- Materials Science and Engineering

## Degree Name

- Doctor of Philosophy (PhD)