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.
There are three kinds of solid states of matter that can exist in physical space: quasicrystalline (quasiperiodic), crystalline (periodic) and amorphous (aperiodic). Herein, we consider the degree of orientational order that develops upon the formation of a solid state to be characterized by the application of quaternion numbers. The formation of icosahedral quasicrystalline solids is considered alongside the development of bulk superfluidity, characterized by a complex order parameter, that occurs by spontaneous symmetry breaking in three-dimensions. Crystalline solid states are viewed as higher-dimensional analogues to phase-coherent topologically-ordered superfluid states of matter that develop in restricted dimensions (Hohenberg-Mermin- Wagner theorem). Lastly, amorphous solid states are viewed as dual to crystalline solids, in analogy to Mott-insulating states of matter that are dual to topologically-ordered superfluids.
In this article, the solidification phase transition between isotropic liquids and icosahedral quasicrystalline structures is classified by the application of a quaternion orientational order parameter. We suggest that the formation of the E8 lattice in 8-dimensions, from which three-dimensional icosahedral quasicrystals derive by projection, occurs without dimensionally restricting the degrees of freedom of the quaternion orientational order parameter to the quaternion plane. This is different from the development of regular crystalline solids in three-dimensions, which occurs in “restricted dimensions” and requires topological-ordering, as previously discussed by the authors. In other words, herein, the formation of lattices in eight-dimensions is considered as a higher-dimensional analogy to the formation of “bulk” superfluidity. It follows that, the solidification of lattices in 8-dimensions (and derivative icosahedral quasicrystalline solids) should occur without the necessity of topological defects.
Herein, fundamentals of topology and symmetry breaking are used to understand crystallization and geometrical frustration in topologically close-packed structures. This frames solidification from a new perspective that is unique from thermodynamic discussions. Crystallization is considered as developing from undercooled liquids in which orientational order is characterized by a surface of a sphere in four-dimensions (quaternion) with the binary polyhedral representation of the preferred orientational order of atomic clustering inscribed on its surface. As a consequence of the dimensionality of the quaternion orientational order parameter, crystallization is seen as occurring in a “restricted dimensions.” Homotopy theory is used to classify all topologically stable defects, and third homotopy group defect elements are considered to be generalized vortices that are available in superfluid ordered systems. This topological perspective approaches the liquid-to-crystal transition from the fundamental concepts of Bose-Einstein condensation, the Mermin-Wagner theorem and Berezinskii-Kosterlitz-Thouless (BKT) topological ordering transitions and thereby generalizes the concepts that apply to superfluidity in “restricted dimensions” to the formation of the solid state.
Recently, generalizations of quantum Hall effects (QHE) have been made from 2D to 4D and 8D by consider- ing their mathematical frameworks within complex (C), quaternion (H) and octonion (O) compact (gauge) Lie algebra domains. Just as QHE in two-dimensional electron gases can be understood in terms of Chern number topological invariants that belong to the first Chern class, QHE in 4D and 8D can be understood in terms of Chern number topological invariants that belong to the 2nd and 4th Chern classes. It has been shown that 2D QHE phenomena are related to topologically-ordered ground states of Josephson junction arrays (JJAs), which map onto an Abelian gauge theory with a periodic topological term that describes charge-vortex coupling. In these 2D JJAs, magnetic point defects and Cooper pair electric charges are dual to one another via electric-magnetic duality (Montonen-Olive). This leads to a quantum phase transition between phase-coherent superconductor and dual phase-incoherent superinsulator ground states, at a “self-dual” critical point. In this article, a framework for topological-ordering of Bose-Einstein condensates is extended to consider four-dimensional quaternion ordered systems that are related to 4D QHE. This is accomplished with the incorporation of a non-Abelian topological term that describes coupling between third homotopy group point defects (as generalized magnetic vortices) and Cooper pair-like charges. Point defects belonging to the third homotopy group are dual to charge excitations, and this leads to the manifestation of a quantum phase transition between orientationally-ordered and orientationally-disordered ground states at a “self-dual” critical point. The frustrated ground state in the vicinity of this “self-dual” critical point, are characterized by global topological invariants belonging to the 2nd Chern class.
Topological origins of the thermal transport properties of crystalline and non-crystalline solid states are considered herein, by the adoption of a quaternion orientational order parameter to describe solidication. Global orientational order, achieved by spontaneous symmetry breaking (SSB), is prevented at nite temperatures for systems that exist in restricted dimensions (Mermin-Wagner theorem). Just as complex ordered systems exist in restricted dimensions in 2D and 1D, owing to the dimensionality of the order parameter, quaternion ordered systems in 4D and 3D exist in restricted dimensions. Just below the melting temperature, misorientational fluctuations in the form of spontaneously generated topological defects prevent the development of the solid state. Such solidifying systems are well-described using O(4) quantum rotor models, and a defect-driven Berezinskii-Kosterlitz-Thouless (BKT) transition is anticipated to separate an undercooled fluid from a crystalline solid state. In restricted dimensions, in addition to orientationally-ordered ground states, orientationally-disordered ground states may be realized by tuning a non-thermal parameter in the relevant O(n) quantum rotor model Hamiltonian. Thus, glassy solid states are anticipated to exist as distinct ground states of O(4) quantum rotor models. Within this topological framework for solidication, the finite Kauzmann temperature marks a first-order transition between crystalline and glassy solid states at a “self-dual” critical point that belongs to O(4) quantum rotor models. This transition is a higher-dimensional analogue to the quantum phase transition that belongs to O(2) Josephson junction arrays (JJAs). The thermal transport properties of crystalline and glassy solid states, above approximately 50 K, are considered alongside the electrical transport properties of JJAs across the superconductor-to-superinsulator transition.
The work presented in this thesis is a topological approach for understanding the formation of structures from the liquid state. The strong difference 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 identified 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 fist 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 phase-coherent 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 fluctuations 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 finite kinetic energy, 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.
We employ the theory of topological phase transitions, of the Berezinskiĭ-Kosterlitz-Thouless (BKT) type, in order to investigate orientational ordering in four spatial dimensions that is characterized by a quaternion n−vector (i.e., n = 4) order parameter. Due to the dimensionality of the quaternion n−vector order parameter, the development of orientational order for systems that exist in four spatial dimensions must be viewed within the context of ordering in restricted dimensions. At finite temperatures, despite the development of a well-defined amplitude of the n−vector order parameter within separate regions, ordered systems that exist in restricted dimensions are prevented from developing global orientational phase-coherence as a consequence of misorientational fluctuations throughout the system. In four dimensions, this gas of misorientational fluctuations takes the form of spontaneously generated topological point defects that belong to the third homotopy group. These topological point defects must become topologically ordered in order to obtain a ground state of aligned order parameters at zero Kelvin; we argue that this topological transition belongs to the BKT universality class. We use standard ‘Metropolis’ Monte Carlo simulations to estimate the thermodynamic response functions, susceptibility and heat capacity, in the vicinity of the transition towards the ground state of perfectly aligned order parameters in our four-dimensional quaternion n−vector ordered model system. On lowering the temperature below a critical value, we identify a transition that results from the minimization of misorientations in the scalar phase angles throughout the system. The thermodynamic response functions obtained by Monte Carlo simulations show characteristic behavior of a topological ordering phase transition.