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.
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.
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.