Abstract
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 solidification. Global orientational order, achieved by spontaneous symmetry breaking, is prevented at finite 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 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 solidification, 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.
Highlights
The origins of the anomalous temperature dependence of the thermal conductivity of non-crystalline materials as compared with crystalline counterparts have remained a matter of great interest over the past century[1,2,3]
Complex 2−vector ordered systems: A most notable example of topological ordering in restricted dimensions is the case of two-dimensional classical Josephson junction arrays, for which conventional longrange order is prevented at finite temperatures (MerminWagner theorem)
Applying the topological framework developed the inverse temperature dependence of the thermal conductivity of crystalline and non-crystalline solid states may be viewed as a consequence of the realization of distinct low-temperature states of O(4) quantum rotor models. It follows that the thermal transport properties of solid states may be viewed as analogous to the well-studied electrical transport properties of charged O(2) Josephson junction arrays[16], which display a singularity at the superconductor-to-superinsulator transition[20] (Fig. 12 (A))
Summary
The origins of the anomalous temperature dependence of the thermal conductivity of non-crystalline materials as compared with crystalline counterparts have remained a matter of great interest over the past century[1,2,3]. Similar ideas[13], replacing disclinations with vortices (each are closed loop topological defects), have led to a theory of the transition to the low-temperature phasecoherent state in two-dimensional complex ordered systems (superfluids) This topological framework leads us to an understanding of the inverse behavior of the thermal transport properties of crystalline and non-crystalline solid states above approximately 50 K. Most notably, two-dimensional complex ordered systems (superfluids) are prevented from developing conventional long-range ordering (by spontaneous symmetry breaking) at finite temperatures[17] and are considered to exist in “restricted dimensions.” This is owing to the possible existence of an abundance of misorientational fluctuations that take the form of spontaneously generated topological point defects; such model systems demonstrate the properties of a topological transition (Berezinskii-Kosterlitz-Thouless13,18) towards a phase-coherent ground state. Just as conventional orientational order is prevented for complex ordered systems in R2, due to the existence of a gas of misorientational fluctuations that take the form of spontaneously generated topological point defects, so to is conventional orientational order prevented for quaternion ordered systems that exist in R4
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