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Topological Characterization of Landau Levels for 2D Massless Dirac Fermions in 3D Layered Systems
Published in Chiun-Yan Lin, Ching-Hong Ho, Jhao-Ying Wu, Thi-Nga Do, Po-Hsin Shih, Shih-Yang Lin, Ming-Fa Lin, Diverse Quantization Phenomena in Layered Materials, 2019
Ching-Hong Ho, Jhao-Ying Wu, Chiun-Yan Lin, Thi-Nga Do, Po-Hsin Shih, Shih-Yang Lin, Ming-Fa Lin
In the paradigmatic grand unification theory, a given thermodynamic equilibrium state of a quantum field or matter in Universe is characterized by the symmetry group (denoted H) of, say, the Hamiltonian [9], which leads to the relevant equation for the considered system. Fundamentally, H is a subgroup of the largest symmetry group G of physical laws. Through cooling downward the ground state, H gets more and more reduced by successive spontaneous symmetry breaking processes. This fact puts the theoretical basis for classifying the states of physical systems in connection to phase transitions as described in, for example, the Landau's theory [10]. That is, as symmetry is reduced, particles in a system tend to organize themselves so that the state can transit into more ordered phases, where the phase transitions occur due to spontaneous symmetry breaking in company with changes of certain local order parameter. In this approach, it is possible that there can appear topological defects [11], such as solitons or domain walls, in real space of an ordered medium, like holes in the surface of an object. The determination of such phases is generally given by the nontrivial elements belonging to the homotopy group πn(H/G), which is defined on Sn, the n-sphere. Also, phase transitions of topological defects can occur with changing πn(H/G), as a result of spontaneous symmetry breaking.
Preface
Published in Liquid Crystals Reviews, 2023
Corrie T. Imrie, Randall D. Kamien, Oleg D. Lavrentovich
The difference between 360° (thick) and 180° (thin) lines manifested a deep connection between the type of ordering and the nature of defects in a medium, which eventually led to the homotopy classification of defects in ordered media by Kleman and Gérard Toulouse in 1976 [14]. To establish which defects are permissible in a medium, one needs to know the ‘degeneracy space’ of the order parameter, also called the ‘order parameter space’ and ‘ground state manifold’ and defined as the manifold of all possible values of the order parameter that do not alter the thermodynamic potentials of the system. For instance, in a recently discovered ferroelectric nematic, the degeneracy space is a sphere, each point of which corresponds to a possible orientation of the spontaneous electric polarization in space. In a conventional quadrupolar nematic, it is a sphere with antipodal points identified, because of the apolar character of the director. Generalizing the Burgers circuit in the construction of dislocations in solids to mappings of Burgers ‘enclosures’ around different defects in ordered media onto the degeneracy space, homotopy theory classifies various defects (such as points, lines, walls, and 3D textures) into separate classes that correspond to the elements of the homotopy groups of the ground state manifold. The stability of each class of defects, their behavior during merging, splitting, mutual entanglement, and crossing is described by topological charges, associated with the elements of these groups. The apolar character of the director in a conventional nematic permits topological stability of 180° disclinations, since a loop encircling such a disclination in real space is mapped onto the degeneracy space as a contour that connects diametrically antipodal points. Such a contour cannot be transformed into a point by any smooth transformation, guaranteeing the topological stability of the disclination and assigning it a nonzero topological charge. In contrast, there are no topologically stable disclination lines in a ferroelectric nematic, as a loop encircling a suspect line in real space is mapped onto a closed loop on the sphere of the ferroelectric degeneracy space; such a loop can always be shrunk into a point, which corresponds to a rearrangement of polarization into a uniformly aligned state. The homotopy classification uncovered a deep connection between liquid crystal defects to numerous topological formations in other types of condensed matter, such as solid crystals, superconductors and superfluids, ferromagnets and ferroelectrics, incompressible fluids, and objects such as magnetic monopoles, instantons, solitons, and cosmic strings.