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[en] The quantization procedure is a necessary tool for a proper understanding of many interesting quantum phenomena in modern physics. In this note, we focus on geometrical framework for such procedures, particularly the group-theoretic approach and their difficulties. Finally we look through the example of Hall effect as a quantized macroscopic phenomenon with group-theoretic quantization approach. (author)
[en] The regularity properties of the integrated density of states and the state density of the Anderson bidimensional tight-binding model, in the presence of a uniform magnetic field, perpendicular to the plane of the system by means of quantum flux with plaques, are studied. (A.C.A.S.)
[pt]Estudam-se propriedades de regularidade da densidade integrada de estados (D.I.E.) e da densidade de estados (D.E.) do modelo tight-binding de Anderson bi-dimensional na presenca de um campo magnetico uniforme perpendicular ao plano do sistema, com meio quantum de fluxo por plaqueta. (A.C.A.S.)
[en] A Hall resistivity formula for the 2DES in graphene is derived from the zero-mass Dirac field model adopting the electron reservoir hypothesis. The formula reproduces perfectly the experimental resistivity data [K.S. Novoselov, et al., Nature 438 (2005) 201]. This perfect agreement cannot be achieved by any other existing models. The electron reservoir is shown to be the 2DES itself. -- Highlights: ► Quantum Hall resistivity formula is derived from the zero-mass Dirac model. ► The formula agrees with the graphene experiment perfectly. ► No existing theories can explain the experiment quantitatively. ► The electron reservoir hypothesis is adopted. ► Mechanism of the electron reservoir is clarified for the first time.
[en] It is shown that a large class of two dimensional Superconductor to Insulator (SC-I), and (Quantum Hall to Insulator (QH-I) transitions can be understood by assuming that the thermodynamic transition in the clean system is first order. The finite correlation lengths at that transition yield a natural separation of the disorder into short and long wavelengths which are then straightforward to incorporate perturbatively and semi classically respectively. This approach reduces problems of disorder+interactions to puddle network models, whose studies have already yielded insight into experiments of QH-I and SC-I. For the CQH-I, the difference between Landauer-Buttiker and Boltzman theories highlights effects of dephasing
[en] Experimental discovery of a quantized Hall state at 5/2 filling factor presented an enigmatic finding in an established field of study that has remained an open issue for more than twenty years. In this review we first examine the experimental requirements for observing this state and outline the initial theoretical implications and predictions. We will then follow the chronology of experimental studies over the years and present the theoretical developments as they pertain to experiments, directed at sets of issues. These topics will include theoretical and experimental examination of the spin properties at 5/2; is the state spin polarized? What properties of the higher Landau levels promote development of the 5/2 state, what other correlation effects are observed there, and what are their interactions with the 5/2 state? The 5/2 state is not a robust example of the fractional quantum Hall effect: what experimental and material developments have allowed enhancement of the effect? Theoretical developments from initial pictures have promoted the possibility that 5/2 excitations are exceptional; do they obey non-abelian statistics? The proposed experiments to determine this and their executions in various forms will be presented: this is the heart of this review. Experimental examination of the 5/2 excitations through interference measurements will be reviewed in some detail, focusing on recent results that demonstrate consistency with the picture of non-abelian charges. The implications of this in the more general physics picture is that the 5/2 excitations, shown to be non-abelian, should exhibit the properties of Majorana operators. This will be the topic of the last review section. (review article)
[en] Topological states of quantum matter have been investigated intensively in recent years in materials science and condensed matter physics. The field developed explosively largely because of the precise theoretical predictions, well-controlled materials processing, and novel characterization techniques. In this Perspective, we review recent progress in topological insulators, the quantum anomalous Hall effect, chiral topological superconductors, helical topological superconductors and Weyl semimetals.