Quantum Hall effect and the relative index for projections

We define the relative index, Index(P,Q), for a pair of infinite-dimensional projections on a Hilbert space to be the integer that is the natural generalization of dim(P)-dim(Q) in finite-dimensional vector spaces. We show that the Hall conductance for independent electrons in the plane is the relative index where P and Q project on the states below the Fermi energy for Hamiltonians that differ by a quantum flux and the Fermi energy is appropriately placed. This approach is closely related to, and sheds light on, Bellissard’s interpretation of the Hall conductance as an index

Charge Deficiency, Charge Transport and Comparison of Dimensions

We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.Comment: 23 pages, no figure

Topological Phases near a Triple Degeneracy

We study the pattern of three state topological phases that appear in systems with real Hamiltonians and wave functions. We give a simple geometric construction for representing these phases. We then apply our results to understand previous work on three state phases. We point out that the ``mirror symmetry'' of wave functions noticed in microwave experiments can be simply understood in our framework.Comment: 4 pages, 1 figure, to appear in Phys. Rev. Let

Modelagem e controle de veículos aéreos híbridos com rotores inclináveis e câmera orientável

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia de Automação e Sistemas, Florianópolis, 2017.Este trabalho apresenta um estudo sobre a utilização de veículos aéreos não tripulados (VANTs) em sistemas de vigilância. O uso de VANTs tem se mostrado eficiente em substituir métodos tradicionais de trabalho, muitas vezes reduzindo custos de operação, riscos à vida, ou mesmo possibilitando atividades antes inviáveis. É realizada a modelagem do quadrirotor convencional e de dois veículos aéreos híbridos (com dois e quatro rotores inclináveis), os quais possuem características construtivas e operacionais desejáveis do ponto de vista da aplicação. Os veículos carregam câmeras acopladas às suas estruturas através de um mecanismo gimbal que lhes proporciona mobilidade adicional para o apontamento e captura de imagens de alvos que se movem em solo. Os modelos são obtidos utilizando a formulação de Euler-Lagrange, considerando tanto a dinâmica do veículo e seus rotores inclináveis quanto a de um gimbal duplo na configuração Az-El, através do qual a câmera é integrada. Para um dos veículos, o VANT híbrido com dois rotores inclináveis, uma lei de controle não linear é empregada, visando o rastreamento de trajetórias relativas à posição e direcionamento do VANT, além de capacitar o sistema a manter um alvo na linha de visão da câmera. A lei de controle consiste em uma estratégia em cascata, onde na malha mais interna é tratada a dinâmica de inclinação dos rotores, seguida de uma malha de controle das variáveis relativas à rotação do VANT e apontamento da câmera, sendo que em ambas é executada uma linearização por realimentação estática de estados, finalizando com o controle do subsistema de translação na malha externa. Verifica-se, perante simulação, o rastreamento das trajetórias desejadas mesmo na presença de distúrbios externos.Abstract : This work presents a study about the use of unmanned aerial vehicles (UAVs) in surveillance systems. The use of UAVs has been shown to be efficient in replacing traditional work methods, often reducing operating costs, risks to life or even making possible formerly impracticable activities. It is performed the modeling of the conventional quadrotor and two hybrid aerial vehicles (with two and four tiltable rotors), which have desirable constructive and operational characteristics from the point of view of the application. The vehicles carry cameras coupled to their structures through a mechanism (gimbal) that provides them additional mobility for pointing and capture images of targets that move on the ground. The models are obtained using the Euler-Lagrange formulation, considering both the vehicle dynamics and its tiltable rotors as well as the double gimbal in the Az-El configuration, through which the camera is integrated. For one of the vehicles, the hybrid UAV with two tiltable rotors, a nonlinear control law is employed, aiming the trajectories tracking related to the UAV position and heading, besides to empower the system to maintain a target in the camera's line of sight. The control law consists of a cascade strategy, where in the innermost loop the rotors tilt dynamics is treated, followed by a control loop of the variables related to the UAV rotation and camera pointing, in which both execute a static feedback linearization, finalizing with the control of the translation subsystem in the external loop. It is verified, through simulation, the tracking of the desired trajectories even in the presence of external disturbances