This thesis is a study of three categories of problems in fermionic systems for which topology plays an important role: (i) The properties of zero modes arising in systems of fermions interacting with a bosonic background, with a special focus on Majorana modes arising in the superconductor state. We propose a method for counting Majorana modes and we study a mechanism for controlling their number parity in lattice systems, two questions that are of relevance to the protection of quantum bits. (ii) The study of dispersionless bands in two dimensions as a platform for correlated physics, where it is shown the possibility of stabilizing the fractional quantum Hall effect in a flat band with Chern number. (iii) The extension of the hierarchy of quantum Hall fluids to the case of time-reversal symmetric incompressible ground states describing a phase of strongly interacting topological insulators in two dimensions. / Physics

<p>The capability of manipulation of the flow of mechanical energy in the form of mechanical waves (including acoustic and elastic waves) has always been a challenge and a critical part in various areas of engineering. The recent advances in topological acoustic/elastic metamaterials certainly open a new pathway to the manipulation of mechanical waves, especially for the novel scattering-immune wave-guiding capability, even in the presence of defects, disorders or sharp bends along the waveguide. In this Dissertation, the theoretical background and experimental evidence of various types of elastic-wave topological metamaterials including analogues to 2D quantum valley Hall effect (QVHE) materials, 2D quantum spin Hall effect (QSHE) topological insulators are presented. First, the formulation the elastic-wave analogue to QVHE materials in a general continuous elastic phononic structure (not limited to local resonant lattices, filling the gap in the literature) is proposed, and a strategy using pressurized cells to actively control the phononic lattice is presented. By finite prestrain and geometric nonlinear effect, the space inversion symmetry of the original hexagonal lattice is broken, resulting in distinct QVHE phases (characterized by valley Chern numbers) in lattice domains with opposite pressurization. With such mechanism, the edge-state path, i.e., the domain wall connecting lattices with distinct QVHE phases, can be real-time configured. Further more, edge states with tunable frequency-wavenumber dispersion can be created at the external boundaries of the lattice by appropriate pressurization of the outermost cells. An aluminum reticular sheet built with water-jet cutting is machined in the pre-deformed pattern with a Z-shape domain wall at the center, which spatially divides the sheet into two domains with opposite QVHE phases. Using piezoelectric transducers and laser Doppler vibrometry, the measured harmonic and transient responses confirm the back-scattering-immunity of the topological edge states, and the frequency-wavenumber dispersion matches the numerical prediction. A strategy is proposed for unidirectionally generating edge states along the domain wall using two off-phase transducers, which is also experimentally demonstrated. For elastic-wave analogue to QSHE topological insulators, we focus on the ``zone-folding'' method and propose a honeycomb 2D elastic beam network with periodically altered thickness with a generalized Kekule distortion pattern. Such framework provides a parametric space with exhaustive control in the topological phase diagram of waves in the lattice compared to earlier works in the literature. The effective Hamiltonian as well as the characterized topological phase are gauge dependent, particularly they change with different reference frames. This lead to ambiguity in the topological phase of such phononic crystal. Based on this argument, it is predicted that edge states could exist at a dislocation interface connecting two piece of phononic structures of the same pattern with relative displacement. Following the same idea, but considering the available fabrication options, a phononic plate with honeycomb groove pattern engraved on both sides is built, which the depth varied according to the Kekule pattern. With proper tuning of the parameters, it realizes an analogue to the QSHE topological insulator. With <em>ab initio</em> calculation of the Berry curvature (without involving any approximations such as the perturbative approach), a new topological invariant <em>local topological charge</em> is defined and evaluated as the counterpart of the Z₂ invariant in the classical-wave-zone-folding analogue. The local topological charge has intrinsic ambiguity and its value depends on the selected reference frame. However, its <em>change </em>according to changes in the parameters, under a consistent reference frame, is well-defined. Given the fact that shifting the reference frame by certain fractions of a lattice constant was equivalent to changing one of the parameters by a certain amount, it also lead to a well-defined change in the local topological charge, which indicates topological phase transition, and one can predict the existence of edge states at the displacement-dislocation interface between two neighboring lattices having the same pattern up to a rigid-body shifting. The phononic plate is machined by a CNC mill, and the experiment is carried out using piezoelectric transducers and laser Doppler vibrometry, which confirms the existence and robustness of the topological edge states at such dislocation interface connecting identical pattern, which was unprecedented in both quantum and classical systems. The final part of this Dissertation focuses on creating classical mechanical analogues to the 1D Kitaev superconducting model and Majorana-like bound states aimed at future acoustic-wave based computation.</p>

The study of topological phases in condensed matter physics has seen remarkable advancements, primarily focusing on systems with a well-defined bulk and boundary. However, the emergence of topological phenomena on self-similar systems, characterized by the absence of a clear distinction between bulk and boundary, presents a fascinating challenge. This thesis focuses on the topological phases on self-similar systems, shedding light on their unique properties through the lens of adiabatic charge pumping. We observe that the spectral flow in these systems exhibits striking qualitative distinctions from that of translationally invariant non-interacting systems subjected to a perpendicular magnetic field. We show that the instantaneous eigenspectra can be used to understand the quantization of the charge pumped over a cycle, and hence to understand the topological character of the system. Furthermore, we establish a correspondence between the local contributions to the Hall conductivity and the spectral flow of edge-like states. We also find that the edge-like states can be approximated as eigenstates of the discrete angular-momentum operator, with their chiral characteristics stemming from this unique perspective. We also investigate the effect of local structure on the topological phases on self-similar structures embedded in two dimensions. We study a geometry dependent model on two self-similar structures having different coordination numbers, constructed from the Sierpinski gasket. For different non-spatial symmetries present in the system, we numerically study and compare the phases on both structures. We characterize these phases by the localization properties of the single-particle states, their robustness to disorder, and by using a real-space topological index. We find that both structures host topologically nontrivial phases and the phase diagrams are different on the two structures, emphasizing the interplay between non-spatial symmetries and the local structure of the self-similar unit in determining topological phases. Furthermore, we demonstrate the presence of topologically ordered chiral spin liquid on fractals by extending the Kitaev model to the Sierpinski Gasket. We show a way to perform the Jordan-Wigner transformation to make this model exactly solvable on the Sierpinski Gasket. This system exhibits a fractal density of states for Majorana modes and showcases a transition from a gapped to a gapless phase. Notably, the gapped phase features symmetry-protected Majorana corner modes, while the gapless phase harbors robust zero-energy and low-energy self-similar Majorana edge-like modes. We also study the vortex excitations, characterized by remarkable localization properties even in small fractal generations. These localized excitations exhibit anyonic behavior, with preliminary calculations hinting at their fundamental differences from Ising anyons observed in the Kitaev model on a honeycomb lattice.

Na busca de um melhor entendimento das propriedades eletrônicas e magnéticas dos isolantes topológicos nos deparamos com uma das suas caraterísticas mais marcantes, a existência de estados de superfície metálicos com textura helicoidal de spin os quais são protegidos de impurezas não magnéticas. Na superfície estes canais de spin possuem um potencial enorme para aplicações em dispositivos spintrônicos. Muito há para se fazer e o tratamento via cálculos de primeiros princípios por simulações permite um caráter preditivo que corrobora na elucidação de fenômenos físicos via análises experimentais. Nesse trabalho analisamos as propriedades eletrônicas de isolantes topológicos tais como: (Bi,Sb)$_2$(Te,Se)$_3$, Germaneno e Germaneno funcionalizado. Cálculos baseados em DFT evidenciam a importância das separações entre as camadas de Van der Waals nos materiais Bi$_2$Se$_3$ e Bi$_2$Te$_3$. Mostramos que devido a falhas de empilhamento, pequenas oscilações no eixo de QLs (\\textit{Quintuple Layers}) podem gerar um desacoplamento dos cones de Dirac, além de criar estados metálicos na fase \\textit{bulk} de Bi$_2$Te$_3$. Em se tratando do Bi$_2$Se$_3$ um estudo sistemático dos efeitos de impurezas de metais de transição foi realizado. Observamos que há quebra de degenerescência do cone de Dirac se houver magnetização em quaisquer dos eixos. Além disso se a magnetização permanecer no plano, além de uma pequena quebra de degenerescência, há um deslocamento do mesmo para outro ponto da rede recíproca. No entanto, se a magnetização apontar para fora do plano a quebra ocorre no próprio ponto $\\Gamma$, porém de maneira mais intensa. Importante enfatizar que além de mapear os sítios com suas orientações magnéticas de menor energia observamos que a quebra da degenerescência está diretamente relacionada com a geometria local da impureza. Isso proporciona imagens de STM distintas para cada sítio possível, permitindo que um experimental localize cada situação no laboratório. Estudamos ainda a transição topológica na liga (Bi$_x$Sb$_{1-x}$)$_2$Se$_3$, onde identificamos um isolante trivial e topológico para $x=0$ e $x=1$. Apesar de óbvia a existência de tal transição, detalhes importantes ainda não estão esclarecidos. Concluímos que a dopagem com impurezas não magnéticas proporciona uma boa técnica para manipulação e engenharia de cone nesta família de materiais, de forma que dependendo da faixa de dopagem podemos eliminar a condutividade que advém do \\textit{bulk}. Finalmente estudamos superfícies de Germaneno e Germaneno funcionalizado com halogênios. Usando uma funcionalização assimétrica e com a avalição do invariante topológico $Z_2$ notamos que o material Ge-I-H é um isolante topológico podendo ser aplicado na elaboração de dispositivos baseados em spin. / In the search of a better understanding of the electronic and magnetic properties of topological insulators we are faced with one of its most striking features, the existence of metallic surface states with helical spin texture which are protected from non-magnetic impurities. On the surface these spin channels allows a huge potential for applications in spintronic devices. There is much to do and treating calculations via \\textit{Ab initio} simulations allows us a predictive character that corroborates the elucidation of physical phenomena through experimental analysis. In this work we analyze the electronic properties of topological insulators such as: (Bi, Sb)$_2$(Te, Se)$_3$, Germanene and functionalized Germanene. Calculations based on DFT show the importance of the separation from interlayers of Van der Waals in materials like Bi$_2$Se$_3$ and Bi$_2$Te$_3$. We show that due to stacking faults, small oscillations in the QLs axis (\\textit{Quintuple Layers}) can generate a decoupling of the Dirac cones and create metal states in the bulk phase Bi$_2$Te$_3$. Regarding the Bi$_2$Se$_3$ a systematic study of the effects of transition metal impurities was performed. We observed that there is a degeneracy lift of the Dirac cone if there is any magnetization on any axis. If the magnetization remains in plane, we observe a small shift to another reciprocal lattice point. However, if the magnetization is pointing out of the plane a lifting in energy occurs at the very $ \\Gamma $ point, but in a more intense way. It is important to emphasize that in addition to mapping the sites with their magnetic orientations of lower energy we saw that the lifting in energy is directly related to the local geometry of the impurity. This provides distinct STM images for each possible site, allowing an experimental to locate each situation in the laboratory. We also studied the topological transition in the alloy (Bi$_x$Sb$_{1-x}$)$_ 2$Se$_3$, where we identify a trivial and topological insulator for $x = 0$ and $x = 1$. Despite the obvious existence of such a transition, important details remain unclear. We conclude that doping with non-magnetic impurities provides a good technique for handling and cone engineering this family of materials so that depending on the range of doping we can eliminate conductivity channels coming from the bulk. Finally we studied a Germanene and functionalized Germanene with halogens. Using an asymmetrical functionalization and with the topological invariant $Z_2$ we noted that the Ge-I-H system is a topological insulator that could be applied in the development of spin-based devices.

Na busca de um melhor entendimento das propriedades eletrônicas e magnéticas dos isolantes topológicos nos deparamos com uma das suas caraterísticas mais marcantes, a existência de estados de superfície metálicos com textura helicoidal de spin os quais são protegidos de impurezas não magnéticas. Na superfície estes canais de spin possuem um potencial enorme para aplicações em dispositivos spintrônicos. Muito há para se fazer e o tratamento via cálculos de primeiros princípios por simulações permite um caráter preditivo que corrobora na elucidação de fenômenos físicos via análises experimentais. Nesse trabalho analisamos as propriedades eletrônicas de isolantes topológicos tais como: (Bi,Sb)$_2$(Te,Se)$_3$, Germaneno e Germaneno funcionalizado. Cálculos baseados em DFT evidenciam a importância das separações entre as camadas de Van der Waals nos materiais Bi$_2$Se$_3$ e Bi$_2$Te$_3$. Mostramos que devido a falhas de empilhamento, pequenas oscilações no eixo de QLs (\\textit{Quintuple Layers}) podem gerar um desacoplamento dos cones de Dirac, além de criar estados metálicos na fase \\textit{bulk} de Bi$_2$Te$_3$. Em se tratando do Bi$_2$Se$_3$ um estudo sistemático dos efeitos de impurezas de metais de transição foi realizado. Observamos que há quebra de degenerescência do cone de Dirac se houver magnetização em quaisquer dos eixos. Além disso se a magnetização permanecer no plano, além de uma pequena quebra de degenerescência, há um deslocamento do mesmo para outro ponto da rede recíproca. No entanto, se a magnetização apontar para fora do plano a quebra ocorre no próprio ponto $\\Gamma$, porém de maneira mais intensa. Importante enfatizar que além de mapear os sítios com suas orientações magnéticas de menor energia observamos que a quebra da degenerescência está diretamente relacionada com a geometria local da impureza. Isso proporciona imagens de STM distintas para cada sítio possível, permitindo que um experimental localize cada situação no laboratório. Estudamos ainda a transição topológica na liga (Bi$_x$Sb$_{1-x}$)$_2$Se$_3$, onde identificamos um isolante trivial e topológico para $x=0$ e $x=1$. Apesar de óbvia a existência de tal transição, detalhes importantes ainda não estão esclarecidos. Concluímos que a dopagem com impurezas não magnéticas proporciona uma boa técnica para manipulação e engenharia de cone nesta família de materiais, de forma que dependendo da faixa de dopagem podemos eliminar a condutividade que advém do \\textit{bulk}. Finalmente estudamos superfícies de Germaneno e Germaneno funcionalizado com halogênios. Usando uma funcionalização assimétrica e com a avalição do invariante topológico $Z_2$ notamos que o material Ge-I-H é um isolante topológico podendo ser aplicado na elaboração de dispositivos baseados em spin. / In the search of a better understanding of the electronic and magnetic properties of topological insulators we are faced with one of its most striking features, the existence of metallic surface states with helical spin texture which are protected from non-magnetic impurities. On the surface these spin channels allows a huge potential for applications in spintronic devices. There is much to do and treating calculations via \\textit{Ab initio} simulations allows us a predictive character that corroborates the elucidation of physical phenomena through experimental analysis. In this work we analyze the electronic properties of topological insulators such as: (Bi, Sb)$_2$(Te, Se)$_3$, Germanene and functionalized Germanene. Calculations based on DFT show the importance of the separation from interlayers of Van der Waals in materials like Bi$_2$Se$_3$ and Bi$_2$Te$_3$. We show that due to stacking faults, small oscillations in the QLs axis (\\textit{Quintuple Layers}) can generate a decoupling of the Dirac cones and create metal states in the bulk phase Bi$_2$Te$_3$. Regarding the Bi$_2$Se$_3$ a systematic study of the effects of transition metal impurities was performed. We observed that there is a degeneracy lift of the Dirac cone if there is any magnetization on any axis. If the magnetization remains in plane, we observe a small shift to another reciprocal lattice point. However, if the magnetization is pointing out of the plane a lifting in energy occurs at the very $ \\Gamma $ point, but in a more intense way. It is important to emphasize that in addition to mapping the sites with their magnetic orientations of lower energy we saw that the lifting in energy is directly related to the local geometry of the impurity. This provides distinct STM images for each possible site, allowing an experimental to locate each situation in the laboratory. We also studied the topological transition in the alloy (Bi$_x$Sb$_{1-x}$)$_ 2$Se$_3$, where we identify a trivial and topological insulator for $x = 0$ and $x = 1$. Despite the obvious existence of such a transition, important details remain unclear. We conclude that doping with non-magnetic impurities provides a good technique for handling and cone engineering this family of materials so that depending on the range of doping we can eliminate conductivity channels coming from the bulk. Finally we studied a Germanene and functionalized Germanene with halogens. Using an asymmetrical functionalization and with the topological invariant $Z_2$ we noted that the Ge-I-H system is a topological insulator that could be applied in the development of spin-based devices.