The subject of the present thesis are some aspects of impurities affecting mesoscopic systems with regard to their topological properties and related effects like Majorana fermions and quantized conductance. A focus is on two-dimensional systems including both topological insulators and superconductors. First, the question of whether individual nonmagnetic impurities can induce zero-energy states in time-reversal invariant superconductors from Altland-Zirnbauer (AZ) symmetry class DIII is addressed, and a class of symmetries which guarantee the existence of such states for a specific value of the impurity strength is defined. These general results are applied to the time-reversal invariant p-wave phase of the doped Kitaev-Heisenberg model, where it is also demonstrated how a lattice of impurities can drive a topologically trivial system into the nontrivial phase. Second, the result about the existence of zero-energy impurity states is generalized to all AZ symmetry classes. This is achieved by considering, for general Hamiltonians H from the respective symmetry classes, the “generalized roots of det H”, which subsequently are used to further explore the opportunities that lattices of nonmagnetic impurities provide for the realization of topologically nontrivial phases. The 1d Kitaev chain model, the 2d px + ipy superconductor, and the 2d Chern insulator are considered to show that impurity lattices generically enable topological phase transitions and, in the case of the 2d models, even provide access to a number of phases with large Chern numbers. Third, elastic backscattering in helical edge modes caused by a magnetic impurity with spin S and random Rashba spin-orbit coupling is investigated. In a finite bias steady state, the impurity induced resistance is found to slightly increase with decreasing temperature for S > 1/2. Since the underlying backscattering mechanism is elastic, interference between different scatterers can explain reproducible conductance fluctuations. Thus, the model is in agreement with central experimental results on edge transport in 2d topological insulators.

Desenvolvemos as ideias centrais da Homologia Simplicial e provamos a invariância topológica dos grupos de homologia para espaços homeomorfos. Discutimos também a invariância topológica da característica de Euler-Poincaré mostrando a sua relação com os grupos de homologia através dos números de Betti. Adicionalmente apresentamos conceitos da Álgebra Abstrata, especificamente da teoria de Grupos, importantes para o entendimento formal da álgebra homológica. Ao final, propomos atividades didáticas com objetivo de trazer as ideias de triangulação e invariância topológica ao contexto da sala de aula. / We develop central ideas of Simplicial Homology and prove the topological invariance of homology groups for homeomorphic spaces. We also discuss topological invariance of Euler- Poincaré characteristic showing its relation with the homology groups through Betti numbers. In addition, we present concepts of abstract algebra, specifically of group theory, which are important to formal understanding of homological algebra. In the end, we propose didactic activities in order to bring the ideas of triangulation and topological invariance to context of math classes on basic education.

The subject of the present thesis are some aspects of impurities affecting mesoscopic systems with regard to their topological properties and related effects like Majorana fermions and quantized conductance. A focus is on two-dimensional systems including both topological insulators and superconductors. First, the question of whether individual nonmagnetic impurities can induce zero-energy states in time-reversal invariant superconductors from Altland-Zirnbauer (AZ) symmetry class DIII is addressed, and a class of symmetries which guarantee the existence of such states for a specific value of the impurity strength is defined. These general results are applied to the time-reversal invariant p-wave phase of the doped Kitaev-Heisenberg model, where it is also demonstrated how a lattice of impurities can drive a topologically trivial system into the nontrivial phase. Second, the result about the existence of zero-energy impurity states is generalized to all AZ symmetry classes. This is achieved by considering, for general Hamiltonians H from the respective symmetry classes, the “generalized roots of det H”, which subsequently are used to further explore the opportunities that lattices of nonmagnetic impurities provide for the realization of topologically nontrivial phases. The 1d Kitaev chain model, the 2d px + ipy superconductor, and the 2d Chern insulator are considered to show that impurity lattices generically enable topological phase transitions and, in the case of the 2d models, even provide access to a number of phases with large Chern numbers. Third, elastic backscattering in helical edge modes caused by a magnetic impurity with spin S and random Rashba spin-orbit coupling is investigated. In a finite bias steady state, the impurity induced resistance is found to slightly increase with decreasing temperature for S > 1/2. Since the underlying backscattering mechanism is elastic, interference between different scatterers can explain reproducible conductance fluctuations. Thus, the model is in agreement with central experimental results on edge transport in 2d topological insulators.

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques.

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques.

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The discovery of topological quantum states of matter has required physicists to look beyond Landau’s theory of symmetry-breaking, previously the main paradigm for<br>studying states of matter. This has led also to the development of new topological theories for describing the novel properties. In this dissertation an investigation in this<br>frontier research area is presented, which looks at the interplay between the quantum geometry of these states, defects and disorder. After a brief introduction to the topological quantum states of matter considered herein, some aspects of my work in this area are described. First, the disorder-induced band structure engineering of topological insulator surface states is considered, which is possible due to their resilience from Anderson localization, and believed to be a consequence of their topological origin.<br>Next, the idiosyncratic behavior of these same surface states is considered, as observed in experiments on thin film topological insulators, in response to competition between<br>hybridization effects and an in-plane magnetic field. Then moving in a very different direction, the uncovering of topological ‘gravitational’ response is explained: the<br>topologically-protected charge response of two dimensional gapped electronic topological states to a special kind of 0-dimensional boundary – a disclination – that encodes spatial curvature. Finally, an intriguing relation between the gravitational response of quantum Hall states, and their response to an apparently unrelated perturbation – nonuniform electric fields is reported. <br>