No description available.

No / In this paper, we argue that two key trends – an unfolding ecological crisis and a reduction in the amount of (cheap) energy available to society – bring into question both the relevance and the resilience of existing educational systems, requiring us to rethink both the content and the form of education in general, and peace education in particular. Against this background, we consider the role education might play in enabling citizens and societies to adapt peacefully to conditions of energy descent and a less benign ecological system, taking seriously the possibility that there will be fewer resources available for education. Drawing on Wes Jackson’s and Wendell Berry’s concept of an education in ‘homecoming’, and on E.F. Schumacher’s concept of ‘appropriate technology’, we suggest a possible vision of peace education. We propose that such education might be focused around ‘appropriate knowledge’, commitment to place, and an understanding of the needs and characteristics of each local context. We then consider an example of what this might mean in practice, particularly under conditions of increasing resource scarcity: Permaculture education in El Salvador, we suggest, illustrates the characteristics and relevance of an education that aims to foster ‘appropriate knowledge’ within a particular and very challenging context. The paper concludes by considering the wider implications of our argument.

No / In this paper, we argue that two key trends – an unfolding ecological crisis and a reduction in the amount of (cheap) energy available to society – bring into question both the relevance and the resilience of existing educational systems, requiring us to rethink both the content and the form of education in general, and peace education in particular. Against this background, we consider the role education might play in enabling citizens and societies to adapt peacefully to conditions of energy descent and a less benign ecological system, taking seriously the possibility that there will be fewer resources available for education. Drawing on Wes Jackson’s and Wendell Berry’s concept of an education in ‘homecoming’, and on E.F. Schumacher’s concept of ‘appropriate technology’, we suggest a possible vision of peace education. We propose that such education might be focused around ‘appropriate knowledge’, commitment to place, and an understanding of the needs and characteristics of each local context. We then consider an example of what this might mean in practice, particularly under conditions of increasing resource scarcity: Permaculture education in El Salvador, we suggest, illustrates the characteristics and relevance of an education that aims to foster ‘appropriate knowledge’ within a particular and very challenging context. The paper concludes by considering the wider implications of our argument.

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.

Em teoria dos números, o teorema de Erdös-Kac, também conhecido como o teorema fundamental de teoria probabilística dos números, diz que se w(n) denota a quantidade de fatores primos distintos de n, então a sequência de funções de distribuições N definidas por FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converge uniformemente sobre R para a distribuição normal padrão. Neste trabalho desenvolvemos todos os teoremas necessários para uma demonstração analítica, que nos permitirá encontrar a ordem de erro da convergência acima. / In number theory, the Erdös-Kac theorem, also known as the fundamental theorem of probabilistic number theory, states that if w(n) is the number of distinct prime factors of n, then the sequence of distribution functions N, defined by FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converges uniformly on R to the standard normal distribution. In this work we developed all theorems needed to an analytic demonstration, which will allow us to find an order of error of the above convergence.

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.

Em teoria dos números, o teorema de Erdös-Kac, também conhecido como o teorema fundamental de teoria probabilística dos números, diz que se w(n) denota a quantidade de fatores primos distintos de n, então a sequência de funções de distribuições N definidas por FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converge uniformemente sobre R para a distribuição normal padrão. Neste trabalho desenvolvemos todos os teoremas necessários para uma demonstração analítica, que nos permitirá encontrar a ordem de erro da convergência acima. / In number theory, the Erdös-Kac theorem, also known as the fundamental theorem of probabilistic number theory, states that if w(n) is the number of distinct prime factors of n, then the sequence of distribution functions N, defined by FN(x) = (1/N) #{n <= N : (w(n) log log N)/(log log N)^(1/2)} <= x}, converges uniformly on R to the standard normal distribution. In this work we developed all theorems needed to an analytic demonstration, which will allow us to find an order of error of the above convergence.

Quelques particularités des polaritons, (quasi) particules-modes normaux du système d'excitons en interaction avec des photons en régime de couplage dit fort, sont théoriquement et numériquement analysés dans les systèmes de dimensionnalité basse. Dans le chapitre 1 est donné un bref aperçu en structure 0D, 1D et 2D semi-conductrices avec une introduction générale au domaine des polaritons. Le chapitre 2 est consacré aux micro / nano fils. Les modes de galerie sifflants sont étudiés dans le cas général d'un système anisotrope ainsi que la formation des polaritons dans les fils de ZnO. Le modèle théorique est comparé à l’expérience. Dans le chapitre 3 la dynamique de type Josephson pour les condensats de Bose-Einstein des polaritons est analysé en prenant en compte le pseudospin. Le chapitre 4 commence par une introduction à l'effet Aharonov-Bohm, qui est la phase géométrique la plus connue. Une autre phase géométrique - phase de Berry, qui existe pour une large classe de systèmes en évolution adiabatique sur un contour fermé, est l'objet principal de cette section. Nous avons examiné une proposition d'un interféromètre en anneau avec exciton-polaritons basé sur l'effet phase de Berry. Le chapitre 5 concerne un système 0D: un exciton d’une boîte quantique fortement couplé avec des photons dans une cavité optique. Nous avons discuté de la possibilité d'obtenir des états intriqués à partir d'une boîte quantique embarquée dans un cristal photonique en régime polaritonique. / Some special features of polaritons, quasi-particles being normal modes of system of excitons interacting with photons in so called strong coupling regime, are theoretically and numerically analyze in low dimensional systems. In Chapter 1 is given a brief overview of 0D, 1D and 2D semiconductor structures with a general introduction to the polariton field. Chapter 2 is devoted to micro / nano wires. The so called whispering gallery modes are studied in the general case of an anisotropic systems as well as polariton formation in ZnO wires. Theoretical model is compared with an experiment. In the Chapter 3 Josephson type dynamics with Bose-Einstein condensates of polaritons is analyzed taking into account pseudospin degree of freedom. Chapter 4 start with an introduction to Aharonov-Bohm effect, as the best known represent of geometrical phases. An another geometrical phase – Berry phase, occurring for a wide class of systems performing adiabatic motion on a closed ring, is main subject of this section. We considered one proposition for an exciton polariton ring interferometer based on Berry phase effect. Chapter 5 concerns one 0D system : strongly coupled quantum dot exciton to cavity photon. We have discussed possibility of obtaining entangled states from a quantum dot embedded in a photonic crystal in polariton regime.

In den letzten Jahren führte die Verbindung zwischen Topologie und kondensierter Materie zur Entdeckung vieler interessanter und exotischer elektronischer Effekte. Während sich die Forschung anfangs auf elektronische Systeme mit einer Bandlücke wie den topologischen Isolator konzentrierte, erhalten in letzter Zeit topologische Halbmetalle viel Aufmerksamkeit. Das bekannteste Beispiel sind Weyl-Halbmetalle, die an beliebigen Punkten in der Brillouin-Zone lineare Kreuzungen von nicht entarteten Bändern aufweist. An diese Punkte ist eine spezielle Quantenzahl namens Chiralität gebunden, die die Existenz von Weyl-Punktpaaren erzwingt. Diese Paare sind topologisch geschützt und wirken als Quellen und Senken der Berry-Krümmung, einem topologischen Feld im reziproken Raum. Diese Berry-Krümmung steht in direktem Zusammenhang mit dem anomalen Hall-Effekt, der die Entstehung einer Querspannung aus einem Längsstrom in einem magnetischen Material beschreibt. Analog existiert auch der anomale Nernst-Effekt, bei dem der longitudinale Strom durch einen thermischen Gradienten ersetzt wird. Dieser Effekt ermöglicht die Umwandlung von Wärme in elektrische Energie und ist zudem stark an die Berry-Krümmung gebunden. In dieser Arbeit werden die anomalen Transporteffekte zunächst in fundamentalen Modellsystemen untersucht. Hier wird eine Kombination aus analytischen und numerischen Methoden verwendet, um Quantisierungen sowohl des Hall- und Nernst- als auch des thermischen Hall-Effekts in zweidimensionalen Systemen mit und ohne externen Magnetfeldern zu zeigen. Eine Erweiterung in drei Dimensionen zeigt eine Quasi-Quantisierung, bei der die Leitfähigkeiten Werte der jeweiligen zweidimensionalen Quanten skaliert durch charakteristische Wellenvektoren annehmen. Im nächsten Schritt werden verschiedene Mechanismen zur Erzeugung starker Berry-Krümmung und damit großer anomaler Hall- und Nernst-Effekte sowohl in Modellsystemen als auch in realen Materialien untersucht. Dies ermöglicht die Identifizierung und Isolierung vielversprechender Effekte in den einfachen Modellen, in denen wichtige Merkmale untersucht werden können. Die Ergebnisse können dann auf die realen Materialien übertragen werden, wo die jeweiligen Effekte erkennbar sind. Hier werden sowohl Weyl-Punkte als auch Knotenlinien in Kombination mit Magnetismus als vielversprechende Eigenschaften identifiziert und Materialrealisierungen in der Klasse der Heusler-Verbindungen vorgeschlagen. Diese Verbindungen sind eine sehr vielseitige Materialklasse, in der unter anderem auch magnetische topologische Metalle zu finden sind. Um ein tieferes Verständnis der anomalen Transporteffekte zu erhalten sowie Faustregeln für Hochleistungsverbindungen abzuleiten, wurde eine High-Throughput-Rechnung von magnetisch-kubischen Voll-Heusler-Verbindungen durchgeführt. Diese Berechnung zeigt die Bedeutung von Spiegelebenen in magnetischen Materialien für große anomale Hall- und Nernst-Effekte und zeigt, dass einige der Heusler-Verbindungen die höchsten bisher berichteten Literaturwerte bei diesen Effekten übertreffen. Auch andere interessante Effekte im Zusammenhang mit Weyl-Punkten werden untersucht. Beim bekannten Weyl-Halbmetall NbP weisen die Weyl-Punkte aufgrund der hohen Symmetrie des Kristalls eine hohe Entartung auf. Die Anwendung von einachsigem Zug reduziert jedoch die Symmetrien und hebt damit die Entartungen auf. Eine theoretische Untersuchung zeigt, dass die Weyl-Punkte bei einachsigem Zug energetisch verschoben werden und, was noch wichtiger ist, dass sie bei realistischen Werten das Fermi-Niveau durchschreiten. Dies macht NbP zu einer vielversprechenden Plattform, um die Weyl-Physik weiter zu untersuchen. Die theoretischen Ergebnisse werden mit experimentellen Messungen von Shubnikov-de-Haas-Oszillationen unter einachsigem Zug kombiniert und es wird eine gute Übereinstimmung mit den theoretischen Ergebnissen gefunden. Als erster Schritt in Richtung neuer Berechnungsmethoden wird die Idee eines Weyl-Halbmetall-basierten Chiralitätsfilters für Elektronen untersucht. An der Grenzfläche zweier Weyl-Halbmetalle kann in Abhängigkeit von den genauen Weyl-Punktparametern nur eine Chiralität übertragen werden. Hier wird ein effektives geometrisches Modell erstellt und zur Untersuchung realer Materialgrenzflächen eingesetzt. Während im Allgemeinen eine Filterwirkung möglich erscheint, zeigten die untersuchten Materialien keine geeignete Kombination. Hier können weitere Studien mit Fokus auf magnetische Weyl-Halbmetalle oder Multifold-Fermion-Materialien durchgeführt werden.:List of publications Preface 1. Theoretical background 1.1. Berry curvature and Weyl semimetals 1.1.1. From the adiabatic evolution to the Berry phase 1.1.2. From the Berry phase to the Berry curvature 1.1.3. Topological phases of condensed matter 1.1.4. Weyl semimetals 1.1.5. Dirac semimetals 1.1.6. Nodal line semimetals 1.2. Density-functional theory 1.2.1. Born-Oppenheimer approximation 1.2.2. Hohenberg-Kohn theorems 1.2.3. Kohn-Sham formalism 1.2.4. Exchange-correlation functional 1.2.5. Pseudopotentials 1.2.6. Basis functions 1.2.7. VASP 1.3. Tight-binding Hamiltonian from Wannier functions 1.3.1. Wannier functions 1.3.2. Constructing Wannier functions from DFT 1.3.3. Generating a Wannier tight-binding Hamiltonian 1.3.4. Necessity of the tight-binding Hamiltonian 1.4. Linear response theory 1.4.1. General introduction to linear response 1.4.2. Anomalous Hall effect 1.4.3. Anomalous Nernst effect 1.4.4. Anomalous thermal Hall effect 1.4.5. Common features of anomalous transport effects 1.4.6. Symmetry considerations for Berry curvature related transport effects 1.4.7. Magneto-optic Kerr effect 1.4.8. About the efficiency of the calculations 2. (Quasi-)Quantization in the Hall, thermal Hall, and Nernst effects 2.1. Quantization with an external magnetic field 2.1.1. Two-dimensional case 2.1.2. Three-dimensional case 2.2. Quantization without an external field 2.2.1. Two-dimensional case 2.2.2. Three-dimensional case . 2.3. A remark on the spin Hall effect 2.4. A remark on the quasi-quantization of the three-dimensional conductivities 2.5. Conclusions 3. Understanding anomalous transport 3.1. Anomalous transport without a net magnetic moment 3.1.1. Toy model 3.1.2. Ti2MnAl and related compounds 3.2. Large Berry curvature enhancement from nodal line gapping 3.2.1. Toy model 3.2.2. Fe2MnP and related compounds 3.2.3. Co2MnGa 3.3. Topological features away from the Fermi level and the anomalous Nernst effect 3.3.1. Toy model . 3.3.2. Co2FeGe and Co2FeSn 3.4. Conclusions 4. Heusler database calculation 4.1. Workflow 4.2. Importance of mirror planes 4.3. The right valence electron count 4.4. Correlation between anomalous Hall and Nernst effects 4.5. Selected special compounds 4.6. Conclusions 5. NbP under uniaxial strain 5.1. NbP and its symmetries 5.2. The influence of strain on the electronic structure 5.2.1. Shifting of the Weyl points 5.2.2. Splitting of the Fermi surfaces 5.3. Comparison with experimental results 5.4. Conclusions 6. A tunable chirality filter 6.1. Concept 6.2. Geometrical simplification and expansion for more Weyl points 6.3. Material selection 6.3.1. Workflow 6.3.2. Results for NbP and TaAs 6.3.3. Results for Ag2Se and Ag2S 6.4. Conclusions and perspective . Summary and outlook A. Numerical tricks A.1. Hamiltonian setup at several k points at once A.2. Precalculating prefactors B. Derivation of the conductivity (quasi-)quanta B.1. Two dimensions B.1.1. General formula and necessary approximations B.1.2. Useful integrals B.1.4. Quantized thermal Hall effect B.1.5. Quantized Nernst effect B.1.6. Flat bands and the Nernst effect B.2. Three dimensions B.2.1. General formula B.2.2. Three-dimensional electron gas B.2.3. Three-dimensional Weyl semimetal C. Heusler database tables D. Details on the NbP strain calculations E. Details on the geometrical matching procedure References List of abbreviations List of Figures List of Tables Acknowledgements Eigenständigkeitserklärung / In recent years, the connection between topology and condensed matter resulted in the discovery of many interesting and exotic electronic effects. While in the beginning, the research was focused on gapped electronic systems like the topological insulator, more recently, topological semimetals are getting a lot of attention. The most well-known example is the Weyl semimetal, which hosts linear crossings of non-degenerate bands at arbitrary points in the Brillouin zone. Tied to these points there is a special quantum number called chirality, which enforces the existence of Weyl point pairs. These pairs are topologically protected and act as sources and sinks of the Berry curvature, a topological field in reciprocal space. This Berry curvature is directly connected to the anomalous Hall effect, which describes the emergence of a transverse voltage from a longitudinal current in a magnetic material. Analogously, there also exists the anomalous Nernst effect, where the longitudinal current is replaced by a thermal gradient. This effect allows for the conversion of heat into electrical energy and is also strongly tied to the Berry curvature. In this work, the anomalous transport effects are at first studied in fundamental model systems. Here, a combination of analytical and numerical methods is used to reveal quantizations in both the Hall, the Nernst, and the thermal Hall effects in two-dimensional systems with and without external magnetic fields. An expansion into three dimensions shows a quasi-quantization, where the conductivities take values of the respective two-dimensional quanta scaled by characteristic wavevectors. In the next step, several mechanisms for the generation of strong Berry curvature and thus large anomalous Hall and Nernst effects are studied in both model systems and real materials. This allows for the identification and isolation of promising effects in the simple models, where important features can be studied. The results can then be applied to the real materials, where the respective effects can be recognized. Here, both Weyl points and nodal lines in combination with magnetism are identified as promising features and material realizations are proposed in the class of Heusler compounds. These compounds are a very versatile class of materials, where among others also magnetic topological metals can be found. To get a deeper understanding of the anomalous transport effects as well as to derive guidelines for high-performance compounds, a high-throughput calculation of magnetic cubic full Heusler compounds was carried out. This calculation reveals the importance of mirror planes in magnetic materials for large anomalous Hall and Nernst effects and shows that some of the Heusler compounds outperform the highest so-far reported literature values in these effects. Also other interesting effects related to Weyl points are investigated. In the well-known Weyl semimetal NbP, the Weyl points have a high degeneracy due to the high symmetry of the crystal. However, the application of uniaxial strain reduces the symmetries and therefore lifts the degeneracies. A theoretical investigation shows, that the Weyl points are moved in energy under uniaxial strain and, more importantly, that at reasonable strain values they cross the Fermi level. This renders NbP a promising platform to further study Weyl physics. The theoretical results are combined with experimental measurements of Shubnikov-de Haas oscillations under uniaxial strain and a good agreement with the theoretical results is found. As a first step in the direction of new ways of computation, an idea of a Weyl semimetal based chirality filter for electrons is investigated. At the interface of two Weyl semimetals, depending on the exact Weyl point parameters, it is possible to transmit only one chirality. Here, an effective geometrical model is established and employed for the investigation of real material interfaces. While in general, a filtering effect seems possible, the investigated materials did not show any suitable combination. Here, further studies can be made with the focus on either magnetic Weyl semimetals of multifold-fermion materials.:List of publications Preface 1. Theoretical background 1.1. Berry curvature and Weyl semimetals 1.1.1. From the adiabatic evolution to the Berry phase 1.1.2. From the Berry phase to the Berry curvature 1.1.3. Topological phases of condensed matter 1.1.4. Weyl semimetals 1.1.5. Dirac semimetals 1.1.6. Nodal line semimetals 1.2. Density-functional theory 1.2.1. Born-Oppenheimer approximation 1.2.2. Hohenberg-Kohn theorems 1.2.3. Kohn-Sham formalism 1.2.4. Exchange-correlation functional 1.2.5. Pseudopotentials 1.2.6. Basis functions 1.2.7. VASP 1.3. Tight-binding Hamiltonian from Wannier functions 1.3.1. Wannier functions 1.3.2. Constructing Wannier functions from DFT 1.3.3. Generating a Wannier tight-binding Hamiltonian 1.3.4. Necessity of the tight-binding Hamiltonian 1.4. Linear response theory 1.4.1. General introduction to linear response 1.4.2. Anomalous Hall effect 1.4.3. Anomalous Nernst effect 1.4.4. Anomalous thermal Hall effect 1.4.5. Common features of anomalous transport effects 1.4.6. Symmetry considerations for Berry curvature related transport effects 1.4.7. Magneto-optic Kerr effect 1.4.8. About the efficiency of the calculations 2. (Quasi-)Quantization in the Hall, thermal Hall, and Nernst effects 2.1. Quantization with an external magnetic field 2.1.1. Two-dimensional case 2.1.2. Three-dimensional case 2.2. Quantization without an external field 2.2.1. Two-dimensional case 2.2.2. Three-dimensional case . 2.3. A remark on the spin Hall effect 2.4. A remark on the quasi-quantization of the three-dimensional conductivities 2.5. Conclusions 3. Understanding anomalous transport 3.1. Anomalous transport without a net magnetic moment 3.1.1. Toy model 3.1.2. Ti2MnAl and related compounds 3.2. Large Berry curvature enhancement from nodal line gapping 3.2.1. Toy model 3.2.2. Fe2MnP and related compounds 3.2.3. Co2MnGa 3.3. Topological features away from the Fermi level and the anomalous Nernst effect 3.3.1. Toy model . 3.3.2. Co2FeGe and Co2FeSn 3.4. Conclusions 4. Heusler database calculation 4.1. Workflow 4.2. Importance of mirror planes 4.3. The right valence electron count 4.4. Correlation between anomalous Hall and Nernst effects 4.5. Selected special compounds 4.6. Conclusions 5. NbP under uniaxial strain 5.1. NbP and its symmetries 5.2. The influence of strain on the electronic structure 5.2.1. Shifting of the Weyl points 5.2.2. Splitting of the Fermi surfaces 5.3. Comparison with experimental results 5.4. Conclusions 6. A tunable chirality filter 6.1. Concept 6.2. Geometrical simplification and expansion for more Weyl points 6.3. Material selection 6.3.1. Workflow 6.3.2. Results for NbP and TaAs 6.3.3. Results for Ag2Se and Ag2S 6.4. Conclusions and perspective . Summary and outlook A. Numerical tricks A.1. Hamiltonian setup at several k points at once A.2. Precalculating prefactors B. Derivation of the conductivity (quasi-)quanta B.1. Two dimensions B.1.1. General formula and necessary approximations B.1.2. Useful integrals B.1.4. Quantized thermal Hall effect B.1.5. Quantized Nernst effect B.1.6. Flat bands and the Nernst effect B.2. Three dimensions B.2.1. General formula B.2.2. Three-dimensional electron gas B.2.3. Three-dimensional Weyl semimetal C. Heusler database tables D. Details on the NbP strain calculations E. Details on the geometrical matching procedure References List of abbreviations List of Figures List of Tables Acknowledgements Eigenständigkeitserklärung

This report discusses higher differential forms with applications in the study of topological phenomena. The integer quantum Hall effect is first discussed, demonstrating a connection between models on a lattice and quantum field theories bridged by a topological invariant, namely the Chern number. Next, for parametrized models on a lattice, the higher Berry curvature is described. This is a rank-(d + 2) differential form on a (d + 2)-dimensional parameter manifold which provides a relation between models in a bulk and on a lower-dimensional interface. Finally, a family of quantum field theories connected to a (d + 1)-dimensional manifold, termed a target space, is constructed. This connection is realized through the incorporation of a set of classical fields, and the effective action of the full field theories all contain a Wess-Zumino-Witten term given by the pullback of a rank-(d + 1) differential form from the target space to spacetime. By performing an extension of spacetime, a (d + 2)-form on a (d + 2)-dimensional target space is constructed in a similar way. Extending a theory in d dimensions thus yields a form on a target space of the same dimension as that of a (d + 1)-dimensional theory without extension, defining a dimensional hierarchy. The dimensional relations inherent in the two higher forms studied indicate the possibility of a relation between them. / Denna rapport beskriver högre ordningens differentialformer med tillämpningar inom topologiska fenomen. Den heltaliga kvantmekaniska Halleffekten beskrivs först, som ett exempel på ett samband mellan modeller på ett gitter och kvantfältteorier som förbindas av topologiska invarianter, specifikt Chern-talet. För parametriserade modeller på ett gitter beskrivs därefter den högre Berrykrökningen. Detta är en differentialform av ordning (d + 2) definierad på en (d + 2)-dimensionell parametermångfald som ger en koppling mellan modeller i en kropps inre och på dens gränsskikt, som är i en lägre dimension. Slutligen konstrueras en familj av kvantfältteorier som är kopplade till en (d + 1)-dimensionell mångfald kallad modellens målrum. Denna koppling realiseras genom introduktionen av ett antal klassiska fält, och den effektiva verkan för den fullständiga teorin innehåller en Wess-Zumino-Witten-term som ges av en tillbakadragen (d + 1)-form från målrummet till rumtiden. Genom att utvidga rumtiden kan även en (d + 2)-form på en (d + 2)-dimensionellt målrum konstrueras på ett motsvarande sätt. Utvidgningen av en teori i d dimensioner ger därmed en differentialform på ett målrum med samma dimension som målrummet för en (d + 1)-dimensionell teori utan utvidning, vilket definierar en dimensionell hierarki. Dimensionsrelationerna inbyggda i dessa två differentialformer indikerar den möjliga existensen av en relation mellan dem.