Weyl-Semimetalle haben bemerkenswerte Eigenschaften. Ihr elektrischer Widerstand steigt linear und unsaturiert mit einem angelegten Magnetfeld, diverse Ergebnisse deuten darauf hin, dass sie einen unordnungsinduzierten Metall-Isolator-Phasenübergang aufweisen und ihre Ladungsträger zeigen die chirale Anomalie, d.h., die Nichtkonservierung der chiralen Ladung. Diese Eigenschaften haben ihren Ursprung in der Niedrigenergiephysik der Weyl- Semimetalle, die von Weyl-Punkten, Berührungspunkten zwischen Leitungs- und Valenzband an der Fermi-Energie mit einer linearen Dispersionsrelation, dominiert wird. Diese Berührungspunkte sind topologisch geschützt, d.h., kleine Störung können ihnen nichts anhaben. Weyl-Semimetalle sind daher Beispiele für topologische Semimetalle, Materialien mit geschützten niedrigdimensionalen Berührungenspunkten, -linien, oder -oberflächen an der Fermi-Energie. In dieser Arbeit zeigen wir, wie die Eigenschaften von Weyl-Semimetallen durch Unordnung, Magnetfelder und Deformationen beeinflusst werden. Wir zeigen außerdem eine Querverbindung zwischen Weyl-Semimetallen und nodal line-Semimetallen, topologisch geschützten Semimetallen mit einer eindimensionalen Fermi-Fläche. Durch die Nutzung von Gitter- und Niedrigenergiekontinuumsmodellen können wir Wege aufzeigen, wie man unsere Ergebnisse sowohl aus einer Festkörperphysik- als auch aus einer Hochenergiephysikperspektive verstehen kann. Insbesondere identifizieren wir eine experimentelle Signatur der chiralen Anomalie: die blaue Note, ein charakteristisches Muster in Form einer Note, das mit Hilfe von winkelaufgelöster Photoelektronenspektroskopie gemessen werden kann. Ein weiteres wichtiges Charakteristikum ist der Magnetwiderstand, der in Weyl-Semimetallen vom Winkel zwischen einem angelegten Magnetfeld und der Transportrichtung abhängt. Durch den Einfluss der chiralen Anomalie ist der longitudinale Magnetwiderstand negativ, der transversale Widerstand hingegen wächst linear und grenzenlos mit dem angelegten Magnetfeld. In dieser Dissertation untersuchen wir beide Charakteristiken analytisch und numerisch. Inspiriert durch Experimente, in denen ein scharfes Leitfähigkeitsmaximum für parallele elektrische und Magnetfelder observiert wurde, zeigen wir, dass die Leitfähigkeit vom Winkel zwischen den angelegten Feldern und dem Abstandsvektor der Weyl-Punkte abhängt und dass sie insbesondere für Felder parallel zum Abstandsvektor ein scharfes Maximum aufweist. Dieser Effekt ist besonders ausgeprägt, wenn nur das niedrigste Landau-Niveau zur Leitfähigkeit beiträgt, er bleibt aber auch bei höheren Energien beobachtbar. Für parallelen Magnettransport untersuchen wir starke Unordnung, die außerhalb des von der Störungstheorie abgedeckten Bereichs liegt, numerisch und beobachten einen positiven Magnetwiderstand, qualitativ ähnlich zu experimentellen Daten. Aus Deformationen in Weyl-Semimetallen entstehen sogenannte chirale oder auch axiale Felder, die ähnliche Konsequenzen wie externe elektromagnetische Felder haben, wobei noch viele Details im Verborgenen liegen. Wir untersuchen Deformationen aus zwei verschiedenen Perspektiven: zunächst zeigen wir, wie zwei widersprüchliche Vorhersagen aus der Quantenfeldtheorie, die konsistenten und kovarianten Anomalien, in einem Gittermodell beobachtbar sind. Dann untersuchen wir elektrischen Transport unter Einfluss von axialen Magnetfeldern und zeigen, dass Moden, die sich in unterschiedliche Richtungen bewegen, räumlich getrennt sind. Diese räumliche Trennung hat eine unübliches Wachstums des elektrischen Leitwerts mit der transversalen Systembreite zur Folge. Des weiteren zeigen wir, wie ein nodal line-Semimetall aus einem Weyl-Semimetall entstehen kann, das einer Supergitterstruktur ausgesetzt ist. Wir interpretieren die Oberflächenzustände mit Hilfe der interzellulären Zak-Phase und zeigen zwei verschiedene Mechanismen, die die Bandstruktur vor der Öffnung einer Bandlücke schützen, auf. Um unsere Diskussion abzuschließen, untersuchen wir Transport in nodal line-Semimetallen in Kürze und stellen ihre Quantenfeldtheorie vor. Schließlich wenden wir uns wechselwirkenden Phasen zu und zeigen, welche Konsequenzen die Symmetrieklassifizierung des Sachdev- Ye-Kitaev-Modells hat – ein Modell von Teilchen mit zufälligen Wechselwirkungsstärken, dessen Topologie von der Anzahl der enthaltenen Teilchen bestimmt wird.:1 Introduction 2 Topological Band Theory 2.1 Geometric Phase and Berry Phase 2.1.1 The Adiabatic Theorem 2.1.2 The Zak Phase 2.2 Tenfold Classification of Topological Insulators and Superconductors 2.3 Topological Semimetals 2.3.1 Weyl Semimetals 2.3.2 Nodal Line Semimetals 2.4 Bulk-boundary Correspondence from the Intercellular Zak Phase 2.4.1 Intra- and Intercellular Zak Phase 2.4.2 Bulk-boundary Correspondence 2.4.3 Conclusion 3 Field Theory Perspective on Topological Phases 3.1 Topological Insulators 3.2 Weyl Fermions and the Chiral Anomaly 3.3 Visualizing the Chiral Anomaly with Photoemission Spectroscopy 3.3.1 The Chiral Anomaly in Condensed Matter Systems 3.3.2 Model and Methods 3.3.3 ARPES Spectra for Weyl and Dirac Semimetals 3.3.4 Experimental Details 3.3.5 Summary and Conclusion 3.4 The Consistent and Covariant Anomalies 3.5 Consistent and Covariant Anomalies on a Lattice 3.5.1 Model and Methods 3.5.2 Lattice Results for Consistent and Covariant Anomalies 3.5.3 Influence of the Mass Term 3.5.4 The Quest for One Third 3.6 The Action of Nodal Line Semimetals 4 Transport in Topological Semimetals 4.1 Longitudinal Magnetoresistance in Weyl Semimetals 4.2 Transversal Magnetoresistance in Weyl Semimetals 4.2.1 Model 4.2.2 Mesoscopic Transport in Clean Samples 4.2.3 Numerical Magnetotransport in the Presence of Disorder 4.2.4 Born-Kubo Analytical Bulk Conductivity 4.2.5 Numerical Results in Disordered Samples 4.2.6 Conclusion 4.3 Transport in the Presence of Axial Magnetic Fields 4.3.1 Model and Methods 4.3.2 Longitudinal Magnetotransport for Axial Fields 4.3.3 Conclusion 4.4 Transport in Nodal Line Semimetals 5 Nodal Line Semimetals from Weyl Superlattices 5.1 Weyl Semimetal on a Superlattice 5.2 Emergent Nodal Phases 5.3 Symmetry Classification of the Nodal Line 5.4 Surface States 5.5 Stability against Wave Vector Mismatch 5.6 Time-reversal Symmetric Weyl Semimetal 5.7 Conclusion 6 Symmetry Classification of the SYK Model 6.1 Model and Topological Classification 6.2 Overlap of Time-reversed Partners 6.2.1 Even Number of Majoranas 6.2.2 Odd Number of Majoranas 6.3 Spectral Function 6.3.1 Zero Temperature 6.3.2 Infinite Temperature 6.4 Symmetry-breaking Terms 6.5 Lattice Model 6.6 Conclusion 7 Conclusion and Outlook Appendix A Zak Phase and Extra Charge Accumulation Appendix B Material-specific Details for ARPES B.1 Relaxation Rates B.2 ARPES in Finite Magnetic Fields B.3 Estimates of the Chiral Chemical Potential Difference Appendix C Weyl Nodes in a Magnetic Field C.1 Scattering between Different Landau Levels C.2 Analytical Born-Kubo Calculation of Transversal Magnetoconductivity C.2.1 Disorder Scattering in Born Approximation C.2.2 Transversal Magnetoconductivity from Kubo Formula Appendix D Transfer Matrix Method D.1 Longitudinal Magnetic Field D.2 Transversal Magnetic Field Bibliography Acknowledgments List of Publications Versicherung / Weyl semimetals have remarkable properties. Their resistance grows linearly and unsaturated with an applied transversal magnetic field, and they are expected to show a disorder-induced metal-insulator transition. Their charge carriers exhibit the chiral anomaly, i.e., the nonconservation of chiral charge. These properties emerge from their low-energy physics, which are dominated by Weyl nodes: zero-dimensional band crossings at the Fermi energy with a linear dispersion. The band crossings are topologically protected, i.e., they cannot be lifted by small perturbations. Thus, Weyl semimetals are examples of topological semimetals, materials with protected lower-dimensional band crossing close to the Fermi surface. In this work, we show how the properties of Weyl semimetals are affected by disorder, magnetic fields, and strain. We further provide a link between Weyl semimetals and nodal line semimetals, topological semimetals with a one-dimensional Fermi surface. By using both lattice and low-energy continuum models, we present ways to understand the results from a condensed-matter and a quantum-field-theory perspective. In particular, we identify an experimental signature of the chiral anomaly: the blue note, a characteristic note-shaped pattern that can be measured in photoemission spectroscopy. Another important signature is the magnetoresistance. In Weyl semimetals, its behavior depends on the angle between the magnetic field and the transport direction. For parallel transport, a negative longitudinal magnetoresistance as a manifestation of the chiral anomaly is observed; for orthogonal transport, the transversal magnetoresistance shows a linear and unsaturated growth. In this thesis, we investigate both regimes analytically and numerically. Inspired by experiments that show a sharply peaked magnetoresistance for parallel fields, we show that the longitudinal magnetoresistance depends on the angle between applied fields and the Weyl node separation, and that it is sharply peaked for fields parallel to the node separation. This effect is especially strong in the limit where only the lowest Landau level contributes to the magnetoresistance, but it survives at higher chemical potentials. For transversal magnetotransport, we numerically investigate the strong-disorder regime that is beyond the reach of perturbation theory and observe a positive magnetoresistance, qualitatively similar to recent experiments. Strain in Weyl semimetals creates so-called axial fields that result in phenomena similar to the ones driven by electric and magnetic fields, but with some yet unknown consequences. We investigate strain from two perspectives: first, we show how two different predictions from quantum field theory, the consistent and covariant anomalies, manifest on a lattice. Second, we investigate transport in the presence of axial magnetic fields and show that counterpropagating modes are spatially separated, resulting in an unusual scaling of the conductance with the system’s width. We further show how a nodal line semimetal can emerge from a Weyl semimetal on a superlattice. We interpret the presence of surface states in terms of the intercellular Zak phase and show two distinct mechanisms that protect the spectrum from opening a gap. To complete our discussion, transport in nodal line semimetals is briefly discussed, as well as the quantum field theory that describes the low-energy features of these materials. Finally, we conclude this work by showing manifestations of the different symmetry classes that can be realized in the Sachdev-Ye-Kitaev model—a model of randomly interacting particles whose topology is deeply connected to the number of particles.:1 Introduction 2 Topological Band Theory 2.1 Geometric Phase and Berry Phase 2.1.1 The Adiabatic Theorem 2.1.2 The Zak Phase 2.2 Tenfold Classification of Topological Insulators and Superconductors 2.3 Topological Semimetals 2.3.1 Weyl Semimetals 2.3.2 Nodal Line Semimetals 2.4 Bulk-boundary Correspondence from the Intercellular Zak Phase 2.4.1 Intra- and Intercellular Zak Phase 2.4.2 Bulk-boundary Correspondence 2.4.3 Conclusion 3 Field Theory Perspective on Topological Phases 3.1 Topological Insulators 3.2 Weyl Fermions and the Chiral Anomaly 3.3 Visualizing the Chiral Anomaly with Photoemission Spectroscopy 3.3.1 The Chiral Anomaly in Condensed Matter Systems 3.3.2 Model and Methods 3.3.3 ARPES Spectra for Weyl and Dirac Semimetals 3.3.4 Experimental Details 3.3.5 Summary and Conclusion 3.4 The Consistent and Covariant Anomalies 3.5 Consistent and Covariant Anomalies on a Lattice 3.5.1 Model and Methods 3.5.2 Lattice Results for Consistent and Covariant Anomalies 3.5.3 Influence of the Mass Term 3.5.4 The Quest for One Third 3.6 The Action of Nodal Line Semimetals 4 Transport in Topological Semimetals 4.1 Longitudinal Magnetoresistance in Weyl Semimetals 4.2 Transversal Magnetoresistance in Weyl Semimetals 4.2.1 Model 4.2.2 Mesoscopic Transport in Clean Samples 4.2.3 Numerical Magnetotransport in the Presence of Disorder 4.2.4 Born-Kubo Analytical Bulk Conductivity 4.2.5 Numerical Results in Disordered Samples 4.2.6 Conclusion 4.3 Transport in the Presence of Axial Magnetic Fields 4.3.1 Model and Methods 4.3.2 Longitudinal Magnetotransport for Axial Fields 4.3.3 Conclusion 4.4 Transport in Nodal Line Semimetals 5 Nodal Line Semimetals from Weyl Superlattices 5.1 Weyl Semimetal on a Superlattice 5.2 Emergent Nodal Phases 5.3 Symmetry Classification of the Nodal Line 5.4 Surface States 5.5 Stability against Wave Vector Mismatch 5.6 Time-reversal Symmetric Weyl Semimetal 5.7 Conclusion 6 Symmetry Classification of the SYK Model 6.1 Model and Topological Classification 6.2 Overlap of Time-reversed Partners 6.2.1 Even Number of Majoranas 6.2.2 Odd Number of Majoranas 6.3 Spectral Function 6.3.1 Zero Temperature 6.3.2 Infinite Temperature 6.4 Symmetry-breaking Terms 6.5 Lattice Model 6.6 Conclusion 7 Conclusion and Outlook Appendix A Zak Phase and Extra Charge Accumulation Appendix B Material-specific Details for ARPES B.1 Relaxation Rates B.2 ARPES in Finite Magnetic Fields B.3 Estimates of the Chiral Chemical Potential Difference Appendix C Weyl Nodes in a Magnetic Field C.1 Scattering between Different Landau Levels C.2 Analytical Born-Kubo Calculation of Transversal Magnetoconductivity C.2.1 Disorder Scattering in Born Approximation C.2.2 Transversal Magnetoconductivity from Kubo Formula Appendix D Transfer Matrix Method D.1 Longitudinal Magnetic Field D.2 Transversal Magnetic Field Bibliography Acknowledgments List of Publications Versicherung

Submitted by ANGEL EDUARDO OBISPO VASQUEZ null (signaux_fonce@hotmail.com) on 2016-03-10T21:30:19Z No. of bitstreams: 1 Tese final.pdf: 3148732 bytes, checksum: dc9e633bbfd74365e11b41baeb143eff (MD5) / Approved for entry into archive by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br) on 2016-03-14T14:09:25Z (GMT) No. of bitstreams: 1 vasquez_aeo_dr_guara.pdf: 3148732 bytes, checksum: dc9e633bbfd74365e11b41baeb143eff (MD5) / Made available in DSpace on 2016-03-14T14:09:25Z (GMT). No. of bitstreams: 1 vasquez_aeo_dr_guara.pdf: 3148732 bytes, checksum: dc9e633bbfd74365e11b41baeb143eff (MD5) Previous issue date: 2016-02-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Desde aproximadamente o nal da década de 1970 efeitos quânticos e topológicos em sistemas da matéria condensada que são mostrados ocorrer a nível teórico em teoria quântica de campos têm atraído a atenção de físicos. Neste contexto, o grafeno representa uma das maiores vertentes de pesquisa dentro dos estudos das ciência dos materiais. O fato das excitações eletrônicas de baixa energia serem descritas por fermions de Dirac, estimulou uma relação frutífera entre a matéria condensada e a física de altas energias, fornecendo cenários propícios para o aparecimento de novos e exóticos fenômenos que são de grande interesse na física da matéria condensada atual. A presente tese aborda particularmente dois tópicos fundamentais da teoria quântica de campos: As Anomalias quânticas e o Fracionamento do número fermiônico. Especí camente, estamos interessados na realização de ambos fenômenos em redes de grafeno com deformações. No grafeno, um potencial vector de gauge axial surge como produto de deformações locais da rede, na forma de defeitos topológicos ou corrugações suaves. Analisaremos a in uência desses campos pseudomagnéticos nos estados eletrônicos para uma partícula, quando interagem com um campo magnético externo, considerando diferentes con gurações para esses campos. Estudamos o papel que desempenham os estados de modo-zero na indução de um número fermiônico fracionário e sua conexão com a anomalia de paridade. / Since approximately the late 1970s, topological quantum effects in condensed matter systems that are shown the occur at a theoretical level in quantum field theory have attracted the attention of physicists. In this context, the graphene is one of the major lines of research within the studies of materials science. The fact that the electronic excitations of low energy are described by Dirac fermions, stimulating a fruitful relationship between condensed matter and high energy physics, providing favorable scenarios for the arising of new and exotic phenomena which are of great interest in the current condensedmatter physics. This thesis addresses particularly two key topics of quantum field theory: Quantum anomalies and the fermion number fractionalization. Specifically, we are interested in performing both phenomena in deformed graphene lattice. In graphene, an axial vector potential arises as the result of local deformations on the lattice, as topological defects or soft corrugations. We analyze the ináuence of these pseudo-magnetic fields on the one-particle states, when interacting with a background magnetic field, for differents conÖguration for the fields. We study the role played by zero-mode states in fractional fermion number induced and its connection with the anomaly of parity.