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We investigate the gauging of higher-form finite Abelian symmetries and their sub-groups in quantum spin models in spatial dimensions $d=2$ and 3. Doing so, we naturally uncover gauged models with dual higher-group symmetries and potential mixed 't Hooft anomalies. We demonstrate that the mixed anomalies manifest as the symmetry fractionalization of higher-form symmetries participating in the mixed anomaly. Gauging is realized as an isomorphism or duality between the bond algebras that generate the space of quantum spin models with the dual generalized symmetry structures. We explore the mapping of gapped phases under such gauging related dualities for 0-form and 1-form symmetries in spatial dimension $d=2$ and 3. In $d=2$, these include several non-trivial dualities between short-range entangled gapped phases with 0-form symmetries and 0-form symmetry enriched Higgs and (twisted) deconfined phases of the gauged theory with possible symmetry fractionalizations. Such dualities also imply strong constraints on several unconventional, i.e., deconfined or topological transitions. In $d=3$, among others, we find, dualities between topological orders via gauging of 1-form symmetries. Hamiltonians self-dual under gauging of 1-form symmetries host emergent non-invertible symmetries, realizing higher-categorical generalizations of the Tambara-Yamagami fusion category., Comment: 90 pages, 19 figures

The localization landscape gives direct access to the localization of bottom-of-band eigenstates in non-interacting disordered systems. We generalize this approach to eigenstates at arbitrary energies in systems with or without internal degrees of freedom by introducing a modified $\mathcal{L}^2$-landscape, and we demonstrate its accuracy in a variety of archetypal models of Anderson localization in one and two dimensions. This $\mathcal{L}^2$-landscape function can be efficiently computed using hierarchical methods that allow evaluating the diagonal of a well-chosen Green function. We compare our approach to other landscape methods, bringing new insights on their strengths and limitations. Our approach is general and can in principle be applied to both studies of topological Anderson transitions and many-body localization., 6.5 + 1 pages, all comments are welcome

We study the dynamics of thermalization following a quantum quench using tensor-network methods. Contrary to the common belief that the rapid growth of entanglement and the resulting exponential growth of the bond dimension restricts simulations to short times, we demonstrate that the long time limit of local observables can be well captured using the time-dependent variational principle. This allows to extract transport coefficients such as the energy diffusion constant from simulations with rather small bond dimensions. We further study the characteristic of the chaotic wave that precedes the emergence of hydrodynamics, to find a ballistic diffusively-broadening wave-front., Comment: New version: 10 pages including appendices. Added an author. Used a different quantity to diagnose chaos, for which results are independent of bond dimension. Leads to a modified conclusion concerning the chaotic dynamics

Clean topological insulators exposed to a magnetic field develop Landau levels accompanied by a nonzero Hall conductivity for the infinite slab geometry. In this work we consider the case of disordered topological insulator nanowires and find, in contrast, that a zero Hall plateau emerges within a broad energy window close to the Dirac point. We numerically calculate the conductance and its distribution for a statistical ensemble of disordered nanowires, and use the conductance fluctuations to study the dependence of the insulating phase on system parameters, such as the nanowire length, disorder strength and the magnetic field., 7 pages, 7 figures

We study superconducting states of doped inversion-symmetric Weyl semimetals. Specifically, we consider a lattice model realizing a Weyl semimetal with an inversion symmetry and study the superconducting instability in the presence of a short-ranged attractive interaction. With a phonon-mediated attractive interaction, we find two competing states: a fully gapped finite-momentum (FFLO) pairing state and a nodal even-parity pairing state. We show that, in a BCS-type approximation, the finite-momentum pairing state is energetically favored over the usual even-parity paired state and is robust against weak disorder. Though energetically unfavorable, the even-parity pairing state provides an electronic analogue of the 3He-A phase in that the nodes of the even-parity state carry non-trivial winding numbers and therefore support a surface flat band. We briefly discuss other possible superconducting states that may be realized in Weyl semimetals., Comment: 9 pages, 2 tables, 2 figures. references are updated and the discussion on the half-qauntum and full-quantum vortices is added

Strongly correlated systems, i.e., quantum materials for which the interactions between its constituents are strong, are good candidates for the development of applications based on quantum-mechanical principles, such as quantum computers. Two paradigmatic models of strongly correlated systems are heavy-fermionic systems and one-dimensional spin-12 systems, with and without quenched disorder. In the past decade, improvement in computational methods and a vast enhancement in computational power has made it possible to study these systems in a a non-perturbative manner. In this thesis we present state-of-the-art numerical methods to investigate the properties of strongly correlated systems, and we apply these methods to solve a couple of selected problems in quantum condensed matter theory. We start by revisiting the phase diagram of the Falicov-Kimball model on the square lattice which can be considered as a heavy-fermionic systems. This model describes an interplay between conduction electrons and heavy electrons and reveals several distinct metal-insulator phase transitions. Using a lattice Monte-Carlo method, we study the transport properties of the model. Our analysis describes the role of temperature and interaction strength on the metal-insulator phase transitions in the Falicov-Kimball model. The second part of the thesis investigate the spatial structure of the entanglement in ground and thermal statesof the transverse-field Ising chain. We use the logarithmic negativity as a measure for the entanglement between two disjoint blocks. We investigate how logarithmic negativity depends on the spatial separation between two blocks, which can be viewed as the entanglement analog of a spatial correlation function. We find sharp entanglement thresholds at a critical distance beyond which the logarithmic negativity vanishes exactly and thus the two blocks become unentangled. Our results hold even in the presence of long-ranged quantum correlations, i.e., at the system’s quantum critical point. Using Time-Evolving Block Decimation (TEBD), we explore this feature as a function of temperature and size of the two blocks. We present a simple model to describe our numerical observations. In the last part of this thesis, we introduce an order parameter for a many-body localized spin-glass (MBL-SG) phase. We show that many-body localized spin-glass order can also be detected from two-site reduced density matrices, which we use to construct an eigenstate spin-glass order parameter. We find that this eigenstate spin-glass order parameter captures spin-glass phases in random Ising chains, both in many-body eigenstates as well as in the nonequilibrium dynamics, from a local in time measurement. We discuss how our results can be used to observe MBL-SG order within current experiments in Rydberg atoms and trapped ion systems.

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