One of the main features of topological phases is the presence of robust boundary states that are protected by a topological invariant. Famous examples of such states are the chiral edge states of a Chern insulator, the helical edge states of a two-dimensional Z2 insulator, and the Fermi arcs of Weyl semimetals. Despite their omnipresence, these topological boundary states can typically only be theoretically investigated through numerical studies due to the lack of analytical solutions for their wave functions. In the rare cases that wave-function solutions are available, they only exist for simple fine-tuned systems or for semi-infinite systems. Exact solutions are, however, common in the field of flat bands physics, where they lead to an understanding of the bulk bands rather than the boundary physics. It is well known that fully-periodic lattices with a frustrated geometry host localized modes that have a constant energy throughout the Brillouin zone. These localized modes appear due to a mechanism referred to as destructive interference, which leads to the disappearance of the wave-function amplitude on certain lattice sites. Making use of this mechanism, it is shown in this licentiate thesis that exact wave-function solutions can also be found on d-dimensional geometrically frustrated lattices that feature (d − 1)-dimensional boundaries. These exact solutions localize to the boundaries when the frustrated lattice hosts a topological phase and correspond to the robust, topological boundary states. This licentiate thesis revolves around the publication, which describes the method to finding these exact, analytical solutions for the topological boundary states on geometrically frustrated lattices, which was authored by the author of this licentiate thesis together with Maximilian Trescher and Emil J. Bergholtz and published in Physical Review B on August 30, 2017 with the title Anatomy of topological surface states: Exact solutions from destructive interference on frustrated lattices. An introduction is given on topological phases in condensed matter systems focussing on those models of which explicit examples are given in the paper: two-dimensional Chern insulators and three-dimensional Weyl semimetals. Moreover, by making use of the kagome lattice as an example the appearance of localized and semi-localized modes on geometrically frustrated lattices is elaborated upon. The chapters in this licentiate thesis thus endeavor to provide the reader with the proper background to comfortably read, understand, place into context and judge the relevance of the work in the accompanying publication. The licentiate thesis finishes with an outlook where it is discussed that the method presented in the paper can be generalized to an even larger class of lattices and can also be applied to find exact solutions for higher-order topological phases such as corner and hinge states.

In this thesis, I examine three very different solid-state systems that are all poor conductors when compared to elemental metals. The physics of canonical metals, such as the alkali and noble metals, is well known and is usually idealized in the free-or nearly free-electron picture. Their electron band structures are characterized by parabolic-like bands that cross the Fermi energy and possibly d-bands with flatter dispersions a few eV away. These well-behaved systems lend themselves to the use of simple analytic relations. Each of the three systems that I examine here differs significantly from the nearly-free parabolic band-picture of the electronic structure and require more complex analyses. In the first system of quasicrystals and approximants, we will discover that the electrons are undergoing anomalous diffusion depending on the size and symmetry of the lattices. Of course, as is well known, the details of these atomic lattice are what determine the nature of electronic band structures and how electrons may propagate in solids. In the second system, I find great agreement between my NbAs measurements and calculations on the closely related NbP compound. Incidental to this, I find that a reading of band structures shows that claims of measuring the linear band dispersion in Weyl/Dirac semimetals are not supported by the experimental and theoretical band structures. Finally, in the metallic regime of Nd_(1−x)TiO_3, we find that the Fermi liquid b coefficient is not within the bounds allowed by present models in samples with x = 0.2 and x = 0.15. It is suggested that the approximations used in current models may be why theory and experiment disagree. / Thesis / Doctor of Philosophy (PhD)

Topological phases of matter represent a new phase which cannot be understood in terms of Landau’s theory of symmetry breaking and are characterized by non-local topological properties emerging from purely local (microscopic) degrees of freedom. It is the non-trivial topology of the bulk band structure that gives rise to topological phases in condensed matter systems. Quantum Hall systems are prominent examples of such topological phases. Different quantum Hall states cannot be distinguished by a local order parameter. Instead, non-local measurements are required, such as the Hall conductance, to differentiate between various quantum Hall states. A signature of a topological phase is the existence of robust properties that do not depend on microscopic details and are insensitive to local perturbations which respect appropriate symmetries. Examples of such properties are the presence of protected gapless edge states at the boundary of the system for topological insulators and the remarkably precise quantization of the Hall conductance for quantum Hall states. The robustness of these properties can be under-stood through the existence of a topological invariant, such as the Chern number for quantum Hall states which is quantized to integer values and can only be changed by closing the bulk gap. Two other examples of topological phases of matter are topological superconductors and Weyl semimetals. The study of transport in various kinds of junctions of these topological materials is highly interesting for their applications in modern electronics and quantum computing. Another intriguing area to study is how to generate new kind of gapless edge modes in topological systems. In this thesis I have studied various aspects of topological phases of matter, such as electronic transport in junctions of topological insulators and topological superconductors, the generation of new kinds of boundary modes in the presence of granularity, and the effects of periodic driving in topological systems. We have studied the following topics. 1. transport across a line junction of two three-dimensional topological insulators, 2. transport across a junction of topological insulators and a superconductor, 3. surface and edge states of a topological insulator starting from a lattice model, 4. effects of granularity in topological insulators, 5. Majorana modes and conductance in systems with junctions of topological superconducting wires and normal metals, and 6. generation of new surface states in a Weyl semimetal in the presence of periodic driving by the application of electromagnetic radiation. A detailed description of each chapter is given below. • In the first chapter we introduce a number of concepts which are used in the rest of the thesis. We will discuss the ideas of topological phases of matter (for example, topological insulators, topological superconductors and Majorana modes, and Weyl semimetals), the renormalization group theory for weak interactions, and Floquet theory for periodically driven systems. • In the second chapter we study transport across a line junction which separates the surfaces of two three-dimensional topological insulators. The velocities of the Dirac electrons on the two surfaces may be unequal and may even have opposite signs. For a time-reversal invariant system, we show that the line junction is characterized by an arbitrary real parameter α; this determines the scattering amplitudes (reflection and transmission) from the junction. The physical origin of α is a potential barrier that may be present at the junction. If the surface velocities have the same sign, edge states exist that propagate along the line junction with a velocity and orientation of the spin which depend on α and the ratio of the velocities. Next, we study what happens if the two surfaces are at an angle φ with respect to each other. We study the scattering and differential conductance across the line junction as functions of φ and α. We also show that there are edge states which propagate along the line junction with a velocity and spin orientation which depend on φ. Finally, if the surface velocities have opposite signs, we find that the electrons must necessarily transmit into the two-dimensional interface separating the two topological insulators. • In the third chapter we discuss transport across a line junction lying between two orthogonal topological insulator surfaces and a superconductor which can have either s-wave (spin-singlet) or p-wave (spin-triplet) pairing symmetry. This junction is more complicated than the line junction discussed in the previous chapter because of the presence of the superconductor. In a topological insulator spin-up and spin-down electrons get coupled while in a superconductor electrons and holes get coupled. Hence we have to use a four-component spinor formalism to describe both spin and particle-hole degrees of freedom. The junction can have three time-reversal invariant barriers on the three sides. We compute the subgap charge conductance across such a junction and study their behaviors as a function of the bias voltage applied across the junction and the three parameters which characterize the barriers. We find that the presence of topological insulators and a superconductor leads to both Dirac and Schrodinger-like features in the charge conductances. We discuss the effects of bound states on the superconducting side on the conductance; in particular, we show that for triplet p-wave superconductors such a junction may be used to determine the spin state of its Cooper pairs. • In the fourth chapter we derive the surface Hamiltonians of a three-dimensional topological insulator starting from a microscopic model. (This description was not discussed in the previous chapters where we directly started from the surface Hamiltonians without deriving them form a bulk Hamiltonian). Here we begin from the bulk Hamiltonian of a three-dimensional topological insulator Bi2Se3. Using this we derive the surface Hamiltonians on various surfaces of the topological insulator, and we find the states which appear on the different surfaces and along the edge between pairs of surfaces. The surface Hamiltonians depend on the orientation of the surfaces and are therefore quite different from the previous chapters. We use both analytical methods based on the surface Hamiltonians (which are derived from the bulk Hamiltonian) and numerical methods based directly on a lattice discretization of the bulk Hamiltonian in order to find surface and edge states. We find that the application of a potential barrier along an edge can give rise to states localized at that edge. These states have an unusual energy-momentum dispersion which can be controlled by applying a potential along the edge; in particular, the velocity of these states can be tuned to zero. The scattering and conductance across the edge are studied as a function of the edge potential. We show that a magnetic field applied in a particular direction can also give rise to zero energy states on certain edges. We point out possible experimental ways of looking for the various edge states. • In the fifth chapter we study a system made of topological insulator (TI) nanocrystals which are coupled to each other. Our theoretical studies are motivated by the following experimental observations. Electrical transport measurements were carried out on thin films of nanocrystals of Bi2Se3 which is a TI. The measurements reveal that the entire system behaves like a single TI with two topological surface states at the two ends of the system. The two surface states are found to be coupled if the film thickness is small and decoupled above a certain film thickness. The surface state penetration depth is found to be unusually large and it decreases with increasing temperature. To explain all these experimental results we propose a theoretical model for this granular system. This consists of multiple grains of Bi2Se3 stacked next to each other in a regular array along the z-direction (the c-axis of Bi2Se3 nanocrystals). We assume translational invariance along the x and y directions. Each grain has top and bottom surfaces on which the electrons are described by Hamiltonians of the Dirac form which can be derived from the bulk Hamiltonian known for this material. We introduce intra-grain tunneling couplings t1 between the opposite surfaces of a single grain and inter-grain couplings t2 between nearby surfaces of two neighboring grains. We show that when t1 < t2 the entire system behaves like a single topological insulator whose outermost surfaces have gapless spectra described by Dirac Hamiltonians. We find a relation between t1, t2 and the surface state penetration depth λ which explains the properties of λ that are seen experimentally. We also present an expression for the surface state Berry phase as a function of the hybridization between the surface states and a Zeeman magnetic field that may be present in the system. At the end we theoretically studied the surface states on one of the side surfaces of the granular system and showed that many pairs of surface states can exist on the side surfaces depending on the length of the unit cell of the granular system. • In the sixth chapter we present our work on junctions of p-wave superconductors (SC) and normal metals (NM) in one dimension. We first study transport in a system where a SC wire is sandwiched between two NM wires. For such a system it is known that there is a Majorana mode at the junction between the SC and each NM lead. If the p-wave pairing changes sign at some point inside the SC, two additional Majorana modes appear near that point. We study the effect of all these modes on the subgap conductance between the leads and the SC. We derive an analytical expression as a function of and the length L of the SC for the energy shifts of the Majorana modes at the junctions due to hybridization between them; the energies oscillate and decay exponentially as L is increased. The energies exactly match the locations of the peaks in the conductance. We find that the subgap conductances do not change noticeably with the sign of . So there is no effect of the extra Majorana modes which appear inside the SC (due to changes in the signs of Δ) on the subgap conductance. Next we study junctions of three p-wave SC wires which are connected to the NM leads. Such a junction is of interest as it is the simplest system where braiding of Majorana modes is possible. Another motivation for studying this system is to see if the subgap transport is affected by changes in the signs of . For sufficiently long SCs, there are zero energy Majorana modes at the junctions between the SCs and the leads. In addition, depending on the signs of the Δ’s in the three SCs, there can also be one or three Majorana modes at the junction of the three SCs. We show that the various subgap conductances have peaks occurring at the energies of all these modes; we therefore get a rich pattern of conductance peaks. Next we study the effects of interactions between electrons (in the NM leads) on the transport. We use a renormalization group approach to study the effect of interactions on the conductance at energies far from the SC gap. Hence the earlier part of this chapter where we studied the transport at an energy E inside the SC gap (so that − < E < Δ) differs from this part where we discuss conductance at an energy E where |E| ≫ . For the latter part we assume the region of three SC wires to be a single region whose only role is to give rise to a scattering matrix for the NM wires; this scattering matrix has both normal and Andreev elements (namely, an electron can be reflected or transmitted as either an electron or a hole). We derive a renormalization group equation for the elements of the scattering matrix by assuming the interaction to be sufficiently weak. The fixed points of the renormalization group flow and their stabilities are studied; we find that the scattering matrix at the stable fixed point is highly symmetric even when the microscopic scattering matrix and the interaction strengths are not symmetric. Using the stability analysis we discuss the dependence of the conductances on the various length scales of the problem. Finally we propose an experimental realization of this system which can produce different signs of the p-wave pairings in the different SCs. • In the seventh chapter we show that the application of circularly polarized electro-magnetic radiation on the surface of a Weyl semimetal can generate states at that surface. The surface states can be characterized by their momenta due to translation invariance. The Floquet eigenvalues of these states come in complex conjugate pairs rather than being equal to ±1. If the amplitude of the radiation is small, we find some unusual bulk-boundary relations: the Floquet eigenvalues of the surface states lie at the extrema of the Floquet eigenvalues of the bulk system when the latter are plotted as a function of the momentum perpendicular to the surface, and the peaks of the Fourier transforms of the surface state wave functions lie at the momenta where the bulk Floquet eigenvalues have extrema. For the case of zero surface momentum, we can analytically derive interesting scaling relations between the decay lengths of the surface states and the amplitude and penetration depth of the radiation. For topological insulators, we again find that circularly polarized radiation can generate states on the surfaces; these states have much larger decay lengths (which can be tuned by the radiation amplitude) than the topological surface states which are present even in the absence of radiation. Finally, we show that radiation can generate surface states even for trivial insulators.

Recently, different studies on Weyl semimetals have shown some great potential for applications in spintronics. Indeed, spin-chiral Weyl nodes are perfect sources or sinks of the Berry curvature, which give new transport properties due to their topological nature, such as the chiral anomaly, and a large anomalous Hall response. Moreover, type-II Weyl semimetals, such as WTe2, have a specific band structure with tilted Weyl cones and overlapping electron/hole bands that can result in a perfect charge compensation and an extremely large magnetoresistance (XMR) . Yet, in WTe2 , Weyl nodes are usually located about 50 meV above the Fermi energy, a situation that questions the observation of both a large positive XMR and a negative magnetoresistance attributed to the chiral anomaly in some studies. In this work, we investigate the magneto-transport properties of WTe2 nanos- tructures obtained by different methods (mechanical exfoliation, chemical vapor transport), considering both the real electronic band structure and scattering by dis- order. Although the XMR amplitude also depends on charge mobilities, it is shown that the subquadratic response is not strongly influenced by the degree of disorder. Taking carrier densities infered from quantum oscillations into account, a three-band model explains this behavior by a large difference in hole mobilities, as confirmed by numerical simulations. At low temperatures and for small magnetic fields, an isotropic negative magneto-resistance is observed and attributed to a topological property of the band structure far away from the Weyl nodes. This new mechanism, different from the chiral anomaly, allows us to reproduce the experimental results by numerical calculations based on the real band structure of WTe2. / In den vergangenen Jahren haben verschiedene Untersuchungen von Weyl Halb- metallen gezeigt, dass sich diese sehr gut als Spintronische Geräte eignen. In der Tat sind die Spin-chiralen Weyl Quasiteilchen perfekte Quellen und Abflüsse der Berrykrümmung, was auf Grund ihrer topologischen Natur neue Transporteigen- schaften hervorruft, wie beispielsweise die chirale Anomalie und einen großen, anomalen Hall Effekt. Außerdem haben Typ II Weyl Halbmetalle wie WTe2 eine spezifische Bandstruktur mit gekippten Weylkegeln und überlappenden Elektronen-/Lochbändern, die dazu führen können, dass die Ladungsträgerkompensation ideal wird und ein sehr starker Magnetowiderstand (XMR) entsteht. Dennoch befinden sich die Weylknoten in WTe2 ca. 50 meV über dem Ferminiveau, eine Beobachtung die sowohl den starken positiven Magnetowiderstand, als auch den negativen Mag- netowiderstand, der meist mit der chiralen Anomalie in Verbindung gebracht wird, in Frage stellt. In dieser Arbeit untersuchen wir die Magnetotransporteigenschaften von WTe2 Nanostrukturen, die durch verschiedene Wachstumsarten hergestellt werden (mech- anische Exfoliation, chemische Gasphasenabscheidung), um sowohl die reale Band- struktur, als auch Streuung an Störstellen in Betracht ziehen zu können. Es wird gezeigt, dass der extrem große Magnetowiderstand nicht direkt vom Grad der Un- ordnung abhängt und dass das typisch subquadratische Verhalten im Rahmen eines Multibandmodells, was über das Zweibandmodell hinaus geht, verstanden wer- den kann und sich auch mit numerischen Simulationen bestätigt lässt. Bei tiefen Temperaturen und für kleine Magnetfelder kann ein isotropisch negativer Magne- towiderstand beobachtet werden, der der topologischen Eigenschaft der Bandstruk- tur weit weg von den Weylknoten geschuldet ist. Dieser neue Mechanismus, der sich von der chiralen Anomalie unterscheidet, erlaubt es uns die experimentellen Ergebnisse mit numerischen Berechnungen, die auf der realen Bandstruktur basieren, zu reproduzieren. / Récemment, différentes études sur les semimétaux de Weyl ont montré leur large potentiel pour des applications en spintronique. En effet, les noeuds de Weyl avec leur chiralité de spin sont des sources ou puits parfaits de la courbure de Berry, ce qui peut conduire à de nouvelles propriétés de transport, dues à la nature topologique de la structure de bande, comme l’anomalie chirale et une large réponse liée à l’effet Hall anormal dit intrinsèque. De plus, les semimétaux de Weyl de type II, comme WTe2, ont une structure de bande particulière avec des cônes de Weyl inclinés et un chevauchement des bandes de trous et d’électrons qui résulte en une forte compensation de charge et une magnétorésistance extrêmement large (XMR) associée. Cependant, dans WTe2, les noeuds de Weyl se trouvent environ 50 meV au-dessus de l’énergie de Fermi, ce qui remet en cause la possibilité d’observer à la fois une XMR positive à fort champ et une magnétorésistance négative à champ faible due à l’anomalie chirale. Dans ce travail, nous étudions les propriétés de magnéto-transport de nanostructures WTe2 obtenues par différentes méthodes (exfoliation mécanique, transport en phase vapeur), avec des degrés de désordre microscopique différents, en considérant à la fois la structure de bande réelle du matériau et les processus de diffusion liés au désordre. Il est montré que la XMR présente un comportement subquadratique, qui peut être compris dans le cadre d’un modèle multi-bandes, au-delà de deux bandes, comme confirmé par des simulations numériques. A très basse température et faible champ magnétique, une magnétorésistance négative et isotrope est observée et attribuée à une propriété topologique de la structure de bandes loin des noeuds de Weyl. Ce nouveau mécanisme, différent de celui de l’anomalie chirale, nous permet de reproduire nos résultats expérimentaux par des simulations numériques basées sur la structure de bande réelle de WTe2.

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.