Les progrès techniques réalisés dans le cadre des expériences sur les gaz de fermions ultrafroids ont engendré une émulation particulièrement importante ces dernières années. En effet, ces dispositifs expérimentaux permettent de produire des systèmes gazeux ≪ à la carte ≫, notamment grâce au phénomène de résonances de Feshbach qui permet de contrôler le signe de la longueur de diffusion a par application d'un champ magnétique extérieur. Il est alors possible de générer aussi bien une interaction attractive (a < 0) que répulsive (a > 0). La résonance de Feshbach en elle-même se trouve en a → ±∞, cette limite correspondant à un régime de fortes corrélations entre les particules. De plus, dans la région où a est positive, des états lies moléculaires (bosoniques car formés de deux fermions) peuvent se former. En-dessous d'une certaine température, une phase superfluide peut alors apparaitre, et une transition de phase continue entre l'état bosonique et l'état fermionique peut être observée (BEC-BCS crossover). En fonction de la position dans le diagramme de phases, les modes collectifs possèderont des caractéristiques (fréquence, amortissement) différentes. En ce sens, ils constituent une sonde de l'état de la matière et une connaissance précise de ces modes est par conséquent très importante. Le travail présenté dans cette thèse comporte une caractérisation détaillée de plusieurs modes collectifs dans la phase normale du système atomique. L'étude repose principalement sur l'équation de Boltzmann, que nous résolvons de deux façons différentes. La première consiste à utiliser une méthode des moments ≪ améliorée ≫ (c'est-à-dire d'ordre supérieur). La seconde est numérique et a nécessité l'écriture d'un programme de simulation permettant l'incorporation de tous les effets de milieu (potentiel de champ moyen et section efficace). Une attention toute particulière a été apportée à la mise en place des simulations afin de reproduire le plus fidèlement possible les conditions expérimentales. Les techniques expérimentales permettent également désormais la création de gaz polarisés. Nous présenterons donc dans ce travail une étude de ces gaz utilisant notre programme de simulation (mise en évidence des différents régimes de collision), puis une étude plus théorique ayant pour principal objectif d'établir le diagramme de phase encore méconnu de ces gaz particuliers, et enfin de proposer une méthode de calcul des effets de milieu, les techniques habituelles utilisées pour les gaz non polarisés n'étant plus valables / Technical progress on ultra-cold Fermi gases experiments induced numerous studies for the last few years. Using these experimental setups, it is effectively possible to generate ultra-cold gases with selected properties, in particular through the Feshbach resonances phenomenon. This allows us to set the sign of the scattering length a using an external magnetic field. It is then possible to have an attractive interaction (a < 0) as well as a repulsive one (a > 0). The Feshbach resonance itself is defined for infinite values of a (positive or negative), which corresponds to a strongly interacting regime. Moreover, when a > 0, molecular bound states (bosonic because they are made with two fermionic atoms) can appear. Thus, below a critical temperature, a superfluid phase can emerge and a crossover can be observed (from the BEC to BCS superfluid states). Depending on the position on the phase diagram, frequency and damping of collective modes will be different. This is why the collective modes are good probes of the system phase. A precise extensive knowledge of their characteristics is thus very important. This thesis presents a complete study of some of these collective modes in the normal phase. This work mainly relies on the Boltzmann equation which will be solved in two different ways: firstly, with an improved (higher order) version of the so-called moments method; secondly with a numerical solution that has required to write a numerical code in order to take into account the in-medium effects (mean field potential and in-medium cross section). Particular attention has been paid to numerical simulations in order to reproduce as closely as possible the experimental conditions. Moreover, experimental procedures now allow to create spin unbalanced gases. We have shown in this work a study of these systems using the numerical resolution of the Boltzmann equation. Moreover, we have developed a theoretical approach in order to build the phase diagram of these polarized gases, which is not fully described yet. Finally, we have suggested a method to determine the in-medium effects, with the aim to solve the problem emerging with the usual method used in the balanced case

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.

Dans cette thèse, nous avons étudié expérimentalement les propriétés physiques des fermions ultra-froids grâce à une machine conçue pour refroidir un mélange fermionique de 6Li et 40K. Après une courte description concernant la construction de l'expérience et quelques améliorations que j'ai implémentées pendant ma thèse (telles que la désorption atomique par lumière ultraviolette dans le 2D-MOT), l'exposé se concentre sur deux observations principales de l'origine fermionique des gaz de potassium et de lithium.La première partie présente la quasithermalization du 6Li dans un potentiel quadrupolaire, créé par un piège magnétique. Malgré l'absence de collisions dans un gaz fermionique polarisé en dessous d'une température donnée, nous observons une redistribution d'énergie dans l'ensemble statistique après une excitation dans le piège linéaire. Une étude expérimentale détaillée ainsi qu'une analyse théorique du phénomène sont présentées. De plus, une transformation canonique de l'hamiltonien du système permet la description de particules sans masses dans un piège harmonique. Les résultats expérimentaux du système réel (gaz 6Li dans un potentiel quadrupolaire) sont donc réinterprétés pour décrire ces particules non massiques, difficiles à observer. Un développement supplémentaire de notre système expérimental permet également la réalisation d'un couplage spin-orbite non-abélien dans le gaz fermionique sans interactions.Dans la deuxième partie, on décrit la réalisation d'un gaz dégénéré de 40K à l'aide du refroidissement évaporatif. Une succession d'étapes d'évaporation, utilisant différentes technologies de piégeage, nous permet d'obtenir 1.5e5 atomes dans l'état fondamental à une température de 62nK, température équivalente à 17% de la température de Fermi. / In this thesis we investigate experimentally the physics of a cold fermionic mixture consisting of 6Li and 40K. After a short description of the experimental apparatus and of a few technical particularities implemented during my PhD, for example the light-induced atomic desorption in the 2D-MOT by UV-light, we focus on two main observations of the fermionic nature of the gas.The first part describes the quasithermalization of 6Li in a magnetic quadrupole potential. Even though collisions are absent in a spin-polarized fermionic gas below a given temperature, the statistical ensemble undergoes energy redistribution after an excitation within the linear potential. We present an extensive experimental study as well as a comprehensive theoretical analysis. Moreover, the studied Hamiltonian can be canonically mapped onto a system of massless, harmonically trapped particles and the previously developed results are re-interpreted in order to describe this experimentally inaccessible system. A further development of the realized experiment allows even for the implementation of spin-orbit coupling in a gas of non-interacting fermions.In the second part, we describe the evaporative cooling of 40K to quantum degeneracy. Through different evaporative cooling stages we reach with a final number of 1.5e5 atoms in the ground-state a temperature of 62nK, which corresponds to 17% of the Fermi temperature.

Au cours de ma thèse, j'ai travaillé à la construction de l'expérience Fermix, consacrée à l'étude d'un mélange de fermions (6Li-40K) à très basses températures où les effets quantiques sont prédominants. Nous présentons ici deux résultats principaux. Premièrement, nous avons développé une nouvelle méthode de refroidissement sub-Doppler qui tire parti de l'existence d'états noirs dans la raie optique D1. Cette mélasse grise permet d'atteindre une densité de l'espace des phases de 10^-4, la valeur la plus élevée rapportée dans la littérature pour le refroidissement laser simultané des deux espèces. Grâce à cette étape, nous avons pu réaliser un gaz fortement dégénéré de 3x10^5 atomes de 40K, répartis dans deux états de spins, à une température de 62 nK, soit 17% de la température de Fermi. D'autre part, nous introduisons une transformation canonique pour montrer l'équivalence formelle entre le comportement de particules ultra-relativistes sans interactions (particules de Weyl) dans un potentiel harmonique et celui de fermions froids confinés dans un piège quadrupolaire. Nous étudions expérimentalement, numériquement et théoriquement la relaxation de tels systèmes vers un état stationnaire, non-Botlzmanien, caractérisé par des températures effectives fortement anisotropes. Cette analogie permet également d'interpréter des propriétés caractéristiques des particules relativistes. Ainsi, nous montrons que le paradoxe de Klein est analogue aux pertes Majorana. Pour finir, nous proposons une étude théorique des interactions médiées à longue distance par un système en dimensions mixtes. / During my PhD, I contributed to the design and construction of the Fermix experiment, dedicated to the study of a 6Li-40K fermionic mixture at ultra low temperatures. Our main results are twofold. First, we developed a new sub-Doppler laser cooling scheme, taking advantage of the existence of dark states in the D1 line of alkali atoms. This so-called \emph{grey molasses} allows for a phase space density up to $10^{-4}$, the highest value reported for the simultaneous laser cooling of those two species. The improvement due to this cooling step enabled the production of a quantum degenerate 40K gas in a dipole trap, with 3x10^5 atoms in two spin states at 62 nK, corresponding to 17% of the Fermi temperature. Second, introducing a canonical mapping, we showed that non-interacting ultra-relativistic particles (Weyl fermions) in a harmonic trap can be simulated by cold fermions confined in a quadrupole potential. We study experimentally, numerically and theoretically the relaxation of these systems towards a steady state which can not be described by a Boltzman distribution, but rather presents strongly anisotropic effective temperatures. This analogy also allows us to translate fundamental properties of relativistic particles in the language of cold atoms. In particular, we demonstrate that the Klein paradox is equivalent to Majorana losses.Finally, we present a theoretical study of the long range interactions between particles confined in two 2D layers immersed in a 3D atomic cloud.

Ce travail rend compte de nouvelles techniques développées pour l'étude expérimentale de gaz ultrafroids de lithium et de potassium fermioniques. Les améliorations de notre expérience 6Li-40K y sont décrites et caractérisées. Nous présentons un laser solide de grande finesse capable d'émettre 5W de puissance à 671 nm. Nous utilisons cette source laser dans le contexte d'une nouvelle technique de refroidissement sub-Doppler, reposant sur la transition atomique D1 des atomes alcalins, pour refroidir des atomes de lithium. Cette melasse D1 nous permet de refroidir simultanément les atomes de 6Li et de 40K à des températures bien inférieures à la limite Doppler, tout en manipulant des grands nombres d'atomes à des densités importantes. Nous avons mesuré une densité dans l'espace des phases après l'étape de mélasse de l'ordre de 10-4 à la fois pour le 6Li et le 40K. Le refroidissement laser D1 ouvre la voie à une évaporation rapide vers la dégénérescence quantique dans un piège magnétique ou optique. Nous présentons le refroidissement évaporatif d'atomes de 40K. L'évaporation débute dans une piège magnétique pluggé et continue dans un piège dipolaire optique. A l'issue de l'évaporation, nous obtenons un mélange de spins dégénéré, avec plus de 7x105 atomes dans chacun des deux états de spin et une température T/TF<0.34. / This thesis reports on novel techniques for experimental studies of ultracold, fermionic lithium and potassium quantum gases. The new parts of our 6Li-40K apparatus are described and characterised. We present a narrow-linewidth, all-solid-state laser source, emitting 5W at 671 nm. We employ the laser source in the context of a novel sub-Doppler cooling mechanism, operating on the D1 atomic transition of alkali atoms, for laser cooling of lithium. This D1 molasses allows us to simultaneously cool a mixture of 6Li and 40K atoms to deep sub-Doppler temperatures, while retaining large atom numbers and high atomic densities. The measured phase space densities after the molasses phase are on the order of 10-4 for both 6Li and 40K. The D1 laser cooling paves the way for fast evaporation to quantum degeneracy in magnetic and optical traps. We present the evaporative cooling of 40K atoms. The evaporation starts in an optically plugged magnetic quadrupole trap and continues in an optical dipole trap. At the end of the evaporation, we obtain a quantum degenerate spin-mixture of 40K atoms, with more than 7x105 atoms in each of the two spin states and T/TF<0.34.

Orientador: Antonio Carlos Ferreira Seridonio / Resumo: Estudou-se teoricamente um modelo composto por um fio de Kitaev na fase topológica com dois pontos quânticos (QDs - Quantum Dots), um em cada extremidade do nanofio. Desta forma, dois casos foram factíveis de análise, um deles com os estados ligados de Majorana (MBSs - Majorana Bound States) das bordas do fio acoplados a um único QD e o segundo em que se tem ambos os MBSs acoplados aos dois QDs. Para a primeira situação três condições foram estudadas, nas quais se verificou, na primeira, os perfis de férmions de Majorana não locais, dados pelo acoplamento entre o MBS e o QD mais próximo e, nas outras duas condições, dois perfis relacionados aos acoplamentos dos dois MBSs a um QD, em que se considerou também a superposição entre os MBS. Estes dois perfis são denominados de bowtie e diamond, já conhecidos na literatura, possuindo também experimentos que validam suas manifestações. No segundo caso, em que se tem o acoplamento dos dois MBSs aos dois QDs e que se considerou também amplitudes de superposição entre os férmions de Majorana, investigou-se a manifestação de estados moleculares mediados por tais férmions, pois o transporte eletrônico entre os QDs, no sistema proposto, se dá por meio do nanofio. Constatou-se padrões condizentes a níveis moleculares ligante e antiligante nas assinaturas dos estados ligados de Andreev (ABSs), originários da superposição dos MBSs, assim como nos níveis dos QDs que foram desdobrados após a formação molecular. / Mestre

Nanoelectronics and spintronics are concerned with writing, transporting, and reading information stored in electronic charge and spin degrees of freedom at the nanoscale. Past few years have shown that two spintronics effects discovered in the 19th century, namely anisotropic magnetoresistance and anomalous Hall effect, can be used also for sensing antiferromagnetism which opened the field of antiferromagnetic spintronics. The more than a century of controversial studies of these effects have shown their relativistic spin-orbit coupling and spin-polarisation symmetry breaking origin. However, a complete understanding of these effects and a fully predictive theory capable of identifying novel suitable antiferromagnetic materials are still lacking. Here, by extending modern symmetry and topology concepts in condensed matter physics, we have further developed the theory of anisotropic magnetoresistance and spontaneous Hall effect. Our approach is based on magnetic symmetry and topology analysis of antiferromagnetic energy bands, Bloch spectral functions, and Berry curvatures calculated from the state-of-the- art first-principle theory. This guided us to the prediction of two novel, previously unanticipated effects: relativistic metal-insulator transition from antiferromagnetic Dirac fermions, and crystal Hall...

Eine Vielzahl von interessanten Phänomenen entsteht durch die quantenmechanischeWechselwirkung einer großen Zahl von Teilchen. In den meisten Fällen ist die Beschreibung der relevanten physikalischen Eigenschaften extrem schwierig, da die Komplexität des Systems exponentiell mit der Anzahl der wechselwirkenden Teilchen anwächst und das Lösen der zugrunde liegenden Schrödingergleichung unmöglich macht. Trotzdem gab es in der Geschichte der Festkörperphysik eine Reihe von bahnbrechenden Entdeckungen, die unser Verständnis von komplexen Phänomenen deutlich voran gebracht haben. Dazu zählt die Entwicklung der Landau’schen Theorie der Fermiflüssigkeit, der BCS-Theorie der Supraleitung, der Theorie der Supraflüssigkeit und der Theorie des fraktionalen Quanten-Hall-Effekts. In all diesen Fällen ist ein theoretisches Verständnis mithilfe sogenannter Quasiteilchen gelungen. Anstatt ein komplexes Phänomen durch das Verhalten von fundamentalen Teilchen wie der Elektronen zu erklären, ist es möglich, die entsprechenden Eigenschaften durch das simple Verhalten von Quasiteilchen zu beschreiben, die allein auf Grund der komplexen kollektiven Wechselwirkung entstehen. Eines der seltenen Beispiele, bei dem ein stark korreliertes quantenmagnetisches Problem analytisch lösbar ist, ist das Kitaev Modell. Es beschreibt wechselwirkende Spins auf einem Sechseck-Gitter und zeichnet sich durch einen Spinflüssigkeits-Grundzustand aus. Auch hier gelang die Lösung mittels spezieller Quasiteilchen, den Majorana Fermionen. Experimentell ist es jedoch noch nicht gelungen eine Spinflüssigkeit eindeutig nachzuweisen, da diese sich gerade durch das Fehlen jeglicher klassischer Ordnung und üblicher experimenteller Kenngrößen auszeichnet. Dagegen kann die Beobachtung von Quasiteilchenanregungen einen Hinweis auf den zugrunde liegenden Zustand liefern. Aber auch der definitive Nachweis von Majorana Fermionen in jeglicher Art System, bleibt ein ausstehendes Ziel in der modernen Festkörperphysik. Diese Arbeit befasst sich daher mit der Frage, wie solche Quasiteilchen experimentell sichtbar gemacht werden könnten. Dazu untersuchen wir den Einfluss von Unordnung auf die Zustände und Messgrößen des Kitaev Modells. Dies ist in zweierlei Hinsicht relevant. Einerseits ist Unordnung in der Natur allgegenwärtig, andererseits kann sie auch strategisch herbeigeführt werden, um die Reaktion eines System gezielt zu testen. Das zentrale Ergebnis dieser Arbeit ist, dass den Majorana Fermionen dabei in der Tat eine physikalische, messbare Bedeutung zukommt. Die Arbeit beginnt mit einer Einführung in frustrierte quantenmagnetische Systeme und Spinflüssigkeiten und diskutiert einige Effekte, die durch Gitterverzerrungen oder Verunreinigungen entstehen können. Anschließend zeigen wir, wie sich durch die frustrierte Wechselwirkung im Kitaev Modell ein Spinflüssigkeits-Grundzustand herausbildet. Die analytische Lösung des Modells gelingt mit Hilfe von Majorana Fermionen, jedoch verdoppelt sich der Hilbertraum pro Spin durch die Einführung dieser Quasiteilchen. Ein zentraler Aspekt dieser Arbeit ist daher die richtige Auswahl der „physikalischen“ Zustände, also solcher, die einem Zustand im ursprünglichen Spin Modell entsprechen. Dabei unterscheiden wir zwischen offenen und periodischen Randbedingungen. Wir konnten beweisen, dass sich, in der Phase ohne Bandlücke und für periodische Systeme, stets ein angeregtes Fermion befindet. Dies führt zu großen Effekten in endlichen Systemen, wie wir anhand der Suszeptibilität und der Anregungslücke für magnetische Flüsse zeigen. Außerdem berechnen wir numerisch die statische und dynamische Suszeptibilität abhängig von der Unordnung in der Wechselwirkungsstärke. Diese Art der Unordnung entsteht beispielsweise durch unregelmäßige Gitterstrukturen oder chemische Verunreinigungen auf den nicht-magnetischen Gitterplätzen. Insbesondere ergibt die Verteilung der lokalen Suszeptibilitäten das Linienspektrum, welches sich in Kernspinresonanz Experimenten messen lässt. Für große Unordnung postulieren wir einen Übergang zu einem Zustand mit einer zufälligen Verteilung magnetischer Flüsse. Ein weiterer Kern der Dissertation ist die Untersuchung eines magnetischen Defekts im Kitaev Modell. Diese Situation beschreibt den ungewöhnlichen Fall eines Kondoeffekts in einer Spinflüssigkeit. In der Majorana Fermionen Darstellung gelingt es uns, das Problem in eine Form zu bringen, die mit Hilfe von Wilson’s numerischer Renormalisierungsgruppe untersucht werden kann. Es zeigt sich, dass dadurch eine Nullpunktsentropie des Defekts entsteht, die durch lokalisierte Majorana Fermionen erklärt werden kann. Durch die Darstellung des Kitaev Modells mithilfe von Quasiteilchen ist es möglich eine elegante Beschreibung eines komplexen, stark wechselwirkenden Systems zu finden. Die Ergebnisse dieser Arbeit zeigen, dass den Majorana Fermionen dabei durchaus eine physikalische Bedeutung zukommt. Gelingt es sie z.B. durch magnetische Störstellen zu lokalisieren, wäre ein direkter experimenteller Nachweis möglich. / Many interesting phenomena in quantum physics arise through the quantum mechanical interaction of a large number of particles. In most cases describing the relevant physical properties is extremely difficult, because the complexity of the system increases exponentially with the number of interacting particles and solving the underlying Schrödinger equation becomes impossible. Nevertheless, our understanding of complex phenomena has progressed through some groundbreaking discoveries in the history of condensed matter physics. Examples include the development of Landau’s theory of Fermi liquids, the BCStheory of superconductivity, the theory of superfluidity and the theory of the fractional quantum Hall effect. In all these cases a theoretical understanding was achieved with so-called quasi-particles. Instead of explaining a phenomenon through the behavior of fundamental particles, such as electrons, the corresponding properties can be described by the simple behavior of quasi-particles, which are themselves a result of the complex collective interaction. One of the rare examples, where a strongly correlated quantum mechanical problem can be solved analytical, is the Kitaev model. It describes interacting spins on a honeycomb lattice and exhibits a spin liquid ground state. Here the solution was achieved by means of certain quasi-particles, called Majorana fermions. However, it has not been possible to clearly identify such a spin liquid experimentally, because its defining feature is the absence of any conventional order, in particular magnetic order. In contrast, the observation of quasiparticle excitations may hint at the nature of the ground state. But also a definite detection of Majorana fermions in any kind of system remains one of the outstanding issues in modern condensed matter physics. Therefore this thesis is devoted to the question how such quasiparticles may be found experimentally. For this reason we study the influence of disorder on the states and observables of the Kitaev model. This is relevant in two respects: Firstly, disorder is ubiquitous in nature and secondly, it may be used strategically to probe the response of a system. The central result of this work is that Majorana fermions hereby indeed obtain a true physical and observable significance. The thesis starts with an introduction of frustrated quantum mechanical systems and spin liquids, and discusses some of the effects that arise through lattice distortions or impurities. Afterwards we show how the frustrated interactions in the Kitaev model lead to a spin liquid ground state. The analytical solution of the model is achieved through the introduction of Majorana fermions. However, resulting from the introduction of these quasi-particles the Hilbert space per spin doubles. A central aspect of this thesis is therefore the right selection of the “physical” states, which correspond to a state of the original spin Hamiltonian. To do this, we distinguish between periodic and open boundary conditions explicitly. We were able to prove that there is always one excited fermion in the gapless phase of the periodic system. This leads to large finite-size effects, as we will illustrate for the susceptibility and the magnetic flux gap. Moreover we compute the static and dynamic spin susceptibilities for finite-size systems subject to disorder in the exchange couplings. In a possible experimental realization, this kind of disorder arises from lattice distortions or chemical disorder on nonmagnetic sites. Specifically, we calculate the distribution of local susceptibilities and extract the lineshape, which can be measured in nuclear-magnetic-resonance experiments. Further, for increasing disorder we predict a transition to a random-flux state. Another core of this dissertation is the investigation of a magnetic impurity in the Kitaev model. This setup represents the unusual case of a Kondo effect in a quantum spin liquid. Utilizing the Majorana representation we are able to formulate the problem in a way that can be analyzed using Wilson’s numerical renormalization group. The numerics reveal an impurity entropy which can be explained by localized Majorana fermions. Through the representation of the Kitaev model in terms of quasi-particles an elegant description of a complex, strongly correlated system is possible. The results of this thesis indicate that these Majorana acquire a relevant physical meaning. If one can localize them, for example with the help of magnetic impurities, a direct experimental observation would be feasible.

Das Auftreten von Supraleitung in Systemen mit schweren Fermionen, erstmals entdeckt in CeCu_2Si_2, wird mit der Nähe zu einem quantenkritischen Punkt in Verbindung gebracht. Daraus ergibt sich ein komplexes Zusammenspiel von Magnetismus und Supraleitung, das in der vorliegenden Arbeit durch Messungen der spezifischen Wärme, der Wechselfeldsuszeptibilität und durch inelastische Neutronenstreuexperimente an verschiedenen Einkristallen von CeCu_2(Si_{1-x}Ge_x)_2 untersucht wird. Der Schwerpunkt liegt auf der genauen Charakterisierung des magnetischen Anregungsspektrums von CeCu_2Si_2 des S-Typs. Die Ergebnisse der Neutronenstreumessungen implizieren stark, dass die Kopplung der supraleitenden Cooper-Paare durch überdämpfte Spinfluktuationen vermittelt wird, die in der Umgebung eines Quantenphasenübergangs gehäuft auftreten. Unter Substitution einiger Si- durch Ge-Atome in CeCu_2Si_2 stabilisiert sich die magnetische Ordnung, und die Supraleitung wird zunehmend unterdrückt. Neutronenstreumessungen ergeben, dass dies bei 2 % Ge-Substitution dazu führt, dass sich Magnetismus und Supraleitung gegenseitig verdrängen, während sie bei 10 % Ge-Substitution mikroskopisch koexistieren. - (Die Dissertation ist veröffentlicht im Logos Verlag Berlin GmbH, Berlin, Deutschland, http://www.logos-verlag.de, ISBN: 978-3-8325-2456-2) / The occurrence of superconductivity in systems with heavy fermions, discovered for the first time in CeCu_2Si_2, is often linked to the vicinity of a quantum critical point. This results in a complex interplay of magnetism and superconductivity, which is studied by means of specific heat and ac susceptibility measurements as well as neutron scattering experiments on different single crystals of CeCu_2(Si_{1-x}Ge_x)_2 in the present thesis. The focus is put on the detailed characterisation of the magnetic excitation spectrum in S-type CeCu_2Si_2. Neutron scattering results strongly imply that the coupling of superconducting Cooper pairs is mediated by overdamped spin fluctuations, which accumulate in the vicinity of a quantum phase transition. By substituting Si by Ge atoms in CeCu_2Si_2 magnetic order is stabilised and superconductivity successively suppressed. Neutron scattering experiments demonstrate that 2 % Ge substitution leads to magnetic order being displaced by superconductivity on decreasing temperature, whereas both coexist microscopically in the case of 10 % Ge substitution.

Die Elektronenspinresonanz (ESR) untersucht die im quasistatischen Magnetfeld resonante Absorption eines an die Probe angelegten Mikrowellenmagnetfeldes. Es wurde das System Yb1-w A1-w (Rh1-x Cox)2 (Si1-y Gey) 2 mit A=La, bzw. Lu, sowie das System YbIr2Si2 mittels ESR untersucht. Unter Kondo-Wechselwirkung vieler Leitungselektronen mit einem lokalen 4f-Moment des Kondo-Ions bildet sich ein nicht-magnetisches Grundzustands-Singlett, was zur Abschirmung des magnetischen Moments führt. YbRh2Si2 ist das erste Schwere-Fermionen-System mit Kondo-Ionen, das ohne Dotierung zusätzlicher ESR-Sonden ein ESR-Signal unterhalb der Kondo-Temperatur aufweist. Es zeigt sich, dass das ESR-Signal nicht mittels gängiger ESR-Theorien konsistent beschrieben werden kann. Die Messungen, die im Rahmen dieser Arbeit angestellt wurden, flossen in die Entwicklung von weiterführenden Theorien (z.B. [1], [2]) ein. Die Temperaturabhängigkeit des ESR-g-Faktors konnte damit erfolgreich beschrieben werden, womit erstmals der Nachweis einer Kondo-Wechselwirkung in Kondo-Gitter-Systemen mittels ESR gelang. Ferner konnte die Bedeutung von ferromagnetischen Fluktuationen für eine kleine, beobachtbare Linienbreite beschrieben werden. Der ESR-Methode ist somit die Kondo-Spindynamik direkt zugänglich. Dieser Zugang ist neu und einzigartig, denn andere Methoden (NMR, inelastische Neutronenstreuung) charakterisieren die Kondo-Spindynamik auf indirekte Weise. [1] P. Wölfle und E. Abrahams. Phenomenology of esr in heavy-fermion systems: Fermi-liquid und nicht-fermi-liquid regimes Phys. Rev. B, 80(23): 235112, 2009. [2] B. I. Kochelaev, S. I. Belov, A. M. Skvortsova, A. S. Kutusov, J. Sichelschmidt, J. Wykhoff, C. Geibel und F. Steglich. Why could electron spin resonance be observed in a heavy fermion kondo lattice? Eur. Phys. J. B, 72(4): 485, 2009.