Cette thèse poursuit deux directions dans le domaine des isolants et supraconducteurs topologiques.Dans la première partie de la thèse nous étudions des isolants en deux dimensions sur réseau, présentant un effet Hall quantique anormal (c'est-à-dire en l'absence d'un champ magnétique externe), induit par la présence d'un flux magnétique inhomogène dans la maille. Le système possède des phase isolantes caractérisés par un invariant topologique, le nombre de Chern, qui est lié à la conductance portée par le bord états. Nous montrons que les modèles à deux bandes admettent des phase à nombre de Chern arbitraire, ou, de façon équivalente, un nombre arbitraire d'états de bord, quand on augmente la portée des couplages sur réseau. Cette compréhension est rendue possible grâce à la démonstration d'une formule montrant que le nombre de Chern d'une bande dépend de certains propriétés d'un ensemble discret de points dans la zone de Brillouin, les points de Dirac en l'absence du gap. Ces idées sont rendues plus concrètes dans l'étude du modèle de Haldane et dans la création d'un modèle artificiel avec cinq phases de Chern dont les états de bord sont déterminés en détail. La deuxième partie de la thèse porte sur les supraconducteurs topologiques unidimensionnels qui exhibent des états exotiques d'énergie zéro: les états liés de Majorana. Nous étudions ici la présence de fermions de Majorana dans des fils de semiconducteurs à fort couplage spin-orbite sous l’effet de proximité d'un supraconducteur d'onde s. Nous montrons que la polarisation de spin des degrés de liberté électroniques dans la fonction d'onde Majorana dépend du poids relatif du couplage spin-orbite Dresselhaus et Rashba. Nous étudions également les fermions de Majorana dans des jonctions linéaires longues supraconducteur-normal et supraconducteur-normal-supraconducteur (SNS) où ils apparaissent comme des états étendus dans la jonction normale. En outre, la géométrie d'anneaux peut être mise en correspondance avec une jonction SNS, et, sous l'action de gradients dans la phase supraconductrice, des fermions Majorana étendus se forment encore à l'intérieur du fil normal. Enfin, un modèle à deux bandes avec des fermions de Majorana multiples est traité. Nous démontrons que les jonctions Josephson construites à partir de ce modèle maintiennent l'une des signatures remarquables des fermions de Majorana, à savoir la périodicité 4π de l'effet Josephson fractionnaire. / This thesis follows two threads in the field of topological insulators and superconductors. The first part of the thesis is devoted to the study of two-dimensional quantum anomalous Hall insulators on a lattice, in the absence of an external magnetic flux, but induced by an inhomogeneous flux in the unit cell. The system possesses several gapped phases characterized by a topological invariant, the Chern number, that is related to the conductance carried by the edge states. Here we show that two-band models admit an arbitrary large number of Chern phases or, equivalently, an arbitrary number of edge states, by adding hopping between distant neighbor sites. This result is based on a formula proving that the Chern number of a band depends on certain properties of a finite set of points in the Brillouin zone, i.e. the Dirac points for the gapless system. These ideas are made more concrete in the study of a modified Haldane model, and also by creating an artificial model with five Chern phases, whose edge states are determined in detail. The second part of the thesis focuses on one-dimensional topological superconductors with exotic zero-energy edge states: the Majorana bound states. Here we investigate the presence of Majorana fermions in spin-orbit coupled semiconducting wire in proximity to an s-wave superconductor. We show that the spin-polarization of the electronic degrees of freedom in the Majorana wave function depends on the relative weight of Dresselhaus and Rashba spin-orbit couplings. We also investigate Majorana fermions in linear superconductor-normal and long superconductor-normal-superconductor (SNS) junctions where they appear as extended states in the normal junction. Furthermore, ring geometries can be mapped to an SNS junction, and, we have shown that under the action of superconducting phases gradients, extended Majorana fermions can form again inside the normal wire. Finally a two-band model with multiple Majorana fermions is treated and we show that Josephson junctions built from this model maintain the 4π periodicity for the fractional Josephson effect, one of Majorana fermions signatures.

<div>The quantum phase transitions and exotic excitations are exciting and important topics of nowadays condensed matter theory. Topologically protected excitations are of great interest for potential applications in quantum computing. This Thesis explores two examples of exotic topologically protected excitations, Majorana fermions and parafermions in hybrid superconductor/semiconductor systems.</div><div><br></div><div>In the first part of the thesis, after a brief review of ideas on Majorana zero modes in solid state systems obtained by researchers over the past decade, I present our study of the emergence of Majorana fermions in charge carrier holes doped quantum wires. Study of Majorana modes in this system requires understanding Luttinger holes in low dimensions, which is also crucial for numerous spin-dependent phenomena, emerging field of spintronics and nanotechnology. We find that hole-doped quantum wires that are proximity coupled to a conventional s-wave superconductor is a promising system for the observation of Majorana fermions. We advanced understanding of Luttinger holes in quantum wells and quantum wires. We have shown that the vast majority of earlier treatments of Luttinger holes ignored an important effect, a mutual transformation of heavy and light holes at the heteroboundaries. We have derived the effective hole Hamiltonians in the ground size-quantized sub-bands of quantum wells and quantum wires. The effect of mutual transformation of holes is crucial for understanding Zeeman and spin-orbit coupling, and results in several spin-orbit terms linear in momentum in hole-doped quantum wires. We discuss the criterion for realizing Majorana modes in charge carrier hole systems and show that GaAs or InSb hole wires shall exhibit stronger topological superconducting pairing, providing additional opportunities for its control compared to intensively studies InSb and InAs electron systems.</div><div><br></div><div>In the second part of the thesis, I first introduce the basic facts of the current theoretical understanding of the fractional quantum Hall effect and a theoretical model of parafermion excitations. Parafermion zero modes are promising for universal quantum computing. However, physical systems that are predicted to host these exotic excitations are rare and difficult to realize in experiments. I present our work on modeling domain walls on the boundary between gate-induced polarized and unpolarized domains of the fractional quantum Hall effect system near the spin transitions, and the emergence of the parafermion zero modes when such domain wall is proximity coupled to an s-wave superconductor. Exact diagonalization of the Hamiltonian in a disk and torus geometries proves formation of the counter-propagating edge states with different spin polarizations at the boundaries between areas of the electron liquid in polarized and unpolarized filling factor $\nu=2/3$ phases. By analytical and numerical methods we find the conditions for emergence of parafermion zero modes in hybrid fractional quantum Hall/s-wave superconductor system. The phase diagram indicates that the parafermionic phase, which is represented by the six-fold ground state degeneracy, is separated from other phases by a topological phase transition. Such parafermion modes are experimentally feasible. They present a vital step toward the realization of Fibonacci anyons that allow a full universal set of quantum operations with topologically protected quasiparticles.</div><div><br></div>

<p>Boundary quantum phases ---a special type of quantum phenomena--- occur in the boundary part of the system. The boundary part can be a surface of a bulk material, an interface between two distinct system, and even it can be a single impurity or a impurity cluster embedded into a bulk system. The properties of the boundary degree of freedom can be affected by many strong electron correlation effects, mesoscopic effects, and topological effects, which, therefore, induce a vast variety of exotic boundary quantum phases. Many techniques for precise fabrication and measurement in nanostructures had been developed,</p><p>which can provide ways to prob, understand, and control those boundary quantum phases.</p><p>In this thesis, we focus on three types of the boundary quantum phases : Kondo effects, boundary quantum phase transitions, and Majorana fermions. Our motivation is to design and prob those effects by using a important type of nanostructures, i.e. quantum dots. A vast variety of models related to quantum dots (QDs) are studied theoretically, which includes a QD coupled to a mesoscopic bath, a quadruple QD system with metallic leads, a QD with dissipative environments, and a QD coupled to a Majorana fermion zero mode.</p><p>Quantum dots provide a way to study the interplay of Kondo effects and mesoscopic fuctuations. In chapter 5, we consider a model including an Anderson impurity (small QD) coupled to a mesoscopic bath (large QD). Both the weak and strong coupling Anderson impurity problems are characterized by Fermi-liquid theories with weakly interacting quasiparticles. We find that the fluctuations of single particle properties in the two limits are highly correlated and universal : The distributions of the spectrum within the Kondo temperature collapse to universal forms; and the strong coupling impurity changes the wave functions corresponding to the spectrum within the Kondo temperature. </p><p>Quantum dots also bring the possibility to study more complex quantum impurities (multi-QDs) and the competition among dierent interactions, which may induce exotic effects: boundary quantum phase transitions and novel Kondo effects. In chapter 7, we design a quadruple quantum dot system to study the competition among three types of interactions: Kondo, Heisenberg, and Ising. We find a rich phase diagram containing two sharp features : a Berezinsky-Kosterlitz-Thouless type quantum phase transition between a charge-ordered phase and a charge liquid phase and a U(1)XU(1) Kondo state with emergent symmetry from Z2 to U(1). In chapter 8, we study a dissipative resonant level model in which the coupling of a fermionc bath competes with a dissipation-induced bosonic bath. we establish an exact mapping from this dissipative resonant level model to a model of a quantum dot embedded into a Luttinger liquid wire, and we also find two kinds of boundary quantum phase transitions (a Berezinsky-Kosterlitz-Thouless type and a second order type).</p><p>Finally, in chapter 9, we propose an experimental system to detect Majorana fermion zero modes. This system consists of a spinless quantum do coupled to a Majorana fermion which exists in the end of a p-wave superconductor wire. The Majorana Fermion strongly infuence the transport properties of the quantum dot. The zero temperature conductance peak value (when the dot is on resonance and symmetrically coupled to the leads) is e^2/2h. In contrast, if the wire is in its topological trivial phase, the result is e^2/h; if the side-coupled mode is a regular fermionic zero mode, the result is zero. Driving the wire through the topological phase transition causes a sharp jump in the conductance by a factor of 1/2. This result can be used to detect the existence of Majorana fermions.</p> / Dissertation

This thesis is a study of three categories of problems in fermionic systems for which topology plays an important role: (i) The properties of zero modes arising in systems of fermions interacting with a bosonic background, with a special focus on Majorana modes arising in the superconductor state. We propose a method for counting Majorana modes and we study a mechanism for controlling their number parity in lattice systems, two questions that are of relevance to the protection of quantum bits. (ii) The study of dispersionless bands in two dimensions as a platform for correlated physics, where it is shown the possibility of stabilizing the fractional quantum Hall effect in a flat band with Chern number. (iii) The extension of the hierarchy of quantum Hall fluids to the case of time-reversal symmetric incompressible ground states describing a phase of strongly interacting topological insulators in two dimensions. / Physics

Orientador: Antonio Carlos Ferreira Seridonio / Resumo: O físico italiano Ettore Majorana propôs, no campo da Física de altas energias, a existência de férmions peculiares que têm como característica serem suas próprias antipartículas. No contexto de Física da matéria condensada, tais férmions emergem como quasipartículas de Majorana (MQPs). Da perspectiva da compu- tação quântica, duas MQPs podem compor um férmion regular e atuar como um qubit protegido, que está desacoplado do ambiente e livre do efeito de decoerência. Até onde sabemos, a verificação experimental de uma MQP ainda é questionável, apesar de alguns resultados experimentais, e desta forma, o objetivo desta tese é de propor formas experimentais a fim de ajudar na busca das assinaturas de tais excitações. Como o efeito Fano é um efeito de interferência na qual canais de tunelamento competem entre si pelo transporte eletrônico, ele torna-se uma forma de capturar tais assinaturas das MQPs em sistemas de matéria condensada. Baseado nisto, a ideia é investigar teoricamente três diferentes interferômetros a fim de obter uma assinatura definitiva das MQPs. O primeiro é um interferômetro do tipo Aharanov-Bohm composto por dois quantum dots, sendo um deles acoplado a uma MQP, que se localiza na borda de um fio de Kitaev semi-infinito na fase topológica. Ajustando o nível de Fermi dos terminais e o detuning simétrico dos níveis dos dots, mostrou-se que regimes Fano opostos resultam em uma transmitância caracterizada por distintas regiões condutoras e isolantes, que são marcas ... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The Italian physicist Ettore Majorana proposed in the field of high-energy Physics the existence of peculiar fermions that constitute their own antiparticles. In the context of condensed matter Physics, these fermions are Majorana quasiparticles (MQPs). From the quantum computing perspective, two MQPs can compose a regular fermion acting as a protected qubit, which is indeed decoupled from the host environment and free of the decoherence effect. To the best of our knowledge, the experimental capture of a MQP up to now is still questionable despite some experimental results, then, the goal of this thesis is to propose helpful experiment manners in revealing signatures from such excitations. As the Fano effect is an interference phenomenon where tunneling paths compete for the electronic transport, it becomes a probe to catch fingerprints of MQPs lying on condensed matter systems. Based on this, the idea is to investigate theoretically three different interferometers in order to obtain a MQP smoking- gun signature. The first one was an Aharonov-Bohm-like interferometer composed by two quantum dots, being one of them coupled to a MQP, which is attached to one of the edges of a semi-infinite Kitaev wire within the topological phase. By changing the Fermi energy of the leads and the symmetric detuning of the levels for the dots, we show that opposing Fano regimes result in a transmittance characterized by distinct conducting and insulating regions, which are fingerprints of an iso... (Complete abstract click electronic access below) / Doutor

Ce manuscrit de thèse aborde les propriétés de transport hors-équilibre des systèmes mésoscopiques supra-conducteurs. Cette étude se décline en deux volets : i) la signature des fermions de Majorana dans les jonctionsJosephson topologiques, et ii) les corrélations du courant dans les jonctions Josephson tri-terminales.Les fermions de Majorana apparaissent aux bords d’un supraconducteur topologique. Lorsque deux supra-conducteurs topologiques sont reliés pour former une jonction Josephson, les états de Majorana d’énergie nullede part et d’autre de jonction forment un état lié d’Andreev. Puisque cet état porteur du supercourant est4π-périodique vis-à-vis de la différence de phase supraconductrice, il a été spéculé un effet Josephson fraction-naire en présence d’une tension de polarisation. On montre qu’une vitesse de phase finie induit un couplagedynamique entre l’état lié et le continuum des états au dessus de l’amplitude du gap supraconducteur. Ce cou-plage intrinsèque constitue un mécanisme inévitable qui altère l’effet Josephson fractionnaire. On discute, enfonction des paramètres du circuit, les signatures expérimentales pertinentes de l’effet Josephson fractionnaire :l’effet pair-impair dans les marches de Shapiro et l’émergence d’un pic à la fréquence fractionnaire dans la den-sité spectrale du bruit en courant. D’autres manifestations de ces états d’énergie nulle dans la caractéristiquecourant-tension, sous l’amplitude du paramètre d’ordre supraconducteur, sont également exposés.Dans un second temps sont abordées les fluctuations du courant dissipatif dans les jonctions Josephsontri-terminales. On montre que, les corrélations croisées du courant peuvent être positives et amplifiées dans unrégime cohérent. Ces résultats ouvrent la possibilité à des études plus élaborées sur l’enchevêtrement quantiquedans ces systèmes. / This PhD thesis manuscript deals with the non equilibrium transport properties of superconducting meso-scopic systems. This study declines in two shutters : i) signatures of Majorana fermions in topological Josephsonjunctions and ii) current-current correlations in three-terminal Josephson junctions.Majorana fermions appears at the boundaries of topological superconductors. When two topological su-perconductors are connected to form a Josephson junction, the zero-energy Majorana bound states localizedon either side of the junction form an Andreev bound state. As this current carrying state is 4π-periodic inthe superconducting phase difference, it was speculated that, at finite dc bias voltage, the junction exhibits afractional Josephson effect. We show that any finite phase velocity induces a dynamic coupling between thebound state and the continuum of states above the superconducting gap amplitude. This intrinsic couplingprovides an unavoidable mechanism that alters the fractional Josephson effect. We discuss, in terms of thecircuit parameters, signatures of the fractional Josephson effect that could be relevant for current experimen-tal investigations : the even-odd effect in Shapiro steps and the emergence of a peak at fractional Josephsonfrequency in the current noise spectrum. Furthermore, other manifestations of the Majorana bound states onthe subgap current-voltage characteristic are discussed.In a second step, we discuss the dissipative current fluctuations in three terminal Josephson junctions. Weshow that, current-current cross correlations can be positive and amplified in a coherent regime. This findingopens the possibility for further investigations on quantum entanglement in those systems

Submitted by Fernando Augusto Dessotti null (fernandodessoti@hotmail.com) on 2018-01-30T17:37:28Z No. of bitstreams: 1 Tese oficial.pdf: 12552592 bytes, checksum: e44289b205ab39ad49d51a2f976df63c (MD5) / Approved for entry into archive by Cristina Alexandra de Godoy null (cristina@adm.feis.unesp.br) on 2018-01-30T18:39:52Z (GMT) No. of bitstreams: 1 dessotti_fa_dr_ilha.pdf: 12552592 bytes, checksum: e44289b205ab39ad49d51a2f976df63c (MD5) / Made available in DSpace on 2018-01-30T18:39:52Z (GMT). No. of bitstreams: 1 dessotti_fa_dr_ilha.pdf: 12552592 bytes, checksum: e44289b205ab39ad49d51a2f976df63c (MD5) Previous issue date: 2017-12-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O físico italiano Ettore Majorana propôs, no campo da Física de altas energias, a existência de férmions peculiares que têm como característica serem suas próprias antipartículas. No contexto de Física da matéria condensada, tais férmions emergem como quasipartículas de Majorana (MQPs). Da perspectiva da compu- tação quântica, duas MQPs podem compor um férmion regular e atuar como um qubit protegido, que está desacoplado do ambiente e livre do efeito de decoerência. Até onde sabemos, a verificação experimental de uma MQP ainda é questionável, apesar de alguns resultados experimentais, e desta forma, o objetivo desta tese é de propor formas experimentais a fim de ajudar na busca das assinaturas de tais excitações. Como o efeito Fano é um efeito de interferência na qual canais de tunelamento competem entre si pelo transporte eletrônico, ele torna-se uma forma de capturar tais assinaturas das MQPs em sistemas de matéria condensada. Baseado nisto, a ideia é investigar teoricamente três diferentes interferômetros a fim de obter uma assinatura definitiva das MQPs. O primeiro é um interferômetro do tipo Aharanov-Bohm composto por dois quantum dots, sendo um deles acoplado a uma MQP, que se localiza na borda de um fio de Kitaev semi-infinito na fase topológica. Ajustando o nível de Fermi dos terminais e o detuning simétrico dos níveis dos dots, mostrou-se que regimes Fano opostos resultam em uma transmitância caracterizada por distintas regiões condutoras e isolantes, que são marcas de uma MQP isolada. O dispositivo proposto aqui constitui uma alternativa experimental para detectar as MQPs. O segundo interferômetro é composto por pontas de STM e AFM próximas a um dímero de Kitaev de átomos adsorvidos supercondutores, na qual o átomo adsorvido localizado abaixo da ponta de AFM, encerra um par de MQPs. Para uma energia de ligação ∆ do par de Cooper delocalizado nos átomos adsorvidos abaixo das pontas coincidente com a amplitude de tunelamento t entre eles, ou seja, ∆ = t, mostrou-se que somente uma MQP abaixo da ponta de AFM hibridiza com o átomo adsorvido abaixo das pontas de STM, e para esta situação, o padrão de Fano permanece como universal. Mas para o caso das duas MQPs conectadas ao átomo adsorvido abaixo das pontas de STM, foi verificado que tal característica universal foi quebrada. O terceiro e último interferômetro é composto por dois quantum dots assimetricamente acoplados a MQPs isoladas que se localizam em duas cadeias de Kitaev na fase topológica. Este dispositivo habilita a medição da MQP em uma forma distinta do pico zero-bias. Mais importante, o sistema se comporta como um seletor de correntes composto por dois caminhos distintos: (i) para o dot superior conectado a ambas as cadeias, o dispositivo percebe ambas as MQPs como um férmion regular e a corrente atravessa somente o dot inferior, pois a corrente no dot superior é impedida devido a presença de um gap supercondutor; e (ii) pela leve supressão da hibridização do dot superior com a cadeia, a corrente é abruptamente trocada para fluir através deste mesmo dot, uma vez que um elétron é armadilhado como um estado ligado ao contínuo (BIC) surge no dot inferior. Tal seletor de corrente entre os dots inferior e superior caracteriza uma transição de fase quântica, que possibilita não somente a revelação de MQPs, mas também produz um seletor de corrente assistido por elas. / The Italian physicist Ettore Majorana proposed in the field of high-energy Physics the existence of peculiar fermions that constitute their own antiparticles. In the context of condensed matter Physics, these fermions are Majorana quasiparticles (MQPs). From the quantum computing perspective, two MQPs can compose a regular fermion acting as a protected qubit, which is indeed decoupled from the host environment and free of the decoherence effect. To the best of our knowledge, the experimental capture of a MQP up to now is still questionable despite some experimental results, then, the goal of this thesis is to propose helpful experiment manners in revealing signatures from such excitations. As the Fano effect is an interference phenomenon where tunneling paths compete for the electronic transport, it becomes a probe to catch fingerprints of MQPs lying on condensed matter systems. Based on this, the idea is to investigate theoretically three different interferometers in order to obtain a MQP smoking- gun signature. The first one was an Aharonov-Bohm-like interferometer composed by two quantum dots, being one of them coupled to a MQP, which is attached to one of the edges of a semi-infinite Kitaev wire within the topological phase. By changing the Fermi energy of the leads and the symmetric detuning of the levels for the dots, we show that opposing Fano regimes result in a transmittance characterized by distinct conducting and insulating regions, which are fingerprints of an isolated MQP. The setup proposed here constitutes an alternative experimental tool to detect MQPs. The second one is composed by STM and AFM tips close to a Kitaev dimer of superconducting adatoms, in which the adatom placed under the AFM tip, encloses a pair of MQPs. For the binding energy ∆ of the Cooper pair delocalized into the adatoms under the tips coincident with the tunneling amplitude t between them, namely ∆ = t, we find that only one MQP beneath the AFM tip hybridizes with the adatom coupled to the STM tips, and for this situation, the Fano pattern is still universal. But for the case of two MQPs connected to the adatom beneath the STM tips, we verify that such a universality is broken. The third and last one is composed by two quantum dots asymmetrically coupled to isolated MQPs, lying on the edges of two topological Kitaev chains. This setup enables us to probe MQPs in a quite distinct way from the zero-bias peak feature. Most importantly, the system behaves as a current switch made by two distinct paths: (i) for the upper dot connected to both chains, the device perceives both MQPs as an ordinary fermion and the current crosses solely the lower dot, since current in the upper dot is prevented due to the presence of the superconducting gap; and (ii) by suppressing slightly the hybridization of the upper dot with one chain, the current is abruptly switched to flow through this dot, once a trapped electron as a bound state in the continuum (BIC) appears in the lower dot. Such a current switch between upper and lower dots characterizes a quantum phase transition, which enables not only the fundamental revealing of MQPs, but also yields a current switch assisted by them.

Thesis advisor: Kevin S. Bedell / We present several results based upon non-trivial extensions of Landau-Fermi liquid theory. First proposed in the mid-20th century, the Fermi liquid approach assumes an adiabatic “switching-on” of the interaction, which allows one to describe the collective excitations of the many-body system in terms of weakly-interacting quasiparticles and quasiholes. At its core, Landau-Fermi liquid theory is often considered a perturbative approach to study the equilibrium thermodynamics and out-of-equilibrium response of weakly-correlated itinerant fermions, and therefore non-trivial extensions and consequences are usually overlooked in the contemporary literature. Instead, more emphasis is often placed on the breakdown of Fermi liquid theory, either due to strong correlations, quantum critical fluctuations, or dimensional constraints. After a brief introduction to the theory of a Fermi liquid, I will first apply the Landau quasiparticle paradigm to the theory of itinerant Majorana-like fermions. Defined as fermionic particles which are their own anti-particle, traditional Majorana zero modes found in topological materials lack a coherent number operator, and therefore do not support a Fermi liquid-like ground state. To remedy this, we will apply a combinatorical approach to build a statistical theory of self-conjugate particles, explicitly showing that, under this definition, a filled Fermi surface exists at zero temperature. Landau-Fermi liquid theory is then used to describe the interacting phase of these Majorana particles, from which we find unique signatures of zero sound in addition to exotic, non-analytic contributions to the specific heat. The latter is then exploited as a “smoking-gun” signature for Majorana-like excitations in the candidate Kitaev material Ag3LiIr2O6, where experimental measurements show good agreement with a sharply-defined, “Majorana-Fermi surface” predicted in the underlying combinatorial treatment. I will then depart from Fermi liquid theory proper to tackle the necessary conditions for the applicability of Luttinger’s theorem. In a nutshell, Luttinger’s theorem is a powerful theorem which states that the volume of phase space contained in the Fermi surface is invariant with respect to interaction strength. In this way, whereas Fermi liquid only describes fermionic excitations near the Fermi surface, Luttinger’s theorem describes the fermionic degrees of freedom throughout the entire Fermi sphere. We will show that Luttinger’s theorem remains valid only for certain frequency and momentum-dependencies of the self-energy, which correlate to the exis- tence of a generalized Fermi surface. In addition, we will show that the existence of a power-law Green’s function (a unique feature of “un-particle” systems and a proposed characteristic of the pseudo-gap phase of the cuprate superconductors) forces Luttinger’s theorem and Fermi liquid theory to be mutually exclusive for any non-trivial power of the Feynman propagator. Finally, we will return to Landau-Fermi liquid theory, and close with novel out-of-equilibrium behavior and stability in unconventional Fermi liquids. First, we will consider a perfectly two- dimensional Fermi liquid. Due to the reduction in dimension, the traditional mode expansion in terms of Legendre polynomials is modified to an expansion in terms of Chebyshev polynomials. The resulting orthogonality conditions greatly modifies the stability and collective modes in the 2D system. Second, we will look at a Fermi liquid in the presence of a non-trivial gauge field. The existence of a gauge field will effectively shift the Fermi surface in momentum space, resulting in, once again, a modified stability condition for the underlying Fermi liquid. Supplemented with a modernized version of Mermin’s condition for the propagation of zero sound, we outline the full effects a spin symmetric or anti-symmetric gauge would have on a Fermi liquid ground state. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

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.

Submitted by ANDRE RAMALHO DOS SANTOS null (ramalho_inf@yahoo.com.br) on 2018-02-14T03:20:16Z No. of bitstreams: 1 Dissertação André Ramalho.final.pdf: 2789018 bytes, checksum: d4170ea3aaec8f302b447a0aac5e5986 (MD5) / Approved for entry into archive by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br) on 2018-02-14T16:54:15Z (GMT) No. of bitstreams: 1 santos_ar_me_rcla.pdf: 2620534 bytes, checksum: cb88b11f48c3fc7b2fef3c938febb8e0 (MD5) / Made available in DSpace on 2018-02-14T16:54:15Z (GMT). No. of bitstreams: 1 santos_ar_me_rcla.pdf: 2620534 bytes, checksum: cb88b11f48c3fc7b2fef3c938febb8e0 (MD5) Previous issue date: 2017-11-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nós estudamos teoricamente como o calor e a eletricidade são afetados pela sobreposição de dois férmions de Majorana (MFs, de Majorana fermions em Inglês), os quais estão isolados nas bordas de um fio topológico de Kitaev, em particular, na forma de “ferradura”. É considerado que esse fio está assimetricamente acoplado a um único ponto quântico (QD, de Quantum dot em Inglês) hibridizado com contatos metálicos. Em baixas temperaturas e dependente do nível de energia desse QD, nós mostramos que ao ajustar a assimetria acima, as respostas ressonantes das condutâncias termoelétricas mudam inesperadamente de forma drástica. Assim, propomos como aplicação, um sintonizador termoelétrico em nanoescala assistido por MFs. / We study theoretically in a topological U-shaped Kitaev wire, with Majorana fermions (MFs) on the edges, how heat and electricity are affected by them when found overlapped. The asymmetric regime of their couplings with a single quantum dot (QD) hybridized with metallic leads is considered. At low temperatures and dependent upon the QD energy level, we show that by tuning this asymmetry, the resonance positions of the thermoelectrical conductances change drastically. Thereby, the tuner of heat and electricity here proposed is constituted.