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

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.

Motivated by the recent experimental progress in the search for Majorana fermions, we identify signatures of topological superconductivity and propose realistic experiments to observe these signatures. In the first part of this thesis, we study charge transport through a topological superconductor with a pair of Majorana end states, coupled to leads via quantum dots with resonant levels. The nonlocality of the Majorana bound states opens the possibility of Cooper pair splitting with nonlocal shot noise. In the space of quantum dot energy levels, we find a characteristic four-peaked cloverlike pattern for the strength of noise due to Cooper pair splitting, distinct from the single ellipsoidal peak found in the absence of Majorana end states. Semiconductor-superconductor hybrid systems are promising candidates for the realiza- tion Majorana fermions and topological order in solid state devices. In the second part, we show that the topological order is mirrored in the excitation spectra and can be observed in nonlinear Coulomb blockade transport through a ring-shaped nanowire. Especially, the ex- citation spectrum is almost independent of magnetic flux in the topologically trivial phase but acquires a characteristic h/e magnetic flux periodicity in the nontrivial phase. The transition between the trivial and nontrivial phase is reflected in the closing and reopening of an excitation gap. In the third part, we investigate characteristic features in the spin response of doped three-dimensional topological insulators with odd-parity unequal-spin superconducting pairing, which are predicted to have gapless Majorana surface modes. These Majorana modes contribute to the spin response, giving rise to a characteristic temperature behavior of the Knight shift and the spin-lattice relaxation time in magnetic resonance experiments.

We perform microscopic mean-field studies of topological order in a two-dimensional topological superconductor in the Bogoliubov-de Gennes (BdG) formalism. By adopting a two-dimensional s-wave topological superconductivity (TSC) model on a minimal tight-binding system, we solve the BdG equations self-consistently to obtain not only the superconducting order parameter, but also the Hartree potential. By computing the Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) number and investigating the bulk-boundary correspondence, we study the nature of Abelian and non-Abelian TSC in terms of self-consistent solutions to the BdG equations. In particular, we examine the effects of temperature and a single non-magnetic impurity deposited in the centre of the system and how they vary depending on topology. We find that the non-Abelian phase exhibits signs of unconventional superconductivity, and by examining the behaviour of this phase under both low and high Zeeman field conditions, we show that the magnitude of the Zeeman field largely dictates the susceptibility of the system to temperature. Furthermore, we investigate the possible interplay of charge density waves (CDW) and TSC. By self-consistently solving for the mean fields, we show that TSC and topological CDW are degenerate ground states---with the same excitation spectrum in the presence of surfaces---and thus can coexist in the Abelian phase. The effects of a non-magnetic impurity, which tends to pin the phase of charge density modulations, are examined in the context of topological CDW.

Recent developments in the study of Majorana fermions through braid theory have shown that there exists a set of interchanges that allow for the realization of true quantum computation. Alongside these developments there have been studies of topological superconductivity which show the existence of states that exhibit non-Abelian exchange statistics. Motivated by these developments we study the differences between Abelian and non-Abelian topological phase in the vortex state through the Bogoliubov de-Gennes (BdG) formalism. Due to our interests in low-energy states we first implement computationally efficient algorithms for calculating the mean fields and computing eigenpairs in an arbitrary energy window. We have shown that these algorithms adequately reproduce results obtained from a variety of other techniques and show that these algorithms retain spatial inhomogeneity information. Our results show topological superconductivity and vortex states can coexist; providing a means to realize zero-energy bound states, the number of which corresponds to the topological phase. With the use of our methods we present results contrasting the differences between Abelian and non-Abelian topological phase. Our calculations show that an increase in Zeeman field affects numerous parameters within topological superconductors. It causes the order parameter to become more sensitive to temperature variations in addition to a reduced rate of recovery to the bulk value from a vortex core. The increased field suppresses spin-up local density of states (LDOS) in close proximity to the vortex core for low-energy states. Further, it narrows the spectral gap at the lattice centre. Both energy spectrum and LDOS calculations confirm that trivial topological phase have no zero-energy bound states, Abelian phases have an even number, while non-Abelian phases have an odd number.

One of the most central concepts in condensed matter physics is the electronic band structure. Although band theory was established more than 80 years ago, recent developments have led to new insights that are formulated in the framework of topological band theory. In this thesis a subset of topological band theory is presented, with particular focus on topological supercon- ductors and accompanying Majorana fermions. While simple models are used to introduce basic concepts, a physically more realistic model is also studied intensely in the papers. Through self- consistent tight-binding calculations it is confirmed that Majorana fermions appear in vortex cores and at wire end points when the superconductor is in the topologically non-trivial phase. Many other properties such as the topological invariant, experimental signatures in the local density of states and spectral function, unconventional and odd-frequency pairing, the precense of spin-polarized currents and spin-polarization of the Majorana fermions, and a local π-phase shift in the order parameter at magnetic impurities are also investigated.

Thesis advisor: Ziqiang Wang / Exotic phases and associated phase transitions in low dimensions have been a fascinating frontier and a driving force in modern condensed matter physics since the 80s. Due to strong correlation effect, they are beyond the description of mean-field theory based on a single-particle picture and Landau's symmetry-breaking theory of phase transitions. These new phases of matter require new physical quantities to characterize them and new languages to describe them. This thesis is devoted to the study on exotic phases of correlated electrons in two spatial dimensions. We present the following efforts in understanding two-dimensional exotic phases: (1) Using Zn vertex algebra, we give a complete classification and characterization of different one-component fractional quantum Hall (FQH) states, including their ground state properties and quasiparticles. (2) In terms of a non-unitary transformation, we obtain the exact form of statistical interactions between composite fermions in the lowest Landau level (LLL) with v=1/(2m), m=1,2... By studying the pairing instability of composite fermions we theoretically explains recently observed FQHE in LLL with v=1/2,1/4. (3) We classify different Z2 spin liquids (SLs) on kagome lattice in Schwinger-fermion representation using projective symmetry group (PSG). We propose one most promising candidate for the numerically discovered SL state in nearest-neighbor Heisenberg model on kagome lattice}. (4) By analyzing different Z2 spin liquids on honeycomb lattice within PSG classification, we find out the nature of the gapped SL phase in honeycomb lattice Hubbard model, labeled sublattice pairing state (SPS) in Schwinger-fermion representation. We also identify the neighboring magnetic phase of SPS as a chiral-antiferromagnetic (CAF) phase and analyze the continuous phase transition between SPS and CAF phase. For the first time we identify a SL called 0-flux state in Schwinger-boson representation with one (SPS) in Schwinger-fermion representation by a duality transformation. (5) We show that when certain non-collinear magnetic order coexists in a singlet nodal superconductor, there will be Majorana bound states in vortex cores/on the edges of the superconductor. This proposal opens a window for discovering Majorana fermions in strongly correlated electrons. (6) Motivated by recent numerical discovery of fractionalized phases in topological flat bands, we construct wavefunctions for spin-polarized fractional Chern insulators (FCI) and time reversal symmetric fractional topological insulators (FTI) by parton approach. We show that lattice symmetries give rise to different FCI/FTI states even with the same filling fraction. For the first time we construct FTI wavefunctions in the absence of spin conservation which preserve all lattice symmetries. The constructed wavefunctions also set up the framework for future variational Monte Carlo simulations. / Thesis (PhD) — Boston College, 2011. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

We investigate quantum resonant transport in two different systems: (i) a ferromagnetic Scanning Tunneling Microscope (STM) tip coupled to an adatom (interacting) on a host surface (metallic or semiconductor), and (ii) a quantum dot connected to source and drain leads and side-coupled to a superconducting nanowire sustaining Majorana zero modes (Kitaev chain). Both problems are studied within the Green’s functions approach, which allows us to determine the transport properties of the system. In the first setup, due to the ferromagnetic and nonmagnetic ‘natures’ of the tip and host, respectively, it is possible to obtain the spin-diode effect, which occurs only in the singly occupied regime. In addition, because of the presence of the adsorbed atom on the surface, Friedel oscillations are observed in the current. The second system differs from the first mainly because it is spinless and there is no Coloumb interaction. Interestingly, we find that the Majorana mode of the wire leaks into the dot thus giving rise to a Majorana (zero mode) resonance in the dot, pinned to the Fermi level of the leads. Surprisingly, this resonance occurs even when the gate-controlled dot level is far above or far below the Fermi level of the leads. We study three possible experimental scenarios to probe unambigoulsy this Majorana mode in wires via these leaked/pinned modes. / Nesta tese investigamos transporte quântico ressonante em dois sistemas diferentes: (i) uma ponta STM ferromagnética acoplada a um átomo (interagente) adsorvido em uma superfície metálica ou semicondutora, e (ii) um ponto quântico conectado a reservatórios de elétrons e lateralmente acoplado a um nanofio supercondutor que possui modos de Majorana (cadeia Kitaev). Ambos os problemas são estudados no contexto de funções de Green, o que nos permite determinar as propriedades de transporte do sistema. Na primeira configuração, devido à natureza ferromagnética e não magnética da ponta STM e da superfície e, respectivamente, é possível obter o efeito diodo de spin, que ocorre apenas no regime em que o adátomo está ocupado com um único elétron. Além disso, por causa da presença do átomo adsorvido sobre a superfície, oscilções de Friedel são observadas na corrente. O segundo sistema é diferente do primeiro, principalmente pela ausência da interação de Coloumb e pelo fato de não ter spin. Curiosamente, vemos que o modo de Majorana do fio vai para o ponto quântico dando origem assim a um modo com energia zero no ponto quântico localizado sempre no nível de Fermi dos contatos. Surpreendentemente, essa ressonância ocorre mesmo quando o nível do ponto quântico, controlado por uma tensão externa, está muito acima ou muito abaixo do nível de Fermi dos contatos. Propomos três possíveis cenários experimentais para identificar de maneira conclusiva este modo de Majorana em fios através do modo que aparece no ponto quântico.

<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>

Orientador: Eduardo Peres Novais de Sá / Dissertação (mestrado) - Universidade Federal do ABC. Programa de Pós-Graduação em Física, 2013